id
stringlengths 20
20
| content
stringlengths 211
8.3M
| dsir_books
float64 -20,679,750.45
-316.68
| fluency_en
listlengths 2
2
| rps_lines_ending_with_terminal_punctution_mark
float64 0
100
| modernbert_cleanliness
listlengths 6
6
| qurater
listlengths 4
4
| rps_doc_num_sentences
int64 1
84.7k
| rps_doc_word_count
float64 9
1.29M
| ad_en
listlengths 2
2
| rps_doc_frac_no_alph_words
float64 15.7
92
| modernbert_reasoning
listlengths 6
6
| rps_doc_frac_chars_top_2gram
float64 0
92.9
| rps_lines_uppercase_letter_fraction
float64 0.06
87.8
| rps_doc_frac_unique_words
float64 1.16
100
| rps_lines_numerical_chars_fraction
float64 0
85.7
| fineweb_edu
listlengths 1
1
| dsir_math
float64 -17,396,368.29
-306.94
| rps_doc_mean_word_length
float64 1.44
61
| dsir_wiki
float64 -20,772,722.61
-313.93
| rps_doc_frac_chars_top_3gram
float64 0
119
| rps_doc_unigram_entropy
float64 1.2
8.5
| modernbert_professionalism
listlengths 6
6
| modernbert_readability
listlengths 6
6
| sub_path
stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
BkiUdK84eIZjvCjkaEGo
|
\section{ Self-Supervised Local Low Rank Approximation}
\subsection{Approach and Related Work}
Among different approaches to denoising, methods based on low-rank approximations have given the most promising results for DWI denoising. Representing the data as low-rank requires accurate estimation of a threshold to truncate the basis set, typically constructed via eigen-decomposition of the data. The only unsupervised approach for doing so in the case of DWI makes use of random-matrix-theory based Marchenko-Pastur distribution to classify the eigenvalues pertaining to noise \cite{veraart_novikov_christiaens_ades-aron_sijbers_fieremans_2016}. Extending the idea of LPCA, it computes a universal signature of noise, in the PCA domain. However, this is constrained by the variability in denoising caused by local patch-sizes and assumption of the noise being homoscedastic. Patch2Self is the first of its kind approach to DWI denoising that leverages the following two properties:
\subsubsection{Statistical Independence of Noise}
Since noise is perceived as random fluctuations in the signal measurements, one can assume that the noise in one measurement is independent of another. Using this, \cite{noise2noise} showed that given two noisy independent images, we could use one measurement to denoise another by posing denoising as a prediction problem. Expanding this idea, \cite{batson19} laid out a theoretically grounded notion of self-supervision for denoising signals from the same corrupted image/ signal measurement. Leveraging this idea of statistical independence, different approaches such as \cite{alex2018noise2void} and \cite{laine2019highquality} have shown competitive performance with varying approximation settings. However, most of these approaches tackle 2D images typically via CNN-based deep learning methods with the motive of improving the visual quality of the image (i.e., they do not account for modeling the signal for scientific analysis like DWI). These approaches are not feasible in 4D DWI data, as the denoising needs to be unsupervised, fast and clinically viable for downstream image-analysis. In Patch2Self, we delineate how one can extrapolate the notion of noise independence purely via patches in a 4D volumetric setting via a regression framework.
\subsubsection{Patches and Local Matrix Approximations}
Patch-based self-supervision has been used to learn representations that are invariant to distortions \cite{discr_dos, dosovitskiy2014discriminative}, for learning relations between patches \cite{doersch2015unsupervised}, for filling in missing data (i.e. image in-painting) \cite{pathak2016context}, etc. Patch2Self abides by a similar patch-based approach where we learn an underlying clean signal representation that is invariant to random fluctuations in the observed signal. Inspired by the local matrix approximation works presented in \cite{llorma, candes_tao_2010}, we formulate a global estimator per 3D volume of the 4D data by training on local patches sampled from the remaining volumes. This estimator function, thus has access to local and non-local information to learn the mapping between corrupted signal and true signal, similar to dictionary learning \cite{golts_elad_2016, scetbon2019deep, Sulam_2016, gramfort2014denoising, bilgic2012accelerated} and non-local block matching \cite{bm3d}. Due to the self-supervised formulation, Patch2Self can be viewed as a non-parametric method that regresses over patches from all other volumes except from the one that is held-out for denoising. Similar to \cite{blindreg}, our experiments demonstrate that a simplistic linear-regression model can be used to denoise noisy matrices using $p$-neighbourhoods and a class of $\mathcal{J}$-invariant functions.
\begin{figure}[h]
\centering
\includegraphics[width=.95\textwidth]{images/flow_final.pdf}
\caption{ Depicts the workflow of Patch2Self in two phases: \textbf{(A)} Is the self-supervised training phase where the 4D DWI data is split into the training $Y_{*, *, -j}$ and target $Y_{*,0,j}$ sets. $p$-neighbourhoods are extracted from each 3D volume from both $Y_{*, *, -j}$ and $Y_{*,0,j}$. $\Phi_J$ is the learnt mapping by regressing over $p$-neighbourhoods of $Y_{*, *, -j}$ to estimate $Y_{*,0,j}$. \textbf{(B)} Depicts the voxel-by-voxel denoising phase where $\hat{\Phi}_J$ predicts the denoised volume $\hat{X}_{*, *, *, j}$ from $Y_{*, *, -j}$.}
\label{fig:flow}
\end{figure}
\subsection{Denoising via Self-Supervised Local Approximations}
\label{sec:neighbourhood}
\paragraph{Extracting 3D patches} In the first phase of Patch2Self, we extract a $p$-neighbourhood for each voxel from the 4D DWI data. To do so, we construct a 3D block of radius $p$ around each voxel, resulting in a local $p$-neighbourhood of dimension $p\times p \times p$. Therefore, if the 4D DWI has $n$ volumes $\{v_1, \ldots, v_n\}$ (each volume corresponding to a different gradient direction) and each 3D volume has $m$ voxels (see Fig.~\ref{fig:flow}), after extracting the $p$-neighbourhoods, we get a $m \times p\times p \times p\times n$ tensor. Next, we flatten this this tensor along the $p^{th}$-dimension to obtain a representation: $m \times (p^3 \times n)$. Thus, we have transformed the data from the original 4D space to obtain $m$ samples of $p^3 \times n$ dimensional 2D feature matrices, which we use to train the denoiser.
\begin{wrapfigure}[13]{R}{0.72\textwidth}
\vspace{-.635cm}
\begin{minipage}{\textwidth}
\begin{algorithm}[H]
\scriptsize
\begin{algorithmic}
\caption*{\textbf{Algorithm:} Patch2Self}
\State Input 4D data $X$ of dimension $l \times w \times h \times n$
\For {volume $j=1,2,\ldots n$} [where $n$ is the number of volumes]
\For {voxel $k=1,2,\ldots m$} [where $m = lwh$ is the number of voxels]
\State Extract a {$p\times p\times p$} neighbourhood of voxel $k$
\EndFor
\State Flatten and concatenate the $p$-neighbourhood of each voxel into a feature vector of length $p^3 \times n$.
\EndFor
\State Stack feature vectors into a matrix of size $m \times (p^3 \times n)$.
\For {volume $j=1,2,\ldots n$}
\State Hold-out features from volume $j$ to get a feature matrix $Y_{*,*,-j}$ of dimension $m \times p^3 \times n - 1$
\State Select the central pixels from volume $j$ to get a target vector $Y_{*,0,j}$ of dimension $m$.
\State Train a linear regressor $\Phi: Y_{*,*,-j} \mapsto Y_{*,0,j}$.
\State Set the denoised volume $\hat{X}_{*,*,*,j}$ to the unraveled output $\hat{\Phi}(Y_{*,*,-j})$.
\EndFor
\Return Denoised 4D data $\hat{X}$
\end{algorithmic}
\label{alg:p2s}
\end{algorithm}
\end{minipage}
\end{wrapfigure}
\\ \textbf{Self-Supervised Regression} \hspace{3mm}In the second phase, using the $p$-neighbourhoods, Patch2Self reformulates the problem of denoising with a predictive approach. The goal is to iterate and denoise each 3D volume of the noisy 4D DWI data ($X$) using the following training and prediction phases:\\
(i) Training: To denoise a particular volume, $v_j$, we train the a regression function $\Phi_J$ using $p$-neighbourhoods of the voxels denoted by the set $Y$. From the first phase, $Y$ is a set containing $m$ training samples with dimension: $p^3 \times n$. Next, we hold out the dimension corresponding to volume $v_j$ from each of the $p$-neighbourhoods and use it as a target for training the regressor function $\Phi_J$ (shown in Fig.~\ref{fig:flow}A). Therefore our training set $Y_{*, *, -j}$ has dimension: $m \times p^3 \times (n-1)$, where $j$ indexes the held out dimension of the $p$-neighbourhoods set. Using the regressor function $\Phi_J$, we use the training set $Y_{*, *, -j}$ to only predict the center voxel of the set of $p$-neighbourhoods in the corresponding target set of dimension $Y_{*, 0, -j}$. The target set, is therefore only an $m$-dimensional vector of the center voxels of the corresponding $p$-neighbourhoods of volume $v_j$. In summary, we use the localized spatial neighbourhood information around each voxel of the set of volumes $v_{-j}$, to train $\Phi_J$ for approximating the center voxel of the target volume $v_j$. To do so, we propose minimizing the self-supervised loss over the described $p$-neighbourhood sets as follows:
\begin{equation}
\mathcal{L}(\Phi_J)=\mathbb{E}\|\Phi_J(Y_{*, *, -j})-Y_{*, 0, j}\|^{2}
\label{eq:ssl}
\end{equation}
Where, $\Phi_J: \mathbb{R}^{p^3\times n} \mapsto \mathbb{R}^{1}$, is trained on $m$ samples of $p$-neighbourhoods.\\
(ii) Predict: After training for $m$ samples, we have now constructed a $\mathcal{J}$-invariant regressor $\hat{\Phi}_J$ that can be used to denoise the held out volume $v_j$. To do so, $p$-neighbourhoods from the set $Y_{*, *, -j}$ are simply fed into $\hat{\Phi}_J$ to obtain the denoised $p$-neighbourhoods corresponding to the denoised volume $\hat{Y}_{*, 0, -j}$. After doing so, for each $j \in \{1 \dots n\}$, we unravel the $p$-neighbourhoods for each volume $v_j \in \{v_1 \dots v_n\}$ (in Fig.~\ref{fig:flow} as $\hat{X}_{*,*,*,j}$) and append them to obtain the denoised 4D DWI data $\hat{X}$.
\vspace{-3mm}
\paragraph{$\mathcal{J}$-Invariance}The reason one might expect the regressors learned using the self-supervised loss above to be effective denoisers is the theory of $\mathcal{J}$-invariance introduced in \cite{batson19}. Consider the partition of the data into volumes, $\mathcal{J} = \{v_1, \dots, v_n\}$. If the noise in each volume is independent from the noise in each other volume, and a denoising function $\Phi$ satisfies the property that the output of $\Phi$ in volume $v_j$ does not depend on the input to $\Phi$ in volume $v_j$, then according to Proposition 1 of \cite{batson19}, the sum over all volumes of the self-supervised losses in equation \ref{eq:ssl} will in expectation be equal to the ground-truth loss of the denoiser $\Phi$, plus an additive constant. This means that $\mathcal{J}$-invariant functions minimizing the self-supervised loss will also minimize the ground-truth loss. This holds by construction for our denoiser $\Phi = (\Phi_1, \dots, \Phi_n)$. Intuitively, each $\Phi_J$ only has access to the signal present in the volumes other than $v_j$, and since the noise in those volumes is irrelevant for predicting the noise in $v_j$, it will learn to suppress the fluctuations due to noise while preserving the signal. Note that, if linear regression is used to fit each $\Phi_J$, then the final denoiser $\Phi$ is a linear function. Unlike methods which work by thresholding the singular values obtained from a local eigen-decomposition \cite{manjon_coupe_concha_buades_collins_robles_2013, veraart_novikov_christiaens_ades-aron_sijbers_fieremans_2016}, which produce denoised data that are locally low-rank, this mapping $\Phi$ can be full-rank.
\label{sec:jinvpatch}
\textbf{Choice of Regressor:}
Any class of regressor can be fit in the above method, from simple linear regression/ordinary least squares to regularized linear models like Ridge and Lasso, to more complex nonlinear models. Our code-base allows for the use of any regression model from \cite{scikit-learn}. Surprisingly, we found that linear regression performed comparably to the more sophisticated models, and was of course faster to train (see supplement for comparisons).
\textbf{Choice of Patch Radius:}
To determine the effect of changing the patch radius on denoising accuracy, we compute the Root Mean Squared Error (RMSE) between the ground truth and Patch2Self denoised estimates at SNR 15 (details of simulation in Sec.~\ref{sec:simulated}). For patch radius zero and one, we show the effect at different number of volumes as depicted in Fig.~\ref{fig:all_phantom}C. The line-plot trend shows that the difference in the RMSE scores between the two patch radii steadily decreases with an increase in number of volumes. However, with lesser number of volumes, a bigger patch-radius must be used. In the remainder of the text, we use and show results with patch radius zero and linear regressors.
\section{Experiments}
\begin{figure}[h]
\centering
\includegraphics[width=1\textwidth]{images/diff_data.pdf}
\caption{Shows the comparison of denoising on 3 different types of datasets: Parkinson's Progression Markers Initiative (PPMI), Stanford HARDI and Sherbrooke 3-Shell HARDI data. The denoising of Patch2Self is compared against the original noisy image and Marchenko-Pastur denoised data along with their corresponding residuals. Notice that Patch2Self suppresses more noise and also does not show any anatomical structure in the corresponding residual plots.}
\label{fig:ppmi}
\end{figure}
\section{Evaluation on Real Data}
\subsection{Evaluation on \textit{in-vivo} data}
We compare the performance of Patch2Self with Marchenko-Pastur on the Parkinson's Progression Markers Initiative (PPMI) \cite{marek2011parkinson}, Stanford HARDI \cite{rokem_2016} and Sherbrooke 3-Shell \cite{garyfallidis_brett_amirbekian_rokem_walt_descoteaux_nimmo-smith__2014} datasets as shown in Fig.~\ref{fig:ppmi}. These datasets represent different commonly used acquisition schemes: (1) Single-Shell (PPMI, $65$ gradient directions), (2) High-Angular Resolution Diffusion Imaging (Stanford HARDI, $160$ gradient directions) and (3) Multi-Shell (Sherbrooke 3-Shell, $193$ gradient directions). For each of the datasets, we show the axial slice of a randomly chosen 3D volume and the corresponding residuals (squared differences between the noisy data and the denoised output). Note that both, Marchenko-Pastur and Patch2Self, do not show any anatomical features in the error-residual maps, so it is likely that neither is introducing structural artifacts. Patch2Self produced more visually coherent outputs, which is important as visual inspection is part of clinical diagnosis.
\subsection{Effect on Tractography}
\label{sec:tractography}
To reconstruct white-matter pathways in the brain, one integrates orientation information of the underlying axonal bundles (streamlines) obtained by decomposing the signal in each voxel using a microstructure model \cite{behrens_jbabdi_2009, novikov2018modeling}. Noise that corrupts the acquired DWI may impact the tractography results, leading to spurious streamlines generated by the tracking algorithm. We evaluate the effects of denoising on probabilistic tracking \cite{girard_whittingstall_deriche_descoteaux_2014} using the Fiber Bundle Coherency (FBC) metric \cite{portegies_fick_sanguinetti_meesters_girard_duits_2015}. To perform the probabilistic tracking, the data was first fitted with the Constant Solid Angle (CSA) model \cite{aganj_lenglet_sapiro_yacoub_ugurbil_harel_2009}. The Generalized Fractional Anisotropy (GFA) metric extracted from this fitting was used as a stopping criterion within the probabilistic tracking algorithm. The fiber orientation distribution information required to perform the tracking was obtained from the Constrained Spherical Deconvolution (CSD) \cite{tournier_calamante_connelly_2007} model fitted to the same data. In Fig.~\ref{fig:meyer}, we show the effect of denoising on tractography for the Optic Radiation (OR) bundle as in \cite{portegies_fick_sanguinetti_meesters_girard_duits_2015}. The OR fiber bundle, which connects the visual cortex:V1 (calcarine sulcus) to the lateral geniculate nucleus (LGN), was obtained by selecting a $3\times3\times3$ Region Of Interest (ROI) using a seeding density of 6. After the streamlines were generated, their coherency was measured with the local FBC algorithm \cite{portegies_fick_sanguinetti_meesters_girard_duits_2015, duits_franken_2010}), with \textit{yellow-orange} representing - spurious/incoherent fibers and \textit{red-blue} representing valid/coherent fibers. In Fig,~\ref{fig:meyer}, OR bundle tracked from original/ raw data contains 3114 streamlines, Marchenko-Pastur denoised data \cite{veraart_novikov_christiaens_ades-aron_sijbers_fieremans_2016} contains 2331 streamlines and Patch2Self denoised data contains 1622 streamlines. Patch2Self outperforms Marchenko-Pastur by reducing the number of incoherent streamlines, as can be seen in the \textit{red-blue} (depicting high coherence) coloring in Fig.~\ref{fig:meyer}.
\begin{figure}[h]
\centering
\includegraphics[width=1\textwidth]{images/fbc.pdf}
\caption{Depicts the Fiber to Bundle Coherency (FBC) density map projected on the streamlines of the optic radiation bundle generated by the probabilistic tracking algorithm. The color of the streamlines depicts the coherency$~-~$\textit{yellow} corresponding to incoherent and \textit{blue} corresponding to coherent. Notice that the number of incoherent streamlines present in the original fiber-bundle is reduced after Marchenko-Pastur denoising. Patch2Self denoising further reduces spurious tracts, resulting in a cleaner representation of the fiber bundle.}
\label{fig:meyer}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=1\textwidth]{images/fitting_box.pdf}
\caption{\textbf{(A)} Shows the mean kurtosis parameter maps obtained by fitting the DKI model to the axial slice of the CFIN dataset \cite{hansen_jespersen_2016}. Notice that Patch2Self \textbf{(A3)} alleviates more degenerecies in model estimation (visible as black voxels in the region highlighted with arrows) as compared to noisy (original) data \textbf{(A1)} and Marchenko-Pastur \textbf{(A2)} denoised data. \textbf{(B)} Box-plots quantifying the increase in $R^2$ metric after fitting downstream DTI and CSD models. The $R^2$ improvements in each case are plotted by subtracting the scores of model fitting on noisy data from $R^2$ of fitting each denoised output. Note that the consistency of microstructure model fitting on Patch2Self denoised data is higher than that obtained from Marchenko-Pastur (see \ref{sec:micro} for details and significance).\vspace{-3mm}}
\label{fig:dki}
\end{figure}
\subsection{Impacts on Microstructure Model Fitting}
\label{sec:micro}
The domain of microstructure modeling employs either mechanistic or phenomenological approaches to resolve tissue structure at a sub-voxel scale. Fitting these models to the data is a hard inverse problem and often leads to degenerate parameter estimates due to the low SNR of DWI acquisitions \cite{novikov_kiselev_jespersen_2018}. We apply two of the most commonly used diffusion microstructure models, Constrained Spherical Deconvolution (CSD) \cite{tournier_calamante_connelly_2007} and Diffusion Tensor Imaging (DTI) \cite{basser_mattiello_lebihan_1994}, on raw and denosied data. DTI is a simpler model that captures the local diffusion information within each voxel by modeling it in the form of a 6-parameter tensor. CSD is a more complex model using a spherical harmonic representation of the underlying fiber orientation distributions. In order to compare the goodness of each fit, we perform a k-fold cross-validation (CV) \cite{hastie_friedman_tisbshirani_2017} at two exemplary voxel locations, corpus callosum (CC), a single-fiber structure, and centrum semiovale (CSO), a crossing-fiber structure. The data is divided into $k=3$ different subsets for the selected voxels, and data from two folds are used to fit the model, which predicts the data on the held-out fold. The scatter plots of CV predictions against the original data are shown in Fig.~\ref{fig:r2} for those two voxels. As measured by $R^2$, Patch2Self has a better goodness-of-fit than Marchenko-Pastur by $22\%$ for CC and $65\%$ for CSO. To show that Patch2Self consistently improves model fitting across all voxels, in Fig.~\ref{fig:dki}B we depict the improvement of the $R^2$ metric obtained from the same procedure for the axial slice (4606 voxels) of masked data (using \cite{rokem_2016} data). This was done by simply subtracting the goodness-of-fit $R^2$ scores of fitting noisy data, from Marchenko-Pastur and Patch2Self denoised data for both CSD and DTI models. Patch2Self shows a significant improvement on both DTI and CSD (two-sided t-test, p < 1e-300, Fig.~\ref{fig:dki}). The Diffusion Kurtosis (DKI) model contrast, uses higher-order moments to quantify the non-gaussianity of the underlying stochastic diffusion process. This can be used to characterize microstructural heterogeneity \cite{jensen_helpern_2010} leading to important biomarkers of axonal fiber density and diffusion tortuosity \cite{fieremans_jensen_helpern_2011}. Models such as DKI are susceptible to noise and signal fluctuations can often lead to estimation degeneracies. In Fig.~\ref{fig:dki}A, we compare the effects of different denoising algorithms on DKI parameter estimation by visualizing the \textit{mean kurtosis} maps. We make use of the CFIN dataset \cite{hansen_jespersen_2016} which was designed to evaluate kurtosis modeling and imaging strategies to depict the effects of denoising. As highlighted by the arrows, Marchenko-Pastur does not add any new artifacts due to noise suppression but also does not help alleviate degeneracies in parameter estimation. Patch2Self reduces the number of degeneracies without adding any artifacts due to denoising.
\begin{figure}[h]
\centering
\includegraphics[width=1\textwidth]{images/kfold.pdf}
\caption{Quantitative comparison of the goodness-of-fit evaluated using a cross-validation approach. \textbf{(A)} and \textbf{(B)} depict the scatter plots of the model predictions obtained by fitting CSD to voxels in the corpus callosum (CC) and centrum semiovale (CSO) for original (noisy), Marchenko-Pastur (denoised) and Patch2Self (denoised) data. Similarly \textbf{(C)} and \textbf{(D)} show the scatter plots of predictions obtained from DTI fitting in the same voxel locations. Top-left of each plot shows the $R^2$ metric computed from each model fit on the corresponding data.}
\label{fig:r2}
\end{figure}
\vspace{-.6cm}
\section{Evaluation on Simulated Data}
We begin with whole brain noise-free DWIs simulated from the framework proposed in \cite{graham_drobnjak_zhang_2016, wen2017pca}, with 2 image volumes at b-value=0 $s/mm^2$ (i.e., b0), 30 diffusion directions at b-value=1000 $s/mm^2$ (b1000) and 30 directions at b-value=2000 $s/mm^2$ (b2000). The real-world noise distribution was simulated with multi-channel acquisition: a realistic 8-channel coil sensitivity map was used and Gaussian noise was added to the real and imaginary part of each channel of the DWIs respectively \cite{wen2017pca}. Finally DWIs were combined with sum-of-square coil combination and signal-to-noise ratio (SNR) was calculated in the white-matter of the b=0 image. All together, 5 datasets were simulated: noise-free, SNR= 10, 15, 20, 25 and 30. We take two different approaches of comparing Patch2Self and the performance gains it provides: (1) Compute the mean squared error (MSE) between the denoised data and the ground truth at each SNR. (2) Using the $R^2$ metric of the denoised data against ground truth. The outcomes from both these evaluation strategies have been compared against the raw noisy data and Marchenko-Pastur denoised data. As shown in table~\ref{tab:metric} the performance gains obtained by using Patch2Self are substantial in realistic SNR ranges, especially in the 5-20 range common for in-vivo imaging. This can also be seen qualitatively in Fig.~\ref{fig:all_phantom}, where Patch2Self is visibly cleaner than with Marchenko-Pastur\cite{veraart_novikov_christiaens_ades-aron_sijbers_fieremans_2016} at each low SNRs. A scatterplot of ground-truth versus denoised voxel value illustrates this performance gain as well (Fig.~\ref{fig:all_phantom}D). Notice that with an increase in SNR, the performance of Patch2Self improves consistently (see table \ref{tab:metric}) and does consistently better than the Marchenko-Pastur method (evaluated via MSE and $R^2$ metrics).
\vspace{-3mm}
\label{sec:simulated}
\begin{figure}[h]
\centering
\includegraphics[width=1.\textwidth]{images/phantom_comb.pdf}
\caption{\textbf{(A)} Qualitative comparison of denoising on simulated data with varying levels of noise (from SNR 5 to 30) [see Sec.~\ref{sec:simulated} for details of simulation]. The top row shows a slice of the simulated noisy data, the middle and the bottom row correspond to the Marchenko-Pastur and Patch2Self denoised outputs, respectively. \textbf{(B)} Plots the ground truth for the same slice. \textbf{(C)} Shows line plots of RMSE values at different number of volumes at patch radii 0 and 1. \textbf{(D)} Scatter plots of ground truth against denoised data of the simulated phantom at SNR 10, 15, 20 and 20. In each case, Patch2Self suppresses more noise and shows a consistent performance gain as the SNR increases (see table \ref{tab:metric}).
}
\label{fig:all_phantom}
\end{figure}
\begin{table}[h]
\vspace{-2mm}
\centering
\resizebox{13cm}{12mm}{%
\begin{tabular}{@{}cllcccccccccc@{}}
\toprule
&
\multicolumn{2}{c}{SNR 5} &
\multicolumn{2}{c}{SNR 10} &
\multicolumn{2}{c}{SNR 15} &
\multicolumn{2}{c}{SNR 20} &
\multicolumn{2}{c}{SNR 25} &
\multicolumn{2}{c}{SNR 30} \\ \cmidrule(l){2-13}
\multirow{-2}{*}{} &
\multicolumn{1}{l|}{$R^2$} &
\multicolumn{1}{l|}{RMSE} &
\multicolumn{1}{c|}{$R^2$} &
\multicolumn{1}{c|}{RMSE} &
\multicolumn{1}{c|}{$R^2$} &
\multicolumn{1}{c|}{RMSE} &
\multicolumn{1}{c|}{$R^2$} &
\multicolumn{1}{c|}{RMSE} &
\multicolumn{1}{c|}{$R^2$} &
\multicolumn{1}{c|}{RMSE} &
\multicolumn{1}{c|}{$R^2$} &
\multicolumn{1}{c|}{RMSE} \\ \midrule
\multicolumn{1}{|c|}{Noisy} &
\multicolumn{1}{l|}{0.04} &
\multicolumn{1}{l|}{14.84} &
\multicolumn{1}{c|}{0.27} &
\multicolumn{1}{c|}{5.57} &
\multicolumn{1}{c|}{0.52} &
\multicolumn{1}{c|}{3.05} &
\multicolumn{1}{c|}{0.69} &
\multicolumn{1}{c|}{2.02} &
\multicolumn{1}{c|}{0.79} &
\multicolumn{1}{c|}{1.47} &
\multicolumn{1}{c|}{0.85} &
\multicolumn{1}{c|}{1.15} \\ \midrule
\multicolumn{1}{|c|}{Marchenko-Pastur} &
\multicolumn{1}{l|}{0.10} &
\multicolumn{1}{l|}{14.40} &
\multicolumn{1}{c|}{0.52} &
\multicolumn{1}{c|}{5.27} &
\multicolumn{1}{c|}{0.73} &
\multicolumn{1}{c|}{2.79} &
\multicolumn{1}{c|}{0.84} &
\multicolumn{1}{c|}{1.79} &
\multicolumn{1}{c|}{0.88} &
\multicolumn{1}{c|}{1.28} &
\multicolumn{1}{c|}{0.91} &
\multicolumn{1}{c|}{0.98} \\ \midrule
\rowcolor[HTML]{F8A102}
Patch2Self &
0.20 &
13.86 &
0.69 &
5.18 &
0.84 &
2.74 &
0.89 &
1.73 &
0.91 &
1.25 &
0.93 &
0.98 \\ \bottomrule
\end{tabular}%
}
\caption{Reports the $R^2$ and the Root Mean Squared Error (RMSE) metrics on the simulated data at SNRs 5 to 30. The metrics have been computed for noisy (simulated phantom) data, Marchenko-Pastur and Patch2Self denoised data by comparing against ground-truth (noise-free) data.}
\label{tab:metric}
\end{table}
\section{Introduction}
Diffusion Weighted Magnetic Resonance Imaging (DWI) \cite{basser_mattiello_lebihan_1994} is a powerful method for measuring tissue microstructure \cite{novikov_fieremans_jespersen_kiselev_2018, novikov2018modeling}. Estimates of the diffusion signal in each voxel can be integrated using mathematical models to reconstruct the white matter pathways in the brain. The fidelity of those inferred structures is limited by the substantial noise present in DWI acquisitions, due to numerous factors including thermal fluctuations. With new acquisition schemes or diffusion-encoding strategies, the sources and distribution of the noise can vary, making it difficult to model and remove. The noise confounds both qualitative (visual) and quantitative (microstructure and tractography) analysis. Denoising is therefore a vital processing step for DWI data prior to anatomical inference
DWI data consist of many 3D acquisitions, in which diffusion along different gradient directions is measured. Simple models of Gaussian diffusion are parametrized by a six-dimensional tensor, for which six measurements would be sufficient, but as each voxel may contain an assortment of tissue microstructure with different properties, many more gradient directions are often acquired. While each of these acquired volumes may be quite noisy, the fact that the same structures are represented in each offers the potential for significant denoising.
The first class of denoising methods used for DWI data were extensions of techniques developed for 2D images, such as non-local means (NL-means \cite{coupe_yger_prima_hellier_kervrann_barillot_2008} and its variants \cite{coupe_manjon._robles_collins_2012, chen2016xq}), total variation norm minimization \cite{knoll_bredies_pock_stollberger_2010}, cosine transform filtering \cite{manjon_coupe_buades_collins_robles_2012}, empirical Bayes \cite{awate2007feature} and correlation based joint filtering \cite{tristan2010dwi}. Other methods take more direct advantage of the fact that DWI measurements have a special 4D structure, representing many acquisitions of the same 3D volume at different b-values and in different gradient directions. Assuming that small spatial structures are more-or-less consistent across these measurements, these methods project to a local low-rank approximation of the data \cite{pai_rapacchi_kellman_croisille_wen_2010, manjon_coupe_concha_buades_collins_robles_2013}. The top performing methods are overcomplete Local-PCA (LPCA) \cite{manjon_coupe_concha_buades_collins_robles_2013} and its Marchenko-Pastur extension \cite{veraart_novikov_christiaens_ades-aron_sijbers_fieremans_2016}.
The current state-of-the-art unsupervised method for denoising DWI is the Marchenko-Pastur PCA, which handles the choice of rank in a principled way by thresholding based on the eigenspectrum of the expected noise covariance matrix. Note that Marchenko-Pastur PCA, like the classical total variation norm and NL-means methods as well, requires a noise model to do the denoising, either as an explicit standard deviation and covariance as in LPCA, or implicitly in the choice of a noise correction method \cite{koay_ozarslan_basser_2009, st-jean_coupe_descoteaux_2016}
We propose a self-supervised denoising method for DWI data that incorporates the key features of successful prior methods while removing the requirement to select or calibrate a noise model. Our method, Patch2Self, learns locally linear relationships between different acquisition volumes on small spatial patches. This regression framework satisfies the $\mathcal{J}$-invariance property described in \cite{batson19}, which, as long as the noise in different acquisitions is statistically independent, will guarantee denoising performance. With thorough comparisons on real and simulated data, we show that Patch2Self outperforms other unsupervised methods for denoising DWI at visual and modeling tasks.
\section{Conclusions}
\vspace{-.1cm}
This paper proposes a new method for denoising Diffusion Weighted Magnetic Resonance Imaging data, which is usually acquired at a low SNR, for the purpose of improving microstructure modeling, tractography, and other downstream tasks. We demonstrated that denoising by Patch2Self outperforms the state-of-the-art random-matrix-theory-based Marchenko-Pastur method on these subsequent analyses. To enable broad adoption of this method by the MRI community, we will incorporate an efficient and unit-tested implementation of Patch2Self into the widely-used open-source library DIPY.
\clearpage
\section*{Broader Impacts}
The broader impacts of this work fall into three categories: the direct impact on medical imaging, the theoretical impact on self-supervised learning more broadly, and the societal impact of improvements to those two technologies. In medical imaging, better denoising allows for higher quality images with fewer or shorter acquisitions, potentially making advanced acquisition schemes clinically viable, allowing for new bio-markers, and visualizing small structures such as the spinal cord in MRI.
Patch2Self provides a method for doing fast local matrix approximations, which could be used for matrix completion, subspace tracking, and subspace clustering, with applications across signal processing domain. To the extent that self-supervision enhances the ability to extract signal from poor measurements, it may expand the reach of state or private surveillance apparatuses allowing people's identities, movements, or disease status to be obtained from a greater distance and at lower cost. If a cache of easily acquired low-quality data can be efficiently used, it may open the door to exploitation by new actors.
\begin{ack}
We sincerely thank Prof. Qiuting Wen (Indiana University School of Medicine) for providing the simulated data used in the above experiment. S.F. and E.G. were supported by the National Institute of Biomedical Imaging and Bioengineering (NIBIB) of the National Institutes of Health (NIH) under Award Number R01EB027585. J.B. was supported by the Chan Zuckerberg Biohub.
\end{ack}
{\small
\bibliographystyle{ieee}
\section*{\LARGE{Appendix}}
\begin{appendix}
Diffusion values at different gradient directions (each corresponding to a 3D volume) are used populate a $q$-space such that one diffusivity value corresponds to a particular gradient direction. From this $q$-space information, one can infer the signal structure within each voxel. To model this signal as a tensor (DTI), one typically measures six values in $q$-space to capture the 3D orientation information of the tensor. By fitting a tensor to each voxel, the spatial correlations between the tensors and their orientations can be used to reconstruct white-matter fibers (tracts). Gaussian assumption of the tensor model to hinders recovery of crossing fibers from the measured signal \cite{farquharson_tournier_calamante_fabinyi_schneider-kolsky_jackson_connelly_2013}. Hence, this $q$-space is often over-sampled (with high coverage and sampling density on the spherical grid) using a High Angular Resolution Diffusion Imaging (HARDI) \cite{descoteaux_2015} acquisition scheme. This paved way for Q-Ball imaging based techniques \cite{tuch_2004, descoteaux_angelino_fitzgibbons_deriche_2007, aganj_lenglet_sapiro_yacoub_ugurbil_harel_2009, tournier_calamante_connelly_2007} which employs spherical harmonic based signal decomposition to perform more accurate tractography with ability to represent crossing information.
\end{appendix}
| -19,876.830932 |
[
-2.81640625,
2.65625
] | 19.248826 |
[
-3.294921875,
0.35498046875,
-1.6376953125,
-4.27734375,
-0.60498046875,
6.96875
] |
[
1.8427734375,
7.80859375,
1.515625,
6.2265625
] | 243 | 4,325 |
[
-2.015625,
2.345703125
] | 24.143061 |
[
-6.31640625,
-4.296875,
-4.5078125,
-2.05078125,
2.55859375,
12.4609375
] | 0.824093 | 10.140486 | 29.433526 | 11.409687 |
[
1.5263367891311646
] | -13,953.328101 | 6.172486 | -19,315.008485 | 0.224753 | 6.07291 |
[
-2.736328125,
-3.552734375,
-3.79296875,
-4.4921875,
2.3984375,
11.5859375
] |
[
-5.41015625,
-1.5693359375,
-2.013671875,
-1.3935546875,
3.673828125,
4.54296875
] | |
BkiUbLPxK6-gDw5AtUZ3
|
\section{Introduction}\label{sec:intro}
For $q = p^r$ a prime power let ${\mathbb F}_q$ denote the finite field of $q$ elements, and let $I_q(n)$ denote the number of monic
irreducible polynomials in ${\mathbb F}_q[x]$ of degree $n$. A classical result due to Gauss~\cite[pp. 602-629]{gauss} states that
\[
I_q(n) = \frac{1}{n} \sum_{d \mid n} \mu(d) q^{n/d}.
\]
A natural problem is to determine the number of monic irreducible polynomials in ${\mathbb F}_q[x]$ of degree $n$ for which
certain coefficients are prescribed, which we refer to in general as the
{\em prescribed coefficients problem}. As Panario has stated~\cite[p. 115]{Panarioquote},
{\em ``The long-term goal here is to provide existence and counting results for
irreducibles with any number of prescribed coefficients to any given values. This
goal is completely out of reach at this time. Incremental steps seem doable, but
it would be most interesting if new techniques were introduced to attack these
problems.''}
Regarding existence, building upon work of Bourgain~\cite{Bourgain} and Pollack~\cite{Pollack}, the best result to date is due
to Ha~\cite{Ha}, who in 2016 proved that there exists a monic irreducible polynomial in ${\mathbb F}_q[x]$ of degree $n$ with up to
$\lfloor{(1/4 - \epsilon)n}\rfloor$ coefficients prescribed in any positions, for any $\epsilon > 0$ and $q$ sufficiently large.
In contrast to Ha's important progress towards the above long-term goal on the existence side, the corpus of counting results
is far less well developed. Nearly all such research has focused on the subproblem of determining the number of monic irreducible polynomials
in ${\mathbb F}_q[x]$ of degree $n$ for which the first $l$ coefficients have the prescribed values $t_1,\ldots,t_l$, which we denote by
$I_q(n,t_1,\ldots,t_l)$. Although asymptotics for such subproblems have been obtained by Cohen~\cite{Cohen}, very few
exact results are known. In 1952 Carlitz gave formulae for $I_q(n,t_1)$~\cite{Carlitz}, while
in 1990 Kuz'min gave formulae for $I_q(n,t_1,t_2)$~\cite{kuzmin1,kuzmin2}; Cattell~{\em et al. } later reproduced Kuz'min's results
for the base field ${\mathbb F}_2$, in 1999~\cite{cattell}. In 2001 the three coefficient case $I_2(n,t_1,t_2,t_3)$ was solved, by Yucas and Mullen
for $n$ even~\cite{yucasmullen} and by Fitzgerald and Yucas for $n$ odd~\cite{fitzyucas}.
Formulae for $I_{2^r}(n,t_1,t_2)$ for all $r \ge 1$ were given again in 2013 by Ri {\em et al. }~\cite{RMR}.
Most recently, in 2016 Ahmadi {\em et al. } gave formulae for $I_{2^r}(n,0,0,0)$ for all $r \ge 1$~\cite{AGGMY}.
Rather than study the above subproblem instances directly, the papers~\cite{cattell,yucasmullen,fitzyucas,RMR,AGGMY} all study a set of equivalent
problems, namely counting the number of elements of ${\mathbb F}_{q^n}$ with correspondingly prescribed traces, which we refer to in general as the
{\em prescribed traces problem}. In particular, for $a \in {\mathbb F}_{q^n}$ the
characteristic polynomial of $a$ with respect to the extension ${\mathbb F}_{q^n}/{\mathbb F}_q$ is defined to be:
\begin{equation*}
\prod_{i = 0}^{n-1} (x - a^{q^i}) = x^n - T_1(a)x^{n-1} + T_2(a)x^{n-2} - \cdots + (-1)^{n-1} T_{n-1}(a)x + (-1)^n T_n(a),
\end{equation*}
with $T_l: {\mathbb F}_{q^n} \rightarrow {\mathbb F}_q$, $1 \le l \le n$ the successive trace functions
\begin{eqnarray*}
T_1(a) &=& \sum_{i = 0}^{n-1} a^{q^i},\\
T_2(a) &=& \sum_{0 \le i_1 < i_2 \le n-1} a^{q^{i_1} + q^{i_2}},\\
T_3(a) &=& \sum_{0 \le i_1 < i_2 < i_3 \le n-1} a^{q^{i_1} + q^{i_2} + q^{i_3}},\\
&\vdots&\\
T_l(a) &=& \sum_{0 \le i_1 < \cdots < i_l \le n-1} a^{q^{i_1} + \cdots + q^{i_l}},\\
&\vdots&\\
T_n(a) &=& a^{1 + q + q^2 + \cdots + q^{n-1}}.
\end{eqnarray*}
For any $n \ge l$ and $t_1,\ldots,t_l \in {\mathbb F}_q$, let $F_q(n,t_1,\ldots,t_l)$ denote the number of elements $a \in {\mathbb F}_{q^n}$ for which $T_1(a) = t_1$, \ldots, $T_l(a) = t_l$.
If for a given $q = p^r$ and $l \ge 1$ one determines $F_q(n,t_1,\ldots,t_l)$ for all $(t_1,\ldots,t_l) \in ({\mathbb F}_q)^l$, then an application of the multinomial theorem and a
generalised M\"obius inversion-type argument gives $I_q(n,t_1,\ldots,t_l)$ for all $(t_1,\ldots,t_l) \in ({\mathbb F}_q)^l$, and vice versa, hence the equivalence.
The formulae for the equivalence follow from the approach of Miers and Ruskey~\cite{miersruskey2},
extending the subcases already proven for binary fields~\cite{cattell,yucasmullen,fitzyucas,RMR,AGGMY}.
For our main case of interest for which $l < p$, each $I_q(n,t_1,\ldots,t_l)$ can be expressed in terms of only one $F_q(n,t_{1}',\ldots,t_{l}')$,
and vice versa, see~\S\ref{sec:transform0}.
As well as proving formulae for $I_{2^r}(n,0,0,0)$ for all $r \ge 1$, the work~\cite{AGGMY} also explained an intriguing phenomenon,
which is that the formulae for $F_2(n,t_1,t_2)$ and $F_2(n,t_1,t_2,t_3)$ proven in~\cite{cattell} and~\cite{yucasmullen,fitzyucas}
depend on $n \bmod 8$ and on $n \bmod 24$, respectively. In particular, by Fourier analysing the formulae the present author showed
that they are related to the number of ${\mathbb F}_{2^n}$-rational affine points of certain genus one and two supersingular curves defined
over ${\mathbb F}_2$. A simple argument gave a new derivation of the formulae for the $t_1 = 0$ cases and the $n$ odd cases, and since the
curves featured are supersingular their normalised Weil numbers are roots of unity, which explains the observed periodicity.
In this paper we greatly extend the curve-based approach of~\cite{AGGMY} in order to prove the following theorem.
Let $\overline{{\mathbb Q}}$ be an algebraic closure of ${\mathbb Q}$ and let $\overline{{\mathbb Z}}$ be the integral closure of ${\mathbb Z}$ in $\overline{{\mathbb Q}}$.
\begin{theorem}\label{thm:maintheorem}
Let $q = p^r$, $l < p$, $\overline{n} \in \{1,\ldots,p-1\}$, and let
\begin{equation*}\label{eq:N}
N = \begin{cases}
\begin{array}{ll}
(q^l(ql - q -l)+q) & \ \text{if} \ r = 1 \\
(q^l(ql - q -l)+q)(p-1) & \ \text{if} \ r > 1.\\
\end{array}
\end{cases}
\end{equation*}
Then there exist $\omega_1,\ldots,\omega_N \in \overline{{\mathbb Z}}$ (not necessarily distinct), all of norm $\sqrt{q}$,
such that for all $(t_1,\ldots,t_l) \in ({\mathbb F}_q)^l$ there exist explicitly determined $\upsilon_1,\ldots, \upsilon_N \in \{0,\pm1\}$
such that for all $n \equiv \overline{n} \pmod{p}$ with $n \ge l$ one has
\begin{equation}\label{eq:maintheorem}
F_q(n,t_1,\ldots,t_l) = \frac{1}{q^l}\Big( q^n + \frac{q-1}{p-1} \sum_{k=1}^N \upsilon_k \omega_{k}^n \Big) = q^{n-l} + \mathcal{O}(q^{n/2}).
\end{equation}
Moreover, there exists an explicit deterministic algorithm for computing $\omega_1,\ldots,\omega_N$
in $q^{l}\cdot\tilde{\mathcal{O}}(l^5 p^4)$ bit operations when $r=1$, and $q^{l+1}\cdot\tilde{\mathcal{O}}(l^5 p^4 r^3)$ bit operations when $r>1$.
\end{theorem}
Here we use the soft-oh notation $\tilde{\mathcal{O}}$, which ignores factors which are logarithmic in the argument. Note that
Theorem~\ref{thm:maintheorem} implies an asymptotic equidistribution result for the prescribed traces problem, as one expects.
The main idea behind the algorithm is to associate to the prescribed traces problem a fibre product of $l$ Artin-Schreier curves defined over ${\mathbb F}_q$,
whose number of ${\mathbb F}_{q^n}$-rational affine points solves the problem. In order to count this number we propose two methods.
The first -- which we refer to as the {\em direct method} -- computes a single $F_q(n,t_1,\ldots,t_l)$ by replacing the fibre product
by an intersection. However, this results in curves of relatively large genus for which we do not know of an efficient point counting algorithm.
The second -- which we refer to as the {\em indirect}, or {\em `batching' method} -- determines all of the $q^l$ counts
$F_q(n,t_1,\ldots,t_l)$ simultaneously, by transforming these counting problems into a set of $q^l$ equivalent counting problems.
Each of the latter problems requires computing the zeta function of an Artin-Schreier curve defined over ${\mathbb F}_q$,
which is accomplished using a $p$-adic algorithm due to Lauder and Wan~\cite{LauderWan}.
While obtaining the relevant curves is immediate, determining their zeta functions is a fundamentally computational problem, so one
should not expect to be able to simply write down general formulae for $F_q(n,t_1,\ldots,t_l)$.
There may of course exist faster algorithms for determining the exact formulae than the approach we take, particularly when only one
$F_q(n,t_1,\ldots,t_l)$ is required, but we emphasise that the one presented here constitutes the first algorithmic approach to solving
the prescribed traces problem exactly, which therefore represents a shift in perspective with regard to its study.
Although the indirect method is very efficient for computing all $q^l$ counts $F_q(n,t_1,\ldots,t_l)$ simultaneously,
a potential disadvantage is that the number $N$ of algebraic integers featuring in Theorem~\ref{thm:maintheorem} may be larger than for
the direct method, even accounting for multiplicities. There may, therefore, be some redundancy in the resulting formulae.
Such redundancy can be eliminated with some postcomputation by identifying, for each $\overline{n}$, cancellations between linear
combinations of $n$-th powers of the $\omega_k$'s which are valid for all $n \equiv \overline{n} \pmod{p}$, see~\S\ref{sec:ternaryzetas} for an example.
One should therefore attempt to use the direct method first, and if it is not computationally feasible then use the indirect method,
noting that some redundancy elimination may be required.
It should be no surprise that the formulae for $F_q(n,t_1,\ldots,t_l)$ arise from the number of points on curves. As noted by Voloch~\cite{voloch},
a method used by Hayes~\cite{hayes}, Hsu~\cite{hsu}, Voloch himself and others to estimate $I_q(n,t_1,\ldots,t_l)$ relates these counts to the number
of points over ${\mathbb F}_{q^n}$ of certain curves defined over ${\mathbb F}_q$ whose function fields are subfields of the so-called cyclotomic function fields. The equivalence to $F_q(n,t_1,\ldots,t_l)$ then implies that these counts are related to the number of points on curves as well.
The prescribed traces problem that we partially solve in this work is similar to a problem studied by Miers and Ruskey~\cite{miersruskey1}.
In particular, using combinatorial techniques, expressions were given for the number of strings over finite rings with prescribed symmetric
function evaluations. However, while this problem is similar the techniques used do not seem applicable to the prescribed traces problem as we
have defined it. More relevant to our case, but still seemingly inapplicable are the expressions given by Miers and Ruskey for the number of equivalence
classes of aperiodic strings over finite rings with prescribed symmetric function evaluations, under rotation~\cite{miersruskey2}. However,
as already mentioned the formulae for the equivalence between the prescribed coefficients problem and the prescribed traces problem that we study
follow easily from this work.
A fundamental limitation of our proposed algorithm is that it breaks for $l \ge p$ due to the failure of Newton's identities in positive characteristic.
Nevertheless, by employing a slightly different but likely limited technique to bypass the $l < p$ constraint, we computed curves for $q = 2$, $l \le 7$
and $n$ odd whose number of ${\mathbb F}_{2^n}$-rational affine points solve the $F_2(n,t_1,\ldots,t_l)$ counting problems.
In fact, the ingredients of our main algorithm were initially developed for $q = 2$, since even the $l = 4$
cases were open problems with the only previous result being an approximation to the counts~\cite{Koma,Komathesis} (there is however a small
error in the transforms used, see~\S\ref{sec:transformPCPPTP}).
We used the computational algebra system Magma~\cite{magma} to compute the corresponding zeta functions for $l = 4$ and $l = 5$, obtaining explicit formulae for these open problems for $n$ odd. Interestingly, for these cases the characteristic polynomial of the Frobenius
endomorphism of the featured curves have factors that arise from ordinary -- rather than supersingular -- abelian varieties, and a simple argument
implies that the set of formulae $\{ F_2(n,t_1,t_2,t_3,t_4) \}_{n \ge 4}$ and $\{ F_2(n,t_1,t_2,t_3,t_4,t_5) \}_{n \ge 5}$ for each
$(t_1,\ldots,t_4)$ and $(t_1,\ldots,t_5)$ respectively, can not be periodic in $n$, as in the $l = 2$ and $l = 3$ cases.
This perhaps explains why there had been little progress in the four coefficients problem for the past $15$ years, since there
is not a finite set of cases to enumerate.
We also give formulae for a subset of $F_2(n,t_1,t_2,t_3,t_4)$ and $F_2(n,t_1,t_2,t_3,t_4,t_5)$ which are valid {\em for all}
$n \ge 4$ and $n \ge 5$, respectively. Somewhat surprisingly, the formulae for $F_2(n,0,0,0,*,t_5)$ for $n \ge 5$ -- where the asterisk
means that we do not mind what the value of the fourth trace function is -- are periodic in $n$ with period $120$. It is therefore conceivable
that these formulae could have been discovered without using our curve-based approach, although no doubt far less efficiently.
The method for $q = 2$ extends to one for any $q = 2^r$ and $l \le 7$, and also extends to $q = 3^r$ and $l \le 3$, which features
non-supersingular Weil numbers for just three prescribed coefficients when $r=1$.
The sequel is organised as follows.
In~\S\ref{sec:generalalg} we present our main algorithm for arbitrary $q = p^r$, $l < p$ and $n$ coprime to $p$, and
present details of a very efficient proof-of-concept demonstration for $q=5$ and $l=4$.
In~\S\ref{sec:char2} we present direct and indirect methods for solving the prescribed traces problem for $q = 2$ and $n$ odd and discuss
its possible limitations.
Then in~\S\ref{sec:binaryzetas} we apply these methods to compute curves for $l \le 7$, some for $n$ odd and some for all relevant $n$, provide explicit formulae for $l = 4$ and $l = 5$, and for $l=4$ detail a connection to binary Kloosterman sums and provide the correct transform from
the prescribed traces problems to the prescribed coefficients problem.
In~\S\ref{sec:ternaryzetas} we present some formulae for the $q = l = 3$ case.
Finally, in~\S\ref{sec:summary} we discuss some of the computational challenges and theoretical questions arising from our
approach in the general case and propose some natural open problems.
We assume the reader is familiar with curves and their zeta functions; should they not be, the relevant definitions
may be found in~\cite[\S3]{AGGMY}, for example. The code used for all interesting computations performed with Magma and Maple~\cite{maple}
is openly available from \url{https://github.com/robertgranger/CountingIrreducibles}, with the relevant files indicated in footnotes in the text.
\section{The Main Algorithm}\label{sec:generalalg}
In this section we present an algorithm for solving the prescribed traces problem -- and thus the prescribed coefficients problem -- exactly for any
prime power $q = p^r$, any $1 \le l < p$ and any $n \ge l$ coprime to $p$. The input traces whose values are prescribed are $T_{1},\ldots,T_{l}$,
i.e., the problem is to compute
\begin{equation}\label{eq:mainalg1}
F_q(n,t_1,\ldots,t_l) = \#\{a \in {\mathbb F}_{q^n} \mid T_1(a) = t_1,\ldots, T_l(a) = t_l\},
\end{equation}
for any $t_1,\ldots,t_l \in {\mathbb F}_q$. We begin with what we refer to as the direct method in~\S\ref{sec:maindirect} and present the indirect,
or batching method in~\S\ref{sec:mainindirect}.
\subsection{Direct method}\label{sec:maindirect}
We begin with the following extremely simple lemma, whose proof follows from~\cite[Theorem 2.25]{lidl}.
\begin{lemma}\label{lem:paramq}
\begin{enumerate}[label={(\arabic*)}]
\item For $a \in {\mathbb F}_{q^n}$ the condition $T_1(a) = 0$ is equivalent to $a = a_{1}^q - a_{1}$, for $q$ different $a_1 \in {\mathbb F}_{q^n}$.
\item For $a \in {\mathbb F}_{q^n}$ and $n \not\equiv 0 \pmod{p}$, the condition $T_1(a) = t_1$ is equivalent to $a = a_{1}^q - a_{1} + t_1/n $,
for $q$ different $a_1 \in {\mathbb F}_{q^n}$.
\end{enumerate}
\end{lemma}
We now recall Newton's identities over ${\mathbb Z}$ (see e.g., \cite[Theorem 1.75]{lidl}) with indeterminates $\alpha_1,\ldots,\alpha_n$.
Abusing notation slightly, we refer to the elementary symmetric polynomials in $\alpha_1,\ldots,\alpha_n$ as
$T_1(\alpha),T_2(\alpha),\ldots$, and to the power sum symmetric polynomials $\alpha_{1}^j + \cdots + \alpha_{n}^j$
as $T_1( \alpha^j)$ for $j \ge 1$, i.e., we work in the ring of symmetric functions, suppressing the dependence on $n$.
We use the convention that $T_0(\alpha) = 1$.
\begin{lemma}
For all $k \ge 1$ and $n \ge k$ we have
\begin{equation}\label{eq:NI}
k\,T_k(\alpha) = \sum_{j=1}^k (-1)^{j-1} T_{k-j}(\alpha)T_1(\alpha^j).
\end{equation}
\end{lemma}
Letting $\alpha = a$ and substituting $T_j(a) = t_j$ for $1 \le j \le k$, Eq.~(\ref{eq:NI}) becomes
\begin{eqnarray}
\nonumber k t_k &=& \sum_{j=1}^k (-1)^{j-1} t_{k-j} T_1(a^j)\\
\label{eq:recurrencespecialised} &=& (-1)^{k-1} T_1(a^k) + \sum_{j=1}^{k-1}(-1)^{j-1}t_{k-j} T_1(a^j),
\end{eqnarray}
since by convention we have $T_0(\cdot)=1$ and so $t_0 = 1$. Eq.~(\ref{eq:recurrencespecialised}) allows one to express each $T_1(a^k)$ as a polynomial
in $t_1,\ldots,t_k$ only, which we refer to as $p_k(t_1,\ldots,t_k)$. In particular, $p_1(t_1) = t_1$, $p_2(t_1,t_2) = t_{1}^2 - 2t_2$,
$p_3(t_1,t_2,t_3) = 3t_3 + t_{1}^3 - 3t_1t_2$, and in general we have
\begin{equation}\label{eq:def:p_k}
p_k(t_1,\ldots,t_k) = T_1(a^k) = (-1)^{k-1}\Big( kt_k - \sum_{j=1}^{k-1}(-1)^{j-1}t_{k-j}p_j(t_1,\ldots,t_j) \Big).
\end{equation}
Since $1 \le k \le l < p$, each $t_k$ features in the condition on $T_1(a^k)$ in Eq.~(\ref{eq:def:p_k}) and does so linearly.
Therefore for each $(t_{1}',\ldots,t_{l}') \in ({\mathbb F}_q)^l$ there is a unique $(t_1,\ldots,t_l) \in ({\mathbb F}_q)^l$ such that for all $1 \le k \le l$ we have
$t_{k}' = p_k(t_1,\ldots,t_k)$, and vice versa. Eq.~(\ref{eq:mainalg1}) thus becomes
\begin{equation}\label{eq:newmain}
\# \{ a \in {\mathbb F}_{q^n} \mid T_{1}(a) = t_{1}', T_1(a^2) = t_{2}',\ldots,T_1(a^l) = t_{l}'\}.
\end{equation}
To evaluate Eq.~(\ref{eq:newmain}) with what we call the direct method, let $\overline{n} \in \{1,\ldots,p-1\}$ and suppose
$n \equiv \overline{n} \pmod{p}$. Then by introducing variables $a_1,\ldots,a_l$
and repeatedly applying Lemma~\ref{lem:paramq}, Eq.~(\ref{eq:newmain}) equals
\begin{equation}\label{eq:main_direct}
\frac{1}{q^l} \# \{ (a,a_1,\ldots,a_l) \in ({\mathbb F}_{q^n})^{l+1} \mid a_{1}^q - a_{1} = a - t_{1}'/\overline{n}, \ldots,
a_{l}^q - a_{l} = a^l - t_{l}'/\overline{n} \}.
\end{equation}
Note that if $t_{1}' = \cdots = t_{l}' = 0$ then one need not introduce $\overline{n}$ at all, and the count~(\ref{eq:main_direct}) is valid for
all $n \ge l$. Rather than work with this intersection one can alternatively define the curves
$C_k/{\mathbb F}_{q}: a_{1}^q - a_{1} = a^k - t_{k}'/\overline{n}$ for $1 \le k \le l$ and consider their fibre product, \`a la~\cite[\S2]{vandegeer}.
In~\S\ref{sec:mainindirect} we take a similar but more elementary approach, working only with an associated set of Artin-Schreier curves
of much smaller genus, since there exists a practical algorithm for computing their zeta functions due to Lauder and Wan~\cite{LauderWan}.
We do so by evaluating the count~(\ref{eq:newmain}) indirectly, which allows one to solve the prescribed traces problem for all $q^l$
such $F_q(n,t_1,\ldots,t_l)$ simultaneously.
\subsection{A transform of the prescribed traces problem}\label{sec:transform1p}
In this subsection we transform the problem of counting the number of elements of ${\mathbb F}_{q^n}$ with prescribed traces to the
problem of counting the number of elements for which linear combinations of the trace functions evaluate to $1$.
The transform is more general than is required for the target problem of interest and may be applied to any number of functions from ${\mathbb F}_{q^n}$
to ${\mathbb F}_q$. Therefore let $m \ge 1$ be the number of such functions. We first fix some notation.
We require a bijection from the integers $\{0,\ldots,q^m-1\}$ to $({\mathbb F}_{q})^m$, the image of an input $i$ being denoted by $\vec{i}$.
One can for instance take the base-$q$ expansion of $i$ to give $i_{m-1}^{'}q^{m-1} + \cdots + i_{0}^{'}$ and then set
$\vec{i} = (i_{m-1},\ldots,i_0) = (\tau(i_{m-1}^{'}),\ldots,\tau(i_{0}^{'}))$, where $\tau: \{0,\ldots,q-1\} \rightarrow {\mathbb F}_q$ is defined by
fixing a degree $r$ monic irreducible $f \in {\mathbb F}_p[x]$ and a polynomial basis for ${\mathbb F}_{p^r}/{\mathbb F}_p$, and mapping the base-$p$ expansion of an
integer in $\{0,\ldots,q-1\}$ to the polynomial with those coefficients. Note that according to this definition, $\vec{0} = (0,\ldots,0)$ is
the all-zero vector in $({\mathbb F}_{q})^m$.
Let $f_0,\ldots,f_{m-1}: {\mathbb F}_{q^n} \rightarrow {\mathbb F}_q$ be any functions and let $\vec{f} = (f_{m-1},\ldots,f_0)$.
For $\vec{i} = (i_{m-1},\ldots,i_0),\vec{j} = (j_{m-1},\ldots,j_0) \in ({\mathbb F}_{q})^m$ let $\vec{i} \cdot \vec{j}$ denote the usual inner product.
For any $\vec{i} \in ({\mathbb F}_q)^m$, let $\vec{i}\cdot \vec{f}$ denote the function
\[
\sum_{k=0}^{m-1} i_k f_k: {\mathbb F}_{q^n} \rightarrow {\mathbb F}_q,
\]
and let $V_1(\vec{i}\cdot \vec{f})$ denote the number of elements of ${\mathbb F}_{q^n}$ for which $\vec{i} \cdot \vec{f}$ evaluates to $1$. We define
$V_1(\vec{0} \cdot \vec{f})$ to be $q^n$. Finally, let $N(\vec{j}) = N(j_{m-1},\ldots,j_{0})$ denote the number of $a \in {\mathbb F}_{q^n}$ such that
$f_k(a) = j_k$, for $k = 0,\ldots,m-1$.
Our goal is to express any $N(\vec{j})$ in terms of the $V_1(\vec{i}\cdot \vec{f})$, but we begin by first solving the inverse problem,
i.e., expressing any $V_1(\vec{i}\cdot \vec{f})$ in terms of the $N(\vec{j})$.
\begin{lemma}
With the notation as above, for $\vec{i} \in ({\mathbb F}_q)^m \setminus \{\vec{0}\}$ we have
\begin{equation}\label{eq:basicrelationcharp}
V_1(\vec{i}\cdot \vec{f}) = \sum_{\vec{i} \cdot \vec{j} \, = \, 1} N(\vec{j}).
\end{equation}
\end{lemma}
\begin{proof}
By definition, we have $V_1(\vec{i}\cdot \vec{f}) = \#\{ a \in {\mathbb F}_{q^n} \mid \vec{i}\cdot \vec{f} (a) = 1\} =
\#\{ a \in {\mathbb F}_{q^n} \mid \sum_{k=0}^{m-1} i_k f_k(a) = 1\}$.
Since $N(\vec{j})$ counts precisely those $a \in {\mathbb F}_{q^n}$ such that $f_k(a) = j_k$, we must count over all those $\vec{j}$ for which
$\sum_{k=0}^{m-1} i_k f_k(a) = 1$, i.e., those such that $\sum_{k=0}^{m-1} i_k j_k = 1$. \qed
\end{proof}
Writing Eq.~(\ref{eq:basicrelationcharp}) in matrix form, for $i,j \in \{0,\ldots,q^m-1\}$ we have
\[
\begin{bmatrix}
V_1(\vec{i}\cdot\vec{f})
\end{bmatrix}^T
=
S_{q,m}
\cdot
\begin{bmatrix}
N(\vec{j})
\end{bmatrix}^T,
\]
where
\begin{equation*}\label{eq:spm}
(S_{q,m})_{i,j} = \begin{cases}
\begin{array}{ll}
1 & \ \text{if} \ \vec{i}\cdot\vec{j} = 1 \ \text{or if} \ \vec{i} = \vec{0} \\
0 & \ \text{otherwise}
\end{array}.
\end{cases}
\end{equation*}
We have the following lemma.
\begin{lemma}\label{lem:spm1}
For all prime powers $q = p^r$ and $m \ge 1$, the $q^m \times q^m$ matrix $S_{q,m}$ is invertible over ${\mathbb Q}$.
\end{lemma}
\begin{proof}
Indexing the rows and columns by $i$ and $j$ for $0 \le i,j \le q^m - 1$, the $0$-th row of $S_{q,m}$ consists of $1$'s only,
while besides the initial $1$, the $0$-th column consists of $0$'s only. Therefore no ${\mathbb Q}$-linear combination of rows $1$ to $q^m-1$ can cancel the $1$
in position $(0,0)$. Hence if one shows that the submatrix
\begin{equation*}
S_{1 \le i,j \le q^m - 1} = \begin{cases}
\begin{array}{ll}
1 & \ \text{if} \ \vec{i}\cdot\vec{j} = 1\\
0 & \ \text{otherwise}
\end{array}
\end{cases}
\end{equation*}
of $S_{q,m}$ is invertible then we are done, since this implies that $S_{q,m}$ has full rank. We claim that the inverse of $S$ is
\begin{equation*}\label{eq:spm}
S_{1 \le i,j \le q^m - 1}^{\text{inv}} = \frac{1}{q^{m-1}} \begin{cases}
\begin{array}{rl}
1 & \ \text{if} \ \vec{i}\cdot\vec{j} = 1\\
-1 & \ \text{if} \ \vec{i}\cdot\vec{j} = 0\\
0 & \ \text{otherwise}
\end{array}
\end{cases}.
\end{equation*}
Let $R = S \cdot S^{\text{inv}}$. Then for $1 \le i,j \le q^m-1$ one has
\begin{equation}\label{eq:rij}
R_{i,j} = \sum_{k = 1}^{q^m - 1} S_{i,k} \cdot S_{k,j}^{\text{inv}} =
\sum_{\substack{k = 1,\\
\vec{i}\cdot\vec{k} = 1, \ \vec{j}\cdot\vec{k} = 1}}^{q^m - 1} 1
\ -\sum_{\substack{k = 1,\\
\vec{i}\cdot\vec{k} = 1, \ \vec{j}\cdot\vec{k} = 0}}^{q^m - 1} 1 \ .
\end{equation}
If $i = j$ then the second term of the r.h.s. of Eq.~(\ref{eq:rij}) is zero, while the first is $q^{m-1}$, since if one chooses a non-zero
component $i_l$ of $\vec{i}$ (of which there is at least one as $i \ne 0$), then one can freely choose the $m-1$ coefficients
of $\vec{k}$ other than $k_l$, while the condition $\vec{i}\cdot\vec{k} = 1$ entails that
\[
k_l = (1 - \sum_{\substack{w = 0,\\w \ne l}}^{m-1} i_w k_w)/i_l,
\]
which is well-defined because $i_l$ is invertible.
Now assume $i \ne j$. If possible choose $l,l'$ such that $l \ne l'$ and $i_l \ne 0$ and $j_{l'} \ne 0$.
Then considering the set of all $k$ for which $\vec{i}\cdot\vec{k} = 1$ as described above, as $k_{l'}$ varies over ${\mathbb F}_q$,
so does $k_{l'} j_{l'}$. Hence both terms of the r.h.s. of Eq.~(\ref{eq:rij}) are precisely $q^{m-2}$, since there are $m-2$ free components of
$\vec{k}$. For the remaining case where $\vec{i}$ and $\vec{j}$ both have only one non-zero component, in position $l$ say, then both terms
of the r.h.s. of Eq.~(\ref{eq:rij}) are zero, since for the first there is no $k_l$ for which $i_l k_l = 1$ and $j_l k_l = 1$ since
$i_l \ne j_l$, while for the second the condition $\vec{j}\cdot\vec{k} = 0$ implies $k_l = 0$, in which case $i_l k_l = 1$ can not hold.
Therefore, $R$ is the $q^m-1 \times q^m-1$ identity matrix.\qed
\end{proof}
To compute $N(\vec{0})$, we claim that
\begin{equation}\label{eq:N0}
N(\vec{0}) = V_1( \vec{0}\cdot\vec{f}) - \frac{1}{q^{m-1}}\sum_{i=1}^{q^m-1} V_1( \vec{i}\cdot\vec{f} ).
\end{equation}
Since $V_1( \vec{0}\cdot\vec{f}) = q^n = \sum_{j = 0}^{q^m-1} N(\vec{j})$ by definition, Eq.~(\ref{eq:N0}) is equivalent to
\begin{equation}\label{eq:N01}
\sum_{j = 1}^{q^m-1} N(\vec{j}) = \frac{1}{q^{m-1}}\sum_{i=1}^{q^m-1} V_1( \vec{i}\cdot\vec{f} ) =
\frac{1}{q^{m-1}} \sum_{i=1}^{q^m-1} \sum_{\vec{i} \cdot \vec{j} \, = \, 1} N(\vec{j}),
\end{equation}
with the latter equality given by Eq.~(\ref{eq:basicrelationcharp}). Considering the r.h.s. of Eq.~(\ref{eq:N01}), the number of occurrences
of $N(\vec{j})$ is $q^{m-1}$ since as we argued previously if one fixes an $l$ for which $j_l \ne 0$, then one can choose the components of
$\vec{i}$ other than $i_l$ freely, while $i_l$ is then fixed by the condition $\vec{i}\cdot\vec{j} = 1$.
We have therefore proven the following:
\begin{equation}\label{eq:Sminus1}
S_{q,m}^{-1} = \frac{1}{q^{m-1}}
\renewcommand\arraystretch{1.3}
\mleft[
\begin{array}{c|ccc}
q^{m-1} & -1 & \cdots & - 1 \\
\hline
0 & & & \\
\vdots & & S_{1 \le i,j \le q^m - 1}^{inv} =\begin{cases}
\begin{array}{rl}
1 & \ \text{if} \ \vec{i}\cdot\vec{j} = 1\\
-1 & \ \text{if} \ \vec{i}\cdot\vec{j} = 0\\
0 & \ \text{otherwise}
\end{array}
\end{cases} & \\
0 & & & \\
\end{array}
\mright]
\end{equation}
Thus in order to compute any of the $q^m$ possible outputs $N(\vec{j})$ of any set of $m$ functions $\vec{f}$, it is sufficient to count the number
of evaluations to $1$ of all the $q^m-1$ non-zero ${\mathbb F}_q$-linear combinations of the functions, and then apply $S_{q,m}^{-1}$.
In particular, one may choose the $f_{0},\ldots,f_{m-1}$ to be any subset of the trace functions $T_1,\ldots,T_{n}$, or in our case of
interest, the trace functions $T_1,\ldots,T_{l}$.
\subsection{Indirect method}\label{sec:mainindirect}
As per the transform of the previous subsection let $\vec{f} = (T_1(a^l),T_1(a^{l-1}),\ldots,T_1(a))$ and for
$1 \le i \le q^{l} - 1$ let $\vec{i} = (i_{l-1},\ldots,i_0)$. Computing formulae for $V_1(\vec{i} \cdot \vec{f})$ for each
such $i$ produces formulae for all $q^l$ possible counts in Eq.~(\ref{eq:newmain}), which then uniquely determine $F_q(n,t_1,\ldots,t_l)$.
For $\overline{n} \in \{1,\ldots,p-1\}$ and all $n \equiv \overline{n} \pmod{p}$, applying Lemma~\ref{lem:paramq} we have
\begin{eqnarray}
\nonumber V_1(\vec{i} \cdot \vec{f}) &=& \#\{ a \in {\mathbb F}_{q^n} \mid \sum_{k=0}^{l-1} i_k f_k(a) = 1\}\\
\label{eq:artin} &=& \frac{1}{q} \#\{ (a,a_1) \in ({\mathbb F}_{q^n})^2 \mid a_{1}^q - a_1 = -\frac{1}{\overline{n}} + \sum_{k=0}^{l-1} i_{k} a^{k+1} \big)\}.
\end{eqnarray}
In order to compute the zeta functions of the Artin-Schreier curves in~(\ref{eq:artin}) we use a $p$-adic point counting algorithm due to
Lauder and Wan that is well suited to this purpose~\cite[Algorithm 25]{LauderWan}.
Let $f \in {\mathbb F}_q[a]$ be of degree $d$ not divisible by $p$, let $C_f/{\mathbb F}_q$ be the affine curve with equation $a_{1}^p - a_1 = f(a)$ and let
$\tilde{C}_f$ denote the unique smooth projective curve that is birational to $C_f$.
By Weil~\cite{weil}, we know that the zeta function $Z(\tilde{C}_f,T)$ of $\tilde{C}_f$ satisfies
\begin{equation}\label{zeta}
Z(\tilde{C}_f,T) = \frac{P(\tilde{C}_f,T)}{(1-T)(1-qT)},
\end{equation}
where $P$ is polynomial of degree $2g$ with $g = (p-1)(d-1)/2$ the genus of $\tilde{C}_f$, and which factorises as
$\prod_{k=1}^{2g} (1 - \omega_k t)$ with $|\omega_k| = \sqrt{q}$. Since $\tilde{C}_f$ is birational to $C_f$ their
respective zeta functions differ only by a constant factor and thus to compute $Z(C_f,T)$ it suffices to compute $Z(\tilde{C}_f,T)$.
An exponential sum approach also allows one to deduce the form of $Z(C_f,T)$ directly~\cite[\S2.2]{LauderWan}.
We have the following.
\begin{theorem}\cite[Theorem 1]{LauderWan}
\label{thm:LW}
The zeta function of the smooth projective curve $\tilde{C}_f$ (and thus the affine curve $C_f$) may be computed deterministically in
$\tilde{\mathcal{O}}(d^5 p^4 r^3)$ bit operations.
\end{theorem}
We note that one does not need to compute the unique smooth projective curve $\tilde{C}_f$ in order to apply the Lauder-Wan
algorithm.
Let the right hand side of the curves in~(\ref{eq:artin}) be denoted by $\overline{p}_{\vec{i},\overline{n}}(a)$ and denote each curve by
\begin{equation}\label{eq:maincurve}
C_{\vec{i},\overline{n}}/{\mathbb F}_{q}: a_{1}^q - a_{1} = \overline{p}_{\vec{i},\overline{n}}(a).
\end{equation}
The degree of $\overline{p}_{\vec{i},\overline{n}}(a)$ is
$k'+1$ where $k'$ is the largest $0 \le k' \le l-1$ such that $i_{k'} \neq 0$. Since $k'+1 \le l < p$, for all $1 \le i \le q^{l}-1$
the degree of $\overline{p}_{\vec{i},\overline{n}}(a)$ is less than $p$ and thus not divisible by $p$, and so this precondition
of the Lauder-Wan algorithm is satisfied.
However, the Lauder-Wan algorithm can only be applied when the left hand side of~(\ref{eq:maincurve}) is $a_{1}^p - a_1$; we therefore need to
reduce to this case. This issue was already addressed in~\cite[Corollary 1]{AGGMY} for the case $p = 2$ (albeit with a sign error) using a theorem of
Kani-Rosen~\cite{kanirosen} and by studying quotients arising from the action of automorphisms of the left hand side only. The correct generalisation
is as follows, where for an affine curve $C/{\mathbb F}_q$ we denote by $\#C({\mathbb F}_{q^n})$ the number of ${\mathbb F}_{q^n}$-rational affine points of $C$.
\begin{lemma}\label{thm:extensionreduction}
Let $p \ge 2$ be prime, for $r \ge 1$ let $q = p^r$, let $f \in {\mathbb F}_q[x]$, let $C/{\mathbb F}_q$ be the Artin-Schreier curve $y^q - y = f(x)$ and for
$\alpha \in {\mathbb F}_{q}^{\times}$ let $C_{\alpha}/{\mathbb F}_q$ be the Artin-Schreier curve $y^p - y = \alpha f(x)$. Then for all $n \ge 1$ we have
\begin{equation}\label{eqn:extensionreduction}
(p-1)\#C({\mathbb F}_{q^n}) - \sum_{\alpha \in {\mathbb F}_{q}^{\times}} \#C_{\alpha}({\mathbb F}_{q^n}) = (p - q)q^n.
\end{equation}
\end{lemma}
Thus computing the zeta function of $C/{\mathbb F}_q$ reduces to computing the zeta functions of the $q-1$ curves $C_{\alpha}/{\mathbb F}_q$.
Lemma~\ref{thm:extensionreduction} may be proven using an exponential sum argument and such a proof can be found (in Persian) in~\cite{omranproof}.
For completeness we include it here and thank Omran Ahmadi for communicating to us its translation.
\begin{proof}
For $m,r \ge 1$ positive integers let $\text{Tr}_{m}:{\mathbb F}_{p^m} \rightarrow {\mathbb F}_p: x \mapsto x + x^p + x^{p^2} + \cdots + x^{p^{m-1}}$ denote the absolute
trace function, and let $\text{Tr}_{rm|r}:{\mathbb F}_{p^{rm}} \rightarrow {\mathbb F}_{p^r}: x \mapsto x + x^{p^r} + x^{p^{2r}} + \cdots + x^{p^{(m-1)r}}$ denote
the trace. By Lemma~\ref{lem:paramq}~(1) if $\text{Tr}_{rn}(\alpha f(a))=0$, then there are $p$ points on $C_\alpha$ with $x$-coordinates equal to $a$,
and otherwise none since $\text{Tr}_{rn}(y^p - y) = 0$ for all $y \in {\mathbb F}_{p^{rn}}$. Therefore the number of points on $C_{\alpha}$ with
$x$-coordinate equal to $a$ is $\sum \psi( \text{Tr}_{rn}(\alpha f(a)))$
where the sum is over the additive characters of ${\mathbb F}_p$. Denoting the set of all additive characters of ${\mathbb F}_p$ by $\Psi$ and its trivial character
by $\psi_0$ we have
\[
\#C_\alpha({\mathbb F}_{p^{rn}})= \sum_{a \in {\mathbb F}_{p^{rn}}} \sum_{\psi \in \Psi} \psi( \text{Tr}_{rn}(\alpha f(a))) = p^{rn} +
\sum_{a \in {\mathbb F}_{p^{rn}}} \sum_{\psi \neq \psi_0} \psi( \text{Tr}_{rn}(\alpha f(a))).
\]
Summing over all $\alpha \in {\mathbb F}_{p^r}^{\times}$ we obtain
\begin{eqnarray}
\nonumber S = \sum_{\alpha \in {\mathbb F}_{p^r}^{\times}} \#C_\alpha({\mathbb F}_{p^{rn}}) &=& p^{rn}(p^r-1) +
\sum_{\alpha \in {\mathbb F}_{p^r}^{\times}} \sum_{a \in {\mathbb F}_{p^{rn}}} \sum_{\psi \neq \psi_0} \psi( \text{Tr}_{rn}(\alpha f(a)))\\
\label{Total-Sum} &=& p^{rn}(p^r-1) +
\sum_{a \in {\mathbb F}_{p^{rn}}} \sum_{\alpha \in {\mathbb F}_{p^r}^{\times}} \sum_{\psi \neq \psi_0} \psi( \text{Tr}_{rn}(\alpha f(a))).
\end{eqnarray}
Since $\alpha \in {\mathbb F}_{p^r}$, we have
\[
\text{Tr}_{rn}(\alpha f(a))=\text{Tr}_{r}(\text{Tr}_{rn|r}(\alpha f(a))) = \text{Tr}_{r}(\alpha\text{Tr}_{rn|r}(f(a))).
\]
This implies that if $\text{Tr}_{rn|r}(f(a)) = 0$ for an $a \in {\mathbb F}_{p^{rn}}$, then
\begin{equation}\label{eq-T}
\sum_{\alpha \in {\mathbb F}_{p^r}^{\times}} \sum_{\psi \neq \psi_0} \psi( \text{Tr}_{rn}(\alpha f(a))) =
\sum_{\alpha \in {\mathbb F}_{p^r}^{\times}} \sum_{\psi \neq \psi_0} \psi( \text{Tr}_{r}(\alpha\text{Tr}_{rn|r}(f(a)))) = (p-1)(p^r - 1),
\end{equation}
and otherwise
\begin{equation}\label{eq-U}
\sum_{\alpha \in {\mathbb F}_{p^r}^{\times}} \sum_{\psi \neq \psi_0} \psi( \text{Tr}_{rn}(\alpha f(a))) =
\sum_{\alpha \in {\mathbb F}_{p^r}^{\times}} \sum_{\psi \neq \psi_0} \psi( \text{Tr}_{r}(\alpha\text{Tr}_{rn|r}(f(a)))) = 1 - p.
\end{equation}
If we let $$T=\#\left\{a \in |\text{Tr}_{rn|r}(f(a)) = 0 \right\}$$ and $$U=\#\left\{a \in |\text{Tr}_{rn|r}(f(a)) \neq 0\right\},$$
then $T+U=p^{rn}$. Using this fact and combining Equations~\eqref{Total-Sum},~\eqref{eq-T} and ~\eqref{eq-U}, we obtain
\begin{eqnarray}
\nonumber S &=& p^{rn}(p^r-1) + (p^r-1)(p-1)T + (1-p)U\\
\nonumber &=& p^{rn}(p^r-1) + (p^r-1)(p-1)T + (1-p)(p^{rn} - T)\\
\label{eq:S} &=& (p-1)p^r T + p^{rn}(p^r - p)
\end{eqnarray}
Applying Lemma~\ref{lem:paramq}~(1) again we have
\[
\#C({\mathbb F}_{p^{rn}})=p^r T,
\]
and hence by Eq.~(\ref{eq:S}) we have
\[
(p-1)\#C({\mathbb F}_{p^{rn}}) - \sum_{\alpha \in {\mathbb F}_{p^r}^{\times}} \#C_\alpha({\mathbb F}_{p^{rn}}) = (p - p^r)p^{rn}.
\]
\flushright \qed
\end{proof}
\noindent {\it Proof of Theorem~\ref{thm:maintheorem}:}
By transform~(\ref{eq:Sminus1}), in order to compute all $F_q(n,t_1,\ldots,t_l)$ for a given $\overline{n}$
one need only compute $V_1(\vec{i} \cdot \vec{f})$ for all $1 \le i \le q^{l}-1$.
For each such $i$ let $d(\vec{i})$ denote the degree of $\overline{p}_{\vec{i},\overline{n}}(a)$,
which we recall is $(k'+1)$ where $k'$ is the largest $0 \le k' \le l-1$ such that $i_{k'} \neq 0$.
The number of $i$'s for which $d(\vec{i})$ is $l,l-1,\ldots,1$ is therefore $(q-1)q^{l-1},(q-1)q^{l-2},\ldots,q-1$, respectively.
We first count the number of roots -- counted with multiplicities, since they may not all be distinct -- which appear in the numerator of the zeta function of the curve $C_{\vec{i},\overline{n}}/{\mathbb F}_q$ featured in $V_1(\vec{i} \cdot \vec{f})$.
For $q = p$ the genus of $C_{\vec{i},\overline{n}}/{\mathbb F}_p$ is $(p-1)(d(\vec{i})-1)/2$ and so there are $(p-1)(d(\vec{i})-1)$ roots.
The total number of roots over all $1 \le i \le p^l-1$ is therefore
\begin{equation*}\label{eq:numberofroots}
(p-1)\sum_{i = 1}^{p^l-1} (d(\vec{i}) - 1) = (p-1) \sum_{d=1}^{l} (p-1) p^{d-1} (d-1) = (p-1)^2 \sum_{d=1}^{l-1} dp^d = p^l(pl - p - l) + p.
\end{equation*}
For $q = p^r$ with $r > 1$, as per Lemma~\ref{thm:extensionreduction} for $\alpha \in {\mathbb F}_{q}^{\times}$ let
\begin{equation}\label{eq:Calpha}
C_{\alpha,\vec{i},\overline{n}}/{\mathbb F}_{q}: a_{1}^p - a_{1} = \alpha\, \overline{p}_{\vec{i},\overline{n}}(a).
\end{equation}
Since the number of roots of $C_{\vec{i},\overline{n}}/{\mathbb F}_q$ is the sum over $\alpha \in {\mathbb F}_{q}^{\times}$ of the number of roots of
$C_{\alpha,\vec{i},\overline{n}}/{\mathbb F}_{q}$, the total number of roots is therefore
\begin{eqnarray*}\label{eq:numberofrootsq}
\nonumber (q-1)(p-1)\sum_{i = 1}^{q^l-1} (d(\vec{i}) - 1) &=& (q-1)(p-1) \sum_{d=1}^{l} (q-1) q^{d-1} (d-1) = (q-1)^2(p-1) \sum_{d=1}^{l-1} dq^d\\
&=& (p-1)(q^l(ql - q - l) + q).
\end{eqnarray*}
Thus $N$ is as stated in the theorem and the (not necessarily distinct) roots $\omega_1,\ldots,\omega_N \in \overline{{\mathbb Z}}$ all have norm $\sqrt{q}$.
The $\upsilon_1,\ldots,\upsilon_N$ corresponding to each $\vec{t} = (t_1,\ldots,t_l)$ follow immediately from transform~(\ref{eq:Sminus1}).
In particular, for $q=p$ we have
\begin{equation*}
F_p(n,\vec{0}) = \frac{1}{p^{l-1}} \Big( p^{n + l - 1} - \sum_{i=1}^{p^l-1} V_1(\vec{i} \cdot \vec{f}) \Big)
= p^n - \frac{1}{p^l} \sum_{i=1}^{p^l-1} \Big( p^n - \sum_{k=1}^{(p-1)(d(\vec{i}) - 1)} \omega_{i,k}^n \Big)
= \frac{1}{p^l} \Big(p^n + \sum_{k=1}^{N} \omega_{k}^n \Big),
\end{equation*}
while for $\vec{t} \ne \vec{0}$ we have
\begin{equation*}
F_p(n,\vec{t}) = \frac{1}{p^{l-1}} \Big( \sum_{\vec{i} \cdot \vec{t} = 1} V_1(\vec{i} \cdot \vec{f})
- \sum_{\vec{i} \cdot \vec{t} = 0, \ \vec{i} \ne \vec{0}} V_1(\vec{i} \cdot \vec{f})\Big)
= \frac{1}{p^{l}} \Big( p^n + \sum_{k=1}^N \upsilon_k \omega_{k}^n \Big),
\end{equation*}
where the $\upsilon_{k}$ are in $\{0,\pm 1 \}$ and the $p^n$ arises as there are $p^{l-1}$ indices $i$ for which $\vec{i} \cdot \vec{t} = 1$,
while there are $p^{l-1}-1$ non-zero indices $i$ for which $\vec{i} \cdot \vec{t} = 0$.
For $q = p^r$ with $r > 1$, we have
\[
\#C_{\alpha,\vec{i},\overline{n}}({\mathbb F}_{q^n}) = q^n - \sum_{k=1}^{(p-1)(d(\vec{i})-1)} \omega_{\alpha,i,k}^n,
\]
and hence
\[
\sum_{\alpha \in {\mathbb F}_{q}^{\times}} \#C_{\alpha,\vec{i},\overline{n}}({\mathbb F}_{q^n}) = (q-1)q^n - \sum_{k=1}^{(q-1)(p-1)(d(\vec{i})-1)} \omega_{i,k}^n,
\]
and by Lemma~\ref{thm:extensionreduction} we have
\begin{eqnarray*}
\#C_{\vec{i},\overline{n}}({\mathbb F}_{q^n}) &=& \frac{p-q}{p-1}\,q^n + \frac{q-1}{p-1}\Big( q^n - \sum_{k=1}^{(q-1)(p-1)(d(\vec{i})-1)} \omega_{i,k}^n \Big)\\
&=& q^n - \frac{q-1}{p-1} \sum_{k=1}^{(q-1)(p-1)(d(\vec{i})-1)} \omega_{i,k}^n,
\end{eqnarray*}
and
\[
V_1(\vec{i} \cdot \vec{f}) = \frac{1}{q} \Big( q^n - \frac{q-1}{p-1} \sum_{k=1}^{(q-1)(p-1)(d(\vec{i})-1)} \omega_{i,k}^n \Big).
\]
As before, by~(\ref{eq:Sminus1}) we have
\begin{eqnarray*}
F_q(n,\vec{0}) = \frac{1}{q^{l-1}} \Big( q^{n + l - 1} - \sum_{i=1}^{q^l-1} V_1(\vec{i} \cdot \vec{f}) \Big)
&=& q^n - \frac{1}{q^l} \sum_{i=1}^{q^l-1} \Big( q^n - \frac{q-1}{p-1} \sum_{k=1}^{(q-1)(p-1)(d(\vec{i}) - 1)} \omega_{i,k}^n \Big)\\
&=& \frac{1}{q^l} \Big(q^n + \frac{q-1}{p-1}\sum_{k=1}^{N} \omega_{k}^n \Big),
\end{eqnarray*}
while for $\vec{t} \ne \vec{0}$ we have
\begin{equation*}
F_q(n,\vec{t}) = \frac{1}{q^{l-1}} \Big( \sum_{\vec{i} \cdot \vec{t} = 1} V_1(\vec{i} \cdot \vec{f})
- \sum_{\vec{i} \cdot \vec{t} = 0, \ \vec{i} \ne \vec{0}} V_1(\vec{i} \cdot \vec{f})\Big)
= \frac{1}{q^{l}} \Big( q^n + \frac{q-1}{p-1}\sum_{k=1}^N \upsilon_k \omega_{k}^n \Big).
\end{equation*}
In all cases, for fixed $p$ and $q$, and $n \rightarrow \infty$ we have
\begin{equation*}
F_q(n,t_1,\ldots,t_l) = q^{n-l} + \mathcal{O}(q^{n/2}).
\end{equation*}
Regarding the complexity claim, for $q = p$, by Theorem~\ref{thm:LW} the cost of computing all the relevant zeta functions in terms of the
number of bit operations is
\begin{equation*}
\sum_{i = 1}^{p^l - 1} \tilde{\mathcal{O}}(d(\vec{i})^5 p^4) =
\sum_{j=1}^{l} (p-1)p^{j-1} \cdot \tilde{\mathcal{O}}(j^5 p^4 ) = p^l \cdot \tilde{\mathcal{O}}(l^5 p^4).
\end{equation*}
For $q = p^r$ with $r > 1$ we employ Lemma~\ref{thm:extensionreduction} and similarly obtain a cost in terms of the number of bit operations of
\begin{equation*}
(q-1) \sum_{i = 1}^{q^l - 1} \tilde{\mathcal{O}}(d(\vec{i})^5 p^4 r^3) =
(q-1) \sum_{j=1}^{l} (q-1)q^{j-1} \cdot \tilde{\mathcal{O}}(j^5 p^4 r^3 ) = q^{l+1} \cdot \tilde{\mathcal{O}}( l^5 p^4 r^3 ).
\end{equation*} \qed
\subsubsection{Some remarks on the algorithm.}
Firstly, note that the `cost per $F_q(n,t_1,\ldots,t_l)$' is just $\tilde{\mathcal{O}}( l^5 p^4 )$ and $q \cdot \tilde{\mathcal{O}}( l^5 p^4 r^3)$ bit operations for $r = 1$ and $r > 1$, respectively. The algorithm is thus very efficient at computing all such $F_q(n,t_1,\ldots,t_l)$.
Regarding practical efficiency, it should be possible to re-use much of the computation in the $q^l-1$ runs of the Lauder-Wan algorithm.
It may also be possible for $r > 1$ to work with the curve~(\ref{eq:maincurve}) rather than the curves~(\ref{eq:Calpha}),
thus bypassing Lemma~\ref{thm:extensionreduction} and possibly reducing the complexity by a factor of $q-1$ in this case.
Furthermore, if only one $F_q(n,t_1,\ldots,t_l)$ is required then there may be a way to exploit the direct method which is more
efficient than computing either $2q^{l-1} - 1$ or $q^l - 1$ zeta functions, as is required by the indirect method.
Secondly, as already noted there may be many repeats amongst the $\omega_k$'s and so the number of summands in~(\ref{eq:maintheorem}) may be far
less than $N$, even for $F_q(n,\vec{0})$ to which they all contribute. Indeed, using the direct approach leads to formulae with less redundancy
than the indirect method.
For example, for $q = 5$ and $l = 4$, for the direct method the curve~(\ref{eq:main_direct}) is absolutely irreducible and has genus $860$, producing
$1720$ roots (not necessarily distinct), whereas the indirect method produces $6880$ roots for $F_q(n,\vec{0})$ (not necessarily distinct)
which is a factor of $p-1$ more.
The advantage of the indirect method is that the genus of the arising curves is far smaller ($0,2,4$ and $6$ in this example),
making the computation of their zeta functions extremely efficient. One may be able to eliminate the redundancy: we provide an example of this
in~\S\ref{sec:ternaryzetas}.
Finally, note that in contrast to the direct method, for the indirect method even if $\vec{t} = \vec{0}$,
one needs to use $1/\overline{n}$ as it features in~(\ref{eq:artin}), so the obtained formulae are only valid for $n$ coprime to $p$.
\subsection{Reducing the prescribed coefficients problem to the prescribed traces problem}\label{sec:transform0}
We now show how to express our titular problem in terms of the prescribed traces problem. For the present case of interest for which $l < p$,
the transform follows almost immediately from the work of Miers and Ruskey~\cite{miersruskey2}; only a small change in the relevant
definitions is needed.
For $a \in {\mathbb F}_{q^n}$ denote by $\overline{a}$ the string $(a,a^q,a^{q^2},\ldots,a^{q^{n-1}})$. Such a string is called periodic if there is a substring whose repeated concatenation
gives $\overline{a}$; otherwise the string is called aperiodic. Observe that $\overline{a}$ is aperiodic if and only if $a$ does not belong to any proper
subfield of ${\mathbb F}_{q^n}$.
Let $A_q(n,t_1,\ldots,t_l)$ denote the number of aperiodic strings corresponding to elements of ${\mathbb F}_{q^n}$ in the above manner for which
$T_1(a) = t_1,\ldots,T_l(a) = t_l$. If $\overline{a}$ is counted by $A_q(n,t_1,\ldots,t_l)$ then all rotations of $\overline{a}$ are distinct and
produce the same trace values. By this and the above observation we thus have
\[
I_q(n,t_1,\ldots,t_l) = \frac{1}{n}A_q(n,t_1,\ldots,t_l).
\]
Now let $f$ denote the minimum polynomial of $a$ over ${\mathbb F}_q$, which has degree $n/d$ for some $d \mid n$.
Note that $T_k(a)$ is the coefficient of $x^{n-k}$ in $f^d$~\cite[Lemma 2]{cattell}, so abusing notation slightly we also write $T_k(a)$ as
$T_k(f^d)$. The multinomial theorem (cf.~\cite[Lemma 2.1]{miersruskey2}) gives the following.
\begin{lemma}\label{lemma:multinomial}
For all $k \ge 1$ and $d \ge 1$ we have
\begin{equation*}\label{eq:multinomial}
T_k(f^d) = \sum_{\nu_1 + 2\nu_2 + \cdots k\nu_k = k} \binom{d}{\nu_1,\ldots,\nu_k,d - (\nu_1 + \cdots + \nu_k)} T_1(f)^{\nu_1} T_2(f)^{\nu_2}\cdots T_k(f)^{\nu_k}.
\end{equation*}
\end{lemma}
This motivates the following definition. For a positive integer $d$ and $\vec{t} = (t_1,\ldots,t_l) \in ({\mathbb F}_q)^l$ define the map
$\theta_d: ({\mathbb F}_q)^l \rightarrow ({\mathbb F}_q)^l: \vec{t} \rightarrow \vec{u} = (u_1,\ldots,u_l)$ by
\begin{eqnarray*}\label{eq:theta}
u_k &=& \sum_{\nu_1 + 2\nu_2 + \cdots + k\nu_k = k} \binom{d}{\nu_1,\ldots,\nu_k,d - (\nu_1 + \cdots + \nu_k)} t_{1}^{\nu_1} t_{2}^{\nu_2}\cdots t_{k}^{\nu_k}\\
&=& \sum_{\nu_1 + 2\nu_2 + \cdots + k\nu_k = k} d^{(\nu_1 + \nu_2 + \cdots + \nu_k)} \frac{t_{1}^{\nu_1}}{\nu_1 !} \frac{t_{2}^{\nu_2}}{\nu_2 !}\cdots
\frac{t_{k}^{\nu_k}}{\nu_k !},
\end{eqnarray*}
where $d^{(m)} = d(d-1)\cdots(d-m+1)$ is the falling factorial.
For a propostion $P$ let $[P]$ denote its truth value. Since every periodic string is the repeated concatenation of an aperiodic string, we have
(cf.~\cite[Eq. (2.5)]{miersruskey2}):
\[
F_q(n,\vec{u}) = \sum_{d \mid n} \sum_{\vec{t} \in ({\mathbb F}_{q})^l} [\theta_d(\vec{t}) = \vec{u}] A_q\big(\frac{n}{d},\vec{t}\big).
\]
For the present case of interest there is precisely one $\vec{t}$ for each $\vec{u}$. Indeed, we have the following theorem -- in which
$\mu$ denotes the usual M\"obius function -- whose proof is identical to that given by Miers and Ruskey~\cite[Theorem 6.1]{miersruskey2}.
\begin{theorem}
Let $q = p^r$ with $p$ an odd prime and let $\vec{t} = (t_1,\ldots,t_l)$ with $l < p$. Then
\begin{equation}\label{eq:transform0}
I_q(n,\vec{t}) = \begin{dcases}
\frac{1}{n} \sum_{\substack{d \mid n\\ p \nmid d}} \mu(d) \Big(F_q\Big(\frac{n}{d},\vec{0}\Big) - [pd \mid n]q^{n/(pd)}\Big) \ &\text{if} \ \vec{t} = \vec{0},\\
\frac{1}{n} \sum_{\substack{d \mid n\\ p \nmid d}} \mu(d)F_q\Big(\frac{n}{d},\theta_{d^{-1}}(\vec{t})\Big) \ &\text{otherwise}.
\end{dcases}
\end{equation}
\end{theorem}
First note that the inverse of $d$ is computed mod $p$. Also note that since our algorithm does not address the $n \equiv 0 \pmod{p}$ cases,
the condition $p \nmid d$ in the summations is vacuous and the second term in each summand of the $\vec{t} = \vec{0}$ case is always zero.
Therefore, for all $\vec{t}$ we have
\[
I_q(n,\vec{t}) = \frac{1}{n} \sum_{d \mid n} \mu(d)F_q\Big(\frac{n}{d},\theta_{d^{-1}}(\vec{t})\Big) = \frac{1}{n}\big(q^{n-l} + \mathcal{O}(q^{n/2})\Big).
\]
Finally, we note that for $q$ even only the first trace is prescribed, and this case was solved by Carlitz~\cite{Carlitz}.
\subsection{Example: $F_5(n,t_1,t_2,t_3,t_4)$}
As a proof-of-concept example, for each $\overline{n} \in \{1,2,3,4\}$ we computed the zeta functions of all $5^4 - 1$
curves~(\ref{eq:maincurve}) that are required by the indirect method in order to compute each $F_5(n,t_1,t_2,t_3,t_4)$ for all
$n \equiv \overline{n} \pmod{5}$ with $n \ge 4$. The computations for each $\overline{n}$ took under a minute using Magma V22.2-3
on a 2.0GHz AMD Opteron computer.
Note that Magma uses brute-force point counting over successive extension fields
in order to compute the zeta functions. For this small example in which the genera are $0,2,4$ or $6$ this is clearly not a concern.
For much larger $p$ the Lauder-Wan algorithm should be used.
For the sake of space, in the following theorem we only give the formula for $F_5(n,0,0,0,0)$, which is the same for all four relevant
residue classes of $n$ and combines all $5^4-1$ zeta functions as per the transform~(\ref{eq:Sminus1});
it thus has the greatest weight amongst all of the $F_5(n,t_1,t_2,t_3,t_4)$ formulae.
For a polynomial $\gamma(X) \in {\mathbb Z}[X]$ let $\rho_n(\gamma)$ denote the sum of the $n$-th
powers of the (complex) roots of $\gamma(X)$.
\begin{theorem}\label{thm:q5l4}
For $n \ge 4$ and $(n,5) = 1$ we have
\begin{eqnarray*}
F_5(n,0,0,0,0) &=& 5^{n-4} - \frac{1}{5^4}\big(
160\rho_n(\gamma_{2,1})
+16\rho_n(\gamma_{2,2})
+164\rho_n(\gamma_{4,1})
+16\rho_n(\gamma_{4,2})
+116\rho_n(\gamma_{4,3})
+25\rho_n(\gamma_{8,1})\\
&+&20\rho_n(\gamma_{8,2})
+16\rho_n(\gamma_{8,3})
+18\rho_n(\gamma_{8,4})
+20\rho_n(\gamma_{8,5})
+20\rho_n(\gamma_{8,6})
+20\rho_n(\gamma_{8,7})
+16\rho_n(\gamma_{8,8})\\
&+& 24\rho_n(\gamma_{8,9})
+16\rho_n(\gamma_{8,10})
+21\rho_n(\gamma_{8,11})
+17\rho_n(\gamma_{8,12})
+16\rho_n(\gamma_{8,13})
+16\rho_n(\gamma_{8,14})\\
&+& 12\rho_n(\gamma_{8,15})
+10\rho_n(\gamma_{8,16})
+8\rho_n(\gamma_{8,17})
+13\rho_n(\gamma_{8,18})
+20\rho_n(\gamma_{12,1})
+20\rho_n(\gamma_{12,2})\\
&+& 20\rho_n(\gamma_{12,3})
+16\rho_n(\gamma_{12,4})
+16\rho_n(\gamma_{12,5})
+16\rho_n(\gamma_{12,6})
+16\rho_n(\gamma_{12,7})
+16\rho_n(\gamma_{12,8})\\
&+& 16\rho_n(\gamma_{12,9})
+16\rho_n(\gamma_{12,10})
+16\rho_n(\gamma_{12,11})
+16\rho_n(\gamma_{12,12})
+12\rho_n(\gamma_{12,13})
+12\rho_n(\gamma_{12,14})\\
&+& 12\rho_n(\gamma_{12,15})
\big),
\end{eqnarray*}
where
{\small
\begin{flalign*}
\gamma_{2,1} &= X^2 - 5, &\\
\gamma_{2,2} &= X^2 + 5,\\
\gamma_{4,1} &= X^4 - 5X^3 + 15X^2 - 25X + 25,\\
\gamma_{4,2} &= X^4 + 5X^2 + 25,\\
\gamma_{4,3} &= X^4 + 5X^3 + 15X^2 + 25X + 25,\\
\gamma_{8,1} &= X^8 - 10X^7 + 45X^6 - 130X^5 + 305X^4 - 650X^3 + 1125X^2 - 1250X + 625,\\
\gamma_{8,2} &= X^8 - 5X^7 + 10X^6 - 25X^5 + 75X^4 - 125X^3 + 250X^2 - 625X + 625,\\
\gamma_{8,3} &= X^8 - 5X^7 + 10X^6 + 5X^5 - 45X^4 + 25X^3 + 250X^2 - 625X + 625,\\
\gamma_{8,4} &= X^8 - 5X^7 + 15X^6 - 35X^5 + 80X^4 - 175X^3 + 375X^2 - 625X + 625,\\
\gamma_{8,5} &= X^8 - 5X^7 + 20X^6 - 65X^5 + 155X^4 - 325X^3 + 500X^2 - 625X + 625,\\
\gamma_{8,6} &= X^8 - 15X^6 + 105X^4 - 375X^2 + 625,\\
\gamma_{8,7} &= X^8 - 10X^6 + 55X^4 - 250X^2 + 625,\\
\gamma_{8,8} &= X^8 - 5X^6 + 25X^4 - 125X^2 + 625,\\
\gamma_{8,9} &= X^8 + 30X^4 + 625,\\
\gamma_{8,10} &= X^8 + 5X^6 - 20X^5 + 5X^4 - 100X^3 + 125X^2 + 625,\\
\end{flalign*}
}
\vspace{-10mm}
{\small
\begin{flalign*}
\gamma_{8,11} &= X^8 + 5X^6 - 10X^5 + 5X^4 - 50X^3 + 125X^2 + 625,\\
\gamma_{8,12} &= X^8 + 5X^6 + 10X^5 + 5X^4 + 50X^3 + 125X^2 + 625,\\
\gamma_{8,13} &= X^8 + 5X^6 + 20X^5 + 5X^4 + 100X^3 + 125X^2 + 625,&\\
\gamma_{8,14} &= X^8 + 5X^7 + 10X^6 - 5X^5 - 45X^4 - 25X^3 + 250X^2 + 625X + 625,\\
\gamma_{8,15} &= X^8 + 5X^7 + 10X^6 + 25X^5 + 75X^4 + 125X^3 + 250X^2 + 625X + 625,\\
\gamma_{8,16} &= X^8 + 5X^7 + 15X^6 + 35X^5 + 80X^4 + 175X^3 + 375X^2 + 625X + 625,\\
\gamma_{8,17} &= X^8 + 5X^7 + 20X^6 + 65X^5 + 155X^4 + 325X^3 + 500X^2 + 625X + 625,\\
\gamma_{8,18} &= X^8 + 10X^7 + 45X^6 + 130X^5 + 305X^4 + 650X^3 + 1125X^2 + 1250X + 625,\\
\gamma_{12,1} &= X^{12} - 5X^{11} + 5X^{10} + 5X^9 + 5X^8 + 75X^7 - 425X^6 + 375X^5 + 125X^4 + 625X^3 + 3125X^2 - 15625X + 15625,\\
\gamma_{12,2} &= X^{12} - 5X^{11} + 10X^{10} + 5X^9 - 20X^8 - 125X^7 + 575X^6 - 625X^5 - 500X^4 + 625X^3 + 6250X^2 - 15625X + 15625,\\
\gamma_{12,3} &= X^{12} - 5X^{11} + 15X^{10} - 45X^9 + 80X^8 - 125X^7 + 325X^6 - 625X^5 + 2000X^4 - 5625X^3 + 9375X^2 - 15625X + 15625,\\
\gamma_{12,4} &= X^{12} - 10X^{10} - 5X^9 + 80X^8 - 425X^6 + 2000X^4 - 625X^3 - 6250X^2 + 15625,\\
\gamma_{12,5} &= X^{12} - 10X^{10} + 10X^9 + 55X^8 - 25X^7 - 175X^6 - 125X^5 + 1375X^4 + 1250X^3 - 6250X^2 + 15625,\\
\gamma_{12,6} &= X^{12} - 5X^{10} - 15X^9 + 5X^8 + 50X^7 + 75X^6 + 250X^5 + 125X^4 - 1875X^3 - 3125X^2 + 15625,\\
\gamma_{12,7} &= X^{12} - 5X^{10} - 10X^9 - 20X^8 + 25X^7 + 325X^6 + 125X^5 - 500X^4 - 1250X^3 - 3125X^2 + 15625,\\
\gamma_{12,8} &= X^{12} - 15X^9 + 30X^8 - 50X^7 + 75X^6 - 250X^5 + 750X^4 - 1875X^3 + 15625,\\
\gamma_{12,9} &= X^{12} + 15X^9 - 20X^8 - 50X^7 + 75X^6 - 250X^5 - 500X^4 + 1875X^3 + 15625,\\
\gamma_{12,10} &= X^{12} + 5X^{10} - 5X^9 + 5X^8 + 50X^7 + 75X^6 + 250X^5 + 125X^4 - 625X^3 + 3125X^2 + 15625,\\
\gamma_{12,11} &= X^{12} + 10X^{10} + 20X^9 + 55X^8 + 175X^7 + 325X^6 + 875X^5 + 1375X^4 + 2500X^3 + 6250X^2 + 15625,\\
\gamma_{12,12} &= X^{12} + 15X^{10} - 10X^9 + 130X^8 - 75X^7 + 825X^6 - 375X^5 + 3250X^4 - 1250X^3 + 9375X^2 + 15625,\\
\gamma_{12,13} &= X^{12} + 5X^{11} + 10X^{10} + 25X^9 + 80X^8 + 225X^7 + 575X^6 + 1125X^5 + 2000X^4 + 3125X^3 + 6250X^2 + 15625X\\
&+ 15625,\\
\gamma_{12,14} &= X^{12} + 5X^{11} + 15X^{10} + 25X^9 + 5X^8 - 125X^7 - 425X^6 - 625X^5 + 125X^4 + 3125X^3 + 9375X^2 + 15625X + 15625,\\
\gamma_{12,15} &= X^{12} + 5X^{11} + 20X^{10} + 75X^9 + 230X^8 + 600X^7 + 1450X^6 + 3000X^5 + 5750X^4 + 9375X^3 + 12500X^2 + 15625X\\
&+ 15625.
\end{flalign*}
}
\end{theorem}
Each of the above polynomials is the characteristic polynomial of the Frobenius endomorphism of an abelian variety.
Regarding the potential periodicity in $n$ of the formula in Theorem~\ref{thm:q5l4}, whether the abelian varieties are supersingular or not can be determined by the following theorem of Stichtenoth and Xing~\cite[Prop.~1]{Stichtenoth}.
\begin{theorem}\label{thm:SS}
Let $A$ be an abelian variety of dimension $g$ over ${\mathbb F}_q = {\mathbb F}_{p^r}$ and let $P(X) = X^{2g} + a_1X^{2g-1} + \cdots + a_g X^g + \cdots + q^g$
be the characteristic polynomial of the Frobenius endomorphism on $A$. Then $A$ is supersingular if and only if for all $1 \le k \le g$ one has
$p^{\lceil kn/2 \rceil} \mid a_k$.
\end{theorem}
Applying Theorem~\ref{thm:SS} we see that $\gamma_{2,1},\gamma_{2,2},\gamma_{4,1},\gamma_{4,2},\gamma_{4,3},\gamma_{8,2},\gamma_{8,8}$ and $\gamma_{8,15}$ are the only supersingular cases.
It is possible in principle to apply Kronecker's theorem to the phases of the non-supersingular Weil numbers to determine whether or not the formula
in Theorem~\ref{thm:q5l4} is periodic in $n$. Note that the product of the powers of these polynomials as per
Theorem~\ref{thm:q5l4} produces (a redundant version of) the characteristic polynomial of Frobenius of the intersection curve of genus $860$
given by the direct method. Although it is not true for this polynomial that $a_{12} \equiv 0 \pmod{5^6}$, as required by Theorem~\ref{thm:SS}
for supersingularity, it is possible (although seemingly unlikely) that the formula for the non-redundant method is periodic in $n$, which would
require that all of the non-supersingular contributions cancel.
\section{An Algorithm for the Prescribed Traces Problem for $q=2$}\label{sec:char2}
When $l \ge p$ the approach of~\S\ref{sec:generalalg} breaks due to the failure of Newton's identities in positive characteristic.
For example, for $l = p$ we have $T_1(a^p) = t_{1}^p$ which does not feature $t_p$ and thus the system of linear trace
conditions~(\ref{eq:newmain}) is underdetermined.
In this section we present an algorithm that obviates this issue and solves the
prescribed traces problem for binary base fields for $l \le 7$ and $n$ odd, which is potentially extendable to larger $l$. We focus here on
the $q = 2$ case, noting that the general $q = 2^r$ case follows {\em mutatis mutandis}.
In contrast to the main algorithm we now permit any subset of the first $l$ traces to be prescribed: for subscripts $1 \le l_0 < \cdots < l_{m-1} \le l$,
$n \ge l_{m-1}$ and $t_{l_0},\ldots,t_{l_{m-1}} \in {\mathbb F}_2$ we denote by $F_2(n,t_{l_0},\ldots,t_{l_{m-1}})$ the number of elements $a \in {\mathbb F}_{2^n}$
for which $T_{l_0}(a) = t_{l_0}$, \ldots, $T_{l_{m-1}}(a) = t_{l_{m-1}}$.
After presenting a specialised version of the transform from~\S\ref{sec:transform1p} in~\S\ref{sec:transform1char2},
we present the indirect method, in~\S\ref{subsec:indirectbinary}, as it is slightly simpler than the direct method,
which we present in~\S\ref{subsec:directbinary}. The algebraic property upon which both methods rely is presented in~\S\ref{sec:Tlab}.
The values $T_l(1) = \binom{n}{l}$ feature repeatedly in the ensuing methods. We therefore begin with two preliminary results,
the first of which is an immediate corollary of a theorem of Zabek~\cite{zabek}.
\begin{lemma}\label{lem:period}
For $p$ a prime and $j \ge 1$ the period of the sequence $\big(\binom{0}{j},\binom{1}{j},\binom{2}{j},\ldots\big)$ mod $p$ is $p^{1 + \lfloor \log_p{j} \rfloor}$.
\end{lemma}
We shall use the following corollary repeatedly.
\begin{corollary}\label{cor:period}
For $p$ a prime and $j \ge 1$ the period in $n$ of the set of vectors $\big(\binom{n}{j},\binom{n}{j-1},\ldots,\binom{n}{1}\big)$ mod $p$ is $p^{1 + \lfloor \log_p{j} \rfloor}$.
\end{corollary}
\begin{proof}
By Lemma~\ref{lem:period} the period in $n$ of $\binom{n}{j}$ mod $p$ is $p^{1 + \lfloor \log_p{j} \rfloor}$. Therefore the period in $n$
of the stated set of vectors is the LCM of periods of each, which is also $p^{1 + \lfloor \log_p{j} \rfloor}$.
\end{proof}
\subsection{A transform of the prescribed traces problem for $q=2$}\label{sec:transform1char2}
We now transform the problem of counting field elements with prescribed traces to the problem of counting the number of zeros
of linear combinations of the trace functions. Note that counting zeros rather than evaluations to one means that one does not
necessarily need to introduce a $1/\overline{n}$ term when defining the relevant curves and as a consequence some results are obtainable
for all $n$, rather than just $n$ odd. We first fix some notation, which is virtually the same as that defined in~\S\ref{sec:transform1p}.
Let $f_0,\ldots,f_{m-1}: {\mathbb F}_{2^n} \rightarrow {\mathbb F}_2$ be any functions and let $\vec{f} = (f_{m-1},\ldots,f_0)$.
For $i,j \in \{0,\ldots,2^m-1\}$ let $\vec{i} = (i_{m-1},\ldots,i_0)$ and $\vec{j} = (j_{m-1},\ldots,j_0)$
denote the binary expansions of $i$ and $j$ respectively, and let $\vec{i} \cdot \vec{j}$ denote their inner product mod $2$.
For any $i \in \{0,\ldots,2^m-1\}$, let $\vec{i}\cdot \vec{f}$ denote the function
\[
\sum_{k=0}^{m-1} i_k f_k: {\mathbb F}_{2^n} \rightarrow {\mathbb F}_2,
\]
and let $V(\vec{i}\cdot \vec{f})$ denote the number of zeros in ${\mathbb F}_{2^n}$ of $\vec{i}\cdot \vec{f}$. We interpret $V(\vec{0} \cdot \vec{f})$ to be
the number of zeros of the empty function, which we define to be $2^n$.
Furthermore, let $N(\vec{j}) = N(j_{m-1},\ldots,j_{0})$ denote the number of $a \in {\mathbb F}_{2^n}$ such that $f_k(a) = j_k$, for
$k = 0,\ldots,m-1$.
As before, our goal is to express any $N(\vec{j})$ in terms of the $V(\vec{i}\cdot \vec{f})$, but we begin by first solving the inverse problem,
i.e., expressing any $V(\vec{i}\cdot \vec{f})$ in terms of the $N(\vec{j})$.
\begin{lemma}
With the notation as above, for $0 \le i \le 2^m-1$ we have
\begin{equation}\label{eq:basicrelation}
V(\vec{i}\cdot \vec{f}) = \sum_{\vec{i} \cdot \vec{j} \, = \, 0} N(\vec{j}).
\end{equation}
\end{lemma}
\begin{proof}
By definition, we have $V(\vec{i}\cdot \vec{f}) = \#\{ a \in {\mathbb F}_{2^n} \mid \vec{i}\cdot \vec{f} (a) = 0\} =
\#\{ a \in {\mathbb F}_{2^n} \mid \sum_{k=0}^{m-1} i_k f_k(a) = 0\}$.
Since $N(\vec{j})$ counts precisely those $a \in {\mathbb F}_{2^n}$ such that $f_k(a) = j_k$, we must count over all those $\vec{j}$ for which
$\sum_{k=0}^{m-1} i_k f_k(a) = 0$, i.e., those such that $\sum_{k=0}^{m-1} i_k j_k = 0$. \qed
\end{proof}
Writing Eq.~(\ref{eq:basicrelation}) in matrix form, for $i,j \in \{0,\ldots, 2^m-1\}$ we have
\[
\begin{bmatrix}
V(\vec{i}\cdot\vec{f})
\end{bmatrix}^T
=
S_m
\cdot
\begin{bmatrix}
N(\vec{j})
\end{bmatrix}^T,
\]
where $(S_m)_{i,j} = 1 - \vec{i}\cdot\vec{j}$ is nothing but Sylvester's construction~\cite{sylvester} of Hadamard matrices~\cite{hadamard},
with the minus ones replaced with zeros.
Now let $H_m$ be the $2^m \times 2^m$ matrix with entries $(H_m)_{i,j} = (-1)^{\vec{i} \cdot \vec{j}}$, i.e., the Walsh-Hadamard
transform~\cite{walsh-orig}, scaled by $2^{m/2}$. Since the Walsh-Hadamard matrix is involutory we have $H_{m}^{-1} = \frac{1}{2^m} H_m$.
Also let $A_m$ be the $2^m \times 2^m$
matrix with $(A_m)_{0,0} = 1$ and all other entries $0$, and let $B_m$ be the $2^m \times 2^m$ matrix with all entries $1$.
Lastly let $\text{Id}_m$ be the $2^m \times 2^m$ identity matrix.
\begin{lemma}
We have
\begin{equation}\label{eq:Sinverse}
S_{m}^{-1} = \frac{1}{2^{m-1}}H_m - A_m.
\end{equation}
\end{lemma}
\begin{proof}
Noting that $2S_m = H_m + B_m$, we have
\begin{eqnarray*}
\Big(\frac{1}{2^{m-1}}H_m - A_m \Big)S_m &=& \frac{1}{2^m} H_m \cdot 2S_m - A_mS_m\\
&=& \frac{1}{2^m} H_m( H_m + B_m) - A_m S_m\\
&=& \text{Id}_m + \frac{1}{2^m} H_mB_m - A_m S_m.
\end{eqnarray*}
Since all but the first row of $H_m$ contains the same number of ones and minus ones, $\frac{1}{2^m} H_m B_m$ consists of the all-one vector in the
first row and the all-zero vector for the others, as does $A_m S_m$. \qed
\end{proof}
Thus in order to compute any of the $2^m$ possible outputs $N(\vec{j})$ of any set of $m$ functions $\vec{f}$, it is sufficient to count the number
of zeros of all the $2^m$ ${\mathbb F}_2$-linear combinations of the functions, and then apply $S_{m}^{-1}$. In particular, one may choose the
$f_{0},\ldots,f_{m-1}$ to be any subset of the trace functions $T_1,\ldots,T_{n}$.
Note that for $q = 2^r$ with $r > 1$ one must revert to using transform~(\ref{eq:Sminus1}).
\subsection{The indirect method}\label{subsec:indirectbinary}
Let the input traces whose values are prescribed be $\vec{f} = (T_{l_{m-1}},\ldots,T_{l_{0}})$ with $l_{m-1} > \cdots > l_0$.
Then by the transform of the previous subsection, for all $i \in \{1,\ldots,2^m-1\}$ one needs to compute
\begin{equation}\label{eq:Zeq}
V(\vec{i} \cdot \vec{f}) = \#\{ a \in {\mathbb F}_{2^n} \mid \sum_{k=0}^{m-1} i_k T_{l_{k}}(a) = 0 \}.
\end{equation}
In general this problem appears to be non-trivial, since the degree of each $T_{l_k}(a)$ in $a$ and its Frobenius powers is $l_k$ and the
approach of~\S\ref{sec:generalalg} can not be applied.
However, it can be obviated -- at least for $n$ odd -- by using the following degree-lowering idea.
Firstly, note that since the input to the trace functions has linear trace either $0$ or $1$, Eq.~(\ref{eq:Zeq}) can be rewritten as
\begin{equation*}
V( \vec{i} \cdot \vec{f} ) = \#\{ a \in {\mathbb F}_{2^n} \mid T_1(a) = 0, \sum_{k=0}^{m-1} i_k T_{l_{k}}(a) = 0 \} +
\#\{ a \in {\mathbb F}_{2^n} \mid T_1(a) = 1, \sum_{k=0}^{m-1} i_k T_{l_{k}}(a) = 0 \}.
\end{equation*}
Secondly, note that for $n$ odd Lemma~\ref{lem:paramq} implies
\begin{equation}\label{eq:Zeq1}
V( \vec{i} \cdot \vec{f} ) = \frac{1}{2} \sum_{r_0 \in {\mathbb F}_2} \#\{ a_0 \in {\mathbb F}_{2^n} \mid \sum_{k=0}^{m-1} i_k T_{l_{k}}(a_{0}^2 + a_0 + r_0) = 0 \}.
\end{equation}
Thirdly, it happens that the functions $T_l(a_{0}^2 + a_{0})$ and $T_l(a_{0}^2 + a_{0} + 1)$ for $2 \le l \le 7$ are all expressible in
characteristic two as polynomials of traces of lower degree whose arguments are polynomials in $a_0$, see~\S\ref{sec:Tlab}.
Hence rather than having a single equation whose zeros one must count (Eq.~(\ref{eq:Zeq})), one now has two equations whose number of zeros
one must add and divide by $2$ (Eq.~(\ref{eq:Zeq1})), both now of lower degree than before.
If after the above three steps there are terms that are not linear, i.e., not of the form $T_1(\cdot)$ for some argument, then
the idea is to pick an argument of a trace function featuring in a non-linear term and apply the above three steps again.
In particular, if the chosen argument is $g(a_0)$ then one introduces a new variable $a_1$ and as before writes
$g(a_0) = a_{1}^2 + a_{1} + r_1$ with $r_1 \in {\mathbb F}_2$ to account for whether the linear trace of $g(a_0)$ is $0$ or $1$,
and expands all those terms in Eq.~(\ref{eq:Zeq1}) which have this argument. This results in four equations whose number of zeros one must
sum and divide by $4$, with the degrees of the terms which feature this argument having been lowered, as before.
By recursively applying this idea and introducing variables $a_0,\ldots,a_{s_{i}-1}$ as necessary,
with corresponding linear trace variables $r_0,\ldots,r_{s_{i}-1}$, since the degrees of the non-linear terms
always decreases one eventually obtains a set of $2^{s_{i}}$ trace equations of the form $T_1(g_{\vec{r}}(a_0,\ldots,a_{s_{i}-1})) = 0$
indexed by $\vec{r} =(r_0,\ldots,r_{s_{i}-1}) \in ({\mathbb F}_2)^{s_{i}}$, the vector of trace values of the $s_{i}$ rewritten arguments.
Each of these can be eliminated by introducing a final variable $a_{s_{i}}$ and writing
$a_{s_{i}}^2 + a_{s_{i}} = g_{\vec{r}}(a_0,\ldots,a_{s_{i}-1})$,
giving a system of $s_{i}$ equations in $s_{i}+1$ variables, with the initial variable $a$ having been completely eliminated
in going from Eq.~(\ref{eq:Zeq}) to Eq.~(\ref{eq:Zeq1}). Observe that since $a_0$ is the only free variable in this system, the affine
algebraic set that it defines is one-dimensional and is thus a curve.
Depending on $i$, the final linear trace equation features a subset of the coefficients $\binom{n}{l_k}$ for $k = 0, \ldots, m-1$,
which by Corollary~\ref{cor:period} have period a divisor of the maximum period $2^\theta := 2^{1 + \lfloor \log_2{l_{m-1}}\rfloor}$.
Therefore let $\overline{n} \in \{1,3,5,\ldots,2^\theta-1\}$ represent the applicable residue classes of $n$ mod $2^\theta$
and substitute each $\binom{n}{l_k}$ featuring in the above intersection by $\binom{\overline{n}}{l_k}$.
We denote by $C_{i,\vec{r},\overline{n}}$ the affine curve defined by the above $s_{i}$
equations whose coefficients are functions of $\vec{r} = (r_0,\ldots,r_{s_{i}-1})$ and $\overline{n}$.
For each $\overline{n}$ we then take the average over all specialisations of $\vec{r}$ of the number of ${\mathbb F}_{2^n}$-rational points
on $C_{i,\vec{r},\overline{n}}$ -- remembering to also divide by $2$ to account for the presence of $a_{s_{i}}$ --
in order to determine $V(\vec{i}\cdot \vec{f})$, i.e., for all $n \equiv \overline{n} \pmod{2^\theta}$
with $n \ge l_{m-1}$ we have
\begin{equation}\label{eq:indirectmethodbinary}
V(\vec{i}\cdot \vec{f}) = \frac{1}{2^{s_{i}+1}} \sum_{\vec{r} \in ({\mathbb F}_2)^{s_{i}}} \#C_{i,\vec{r},\overline{n}}({\mathbb F}_{2^n}).
\end{equation}
\subsection{The direct method}\label{subsec:directbinary}
By definition, we have
\[
F_2(n,t_{l_0},\ldots,t_{l_{m-1}}) = \#\{ a \in {\mathbb F}_{2^n} \mid T_{l_{0}}(a) = t_{l_0},\ldots, T_{l_{m-1}}(a) = t_{l_{m-1}}\}.
\]
By using the same recursive procedure described in the previous subsection, for $n \ge l_{m-1}$ and odd suppose that for $0 \le k \le m-1$ one
linearises $T_{l_k}(a)$, i.e., reduces it to a $T_1$ expression, by introducing variables $a_{0},a_{k,1},\ldots,a_{k,s_k-1}$ with corresponding linear
traces $r_{0},r_{k,1},\ldots,r_{k,s_k-1}$, resulting in $2^{s_k}$ trace equations of the form
\begin{equation}\label{eq:traceequationbinary}
T_1(g_{\vec{r}_k}(a_0,a_{k,1},\ldots,a_{k,s_k-1})) = t_{l_k}.
\end{equation}
Observe that amongst all of the $a_{k,i}$'s there may be definitional repetitions.
Therefore, let $U$ be the union of all of the equations arising from the linearisation of
each of the featured trace functions, once repetitions have been eliminated. Let $s \le \sum_{k=0}^{m-1} s_k$ be the number of variables in this union,
which we relabel as $a_0,a_1,\ldots,a_{s-1}$, and let $\vec{r} = (r_0,\ldots,r_{s-1})$ be the corresponding vector of possible linear traces of the
rewritten arguments. Note that since $a$ was completely eliminated by the introduction of $a_0$ there are only
$s-1$ corresponding equations in addition to the $m$ trace conditions in~(\ref{eq:traceequationbinary}).
Each of these $m$ trace equations can be parameterised by introducing for $0 \le k \le m-1$ a variable $a_{s_k}$
and writing $a_{s_k}^2 + a_{s_k} + t_{l_k} = g_{\vec{r}_k}(a_{0},a_1\ldots,a_{s-1})$, giving a system of $m+s-1$ equations in $m+s$ variables.
As in the indirect method observe that $a_0$ is the only free variable in this system and so the affine
algebraic set that it defines is one-dimensional and is thus a curve.
Also as before, this system features the coefficients $\binom{n}{l_k}$, which by Corollary~\ref{cor:period} have period
a divisor of $2^{\theta}$, so again let $\overline{n} \in \{1,3,5,\ldots,2^{\theta}-1\}$ represent the applicable residue classes of
$n$ mod $2^{\theta}$, and substitute each $\binom{n}{l_k}$ by $\binom{\overline{n}}{l_k}$.
We denote by $C_{\vec{r},\overline{n}}$ the affine curve defined by the $m+s-1$
above equations whose coefficients are functions of the prescribed values, $\vec{r} = (r_0,\ldots,r_{s-1})$ and $\overline{n}$.
For each $\overline{n}$ we take the average over all specialisations of $\vec{r}$ of the number of ${\mathbb F}_{2^n}$-rational points
on $C_{\vec{r},\overline{n}}$ and divide by $2^{m}$ in order to determine $F_2(n,t_{l_0},\ldots,t_{l_{m-1}})$, i.e.,
for all $n \equiv \overline{n} \pmod{2^{\theta}}$ with $n \ge l_{m-1}$ we have
\begin{equation}\label{eq:directmethodbinary}
F_2(n,t_{l_0},\ldots,t_{l_{m-1}}) = \frac{1}{2^{m+s}} \sum_{\vec{r} \in ({\mathbb F}_2)^s} \#C_{\vec{r},\overline{n}}({\mathbb F}_{2^n}).
\end{equation}
Since in general there will be more variables and equations than for the indirect method, computing the zeta functions of the arising curves
will likely be more costly for this method.
\subsection{Computing $T_{l}(\alpha - \beta)$}\label{sec:Tlab}
We now explain the how one can obtain expressions for $T_l(a_{0}^2 + a_0)$ and $T_l(a_{0}^2 + a_0 + 1)$ for $2 \le l \le 7$.
Expressions for $T_2(\alpha - \beta)$ and $T_3(\alpha - \beta)$ in characteristic two were given by Fitgerald and Yucas~\cite[Lemma 1.1]{fitzyucas} and
proven by expanding the bilinear forms $T_l(\alpha + \beta) + T_l(\alpha) + T_l(\beta)$ in terms of the Frobenius powers of $\alpha$ and $\beta$,
and deducing the correct function of the lower degree traces. It is possible -- though laborious -- to continue in this manner (we did so for $l = 4$),
so instead we present an easier method.
We begin by recalling Eq~(\ref{eq:NI}):
\begin{equation*}
k \,T_k(\alpha) = \sum_{j=1}^k (-1)^{j-1} T_{l-j}(\alpha)T_1(\alpha^j).
\end{equation*}
In order to use the argument $\alpha - \beta$ we need to work instead in the ring ${\mathbb Z}[\alpha_1,\ldots,\alpha_n,\beta_1,\ldots,\beta_n]$ and with
the ring of multisymmetric functions in two variables, with the symmetric group $S_n$ acting on $\alpha_1,\ldots,\alpha_n$ and $\beta_1,\ldots,\beta_n$ independently (see~\cite{dalbec} for a formal definition). Abusing notation slightly, in this ring Eq.~(\ref{eq:NI}) becomes:
\begin{equation}\label{eq:NIab}
l \,T_l(\alpha -\beta) = \sum_{k=1}^l (-1)^{k-1} T_{l-k}(\alpha - \beta)T_1( (\alpha - \beta)^k).
\end{equation}
If one works over ${\mathbb Q}$ rather than ${\mathbb Z}$ then Eq.~(\ref{eq:NIab}) leads to expressions for $T_{l}(\alpha - \beta)$ for any $l \ge 1$ as a sum of
products of $T_1$ terms with arguments being various powers of $\alpha - \beta$.
However, in positive characteristic this is not so useful. For instance, computing $T_2(\alpha- \beta)$ in
characteristic two in this way is not possible as the left hand side vanishes.
Nevertheless, working inductively for $2 \le l \le 7$ and applying Newton's identities evaluated at various products of powers of $\alpha$
and $\beta$ so that no trace occurs to any power larger than one, all of the coefficients become divisible by $l$. Upon dividing by $l$ one obtains an equation for $T_l(\alpha - \beta)$ over ${\mathbb Z}$,
which can then be substituted into Eq.~(\ref{eq:NIab}) in order to attempt to compute $T_{l+1}(\alpha - \beta)$.
For example, from Eq.~(\ref{eq:NIab}) we have
\begin{eqnarray*}
2T_2(\alpha - \beta) &=& T_1(\alpha - \beta)^2 - T_1( (\alpha - \beta)^2 ) \\
&=& (T_1(\alpha) - T_1(\beta))^2 - T_1(\alpha^2) + 2T_1(\alpha\beta) - T_1(\beta^2) \\
&=& T_1(\alpha)^2 - 2T_1(\alpha)T_1(\beta) + T_1(\beta)^2 - T_1(\alpha^2) + 2T_1(\alpha\beta) - T_1(\beta^2) \\
&=& 2T_2(\alpha) + 2T_2(\beta) - 2T_1(\alpha)T_1(\beta) + 2T_1(\alpha\beta),
\end{eqnarray*}
where in the final line we have used Eq.~(\ref{eq:NI}) for $l=2$ for $\alpha$ and $\beta$ separately. We have therefore proven that
\[
T_2(\alpha - \beta) = T_2(\alpha) + T_2(\beta) - T_1(\alpha)T_1(\beta) + T_1(\alpha\beta).
\]
The parts of the following lemma can either be proven by induction on $n$ using the identity
\[
T_{l}(\alpha_1,\ldots,\alpha_n) = T_{l-1}(\alpha_1,\ldots,\alpha_{n-1}) \, \alpha_n + T_{l}(\alpha_1,\ldots,\alpha_{n-1}),
\]
or via sequences of manipulations as described above\footnote[2]{See~\url{NewtonApproach_l_le_7.mw} for a derivation of these expressions.}.
\begin{lemma}\label{lem:T}
For all $n \ge l$ we have
\begin{enumerate}[label={(\arabic*)}]
\item $\displaystyle\begin{aligned}[t]
T_1(\alpha - \beta) = T_1(\alpha) - T_1(\beta),
\end{aligned}$
\item $\displaystyle\begin{aligned}[t]
T_2(\alpha - \beta) = T_2(\alpha) + T_2(\beta) - T_1(\alpha)T_1(\beta) + T_1(\alpha\beta),
\end{aligned}$
\item $\displaystyle\begin{aligned}[t]
T_3(\alpha - \beta) &= T_3(\alpha)-T_3(\beta) + T_1(\alpha)T_2(\beta) - T_1(\beta)T_2(\alpha) +T_1(\alpha)T_1(\alpha\beta) -T_1(\beta)T_1(\alpha\beta)\\
& + T_1(\alpha\beta^2) - T_1(\alpha^2\beta),
\end{aligned}$
\item $\displaystyle\begin{aligned}[t]
T_4(\alpha - \beta) &= T_4(\alpha)+T_4(\beta) -T_1(\alpha)T_3(\beta)-T_1(\beta)T_3(\alpha) +T_2(\alpha)T_2(\beta) -T_1(\alpha)T_1(\beta)T_1(\alpha\beta)\\
&+T_1(\alpha\beta)T_2(\alpha)+T_1(\alpha\beta)T_2(\beta)-T_1(\alpha)T_1(\alpha^2\beta)+T_1(\alpha)T_1(\alpha\beta^2)+T_1(\beta)T_1(\alpha^2\beta)\\
& -T_1(\beta)T_1(\alpha\beta^2) +T_1(\alpha^3\beta)-T_1(\alpha^2\beta^2)+T_1(\alpha\beta^3)+T_2(\alpha\beta),
\end{aligned}$
\item $\displaystyle\begin{aligned}[t]
T_5(\alpha - \beta) &= T_5(\alpha)-T_5(\beta) +T_1(\alpha)T_4(\beta) -T_1(\beta)T_4(\alpha) +T_2(\beta)T_3(\alpha)-T_2(\alpha)T_3(\beta) + T_1(\alpha)T_1(\beta)T_1(\alpha^2\beta)\\
& -T_1(\beta)T_1(\alpha\beta)T_2(\alpha)+T_1(\alpha)T_1(\alpha\beta)T_2(\beta)-T_1(\alpha)T_1(\beta)T_1(\alpha\beta^2) -T_1(\alpha^2 \beta)T_2(\alpha)\\
& -T_1(\alpha^2 \beta)T_2(\beta)+T_1(\alpha\beta^2)T_2(\alpha)+T_1(\alpha \beta^2)T_2(\beta)+T_1(\alpha\beta)T_3(\alpha)-T_1(\alpha\beta)T_3(\beta)\\
& -T_1(\alpha\beta)T_1(\alpha^2 \beta)+T_1(\alpha\beta)T_1(\alpha \beta^2)+T_1(\alpha)T_1(\alpha^3 \beta)-T_1(\alpha)T_1(\alpha^2 \beta^2)+T_1(\alpha)T_1(\alpha \beta^3)\\
& -T_1(\beta)T_1(\alpha^3 \beta)+T_1(\beta)T_1(\alpha^2 \beta^2) -T_1(\beta)T_1(\alpha \beta^3)+T_1(\alpha)T_2(\alpha\beta)-T_1(\beta)T_2(\alpha\beta)\\
& -T_1(\alpha^4 \beta)+2T_1(\alpha^3 \beta^2)-2T_1(\alpha^2 \beta^3)+T_1(\alpha \beta^4),\\
\end{aligned}$
\item $\displaystyle\begin{aligned}[t]
T_6(\alpha - \beta) &= T_6(\alpha)+T_6(\beta) -T_1(\alpha)T_5(\beta)-T_1(\beta)T_5(\alpha) +T_2(\alpha)T_4(\beta)+T_2(\beta)T_4(\alpha)-T_3(\beta)T_3(\alpha)\\
& +T_2(\alpha)T_2(\alpha\beta)+T_1(\alpha^5\beta)-2T_1(\alpha^4\beta^2)+2T_1(\alpha^3\beta^3)-2T_1(\alpha^2\beta^4)+T_1(\alpha\beta^5)\\
& +T_1(\alpha\beta)T_2(\alpha)T_2(\beta)+T_2(\alpha\beta)T_2(\beta)-T_1(\alpha)T_1(\alpha\beta)T_1(\alpha^2\beta)-T_1(\beta)T_1(\alpha\beta)T_1(\beta^2\alpha)\\
& -T_1(\alpha)T_1(\beta)T_2(\alpha\beta)+T_1(\beta)T_1(\alpha^2\beta)T_2(\alpha)+T_1(\alpha)T_1(\alpha\beta^2)T_2(\beta)-T_1(\alpha)T_1(\beta)T_1(\alpha\beta^3)\\
& +T_1(\alpha)T_1(\beta)T_1(\alpha^2\beta^2)+T_1(\alpha)T_1(\alpha\beta)T_1(\alpha\beta^2)-T_1(\beta)T_1(\alpha\beta^2)T_2(\alpha)-T_1(\alpha)T_1(\beta)T_1(\alpha^3\beta)\\
& -T_1(\beta)T_1(\alpha\beta)T_3(\alpha)-T_1(\alpha)T_1(\alpha^2\beta)T_2(\beta)-T_1(\alpha)T_1(\alpha\beta)T_3(\beta)+T_1(\beta)T_1(\alpha\beta)T_1(\alpha^2\beta)\\
& +4T_3(\alpha\beta)+T_2(\alpha\beta^2)-T_1(\alpha)T_1(\alpha^4\beta)+2T_1(\alpha)T_1(\alpha^3\beta^2)-2T_1(\alpha)T_1(\alpha^2\beta^3)+T_1(\alpha)T_1(\alpha\beta^4)\\
& +T_1(\beta)T_1(\alpha^4\beta)-2T_1(\beta)T_1(\alpha^3\beta^2)+2T_1(\beta)T_1(\alpha^2\beta^3)-T_1(\beta)T_1(\alpha\beta^4)+T_2(\alpha^2\beta)\\
& +T_1(\alpha\beta)T_4(\alpha)+T_1(\alpha\beta)T_4(\beta)+T_1(\alpha\beta)T_1(\alpha^3\beta)+T_1(\alpha\beta)T_1(\alpha\beta^3)-T_1(\alpha\beta)T_2(\alpha\beta)\\
& -T_1(\alpha^2\beta)T_3(\alpha)+T_1(\alpha^2\beta)T_3(\beta)-T_1(\alpha^2\beta)T_1(\alpha\beta^2)+T_1(\alpha\beta^2)T_3(\alpha)-T_1(\alpha\beta^2)T_3(\beta)\\
& +T_1(\alpha^3\beta)T_2(\alpha)+T_1(\alpha^3\beta)T_2(\beta)-T_1(\alpha^2\beta^2)T_2(\alpha)-T_1(\alpha^2\beta^2)T_2(\beta)+T_1(\alpha\beta^3)T_2(\alpha)\\
& +T_1(\alpha\beta^3)T_2(\beta),
\end{aligned}$
\item $\displaystyle\begin{aligned}[t]
T_7(\alpha - \beta) &=
T_7(\alpha) -T_7(\beta) +T_1(\alpha)T_6(\beta) -T_6(\alpha)T_1(\beta) -T_2(\alpha)T_5(\beta) +T_2(\beta)T_5(\alpha)+T_3(\alpha)T_4(\beta)\\
&-T_3(\beta)T_4(\alpha) + T_1(\alpha\beta)T_5(\alpha)
-T_1(\alpha\beta)T_5(\beta)
+T_1(\alpha\beta^6) - T_1(\alpha^6\beta)
-T_1(\alpha\beta)T_1(\alpha^4\beta)\\
&+2T_1(\alpha\beta)T_1(\alpha^3\beta^2)
-2T_1(\alpha\beta)T_1(\alpha^2\beta^3)
+T_1(\alpha\beta)T_1(\alpha\beta^4)
-T_1(\alpha^2\beta)T_1(\alpha^3\beta)\\
&+T_1(\alpha^2\beta)T_1(\alpha^2\beta^2)
-T_1(\alpha^2\beta)T_1(\alpha\beta^3)
-T_2(\alpha\beta)T_3(\beta)
-T_1(\beta)T_1(\alpha\beta^2)T_3(\alpha)\\
&-T_1(\beta)T_1(\alpha^3\beta)T_2(\alpha)
+T_1(\beta)T_1(\alpha^2\beta^2)T_2(\alpha)
-T_1(\beta)T_2(\alpha\beta)T_2(\alpha)
-T_1(\beta)T_1(\alpha\beta^3)T_2(\alpha)\\
&+3T_1(\alpha^5\beta^2)
-5T_1(\alpha^4\beta^3)
+5T_1(\alpha^3\beta^4)
-3T_1(\alpha^2\beta^5)
+T_2(\alpha\beta)T_3(\alpha)
+T_1(\alpha)T_1(\alpha\beta)T_1(\alpha^3\beta)\\
&+2T_1(\alpha)T_1(\alpha\beta)T_1(\alpha^2\beta^2)
+T_1(\alpha)T_1(\alpha\beta)T_1(\alpha\beta^3)
+T_1(\alpha)T_1(\alpha\beta)T_4(\beta)\\
&-3T_1(\alpha)T_1(\alpha\beta)T_2(\alpha\beta)
+T_1(\alpha)T_1(\alpha^2\beta)T_3(\beta)
-T_1(\alpha)T_1(\alpha^2\beta)T_1(\alpha\beta^2)\\
&-T_1(\alpha)T_1(\alpha\beta^2)T_3(\beta)
+T_1(\alpha)T_1(\alpha^3\beta)T_2(\beta)
-T_1(\alpha)T_1(\alpha^2\beta^2)T_2(\beta)
+T_1(\alpha)T_1(\beta)T_1(\alpha^4\beta)\\
&-2T_1(\alpha)T_1(\beta)T_1(\alpha^3\beta^2)
+2T_1(\alpha)T_1(\beta)T_1(\alpha^2\beta^3)
-T_1(\alpha)T_1(\beta)T_1(\alpha\beta^4)\\
&+T_1(\alpha)T_2(\alpha\beta)T_2(\beta)
+T_1(\alpha)T_1(\alpha\beta^3)T_2(\beta)
-T_1(\beta)T_1(\alpha\beta)T_1(\alpha^3\beta)\\
&-2T_1(\beta)T_1(\alpha\beta)T_1(\alpha^2\beta^2)
-T_1(\beta)T_1(\alpha\beta)T_1(\alpha\beta^3)
-T_1(\beta)T_1(\alpha\beta)T_4(\alpha)\\
&+3T_1(\beta)T_1(\alpha\beta)T_2(\alpha\beta)
+T_1(\beta)T_1(\alpha^2\beta)T_3(\alpha)
+T_1(\beta)T_1(\alpha^2\beta)T_1(\alpha\beta^2)\\
&-T_1(\alpha^2\beta)T_2(\beta)T_2(\alpha)
+T_1(\alpha\beta^2)T_2(\beta)T_2(\alpha)
+T_1(\alpha\beta)T_2(\beta)T_3(\alpha)
+T_1(\alpha)T_2(\alpha\beta^2)\\
\end{aligned}$
\end{enumerate}
$\begin{aligned}
\hspace{22.5mm} &+T_1(\alpha)T_2(\alpha^2\beta)
+10T_1(\alpha)T_3(\alpha\beta)
-T_1(\beta)T_2(\alpha\beta^2)
-T_1(\beta)T_2(\alpha^2\beta)
-10T_1(\beta)T_3(\alpha\beta)\\
&+T_1(\alpha)T_1(\alpha^5\beta)
-2T_1(\alpha)T_1(\alpha^4\beta^2)
-2T_1(\alpha)T_1(\alpha^2\beta^4)
+T_1(\alpha)T_1(\alpha\beta^5)
-T_1(\beta)T_1(\alpha^5\beta)\\
&+2T_1(\beta)T_1(\alpha^4\beta^2)
+2T_1(\beta)T_1(\alpha^2\beta^4)
-T_1(\beta)T_1(\alpha\beta^5)
-T_1(\alpha^4\beta)T_2(\alpha)
-T_1(\alpha^4\beta)T_2(\beta)\\
&+2T_1(\alpha^3\beta^2)T_2(\alpha)
+2T_1(\alpha^3\beta^2)T_2(\beta)
-2T_1(\alpha^2\beta^3)T_2(\alpha)
-2T_1(\alpha^2\beta^3)T_2(\beta)\\
&+T_1(\alpha\beta^4)T_2(\alpha)
+T_1(\alpha\beta^4)T_2(\beta)
+T_1(\alpha^3\beta)T_3(\alpha)
-T_1(\alpha^3\beta)T_3(\beta)
-T_1(\alpha^2\beta^2)T_3(\alpha)\\
&+T_1(\alpha^2\beta^2)T_3(\beta)
+T_1(\alpha\beta^3)T_3(\alpha)
-T_1(\alpha\beta^3)T_3(\beta)
-T_1(\alpha^2\beta)T_4(\alpha)
-T_1(\alpha^2\beta)T_4(\beta)\\
&-T_1(\alpha^2\beta)T_2(\alpha\beta)
+T_1(\alpha\beta^2)T_1(\alpha^3\beta)
-T_1(\alpha\beta^2)T_1(\alpha^2\beta^2)
+T_1(\alpha\beta^2)T_1(\alpha\beta^3)\\
&+T_1(\alpha\beta^2)T_4(\alpha)
+T_1(\alpha\beta^2)T_4(\beta)
+T_1(\alpha\beta^2)T_2(\alpha\beta)
-T_1(\alpha\beta)T_1(\alpha^2\beta)T_2(\alpha)\\
&-T_1(\alpha\beta)T_1(\alpha^2\beta)T_2(\beta)
+T_1(\alpha\beta)T_1(\alpha\beta^2)T_2(\alpha)
+T_1(\alpha\beta)T_1(\alpha\beta^2)T_2(\beta)\\
&-T_1(\alpha\beta)T_2(\alpha)T_3(\beta)
+T_1(\alpha)T_1(\beta)T_1(\alpha\beta)T_1(\alpha^2\beta)
-T_1(\alpha)T_1(\beta)T_1(\alpha\beta)T_1(\alpha\beta^2).
\end{aligned}$
\end{lemma}
Observe that for each $1 \le l \le 7$ all of the terms appearing in Lemma~\ref{lem:T} part (l) have total degree $l$, when
counted in the natural way. Hence when one reduces mod $2$ and sets
$\beta = \alpha^2$, the two terms $T_l(\alpha)$ and $T_l(\beta)$ cancel, leaving only $T_l$'s of degree $< l$, as claimed earlier.
Unfortunately, our approach of using Newton's identities evaluated for various $l$ at products of powers of $\alpha$ and $\beta$ so that no
trace occurs to any power larger than one, fails for $l = 8$ due to the presence of the term $T_2(xy)^2$, which can not be eliminated while
keeping the remaining terms' coefficients divisible by $8$. Whether or not there exist such expressions for $T_l(\alpha - \beta)$ over ${\mathbb Z}$
for $l \ge 8$, we leave as an open problem. Note that it is known that the ring of multisymmetric functions in two sets of variables
$\alpha_1,\ldots,\alpha_n$ and $\beta_1,\ldots,\beta_n$ is not generated over ${\mathbb Z}$ by the elementary multisymmetric functions that we are using, unless
$n=2$~\cite{dalbec}. However, since we are only interested in a particular family of multisymmetric functions -- namely $T_l(\alpha - \beta)$ -- and not all of them, it is possible that such expressions exist.
\section{Curves and Explicit Formulae for $q = 2$ and $l \le 7$}\label{sec:binaryzetas}
In this section we apply the indirect method and for some cases the direct method to determine relevant curves for $l \le 7$ and $n$ odd.
Using Magma we provide explicit formulae for $l \le 5$ and $n$ odd using the indirect method, and provide explicit formulae for a subset of these cases using the direct method, for all relevant $n$.
In practice, rather than obtain a curve as sketched in the previous section for each $i \in \{0,\ldots,2^l-1\}$, it is more efficient
to compute a linearisation for each featured $T_{l_k}(a)$ and then combine them as appropriate according to whether $i_k$ is $0$ or $1$ for a
given $i$, as we do in the examples that follow.
Observe that for the indirect method, once $V(\vec{i}\cdot\vec{f})$ has been obtained for
$\vec{f} = (T_l,\ldots,T_1)$ and $i \in \{0,\ldots,2^l-1\}$, these functions need not be recomputed for the subsequent $\vec{f} = (T_{l+1},\ldots,T_1)$.
\subsection{Computing $F_2(n,t_1,t_2,t_3)$}
The formulae for $l = 3$ were presented in~\cite[\S\S4\&5]{AGGMY} -- although obtained there in a slightly different
manner -- but we include them here for demonstration purposes and completeness.
\subsubsection{Indirect method.} Setting $\vec{f} = (T_3,T_2,T_1)$, by the transform~(\ref{eq:Sinverse}) we have
\begin{equation}\label{eq:transform1_3}
{\small
\begin{bmatrix}
F_2(n,0,0,0)\\
F_2(n,1,0,0)\\
F_2(n,0,1,0)\\
F_2(n,1,1,0)\\
F_2(n,0,0,1)\\
F_2(n,1,0,1)\\
F_2(n,0,1,1)\\
F_2(n,1,1,1)\\
\end{bmatrix}
=
\begin{bmatrix}
N(\vec{0})\\
N(\vec{1})\\
N(\vec{2})\\
N(\vec{3})\\
N(\vec{4})\\
N(\vec{5})\\
N(\vec{6})\\
N(\vec{7})\\
\end{bmatrix}
=
\frac{1}{4}
\begin{bmatrix*}[r]
-3 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\
1 & -1 & 1 & -1 & 1 & -1 & 1 & -1\\
1 & 1 & -1 & -1 & 1 & 1 & -1 & -1\\
1 & -1 & -1 & 1 & 1 & -1 & -1 & 1\\
1 & 1 & 1 & 1 & -1 & -1 & -1 & -1\\
1 & -1 & 1 & -1 & -1 & 1 & -1 & 1\\
1 & 1 & -1 & -1 & -1 & -1 & 1 & 1\\
1 & -1 & -1 & 1 & -1 & 1 & 1 & -1\\
\end{bmatrix*}
\begin{bmatrix}
V(\vec{0} \cdot \vec{f})\\
V(\vec{1} \cdot \vec{f})\\
V(\vec{2} \cdot \vec{f})\\
V(\vec{3} \cdot \vec{f})\\
V(\vec{4} \cdot \vec{f})\\
V(\vec{5} \cdot \vec{f})\\
V(\vec{6} \cdot \vec{f})\\
V(\vec{7} \cdot \vec{f})\\
\end{bmatrix}.}
\end{equation}
By definition we have $V(\vec{0} \cdot \vec{f}) = 2^n$, while $V(\vec{1} \cdot \vec{f}) = V(T_1) = \#\{a \in {\mathbb F}_{2^n} \mid T_1(a) = 0\} = 2^{n-1}$.
To determine $V( \vec{i} \cdot \vec{f})$ for $2 \le i \le 7$, we use Lemma~\ref{lem:T} parts (1) to (3). In particular, setting
$\alpha = a_{0}^2$ and $\beta = a_{0}$ for $r_0 = 0$, and $\alpha = a_{0}^2 + a_0$ and $\beta = 1$ for $r_0 = 1$, and evaluating mod $2$
gives the following:
\begin{eqnarray}
\label{eq:T1linear} T_1(a_{0}^2 + a_0 + r_0) &=& T_1(r_0),\\
\label{eq:T2linear} T_2(a_{0}^2 + a_0 + r_0) &=& T_1\Big(a_{0}^3 + a_{0} + r_0\binom{n}{2}\Big),\\
\label{eq:T3linear} T_3(a_{0}^2 + a_0 + r_0) &=& T_1\Big(a_{0}^5 + a_{0} + r_0\Big(a_{0}^3 + a_0 + \binom{n}{3} \Big)\Big).
\end{eqnarray}
For $2 \le i \le 7$ let $\vec{i} = (i_2,i_1,i_0)$. The curves we are interested in for $a$ of trace $r_0$
are therefore:
\begin{eqnarray}
\label{3trace1} a_{1}^2 + a_{1} &=& i_2\Big(a_{0}^5 + a_{0} + r_0\Big(a_{0}^3 + a_0 + \binom{n}{3} \Big)\Big)
+ i_1\Big(a_{0}^3 + a_{0} + r_0\binom{n}{2}\Big) + i_0 r_0 \binom{n}{1}.
\end{eqnarray}
These curves have genus $1$ if $i_2 = 0$ and genus $2$ if $i_2 = 1$, and are all supersingular. As pointed out in~\cite{AGGMY}, this is why the formulae are periodic in $n$.
By Corollary~\ref{cor:period} the vector $(\binom{n}{3},\binom{n}{2},\binom{n}{1})$ mod $2$ has period $4$ and is equal to $(0,0,1)$ if
$n \equiv 1 \pmod{4}$, and $(1,1,1)$ if $n \equiv 3 \pmod{4}$. Hence there are two cases to consider when computing the zeta functions of the
curves specified in Eq.~(\ref{3trace1}).
In order to express $F_2(n,t_1,t_2,t_3)$ compactly, we define the following polynomials:
\begin{eqnarray*}
\delta_{2,1} &=& X^2 + 2X + 2,\\
\delta_{4,1} &=& X^4 + 2X^3 + 2X^2 + 4X + 4,
\end{eqnarray*}
which are the characteristic polynomials of Frobenius of the following two supersingular curves, respectively:
\begin{eqnarray*}
C_{2,1}/{\mathbb F}_2: && y^2 + y = x^3 + x,\\
C_{4,1}/{\mathbb F}_2: && y^2 + y = x^5 + x^3.
\end{eqnarray*}
Using Magma to compute the zeta functions of the curves~(\ref{3trace1}) and applying~(\ref{eq:transform1_3}) gives the following.
\begin{theorem}\label{thm:3traces}
For $n \ge 3$ we have
{\small
\begin{eqnarray*}
F_2(n,0,0,0) &=& 2^{n-3} - \frac{1}{8} \big( 2\rho_n(\delta_{2,1}) + \rho_n(\delta_{4,1}) \big) \ \ \text{if} \ n \equiv 1,3 \pmod{4}\\
F_2(n,1,0,0) &=& 2^{n-3} - \frac{1}{8} \cdot \begin{cases}
\begin{array}{lr}
2\rho_n(\delta_{2,1}) + \rho_n(\delta_{4,1}) & \ \text{if} \ n \equiv 1 \pmod{4}\\
-\rho_n(\delta_{4,1}) & \ \text{if} \ n \equiv 3 \pmod{4}\\
\end{array}
\end{cases}\\
F_2(n,0,1,0) &=&2^{n-3} - \frac{1}{8} \big( - \rho_n(\delta_{4,1}) \big) \ \ \text{if} \ n \equiv 1,3 \pmod{4}\\
F_2(n,1,1,0) &=&2^{n-3} - \frac{1}{8} \cdot \begin{cases}
\begin{array}{lr}
-2\rho_n(\delta_{2,1}) + \rho_n(\delta_{4,1}) & \ \text{if} \ n \equiv 1 \pmod{4}\\
-\rho_n(\delta_{4,1}) & \ \text{if} \ n \equiv 3 \pmod{4}\\
\end{array}
\end{cases}\\
F_2(n,0,0,1) &=&2^{n-3} - \frac{1}{8} \big( - \rho_n(\delta_{4,1}) \big) \ \ \text{if} \ n \equiv 1,3 \pmod{4}\\
F_2(n,1,0,1) &=&2^{n-3} - \frac{1}{8} \cdot \begin{cases}
\begin{array}{lr}
- \rho_n(\delta_{4,1}) & \ \text{if} \ n \equiv 1 \pmod{4}\\
-2\rho_n(\delta_{2,1}) + \rho_n(\delta_{4,1}) & \ \text{if} \ n \equiv 3 \pmod{4}\\
\end{array}
\end{cases}\\
F_2(n,0,1,1) &=&2^{n-3} - \frac{1}{8} \big( -2\rho_n(\delta_{2,1}) + \rho_n(\delta_{4,1}) \big) \ \ \text{if} \ n \equiv 1,3 \pmod{4}\\
F_2(n,1,1,1) &=&2^{n-3} - \frac{1}{8} \cdot \begin{cases}
\begin{array}{lr}
- \rho_n(\delta_{4,1}) & \ \text{if} \ n \equiv 1 \pmod{4}\\
2\rho_n(\delta_{2,1}) + \rho_n(\delta_{4,1}) & \ \text{if} \ n \equiv 3 \pmod{4}\\
\end{array}
\end{cases}\\
\end{eqnarray*}
}
\end{theorem}
The roots of $\delta_{2,1}$ are $\sqrt{2}\omega_{8}^3,\sqrt{2}\omega_{8}^5$, with $\omega_8 = e^{i \pi/4} = (1 + i)/\sqrt{2}$,
while the roots of $\delta_{4,1}$ are $\sqrt{2}\omega_{24}^{5},\sqrt{2}\omega_{24}^{11}$,
$\sqrt{2}\omega_{24}^{13},\sqrt{2}\omega_{24}^{19}$,
with $\omega_{24} = e^{i \pi /12} = ((1+\sqrt{3}) + (-1+\sqrt{3})i)/2\sqrt{2}$.
Observe that $F_2(n,t_1,t_2)$ can be obtained similarly, or by adding $F_2(n,t_1,t_2,0)$ and $F_2(n,t_1,t_2,1)$ as given in
Theorem~\ref{thm:3traces}. Likewise $F_2(n,t_1)$ can be obtained as $F_2(n,t_1,0,0) + F_2(n,t_1,0,1)+ F_2(n,t_1,1,0) + F_2(n,t_1,1,1)$,
and summing all the expressions gives $2^n$, as they must.
\subsubsection{Direct method.} For odd $n$, applying Equations~(\ref{eq:T1linear}) to~(\ref{eq:T3linear}) we have
\begin{eqnarray}
\nonumber F_2(n,t_1,t_2,t_3) &=& \#\{a \in {\mathbb F}_{2^n} \mid T_1(a) = t_1, \ T_2(a) = t_2, \ T_3(a) = t_3\}\\
\nonumber &=& \frac{1}{2} \, \#\{a_0 \in {\mathbb F}_{2^n} \mid T_2(a_{0}^2 + a_{0} + t_1) = t_2, \ T_3(a_{0}^2 + a_{0} + t_1) = t_3 \}\\
\nonumber &=& \frac{1}{2} \, \#\{a_0 \in {\mathbb F}_{2^n} \mid T_1\Big(a_{0}^3 + a_{0} + t_1\binom{n}{2}\Big) = t_2, \ T_1\Big(a_{0}^5 + a_{0} + t_1\Big(a_{0}^3 +
a_0+\binom{n}{3}\Big)\Big) = t_3 \}\\
\nonumber &=& \frac{1}{8} \, \#\{(a_0,a_1,a_2) \in ({\mathbb F}_{2^n})^3 \mid a_{1}^2 + a_1 = a_{0}^3 + a_{0} + t_1\binom{n}{2} + t_2,\\
\nonumber && \ \ \ \ a_{2}^2 + a_2 = a_{0}^5 + a_{0} + t_1\Big(a_{0}^3 + a_0+\binom{n}{3}\Big) + t_3\}.
\end{eqnarray}
These supersingular curves are all absolutely irreducible and are of genus $5$, and for $n$ odd their zeta functions reproduce Theorem~\ref{thm:3traces}.
Note that this method is slightly different to that used in~\cite[\S5]{AGGMY}, since it counts the number of points on a curve defined by
an intersection of two curves (with one variable in common), rather than the sum of the number of points on three curves.
Letting $\delta_{2,2} = X^2 + 2$ (corresponding to the curve $C_{2,2}/{\mathbb F}_2: y^2 + y = x^3 + 1$),
further note that for all $n \ge 3$ we have
\begin{eqnarray}
\nonumber F_2(n,0,0,0) &=& \frac{1}{8} \, \#\{ (a_0,a_1,a_2) \in ({\mathbb F}_{2^n})^3 \mid a_{1}^2 + a_{1}=a_{0}^3 + a_{0}, \ a_{2}^2 + a_{2} = a_{0}^5 + a_{0}\}\\
\nonumber &=& \frac{1}{8}\big(2^{n} - 2\rho_n(\delta_{2,1}) - \rho_n(\delta_{2,2}) - \rho_n(\delta_{4,1})\big),
\end{eqnarray}
since one does not need to parameterise any `linear trace $= 1$' conditions. Using the same basic observations from~\cite[\S\S4\&5]{AGGMY} one can
determine the formulae, valid for all $n \ge 3$, for all four $F_2(n,0,t_2,t_3)$.
\subsection{Computing $F_2(n,t_1,t_2,t_3,t_4)$}
We begin by applying the indirect method.
\subsubsection{Indirect method.} Setting $\vec{f} = (T_4,T_3,T_2,T_1)$, by the transform~(\ref{eq:Sinverse}) we have
\begin{equation}\label{eq:transform1_4}
{\small
\begin{bmatrix}
F_2(n,0,0,0,0)\\
F_2(n,1,0,0,0)\\
F_2(n,0,1,0,0)\\
F_2(n,1,1,0,0)\\
F_2(n,0,0,1,0)\\
F_2(n,1,0,1,0)\\
F_2(n,0,1,1,0)\\
F_2(n,1,1,1,0)\\
F_2(n,0,0,0,1)\\
F_2(n,1,0,0,1)\\
F_2(n,0,1,0,1)\\
F_2(n,1,1,0,1)\\
F_2(n,0,0,1,1)\\
F_2(n,1,0,1,1)\\
F_2(n,0,1,1,1)\\
F_2(n,1,1,1,1)\\
\end{bmatrix}
=
\begin{bmatrix}
N(\vec{0})\\
N(\vec{1})\\
N(\vec{2})\\
N(\vec{3})\\
N(\vec{4})\\
N(\vec{5})\\
N(\vec{6})\\
N(\vec{7})\\
N(\vec{8})\\
N(\vec{9})\\
N(\vec{10})\\
N(\vec{11})\\
N(\vec{12})\\
N(\vec{13})\\
N(\vec{14})\\
N(\vec{15})\\
\end{bmatrix}
=
\frac{1}{8}
\begin{bmatrix*}[r]
-7 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\
1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1\\
1 & 1 & -1 & -1 & 1 & 1 & -1 & -1 & 1 & 1 & -1 & -1 & 1 & 1 & -1 & -1\\
1 & -1 & -1 & 1 & 1 & -1 & -1 & 1 & 1 & -1 & -1 & 1 & 1 & -1 & -1 & 1\\
1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1\\
1 & -1 & 1 & -1 & -1 & 1 & -1 & 1 & 1 & -1 & 1 & -1 & -1 & 1 & -1 & 1\\
1 & 1 & -1 & -1 & -1 & -1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & 1 & 1\\
1 & -1 & -1 & 1 & -1 & 1 & 1 & -1 & 1 & -1 & -1 & 1 & -1 & 1 & 1 & -1\\
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1\\
1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & 1\\
1 & 1 & -1 & -1 & 1 & 1 & -1 & -1 & -1 & -1 & 1 & 1 & -1 & -1 & 1 & 1\\
1 & -1 & -1 & 1 & 1 & -1 & -1 & 1 & -1 & 1 & 1 & -1 & -1 & 1 & 1 & -1\\
1 & 1 & 1 & 1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & -1 & 1 & 1 & 1 & 1\\
1 & -1 & 1 & -1 & -1 & 1 & -1 & 1 & -1 & 1 & -1 & 1 & 1 & -1 & 1 & -1\\
1 & 1 & -1 & -1 & -1 & -1 & 1 & 1 & -1 & -1 & 1 & 1 & 1 & 1 & -1 & -1\\
1 & -1 & -1 & 1 & -1 & 1 & 1 & -1 & -1 & 1 & 1 & -1 & 1 & -1 & -1 & 1\\
\end{bmatrix*}
\begin{bmatrix}
V(\vec{0} \cdot \vec{f})\\
V(\vec{1} \cdot \vec{f})\\
V(\vec{2} \cdot \vec{f})\\
V(\vec{3} \cdot \vec{f})\\
V(\vec{4} \cdot \vec{f})\\
V(\vec{5} \cdot \vec{f})\\
V(\vec{6} \cdot \vec{f})\\
V(\vec{7} \cdot \vec{f})\\
V(\vec{8} \cdot \vec{f})\\
V(\vec{9} \cdot \vec{f})\\
V(\vec{10} \cdot \vec{f})\\
V(\vec{11} \cdot \vec{f})\\
V(\vec{12} \cdot \vec{f})\\
V(\vec{13} \cdot \vec{f})\\
V(\vec{14} \cdot \vec{f})\\
V(\vec{15} \cdot \vec{f})\\
\end{bmatrix}.
}
\end{equation}
To determine $V( \vec{i} \cdot \vec{f})$ for $8 \le i \le 15$, we use Lemma~\ref{lem:T} part (4).
In particular, setting $\alpha = a_{0}^2$ and $\beta = a_{0}$ for $r_0 = 0$, and $\alpha = a_{0}^2 + a_0$ and $\beta = 1$ for $r_0 = 1$,
and evaluating mod $2$ gives the following:
\begin{eqnarray}
\nonumber T_4(a_{0}^2 + a_0 + r_0) &=& T_2(a_{0}^3) + T_2(a_{0}) + T_1(a_{0}^3)T_1(a_0)\\
&+& T_1\Big(a_{0}^7 + a_{0}^5 + a_{0}^3 + r_0\Big(a_{0}^3 + a_0 + (a_{0}^3 + a_0)\binom{n}{2} + \binom{n}{4}\Big)\Big).\label{eq:4coeffsparam1}
\end{eqnarray}
This can be linearised using the substitutions $a_0 = a_{1}^2 + a_1 + r_1$ and $a_{0}^3 = a_{2}^2 + a_2 + r_2$,
where $r_1,r_2 \in {\mathbb F}_2$ are the traces of $a_0$ and $a_{0}^3$ respectively. This results in\footnote[2]{See~\url{NewtonApproach_l_le_7.mw} to verify the linearised expressions for $4,5,6$ and $7$ coefficients.}
\begin{eqnarray*}
\nonumber T_4(a_{0}^2 + a_0 + r_0) &=& T_1\Big(a_{2}^3 + a_2 + a_{1}^3 + a_1 + a_{0}^7 + a_{0}^5 + r_0r_1 + r_0r_2 + r_1r_2 + r_2\\
&+&(r_0 + 1)(r_1 + r_2)\binom{n}{2} + r_0\binom{n}{4} \Big).
\end{eqnarray*}
For $8 \le i \le 15$ let $\vec{i} = (i_3,i_2,i_1,i_0)$. The curves we are interested in are given by the following intersections:
\begin{eqnarray}
\nonumber a_{3}^2 + a_{3} &=& i_3\Big(a_{2}^3 + a_2 + a_{1}^3 + a_1 + a_{0}^7 + a_{0}^5 + r_0r_1 + r_0r_2 + r_1r_2 + r_2 + (r_0 + 1)(r_1 + r_2)\binom{n}{2} + r_0\binom{n}{4}\Big)\\
\nonumber &+& i_2\Big(a_{0}^5 + a_{0} + r_0\Big(a_{0}^3 + a_0 + \binom{n}{3} \Big)\Big) + i_1\Big(a_{0}^3 + a_{0} + r_0\binom{n}{2}\Big) + i_0r_0,\\
\label{eq:4trace1} a_0 &=& a_{1}^2 + a_1 + r_1,\\
\nonumber a_{0}^3 &=& a_{2}^2 + a_2 + r_2.
\end{eqnarray}
For $i_3 = 1$ the genus of all of these absolutely irreducible curves is $14$. Corollary~\ref{cor:period} implies that mod $2$ one has
\begin{eqnarray*}
\Big(\binom{n}{4},\binom{n}{3},\binom{n}{2},\binom{n}{1}\Big) \equiv \begin{cases}
\begin{array}{lr}
(0,0,0,1) & \ \text{if} \ n \equiv 1 \pmod{8}\\
(0,1,1,1) & \ \text{if} \ n \equiv 3 \pmod{8}\\
(1,0,0,1) & \ \text{if} \ n \equiv 5 \pmod{8}\\
(1,1,1,1) & \ \text{if} \ n \equiv 7 \pmod{8}\\
\end{array},
\end{cases}
\end{eqnarray*}
and hence there are four cases to consider when computing the zeta functions of each of the curves~(\ref{eq:4trace1}).
In order to express $F_2(n,t_1,t_2,t_3,t_4)$ compactly, we further define the following polynomials:
\begin{eqnarray*}
\delta_{8,1} &=& X^8 + 4X^7 + 6X^6 + 4X^5 + 2X^4 + 8X^3 + 24X^2 + 32X + 16,\\
\delta_{8,2} &=& X^8 + 2X^6 + 4X^5 + 2X^4 + 8X^3 + 8X^2 + 16.
\end{eqnarray*}
Note that by~\cite[Prop. 1]{Stichtenoth}, neither $\delta_{8,1}$ or $\delta_{8,2}$ are the
characteristic polynomials of the Frobenius endomorphism of supersingular abelian varieties.
There are two other polynomials which occur as factors of the characteristic polynomial of Frobenius of the above curves,
but they are even polynomials and hence can be ignored for $n$ odd.
Using Magma to compute the zeta functions of the curves~(\ref{eq:4trace1}) and applying~(\ref{eq:transform1_4}) gives the following result.
\begin{theorem}\label{thm:4traces}
For $n \ge 4$ we have
{\small
\begin{eqnarray*}
F_2(n,0,0,0,0) &=& 2^{n-4} - \frac{1}{16} \big( 4\rho_n(\delta_{2,1}) + \rho_n(\delta_{4,1}) + \rho_n(\delta_{8,1}) + \rho_n(\delta_{8,2}) \big) \ \ \text{if} \ n \equiv 1,3,5,7 \pmod{8}\\
F_2(n,1,0,0,0) &=& 2^{n-4} - \frac{1}{16} \cdot \begin{cases}
\begin{array}{lr}
4\rho_n(\delta_{2,1}) + \rho_n(\delta_{4,1}) + \rho_n(\delta_{8,1}) + \rho_n(\delta_{8,2}) & \ \text{if} \ n \equiv 1 \pmod{8}\\
2\rho_n(\delta_{2,1}) - \rho_n(\delta_{4,1}) - \rho_n(\delta_{8,1}) + \rho_n(\delta_{8,2}) & \ \text{if} \ n \equiv 3 \pmod{8}\\
\rho_n(\delta_{4,1}) - \rho_n(\delta_{8,1}) - \rho_n(\delta_{8,2}) & \ \text{if} \ n \equiv 5 \pmod{8}\\
- 2\rho_n(\delta_{2,1}) - \rho_n(\delta_{4,1}) + \rho_n(\delta_{8,1}) - \rho_n(\delta_{8,2}) & \ \text{if} \ n \equiv 7 \pmod{8}\\
\end{array}
\end{cases}\\
F_2(n,0,1,0,0) &=& 2^{n-4} - \frac{1}{16} \cdot \begin{cases}
\begin{array}{lr}
- 2\rho_n(\delta_{2,1}) - \rho_n(\delta_{4,1}) + \rho_n(\delta_{8,1}) - \rho_n(\delta_{8,2}) & \ \text{if} \ n \equiv 1,5 \pmod{8}\\
2\rho_n(\delta_{2,1}) - \rho_n(\delta_{4,1}) - \rho_n(\delta_{8,1}) + \rho_n(\delta_{8,2}) & \ \text{if} \ n \equiv 3,7 \pmod{8}\\
\end{array}
\end{cases}\\
F_2(n,1,1,0,0) &=& 2^{n-4} - \frac{1}{16} \cdot \begin{cases}
\begin{array}{lr}
-4\rho_n(\delta_{2,1}) + \rho_n(\delta_{4,1}) + \rho_n(\delta_{8,1}) + \rho_n(\delta_{8,2}) & \ \text{if} \ n \equiv 1 \pmod{8}\\
-2\rho_n(\delta_{2,1}) - \rho_n(\delta_{4,1}) + \rho_n(\delta_{8,1}) - \rho_n(\delta_{8,2}) & \ \text{if} \ n \equiv 3 \pmod{8}\\
\rho_n(\delta_{4,1}) - \rho_n(\delta_{8,1}) - \rho_n(\delta_{8,2}) & \ \text{if} \ n \equiv 5 \pmod{8}\\
2\rho_n(\delta_{2,1}) - \rho_n(\delta_{4,1}) - \rho_n(\delta_{8,1}) + \rho_n(\delta_{8,2}) & \ \text{if} \ n \equiv 7 \pmod{8}\\
\end{array}
\end{cases}\\
F_2(n,0,0,1,0) &=& 2^{n-4} - \frac{1}{16} \big(-2\rho_n(\delta_{2,1}) - \rho_n(\delta_{4,1}) + \rho_n(\delta_{8,1}) - \rho_n(\delta_{8,2} )\big) \ \ \text{if} \ n \equiv 1,3,5,7 \pmod{8}\\
F_2(n,1,0,1,0) &=& 2^{n-4} - \frac{1}{16} \cdot \begin{cases}
\begin{array}{lr}
-2\rho_n(\delta_{2,1}) - \rho_n(\delta_{4,1}) + \rho_n(\delta_{8,1}) - \rho_n(\delta_{8,2}) & \ \text{if} \ n \equiv 1 \pmod{8}\\
-4\rho_n(\delta_{2,1}) + \rho_n(\delta_{4,1}) + \rho_n(\delta_{8,1}) + \rho_n(\delta_{8,2}) & \ \text{if} \ n \equiv 3 \pmod{8}\\
2\rho_n(\delta_{2,1}) - \rho_n(\delta_{4,1}) - \rho_n(\delta_{8,1}) + \rho_n(\delta_{8,2}) & \ \text{if} \ n \equiv 5 \pmod{8}\\
\rho_n(\delta_{4,1}) - \rho_n(\delta_{8,1}) - \rho_n(\delta_{8,2}) & \ \text{if} \ n \equiv 7 \pmod{8}\\
\end{array}
\end{cases}\\
F_2(n,0,1,1,0) &=& 2^{n-4} - \frac{1}{16} \cdot \begin{cases}
\begin{array}{lr}
\rho_n(\delta_{4,1}) - \rho_n(\delta_{8,1}) - \rho_n(\delta_{8,2}) & \ \text{if} \ n \equiv 1,5 \pmod{8}\\
-4\rho_n(\delta_{2,1}) + \rho_n(\delta_{4,1}) + \rho_n(\delta_{8,1}) + \rho_n(\delta_{8,2}) & \ \text{if} \ n \equiv 3,7 \pmod{8}\\
\end{array}
\end{cases}\\
F_2(n,1,1,1,0) &=& 2^{n-4} - \frac{1}{16} \cdot \begin{cases}
\begin{array}{lr}
2\rho_n(\delta_{2,1}) - \rho_n(\delta_{4,1}) - \rho_n(\delta_{8,1}) + \rho_n(\delta_{8,2}) & \ \text{if} \ n \equiv 1 \pmod{8}\\
4\rho_n(\delta_{2,1}) + \rho_n(\delta_{4,1}) + \rho_n(\delta_{8,1}) + \rho_n(\delta_{8,2}) & \ \text{if} \ n \equiv 3 \pmod{8}\\
- 2\rho_n(\delta_{2,1}) - \rho_n(\delta_{4,1}) + \rho_n(\delta_{8,1}) - \rho_n(\delta_{8,2}) & \ \text{if} \ n \equiv 5 \pmod{8}\\
\rho_n(\delta_{4,1}) - \rho_n(\delta_{8,1}) - \rho_n(\delta_{8,2}) & \ \text{if} \ n \equiv 7 \pmod{8}\\
\end{array}
\end{cases}\\
F_2(n,0,0,0,1) &=& 2^{n-4} - \frac{1}{16} \big( \rho_n(\delta_{4,1}) - \rho_n(\delta_{8,1}) - \rho_n(\delta_{8,2}) \big) \ \ \text{if} \ n \equiv 1,3,5,7 \pmod{8}\\
F_2(n,1,0,0,1) &=& 2^{n-4} - \frac{1}{16} \cdot \begin{cases}
\begin{array}{lr}
\rho_n(\delta_{4,1}) - \rho_n(\delta_{8,1}) - \rho_n(\delta_{8,2}) & \ \text{if} \ n \equiv 1 \pmod{8}\\
-2\rho_n(\delta_{2,1}) - \rho_n(\delta_{4,1}) + \rho_n(\delta_{8,1}) - \rho_n(\delta_{8,2}) & \ \text{if} \ n \equiv 3 \pmod{8}\\
4\rho_n(\delta_{2,1}) + \rho_n(\delta_{4,1}) + \rho_n(\delta_{8,1}) + \rho_n(\delta_{8,2}) & \ \text{if} \ n \equiv 5 \pmod{8}\\
2\rho_n(\delta_{2,1}) - \rho_n(\delta_{4,1}) - \rho_n(\delta_{8,1}) + \rho_n(\delta_{8,2}) & \ \text{if} \ n \equiv 7 \pmod{8}\\
\end{array}
\end{cases}\\
F_2(n,0,1,0,1) &=& 2^{n-4} - \frac{1}{16} \cdot \begin{cases}
\begin{array}{lr}
2\rho_n(\delta_{2,1}) - \rho_n(\delta_{4,1}) - \rho_n(\delta_{8,1}) + \rho_n(\delta_{8,2}) & \ \text{if} \ n \equiv 1,5 \pmod{8}\\
-2\rho_n(\delta_{2,1}) - \rho_n(\delta_{4,1}) + \rho_n(\delta_{8,1}) - \rho_n(\delta_{8,2}) & \ \text{if} \ n \equiv 3,7 \pmod{8}\\
\end{array}
\end{cases}\\
F_2(n,1,1,0,1) &=& 2^{n-4} - \frac{1}{16} \cdot \begin{cases}
\begin{array}{lr}
\rho_n(\delta_{4,1}) - \rho_n(\delta_{8,1}) - \rho_n(\delta_{8,2}) & \ \text{if} \ n \equiv 1 \pmod{8}\\
2\rho_n(\delta_{2,1}) - \rho_n(\delta_{4,1}) - \rho_n(\delta_{8,1}) + \rho_n(\delta_{8,2}) & \ \text{if} \ n \equiv 3 \pmod{8}\\
-4\rho_n(\delta_{2,1}) + \rho_n(\delta_{4,1}) + \rho_n(\delta_{8,1}) + \rho_n(\delta_{8,2}) & \ \text{if} \ n \equiv 5 \pmod{8}\\
-2\rho_n(\delta_{2,1}) - \rho_n(\delta_{4,1}) + \rho_n(\delta_{8,1}) - \rho_n(\delta_{8,2}) & \ \text{if} \ n \equiv 7 \pmod{8}\\
\end{array}
\end{cases}\\
F_2(n,0,0,1,1) &=& 2^{n-4} - \frac{1}{16} \big( 2\rho_n(\delta_{2,1}) - \rho_n(\delta_{4,1}) - \rho_n(\delta_{8,1}) + \rho_n(\delta_{8,2}) \big) \ \ \text{if} \ n \equiv 1,3,5,7 \pmod{8}\\
F_2(n,1,0,1,1) &=& 2^{n-4} - \frac{1}{16} \cdot \begin{cases}
\begin{array}{lr}
2\rho_n(\delta_{2,1}) - \rho_n(\delta_{4,1}) - \rho_n(\delta_{8,1}) + \rho_n(\delta_{8,2}) & \ \text{if} \ n \equiv 1 \pmod{8}\\
\rho_n(\delta_{4,1}) - \rho_n(\delta_{8,1}) - \rho_n(\delta_{8,2}) & \ \text{if} \ n \equiv 3 \pmod{8}\\
- 2\rho_n(\delta_{2,1}) - \rho_n(\delta_{4,1}) + \rho_n(\delta_{8,1}) - \rho_n(\delta_{8,2}) & \ \text{if} \ n \equiv 5 \pmod{8}\\
- 4\rho_n(\delta_{2,1}) + \rho_n(\delta_{4,1}) + \rho_n(\delta_{8,1}) + \rho_n(\delta_{8,2}) & \ \text{if} \ n \equiv 7 \pmod{8}\\
\end{array}
\end{cases}\\
F_2(n,0,1,1,1) &=& 2^{n-4} - \frac{1}{16} \cdot \begin{cases}
\begin{array}{lr}
-4\rho_n(\delta_{2,1}) + \rho_n(\delta_{4,1}) + \rho_n(\delta_{8,1}) + \rho_n(\delta_{8,2}) & \ \text{if} \ n \equiv 1,5 \pmod{8}\\
\rho_n(\delta_{4,1}) - \rho_n(\delta_{8,1}) - \rho_n(\delta_{8,2}) & \ \text{if} \ n \equiv 3,7 \pmod{8}\\
\end{array}
\end{cases}\\
F_2(n,1,1,1,1) &=& 2^{n-4} - \frac{1}{16} \cdot \begin{cases}
\begin{array}{lr}
- 2\rho_n(\delta_{2,1}) - \rho_n(\delta_{4,1}) + \rho_n(\delta_{8,1}) - \rho_n(\delta_{8,2}) & \ \text{if} \ n \equiv 1 \pmod{8}\\
\rho_n(\delta_{4,1}) - \rho_n(\delta_{8,1}) - \rho_n(\delta_{8,2}) & \ \text{if} \ n \equiv 3 \pmod{8}\\
2\rho_n(\delta_{2,1}) - \rho_n(\delta_{4,1}) - \rho_n(\delta_{8,1}) + \rho_n(\delta_{8,2}) & \ \text{if} \ n \equiv 5 \pmod{8}\\
4\rho_n(\delta_{2,1}) + \rho_n(\delta_{4,1}) + \rho_n(\delta_{8,1}) + \rho_n(\delta_{8,2}) & \ \text{if} \ n \equiv 7 \pmod{8}\\
\end{array}
\end{cases}\\
\end{eqnarray*}
}
\end{theorem}
One can check that the roots of $\delta_{8,1}$ are $\alpha_1,\alpha_2,\alpha_3,\alpha_4$ and their complex conjugates
$\overline{\alpha_1},\overline{\alpha_2},\overline{\alpha_3},\overline{\alpha_4}$, where:
\begin{eqnarray*}
\alpha_1 &=& -\frac{1}{2}+\frac{\sqrt{2}}{4}\Big(1+\sqrt{7+4\sqrt{2}}\Big)-\frac{\sqrt{2}}{4}\Big(1+\sqrt{5-2\sqrt{2}}\Big)i,\\
\alpha_2 &=& -\frac{1}{2}+\frac{\sqrt{2}}{4}\Big(1-\sqrt{7+4\sqrt{2}}\Big)-\frac{\sqrt{2}}{4}\Big(1-\sqrt{5-2\sqrt{2}}\Big)i,\\
\alpha_3 &=& -\frac{1}{2}-\frac{\sqrt{2}}{4}\Big(1 + \frac{1}{\sqrt{17}}(3\sqrt{2}-1)\sqrt{5-2\sqrt{2}}\Big) + \frac{\sqrt{2}}{4}\Big(1 - \frac{1}{\sqrt{17}}(3\sqrt{2}-1)\sqrt{7 + 4\sqrt{2}}\Big)i,\\
\alpha_4 &=& -\frac{1}{2}-\frac{\sqrt{2}}{4}\Big(1 - \frac{1}{\sqrt{17}}(3\sqrt{2}-1)\sqrt{5-2\sqrt{2}}\Big) + \frac{\sqrt{2}}{4}\Big(1 + \frac{1}{\sqrt{17}}(3\sqrt{2}-1)\sqrt{7 + 4\sqrt{2}}\Big)i.\\
\end{eqnarray*}
One can also check that the roots of $\delta_{8,2}$ are $i\alpha_1,i\alpha_2,i\alpha_3,i\alpha_4$ and $\overline{i\alpha_1},\overline{i\alpha_2},\overline{i\alpha_3},\overline{i\alpha_4}$.
In Theorem~\ref{thm:4traces} the formulae for each $F_2(n,t_1,t_2,t_3,t_4)$ and each odd $n$ mod $8$ have non-supersingular terms of the form $\pm \rho_n(\delta_{8,1}) \pm \rho_n(\delta_{8,2})$ or $\pm \rho_n(\delta_{8,1}) \mp \rho_n(\delta_{8,2})$.
A simple application of Kronecker's theorem to the phases of these non-supersingular Weil numbers allows one to deduce that the formulae are not periodic in $n$. We leave stating what abelian varieties $\delta_{8,1}$ and $\delta_{8,2}$ are the characteristic polynomials of Frobenius for as an open problem, but note that they must be isogenous over ${\mathbb F}_{2^4}$.
\subsubsection*{An alternative parameterisation.}\label{sec:4coeffsparam2}
We now present an alternative parameterisation of Eq.~(\ref{eq:4coeffsparam1}) which requires one less variable to linearise.
In particular, we have
\begin{eqnarray}
\nonumber T_4(a_{0}^2 + a_0 + r_0) &=& T_2(a_{0}^3) + T_2(a_{0}) + T_1(a_{0}^3)T_1(a_0)\\
\nonumber &+& T_1\Big(a_{0}^7 + a_{0}^5 + a_{0}^3 + r_0\Big(a_{0}^3 + a_0 + (a_{0}^3 + a_0)\binom{n}{2} + \binom{n}{4}\Big)\Big)\\
\label{eq:T4linearalt} &=& T_2(a_{0}^3 + a_{0}) + T_1\Big(a_{0}^7 + a_{0}^5 + a_{0}^3 + a_{0} + r_0\Big(a_{0}^3 + a_0 + (a_{0}^3 + a_0)\binom{n}{2}+\binom{n}{4}\Big)\Big),
\end{eqnarray}
where the second equality follows from Lemma~\ref{lem:T} part (2), by setting $\alpha = a_{0}^3$ and $\beta = a_{0}$.
This can be linearised using the substitution $a_{0}^3 + a_{0} = a_{1}^2 + a_{1} + r_1$, where $r_1$ is the trace of $a_{0}^3 + a_{0}$.
This results in
\begin{eqnarray*}
\nonumber T_4(a_{0}^2 + a_0 + r_0) &=& T_1\Big( a_{1}^3 + a_{1} + a_{0}^7 + a_{0}^5 + a_{0}^3 + a_{0}\\
&+& r_0\Big(a_{0}^3 + a_0 + (a_{0}^3 + a_0)\binom{n}{2} + \binom{n}{4}\Big) + r_1\binom{n}{2}\Big).
\end{eqnarray*}
For $8 \le i \le 15$ let $\vec{i} = (i_3,i_2,i_1,i_0)$. The curves we are interested in are given by the following intersections:
\begin{eqnarray}
\nonumber a_{2}^2 + a_{2} &=& i_3\Big( a_{1}^3 + a_{1} + a_{0}^7 + a_{0}^5 + a_{0}^3 + a_{0} + r_0\Big(a_{0}^3 + a_0 + (a_{0}^3 + a_0)\binom{n}{2} + \binom{n}{4}\Big) + r_1\binom{n}{2}\Big)\\
\label{4coeffsaltparamintersection} &+& i_2\Big(a_{0}^5 + a_{0} + r_0\Big(a_{0}^3 + a_0 + \binom{n}{3} \Big)\Big) + i_1\Big(a_{0}^3 + a_{0} + r_0\binom{n}{2}\Big) + i_0r_0,\\
\nonumber a_{0}^3 + a_{0} &=& a_{1}^2 + a_1 + r_1.
\end{eqnarray}
For $i_3 = 1$ the genus of all of these absolutely irreducible curves is $7$ -- rather than $14$ as in the first parameterisation -- and
therefore the characteristic polynomials of Frobenius have lower degrees than before. Indeed, there are fewer even polynomials appearing as
factors and those that are not even occur to lower powers. Nevertheless, once~(\ref{eq:transform1_4}) is applied one again obtains the formulae
given in Theorem~\ref{thm:4traces}.
\subsubsection{Direct method.}\label{sec:4coeffsdirect}
For odd $n$, applying Equations~(\ref{eq:T1linear}) to~(\ref{eq:T3linear}) and~(\ref{eq:T4linearalt}) we have
\begin{eqnarray}
\nonumber F_2(n,t_1,t_2,t_3,t_4) &=& \#\{a \in {\mathbb F}_{2^n} \mid T_1(a) = t_1, \ T_2(a) = t_2, \ T_3(a) = t_3, \ T_4(a) = t_4\}\\
\nonumber &=& \frac{1}{2} \, \#\{a_0 \in {\mathbb F}_{2^n} \mid T_2(a_{0}^2 + a_{0} + t_1) = t_2, \ T_3(a_{0}^2 + a_{0} + t_1) = t_3,
\ T_4(a_{0}^2 + a_{0} + t_1) = t_4\}\\
\nonumber &=& \frac{1}{2} \, \#\{a_0 \in {\mathbb F}_{2^n} \mid T_1\Big(a_{0}^3 + a_{0} + t_1\binom{n}{2}\Big) = t_2, \ T_1\Big(a_{0}^5 + a_{0} + t_1\Big(a_{0}^3 +
a_0+\binom{n}{3}\Big)\Big) = t_3,\\
\nonumber && \ \ \ \ T_2(a_{0}^3 + a_{0}) + T_1\Big(a_{0}^7 + a_{0}^5 + a_{0}^3 + a_{0} + t_1\Big(a_{0}^3 + a_0 + (a_{0}^3 + a_0)\binom{n}{2}+\binom{n}{4}\Big)\Big) = t_4\}\\
\nonumber &=& \frac{1}{16} \, \#\{(a_0,a_1,a_2,a_3) \in ({\mathbb F}_{2^n})^4 \mid a_{1}^2 + a_1 = a_{0}^3 + a_{0} + t_1\binom{n}{2} + t_2,\\
\nonumber && \ \ \ \ a_{2}^2 + a_2 = a_{0}^5 + a_{0} + t_1\Big(a_{0}^3 + a_0+\binom{n}{3}\Big) + t_3, \ a_{3}^2 + a_3 = a_{1}^3 + a_1 + (t_1 + t_2)\binom{n}{2} + \\
\nonumber && \ \ \ \ a_{0}^7 + a_{0}^5 + a_{0}^3 + a_{0} + t_1\Big(a_{0}^3 + a_0
+ (a_{0}^3 + a_{0})\binom{n}{2} +\binom{n}{4}\Big) + t_4\},
\end{eqnarray}
where we have used Lemma~\ref{lem:T} part (2) and the parameterisation of $T_2(a) = t_2$ to compute $T_2(a_{0}^3 + a_{0}) = T_2(a_{1}^2 + a_{1} + t_1\binom{n}{2} + t_2) = T_2(a_{1}^2 + a_{1}) + T_2\Big(t_1\binom{n}{2} + t_2\Big) + T_1(a_{1}^2 + a_{1})T_1\Big(t_1\binom{n}{2} + t_2\Big) + T_1\Big(\Big(t_1\binom{n}{2} + t_2\Big)(a_{1}^2 + a_1)\Big) = T_1(a_{1}^3 + a_{1}) + (t_1 + t_2)\binom{n}{2}$, since the last two terms are zero.
These curves are all absolutely irreducible and of genus $17$, and for $n$ odd their zeta functions reproduce Theorem~\ref{thm:4traces}.
\subsubsection{General formulae.}\label{subsubsec:generalformula}
We have the following result.
\begin{theorem}\label{thm:4coeffsgeneral}
For $n \ge 4$ we have
\begin{eqnarray*}
F_2(n,0,0,0,0) &=& 2^{n-4} - \frac{1}{16}\big(4\rho_n(\delta_{2,1}) + 3\rho_n(\delta_{2,2}) + \rho_n(\delta_{4,1}) + \rho_n(\delta_{8,1}) + \rho_n(\delta_{8,2})\big),\\
F_2(n,0,0,0,1) &=& 2^{n-4} - \frac{1}{16}\big(-\rho_n(\delta_{2,2}) + \rho_n(\delta_{4,1}) - \rho_n(\delta_{8,1}) - \rho_n(\delta_{8,2})\big),\\
F_2(n,0,0,1,0) &=& 2^{n-4} - \frac{1}{16}\big(-2\rho_n(\delta_{2,1}) + \rho_n(\delta_{2,2}) - \rho_n(\delta_{4,1}) + \rho_n(\delta_{8,1}) - \rho_n(\delta_{8,2})\big),\\
F_2(n,0,0,1,1) &=& 2^{n-4} - \frac{1}{16}\big(2\rho_n(\delta_{2,1}) - 3\rho_n(\delta_{2,2}) - \rho_n(\delta_{4,1}) - \rho_n(\delta_{8,1}) + \rho_n(\delta_{8,2})\big).
\end{eqnarray*}
\end{theorem}
\begin{proof}
Using the direct method, for all $n \ge 4$ we have
\begin{eqnarray}
\nonumber F_2(n,0,0,0,0) &=& \frac{1}{16} \, \#\{ (a_{0},a_{1},a_{2},a_{3}) \in ({\mathbb F}_{2^n})^4 \mid a_{1}^2 + a_{1} = a_{0}^3 + a_{0},\ a_{2}^2 + a_{2} =a_{0}^5 + a_{0},\\
\nonumber && \ \ \ \ a_{3}^2 + a_{3} = a_{1}^3 + a_{1} + a_{0}^7 + a_{0}^5 + a_{0}^3 + a_0\}\\
\nonumber &=& 2^{n-4} - \frac{1}{16}\big(4\rho_n(\delta_{2,1}) + 3\rho_n(\delta_{2,2}) + \rho_n(\delta_{4,1}) + \rho_n(\delta_{8,1}) + \rho_n(\delta_{8,2})\big),
\end{eqnarray}
since one does not need to parameterise any `linear trace $= 1$' conditions. Since $F_2(n,0,0,0) = F_2(n,0,0,0,0) + F_2(n,0,0,0,1)$, for all
$n \ge 4$ we also have:
\begin{eqnarray}
\nonumber F_2(n,0,0,0,1) &=& 2^{n-4} - \frac{1}{16}\big(-\rho_n(\delta_{2,2}) + \rho_n(\delta_{4,1}) - \rho_n(\delta_{8,1}) - \rho_n(\delta_{8,2})\big).
\end{eqnarray}
Omitting the $T_3(a)$ condition, we obtain:
\begin{eqnarray}
\nonumber F_2(n,0,0,*,0) &=&
\frac{1}{8} \, \#\{ (a_{0},a_{1},a_{2}) \in ({\mathbb F}_{2^n})^3 \mid a_{1}^2 + a_{1} = a_{0}^3 + a_{0}, \ a_{2}^2 + a_{2} = a_{1}^3 + a_{1} + a_{0}^7 + a_{0}^5 + a_{0}^3 + a_0\}\\
&=& 2^{n-3} - \frac{1}{8}\big(\rho_n(\delta_{2,1}) + 2\rho_n(\delta_{2,2}) + \rho_n(\delta_{8,1})\big),\label{eq:kloosterman}
\end{eqnarray}
this intersection describing an absolutely irreducible curve of genus $7$. Since $F_2(n,0,0,*,0) = F_2(n,0,0,0,0) + F_2(n,0,0,1,0)$ we also have:
\begin{eqnarray}
\nonumber F_2(n,0,0,1,0) &=& 2^{n-4} - \frac{1}{16}\big(-2\rho_n(\delta_{2,1}) + \rho_n(\delta_{2,2}) - \rho_n(\delta_{4,1}) + \rho_n(\delta_{8,1}) - \rho_n(\delta_{8,2})\big).
\end{eqnarray}
Moreover, since $F_2(n,0,0) = F_2(n,0,0,0,0) + F_2(n,0,0,0,1) + F_2(n,0,0,1,0) + F_2(n,0,0,1,1)$ we have
\begin{eqnarray}
\nonumber F_2(n,0,0,1,1) &=& 2^{n-4} - \frac{1}{16}\big(2\rho_n(\delta_{2,1}) - 3\rho_n(\delta_{2,2}) - \rho_n(\delta_{4,1}) - \rho_n(\delta_{8,1}) + \rho_n(\delta_{8,2})\big).
\end{eqnarray}\qed
\end{proof}
Note that only the $F_2(n,0,0,0,0)$ formula comes from the characteristic polynomial of Frobenius of a curve, since the other three have terms with
the wrong sign. Note also that the total degree of the corresponding polynomials (numerator degree plus denominator degree) is $2 \cdot 17 = 34$ only
for $F_2(n,0,0,0,0)$. This does not contradict the fact that the direct method always produces curves of genus $17$, since the direct method in general
only represents $F_2(n,t_1,t_2,t_3,t_4)$ for $n$ odd, so there are cancellations, and furthermore all the featured `+' signs in the formulae for
$F_2(n,0,0,t_3,t_4)$ may be replaced by `-' signs once the featured polynomials $\delta(X)$ are replaced with $\delta(-X)$.
If one similarly tries to omit the $T_2(a)$ condition then one can not automatically simplify the $T_2(a_{0}^3 + a_0)$ term which arises from the condition
$T_4(a_{0}^2 + a_{0}) = 0$; one is forced to condition on whether the linear trace of $a_{0}^3 + a_0$ is $0$ or $1$, in which case one needs $n$ to be
odd in order to parameterise the latter condition. Therefore, it is apparently not possible to find formulae for all $n \ge 4$ for
$F_2(n,0,1,t_3,t_4)$ with this approach.
Nevertheless, we expect that similar formulae hold for all $n \ge 4$ for each $F_2(n,t_1,t_2,t_3,t_4)$, with additional terms arising from the $n$-th
powers of roots of a set of even polynomials. Furthermore, if for a given $F_2(n,t_1,t_2,t_3,t_4)$ the coefficients of the various $\rho_n(\delta_i)$
also depend on the residue of $n$ mod $8$, as they do for $n$ odd, then one can Fourier analyse the coefficients in order to express them in terms
of the complex $8$-th roots of unity to obtain a single formula, as in~\cite[Prop.~3\&5]{AGGMY}.
So while the formulae themselves are not periodic in $n$, it may be that the coefficients of each $\rho_n(\delta_i)$ featured in each
$F_2(n,t_1,t_2,t_3,t_4)$ are periodic in $n$.
\subsubsection{An alternative proof of general formulae.}\label{subsubsec:altgeneralformulae}
We now provide another proof of Theorem~\ref{thm:4coeffsgeneral} which relies on the alternative parameterisation
of $T_4(a_{0}^2 + a_{0})$ and a generalisation of the transform~(\ref{eq:Sinverse}).
Observe that the functions $f_0,\ldots,f_{m-1}: {\mathbb F}_{2^n} \rightarrow {\mathbb F}_2$ in~\S\ref{sec:transform1char2} may be replaced by functions
$f_0,\ldots,f_{m-1}: A \rightarrow {\mathbb F}_2$ where $A$ is any relevant domain -- and thus for instance any algebraic set -- and exactly the same
argument holds, provided that $V(\vec{0} \cdot \vec{f})$ is redefined to be $|A|$.
The conditions $T_1(a) = T_2(a) = 0$ imply that we should set $A = \{(a_0,a_1) \in ({\mathbb F}_{2^n})^2 \mid a_{0}^3 + a_{0} = a_{1}^2 + a_{1}\}$.
Furthermore let $\vec{f} = (T_4(a_{0}^2 + a_{0}), T_3(a_{0}^2 + a_{0})) = (T_1(a_{1}^3 + a_1 + a_{0}^7 + a_{0}^5 + a_{0}^3 + a_{0}), T_1(a_{0}^5 + a_0))$.
Then by the stated generalisation we have
\begin{equation}\label{eq:alternativeproof}
{\small
\begin{bmatrix}
F_2(n,0,0,0,0)\\
F_2(n,0,0,1,0)\\
F_2(n,0,0,0,1)\\
F_2(n,0,0,1,1)\\
\end{bmatrix}
=
\begin{bmatrix}
N(\vec{0})\\
N(\vec{1})\\
N(\vec{2})\\
N(\vec{3})\\
\end{bmatrix}
=
\frac{1}{8}
\begin{bmatrix*}[r]
-1 & 1 & 1 & 1\\
1 & -1 & 1 & -1\\
1 & 1 & -1 & -1\\
1 & -1 & -1 & 1\\
\end{bmatrix*}
\begin{bmatrix}
V(\vec{0} \cdot \vec{f})\\
V(\vec{1} \cdot \vec{f})\\
V(\vec{2} \cdot \vec{f})\\
V(\vec{3} \cdot \vec{f})\\
\end{bmatrix},
}
\end{equation}
where we have a factor of $1/8$ rather than $1/2$ because of the two factors of $1/2$ arising from the introduction of the variables $a_0$ and $a_1$ defining $A$. Note that $V(\vec{0} \cdot \vec{f}) = |A| = 2^n - \rho_n(\delta_{2,1})$. For $1 \le i \le 3$ let $\vec{i} = (i_1,i_0)$. We thus have:
\begin{equation*}
V(\vec{i} \cdot \vec{f}) = \frac{1}{2}\#\{(a_0,a_1,a_2) \mid a_{0}^3 + a_{0} = a_{1}^2 + a_{1}, \ a_{2}^2 + a_{2} = i_1(a_{1}^3 + a_1 +
a_{0}^7 + a_{0}^5 + a_{0}^3 + a_{0}) + i_0(a_{0}^5 + a_{0}) \}.
\end{equation*}
For $i = 1,2$ and $3$ these are absolutely irreducible curves of genus $5,7$ and $7$ respectively, and thus their zeta functions
are easier to compute than for those curves arising from the direct method, which have genus $17$. Combining them as per Eq.~(\ref{eq:alternativeproof})
gives the formulae in~\S\ref{subsubsec:generalformula}; it is therefore a slightly more streamlined argument than the one given there.
Another way of viewing this approach is to take the intersections of~(\ref{4coeffsaltparamintersection}) and set $r_0 = r_1 = i_0 = i_1 = 0$ and observe
that the $i_2 = i_3 = 0$ condition gives $V(\vec{0} \cdot \vec{f}) = |A|$, without the factor $1/2$ present for the other three $V(\vec{i} \cdot \vec{f})$
because there is no $a_2$ variable in this case. The approach is therefore more useful in these cases than the direct method, which can not
produce formulae for all $F_2(n,0,0,t_3,t_4)$ for all $n \ge 4$, as it only works for $n$ odd in general.
\subsubsection{Connection with binary Kloosterman sums.}
The binary Kloosterman sum $\mathcal{K}_{2^n}: {\mathbb F}_{2^n} \rightarrow {\mathbb Z}$ can be defined by
\[
\mathcal{K}_{2^n}(a) = 1 + \sum_{x \in {\mathbb F}_{2^n}^{\times}} (-1)^{T_1(x^{-1} + ax)}.
\]
Kloosterman sums have applications in cryptography and coding theory, see for example~\cite{gong,moisiocode}.
In particular, zeros of $\mathcal{K}_{2^{n}}$ lead to bent functions from ${\mathbb F}_{2^{2n}} \rightarrow {\mathbb F}_{2}$~\cite{dillon}.
The following elementary lemma connects Kloosterman sums to a family of elliptic curves.
\begin{lemma}[\cite{wolfmann}]\label{lis1}
Let $a \in {\mathbb F}_{2^n}^{\times}$ and define the elliptic curve $E_{2^n}(a)$ over
${\mathbb F}_{2^n}$ by
\[
E_{2^n}(a): y^2 + xy = x^3 + a.
\]
Then $\#E_{2^n}(a) = 2^n + \mathcal{K}_{2^n}(a)$.
\end{lemma}
Computing Kloosterman sum zeros is generally regarded as being difficult, currently taking exponential time (in $n$) to find a single non-trivial
($a \ne 0$) zero.
Besides the deterministic test due to Ahmadi and Granger~\cite{KloostermanAG}, which computes the cardinality of the Sylow $2$-subgroup of any
$E_{2^n}(a)$ via point-halving, and thus by Lemma~\ref{lis1} the maximum power of $2$ dividing $\mathcal{K}_{2^n}(a)$,
research has focused on characterising Kloosterman sums modulo small integers~\cite{moisio2,lisonek,lisonek2,helleseth2,lisonek3,faruk1,faruk2,faruk3}.
In order to analyse the expected running time of the algorithm of Ahmadi and Granger, it is necessary to know the distribution of Kloosterman sums
which are divisible by successive powers of $2$. Table~\ref{dist2} presents this distribution for $n \le 13$, which was also presented
in~\cite{KloostermanAG}.
Let $T(n,k)$ denote the $(n,k)$-th entry of Table~\ref{dist2}, i.e., the number of $a \in {\mathbb F}_{2^n}^{\times}$ for which $\#E_{2^n}(a)$ is divisible
by $2^{k}$.
By using a result of Katz and Livn\'e~\cite{katz} it is possible to express $T(n,k)$ in terms of the class numbers of certain imaginary
quadratic fields. However, it remains an open problem to give exact formulae for $k > 4$, with the formulae for the first four columns being as follows.
Since the orders of all of the elliptic curves in Lemma~\ref{lis1} are divisible by $4$, one has $T(n,1) = T(n,2) = 2^n - 1$.
One can show that $E_{2^n}(a)$ has a point of order $8$ if and only if $T_1(a) = 0$ (see e.g.~\cite{geer}), hence $T_{2^n}(3) = 2^{n-1}-1$.
Finally, Lison\v{e}k and Moisio proved that $T(n,4) = (2^n - (-1 + i)^n - (-1 - i)^n)/4 - 1$, connecting it with the number of points on a supersingular
elliptic curve~\cite[Theorem 3.6]{lisonek3}.
\begin{table}\caption{$T(n,k) = \#\{a \in {\mathbb F}_{2^n}^{\times} \mid \#E_{2^n}(a) \equiv 0 \pmod{2^k} \}$}\label{dist2}
\begin{center}
\begin{tabularx}{0.75\textwidth}{c *{13}{|Y}}
$n \backslash k$ & $1$ & $2$ & $3$ & $4$ & $5$ & $6$ & $7$ & $8$ &
$9$ & $10$ & $11$ & $12$ & $13$\\
\hline
1 & 1 & 1 & & & & & & & & & & & \\
2 & 3 & 3 & & & & & & & & & & & \\
3 & 7 & 7 & 3 & & & & & & & & & & \\
4 & 15 & 15 & 7 & 5 & & & & & & & & & \\
5 & 31 & 31 & 15 & 5 & 5 & & & & & & & & \\
6 & 63 & 63 & 31 & 15 & 12 & 12 & & & & & & & \\
7 & 127 & 127 & 63 & 35 & 14 & 14 & 14 & & & & & & \\
8 & 255 & 255 & 127 & 55 & 21 & 16 & 16 & 16 & & & & & \\
9 & 511 & 511 & 255 & 135 & 63 & 18 & 18 & 18 & 18 & & & &\\
10 & 1023 & 1023 & 511 & 255 & 125 & 65 & 60 & 60 & 60 & 60 & & &\\
11 & 2047 & 2047 & 1023 & 495 & 253 & 132 & 55 & 55 & 55 & 55 & 55 & & \\
12 & 4095 & 4095 & 2047 & 1055 & 495 & 252 & 84 & 72 & 72 & 72 & 72 & 72 &\\
13 & 8191 & 8191 & 4095 & 2015 & 1027 & 481 & 247 & 52 & 52 & 52
& 52 & 52 & 52\\
\end{tabularx}
\end{center}
\end{table}
The following theorem connects the distribution of binary Kloosterman sums mod $32$ to the distribution of the first four coefficients of
the characteristic polynomial.
\begin{theorem}[\cite{faruk3}]\label{thm:Kloosterman256}
Let $a \in {\mathbb F}_{2^n}$ with $n \ge 4$ and let $e_1,\ldots,e_4$ be the coefficients of the characteristic polynomial of $a$, regarded as integers.
Then
\[
\mathcal{K}_{2^n}(a) \equiv 28e_1 +8e_2 + 16(e_1 e_2 + e_1 e_3 + e_4) \pmod{32}.
\]
\end{theorem}
Combining Theorems~\ref{thm:4traces} and~\ref{thm:Kloosterman256} therefore provides explicit formulae for the distribution of binary Kloosterman sums
mod $32$, for $n$ odd. Furthermore, combining Theorem~\ref{thm:Kloosterman256} with Eq.~(\ref{eq:kloosterman}) provides an explicit formula for
$\#\{a \in {\mathbb F}_{2^n} \mid \mathcal{K}_{2^n}(a) \equiv 0 \pmod{32}\} = T(n,5) + 1$; indeed, this connection was our original motivation for considering
the first-four prescribed traces problem.
\begin{corollary}
For $n \ge 5$ we have
\begin{eqnarray}
\nonumber \#\{a \in {\mathbb F}_{2^n} \mid \mathcal{K}_{2^n}(a) \equiv 0 \pmod{32}\} &=& F_2(n,0,0,0,0) + F_2(n,0,0,1,0)\\
\nonumber &=& 2^{n-3} - \frac{1}{8}\big(\rho_n(\delta_{2,1}) + 2\rho_n(\delta_{2,2}) + \rho_n(\delta_{8,1})\big).
\end{eqnarray}
\end{corollary}
We also have:
\begin{theorem}[\cite{faruk3}]\label{thm:Kloosterman64}
Let $a \in {\mathbb F}_{2^n}$ with $n \ge 6$ and let $e_1,\ldots,e_8$ be the coefficients of the characteristic polynomial of $a$, regarded as integers.
Then
\begin{eqnarray*}
\mathcal{K}_{2^n}(a) &\equiv& 28e_1 + 40e_2 + 16(e_1 e_2 + e_1 e_3 + e_4)\\
&+& 32(e_1e_4 + e_1e_5 + e_1e_6 + e_1e_7 + e_2e_3 + e_2e_4 + e_2e_6 +
e_3e_5 + e_1e_2e_3 + e_1e_2e_4 + e_8) \pmod{64}.
\end{eqnarray*}
\end{theorem}
Therefore, if one could solve the first-eight prescribed traces problem, then one could also determine a formula for the entries of the sixth column of
Table~\ref{dist2}.
\subsection{Computing $F_2(n,t_1,t_2,t_3,t_4,t_5)$}
We begin by applying the indirect method.
\subsubsection{Indirect method.} Setting $\vec{f} = (T_5,T_4,T_3,T_2,T_1)$, by the transform~(\ref{eq:Sinverse}) we have
\[
{\tiny
\begin{bmatrix}
F_2(n,0,0,0,0,0)\\
F_2(n,1,0,0,0,0)\\
F_2(n,0,1,0,0,0)\\
F_2(n,1,1,0,0,0)\\
F_2(n,0,0,1,0,0)\\
F_2(n,1,0,1,0,0)\\
F_2(n,0,1,1,0,0)\\
F_2(n,1,1,1,0,0)\\
F_2(n,0,0,0,1,0)\\
F_2(n,1,0,0,1,0)\\
F_2(n,0,1,0,1,0)\\
F_2(n,1,1,0,1,0)\\
F_2(n,0,0,1,1,0)\\
F_2(n,1,0,1,1,0)\\
F_2(n,0,1,1,1,0)\\
F_2(n,1,1,1,1,0)\\
F_2(n,0,0,0,0,1)\\
F_2(n,1,0,0,0,1)\\
F_2(n,0,1,0,0,1)\\
F_2(n,1,1,0,0,1)\\
F_2(n,0,0,1,0,1)\\
F_2(n,1,0,1,0,1)\\
F_2(n,0,1,1,0,1)\\
F_2(n,1,1,1,0,1)\\
F_2(n,0,0,0,1,1)\\
F_2(n,1,0,0,1,1)\\
F_2(n,0,1,0,1,1)\\
F_2(n,1,1,0,1,1)\\
F_2(n,0,0,1,1,1)\\
F_2(n,1,0,1,1,1)\\
F_2(n,0,1,1,1,1)\\
F_2(n,1,1,1,1,1)\\
\end{bmatrix}
=
\begin{bmatrix}
N(\vec{0})\\
N(\vec{1})\\
N(\vec{2})\\
N(\vec{3})\\
N(\vec{4})\\
N(\vec{5})\\
N(\vec{6})\\
N(\vec{7})\\
N(\vec{8})\\
N(\vec{9})\\
N(\vec{10})\\
N(\vec{11})\\
N(\vec{12})\\
N(\vec{13})\\
N(\vec{14})\\
N(\vec{15})\\
N(\vec{16})\\
N(\vec{17})\\
N(\vec{18})\\
N(\vec{19})\\
N(\vec{20})\\
N(\vec{21})\\
N(\vec{22})\\
N(\vec{23})\\
N(\vec{24})\\
N(\vec{25})\\
N(\vec{26})\\
N(\vec{27})\\
N(\vec{28})\\
N(\vec{29})\\
N(\vec{30})\\
N(\vec{31})\\
\end{bmatrix}
=
\frac{1}{16}
\begin{bmatrix*}[c]
-15& 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\
1 & - & 1 & - & 1 & - & 1 & - & 1 & - & 1 & - & 1 & - & 1 & - & 1 & - & 1 & - & 1 & - & 1 & - & 1 & - & 1 & - & 1 & - & 1 & -\\
1 & 1 & - & - & 1 & 1 & - & - & 1 & 1 & - & - & 1 & 1 & - & - & 1 & 1 & - & - & 1 & 1 & - & - & 1 & 1 & - & - & 1 & 1 & - & -\\
1 & - & - & 1 & 1 & - & - & 1 & 1 & - & - & 1 & 1 & - & - & 1 & 1 & - & - & 1 & 1 & - & - & 1 & 1 & - & - & 1 & 1 & - & - & 1\\
1 & 1 & 1 & 1 & - & - & - & - & 1 & 1 & 1 & 1 & - & - & - & - & 1 & 1 & 1 & 1 & - & - & - & - & 1 & 1 & 1 & 1 & - & - & - & -\\
1 & - & 1 & - & - & 1 & - & 1 & 1 & - & 1 & - & - & 1 & - & 1 & 1 & - & 1 & - & - & 1 & - & 1 & 1 & - & 1 & - & - & 1 & - & 1\\
1 & 1 & - & - & - & - & 1 & 1 & 1 & 1 & - & - & - & - & 1 & 1 & 1 & 1 & - & - & - & - & 1 & 1 & 1 & 1 & - & - & - & - & 1 & 1\\
1 & - & - & 1 & - & 1 & 1 & - & 1 & - & - & 1 & - & 1 & 1 & - & 1 & - & - & 1 & - & 1 & 1 & - & 1 & - & - & 1 & - & 1 & 1 & -\\
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & - & - & - & - & - & - & - & - & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & - & - & - & - & - & - & - & -\\
1 & - & 1 & - & 1 & - & 1 & - & - & 1 & - & 1 & - & 1 & - & 1 & 1 & - & 1 & - & 1 & - & 1 & - & - & 1 & - & 1 & - & 1 & - & 1\\
1 & 1 & - & - & 1 & 1 & - & - & - & - & 1 & 1 & - & - & 1 & 1 & 1 & 1 & - & - & 1 & 1 & - & - & - & - & 1 & 1 & - & - & 1 & 1\\
1 & - & - & 1 & 1 & - & - & 1 & - & 1 & 1 & - & - & 1 & 1 & - & 1 & - & - & 1 & 1 & - & - & 1 & - & 1 & 1 & - & - & 1 & 1 & -\\
1 & 1 & 1 & 1 & - & - & - & - & - & - & - & - & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & - & - & - & - & - & - & - & - & 1 & 1 & 1 & 1\\
1 & - & 1 & - & - & 1 & - & 1 & - & 1 & - & 1 & 1 & - & 1 & - & 1 & - & 1 & - & - & 1 & - & 1 & - & 1 & - & 1 & 1 & - & 1 & -\\
1 & 1 & - & - & - & - & 1 & 1 & - & - & 1 & 1 & 1 & 1 & - & - & 1 & 1 & - & - & - & - & 1 & 1 & - & - & 1 & 1 & 1 & 1 & - & -\\
1 & - & - & 1 & - & 1 & 1 & - & - & 1 & 1 & - & 1 & - & - & 1 & 1 & - & - & 1 & - & 1 & 1 & - & - & 1 & 1 & - & 1 & - & - & 1\\
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & -\\
1 & - & 1 & - & 1 & - & 1 & - & 1 & - & 1 & - & 1 & - & 1 & - & - & 1 & - & 1 & - & 1 & - & 1 & - & 1 & - & 1 & - & 1 & - & 1\\
1 & 1 & - & - & 1 & 1 & - & - & 1 & 1 & - & - & 1 & 1 & - & - & - & - & 1 & 1 & - & - & 1 & 1 & - & - & 1 & 1 & - & - & 1 & 1\\
1 & - & - & 1 & 1 & - & - & 1 & 1 & - & - & 1 & 1 & - & - & 1 & - & 1 & 1 & - & - & 1 & 1 & - & - & 1 & 1 & - & - & 1 & 1 & -\\
1 & 1 & 1 & 1 & - & - & - & - & 1 & 1 & 1 & 1 & - & - & - & - & - & - & - & - & 1 & 1 & 1 & 1 & - & - & - & - & 1 & 1 & 1 & 1\\
1 & - & 1 & - & - & 1 & - & 1 & 1 & - & 1 & - & - & 1 & - & 1 & - & 1 & - & 1 & 1 & - & 1 & - & - & 1 & - & 1 & 1 & - & 1 & -\\
1 & 1 & - & - & - & - & 1 & 1 & 1 & 1 & - & - & - & - & 1 & 1 & - & - & 1 & 1 & 1 & 1 & - & - & - & - & 1 & 1 & 1 & 1 & - & -\\
1 & - & - & 1 & - & 1 & 1 & - & 1 & - & - & 1 & - & 1 & 1 & - & - & 1 & 1 & - & 1 & - & - & 1 & - & 1 & 1 & - & 1 & - & - & 1\\
1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\
1 & - & 1 & - & 1 & - & 1 & - & - & 1 & - & 1 & - & 1 & - & 1 & - & 1 & - & 1 & - & 1 & - & 1 & 1 & - & 1 & - & 1 & - & 1 & -\\
1 & 1 & - & - & 1 & 1 & - & - & - & - & 1 & 1 & - & - & 1 & 1 & - & - & 1 & 1 & - & - & 1 & 1 & 1 & 1 & - & - & 1 & 1 & - & -\\
1 & - & - & 1 & 1 & - & - & 1 & - & 1 & 1 & - & - & 1 & 1 & - & - & 1 & 1 & - & - & 1 & 1 & - & 1 & - & - & 1 & 1 & - & - & 1\\
1 & 1 & 1 & 1 & - & - & - & - & - & - & - & - & 1 & 1 & 1 & 1 & - & - & - & - & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & - & - & - & -\\
1 & - & 1 & - & - & 1 & - & 1 & - & 1 & - & 1 & 1 & - & 1 & - & - & 1 & - & 1 & 1 & - & 1 & - & 1 & - & 1 & - & - & 1 & - & 1\\
1 & 1 & - & - & - & - & 1 & 1 & - & - & 1 & 1 & 1 & 1 & - & - & - & - & 1 & 1 & 1 & 1 & - & - & 1 & 1 & - & - & - & - & 1 & 1\\
1 & - & - & 1 & - & 1 & 1 & - & - & 1 & 1 & - & 1 & - & - & 1 & - & 1 & 1 & - & 1 & - & - & 1 & 1 & - & - & 1 & - & 1 & 1 & -\\
\end{bmatrix*}
\begin{bmatrix}
V(\vec{0} \cdot \vec{f})\\
V(\vec{1} \cdot \vec{f})\\
V(\vec{2} \cdot \vec{f})\\
V(\vec{3} \cdot \vec{f})\\
V(\vec{4} \cdot \vec{f})\\
V(\vec{5} \cdot \vec{f})\\
V(\vec{6} \cdot \vec{f})\\
V(\vec{7} \cdot \vec{f})\\
V(\vec{8} \cdot \vec{f})\\
V(\vec{9} \cdot \vec{f})\\
V(\vec{10} \cdot \vec{f})\\
V(\vec{11} \cdot \vec{f})\\
V(\vec{12} \cdot \vec{f})\\
V(\vec{13} \cdot \vec{f})\\
V(\vec{14} \cdot \vec{f})\\
V(\vec{15} \cdot \vec{f})\\
V(\vec{16} \cdot \vec{f})\\
V(\vec{17} \cdot \vec{f})\\
V(\vec{18} \cdot \vec{f})\\
V(\vec{19} \cdot \vec{f})\\
V(\vec{20} \cdot \vec{f})\\
V(\vec{21} \cdot \vec{f})\\
V(\vec{22} \cdot \vec{f})\\
V(\vec{23} \cdot \vec{f})\\
V(\vec{24} \cdot \vec{f})\\
V(\vec{25} \cdot \vec{f})\\
V(\vec{26} \cdot \vec{f})\\
V(\vec{27} \cdot \vec{f})\\
V(\vec{28} \cdot \vec{f})\\
V(\vec{29} \cdot \vec{f})\\
V(\vec{30} \cdot \vec{f})\\
V(\vec{31} \cdot \vec{f})\\
\end{bmatrix},}
\]
where for reasons of space and in common with Hadamard matrix notation, we represent each $-1$ simply as a `$-$'.
To determine $V( \vec{i} \cdot \vec{f})$ for $16 \le i \le 31$, we use Lemma~\ref{lem:T} part (5).
In particular, setting $\alpha = a_{0}^2$ and $\beta = a_{0}$ for $r_0 = 0$, and $\alpha = a_{0}^2 + a_0$ and $\beta = 1$ for $r_0 = 1$,
and evaluating mod $2$ gives the following:
\begin{eqnarray}
\nonumber T_5(a_{0}^2 + a_0 + r_0) &=& T_1(a_{0}^5)T_1(a_{0}^3) + T_1(a_{0}^5)T_1(a_{0}) + T_1(a_{0}^3)T_1(a_{0}) + r_0(T_2(a_{0}^3) + T_2(a_{0}) + T_1(a_{0}^3)T_1(a_{0}))\\
&+& T_1\Big(a_{0}^9 + a_{0}^3 + a_{0} + r_0 \Big(a_{0}^7 + (a_{0}^5 + a_0)\binom{n}{2} + (a_{0}^3 + a_0)\binom{n}{3} + \binom{n}{5}\Big)\Big).\label{eq:5coeffsparam1}
\end{eqnarray}
This can be linearised using the same the substitutions that were used for the four coefficient case, namely,
$a_0 = a_{1}^2 + a_1 + r_1$ and $a_{0}^3 = a_{2}^2 + a_2 + r_2$, where $r_1,r_2 \in {\mathbb F}_2$ are the traces of $a_0$ and $a_{0}^3$ respectively.
This results in
\begin{eqnarray*}
T_5(a_{0}^2 + a_0 + r_0) &=& T_1\Big(r_0(a_{2}^3 + a_{2} + a_{1}^3 + a_{1}) + a_{0}^9 + r_0a_{0}^7+ (r_1 + r_2)a_{0}^5
+ r_1 + r_2 + r_1r_2 + r_0r_1r_2\\
&+& (r_0a_{0}^5 + r_0r_2)\binom{n}{2} + (r_0r_1 + r_0r_2)\binom{n}{3} + r_0\binom{n}{5}\Big).
\end{eqnarray*}
For $16 \le i \le 31$ let $\vec{i} = (i_4,i_3,i_2,i_1,i_0)$. The curves we are interested in are given by the following intersections:
\begin{eqnarray}
\nonumber a_{3}^2 + a_{3} &=&
\nonumber i_4\Big(r_0(a_{2}^3 + a_{2} + a_{1}^3 + a_{1}) + a_{0}^9 + r_0a_{0}^7 + (r_1 + r_2)a_{0}^5 + r_1 + r_2 + r_1r_2 + r_0r_1r_2\\
\nonumber &+& (r_0a_{0}^5 + r_0r_2)\binom{n}{2} + (r_0r_1 + r_0r_2)\binom{n}{3} + r_0\binom{n}{5}\Big)\\
\nonumber &+& i_3\Big(a_{2}^3 + a_2 + a_{1}^3 + a_1 + a_{0}^7 + a_{0}^5 + (r_0 + 1)(r_1 + r_2)\binom{n}{2} + r_0r_1 + r_0r_2 + r_1r_2 + r_2 + r_0\binom{n}{4}\Big)\\
\nonumber &+& i_2\Big(a_{0}^5 + a_{0} + r_0\Big(a_{0}^3 + a_0 + \binom{n}{3} \Big)\Big) + i_1\Big(a_{0}^3 + a_{0} + r_0\binom{n}{2}\Big) + i_0r_0,\\
\label{eq:5trace1} a_0 &=& a_{1}^2 + a_1 + r_1,\\
\nonumber a_{0}^3 &=& a_{2}^2 + a_2 + r_2.
\end{eqnarray}
For each $16 \le i \le 31$ the genus of all of these absolutely irreducible curves is $18$.
Corollary~\ref{cor:period} implies that mod $2$ one has
\begin{eqnarray*}
\Big(\binom{n}{5},\binom{n}{4},\binom{n}{3},\binom{n}{2},\binom{n}{1}\Big) \equiv \begin{cases}
\begin{array}{lr}
(0,0,0,0,1) & \ \text{if} \ n \equiv 1 \pmod{8}\\
(0,0,1,1,1) & \ \text{if} \ n \equiv 3 \pmod{8}\\
(1,1,0,0,1) & \ \text{if} \ n \equiv 5 \pmod{8}\\
(1,1,1,1,1) & \ \text{if} \ n \equiv 7 \pmod{8}\\
\end{array},
\end{cases}
\end{eqnarray*}
and hence there are four cases to consider when computing the zeta functions of each of the curves~(\ref{eq:5trace1}).
In order to express $F_2(n,t_1,t_2,t_3,t_4,t_5)$ compactly, we define the following polynomial:
\begin{eqnarray*}
\delta_{8,3} &=& X^8 + 2X^7 + 2X^6 - 4X^4 + 8X^2 + 16X + 16,
\end{eqnarray*}
which is the characteristic polynomial of Frobenius of the supersingular curve
\[
C_{8,3}/{\mathbb F}_2: y^2 + y = x^9 + x^5.
\]
As with the four coefficient case there are several other even polynomials which occur as factors of the characteristic polynomial of Frobenius
of the above curves, which can hence be ignored for $n$ odd.
We used Magma V22.2-3 to compute the zeta functions of the curves~(\ref{eq:5trace1}), which took just under $15$ minutes on a
2.0GHz AMD Opteron computer and leads to the following theorem\footnote[2]{See~\url{F2(n,t1,t2,t3,t4,t5).m}.}.
\begin{theorem}\label{thm:5traces}
For $n \ge 5$ we have
{\small
\begin{eqnarray*}
F_2(n,0,0,0,0,0) &=& 2^{n-5} - \frac{1}{32} \big( 5\rho_n(\delta_{2,1}) + 5\rho_n(\delta_{4,1}) + 2\rho_n(\delta_{8,1}) + \rho_n(\delta_{8,3}) \big) \ \ \text{if} \ n \equiv 1,3,5,7 \pmod{8}\\
F_2(n,1,0,0,0,0) &=& 2^{n-5} - \frac{1}{32} \cdot \begin{cases}
\begin{array}{lr}
5\rho_n(\delta_{2,1}) + 5\rho_n(\delta_{4,1}) + 2\rho_n(\delta_{8,1}) + \rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 1 \pmod{8}\\
3\rho_n(\delta_{2,1}) - 3\rho_n(\delta_{4,1}) + 2\rho_n(\delta_{8,2}) - \rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 3 \pmod{8}\\
\rho_n(\delta_{2,1}) + \rho_n(\delta_{4,1}) - 2\rho_n(\delta_{8,1}) + \rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 5 \pmod{8}\\
-\rho_n(\delta_{2,1}) + \rho_n(\delta_{4,1}) - 2\rho_n(\delta_{8,2}) - \rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 7 \pmod{8}\\
\end{array}
\end{cases}\\
F_2(n,0,1,0,0,0) &=& 2^{n-5} - \frac{1}{32} \cdot \begin{cases}
\begin{array}{lr}
-\rho_n(\delta_{2,1}) + \rho_n(\delta_{4,1}) -2\rho_n(\delta_{8,2}) - \rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 1,5 \pmod{8}\\
3\rho_n(\delta_{2,1}) - 3\rho_n(\delta_{4,1}) + 2\rho_n(\delta_{8,2}) - \rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 3,7 \pmod{8}\\
\end{array}
\end{cases}\\
F_2(n,1,1,0,0,0) &=& 2^{n-5} - \frac{1}{32} \cdot \begin{cases}
\begin{array}{lr}
-3\rho_n(\delta_{2,1}) - 3\rho_n(\delta_{4,1}) + 2\rho_n(\delta_{8,1}) + \rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 1 \pmod{8}\\
-\rho_n(\delta_{2,1}) + \rho_n(\delta_{4,1}) - 2\rho_n(\delta_{8,2}) - \rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 3 \pmod{8}\\
\rho_n(\delta_{2,1}) + \rho_n(\delta_{4,1}) - 2\rho_n(\delta_{8,1}) + \rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 5 \pmod{8}\\
3\rho_n(\delta_{2,1}) - 3\rho_n(\delta_{4,1}) + 2\rho_n(\delta_{8,2}) - \rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 7 \pmod{8}\\
\end{array}
\end{cases}\\
F_2(n,0,0,1,0,0) &=& 2^{n-5} - \frac{1}{32} \big(-\rho_n(\delta_{2,1})+ \rho_n(\delta_{4,1}) -2\rho_n(\delta_{8,2}) -\rho_n(\delta_{8,3} )\big) \ \ \text{if} \ n \equiv 1,3,5,7 \pmod{8}\\
F_2(n,1,0,1,0,0) &=& 2^{n-5} - \frac{1}{32} \cdot \begin{cases}
\begin{array}{lr}
-3\rho_n(\delta_{2,1}) - 3\rho_n(\delta_{4,1}) + 2 \rho_n(\delta_{8,1})+ \rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 1 \pmod{8}\\
-5\rho_n(\delta_{2,1}) +5\rho_n(\delta_{4,1}) + 2\rho_n(\delta_{8,2}) - \rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 3 \pmod{8}\\
\rho_n(\delta_{2,1}) + \rho_n(\delta_{4,1}) - 2\rho_n(\delta_{8,1}) + \rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 5 \pmod{8}\\
-\rho_n(\delta_{2,1}) + \rho_n(\delta_{4,1}) - 2\rho_n(\delta_{8,2}) - \rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 7 \pmod{8}\\
\end{array}
\end{cases}\\
F_2(n,0,1,1,0,0) &=& 2^{n-5} - \frac{1}{32} \cdot \begin{cases}
\begin{array}{lr}
- \rho_n(\delta_{2,1}) + \rho_n(\delta_{4,1}) - 2\rho_n(\delta_{8,2}) - \rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 1,5 \pmod{8}\\
-5\rho_n(\delta_{2,1}) + 5\rho_n(\delta_{4,1}) + 2\rho_n(\delta_{8,2}) - \rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 3,7 \pmod{8}\\
\end{array}
\end{cases}\\
F_2(n,1,1,1,0,0) &=& 2^{n-5} - \frac{1}{32} \cdot \begin{cases}
\begin{array}{lr}
3\rho_n(\delta_{2,1}) - 3\rho_n(\delta_{4,1}) + 2\rho_n(\delta_{8,2}) - \rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 1 \pmod{8}\\
5\rho_n(\delta_{2,1}) + 5\rho_n(\delta_{4,1}) + 2\rho_n(\delta_{8,1}) + \rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 3 \pmod{8}\\
- \rho_n(\delta_{2,1}) + \rho_n(\delta_{4,1}) - 2\rho_n(\delta_{8,2}) - \rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 5 \pmod{8}\\
\rho_n(\delta_{2,1}) + \rho_n(\delta_{4,1}) - 2\rho_n(\delta_{8,1}) + \rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 7 \pmod{8}\\
\end{array}
\end{cases}\\
F_2(n,0,0,0,1,0) &=& 2^{n-5} - \frac{1}{32} \big( \rho_n(\delta_{2,1}) + \rho_n(\delta_{4,1}) - 2\rho_n(\delta_{8,1}) + \rho_n(\delta_{8,3}) \big) \ \ \text{if} \ n \equiv 1,3,5,7 \pmod{8}\\
F_2(n,1,0,0,1,0) &=& 2^{n-5} - \frac{1}{32} \cdot \begin{cases}
\begin{array}{lr}
-\rho_n(\delta_{2,1}) + \rho_n(\delta_{4,1}) -2\rho_n(\delta_{8,2}) - \rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 1 \pmod{8}\\
-3\rho_n(\delta_{2,1}) - 3\rho_n(\delta_{4,1}) + 2\rho_n(\delta_{8,1}) + \rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 3 \pmod{8}\\
3\rho_n(\delta_{2,1}) - 3\rho_n(\delta_{4,1}) + 2\rho_n(\delta_{8,2}) -\rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 5 \pmod{8}\\
\rho_n(\delta_{2,1}) + \rho_n(\delta_{4,1}) - 2\rho_n(\delta_{8,1}) + \rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 7 \pmod{8}\\
\end{array}
\end{cases}\\
F_2(n,0,1,0,1,0) &=& 2^{n-5} - \frac{1}{32} \cdot \begin{cases}
\begin{array}{lr}
3\rho_n(\delta_{2,1}) -3\rho_n(\delta_{4,1}) + 2\rho_n(\delta_{8,2}) -\rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 1,5 \pmod{8}\\
-\rho_n(\delta_{2,1}) + \rho_n(\delta_{4,1}) - 2\rho_n(\delta_{8,2}) -\rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 3,7 \pmod{8}\\
\end{array}
\end{cases}\\
F_2(n,1,1,0,1,0) &=& 2^{n-5} - \frac{1}{32} \cdot \begin{cases}
\begin{array}{lr}
- \rho_n(\delta_{2,1}) + \rho_n(\delta_{4,1}) - 2\rho_n(\delta_{8,2}) - \rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 1 \pmod{8}\\
\rho_n(\delta_{2,1}) + \rho_n(\delta_{4,1}) - 2\rho_n(\delta_{8,1}) + \rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 3 \pmod{8}\\
-5\rho_n(\delta_{2,1}) +5\rho_n(\delta_{4,1}) + 2\rho_n(\delta_{8,2}) - \rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 5 \pmod{8}\\
-3\rho_n(\delta_{2,1}) -3\rho_n(\delta_{4,1}) + 2\rho_n(\delta_{8,1}) + \rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 7 \pmod{8}\\
\end{array}
\end{cases}\\
F_2(n,0,0,1,1,0) &=& 2^{n-5} - \frac{1}{32} \big( 3\rho_n(\delta_{2,1}) - 3\rho_n(\delta_{4,1}) + 2\rho_n(\delta_{8,2}) - \rho_n(\delta_{8,3}) \big) \ \ \text{if} \ n \equiv 1,3,5,7 \pmod{8}\\
F_2(n,1,0,1,1,0) &=& 2^{n-5} - \frac{1}{32} \cdot \begin{cases}
\begin{array}{lr}
3\rho_n(\delta_{2,1}) - 3\rho_n(\delta_{4,1}) + 2\rho_n(\delta_{8,2}) - \rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 1 \pmod{8}\\
\rho_n(\delta_{2,1}) + \rho_n(\delta_{4,1}) - 2\rho_n(\delta_{8,1}) + \rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 3 \pmod{8}\\
- \rho_n(\delta_{2,1}) + \rho_n(\delta_{4,1}) - 2\rho_n(\delta_{8,2}) - \rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 5 \pmod{8}\\
- 3\rho_n(\delta_{2,1}) - 3\rho_n(\delta_{4,1}) + 2\rho_n(\delta_{8,1}) + \rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 7 \pmod{8}\\
\end{array}
\end{cases}\\
F_2(n,0,1,1,1,0) &=& 2^{n-5} - \frac{1}{32} \cdot \begin{cases}
\begin{array}{lr}
-5\rho_n(\delta_{2,1}) + 5\rho_n(\delta_{4,1}) + 2\rho_n(\delta_{8,2}) - \rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 1,5 \pmod{8}\\
- \rho_n(\delta_{2,1}) + \rho_n(\delta_{4,1}) - 2\rho_n(\delta_{8,2}) - \rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 3,7 \pmod{8}\\
\end{array}
\end{cases}\\
F_2(n,1,1,1,1,0) &=& 2^{n-5} - \frac{1}{32} \cdot \begin{cases}
\begin{array}{lr}
- 3\rho_n(\delta_{2,1}) - 3\rho_n(\delta_{4,1}) + 2\rho_n(\delta_{8,1}) + \rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 1 \pmod{8}\\
- \rho_n(\delta_{2,1}) + \rho_n(\delta_{4,1}) - 2\rho_n(\delta_{8,2}) - \rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 3 \pmod{8}\\
\rho_n(\delta_{2,1}) + \rho_n(\delta_{4,1}) - 2\rho_n(\delta_{8,1}) + \rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 5 \pmod{8}\\
3\rho_n(\delta_{2,1}) - 3\rho_n(\delta_{4,1}) + 2\rho_n(\delta_{8,2}) - \rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 7 \pmod{8}\\
\end{array}
\end{cases}\\
\end{eqnarray*}
}
{\small
\begin{eqnarray*}
F_2(n,0,0,0,0,1) &=& 2^{n-5} - \frac{1}{32} \big( 3\rho_n(\delta_{2,1})- 3\rho_n(\delta_{4,1}) + 2\rho_n(\delta_{8,2}) - \rho_n(\delta_{8,3}) \big) \ \ \text{if} \ n \equiv 1,3,5,7 \pmod{8}\\
F_2(n,1,0,0,0,1) &=& 2^{n-5} - \frac{1}{32} \cdot \begin{cases}
\begin{array}{lr}
3\rho_n(\delta_{2,1}) - 3\rho_n(\delta_{4,1}) + 2\rho_n(\delta_{8,2}) - \rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 1 \pmod{8}\\
\rho_n(\delta_{2,1}) + \rho_n(\delta_{4,1}) - 2\rho_n(\delta_{8,1}) + \rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 3 \pmod{8}\\
- \rho_n(\delta_{2,1}) + \rho_n(\delta_{4,1})- 2\rho_n(\delta_{8,2}) - \rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 5 \pmod{8}\\
- 3\rho_n(\delta_{2,1}) - 3\rho_n(\delta_{4,1}) + 2\rho_n(\delta_{8,1}) + \rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 7 \pmod{8}\\
\end{array}
\end{cases}\\
F_2(n,0,1,0,0,1) &=& 2^{n-5} - \frac{1}{32} \cdot \begin{cases}
\begin{array}{lr}
- 3\rho_n(\delta_{2,1}) - 3\rho_n(\delta_{4,1}) + 2\rho_n(\delta_{8,1}) + \rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 1,5 \pmod{8}\\
\rho_n(\delta_{2,1}) + \rho_n(\delta_{4,1}) - 2\rho_n(\delta_{8,1}) + \rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 3,7 \pmod{8}\\
\end{array}
\end{cases}\\
F_2(n,1,1,0,0,1) &=& 2^{n-5} - \frac{1}{32} \cdot \begin{cases}
\begin{array}{lr}
-5\rho_n(\delta_{2,1}) + 5\rho_n(\delta_{4,1}) + 2\rho_n(\delta_{8,2}) - \rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 1 \pmod{8}\\
-3\rho_n(\delta_{2,1}) - 3\rho_n(\delta_{4,1}) + 2\rho_n(\delta_{8,1}) + \rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 3 \pmod{8}\\
-\rho_n(\delta_{2,1}) + \rho_n(\delta_{4,1}) - 2\rho_n(\delta_{8,2}) - \rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 5 \pmod{8}\\
\rho_n(\delta_{2,1}) + \rho_n(\delta_{4,1}) - 2\rho_n(\delta_{8,1}) + \rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 7 \pmod{8}\\
\end{array}
\end{cases}\\
F_2(n,0,0,1,0,1) &=& 2^{n-5} - \frac{1}{32} \big( - 3\rho_n(\delta_{2,1}) - 3\rho_n(\delta_{4,1}) + 2\rho_n(\delta_{8,1}) + \rho_n(\delta_{8,3}) \big) \ \ \text{if} \ n \equiv 1,3,5,7 \pmod{8}\\
F_2(n,1,0,1,0,1) &=& 2^{n-5} - \frac{1}{32} \cdot \begin{cases}
\begin{array}{lr}
- \rho_n(\delta_{2,1}) + \rho_n(\delta_{4,1}) - 2\rho_n(\delta_{8,2}) - \rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 1 \pmod{8}\\
- 3\rho_n(\delta_{2,1}) - 3\rho_n(\delta_{4,1}) + 2\rho_n(\delta_{8,1}) + \rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 3 \pmod{8}\\
3\rho_n(\delta_{2,1}) -3\rho_n(\delta_{4,1}) + 2\rho_n(\delta_{8,2}) - \rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 5 \pmod{8}\\
\rho_n(\delta_{2,1}) + \rho_n(\delta_{4,1}) - 2\rho_n(\delta_{8,1}) + \rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 7 \pmod{8}\\
\end{array}
\end{cases}\\
F_2(n,0,1,1,0,1) &=& 2^{n-5} - \frac{1}{32} \cdot \begin{cases}
\begin{array}{lr}
\rho_n(\delta_{2,1})+ \rho_n(\delta_{4,1}) - 2\rho_n(\delta_{8,1}) + \rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 1,5 \pmod{8}\\
- 3\rho_n(\delta_{2,1}) - 3\rho_n(\delta_{4,1}) + 2\rho_n(\delta_{8,1}) + \rho_n(\delta_{8,3})& \ \text{if} \ n \equiv 3,7 \pmod{8}\\
\end{array}
\end{cases}\\
F_2(n,1,1,1,0,1) &=& 2^{n-5} - \frac{1}{32} \cdot \begin{cases}
\begin{array}{lr}
\rho_n(\delta_{2,1}) + \rho_n(\delta_{4,1}) - 2\rho_n(\delta_{8,1}) + \rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 1 \pmod{8}\\
3\rho_n(\delta_{2,1}) - 3\rho_n(\delta_{4,1}) + 2\rho_n(\delta_{8,2}) - \rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 3 \pmod{8}\\
- 3\rho_n(\delta_{2,1}) - 3\rho_n(\delta_{4,1}) + 2\rho_n(\delta_{8,1}) + \rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 5 \pmod{8}\\
- \rho_n(\delta_{2,1}) + \rho_n(\delta_{4,1}) - 2\rho_n(\delta_{8,2}) - \rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 7 \pmod{8}\\
\end{array}
\end{cases}\\
F_2(n,0,0,0,1,1) &=& 2^{n-5} - \frac{1}{32} \big(-\rho_n(\delta_{2,1}) + \rho_n(\delta_{4,1}) - 2\rho_n(\delta_{8,2}) -\rho_n(\delta_{8,3}) \big) \ \ \text{if} \ n \equiv 1,3,5,7 \pmod{8}\\
F_2(n,1,0,0,1,1) &=& 2^{n-5} - \frac{1}{32} \cdot \begin{cases}
\begin{array}{lr}
\rho_n(\delta_{2,1}) + \rho_n(\delta_{4,1}) - 2\rho_n(\delta_{8,1}) + \rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 1 \pmod{8}\\
- \rho_n(\delta_{2,1}) + \rho_n(\delta_{4,1}) - 2\rho_n(\delta_{8,2}) - \rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 3 \pmod{8}\\
5 \rho_n(\delta_{2,1}) + 5 \rho_n(\delta_{4,1}) + 2 \rho_n(\delta_{8,1}) + \rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 5 \pmod{8}\\
3 \rho_n(\delta_{2,1}) - 3\rho_n(\delta_{4,1}) + 2\rho_n(\delta_{8,2}) - \rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 7 \pmod{8}\\
\end{array}
\end{cases}\\
F_2(n,0,1,0,1,1) &=& 2^{n-5} - \frac{1}{32} \cdot \begin{cases}
\begin{array}{lr}
\rho_n(\delta_{2,1}) + \rho_n(\delta_{4,1}) - 2\rho_n(\delta_{8,1}) + \rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 1,5 \pmod{8}\\
- 3\rho_n(\delta_{2,1}) - 3\rho_n(\delta_{4,1}) + 2\rho_n(\delta_{8,1}) + \rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 3,7 \pmod{8}\\
\end{array}
\end{cases}\\
F_2(n,1,1,0,1,1) &=& 2^{n-5} - \frac{1}{32} \cdot \begin{cases}
\begin{array}{lr}
\rho_n(\delta_{2,1}) + \rho_n(\delta_{4,1}) - 2\rho_n(\delta_{8,1}) + \rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 1 \pmod{8}\\
3 \rho_n(\delta_{2,1}) - 3\rho_n(\delta_{4,1}) + 2\rho_n(\delta_{8,2}) - \rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 3 \pmod{8}\\
-3 \rho_n(\delta_{2,1}) - 3 \rho_n(\delta_{4,1}) + 2\rho_n(\delta_{8,1}) + \rho_n(\delta_{8,3})& \ \text{if} \ n \equiv 5 \pmod{8}\\
- \rho_n(\delta_{2,1}) + \rho_n(\delta_{4,1}) - 2\rho_n(\delta_{8,2}) - \rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 7 \pmod{8}\\
\end{array}
\end{cases}\\
F_2(n,0,0,1,1,1) &=& 2^{n-5} - \frac{1}{32} \big( \rho_n(\delta_{2,1}) + \rho_n(\delta_{4,1}) - 2\rho_n(\delta_{8,1}) + \rho_n(\delta_{8,3}) \big) \ \ \text{if} \ n \equiv 1,3,5,7 \pmod{8}\\
F(n,1,0,1,1,1) &=& 2^{n-5} - \frac{1}{32} \cdot \begin{cases}
\begin{array}{lr}
\rho_n(\delta_{2,1}) + \rho_n(\delta_{4,1}) - 2\rho_n(\delta_{8,1}) + \rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 1 \pmod{8}\\
- \rho_n(\delta_{2,1}) + \rho_n(\delta_{4,1}) - 2 \rho_n(\delta_{8,2}) - \rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 3 \pmod{8}\\
-3 \rho_n(\delta_{2,1}) -3 \rho_n(\delta_{4,1}) + 2 \rho_n(\delta_{8,1}) + \rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 5 \pmod{8}\\
-5 \rho_n(\delta_{2,1}) + 5 \rho_n(\delta_{4,1}) + 2 \rho_n(\delta_{8,2}) - \rho_n(\delta_{8,3})& \ \text{if} \ n \equiv 7 \pmod{8}\\
\end{array}
\end{cases}\\
F_2(n,0,1,1,1,1) &=& 2^{n-5} - \frac{1}{32} \cdot \begin{cases}
\begin{array}{lr}
-3 \rho_n(\delta_{2,1}) -3 \rho_n(\delta_{4,1})+ 2 \rho_n(\delta_{8,1}) + \rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 1,5 \pmod{8}\\
\rho_n(\delta_{2,1}) + \rho_n(\delta_{4,1}) - 2\rho_n(\delta_{8,1}) + \rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 3,7 \pmod{8}\\
\end{array}
\end{cases}\\
F_2(n,1,1,1,1,1) &=& 2^{n-5} - \frac{1}{32} \cdot \begin{cases}
\begin{array}{lr}
- \rho_n(\delta_{2,1})+ \rho_n(\delta_{4,1}) -2 \rho_n(\delta_{8,2}) - \rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 1 \pmod{8}\\
\rho_n(\delta_{2,1}) + \rho_n(\delta_{4,1}) - 2\rho_n(\delta_{8,1}) + \rho_n(\delta_{8,3})& \ \text{if} \ n \equiv 3 \pmod{8}\\
3 \rho_n(\delta_{2,1}) -3 \rho_n(\delta_{4,1}) + 2 \rho_n(\delta_{8,2}) - \rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 5 \pmod{8}\\
5 \rho_n(\delta_{2,1})+ 5 \rho_n(\delta_{4,1})+ 2 \rho_n(\delta_{8,1}) + \rho_n(\delta_{8,3}) & \ \text{if} \ n \equiv 7 \pmod{8}\\
\end{array}
\end{cases}\\
\end{eqnarray*}
}
\end{theorem}
One can check that the roots $\delta_{8,3}$ are $\sqrt{2}\omega_{40}^3,\sqrt{2}\omega_{40}^{11},\sqrt{2}\omega_{40}^{13},\sqrt{2}\omega_{40}^{19},
\sqrt{2}\omega_{40}^{21},\sqrt{2}\omega_{40}^{27},\sqrt{2}\omega_{40}^{29},\sqrt{2}\omega_{40}^{37}$, where
\begin{eqnarray*}
\omega_{40} = \frac{1}{\sqrt{2}}\Big( \frac{1+\sqrt{5}}{4} - \frac{1-\sqrt{5}}{8}\sqrt{10 + 2\sqrt{5}}\Big)
+ \frac{1}{\sqrt{2}}\Big( \frac{1+\sqrt{5}}{4} + \frac{1-\sqrt{5}}{8}\sqrt{10 + 2\sqrt{5}}\Big)i
\end{eqnarray*}
is the primitive $40$-th root of unity with smallest (positive) argument.
Due to the presence of the roots of either $\delta_{8,1}$ or $\delta_{8,2}$ in each formula, they can not be periodic in $n$.
\subsubsection*{An alternative parameterisation.}\label{sec:5coeffsparam2}
We now present an alternative parameterisation of Eq.~(\ref{eq:5coeffsparam1}) which although requires the same number of
variables to linearise, is perhaps more natural given the form of $T_2(a_{0}^2 + a_0 + r_0)$ and $T_3(a_{0}^2 + a_0 + r_0)$,
see~\S\ref{sec:5direct}. In particular,
using Lemma~\ref{lem:T} part (2) three times with $(\alpha,\beta) = (a_{0}^5, a_{0}^3)$, $(a_{0}^5, a_{0})$ and $(a_{0}^3, a_{0})$,
we have
\begin{eqnarray}
\nonumber T_5(a_{0}^2 + a_0 + r_0) &=& T_1(a_{0}^5)T_1(a_{0}^3) + T_1(a_{0}^5)T_1(a_{0}) + T_1(a_{0}^3)T_1(a_{0}) + r_0(T_2(a_{0}^3) + T_2(a_{0}) + T_1(a_{0}^3)T_1(a_{0}))\\
\nonumber &+& T_1\Big(a_{0}^9 + a_{0}^3 + a_{0} + r_0 \Big(a_{0}^7 + (a_{0}^5 + a_0)\binom{n}{2} + (a_{0}^3 + a_0)\binom{n}{3} + \binom{n}{5}\Big)\Big)\\
\nonumber &=& T_2(a_{0}^5 + a_{0}^3) + T_2(a_{0}^5 + a_{0}) + (r_0 + 1)T_2(a_{0}^3 + a_{0}) \\
\label{eq:T5linearalt} &+& T_1\Big(a_{0}^9 + a_{0} + r_0 \Big(a_{0}^7 + a_0 +(a_{0}^5 + a_0)\binom{n}{2} + (a_{0}^3 + a_0)\binom{n}{3} + \binom{n}{5}\Big)\Big).
\end{eqnarray}
This can be linearised using the substitutions $a_{0}^3 + a_{0} = a_{1}^2 + a_{1} + r_1$ and $a_{0}^5 + a_{0} = a_{2}^2 + a_{2} + r_2$,
where $r_1$ and $r_2$ are the traces of $a_{0}^3 + a_{0}$ and $a_{0}^5 + a_{0}$ respectively, which when added together imply that
$a_{0}^5 + a_{0}^3 = (a_1 + a_2)^2 + (a_1 + a_2) + (r_1 + r_2)$.
This results in
\begin{eqnarray*}
\nonumber T_5(a_{0}^2 + a_0 + r_0) &=& T_1\Big( a_{1}^2 a_2 + a_1 a_{2}^2 + a_{0}^9 + a_{0} \\
&+& r_0\Big(a_{1}^3 + a_{1} + a_{0}^7 + a_0 +(a_{0}^5 + a_0 + r_1)\binom{n}{2} + (a_{0}^3 + a_0)\binom{n}{3} + \binom{n}{5}\Big)\Big).
\end{eqnarray*}
For $16 \le i \le 31$ let $\vec{i} = (i_4,i_3,i_2,i_1,i_0)$. The curves we are interested in are given by the following intersections:
\begin{eqnarray}
\nonumber a_{3}^2 + a_{3} &=& i_4\Big(a_{1}^2 a_2 + a_1 a_{2}^2 + a_{0}^9 + a_{0} + r_0\Big(a_{1}^3 + a_{1} + a_{0}^7 + a_0 +(a_{0}^5 + a_0 + r_1)\binom{n}{2} + (a_{0}^3 + a_0)\binom{n}{3} + \binom{n}{5}\Big)\Big)\\
\nonumber &+& i_3\Big( a_{1}^3 + a_{1} + a_{0}^7 + a_{0}^5 + a_{0}^3 + a_{0} + r_0\Big(a_{0}^3 + a_0 + (a_{0}^3 + a_0)\binom{n}{2} + \binom{n}{4}\Big) + r_1\binom{n}{2}\Big)\\
\nonumber &+& i_2\Big(a_{0}^5 + a_{0} + r_0\Big(a_{0}^3 + a_0 + \binom{n}{3} \Big)\Big) + i_1\Big(a_{0}^3 + a_{0} + r_0\binom{n}{2}\Big) + i_0r_0,\\
\nonumber a_{0}^3 + a_{0} &=& a_{1}^2 + a_1 + r_1\\
\nonumber a_{0}^5 + a_{0} &=& a_{2}^2 + a_2 + r_2.
\end{eqnarray}
For $i_4 = 1$ the genus of all of these absolutely irreducible curves is $21$ -- rather than $18$ as in the first parameterisation.
Nevertheless, once the transform~(\ref{eq:Sinverse}) is applied one again obtains the formulae given in Theorem~\ref{thm:5traces}.
\subsubsection{Some examples from the direct method.}\label{sec:5direct}
For $n$ odd, using Equations~(\ref{eq:T1linear}) to~(\ref{eq:T3linear}) and the alternative parametrisations~(\ref{eq:T4linearalt})
and~(\ref{eq:T5linearalt}), one can write down curves arising from the direct method for $F_2(n,t_1,t_2,t_3,t_4,t_5)$, as in~\S\ref{sec:4coeffsdirect}.
Here we give the two simplest examples, valid for all $n \ge 5$.
\begin{eqnarray}
\nonumber F_2(n,0,0,0,0,0) &=& \#\{a \in {\mathbb F}_{2^n} \mid T_1(a) = 0, \ T_2(a) = 0, \ T_3(a) = 0, \ T_4(a) = 0, \ T_5(a) = 0\}\\
\nonumber &=& \frac{1}{2} \, \#\{ a_{0} \in {\mathbb F}_{2^n} \mid T_2(a_{0}^2 + a_{0}) = 0, \ T_3(a_{0}^2 + a_{0}) = 0, \ T_4(a_{0}^2 + a_{0}) = 0, \
T_5(a_{0}^2 + a_{0}) = 0\}\\
\nonumber &=& \frac{1}{2} \, \#\{ a_{0} \in {\mathbb F}_{2^n} \mid T_1(a_{0}^3 + a_{0}) = 0, \ T_1(a_{0}^5 + a_{0}) = 0,\\
\nonumber && \ \ \ \ T_2(a_{0}^3 + a_{0}) + T_1( a_{0}^7 + a_{0}^5 + a_{0}^3 + a_0) = 0,\\
\nonumber && \ \ \ \ T_1(a_{0}^5)T_1(a_{0}^3) + T_1(a_{0}^5)T_1(a_{0}) + T_1(a_{0}^3)T_1(a_{0}) + T_1(a_{0}^9 + a_{0}^3 + a_{0}) = 0\}\\
\nonumber &=& \frac{1}{2} \, \#\{ a_{0} \in {\mathbb F}_{2^n} \mid T_1(a_{0}^3 + a_{0}) = 0, \ T_1(a_{0}^5 + a_{0}) = 0,\\
\nonumber && \ \ \ \ T_2(a_{0}^3 + a_{0}) + T_1( a_{0}^7 + a_{0}^5 + a_{0}^3 + a_0) = 0,\\
\nonumber && \ \ \ \ T_2(a_{0}^5 + a_{0}^3) + T_2(a_{0}^5 + a_{0}) + T_2(a_{0}^3 + a_{0}) + T_1(a_{0}^9 + a_0) = 0\}\\
\nonumber &=& \frac{1}{8} \, \#\{ (a_{0},a_{1},a_{2}) \in ({\mathbb F}_{2^n})^3 \mid a_{1}^2 + a_{1} = a_{0}^3 + a_{0}, \ a_{2}^2 + a_{2} = a_{0}^5+a_{0},\\
\nonumber && \ \ \ \ T_1( a_{1}^3 + a_{1} + a_{0}^7 + a_{0}^5 + a_{0}^3 + a_0) = 0,\\
\nonumber && \ \ \ \ T_2((a_{1} + a_{2})^2 + (a_{1} + a_{2})) + T_1(a_{2}^3 + a_{2}) + T_1(a_{1}^3 + a_{1}) + T_1(a_{0}^9 + a_0) = 0\}\\
\nonumber &=& \frac{1}{32} \, \#\{ (a_{0},a_{1},a_{2},a_{3},a_{4}) \in ({\mathbb F}_{2^n})^5 \mid a_{1}^2 + a_{1} = a_{0}^3 + a_{0}, \ a_{2}^2 + a_{2} =
a_{0}^5+a_{0},\\
\label{genus49} && \ \ \ \ a_{3}^2 + a_{3} = a_{1}^3 + a_{1} + a_{0}^7 + a_{0}^5 + a_{0}^3 + a_0, \ a_{4}^2 + a_{4} = a_{1}^2a_{2} + a_{1}a_{2}^2 + a_{0}^9
+ a_0\}.
\end{eqnarray}
The genus of the absolutely irreducible curve this intersection describes is $49$, and thus one can not easily brute-force compute
the characteristic polynomial of Frobenius by computing the number of points over ${\mathbb F}_{2^n}$ for $n \le 49$, as for
the four coefficient direct method cases. The state-of-the-art $p$-adic point counting
algorithms of Tuitman~\cite{tuitman1,tuitman2} are unfortunately not easily adaptable to such intersections, while the prime $2$ is problematic.
However, by ignoring the fourth trace, letting $\delta_{2,3}(X) = X^2 - 2X +2$ and $\delta_{4,2} = X^4 + 2X^2 + 4$, we have
\begin{eqnarray}
\nonumber F_2(n,0,0,0,*,0) &=& \#\{a \in {\mathbb F}_{2^n} \mid T_1(a) = 0, \ T_2(a) = 0, \ T_3(a) = 0, \ T_5(a) = 0\}\\
\nonumber &=& \frac{1}{2} \, \#\{ a_{0} \in {\mathbb F}_{2^n} \mid T_2(a_{0}^2 + a_{0}) = 0, \ T_3(a_{0}^2 + a_{0}) = 0, \ T_5(a_{0}^2 + a_{0}) = 0\}\\
\nonumber &=& \frac{1}{2} \, \#\{ a_{0} \in {\mathbb F}_{2^n} \mid T_1(a_{0}^3 + a_{0}) = 0, \ T_1(a_{0}^5 + a_{0}) = 0,\\
\nonumber && \ \ \ \ T_2(a_{0}^5 + a_{0}^3) + T_2(a_{0}^5 + a_{0}) + T_2(a_{0}^3 + a_{0}) + T_1(a_{0}^9 + a_0) = 0\}\\
\nonumber &=& \frac{1}{16} \, \#\{ (a_{0},a_{1},a_{2},a_{3}) \in ({\mathbb F}_{2^n})^4 \mid a_{1}^2 + a_{1} = a_{0}^3 + a_{0}, \ a_{2}^2 + a_{2} = a_{0}^5+a_{0},\\
\nonumber && \ \ \ \ a_{3}^2 + a_{3} = a_{1}^2a_{2} + a_{1}a_{2}^2 + a_{0}^9 + a_0\}\\
\nonumber &=& 2^{n-4} - \frac{1}{16}\big(5\rho_n(\delta_{2,1}) + 2\rho_n(\delta_{2,2}) + 2\rho_n(\delta_{2,3}) + 3\rho_n(\delta_{4,1}) + \rho_n(\delta_{4,2}) + \rho_n(\delta_{8,3})\big),
\end{eqnarray}
where in the final equality we have used Magma to compute the characteristic polynomial of Frobenius of the genus $21$ curve
described by the intersection. Since neither $\delta_{8,1}$ nor $\delta_{8,2}$ feature in this formula, we deduce that
the set of formulae for $F_2(n,0,0,0,*,0)$ are periodic in $n$ with period $\text{LCM}(8,8,4,24,6,40) = 120$, in similarity with the two and
three coefficient cases for which the periods are $8$ and $24$ in $n$, respectively.
\subsubsection{Further general formulae.}\label{subsubsec:5coeffgeneral}
While one can not easily compute the zeta function for $F_2(n,0,0,0,0,0)$ using the direct method, one can use the approach
of~\S\ref{subsubsec:altgeneralformulae} and the alternative parameterisations~(\ref{eq:T4linearalt}) and~(\ref{eq:T5linearalt}) to compute
$F_2(n,0,0,0,t_4,t_5)$ for all $n \ge 5$, as follows.
The conditions $T_1(a) = T_2(a) = T_3(a) = 0$ imply that we should set $A = \{(a_0,a_1,a_2) \in ({\mathbb F}_{2^n})^3 \mid a_{0}^3 + a_{0} = a_{1}^2 + a_{1}
\ a_{0}^5 + a_{0} = a_{2}^2 + a_{2}\}$.
Furthermore let $\vec{f} = (T_5(a_{0}^2 + a_{0}), T_4(a_{0}^2 + a_{0})) = (T_1(a_{1}^2 a_2 + a_{1}a_{2}^2 + a_{0}^9 + a_0), T_1(a_{1}^3 + a_1 + a_{0}^7 + a_{0}^5 + a_{0}^3 + a_{0}))$.
Then by the generalisation of the transform~(\ref{eq:Sinverse}) we have
\begin{equation}\label{eq:general5}
{\small
\begin{bmatrix}
F_2(n,0,0,0,0,0)\\
F_2(n,0,0,0,1,0)\\
F_2(n,0,0,0,0,1)\\
F_2(n,0,0,0,1,1)\\
\end{bmatrix}
=
\begin{bmatrix}
N(\vec{0})\\
N(\vec{1})\\
N(\vec{2})\\
N(\vec{3})\\
\end{bmatrix}
=
\frac{1}{16}
\begin{bmatrix*}[r]
-1 & 1 & 1 & 1\\
1 & -1 & 1 & -1\\
1 & 1 & -1 & -1\\
1 & -1 & -1 & 1\\
\end{bmatrix*}
\begin{bmatrix}
V(\vec{0} \cdot \vec{f})\\
V(\vec{1} \cdot \vec{f})\\
V(\vec{2} \cdot \vec{f})\\
V(\vec{3} \cdot \vec{f})\\
\end{bmatrix},}
\end{equation}
where we have a factor of $1/16$ rather than $1/2$ because of the three factors of $1/2$ arising from the introduction of the variables $a_0, a_1$
and $a_2$ defining $A$. Note that $V(\vec{0} \cdot \vec{f}) = |A| = 2^n - 2\rho_n(\delta_{2,1}) - \rho_n(\delta_{2,2}) - \rho_n(\delta_{4,1})$.
For $1 \le i \le 3$ let $\vec{i} = (i_1,i_0)$. We thus have:
\begin{eqnarray*}
V(\vec{i} \cdot \vec{f}) &=& \frac{1}{2}\#\{(a_0,a_1,a_2,a_3) \mid a_{0}^3 + a_{0} = a_{1}^2 + a_{1}, \ a_{0}^5 + a_{0} = a_{2}^2 + a_{2}\\
&& \ \ \ \ a_{3}^2 + a_{3} = i_1(a_{1}^2 a_2 + a_{1}a_{2}^2 + a_{0}^9 + a_0) + i_0(a_{1}^3 + a_1 + a_{0}^7 + a_{0}^5 + a_{0}^3 + a_{0}) \}.
\end{eqnarray*}
For $i = 1,2$ and $3$ these are absolutely irreducible curves of genus $5,17,21$ and $21$ respectively, and thus their zeta functions
are far easier to compute than for those curves arising from the direct method. Let $\delta_{8,4}(X) = X^8 + 2X^6 - 4X^5 + 2X^4 - 8X^3 + 8X^2 + 16$.
Combining the $V(\vec{i} \cdot \vec{f})$ as per Eq.~(\ref{eq:general5}) gives the following theorem.
\begin{theorem}
For $n \ge 5$ we have
\begin{eqnarray*}
F_2(n,0,0,0,0,0) &=& 2^{n-5} - \frac{1}{32}\big(7\rho_n(\delta_{2,1}) + 4\rho_n(\delta_{2,2}) + 2\rho_n(\delta_{2,3})+ 5\rho_n(\delta_{4,1}) + 3\rho_n(\delta_{4,2}) + 2\rho_n(\delta_{8,1})\\
&& \ \ \ \ \ \ \ + \rho_n(\delta_{8,2}) + \rho_n(\delta_{8,3}) + \rho_n(\delta_{8,4})\big),\\
F_2(n,0,0,0,1,0) &=& 2^{n-5} - \frac{1}{32}\big(3\rho_n(\delta_{2,1}) + 2\rho_n(\delta_{2,3}) + \rho_n(\delta_{4,1}) -\rho_n(\delta_{4,2})
- 2\rho_n(\delta_{8,1}) - \rho_n(\delta_{8,2})\\
&& \ \ \ \ \ \ \ + \rho_n(\delta_{8,3}) - \rho_n(\delta_{8,4}) \big),\\
F_2(n,0,0,0,0,1) &=& 2^{n-5} - \frac{1}{32}\big(\rho_n(\delta_{2,1}) + 2\rho_n(\delta_{2,2}) - 2\rho_n(\delta_{2,3}) -3\rho_n(\delta_{4,1}) - 3\rho_n(\delta_{4,2})
+ \rho_n(\delta_{8,2})\\
&& \ \ \ \ \ \ \ - \rho_n(\delta_{8,3}) - \rho_n(\delta_{8,4}) \big),\\
F_2(n,0,0,0,1,1) &=& 2^{n-5} - \frac{1}{32}\big(-3\rho_n(\delta_{2,1}) - 2\rho_n(\delta_{2,2}) - 2\rho_n(\delta_{2,3}) + \rho_n(\delta_{4,1}) + \rho_n(\delta_{4,2})
- \rho_n(\delta_{8,2})\\
&& \ \ \ \ \ \ \ - \rho_n(\delta_{8,3}) + \rho_n(\delta_{8,4}) \big).
\end{eqnarray*}
\end{theorem}
Note that the formula for $F_2(n,0,0,0,0,0)$ arises from the curve~(\ref{genus49}) of genus $49$. This approach therefore provides a much more efficient way to compute the zeta function than the direct method does.
Further note that the terms involving $\delta_{8,1}$, $\delta_{8,2}$ and $\delta_{8,4}$ all cancel in
$F_2(n,0,0,0,0,0) + F_2(n,0,0,0,1,0) = F_2(0,0,0,*,0)$, as well as in
$F_2(n,0,0,0,0,1) + F_2(n,0,0,0,1,1) = F_2(0,0,0,*,1)$, which has period $120$ in $n$ as well.
\subsection{Computing $F_2(n,t_1,t_2,t_3,t_4,t_5,t_6)$}
In this subsection and the next we shall only detail the indirect method, since the curves produced have genera which are already very large,
making the brute force computation of the characteristic polynomials of Frobenius prohibitive, while the direct method produce curves of even
larger genus.
Let $\vec{f} = (T_6,T_5,T_4,T_3,T_2,T_1)$. To determine $V( \vec{i} \cdot \vec{f})$ for $32 \le i \le 63$, we use Lemma~\ref{lem:T} part (6).
In particular, setting $\alpha = a_{0}^2$ and $\beta = a_{0}$ for $r_0 = 0$, and $\alpha = a_{0}^2 + a_0$ and $\beta = 1$ for $r_0 = 1$,
and evaluating mod $2$ gives the following:
\begin{eqnarray*}
T_6(a_{0}^2 + a_{0} + r_0) &=& T_3(a_0) + T_2(a_{0}^3)T_1(a_{0}^3) + T_2(a_{0}^3)T_1(a_0) + T_2(a_0)T_1(a_{0}^3) + T_2(a_{0}^5) + T_2(a_0)\\
&+& T_1(a_{0}^7)T_1(a_{0}^3) + T_1(a_{0}^7)T_1(a_{0}) + T_1(a_{0}^5)T_1(a_{0}^3) + T_1(a_{0}^3)T_1(a_{0})\\
&+& r_0\big(T_2(a_{0}^3) + T_2(a_0) + T_1(a_{0}^3)T_1(a_0) + T_1(a_{0}^7)\big)\binom{n}{2}\\
&+& T_1\Big(a_{0}^{11} + a_{0}^7 + r_0(a_{0}^3 + a_0) + r_0(a_{0}^5 + a_0)\binom{n}{3} + r_0(a_{0}^3 + a_0)\binom{n}{4} + r_0\binom{n}{6}\Big).
\end{eqnarray*}
This can be reduced to a $T_1$ expression using the substitutions $a_0 = a_{1}^2 + a_1 + r_1$, $a_{0}^3 = a_{2}^2 + a_2 + r_2$ and
$a_{0}^5 = a_{3}^2 + a_3 + r_3$, where $r_1,r_2,r_3 \in {\mathbb F}_2$ are the traces of $a_0,a_{0}^3$ and $a_{0}^5$ respectively.
This results in
\begin{eqnarray*}
T_6(a_{0}^2 + a_0 + r_0) &=& T_1\Big( a_{3}^3 + a_3 + (r_1 + r_2)(a_{2}^3 + a_2 + a_{1}^3 + a_1 + a_{0}^7) + a_{1}^5 + a_{1}^3 + a_{0}^{11} + a_{0}^7
+ r_0r_1 + r_0r_2 \\
&+& r_1r_2 + r_2r_3 + \big(r_0(a_{2}^3 + a_2 + a_{1}^3 + a_1 + a_{0}^7 + r_1 + r_2 + r_1r_2 +1) +r_1 + r_2 + r_3\big)\binom{n}{2}\\
&+& (r_0r_1 + r_0r_3 + r_1)\binom{n}{3} + r_0(r_1 + r_2)\binom{n}{4} + r_0\binom{n}{6}\Big).
\end{eqnarray*}
For $32 \le i \le 63$ let $\vec{i} = (i_5,i_4,i_3,i_2,i_1,i_0)$. The curves we are interested in are given by the following intersections:
\begin{eqnarray}
\nonumber a_{4}^2 + a_{4} &=& i_5\Big( a_{3}^3 + a_3 + (r_1 + r_2)(a_{2}^3 + a_2 + a_{1}^3 + a_1 + a_{0}^7) + a_{1}^5 + a_{1}^3 + a_{0}^{11} + a_{0}^7
+ r_0r_1 + r_0r_2 \\
\nonumber &+& r_1r_2 + r_2r_3 + \big(r_0(a_{2}^3 + a_2 + a_{1}^3 + a_1 + a_{0}^7 + r_1 + r_2 + r_1r_2 +1) +r_1 + r_2 + r_3\big)\binom{n}{2}\\
\nonumber &+& (r_0r_1 + r_0r_3 + r_1)\binom{n}{3} + r_0(r_1 + r_2)\binom{n}{4} + r_0\binom{n}{6}\Big)\\
\nonumber &+& i_4\Big(r_0(a_{2}^3 + a_{2} + a_{1}^3 + a_{1}) + a_{0}^9 + r_0a_{0}^7 + (r_1 + r_2)a_{0}^5 + r_1 + r_2 + r_1r_2 + r_0r_1r_2\\
\nonumber &+& (r_0a_{0}^5 + r_0r_2)\binom{n}{2} + (r_0r_1 + r_0r_2)\binom{n}{3} + r_0\binom{n}{5}\Big)\\
\nonumber &+& i_3\Big(a_{2}^3 + a_2 + a_{1}^3 + a_1 + a_{0}^7 + a_{0}^5 + (r_0 + 1)(r_1 + r_2)\binom{n}{2} + r_0r_1 + r_0r_2 + r_1r_2 + r_2 + r_0\binom{n}{4}\Big)\\
\nonumber &+& i_2\Big(a_{0}^5 + a_{0} + r_0\Big(a_{0}^3 + a_0 + \binom{n}{3} \Big)\Big) + i_1\Big(a_{0}^3 + a_{0} + r_0\binom{n}{2}\Big) + i_0r_0,\\
\nonumber a_0 &=& a_{1}^2 + a_1 + r_1,\\
\label{eq:6coefficients} a_{0}^3 &=& a_{2}^2 + a_2 + r_2.\\
\nonumber a_{0}^5 &=& a_{3}^2 + a_3 + r_3.
\end{eqnarray}
Corollary~\ref{cor:period} implies that mod $2$ one has
\begin{eqnarray*}
\Big(\binom{n}{6},\binom{n}{5},\binom{n}{4},\binom{n}{3},\binom{n}{2},\binom{n}{1}\Big) \equiv \begin{cases}
\begin{array}{lr}
(0,0,0,0,0,1) & \ \text{if} \ n \equiv 1 \pmod{8}\\
(0,0,0,1,1,1) & \ \text{if} \ n \equiv 3 \pmod{8}\\
(0,1,1,0,0,1) & \ \text{if} \ n \equiv 5 \pmod{8}\\
(1,1,1,1,1,1) & \ \text{if} \ n \equiv 7 \pmod{8}\\
\end{array},
\end{cases}
\end{eqnarray*}
and hence there are four cases to consider when computing the zeta functions of each of the curves~(\ref{eq:6coefficients}).
For $i_5 = 1$ the genus of all of the above curves is $50$\footnote[2]{See~\url{6CoefficientsGenus.m}.} and
therefore the brute-force computation of their zeta functions is non-trivial.
A curve-specific analysis may yield the zeta functions more efficiently,
but since our algorithm is arguably more interesting than the explicit formulae, we leave this as an open problem.
\subsection{Computing $F_2(n,t_1,t_2,t_3,t_4,t_5,t_6,t_7)$}\label{subsec:F7}
Let $\vec{f} = (T_7,T_6,T_5,T_4,T_3,T_2,T_1)$. To determine $V( \vec{i} \cdot \vec{f})$ for $64 \le i \le 127$, we use Lemma~\ref{lem:T} part (7).
In particular, setting $\alpha = a_{0}^2$ and $\beta = a_{0}$ for $r_0 = 0$, and $\alpha = a_{0}^2 + a_0$ and $\beta = 1$ for $r_0 = 1$,
and evaluating mod $2$ gives the following:
\begin{eqnarray*}
T_7(a_{0}^2 + a_{0} + r_0) &=& r_0T_3(a_{0}) + T_1(a_{0}^5)T_1(a_{0}^3)T_1(a_{0}) + r_0(T_2(a_{0}^3)T_1(a_{0}^3) + T_2(a_{0}^3)T_1(a_{0}) + T_2(a_{0})T_1(a_{0}^3))\\
&+& T_2(a_{0}^3)T_1(a_{0}^5) + T_2(a_{0}^3)T_1(a_{0}) + T_2(a_{0})T_1(a_{0}^5) + T_2(a_{0})T_1(a_{0}) + r_0(T_2(a_{0})\\
&+& T_2(a_{0}^3))\binom{n}{3} + r_0T_2(a_{0})+ r_0T_2(a_{0}^5) + T_1(a_{0}^9)T_1(a_{0}^3) + T_1(a_{0}^9)T_1(a_{0}) + T_1(a_{0}^7)T_1(a_{0}^5)\\
&+& r_0T_1(a_{0}^7)T_1(a_{0}^3) + (r_0 + 1)T_1(a_{0}^7)T_1(a_{0}) + \Big(r_0 + 1 + r_0\binom{n}{2}\Big)T_1(a_{0}^5)T_1(a_{0}^3)\\
&+& \Big(1 + r_0\binom{n}{2}\Big)T_1(a_{0}^5)T_1(a_{0}) + \Big(r_0 + 1 + r_0\binom{n}{2} + r_0\binom{n}{3}\Big)T_1(a_{0}^3)T_1(a_{0})\\
&+& T_1\Big(a_{0}^{13} + (r_0+1)a_{0}^{11} + a_{0}^9 + r_0(a_{0}^7 + a_{0}^5 + a_{0}^3) + a_{0} + r_0(a_{0}^9 + a_{0}^3 + a_{0})\binom{n}{2}\\
&+& r_0a_{0}^7\binom{n}{3} + r_0(a_{0}^5 + a_{0})\binom{n}{4} + r_0(a_{0}^3 + a_0)\binom{n}{5} + r_0\binom{n}{7}\Big).
\end{eqnarray*}
As in the six coefficient case, this can be reduced to a $T_1$ expression using the substitutions $a_0 = a_{1}^2 + a_1 + r_1$,
$a_{0}^3 = a_{2}^2 + a_2 + r_2$ and $a_{0}^5 = a_{3}^2 + a_3 + r_3$, where $r_1,r_2,r_3 \in {\mathbb F}_2$ are the traces of $a_0,a_{0}^3$
and $a_{0}^5$ respectively. This results in
\begin{eqnarray*}
T_7(a_{0}^2 + a_0 + r_0) &=& T_1\Big(r_0(a_{3}^3 + a_{3}) + (r_0r_1 + r_0r_2 + r_1 + r_3 + r_0\binom{n}{3})(a_{2}^3 + a_2) + r_0(a_{1}^5 + a_{1}^3)\\
&+& (r_0r_1 + r_0r_2 + r_1 + r_3 + r_0\binom{n}{3})(a_{1}^3 + a_1) + a_{0}^{13} + (r_0 + 1)a_{0}^{11}\\
&+& \Big(r_1 + r_2 + 1 + r_0\binom{n}{2}\Big)a_{0}^9 + \Big(r_0 + r_1 + r_3 + r_0r_1 + r_0r_2 + r_0\binom{n}{3} \Big)a_{0}^7\\
&+& r_1 + r_0r_2 + r_0r_3 + r_1r_2 + r_1r_3 + r_2r_3 + r_0r_1r_2 + r_0r_2r_3 + r_1r_2r_3\\
&+& \Big(r_1 + r_0r_3 + r_1r_2 + r_1r_3 + r_2r_3 + r_0r_1r_2 + r_0r_1r_3 + r_0r_2r_3\Big)\binom{n}{2}\\
&+& \Big( r_0r_1 + r_0r_1r_2 \Big)\binom{n}{3} + \Big(r_0r_1 + r_0r_2 \Big)\binom{n}{2}\binom{n}{3} + \Big( r_0r_1 + r_0r_3\Big)\binom{n}{4}\\
&+& \Big( r_0r_1 + r_0r_2 \Big)\binom{n}{5} + r_0\binom{n}{7}\Big).
\end{eqnarray*}
For $64 \le i \le 127$ let $\vec{i} = (i_6,i_5,i_4,i_3,i_2,i_1,i_0)$.
Applying Transform 3, the curves we are interested in are given by the following intersections:
\begin{eqnarray*}
a_{4}^2 + a_{4} &=&
i_6\Big(r_0(a_{3}^3 + a_{3}) + (r_0r_1 + r_0r_2 + r_1 + r_3 + r_0\binom{n}{3})(a_{2}^3 + a_2) + r_0(a_{1}^5 + a_{1}^3)\\
&+& (r_0r_1 + r_0r_2 + r_1 + r_3 + r_0\binom{n}{3})(a_{1}^3 + a_1) + a_{0}^{13} + (r_0 + 1)a_{0}^{11}\\
&+& \Big(r_1 + r_2 + 1 + r_0\binom{n}{2}\Big)a_{0}^9 + \Big(r_0 + r_1 + r_3 + r_0r_1 + r_0r_2 + r_0\binom{n}{3} \Big)a_{0}^7\\
&+& r_1 + r_0r_2 + r_0r_3 + r_1r_2 + r_1r_3 + r_2r_3 + r_0r_1r_2 + r_0r_2r_3 + r_1r_2r_3\\
&+& \Big(r_1 + r_0r_3 + r_1r_2 + r_1r_3 + r_2r_3 + r_0r_1r_2 + r_0r_1r_3 + r_0r_2r_3\Big)\binom{n}{2}\\
&+& \Big( r_0r_1 + r_0r_1r_2 \Big)\binom{n}{3} + \Big(r_0r_1 + r_0r_2 \Big)\binom{n}{2}\binom{n}{3} + \Big( r_0r_1 + r_0r_3\Big)\binom{n}{4}\\
\end{eqnarray*}
\begin{eqnarray}
\nonumber &+& \Big( r_0r_1 + r_0r_2 \Big)\binom{n}{5} + r_0\binom{n}{7}\Big)\\
\nonumber &+& i_5\Big( a_{3}^3 + a_3 + (r_1 + r_2)(a_{2}^3 + a_2 + a_{1}^3 + a_1 + a_{0}^7) + a_{1}^5 + a_{1}^3 + a_{0}^{11} + a_{0}^7
+ r_0r_1 + r_0r_2 \\
\nonumber &+& r_1r_2 + r_2r_3 + \big(r_0(a_{2}^3 + a_2 + a_{1}^3 + a_1 + a_{0}^7 + r_1 + r_2 + r_1r_2 +1) +r_1 + r_2 + r_3\big)\binom{n}{2}\\
\nonumber &+& (r_0r_1 + r_0r_3 + r_1)\binom{n}{3} + r_0(r_1 + r_2)\binom{n}{4} + r_0\binom{n}{6}\Big)\\
\nonumber &+& i_4\Big(r_0(a_{2}^3 + a_{2} + a_{1}^3 + a_{1}) + a_{0}^9 + r_0a_{0}^7 + (r_1 + r_2)a_{0}^5 + r_1 + r_2 + r_1r_2 + r_0r_1r_2\\
\nonumber &+& (r_0a_{0}^5 + r_0r_2)\binom{n}{2} + (r_0r_1 + r_0r_2)\binom{n}{3} + r_0\binom{n}{5}\Big)\\
\nonumber &+& i_3\Big(a_{2}^3 + a_2 + a_{1}^3 + a_1 + a_{0}^7 + a_{0}^5 + (r_0 + 1)(r_1 + r_2)\binom{n}{2} + r_0r_1 + r_0r_2 + r_1r_2 + r_2 + r_0\binom{n}{4}\Big)\\
\nonumber &+& i_2\Big(a_{0}^5 + a_{0} + r_0\Big(a_{0}^3 + a_0 + \binom{n}{3} \Big)\Big) + i_1\Big(a_{0}^3 + a_{0} + r_0\binom{n}{2}\Big) + i_0r_0,\\
\nonumber a_0 &=& a_{1}^2 + a_1 + r_1,\\
\label{eq:7coefficients} a_{0}^3 &=& a_{2}^2 + a_2 + r_2.\\
\nonumber a_{0}^5 &=& a_{3}^2 + a_3 + r_3.
\end{eqnarray}
Corollary~\ref{cor:period} implies that mod $2$ one has
\begin{eqnarray*}
\Big(\binom{n}{7},\binom{n}{6},\binom{n}{5},\binom{n}{4},\binom{n}{3},\binom{n}{2},\binom{n}{1}\Big) \equiv \begin{cases}
\begin{array}{lr}
(0,0,0,0,0,0,1) & \ \text{if} \ n \equiv 1 \pmod{8}\\
(0,0,0,0,1,1,1) & \ \text{if} \ n \equiv 3 \pmod{8}\\
(0,0,1,1,0,0,1) & \ \text{if} \ n \equiv 5 \pmod{8}\\
(1,1,1,1,1,1,1) & \ \text{if} \ n \equiv 7 \pmod{8}\\
\end{array},
\end{cases}
\end{eqnarray*}
and hence there are four cases to consider when computing the zeta functions of each of the curves~(\ref{eq:7coefficients}).
For $i_6 = 1$, the genus of each of the above curves is $58$\footnote[2]{See~\url{7CoefficientsGenus.m}.}.
Therefore, we again leave it as an open problem to determine their zeta functions.
\subsection{Reducing the prescribed coefficients problem to the prescribed traces problem}\label{sec:transformPCPPTP}
In this section we give formulae for all $I_2(n,t_1,t_2,t_3,t_4)$ in terms of the $F_2(n,t_1,t_2,t_3,t_4)$.
In the work~\cite{Koma}, Koma and Panario gave explicit formulae for the transform between $I_2(n,t_1,\ldots,t_l)$ and $F_2(n,t_1,\ldots,t_l)$
for $l = 4$ and $l = 5$; the extension to the $l = 6$ and $l = 7$ cases was also explained and a sketch of how to compute the
transform for arbitrary $l \ge 8$ was provided.
However, there is a small error in the application of the multinomial theorem which unfortunately invalidates several lemmas, propositions and
some of the formulae. Since in this work we have provided formulae for $F_2(n,t_1,t_2,t_3,t_4)$ we now provide the correct formulae for
$I_2(n,t_1,t_2,t_3,t_4)$.
The following lemma is simply a subcase of Lemma~\ref{lemma:multinomial}.
\begin{lemma}\label{lemma:multinomial2}
For each integer $d \ge 1$ and $f(x) \in {\mathbb F}_2[x]$,
\begin{enumerate}
\item $T_1(f^d) = d T_1(f)$
\item $T_2(f^d) = \binom{d}{2} T_1(f) + d T_2(f)$
\item $T_3(f^d) = \binom{d}{3} T_1(f) + d T_3(f)$
\item $T_4(f^d) = \binom{d}{4} T_1(f) + \binom{d}{2} T_2(f) + dT_4(f) + (d-2)\binom{d}{2} T_1(f)T_2(f)$
\end{enumerate}
\end{lemma}
It is the final term of Lemma~\ref{lemma:multinomial2} part (4) that is missing from~\cite[Prop. 2.1]{Koma}, which has coefficient $1$ for $d \equiv 3 \pmod{4}$.
Recall from Corollary~\ref{cor:period} that the featured binomial coefficients have maximal period $8$ and hence we need to consider the residue
classes of $d$ mod $8$.
For brevity we write $d \equiv a \, (8)$ to denote $d \equiv a \pmod{8}$ and similarly write $d \equiv a \, (4)$ to denote $d \equiv a \pmod{4}$.
As in~\cite{cattell} let ${\bf Irr}(n)$ denote the set of all irreducible polynomials of degree $n$ over ${\mathbb F}_2$,
and let $a \cdot {\bf Irr}(n)$ denote the multiset consisting of $a$ copies of ${\bf Irr}(n)$.
Following~\cite{cattell} and~\cite{yucasmullen}, applying Lemma~\ref{lemma:multinomial2} gives the following result.
\begin{lemma}\label{lemma:4coefftransform0}
For $n \ge 4$ we have
\begin{eqnarray}
\nonumber F_2(n,t_1,t_2,t_3,t_4) &=& \Big\vert \bigcup_{\beta \in {\mathbb F}_{2^n}, T_1(\beta) = t_1, T_2(\beta) = t_2, T_3(\beta) = t_3, T_4(\beta) = t_4} Min(\beta) \Big\vert\\
\nonumber &=& \Big\vert \bigcup_{d \mid n} \frac{n}{d} \big\{ f \in {\bf Irr}(\frac{n}{d}): dT_1(f) = t_1, \binom{d}{2}T_1(f) + d T_2(f) = t_2,
\binom{d}{3}T_1(f) + d T_3(f) = t_3\\
\nonumber && \binom{d}{4} T_1(f) + \binom{d}{2} T_2(f) + dT_4(f) + (d-2)\binom{d}{2} T_1(f)T_2(f) = t_4 \big\} \Big\vert \\
\nonumber &=& \Big\vert \bigcup_{d \mid n, \ d \equiv 0 \, (8)} \frac{n}{d} \big\{ f \in {\bf Irr}(\frac{n}{d}):
0 = t_1, 0 = t_2, 0 = t_3, 0 = t_4 \big\} \Big\vert \\
\nonumber &+& \Big\vert \bigcup_{d \mid n, \ d \equiv 1 \, (8)} \frac{n}{d} \big\{ f \in {\bf Irr}(\frac{n}{d}):
T_1(f) = t_1, T_2(f) = t_2, T_3(f) = t_3, T_4(f) = t_4 \big\} \Big\vert \\
\nonumber &+& \Big\vert \bigcup_{d \mid n, \ d \equiv 2 \, (8)} \frac{n}{d} \big\{ f \in {\bf Irr}(\frac{n}{d}):
0 = t_1, T_1(f) = t_2, 0 = t_3, T_2(f) = t_4 \big\} \Big\vert \\
\nonumber &+& \Big\vert \bigcup_{d \mid n, \ d \equiv 3 \, (8)} \frac{n}{d} \big\{ f \in {\bf Irr}(\frac{n}{d}):
T_1(f) = t_1, T_1(f) + T_2(f) = t_2, T_1(f) + T_3(f) = t_3,\\
\nonumber && T_2(f) + T_4(f) + T_1(f)T_2(f) = t_4 \big\} \Big\vert \\
\nonumber &+& \Big\vert \bigcup_{d \mid n, \ d \equiv 4 \, (8)} \frac{n}{d} \big\{ f \in {\bf Irr}(\frac{n}{d}):
0 = t_1, 0 = t_2, 0 = t_3, T_1(f) = t_4 \big\} \Big\vert \\
\nonumber &+& \Big\vert \bigcup_{d \mid n, \ d \equiv 5 \, (8)} \frac{n}{d} \big\{ f \in {\bf Irr}(\frac{n}{d}):
T_1(f) = t_1, T_2(f) = t_2, T_3(f) = t_3, T_1(f) + T_4(f) = t_4 \big\} \Big\vert \\
\nonumber &+& \Big\vert \bigcup_{d \mid n, \ d \equiv 6 \, (8)} \frac{n}{d} \big\{ f \in {\bf Irr}(\frac{n}{d}):
0 = t_1, T_1(f) = t_2, 0 = t_3, T_1(f) + T_2(f) = t_4 \big\} \Big\vert \\
\nonumber &+& \Big\vert \bigcup_{d \mid n, \ d \equiv 7 \, (8)} \frac{n}{d} \big\{ f \in {\bf Irr}(\frac{n}{d}):
T_1(f) = t_1, T_1(f) + T_2(f) = t_2, T_1(f) + T_3(f) = t_3,\\
\nonumber && T_1(f) + T_2(f) + T_4(f) + T_1(f)T_2(f) = t_4 \big\} \Big\vert.
\end{eqnarray}
\end{lemma}
Evaluating Lemma~\ref{lemma:4coefftransform0} at each set of traces and applying the same arguments as given in~\cite{Koma}
and~\cite[Chap. 5]{Komathesis} {\em mutatis mutandis} leads to Theorem~\ref{thm:transform0binary4}. The impact of the extra term of
Lemma~\ref{lemma:multinomial2} part (4) affects only parts $(2)$, $(4)$, $(6)$ and $(8)$ -- switching the $d \equiv 3$ and $d \equiv 7$ terms --
but we include all of them for the sake of completeness.
\begin{theorem}\label{thm:transform0binary4}
For $n \ge 4$ we have
\begin{enumerate}[label={(\arabic*)}]
\item $\displaystyle\begin{aligned}[t]
n I_2(n,1,1,1,0) &=& \sum_{\substack{d \mid n \\ d \equiv 1 \, (8)}} \mu(d) F_2(n/d,1,1,1,0) + \sum_{\substack{d \mid n \\ d \equiv 3 \, (8)}} \mu(d) F_2(n/d,1,0,0,0)\\
&+& \sum_{\substack{d \mid n \\ d \equiv 5 \, (8)}} \mu(d) F_2(n/d,1,1,1,1) + \sum_{\substack{d \mid n \\ d \equiv 7 \, (8)}} \mu(d) F_2(n/d,1,0,0,1),
\end{aligned}$
\item $\displaystyle\begin{aligned}[t]
n I_2(n,1,0,0,0) &=& \sum_{\substack{d \mid n \\ d \equiv 1 \, (8)}} \mu(d) F_2(n/d,1,0,0,0) + \sum_{\substack{d \mid n \\ d \equiv 3 \, (8)}} \mu(d) F_2(n/d,1,1,1,0)\\
&+& \sum_{\substack{d \mid n \\ d \equiv 5 \, (8)}} \mu(d) F_2(n/d,1,0,0,1) + \sum_{\substack{d \mid n \\ d \equiv 7 \, (8)}} \mu(d) F_2(n/d,1,1,1,1),
\end{aligned}$
\item $\displaystyle\begin{aligned}[t]
n I_2(n,1,1,1,1) &=& \sum_{\substack{d \mid n \\ d \equiv 1 \, (8)}} \mu(d) F_2(n/d,1,1,1,1) + \sum_{\substack{d \mid n \\ d \equiv 3 \, (8)}} \mu(d) F_2(n/d,1,0,0,1)\\
&+& \sum_{\substack{d \mid n \\ d \equiv 5 \, (8)}} \mu(d) F_2(n/d,1,1,1,0) + \sum_{\substack{d \mid n \\ d \equiv 7 \, (8)}} \mu(d) F_2(n/d,1,0,0,0),\\
\end{aligned}$
\item $\displaystyle\begin{aligned}[t]
n I_2(n,1,0,0,1) &=& \sum_{\substack{d \mid n \\ d \equiv 1 \, (8)}} \mu(d) F_2(n/d,1,0,0,1) + \sum_{\substack{d \mid n \\ d \equiv 3 \, (8)}} \mu(d) F_2(n/d,1,1,1,1)\\
&+& \sum_{\substack{d \mid n \\ d \equiv 5 \, (8)}} \mu(d) F_2(n/d,1,0,0,0) + \sum_{\substack{d \mid n \\ d \equiv 7 \, (8)}} \mu(d) F_2(n/d,1,1,1,0),\\
\end{aligned}$
\item $\displaystyle\begin{aligned}[t]
n I_2(n,1,1,0,0) &=& \sum_{\substack{d \mid n \\ d \equiv 1 \, (8)}} \mu(d) F_2(n/d,1,1,0,0) + \sum_{\substack{d \mid n \\ d \equiv 3 \, (8)}} \mu(d) F_2(n/d,1,0,1,0)\\
&+& \sum_{\substack{d \mid n \\ d \equiv 5 \, (8)}} \mu(d) F_2(n/d,1,1,0,1) + \sum_{\substack{d \mid n \\ d \equiv 7 \, (8)}} \mu(d) F_2(n/d,1,0,1,1),\\
\end{aligned}$
\item $\displaystyle\begin{aligned}[t]
n I_2(n,1,0,1,0) &=& \sum_{\substack{d \mid n \\ d \equiv 1 \, (8)}} \mu(d) F_2(n/d,1,0,1,0) + \sum_{\substack{d \mid n \\ d \equiv 3 \, (8)}} \mu(d) F_2(n/d,1,1,0,0)\\
&+& \sum_{\substack{d \mid n \\ d \equiv 5 \, (8)}} \mu(d) F_2(n/d,1,0,1,1) + \sum_{\substack{d \mid n \\ d \equiv 7 \, (8)}} \mu(d) F_2(n/d,1,1,0,1),\\
\end{aligned}$
\item $\displaystyle\begin{aligned}[t]
n I_2(n,1,1,0,1) &=& \sum_{\substack{d \mid n \\ d \equiv 1 \, (8)}} \mu(d) F_2(n/d,1,1,0,1) + \sum_{\substack{d \mid n \\ d \equiv 3 \, (8)}} \mu(d) F_2(n/d,1,0,1,1)\\
&+& \sum_{\substack{d \mid n \\ d \equiv 5 \, (8)}} \mu(d) F_2(n/d,1,1,0,0) + \sum_{\substack{d \mid n \\ d \equiv 7 \, (8)}} \mu(d) F_2(n/d,1,0,1,0),\\
\end{aligned}$
\item $\displaystyle\begin{aligned}[t]
n I_2(n,1,0,1,1) &=& \sum_{\substack{d \mid n \\ d \equiv 1 \, (8)}} \mu(d) F_2(n/d,1,0,1,1) + \sum_{\substack{d \mid n \\ d \equiv 3 \, (8)}} \mu(d) F_2(n/d,1,1,0,1)\\
&+& \sum_{\substack{d \mid n \\ d \equiv 5 \, (8)}} \mu(d) F_2(n/d,1,0,1,0) + \sum_{\substack{d \mid n \\ d \equiv 7 \, (8)}} \mu(d) F_2(n/d,1,1,0,0),\\
\end{aligned}$
\item $\displaystyle\begin{aligned}[t]
n I_2(n,0,0,1,0) &=& \sum_{\substack{d \mid n \\ d \ \text{odd}}} \mu(d) F_2(n/d,0,0,1,0),\\
\end{aligned}$
\item $\displaystyle\begin{aligned}[t]
n I_2(n,0,0,1,1) &=& \sum_{\substack{d \mid n \\ d \ \text{odd}}} \mu(d) F_2(n/d,0,0,1,1),\\
\end{aligned}$
\item $\displaystyle\begin{aligned}[t]
n I_2(n,0,1,1,1) &=& \sum_{\substack{d \mid n \\ d \equiv 1 \, (4)}} \mu(d) F_2(n/d,0,1,1,1) + \sum_{\substack{d \mid n \\ d \equiv 3 \, (4)}} \mu(d) F_2(n/d,0,1,1,0),\\
\end{aligned}$
\item $\displaystyle\begin{aligned}[t]
n I_2(n,0,1,1,0) &=& \sum_{\substack{d \mid n \\ d \equiv 1 \, (4)}} \mu(d) F_2(n/d,0,1,1,0) + \sum_{\substack{d \mid n \\ d \equiv 3 \, (4)}} \mu(d) F_2(n/d,0,1,1,1),\\
\end{aligned}$
\item $\displaystyle\begin{aligned}[t]
n I_2(n,0,0,0,0) &=& \sum_{\substack{d \mid n \\ d \ \text{odd}}} \mu(d) F_2(n/d,0,0,0,0) - \sum_{\substack{d \mid n, \ \frac{n}{d} \ \text{even} \\ d \ \text{odd}}} \mu(d) F_2(n/2d,0,0),\\
\end{aligned}$
\item $\displaystyle\begin{aligned}[t]
n I_2(n,0,0,0,1) &=& \sum_{\substack{d \mid n \\ d \ \text{odd}}} \mu(d) F_2(n/d,0,0,0,1) - \sum_{\substack{d \mid n, \ \frac{n}{d} \ \text{even} \\ d \ \text{odd}}} \mu(d) F_2(n/2d,0,1),\\
\end{aligned}$
\item $\displaystyle\begin{aligned}[t]
n I_2(n,0,1,0,0) &=& \sum_{\substack{d \mid n \\ d \equiv 1 \, (4)}} \mu(d) F_2(n/d,0,1,0,0) - \sum_{\substack{d \mid n, \ \frac{n}{d} \ \text{even} \\ d \equiv 1 \, (4)}} \mu(d) F_2(n/2d,1,0)\\
&+& \sum_{\substack{d \mid n \\ d \equiv 3 \, (4)}} \mu(d) F_2(n/d,0,1,0,1) - \sum_{\substack{d \mid n, \ \frac{n}{d} \ \text{even} \\ d \equiv 3 \, (4)}} \mu(d) F_2(n/2d,1,1),\\
\end{aligned}$
\item $\displaystyle\begin{aligned}[t]
n I_2(n,0,1,0,1) &=& \sum_{\substack{d \mid n \\ d \equiv 1 \, (4)}} \mu(d) F_2(n/d,0,1,0,1) - \sum_{\substack{d \mid n, \ \frac{n}{d} \ \text{even} \\ d \equiv 1 \, (4)}} \mu(d) F_2(n/2d,1,1)\\
&+& \sum_{\substack{d \mid n \\ d \equiv 3 \, (4)}} \mu(d) F_2(n/d,0,1,0,0) - \sum_{\substack{d \mid n, \ \frac{n}{d} \ \text{even} \\ d \equiv 3 \, (4)}} \mu(d) F_2(n/2d,1,0).\\
\end{aligned}$
\end{enumerate}
\end{theorem}
\subsubsection{Formulae for $l \ge 5$.}
The formulae for five coefficients follow from an argument analogous to that given in Lemma~\ref{lemma:4coefftransform0},
using the identity
\begin{equation}\label{eq:5coeffsT}
T_5(f^d) = \binom{d}{5}T_1(f) + dT_5(f) + (d-2)\binom{d}{2}\big( T_1(f)T_2(f) + T_1(f)T_3(f)\big).
\end{equation}
We omit the details since they are not part of the primary contribution of this work and follow easily.
Note that Eq.~(\ref{eq:5coeffsT}) is at odds with Eq.~(3) of~\cite{Koma} which claims that for any $l \ge 1$ and $d \ge 1$ the following holds mod $2$:
\begin{equation*}\label{eq:badmultinomial}
T_l(f^d) = \sum_{k \mid l} \binom{d}{k} T_{l/k}(f),
\end{equation*}
which contradicts Lemma~\ref{lemma:multinomial}. If combined with Lemma~\ref{lemma:multinomial}, the approach sketched in~\cite[\S4.2]{Koma} for
arbitrary $l$, which uses the generalised M\"obius inversion of Miers and Ruskey~\cite[Theorem 3.2]{miersruskey2}, can no doubt be developed into a
general algorithm for computing the required transforms. However, we leave describing such an algorithm, for $q=2$ and for arbitrary $q$ and $l$,
as an open problem.
\section{Curves and Explicit Formulae for $q = l = 3$}\label{sec:ternaryzetas}
In this section we determine curves and explicit formulae for all $F_3(n,t_1,t_2,t_3)$ for $n \ge 3$ but coprime to $3$ and for $F_3(n,0,0,0)$ for all
$n \ge 3$. We do this using the indirect and direct methods, and one other method, in order to highlight some redundancy
in the formulae arising from the indirect method, i.e., linear relations between powers of the roots of the featured characteristic polynomials of Frobenius.
Since the Galois groups of all the featured polynomials are soluble, eliminating such redundancy
is feasible, by identifying cancellations between their roots for various residues of $n$ mod $9$.
On the other hand the direct method, although producing curves with harder-to-compute zeta functions, seems to have no such redundancy.
This implies that there is a trade-off between the ease of computation and the compactness of the formulae.
The only previous result on such counts over ${\mathbb F}_3$ are due to Sharma~{\em et al. }~\cite{Sharma}, who gave approximations to $F_3(n,t_1,*,t_3)$ using
analogous simplifications to those used in~\cite{Koma,Komathesis}. Note that~\cite{Sharma} contains the transforms to $I_3(n,t_1,*,t_3)$ and
hence one can use the results of this section to compute these counts.
The reason we do not compute formulae for four coefficients is that the terms $T_4(\alpha)$ and $T_4(\beta)$ in Lemma~\ref{lem:T} part (4)
do not cancel mod $3$ and the method of introducing new variables in order to linearise $T_4(a_{0}^3 - a_{0} + r_0)$ fails as a result.
\noindent In order to express $F_3(n,t_1,t_2,t_3)$ compactly, we define the following eight polynomials:
\begin{eqnarray*}
\epsilon_{2,1} &=& X^2 - 3X + 3,\\
\epsilon_{2,2} &=& X^2 + 3X + 3,\\
\epsilon_{2,3} &=& X^2 + 3,\\
\epsilon_6 &=& X^6 + 3X^5 + 9X^4 + 15X^3 + 27X^2 + 27X + 27,\\
\epsilon_{12,1} &=& X^{12} - 3X^{11} + 3X^9 + 9X^8 - 45X^6 + 81X^4 + 81X^3 - 729X + 729,\\
\epsilon_{12,2} &=& X^{12} - 3X^{11} + 12X^9 - 18X^8 - 27X^7 + 117X^6 - 81X^5 - 162X^4 + 324X^3 - 729X + 729,\\
\epsilon_{12,3} &=& X^{12} - 3X^{11} + 9X^{10} - 15X^9 + 36X^8 - 54X^7 + 117X^6 - 162X^5 + 324X^4 - 405X^3 + 729X^2\\
&&- 729X + 729,\\
\epsilon_{12,4} &=& X^{12} + 6X^{11} + 18X^{10} + 39X^9 + 63X^8 + 81X^7 + 117X^6 + 243X^5 + 567X^4 + 1053X^3 + 1458X^2\\
&& + 1458X + 729.
\end{eqnarray*}
Observe that by~\cite[Prop. 1]{Stichtenoth}, $\epsilon_{6}, \epsilon_{12,1}, \epsilon_{12,2}, \epsilon_{12,3}$ and $\epsilon_{12,4}$ are not the
characteristic polynomials of the Frobenius endomorphism of supersingular abelian varieties.
As in~\S\ref{sec:binaryzetas} we use Lemma~\ref{lem:T} parts (1) to (3). Also as
in~\S\ref{sec:binaryzetas}, it is more efficient to compute linearised forms for $T_{2}(a)$ and $T_3(a)$ and to combine them as appropriate for
each $i \in \{1,\ldots,3^3-1\}$.
\subsection{Indirect method}
Setting $\vec{f} = (T_3,T_2,T_1)$, by the transform~(\ref{eq:Sminus1}) we have
\begin{equation}\label{eq:transform1char3}
{\tiny
\begin{bmatrix}
F_3(n,0,0,0)\\
F_3(n,1,0,0)\\
F_3(n,2,0,0)\\
F_3(n,0,1,0)\\
F_3(n,1,1,0)\\
F_3(n,2,1,0)\\
F_3(n,0,2,0)\\
F_3(n,1,2,0)\\
F_3(n,2,2,0)\\
F_3(n,0,0,1)\\
F_3(n,1,0,1)\\
F_3(n,2,0,1)\\
F_3(n,0,1,1)\\
F_3(n,1,1,1)\\
F_3(n,2,1,1)\\
F_3(n,0,2,1)\\
F_3(n,1,2,1)\\
F_3(n,2,2,1)\\
F_3(n,0,0,2)\\
F_3(n,1,0,2)\\
F_3(n,2,0,2)\\
F_3(n,0,1,2)\\
F_3(n,1,1,2)\\
F_3(n,2,1,2)\\
F_3(n,0,2,2)\\
F_3(n,1,2,2)\\
F_3(n,2,2,2)\\
\end{bmatrix}
=
\begin{bmatrix}
N(\vec{0})\\
N(\vec{1})\\
N(\vec{2})\\
N(\vec{3})\\
N(\vec{4})\\
N(\vec{5})\\
N(\vec{6})\\
N(\vec{7})\\
N(\vec{8})\\
N(\vec{9})\\
N(\vec{10})\\
N(\vec{11})\\
N(\vec{12})\\
N(\vec{13})\\
N(\vec{14})\\
N(\vec{15})\\
N(\vec{16})\\
N(\vec{17})\\
N(\vec{18})\\
N(\vec{19})\\
N(\vec{20})\\
N(\vec{21})\\
N(\vec{22})\\
N(\vec{23})\\
N(\vec{24})\\
N(\vec{25})\\
N(\vec{26})\\
\end{bmatrix}
=
\frac{1}{9}
\begin{bmatrix*}[r]
9 & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & - & -\\
0 & 1 & 0 & - & 1 & 0 & - & 1 & 0 & - & 1 & 0 & - & 1 & 0 & - & 1 & 0 & - & 1 & 0 & - & 1 & 0 & - & 1 & 0\\
0 & 0 & 1 & - & 0 & 1 & - & 0 & 1 & - & 0 & 1 & - & 0 & 1 & - & 0 & 1 & - & 0 & 1 & - & 0 & 1 & - & 0 & 1\\
0 & - & - & 1 & 1 & 1 & 0 & 0 & 0 & - & - & - & 1 & 1 & 1 & 0 & 0 & 0 & - & - & - & 1 & 1 & 1 & 0 & 0 & 0\\
0 & 1 & 0 & 1 & 0 & - & 0 & - & 1 & - & 1 & 0 & 1 & 0 & - & 0 & - & 1 & - & 1 & 0 & 1 & 0 & - & 0 & - & 1\\
0 & 0 & 1 & 1 & - & 0 & 0 & 1 & - & - & 0 & 1 & 1 & - & 0 & 0 & 1 & - & - & 0 & 1 & 1 & - & 0 & 0 & 1 & -\\
0 & - & - & 0 & 0 & 0 & 1 & 1 & 1 & - & - & - & 0 & 0 & 0 & 1 & 1 & 1 & - & - & - & 0 & 0 & 0 & 1 & 1 & 1\\
0 & 1 & 0 & 0 & - & 1 & 1 & 0 & - & - & 1 & 0 & 0 & - & 1 & 1 & 0 & - & - & 1 & 0 & 0 & - & 1 & 1 & 0 & -\\
0 & 0 & 1 & 0 & 1 & - & 1 & - & 0 & - & 0 & 1 & 0 & 1 & - & 1 & - & 0 & - & 0 & 1 & 0 & 1 & - & 1 & - & 0\\
0 & - & - & - & - & - & - & - & - & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 1 & 0 & - & 1 & 0 & - & 1 & 0 & 1 & 0 & - & 1 & 0 & - & 1 & 0 & - & 0 & - & 1 & 0 & - & 1 & 0 & - & 1\\
0 & 0 & 1 & - & 0 & 1 & - & 0 & 1 & 1 & - & 0 & 1 & - & 0 & 1 & - & 0 & 0 & 1 & - & 0 & 1 & - & 0 & 1 & -\\
0 & - & - & 1 & 1 & 1 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 & - & - & - & 0 & 0 & 0 & - & - & - & 1 & 1 & 1\\
0 & 1 & 0 & 1 & 0 & - & 0 & - & 1 & 1 & 0 & - & 0 & - & 1 & - & 1 & 0 & 0 & - & 1 & - & 1 & 0 & 1 & 0 & -\\
0 & 0 & 1 & 1 & - & 0 & 0 & 1 & - & 1 & - & 0 & 0 & 1 & - & - & 0 & 1 & 0 & 1 & - & - & 0 & 1 & 1 & - & 0\\
0 & - & - & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & - & - & - & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & - & - & -\\
0 & 1 & 0 & 0 & - & 1 & 1 & 0 & - & 1 & 0 & - & - & 1 & 0 & 0 & - & 1 & 0 & - & 1 & 1 & 0 & - & - & 1 & 0\\
0 & 0 & 1 & 0 & 1 & - & 1 & - & 0 & 1 & - & 0 & - & 0 & 1 & 0 & 1 & - & 0 & 1 & - & 1 & - & 0 & - & 0 & 1\\
0 & - & - & - & - & - & - & - & - & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\
0 & 1 & 0 & - & 1 & 0 & - & 1 & 0 & 0 & - & 1 & 0 & - & 1 & 0 & - & 1 & 1 & 0 & - & 1 & 0 & - & 1 & 0 & -\\
0 & 0 & 1 & - & 0 & 1 & - & 0 & 1 & 0 & 1 & - & 0 & 1 & - & 0 & 1 & - & 1 & - & 0 & 1 & - & 0 & 1 & - & 0\\
0 & - & - & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & - & - & - & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & - & - & -\\
0 & 1 & 0 & 1 & 0 & - & 0 & - & 1 & 0 & - & 1 & - & 1 & 0 & 1 & 0 & - & 1 & 0 & - & 0 & - & 1 & - & 1 & 0\\
0 & 0 & 1 & 1 & - & 0 & 0 & 1 & - & 0 & 1 & - & - & 0 & 1 & 1 & - & 0 & 1 & - & 0 & 0 & 1 & - & - & 0 & 1\\
0 & - & - & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 0 & 0 & 1 & 1 & 1 & - & - & - & 1 & 1 & 1 & - & - & - & 0 & 0 & 0\\
0 & 1 & 0 & 0 & - & 1 & 1 & 0 & - & 0 & - & 1 & 1 & 0 & - & - & 1 & 0 & 1 & 0 & - & - & 1 & 0 & 0 & - & 1\\
0 & 0 & 1 & 0 & 1 & - & 1 & - & 0 & 0 & 1 & - & 1 & - & 0 & - & 0 & 1 & 1 & - & 0 & - & 0 & 1 & 0 & 1 & -\\
\end{bmatrix*}
\begin{bmatrix}
V_1(\vec{0} \cdot \vec{f})\\
V_1(\vec{1} \cdot \vec{f})\\
V_1(\vec{2} \cdot \vec{f})\\
V_1(\vec{3} \cdot \vec{f})\\
V_1(\vec{4} \cdot \vec{f})\\
V_1(\vec{5} \cdot \vec{f})\\
V_1(\vec{6} \cdot \vec{f})\\
V_1(\vec{7} \cdot \vec{f})\\
V_1(\vec{8} \cdot \vec{f})\\
V_1(\vec{9} \cdot \vec{f})\\
V_1(\vec{10} \cdot \vec{f})\\
V_1(\vec{11} \cdot \vec{f})\\
V_1(\vec{12} \cdot \vec{f})\\
V_1(\vec{13} \cdot \vec{f})\\
V_1(\vec{14} \cdot \vec{f})\\
V_1(\vec{15} \cdot \vec{f})\\
V_1(\vec{16} \cdot \vec{f})\\
V_1(\vec{17} \cdot \vec{f})\\
V_1(\vec{18} \cdot \vec{f})\\
V_1(\vec{19} \cdot \vec{f})\\
V_1(\vec{20} \cdot \vec{f})\\
V_1(\vec{21} \cdot \vec{f})\\
V_1(\vec{22} \cdot \vec{f})\\
V_1(\vec{23} \cdot \vec{f})\\
V_1(\vec{24} \cdot \vec{f})\\
V_1(\vec{25} \cdot \vec{f})\\
V_1(\vec{26} \cdot \vec{f})\\
\end{bmatrix}.}
\end{equation}
By definition we have $V_1(\vec{0} \cdot \vec{f}) = 3^n$, while $V_1(\vec{1} \cdot \vec{f}) = V_1(T_1) = \#\{a \in {\mathbb F}_{3^n} \mid T_1(a) = 1\} = 3^{n-1}$,
as does $V_1(\vec{2} \cdot \vec{f}) = V_1(2T_1) = \#\{a \in {\mathbb F}_{3^n} \mid 2T_1(a) = 1\}$.
Note that for $r_0 \in {\mathbb F}_3$ one has $T_l(r_0) = \binom{n}{l}r_0$. To determine $V_1( \vec{i} \cdot \vec{f})$ for $3 \le i \le 26$, setting
$\alpha = a_{0}^3 - a_{0}$ and $\beta = -r_0$, and using Lemma~\ref{lem:T} parts (1) to (3) mod $3$ gives the
following\footnote[2]{See~\url{Ternary3Coefficients.mw}.}:
\begin{eqnarray}
\label{eq:ternary1} T_1(a_{0}^3 - a_0 + r_0) &=& T_1(r_0),\\
\label{eq:ternary2} T_2(a_{0}^3 - a_0 + r_0) &=& T_1\Big(a_{0}^4 - a_{0}^2 - r_0\binom{n}{2}/n\Big),\\
\label{eq:ternary3} T_3(a_{0}^3 - a_0 + r_0) &=& T_1\Big(a_{0}^7 - a_{0}^5 + r_0(n+1)(a_{0}^4 - a_{0}^2) + r_0\binom{n}{3}/n\Big).
\end{eqnarray}
For $3 \le i \le 26$ let $\vec{i} = (i_2,i_1,i_0)$. The curves we are interested in for $a$ of trace $r_0$ are
\begin{eqnarray}\label{eq:curvesternary}
a_{1}^3 - a_{1} + 1/n &=& i_2\Big(a_{0}^7 - a_{0}^5 - r_0(n+1)(a_{0}^4 - a_{0}^2)
- r_0\binom{n}{3}/n\Big) + i_1\Big(a_{0}^4 - a_{0}^2 + r_0\binom{n}{2}/n\Big) + i_0 r_0.
\end{eqnarray}
These curves have genus $3$ if $i_2 = 0$ and genus $6$ if $i_2 \neq 0$. Corollary~\ref{cor:period} implies that mod $3$ one has
\begin{eqnarray*}
\Big(\binom{n}{3},\binom{n}{2},\binom{n}{1}\Big) \equiv \begin{cases}
\begin{array}{lr}
(0,0,1) & \ \text{if} \ n \equiv 1 \pmod{9}\\
(0,1,-1) & \ \text{if} \ n \equiv 2 \pmod{9}\\
(1,0,1) & \ \text{if} \ n \equiv 4 \pmod{9}\\
(1,1,-1) & \ \text{if} \ n \equiv 5 \pmod{9}\\
(-1,0,1) & \ \text{if} \ n \equiv 7 \pmod{9}\\
(-1,1,-1) & \ \text{if} \ n \equiv 8 \pmod{9}\\
\end{array},
\end{cases}
\end{eqnarray*}
and hence there are six cases to consider when computing the zeta functions of each of the relevant curves.
Using Magma to compute the zeta functions of the curves~(\ref{eq:curvesternary}) and applying~(\ref{eq:transform1char3}) gives the following theorem\footnote[3]{See~\url{F3(n,t1,t2,t3).m}
for generation and counting code, and~\url{F3(n,t1,t2,t3)verify.mw} for verification of $F_3(n,0,0,0)$, which can easily be adapted for the other cases.},
where $\vec{v} = (\rho_n(\epsilon_{2,1}),\rho_n(\epsilon_{2,2}),\rho_n(\epsilon_{2,3}),\rho_n(\epsilon_{6}),\rho_n(\epsilon_{12,1}),\rho_n(\epsilon_{12,2}),\rho_n(\epsilon_{12,3}),\rho_n(\epsilon_{12,4}))$.
\begin{theorem}\label{thm:char3_3traces}
For $n \ge 3$ we have
{\small
\begin{flalign*}
F_3(n,0,0,0) &= 3^{n-3} - \frac{1}{81}
(-21, -15, -18, -8, -14, -14, -14, -8)\cdot \vec{v}
\ \ \text{if} \ n \equiv 1,2,4,5,7,8 \pmod{9}\\
F_3(n,1,0,0) &= 3^{n-3} - \frac{1}{81} \cdot \begin{cases}
\begin{array}{lr}
(-3, 3, 0, 4, -2, -2, -2, 4)\cdot\vec{v} & \ \ \text{if} \ n \equiv 1 \pmod{9}\\
( 3, 0, -3, 4, -2, 4, -2, -2)\cdot\vec{v} & \ \ \text{if} \ n \equiv 2 \pmod{9}\\
(-3, 3, 0, -2, 1, 1, 1, -2)\cdot\vec{v} & \ \ \text{if} \ n \equiv 4,7 \pmod{9}\\
( 3, 0, -3, -2, 1, -2, 1, 1)\cdot\vec{v} & \ \ \text{if} \ n \equiv 5,8 \pmod{9}\\
\end{array}
\end{cases}\\
F_3(n,2,0,0) &= 3^{n-3} - \frac{1}{81} \cdot \begin{cases}
\begin{array}{lr}
( -3, 3, 0, 4, -2, -2, -2, 4)\cdot\vec{v} & \ \text{if} \ n \equiv 1 \pmod{9}\\
(0, -3, 3, 4, -2, -2, 4, -2)\cdot\vec{v} & \ \text{if} \ n \equiv 2 \pmod{9}\\
( -3, 3, 0, -2, 1, 1, 1, -2 )\cdot\vec{v} & \ \text{if} \ n \equiv 4,7 \pmod{9}\\
( 0, -3, 3, -2, 1, 1, -2, 1)\cdot\vec{v} & \ \ \text{if} \ n \equiv 5,8 \pmod{9}\\
\end{array}
\end{cases}\\
F_3(n,0,1,0) &= 3^{n-3} - \frac{1}{81} \cdot \begin{cases}
\begin{array}{lr}
( 12, 9, 6, 4, -2, 4, -2, -2)\cdot\vec{v} & \ \text{if} \ n \equiv 1,4,7 \pmod{9}\\
( 9, 6, 12, -8, -14, 4, 10, 4)\cdot\vec{v} & \ \text{if} \ n \equiv 2,5,8 \pmod{9}\\
\end{array}
\end{cases}\\
F_3(n,1,1,0) &= 3^{n-3} - \frac{1}{81} \cdot \begin{cases}
\begin{array}{lr}
(3, 0, -3, -2, 1, -2, 1, 1)\cdot\vec{v} & \ \text{if} \ n \equiv 1,7 \pmod{9}\\
( -3, 3, 0, 4, -2, -2, -2, 4 )\cdot\vec{v} & \ \text{if} \ n \equiv 2 \pmod{9}\\
(3, 0, -3, 4, -2, 4, -2, -2)\cdot\vec{v} & \ \text{if} \ n \equiv 4 \pmod{9}\\
( -3, 3, 0, -2, 1, 1, 1, -2)\cdot\vec{v} & \ \text{if} \ n \equiv 5,8 \pmod{9}\\
\end{array}
\end{cases}\\
F_3(n,2,1,0) &= 3^{n-3} - \frac{1}{81} \cdot \begin{cases}
\begin{array}{lr}
( 3, 0, -3, -2, 1, -2, 1, 1 )\cdot\vec{v} & \ \text{if} \ n \equiv 1,5,7,8 \pmod{9}\\
( 3, 0, -3, 4, -2, 4, -2, -2)\cdot\vec{v} & \ \text{if} \ n \equiv 2,4 \pmod{9}\\
\end{array}
\end{cases}\\
F_3(n,0,2,0) &= 3^{n-3} - \frac{1}{81} \cdot \begin{cases}
\begin{array}{lr}
( 9, 6, 12, 4, -2, -2, 4, -2)\cdot\vec{v} & \ \text{if} \ n \equiv 1,4,7 \pmod{9}\\
( 12, 9, 6, -8, -14, 10, 4, 4)\cdot\vec{v} & \ \text{if} \ n \equiv 2,5,8 \pmod{9}\\
\end{array}
\end{cases}\\
F_3(n,1,2,0) &= 3^{n-3} - \frac{1}{81} \cdot \begin{cases}
\begin{array}{lr}
( 0, -3, 3, -2, 1, 1, -2, 1 )\cdot\vec{v} & \ \text{if} \ n \equiv 1,4,5,8 \pmod{9}\\
( 0, -3, 3, 4, -2, -2, 4, -2)\cdot\vec{v} & \ \text{if} \ n \equiv 2,7 \pmod{9}\\
\end{array}
\end{cases}\\
F_3(n,2,2,0) &= 3^{n-3} - \frac{1}{81} \cdot \begin{cases}
\begin{array}{lr}
( 0, -3, 3, -2, 1, 1, -2, 1)\cdot\vec{v} & \ \text{if} \ n \equiv 1,4 \pmod{9}\\
( -3, 3, 0, 4, -2, -2, -2, 4)\cdot\vec{v} & \ \text{if} \ n \equiv 2 \pmod{9}\\
( -3, 3, 0, -2, 1, 1, 1, -2)\cdot\vec{v} & \ \text{if} \ n \equiv 5,8 \pmod{9}\\
( 0, -3, 3, 4, -2, -2, 4, -2 )\cdot\vec{v} & \ \text{if} \ n \equiv 7 \pmod{9}\\
\end{array}
\end{cases}\\
F_3(n,0,0,1) &= 3^{n-3} - \frac{1}{81}
( -21, -15, -18, 4, 7, 7, 7, 4)\cdot\vec{v}
\ \ \text{if} \ n \equiv 1,2,4,5,7,8 \pmod{9}\\
\end{flalign*}
}
{\small
\begin{flalign*}
F_3(n,1,0,1) &= 3^{n-3} - \frac{1}{81} \cdot \begin{cases}
\begin{array}{lr}
(-3, 3, 0, -2, 1, 1, 1, -2 )\cdot\vec{v} & \ \text{if} \ n \equiv 1,7 \pmod{9}\\
(3, 0, -3, -2, 1, -2, 1, 1)\cdot\vec{v} & \ \text{if} \ n \equiv 2,5 \pmod{9}\\
( -3, 3, 0, 4, -2, -2, -2, 4)\cdot\vec{v} & \ \text{if} \ n \equiv 4 \pmod{9}\\
( 3, 0, -3, 4, -2, 4, -2, -2 )\cdot\vec{v} & \ \text{if} \ n \equiv 8 \pmod{9}\\
\end{array}
\end{cases}\\
F_3(n,2,0,1) &= 3^{n-3} - \frac{1}{81} \cdot \begin{cases}
\begin{array}{lr}
( -3, 3, 0, -2, 1, 1, 1, -2)\cdot\vec{v} & \ \text{if} \ n \equiv 1,4 \pmod{9}\\
( 0, -3, 3, -2, 1, 1, -2, 1 )\cdot\vec{v} & \ \text{if} \ n \equiv 2,8 \pmod{9}\\
( 0, -3, 3, 4, -2, -2, 4, -2 )\cdot\vec{v} & \ \text{if} \ n \equiv 5 \pmod{9}\\
(-3, 3, 0, 4, -2, -2, -2, 4)\cdot\vec{v} & \ \text{if} \ n \equiv 7 \pmod{9}\\
\end{array}
\end{cases}\\
F_3(n,0,1,1) &= 3^{n-3} - \frac{1}{81} \cdot \begin{cases}
\begin{array}{lr}
( 12, 9, 6, -2, 1, -2, 1, 1 )\cdot\vec{v} & \ \text{if} \ n \equiv 1,4,7 \pmod{9}\\
(9, 6, 12, 4, 7, -2, -5, -2)\cdot\vec{v} & \ \text{if} \ n \equiv 2,5,8 \pmod{9}\\
\end{array}
\end{cases}\\
F_3(n,1,1,1) &= 3^{n-3} - \frac{1}{81} \cdot \begin{cases}
\begin{array}{lr}
( 3, 0, -3, -2, 1, -2, 1, 1)\cdot\vec{v} & \ \text{if} \ n \equiv 1,4 \pmod{9}\\
(-3, 3, 0, -2, 1, 1, 1, -2 )\cdot\vec{v} & \ \text{if} \ n \equiv 2,5 \pmod{9}\\
(3, 0, -3, 4, -2, 4, -2, -2)\cdot\vec{v} & \ \text{if} \ n \equiv 7 \pmod{9}\\
(-3, 3, 0, 4, -2, -2, -2, 4)\cdot\vec{v} & \ \text{if} \ n \equiv 8 \pmod{9}\\
\end{array}
\end{cases}\\
F_3(n,2,1,1) &= 3^{n-3} - \frac{1}{81} \cdot \begin{cases}
\begin{array}{lr}
(3, 0, -3, 4, -2, 4, -2, -2)\cdot\vec{v} & \ \text{if} \ n \equiv 1,5 \pmod{9}\\
( 3, 0, -3, -2, 1, -2, 1, 1)\cdot\vec{v} & \ \text{if} \ n \equiv 2,4,7,8 \pmod{9}\\
\end{array}
\end{cases}\\
F_3(n,0,2,1) &= 3^{n-3} - \frac{1}{81} \cdot \begin{cases}
\begin{array}{lr}
( 9, 6, 12, -2, 1, 1, -2, 1)\cdot\vec{v} & \ \text{if} \ n \equiv 1,4,7 \pmod{9}\\
(12, 9, 6, 4, 7, -5, -2, -2)\cdot\vec{v} & \ \text{if} \ n \equiv 2,5,8 \pmod{9}\\
\end{array}
\end{cases}\\
F_3(n,1,2,1) &= 3^{n-3} - \frac{1}{81} \cdot \begin{cases}
\begin{array}{lr}
(0, -3, 3, 4, -2, -2, 4, -2)\cdot\vec{v} & \ \text{if} \ n \equiv 1,8 \pmod{9}\\
( 0, -3, 3, -2, 1, 1, -2, 1)\cdot\vec{v} & \ \text{if} \ n \equiv 2,4,5,7 \pmod{9}\\
\end{array}
\end{cases}\\
F_3(n,2,2,1) &= 3^{n-3} - \frac{1}{81} \cdot \begin{cases}
\begin{array}{lr}
( 0, -3, 3, -2, 1, 1, -2, 1 )\cdot\vec{v} & \ \text{if} \ n \equiv 1,7 \pmod{9}\\
( -3, 3, 0, -2, 1, 1, 1, -2 )\cdot\vec{v} & \ \text{if} \ n \equiv 2,8 \pmod{9}\\
( 0, -3, 3, 4, -2, -2, 4, -2 )\cdot\vec{v} & \ \text{if} \ n \equiv 4 \pmod{9}\\
( -3, 3, 0, 4, -2, -2, -2, 4 )\cdot\vec{v} & \ \text{if} \ n \equiv 5 \pmod{9}\\
\end{array}
\end{cases}\\
F_3(n,0,0,2) &= 3^{n-3} - \frac{1}{81}
( -21, -15, -18, 4, 7, 7, 7, 4 )\cdot \vec{v}
\ \ \text{if} \ n \equiv 1,2,4,5,7,8 \pmod{9}\\
F_3(n,1,0,2) &= 3^{n-3} - \frac{1}{81} \cdot \begin{cases}
\begin{array}{lr}
( -3, 3, 0, -2, 1, 1, 1, -2)\cdot\vec{v} & \ \text{if} \ n \equiv 1,4 \pmod{9}\\
( 3, 0, -3, -2, 1, -2, 1, 1 )\cdot\vec{v} & \ \text{if} \ n \equiv 2,8 \pmod{9}\\
( 3, 0, -3, 4, -2, 4, -2, -2 )\cdot\vec{v} & \ \text{if} \ n \equiv 5 \pmod{9}\\
( -3, 3, 0, 4, -2, -2, -2, 4 )\cdot\vec{v} & \ \text{if} \ n \equiv 7 \pmod{9}\\
\end{array}
\end{cases}\\
F_3(n,2,0,2) &= 3^{n-3} - \frac{1}{81} \cdot \begin{cases}
\begin{array}{lr}
( -3, 3, 0, -2, 1, 1, 1, -2 )\cdot\vec{v} & \ \text{if} \ n \equiv 1,7 \pmod{9}\\
( 0, -3, 3, -2, 1, 1, -2, 1)\cdot\vec{v} & \ \text{if} \ n \equiv 2,5 \pmod{9}\\
( -3, 3, 0, 4, -2, -2, -2, 4)\cdot\vec{v} & \ \text{if} \ n \equiv 4 \pmod{9}\\
( 0, -3, 3, 4, -2, -2, 4, -2 )\cdot\vec{v} & \ \text{if} \ n \equiv 8 \pmod{9}\\
\end{array}
\end{cases}\\
F_3(n,0,1,2) &= 3^{n-3} - \frac{1}{81} \cdot \begin{cases}
\begin{array}{lr}
( 12, 9, 6, -2, 1, -2, 1, 1 )\cdot\vec{v} & \ \text{if} \ n \equiv 1,4,7 \pmod{9}\\
( 9, 6, 12, 4, 7, -2, -5, -2)\cdot\vec{v} & \ \text{if} \ n \equiv 2,5,8 \pmod{9}\\
\end{array}
\end{cases}\\
F_3(n,1,1,2) &= 3^{n-3} - \frac{1}{81} \cdot \begin{cases}
\begin{array}{lr}
( 3, 0, -3, 4, -2, 4, -2, -2)\cdot\vec{v} & \ \text{if} \ n \equiv 1 \pmod{9}\\
(-3, 3, 0, -2, 1, 1, 1, -2)\cdot\vec{v} & \ \text{if} \ n \equiv 2,8 \pmod{9}\\
(3, 0, -3, -2, 1, -2, 1, 1)\cdot\vec{v} & \ \text{if} \ n \equiv 4,7 \pmod{9}\\
( -3, 3, 0, 4, -2, -2, -2, 4 )\cdot\vec{v} & \ \text{if} \ n \equiv 5 \pmod{9}\\
\end{array}
\end{cases}\\
F_3(n,2,1,2) &= 3^{n-3} - \frac{1}{81} \cdot \begin{cases}
\begin{array}{lr}
( 3, 0, -3, -2, 1, -2, 1, 1 )\cdot\vec{v} & \ \text{if} \ n \equiv 1,2,4,5 \pmod{9}\\
( 3, 0, -3, 4, -2, 4, -2, -2 )\cdot\vec{v} & \ \text{if} \ n \equiv 7,8 \pmod{9}\\
\end{array}
\end{cases}\\
F_3(n,0,2,2) &= 3^{n-3} - \frac{1}{81} \cdot \begin{cases}
\begin{array}{lr}
( 9, 6, 12, -2, 1, 1, -2, 1 )\cdot\vec{v} & \ \text{if} \ n \equiv 1,4,7 \pmod{9}\\
(12, 9, 6, 4, 7, -5, -2, -2)\cdot\vec{v} & \ \text{if} \ n \equiv 2,5,8 \pmod{9}\\
\end{array}
\end{cases}\\
\end{flalign*}
}
{\small
\begin{flalign*}
F_3(n,1,2,2) &= 3^{n-3} - \frac{1}{81} \cdot \begin{cases}
\begin{array}{lr}
( 0, -3, 3, -2, 1, 1, -2, 1 )\cdot\vec{v} & \ \text{if} \ n \equiv 1,2,7,8 \pmod{9}\\
( 0, -3, 3, 4, -2, -2, 4, -2 )\cdot\vec{v} & \ \text{if} \ n \equiv 4,5 \pmod{9}\\
\end{array}
\end{cases}\\
F_3(n,2,2,2) &= 3^{n-3} - \frac{1}{81} \cdot \begin{cases}
\begin{array}{lr}
( 0, -3, 3, 4, -2, -2, 4, -2 )\cdot\vec{v} & \ \text{if} \ n \equiv 1 \pmod{9}\\
( -3, 3, 0, -2, 1, 1, 1, -2)\cdot\vec{v} & \ \text{if} \ n \equiv 2,5 \pmod{9}\\
( 0, -3, 3, -2, 1, 1, -2, 1 )\cdot\vec{v} & \ \text{if} \ n \equiv 4,7 \pmod{9}\\
( -3, 3, 0, 4, -2, -2, -2, 4 )\cdot\vec{v} & \ \text{if} \ n \equiv 8 \pmod{9}\\
\end{array}
\end{cases}\\
\end{flalign*}
}
\end{theorem}
\subsubsection{Less redundant formulae.}\label{sec:lessredundant}
An alternative and more representationally efficient way to evaluate these formulae is to write:
\begin{eqnarray}
\nonumber F_3(n,t_1,t_2,t_3) &=& \frac{1}{3}\#\{a_0 \in {\mathbb F}_{3^n} \mid T_1\big(a_{0}^4 - a_{0}^2
- t_1\binom{n}{2}/n^2\big) = t_2, \\
\label{eq:altF3t1t2t3} && \ \ T_1\big( a_{0}^7 - a_{0}^5 + t_1(n+1)(a_{0}^4 - a_{0}^2)/n
+ t_1\binom{n}{3}/n^2\big) = t_3\},
\end{eqnarray}
and then for each $t_1$ apply the indirect method to compute all nine $F_3(n,t_1,t_2,t_3)$ simultaneously.
In particular, we instead let $\vec{f} = (a_{0}^7 - a_{0}^5 + t_1(n+1)(a_{0}^4 - a_{0}^2)/n
+ t_1\binom{n}{3}/n^2, a_{0}^4 - a_{0}^2 - t_1\binom{n}{2}/n^2)$.
Since there are now fewer summands, the total degree of the expressions is $84$ in the worst case: for $n \ge 3$ and coprime to $3$ we have
\[
F_3(n,0,0,0) = 3^{n-3} + \frac{1}{27}\big(3\rho_n(\epsilon_{2,1}) + \rho_n(\epsilon_{2,2}) + 2\rho_n(\epsilon_{2,3}) + 2\rho_n(\epsilon_{12,1}) + 2\rho_n(\epsilon_{12,2})+ 2\rho_n(\epsilon_{12,3}) \big).
\]
Comparing this with Theorem~\ref{thm:char3_3traces} implies the identity, valid for all $n \equiv 1,2,4,5,7,8 \pmod{9}$:
\begin{equation*}\label{rootidentity}
3\big(\rho_n(\epsilon_{2,1}) + \rho_n(\epsilon_{2,2}) + \rho_n(\epsilon_{2,3})\big) + 2\big(\rho_n(\epsilon_{6}) + \rho_n(\epsilon_{12,1}) + \rho_n(\epsilon_{12,2}) + \rho_n(\epsilon_{12,3}) +\rho_n(\epsilon_{12,4})\big) = 0.
\end{equation*}
This is therefore an example of a linear relation between the powers of the roots
of the featured characteristic polynomials of Frobenius.
Furthermore, since the roots of $\epsilon_{2,1},\epsilon_{2,2},\epsilon_{2,3}$ are $12$-th roots of unity, one can check that
\begin{equation*}
\rho_n(\epsilon_{2,1}) + \rho_n(\epsilon_{2,2}) + \rho_n(\epsilon_{2,3}) = \begin{cases}
\ 6 \cdot 3^{n/2} \ \text{if} \ n \equiv 0 \pmod{12}\\
-6 \cdot 3^{n/2} \ \text{if} \ n \equiv 6 \pmod{12}\\
0 \ \text{otherwise}.
\end{cases}
\end{equation*}
We therefore deduce that for all $n \equiv 1,2,4,5,7,8 \pmod{9}$, we have
\[
\rho_n(\epsilon_{6}) + \rho_n(\epsilon_{12,1}) + \rho_n(\epsilon_{12,2}) + \rho_n(\epsilon_{12,3}) +\rho_n(\epsilon_{12,4}) = 0,
\]
which does not seem obvious from the polynomials themselves.
\subsection{Direct method}\label{subsec:directF3}
For $n \ge 3$ and coprime to $3$ applying Equations~(\ref{eq:ternary1}) to~(\ref{eq:ternary3}) we have
\begin{eqnarray}
\nonumber F_3(n,t_1,t_2,t_3) &=& \#\{a \in {\mathbb F}_{3^n} \mid T_1(a) = t_1, \ T_2(a) = t_2, \ T_3(a) = t_3 \}\\
\nonumber &=& \frac{1}{3}\#\{a_0 \in {\mathbb F}_{3^n} \mid T_2(a_{0}^3 - a_{0} + t_1/\overline{n}) = t_2, \ T_3(a_{0}^3 - a_{0} + t_1/n) = t_3\}\\
\nonumber &=& \frac{1}{3^3}\#\{(a_0,a_1,a_2) \in ({\mathbb F}_{3^n})^3 \mid a_{1}^3 - a_{1} + t_2/n = a_{0}^4 - a_{0}^2
- t_1\binom{n}{2}/n^2,\\
\label{eq:directF3t1t2t3} && a_{2}^3 - a_{2} + t_3/n = a_{0}^7 - a_{0}^5 + t_1(n+1)(a_{0}^4 - a_{0}^2)/n +
t_1\binom{n}{3}/n^2\}.
\end{eqnarray}
These curves are all absolutely irreducible and of genus $21$, which is much less than half of the total degree of each expression in Theorem~\ref{thm:char3_3traces} and precisely half the degree of the worst case of the alternative indirect method of~\S\ref{sec:lessredundant}.
Based on some preliminary experiments, Magma can compute the zeta functions of the curves~(\ref{eq:directF3t1t2t3}) in a
matter of days.
However, in order to compute these functions more efficiently, since there are only two linear conditions one can apply~\cite[Lemma 6]{AGGMY}.
In particular, we have
\[
F_3(n,t_1,t_2,t_3) = \frac{1}{9}\big( V( (0,1)\cdot \vec{f}) + \sum_{\alpha \in {\mathbb F}_3} V( (1,\alpha) \cdot \vec{f}) - 3^n),
\]
where $\vec{f} = (a_{0}^7 - a_{0}^5 + t_1(n+1)(a_{0}^4 - a_{0}^2)/n
+ t_1\binom{n}{3}/n^2, a_{0}^4 - a_{0}^2 - t_1\binom{n}{2}/n^2)$.
Since this uses the zero count $V(\vec{i} \cdot \vec{f})$ the resulting formula for $F_3(n,0,0,0)$ is valid for all $n \ge 3$.
This leads to the following refinement of Theorem~\ref{thm:char3_3traces}.
\begin{theorem}\label{thm:char3_3traces_new}
For $n \ge 3$ we have
{\small
\begin{flalign*}
F_3(n,0,0,0) &= 3^{n-3} - \frac{1}{27}
(0, 2, 1, 2, 0, 0, 0, 2)\cdot \vec{v}
\ \ \text{for all} \ n \ge 3\\
F_3(n,1,0,0) &= 3^{n-3} - \frac{1}{27} \cdot \begin{cases}
\begin{array}{lr}
( 0, 2, 1, 2, 0, 0, 0, 2)\cdot\vec{v} & \ \ \text{if} \ n \equiv 1 \pmod{9}\\
( 2, 1, 0, 2, 0, 2, 0, 0)\cdot\vec{v} & \ \ \text{if} \ n \equiv 2 \pmod{9}\\
( 0, 2, 1, 0, 1, 1, 1, 0)\cdot\vec{v} & \ \ \text{if} \ n \equiv 4,7 \pmod{9}\\
( 2, 1, 0, 0, 1, 0, 1, 1)\cdot\vec{v} & \ \ \text{if} \ n \equiv 5,8 \pmod{9}\\
\end{array}
\end{cases}\\
F_3(n,2,0,0) &= 3^{n-3} - \frac{1}{27} \cdot \begin{cases}
\begin{array}{lr}
( 0, 2, 1, 2, 0, 0, 0, 2)\cdot\vec{v} & \ \text{if} \ n \equiv 1 \pmod{9}\\
( 1, 0, 2, 2, 0, 0, 2, 0)\cdot\vec{v} & \ \text{if} \ n \equiv 2 \pmod{9}\\
( 0, 2, 1, 0, 1, 1, 1, 0 )\cdot\vec{v} & \ \text{if} \ n \equiv 4,7 \pmod{9}\\
( 1, 0, 2, 0, 1, 1, 0, 1)\cdot\vec{v} & \ \ \text{if} \ n \equiv 5,8 \pmod{9}\\
\end{array}
\end{cases}\\
F_3(n,0,1,0) &= 3^{n-3} - \frac{1}{27} \cdot \begin{cases}
\begin{array}{lr}
( 2, 1, 0, 2, 0, 2, 0, 0 )\cdot\vec{v} & \ \text{if} \ n \equiv 1,4,7 \pmod{9}\\
( 1, 0, 2, 2, 0, 0, 2, 0 )\cdot\vec{v} & \ \text{if} \ n \equiv 2,5,8 \pmod{9}\\
\end{array}
\end{cases}\\
F_3(n,1,1,0) &= 3^{n-3} - \frac{1}{27} \cdot \begin{cases}
\begin{array}{lr}
( 2, 1, 0, 0, 1, 0, 1, 1 )\cdot\vec{v} & \ \text{if} \ n \equiv 1,7 \pmod{9}\\
( 0, 2, 1, 2, 0, 0, 0, 2)\cdot\vec{v} & \ \text{if} \ n \equiv 2 \pmod{9}\\
( 2, 1, 0, 2, 0, 2, 0, 0 )\cdot\vec{v} & \ \text{if} \ n \equiv 4 \pmod{9}\\
( 0, 2, 1, 0, 1, 1, 1, 0)\cdot\vec{v} & \ \text{if} \ n \equiv 5,8 \pmod{9}\\
\end{array}
\end{cases}\\
F_3(n,2,1,0) &= 3^{n-3} - \frac{1}{27} \cdot \begin{cases}
\begin{array}{lr}
( 2, 1, 0, 0, 1, 0, 1, 1)\cdot\vec{v} & \ \text{if} \ n \equiv 1,5,7,8 \pmod{9}\\
( 2, 1, 0, 2, 0, 2, 0, 0 )\cdot\vec{v} & \ \text{if} \ n \equiv 2,4 \pmod{9}\\
\end{array}
\end{cases}\\
F_3(n,0,2,0) &= 3^{n-3} - \frac{1}{27} \cdot \begin{cases}
\begin{array}{lr}
( 1, 0, 2, 2, 0, 0, 2, 0 )\cdot\vec{v} & \ \text{if} \ n \equiv 1,4,7 \pmod{9}\\
( 2, 1, 0, 2, 0, 2, 0, 0)\cdot\vec{v} & \ \text{if} \ n \equiv 2,5,8 \pmod{9}\\
\end{array}
\end{cases}\\
F_3(n,1,2,0) &= 3^{n-3} - \frac{1}{27} \cdot \begin{cases}
\begin{array}{lr}
( 1, 0, 2, 0, 1, 1, 0, 1 )\cdot\vec{v} & \ \text{if} \ n \equiv 1,4,5,8 \pmod{9}\\
( 1, 0, 2, 2, 0, 0, 2, 0 )\cdot\vec{v} & \ \text{if} \ n \equiv 2,7 \pmod{9}\\
\end{array}
\end{cases}\\
F_3(n,2,2,0) &= 3^{n-3} - \frac{1}{27} \cdot \begin{cases}
\begin{array}{lr}
(1, 0, 2, 0, 1, 1, 0, 1 )\cdot\vec{v} & \ \text{if} \ n \equiv 1,4 \pmod{9}\\
( 0, 2, 1, 2, 0, 0, 0, 2 )\cdot\vec{v} & \ \text{if} \ n \equiv 2 \pmod{9}\\
( 0, 2, 1, 0, 1, 1, 1, 0)\cdot\vec{v} & \ \text{if} \ n \equiv 5,8 \pmod{9}\\
( 1, 0, 2, 2, 0, 0, 2, 0 )\cdot\vec{v} & \ \text{if} \ n \equiv 7 \pmod{9}\\
\end{array}
\end{cases}\\
F_3(n,0,0,1) &= 3^{n-3} - \frac{1}{27}
( 0, 2, 1, 0, 1, 1, 1, 0)\cdot\vec{v}
\ \ \text{if} \ n \equiv 1,2,4,5,7,8 \pmod{9}\\
F_3(n,1,0,1) &= 3^{n-3} - \frac{1}{27} \cdot \begin{cases}
\begin{array}{lr}
( 0, 2, 1, 0, 1, 1, 1, 0 )\cdot\vec{v} & \ \text{if} \ n \equiv 1,7 \pmod{9}\\
( 2, 1, 0, 0, 1, 0, 1, 1)\cdot\vec{v} & \ \text{if} \ n \equiv 2,5 \pmod{9}\\
( 0, 2, 1, 2, 0, 0, 0, 2)\cdot\vec{v} & \ \text{if} \ n \equiv 4 \pmod{9}\\
( 2, 1, 0, 2, 0, 2, 0, 0 )\cdot\vec{v} & \ \text{if} \ n \equiv 8 \pmod{9}\\
\end{array}
\end{cases}\\
F_3(n,2,0,1) &= 3^{n-3} - \frac{1}{27} \cdot \begin{cases}
\begin{array}{lr}
( 0, 2, 1, 0, 1, 1, 1, 0 )\cdot\vec{v} & \ \text{if} \ n \equiv 1,4 \pmod{9}\\
( 1, 0, 2, 0, 1, 1, 0, 1 )\cdot\vec{v} & \ \text{if} \ n \equiv 2,8 \pmod{9}\\
( 1, 0, 2, 2, 0, 0, 2, 0 )\cdot\vec{v} & \ \text{if} \ n \equiv 5 \pmod{9}\\
( 0, 2, 1, 2, 0, 0, 0, 2)\cdot\vec{v} & \ \text{if} \ n \equiv 7 \pmod{9}\\
\end{array}
\end{cases}\\
\end{flalign*}
}
{\small
\begin{flalign*}
F_3(n,0,1,1) &= 3^{n-3} - \frac{1}{27} \cdot \begin{cases}
\begin{array}{lr}
( 2, 1, 0, 0, 1, 0, 1, 1 )\cdot\vec{v} & \ \text{if} \ n \equiv 1,4,7 \pmod{9}\\
( 1, 0, 2, 0, 1, 1, 0, 1 )\cdot\vec{v} & \ \text{if} \ n \equiv 2,5,8 \pmod{9}\\
\end{array}
\end{cases}\\
F_3(n,1,1,1) &= 3^{n-3} - \frac{1}{27} \cdot \begin{cases}
\begin{array}{lr}
( 2, 1, 0, 0, 1, 0, 1, 1 )\cdot\vec{v} & \ \text{if} \ n \equiv 1,4 \pmod{9}\\
( 0, 2, 1, 0, 1, 1, 1, 0 )\cdot\vec{v} & \ \text{if} \ n \equiv 2,5 \pmod{9}\\
( 2, 1, 0, 2, 0, 2, 0, 0)\cdot\vec{v} & \ \text{if} \ n \equiv 7 \pmod{9}\\
( 0, 2, 1, 2, 0, 0, 0, 2 )\cdot\vec{v} & \ \text{if} \ n \equiv 8 \pmod{9}\\
\end{array}
\end{cases}\\
F_3(n,2,1,1) &= 3^{n-3} - \frac{1}{27} \cdot \begin{cases}
\begin{array}{lr}
( 2, 1, 0, 2, 0, 2, 0, 0)\cdot\vec{v} & \ \text{if} \ n \equiv 1,5 \pmod{9}\\
( 2, 1, 0, 0, 1, 0, 1, 1 )\cdot\vec{v} & \ \text{if} \ n \equiv 2,4,7,8 \pmod{9}\\
\end{array}
\end{cases}\\
F_3(n,0,2,1) &= 3^{n-3} - \frac{1}{27} \cdot \begin{cases}
\begin{array}{lr}
( 1, 0, 2, 0, 1, 1, 0, 1 )\cdot\vec{v} & \ \text{if} \ n \equiv 1,4,7 \pmod{9}\\
( 2, 1, 0, 0, 1, 0, 1, 1 )\cdot\vec{v} & \ \text{if} \ n \equiv 2,5,8 \pmod{9}\\
\end{array}
\end{cases}\\
F_3(n,1,2,1) &= 3^{n-3} - \frac{1}{27} \cdot \begin{cases}
\begin{array}{lr}
( 1, 0, 2, 2, 0, 0, 2, 0 )\cdot\vec{v} & \ \text{if} \ n \equiv 1,8 \pmod{9}\\
( 1, 0, 2, 0, 1, 1, 0, 1 )\cdot\vec{v} & \ \text{if} \ n \equiv 2,4,5,7 \pmod{9}\\
\end{array}
\end{cases}\\
F_3(n,2,2,1) &= 3^{n-3} - \frac{1}{27} \cdot \begin{cases}
\begin{array}{lr}
( 1, 0, 2, 0, 1, 1, 0, 1 )\cdot\vec{v} & \ \text{if} \ n \equiv 1,7 \pmod{9}\\
( 0, 2, 1, 0, 1, 1, 1, 0 )\cdot\vec{v} & \ \text{if} \ n \equiv 2,8 \pmod{9}\\
( 1, 0, 2, 2, 0, 0, 2, 0)\cdot\vec{v} & \ \text{if} \ n \equiv 4 \pmod{9}\\
( 0, 2, 1, 2, 0, 0, 0, 2 )\cdot\vec{v} & \ \text{if} \ n \equiv 5 \pmod{9}\\
\end{array}
\end{cases}\\
F_3(n,0,0,2) &= 3^{n-3} - \frac{1}{27}
( 0, 2, 1, 0, 1, 1, 1, 0 )\cdot \vec{v}
\ \ \text{if} \ n \equiv 1,2,4,5,7,8 \pmod{9}\\
F_3(n,1,0,2) &= 3^{n-3} - \frac{1}{27} \cdot \begin{cases}
\begin{array}{lr}
( 0, 2, 1, 0, 1, 1, 1, 0 )\cdot\vec{v} & \ \text{if} \ n \equiv 1,4 \pmod{9}\\
( 2, 1, 0, 0, 1, 0, 1, 1)\cdot\vec{v} & \ \text{if} \ n \equiv 2,8 \pmod{9}\\
( 2, 1, 0, 2, 0, 2, 0, 0 )\cdot\vec{v} & \ \text{if} \ n \equiv 5 \pmod{9}\\
( 0, 2, 1, 2, 0, 0, 0, 2)\cdot\vec{v} & \ \text{if} \ n \equiv 7 \pmod{9}\\
\end{array}
\end{cases}\\
F_3(n,2,0,2) &= 3^{n-3} - \frac{1}{27} \cdot \begin{cases}
\begin{array}{lr}
( 0, 2, 1, 0, 1, 1, 1, 0 )\cdot\vec{v} & \ \text{if} \ n \equiv 1,7 \pmod{9}\\
(1, 0, 2, 0, 1, 1, 0, 1 )\cdot\vec{v} & \ \text{if} \ n \equiv 2,5 \pmod{9}\\
( 0, 2, 1, 2, 0, 0, 0, 2 )\cdot\vec{v} & \ \text{if} \ n \equiv 4 \pmod{9}\\
( 1, 0, 2, 2, 0, 0, 2, 0)\cdot\vec{v} & \ \text{if} \ n \equiv 8 \pmod{9}\\
\end{array}
\end{cases}\\
F_3(n,0,1,2) &= 3^{n-3} - \frac{1}{27} \cdot \begin{cases}
\begin{array}{lr}
( 2, 1, 0, 0, 1, 0, 1, 1 )\cdot\vec{v} & \ \text{if} \ n \equiv 1,4,7 \pmod{9}\\
( 1, 0, 2, 0, 1, 1, 0, 1)\cdot\vec{v} & \ \text{if} \ n \equiv 2,5,8 \pmod{9}\\
\end{array}
\end{cases}\\
F_3(n,1,1,2) &= 3^{n-3} - \frac{1}{27} \cdot \begin{cases}
\begin{array}{lr}
( 2, 1, 0, 2, 0, 2, 0, 0 )\cdot\vec{v} & \ \text{if} \ n \equiv 1 \pmod{9}\\
( 0, 2, 1, 0, 1, 1, 1, 0 )\cdot\vec{v} & \ \text{if} \ n \equiv 2,8 \pmod{9}\\
( 2, 1, 0, 0, 1, 0, 1, 1)\cdot\vec{v} & \ \text{if} \ n \equiv 4,7 \pmod{9}\\
([ 0, 2, 1, 2, 0, 0, 0, 2 )\cdot\vec{v} & \ \text{if} \ n \equiv 5 \pmod{9}\\
\end{array}
\end{cases}\\
F_3(n,2,1,2) &= 3^{n-3} - \frac{1}{27} \cdot \begin{cases}
\begin{array}{lr}
( 2, 1, 0, 0, 1, 0, 1, 1 )\cdot\vec{v} & \ \text{if} \ n \equiv 1,2,4,5 \pmod{9}\\
( 2, 1, 0, 2, 0, 2, 0, 0 )\cdot\vec{v} & \ \text{if} \ n \equiv 7,8 \pmod{9}\\
\end{array}
\end{cases}\\
F_3(n,0,2,2) &= 3^{n-3} - \frac{1}{27} \cdot \begin{cases}
\begin{array}{lr}
( 1, 0, 2, 0, 1, 1, 0, 1 )\cdot\vec{v} & \ \text{if} \ n \equiv 1,4,7 \pmod{9}\\
( 2, 1, 0, 0, 1, 0, 1, 1 )\cdot\vec{v} & \ \text{if} \ n \equiv 2,5,8 \pmod{9}\\
\end{array}
\end{cases}\\
F_3(n,1,2,2) &= 3^{n-3} - \frac{1}{27} \cdot \begin{cases}
\begin{array}{lr}
( 1, 0, 2, 0, 1, 1, 0, 1 )\cdot\vec{v} & \ \text{if} \ n \equiv 1,2,7,8 \pmod{9}\\
( 1, 0, 2, 2, 0, 0, 2, 0 )\cdot\vec{v} & \ \text{if} \ n \equiv 4,5 \pmod{9}\\
\end{array}
\end{cases}\\
F_3(n,2,2,2) &= 3^{n-3} - \frac{1}{27} \cdot \begin{cases}
\begin{array}{lr}
( 1, 0, 2, 2, 0, 0, 2, 0)\cdot\vec{v} & \ \text{if} \ n \equiv 1 \pmod{9}\\
( 0, 2, 1, 0, 1, 1, 1, 0)\cdot\vec{v} & \ \text{if} \ n \equiv 2,5 \pmod{9}\\
( 1, 0, 2, 0, 1, 1, 0, 1 )\cdot\vec{v} & \ \text{if} \ n \equiv 4,7 \pmod{9}\\
( 0, 2, 1, 2, 0, 0, 0, 2 )\cdot\vec{v} & \ \text{if} \ n \equiv 8 \pmod{9}\\
\end{array}
\end{cases}\\
\end{flalign*}
}
\end{theorem}
Observe that the total degrees of the featured polynomials in Theorem~\ref{thm:char3_3traces_new} is always $42$, which means these are the
characteristic polynomials of Frobenius arising from the direct method and should have optimal representational efficiency.
If one compares for instance the formulae for $F_3(n,0,1,2)$ in Theorems~\ref{thm:char3_3traces} and~\ref{thm:char3_3traces_new}
one can further deduce that for all $n \equiv 2,5,8 \pmod{9}$, we have
$\rho_n(\epsilon_{6}) + \rho_n(\epsilon_{12,1}) = 0$ and $\rho_n(\epsilon_{12,2}) + \rho_n(\epsilon_{12,3}) +\rho_n(\epsilon_{12,4}) = 0$, which again are seemingly not
obvious from the polynomials alone. The three observed identities and possibly any others arising by such a comparison would probably
allow one to transform Theorem~\ref{thm:char3_3traces} into Theorem~\ref{thm:char3_3traces_new} without carrying out any zeta
function computations.
Conversely, although our goal was to compute formulae for $F_3(n,t_1,t_2,t_3)$, by using various approaches to do so one can also deduce identities
between the roots of the featured characteristic polynomials of Frobenius for residues of $n \bmod{9}$ without computing any of the roots
explicitly.
\section{Final Remarks and Open Problems}\label{sec:summary}
We have presented the first algorithmic approaches to solving the prescribed traces problem and therefore the prescribed coefficients problem.
While our main algorithm for $l < p$ is extremely simple to state and very efficient, the $l \ge p$ case in full generality remains open.
There are many properties and consequences of our approaches that have yet to be determined or explored.
We now present some open problems, in what may be considered to be in an approximately increasing order of interest.
\vspace{3mm}
\noindent {\bf Problem 1:} Provide an algorithm which for any $q = p^r$ and $l$ outputs the transforms between $I_q(n,t_1,\ldots,t_l)$ and $F_q(n,t_1,\ldots,t_l)$, and deduce its complexity.
\vspace{3mm}
\noindent {\bf Problem 2:} Compute the characteristic polynomials of Frobenius of the curves arising for $F_2(n,t_1,\ldots,t_6)$ and
$F_2(n,t_1,\ldots,t_7)$, for $n$ odd. Although it is feasible to compute these by brute force point counting, it would be preferable to employ a more
elegant approach, perhaps by computing the quotient curves arising from various curve automorphisms and then applying a theorem of Kani and
Rosen~\cite{kanirosen} to infer the decomposition of their Jacobians.
\vspace{3mm}
\noindent {\bf Problem 3:} Compute formulae for $F_2(n,t_1,\ldots,t_l)$ with $4 \le l \le 7$ for all $n \ge l$.
\vspace{3mm}
\noindent {\bf Problem 4:} More generally, for the main algorithm provide a method for solving the $n \equiv 0 \pmod{p}$ cases, for any $q \ge 2$.
Note that if solved then the formulae for $F_q(n,t_1,\ldots,t_l)$ for each of the $p$ residue classes of $n$ can be unified via Fourier analysis using the complex $p$-th roots of unity, as in~\cite{AGGMY}.
Similarly, if $l \ge p$ then as per Corollary~\ref{cor:period} the complex $p^{1+ \lfloor \log_{p} l \rfloor}$-th roots of unity can be employed.
However, it may be preferable to retain the residue class distinctions for simplicity (cf.~\cite[Prop. 4 and 5]{AGGMY}).
\vspace{3mm}
\noindent {\bf Problem 5:} Determine if it is possible to express $T_l(\alpha - \beta)$ in terms of lower degree traces, over ${\mathbb Z}$, for any or all
$l > 7$, so that one may extend the $q = 2$ approach.
\vspace{3mm}
\noindent {\bf Problem 6:} For the indirect method in the main algorithm when $q > p$, remove the reliance on Lemma~\ref{thm:extensionreduction}
so that the $\frac{q-1}{p-1}$ factor in the sum in~(\ref{eq:maintheorem}) in Theorem~\ref{thm:maintheorem} may be removed.
\vspace{3mm}
\noindent {\bf Problem 7:} When $l < p$ provide an algorithm to compute the zeta function of the curve~(\ref{eq:main_direct}) arising
from the direct method, or indeed of the fibre product~(\ref{eq:newmain}), which is more efficient for a single $F_q(n,t_1,\ldots,t_l)$ than
executing the indirect method. This is also desirable since the direct method seems to produce the most compact formulae.
\vspace{3mm}
\noindent {\bf Problem 8:} As is evident from the examples we have given for $q = 2$, there are often several ways to associate an affine curve
$C_{\vec{r},\overline{n}}$ to a given counting problem $F_q(n,t_{l_0},\ldots,t_{l_{m-1}})$ with differing numbers of auxiliary variables
$(a_0,\ldots,a_{s-1})$ with corresponding linear traces $\vec{r} = (r_0,\ldots,r_{s-1})$. Not only does this affect the complexity of determining
the relevant
characteristic values, but each such curve may lead to different (but necessarily numerically identical) formulae. One can associate a generating,
or zeta function to each prescribed traces problem, by letting
$N_n = q^m \cdot F_q(n,t_{l_0},\ldots,t_{l_{m-1}}) + 1$ and as usual defining
\[
Z( F_q(n,t_{l_0},\ldots,t_{l_{m-1}}); t) = \exp{\left( \sum_{n=1}^{\infty} \frac{N_n}{n} t^n \right)}.
\]
If $C_{\vec{r},\overline{n}}$ is absolutely irreducible for each $\vec{r}$ and assuming Problem 4 has been solved then for all $\overline{n} \in \{0,\ldots,p-1\}$ one has:
\begin{equation}\label{eq:zetafunction}
Z( F_q(n,t_{l_0},\ldots,t_{l_{m-1}}); t) = \frac{L_{\overline{n}}(t)^{1/q^s}}{(1-t)(1-qt)}, \ \text{for all} \ \ n \equiv \overline{n}\pmod{p} \
\ \text{with} \ \ n \ge l_{m-1},
\end{equation}
where $L_{\overline{n}}(t) = c_0 + c_1t + \cdots + c_{2g}t^{2g} \in {\mathbb Z}[t]$ with $c_0 = 1$ is the product of the characteristic polynomials of Frobenius
of the $q^s$ specialisations of $C_{\vec{r},\overline{n}}$.
Since $L_{\overline{n}}(t)$ and $s$ in Eq.~(\ref{eq:zetafunction}) are non-canonical, this is less than aesthetically pleasing in comparison
to the rational zeta functions of smooth projective curves. Is there a canonical way to associate such a curve, which also avoids all redundancy?
Perhaps one can associate a motivic zeta function to each $F_q(n,t_{l_0},\ldots,t_{l_{m-1}})$ \`a la~\cite{duSautoy1,duSautoy2}, which resolve
a similar non-canonicality issue when associating varieties to counting problems in the theory of $p$-groups and nilpotent groups?
\vspace{3mm}
\noindent {\bf Problem 9:} Regarding the general $l \ge p$ case, another natural question is whether or not it is possible to obviate the
failure of Newton's identities by working $p$-adically and in Galois rings, \`a la Fan and Han's refinement~\cite{fanhan} of Han's work
on Cohen's problem~\cite{han}? See also the exposition of Cohen~\cite{cohenexpo}.
If so, this could lead to an algorithm for solving the prescribed traces problem for any number of coefficients in any positions.
Assuming this can be done, for the sake of generality we make the following conjecture.
\begin{conjecture}\label{conj:generalconj}
For every $q = p^r$, $1 \le l_0 < \cdots < l_{m-1}$, $(t_{l_0},\ldots,t_{l_{m-1}}) \in ({\mathbb F}_q)^m$ and
$\overline{n} \in \{0,\ldots,p^{1 + \lfloor \log_p{l_{m-1}} \rfloor}-1\}$,
there exists $\omega_1,\ldots,\omega_N \in \overline{{\mathbb Z}}$, all of norm $\sqrt{q}$, $\upsilon_1,\ldots,\upsilon_N \in {\mathbb Z}$ and an integer $s \ge 0$ such that for
all $n \equiv \overline{n} \pmod{p^{1 + \lfloor \log_p{l_{m-1}} \rfloor}}$ with $n \ge l_{m-1}$ one has
\begin{equation*}\label{eq:mainconj}
F_q(n,t_{l_0},\ldots,t_{l_{m-1}}) = \frac{1}{q^m}\Big( q^n + \frac{1}{q^s}\sum_{i=1}^N \upsilon_{i} \alpha_{i}^n \Big) = q^{n-m} + O(q^{n/2}).
\end{equation*}
\end{conjecture}
Note that the sum is missing the factor $\frac{q-1}{p-1}$ from Theorem~\ref{thm:maintheorem}, which will be avoidable should Problem 6 be solved.
Also note that it may always be possible to set $s = 0$.
\vspace{3mm}
\noindent {\bf Problem 10:}
Should sufficiently general versions of Problems $1$, $4$ and $9$ be resolved positively and bounds
on the genera of the resulting curves be sufficiently small, then can one prove interesting existence results -- in the best case
with up to $\lfloor (1/2 - \epsilon)n \rfloor$ coefficients prescribed for any $\epsilon > 0$ -- when $n$ is sufficiently large for the main
term to dominate the error term in Conjecture~\ref{eq:mainconj}?
\section*{Acknowledgements}
This work was supported by the Swiss National Science Foundation via grant number 200021-156420. Some of the ideas were developed
while the author was visiting Steven Galbraith at The University of Auckland in April 2016; I would like to thank him for his excellent
hospitality and for some enlightening discussions on intersections. I would also like to thank: Wouter Castryck and Jan Tuitman for explaining to me
the state-of-the-art in $p$-adic point counting techniques for curves; Daniel Panario for his useful comments on an early draft of this work;
Alan Lauder for answering my questions regarding his and Wan's algorithm; Claus Diem and Benjamin Wesolowski for discussions; and Omran Ahmadi
for providing the translation of Lemma~\ref{thm:extensionreduction}. Finally, I would very much like to thank Thorsten Kleinjung for numerous
very helpful discussions and suggestions, and for acting as a sounding board throughout much of this work.
\bibliographystyle{plain}
| -389,254.8523 |
[
-1.865234375,
1.61328125
] | 18.553092 |
[
-2.859375,
0.73779296875,
-2.01171875,
-5.85546875,
-1.087890625,
8.1875
] |
[
3.212890625,
8.8828125,
2.54296875,
7.29296875
] | 725 | 25,036 |
[
-3.46875,
3.90234375
] | 50.163624 |
[
-5.81640625,
-4.0625,
-4.72265625,
-2.73046875,
1.8115234375,
13.078125
] | 1.656825 | 8.217409 | 14.155616 | 13.977909 |
[
1.8647780418395996
] | -229,908.893223 | 5.086755 | -381,228.503795 | 2.19549 | 6.244082 |
[
-2.220703125,
-3.2421875,
-3.427734375,
-4.98828125,
2.056640625,
11.7265625
] |
[
-5.75390625,
-2.134765625,
-1.896484375,
-1.474609375,
3.763671875,
4.578125
] | |
BkiUd1_xaKgQMdMtQloi
|
\section{Introduction}
\textit{Background}: Multi-agent systems (MAS) or agent-based models (ABM) have had a long history of statistical inference in quantitative finance research. In particular, they have been used to study phenomena leading to price formation and hence general market microstructure, such as: the law of supply and demand~\cite{Benzaquen2018}, game theory~\cite{ErevRoth2014}, order books~\cite{Huang2015}, high-frequency trading~\cite{Wah2013,Aloud2014}, cross-market structure~\cite{Xu2014}, quantitative easing~\cite{Westerhoff2008}, market regulatory impact~\cite{Boero2015}, or other exogenous effects~\cite{Gualdi2015}. Remarkably, financial MAS have highlighted over the years certain specific patterns that are proper to virtually almost all asset classes and time scales, called \textit{stylised facts}. These have since been an active topic of quantitative research~\cite{Lipski2013,Barde2015}, and can be grouped in three main neighbouring categories: i- distributions of price returns are non-gaussian~\cite{Cristelli2014,Cont2001,Potters2001} (they are asymmetric, negatively skewed, and platykurtic), ii- volatilities and volumes are clustered~\cite{Lipski2013} (large jumps in prices and volumes are more likely to be followed by the same), iii- price auto-correlations decay~\cite{Cont2001,Cont2005} (there is a loss of arbitrage opportunity). In particular, the second stylised fact has long-range implications on the dynamics of meta-orders, and comprises the \textit{square-root impact law}~\cite{Bouchaud2018} (growth in square-root of orders impact with traded volumes). Implicit consequences of these stylised facts have fed numerous discussions pertaining to the validity of market memory~\cite{Cont2005,Cristelli2014} and the extension of the efficient market hypothesis~\cite{Fama1970,Bera2015}.
\vspace{1mm}
\textit{Epistemology}: In the past, the epistemological pertinence of such models was sometimes put into question, because of the general conceptual challenge of framing realistic agents. Unlike other disciplines from hard science, financial MAS indeed had to justify their proper bottom-up approach to complex system inference~\cite{Gao2011} in the light of a specific challenge of their field, namely the difficulty to realistically model and emulate human agents. Such critics especially carried greater weight by the fact that previous generations of financial MAS relied on so-called zero-intelligence agents~\citep{Gode1993}: these would trade according to specific and imbedded rules of trading, thereby overshadowing certain dynamics proper to game and decision theory that are crucial to real market activity. Yet, if compared with other famous types of models used in quantitative finance, like econometrics~\cite{Greene2017}, MAS have two major advantages: i- they naturally display specific emergent phenomena proper to complex systems~\cite{Bouchaud2019}, and ii- they require fewer model assumptions (no gaussian distributions, no efficient market hypothesis~\cite{Fama1970,Bera2015}, etc.). As for their shortcomings, we can mention specific conceptual challenges pertaining to: i- modelling complex system heterogeneity~\cite{Chen2017}, ii- discretionary framing of certain model parameters (e.g. number of agents) apart from possible empiricism~\cite{Platt2018}.
\vspace{1mm}
\textit{Prospects}: These general conceptual and epistemological considerations have been upset in the past few years, by the notable progress of two fields. The first, namely machine learning and especially multi-agent reinforcement learning~\cite{Silver2018} (RL), has produced spectacular results that have far-reaching applications to other domains of interest to quantitative finance, like decision theory and game theory. The second, neuroscience and neurofinance especially, has benefitted from the wide use of brain-imaging devices~\cite{Eickhoff2018} and peer data curation~\cite{Smith2018}. On both ends, some sort of technological emergence within these two fields is of special interest and relevance to financial MAS research, in that machine learning can imbed results from the latter~\cite{Lefebvre2017,Palminteri2015}, and vice-versa~\cite{Duncan2018,Momennejad2017}. Among other machine learning applications to finance~\citep{Hu2019,Neuneier1997,Deng2017}, and in a way similar to what has been done for recent order book models~\citep{Spooner2018,Biondo2019,Sirignano2019}, next-generation MAS stock market simulators can now be designed, and outstrip the former epistemological considerations pertaining to agent design, with a whole new degree of realism in market microstructure emulation. More importantly, these can address issues never computationally studied before, such as the crucial issues of agent information and learning, central to price formation~\cite{Dodonova2018,Naik2018} and hence to all market activity.
\vspace{1mm}
\textit{Our study}: We have designed such a MAS stock market simulator, in which agents autonomously learn to do price forecasting and stock trading by reinforcement learning, via a centralised double-auction limit order book, and shown in a previous work~\cite{Lussange2019} its calibration performances with respect to real market data from the London Stock Exchange daily quotes between $2008$ and $2018$. We shall not review here its design in full details, but will recall its general architecture in Section II. Then our study will focus in Section III on agent learning: how can we gauge how heterogenous are the agent policies as compared to one another, and what are the trading characteristics of successful agents. Finally in Section IV, we will study the impact of such agent learning on the whole market at the mesoscale, and in particular with respect to herding or reflexivity effects when agents emulate the investments of the best (or worst) agent, and when increasing proportions of agents trade randomly as ``noise traders." As said, the greatest interest of financial MAS is to study and quantitatively gauge the impact of agent learning, which is at the heart of the price formation processes and hence of all market activity, and the motivation for multi-agent reinforcement learning for such a framework is motivated by the important parallels between reinforcement learning and decision processes in the brain~\cite{Dayan2008}.
\section{Model}
\textit{Architecture}: We here briefly sketch the general architecture of our MAS simulator. Our model relies on a number $I$ of economic agents autonomously trading a number $J$ of stocks, over a number $T$ of simulation time steps, where we set $T_{y}=286, T_{m}=21, T_{w}=5$ as the number of annual trading days. All $I$ agents and their individual parameters are initialised at $t=0$, with a portfolio made of specific stock holdings of value $A_{equity}^{i}(t)=\sum_{j=0}^J Q^{i,j}(t)P^{j}(t)$, where $Q^{i,j}(t)$ is the number of stocks $j$ of agent $i$ and $P^{j}(t)$ the market price of stock $j$, together with risk-free assets (e.g. a bank account) of value $A_{bonds}^{i}(t)$. Then all market prices are initialised at $P^{j}(t=0)=\pounds 100$, and as in other models~\cite{Franke2011,Chiarella2007}, the simulation generates $J$ time series $\mathcal{T}^{j}(t)$, which correspond to the fundamental values of the stocks. These are not fully known by the $I$ agents. Instead, each agent $i$ approximates the values $T^j(t)$ of stock $j$ according to a proprietary rule~\citep{Murray1994} of cointegration $\kappa ^{i,j} [ \mathcal{T}^{j}(t) ]=\mathcal{B}^{i,j}(t)$. The time series $\mathcal{B}^{i,j}(t)$ are hence the approximation of the fundamental values of stock $j$ over time $t$ according to agent $i$. Each agent thus relies on these two sources of information for its stock pricing strategy: one that is chartist, and one that is fundamental. At $t=0$, the agent are agnostic with respect to trading. They are then allowed to learn over the course of $1000$ time steps, after which their stock holdings and risk-free assets are reset back to their first value. The simulation and following results are hence applied to agents after this learning phase of $1000$ time steps.
\textit{Initialisation}: Let $\mathcal{U} ()$ and $\mathcal{U} \{ \}$ denote the continuous and discrete uniform distributions, respectively. Each agent is then initialised with the following parameters: a drawdown limit $l^{i} \sim \mathcal{U} (50 \%, 60\%)$ (which is the maximum year-to-date loss in net asset value below which the agent is set as bankrupt), a reflexivity parameter $\rho^{i} \sim \mathcal{U} (0, 100\%)$ (which gauges how fundamental or chartist the agent is via a weighted average of its price forecast), an investment horizon $\tau^{i} \sim \mathcal{U} \{T_w, 6T_m \}$ (which is the number of time steps after which the agent liquidates its position), a trading window $w^{i} \sim \mathcal{U} \{T_w, \tau^{i} \}$ (which assesses the optimal trading time for sending an order), a memory interval $h^{i} \sim \mathcal{U} \{T_w, T-\tau^{i}-2T_w \}$ (which is the size of the past lag interval used by the agent for its learning process), a transaction gesture $g^{i} \sim \mathcal{U} (0.2, 0.8)$ (which scales with bid-ask spread to set how far above or below the value of its own stock pricing the agent is willing to deal the transaction), and a reinforcement learning rate $\alpha \sim \mathcal{U} (0.05, 0.20)$ proper to both reinforcement learning algorithms $\mathcal{F}^{i}$ and $\mathcal{T}^{i}$ (see below).
\textit{Order book}: At each time step, the agents may send transaction orders to the order book of each stock, whose function is to match these orders and process associated business transactions. More specifically, a number $J$ of order books are filled with all the agents' trading limit orders for each stock $j$ at time step $t$. All buy orders are there sorted by descending bid prices, all sell orders are sorted by ascending ask prices, each with their own associated number of stocks to trade. Then the order book clears these matching orders at same time step $t$, with each transaction set at mid-price between buy and sell-side, starting from the top of the order book to the lowest level where the bid price still exceeds the ask price. Importantly, we then define the market price $P^{j}(t+1)$ of stock $j$ at the next time step $t$ as that last and lowest level mid-price cleared by the order book. We also define the trading volume $V^{j}(t+1)$ as the number of stocks $j$ traded during that same time $t$. We also model the friction costs via broker fees~\citep{IG} applied to each transaction set at $0.1 \%$, an annual risk-free rate of $1 \%$ applied to $A_{bonds}^{i}(t)$, and an annual stock dividend yield of $2 \%$ according to~\citep{DividendYield} applied to $A_{equity}^{i}(t)$.
\textit{Agents}: Each agent autonomously uses two distinct reinforcement learning algorithms to interact with the market. For a brief introductory sum up of reinforcement learning, we refer the reader to~\cite{Lussange2019}, and to~\cite{SuttonBarto,Wiering2012,Csaba2010} for a thorough study of the subject. A first algorithm $\mathcal{F}^{i}$ learns the optimal econometric prediction function for the agent's investment horizon, depending on specific local characteristics of the market microstructure and the agent's fundamental valuation $\mathcal{B}^{i,j}(t)$. It thus outputs this price forecast, which will in turn enter as input the second reinforcement learning algorithm $\mathcal{T}^{i}$. This second algorithm is in charge of sending an optimal limit order to a double auction order book~\citep{Mota2016} at this same time step, based on this prediction and a few other market microstructure and agent portfolio indicators. Each reinforcement learning algorithm is individually ran by each agent $i$ following a direct policy search, for each stock $j$, and at each time step $t$. Each algorithm has $27 \times 27=729$ and $108 \times 9=972$ potential action-state pairs, respectively. We define the sets of states $\mathcal{S}$, actions $\mathcal{A}$, and returns $\mathcal{R}$ of these two algorithms according to the following:
i- \textit{Forecasting}: In the first algorithm $\mathcal{F}^{i}$, which is used for price forecasting, the agent continuously monitors the longer-term volatility of the stock prices $s_0^{\mathcal{F}}=\{0, 1, 2\}$, their shorter-term volatility $s_1^{\mathcal{F}}=\{0, 1, 2\}$, and the gap between its own present fundamental valuation and the present market price $s_2^{\mathcal{F}}=\{0, 1, 2\}$. This allows the agent to retrieve useful information on the microstructure and topology of the volatility, while avoiding dimensionality issues. Out of this state, it learns to optimise its price prediction at its investment horizon $\tau^{i}$ by selecting from a direct policy search three possible actions: choosing a simple forecasting econometric tool based on mean-reverting, averaging, or trend-following market prices $a_0^{\mathcal{F}}=\{0, 1, 2\}$, choosing the size of the historical lag interval for this forecast $a_1^{\mathcal{F}}=\{0, 1, 2\}$, and choosing the weight of its own fundamental stock pricing in an overall future price estimation, that is both fundamentalist and chartist $a_2^{\mathcal{F}}=\{0, 1, 2\}$. This is done in proportion to the agent reflexivity parameter $\rho^{i}$. Via this action $a_2^{\mathcal{F}}$, the agent thus learns to gauge how fundamental or chartist it should be is in its price valuation, via a weighted average involving $\rho^{i}$ in the agent's technical forecast of the market price $\hat{P}^{i,j}(t)$ and its fundamental pricing $\mathcal{B}^{i,j}(t)$. At each time step, the rewards $r^{\mathcal{F}}=\{-4, -2, -1, 1, 2, 4 \}$ are defined according to percentiles in the distribution of the agent's mismatches between past forecasts at time $t-\tau^{i}$ and their eventual price realisation at time $t$. In parallel, an off-policy method computes the optimal action that was to be performed $t-\tau^{i}$ time steps ago, now that the market price $P^{j}(t)$ is realised, and updates the agent policy accordingly.
ii- \textit{Trading}: In the second algorithm $\mathcal{T}^{i}$, which is used for stock trading, the agent continuously monitors whether the stock prices are increasing or decreasing according to the output of the former algorithm $s_0^{\mathcal{T}}=\{0, 1, 2\}$, their volatility $s_1^{\mathcal{T}}=\{0, 1, 2\}$, its risk-free assets $s_2^{\mathcal{T}}=\{0, 1\}$, its quantity of stock holdings $s_3^{\mathcal{T}}=\{0, 1\}$, and the traded volumes of stock $j$ at former time step $s_4^{\mathcal{T}}=\{0, 1, 2\}$. Out of this state, it learns to optimise its investments by selecting two possible actions via a direct policy search: sending a transaction order to the order book as holding, buying, or selling a position in a given amount proportional to its risk-free assets and stock holdings $a_0^{\mathcal{T}}=\{0, 1, 2\}$, and at what price wrt. the law of supply and demand $a_1^{\mathcal{T}}=\{0, 1, 2\}$. The cashflow difference between the profit or loss consequent to the agent's action, and that without action having been taken, are then computed $\tau^{i}$ time steps after each transaction. The rewards $r^{\mathcal{T}}=\{-4, -2, -1, 1, 2, 4 \}$ are defined according to percentiles in the distribution of these agent's past cashflow differences. In parallel, an off-policy method computes the optimal action that was to be performed $t-\tau^{i}$ time steps ago, now that the market price $P^{j}(t)$ is realised, and updates the agent policy according to it.
\section{Agent learning}
\subsection{Agent policy heterogeneity}
We first want to measure the heterogeneity of the individual policies of these best and worse agents, not only as compared to one another as groups, but also to all other agents in the whole market population, and dynamically as a function of time. For this we compute matrices $\mathcal D^{\mathcal{F}}$ and $\mathcal D^{\mathcal{T}}$ of dimensions $I \times I$, where each matrix element $d^{\mathcal{F}}_{m, n}$ and $d^{\mathcal{T}}_{m, n}$ respectively, is computed as the average of the absolute differences between the probabilities $p_{i_m}$ and $p_{i_n}$ within the policy of agents $i_m$ and $i_n$, for each policy $\pi^{\mathcal{F}}(s,a)$ and $\pi^{\mathcal{T}}(s,a)$:
\begin{eqnarray}
\label{Equ1}
d_{m, n}=\frac{1}{|\mathcal{S}| |\mathcal{A}|} \sum_{k_s=1}^{\mathcal{S}} \sum_{k_a=1}^{\mathcal{A}} | p_{i_m}(s_{k_s}, a_{k_a}) - p_{i_n}(s_{k_s}, a_{k_a}) |
\end{eqnarray}
\begin{figure}[!htbp]
\begin{centering}
\includegraphics[scale=0.53]{policydistance.png}
\caption{\label{K1} Policy distances as defined by equation \ref{Equ1} for the first algorithm $\mathcal{F}()$ (continuous curves) and the second algorithm $\mathcal{T}()$ (dashed curves), scaled to $\mathcal{F}()$ values to account for different numbers of state-action pairs, as a function of simulation time in years, expressing in percentage the heterogeneity between different groups of agents : best $10\%$ agents with themselves (blue), best $10\%$ agents with all other agents (red), best $10\%$ agents with with worst $10\%$ agents (yellow), worst $10\%$ agents with all other agents (green), and worst $10\%$ agents with themselves (brown). The simulations are generated with parameters $I=500$, $J=1$, $T=2875$, $S=20$.}
\end{centering}
\end{figure}
The results are shown on Fig. \ref{K1}, where the values corresponding to $\mathcal{T}()$ were scaled to those corresponding to $\mathcal{F}()$, in order to account for the different numbers of state-action pairs. For the first forecasting algorithm $\mathcal{F}()$, we see that the best performing agents together converge to a pool of more diverse forecasting strategies, as compared to one another, but also to the worst performing agents and the rest of all market agents. Interestingly, the worst performing agents are less disparate in their policies among themselves, even more so than with regards to the rest of the market population. We see similar dynamics with the trading algorithm $\mathcal{T}()$, with an even more pronounced heterogeneity among groups. One can say that as simulation time passes, worst performing agents have more in common among themselves, than best performing agents among themselves. This is a remarkable prediction under our model assumptions, because of its implication to trading strategies: there are more ways to succeed than to fail. From a regulation point of view, this also potentially implies that financial stock markets benefit in stability from the multitudes of available trading instruments, structured products, and a diversification of investment strategies. Finally, we also note that the curves corresponding to the forecasting algorithm $\mathcal{F}()$ are sorted like those of the trading algorithm $\mathcal{T}()$, and nearly overlap one another. We posit this to be a consequence of the fact that each agent has identical reinforcement learning parameters (learning rate, rolling intervals, etc.) for both algorithms $\mathcal{F}()$ and $\mathcal{T}()$.
\subsection{Agent trading strategy}
One of the first valuable statistical inference from the simulation is to gauge the agent propensity to engage in fundamentalist or chartist stock valuation for trading. At the mesoscale, this is one of the greatest known source of bubble formation and other so-called reflexivity effects~\cite{Hardiman2013}. Recall from Section II that each agent was initialised with a reflexivity parameter $\rho^{i} \sim \mathcal{U} (0, 100\%)$, reflecting how fundamental or chartist the agent is in its asset valuation. This reflexivity parameter $\rho^{i}$ works with a weighted average between the agent's technical price forecast and its cointegrated estimation of the fundamental value of the stock. The agent learns to optimise this parameter through action $a_2^{\mathcal{F}}$, and hence learns to be more chartist or fundamentalist, depending on the market dynamics, represented by its states $s_0^{\mathcal{F}}$ (longer-term volatility of the stock prices), $s_1^{\mathcal{F}}$ (shorter-term volatility), and $s_2^{\mathcal{F}}$ (gap between the agent's own present fundamental valuation and present market price). We show the results on Fig. \ref{I1}, where we can see at the end $t=T$ of the simulation a trend for the $10 \%$ best agents to be fundamentalists and the $10 \%$ worse agents to be chartists. We see that best performing agents have a tendency for being more fundamentalist, while the worst performing agents have for being more chartists. This can be linked with assumptions of the efficient market hypothesis of~\cite{Fama1970}, if we would consider that all the information that is endogenous to the market is retrieved by the agents (in the sense that it is available to all), but that exogenous information would be less accessible, and hence a filter to screen agent trading performance.
\begin{figure}[!htbp]
\includegraphics[scale=0.53]{agent_reflexivity.png}
\caption{\label{I1} Distribution of the reflexivity parameter $\rho^{i}$ of the $10 \%$ best (blue curve) and $10 \%$ worse (red curve) agents, at the end of the simulation set with parameters $I=500$, $J=1$, $T=2875$, $S=20$.}
\end{figure}
Another minor trait of reflexivity at the agent level is how much agents differ in their price estimation, and hence in the bid-ask spread formation. We can have a look into this spread and price formation process by checking the propensity for best performing agents to have a large gesture, as compared to worst performing ones. Such study has many parallels with market making and other strategies based on scalping the bid-ask spread. Recall each agent was initialised with a transaction gesture $g^{i} \sim \mathcal{U} (0.2, 0.8)$, reflecting how far above or below its own asset pricing the agent is willing to trade. The results are shown on Fig. \ref{I2}, where we can see that best performing agents have a propensity for having a smaller transaction gesture (propensity to bid larger prices and ask smaller prices in transaction orders), while worst performing agents have a propensity for having a larger one (propensity to bid smaller prices and ask larger prices in transaction orders). Interestingly, such results indicate that ``tougher" negotiators would thus statistically not be prone to better trading performance necessarily.
\begin{figure}[!htbp]
\includegraphics[scale=0.53]{agent_gesture.png}
\caption{\label{I2} Distribution of the gesture parameter $g^{i}$ of the $10 \%$ best (blue curve) and $10 \%$ worse (red curve) agents, at the end of the simulation set with parameters $I=500$, $J=1$, $T=2875$, $S=20$.}
\end{figure}
\section{Market impact}
\subsection{Agent learning rate}
We then want to see the impact of the reinforcement learning rate on agent performance and overall market dynamics. Such study is related to the market impact over the years of ever higher frequency and lower latency strategies of trading. Recall each agent is initialised with a learning rate modelled by a parameter $\beta \sim \mathcal{U} (0.05, 0.20)$ for both reinforcement algorithms $\mathcal{F}^{i}$ and $\mathcal{T}^{i}$. In order to do this, we first vary the percentage $p=0\%, 20\%, 40\%, 60\%, 80\%$ of agents with such a learning rate multiplied by a scalar $\zeta=2$, so that it is statistically twice larger than that of the other agents.Then we study the impact of the learning rate when it is scaled by an increasing value of $\zeta=0.5, 1.0, 1.5, 2.0, 2.5$ for the entire agent population. For both such variations in quantity $p$ and quality $\zeta$ of agent learning rates, we can observe the following:
\begin{itemize}
\item[--] We see on Fig. \ref{N1} rather stable price volatilities at different time-scales.
\item[--] We see on Fig. \ref{N2} a strong increase in the number of market crashes.
\item[--] We see on Fig. \ref{N3} mildly increasing percentages of agent bankruptcies.
\end{itemize}
It is thus interesting to note that both types of variations in agent learning rates in our model do not affect general market volatility, except in tail events with statistically much greater numbers of market crashes. Means of agent bankruptcies are mildly affected.
\begin{figure}[!htbp]
\begin{centering}
\includegraphics[scale=0.1]{a.png}
\includegraphics[scale=0.53]{learningrate_volh.png}\\
\includegraphics[scale=0.1]{b.png}
\includegraphics[scale=0.53]{learningratescale_volh.png}
\caption{\label{N1} (a) Means of volatilities (defined as standard deviations of price normalised to price itself $\sigma/P(t)$) computed over lags of one week (blue), one month (red), and six months (yellow) intervals, for a percentage $p$ of agents corresponding to $p=0\%, 20\%, 40\%, 60\%, 80\%$ of the total agent population with a learning rate scaled by a factor $\zeta=2$ (the remainder $100-p$ being agents initialised with a learning rate $\beta \sim \mathcal{U} (0.05, 0.20)$). (b) Means of volatilities (defined as standard deviations of price normalised to price itself $\sigma/P(t)$) computed over lags of one week (blue), one month (red), and six months (yellow) intervals, for simulations where the entire agent population has a learning rate scaled with factor $\zeta=0.5, 1.0, 1.5, 2.0, 2.5$. The simulations are generated with parameters $I=500$, $J=1$, $T=2875$, $S=20$.}
\end{centering}
\end{figure}
\begin{figure}[!htbp]
\begin{centering}
\includegraphics[scale=0.1]{a.png}
\includegraphics[scale=0.53]{learningrate_crashesh.png}\\
\includegraphics[scale=0.1]{b.png}
\includegraphics[scale=0.53]{learningratescale_crashesh.png}
\caption{\label{N2} (a) Number of market crashes (defined as a drop of more than $20\%$ in market price), for a percentage $p$ of agents corresponding to $p=0\%, 20\%, 40\%, 60\%, 80\%$ of the total agent population with a learning rate scaled by a factor $\zeta=2$ (the remainder $100-p$ being agents initialised with a learning rate $\beta \sim \mathcal{U} (0.05, 0.20)$). (b) Number of market crashes for simulations where the entire agent population has a learning rate scaled with factor $\zeta=0.5, 1.0, 1.5, 2.0, 2.5$. The simulations are generated with parameters $I=500$, $J=1$, $T=2875$, $S=20$.}
\end{centering}
\end{figure}
\begin{figure}[!htbp]
\begin{centering}
\includegraphics[scale=0.1]{a.png}
\includegraphics[scale=0.53]{learningrate_bankruptcyh.png}\\
\includegraphics[scale=0.1]{b.png}
\includegraphics[scale=0.53]{learningratescale_bankruptcyh.png}
\caption{\label{N3} (a) Means of all percentage of bankrupt agents at each time step $t$, for simulations with a percentage $p$ of agents corresponding to $p=0\%, 20\%, 40\%, 60\%, 80\%$ of the total agent population with a learning rate scaled by a factor $\zeta=2$ (the remainder $100-p$ being agents initialised with a learning rate $\beta \sim \mathcal{U} (0.05, 0.20)$). (b) Means of all percentage of bankrupt agents at each time step $t$, for simulations where the entire agent population has a learning rate scaled with factor $\zeta=0.5, 1.0, 1.5, 2.0, 2.5$. The simulations are generated with parameters $I=500$, $J=1$, $T=2875$, $S=20$.}
\end{centering}
\end{figure}
\subsection{Impact of best agent herding}
The importance of reflexivity in the agent proprietary price estimation and hence trading yield an open question, namely as to what or who is the source of the reflexivity trend endogenous to the market. A first natural and logical answer to this would be renown investors, traders, analysts, etc. that often publish investment recommendations or reviews. We thus can study the impact of agent reflexivity or herding on the market as a whole, as we introduce increasing percentages $p$ of agents sending (when possible) the same transaction order to the order book at time $t+1$ that was sent by the agent with \textit{best} trading performance or track record at time $t$. For the sake of simplicity, we consider this agent with best trading performance as the one with largest net asset value at time $t$. Therefore, the rest of the herding agents may follow and emulate different agents over time (just as in real markets). In particular, for larger percentages of such best herding agents, we can mention the following:
\begin{itemize}
\item[--] We see on Fig. \ref{L1} a strong increase in price volatilities, especially for higher percentages. Notice that this trend is almost imperceptible for $p<50$.
\item[--] We see on Fig. \ref{L2} extremely decreasing trading volumes.
\item[--] We see on Fig. \ref{L3} extremely increasing numbers of market crashes.
\item[--] We see on Fig. \ref{L4} steadily decreasing market bid-ask spreads, until $p>60\%$, after which they slightly increase again.
\item[--] Remarkably, the rates of agent bankruptcy remain stable regardless of these varying percentages, with average means of $22.76 \pm 3.25\%$ for all values of $p$.
\end{itemize}
This may be counter-intuitive, but following a renown investor according to this model is extremely averse to market stability.
\begin{figure}[!htbp]
\begin{centering}
\includegraphics[scale=0.1]{a.png}
\includegraphics[scale=0.53]{best_logreturns.png}\\
\includegraphics[scale=0.1]{b.png}
\includegraphics[scale=0.53]{worst_logreturns.png}\\
\includegraphics[scale=0.1]{c.png}
\includegraphics[scale=0.53]{noise_logreturns.png}
\caption{\label{L1} Distribution of logarithmic returns of prices $\log [P(t)/P(t-1)]$ of real (dashed black curve) and simulated (continuous curves) data. The simulations are for a percentage $p$ of agents corresponding to $p=0\%$ (red), $p=20\%$ (yellow), $p=40\%$ (green), $p=60\%$ (brown), and $p=80\%$ (light blue) of the total agent population at time $t$ following the best agent (a), following the worst agent (b), or trading randomly (c), while the remainder $100-p$ engaging in proprietary trading strategies. The simulations are generated with parameters $I=500$, $J=1$, $T=2875$, $S=20$.}
\end{centering}
\end{figure}
\begin{figure}[!htbp]
\begin{centering}
\includegraphics[scale=0.53]{volumesh.png}
\caption{\label{L2} Means of all trading volumes, for a percentage $p$ of agents corresponding to $p=0\%, 20\%, 40\%, 60\%, 80\%$ of the total agent population at time $t$ following the best agent (blue), following the worst agent (red), or trading randomly (grey), while the remainder $100-p$ engaging in proprietary trading strategies. The simulations are generated with parameters $I=500$, $J=1$, $T=2875$, $S=20$.}
\end{centering}
\end{figure}
\begin{figure}[!htbp]
\begin{centering}
\includegraphics[scale=0.53]{crashesh.png}
\caption{\label{L3} Means of market crashes (defined as a drop of more than $20\%$ in market price), for a percentage $p$ of agents corresponding to $p=0\%, 20\%, 40\%, 60\%, 80\%$ of the total agent population at time $t$ following the best agent (blue), following the worst agent (red), or trading randomly (grey), while the remainder $100-p$ engaging in proprietary trading strategies. The simulations are generated with parameters $I=500$, $J=1$, $T=2875$, $S=20$.}
\end{centering}
\end{figure}
\begin{figure}[!htbp]
\begin{centering}
\includegraphics[scale=0.53]{spreadh.png}
\caption{\label{L4} Means of bid-ask spread in percentages of the price, for a percentage $p$ of agents corresponding to $p=0\%, 20\%, 40\%, 60\%, 80\%$ of the total agent population at time $t$ following the best agent (blue), following the worst agent (red), or trading randomly (grey), while the remainder $100-p$ engaging in proprietary trading strategies. The simulations are generated with parameters $I=500$, $J=1$, $T=2875$, $S=20$.}
\end{centering}
\end{figure}
\subsection{Impact of worst agent herding}
We then want to study the impact of agent reflexivity or herding on the whole market, as we introduce an increasing percentage of agents sending (when possible) the same transaction order to the order book at time $t+1$ that was sent by the agent with \textit{worst} trading performance at time $t$. The interest of this is to study how asymmetric market dynamics become to best agent herding. Here we consider this agent with worst trading performance as the one with lowest, non-bankrupt, net asset value at time $t$. Therefore, the rest of the herding agents may follow and emulate different agents over time. In particular, for larger percentages of such worst herding agents, we can mention the following:
\begin{itemize}
\item[--] We see on Fig. \ref{L1} a very strong increase in price volatilities, especially for higher percentages. Notice that this trend is almost imperceptible for $p<50$.
\item[--] We see on Fig. \ref{L2} a very strong decrease in trading volumes.
\item[--] We see on Fig. \ref{L3} an extreme increase in market crashes, especially for higher percentages.
\item[--] We see on Fig. \ref{L4} steadily decreasing bid-ask spreads, except for a strong surge for higher percentages $p>80\%$.
\item[--] As one could expect, the rates of agent bankruptcy greatly increase with these varying percentages, staying above $70\%$ of agent bankruptcy for $p>20\%$.
\end{itemize}
We conclude that market instability explodes with increasing proportions of such somewhat unrealistic agents, since no real investor will try and emulate the worst agent. Nevertheless, this shows and validate the previous observations with increasing percentages of agents following the best investor at time $t$.
\subsection{Impact of noise traders}
We then want to study the whole impact of agent learning on the market as we introduce an increasing percentage of ``noise traders", i. e. agents trading randomly~\cite{Schmitt2012}. \textit{A priori}, an ever increasing number of noise agents should bring a certain financial stability to the whole market, by providing more liquidity and higher trading volumes in both bid and offer. In particular, for larger percentages of such ``noise" agents, we can observe the following:
\begin{itemize}
\item[--] We see on Fig. \ref{L1} a very strong decrease in price volatilities, as seen from price returns distributions, and as we see on Fig. \ref{K5}, a general decrease in volatilities at all time scales.
\item[--] We see on Fig. \ref{L2} a very strong increase in trading volumes.
\item[--] We see on Fig. \ref{L3} a very sharp decrease in market crashes, which virtually almost vanish for $p>50\%$.
\item[--] We see on Fig. \ref{L4} a slow and steady increase in bid-ask spreads.
\item[--] We see on Fig. \ref{K10} a steady increase in length of both bull and bear market regimes, especially the former.
\item[--] Bankruptcy rates steadily decrease with such higher proportion of noise traders from means of $23.22\%$ for $p=0\%$, to $18.84\%$ for $p=80\%$. This is remarkable, as one could have posited that agent survival rates would decrease because of such random trading.
\end{itemize}
We conclude that counter-intuitively, larger numbers of agents trading randomly is beneficial to market stability and performance.
\begin{figure}[!htbp]
\begin{centering}
\includegraphics[scale=0.53]{noise_vol.png}
\caption{\label{K5} Means of volatilities (defined as standard deviations of price normalised to price itself $\sigma/P(t)$) computed over lags of one week (green), one month (red), and six months (blue) intervals, for a percentage $p$ of agents corresponding to $p=0\%, 20\%, 40\%, 60\%, 80\%$ of the total agent population trading randomly (the remainder $100-p$ engaging in proprietary trading strategies). The simulations are generated with parameters $I=500$, $J=1$, $T=2875$, $S=20$.}
\end{centering}
\end{figure}
\begin{figure}[!htbp]
\begin{centering}
\includegraphics[scale=0.53]{noise_trend.png}
\caption{\label{K10} Distribution of the number of consecutive days of rising prices (positive values) and dropping prices (negative values). This is for both real (dashed black curve) and simulated (continuous curves) data, the latter being for a percentage $p$ of agents corresponding to $p=0\%$ (red), $p=20\%$ (yellow), $p=40\%$ (green), $p=60\%$ (brown), and $p=80\%$ (light blue) of the total agent population trading randomly (the remainder $100-p$ engaging in proprietary trading strategies). The simulations are generated with parameters $I=500$, $J=1$, $T=2875$, $S=20$.}
\end{centering}
\end{figure}
\section{Conclusion}
Following calibration performances shown in a previous work~\cite{Lussange2019}, we have used a multi-agent reinforcement learning system to model stock market price microstructure. The advantage of such a framework is that it allows to gauge and quantify agent learning, which is at the source of the price formation process, itself at the foundation of all market activity. We first studied agent learning, and then its mesoscale impact on market stability and agent performance.
According to our results on policy learning, we posit that there are more trading strategies that yield successful portfolio performance, than unsuccessful.
We also found that best performing agents have a propensity for being fundamentalists rather than chartists in their approach to asset price valuation, and to be less stringent in their choices of transaction orders (i. e. willing to transact orders with larger bids or lower asks).
Next, we studied the impact on the market of agent learning rates, and found that market volatilities at all time-scales did not vary much (with more agents with larger learning rates, or when all agent collectively have larger learning rates), except for tail events, with average numbers of crashes greatly increasing. We also found that agent bankruptcy rates were not much impacted by such variations in reinforcement learning rates.
Then we studied the effect of herding or reflexivity, when increasing percentages of agents follow and emulate the investments of the best (and worst) performing agent at each simulation time step. As expected, we found that both such behaviours greatly increase market instability. Yet remarkably, bankruptcy rates of simulations with a best agent herding set up remain quite stable, regardless of the percentages of such agents, and regardless of the yet strongly increasing market volatilities and numbers of crashes.
Finally, we sought to explore the impact of agent trading information on the price formation process, with larger proportions of ``noise traders" (i. e. agents trading randomly), and found a much greater market stability with increasing percentages of such agents, with a number of crashes virtually vanishing. We also found such markets to be more prone to display bull regimes, and that agents bankruptcy rates slightly diminished.
We trust such predictions under our model assumptions would be of interest not only to academia, but to industry practitioners and market regulators alike. A natural extension of our model would be to endow agents with a short selling ability (in order to account for specific microstructure effects in times of bubbles for instance). One could also add to each agent to capacity to perform proper portfolio diversification in the model's multivariate framework. Finally, we graciously acknowledge this work was supported by the RFFI grant nr. 16-51-150007 and CNRS PRC nr. 151199.
\medskip
| -25,273.443327 |
[
-2.759765625,
2.578125
] | 57.249071 |
[
-3.625,
0.271728515625,
-2.0859375,
-6.0703125,
0.0228271484375,
8.375
] |
[
4.0546875,
8.0546875,
3.921875,
6.76171875
] | 311 | 5,460 |
[
-2.544921875,
2.78125
] | 25.660656 |
[
-6.1875,
-4.02734375,
-4.6953125,
-2.162109375,
2.419921875,
12.578125
] | 0.913652 | 42.21663 | 21.227106 | 2.486581 |
[
1.970650553703308
] | -18,687.632042 | 5.512637 | -24,982.881829 | 0.395362 | 5.885749 |
[
-3.4296875,
-3.1953125,
-3.076171875,
-3.9375,
2.693359375,
10.375
] |
[
-6.08984375,
-2.35546875,
-2.1875,
-1.2236328125,
4.0234375,
4.90234375
] | |
BkiUaafxK1Thg9qFa1Qs
|
\section{Introduction} \label{sec: Introduction}
\IEEEPARstart{V}{ISIBLE} light communications (VLC) is an emerging technology for indoor wireless networking that can offer energy efficient Gbps streaming through the lighting system. The idea is to transmit downlink data by modulating white light emitting diodes (LED) that are already being used by energy efficient and cost effective lighting systems. The huge unregulated bandwidth available in VLC technology can relieve the traffic on radio-frequency (RF) communications. However, using LEDs as sources adds restrictions on the modulation schemes and codes that can be used. Main limitations of these LEDs are their limited peak optical power, nonlinear transfer function, and limited modulation bandwidth \cite{VLC-Ghassemlooy}. Therefore, modulation and coding schemes with high spectral efficiencies are required to provide a high data-rate connection.
In this work we propose a new modulation technique to achieve reliable and high-speed data transmission in nonlinear-LED-based VLC systems.
Orthogonal frequency-division multiplexing (OFDM) is an efficient modulation technique for high speed data communication through bandlimited channels, and is being widely used in modern systems because of its high spectral efficiency and robustness against narrow-band interference \cite{OFDM-Book}. OFDM signals are generated by applying an inverse fast Fourier transform (IFFT) on the data stream at the transmitter and decoded using a fast Fourier transform (FFT) at the receiver.
OFDM has been adapted to work in energy efficient optical communications because of its better average-power efficiency compared to other schemes \cite{Optical-OFDM-Armstrong-13}.
Asymmetrically clipped optical OFDM (ACO-OFDM), DC-biased optical OFDM (DCO-OFDM) and asymmetrically clipped DC biased optical OFDM (ADO-OFDM) are modified forms of OFDM proposed for intensity-modulation direct-detection (IM/DD) optical communication systems \cite{ACO-OFDM-06, DCO-OFDM-06, ADO-OFDM-11}. These OFDM techniques use Hermitian symmetry to generate real signals from the data sequence, trading-off loss in the encoding rate. Due to the low rate of these techniques, high order quadratic amplitude modulation (QAM) has to be employed to achieve high spectral efficiencies, which degrades the energy efficiency.
As in the original OFDM, these optical communications techniques generate signals with large peaks, which can end up clipped by the peak optical power constraint of the optical sources. This clipping causes a distortion of the OFDM signals that becomes larger by increasing the transmitted average power. Consequently, in VLC systems where high average optical powers are required for illumination, OFDM is of limited use. This problem can be alleviated by using peak to average power ratio (PAPR) reduction techniques that trade-off complexity and energy inefficiency \cite{OFDM-PAPR-Reduction-VLC-Zabih-14}. For example, Hadamard matrices can be used as precoders in OFDM systems to decrease the PAPR \cite{OFDM-Hadamard-11, OFDM-Hadamard-12}, reduce the BER \cite{OFDM-Hadamard-BER-03} and increase the resistance of the signals against frequency selective fading \cite{OFDM-Hadamard-Fading-10}.
The challenge of supporting a wide range of dimming levels is another big drawback in the application of these modified forms of OFDM to VLC systems. There have been significant efforts to address this problem \cite{OFDM-for-VLC-Wang-14}. Reverse polarity optical-OFDM (RPO-OFDM) combines pulse width modulation (PWM) with OFDM to change the dimming level of the transmitted signals \cite{Dimming-for-OFDM-13}. Although this technique can transmit signals with high average optical power without being significantly clipped by the LED peak-power, the lowest BER it can achieve is higher than that of ACO-OFDM and DCO-OFDM.
Pulsed modulation techniques are another solution to achieve reliable high-speed communication at high optical average power levels in VLC systems \cite{PWM-PPM-VLC-Zabih-14}. Among these techniques, those that use the optical sources in their on/off mode are preferred since they avoid the nonlinear effects of the LEDs \cite{VLC-Networking13}. Although using sources only in their on/off mode limits the spectral efficiency of the system in single-LED systems \cite{EPPM12}, systems with multiple LED sources have the potential to use multilevel signaling, which can be employed to use the available bandwidth more efficiently \cite{Multilevel-EPPM12}.
In \cite{HCM-Globecom-14} we introduce a multilevel modulation technique named HCM that uses the Hadamard matrices as a modulation technique (rather than a precoder). In this technique, the data is modulated using a fast Walsh-Hadamard transform (FWHT) and the receiver uses an inverse fast Walsh-Hadamard transform (IFWHT) to decode the received signals. Therefore, it has a low complexity and can exploit the bandwidth effectively. We also propose a modification referred to as the DC-reduced HCM (DCR-HCM) technique. Because of their low PAPR, HCM and DCR-HCM can provide high illumination levels in VLC systems without being too affected by LED-induced distortion. The technique is reminiscent of the approach in \cite{Hadamard-in-CDMA-04} to use the bipolar Hadamard transform for channel orthogonalization in code division multiple access (CDMA) systems because of their low PAPR.
In this paper we analyze these modulations and study their performance in LED-based VLC systems. We propose using sinc pulses for efficient use of the available LED bandwidth bandwidth and for fair comparison with OFDM. We show that using DCR-HCM the energy efficiency of HCM can be improved. This improvement becomes more significant by increasing the size of the FWHT. DCR-HCM is also able to achieve lower BER levels compared to HCM and OFDM due to its reduced DC level, which decreases the amplitudes of the transmitted signals and makes them less likely to be clipped by the peak-power limit of the LEDs. As in OFDM systems, a cyclic prefix is used to avoid interference between adjacent symbols in bandlimited channels, and then symbol-length interleaving is applied on the HCM signals, as was proposed in \cite{VLC-JLT-I-13}, to decrease the effect of the intra-symbol interference. This approach is shown to lower the error probability of high data-rate transmissions through VLC channels.
The rest of the paper is organized as follows. Section~\ref{sec:System Description} describes the VLC system model including the LED nonlinear transfer function and VLC channel characteristics. Section~\ref{sec:HCM} introduces the principles of HCM and presents modified forms of HCM to increase its energy efficiency and make it renitent against inter-symbol interference (ISI) in dispersive VLC channels. Numerical results are presented in Section~\ref{sec:Numerical Results} that compare the performance of the HCM and its modified forms to OFDM in VLC systems. Finally, conclusions are drawn in Section~\ref{sec:Conclusion}.
\section{Problem Description} \label{sec:System Description}
This section describes the principles of a VLC system. Models for the optical sources and VLC channel are discussed.
In this work we represent vectors with boldfaced lower-case letters, and boldfaced upper-case letters are reserved for matrices. The identity matrix is represented by $\mathbf{I}$. The notation $\mathbf{A}^{\text{T}}$ denotes the transpose of the matrix $\mathbf{A}$, and $x^*$ indicates the conjugate of the complex number $x$. The notations $\mathbf{A}-x$ and $\mathbf{a}-x$ are respectively used to show the matrix and vector that are obtained by subtracting a scalar $x$ from all elements of the matrix $\mathbf{A}$ and vector $\mathbf{a}$. We define the complement of the vector $\mathbf{a}$ as $\overline{\mathbf{a}}:=(1-\mathbf{a})$, and that of the matrix $\mathbf{A}$ as $\overline{\mathbf{A}}:=(1-\mathbf{A})$.
In this paper, $\mathbf{a}{^{(\ell)}}$ and $\mathbf{A}{^{(\ell)}}$ represent the $\ell$th right cyclic-shifts of vector $\mathbf{a}$ and matrix $\mathbf{A}$, respectively.
\subsection{VLC System Description} \label{sec:VLC Config}
According to the Illuminating Engineering Society of North America (IESNA), the standard illumination level for most indoor environments (classroom, conference-room, lecture hall, offices, etc.) is between 300 and 500 lux at 0.8 m height from the floor \cite[Table 32.I]{Illuminance-Standards}. In the daytime, a portion of the indoor illumination needs could be provided by daylight, and the lights can then be dimmed to reduce the energy consumed. The dimming level has a nonlinear relation to the average optical power \cite{VLC-Lighting-Requirements-13}, and affects the performance of the VLC system.
In order to evaluate the performance of a VLC link, it is essential to determine the optical power level that corresponds to a desired illumination level. This can be done using luminance efficiency of radiation (LER), which is defined as the luminous flux per unit optical power. Although the theoretical limit of LER for white LEDs is 260-300 lm/W \cite{LED-Efficiency-Limit-10}, commercially available white LEDs have an LER of 50-150 lm/W. One of the most efficient currently available white LEDs is Cree's XLamp XT-E white LED with an LER of 148 lm/W. Since 1 lux = 1 lm/m$^2$, an illumination level of 500 lux from a light source with an LER of 148 lm/W corresponds to 33.8 $\mu$W average optical power on a photo-detector with an effective area of 0.1 cm$^2$.
\subsection{Nonlinearity in Optical Sources} \label{sec:LED Nonlinearity}
The optical source is a key component of any optical communication system, as it generates optical power as a function of the modulated input electrical signal and converts the information into an optical beam. Because of the structure of LEDs, the output optical power and the forward current are related by a nonlinear function. The maximum optical power in these sources is limited to a peak-power, and this can result in the clipping of large peaks in the modulated signal.
In optical communication systems using multilevel or continuous valued signaling, the nonlinearity of the optical sources introduces a distortion on the transmitted optical signal.
Predistorion is a solution to linearize the relation between the output optical power and the forward current over a range. This technique requires an accurate model for the design of the predistortion and linearization over the dynamic range of the optical source. A polynomial model is presented in \cite{Optical-OFDM-Nonlinear-LED-13} to describe the nonlinear transfer function of the optical sources, through which a predistortion function can be designed to linearize the output optical power in terms of the input current.
A problem with the predistorion technique is that the nonlinear transfer function of optical sources can change due to many factors, one of which is the temperature of the transmitter. LEDs and lasers tend to dissipate a portion of the input energy as heat, which increases the temperature of the device over time and changes the nonlinear relation between the output power and forward current. This means that one predistortion function is not able to keep the device linear over time, and dynamic feedback is needed to modify the model of the instantaneous nonlinear transfer function of the optical source \cite{OWC-IR-Book-08} and actively match the predistortion to that model in order to keep the relation between the optical power and input current linear. This makes the design of the transmitter more complex. In this paper we assume no predistortion is employed.
As discussed in \cite{VLC-JLT-I-13}, in VLC systems that employ arrays of LEDs as sources, pulsed modulation techniques can solve the problem by using the LEDs in an on/off mode.
In these modulation techniques, multilevel signals can be generated by independently turning on and off each element of the LED-array. In this way multilevel signaling can be used without concern for the effects of the LED nonlinearity, and the optical signal level remains proportional to the intended modulating signal \footnote{Note that the LED nonlinearity could still affect the pulse-shape, an effect that is ignored in this paper.}. Based on a similar idea, quantized OFDM is proposed to utilize the full dynamic range of LEDs by using LED arrays and employing discrete power level stepping \cite{LED-Level-Quantization-Haas-13}. Below we show that HCM symbols can also be generated using an LED-array without being affected by the LED nonlinearity.
In this paper we model the LED as an ideal peak-power limited source, i.e., a hard limiter, which generates a power ranging from 0 to the LED peak power, $P_{\max}$, proportional to the forward input current (Fig.~\ref{Fig:LED-Transfer Function}-(b)). Ignoring the bandwidth limit of the LED, the only distortion on the transmitted signals is assumed to be caused by clipping the transmitted signals at 0 and the peak-power of the LED. We model the clipping induced distortion by an attenuation and an additive Gaussian noise with variance \cite{Amp-Distortion-Bussgang}
\begin{align}\label{Variance of Clipping Distortion}
\sigma{^2_{\text{clip}}} = \int_{-\infty}^{0} x^2 f(x) dx + \int_{P_{\max}}^{\infty} \left(x-P_{\max}\right)^2 f(x) dx ,
\end{align}
where $f(\cdot)$ is the probability distribution function (pdf) of the amplitude of the signal sent to the LED.
\begin{figure} [!t]
\begin{center}
{\includegraphics[width=3.4in]{LED-Characteristics}}
\end{center}
\vspace{-0.2in}
\caption{(a) Nonlinear transfer function of an LED (solid) and its linearization after pre-distortion (dashed), and (b) transfer function of and ideal LED with a limited peak-power.}
\label{Fig:LED-Transfer Function}
\vspace{-0.25in}
\end{figure}
\subsection{VLC Channel Model} \label{sec:Channel Model}
The impulse-response of a VLC channel consists of line-of-sight (LOS) and non-line-of-sight (NLOS) parts. In VLC systems, the NLOS part of the impulse response is due to reflections of the light from the walls and other objects and usually causes inter-symbol interference at symbol-rates higher than 50 Msps. The results of \cite{OWC-Channel05} and \cite{OWC-Channel11} can be used to find the impulse-response of a VLC channel with a given room geometry. Given the sampling period, an equivalent discrete impulse-response of the VLC channel, $\mathbf{h} = \{h_{\ell} \}$, can be calculated from the continuous impulse-response.
For mathematical simplicity, in this work we assume the normalized discrete impulse-response of a VLC channel for which we have $\sum\limits_{\ell=-\infty}^{\infty} {h_{\ell}} = 1$, and the channel loss is ignored for notational convenience. We model the noise in the system as an additive white Gaussian noise (AWGN) source, which is a good approximation for high background light scenarios. The front-end of our VLC system model is depicted in Fig.~\ref{Fig:VLC-Model}.
\begin{figure} [!t]
\begin{center}
{\includegraphics[width=3.4in]{VLC-Model}}
\end{center}
\vspace{-0.2in}
\caption{VLC system front end.}
\label{Fig:VLC-Model}
\vspace{-0.2in}
\end{figure}
\section{Hadamard Coded Modulation} \label{sec:HCM}
Hadamard coded modulation (HCM), which uses a binary Hadamard matrix to modulate the input data, is introduced in \cite{HCM-Globecom-14} as an alternative to OFDM. As described in \cite{HCM-Globecom-14}, the HCM signal $\mathbf{x}=[x_0,x_1,\cdots,x_{N-1}]^{\textmd{T}}$ is generated from the data sequence $\mathbf{u}=[0,u_1,\cdots,u_{N-1}]^{\textmd{T}}$ as
\begin{align}\label{HCM Encoder}
\mathbf{x} = \Bigg( \mathbf{H}{_N} \mathbf{u} + \overline{\mathbf{H}}{_N} \overline{\mathbf{u}} \Bigg).
\end{align}
where $\mathbf{H}_N$ is the binary Hadamard matrix of order $N$ \cite{Hadamard-Book}, $\mathbf{\overline{H}}_N$ is the complement of $\mathbf{H}_N$, and the matrix $( \mathbf{H}{_N} - \overline{\mathbf{H}}{_N})$ is the bipolar Hadamard matrix.
The components of $\mathbf{u}$ are assumed to be $M$-ary pulse amplitude modulated (PAM), where $u_n \in \left\{0,\frac{1}{M-1},\frac{2}{M-1},\dots,1\right\}$ for $n=0,1,\dots,N-1$. The complexity of HCM is the same as OFDM since an $N$-size FWHT also has a computational complexity of order $N \log_2 N$.
\begin{figure} [!t]
\begin{center}
{\includegraphics[width=3.0in]{HCM-Transmitter}}
\end{center}
\vspace{-0.2in}
\caption{Block diagram of the HCM transmitter using FWHT.}
\label{Fig:HCM-Transmitter}
\vspace{-0.2in}
\end{figure}
\begin{figure} [!b]
\vspace{-0.25in}
\begin{center}
{\includegraphics[width=2.8in]{HCM-Receiver}}
\end{center}
\vspace{-0.2in}
\caption{Block diagram of the HCM receiver using IFWHT.}
\label{Fig:HCM-Receiver}
\end{figure}
Similar to \cite{Multilevel-EPPM12}, two structures can be used for the HCM transmitter. In the first structure, shown in Fig.~\ref{Fig:HCM-Transmitter}-(a), the HCM symbols generated are sent to an amplitude modulator that then modulates the optical source. This structure, which we call the single-source structure, can be used with the power-line communication (PLC) integrated VLC networks, where the data is send to the LED bulbs via the power lines and each component of the LED array cannot be modulated separately. In the single-source structure, as mentioned earlier in Section~\ref{sec:LED Nonlinearity}, the nonlinear transfer function of the optical source causes unequal spacing between the transmitted power levels, which makes the symbols more susceptible to noise, and therefore, a predistorter is required to make the power levels equal. A control circuit is also needed to compensate for the drift due to the thermal changes, which leads to an increased complexity of the transmitter.
In the second structure, the nonlinearity problem of the optical sources is solved by using each LED in an array in its on or off mode. The second structure, which is referred to as the LED-array structure, directly modulates a set of LEDs with one Hadamard code as shown in Fig.~\ref{Fig:HCM-Transmitter}-(b). Given that the $u_n$'s are $M$-PAM modulated, the LED-array structure uses a total of $N\times (M-1)$ LEDs to modulated the data, i.e., $(M-1)$ LEDs for each Hadamard code. This structure can be used in VLC systems in which each LED can be modulated independently. The LED-array structure guarantees equal spacing between the output optical power levels, and avoids the effect of the LED nonlinearity on the transmitted optical signal. Using HCM this structure is able to transmit only average powers less than $P_{\max}/2$, where $P_{\max}$ is the peak optical power of the LED array.
In either case, the decoded vector $\mathbf{v}$ is obtained from the received vector $\mathbf{y}$ as
\begin{align}\label{HCM Decoder}
\mathbf{v} = \frac{1}{N} \Big( \mathbf{H}{^\textmd{T}_N} \mathbf{y} - \mathbf{\overline{H}}{^\textmd{T}_N} \mathbf{y} \Big) + \frac{P}{2}[1-N,1,1\dots,1]^{\textmd{T}} ,
\end{align}
which can be realized by an inverse FWHT (IFWHT) as shown in Fig.~\ref{Fig:HCM-Receiver}.
The noise due to the channel, $\mathbf{n}$, is assumed to be an additive white Gaussian noise (AWGN) vector with auto-covariance matrix $\sigma{^2_N}\mathbf{I}$, and hence, in the absence of nonlinearity the output signal of a non-dispersive channel, i.e., $h_{\ell}=0$ for $\ell \neq 0$, is given by $\mathbf{y}=\left( P/N \right) \mathbf{x}+\mathbf{n}$, where $P$ is the unclipped peak transmitted power.
Assuming a photo-detector with responsivity equal to 1, the decoded data can be rewritten as
\begin{align}\label{Decoded Signal with Equivalent Noise}
\mathbf{v} = \frac{P}{N} \mathbf{u} + \mathbf{\tilde{n}},
\end{align}
where $\mathbf{\tilde{n}} = \frac{1}{N} \Big( \mathbf{H}{^\textmd{T}_N} - \mathbf{\overline{H}}{^\textmd{T}_N}\Big) \mathbf{n} $ is a $N \times 1$ noise vector with independent components.
\begin{figure} [!b]
\vspace{-0.2in}
\begin{center}
{\includegraphics[width=2.6in]{HCM-Sinc}}
\end{center}
\vspace{-0.15 in}
\caption{(a) An HCM signal, and (b) the transmitted pulses using sinc pulse-shaping.}
\label{Fig:HCM-Sinc}
\end{figure}
\subsection{Pulse Shaping to Increase Spectral Efficiency}
In practice, transmitting rectangular pulses requires a large bandwidth and is not spectrally efficient. In order to overcome this problem, we use sinc pulses instead of rectangular ones to transmit data. But since negative signals cannot be sent over the optical link, we add a DC bias to the signals to make them positive. Fig.~\ref{Fig:HCM-Sinc}-(b) illustrates the transmitted pulses for the three rectangular pulses shown in Fig.~\ref{Fig:HCM-Sinc}-(a). Replacing rectangular pulses with sinc pulses reduces the SNR by 0.83 dB.
\subsection{BER Calculation}
Through (\ref{Decoded Signal with Equivalent Noise}), the BER of $M$-PAM HCM can be calculated from \cite{BER-QAM-02} as
\begin{align}\label{BER Expression for HCM}
\textmd{BER}_{\textmd{HCM}} \approx \frac{(M-1)}{M\log_2 M} Q\Bigg( \sqrt{\frac{3}{\gamma(M^2-1)} \frac{P{^2}/N}{\sigma{^2_N}+\sigma{^2_{\text{clip}}}} } \Bigg) .
\end{align}
where $\gamma$ represents the penalty in SNR due to the pulse-shaping, which is 1.21 in this work, $\sigma{^2_N}$ is the variance of the additive Gaussian noise at the receiver and $\sigma{^2_{\text{clip}}}$ is the variance of the clipping noise.
For the LED-array transmitter structure in Fig.~\ref{Fig:HCM-Transmitter}-(b), $\sigma{^2_{\text{clip}}}=0$ and no further analysis is needed.
Since each of the $N-1$ columns of $\mathbf{H}{_N}$ that are used to modulate the data has an equal number of zeros and ones, HCM signals have a PAPR of 2, and therefore, for the single-source transmitter structure in Fig.~\ref{Fig:HCM-Transmitter}-(a), its signals are not clipped by LEDs for average power levels less than $P_{\max}/2$ and the clipping noise is zero, i.e., $\sigma{^2_{\text{clip}}}=0$.
For average powers larger than $P_{\max}/2$, we use (\ref{Variance of Clipping Distortion}) to find $\sigma{^2_{\text{clip}}}$. In order to find the pdf of $\mathbf{x}$, we first consider $\mathbf{u}$ to be a binary vector, and then we generalize the results to the case when the components of $\mathbf{u}$ are $M$-PAM.
For the binary case, $x_n \in \{0,1,2,\dots,N-1,N\}$ for $n=1,2,\dots,N$, and the probability that $x_n=k$ is equal to
\begin{align}\label{}
{\text{Pr}}\left(x_n=k\right) = {\binom{N}{k}} \left(\frac{1}{2}\right)^N, \hspace{0.3cm} k=0,1,\dots,N.
\end{align}
Through (\ref{Variance of Clipping Distortion}), the clipping noise for binary HCM is
\begin{align}\label{}
\sigma{^2_{\text{clip}}} = \left(\frac{1}{2}\right)^N \sum_{k=\lceil N\frac{P_{\max}}{P}\rceil}^N \left(k\frac{P}{N}-P_{\max}\right)^2 {\binom{N}{k}} .
\end{align}
where $\lceil x \rceil$ is the smallest integer larger than $x$.
For $M$-PAM HCM, the components of $\mathbf{x}$ take values from a larger set as $x_n \in \{0,\frac{1}{M-1},\frac{2}{M-1},\dots,N\}$ for $n=1,2,\dots,N$, and the probability of $x_n=\frac{k}{M-1}$ is
\begin{align}\label{}
{\text{Pr}}\left(x_n=\frac{k}{m-1}\right) = C(m,N,k) \left(\frac{1}{m}\right)^N, \hspace{0.3cm} k=0,1,\dots,N.
\end{align}
where $C(m,N,k)$'s are the \emph{extended binomial coefficients} defined as the coefficients of $x^k$ in the expansion of $(1+x+x^2+\dots,x^{(m-1)})^N$ \cite{Extended-Binom-13}.
\begin{figure} [!b]
\vspace{-0.2 in}
\begin{center}
{\includegraphics[width=3.4in]{DC-Removal}}
\end{center}
\vspace{-0.25 in}
\caption{(a) An HCM signal, and (b) its corresponding DC reduced signal.}
\label{Fig:DC-Removal}
\end{figure}
\begin{figure} [!t]
\vspace{-1.8in}
\hspace{-0.5in}{\includegraphics[width=4.6in]{PMF-HCM}}
\vspace{-2.0in}
\begin{center}
{\small (a)}
\end{center}
\vspace{-1.7in}
\hspace{-0.5in}\includegraphics[width=4.6in]{PMF-DCR-HCM}
\vspace{-2.0in}
\begin{center}
{\small (b)}
\end{center}
\vspace{-0.15 in}
\caption{Probability mass function for (a) binary HCM, and (b) binary DCR-HCM.}
\label{Fig:pdf}
\vspace{-0.25 in}
\end{figure}
\subsection{Increasing Energy Efficiency Using DC-Reduced HCM} \label{sec:Removing DC Bias}
As shown in \cite{HCM-Globecom-14}, the DC part of HCM signals (before the pulse-shaping) can be reduced without losing any information, making HCM more average power efficient. This is important when for illumination reasons the light should be dimmed, i.e., not operated at its brightest level. Let the first component of $\mathbf{u}$ be set to zero and only $N-1$ codewords of the Hadamard matrix be modulated, as proposed in \cite{HCM-Globecom-14}. In this scheme, which is called DC-reduced HCM (DCR-HCM), the average transmitted power is reduced by sending $(\mathbf{x} - \min \mathbf{x}) $ instead of $\mathbf{x}$. The reduced DC level is per HCM symbol and its value can be different for each symbol. The same receiver structure as in Fig.~\ref{Fig:HCM-Receiver} can be used to decode the received signals.
Fig.~\ref{Fig:DC-Removal} shows an example of DC reduction in an HCM symbol of size $N=8$, where the transmitted energy of the HCM symbol in Fig.~\ref{Fig:DC-Removal}-(a) is reduced by a factor of $3/7$ in its corresponding DCR-HCM symbol in Fig.~\ref{Fig:DC-Removal}-(b). This technique can only be easily implemented for the single-source transmitter structure, and an intermediate circuit is required to apply this technique to the LED-array structure.
The DC reduction technique decreases the probability of large amplitudes of $\mathbf{x}$, which makes the signals less likely to be clipped by the optical source, and therefore, DCR-HCM can achieve lower BERs at lower average power levels compared to HCM in peak-power limited systems. The probability mass function of the transmitted signal, ${\text{Pr}}\left(x=k\right)$, is plotted using Monte-Carlo simulation for binary HCM and DCR-HCM respectively in Fig.~\ref{Fig:pdf}-(a) and Fig.~\ref{Fig:pdf}-(b) for $N=128$. According to these results, the peak of ${\text{Pr}}\left(x=k\right)$ is shifted to lower $x$'s for DCR-HCM and the high amplitudes in DCR-HCM signals have lower probabilities compared to that of HCM \footnote{Note that the DC value could instead be increased to its maximum value for scenarios that require even brighter illumination. This idea is entirely analogous to DCR-HCM and is therefore not discussed further here.}.
The energy efficiency of DCR-HCM, denoted $\eta$, is defined as
\begin{align}\label{}
\eta = \frac{ E\left\{ x_n \right\} }{ E\left\{ x_n \right\} - E \{\min \mathbf{x}\} },
\end{align}
In this definition we have used the fact that all components of $\mathbf{x}$ have the same mean, i.e., $E\{x_n\}$ is fixed for all $n=0,1,\dots,N-1$.
Fig.~\ref{Fig:DCR-HCM Energy Efficiency} shows $\eta$ as a function of the order of the Hadamard transform. In this figure, the data sequence is assumed to be $M$-ary PAM modulated. According to these results, the energy efficiency of DCR-HCM increases almost linearly with $\log_2 N$, and also increases slightly with increasing $M$.
\begin{figure} [!t]
\vspace*{-1.5 in}
\hspace{-0.4 in}{\includegraphics[width=4.4in]{Energy-Efficiency-of-DCR-HCM}}
\vspace*{-1.7 in}
\caption{Energy efficiency of DCR-HCM versus the order of the Hadamard transform, $\log_2 N$, for $M=2$, 4, 16 and 64.}
\label{Fig:DCR-HCM Energy Efficiency}
\vspace{-0.15 in}
\end{figure}
\subsection{Dispersive Channels} \label{sec:Interleaved HCM}
VLC experience dispersive channels that create inter-symbol interference (ISI) on the transmitted signals, and therefore, any practical modulation technique must be resistant against ISI. In OFDM, a cyclic prefix is used as a guard interval in order to eliminate the intersymbol interference from adjacent symbols. It also allows the linear convolution of a frequency-selective multipath channel to be modelled as a circular convolution by repeating the end of the symbol, which simplifies channel estimation and equalization at the receiver. Likewise, a cyclic prefix is used for HCM symbols to avoid interference from other symbols, and therefore, the interference on a symbol is intra-symbol interference that is caused by its own pulses. This also allows us to use cyclic shifts of transmitted vectors instead of their right shifts in our analysis. Under these assumptions, given $\mathbf{x}$ is sent, and ignoring background light and other noise, the received signal is proportional to the nonlinearly distorted (clipped) version of $\sum\limits_{\ell} {h_{\ell}} \mathbf{x}{^{(\ell)}}$. In this section two techniques are proposed to handle the dispersive nature of the channel: interleaving and equalization.
Hadamard matrices consist of rows that are cyclic shifts, which increases the similarity between Hadamard codewords in dispersive channels and makes HCM vulnerable to ISI. Interleaving is an effective solution that reduces the ISI by decreasing the cross-correlation of the codewords with their cyclic shifts \cite{VLC-JLT-I-13}.
In this technique, as shown in Fig.~\ref{Fig:Interleaved-HCM}, a symbol-length interleaver and a deinterleaver are used at the transmitter and receiver, respectively, to reduce the effects of intra-symbol ISI due to a dispersive channel. The interleaver is a permutation matrix, $\mathbf{\pi}$, and the deinterleaver is its inverse, $\mathbf{\pi}^{-1}$. Hence, $\mathbf{x} \mathbf{\pi}$ is sent instead of $\mathbf{x}$. For a non-equalizing receiver, the best interleaver matrix is the one that evenly distributes the interference over all symbols, and can be found using binary linear programming \cite{VLC-JLT-I-13}. In non-dispersive channels, the performance of interleaved HCM is the same as HCM since $\mathbf{\pi} \mathbf{\pi}^{-1}=\mathbf{I}$.
\begin{figure} [!t]
\hspace{-0.1in}{\includegraphics[width=3.7in]{Interleaved-HCM}}
\vspace{-0.25in}
\caption{Schematic view of an interleaved HCM system using an for dispersive channels.}
\label{Fig:Interleaved-HCM}
\vspace{-0.25in}
\end{figure}
Assuming $\mathbf{x}$ is transmitted, the noiseless output of the channel is proportional to $\sum\limits_{\ell} {h_{\ell}} \left( \pi \mathbf{x}\right)^{(\ell)}$. Then the decoded signal can be written as
\begin{align}\label{Received Signal with Interleaver}
\mathbf{v} = &\frac{ P}{N} \sum_{\ell} {h_{\ell}} \Big(\mathbf{H}{_N} - \overline{\mathbf{H}}{_N} \Big) \pi^{-1} \mathbf{I}^{(\ell)} \pi \Big( \mathbf{u} \mathbf{H}{_N} + \overline{\mathbf{u}} \overline{\mathbf{H}}{_N} \Big) \nonumber\\ &+ \frac{P}{2}[1-N,1,1\dots,1]^{\textmd{T}} + \mathbf{\tilde{n}}.
\end{align}
Defining $\mathbf{G}:= \sum\limits_{\ell} {h_{\ell}} \mathbf{I}^{(\ell)} $,
(\ref{Received Signal with Interleaver}) can be written as
\begin{align}\label{Interference on the Received Signal with Interleaver - Simplified}
\mathbf{v} = P \Big( \mathbf{H}{_N} - \overline{\mathbf{H}}{_N} \Big) \pi^{\text{T}} \mathbf{G} \pi \Big(\mathbf{H}{_N} - \overline{\mathbf{H}}{_N} \Big)\frac{\left( 2\mathbf{u} -1 \right)}{2N} + \frac{P}{2} + \mathbf{\tilde{n}}.
\end{align}
In addition, minimum mean square error (MMSE) equalization is an effective technique to estimate the data at the output of a noisy dispersive channel. The goal in this equalizer is to find the matrix $\mathbf{W}$ that minimizes the trace of $E\left\{ (\mathbf{u}-\hat{\mathbf{u}}) (\mathbf{u}-\hat{\mathbf{u}})^{\textmd{T}} \right\}$, where $\hat{\mathbf{u}}= \mathbf{W} (\mathbf{v}-P/2) + 1/2$ is an estimate of the data sent, $\mathbf{u}$, based on the observation of the decoded vector $\mathbf{v}$. According to \cite{Proakis}, the optimum $\mathbf{W}$ is given by
\begin{align}\label{Optimal W Matrix for MMASE}
\mathbf{W}= \mathbf{C}_{\mathbf{u}\mathbf{v}} \mathbf{C}{_{\mathbf{v}}^{-1}},
\end{align}
where $\mathbf{C}_{\mathbf{u}\mathbf{v}}$ is cross-covariance matrix between $\mathbf{u}$ and $\mathbf{v}$, and $\mathbf{C}{_{\mathbf{v}}}$ is auto-covariance matrix of $\mathbf{v}$. For (\ref{Optimal W Matrix for MMASE}), the error is
\begin{align}\label{MMSE Error}
\text{LMMSE}=\text{tr}\left\{ \mathbf{C}_{\mathbf{u}} - \mathbf{C}_{\mathbf{u}\mathbf{v}} \mathbf{C}{_{\mathbf{v}}^{-1}} \mathbf{C}_{\mathbf{v}\mathbf{u}} \right\}.
\end{align}
For HCM, using (\ref{HCM Encoder}) and (\ref{HCM Decoder}) we get
\begin{align}\label{}
\mathbf{C}_{\mathbf{u}\mathbf{v}} = \frac{P}{4N} \Big( \mathbf{H}{_N} - \overline{\mathbf{H}}{_N} \Big) \pi^{\text{T}} \mathbf{G} \pi \Big(\mathbf{H}{_N} - \overline{\mathbf{H}}{_N} \Big),
\end{align}
and
\begin{align}\label{}
\mathbf{C}_{\mathbf{v}} = \frac{\sigma{_N^2}}{N} \mathbf{I} + \frac{P{^2}}{4 N} \Big( \mathbf{H}{_N} - \overline{\mathbf{H}}{_N} \Big) \pi^{\text{T}} \mathbf{G} \mathbf{G}^{\textmd{T}} \pi \Big(\mathbf{H}{_N} - \overline{\mathbf{H}}{_N} \Big) ,
\end{align}
where we have used the symmetry property of Hadamard matrices, i.e., $\mathbf{H}_N = \mathbf{H}{^\textmd{T}_N}$.
\begin{figure} [!t]
\vspace*{-1.2 in}
\hspace{-0.2 in}{\includegraphics[width=3.7in]{BER-OFDM-vs-HCM}}
\vspace*{-1.3 in}
\caption{Analytical (dashed lines) and simulated (solid lines) BER of HCM, DRC-HCM, ACO-OFDM and DCO-OFDM versus the average received power in an ideal channel, i.e., $h_0=1$.}
\label{Fig:BER-of-OFDM-vs-HCM}
\vspace{-0.2 in}
\end{figure}
\begin{figure} [!b]
\vspace*{-1.45 in}
\hspace{-0.1 in}{\includegraphics[width=3.8in]{SNR-vs-Spectral-Efficiency}}
\vspace*{-1.4 in}
\caption{Maximum achievable SNR versus the spectral efficiency for HCM, DCR-HCM, ACO-OFDM and DCO-OFDM.}
\label{Fig:SNR-vs-Spectral-Efficiency}
\end{figure}
\begin{figure} [!t]
\vspace*{-1.4 in}
\hspace{-0.2 in}{\includegraphics[width=4.0in]{MMSE}}
\vspace*{-1.55 in}
\caption{Simulated BER of ACO-OFDM, DCR-HCM, and interleaved DCR-HCM in a dispersive channel with impulse response $h_0=0.5$, $h_1=0.3$ and $h_2=0.2$ using an MMSE equalizer for DCR-HCM and one-tap equalizer for ACO-OFDM.}
\label{Fig:MMSE}
\vspace{-0.25 in}
\end{figure}
\section{Numerical Results} \label{sec:Numerical Results}
In this section, numerical results using simulation and analysis are presented to compare the performance of HCM and DCR-HCM to ACO-OFDM and DCO-OFDM.
In the simulations, the sources are assumed to be ideal peak-power limited sources as shown in Fig.~\ref{Fig:LED-Transfer Function}-(b) and the peak received power is assumed to be 0.1 mW. The optical source is modulated as in Fig.~\ref{Fig:HCM-Transmitter}-(a). The transmitter and receiver are assumed to be perfectly synchronized. These results are generated using sinc pulse-shaping, shown in Fig.~\ref{Fig:HCM-Sinc}-(b).
The BERs of HCM, DCR-HCM, ACO-OFDM and DCO-OFDM are plotted versus the average received optical power in Fig.~\ref{Fig:BER-of-OFDM-vs-HCM} for $N=128$. The parameters of these techniques are chosen such that all have spectral efficiency of 1, i.e., the same data-rate. The analytical BERs are in good agreement with Monte-Carlo simulation results. For HCM and DCR-HCM, (\ref{BER Expression for HCM}) and (\ref{Variance of Clipping Distortion}) are used to plot the analytical results, while those of ACO-OFDM and DCO-OFDM are plotted using the results of \cite{Optical-OFDM-Clipping-Hass-11, Optical-OFDM-Armstrong-13}.
ACO-OFDM and DCO-OFDM use 16-QAM and QPSK, respectively, to modulate the data, while on-off keying (OOK) is used to modulate the data in HCM and DCR-HCM. The results are plotted assuming an AWGN channel for a noise level of $\sigma{^2_N}=2$ $\mu$W. The BER of ACO-OFDM decreases by increasing the average optical power until it reaches an optimum point, and increases afterwards due to the clipping imposed distortion. DCO-OFDM has the same behavior but for higher average powers. The average optical power of DCO-OFDM is proportional to its DC level, and hence, it reaches its lowest BER for an average power of half of the peak power. According to these results, DCO-OFDM, HCM and DCR-HCM are better choices for VLC systems since they have a better performance at high average optical powers. As one can see, HCM and DCR-HCM can achieve lower BERs compared to OFDM techniques. Between these two, DCR-HCM uses signals with lower amplitudes and is therefore more resistant to clipping noise.
The maximum {\em achievable SNR } is plotted versus the spectral efficiency for HCM, DCR-HCM, ACO-OFDM and DCO-OFDM for a noise level of $\sigma{^2_N}=0.5$ $\mu$W. Analytical BER expressions of the form $\alpha Q\left( \sqrt{\text{SNR}} \right)$ taken from (\ref{BER Expression for HCM}) for HCM and DCR-HCM and from \cite{Optical-OFDM-Clipping-Hass-11, Optical-OFDM-Armstrong-13} for ACO-OFDM and DCO-OFDM are used to plot these results. For each technique, the optimum average power that minimizes the BER is found analytically and then the SNR is calculated for that average power. For DCO-OFDM, the DC level is set to $P_{\max}/2$ since it minimizes the clipping noise, and hence, corresponds to the maximum SNR among all possible DC levels. Monte-Carlo simulation is used to find the probability mass function of DCR-HCM, and then it is used in (\ref{Variance of Clipping Distortion}) to find the clipping noise power. Note that ACO-OFDM is only better than DCO-OFDM at low spectral efficiencies, and it requires high-order QAM to get spectral efficiencies larger than 2, which is impractical. Hence, ACO-OFDM is not suitable for spectrally efficient VLC systems. HCM and DCR-HCM are able to provide a higher achievable SNR for all spectral efficiencies tested.
The simulated BER of 16-QAM ACO-OFDM is compared to that of OOK DCR-HCM in Fig.~\ref{Fig:MMSE} for a highly dispersive channel with a discrete-time equivalent impulse response of ${h_0}=0.4$, $h_1=0.3$ and ${h_2}=0.3$ (using $N=128$ samples per symbol) using an MMSE equalizer for DCR-HCM and one-tap equalizer for ACO-OFDM. Both techniques use a cyclic prefix of length 4. The noise is assumed to be $\sigma{^2_N}=6$ $\mu$W. According to these results, DCR-HCM can achieve lower BERs compared to ACO-OFDM. The BER of the system using MMSE equalization on the interleaved DCR-HCM is also plotted versus the average optical power. The interleaver is found using the binary linear program described in \cite{VLC-JLT-I-13}. As shown in Fig.~\ref{Fig:MMSE}, interleaving improves the performance of the MMSE equalizer by almost an order of magnitude by spreading the interference equivalently on all Hadamard codewords.
\section{Conclusion} \label{sec:Conclusion}
In this paper, HCM and its modified form DCR-HCM are proposed as alternative techniques to OFDM for LED-based VLC systems that require high illumination levels. HCM and DCR-HCM achieve lower BERs compared to ACO-OFDM for high average power since they transmit signals with lower peak amplitudes. The energy efficiency of HCM can be improved by reducing the DC part of the transmitted signals without losing any information. This efficiency is shown to increase with the size of the Hadamard transform and the size of the PAM constellation used.
The performance of HCM and DCR-HCM is shown to surpass that of ACO-OFDM and DCO-OFDM, and they are able to achieve 2 to 3 orders of magnitude lower BERs.
Interleaving along with MMSE equalization can effectively decrease the BER of HCM by an order of magnitude in dispersive VLC channels.
\bibliographystyle{IEEEtran}
\balance
| -29,358.621029 |
[
-3.23828125,
3.0078125
] | 47.941176 |
[
-3.634765625,
0.5869140625,
-2.3359375,
-6.01953125,
-0.303955078125,
8.4140625
] |
[
2.16796875,
6.30859375,
0.6826171875,
4.9765625
] | 331 | 5,169 |
[
-2.423828125,
2.404296875
] | 26.447233 |
[
-6.1640625,
-3.326171875,
-3.490234375,
-1.8974609375,
2.09765625,
10.765625
] | 1.242445 | 32.457347 | 23.002515 | 3.729452 |
[
2.297703742980957
] | -20,122.040898 | 5.761269 | -28,387.413568 | 0.443251 | 5.83066 |
[
-3.267578125,
-4,
-3.55859375,
-4.37109375,
2.689453125,
11.75
] |
[
-5.8828125,
-2.76953125,
-3.005859375,
-2.330078125,
3.921875,
6.21875
] | |
BkiUbRw4ubnjorKKr43C
|
\section{Introduction}
The largest eigenvalue of network adjacency matrices plays a bridge between dynamical and
structural properties of an underlying system. For example, the
inverse of the largest eigenvalue of a network characterizes the
threshold for phase transition of the virus spread \cite{Mieghem2009}. Recently
Goltsev et. al. have demonstrated the importance of the largest eigenvalue in determining
disease spread in complex networks \cite{Mendes2012}. Furthermore, in coupled oscillators the threshold for
phase transition to synchronized behaviour is determined by the inverse of the largest eigenvalue
\cite{Restrepo}. The dynamical properties of neurons have been shown to be highly influenced by
a change in the spectra of underlying synaptic matrices constructed from randomly distributed numbers
\cite{Sompolinsky}.
A remarkable, fundamental direction to analyze the stability of ecological systems
was put forward by May \cite{MayNature1972}, where
the largest real part of the eigenvalues ($R_{\mathrm max}$) establishes a
relationship between the stability and complexity of the underlying
system. Later, the impact of various types of interactions was demonstrated to
deduce stability criteria in terms of $R_{\mathrm max}$ \cite{Allesina}.
Mathematically, matrices obeying some constraints satisfy the stability criteria
\cite{Quirck1965}, but real-world systems have an underlying interaction matrix that is
too complicated to obey these constraints; hence, the study of fluctuations in
$R_{\mathrm max}$ is crucial to understanding stability of a
system as well as the stability properties of an individual network
in that ensemble. Recent efforts in this direction reveal the similarity of the
maximal Lyapunov exponent of synaptic matrices defined for neural networks with their topological complexities
\cite{Gilles}. A very recent work investigates the
statistical properties of random matrices within the framework of extreme value
theory, thereby providing an estimation about the resolution in complex dynamics for a
finite system size \cite{Luis}.
\subsection{Balanced condition and its role in stability}
The balanced condition in the brain refers to a situation in which for each neuron the weight of
the inhibitory signal is equal to the excitatory signal \cite{Shadlen,Troyer}. Ref. \cite{Rajan}
demonstrates that this condition
forces outliers of the spectra to appear inside the bulk, leading
to a stable underlying neural system.
Further analysis of a dynamical model of cortical networks with the balanced condition for various ratios
of inhibitory and excitatory
neurons reveals a connection between the spectra of connectivity matrix and the
dynamical response \cite{Non-normal}. Balance between recurrent excitation and inhibition generates stable periods of
activity\cite{Compte}. There have been several discussions on how synaptic matrices in the brain
achieve the balanced condition; for instance, it has been demonstrated that the balanced condition
in sensory pathways and memory networks is maintained through a plasticity mechanism at inhibitory synapses \cite{Vogels2}.
In addition to the research emphasizing the importance of the balanced condition, there exist
papers discussing the relevance of different ratios of inhibitory and excitatory neurons in brain; for example,
cortical neurons consist of only $20-30\%$ inhibitory neurons \cite{DiverseInhibitory}.
\subsection{Extreme value theory and its relevance}
The extremal eigenvalue statistics are widely used in various
disciplines of science.
The generalized extreme value distribution (GEV) is applied to model
extrema of independent, identically distributed random variables.
GEV statistics have been realized in many real-world and model systems.
For example, the radius of the bulk of complex eigenvalues of non-Hermitian
random matrices has been shown to follow the Gumbel distribution \cite{Rider}.
Recent research revealing a rich network architecture has given way to the
spectral studies of matrices deviating from a random structure. One of these
studies demonstrates that statistics of largest
eigenvalue of matrices with entries following the power-law
distribution displays a transition
from the Fr\'echet to the Tracy-Widom distribution at a threshold governed by the
power-law exponent \cite{Biroli}.
The statistics of the inverse of the largest eigenvalue for an ensemble of scalefree networks follows the Weibull
distribution \cite{SF_inverse}. Some of the studies pertaining to
sparse random graphs,
and gain matrices in the context of brain networks, are shown to deviate from GEV
statistics and follow normal distribution instead \cite{Komlos}.
The statistical properties of $R_{\mathrm max}$ of synaptic matrices capturing
inhibitory and excitatory couplings reveal a
transition to the extreme value distribution\cite{Extreme}. However, the extreme value
distribution is not observed for a larger parameter regime, thereby restricting the applicability of
the results for a more realistic underlying network construction.
Extreme value statistics for independent,
identically distributed random variables can be formulated entirely in terms of
three universal types of probability functions: the Fr\'echet, Gumbel and
Weibull distributions, also known as GEV statistics depending
upon whether the tail of the density
is power-law, faster than any power-law, and bounded or unbounded respectively
\cite{book_gev}.
GEV statistics
with a location parameter $\mu$, scale parameter $\sigma$ and shape parameter $\xi$
have often been used to model unnormalized data
from a given system.
The probability density function for extreme value statistics
with these parameters is given by \cite{book_gev},
\makeatletter
\def\@eqnnum{{\normalsize \normalcolor (\theequation)}}
\makeatother
{ \small
\begin{equation}
\rho(x) = \begin{cases} \frac{1}{\sigma}\big[1+\big(\xi\frac{(x-\mu)}{\sigma}\big)^{-1-\frac{1}{\xi}}\big]\exp\big[-\big(1+\big(\xi\frac{(x-\mu)}{\sigma}\big)^{-\frac{1}{\xi}}\big)\big]&\\
\hspace*{5.6cm}\mbox{if } \xi\not=0 \\
\frac{1}{\sigma}\exp\big(-\frac{x-\mu}{\sigma}\big)\exp\big[-\exp\big(-\frac{x-\mu}{\sigma}\big)\big]
~~ \mbox{if } \xi=0. \end{cases}
\label{eq_gev}
\end{equation}
}
Distributions associated with $\xi > 0$ , $= 0$ and $< 0$ are characterized by the Fr\'echet, Gumbel,
and Weibull distributions respectively.
In this Letter, we investigate the statistical properties of $R_{\mathrm max}$ for
networks in the balanced condition. The factors affecting the balanced condition
are monitored. We witness the Weibull distribution for the strictly balanced condition.
Observed behaviour is not much affected by the change in underlying architecture; rather
it depends more on the denseness of connections. We present
results for Erd\"os-R\'enyi random networks, small-world networks and scalefree networks.
\begin{figure}[t]
\centerline{\includegraphics[width=0.9\columnwidth,height=4cm]{FigNew1.eps}}
\caption{(Colour online) Statistics of $R_{\mathrm max}$ at different values
of $p_{\mathrm in}$. The histograms are
numerical results; blue and red lines correspond to normal
and GEV fit, respectively. For each case, the balanced matrix
is constructed for network parameters $N=50$ and $<k>=10$.
All plots are obtained for 5000 realizations of the network in that ensemble.}
\label{Fig_ER_Stat}
\end{figure}
\section{Model}
The balanced condition is attained by assigning a fixed weight to inhibitory
and excitatory connections
in the following manner \cite{Non-normal}.
When a node is defined as inhibitory with probability $p_{\mathrm in}$, the
corresponding entry in that row of the matrix $A$ is replaced by
$1-1/p_{\mathrm in}$.
In the matrices constructed as above, most of the column sum would be
fluctuating closely about the zero value for $p_{\mathrm in}$, lying in the vicinity of
0.50. However, for
lesser values of
$p_{\mathrm in}$ there may be some columns that have only excitatory connections,
yielding only $zero$ or $+1$ entries, which hinder the achievement of
the balanced condition.
Furthermore, we achieve a strictly balanced condition by subtracting a constant term
from each non-zero element of a column, which restricts the sum of the
column entries to a zero value. The strictly balanced condition
is defined to resolve situations where the arrangement of inhibitory and excitatory
couplings leads to a fluctuation around the zero value for the column sum,
even after imposing
the balanced condition.
\section{Random Networks}
Erd\"os-R\'enyi random networks of size $N$ are constructed where pairs
of nodes are connected with a probability $p$.
Figure \ref{Fig_ER_Stat} plots the statistical properties of networks with $0$, $1$
and $1-1/p_{\mathrm in}$ entries.
The data is fitted with the Gaussian and GEV distributions (Eq.~\ref{eq_gev}).
Figure~\ref{Fig_ER_Stat} is plotted for various values of $p_{\mathrm in}$ while keeping other network parameters
the same.
The nature of the distribution is normal for
$p_{\mathrm in}$ = 0.
As inhibitory connections are introduced, thereby inducing directionality into the
underlying network, the
complex eigenvalues start appearing in conjugate pairs. The
statistics of $R_{\mathrm max}$
are deformed as compared to that of the undirected network, which can not be characterized by any well-known
statistics for regime
$0 \lesssim p_{\mathrm in} \lesssim 0.40$.
For values of $p_{\mathrm in}$ lying between $0.40$ and $0.50$,
the statistics has an $\xi$ parameter value
close to the zero, indicating a convergence to the Gumbel distribution.
Calculations of the shape parameter and detailed discussions
on fitting have been exemplified in \cite{supp}.
\begin{figure}[t]
\centerline{\includegraphics[width=0.9\columnwidth,height=4cm]{Fig2.eps}}
\caption{(Colour online) Statistics of $R_{\mathrm max}$ at different values
of $C_{max}$ and $C_{\mathrm min}$ for $p_{\mathrm in} = 0.5$.
The histograms are numerical results; blue and red lines correspond to the normal
and GEV fit, respectively. For each case $N=50$ and $<k>=15$.
All plots are obtained for 5000 realizations of the network in that ensemble.}
\label{Fig2}
\end{figure}
\begin{figure}[H]
\centerline{\includegraphics[width=0.9\columnwidth,height=4cm]{Fig3.eps}}
\caption{(Colour online) Statistics of $R_{\mathrm max}$ at different values
of $p_{\mathrm in}$ for Erd\"os-R\'enyi
networks with a strictly balanced condition.
The histograms are numerical results; blue and red lines correspond to normal
and GEV fit respectively. For each case, the statistics displays the Weibull distribution,
except $p_{\mathrm in}$ = 0
All plots are obtained for 5000 realizations of matrices with size 50 and $\langle k \rangle$ = 10.}
\label{Fig3}
\end{figure}
The behaviour of the column sum statistics provides an understanding of
the impact of network structure on the shape parameter of a GEV distribution. For the
balanced condition,
the mean and variance of the column sum are zero and
$Np(1/p_{\mathrm in} - 1)$ respectively. The maximum and minimum values of the
column sum for a particular matrix in the ensemble are denoted by $C_{\mathrm max}$ and $C_{\mathrm min}$.
The Weibull, Gumbel, and Fr\'echet distributions are observed in Fig.\ref{Fig2}(a), (b) and
(c), respectively
for an ensemble consisting of realizations of matrices generated by imposing
three different restrictions on $C_{\mathrm max}$ by keeping $p_{\mathrm in}$, $p$
and $N$ the same. An additional limitation on $C_{\mathrm min}$ to a particular value
shifts the shape parameter $\xi$ towards a positive value yielding Gumbel and Fr\'echet statistics
as illustrated by Fig.\ref{Fig2} (d), (e) and (f). The implications of restricting $C_{\mathrm min}$
and $C_{\mathrm max}$ are that they characterize the deviation from
the strictly balanced condition, and interestingly, decide the shape parameter
of the statistics regardless of the denseness of underlying networks.
\begin{figure}[t]
\centerline{\includegraphics[width=0.65\columnwidth,height=3.5cm]{Fig4.eps}}
\caption{(Colour online) The tail behaviour of the real part of the eigenvalues at different values
of $p_{\mathrm in}$ for Erd\"os-R\'enyi networks with the strictly balanced condition.
For each case, $N=50$ and $\langle k \rangle=10$. }
\label{Fig4}
\end{figure}
Note that for the lower $p_{\mathrm in}$ values, the network has a significant
number of nodes connected to only excitatory nodes, thus failing to
satisfy the strictly balanced condition.
In order to avoid this situation, only those realizations are chosen that lead to columns
with at least one negative entry. Fig.~\ref{Fig3} depicts the statistics under
the strictly balanced condition for a network of size $N=50$ and $p = 0.20$. The Weibull distribution is observed
in the regime $0.20 \lesssim p_{\mathrm in} \lesssim 0.50$.
The statistics witness a
sharp transition from the Gaussian to the Weibull at $p_{\mathrm in} = 0.2$. The estimated parameters of
both the types of statistics and the detailed information of fitting
are addressed in \cite{supp}.
\subsection{Tail behaviour}
The nature of extreme value distribution can further be explained by the tail behaviour
of the parent distribution\cite{book_gev}. In the case of the Weibull distribution the tail of the
parent distribution follows a power-law with bounded maxima. Fig.~\ref{Fig4}
plots tail behaviour extracted from the real part of the eigenvalues for the matrices
associated with the Erd\"os-R\'enyi networks at different values of
$p_{\mathrm in}$, which confirms the Weibull statistics
as expected from the extremal eigenvalues
of this ensemble.
\subsection{Random matrices}
Fig.~\ref{RandomMatricesN400} plots $\rho(R_{\mathrm max})$ for random matrices generated
using Gaussian distributed random numbers under the strictly balanced condition. This matrix represents the case of when coupling weights of
inhibitory and excitatory connections are taken
from Gaussian distributed random numbers with mean $1-1/p_{\mathrm in}$
and 1, and standard deviation 0.05, as considered in the Ref.\cite{Rajan}.
The nature of the $R_{\mathrm max}$ distribution remains normal for $p_{\mathrm in}$ = 0,
whereas the Weibull distribution is observed for
$0.10 \lesssim p_{\mathrm in} \lesssim 0.50$. The robustness of the Weibull
statistics in this parameter regime is
indicated by a fixed value for the $\xi$ parameter. The mean and variance of
the data remains constant for the strictly balanced condition in
the range $0.10 \le p_{\mathrm in} \le 0.50$.
\begin{figure}[t]
\centerline{\includegraphics[width=0.9\columnwidth,height=4cm]{RandomMatricesN400.eps}}
\caption{(Colour online) Statistics of $R_{\mathrm max}$ at different values
of $p_{\mathrm in}$ for random matrices. The histograms are
numerical results; blue and red lines correspond to normal
and GEV fit, respectively. For each cases the statistics display the Weibull distribution,
except $p_{\mathrm in}=0$ for which normal distribution is observed.
All plots are obtained for 5000 realizations of matrices with $N=400$.}
\label{RandomMatricesN400}
\end{figure}
\begin{figure}[H]
\centerline{\includegraphics[width=0.9\columnwidth,height=4cm]{SwExtremeAv20N500.eps}}
\caption{(Colour online) Statistics of $R_{\mathrm max}$ at different values
$p_{\mathrm in}$ for small-world networks with the strictly balanced condition.
The histograms are numerical results; blue and red lines correspond to normal
and GEV fit, respectively. For each case, the statistics exhibit the Weibull distribution.
The rewiring probability is fixed at $p_{\mathrm r}$ = 0.0256 characterizing small-world
transition.
All plots are obtained for 5000 realizations of matrices with $N=500$ and $\langle k \rangle=20$.}
\label{SwExtremeAv20N500}
\end{figure}
This result confirms the robustness of the Weibull distribution for the strictly
balanced condition, as it leads to this distribution
independent of whether the matrix is modelled using a
random network, i.e. entries being 0 and 1, or
a random matrix, where entries are the Gaussian distributed random numbers.
The left panel of the phase diagram in Fig.~\ref{PhageSWregion}
illustrates this behaviour for various values
of $p$ and $p_{\mathrm in}$ for Erd\"os-R\'enyi networks under the strictly balanced condition.
Very small values of $p_{\mathrm in}$ may yield a situation in which some
columns have only positive entries, and which do not allow the strictly balanced condition
to be imposed, thereby making these values of $p_{\mathrm in}$ out of the scope
of the present study.
For some cases, the KS test accepts the normal as well as the Weibull distribution.
This happens because a particular shape parameter range, the Weibull distribution complies closely with
the normal distribution \cite{Dubey}.
For very high values of $p$, the conformation space of a network's structure is reduced,
which results in a lack of variation in network topology for an
ensemble. This might be a reason for the undefined shape of the statistics for
higher $p$ values.
Model systems having a lower average degree
could be modelled by the Weibull distribution.
which might be due to many configuration possibilities in
the ensemble leading to a more fluctuations in the $R_{\mathrm max}$, leading
to a smooth shape of the statistics.
The fact that most of the real-world networks are sparse \cite{sparse},
implies that they too can be modelled by this ensemble, which exhibits the Weibull distribution.
We further analyze the effects of different network configurations on the statistics of extremal
eigenvalues under the strictly balanced condition.
\begin{figure}
\centerline{
\includegraphics[width=.45\columnwidth,height=2.7cm]{PhageAllp.eps}
\includegraphics[width=0.45\columnwidth,height=2.7cm]{PhageSWregion.eps}}
\caption{(Left panel) Statistical behaviour of $R_{\mathrm max}$ for Erd\"os-R\'enyi networks with
$N=50$ at different values of $p_{\mathrm in}$ and $p$. Region 1 denotes the parameter region for which the
strictly balanced
condition cannot be defined, because some columns have only non-negative entries.
Region 2 corresponds to the distribution that is Weibull or close to Weibull. Region 3 stands for
the undefined distribution.
(Right panel) For small-world networks with $N=500$ and $\langle k \rangle=20$ at different values of $p_{\mathrm in}$ and rewiring
probability $p_{\mathrm r}$. Regions 4, 5 and 6
represent Gumbel (or close to it), Weibull
and normal distributions, respectively.
All plots are obtained for 5000 realizations of the networks.}
\label{PhageSWregion}
\end{figure}
\begin{figure}
\centerline{\includegraphics[width=0.9\columnwidth,height=2.2cm]{tailSW.eps}}
\caption{(Colour online) The tail behaviour of the real part of the eigenvalues at various values
of $p_{\mathrm r}$ for small-world networks under the strictly balanced condition.
For each case, $N=100$ and $p_{\mathrm in}=0.50$.}
\label{tailSW}
\end{figure}
\section{Small-World Networks}
First we consider small-world networks generated using the Watts-Strogatz algorithm.
Properties of many real-world networks, including the brain, are prescribed by this small-world
model \cite{Watts}. This type of network maintains the clustering coefficient close to that of the regular
lattices, whereas the average diameter is close to that of the random networks. Small-world networks have been found in C. elegans, cat cortex, and macaque
cortex and it has been shown that the efficiency of the brain to rapidly integrate information
from both locally and distantly specialized brain areas increases with the
organization of small-world topology \cite{sparse}.
We generate small-world networks using the Watts-Strogatz model \cite{Watts}.
For $N=500$ and $\langle k \rangle=20$,
the rewiring probability is chosen as $p_{\mathrm r}$ = 0.0256, which
corresponds to the small-world transition.
Fig.~\ref{SwExtremeAv20N500} confirms that
for all the $p_{\mathrm in}$ values, the statistics remains Weibull. All the
plots of Fig.\ref{SwExtremeAv20N500} except that which corresponds to
$p_{\mathrm in}$ = 0.0, satisfy the strictly balanced
condition.
The mean and variance of the $R_{\mathrm max}$ decrease monotonically with $p_{\mathrm in}$
for the strictly balanced condition.
However, the shape parameter ($\xi$) is most negative for $p_{\mathrm in}=0.20$, which
corresponds closely to a real brain situation \cite{DiverseInhibitory}, reflecting
a less right-skewed Weibull statistics.
Information pertaining to the parameters estimated in the statistics are referred in \cite{supp}.
\subsection{Phase diagram for small-world networks}
The phase diagram in Fig.~\ref{PhageSWregion} (right panel) demonstrates the behaviour of
the statistics for various values
of $p_{\mathrm r}$ and $p_{\mathrm in}$. For $p_{\mathrm r} = 0$, only
inhibitory couplings contribute to the statistics, and as a result the statistics
of $R_{\mathrm max}$ is found close to the
Gumbel.
The tail behaviour displays an exponential decay, thereby
supporting the observed Gumbel distribution (Fig.~\ref{tailSW}).
As $p_{\mathrm r}$ increases, the contribution of
structural variation also increases yielding more variation in the
$R_{\mathrm max}$ statistics. Because the nature of the statistics is
determined by the shape parameter $\xi$, for a fixed
value of disorder ($p_{\mathrm r}$), the occurrence of all the three statistics are possible.
Increased disorder in network structure leads to an enhancement of the
value of the $\sigma$ parameter (Eq.\ref{eq_gev}).
Interestingly, at the small-world transition, the statistics display a
transition from the Gumbel to the Weibull.
At the small-world transition,
the underlying network has sufficient randomness \cite{SJ_EPL2009},
which might be one of the reasons behind a drastic change in spectral behaviour
from the exponential. This indicates that the small-world
transition appears as a critical point in terms of the stability of
the underlying network in that ensemble.
The results for Erd\"os-R\'enyi networks differ slightly from
the networks generated using the small-world algorithm at $p_{\mathrm r}=$ 1.
For lesser $p_{\mathrm in}$ values, instead of the Weibull, the normal distribution is observed.
This might be due to the fixed total number of degrees occurring
in all the realizations for networks generated using the small-world model.
Note that for Erd\"os-R\'enyi networks,
at a particular $p$ value
there exists a fluctuation in the total degree for the different network
realizations.
\section{Scalefree Networks}
In this section we present results for the scalefree network architecture, generated
\begin{figure}[t]
\centerline{\includegraphics[width=0.9\columnwidth,height=4cm]{SFAv8st_ba.eps}}
\caption{(Colour online) Statistics of $R_{\mathrm max}$ for different values
of $p_{\mathrm in}$ for scalefree networks with the strictly balanced condition.
The histograms are numerical results; blue and red lines correspond to normal
and GEV fit, respectively. For each case, the statistics show the Weibull distribution,
except $p_{\mathrm in}$ = 0.
All plots are obtained for 5000 realizations of networks with $N=100$ and $\langle k \rangle=8$.}
\label{SFAv8st_ba}
\end{figure}
using the preferential growth algorithm \cite{Albert}. After introducing inhibitory
connections with the probability $p_{\mathrm in}$, the strictly balanced
condition is imposed.
Fig.~\ref{SFAv8st_ba} demonstrates that for all
$p_{\mathrm in}$ values, the statistics remain Weibull. However, the KS test accepts the
normal distribution as well for $p_{\mathrm in}$ =0. The mean and variance of the data
remains constant for the strictly balanced condition and for
the different $p_{\mathrm in}$ values (i.e. 0.30, 0.35, 0.40, 0.45 and 0.50).
The reason for discarding cases with $p_{\mathrm in} \le 0.20$ is that these values
do not yield enough realizations that satisfy the strictly balanced condition.
It happens due to the presence of a large number of nodes having a lesser degree,
as compared to the Erd\"os-R\'enyi networks. The observed statistics can further be explained from the
tail behaviour of the parent distributions. Fig.~\ref{tailSF} displays
a consistent power-law behaviour with the increase in
$\langle k \rangle$, whereas the shape
parameter of GEV monotonically decreases with an increase
in $p_{\mathrm in}$. The estimated parameters information is
referred to \cite{supp}.
\begin{figure}[t]
\centerline{\includegraphics[width=0.9\columnwidth,height=2.2cm]{tailSF.eps}}
\caption{(Colour online) The tail behaviour of the real part of the eigenvalues at different values
of $\langle k \rangle$ for scalefree networks with the strictly balanced condition.
For each case, $N=100$ and $p_{\mathrm in}=0.50$.}
\label{tailSF}
\end{figure}
\section{Discussions and Conclusion}
The nature of extreme values distribution for many real-world systems
is associated with various shape parameters.
The right skewness reflects the chances of occurrence of higher values.
The effect of the strictly balanced condition dominates the behaviour of
extreme value statistics,
in particular to a fixed Weibull statistics. This condition is so strong that
even changes in the interaction patterns do not affect the distribution behaviour.
Origin of the Weibull distribution for the strictly balanced condition could be explained
by the fact that the strictly balanced condition shifts
the outliers into the bulk of spectra,
i.e. $R_{\mathrm max}$ becomes bounded \cite{Rajan}.
The observed Weibull statistics is supported further by the tail behaviour of the parent
distribution which follows a power-law with bounded maxima.
The strictly balanced condition yields the Weibull distribution for networks
with structural variations such as Erd\"os-R\'enyi random networks and
scalefree networks. The 1-d lattice
structure lacks any structural variation in the ensemble leading to a deviation from
the Weibull distribution even for the strictly balanced condition.
We demonstrated that at the small-world transition, a network has sufficient
structural variations or {\it randomness} leading to a
less right-skewed statistics governed by the
Weibull distribution, consequently making the system more stable.
The Weibull distribution does not display any significant change with the change
in the average degree of the
network in the balanced condition, whereas previous work \cite{Extreme}
demonstrates a deviation from the Weibull to the Fr\'echet
distribution via Gumbel as connectivity of the network increases by keeping
$p_{\mathrm in}$ fixed at $0.5$.
The reasons for the networks with a lower $p$ value (corresponding to a lower average degree)
following the Weibull distribution is that
such matrices have fewer fluctuations around the strictly balanced condition and
exhibit similar statistics to that observed for the strictly balanced condition.
However, higher values of
$p$ yield matrices with more deviations than those satisfying
the strictly balanced condition, and as a result lead to an increased number
of outliers from the bulk part of the eigenvalues, consequently
resulting in a transition from the Weibull statistics.
The extreme value theory might enhance our understanding of stability properties of
real brain systems.
For instance, model
networks capturing real brain network properties, such as small-world architecture and
a 20-80\% inhibitory to excitatory ratio, tend to witness fewer right-skewed $R_{\mathrm max}$
statistics compared to other possible
values of $p_{\mathrm in}$. The higher values of $R_{\mathrm max}$ are more
likely to generate right-skewed statistics with higher variances. The variances can be managed
by weight scaling the connections.
The combined framework of the network
architecture and the strictly balanced situation thus emulates the existence of stable statistics
upon capturing realistic brain scenario.
Future studies will incorporate other network architectures, particularly those
having community structures \cite{community}. Recently the stability of eco-systems has been analyzed using
the spectral properties of
underlying matrices \cite{Allesina}. The framework presented in this Letter
can be extended to have a proper understanding
of other such complex systems \cite{unpublished}.
\section{Acknowledgment} SJ thanks the DST and CSIR for funding.
SKD acknowledges the University grants commission, India for
financial support.
| -19,495.723493 |
[
-3.431640625,
3.083984375
] | 25.473684 |
[
-2.146484375,
0.6806640625,
-2.353515625,
-5.109375,
-1.2412109375,
8.203125
] |
[
3.498046875,
8.46875,
3.2578125,
7.15625
] | 254 | 3,836 |
[
-3.3984375,
4.0859375
] | 22.292971 |
[
-6.171875,
-4.15625,
-4.296875,
-2.6953125,
1.8212890625,
12.7421875
] | 1.620494 | 5.728785 | 21.611053 | 1.42179 |
[
1.9344799518585205
] | -15,425.784535 | 5.871741 | -18,933.267387 | 3.551767 | 5.43753 |
[
-2.857421875,
-3.8046875,
-3.421875,
-4.41796875,
2.541015625,
11.546875
] |
[
-5.171875,
-2.09765625,
-2.0546875,
-1.4072265625,
3.455078125,
4.57421875
] | |
BkiUfgM5qhDCiwNLy3CU
|
\section{Introduction}
\let\thefootnote\relax\footnotetext{
This material is based upon work supported by the Assistant Secretary of Defense for Research and Engineering under Air Force Contract No. FA8702-15-D-0001, National Science Foundation CCF-1533644, and United States Air Force Research Laboratory Cooperative Agreement Number FA8750-19-2-1000. Any opinions, findings, conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the Assistant Secretary of Defense for Research and Engineering, the National Science Foundation, or the United States Air Force. The U.S. Government is authorized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation herein.
}
AI challenges developed by AI experts are a primary tool for advancing AI and engaging the AI ecosystem on important problems. AI challenges are also a key driver of AI computational requirements. Challenges such as YOHO~\cite{yoho}, MNIST~\cite{mnist}, HPC Challenge~\cite{hpcc}, Graph Challenge \cite{samsi2017static,kao2017streaming,kepner2019sparse}, ImageNet~\cite{imagenet} and VAST~\cite{vast1,vast2} have played important roles in driving progress in fields as diverse as machine learning, high performance computing, and visual analytics. YOHO is the Linguistic Data Consortium database for voice verification systems and has been a critical enabler of speech research. The MNIST database of handwritten letters has been a bedrock of the computer vision research community for two decades. HPC Challenge has been used by the supercomputing community to benchmark and test the largest supercomputers in the world as well as stimulate research on the new parallel programming environments. Graph Challenge has led to the development of award winning software and hardware \cite{kepner2016mathematical,davis2019algorithm,bulucc2017design}.
\begin{figure}[htb]
\centering
\includegraphics[width=\columnwidth]{figures/ChallengeMotivation.pdf}
\caption{Challenging AI problems are best solved by enabling diverse teams to work on them. The MIT Air Force AI Accelerator teams are creating many challenges to engage the entire AI ecosystem and maximize the number of solutions.}
\label{fig:ChallengeMotivation}
\end{figure}
ImageNet populated an image dataset according to the WordNet hierarchy consisting of over 100,000 meaningful concepts (called synonym sets or synsets)~\cite{imagenet}, with an average of 1000 images per synset, and has become a critical enabler of vision research. The VAST Challenge is an annual visual analytics challenge that has been held every year since 2006; each year, VAST offers a new topic and submissions are processed like conference papers. With more AI teams working on more challenges, the gathered solutions significantly increase the probability of success (Figure~\ref{fig:ChallengeMotivation}).
The Maneuver Identification Challenge is part of the MIT Air Force AI Accelerator. The AI Accelerator is designed to make fundamental advances in artificial intelligence to improve Department of the Air Force operations while also addressing broader societal needs. AI Accelerator research involves interdisciplinary teams, including Air Force personnel, who collaborate across disparate fields of AI to create new algorithms, technologies, and solutions. A key element of the AI Accelerator is the production of AI challenges to engage the broader AI community and to institutionalize AI data readiness (aia.mit.edu/challenges). The AI Accelerator includes challenges for optimal scheduling of training \cite{CDO2020}, magnetic navigation \cite{gnadt2020signal}, and weather prediction \cite{veillette2020sevir}. A principal benefit of AI challenges is creating AI ready data on behalf of a community. Transforming raw logs or observations into AI ready data is often the majority of the effort in an AI project. A key aspect of becoming an effective AI organization is the regular production of AI ready data.
\begin{table*}
\caption{Example time, position, velocity, and orientation trajectory data for a pilot training session from a flight simulator.}
\centering
\includegraphics[width=1.5\columnwidth]{figures/TSVexample.pdf}
\label{tab:TSVexample}
\end{table*}
AI algorithms that identify maneuvers from trajectory data could play an important role in improving flight safety and pilot training. The Maneuver Identification (ID) Challenge has the potential to enhance and personalize simulator training by providing mid-flight and post-flight feedback to students. The Air Force continues to face a pilot shortage. Nascent technologies to individualize and automate some aspects of training present an opportunity to mitigate that shortage. The Air Force began Pilot Training Next (PTN) to explore the use of nascent technologies such as virtual reality flight simulators and novel instructional tools in small test cohorts for personalizing training and improving student access to training resources \cite{SAFCOPTNtechsummary}. Specifically, the value of each flight hour in a training aircraft or virtual reality simulator can be improved by providing additional teaching tools to instructors. For example, automating flight maneuver recognition and recommending grades for each maneuver would make each debrief more efficient and effective. On the other hand, fully-automated basic maneuver recognition and grading may enable simple flight training for all levels of potential pilot candidates prior to arriving at formal pilot training.
Maneuver identification from trajectory data is a significant technical challenge. It is difficult to simulate realistic flight tracks of simple maneuvers, such as beelines, racetracks, circles, s-curves, figure eights, and dives \cite{6643201}. Determination of physical feasibility of simulated tracks presents its own challenges \cite{ure2009feasible}, as does identifying types of aircraft from tracks \cite{huang2018aircraft}. A logistically difficult, but practical solution to obtaining realistic trajectory data is to collect it from real pilots in flight simulators. Working with PTN, maneuver-id.mit.edu has compiled thousands of trajectories collected from pilots practicing in flight simulators, descriptions of maneuvers, and examples of these maneuvers performed by experienced pilots. Each trajectory consists of times, positions, velocities, and aircraft orientations normalized to a common coordinate system. Construction of the data set required significant data architecture to transform flight simulator logs into AI ready data, which included using a supercomputer for deduplication and data conditioning. These data can be use to support a variety of challenges. The first challenge is separating physically plausible (good) trajectories from unfeasible (bad) trajectories. Human labeled good and bad trajectories are provided to aid in this task. Subsequent challenges are to label trajectories with their intended maneuvers and to assess the quality of those maneuvers.
\section{Data}
The Maneuver Identification Challenge must provide a significant amount of data to train an AI. PTN has gathered thousands of distinct pilot training sessions from hundreds of hours on flight simulators, flown by students, trained pilots, and instructors. The sessions were flown using the Lockheed Martin Prepar3d flight simulator software (prepar3d.com). Commercial flight simulators of this type are not designed to dynamically emit data in a multi-pilot training context. PTN has developed a novel state-of-the-art data logging system that aggregates flight simulator data from multiple simultaneous students.
\begin{figure}[htb]
\centering
\includegraphics[width=\columnwidth]{figures/12000114002.png}
\includegraphics[width=\columnwidth]{figures/13000045001.png}
\caption{Examples of PNG files showing an aerial view of the flight trajectory during the session. Trajectories begin with a $\times$ symbol and end with a $\bigtriangleup$.}
\label{fig:PNGexample}
\end{figure}
In general, log data is not AI ready, and requires significant transformation. The PTN log data is stored in diverse SQL database tables that were periodically check-pointed onto cloud storage and then transferred to the MIT SuperCloud \cite{reuther2018interactive}. The SQL checkpoints were launched using the MIT SuperCloud interactive database management system \cite{prout2015enabling}. Each checkpoint consisted of hundreds of tables, some containing millions of records, with dozens of distinct columns. Determination of the relevant data was accomplished by extracting the data in SQL tables using the D4M system \cite{gadepally2018d4m} and performing a subsequent dimensional data analysis \cite{gadepally2014bigdata}.
The extracted trajectory data consists of co-blended information from many pilot sessions. For redundancy, the logging system created duplicate entries and data from idle sessions. Deduplication involved extracting each session and then doing a full pairwise comparison of every record with every other record in the session. Using 256 64-core compute nodes (16,384 cores total) on the MIT SuperCloud all the data could be deduplicated in a few hours. The code was implemented using Matlab/Octave with the pMatlab parallel library \cite{Kepner2009}. A typical run could be launched in a few seconds using the MIT SuperCloud triples-mode hierarchical launching system \cite{reuther2018interactive}. Typical launch parameters were [256 16 4], corresponding to 256 nodes, 16 Matlab/Octave processes per node, and 4 OpenMP threads per process. On each node, the 16 processes were pinned to 4 adjacent cores to minimize interprocess contention and maximize cache locality for the OpenMP threads \cite{byun2019optimizing}. Each Matlab/Octave process was assigned a subset of the data to deduplicate. Within each Matlab/Octave process, the underlying OpenMP parallelism was used on 4 cores to accelerate the processing. After deduplication, sessions with less than 1000 contiguous valid entries (approximately 200 seconds) were excised, and the data was normalized and saved as tabular data files for each session.
The trajectory data has been made available through maneuver-id.mit.edu as tab separated value (TSV) files. In addition, top-down views of the trajectories are stored as portable network graphics (PNG) files. The TSV files consist of a plain text table containing the times, positions, velocities, and orientations of the aircraft throughout the flight session. The files can be read by most data processing systems and viewed in any spreadsheet program. Table~\ref{tab:TSVexample} shows an example of a TSV file for a single session. The PNG files contain images with an aerial view of the position of the aircraft in kilometers during the session. Figure~\ref{fig:PNGexample} shows examples of a PNG file for two sessions.
The data is being shared using standard trusted data sharing best practices \cite{kepner2021zero}
\begin{itemize}
\item Data is made available in curated repositories
\item Using standard anonymization methods where needed: hashing, sampling, and/or simulation
\item Registration with a repository and demonstration of legitimate research need
\item Recipients legally agree to neither repost a corpus nor deanonymize data
\item Recipients can publish analysis and data examples necessary to review research
\item Recipients agree to cite the repository and provide publications back to the repository
\item Repositories can curate enriched products developed by researchers
\end{itemize}
In accordance with the above practices, the trajectory data has been anonymized and a data sharing agreement is required by researchers who want to obtain the data to confirm the legitimacy of the research and to ensure that they agree to respect the anonymity. Once participants confirm the agreement they gain access to the data. Currently, there are two data sets containing thousands of files provided by PTN labeled 12000000000 and 13000000000, as well as close to 30 data files from sessions conducted at Vance Air Force Base. More may be added in the future.
\begin{figure}[htb]
\centering
\includegraphics[width=0.75\columnwidth]{figures/DataverPlotX.jpg}
\includegraphics[width=0.75\columnwidth]{figures/DataverPlotY.jpg}
\includegraphics[width=0.75\columnwidth]{figures/DataverPlotZ.jpg}
\caption{Plots comparing the (vx,vy,vz) velocities with the derivatives (dx/dt,dy/dt,dz/dt) computed from the (x,y,z) positions of the trajectory data.}
\label{fig:DataverPlot}
\end{figure}
\subsection{Normalization}
The data has been normalized to make it accessible to the broadest range of AI researchers. Each session has been set to start at one second because some data processing systems treat 0 as empty. For the same reason, the minimum spatial coordinate values for each session has been set to (1,1,1). All sessions were placed in a common (x,y,z) Cartesian coordinate system. The longitude, latitude, and altitude have been converted to units of meters. The conversion was done with MATLAB using the {\tt wgsEllipsoid} and {\tt Geodetic2enu} functions. The following mathematical conversions were done to set the minimum coordinate values
\begin{verbatim}
xEast = xEast - min(xEast) + 1
yNorth = yNorth - min(yNorth) + 1
zUp = zUp - min(zUp) + 1
\end{verbatim}
A caveat of the altitude normalization is that some maneuvers are altitude dependent, which is not recoverable from this normalization. The velocities are in units of meters/second and the aircraft orientations are in degrees.
\subsection{Verification}
Flight simulators use a variety of coordinate systems that need to be deduced from the data. For example, some simulators may define `y' as up and others may define `z' as up. Converting from longitude, latitude, and altitude into (x,y,z) Cartesian coordinates allows for cross validation of coordinates, velocities, and orientations. The velocities (vx,vy,vz) were verified by comparing with the derivatives (dx/dt,dy/dt,dz/dt) computed from the (x,y,z) positions (Figure~\ref{fig:DataverPlot}). The heading was verified by comparing with
$$
{\rm tan}^{-1}({\rm vx/vy})
$$
Likewise, the pitch was verified by comparing with
$$
{\rm tan}^{-1}\left(\frac{{\rm vz}}{\sqrt{{\rm vx}^2 + {\rm vy}^2}}\right)
$$
\section{Maneuvers}
\begin{figure}[htb]
\centering
\includegraphics[width=\columnwidth]{figures/SampleDescPhoto.jpg}
\caption{Sample description photo of a loop maneuver (reproduced from \cite{AETC-11-248}).}
\label{fig:SampleDescPhoto}
\end{figure}
\begin{figure}[htb]
\centering
\includegraphics[width=\columnwidth]{figures/SampleIntVid.jpg}
\caption{Screenshot from a video of an experienced pilot flying in the simulator taken from the vantage point of the virtual cockpit.}
\label{fig:SampleIntVid}
\end{figure}
\begin{figure}[htb]
\centering
\includegraphics[width=\columnwidth]{figures/SampleExtVid.jpg}
\caption{Screenshot from a video of an experienced pilot flying in the simulator taken from the vantage point of an outside observer.}
\label{fig:SampleExtVid}
\end{figure}
\begin{figure}[htb]
\centering
\includegraphics[width=\columnwidth]{figures/ManeuverTable.jpg}
\caption{Sample table listing various maneuvers. The airspeed is in both knots, the standard for aviators, and meters/second, the standard for scientists. The torque settings are provided for additional context. The altitude required gives the altitude needed to execute the maneuver, where '+' indicates 'above' and '-' indicates 'below'. For example, the Barrel Roll requires 2,000 feet of clearance above and 1,000 feet below.}
\label{fig:ManeuverTable}
\end{figure}
Maneuver identification requires a representative set of maneuvers for AI researchers to draw upon. A database of maneuvers has been compiled drawing from pilot training documentation \cite{AETC-11-248}. This database contains maneuver descriptions for 18 categories of maneuvers detailing both the trajectory of the maneuver and how to execute it as a pilot. Sample trajectory data, as well as photos and videos of expert pilots in flight simulations, have also been made available.
Maneuver-id.mit.edu provides details on common pilot training maneuvers. For selected maneuvers, there are trajectory descriptions (Figure~\ref{fig:SampleDescPhoto}) and how-to instructions, as well as external and internal videos of experienced pilots performing the maneuvers. Viewers can watch experienced pilots performing maneuvers from the pilot vantage point inside the simulator cockpit (Figure~\ref{fig:SampleIntVid}) as well as from a third person view outside of the aircraft (Figure~\ref{fig:SampleExtVid}). There is also trajectory data for these maneuvers in the form of TSV plain text tables and PNG files. These maneuvers are organized in a table that lists all of the maneuvers with links to descriptions and examples (Figure~\ref{fig:ManeuverTable}).
\begin{figure*}[htb]
\centering
\includegraphics[width=1.5\columnwidth]{figures/Challenge1.jpg}
\caption{Examples of good data (green) and bad data (red), sorted manually. The bad data includes straight lines, jumps, and impossible maneuvers.}
\label{fig:Challenge1}
\end{figure*}
\section{Challenges}
PTN has begun the process of creating and testing algorithms for maneuver detection and scoring. In their initial work, PTN researchers have focused on loops because of their discernible trajectory both visually and based on time series data parameters \cite{SAFCOPTNtechsummary}. The broad methodology consists of comparing the flight simulator data to a convolution kernel of a known maneuver example, and determining whether the two sets matched. The convolution kernel was obtained through the virtual instructor pilot that PTN developed, and the selected example is a demonstration loop and its mirror image. Matching the flight simulator data to the kernel, each row is compared and 'time-matched', the distance between the time-matched points is found and corrected for variability, and the total distance is acquired by summing up the individual distances. This value is a measure of similarity between the flight data and the maneuver. While these methods were generally successful, more could be done to identify and account for false negatives and false positives.
Similar work has been done outside PTN to identify maneuvers which can provide researchers looking to participate in the challenge with sample approaches. One such project was motivated by the importance of maneuver detection for load analyses on aircraft. Researchers analyzed the flight data and compared it to a library of maneuvers \cite{WANG2015133}. Other methods of maneuver identification, such as neural networks, have also been studied \cite{RODIN199295}. Neural networks are attractive because of their fault tolerance, which means the occasional wrong or approximated data point will not skew the results too significantly. Neural networks have been used for classification of underwater sonar tracks and radar tracks, and for target track estimation \cite{RODIN199295,GORMAN198875,farhat1986optical,farhat1989echo}.
Automated maneuver grading is another objective of this challenge. The current method of student grading is labor-intensive and subjective as it relies on a very limited number of instructor pilots manually taking notes on each maneuver performed by a student. Differences in instructor pilot techniques, preferences, and grading inject some subjectivity into the grading \cite{SAFCOPTNtechsummary}. As the number of students increases, it is harder for instructor pilots to give each student proper attention for them to improve and learn. With automated maneuver scoring, novice student pilots could achieve a basic level of proficiency through automated feedback and grading, then rely on instructor pilots for the more advanced ``art'' of flying each maneuver. PTN has outlined a general methodology on maneuver scoring which would require having time series data and accounting for co-variance between the selected trajectory dimensions \cite{PTNv2}. The methodology consists of three main components: characterizing each dimension's behavior, characterizing each dimension's variability, and combining the first two into a single score. Scoring would consist of finding a best fit function and the corresponding acceptable variance according to expert pilot data in comparison with student runs.
Based on this prior work, maneuver-id.mit.edu has identified three challenges for the community. The first challenge deals with sorting the physically feasible (good) and physically infeasible (bad) data into separate sets based on the presence of unbroken trajectories (good) and identifiable maneuvers (good) and straight lines (bad), jumps (bad), and impossible maneuvers (bad). Examples of these good and bad data are shown in Figure~\ref{fig:Challenge1}. Currently, there is good and bad trajectory truth data that has been sorted and verified manually. The manually sorted truth data is included as part of the data set. The second challenge deals with identifying which maneuver the pilot is attempting to execute. Truth data for this challenge is not currently available and may be released later. The third challenge deals with scoring the pilot once the maneuver has been identified. This could greatly benefit the efficiency and quality of the pilot training education. Truth data for this challenge is not currently available and may be released later.
\section{Summary}
AI challenges and data sets are important tools for applying AI to pilot training and offer many potential benefits. Effectively identifying and assessing maneuvers from aircraft data is one such challenge. Maneuver-id.mit.edu has compiled a multitude of resources that can be used to construct relevant challenges. These include flight simulator data containing positions, velocities, and orientations; classifications of good and bad simulator data; maneuver descriptions; and flight simulator data and videos of example maneuvers performed by experienced pilots. The AI community can use these resources to highlight their innovations for classifying good and bad trajectory data, identifying maneuvers, and assessing maneuvers.
\section*{Acknowledgments}
The authors wish to acknowledge the following individuals for their contributions and support: Sean Anderson, Ross Allen, Bob Bond, Jeff Gottschalk, Tucker Hamilton, Chris Hill, Mike Kanaan, Tim Kraska, Charles Leiserson, Mimi McClure, Christian Prothmann, John Radovan, Steve Rejto, Daniela Rus, Allan Vanterpool, Marc Zissman, and the MIT SuperCloud team: Bill Arcand, Bill Bergeron, David Bestor, Chansup Byun, Nathan Frey, Michael Houle, Matthew Hubbell, Hayden Jananthan, Anna Klein, Joseph McDonald, Peter Michaleas, Julie Mullen, Andrew Prout, Antonio Rosa, Albert Reuther, Matthew Weiss.
\bibliographystyle{ieeetr}
| -11,093.678331 |
[
-0.5615234375,
0.94091796875
] | 40.25974 |
[
-3.208984375,
0.73486328125,
-1.048828125,
-3.16796875,
-0.11773681640625,
4.69921875
] |
[
1.462890625,
4.74609375,
1.818359375,
4.6171875
] | 180 | 3,072 |
[
-1.4931640625,
1.0732421875
] | 20.240507 |
[
-4.93359375,
-1.0361328125,
-2.072265625,
-1.490234375,
0.291748046875,
7.3125
] | 0.6384 | 28.435732 | 35.839844 | 1.335505 |
[
1.9697891473770142
] | -9,662.771752 | 6.118815 | -10,750.563899 | 0.2128 | 5.985461 |
[
-3.798828125,
-2.986328125,
-2.390625,
-3.326171875,
2.80859375,
9.0234375
] |
[
-6.171875,
-3.16015625,
-3.0078125,
-2.177734375,
4.25,
6.98046875
] | |
BkiUa2nxK4sA-5Y3qUsP
|
\chapter{Introduction}
\section{Historical context and abstract of the thesis}
Nahm transform is a non-linear analog for instantons of the usual Fourier transform on functions.
It has been extensively studied starting from the beginning of the 1980's, inspired by the seminal
work of M. F. Atiyah, V. Drinfeld, N. J. Hitchin and Yu. I. Manin on a correspondence
(the \emph{ADHM-transform}) between finite-energy solutions of the \emph{Yang-Mills equations}
and some algebraic data (see \cite{adhm}, \cite{Don-Kronh}).
The Yang-Mills equations are the anti-self-duality equations for a unitary connection on a
Hermitian vector bundle defined over $\R^{4}$; their finite-energy solutions are called \emph{instantons}.
Since then, it turned out that the general picture concerning this correspondence is as follows: let $X$ be
any manifold obtained as a quotient of $\R^{4}$ by a closed additive subgroup $\Lambda$. The solutions of the
Yang-Mills equations invariant by $\Lambda$ (that are clearly not of finite energy in the case $\Lambda \neq \{ 0 \}$) can
be identified in an obvious manner to solutions of a system of differential equations on $X$, called the
\emph{reduction} of the Yang-Mills equations. On the other hand, denoting by $(\R^{4})^{\ast }$ the
dual of the vector space $\R^{4}$, $\Lambda$ determines a closed additive subgroup $\Lambda^{\ast}$ called the
\emph{dual subgroup} by saying that an element $\xi \in (\R^{4})^{\ast }$ is in $\Lambda^{\ast}$ if and only if $\xi (\lambda ) \in \Z$
for all $\lambda \in \Lambda$. Hence, we can form the \emph{dual manifold} $X^{\ast } = (\R^{4})^{\ast }/\Lambda^{\ast }$ of $X$, that also
admits a reduction of the Yang-Mills equations. Nahm transform is then a procedure that maps
solutions of the reduced equations on $X$ to solutions of the reduced equations on $X^{\ast}$ bijectively up to
overall gauge transformations on both sides. One remarks that there is a canonical isomorphism between
$((\R^{4})^{\ast })^{\ast }$ and $\R^{4}$, as well as between $(\Lambda^{\ast })^{\ast }$ and $\Lambda$. Therefore, if we start from
a solution of the reduced equations on $X$ and iterate Nahm transform twice, we again get a solution of the
reduced equations on $X$. One important property analogous to usual Fourier transform is that in some cases
the solution we get this way is, up to a coordinate change $x \mapsto -x$, known to be the solution we started
with; that is, Nahm transform is (up to a sign) \emph{involutive}. Moreover, in some cases one knows that
the moduli spaces of solutions of the reduced equations modulo gauge transformations on $X$ and on $X^{\ast}$
are smooth hyper-K\"ahler manifolds with respect to the metric induced by $L^{2}$-norm and the
complex structures induced by $\R^{4}$; Nahm transform is then a hyper-K\"ahler isometry between these moduli
spaces. This is to be compared with Parseval's theorem which states that usual Fourier transform defines an
isometry between $L^{2}$-spaces of functions.
Putting $\Lambda = \{ 0 \}$, one gets $X=\R^{4}$ and $\Lambda^{\ast}=\R^{4}$, so $X^{\ast}=\{ 0 \}$. In this case,
Nahm transform reduces to the ADHM-transform.
There are several other examples of Nahm transform in the literature for different subgroups of $\R^{4}$;
for a nice exposition of these, see the survey paper \cite{jarsur} of M. Jardim. In this work, we are
concerned with the case $\Lambda=\R^{2}$. In this case, the base manifold is $X=\R^{2}$, and its dual $X^{\ast}$ is
another copy of the real plane that we shall denote by $\hat{\R}^{2}$. These are non-compact manifolds,
with compactifications the Riemann spheres $\CP$ and $\CPt$ respectively. The reduction of the original
(Yang-Mills) equations can be viewed in two different ways depending on the complex structure that we choose:
they are the equations defining an integrable connection with harmonic metric, or equivalently, those defining
a Higgs bundle with Hermitian-Einstein metric. Now, it turns out that there are no smooth
solutions on the Riemann sphere of either one of these equations except for the trivial ones (c.f. \cite{Hit}).
However, there are solutions having prescribed singularities in some points, and the solutions of one equation
are still in correspondence with those of the other: this is proved by O. Biquard and Ph. Boalch in
\cite{Biq-Boa}. We establish, under some hypotheses on the singularity behavior, Nahm transform for singular
integrable connections (or equivalently, singular Higgs bundles) on the Riemann sphere.
Note that Nahm transform for singular objects
have already been studied by O. Biquard and M. Jardim in \cite{Biq-Jar} and by S. A. Cherkis and A. Kapustin in
\cite{Cher-Kap}. On the other hand, using different techniques, B. Malgrange has defined in \cite{Mal} a so-called
Fourier-Laplace transform for integrable connections with singularities on the Riemann sphere behaving in the
same manner on the level of singularity data as the one we define here. It is therefore very natural to
believe that these two transforms actually agree. One difference between these works is, however, the
transformation of a parabolic structure and an adapted harmonic metric at the singularities in our case;
for details, see Section \ref{sectharmmetr}.
The construction follows the main ideas of other Nahm transforms found in literature. Namely,
in Section \ref{sectFr} we define positive and negative spinor bundles $S^{\pm}$ over $\CP$, as well as a Dirac operator
$$
\Dir: S^{+} \otimes E \lra S^{-} \otimes E .
$$
We then let $\xi \in \hat{\C} \setminus \hat{P}$ be a parameter, where $\hat{P}$ is the singular locus of the
transformed objects, and for all $\xi$ twist the operator $\Dir$ by some flat connection to
obtain a family of operators $\Dir_{\xi}$. In Section \ref{sectpfFr} we prove that the kernel of these twisted
operators vanish and that the cokernels form a finite-dimensional space. Furthermore, this dimension is
independent of $\xi$; we then define the transformed vector bundle $\hat{E}$ on $\hat{\C}$ as the vector bundle
with fiber over $\xi$ given by $coKer(\Dir_{\xi})$. In Section \ref{l2coh} we carry out an analog of $L^{2}$-Hodge
theory of a compact K\"ahlerian manifold in this case; namely we establish an isomorphism between this cokernel
and the first $L^{2}$-cohomology of an elliptic complex, as well as harmonic $1$-forms with respect to the
Laplacian of the Dirac operator. We then go on to define the transformed flat bundle
and the transformed Hermitian metric in Section \ref{flattr}, and we extend the flat bundle over
the singularities -- so defining the transformed meromorphic integrable connection -- in Section \ref{flatext}.
The transformed metric is then shown to be Hermitian-Einstein in Section \ref{sectharm}.
Next, in Section \ref{ident} we give a completely explicit description of the fibers of the transformed
bundle, first in terms of hypercohomology of a sheaf map, then in terms of the corresponding spectral set.
Then come the constructions of the extensions of the
transformed Higgs bundle to the singular points (Section \ref{ext}). This allows us to
obtain the singularity data of the transformed Higgs bundle in Sections \ref{sing} and \ref{parweightsect},
and we complete the transform by computing the topology of the transformed Higgs bundle in Section
\ref{secttop}. Finally, Chapter \ref{chinv} deals with the involutivity property of the transform.
\section{Integrable connection point of view} \label{flatintro}
Let $\C$ be the complex line, with its natural holomorphic coordinate $z=x+iy$ and
Euclidean metric $\vert\mbox{d}z \vert^{2}$; and let $\CP$ be the complex projective line.
Let $E \to \CP$ be a rank $r$ holomorphic vector bundle on the Riemann sphere,
and $D$ be a meromorphic integrable connection on it, with \emph{first order} or \emph{logarithmic singularities}
at the points of a finite set $\{ p_1,\ldots,p_{n} \} =P \subset \C$ and a second
order singularity at infinity.
In other words, on a small disk $\Delta(p_{j},\varepsilon)$ centered at
$p_j \in P$ in a holomorphic basis $\{ \tau^j_k \}_{k=1,...,r}$ of $E$, $D$ is
of the form $D^{j}+b^{j}$ where $b^{j}$ is a holomorphic $1$-form on the disk and
\begin{eqnarray} \label{dpj}
D^{j}=\mbox{d}+ \frac{A^{j}}{z-p_{j}} \d z \land .
\end{eqnarray}
We suppose furthermore that $A^{j}$ is diagonal:
$$
A^{j}= \begin{pmatrix}
0 & & & & & \\
& \ddots & & & & \\
& & 0 & & & \\
& & & \mu^{j}_{r_{j}+1} & & \\
& & & & \ddots & \\
& & & & & \mu^{j}_{r}
\end{pmatrix};
$$
it is called the \emph{residue} of $D$ at $p_{j}$, and $1 \leq
r-r_{j}\leq r$ is the rank of $A^{j}$. For convenience, we put
$\mu^{j}_{1}=\ldots=\mu^{j}_{r_{j}}=0$, so that
$A^{j}=diag(\mu^{j}_{k})_{k=1,\ldots r}$. We will often make use of the holomorphic local
decomposition
\begin{eqnarray} \label{regsing}
E^{j} = E^{j}_{reg} \oplus E^{j}_{sing},
\end{eqnarray}
into the \emph{regular} and \emph{singular components} of $E$ near $p_{j}$; here by definition $E^{j}_{reg}$ is the
holomorphic subbundle of $E^{j}= E|_{\Delta(p_{j},\varepsilon)}$ spanned by $\{ \tau^j_k \}_{k=1,...,r_{j}}$,
and $E^{j}_{sing}$ is the one spanned by $\{ \tau^j_k \}_{k=r_{j}+1,...,r}$. Intrinsically,
$E^{j}_{sing}$ is the sum of the generalized eigenspaces corresponding to all eigenvalues converging to
infinity of the integrable connection, whereas $E^{j}_{reg}$ is the sum of the generalized eigenspaces
corresponding to the eigenvalues that remain bounded.
In a similar manner, at infinity $D$ is supposed to be equal (up to a holomorphic term)
to a meromorphic local model having a \emph{second order pole},
so that in a holomorphic basis $\{ \tau_{k}^{\infty} \}_{k=1,\ldots,r}$ on a disk $\C \setminus \Delta(0,R)$
corresponding to a standard neighborhood of infinity in $\CP$,
it is of the form $D=D^{\infty } + b^{\infty }$ where $b^{\infty }$ is now a holomorphic $1$-form in the given neighborhood
of infinity, and
\begin{eqnarray} \label{dinf}
D^{\infty }= \mbox{d} + \left( A + \frac{C}{z} \right) \d z \land
\end{eqnarray}
is the second order model with diagonal leading term
$$
A = \begin{pmatrix}
\xi_{1} & & & & & & & \\
& \ddots & & & & & & \\
& & \xi_{1} & & & & & \\
& & & \ddots & & & & \\
& & & & \ddots & & & \\
& & & & & \xi_{n'} & & \\
& & & & & & \ddots & \\
& & & & & & & \xi_{n'}
\end{pmatrix}
$$
and residue
$$
C = \begin{pmatrix}
\mu_1^{\infty} & & \\
& \ddots & \\
& & \mu_r^{\infty}
\end{pmatrix}.
$$
Here $\{ \xi_{l} \}_{l=1}^{n'}$ are the distinct eigenvalues of $A$. Each $\xi_{l }$ appears in neighboring positions
$k=1+a_{l},\ldots,a_{l+1}$, in particular its multiplicity is $m_{l}=a_{l+1}-a_{l}$. Of course, we must then have
$a_{1}=0$ and $a_{n'+1}=r$. In line with the above notation, we set $r_{\infty }=0$ and
$C=diag( \mu_{k}^{\infty})_{k=1,\ldots,r}$. Furthermore, we will write
$$
A=diag( \{ \xi_{l}, m_{l} \} )_{l=1,\ldots,n'}
$$
for the diagonal matrix $A$ as given above, meaning that $A$ is diagonal with $m_{l}$ neighboring eigenvalues
equal to $\xi_{l}$.
\begin{defn}
The integrable connections having singularities near the points of $P \cup \{ \infty \}$ as described above will
be called \emph{meromorphic integrable connections with logarithmic singularities in $P$ and a second-order
singularity at infinity}, or for simplicity \emph{meromorphic integrable connections} although they are by far not
all the meromorphic integrable connections.
\end{defn}
\section[The transform of the integrable connection]{The transform of the meromorphic integrable connection}\label{transmer}
Let $(E,D)$ be a stable vector bundle with a meromorphic integrable connection on the sphere.
Our aim in this paper is to define another complex bundle $\hat{E}$ with a meromorphic connection $\hat{D}$ on
the sphere out of $(E,D)$, which we call the \emph{transformed meromorphic integrable connection}.
Just as the initial connection, the transformed one will also admit a finite number of simple poles
in points of the line and a second-order pole at infinity.
In order to define the transformed vector bundle $\hat{E}$, first we need to set some notation.
Let $\hat{\C}$ be another copy of $\C$. (The importance of distinguishing the two copies of $\C$ is to help us
avoid confusions.) For a parameter $\xi \in \hat{\C}$, consider the following deformation of $D$:
\begin{eqnarray} \label{intdef}
D^{int}_{\xi } = D - \xi \d z \land ,
\end{eqnarray}
where $\xi :E \ra E$ stands for multiplication by $\xi$.
Since we only change the $(1,0)$-part of $D$, and by an endomorphism that is independent of $z$, this is then
another meromorphic integrable connection, with the same underlying
holomorphic bundle as for $D$. Furthermore, its unitary and self-adjoint parts are given by
\begin{align}
D^{+}_{\xi } & = D^{+} - \frac{\xi}{2} \d z + \frac{\bar{\xi}}{2} \d \bar{z} \label{intdefunit}\\
\Phi^{int}_{\xi } & = \Phi - \frac{\xi}{2} \d z - \frac{\bar{\xi}}{2} \d \bar{z} \label{intdefsa}.
\end{align}
Consider the following family in $\xi$ of elliptic complexes $\EC^{int}_{\xi }$ over $\C \setminus P$:
\begin{equation}\label{ellcompl}
\Omega^{0}\otimes E \xrightarrow{\text{$D^{int}_{\xi}$}} \Omega^1 \otimes E \xrightarrow{\text{$D^{int}_{\xi}$}} \Omega^2 \otimes E.
\end{equation}
Fix a Hermitian metric $h$ on $E$ for which the holomorphic sections of the extension at the singularities are
bounded (above and below) by a positive constant, and denote by $\hat{E}^{int}_{\xi }$ the first
$L^{2}$-cohomology of the complex (\ref{ellcompl}) for this metric. In Theorems \ref{Fredholm}
and \ref{laplker} we show that there exists a finite set $\hat{P} \subset \hat{\C}$ such that for
$\xi \in \hat{\C}\setminus \hat{P}$ the first $L^{2}$-cohomologies of this complex are finite-dimensional of the
same dimension for all $\xi$.
\begin{defn} The \emph{transformed vector bundle} $\hat{E}$ is then the vector bundle over
$\hat{\C} \setminus \hat{P}$ whose fiber over $\xi \in \hat{\C} \setminus \hat{P}$ is the first $L^{2}$-cohomology
$L^{2}H^{1}(D^{int}_{\xi})$ of $\EC^{int}_{\xi }$.
\end{defn}
Let $\xi_{0} \in \hat{\C} \setminus \hat{P}$, and let $f(z) \in \hat{E}_{\xi_{0}}$ be a class in the first cohomology
of $\EC^{int}_{\xi_{0}}$.
\begin{defn}
The \emph{transformed flat connection} $\hat{D}$ is by definition the flat connection whose parallel section
$f(\xi; z)$ extending $f$ in some neighborhood of $\xi_{0}$ is given by the first $L^{2}$-cohomology classes in
$\EC^{int}_{\xi }$ of
$$
e^{(\xi - \xi_{0})z }f(z).
$$
\end{defn}
Finally, $h$ induces a natural Hermitian metric $\hat{h}$ on $\hat{E}$ as follows: in Theorem \ref{laplker}
we show that any class in $L^{2}H^{1}(D_{\xi})$ can be represented by a unique harmonic $1$-form with respect to
the Laplacian of the Dirac operator.
\begin{defn}
The \emph{transformed Hermitian metric} $\hat{h}$ on $\hat{E}$ is defined by the $L^{2}$-norm of harmonic
representatives.
\end{defn}
All this will be explained in more detail in Section \ref{flattr} and in Definition \ref{deftrhm}.
When one considers an integrable connection, there exists sometimes a privileged fiber metric on the bundle,
namely a harmonic one. In order to be able to define harmonicity, decompose as usual $D$ into its
unitary and self-adjoint part
\begin{equation}\label{decompsaunit}
D = D^{+} + \Phi,
\end{equation}
put $\nabla_{D^{+}}$ or simply $\nabla^{+}$ for the covariant derivative associated
to the connection $D^{+}$ (so that $\nabla^{+}t$ makes sense for a tensor $t$ of
arbitrary type $(T\CP)^{p} \otimes (T^{\ast }\CP)^{q} \otimes E^{r}\otimes (E^{\ast })^{s}$)
and denote by $(\nabla^{+})_{h}^{\ast}$ the adjoint operator of $\nabla^{+}$ with
respect to $h$.
\begin{defn}
The Hermitian metric $h$ is called \emph{harmonic}, if it satisfies the equation
\begin{eqnarray} \label{harm}
(\nabla^{+})_{h}^{\ast}\Phi =0.
\end{eqnarray}
\end{defn}
This is a second-order non-linear partial differential equation in $h$.
Here is the main result of this thesis in a special case (the one without parabolic structures, see Definition
\ref{parstr}).
\begin{thm}\label{naivethm}
Let $(E,D,h)$ be any meromorphic integrable connection with logarithmic singularities in $P$ as in (\ref{dpj}),
and a double pole (\ref{dinf}) at infinity, endowed with a harmonic metric $h$.
Suppose that the eigenvalues of the polar part of $D$ in the punctures satisfy the following assumptions:
\begin{enumerate}
\item{for fixed $j \in \{ 1,\ldots,n\}$, the complex numbers $\mu^{j}_{k}$ for $k=r_{j}+1,\ldots,r$ are all different,
and $\Re \mu^{j}_{k} \notin \Z$}
\item{for fixed $l \in \{1,\ldots,n'\}$, the complex numbers $\mu^{\infty}_{k}$ for $k=1+a_{l},\ldots,a_{l+1}$ are all different,
and $\Re \mu^{\infty}_{k} \notin \Z$}
\end{enumerate}
Then the set of punctures $\hat{P}\in \hat{\C}$ of the transformed bundle is the set $\{ \xi_{1},\ldots,\xi_{n'} \}$
of distinct eigenvalues of the leading order term $A$ of $D$ at infinity.
For $\xi \notin \hat{P}$, the first $L^{2}$-cohomologies of (\ref{ellcompl}) are finite dimensional vector
spaces of the same dimension. They match up to define a smooth vector bundle $\hat{E}$ of rank
\begin{equation} \label{trrank}
\hat{r}=\sum_{j=1}^{n} (r-r_{j})
\end{equation}
over $\hat{\C} \setminus \hat{P}$. $\hat{D}$ is a flat connection on $\hat{E}$.
It underlies a meromorphic integrable connection (that we continue to denote $(\hat{E},\hat{D})$) of degree
$deg(\hat{E})=deg(E)$, called the \emph{transformed meromorphic connection}. It has logarithmic singularities
in $\hat{P}$ and a double pole at infinity. The non-vanishing eigenvalues of the residue in $\xi_{l} \in \hat{P}$ are
$\{ -\mu_{1+a_{l}}^{\infty},\ldots,-\mu_{a_{l+1}}^{\infty} \}$. The eigenvalues of the second-order term of the transformed
meromorphic connection are $\{-p_{1},\ldots,-p_{n}\}$, the multiplicity of $-p_{j}$ being $(r-r_{j})$;
the eigenvalues of its residue at infinity on the eigenspace of the second-order term corresponding
to $-p_{j}$ are $\{-\mu_{r_{j}+1}^{j},\ldots,-\mu_{r}^{j}\}$. Finally, $\hat{h}$ is harmonic for $\hat{D}$.
\end{thm}
\begin{rk}
The assumptions (1) and (2) of the theorem are clearly generic in the parameter space of all
possible eigenvalues.
\end{rk}
This theorem actually follows from the more general statement \ref{mainthm}. In order to understand the
more general setup, one needs to consider meromorphic connections endowed with a parabolic structure.
\section{Parabolic structure and adapted harmonic metric}\label{sectharmmetr}
Actually, we can suppose more structure on the integrable connection: namely, that it comes
with a parabolic structure on $P$ and at infinity.
\begin{defn} \label{parstr}
A \emph{parabolic structure} on $(E,D)$ is the data of a strictly decreasing filtration by vector subspaces
$$
E_{p} = F_{0}E_{p} \supset F_{1}E_{p} \supset \ldots \supset F_{b_{p}-1} E_{p} \supset F_{b_{p}} E_{p} = \{ 0 \}
$$
(where $1\leq b_{p}\leq r$) of the fiber $E_{p}$ of $E$ in each singular point $p \in P \cup \{ \infty \}$,
called the \emph{parabolic flag}, such that each $F_{m}$ is spanned by some of the restrictions
$\{ \tau^{j}_{k}(p) \}_{k=1}^{r}$ of the holomorphic basis to the singularity $p=p_{j}$ or $\infty$,
together with a sequence of corresponding real numbers
$$
0\leq \tilde{\beta}^{j}_{1}< \ldots < \tilde{\beta}^{p}_{b_{p}} < 1
$$
called the \emph{parabolic weights}.
\end{defn}
\begin{rk}
All parabolic weights can be assigned a natural multiplicity, namely the dimension of the corresponding
graded of the filtration: more precisely, the multiplicity of $\tilde{\beta}^{p}_{k}$ for any $p \in P \cup \{ \infty \}$
and any $k\in \{1,\ldots,b_{p} \}$ is by definition
$$
\dim (F_{k-1}E_{p}/F_{k}E_{p}).
$$
We will write
$$
0\leq {\beta}^{p}_{1}\leq \ldots \leq {\beta}^{p}_{r} < 1
$$
for the parabolic weights repeated according to their multiplicities, and use this numbering of the weights
throughout the whole paper instead of the one in their definition. Moreover, we write $\beta^{j}_{k}$ instead of
$\beta^{p_{j}}_{k}$.
\end{rk}
\begin{rk} The order of the $\tau^{\infty}_{k}$ spanning $F_{m}E_{\infty}$ in the above definition is not necessarily the same
as the one in which the eigenvalues of the second-order term A at infinity appear in one
group, as supposed in (\ref{dinf}). However, this will not cause any confusion in the sequel, because
the basis vectors at infinity in this latter order still have well-defined parabolic weights
(which are then not necessarily increasing).
\end{rk}
\begin{defn} \label{parintcon}
A meromorphic integrable connection $(E,D)$ with described local models and parabolic
structures at the punctures will be called \emph{parabolic integrable connection}. The \emph{parabolic degree}
of $E$ with respect to the given parabolic structure is the real number
\begin{equation} \label{pardeg}
deg_{par} (E) = deg(E) + \sum_{j \in \{1,\ldots,n,\infty \}} \sum_{k=1}^{r} \beta^{j}_{k},
\end{equation}
where $deg(E)$ is the standard (algebraic geometric) degree of $E$, and the sum is taken over all
parabolic weights for all punctures $p$. The \emph{slope} of the parabolic integrable connection
is the real number
\begin{equation} \label{slope}
\mu_{par}(E) = \frac{deg_{par} (E)}{rk(E)},
\end{equation}
and $(E,D)$ is said to be \emph{parabolically stable} (resp. \emph{semi-stable}) if for any subbundle $F$
invariant with respect to $D$ and endowed with the induced parabolic structure over the singularities,
the inequality
\begin{equation} \label{stable}
\mu_{par}(F) < \mu_{par}(E)
\end{equation}
(respectively $\mu_{par}(F) \leq \mu_{par}(E)$) holds. Finally, $(E,D)$ is said to be \emph{parabolically polystable}
if it is a direct sum of parabolically stable bundles that are all invariant by $D$ and of the same slope as
$E$.
\end{defn}
\begin{rk}\label{pardegvanish}
The notions of stability, semi-stability and polystability make sense for meromorphic integrable
connections without a parabolic structure as well: in the corresponding definitions,
one only needs to set all parabloic weights equal to $0$. Notice however that by the residue theorem
we have
\begin{align*}
deg(E) & = - \Re tr(Res(D,\infty)) -\sum_{j \in \{1,\ldots,n \}} \Re tr(Res(D,p_{j})) \\
& = \sum_{k=1}^{r} \Re \mu^{\infty}_{k}-\sum_{j \in \{1,\ldots,n\}} \sum_{k=1}^{r} \Re \mu^{j}_{k},
\end{align*}
(the change of sign coming from the fact that the eigenvalues of the residue at infinity are $-\mu^{\infty}_{k}$
because in the local coordinate $w=1/z$ we have $\d z/z=-\d w/w$.)
Therefore (\ref{pardeg}) is in fact equal to
\begin{align*}
\sum_{k=1}^{r} (\beta^{\infty}_{k} + \Re \mu^{\infty}_{k}) + \sum_{j \in \{1,\ldots,n\}} \sum_{k=1}^{r} (\beta^{j}_{k} - \Re \mu^{j}_{k}) =
\sum_{j \in \{1,\ldots,n, \infty\}} \sum_{k=1}^{r} \gamma^{j}_{k},
\end{align*}
where $\gamma^{j}_{k}$ are the parabolic weights of the local system at $p_{j}$ (Proposition 11.1, \cite{Biq97}).
On the other hand, the parabolic degree of an integrable connection is always equal to $0$: this
follows from the Gauss-Chern formula 2.9 of \cite{Biq91}.
Therefore, the case where the parabolic weights $\beta^{j}_{k}$ of the integrable connection vanish is not the one
where the parabolic weights $\gamma^{j}_{k}$ of the representation of the fundamental group vanish,
and where by Remark 8.2 of \cite{Biq-Boa} stability reduces to irreducibility of the corresponding representation.
\end{rk}
\begin{defn}\label{adaptdef}
A Hermitian fiber metric $h$ on $E$ is said to be \emph{adapted} to the parabolic structure if
near the logarithmic punctures in the holomorphic bases $\tau^{j}_{k}$ it is mutually bounded with
the diagonal model
\begin{equation} \label{parmatr}
diag(|z-p_{j}|^{2 \beta^{j}_{k}})_{k=1}^{r},
\end{equation}
and at infinity in the holomorphic basis $\tau^{\infty}_{k}$ it is mutually bounded with
\begin{equation} \label{parmatrinf}
diag(|z|^{-2 \beta^{\infty}_{k}})_{k=1}^{r}.
\end{equation}
\end{defn}
\begin{rk}
In general, without the hypothesis of semisimplicity of the residue in the puctures made in Section
\ref{flatintro}, the local models of the metric near the punctures are more complicated than in the
above definition: e.g. for the regular singularities one has to take into account an extra filtration
induced by the nilpotent part of the residue, and add a factor $|\ln(r)|^{k}$ on the
corresponding $k$-th graded, see the Synopsis of \cite{Sim}.
\end{rk}
Here is the important existence result of the theory:
\begin{thm}[Sabbah C. \cite{Sab-harm}; Biquard O., Boalch Ph. \cite{Biq-Boa}] \label{harmexistthm}
Let $(E,D)$ be a parabolically stable parabolic integrable connection. Then there exists a unique
harmonic Hermitian metric $h$ adapted to the parabolic structure.
\end{thm}
\begin{rk}
Actually, in the above articles this theorem is proved to hold for parabolic integrable connections having
poles of arbitrary order in the punctures. On the other hand, for integrable connections with only regular
singularities, it had already been shown by C. Simpson, see \cite{Sim}.
\end{rk}
We are now ready to describe the more general version of Nahm transform: that for parabolic integrable connections.
\begin{thm} \label{mainthm}
Let $(E,D)$ be any parabolic integrable connection on the sphere with logarithmic singularities in $P$ as in
(\ref{dpj}), and a double pole (\ref{dinf}) at infinity. Suppose that the eigenvalues of its polar parts $\mu$ and
the parabolic weights $\beta$ in the punctures satisfy the following assumptions:
\begin{enumerate}
\item{for fixed $j \in \{ 1,\ldots,n\}$, the complex numbers $\mu^{j}_{k}-\beta^{j}_{k}$ for $k=r_{j}+1,\ldots,r$ are distinct
and different from $0$, the parabolic weights $\beta^{j}_{k}$ for $k=1,\ldots,r_{j}$ are $0$ and finally
$\Re \mu^{j}_{k} \notin \Z$ for $k=r_{j}+1,\ldots,r$}
\item{for fixed $l \in \{1,\ldots,n'\}$, the complex numbers $\mu^{\infty}_{k}-\beta^{\infty}_{k}$ for $k=1+a_{l},\ldots,a_{l+1}$ are
distinct and different from $0$, and $\Re \mu^{\infty}_{k} \notin \Z$}
\end{enumerate}
Then, in addition to the conclusions of Theorem \ref{naivethm}, the transformed bundle $(\hat{E},\hat{D})$
carries a natural parabolic structure in the punctures (that we will call
\emph{transformed parabolic structure}), such that the transformed metric of the harmonic metric is adapted to
it. Moreover, the set of its non-vanishing parabolic weights in $\xi_{l} \in \hat{P}$
is equal to the set of parabolic weights $\{ \beta^{\infty }_{1+a_{l}},\ldots,\beta^{\infty }_{a_{l+1}} \}$
of $E $ at infinity, restricted to the eigenspace of $A$ corresponding to the eigenvalue $\xi_{l}$;
whereas the parabolic weights of $\hat{E}$ at infinity restricted to the eigenspace of the second-order term of $\hat{D}$
corresponding to the eigenvalue $-p_{j}$ are equal to the parabolic weights
$\{ \beta^{j}_{r_{j}+1},\ldots, \beta^{j}_{r}\}$ of $E $ at $p_{j}$. All these statements are to be understood
with multiplicities.
\end{thm}
\begin{rk}
Again, the conditions (1) and (2) of the theorem are generic in the parameter space of all
possible eigenvalues and parabolic weights. They will regularly appear along this paper,
both in analytical and geometric arguments.
\end{rk}
This theorem is a consequence of Theorem \ref{mainthmhiggs}.
\begin{defn}
The map
\begin{align} \label{nahm}
\Nahm: (E,D) \mapsto (\hat{E},\hat{D})
\end{align}
described in Theorems \ref{naivethm} and \ref{mainthm} will be called \emph{Nahm transform}.
\end{defn}
Finally, as we have already mentioned, Nahm transform has an involutibility property:
\begin{thm}
Let $(E,D)$ be a parabolic integrable connection on the sphere satisfying the assumptions of Theorem
\ref{mainthm}. Then
$$
\Nahm^{2}(E,D) = (-1)^{\ast} (E,D),
$$
where $-1:\C \ra \C$ is the map $z \mapsto -z$,
and $(-1)^{\ast}$ the induced map on fiber bundles with connection. In particular, Nahm transform is invertible.
\end{thm}
This will be proved in Theorem \ref{invtr}, using arguments of the same type as S. K. Donaldson and P. B. Kronheimer in
\cite{Don-Kronh}, namely the study of the spectral sequence of a suitable double complex.
\section{Local model for parabolic integrable connections}\label{locmodint}
We suppose in this section that near each singularity, $h$ coincides with the diagonal
models $h^{j}$ and $h^{\infty }$ given in Definition \ref{adaptdef} (that is, without the extra
$O(|z-p_{j}|^{2 (\beta^{j}_{k} - \beta^{j}_{k'})})$ and $O(|z|^{-2 (\beta^{\infty}_{k} - \beta^{\infty}_{k'})})$ factors in
(\ref{parmatr}) and (\ref{parmatrinf}); in particular, this metric is then not harmonic).
For computations, it will be useful to express the local models of the
integrable connection near the singularities in some orthonormal bases. As in \cite{Biq-Boa},
we consider the orthonormal basis defined by
\begin{equation} \label{ebasisj}
e^{j}_{k} = |z|^{-\beta^{j}_{k}-i\Im \mu^{j}_{k}} {\tau^{j}_{k}} \qquad {k=1,\ldots,r}
\end{equation}
around $p_{j}$. The $h$-unitary part $(D^{+})^{j}$ of $D^{j}$ becomes
\begin{align}
(D^{+})^{j} =
\mbox{d}+ i \Re (A^{j}) \d \theta \label{polarunit}
\end{align}
where
$\Re(A^{j}) = \frac{A^{j}+(A^{j})^{\ast}}{2}= diag(\Re \mu^{j}_{k})_{k=1,\ldots,r}$ (and
$ \Im (A^{j}) = \frac{A^{j}-(A^{j})^{\ast}}{2i} = diag(\Im
\mu^{j}_{k})_{k=1,\ldots,r}$) stands for the real (imaginary) part of
$A^{j}$, and $r$ and $\theta $ are local polar coordinates around
$p_{j}$ such that we have $z-p_{j}=r e^{i\theta}$.
For the self-adjoint part $\Phi^{j}$ of $D^{j}$ in this basis we get:
\begin{align}
\Phi^{j} & = \frac{A^{j}}{2} \frac{\d z}{z-p_{j}} + \frac{(A^{j})^{\ast }}{2} \frac{\d
\bar{z}}{\bar{z}-\bar{p_{j}}} - \beta^{j} \frac{\mbox{d}r}{r} \notag \\
& = [ \Re(A^{j}) - \beta^{j} ] \frac{\mbox{d}r}{r} - \Im(A^{j}) \mbox{d}\theta \label{polarphi} ,
\end{align}
where $\beta^{j}=diag(\beta^{j}_{k})_{k=1,\ldots,r}$. These together imply that with
respect to this basis, the model for the operator $D$ in polar coordinates is
\begin{eqnarray}
D^{j} = \d + i A^{j} \d \theta + [ \Re (A^{j}) - \beta^{j} ] \frac{\mbox{d}r}{r}. \label{polard}
\end{eqnarray}
In an analogous way, in the orthonormal basis $ \left\{ e^{\infty}_{k} \right\}_{k=1,\ldots,r} $ given by
\begin{equation} \label{ebasisinf}
e^{\infty }_{k} = |z|^{\beta^{\infty }_{k}+i\Im \mu^{\infty }_{k}} \exp{[({\xi_{k}
z}-{\bar{\xi_{k}}\bar{z}})/2]} \tau_{k}^{\infty}
\end{equation}
near infinity the unitary part of the model connection $D^{\infty }$ is given by
\begin{eqnarray*}
(D^{+})^{\infty} = \d + i \Re (C) \d \theta,
\end{eqnarray*}
where we have put again
$\Re (C)= \frac{C+C^{\ast }}{2} = diag (\Re \mu^{\infty}_{k})_{k=1,\ldots,r}$ and $z=r e^{i\theta }$. Moreover, putting
$ \Re (z A)= diag \Re (\{ z \xi_{l}, m_{l} \})_{l=1,\ldots n'} $ and
$\Im (z A)= diag \Im (\{ z \xi_{l}, m_{l} \} )_{l=1,\ldots n'} $, the self-adjoint part of $D^{\infty }$ has the form
\begin{align} \label{polarphiinf}
\Phi^{\infty } & = \frac{1}{2}\left( A + \frac{C}{z} \right) \d z + \frac{1}{2}\left( A^{\ast} +
\frac{C^{\ast }}{\bar{z}} \right) \d \bar{z} + \beta^{\infty}
\frac{\mbox{d}r}{r} \notag \\
& = [ \Re (z A + C) + \beta^{\infty} ] \frac{\mbox{d}r}{r} +
\Im ( z A + C) \d \theta
\end{align}
(the inversion of the sign of $\beta$ comes from the fact that if we make a
coordinate change $w=1/z$, $\vert w \vert = \rho = 1/r = 1/ \vert z \vert$,
then $\d \rho / \rho = - \d r / r$). Remark that in these expressions the terms
in $\d \theta, \d r / r , \d z / z, \d \bar{z}/ \bar{z}$ are of lower order then
the ones in $\d z, \d \bar{z}, z \d r /r, z \d \theta $; hence the leading order
term of the singular part of $D$ in this basis is just
\begin{eqnarray} \label{singpart}
d + \frac{A}{2} \d z + \frac{A^{\ast }}{2} \d \bar{z}.
\end{eqnarray}
\section{Higgs bundle point of view}\label{higgsintro}
The idea of the proofs of Theorems \ref{naivethm} and \ref{mainthm} will be to exploit the
correspondence known as nonabelian Hodge theory between parabolic integrable connections
on one side and parabolic Higgs bundles on the other side.
Let us recall the definition of the latter notion:
\begin{defn} \label{parhiggsdef}
A \emph{parabolic Higgs bundle} is given by:
\begin{enumerate}
\item{a holomorphic bundle $\E$ with holomorphic structure $\bar{\partial}^{\E}$ over $\CP$
called the \emph{holomorphic bundle underlying the Higgs bundle},
and with underlying smooth vector bundle $V$;}
\item{in each point $p \in P \cup \{\infty \}$ a strictly decreasing parabolic flag
$$
V_{p} = F_{0}V_{p} \supset F_{1}V_{p} \supset \ldots \supset F_{c_{p}-1} V_{p} \supset F_{c_{p}} V_{p} = \{ 0 \}
$$
for some $1\leq c_{p}\leq r$, with parabolic weights
$$
0\leq \tilde{\alpha}^{p}_{1} <\ldots<\tilde{\alpha}^{p}_{c_{p}}<1;
$$}
\item{a $\bar{\partial}^{\E}$-meromorphic section $\theta \in \Omega^{1,0}(\CP, End(V))$ (called the \emph{Higgs field}),
having a simple pole at the points of $P$ with semi-simple residue respecting the parabolic flag
(that is, such that $Res(\theta,p_{j})$ leaves $F_{k}V_{p_{j}}$ invariant for each $p_{j}\in P$ and
all $0 \leq k \leq c_{p}$), and a second-order pole at infinity, such that there exists a holomorphic
basis of $\E$ near infinity compatible with the parabolic structure in which the residue
and second-order term are both diagonal.}
\end{enumerate}
Again, we write
$$
0\leq {\alpha}^{p}_{1} \leq \ldots\leq {\alpha}^{p}_{r}<1
$$
for the parabolic weights repeated according to their multiplicities
$$
\dim (F_{k-1}V_{p}/F_{k}V_{p}),
$$
and we shorten ${\alpha}^{p_{j}}_{k}$ to ${\alpha}^{j}_{k}$. Finally, we set
\begin{equation} \label{dsecond}
D''=\dbar^{\E}+ \theta,
\end{equation}
that we call the \emph{$D''$-operator} associated to the Higgs bundle.
\end{defn}
The notions of parabolic degree, slope and (poly/semi-)stability of parabolic Higgs bundles are defined
analogously to the case of integrable connections, see Definition \ref{parintcon}.
Theorem 6.1 of \cite{Biq-Boa} then says the following.
\begin{thm}[Biquard O.-Boalch Ph., 2004] \label{wnht}
There exists an isomorphism between the moduli space of parabolically stable rank $r$ integrable
connections with fixed diagonal polar part and parabolic structures up to complex holomorphic
gauge transformations respecting the parabolic flags, and the moduli space of parabolically stable rank $r$
Higgs bundles with fixed diagonal polar part and parabolic structures up to complex holomorphic gauge
transformations respecting the parabolic flags.
\end{thm}
\begin{rk}
Actually, this is a consequence of the existence of a harmonic metric (Theorem \ref{harmexistthm}),
and hence also proved for parabolic integrable
connections with poles of arbitrary fixed order and diagonal polar part in the punctures and parabolic Higgs
bundles with poles of the same order with diagonal polar part.
\end{rk}
The transition from integrable connections to Higgs bundles is given as follows: first,
the underlying smooth vector bundle of the integrable connection and the Higgs bundle are the same.
Furthermore, suppose $h$ is the harmonic metric, consider the decomposition (\ref{decompsaunit})
of the integrable connection into its unitary and self-adjoint part, and decompose the terms further
according to bidegree
\begin{align} \label{decompbidegree}
D^{+} & = (D^{+})^{1,0} + (D^{+})^{0,1} \\
\Phi & = \Phi^{1,0} + \Phi^{0,1}. \notag
\end{align}
The partial connection $(D^{+})^{0,1}$ defines then the holomorphic structure of $\E$, and $\Phi^{1,0}$ will
be the Higgs field $\theta$. The $D''$-operator is of course $(D^{+})^{0,1}+\Phi^{1,0}$.
Harmonicity of the metric implies that $\theta$ is holomorphic.
The transition in the other direction is also established using a privileged metric.
\begin{defn} \label{HE}
Let $(\E,\theta)$ be a Higgs bundle. We say that $h$ is a \emph{Hermitian-Einstein metric} for $(\E,\theta)$ if,
denoting by $D^{+}_{h}$ the Chern connection (the unique $h$-unitary
connection compatible with $\dbar^{\E}$), by $F_{D^{+}_{h}}$ its curvature, and by
$\theta^{\ast}_{h}$ the adjoint of $\theta$ with respect to $h$, then these objects satisfy the real Hitchin equation
\begin{eqnarray*}
F_{D^{+}_{h}}+[\theta,\theta^{\ast}_{h}] = 0,
\end{eqnarray*}
where $[.,.]$ stands for graded commutator of forms.
\end{defn}
Let $(\E,\theta)$ be a parabolically stable parabolic Higgs bundle. By \cite{Biq-Boa}, there exists
a unique Hermitian-Einstein metric $h$ adapted to the parabolic structure. The connection
\begin{equation}
D = D^{+}_{h} + ( \theta + \theta_{h}^{\ast } )
\end{equation}
on $V$ is then integrable, and $h$ is the corresponding harmonic metric adapted to the parabolic structure.
In what follows, in order to simplify notations, we are often going to omit the subscript $h$ in the
notation of the Chern connection and adjoints.
Let now $(E,D)$ be a parabolically stable parabolic integrable connection and $(\E,\theta)$ the associated
parabolic Higgs bundle. An important result we will be constantly using is the following
\begin{thm}[Simpson C. \cite{Sim}] \label{simpson}
Suppose the metric $h$ is harmonic. Then, with the previous notations, the Laplace
operators $\Delta_{D}=DD^{\ast}+D^{\ast}D$ and $\Delta_{D''}=D''(D'')^{\ast}+(D'')^{\ast}D''$ satisfy
$$
\Delta_{D}=2\Delta_{D''}.
$$
In particular, their domain and kernel coincide.
\end{thm}
\section{Local model for Higgs bundles}
In this section, we give the eigenvalues of the residue of the Higgs field and the parabolic weights of the
Higgs bundle in the punctures that correspond to those of the integrable connection via the Theorem \ref{wnht}.
To obtain local models for the operators in this setup, suppose again that
near $p_{j}$ the metric $h$ coincides with the diagonal model $h^{j}$ given by (\ref{parmatr})
(without the correcting $O(|z-p_{j}|^{2(\beta^{j}_{k}-\beta^{j}_{k'})})$ term; in particular, it does not satisfy
Hitchin's equation). Then, according to \cite{Biq-Boa}, in the local $\dbar^{\E}$-holomorphic trivialisation
\begin{align}\label{holtriv}
\sigma^{j}_{k} = |z-p_{j}|^{\Re \mu^{j}_{k}} \frac{e^{j}_{k}}{(z-p_{j})^{[\Re \mu^{j}_{k}]}}
&& (k=1,\ldots,r)
\end{align}
around $p_{j}$, the Higgs field is equal up to a perturbation term to the model Higgs field given by
\begin{align} \label{modelhiggs}
\theta^{j} & = \frac{A^{j}-\beta^{j}}{2} \frac{\d z}{z-p_{j}}\notag \\
& = diag\left( \frac{\mu^{j}_{k}-\beta^{j}_{k}}{2} \frac{\d z}{z-p_{j}} \right)_{k=1,\ldots,r}\notag \\
& = diag\left(\lambda^{j}_{k} \frac{\d z}{z-p_{j}} \right)_{k=1,\ldots,r} ,
\end{align}
where we have put $\lambda^{j}_{k}=(\mu^{j}_{k}-\beta^{j}_{k})/2$.
Moreover, in the same trivialisation, the parabolic weights are
\begin{equation} \label{higgsparweights}
\alpha^{j}_{k}=\Re (\mu^{j}_{k}) -[\Re (\mu^{j}_{k})],
\end{equation}
where $[.]$ denotes integer part.
\begin{rk}
In fact, this formula is not completely correct, because the $\alpha^{j}_{k}$ defined by it are not
necessarily in increasing order, although they should be by definition.
One should instead write the same formula for $\alpha^{j}_{s(k)}$,
where $s$ is a permutation of $\{ 1,\ldots,r\}$. However, in the sequel we discard this minor technical
detail for the sake of simplicity of the notation.
\end{rk}
\begin{rk} Since the gauge transformations between the bases
$\{ \tau^{j}_{k} \}_{k=1,\ldots,r}$ and $\{ \sigma^{j}_{k} \}_{k=1,\ldots,r}$
are just multiplications by some functions (in particular diagonal
matrices), it follows that the smooth subbundle
spanned by the sections $\{ \sigma^{j}_{k} \}_{k=r_{j}+1,\ldots,r}$ is the same as
the one spanned by $\{ \tau^{j}_{k} \}_{k=r_{j}+1,\ldots,r}$, which is by
definition the underlying smooth vector bundle of $E^{j}_{sing}$; and similarly, the subbundle spanned by
$\{\sigma^{j}_{k} \}_{k=1,\ldots,r_{j}}$ is equal to the underlying smooth bundle of $E^{j}_{reg}$. The same remark also
holds for the bases $\{ e^{j}_{k} \}$ instead of $\{ \sigma^{j}_{k} \}$.
In particular, the residue of the model Higgs field $\theta^{j} $ in the
point $p_{j} \in P$ belongs to $End(E^{j}_{sing}|_{p_{j}})$.
\end{rk}
Near infinity, the situation is slightly different: for $h = h^{\infty }$ the diagonal model,
in the local $\dbar^{\E}$-holomorphic frame
\begin{align} \label{holtrivinf}
\sigma_{k}^{\infty} = |z|^{-\Re \mu^{\infty }_{k}} z^{[\Re \mu^{\infty }_{k}]} e^{\infty }_{k} && (k=1,\ldots,r)
\end{align}
the Higgs field is equal up to a perturbation term to the model Higgs field given by
\begin{align} \label{modelhiggsinf}
\theta^{\infty } & = \frac{1}{2} A\d z + \frac{\mu^{\infty} - \beta^{\infty}}{2} \frac{\d z}{z}\notag \\
& = \left( \frac{1}{2} diag(\{ \xi_{l}, m_{l} \})_{l=1,\ldots,n'}
+ \frac{1}{z} diag(\lambda_{k}^{\infty})_{k=1,\ldots,r} \right) \d z,
\end{align}
where we have put again $\lambda_{k}^{\infty}=(\mu_{k}^{\infty}-\beta_{k}^{\infty})/2$, with parabolic weights
being, as in the case of simple poles,
\begin{equation} \label{higgsparweightsinf}
\alpha_{k}^{\infty}=\Re (\mu_{k}^{\infty}) -[\Re (\mu_{k}^{\infty})].
\end{equation}
From these data, as above, one can form the model $D''$-operator
\begin{equation} \label{modeldsecond}
(D'')^{j} = \dbar^{\E} + \theta^{j} \qquad (j \in \{ 1, \ldots n, \infty \} ).
\end{equation}
Notice that since we considered holomorphic trivialisations of $\E^{j}$, the
partial connection part of the model coincides with the usual $\dbar$-operator.
We are now ready to write out the assumptions made in Theorem \ref{mainthm} on the parameters of the
integrable connection, translated to those of the Higgs bundle:
\begin{hyp} \label{main}
We suppose that $(\E ,\theta )$ is a parabolically stable Higgs bundle with diagonal polar part of the
Higgs field in some local holomorphic frame near each puncture, satisfying the properties
\begin{enumerate}
\item{for fixed $j\in \{ 1,\ldots,n \}$ the residues $\lambda^{j}_{k} $ for $k \in \{ r_{j}+1,\ldots,r \} $ are
non-vanishing and distinct, $\lambda^{j}_{k}$ vanish for $k=1,\ldots,r_{j}$ and finally $\alpha_{k}^{j} \neq 0$ if and only
if $\lambda^{j}_{k} \neq 0$;}\label{maini}
\item{for fixed $l \in \{ 1, \ldots, n' \} $ the complex numbers $\lambda^{\infty }_{k}$ for $k \in \{ 1+a_{l}, \ldots a_{l+1} \}$ are
non-vanishing and distinct, and $\alpha^{\infty }_{k} \neq 0$.} \label{mainii}
\end{enumerate}
\end{hyp}
Diagonality of the polar parts has already been assumed when writing the local models (\ref{modelhiggs}) and
(\ref{modelhiggsinf}). The first condition says that no parabolic weight and no eigenvalue of the residue of $\theta$
vanishes on the singular component at any singularity, and that on the singular component near a puncture all
eigenvalues are different; whereas the eigenvalues of the residue and parabolic weights vanish on the regular
component. One more way to say the same thing is: for all $j \in \{ 1,\ldots n \}$, the residue of $\theta $ defines an
automorphism of $E^{j}_{sing}|_{p_{j}}$, and the parabolic weights corresponding to the holomorphic
trivialisation (\ref{holtriv}) are non-vanishing exactly on this subspace.
The second one imposes that on the eigenspace corresponding to a fixed eigenvalue of the second-order term
at infinity, all the eigenvalues of the residue be non-vanishing and distinct, furthermore that no parabolic
weight vanish at infinity. Note that these conditions are generic among all possible choices of
singularity parameters.
\section{The transformation of the Higgs bundle} \label{transf}
Let $(\E ,\theta )$ be a parabolic Higgs bundle and $\xi \in \hat{\C} \setminus \hat{P}$ a parameter.
The natural deformation of the Higgs field is
\begin{align} \label{deformhiggs}
\theta_{\xi} &= \theta - \frac{\xi}{2} \mbox{d}z
\end{align}
with fixed underlying holomorphic bundle $\E$. It is clear that $\theta_{\xi}$ is then also holomorphic with respect
to $\dbar^{\E}$ with the same local models at the logarithmic punctures as $\theta$, but its local model near
infinity is different. If moreover a Hermitian metric is fixed, then we also have
$$
\theta_{\xi}^{\ast} = \theta^{\ast } - \frac{\bar{\xi}}{2} \mbox{d}\bar{z}.
$$
Therefore, the integrable connection corresponding to the deformed Higgs bundle is given by
\begin{eqnarray} \label{higgsdef}
D^{H}_{\xi} = D - \frac{\xi}{2} \mbox{d} z- \frac{\bar{\xi}}{2} \mbox{d}\bar{z},
\end{eqnarray}
and the crucial observation is that via the unitary gauge transformation
\begin{equation} \label{gauge}
\exp[ ( \bar{\xi } \bar{z} - \xi z )/2]
\end{equation}
on $\C$ this is equivalent to the deformation (\ref{intdef}). The self-dual part of this deformation is
\begin{align}
\Phi^{H}_{\xi } & = \Phi - \frac{{\xi}}{2} \d z - \frac{\bar{\xi}}{2} \d \bar{z} \label{higgsdefsa},
\end{align}
the same deformation as in (\ref{intdefsa}). Therefore it will not make any
confusion to refer to $\Phi_{\xi }$ without mentioning the adopted point of
view; consequently, we drop the corresponding upper indices. The connection defined by (\ref{higgsdef})
is still flat, but the underlying holomorphic structure is different from
the one of $D$ (because of the term in $\mbox{d}\bar{z}$). Notice also that the gauge transformation
(\ref{gauge}) between these deformations has an exponential singularity at infinity. Denote by $\EC^{H}_{\xi}$ the
elliptic complex
\begin{equation}\label{merellcompl}
\Omega^{0}\otimes E \xrightarrow{\text{${D}^{H}_{\xi}$}} \Omega^1 \otimes E \xrightarrow{\text{${D}^{H}_{\xi}$}} \Omega^2 \otimes E.
\end{equation}
\begin{defn}
The smooth vector bundle $\hat{V}$ underlying the transformed Higgs bundle is the vector bundle
whose fiber over $\xi \in \hat{\C} \setminus \hat{P}$ is the first $L^{2}$-cohomology $L^{2}H^{1}(\EC^{H}_{\xi})$
of $\EC^{H}_{\xi}$.
\end{defn}
In Proposition \ref{trvbprop} we prove that these vector spaces indeed define a finite rank smooth bundle.
Furthermore, by Theorem \ref{laplker}, any class in $L^{2}H^{1}(\EC^{H}_{\xi})$ admits a unique
${D}^{H}_{\xi}$-harmonic representative.
\begin{defn}
The \emph{transformed holomorphic structure} on $\hat{V}$ is the one induced by the orthogonal projection
$\dbar^{\E}$ of the trivial partial connection with respect to the variable $\xi$ on the trivial $L^{2}$-bundle
over $\CPt$ to $D^{H}_{\xi}$-harmonic $1$-forms.
The \emph{transformed Higgs field} is multiplication by $-z\d \xi /2$ followed by projection onto harmonic
$1$-forms. Finally, the \emph{transformed Hermitian metric} is the $L^{2}$-metric of the harmonic
representative.
\end{defn}
By virtue of Theorems \ref{laplker} and \ref{simpson}, the transformed smooth bundle $\hat{V}$ can also be
computed in this case as the first cohomology of the elliptic complex $\EC_{\xi}''$ given by:
$$
\Omega^{0}\otimes E \xrightarrow{\text{$D_{\xi}''$}} \Omega^1 \otimes E
\xrightarrow{\text{$D_{\xi}''$}} \Omega^2 \otimes E,
$$
where the maps are the corresponding deformations of (\ref{dsecond}) in the
Higgs-bundle point of view. Explicitly, $D_{\xi}''$ reads
$$
(D^{H}_{\xi })''= \dbar^{\E} + \theta_{\xi }.
$$
We use this description of the transformed bundle in Section \ref{sectharm} to show the statement
of Theorem \ref{naivethm} on the transformed metric:
\begin{thm}
If the original metric harmonic then the same thing holds for the transformed metric.
\end{thm}
For this purpose, we prove in fact that the candidate Higgs field $\hat{\theta}$ corresponding to
$\hat{D}$ and $\hat{h}$ is meromorphic with respect to the transformed holomorphic structure.
Furthermore, in this interpretation, the remaining part of Theorems \ref{naivethm} and \ref{mainthm} can be written:
\begin{thm} \label{mainthmhiggs} Suppose $(\E,\theta )$ is a parabolic Higgs bundle with logarithmic singularities
in the points of $P$ and a double pole at infinity, as described in Section \ref{higgsintro}, such that
its singularity parameters satisfy Hypothesis \ref{main}. Then the transformed Higgs bundle
$(\dbar^{\hat{\E}},\hat{\theta} )$ is of the same type (that is, it has a finite number of logarithmic
singularities in points of $\hat{\C}$ and a double pole at infinity, with a parabolic structure in these
points). Furthermore, its topological and singularity parameters are as follows:
\begin{enumerate}
\item{the rank of $\hat{\E}$ is the sum (\ref{trrank}) of the ranks of the residues of $\theta $ in $P$} \label{i}
\item{its degree is the same as that of $\E$} \label{ii}
\item{the logarithmic singularities are located in the set $\hat{P} $, and for all $l \in \{ 1,\ldots,n' \}$
the rank of the transformed Higgs field in the point $\xi_{l }$ is equal to the multiplicity $m_{l}$
of the eigenvalue $\xi_{l }$ of $A$} \label{iii}
\item{the set of non-vanishing eigenvalues of the residue of $\hat{\theta }$ in the point $\xi_{l }$ is
$\{ -\lambda^{\infty }_{1+a_{l}}, \ldots , -\lambda^{\infty }_{a_{l+1}} \}$, where $\{ \lambda^{\infty }_{a_{l}+1}, \ldots, \lambda^{\infty }_{a_{l+1}} \}$ are the
eigenvalues of the residue of the original Higgs field $\theta $ at infinity,
restricted to the eigenspace of $A$ corresponding to the eigenvalue $\xi_{l }$ } \label{iv}
\item{the non-vanishing parabolic weights of $\hat{\E }$ in $\xi_{l }$ is the set of parabolic weights
$\{ \alpha^{\infty }_{1+a_{l }},\ldots,\alpha^{\infty }_{a_{l+1 }} \}$ of $\E $ at infinity, restricted to the same subspace} \label{v}
\item{the eigenvalues of the second-order term of $\hat{\theta} $ at infinity are $\{ -p_{1}/2,\ldots,-p_{n}/2 \}$, and
the multiplicity of $-p_{j}/2$ is equal to the rank $r-r_{j}$ of the residue of $\theta $ in $p_{j}$} \label{vi}
\item{on the eigenspace corresponding to $-p_{j}/2$ of the second-order term at infinity,
the eigenvalues of the residue of $\hat{\theta} $ are \newline $\{ -\lambda^{j}_{r_{j}+1},\ldots,- \lambda^{j}_{r} \}$} \label{vii}
\item{the parabolic weights on the same eigenspace at infinity are the parabolic weights
$\{ \alpha^{j}_{r_{j}+1},\ldots, \alpha^{j}_{r} \}$ of $\E $ at $p_{j}$} \label{viii}
\end{enumerate}
\end{thm}
The proof of this theorem is the object of Chapter \ref{algint}.
\chapter{Analysis of the Dirac operator} \label{chFr}
In this chapter, we study the analytical theory needed for our construction along the lines of
Donaldson-Kronheimer \cite{Don-Kronh}, Jardim \cite{Jardim} and others. First, in Section \ref{sectFr} we
define spinor spaces and Dirac operators that permit us to study the problem.
We also define a suitable functional space $H^{1}$ and state a Fredholm theorem
valid for all deformations of the initial connection.
Then it is natural to define the fibers of the transformed bundle
as the cokernel of the deformed Dirac operator. The Fredholm theorem
is then proved in Section \ref{sectpfFr}.
In Section \ref{l2coh}, we carry out an identification of this cokernel with the first $L^{2}$-cohomology
$L^{2}H^{1}(D^{int}_{\xi})$ of the complex $\EC_{\xi}^{int}$ given in (\ref{ellcompl}), similar in vein to
the Hodge isomorphism between the cokernel of the operator
$\d + \d^{\ast }$ on a compact manifold and the $L^{2}$-cohomology of the operator $\d$.
However, since the manifold we are working on is non-compact, in proving these results we need a careful
study of the singularities.
In all what follows, we fix a parabolic integrable connection with adapted metric $(E,D,h)$ and choose to study the analytic
properties of the deformation from the point of view of integrable connections, hence we set
for simplicity $D_{\xi }= D^{int}_{\xi }$ until further notification.
\section{Statement of the Fredholm theorem} \label{sectFr}
\begin{defn}
The positive and negative \emph{spinor bundles} are the vector bundles over $\C \setminus P$ given by
\begin{align*}
S^{+} &= \Lambda^{0}T^{\ast }(\C \setminus P) \oplus \Lambda^{2}T^{\ast }(\C \setminus P) & S^{-} = \Lambda^{1}T^{\ast }(\C \setminus P)
\end{align*}
\end{defn}
Recall that we have defined $\hat{P}$ as the set $\{ \xi_{1},\ldots,\xi_{r} \}$ of all
eigenvalues of the second order term of $D $ at infinity.
\begin{defn}
For $\xi \in \hat{\C} \setminus \hat{P}$ the \emph{Dirac operator } is the first-order differential operator
\begin{align*}
\Dir_{\xi} &=D_{\xi} - D_{\xi}^{\ast}: \Gamma (S^{+}\otimes E) \lra \Gamma (S^{-}\otimes E)
\end{align*}
where $\Gamma$ is used to denote smooth sections with compact support in $\C \setminus P$.
Its formal adjoint
\begin{align*}
\Dir_{\xi}^{\ast} & = D_{\xi}^{\ast} - D_{\xi} : \Gamma (S^{-}\otimes E) \lra \Gamma
(S^{+}\otimes E),
\end{align*}
is called the \emph{adjoint Dirac operator. }
\end{defn}
For any $\xi\in \hat{\C}$ let us introduce the following norm on sections $f$ of $S^{+}\otimes E$:
\begin{align} \label{norm}
\Vert f \Vert_{H^{1}_{\xi}}^{2} = \int_{\C } & \vert f
\vert^{2} + \vert \nabla_{\xi }^{+}f \vert^{2} + \vert \Phi_{\xi} \otimes f \vert^{2} ,
\end{align}
where $\nabla_{\xi }^{+}$ and $\Phi_{\xi}$ are defined in (\ref{intdefunit}) and (\ref{intdefsa}).
Here and in all what follows, we integrate with respect to the
Euclidean volume form $|\d z|^{2}$, and $\vert x \vert^{2}$ denotes $h(x,x)$, unless the
contrary is explicitly stated. Our convention is furthermore to write
$(x,y)$ for $h(x,y)$, and for sections $x$ and $y$, we write $\langle x,y\rangle $
instead of $\int_{\C } (x,y) $.
Define the space of sections
\begin{equation} \label{h1space}
H^{1}_{\xi}(S^{+}\otimes E) = \{ f \in L^{2}_{loc}(S^{+}\otimes E): \Vert f \Vert_{H^{1}_{\xi}} <
\infty \},
\end{equation}
where in $L^{2}$ we refer to the metric $h$ on the fibers. We will often
write $H^{1}_{\xi}$ instead of $H^{1}_{\xi}(S^{+}\otimes E)$.
As we will see by the end of this chapter, this is the appropriate space to regard the Dirac
operators. First we establish the simple
\begin{lem} \label{lemmaone} The norm $\Vert . \Vert_{H^{1}_{\xi}}$
depends (up to equivalence of norms) neither on $\xi \in \hat{\C}$,
nor on the particular connection $D$ having behavior as in
(\ref{dpj}) and (\ref{dinf}).
\end{lem}
\begin{proof} We begin by showing that the norm is independent of $\xi $.
In order to simplify notations, we let $H^{1}$ stand for $H^{1}_{0}$ from now on.
It is obviously sufficient to prove that for an arbitrary
$\xi \in \hat{\C}$, the $H^{1}_{\xi}$-norm is equivalent to the $H^{1}$-norm. From the point-wise identity
$$
|\Phi_{\xi } \otimes f| = 2 |\theta_{\xi } \otimes f| = 2 |\theta_{\xi }^{\ast } \otimes f| ,
$$
and the point-wise estimation
\begin{align} \label{punctestunit}
|\nabla^{+}_{\xi }f| & \leq |\nabla^{+} f| + 2|\xi || f| ,
\end{align}
one sees that for any section $f=(f_{0},f_{2}) \in \Gamma(S^{+} \otimes E)$ the estimates
$$
\Vert f \Vert^{2}_{H^{1}_{\xi}} \leq (1+ 8\vert \xi \vert^{2}) \Vert f \Vert^{2}_{H^{1}}
$$
and
$$
\Vert f \Vert^{2}_{H^{1}} \leq (1+ 8\vert \xi \vert^{2}) \Vert f \Vert^{2}_{H^{1}_{\xi}}
$$
hold; the first statement of the Lemma follows at once.
Now we show independence of the particular connection $D$ with right
singularity behavior. Introduce the model norm
\begin{eqnarray} \label{modelnorm}
\Vert f \Vert^{2}_{H^{1}_{mod}(\Delta(p_{j}, \varepsilon))} =
\int_{\Delta(p_{j}, \varepsilon)} \vert f \vert^{2} + \vert (D^{+})^{j}f \vert^{2} +
\vert \Phi^{j} f \vert^{2}
\end{eqnarray}
around points of $P$ and the model norm
\begin{eqnarray} \label{modelnorminf}
\Vert f \Vert^{2}_{H^{1}_{mod}(\C \setminus \Delta(0, R))} = \int_{\C \setminus \Delta(0,R)} \vert f \vert^{2} +
\vert (D^{+})^{\infty } f \vert^{2} + \vert \Phi^{\infty} f \vert^{2}
\end{eqnarray}
near infinity.
Then it is sufficient to prove the following:
\begin{clm} \label{equivclm}
If $\varepsilon >0$ is chosen sufficiently small and $R>0$ sufficiently
large, then for any smooth section $f \in H^{1}$ we have
\begin{eqnarray} \label{equiv}
c \Vert f^{j} \Vert^{2}_{H^{1}(\Delta(p_{j}, \varepsilon))} <
\Vert f^{j} \Vert^{2}_{H^{1}_{mod}(\Delta(p_{j}, \varepsilon))}
< C \Vert f^{j} \Vert^{2}_{H^{1}(\Delta(p_{j}, \varepsilon))}
\end{eqnarray}
and similarly
\begin{eqnarray} \label{equivinf}
c \Vert f^{j} \Vert^{2}_{H^{1}(\C \setminus \Delta(0, R))} <
\Vert f^{j} \Vert^{2}_{H^{1}_{mod}(\C \setminus \Delta(0, R))}
< C \Vert f^{j} \Vert^{2}_{H^{1}(\C \setminus \Delta(0, R))}
\end{eqnarray}
with some constants $0<c<C$ independent of $f$.
\end{clm}
\begin{proof}
Consider first the case of $p_{j} \in P$. Decompose $f^{j}=f^{j}_{reg}+f^{j}_{sing}$ corresponding to the
splitting (\ref{regsing}). Write also
\begin{align}
f^{j}_{reg} & = \sum_{k=1}^{r_{j}} \phi^{j}_{k} e^{j}_{k} \label{regdecomp} \\
f^{j}_{sing} & = \sum_{k=r_{j}+1}^{r} \phi^{j}_{k} e^{j}_{k} \label{singdecomp}
\end{align}
with respect to the orthonormal basis $\{ e^{j}_{k} \}$ introduced in (\ref{ebasisj}),
where the $\phi^{j}_{k}$ are functions.
Formulas (\ref{polarunit}) and (\ref{polarphi}) and Hypothesis \ref{main} imply that (\ref{modelnorm})
is equivalent to the weighted Sobolev space of sections satisfying
\begin{align} \label{weighted}
\sum_{k=1}^{r_{j}} \int_{\Delta(p_{j}, \varepsilon)} & \vert \phi^{j}_{k} \vert^{2} + \vert \d \phi^{j}_{k} \vert^{2} \\
& + \sum_{k=r_{j}+1}^{r} \int_{\Delta(p_{j}, \varepsilon)}
\absl \frac{\phi^{j}_{k}}{r} \absr^{2} + \vert \d \phi^{j}_{k} \vert^{2} < \infty ,\notag
\end{align}
where $\d $ stands for the trivial connection on functions. Notice that we only add
weights on the singular component.
By \cite{Sim}, Theorem 1 it follows that in $\Delta(p_{j}, \varepsilon)$ the difference between
$(D^{+})^{j}$ and $D^{+}$ is
\begin{equation} \label{pertj}
a^{j} = O(r^{-1+\delta} )
\end{equation}
for some $\delta >0$, and the same estimation holds for the difference between $\Phi^{j}$ and $\Phi$.
It is then immediate that for any $c>0$, the estimation
$$
\int_{\Delta(p_{j}, \varepsilon)} \absl \frac{\phi^{j}_{k}}{r} \absr^{2} > c
\int_{\Delta(p_{j}, \varepsilon)} | a^{j} \phi^{j}_{k} |^{2}
$$
holds for $k =r_{j}+1,\ldots,r$ and for $\varepsilon >0$ sufficiently small.
We therefore have (\ref{equiv}) for $f_{sing}$.
On the other hand, for a function $g$ defined in $\Delta(0,1)$ and for $\delta >0$ fixed, from the claim in
the proof of Theorem 5.4 in \cite{Biq-Boa} we have
$$
\int_{\Delta(p_{j}, 1)} | r^{-1+\delta }g |^{2} \leq c\left( \int_{\Delta(p_{j}, 1)} | \d g |^{2} +
\int_{\Delta(p_{j}, 1) \setminus \Delta(p_{j}, 1/2) } |g|^{2} \right).
$$
Rescaling this inequality to the disk $ \Delta(p_{j}, \varepsilon) $ one easily checks
that it implies
\begin{align} \label{regineq}
\varepsilon^{-2\delta } \int_{\Delta(p_{j}, \varepsilon)} & | r^{-1+\delta } g|^{2} \notag \\
\leq c & \left( \int_{\Delta(p_{j}, \varepsilon)} | \d g |^{2} + \varepsilon^{-2}
\int_{\Delta(p_{j}, \varepsilon) \setminus \Delta(p_{j}, \varepsilon/2) } |g|^{2} \right).
\end{align}
Choosing $\varepsilon$ sufficiently small, applying this to $\phi^{j}_{k}$ for $k=1,\ldots,r_{j}$, and
recalling that on the regular component $(D^{+})^{j}$ is the trivial connection $\d $ and
$\Phi^{j} =0$, we obtain (\ref{equiv}) for $f_{reg}$ as well. This establishes the
equivalence of the norms $\Vert . \Vert^{2}_{H^{1}_{mod}}$ and
$ \Vert . \Vert^{2}_{H^{1}} $ around a finite singularity.
Around infinity, by \cite{Biq-Boa} Lemma 4.6 the difference between $(D^{+})^{\infty}$ and $D^{+}$
is bounded above by a term
\begin{equation} \label{pertinf}
a^{\infty } = O(r^{-1 - \delta })
\end{equation}
for some $\delta >0$, and again the same holds for $\Phi^{\infty} - \Phi$.
The equivalence (\ref{equivinf}) follows immediately from the estimation
$$
|r^{-1 - \delta }f|\leq c |f|
$$
for any $c>0$, whenever $r>R$ with $R$ sufficiently large.
\end{proof}
This then finishes the proof of Lemma \ref{lemmaone} as well.
\end{proof}
From the previous discussion, we bring out as consequence:
\begin{cor} \label{weightedcor}
The Hilbert space $H^{1}(E)$ is the set of sections $f \in L^{2,1}_{loc}(E)$ such that near a logarithmic
singularity $p_{j}$, in the decompositions (\ref{regdecomp}) and (\ref{singdecomp}) we have
$\phi^{j}_{k} \in L^{2,1}$ for $k=1,\ldots,r_{j}$ and $\phi^{j}_{k}/r, \d \phi^{j}_{k} \in L^{2}$ for $k=r_{j}+1,\ldots,r$;
whereas at infinity, the coordinates $\phi^{\infty }_{k}$ of $f$ in the basis (\ref{ebasisinf}) are $L^{2,1}$;
equipped with the norm
\begin{align*}
& \int_{\C \setminus \cup_{j} \Delta(p_{j}, \varepsilon)} |f|^{2} + |\nabla f|^{2} \\
& + \sum_{j=1}^{n} \left\{ \sum_{k=1}^{r_{j}} \int_{\Delta(p_{j}, \varepsilon)} \vert \phi^{j}_{k} \vert^{2} +
\vert \d \phi^{j}_{k} \vert^{2} + \sum_{k=r_{j}+1}^{r} \int_{\Delta(p_{j}, \varepsilon)}
\absl \frac{\phi^{j}_{k}}{r} \absr^{2} + \vert \d \phi^{j}_{k} \vert^{2} \right\}
\end{align*}
The same result holds for sections of $\Omega^{2} \otimes E$, coordinates being expressed in the basis
$\d z \land \d \bar{z}$.
\end{cor}
\begin{proof}
For sections of $\Omega^{0}$, this follows from Claim \ref{equivclm}, (\ref{weighted}) and
$$
|\Phi \otimes f| \leq K|f|.
$$
We then obtain the case of $\Omega^{2}$ by duality.
\end{proof}
We now come back to the analysis of the Dirac operator.
From the definitions of $H^{1}( S^{+} \otimes E )$ and $\Dir_{\xi}$ we see that
this latter admits a bounded extension
\begin{eqnarray} \label{Dirac}
\Dir_{\xi}: H^{1}( S^{+} \otimes E ) \lra L^{2}(S^{-} \otimes E).
\end{eqnarray}
We are now able to announce the first main result of this chapter:
\begin{thm} \label{Fredholm}
The operator (\ref{Dirac}) is Fredholm; if $h$ is harmonic, its kernel vanishes.
\end{thm}
\begin{cor} \label{cokerbundle}
The bundle over $\hat{C} \setminus \hat{P}$ whose fiber over $\xi$ is the cokernel of (\ref{Dirac}) is
a smooth vector bundle.
\end{cor}
\begin{proof}
We recall the well-known fact that the index of a continuous family of Fredholm operators is constant.
On the other hand, if the kernel of a Fredholm operator vanishes, then its index is equal to the
opposite of the dimension of its cokernel. It then follows immediately from the Fredholm theorem that if
the metric is harmonic, then the dimension of the cokernel of the operator $\Dir_{\xi }$ is a finite
constant independent of $\xi $. Moreover, by standard implicit function theorem arguments in Hilbert space
it follows that the cokernels of these Dirac operators in $L^{2}(S^{-}\otimes E)$ vary smoothly with $\xi $.
\end{proof}
Therefore, we may set the following.
\begin{defn} \label{deftrvb}
The \emph{transformed vector bundle} $\hat{E}$ of $(E,D,h)$ of a singular integrable
connection with harmonic metric is the smooth vector bundle
over $\hat{\C } \setminus \hat{P}$ whose fiber over $\xi $ is the finite-dimensional vector space
$\hat{E}_{\xi } = coKer(\Dir_{\xi } ) \subset L^{2}(S^{-}\otimes E)$.
\end{defn}
In the remaining of this section, we prove vanishing of the kernel.
The proof of the first statement of Theorem \ref{Fredholm} is left for the
next section. For the rest of the discussion in this section, we drop the index $\xi $.
\begin{lem} \label{imorth} The subspaces $Im(\Dir \vert_{H^{1}(\Omega^{0})})$ and
$Im(\Dir \vert_{H^{1}(\Omega^{2})})$ of $L^{2}(\Omega^{1})$ are orthogonal.
\end{lem}
\begin{proof}
Let $f_{0} \in H^{1}(\Omega^{0})$ and $f_{2}=g \d z \land \d\bar{z} \in H^{1}(\Omega^{2})$. Suppose first
that $f_{0}$ is smooth and supported on a compact subset of $\C$, and such that near any
singularity $p_{j } \in P$ its singular part is supported away from $p_{j}$. Then
in a neighborhood of any $p_{j}$ in a holomorphic basis $Df_{0}=(\d +a)f$ for some bounded section
$a \in \Om^{1}(End (\E))$, and so we have by partial integration
\begin{equation}\label{ddstarint}
\int_{\C \setminus P} ( Df_{0},D^{\ast}f_{2} ) =
\int_{\C \setminus P} ( D^{2}f_{0},f_{2} ) = 0,
\end{equation}
since $D$ is flat.
Therefore, in order to finish the proof it is sufficient to show the following:
\begin{clm} \label{dense}
The set of compactly supported smooth sections of $S^{+} \otimes E$ on $\C$
with singular part compactly supported away from any singularity is dense in $H^{1}$.
\end{clm}
\begin{proof}
It is sufficient to show the statement for $\Omega^{0}$, the case of $\Omega^{2 }$ being analogous.
First we concentrate on infinity.
Let $f \in H^{1}(E)$, and define cut-off functions $\rho_{R } (r) $ supported in $[0,2R]$
and equal to $1$ on $[0,R]$, such that $\rho_{R }'$ is supported in $[R,2R]$ with
$$
max| \rho_{R }' | \leq 2/R.
$$
Then we need to check that
$$
\rho_{R} (r) f \lra f
$$
in $H^{1}(E)$ as $R \ra \infty $. In view of Corollary \ref{weightedcor}, this boils down
to the classical calculations
$$
\| (1-\rho_{R} (r)) f \| \leq \int_{R\leq r} |f|^{2}
$$
and
\begin{align*}
\| \nabla^{+} ((1-\rho_{R} (r)) f) \| & \leq \int_{R \leq r \leq 2 R} |\rho_{R }'(r)|^{2} |f |^{2} + K \int_{R \leq r} |\nabla^{+} f|^{2} \\
& \leq K' \int_{R \leq r \leq 2 R} |f |^{2} + K \int_{R \leq r} |\nabla^{+} f|^{2},
\end{align*}
where $K,K'$ are constants independent of $R$ and $f$.
Next, let us consider a logarithmic singularity $p_{j}$, and define cut-off functions $\rho_{\varepsilon }$
supported in $[0,\varepsilon ] $, equal to $1$ on $[0, \varepsilon /2]$, and such that
$$
\max |\rho_{\varepsilon }' | \leq \frac{4}{{\varepsilon }}.
$$
We need to show that
$$
(1- \rho_{\varepsilon }) f^{sing} \lra f^{sing}
$$
in $H^{1}(E)$ as $\varepsilon \ra 0$. One sees that
$$
\int_{\C } |\rho_{\varepsilon }f^{sing}|^{2} \leq \int_{r < \varepsilon }|f^{sing}|^{2} \ra 0,
$$
since $f^{sing} \in L^{2}$. In the same way,
$$
\int_{\C } \frac{|\rho_{\varepsilon }f^{sing}|^{2}}{r^{2}} \leq \int_{r < \varepsilon }
\frac{|f^{sing}|^{2}}{r^{2}}\ra 0,
$$
since $f^{sing}/r \in L^{2}$. Finally, we also see that
\begin{align*}
\int_{\C } | \nabla^{+} (\rho_{\varepsilon } f^{sing})|^{2} & \leq \frac{16}{\varepsilon^{2} }
\int_{\varepsilon /2 < r < \varepsilon } |f^{sing}|^{2} + \int_{r < \varepsilon } | \nabla^{+} f^{sing}|^{2} \\
& \leq \int_{\varepsilon /2 < r < \varepsilon } \frac{16 |f^{sing}|^{2}}{r^{2} }
+ \int_{r < \varepsilon } | \nabla^{+} f^{sing}|^{2}
\end{align*}
and all of these expressions converge to zero as well.
\end{proof}
Applying the claim to approximate $f_{0}$ and $f_{2}$ in $H^{1}$ by sections with compactly supported singular
component combined with (\ref{ddstarint}), we immediately get the lemma.
\end{proof}
Now we can come to vanishing of the kernel of (\ref{Dirac}):
by Lemma \ref{imorth}, we have
$$
Ker(\Dir_{\xi}) = Ker(D_{\xi} \vert_{H^{1}(\Omega^{0})}) \oplus Ker(D^{\ast }_{\xi} \vert_{H^{1}(\Omega^{2})}),
$$
it is therefore sufficient to prove vanishing of the kernels of $D$ and of
$D^{\ast }$. By duality, we only need to treat the case of $D$.
Harmonicity of the metric implies the Weitzenb\"ock formula:
\begin{equation} \label{weitzen}
\Dir^{\ast }_{\xi} \Dir_{\xi} = (\nabla^{+}_{\xi})^{\ast } (\nabla^{+}_{\xi}) + (\Phi_{\xi} \otimes )^{\ast } \Phi_{\xi} \otimes
\end{equation}
(see \cite{Biq97}, Thm 5.4.), which then gives by partial integration and Claim \ref{dense} the identity
\begin{eqnarray} \label{trianglezero}
\Vert \Dir_{\xi} f \Vert_{L^{2}}^{2} = \Vert D^{+}_{\xi}f \Vert_{L^{2}}^{2} +
\Vert \Phi_{\xi} f \Vert_{L^{2}}^{2}
\end{eqnarray}
for any $f \in H^{1}(\Omega^{0})$. Suppose now that $f $ is in the kernel of $\Dir_{\xi}$.
Then (\ref{trianglezero}) implies $\Phi_{\xi} f=0$, and
since $\Phi_{\xi}$ is an isomorphism near infinity because of the choice $\xi\notin \hat{P}$,
we also have there $f=0$.
Again by (\ref{trianglezero}), $f$ is covariant constant. This gives
the result, since a covariant constant section vanishing on an open set
vanishes everywhere.
\section{Proof of the Fredholm Theorem }\label{sectpfFr}
A modification of the usual gluing argument of Fredholm-type theorems
works in this case as well. One lets $\phi_{1}$ be a cut-off
function supported in a compact region $R$ outside a neighborhood of the
singularities, and puts $\phi_{2}=1-\phi_{1}$. Since $\Dir $ is a non-singular
first-order elliptic operator in $R$, elliptic theory of a compact
manifold implies that a parametrix $P_{1}$ exists for
$\Dir$ in this region. Next, one considers the problem in neighborhoods of the
singularities. First, one studies the model operators $\Dir^{j} = D^{j} +
(D^{j})^{\ast }$ instead of the Dirac operator itself. There are two different ways of treating these:
\begin{enumerate}
\item{either one extends the functional spaces and the model Dirac operator onto
a natural completion of the neighborhood, which can be either a conformal cylinder or
a complex line (depending on the form of the metric and the functional
spaces), and defines a two-sided inverse of $\Dir^{j}$ on this completion} \label{compl}
\item{or one finds directly a two-sided inverse of $\Dir^{j}$ on a small disk around the
singularity, with a boundary condition verified by any section supported
outside a neighborhood of the boundary.} \label{boundcond}
\end{enumerate}
Let us see how these allow to deduce the Fredholm theorem:
if we take $R$ sufficiently
large, then on the support of $\phi_{2}$ all of these inverses $(\Dir^{j})^{-1}$ are defined.
One then sets
\begin{align*}
& P : L^{2}(S^{-} \otimes E) \lra H^{1}(S^{+} \otimes E) \\
& P(u)= \phi_{1}P_{1}(\phi_{1}u) + \sum_{j} \phi_{2} (\Dir^{j})^{-1}(\phi_{2}u),
\end{align*}
and shows that this operator is a two-sided parametrix of $\Dir$ on all
$\C$. This can be done along classical lines, the only difference being
that near the singularities we have inverses of the local models of the
operator and not inverses of the operator itself.
Therefore, we proceed as follows: first, we study the local models of the
Dirac operator around the singularities, and establish the isomorphisms as
in (\ref{compl}) or in (\ref{boundcond}).
Then we prove that the effect of passing to the model operators
from the global ones at the singularities only amounts to
adding a compact operator $H^{1}(S^{+} \otimes E) \ra L^{2}(S^{-} \otimes E)$, which then gives the theorem.
\subsection{Logarithmic singularities} Let $\Delta(p, \varepsilon )$ be a small
neighborhood of $p \in P$. Up to a change of coordinates, we may suppose
$ \varepsilon =1$. Identify $ \Delta(p, 1 ) \setminus \{ p \} = S^{1} \times ]0,1] $ via
polar coordinates $(r, \theta )$. Since the local model (\ref{polard}) is
diagonal in the basis $\{ e^{j}_{k} \}$, we see that the model Dirac operator on this disk
$$
\Dir^{j}_{0} = D^{j} - ( D^{j} )^{\ast }: (\Omega^{0} \oplus \Omega^{2} ) \otimes
E|_{ \Delta(p, 1 ) } \lra \Omega^{1} \otimes E|_{ \Delta(p, 1)}
$$
splits into the direct sum of its restrictions to the rank-one
components generated by one of the $\{ e^{j}_{k} \}$. Again, we have two
cases: first, $k \in \{ 1, \ldots r_{j} \}$ (regular case) and secondly $k \in \{
r_{j}+1, \ldots r \}$ (singular case).
In the regular case, by definition the model Dirac operator on a rank-one
component is just the operator
$$
\Dir = \d - \d^{\ast} : S^{+} = \Omega^{0} \oplus \Omega^{2} \lra \Omega^{1} = S^{-},
$$
which identifies to a projection of the real part of the usual Dirac operator
on a product of two disks in $\C^{2}$ given by
$$
\dbar - \dbar^{\ast } : \Omega^{0,0} \oplus \Omega^{0,2} \lra \Omega^{0,1}.
$$
Since this is known to have an inverse for the Atiyah-Patodi-Singer
boundary condition, the case of the regular part at a finite singularity
follows.
On the singular component near a finite singularity, consider again the
coordinate change $t = - \ln r \in \R^{+} $. The local model of $D$
with respect to $t$ is given by
\begin{align*}
D^{j} & = d + i \bar{\mu}^{j}_{k} \d \theta + [\Re \mu^{j}_{k}-\beta^{j}_{k}]
\frac{\d r} {r}
\end{align*}
(see (\ref{polard})).
Notice that the rank of $S^{+}$ and that of $S^{-}$ are both equal to $2$:
we trivialize them using the unit-norm sections $(1,r \ \d r \land \d \theta )$ and
$(\d r , r \d \theta)$ respectively, so that both $S^{+}\otimes E_{sing}$ and $S^{-}\otimes E_{sing}$ become isomorphic
to $E_{sing}\oplus E_{sing}$ as Hermitian bundles.
As we have seen in Lemma \ref{lemmaone}, the space $H^{1}( \Delta(p, 1),E_{sing})$ is equal to the model
space of all sections $\phi$ having
$$
\int_{\Delta (p, 1 )} \left( | \nabla \phi |^{2} + \absl \frac{\phi }{r} \absr^{2} \right) \ r \d r
\d \theta < \infty.
$$
By conformal invariance of the norm of $1$-forms and $\d t=\d r/r$, this is
$$
\int_{ S^{1} \times \R^{+} } \left( | \nabla \phi |^{2} + | \phi |^{2} \right)\ \d t \d \theta < \infty,
$$
with the norm of the $1$-form $\nabla \phi $ measured with respect to the volume form $\d t \d \theta$.
This latter is just the definition of the weighted Sobolev space $L^{2,1}_{0}(S^{1}\times \R^{+},E_{sing})$
with one derivative in $L^{2}$and weight $0$. In a similar way, the usual
$L^{2}$-space of sections of $E_{sing}$ on the disk is identified with the space
$L^{2}_{-1}(S^{1} \times \R^{+},E_{sing})$ of $L^{2}$-sections with weight $-1$ on the half cylinder, for
$$
\int_{\Delta (p, 1 )} | \phi |^{2} \ r \d r \d \theta = \int_{S^{1} \times \R^{+}}|\phi e^{-t} |^{2} \d t \d \theta.
$$
Hence in the trivialisation $(\d r , r \d \theta)$ of $S^{-}$, the usual $L^{2}$-space of $1$-forms
on the disk is identified with the weighted space
$$
L^{2}_{-1}(S^{1} \times \R^{+}, E_{sing}\oplus E_{sing}).
$$
\begin{clm}
Let $(r, \theta )$ be polar coordinates around $p=p^{j}$ as above. Let $k \in \{ r_{j}+1,\ldots,r \}$ and
$$
(f, g (r \d r \land \d \theta) ) \otimes e_{k}^{j} \in C^{\infty}(\Delta \setminus \{ 0 \},S^{+} \otimes E_{sing}).
$$
Then the value of the model Dirac operator $\Dir^{j}$ on this section is
\begin{align*}
& \left( \partial_{r}f + \frac{\Re \mu_{k}^{j}-\beta_{k}^{j}}{r} f -
\frac{\partial_{\theta }+i \mu_{k}^{j}}{r} g \right) dr \\ & +
\left( \frac{\partial_{\theta }+i \bar{\mu}_{k}^{j}}{r} f +
\partial_{r} g - \frac{\Re \mu_{k}^{j} - \beta_{k}^{j}}{r} g \right) r \d \theta .
\end{align*}
In particular, in the unitary trivialisations $(1,r \ \d r \land \d \theta )$ and $(\d r , r \d \theta)$ of $S^{+}$ and
$S^{-}$, the operator
$$
r\Dir^{j}=e^{-t}\Dir^{j}
$$
is translation-invariant with respect to the cylindrical coordinate $t$.
\end{clm}
\begin{proof}
This is a direct computation: for $f \otimes e_{k}^{j}$ it follows
immediately from (\ref{polard}). For the image of
$g (r \d r \land \d \theta) \otimes e_{k}^{j}$, consider first the smooth form $\varphi
\d r \otimes e_{k}^{j} $ supported in a compact region of $\Delta \setminus \{
0 \}$; then by the same formula we have
\begin{align*}
\langle \varphi \d r \otimes e_{k}^{j}, (D^{j})^{\ast } g (r \d r \land \d \theta) \otimes e_{k}^{j} \rangle & = \langle D^{j} (\varphi \d
r),g(r\d r \land \d \theta) \rangle \\
& = -\langle (\partial_{\theta }+ i \bar{\mu}_{k}^{j} ) \varphi \d r
\land \d \theta , g (r \d r \land \d \theta) \rangle \\
& = - \frac{1}{r} \langle (\partial_{\theta }+ i
\bar{\mu}_{k}^{j} ) \varphi , g \rangle \\
& = \frac{1}{r} \langle \varphi , (\partial_{\theta } + i \mu_{k}^{j} )g \rangle
\end{align*}
and thus the projection of $(D^{j})^{\ast } g (r \d r \land \d \theta) \otimes e_{k}^{j} $ on the
$\d r$-component is $(\partial_{\theta } + i\mu_{k}^{j} )g \d r \otimes e_{k}^{j}$.
The other component is obtained taking a compactly supported smooth
form $\psi r \d \theta \otimes e_{k}^{j}$:
\begin{align*}
\langle \psi r \d \theta \otimes e_{k}^{j}, (D^{j})^{\ast } g ( r \d r \land \d \theta ) \otimes e_{k}^{j} \rangle & =
\langle D^{j} (\psi r \d \theta), g(r\d r \land \d \theta) \rangle \\
& = \left\langle \left( \partial_{r} + \frac{\Re \mu_{k}^{j} - \beta_{k}^{j}}{r} \right) \psi , g \right\rangle \\
& =\left\langle \psi , \left(-\partial_{r} + \frac{\Re \mu_{k}^{j} - \beta_{k}^{j}}{r}
\right)g \right\rangle,
\end{align*}
and the formula of the claim follows. It implies that $r\Dir^{j}$ is translation-invariant
because $\partial_{r}=-\partial_{t}/r$.
\end{proof}
By definition, the weight $0$ is \emph{critical} for $r\Dir^{j}$ if and only if
there exists a non-trivial solution of
$$
e^{-t}\Dir^{j} (A e^{ -\nu t+in \theta}, B e^{-\nu t + in \theta } (r \ \d r \land \d \theta)) = 0
$$
with some constants $A,B \in \C $ and a constant $\nu \in \C$ such that $\Re \nu=0$.
Turning back to the coordinate $r$ again, this is equivalent to having
\begin{eqnarray} \label{critval}
r\Dir^{j} (A r^{\nu } e^{in \theta }, B r^{\nu } e^{in \theta } (r \ \d r \land \d \theta)) = 0.
\end{eqnarray}
By \cite{Lock-McO}, if $0$ is not a critical weight, then the translation-invariant elliptic
differential operator
$$
e^{-t}\Dir^{j} : L^{2,1}_{0}(S^{1} \times \R^{+}, S^{+} ) \lra L^{2}_{0} (S^{1}\times \R^{+},S^{-})
$$
is invertible, and thus so is
$$
\Dir^{j} : L^{2,1}_{0}( S^{1} \times \R^{+}, S^{+} ) \lra L^{2}_{-1} (S^{1}\times \R^{+},S^{-})
$$
since
$$
e^{t}:L^{2}_{0} \lra L^{2}_{-1}
$$
is an isomorphism. Therefore, in order to establish the desired isomorphism in the singular
case, we only need to check the weight $0$ is not critical for $r\Dir^{j}$.
Applying the claim to the equation (\ref{critval}), we see that $0$ is a
critical weight if and only if the system of linear equations
\begin{align*}
(\nu + \Re \mu -\beta )A - i (n + \mu )B & = 0 \\
i (n + \bar{\mu} ) A + (\nu + \beta - \Re \mu )B &= 0
\end{align*}
has a non-trivial solution $(A,B)\in \C^{2}$ for some $\nu \in \C$ with $\Re \nu=0$
(here we have omitted indices $j$ and $k$ of $\mu$ and $\beta$ for simplicity).
This system has a non-trivial solution if and only if the determinant formed by the
coefficients is equal to $0$:
$$
\nu ^{2} - (\Re \mu - \beta )^{2} - \vert n + \mu \vert ^{2} =0.
$$
Since $\Re \nu $ must be $0$, this can only be the case if $\nu = \Re \mu - \beta = n+ \mu
=0$. By assumption $0 \leq \beta < 1$, and $n$ is an integer, therefore the
only case this can hold is when $\beta = \mu = 0$, which is impossible, since we
are looking at the singular component of the bundle. Therefore, there are
no non-trivial solutions to (\ref{critval}), and $0$ is not a critical
weight.
\subsection{Singularity at infinity}
In this section the importance of the condition $\xi \notin \hat{P}$ will come out;
therefore we write out the index $\xi $ of our operators.
A neighborhood of infinity in $\C \setminus P$ is given by the complementary
$\C \setminus \Delta(R) $ of a large disk around $0$. A natural choice of
completion of this manifold is of course $\C $, with its standard metric
$|\d z|^{2}$. We choose to study the local model in the orthonormal basis $\{ e^{\infty }_{k}
\}$ defined in (\ref{ebasisinf}). This
allows us to think of $E$ as the trivial bundle $\C^{r}$ over $\C \setminus \Delta(R)$, with
standard hermitian metric on the fibers. By (\ref{holtrivinf}) this basis (up to a
polynomial scaling factor) is a natural one for the Higgs-bundle point of
view, so the deformation is that considered in (\ref{higgsdef}), and
the operator $D_{\xi }$ near infinity is given (up to terms of order $r^{-1}$) by
$$
D_{\xi }^{\infty } = \d + \frac{A- \xi \Id }{2} \d z + \frac{(A- \xi \Id )^{\ast }}{2}
\d \bar{z}
$$
(see (\ref{singpart})), and a natural extension of it to all of $\C $ can be
given by the same formula. This implies immediately that
$$
\Phi_{\xi }^{\infty } = \frac{A- \xi \Id }{2} \d z + \frac{A^{\ast }- \bar{\xi} \Id }{2} \d \bar{z}
$$
and $(D^{\infty })^{+} = \nabla $ (the trivial connection) on all of $\C$.
For a section $\phi \in L^{2}(\Omega^{0})$ supported in $\C \setminus \Delta(R)$, the condition
$\Phi_{\xi } \phi \in L^{2}( \Omega^{0} )$ then automatically holds, and $(D_{\xi }^{\infty })^{+} \phi \in L^{2}$ is
equivalent to $\nabla \phi \in L^{2}$. Therefore, on sections of $\Omega^{0}$ supported on the
complementary of $\Delta(R)$, the $H^{1}$-norm is equivalent to the usual
Sobolev $L^{2,1}$-norm. A similar argument shows that for sections of $\Omega^{2}$, the
$H^{1}$-norm is also equivalent to the usual $L^{2,1}$-norm.
Therefore, on all of $\C $, we must consider a natural extension of these
functional spaces, namely $L^{2,1} (\C , \Omega^{0} \oplus \Omega^{2})$. In an analogous manner, on $S^{-}$ we
consider the extension $L^{2}(\C, \Omega^{1})$ of $L^{2}( \C \setminus \Delta(R) ,
\Omega^{1})$. Therefore, we need to prove the
\begin{lem} On $\C $, the Dirac operator
\begin{eqnarray} \label{diracinf}
\Dir_{\xi }^{\infty }= D_{\xi }^{\infty } - (D_{\xi }^{\infty })^{\ast } : L^{2,1}( \Omega^{0} \oplus \Omega^{2} ) \lra
L^{2} ( \Omega^{1} )
\end{eqnarray}
is an isomorphism.
\end{lem}
\begin{proof}
Since $A$ is supposed to be diagonal in this basis with eigenvalues
$\xi_{l} \ (l=1,\ldots , n')$, we may restrict ourselves to the study of the
operator $D^{\infty } = \d + (\xi_{l}-\xi )/2 \d z + (\bar{\xi }_{l}-\bar{\xi })/2 \d \bar{z}$. We need the
following:
\begin{clm} Denote by $\Delta$ the plain Laplace operator $\nabla^{\ast } \nabla$ on forms. Then we have
\begin{eqnarray} \label{weitz}
\Dir_{\xi }^{\infty} (\Dir_{\xi }^{\infty})^{\ast } = - \Delta - \frac{\vert \xi_{l }-\xi \vert^{2}}{4}.
\end{eqnarray}
\end{clm}
\begin{proof}
This is an easy computation.
\end{proof}
Now recall that by the classical theory of the
Laplace operator, $ \Delta + \lambda^{2} $ with $ \lambda > 0$ is an
isomorphism
\begin{equation} \label{isominf}
L^{2,2}(\C , \Omega^{j} ) \lra L^{2} (\C , \Omega^{j} ).
\end{equation}
This statement can be for example obtained passing to the Fourier transform $|\hat{x}|^{2}+ \lambda^{2}$ of this
operator.
Coming back to our situation, the condition $\xi \notin \hat{P}$ means exactly that $\xi_{l}-\xi \neq 0$ for any $l=1,\ldots,n'$.
This immediately implies that (\ref{diracinf}) is surjective: indeed,
clearly $Im( (\Dir_{\xi }^{\infty})^{\ast } ) \subset L^{2,1}( \Omega^{0} \oplus \Omega^{2} )$, and
$ \Dir_{\xi }^{\infty} (\Dir_{\xi }^{\infty})^{\ast } $ is surjective by the isomorphism
(\ref{isominf}). For injectivity, note that a formula similar to
(\ref{weitz}) holds for the Laplace operator $ (\Dir_{\xi }^{\infty})^{\ast } \Dir_{\xi }^{\infty} $ as well.
This in turn implies that the $L^{2,2}$-kernel of $\Dir_{\xi }^{\infty}$ vanishes. Elliptic regularity then shows
that the $L^{2,1}$-kernel vanishes as well.
\end{proof}
\subsection{Compact perturbation}
We wish to prove that near each one of the singularities the effect of
passing from the global operator to its local model, i.e. subtracting
the perturbation term only amounts to a
compact operator $H^{1}(S^{+}\otimes E ) \ra L^{2}(S^{-}\otimes E )$. This then
finishes the proof of the Fredholm theorem, because the sum of a Fredholm operator
and a finite number of compact operators is Fredholm.
Consider first the case of a singularity at a finite point. Recall from
Lemma \ref{lemmaone} that near $p_{j}$ the space $H^{1}(S^{+} \otimes E)$
is equal to the sum
$$
L^{2,1}_{eucl} (S^{+} \otimes E_{reg}) \oplus L^{2,1}_{0} (S^{+} \otimes E_{sing}),
$$
where $L^{2,1}_{eucl}$ is the usual Sobolev space on the disk of $L^{2}$-functions
with one derivative in $L^{2}$ with respect to Euclidean metric, whereas $L^{2,1}_{0}$
is the weighted Sobolev space defined by
$$
\int_{\Delta(p_{j},\varepsilon )} \left( \absl \frac{\phi }{r} \absr^{2} + |\nabla \phi
|^{2} \right) |\d z |^{2} \leq \infty .
$$
Also, the order of growth of the $1$-form perturbation term $a^{j}$ with
respect to Euclidean metric is by (\ref{pertj}) at most $O(r^{-1+\delta })$,
with $\delta >0$. We need to prove that we have compact Sobolev multiplications for
functions on the disk
\begin{equation} \label{mult1}
L^{2,1}_{eucl} \xrightarrow{a^{j}} L^{2}_{eucl}
\end{equation}
and
\begin{equation} \label{mult2}
L^{2,1}_{0} \xrightarrow{a^{j}} L^{2}_{eucl}.
\end{equation}
Consider first (\ref{mult1}): since the disk is a compact manifold, for any
$2<p<\infty $ the inclusion $L^{2,1}_{eucl} \inc L^{p}_{eucl}$ is compact. On the other hand,
$O(r^{-1+\delta }) \d r + O(r^{-1+\delta }) r \d \theta $ is in $L^{2+\varepsilon }_{eucl}$
for some $\varepsilon>0$. Choose $p$ such that $1/2=1/(2+\varepsilon) + 1/p$;
(\ref{mult1}) then follows immediately from the
continuous multiplication $L^{2+\varepsilon}_{eucl} \times L^{p}_{eucl} \ra L^{2}_{eucl} $.
Now, we come to (\ref{mult2}): this is an immediate consequence of the
previous, for the weighted norm $L^{2,1}_{0}$ is stronger then
$L^{2,1}_{eucl}$.
Next, let us treat the case of the singularity at infinity. In the
coordinate $w=1/z$ we have a second-order singularity on the disk
$\Delta(0,1/R)$. Let $w= \rho e^{i\vartheta }$; by (\ref{pertinf}) the perturbation is $O(\rho^{-1-\delta})$, and
the $H^{1}$-norm of a function $\phi $ supported near infinity is given by
$$
\int_{ \C \setminus \Delta (0,R)} \left( |\phi |^{2} + |\nabla \phi |^{2} \right) | \d z |^{2} =
\int_{ \Delta (0,1/R)} \left( \absl \frac{\phi }{\rho^{2} } \absr^{2} + |\nabla \phi |^{2} \right) |\d w|^{2}.
$$
In particular, in the coordinate $w$ this norm is also stronger then $L^{2,1}_{eucl}$, so we
conclude from (\ref{mult1}).
\section{$L^{2}$-cohomology and Hodge theory}\label{l2coh}
In this section we keep on supposing that we have on one side an integrable
connection $D$ with singularities in $P \cup \{ \infty \}$, with prescribed behaviors
at these points, given in regular singularities by (\ref{pertj}) and at infinity by (\ref{pertinf}).
In Theorem \ref{Fredholm} we proved that the deformed operators $\Dir_{\xi}$
are Fredholm between the spaces $H^{1}$ and $L^{2}$; in particular their indices agree.
We also showed that if the metric is harmonic then the kernel of the Dirac operator vanishes, hence the
index of $\Dir_{\xi}$ is equal to the opposite of the dimension of the cokernel
$Coker(\Dir_{\xi } )$, this operator being considered between functional
spaces as in (\ref{Dirac}). This dimension is therefore a constant independent of $\xi $, and it follows
from the implicit function theorem that the spaces $\hat{E}_{\xi }=Coker(\Dir_{\xi } )$ define a finite-rank smooth
vector bundle $\hat{E}$ over $\hat{\C} \setminus \hat{P} $, the rank being
equal to the opposite of the index of (\ref{Dirac}).
Here we wish to interpret this cokernel as the first cohomology of the elliptic complex
\begin{eqnarray} \label{ltwocompl}
L^{2}(\Omega^{0} \otimes E) \xrightarrow{\text{$D_{\xi}$}} L^{2}(\Omega^{1} \otimes E)
\xrightarrow{\text{$D_{\xi}$}} L^{2}(\Omega^{2} \otimes E),
\end{eqnarray}
(see Theorem \ref{coker}), and also as the space of harmonic sections with respect to the Laplace operator
of the adjoint Dirac operator $\Dir_{\xi }^{\ast }$ (Theorem \ref{laplker}).
Since the operators in (\ref{ltwocompl}) are unbounded, we need to define their domains.
In this chapter $C^{\infty }_{0}$ stands for smooth sections supported in a compact subset of $\C \setminus P$.
\begin{defn} The \emph{maximal domain} of $D|_{\Omega^{i}}$ is
$$
\Dm(D|_{\Omega^{i}}) = \{ u \in L^{2}(\Omega^{i}): Du \in L^{2}(\Omega^{i+1}) \} ,
$$
where $Du \in L^{2} $ is understood in the sense of currents, i.e.
the functional $v \in C^{\infty }_{0}(\Omega^{i+1}) \mapsto \langle u, D^{\ast }v \rangle $ is
continuous in the $L^{2}$-topology.
\end{defn}
By local elliptic regularity, this amounts to the same thing as $Du$ being
an $L^{2}$-section. When it does not cause any confusion,
we will simply write $\Dm (\Omega^{i})$ for $\Dm(D|_{\Omega^{i}})$.
It is easy to see that if we consider $D$
on its maximal domain, then the kernel is a closed subspace of $L^{2}$, and
the image of $D$ on $\Omega^{i-1} $ is contained in the kernel of $D$ on $\Omega^{i}$.
The image of a general differential operator is however not always a closed subspace of the kernel.
\begin{defn} For $i \in \{ 0,1,2 \}$, the $i^{th}$ \emph{$L^{2}$-cohomology} of $D$ is
$Ker(D\vert _{\Omega^{i} \otimes E})/ {Im( D\vert _{\Omega^{i-1} \otimes E})} $, where both
of these operators are considered with maximal domain, and the
operators not shown in (\ref{ltwocompl}) are trivial. It is denoted by $L^{2}H^{1}(D)$.
\end{defn}
Our aim is to obtain the following:
\begin{thm} \label{coker} The cokernel of $\Dir $ defined on $H^{1}(S^{+} \otimes E)$ is equal to the first
$L^{2}$-cohomology of $D$.
\end{thm}
\begin{proof}
Recall that by definition
\begin{align}
Coker(\Dir \vert_{H^{1}(S^{+} \otimes E)} ) & = (Im(\Dir \vert_{H^{1}(S^{+} \otimes E)} ))^{\bot }\notag \\
& = (Im(D \vert_{H^{1}(\Omega^{0} \otimes E)} ))^{\bot } \cap (Im(D^{\ast } \vert_{H^{1}(\Omega^{2} \otimes E)} ))^{\bot },\label{descrcoker}
\end{align}
where $A^{\bot }$ stands for the $L^{2}$-orthogonal of the subspace $A \subset
L^{2}$. Therefore, it is sufficient to prove the following lemmas:
\begin{lem} \label{maxdom}
The maximal domain of
$$
D:L^{2}(\Omega ^{0} \otimes E) \lra L^{2}(\Omega ^{1} \otimes E)
$$
is $H^{1}(\Omega ^{0} \otimes E)$. Similarly,
the maximal domain of
$$
D^{\ast }:L^{2}(\Omega ^{2} \otimes E) \lra L^{2}(\Omega ^{1} \otimes E)
$$
is $H^{1}(\Omega ^{2} \otimes E)$. In particular, the maximal domain of
$$
\Dir :L^{2}(S^{+} \otimes E) \lra L^{2}(S^{-} \otimes E)
$$
is $H^{1}(S^{+} \otimes E)$. Moreover, if this latter space is equipped with the norm $\| . \|_{H^{1}}$ defined in (\ref{norm}),
then $\Dir$ is a bounded operator from $H^{1}(S^{+} \otimes E)$ to $L^{2}(S ^{-} \otimes E)$.
\end{lem}
\begin{lem} \label{imker} We have
$$
(Im(D^{\ast } \vert_{H^{1}(\Omega^{2} \otimes E)} ))^{\bot } = Ker( D \vert_{\Dm(\Omega ^{1}
\otimes E)}).
$$
\end{lem}
\begin{lem} \label{closed} The image of $D: H^{1}(\Omega ^{0} \otimes E) \ra L^{2}(\Omega ^{1} \otimes E)$ is
closed.
\end{lem}
Indeed, Lemmas \ref{maxdom} and \ref{imker} together with (\ref{descrcoker}) imply
that the cokernel is equal to
$$
(Im(D \vert_{\Dm(\Omega^{0} \otimes E)} ))^{\bot } \cap Ker( D \vert_{\Dm(\Omega ^{1}\otimes E)}),
$$
which in turn is identified to the first reduced $L^{2}$-cohomology of
(\ref{ltwocompl}), i.e. to
$$
Ker(D\vert _{\Dm(\Omega^{1} \otimes E)})/ \overline{Im( D\vert _{\Dm(\Omega^{0} \otimes E)})},
$$
where the bar over the image stands for the $L^{2}$-closure of that
space. Lemma \ref{closed} now concludes the proof of Theorem \ref{coker}.
\begin{proof} (Lemma \ref{imker}) We first show the
\begin{clm} The adjoint of the unbounded operator
\begin{eqnarray} \label{Dstar}
D^{\ast } : L^{2} (\Omega^{2} \otimes E) \lra L^{2}(\Omega^{1} \otimes E)
\end{eqnarray}
with domain $H^{1}(\Omega^{2} \otimes E)$ is the unbounded operator
\begin{eqnarray} \label{Dstaradj}
D: L^{2}(\Omega^{1} \otimes E) \lra L^{2} (\Omega^{2} \otimes E)
\end{eqnarray}
with domain $\Dm(\Omega^{1} \otimes E)$.
\end{clm}
\begin{proof} (Claim)
It is clear that the formal adjoint of (\ref{Dstar})
is (\ref{Dstaradj}), we only need to prove its domain is $\Dm$. By
definition, a section $u \in L^{2}( \Omega ^{1})$
is in the domain of the adjoint operator $\mbox{Dom}((D^{\ast })^{\ast })$ if and
only if for all $v \in H^{1} (\Omega^{2} \otimes E)$ we have
$$
\vert \langle u,D^{\ast }v \rangle \vert \leq K \| v \|
$$
with a constant $K$ only depending on $u$. Now, since $v \in H^{1}$ and $u \in
L^{2}$, by Claim \ref{dense} we can perform partial integration to the left-hand side
of this formula. Therefore, $u$ is
in the domain of the adjoint operator if and only if the functional
$$
v \mapsto \langle Du, v \rangle
$$
is bounded in $L^{2}(\Omega^{2} \otimes E)$. But this condition is equivalent to $Du \in
L^{2}(\Omega^{2} \otimes E)$, and the claim follows.
\end{proof}
Lemma \ref{imker} now directly follows from the claim and the general fact
that the cokernel of an unbounded operator is equal to the kernel of its
adjoint.
\end{proof}
\begin{proof} (Lemma \ref{maxdom}) First we need to prove that for
a section $u$ of $ L^{2}(\Omega ^{0} \otimes E) $ we have $Du \in L^{2}$ if and only if
both $D^{+}u \in L^{2}$ and $\Phi u \in L^{2}$. The ``if '' direction being obvious, we
concentrate ourselves on the opposite statement, and suppose in what
follows that $u$ is an $ L^{2}$-function with $Du \in L^{2}$.
We first study the singularity at infinity. For $|u|$ sufficiently large, we have the point-wise
estimate
$$
\vert \Phi u \vert \leq 2 K \vert u \vert,
$$
where $K$ is the maximal modulus of the eigenvalues of the matrix
$A$. Therefore, $u \in L^{2}$ at infinity implies $\Phi u \in L^{2}$ at infinity,
and consequently $D^{+}u=Du - \Phi u \in L^{2}$ at infinity, and we are done.
Next, consider the case of a singularity at a finite point.
In the orthonormal basis (\ref{ebasisj}), the operators we study are equal, up to a perturbation
term, to the local models (see (\ref{polarunit}), (\ref{polarphi}), (\ref{polard}))
\begin{align*}
(D^{+})^{j} \phi & =(\mbox{d} + i \Re \mu^{j}_{k} \mbox{d} \theta) \phi \\
\Phi^{j} \phi & = [(\Re \mu^{j}_{k} - \beta^{j}_{k}) \frac{\d r}{r} + \Im
\mu^{j}_{k} \d \theta ] \phi \\
D^{j}\phi & = [\mbox{d} + i \bar{\mu}^{j}_{k} \d \theta + (\Re \mu^{j}_{k} -
\beta^{j}_{k}) \frac{\d r}{r} ]\phi
\end{align*}
To simplify notation, from now on we drop the indices $j$ and $k$. Note
that because of Lemma \ref{lemmaone}, it is sufficient to prove that $\Phi^{j}
\phi $ and $(D^{+})^{j} \phi$ are in $L^{2}$.
Notice also that since the perturbation $a^{j}$ may mix the regular and singular
components, a priori it is not sufficient to prove for example that
$ \phi_{reg } \in L^{2}$ and $D \phi_{reg } \in L^{2} $ imply $(D^{+})^{j} \phi_{reg } \in
L^{2}$, because $D \phi \in L^{2}$ does not imply directly $D \phi_{reg } \in L^{2}$
in the presence of a mixing perturbation term. However, remark that
denoting by $a^{j}_{r,r}$ the part of the endomorphism $a^{j}$ that takes
the regular component into the regular one, and $a^{j}_{r,s},a^{j}_{s,r},a^{j}_{s,s}$
the other parts, we have
\begin{align} \label{mix}
\int_{\Delta(p_{j}, \varepsilon ) } |(D^{j}+a^{j})\phi |^{2} = & \int_{\Delta (p_{j}, \varepsilon )} |(D^{j}+a^{j}_{r,r})
\phi_{reg}+a^{j}_{s,r}\phi_{sing}|^{2}\notag \\
& + \int_{\Delta(p_{j}, \varepsilon ) } |(D^{j}+a^{j}_{s,s})
\phi_{sing}+a^{j}_{r,s}\phi_{reg}|^{2} \\
\geq & \int_{\Delta(p_{j}, \varepsilon ) } |D^{j} \phi_{reg}|^{2} + |D^{j} \phi_{sing}|^{2}\notag \\
& - |a^{j}\phi_{reg}|^{2} - |a^{j}\phi_{sing}|^{2}, \notag
\end{align}
and this estimate shows that we can treat the two components separately: the left-hand side
is finite by hypothesis, whereas the integrals of $|a^{j}\phi_{reg}|^{2}$ and $|a^{j}\phi_{sing}|^{2}$
by Kato's inequality and (\ref{regineq}); hence the same thing holds for the integrals of
$|D^{j} \phi_{reg}|^{2}$ and $|D^{j} \phi_{reg}|^{2}$.
On the regular component, the above expressions simplify to $D^{j}=(D^{+})^{j}=\nabla $ (the
trivial connection), and $\Phi^{j} =0$. What we need to show is that $\phi_{reg} ,D \phi_{reg} \in
L^{2}$ implies $\nabla \phi_{reg} \in L^{2}$, if $D =\nabla + a^{j}$ with $a^{j} =O(r^{-1+\delta })$.
Recall that by Kato's inequality and (\ref{regineq}) with $\varepsilon >0$ chosen sufficiently small
we have
$$
\int_{\Delta(p_{j}, \varepsilon )} \vert a^{j} \phi_{reg} \vert^{2} \leq
\int_{\Delta(p_{j}, \varepsilon )} \vert D \phi_{reg} \vert^{2} +
\int_{\Delta(p_{j}, \varepsilon )\setminus \Delta(p_{j}, \varepsilon /2 )} \vert \phi_{reg} \vert^{2}.
$$
It follows that
\begin{align*}
\int_{\Delta(p_{j}, \varepsilon )} \vert \nabla \phi_{reg} \vert^{2}
& \leq \int_{\Delta(p_{j}, \varepsilon )} \vert D \phi_{reg} \vert^{2} +
\int_{\Delta(p_{j}, \varepsilon )} \vert a^{j} \phi_{reg} \vert^{2} \\
& < 2 \int_{\Delta(p_{j}, \varepsilon )} \vert D \phi_{reg} \vert^{2} +
2\int_{\Delta(p_{j}, \varepsilon )} \vert \phi_{reg} \vert^{2}.
\end{align*}
Now by the hypothesis $\phi, D \phi \in L^{2}$, the right-hand side
is finite. Therefore $\nabla \phi \in L^{2}$ as we wished to show.
Consider now the singular case: again, we need to show that if we have a
section $\phi \in L^{2}$ such that $D\phi \in L^{2}$, then $D^{+}\phi_{sing}, \Phi \phi_{sing} \in L^{2}$.
Here, usual elliptic regularity does not give the claim,
because we need to deduce that $\phi_{sing} /r \in L^{2}$. From now on, we write
$\phi = \phi_{sing}$ to lighten notation. Decompose $\phi$ into its Fourier-series near $p^{j}$:
$$
\phi(r,\theta )= \sum_{n=-\infty }^{\infty } \phi_{n}(r)e^{in\theta }
$$
Choosing $\varepsilon$ sufficiently small, we can make the perturbation
term $a^{j}$ be smaller on $\Delta(p_{j}, \varepsilon )$ then $\nu/r$ for
any $\nu>0$.
Write first the $\d \theta $-term of $D^{j}\phi$:
$$
D_{\theta }^{j}\phi = (\partial_{\theta }+ i \bar{\mu })\phi \d \theta = i \d \theta \sum_{n=-\infty }^{\infty } (n + \bar{\mu })
\phi_{n}(r) e^{in\theta }.
$$
By this and the estimate on the perturbation, we infer that
\begin{align} \label{mixsing}
\Vert (D^{j}_{\theta } +a^{j}) \phi \Vert_{L^{2}(\Delta(p_{j}, \varepsilon ))}^{2} \geq
\Vert D^{j}_{\theta } \phi \Vert_{L^{2}(\Delta(p_{j}, \varepsilon ))}^{2} -
\Vert \nu \phi /r \Vert_{L^{2}(\Delta(p_{j}, \varepsilon ))}^{2}& \notag \\
= \int_{\Delta(p,\varepsilon )} \sum_{n=-\infty }^{\infty } (\vert n + \bar{\mu } \vert^{2}
-\nu^{2})\frac{\vert \phi_{n}(r) \vert^{2}}{r^{2}}& \\
= \int_{\Delta(p,\varepsilon )} \sum_{n=-\infty }^{\infty } (\vert n + \Re \mu \vert^{2} -
\nu^{2} + \vert \Im \mu \vert^{2}) \frac{\vert \phi_{n}(r) \vert^{2}}{r^{2}}&. \notag
\end{align}
By Hypothesis \ref{main} we have $\Re \mu \notin \Z$, and so if $\nu $ is sufficiently small, then
the last expression can be bounded from below by
\begin{align} \label{est}
\frac{1}{2} \int_{\Delta(p,\varepsilon )}& \sum_{n=-\infty }^{\infty } (\vert n + \Re \mu \vert^{2} +
\vert \Im \mu \vert^{2}) \frac{\vert \phi_{n}(r) \vert^{2}}{r^{2}} \\
= &\frac{1}{2} \int_{\Delta(p,\varepsilon )} |(D_{\theta }^{+})^{j}\phi|^{2}+|\Phi_{\theta }^{j} \phi|^{2}.\notag
\end{align}
As in the regular case, by (\ref{regineq}) the left-hand side of (\ref{mixsing}) is finite,
so we see that $(D_{\theta }^{+})^{j}\phi \in L^{2}$ and $\Phi_{\theta }^{j} \phi \in L^{2} $.
The $\d r$-part $\Phi_{r}^{j} \phi $ of $\Phi^{j} \phi $ is in $L^{2} $ if and only if
$$
\int_{\Delta(p,\varepsilon )} \vert \Re \mu - \beta \vert^{2} \frac{\vert \phi (r)
\vert^{2}}{r^{2}} < \infty .
$$
Again by our main hypothesis $\Re \mu \notin \Z$ there exists a constant
$K >0$ such that
$$
\sum_{n=-\infty }^{\infty } \vert \Re \mu - \beta \vert^{2} \frac{ \vert \phi_{n} (r)
\vert^{2}}{r^{2}} \leq
K \sum_{n=-\infty }^{\infty } \vert n+ \Re \mu \vert^{2} \frac{\vert \phi_{n} (r) \vert^{2}}{r^{2}}.
$$
As we have already seen, this last expression is integrable, therefore
$\Phi^{j} \phi \in L^{2}$. Since the perturbation is negligible compared to the
behavior $O(r^{-1})$ of (\ref{est}), we then also have $\Phi \phi \in L^{2}$.
We conclude using $D^{+}\phi = D\phi - \Phi\phi$.
By duality, the case of a $2$-form $v \d z \land \d \bar{z}$ is settled the same way.
The general case (that of $S^{+} \otimes E$) then follows from Lemma \ref{imorth}.
The fact that
$$
\Dir : H^{1}(S^{+} \otimes E) \lra L^{2} (S^{-} \otimes E)
$$
is bounded, is then immediate (and has already been pointed out, see (\ref{Dirac})).
\end{proof}
\begin{proof} (Lemma \ref{closed}) This is immediate
from Theorem 1 and Claim \ref{imorth}. \end{proof}
We have established lemmata \ref{imker}, \ref{closed} and \ref{maxdom}, hence we
finished the proof of Theorem \ref{coker}.
\end{proof}
\begin{thm} \label{laplker} The first $L^{2}$-cohomology of the complex (\ref{ltwocompl})
is canonically isomorphic to the kernel of the adjoint Dirac operator
\begin{equation} \label{diradj}
\Dir_{\xi}^{\ast }: L^{2}(S^{-} \otimes E) \lra L^{2}(S^{+} \otimes E)
\end{equation}
on its domain, or alternatively to the kernel of the Laplace operator
\begin{equation} \label{lapldirac}
\Delta_{\xi } = \Dir_{\xi } \Dir_{\xi}^{\ast } = - D_{\xi } D_{\xi }^{\ast } - D_{\xi }^{\ast } D_{\xi } :
L^{2}(S^{-} \otimes E) \lra L^{2}(S^{-} \otimes E)
\end{equation}
on its domain.
\end{thm}
\begin{proof}
By duality, we get from Lemma \ref{imker} that
$$
(Im(D|_{H^{1}(\Omega^{0} \otimes E)}))^{\bot } = ker(D^{\ast }|_{\Dm(\Omega^{1} \otimes E)}),
$$
and this implies
\begin{align*}
coKer(\Dir|H^{1}(S^{+} \otimes E )) & = ker( D^{\ast }|_{\Dm(\Omega^{1} \otimes E)} ) \cap
ker( D|_{\Dm(\Omega^{1} \otimes E)} ) \\
& = ker( \Dir^{\ast }|_{\Dm(\Omega^{1} \otimes E)} ) .
\end{align*}
It remains to show that this latter is equal to $ker( \Dir \Dir^{\ast }|_{\Dm(\Omega^{1} \otimes E)} )$.
It is clear that
$$
ker( \Dir \Dir^{\ast }|_{\Dm(\Omega^{1} \otimes E)} ) \supseteq ker( \Dir^{\ast }|_{\Dm(\Omega^{1} \otimes E)}).
$$
Suppose now $u \in L^{2}(\Omega^{1} \otimes E)$ satisfies $\Dir \Dir^{\ast }u=0$. This means that
$$
\Dir^{\ast }u\in Ker(\Dir)\subset \Dm(\Dir)=H^{1}(S^{+} \otimes E)
$$
by Lemma \ref{maxdom}. Vanishing of the $L^{2}$-kernel of $\Dir$ on $H^{1}(S^{+} \otimes E)$
(c.f. Theorem \ref{Fredholm}) gives $\Dir^{\ast }u=0$, that is $u \in Ker(\Dir^{\ast })$, whence
$$
ker( \Dir \Dir^{\ast }|_{\Dm(\Omega^{1} \otimes E)} ) \subseteq ker( \Dir^{\ast }|_{\Dm(\Omega^{1} \otimes E)}).
$$
\end{proof}
Finally, let us introduce the norm
$$
\| f \|_{H^{2} (S^{+} \otimes E) } = \int_{\C } |f|^{2} + |(\nabla^{+})^{\ast } \nabla^{+} f|^{2} +
|(\Phi \otimes )^{\ast } \Phi \otimes f|^{2}
$$
and the corresponding function space
$$
H^{2} (S^{+} \otimes E) = \{ f : \hspace{5mm} \| f \|_{H^{2} (S^{+} \otimes E) } < \infty \}
$$
Then we have the following.
\begin{thm} \label{Hilbertisom}
The domain of the Laplace operator $\Delta_{\xi }=\Dir_{\xi}^{\ast } \Dir_{\xi }$ is $H^{2} (S^{+} \otimes E)$. It defines
a Hilbert-space isomorphism
$$
H^{2} (S^{+} \otimes E) \lra L^{2} (S^{+} \otimes E).
$$
\end{thm}
\begin{proof}
The fact that $\Delta_{\xi }$ is a well-defined bounded operator on $H^{2} (S^{+} \otimes E)$ follows from the
Weitzenb\"ock formula (\ref{weitzen}). Its maximal domain is the set of $u\in L^{2} (S^{+} \otimes E)$
such that $\Dir_{\xi }u\in \Dm(\Dir_{\xi}^{\ast })$. This latter is, by computations similar to Lemma \ref{maxdom},
the Sobolev space $H^{1}(S^{-} \otimes E)$ is with $1$ derivative in $L^{2}$, and weight $-1$ on the irregular
component near logarithmic singularities like in Corollary \ref{weightedcor}.
We deduce that the maximal domain of $\Delta_{\xi }$ is $H^{2}(S^{+} \otimes E)$, and that it splits as
$$
H^{2}(S^{+} \otimes E) \xrightarrow{\Dir_{\xi }} H^{1}(S^{-} \otimes E) \xrightarrow{\Dir_{\xi }^{\ast }} L^{2} (S^{+} \otimes E).
$$
Exactly as in Theorem \ref{Fredholm}, the first map is Fredholm with vanishing kernel from the
Sobolev space $H^{2}(S^{+} \otimes E)$ into $H^{1}(S^{-} \otimes E)$, both space being endowed with the $L^{2}$-inner product.
This with the identity $Im(\Dir_{\xi })^{\bot} = Ker(\Dir_{\xi }^{\ast })$ implies that $Ker(\Delta_{\xi })=\{ 0 \}$ and
that $Im(\Delta_{\xi })=Im(\Dir_{\xi }^{\ast })=Ker(\Dir_{\xi })^{\bot }=L^{2}(S^{+} \otimes E)$. Therefore,
$\Delta_{\xi }$ is a bounded bijective operator from $H^{2}(S^{+} \otimes E)$ to $L^{2} (S^{+} \otimes E)$.
By the closed graph theorem, we conclude that its inverse is also bounded.
\end{proof}
\section{Properties of the Green's operator} \label{greensect}
\begin{defn} \label{green}
Let us call the bounded linear inverse of $\Dir_{\xi}^{\ast } \Dir_{\xi }$ provided by Theorem \ref{Hilbertisom}
the \emph{Green's operator} of the Dirac-Laplace operator, and denote it by
$$
G_{\xi }: L^{2} (S^{+} \otimes E) \lra H^{2} (S^{+} \otimes E).
$$
\end{defn}
In this section we list the properties of this operator that we will need in later chapters.
\begin{lem} \label{diaglem}
$G_{\xi }$ is diagonal with respect to the decomposition $S^{+}\otimes E= (\Omega^{0} \otimes E) \oplus (\Omega^{2}\otimes E)$.
\end{lem}
\begin{proof}
Since $G_{\xi }$ is the inverse of $\Delta_{\xi }$, it is sufficient to prove the statement for this latter
operator. This comes from the identity
$$
\Delta_{\xi } = \Dir_{\xi }^{\ast } \Dir_{\xi } = (D_{\xi }^{\ast }-D_{\xi }) (D_{\xi }-D_{\xi }^{\ast })=
-D_{\xi }^{\ast } D_{\xi } - D_{\xi } D_{\xi }^{\ast },
$$
which is satisfied since $D_{\xi }$ is flat.
\end{proof}
\begin{lem} \label{estgreen}
There exist $K,K'>0$ such that for $|\xi |$ sufficiently large and for any positive spinor $\psi \in H^{1}(S^{+} \otimes E)$,
the following estimates hold:
\begin{align}
\left\Vert G_{\xi } \psi \right\Vert_{L^{2}(\C )} & \leq K |\xi |^{-2} \Vert \psi \Vert_{L^{2}(\C )} \label{estgreen1} \\
\left\Vert G_{\xi } \psi \right\Vert_{H^{1}(\C )} & \leq K' |\xi |^{-1} \Vert \psi \Vert_{L^{2}(\C )} \label{estgreen2}
\end{align}
\end{lem}
\begin{proof}
Since by definition, for any $\psi $ the positive spinor $G_{\xi } \psi $ is the solution $\varphi $ of
$$
\Delta_{\xi } \varphi = \psi ,
$$
the estimates (\ref{estgreen1}) and (\ref{estgreen2}) can be rewritten respectively as
\begin{align}
\left\Vert \varphi \right\Vert_{L^{2}(\C )} & \leq K |\xi |^{-2} \Vert \Delta_{\xi } \varphi \Vert_{L^{2}(\C )} \label{estlapl1} \\
\left\Vert \varphi \right\Vert_{H^{1}(\C )} & \leq K' |\xi |^{-1} \Vert \Delta_{\xi } \varphi \Vert_{L^{2}(\C )}. \label{estlapl2}
\end{align}
Call \emph{$\xi $-energy} of $\varphi $ over all $\C $ the quantity
\begin{equation} \label{energyglobaldef}
E( \xi ; \varphi ) = \int_{\C } |\nabla^{+}_{\xi } \varphi |^{2} + |\Phi_{\xi } \otimes \varphi |^{2} |\d z |^{2}.
\end{equation}
By partial integration, the Weitzenb\"ock formula (\ref{weitzen}) and Cauchy's inequality we have
\begin{align} \label{laplest}
E( \xi ; \varphi ) & = \int_{\C } \langle \varphi , \Delta_{\xi } \varphi \rangle |\d z |^{2} \\
& \leq \Vert \varphi \Vert_{L^{2}} \Vert \Delta_{\xi } \varphi \Vert_{L^{2}} . \notag
\end{align}
Now, as we will see from (\ref{confhiggs}), on the complementary of a finite union of disks
$\Delta(q_{k}(\xi), \varepsilon_{0}|\xi |^{-1})$ we have the point-wise lower bound
\begin{equation} \label{higgsptwise}
\absl \Phi_{\xi } \otimes \varphi \absr^{2} \geq c |\xi |^{2} | \varphi |^{2}
\end{equation}
for some $c>0$. Furthermore, we can choose $\varepsilon_{0}$ sufficiently small so that the balls
$\Delta(q(\xi ), 2\varepsilon_{0}|\xi |^{-1})$ are disjoint and do not meet $P$ for $|\xi|$ large. Setting
$$
B_{\xi }:= \bigcup_{q(\xi ) \in \Sigma_{\xi }} \Delta(q(\xi ), \varepsilon_{0}|\xi |^{-1})
$$
we then deduce the estimation
\begin{equation} \label{higgsint}
\int_{\C \setminus B_{\xi } } \absl \Phi_{\xi } \otimes \varphi \absr^{2} |\d z |^{2} \geq
c |\xi |^{2} \int_{\C \setminus B_{\xi } } \absl \varphi \absr^{2} |\d z |^{2} .
\end{equation}
Of course, extending this inequality over the disks $\Delta(q(\xi ), \varepsilon_{0}|\xi |^{-1}) $ is not possible,
since $\Phi_{\xi }$ has a zero in $q(\xi )$. However, the integral of $|\Phi_{\xi } \otimes \varphi |^{2} + |\nabla^{+}_{\xi } \varphi|^{2}$ does
control $|\xi |^{2}$ times that of $|\varphi |^{2}$ on the whole plane; that is, we have:
\begin{clm} \label{estgreenclm}
There exists $c >0$ such that for $| \xi |$ sufficiently large and for any spinor $\varphi $ we have
\begin{equation} \label{globalxi2est}
E(\xi ; \varphi ) \geq c |\xi |^{2} \int_{\C } \absl \varphi \absr^{2} |\d z |^{2}
\end{equation}
\end{clm}
\begin{proof}
By Kato's inequality $E(\xi ; \varphi )$ can be bounded from below by
$$
\int_{\C } \absl \Phi_{\xi } \otimes \varphi \absr^{2} + \absl \d | \varphi | \absr^{2} |\d z |^{2} .
$$
By (\ref{higgsint}), it only remains to show that for any $q(\xi ) \in \Sigma_{\xi }$ this integral bounds from above
$c |\xi |^{2} \int_{\Delta(q(\xi ), \varepsilon_{0}|\xi |^{-1})} |\varphi |^{2} |\d z |^{2} $, for some $c>0$
(not necessarily the same as before). But since on the annulus
$$
\Delta(q(\xi ), 2\varepsilon_{0}|\xi |^{-1}) \setminus \Delta(q(\xi ), \varepsilon_{0}|\xi |^{-1})
$$
we already have the estimation (\ref{higgsptwise}), this is just a consequence of (\ref{regineq}) applied at
the point $q(\xi )$ instead of $p_{j}$ to the function $g=| \varphi |$, with $\varepsilon = \varepsilon_{0}|\xi |^{-1}$
and $\delta =0$.
\end{proof}
By the claim and (\ref{laplest}), we have
$$
c |\xi |^{2} \Vert \varphi \Vert_{L^{2}(\C )}^{2} \leq \Vert \varphi \Vert_{L^{2}(\C )} \Vert \Delta_{\xi } \varphi \Vert_{L^{2}(\C )},
$$
and after dividing both sides by $\Vert \varphi \Vert_{L^{2}(\C )}$, we get (\ref{estlapl1}).
Plugging (\ref{estlapl1}) into (\ref{laplest}), we obtain
\begin{equation} \label{energylaplineq}
E(\xi ; \varphi ) \leq K | \xi |^{-2} \Vert \Delta_{\xi } \varphi \Vert_{L^{2}(\C )}^{2}.
\end{equation}
On the other hand, by the definitions
\begin{align*}
\nabla_{\xi }^{+} = & \nabla^{+} - \frac{\xi }{2} \d z + \frac{\bar{\xi }}{2} \d \bar{z} \\
\Phi_{\xi } = & \Phi - \frac{\xi }{2} \d z - \frac{\bar{\xi }}{2} \d \bar{z}
\end{align*}
we obtain the point-wise bounds
\begin{align*}
\frac{1}{2} \absl \Phi \otimes \varphi \absr^{2} - \frac{3}{2}|\xi |^{2} | \varphi |^{2} \leq & \absl \Phi_{\xi } \otimes \varphi \absr^{2} \leq
2\absl \Phi \otimes \varphi \absr^{2} + |\xi |^{2} | \varphi |^{2} \\
\frac{1}{2} \absl \nabla^{+} \varphi \absr^{2} - \frac{3}{2}|\xi |^{2} | \varphi |^{2} \leq & \absl \nabla_{\xi }^{+} \otimes \varphi \absr^{2} \leq
2\absl \nabla^{+} \varphi \absr^{2} + |\xi |^{2} | \varphi |^{2}
\end{align*}
and therefore
\begin{equation} \label{energyh1ineq}
\frac{1}{2}\Vert \varphi \Vert_{H^{1}(\C )}^{2} - (3|\xi |^{2} +1) \Vert \varphi \Vert_{L^{2}(\C )}^{2} \leq E(\xi ; \varphi ) \leq
2\Vert \varphi \Vert_{H^{1}(\C )}^{2} + (2|\xi |^{2} +1) \Vert \varphi \Vert_{L^{2}(\C )}^{2}.
\end{equation}
Putting together this with (\ref{energylaplineq}) and (\ref{estlapl1}), we get
\begin{align*}
\Vert \varphi \Vert_{H^{1}(\C )}^{2} \leq & 2E(\xi ; \varphi ) + (6|\xi |^{2} +2) \Vert \varphi \Vert_{L^{2}(\C )}^{2} \\
\leq & 2E(\xi ; \varphi ) + 7|\xi |^{2} \Vert \varphi \Vert_{L^{2}(\C )}^{2} \\
\leq & ( 2K + 7K^{2} ) | \xi |^{-2} \Vert \Delta_{\xi } \varphi \Vert_{L^{2}(\C )}^{2},
\end{align*}
whence (\ref{estlapl2}).
\end{proof}
We now investigate what happens to the Green's operator when $\xi$ is close to one of the points of $\hat{P}$.
\begin{lem}\label{estgreenlog}
There exist $K,K'>0$ such that for $|\xi -\xi_{l}|$ sufficiently small and for any positive spinor $\psi \in H^{1}(S^{+} \otimes E)$,
the following estimates hold:
\begin{align}
\left\Vert G_{\xi } \psi \right\Vert_{L^{2}(\C )} & \leq K |\xi -\xi_{l}|^{-2} \Vert \psi \Vert_{L^{2}(\C )} \label{estgreenlog1} \\
\left\Vert \Dir_{\xi } G_{\xi } \psi \right\Vert_{L^{2}(\C )} & \leq K'' |\xi -\xi_{l}|^{-1} \Vert \psi \Vert_{L^{2}(\C )} \label{estgreenlog2}
\end{align}
\end{lem}
\begin{proof}
Analogous to Lemma \ref{estgreen}. Notice that by partial integration and the Weitzenb\"ock formula
(\ref{weitzen}) one has
\[
\left\Vert \Dir_{\xi }\varphi \right\Vert_{L^{2}(\C )}^{2} = E(\xi;\varphi)
\]
for any positive spinor $\varphi$. Using this and setting $G_{\xi}\psi=\varphi$ the inequalities to prove can be rewritten as
\begin{align}
\left\Vert \varphi \right\Vert_{L^{2}(\C )} & \leq K |\xi -\xi_{l}|^{-2} \Vert \Delta_{\xi } \varphi \Vert_{L^{2}(\C )} \label{estlapllog1} \\
E(\xi;\varphi) & \leq K'' |\xi -\xi_{l}|^{-2} \Vert \Delta_{\xi } \varphi \Vert_{L^{2}(\C )}^{2}. \label{estlapllog2}
\end{align}
The behavior (\ref{confhiggslog}) of the Higgs field shows that outside of a finite union of disks
$\Delta(q_{k}(\xi), \varepsilon_{0}|\xi -\xi_{l}|^{-1})$ there exists $c>0$ for which we have the point-wise lower bound
\begin{equation} \label{higgsptwiselog}
\absl \Phi_{\xi } \otimes \varphi \absr^{2} \geq c |\xi -\xi_{l}|^{2} | \varphi |^{2} .
\end{equation}
It follows that denoting by $B_{\xi}$ the union of all the above mentioned disks where this estimate may fail,
we have the inequality
\begin{equation} \label{higgsintlog}
\int_{\C \setminus B_{\xi } } \absl \Phi_{\xi } \otimes \varphi \absr^{2} |\d z |^{2} \geq
c |\xi -\xi_{l}|^{2} \int_{\C \setminus B_{\xi } } \absl \varphi \absr^{2} |\d z |^{2}.
\end{equation}
It is not possible to extend this inequality to the whole plane; however, we have again
\begin{clm}
There exists $c >0$ such that for $|\xi -\xi_{l}|$ sufficiently small and for any spinor $\varphi$ we have
\begin{equation}
E(\xi ; \varphi ) \geq c |\xi -\xi_{l}|^{2} \int_{\C } \absl \varphi \absr^{2} |\d z |^{2}
\end{equation}
\end{clm}
\begin{proof}
Similar to Claim \ref{estgreenclm}, using Kato's inequality and (\ref{regineq}) rescaled conveniently by the
homothety $w=(\xi -\xi_{l})z$.
\end{proof}
This together with (\ref{laplest}) then shows
\[
c |\xi -\xi_{l}|^{2} \Vert \varphi \Vert_{L^{2}(\C )}^{2} \leq \Vert \varphi \Vert_{L^{2}(\C )} \Vert \Delta_{\xi } \varphi \Vert_{L^{2}(\C )},
\]
which gives us (\ref{estlapllog1}). Plugging this back into (\ref{laplest}), we obtain (\ref{estlapllog2}).
\end{proof}
\section[Exponential decay for harmonic spinors]{Exponential decay results for harmonic spinors}\label{harmspinsect}
In this section we give some analytic properties of $\Delta_{\xi }$-harmonic spinors. They will
be needed in Section \ref{flattr}, where we study the transformed flat connection. More precisely, they
will allow us to multiply any $L^{2}$ harmonic section by exponential factor so that the result remains
in $L^{2}$. They will also be of use in the computation of the parabolic weights of the transform in
Section \ref{parweightsect}.
First we set some further notation. Fix $\xi \in \hat{\C} \setminus \hat{P}$, and let $\varphi $ be a harmonic
negative spinor with respect to $\Dir_{\xi } \Dir_{\xi }^{\ast }$ and $p \in \C \setminus P$ any point of the plane.
Finally, for any spinor $\psi $ (not necessarily harmonic), call \emph{$\xi $-energy of $\psi $ in the disk
$\Delta(p,\varepsilon )$} the quantity
\begin{equation} \label{energydef}
E(p,\varepsilon, \xi ; \psi ) = \int_{\Delta(p,\varepsilon )} |\nabla^{+}_{\xi } \psi |^{2} + |\Phi_{\xi } \otimes \psi |^{2} .
\end{equation}
\begin{lem} \label{basicest}
Suppose that there exists $\varepsilon_{0} > 0$, $R>0$ and $c>0$ such that the disk
$\Delta(p,(R+1)\varepsilon_{0} )$ is disjoint from $P$, and all of the eigenvalues of $\theta_{\xi }$ in any
point of this disk are bounded below in absolute value by $c>0$. Under these assumptions,
we have the inequality
\begin{equation}
E(p,\varepsilon_{0}, \xi ; \varphi ) \leq e^{-2 c R \varepsilon_{0} } \left( 2\Vert \varphi \Vert_{H^{1}(\C)}^{2}
+ (2|\xi |^{2}+1) \Vert \varphi \Vert_{L^{2}(\C)}^{2} \right) .
\end{equation}
\end{lem}
\begin{proof}
Denote by $C(p,r )$ the boundary of $\Delta(p,r )$, and by $\frac{\partial }{\partial n}$ an outward-pointing
unit normal vector to it. Stokes' formula gives
\begin{align*}
E(p,r, \xi ; \varphi )= \int_{\Delta(p,r )} &
\left( (\nabla^{+}_{\xi })^{\ast } \nabla^{+}_{\xi }\varphi + (\Phi_{\xi } \otimes)^{\ast } \Phi_{\xi } \otimes \varphi ,\varphi \right) \\ +
& \int_{C(p,r )} \left( \left(\nabla^{+}_{\xi } \right)_{\frac{\partial }{\partial n}} \varphi ,\varphi \right) r \d \theta .
\end{align*}
Since $\varphi $ is $\Delta_{\xi }$-harmonic, the Weitzenb\"ock formula (\ref{weitzen}) implies that the first term on
the right-hand side vanishes. Therefore, by the tic-tac-toe inequality, we have
$$
E(p,r, \xi ; \varphi ) \leq \frac{1}{2} \int_{C(p,r )} \frac{1}{c} \absl \nabla^{+} \varphi \absr^{2}
+ c|\varphi |^{2} r \d \theta .
$$
On the other hand, we have
$$
\frac{\d E(p,r, \xi ; \varphi )}{\d r} = \int_{C(p,r )} \absl \nabla^{+}_{\xi } \varphi \absr^{2}
+ |\Phi_{\xi } \otimes \varphi|^{2} r \d \theta.
$$
By assumption, for $r \leq (R+1)\varepsilon_{0}$ we have the estimate
$$
\int_{C(p,r )} |\Phi_{\xi } \otimes \varphi|^{2} r \d \theta \geq c^{2} \int_{C(p,r )} | \varphi|^{2} r \d \theta .
$$
Putting together these estimates, we see that
$$
\frac{\d E(p,r, \xi ; \varphi )}{\d r} \geq 2c E(p,r, \xi ; \varphi ),
$$
whence
$$
\frac{\d \log E(p,r, \xi ; \varphi )}{\d r} \geq 2c.
$$
Integrating this inequality from $r=\varepsilon_{0}$ to $r= (R+1)\varepsilon_{0}$,
we obtain
\begin{align*}
\log E(p,\varepsilon_{0} , \xi ; \varphi ) & \leq 2c [\varepsilon_{0} - (R+1)\varepsilon_{0}]
+ \log E(p,(R+1)\varepsilon_{0} , \xi ; \varphi ).
\end{align*}
Taking exponential of both sides, we get
\begin{align*}
E(p,\varepsilon_{0} , \xi ; \varphi ) & \leq e^{-2c R \varepsilon_{0}}
E(p,(R+1)\varepsilon_{0} , \xi ; \varphi ) \\
& \leq e^{-2c R \varepsilon_{0}} E( \xi ; \varphi ),
\end{align*}
and we conclude using (\ref{energyh1ineq}).
\end{proof}
Next, we use the above lemma to obtain exponential decay results in terms of $\xi$ for the energy of
harmonic spinors when $\xi$ is large, first in a fixed disk of $\C$ away from the singularities $P$,
then near infinity in $\C$. In the first case, the statement is as follows.
\begin{lem} \label{expdecord}
Let $p \in {\C } \setminus P$ be arbitrary, and let $\varepsilon_{0}>0$ be such that the distance between
$p$ and $P$ is at least $3\varepsilon_{0}$. Then for $|\xi |$ sufficiently large we have the estimate
$$
\Vert \varphi \Vert^{2}_{H^{1}(\Delta(p, \varepsilon_{0}) )} \leq e^{- \varepsilon_{0} |\xi |/3 } \Vert \varphi \Vert_{H^{1}(\C)}^{2}
$$
for any $\Delta_{\xi }$-harmonic spinor $\varphi $.
\end{lem}
\begin{proof}
Since $p$ is away from $P$, in the Higgs field $\theta_{\xi } = \theta - \xi\d z/2$ the term $\theta$ is bounded on $\Delta(p,2\varepsilon_{0})$.
Therefore, if $|\xi |$ is sufficiently large, then the eigenvalues of $\theta_{\xi }$ on this disk are bounded below in absolute
value by $|\xi | /4$. Apply Lemma \ref{basicest} with $R=1$ and $c=|\xi | /4$ to get
\begin{align*}
E(p,\varepsilon_{0}, \xi ; \varphi ) & \leq e^{- \varepsilon_{0} |\xi |/2 } \left( 2\Vert \varphi \Vert_{H^{1}(\C)}^{2} +
(2|\xi |^{2} +1) \Vert \varphi \Vert_{L^{2}(\C)}^{2} \right) \\
& \leq 5 e^{- \varepsilon_{0} |\xi |/2 } |\xi |^{2} \Vert \varphi \Vert_{H^{1}(\C)}^{2} \\
& \leq \frac{1}{33} e^{- \varepsilon_{0} |\xi |/3 } \Vert \varphi \Vert_{H^{1}(\C)}^{2}
\end{align*}
for $\xi $ sufficiently large. On the other hand, we have
\begin{align} \label{h1energyineq}
\Vert \varphi \Vert^{2}_{H^{1}(\Delta(p, \varepsilon_{0}) )} = &
\int_{\Delta(p, \varepsilon_{0})} |\varphi |^{2} + \absl \nabla^{+} \varphi \absr^{2} + \absl \Phi \otimes \varphi \absr^{2} \notag \\
\leq & \int_{\Delta(p, \varepsilon_{0})} 2 |\xi |^{2} |\varphi |^{2} + \absl \nabla^{+}_{\xi } \varphi \absr^{2} + \absl \Phi_{\xi } \otimes \varphi \absr^{2}\\
\leq & 33 \; E(p,\varepsilon_{0}, \xi ; \varphi ) \notag,
\end{align}
where the last line is a consequence of $|\Phi_{\xi } \otimes \varphi |^{2} \geq |\xi |^{2}| \varphi |^{2} /16 $ in
$\Delta(p,\varepsilon_{0})$. Putting together these two estimates, we get the lemma.
\end{proof}
In the second case, we have the following statement.
\begin{lem} \label{expdecinf}
For any $\xi \notin \hat{P}$ there exists $R_{0}= R_{0}(\xi )>0$, $K= K(\xi )>0$ and $c=c(\xi )>0$ such that
for any $\Delta_{\xi }$-harmonic spinor $\varphi $ and all $R>R_{0}$ the following estimate holds:
\begin{equation*}
\Vert \varphi \Vert_{H^{1}(\C \setminus \Delta(0,2R ))}^{2} \leq K e^{-R c} \Vert \varphi \Vert_{H^{1}(\C )}^{2}.
\end{equation*}
Furthermore, if $|\xi |$ is sufficiently large, we can choose $c= |\xi |/3$ and $R_{0},K$ constants
independent of $\xi $.
\end{lem}
\begin{proof}
The proof is an amalgam of that of Lemmata \ref{basicest} and \ref{expdecord}.
Define the \emph{$\xi $-energy at infinity} of a spinor by the integral
\begin{equation} \label{energyinfdef}
E(\infty , R , \xi ; \varphi ) = \int_{ \C \setminus \Delta(0,R )} |\nabla^{+}_{\xi } \varphi |^{2} + |\Phi_{\xi } \otimes \varphi |^{2} .
\end{equation}
Choose $R_{0}>0$ and $c_{0}$ such that for $|z|>R_{0}$ the eigenvalues of $\theta_{\xi }(z)$ are all bigger in
absolute value then $c_{0}$. Clearly, such a choice is possible because $\xi \notin \hat{P}$. Moreover,
for $|\xi |$ sufficiently large one can put $c_{0} = |\xi | /4$ and $R_{0}$ a constant only depending on
the initial data $\theta $. For $r \geq R_{0}$, we have the estimate
$$
- E(\infty , r , \xi ; \varphi ) \geq -\frac{1}{2} \int_{C(0,r )} \frac{1}{c_{0} } \absl \nabla^{+}_{\xi } \varphi \absr^{2} +
c_{0} |\varphi |^{2} r \d \theta .
$$
On the other hand, we have
$$
\frac{\d E(\infty ,r, \xi ; \varphi )}{\d r} = - \int_{C(0,r )} \absl \nabla^{+}_{\xi } \varphi \absr^{2}
+ |\Phi_{\xi } \otimes \varphi|^{2} r \d \theta.
$$
By assumption, we have also
$$
\int_{C(0,r )} |\Phi_{\xi } \otimes \varphi|^{2} r \d \theta \geq c_{0}^{2} \int_{C(0,r )} | \varphi|^{2} r \d \theta .
$$
Putting together these estimates, we see that for $r \geq R_{0}$
$$
\frac{\d E(\infty ,r , \xi ; \varphi )}{\d r} \leq - 2c_{0} E(\infty ,r , \xi ; \varphi ),
$$
whence
$$
\frac{\d \log E (\infty ,r, \xi ; \varphi )}{\d r} \leq - 2c_{0}.
$$
Integrating this inequality from $R$ to $2R$ and using (\ref{energyh1ineq}), we obtain
\begin{align*}
E(\infty , 2R, \xi ; \varphi ) \leq & E( \xi ; \varphi ) e^{-R c_{0}} \\
\leq & (|\xi |^{2}+3) e^{- R c_{0} } \Vert \varphi \Vert_{H^{1}(\C)}^{2}.
\end{align*}
On the other hand,
\begin{align*}
E(\infty , 2R, \xi ; \varphi ) & \geq \int_{\C \setminus \Delta(0,2R)} \absl \Phi_{\xi } \otimes \varphi \absr^{2} \\
& \geq c_{0}^{2} \int_{\C \setminus \Delta(0,2R)} |\varphi |^{2}
\end{align*}
implies
$$
K_{0} E(\infty , 2R, \xi ; \varphi ) \geq \Vert \varphi \Vert_{H^{1}(\C \setminus \Delta(0,2R ))}^{2}
$$
for some $K_{0}>0$. This gives the lemma for $\xi $ in a finite region. The case of $|\xi |$ large also
follows noting that $K$ depends at most polynomially on $\xi $.
\end{proof}
Since a $\Delta_{\xi }$-harmonic spinor is subharmonic in the usual sense, the above results also imply point-wise
exponential decay on harmonic spinors:
\begin{lem} \label{ptwiseexpdec}
Suppose $R>R_{0}$. Then there exists $K,c>0$ such that for any $|z | > 2R+1$ and any $\Delta_{\xi }$-harmonic spinor
$\varphi $ we have
$$
|\varphi (z) | \leq K e^{-Rc} \Vert \varphi \Vert_{H^{1}(\C )}^{2}.
$$
\end{lem}
\begin{proof}
Because of the condition $|z | > 2R+1$, the disk $\Delta(z,1) $ centered at $z$ of radius $1$ is contained
in $\C \setminus \Delta(0,2R)$. On the other hand, by subharmonicity of $\varphi $ with respect to the usual
Laplace operator, we have
\begin{align*}
|\varphi (z) | & \leq K_{0} \int_{\Delta(z,1)} |\varphi (w)| |\d w |^{2} \\
& \leq K_{1} \left( \int_{\Delta(z,1)} |\varphi (w)|^{2} |\d w |^{2} \right)^{1/2} \\
& \leq K_{1} \left( \int_{\C \setminus \Delta(0,2R)} |\varphi (w)|^{2} |\d w |^{2} \right)^{1/2}
\end{align*}
We conclude using Lemma \ref{expdecinf}.
\end{proof}
\chapter{The transform of the integrable connection}
In this chapter, we define the transformed parabolic integrable connection induced by the deformation $D_{\xi }$.
First, in Section \ref{flattr}, we define the underlying flat bundle; then in Section \ref{flatext}
we show that its behavior at infinity verifies appropriate asymptotic conditions. This then allows us
to apply the results of \cite{Biq-Boa} in order to define an extension into a parabolic integrable connection over
the singularity at infinity; the same thing for other singularities follows from \cite{Sim}.
Before starting these points, we need however
to introduce some notation. Recall first that $\hat{P}$ was defined as the set $\{ \xi_{1} , \ldots , \xi_{n'} \}$
of eigenvalues of the second-order term of $D$ at infinity. Let $\hat{H} \ra \hat{\C} \setminus \hat{P}$
denote the trivial Hilbert bundle with fibers $L^{2}(\C , S^{-}\otimes E)$.
By Theorem \ref{laplker}, the transformed bundle $\hat{E}$
can be given as the vector bundle whose fiber over $\xi \in \C
\setminus \hat{P}$ is the kernel of the adjoint Dirac operator
$(\Dir_{\xi })^{\ast }$. By the same theorem, such an element is also $\Delta_{\xi }$-harmonic.
Now remark that on the bundle $\hat{H}$ there exists a
hermitian metric $\langle.,.\rangle $ which is canonical once a hermitian metric
$h(.,.)$ is fixed on $E$: for any two elements
$\hat{f}_{1},\hat{f}_{2} \in \hat{H}_{\xi }=L^{2}(\C , S^{-}\otimes E)$,
it is defined by the $L^{2}$ inner product
$$
\langle \hat{f}_{1},\hat{f}_{2}\rangle= \int_{\C } h(\hat{f}_{1},\hat{f}_{2}) |\d z|^{2}.
$$
Moreover, the trivial connection $\hat{\d}$ on the bundle $\hat{H}$ is
unitary with respect to this metric. Let $\hat{\pi}_{\xi }$ denote orthogonal
projection of $\hat{H}_{\xi }$ onto the subspace $\hat{E}_{\xi }$, and $ i $ the
inclusion $\hat{E} \inc \hat{H}$.
\begin{defn} \label{deftrhm}
We call \emph{transformed hermitian metric} the fiber metric $\hat{h}$ on $\hat{E}$ which
is equal on the fiber $\hat{E}_{\xi }$ to the restriction of the above defined $L^{2}$ scalar product $\langle.,.\rangle$
to the subspace $\hat{E}_{\xi } \subset L^{2}(\C , S^{-}\otimes E)$.
\end{defn}
\section[Construction]{Construction of the transformed flat connection}\label{flattr}
In this section we show that the transformed bundle admits an integrable
connection, which is determined only by the deformation $D_{\xi }$ and the .
First, we describe its intrinsic construction, then we give it in terms
of an explicit formula.
\subsection{Intrinsic definition}
Defining a flat connection is equivalent to giving a basis of parallel
sections on a disk $B_{0}$ around each point $\xi_{0} \in \hat{\C} \setminus
\hat{P} $. Given this, in order to see that it defines
indeed a flat connection, one only needs to prove that the transition
matrices on $B_{0} \cap B_{1}$ between two such bases (corresponding to
points $\xi_{0}$ and $\xi_{1}$) are constant.
So suppose
$\xi_{0} \in \hat{\C} \setminus \hat{P}$, and let $\hat{f}_{1}( z),\ldots,\hat{f}_{\hat{r}}( z)$
be a basis of the vector space $\hat{E}_{\xi_{0}}$. On the basis of Lemma \ref{ptwiseexpdec},
for $\varepsilon_{0} = \varepsilon_{0} (\xi _{0}) >0 $ sufficiently small, the expressions
\begin{align}
\hat{f}_{j}(\xi;z) & = \hat{\pi}_{\xi}(e^{(\xi - \xi_{0})z }\hat{f}_{j}(z)) \in \hat{E}_{\xi } \label{extf}
\end{align}
make sense for $\xi $ on the ball $B_{0}=B(\xi _{0}, \varepsilon_{0})$ of radius $\varepsilon_{0}$ centered
at $\xi_{0}$. Therefore, (restricting $\varepsilon_{0}$ if necessary), they define an extension of the basis
$\hat{f}_{1},\ldots,\hat{f}_{\hat{r}}$ of the vector space $\hat{E}_{\xi_{0}}$ to a trivialisation of the bundle
$\hat{E}$ over $B_{0}$.
\begin{prop} The family of sections (\ref{extf}) for all $\xi_{0} \in \hat{\C} \setminus \hat{P}$,
for $j \in \{ 1,\ldots,\hat{r} \}$, and for all $\xi \in B_{0}$ define a local system for a flat connection
$\hat{D}$ on $\hat{E} \ra \hat{\C} \setminus \hat{P}$.
\end{prop}
\begin{defn} \label{deftrflat}
We will call $\hat{D}$ the \emph{transformed flat connection} on $\hat{\C} \setminus \hat{P}$.
\end{defn}
\begin{proof} (Proposition)
Let $\tilde{\xi}_{0} \neq \xi_{0}$ be another point of $ \hat{\C} \setminus \hat{P}$, and
$\hat{g}_{1}(z),\ldots,\hat{g}_{\hat{r}}(z)$ be a basis for the vector space $\hat{E}_{\tilde{\xi}_{0}}$.
According to (\ref{extf}), the local trivialisation of $\hat{E}$ near $\tilde{\xi}_{0}$ we need to
consider is then $\hat{g}_{1}(\xi ), \ldots,\hat{g}_{\hat{r}}(\xi )$, with
\begin{align}
\hat{g}_{l}(\xi;z) & = \hat{\pi}_{\xi}(e^{(\xi - \tilde{\xi}_{0})z }\hat{g}_{l}(z)) \label{extg}
\end{align}
for $\xi $ in a small disk $\tilde{B}_{0}$ around $\tilde{\xi}_{0}$. In order to show that the local
bases (\ref{extf}) and (\ref{extg}) define indeed a local system,
we need to show that the transition matrices $m(\xi )$ between them are
independent of the point $\xi \in B_{0} \cap \tilde{B}_{0}$.
We will make use of the following:
\begin{lem} \label{trans} For any $\xi, \xi' \in B_{0}$, and any $k_{0} \in
ker(D_{\xi_{0} }|S^{-}\otimes E)$ we have
\[
\hat{\pi}_{\xi' } \left( e^{(\xi' - \xi )z } \hat{\pi}_{\xi}( e^{(\xi -\xi_{0})z} k_{0}(z) )
\right) = \hat{\pi}_{\xi' } ( e^{(\xi' -\xi_{0})z} k_{0}(z) ).
\]
\end{lem}
\begin{proof} (Lemma) Set $k_{\xi }(z)=e^{(\xi -\xi_{0})z} k_{0}(z)$;
we need to prove that
\[
\hat{\pi}_{\xi' }[ e^{(\xi' - \xi )z} \hat{\pi}_{\xi}( k_{\xi }(z) ) ] = \hat{\pi}_{\xi' }( e^{(\xi' - \xi )z} k_{\xi }(z) ),
\]
or equivalently that
\[
\hat{\pi}_{\xi' }[ e^{(\xi' - \xi )z} ( \Id - \hat{\pi}_{\xi} ) ( k_{\xi } ) ] = 0,
\]
which is still equivalent to
\begin{equation} \label{orthrel}
e^{(\xi' - \xi )z} ( \Id - \hat{\pi}_{\xi} ) ( k_{\xi } ) \bot \hat{E}_{\xi '}.
\end{equation}
Since $\hat{\pi}_{\xi}$ is orthogonal projection to $\hat{E}_{\xi }$, we have
\begin{equation} \label{eorth}
( \Id - \hat{\pi}_{\xi} ) ( k_{\xi } ) \in \hat{E}_{\xi }^{\bot }.
\end{equation}
Moreover, observe that for $\xi_{0}$ and $\xi$ fixed, the relation
\begin{equation} \label{imgauge}
e^{(\xi - \xi_{0})z}.D_{\xi_{0} } = D_{\xi_{0} } - (\xi - \xi_{0} )\d z \land
= D_{\xi },
\end{equation}
holds, and so
\begin{equation} \label{impluse}
k_{\xi } = e^{(\xi -\xi_{0})z} k_{0} \in e^{(\xi -\xi_{0})z} ker(D_{\xi_{0} }) \subset
ker(D_{\xi }) = Im(D_{\xi }^{\ast })^{\bot }= Im(D_{\xi }) \oplus \hat{E}_{\xi }.
\end{equation}
From (\ref{eorth}) and (\ref{impluse}) it follows that $( \Id - \hat{\pi}_{\xi} ) k_{\xi } \in Im(D_{\xi })$.
Now using (\ref{imgauge}) for $(\xi'- \xi ) $ instead of $(\xi-\xi_{0})$,
we deduce that $e^{(\xi' - \xi )z}( \Id - \hat{\pi}_{\xi} ) k_{\xi } \in Im( D_{\xi' } )$,
whence (\ref{orthrel}). This finishes the proof of the lemma.
\end{proof}
Let us now come back to the study of the transition matrix: let $\xi , \xi' \in B_{0} \cap \tilde{B}_{0} $,
and suppose we have
\begin{equation} \label{tm}
\hat{f}_{j}(\xi ) = \sum_{l=1}^{\hat{r}} m_{jl} \hat{g}_{l}(\xi ),
\end{equation}
where $(m_{jl})$ is the transition matrix between the two bases at the point $\xi $.
Lemma \ref{trans} means that for $|\xi - \xi '|$ sufficiently small, we have
\begin{align}
\hat{f}_{j}(\xi' ) & = \hat{\pi}_{\xi '}( e^{(\xi' - \xi )z} \hat{f}_{j}(\xi ) ) \label{bpch} \\
\hat{g}_{l}(\xi' ) & = \hat{\pi}_{\xi '}( e^{(\xi' - \xi )z} \hat{g}_{l}(\xi ) ). \label{bpchg}
\end{align}
Now plugging (\ref{tm}) into (\ref{bpch}), then using (\ref{bpchg}) we obtain
\begin{align*}
\hat{f}_{j}(\xi' ) & = \hat{\pi}_{\xi '} \left( e^{(\xi' - \xi )z} \sum_{l=1}^{\hat{r}} m_{jl}
\hat{g}_{l}(\xi ) \right) \\
& = \sum_{l=1}^{\hat{r}} m_{jl} \hat{\pi}_{\xi '}( e^{(\xi' - \xi )z}\hat{g}_{l}(\xi )) \\
& = \sum_{l=1}^{\hat{r}} m_{jl} \hat{g}_{l}(\xi' ),
\end{align*}
so the transition matrix at the point $\xi '$ is the same as the one at $\xi $,
whence we obtain the Proposition.
\end{proof}
\subsection{Explicit description}
We now give an explicit formula for the flat connection constructed above.
In the sequel we follow \cite{Jardim}. First define a unitary connection on $\hat{E}$ with respect to
the transformed hermitian metric by
\begin{equation} \label{trunit}
\hat{\nabla } = \hat{\pi}_{\xi} \circ \hat{\d} \circ i .
\end{equation}
The fact that this connection is indeed $\hat{h}$-unitary can be seen as follows:
let $f,g \in \Gamma(\hat{E})$ be local sections around $\xi_{0}$, then from
orthogonality of $\hat{\pi}_{\xi}$ to $\hat{E}$ with respect to the norm $\langle .,. \rangle $
we have in $\xi_{0}$
\begin{align*}
\hat{\d} (\hat{h}(\hat{f},\hat{g})) & = \hat{\d} \langle \hat{f},\hat{g} \rangle = \langle \hat{\d } \hat{f},\hat{g} \rangle +
\langle \hat{f},\hat{\d }\hat{g} \rangle \\ & = \langle \hat{\nabla } \hat{f},\hat{g} \rangle + \langle \hat{f},\hat{\nabla } \hat{g} \rangle
= \hat{h}(\hat{\nabla } \hat{f},\hat{g}) + \hat{h}( \hat{f},\hat{\nabla } \hat{g}),
\end{align*}
where $\hat{\d}$ stands for exterior differentiation of functions along the coordinate $\xi$ as
well as for the trivial connection with respect to $\xi$ on the trivial Hilbert bundle $\hat{H}$.
Finally, we define an endomorphism-valued $(1,0)$-form
(a candidate to be a transformed Higgs field) by mapping a
$\Delta_{\xi }$-harmonic section $\hat{f}(\xi;z)$ to
\begin{equation} \label{trhiggs}
\hat{\theta }_{\xi }(\hat{f}(\xi;z))= -\frac{1}{2}\hat{\pi}_{\xi}(z\hat{f}(\xi;z)) \d \xi,
\end{equation}
where $\d \xi $ stands for the standard generator of the holomorphic
$(1,0)$-forms on $\hat{\C}$. This field will indeed be holomorphic provided
that the original metric $h$ is harmonic (see Section \ref{sectharm}).
\begin{prop} \label{explflattr}
The connection $\hat{\nabla }+ 2\hat{\theta }$ is equal to the transformed flat connection $\hat{D}$ defined above.
\end{prop}
\begin{proof}
We need to show that for all $\xi_{0}$ and all $f(z) \in \hat{E}_{\xi_{0}}$, the
local $\hat{D}$-parallel section in $\xi \in B_{0}$ given by
\begin{equation} \label{parallel}
\hat{f}(\xi;z)= \hat{\pi}_{\xi}(e^{(\xi - \xi_{0})z }\hat{f}(z))
\end{equation}
is parallel in $B_{0}$ with respect to $\hat{\nabla }+ 2\hat{\theta }$. First, let us check it in $\xi_{0}$:
\begin{align*}
((\hat{\nabla }+ 2\hat{\theta })\hat{f})(\xi_{0}) =
\hat{\pi}_{\xi_{0}}[(\hat{\d}\hat{f})(\xi_{0} )- z\hat{f}(\xi_{0} )\d \xi ] .
\end{align*}
We observe that by (\ref{parallel}) we have
\begin{align*}
(\hat{\d} \hat{f})(\xi_{0} )= (\hat{\d} \hat{\pi}_{\xi})_{\xi_{0}} \hat{f}(\xi_{0} ) +
\hat{\pi}_{\xi_{0} }( z \hat{f}(\xi_{0}) \d \xi ),
\end{align*}
hence
\[
((\hat{\nabla }+ 2\hat{\theta })\hat{f})(\xi_{0}) = \hat{\pi}_{\xi_{0}} [ (\hat{\d} \hat{\pi}_{\xi})_{\xi_{0}} \hat{f}(\xi_{0} )] .
\]
Now $\hat{\pi}_{\xi} \circ \hat{\pi}_{\xi} = \hat{\pi}_{\xi}$ implies
\[
\hat{\d} \hat{\pi}_{\xi} \circ \hat{\pi}_{\xi} + \hat{\pi}_{\xi} \circ \hat{\d} \hat{\pi}_{\xi} = \hat{\d} \hat{\pi}_{\xi},
\]
therefore
\[
\hat{\pi}_{\xi_{0}} [ (\hat{\d} \hat{\pi}_{\xi})_{\xi_{0}} \hat{f}(\xi_{0} )] = (\hat{\d} \hat{\pi}_{\xi})_{\xi_{0}}
\circ ( \Id - \hat{\pi}_{\xi_{0}} ) \hat{f}(\xi_{0} ) =0,
\]
since $\hat{\pi}_{\xi_{0}}$ is the projection to $ \hat{E}_{\xi_{0}} $ and
$\hat{f}(\xi_{0}) \in \hat{E}_{\xi_{0}}$.
Next, fix an arbitrary $\xi \in B_{0}$. Then, as we have just shown,
the local section defined for $|\xi ' - \xi |$ sufficiently small by
$$
\hat{f}'(\xi ') = \hat{\pi}_{\xi '} (e^{(\xi' - \xi )z}\hat{f}(\xi;z ))
$$
is parallel in $\xi $ (compare with (\ref{parallel}), setting $\xi_{0}= \xi , \xi = \xi '$).
But Lemma \ref{trans} tells us that the local sections $\hat{f}'$ and $\hat{f}$ coincide in a
neighborhood of $\xi$; in particular $\hat{f}$ is parallel in $\xi $.
\end{proof}
The following is now immediate:
\begin{prop} \label{unittr} The unitary part of the transformed flat connection $\hat{D}$ is
$$
\hat{D}^{+}= \hat{\nabla } + \hat{\theta} - \hat{\theta}^{\ast } =
\hat{\pi }_{\xi } \circ (\hat{\d } - \frac{1}{2}z \d \xi \land + \frac{1}{2} \bar{z} \d \bar{\xi} \land ).
$$
\end{prop}
\begin{defn} We will call the above unitary connection $\hat{D}^{+}$ the \emph{transformed
unitary connection}. The covariant derivative associated to it will be denoted $\hat{\nabla }^{+} $.
\end{defn}
\begin{rk} The fact that the formula for the transformed unitary connection involves
extra multiplication terms by $z$ and $\bar{z}$ compared to the usual formulae
of other Nahm transforms is an artifact: as we will see in the next chapter, the transform admits
an interpretation from the point of view of Higgs bundles, in which the formula for
the transformed unitary connection agrees with the usual one.
\end{rk}
\section[Extension over the singularities]{Extension over the singularities}\label{flatext}
At this point, it should be pointed out that a priori we have no guarantee that the constructed flat
connection is indeed of the form given in \cite{Biq-Boa} (and therefore extends nicely over the singularities);
that is, in an orthonormal basis with respect to its harmonic metric it is not necessarily the model
(\ref{polard}) up to a perturbation described in (\ref{pertj}) and (\ref{pertinf}). However, there is a theorem of O. Biquard
and M. Jardim which allows us to show that this is the case. Namely, Theorem 0.1 of \cite{Biq-Jar} states the following:
\begin{thm} \label{biqjar}
Let $\tilde{A}$ be an $SU(2)$-instanton on $\R^{4}$, invariant with respect to
the additive subgroup $\Z \frac{\partial}{\partial x_{3}} \oplus \Z \frac{\partial}{\partial x_{4}}$, and suppose that its curvature
$F_{\tilde{A}}$ has quadratic decay at infinity (that is, $\absl F_{\tilde{A}} \absr = O(r^{-2})$, where
$r^{2}=x_{1}^{2}+x_{2}^{2}$). Then there exists a gauge near infinity in which $\tilde{A}$ is asymptotic
to the following model:
\begin{align*}
\tilde{A}_{0} = \d + i \Big{(} \lambda_{1} \d x_{3} + & \lambda_{2} \d x_{4} + (\mu_{1} \cos \theta -\mu_{2} \sin \theta ) \frac{\d x_{3}}{r} \\
& + (\mu_{1} \sin \theta + \mu_{2} \cos \theta ) \frac{\d x_{4}}{r} + \alpha \d \theta
\Big{)},
\end{align*}
where $z=r e^{i\theta} $ are coordinates for the $(x_{1},x_{2})$-plane. Moreover, the difference $a$ between
$\tilde{A}$ and this model satisfies
$$
|a|=O(r^{-1-\delta }), \; \absl \nabla_{\tilde{A}_{0}} a \absr = O(r^{-2-\delta }).
$$
\end{thm}
In order to be able to apply this result to our case, consider the Euclidean space $(\R^{4})^{\ast }$ spanned by
orthonormal vectors $\frac{\partial}{\partial x^{\ast }_{j}}$ for $j=1,2,3,4$, and identify the subspace spanned by
$\frac{\partial}{\partial x^{\ast }_{1}}$ and $\frac{\partial}{\partial x^{\ast }_{2}}$ with the line $\hat{\C}$ with complex
coordinate $\xi$ underlying $\hat{\D}$. By the results of \cite{Hit}, $\hat{D}$ then induces
an instanton $\tilde{A}$ on $(\R^{4})^{\ast }$ with singularities, invariant with respect to the subspace
$\R \frac{\partial}{\partial x^{\ast }_{3}} \oplus \R \frac{\partial}{\partial x^{\ast }_{4}}$.
In particular, $\tilde{A}$ is invariant with respect to $\Z \frac{\partial}{\partial x^{\ast }_{3}} \oplus \Z \frac{\partial}{\partial x^{\ast }_{4}}$,
so Theorem \ref{biqjar} can be applied to it, provided that its curvature has quadratic decay.
In order to have an explicit description of $\tilde{A}$ and its curvature, remember that $\hat{D}$ decomposes as
$$
\hat{D} = \hat{\nabla}^{+} + \hat{\theta} + \hat{\theta}^{\ast},
$$
where $\hat{\nabla}^{+}$ is the transformed unitary connection, $\hat{\theta}$ the field defined in (\ref{trhiggs}) and
$\hat{\theta}^{\ast}$ its adjoint with respect to the harmonic metric of $\hat{D}$. Now as we will see in Section
\ref{sectharm}, this harmonic metric is in fact the transformed hermitian metric $\hat{h}$ given in Definition
\ref{deftrhm}. The unitary part of $\hat{D}$ decomposes further into its $(1,0)$- and $(0,1)$-part:
$$
\hat{\nabla}^{+} = (\hat{\nabla}^{+})^{1,0} + (\hat{\nabla}^{+})^{0,1}.
$$
Finally, we write $\hat{\vartheta}$ for the endomorphism-part of $\hat{\theta}$:
$$
\hat{\theta} = \hat{\vartheta} \d \xi .
$$
The instanton over $(\R^{4})^{\ast}$ corresponding to $\hat{D}$ is then given by the formula
$$
\tilde{A} = \hat{\nabla}^{+} + \Re \hat{\vartheta} \d x^{\ast}_{3} + \Im \hat{\vartheta} \d x^{\ast}_{4},
$$
where we recall that
$$
\frac{\partial}{\partial \xi} = \frac{1}{2} \left( \frac{\partial}{\partial x^{\ast}_{1}} - \frac{\partial}{\partial x^{\ast}_{2}} \right)
$$
is the natural complex coordinate of $\hat{\C}$, and the connection $\hat{\nabla}^{+}$ on $(\R^{4})^{\ast}$ acts as
$\hat{\nabla}^{+}$ along $\hat{\C}$ and as the trivial connection along $\R \frac{\partial}{\partial x^{\ast }_{3}} \oplus \R \frac{\partial}{\partial x^{\ast }_{4}}$.
Furthermore, as it can be seen from the results in Section 1 of \cite{Hit}, we then have the formula
\begin{align}
F_{\tilde{A}} = - & [ \hat{\vartheta} , \hat{\vartheta}^{\ast }] (\d x^{\ast}_{1} \land \d x^{\ast}_{2} + \d x^{\ast}_{3} \land \d x^{\ast}_{4}) \notag \\
& + (\hat{\nabla}^{+})_{x^{\ast}_{1}} \Re \hat{\vartheta} (\d x^{\ast}_{1} \land \d x^{\ast}_{3} - \d x^{\ast}_{2} \land \d x^{\ast}_{4}) \\
& + (\hat{\nabla}^{+})_{x^{\ast}_{1}} \Im \hat{\vartheta} (\d x^{\ast}_{1} \land \d x^{\ast}_{4} + \d x^{\ast}_{2} \land \d x^{\ast}_{3}) \notag,
\end{align}
where we have written $(\hat{\nabla}^{+})_{x^{\ast}}$ to denote the action of the unitary connection in the
$\frac{\partial}{\partial x^{\ast}}$-direction. Hence, before we can apply Theorem \ref{biqjar} we need to check the following:
\begin{thm} \label{comdec} There exists a constant $K>0$ such that the commutator $ [\hat{\vartheta},\hat{\vartheta}^{\ast}]$
is bounded by $K |\xi |^{-2}$ as $\xi \ra \infty$. The same estimation holds for $\hat{\nabla }^{+} \hat{\vartheta}$.
\end{thm}
\begin{proof}
We start with the case of the commutator.
Let $\hat{f}(\xi;z) \in \hat{E}_{\xi } = Ker (\Dir_{\xi })^{\ast } $ be arbitrary; we wish to show the estimate
$$
\absl [\hat{\vartheta }, \hat{\vartheta }^{\ast } ] \hat{f}(\xi) \absr_{\hat{h}} \leq K |\xi |^{-2} | \hat{f}(\xi)|_{\hat{h}},
$$
with $K$ independent of $\hat{f}$ and of $\xi $. Recall the well-known formula from Hodge theory:
\begin{equation} \label{projform}
\hat{\pi }_{\xi } = \Id - \Dir_{\xi } G_{\xi } \Dir_{\xi }^{\ast }.
\end{equation}
Using this, we obtain
\begin{align}
[\hat{\vartheta }, \hat{\vartheta }^{\ast } ] \hat{f}(\xi) & =
-\frac{1}{2}\hat{\pi }_{\xi }(z \hat{\pi }_{\xi }(\bar{z} \hat{f}(\xi)) -
\bar{z} \hat{\pi }_{\xi }(z \hat{f}(\xi))) \notag \\
& = \frac{1}{2} \hat{\pi }_{\xi } ( z \Dir_{\xi } G_{\xi } \Dir_{\xi }^{\ast } (\bar{z} \hat{f}(\xi))
- \bar{z} \Dir_{\xi } G_{\xi } \Dir_{\xi }^{\ast } (z \hat{f}(\xi)) ). \label{higgscomm}
\end{align}
Since $D_{\xi }$ is a connection, the following commutation relations hold:
\begin{align*}
& [D_{\xi }, z] = \d z \land & & [D_{\xi }, \bar{z} ] = \d \bar{z} \land \\
& [D_{\xi }^{\ast }, z] = \frac{\partial }{\partial \bar{z}} \contr & &
[D_{\xi }^{\ast }, \bar{z}] =\frac{\partial }{\partial {z}},
\end{align*}
where $\contr$ stands for contraction of a differential form by a vector field.
It follows immediately
\begin{align}
[\Dir_{\xi }, z] & = -[\Dir_{\xi }^{\ast }, z] = \d z \land - \frac{\partial }{\partial \bar{z}} \contr = \d z \cdot \label{commrelz} \\
[\Dir_{\xi }, \bar{z}] & = -[\Dir_{\xi }^{\ast }, \bar{z}] = \d \bar{z} \land - \frac{\partial }{\partial {z}} \contr = \d
\bar{z} \cdot \label{commrelzbar}
\end{align}
where the Clifford multiplication $\cdot $ is defined by these formulae.
Plugging these in the expression (\ref{higgscomm}), using $\Dir_{\xi }^{\ast }\hat{f}(\xi;z)=0$ and
$\hat{\pi }_{\xi } |_{ Im \Dir_{\xi }^{\ast }} =0$ together with the definition of $\hat{h}$, we get
\begin{align} \label{commineq}
\absl [\hat{\vartheta }, \hat{\vartheta }^{\ast } ] \hat{f}(\xi) \absr_{\hat{h}} & = \frac{1}{2}
\left\Vert \hat{\pi }_{\xi } \left( \d z \cdot G_{\xi } \d \bar{z}\cdot \hat{f}(\xi) -
\d \bar{z} \cdot G_{\xi } \d z \cdot \hat{f}(\xi) \right) \right\Vert_{L^{2}(\C )} \notag \\
& \leq \frac{1}{2} \left\Vert G_{\xi } \d \bar{z} \cdot \hat{f}(\xi) \right\Vert_{L^{2}(\C )} +
\frac{1}{2} \left\Vert G_{\xi } \d z \cdot \hat{f}(\xi) \right\Vert_{L^{2}(\C )},
\end{align}
since the norm of the orthogonal projection of a vector to a subspace is at most the norm of the vector and
the action of Clifford multiplication by $\d z$ and $\d \bar{z}$ is point-wise bounded.
We conclude by the first statement of Lemma \ref{estgreen}.
Next, let us come to $\hat{\nabla }^{+} \hat{\vartheta}$. Similarly to the above, using (\ref{projform}) and the
commutation formulae (\ref{commrelz})-(\ref{commrelzbar}) we obtain
\begin{align*}
\left( \hat{\nabla }^{+} \hat{\vartheta} \right) \hat{f}(\xi) = &
\left( \hat{D}^{+} \circ \hat{\vartheta } - \hat{\vartheta } \circ \hat{D}^{+} \right) \hat{f}(\xi) \\
= & \hat{\pi }_{\xi } \left( \hat{\d} - \frac{z}{2} \d \xi + \frac{\bar{z}}{2} \d \bar{\xi } \right)
\hat{\pi }_{\xi } \left( - \frac{z}{2} \right) \hat{f}(\xi) \\
& - \hat{\pi }_{\xi } \left( - \frac{z}{2} \right)
\hat{\pi }_{\xi } \left( \hat{\d} - \frac{z}{2} \d \xi + \frac{\bar{z}}{2} \d \bar{\xi } \right) \hat{f}(\xi)\\
= & \hat{\pi }_{\xi } \Big{[} \left( \hat{\d} - \frac{z}{2} \d \xi + \frac{\bar{z}}{2} \d \bar{\xi } \right)
\Dir_{\xi } G_{\xi } \Dir_{\xi }^{\ast } \left( \frac{z}{2} \hat{f}(\xi) \right) \\
& - \frac{z}{2} \Dir_{\xi } G_{\xi } \Dir_{\xi }^{\ast }
\left( \hat{\d} - \frac{z}{2} \d \xi + \frac{\bar{z}}{2} \d \bar{\xi } \right) \hat{f}(\xi) \Big{]} \\
= & \hat{\pi }_{\xi } \Big{[} \left( \frac{1}{2} \d \xi \land \d z - \frac{1}{2} \d \bar{\xi } \land \d \bar{z } \right) \cdot
G_{\xi } \frac{\d z}{2} \cdot \hat{f}(\xi) \\
& \hspace{.8cm} - \frac{\d z}{2} \cdot G_{\xi } \left( \frac{1}{2} \d \xi \land \d z -
\frac{1}{2} \d \bar{\xi } \land \d \bar{z } \right) \cdot \hat{f}(\xi)
\Big{]} \\
& +
\hat{\pi }_{\xi } \left[ \hat{\d} \Dir_{\xi } G_{\xi } \frac{\d z}{2} \cdot \hat{f}(\xi) - \frac{\d z}{2} \cdot G_{\xi }
\Dir_{\xi }^{\ast } \hat{\d} \hat{f}(\xi) \right]
\end{align*}
(here $\d z$ and $\d \bar{z}$ act on the spinors by Clifford multiplication, whereas $\d \xi $ and
$\d\bar{\xi }$ by wedge product). Noticing that $| \d \xi |= | \d \bar{\xi }| =2$, the first term in the last
expression can be treated exactly as in (\ref{commineq}). For the second term, one only needs to remark
that the commutation relations
\begin{align*}
\left[ \hat{\d} , D_{\xi } \right] = & \left[ \hat{\d} , D - \frac{\xi }{2} \d z +
\frac{\bar{\xi }}{2} \d \bar{z} \right] \\
= & - \frac{\d \xi \land \d z \land }{2} + \frac{\d \bar{\xi } \land \d \bar{z} \land}{2}
\end{align*}
and
\begin{align*}
\left[ \hat{\d} , D_{\xi }^{\ast } \right] = & - \frac{\d \xi \land }{2} \frac{\partial }{\partial \bar{z}} \contr
+ \frac{\d \bar{\xi } \land }{2} \frac{\partial }{\partial {z}} \contr
\end{align*}
show that
\begin{align*}
\left[ \hat{\d} , \Dir_{\xi } \right] = - \left[ \hat{\d} , \Dir_{\xi }^{\ast } \right] =
- \frac{1}{2} \d \xi \land \d z \cdot + \frac{1}{2} \d \bar{\xi } \land \d \bar{z } \cdot
\end{align*}
holds. Therefore we can proceed again as in (\ref{commineq}).
\end{proof}
On the basis of Theorem \ref{biqjar}, the behavior of the transformed flat connection at infinity satisfies
the hypothesis considered in \cite{Biq-Boa}. Namely, in a suitable gauge its difference from a model
with second-order pole is in the weighted Sobolev-space $L^{1,2}_{-2+\delta}(\Omega^{1}\otimes E)$ considered in Section 2
of that article. Indeed, passing to a coordinate $w=z^{-1}$, $|w|=\rho$ in which the double pole is in $0$, the norm of
the perturbation is $O(\rho^{1+\delta})$, whereas that of its derivative is also $O(\rho^{1+\delta})$ (because the norm of $1$-forms
near infinity is $|\d z|=|\d w|/|w|=1$), and we conclude since $\rho^{1+\delta}/ \rho^{2} \in L^{2}_{\delta -2}$.
It follows from the results of its Sections 7 and 8 that the analytic flat connection
$\hat{D}$ defined outside infinity extends to an algebraic integrable connection with a parabolic structure
on the singular fiber at infinity. On the other hand, such an extension over logarithmic singularities
(that is, singularities in which the eigenvalues of $\hat{D}$ or equivalently those of $\hat{\vartheta}$ have at most
first-order poles) is ensured by Theorem 2 of \cite{Sim}. Therefore, by Theorem \ref{eigenlog} the flat
connection $\hat{D}$ on $\hat{\C} \setminus \hat{P}$ can be extended into a meromorphic integrable connection
on $\CPt$ with parabolic structures at the singularities.
\begin{defn} \label{transfext}
The \emph{transformed meromorphic integrable connection} is the meromorphic integrable connection with
parabolic structure in the singularities induced by the above extension procedures, subject to local changes
of holomorphic trivialisations near the singularities to take all weights between $0$ and $1$.
We will continue to denote it by $(\hat{E},\hat{D})$. The underlying extension will be
called \emph{transformed extension} of the transformed bundle.
\end{defn}
\begin{rk}
We will see in Section \ref{parweightsect} that the parabolic structures are adapted to the harmonic metric;
namely, the weight $0\leq \hat{\alpha}_{k} <1$ of a subspace $F_{k}\hat{E}|_{p}$ of a singular fiber corresponds in
local coordinate $z$ vanishing at the puncture to a decay bounded above by $|z|^{2\hat{\alpha}_{k}}$ of the norm
of a parallel section extending an element of $F_{k}\hat{E}|_{p}$, as measured by the harmonic metric.
However, in Sections \ref{logext} and \ref{infext} we will construct a different
extension over the punctures -- more suited to analytical study --, where the behavior of the norm of
parallel sections near the singular points will no longer be bounded. We then pass back to the transformed
extension in Corollary \ref{pardegcor}, where we remark that it is the one that establishes a "good" correspondence.
\end{rk}
\chapter[Higgs bundle interpretation]{Interpretation from the point of view of Higgs bundles} \label{algint}
Let $(E,D,h)$ be a Hermitian bundle with integrable connection.
Throughout this chapter, we suppose that the original metric $h$ is harmonic. This metric then defines a Higgs
bundle $(\E, \theta )$ starting from the integrable connection, via the procedure described in Section \ref{higgsintro}.
We first prove that the transformed metric $\hat{h}$ is then harmonic for $\hat{D}$.
Next, we give an interpretation of the transformed Higgs bundle of $(\E, \theta )$ in terms of the hypercohomology
of a sheaf map over $\CP$. These results
will then be used to define the \emph{induced extension} $^{i}\hat{\E}$ of the
transformed bundle over the punctures $\hat{P} \cup \{ \infty \}$, and to compute the topology and
the singularity parameters of this extension of the transformed Higgs bundle.
This will enable us to eventually compute the topology and
the singularity parameters of the transformed Higgs bundle with respect to its transformed extension
given in Definition \ref{transfext}.
\section[Integrable connection and Higgs bundle]{The link with the transformed integrable connection }
Recall that we have defined the deformation of the Higgs bundle by the
formula (\ref{higgsdef}), and we write $D''_{\xi }$ for the $D''$-operator of
this deformation. Explicitly, we have
\[
D''_{\xi } = D''+ \theta_{\xi },
\]
where $\theta_{\xi }= \theta - \xi/2 \d z$. Moreover, as we have noticed in Section \ref{transf}, nonabelian Hodge theory
identifies the deformation of the Higgs bundle structure (\ref{higgsdef}) and that of the integrable
connection via the unitary gauge transformation
\begin{equation*}
g(z,\xi ) =e^{ [\bar{\xi} \bar{z}- \xi z]/2}.
\end{equation*}
In other words, writing $g_{\xi }=g(.,\xi )$ for the gauge transformation restricted to
the fiber $\hat{H}_{\xi }$, we have
\begin{equation} \label{gaugetr}
g_{\xi }.D_{\xi }=D_{\xi }^{H} = D - \frac{\xi}{2} \d z \land - \frac{\bar{\xi}}{2} \d \bar{z} \land.
\end{equation}
Since the gauge transformation $g_{\xi }$ is unitary, in addition to (\ref{gaugetr}) we have as well
\begin{equation} \label{gaugetradj}
g_{\xi }.D_{\xi }^{\ast }=(D_{\xi }^{H})^{\ast }.
\end{equation}
\begin{defn} The operator $\Dir_{\xi }^{H}= D_{\xi }^{H} - (D_{\xi }^{H})^{\ast }$ will be
referred to as the \emph{Higgs Dirac operator}. In the same way, we let $\Dir_{\xi }''$ stand
for the Dirac operator $D_{\xi }'' - (D_{\xi }'')^{\ast }$.
The \emph{transformed smooth bundle underlying the Higgs bundle}
is the bundle $\hat{V}$ over $\hat{\C} \setminus \hat{P}$ whose fiber over $\xi$
is the first $L^{2}$-cohomology space $L^{2}H^{1}(\EC_{\xi }^{H})$ of the operator $D_{\xi }^{H}$.
\end{defn}
\begin{prop}\label{trvbprop}
This way we define a smooth vector bundle $\hat{V}$. Furthermore, there exists
a canonical bundle isomorphism between the smooth bundle $\hat{E}$ underlying the
transformed integrable connection and the smooth bundle $\hat{V}$ underlying the transformed
Higgs bundle.
\end{prop}
\begin{proof}
Theorem \ref{coker} tells us that the transformed bundle underlying the integrable connection
is the bundle of first $L^{2}$-cohomologies of $D_{\xi}^{int}$.
For any $\xi$, the gauge transformation $g_{\xi }$ of $E$ induces a
natural isomorphism between the $L^{2}$-cohomology spaces of the complexes
(\ref{ellcompl}) and
\begin{equation}\label{unittrans}
\Omega^{0}\otimes E \xrightarrow{\text{$g_{\xi }.D^{int}_{\xi}$}} \Omega^1 \otimes E
\xrightarrow{\text{$g_{\xi }.D^{int}_{\xi}$}} \Omega^2 \otimes E.
\end{equation}
which is just $\EC_{\xi }^{H}$. In Theorem \ref{Fredholm} we have shown that the $0$-th and $2$-nd
cohomology of $\EC_{\xi }$ vanishes for all $\xi\in \hat{\C} \setminus \hat{P}$, whereas Corollary \ref{cokerbundle}
implies that the cohomology spaces $L^{2}H^{1}(\EC_{\xi })$ define a smooth vector bundle over
$\hat{\C} \setminus \hat{P}$.
This then implies the same thing for $\EC_{\xi }^{H}$, whence the bundle isomorphism
between the bundles over $\hat{\C} \setminus \hat{P}$ in question.
\end{proof}
Theorem \ref{laplker} has the following interpretation:
\begin{thm} \label{laplkerhiggs} The first $L^{2}$-cohomology $\hat{V}_{\xi}=L^{2}H^{1}(\EC_{\xi}^{H})$ of the operator
$D_{\xi }^{H}$ is canonically isomorphic to the kernel of the adjoint Dirac operator
\begin{equation} \label{diradjhiggs}
(\Dir_{\xi}^{H})^{\ast }: L^{2}(S^{-} \otimes E) \lra L^{2}(S^{+} \otimes E)
\end{equation}
on its domain, or alternatively to the kernel of the Laplace operator
\begin{equation} \label{lapldirachiggs}
\Delta_{\xi }^{H} = \Dir_{\xi }^{H} (\Dir_{\xi}^{H})^{\ast } :
L^{2}(S^{-} \otimes E) \lra L^{2}(S^{-} \otimes E)
\end{equation}
on its domain.
\end{thm}
\begin{proof}
Apply the gauge transformation $g$ to Theorem \ref{laplker} and notice that
(\ref{gaugetr}) and (\ref{gaugetradj}) imply
\begin{equation} \label{gaugetrdir}
g_{\xi }.\Dir_{\xi }^{\ast } =(\Dir_{\xi }^{H} )^{\ast }
\end{equation}
and
\begin{equation} \label{gaugetrlapl}
g_{\xi }.\Delta_{\xi }=\Delta_{\xi }^{H};
\end{equation}
and in particular that
\begin{equation} \label{gaugetrcoker}
g_{\xi } (Ker (\Dir_{\xi }^{\ast })) = Ker ((\Dir_{\xi }^{H})^{\ast })
\end{equation}
and
\begin{equation} \label{gaugetrker}
g_{\xi } (Ker (\Delta_{\xi })) = Ker (\Delta_{\xi }^{H}).
\end{equation}
\end{proof}
This result enables us to put similar definitions as in the integrable deformation case.
\begin{defn}
The hermitian bundle metric on $\hat{V}$ given by $L^{2}$ scalar product of the $(\Dir^{H}_{\xi})^{\ast}$-harmonic
representative will be called the \emph{transformed hermitian metric}, and will be denoted by $\hat{h}$.
Also, $\hat{\pi }_{\xi }^{H}$ will stand for $\hat{h}$-orthogonal projection of $L^{2}(S^{-} \otimes E)$ onto $\hat{V}$.
\end{defn}
\begin{rk}
Starting from a Higgs bundle with any Hermitian metric (not necessary harmonic), we can define in the same
way its transform on the transformed bundle $\hat{V}$.
\end{rk}
Next, we recollect the above considerations in terms of the transformed bundles.
\begin{prop} \label{gaugeprop} The family of gauge transformations $g$ induce a Hermitian bundle isomorphism between $\hat{E}$
and $\hat{V}$.
Furthermore, the fiber $\hat{V}_{\xi }$ can be identified with
the first $L^{2}$-cohomology of the single complex associated to
the following double complex, denoted by $\mathcal{D}_{\xi }$:
\[
\xymatrix{
\Omega^{0,1}\otimes E \ar[r]^{\theta_{\xi } \land} & \Omega^{2}\otimes E\\
\Omega^{0}\otimes E \ar[r]^{\theta_{\xi } \land} \ar[u]^{\dbar^{\E}} & \Omega^{1,0}\otimes E. \ar[u]_{\dbar^{\E}}
}
\]
\end{prop}
\begin{rk} Notice that commutativity of this diagram follows from the hypothesis
$\dbar^{\E}\theta =0$, which is just the definition of the harmonicity of $h$.
\end{rk}
\begin{proof}
By (\ref{gaugetrker}),the $D_{\xi}$-harmonic representative of a class is mapped by $g$ into a
$D_{\xi}^{H}$-harmonic class. Since the transformed metric from both points of view is
induced by $L^{2}$-norm of the harmonic representatives, and $g$ is unitary, this gives
the first statement.
For the second, remark that by Theorem \ref{simpson}, the Laplace operator
$\Delta_{\xi }^{H} $ is equal (up to a factor of 2)
to the Laplace operator $\Delta_{\xi }''= \Dir''_{\xi } (\Dir''_{\xi})^{\ast } $, therefore their
kernels coincide. This then identifies $\hat{V}$ with the first $L^{2}$-cohomology of the
complex
\begin{equation}\label{higgsellcompl}
\Omega^{0}\otimes E \xrightarrow{\text{$D_{\xi}''$}} \Omega^1 \otimes E
\xrightarrow{\text{$D_{\xi}''$}} \Omega^2 \otimes E.
\end{equation}
Finally, recall that the formula
\[
D''_{\xi } = \dbar^{\E} + \theta_{\xi }
\]
gives the decomposition of $D''_{\xi } $ into its $(0,1)$- and $(1,0)$-part respectively.
This means that the complex (\ref{higgsellcompl}) is the single complex associated to the double
complex $\mathcal{D}_{\xi }$. However, it is not necessarily true that the domain of $D''_{\xi }$ is the sum
of the domain of $\dbar^{\E}$ and that of $\theta_{\xi}$, it could in principle be larger.
Still, the two $L^{2}$-cohomologies are the same.
Indeed, suppose $f=f^{1,0} \d z + f^{0,1} \d \bar{z}\in L^{2}(\Omega^{1}\otimes E)$ is in the kernel of $D''_{\xi}$, that is
\begin{equation}\label{dbarplustheta}
\dbar^{\E}f^{1,0}\d z + \theta_{\xi }\land f^{0,1} \d \bar{z}=0.
\end{equation}
We wish to represent the $D''_{\xi}$-cohomology class of $f$ by a class
$\tilde{f}^{1,0} \d z +\tilde{f}^{0,1}\d \bar{z}$ such that $\dbar \tilde{f}^{1,0}\in L^{2}$ and
$\theta_{\xi}\tilde{f}^{0,1}\in L^{2}$. Away from logarithmic singularities,
one can simply choose $f$ itself, for there locally ${f}^{0,1}\in L^{2}$
implies $\theta_{\xi}{f}^{0,1}\in L^{2}$ and by (\ref{dbarplustheta}) then $\dbar^{\E}f^{1,0}\in L^{2}$ as well.
Thus we only need to modify $f$ in a neighborhood of the logarithmic punctures.
By Claim \ref{crsol} near any such puncture we can find $g\in L^{2}(E)$ such that $\theta_{\xi}g\in L^{2}(\Omega^{1,0}\otimes E)$ and
$$
f^{0,1} \d \bar{z} + \dbar^{\E}g=0.
$$
Using $\dbar^{\E}\theta_{\xi}=0$, the last two identities then also imply
$$
\dbar^{\E} (f^{1,0}\d z + \theta_{\xi }g)=0.
$$
Put $\tilde{f}^{1,0}\d z=f^{1,0}\d z + \theta_{\xi }g$;
as both $f^{1,0}$ and $\theta_{\xi }g$ are supposed to be in $L^{2}$, so is $\tilde{f}^{1,0}\d z$.
This then shows that $f$ is cohomologuos in the $L^{2}$ complex of
(\ref{higgsellcompl}) to a class locally represented by a section $\tilde{f}^{1,0} \d z$,
where $\tilde{f}^{1,0}\in L^{2}$ and $\dbar^{\E}\tilde{f}^{1,0}\in L^{2}$. In different terms
$\tilde{f}^{1,0}\d z\in\Dm (\dbar^{\E})$, and this shows that the first $L^{2}$-cohomology of
(\ref{higgsellcompl}) is indeed equal to that of $\mathcal{D}_{\xi }$.
\end{proof}
Next, let us investigate what the transformed integrable connection $\hat{D} $ and
its unitary part $\hat{D}^{+}$ become under this gauge transformation.
Notice that since the gauge transformation $g$ is
unitary, the orthogonal projection $\hat{\pi}$ onto $\hat{E}$ is transformed into the
orthogonal projection $\hat{\pi}^{H}$ onto $\hat{V}$, with respect to
the same $L^{2}$-metric on the fibers; in different terms $g_{\xi }.\hat{\pi}_{\xi}=\hat{\pi}_{\xi}^{H}$.
The image of the transformed integrable connection $\hat{D}$ under the
gauge transformation $g$ in the point $\xi $ is given by
\begin{align} \label{htrint}
\hat{D}^{H}& = g.\hat{D}\notag \\
& = g.(\hat{\pi}_{\xi}\circ (\hat{\d}-z \d \xi \land ))\\
& = \hat{\pi}_{\xi}^{H} \left( \hat{\d} - \frac{1}{2}(z\d \xi \land + \bar{z} \d\bar{\xi } \land ) \right),\notag
\end{align}
(see (\ref{trunit}), (\ref{trhiggs}) and Proposition \ref{explflattr}),
and that of the candidate Higgs field is the endomorphism
\begin{align} \label{htrhiggs}
\hat{\theta }^{H} & = g.\hat{\theta }\notag \\
&= g.(\hat{\pi}_{\xi}\circ (-z/2 \d \xi \land ) ) \\
&= - \frac{1}{2} \hat{\pi}_{\xi}^{H} (z \d \xi \land). \notag
\end{align}
Therefore, if we decompose the transformed flat connection in the point of view of Higgs bundles into its
unitary and self-adjoint part, we obtain
\begin{align} \label{hdecomp}
(\hat{D}^{H})^{+}= \hat{\pi}_{\xi}^{H} ( \hat{\d}) &&
(\hat{D}^{H})^{sa}= \hat{\theta }^{H}+(\hat{\theta }^{H})^{\ast}
\end{align}
(these formulae can also be deduced directly from Proposition \ref{unittr}). This then gives the
desired interpretation of the transformed unitary connection $\hat{D}^{+}$ in this point of view.
\begin{defn} \label{defholtrafo}
We let $\dbar^{\hat{\E}}$ stand for the $(0,1)$-part of $(\hat{D}^{H})^{+}$.
Moreover, we call the holomorphic bundle $\hat{V}$ with partial connection $\dbar^{\hat{\E}}$
the \emph{transformed holomorphic bundle} and we denote it by $\hat{\E}$.
\end{defn}
\section{Harmonicity of the transformed metric} \label{sectharm}
In this section we prove the following result:
\begin{thm}
If the original metric $h$ is harmonic, then the same thing is true for $\hat{h}$.
\end{thm}
\begin{proof}
First remark that by (\ref{hdecomp}), the formula for $\dbar^{\hat{\E}}$ is $\hat{\pi}_{\xi}^{H}(\hat{\d}^{0,1})$.
Also, the $(1,0)$-part of $(\hat{D}^{H})^{sa}$ is just $\hat{\theta }^{H}$.
By definition, harmonicity of the transformed metric $\hat{h}$ resumes then in the
equation
\begin{equation} \label{hharm}
\dbar^{\hat{\E}} \hat{\theta }^{H}=0.
\end{equation}
By Proposition \ref{gaugeprop} we have $\hat{V}_{\xi } = L^{2}H^{1}(D_{\xi }'')$,
with $D''_{\xi } = D''- \xi/2 \d z $. From this formula it is clear
that $D_{\xi }''$ depends holomorphically on $\xi $, so we are in the situation
described in part 3.1.3 of \cite{Don-Kronh} of chain complexes
$$
\Omega^{0}\otimes E \xrightarrow{\text{$D_{\xi}''$}} \Omega^1 \otimes E
\xrightarrow{\text{$D_{\xi}''$}} \Omega^2 \otimes E
$$
varying holomorphically with $\xi $. There it is shown that if the
first cohomology spaces $\hat{V}_{\xi }$ of these complexes are all finite
dimensional, of the same dimension, then the bundle $\hat{V}$ constructed out of them
over the parameter space of $\xi$ carries a natural holomorphic structure.
Explicitly, this is given by by saying that a section $f \in \Gamma(\hat{V})$ in a
neighborhood of $\xi_{0}$ is holomorphic if and only if it admits a lift
$\tilde{f} \in \Gamma(Ker(D_{\xi}''|_{\Omega^{1}}))$ which is itself holomorphic with respect to
the holomorphic structure induced by the $(0,1)$-part $\hat{\d }^{0,1}$ of
the trivial connection $\hat{\d }$ on the Hilbert bundle $\hat{H}$.
This holomorphic structure is the same as the
one defined by the operator $ \dbar^{\hat{\E}}$, since both are induced
by $\hat{\d}^{0,1}$ and $ \hat{\pi}^{H } $. The section $\hat{\theta }^{H} \in End( \hat{V}) \otimes \Omega_{\hat{\C}}^{1,0}$
is then holomorphic for this holomorphic structure if
and only if it maps each holomorphic section $f$ into a holomorphic
section. In particular, this is the case if it admits a lift
\[ \xymatrix{ Ker(D_{\xi}''|_{\Omega^{1}}) \ar[r]^(0.4){\Theta} & Ker(D_{\xi}''|_{\Omega^{1}}) \otimes
\Omega_{\hat{\C}}^{1,0} \\
\hat{V}_{\xi } \ar[u] \ar[r]^(0.4){\hat{\theta }^{H}} & \hat{V}_{\xi } \otimes
\Omega_{\hat{\C}}^{1,0}, \ar[u]
}
\]
such that
\begin{enumerate}
\item{$\Theta$ passes to the quotient $Ker(D_{\xi}''|_{\Omega^{1}}) \ra
Ker(D_{\xi}''|_{\Omega^{1}})/Im(D_{\xi}''|_{\Omega^{0}})=\hat{V}_{\xi }$, the quotient
being $\hat{\theta }^{H}$, and } \label{hol1}
\item{$\Theta$ is holomorphic with respect to the holomorphic structure induced
by $\hat{\d}^{0,1} $.} \label{hol2}
\end{enumerate}
Recall from Section \ref{l2coh} that $Ker(D_{\xi}''|_{\Omega^{1}})$ is a closed Hilbert subspace of
$\hat{H}_{\xi }$; call $\pi_{Ker(D_{\xi}'')}$ orthogonal projection of $\hat{H}_{\xi }$ to it. We now claim that the map
\begin{align*}
\Theta : Ker(D_{\xi}''|_{\Omega^{1}}) & \lra Ker(D_{\xi}''|_{\Omega^{1}}) \otimes
\Omega_{\hat{\C}}^{1,0} \\
\tilde{f}_{\xi } & \mapsto -\frac{1}{2} \pi_{Ker(D_{\xi}'')}(z\tilde{f}_{\xi }(z)) \d \xi
\end{align*}
verifies the hypotheses needed.
For (\ref{hol1}), we need to show $\Theta ( Im ( D_{\xi}''|_{\Omega^{0}} ) ) \subseteq Im (D_{\xi}''|_{\Omega^{0}} ) $.
Let $g_{\xi }$ be a local section of the trivial Hilbert bundle $ L^{2}(E) \ra \hat{\C} $. Then we have
\begin{align*}
\Theta (D_{\xi}''g) &= -\frac{1}{2}\pi_{Ker(D_{\xi}'')}(z D_{\xi}''g_{\xi }) \d \xi \\
& = -\frac{1}{2} \pi_{Ker(D_{\xi}'')}( D_{\xi}''(zg_{\xi }(z))) \d \xi \\
& = -\frac{1}{2} D_{\xi}''(zg_{\xi }(z)) \d \xi,
\end{align*}
because the operator $D_{\xi}''=\dbar^{\E}+\theta_{\xi }$ commutes with multiplication by
$z$, and $Im( D_{\xi}''|_{\Omega^{0}} ) \subseteq Ker ( D_{\xi}''|_{\Omega^{1}} )$. This shows
that $Im( D_{\xi}''|_{\Omega^{0}} )$ is invariant by $\Theta $; the quotient is
clearly $\hat{\theta }^{H}$.
Next come to (\ref{hol2}): we remark that the formula defining $\Theta $ only
depends on $\xi$ via the projection $\pi_{Ker(D_{\xi}'')}$. But since the
operator $D_{\xi}''$ depends holomorphically in $\xi $, so do the subspaces
$Ker(D_{\xi}'')$, and since the metric is independent of $\xi $, the same thing
is true for the projections $\pi_{Ker(D_{\xi}'')}$. This shows that $\Theta $, and so
$\hat{\theta }^{H}$ is holomorphic in $\xi $.
\end{proof}
\section{Identification with hypercohomology}\label{ident}
In this section we will often use basic properties of hypercohomology; for an introduction to this topic,
we refer to \cite{Griff-Harr} and \cite{Dem}.
Before we start, we need to define the functional spaces
\begin{align*}
\tilde{L}_{\xi }^{2}(E) & = \Dm(D_{\xi }''|_{ \Omega^{0}\otimes E }) \\
& = \{ u \in L^{2}( E ) \ : \ \theta_{\xi } \land u, \dbar^{\E} u \in L^{2} \}\\
\tilde{L}_{\xi }^{2}(\Omega^{0,1} \otimes E) & = \Dm(D_{\xi }''|_{ \Omega^{0,1}\otimes E }) \\
& = \{ v \d \bar{z} \in L^{2}( \Omega^{0,1}\otimes E ) \ : \ \theta_{\xi } \land v \d \bar{z} \in L^{2} \} \\
\tilde{L}^{2}(\Omega^{1,0} \otimes E) & = \Dm(D_{\xi }''|_{ \Omega^{1,0}\otimes E }) \\
& = \{ u \d z \in L^{2}( \Omega^{1,0}\otimes E ) \ : \ \dbar^{\E} (u \d z) \in L^{2} \},
\end{align*}
for the Euclidean metric $|\d z|^{2}$ on $\C$ and the hermitian metric $h$ on the fibers,
adapted to the parabolic structure with weights $\{\alpha_{1}^{j},\ldots,\alpha_{r}^{j} \}$.
Notice that we may drop the index $\xi $ of these spaces, since they all coincide: indeed, in a
logarithmic singularity the deformation $ \xi \d z $ is bounded, and at infinity the
condition $\xi \notin \hat{P} $ implies that no eigenvalues of $\theta_{\xi }$ vanish,
and this gives equivalence of the corresponding norms exactly as in Lemma
\ref{lemmaone}. We identify these functional spaces to the sheaves of their local
sections.
In what follows, we are going to define sheaves $\E$ and $\F$ of sections of
$\Omega^{0} \otimes E $ and $\Omega^{1,0} \otimes E $ respectively on $\CP$ with the property that
the $L^{2}$-cohomology $L^{2}H^{\bullet }(D_{\xi }'')$ of (\ref{higgsellcompl})
identifies to the hypercohomology $\H^{\bullet }(\E \xrightarrow{\theta_{\xi } \land } \F)$ of
the sheaf map $\E \xrightarrow{\theta_{\xi } \land } \F$. This latter is then
explicitly given in terms of a sky-scraper sheaf over the zero set $\Sigma_{\xi }$
of $det(\theta_{\xi })$ by a simple use of the spectral sequence of the double
complex.
\subsection{Definition and resolution of the sheaves} \label{sheaves}
Recall that the parabolic structure on $\E$ with adapted Hermitian fiber metric means that the holomorphic
bundle $\E$ on $\C \setminus P$ has a natural extension to all $\CP$:
the holomorphic sections at a singular point are the holomorphic sections
outside the singularity which are bounded with respect to the metric.
By an abuse of language, for $U \subset \CP $ an open set let $\E|_{U}$ be the set of
holomorphic sections of the bundle $\E$ in U. In other words, we denote
by $\E$ the sheaf of local holomorphic sections of $\E$ (extended over the punctures as above).
Next, let us define $\F$: for an open set $U \subset \CP$ containing no singular point,
let $\F|_{U}$ be the set of $\dbar^{\E}$-holomorphic sections of
$\Omega^{1,0} \otimes E$. If $U$ contains exactly one singular point $p_{j} \in P $ (and does
not contain the infinity), then let
$\F|_{U}$ be the set of $\dbar^{\E}$-meromorphic sections $\sigma \d z$ of
$\Omega^{1,0} \otimes E$ such that $\sigma$ be $\dbar^{\E}$-meromorphic in $U$ with only one simple
pole at $p_{j}$, and such that its residue in this point be contained in the subspace
$Im(Res(\theta,p_{j}))$. Finally, if $U$ contains the infinity (but no other singular
points), then let $\F|_{U}$ be
the set of all $\dbar^{\E}$-meromorphic sections $\sigma \d z$ of $\Omega^{1,0} \otimes E$ with a
double pole at infinity, and no other poles in $U$. Notice that since in
the coordinate $w=1/z$ of $\CP$ the section $\d z$ has a double pole at infinity,
this amounts to say that $\sigma $ is a $\dbar^{\E}$-holomorphic section of $E$ in
$U$. Writing $\sigma = \sum_{k} f_{k}^{\infty } \sigma_{k}^{\infty }$ in the holomorphic basis
(\ref{holtrivinf}) at infinity, it is still the same thing to say that $f_{k}^{\infty }$ be a
holomorphic function in $U$ for all $k$ (in particular bounded at infinity).
It is easy to check that this way we defined a sheaf.
We introduce some further notation: set $\tilde{r} = \sqrt{1+|z|^{2}} $ on $\C$;
then for $a \in \{ 0,1 \}$ we denote by
$\tilde{r} \tilde{L}^{2}(\Omega^{a,0}\otimes E)$ the space of sections $u$ of $\Omega^{a,0}\otimes E$ such
that $\tilde{r}^{-1}u \in \tilde{L}^{2}$. This way we only loosen the condition on
the behavior of $u$ at infinity with respect to $\tilde{L}^{2}$, namely that $r^{-1}u$ be in
$\tilde{L}^{2}$ in a neighborhood of infinity. It is immediate that there exist an inclusion
of vector spaces
\begin{equation}\label{inclvsp}
\tilde{L}^{2}(\Omega^{a,0}\otimes E) \hookrightarrow \tilde{r} \tilde{L}^{2}(\Omega^{a,0}\otimes E).
\end{equation}
\begin{lem} \label{lemrese}
The sequence
\begin{eqnarray} \label{resole}
\E \inc \tilde{r} \tilde{L}^{2}(E) \xrightarrow{\dbar^{\E}} \tilde{L}^{2}(\Omega^{0,1} \otimes E)
\end{eqnarray}
is a resolution of $\E$.
\end{lem}
\begin{proof}
It is known that away from the singularities, the sequence of usual $L^{2}$-sections with respect to
Euclidean metric gives a resolution of the sheaf of holomorphic sections.
Therefore, we only need to show that (\ref{resole}) is a resolution at the
singularities.
Consider first $p_{j} \in P$. We first prove that (\ref{resole}) is
locally exact in $\tilde{r} \tilde{L}^{2}(E)$.
Let $E$ be trivialized in $ \Delta(p_{j},\varepsilon ) $ by the local
sections $\{\sigma_{k}^{j} \}$ given in (\ref{holtriv}). As we have seen in
(\ref{modelhiggs}), in this trivialisation up to a perturbation term
$\theta = diag (\lambda_{k}^{j}) {\d z }/{z}$, with
$\lambda^{j}_{k}=(\mu^{j}_{k}-\beta^{j}_{k})/2$, and the parabolic weights are given by
$\alpha^{j}_{k}=\Re (\mu^{j}_{k}) -[\Re (\mu^{j}_{k})]$. By definition,
any holomorphic section $\sigma $ of $E^{j}$ can be given as a sum $\sum_{k}\phi^{j}_{k}
\sigma_{k}^{j}$, where $\phi^{j}_{k}$ are holomorphic functions defined in
$\Delta(p_{j},\varepsilon )$, in particular bounded by a constant $K$.
This implies that $\sigma \in L^{2}(E)$, so that
$\sigma \in \tilde{L}^{2}(E)$ if and only if $\theta \land \sigma \in L^{2}$.
Recall that $L^{2}$ is defined with respect
to the parabolic structure $\{ \alpha_{k}^{j} \}$, and that the perturbation
term in $\theta$ behaves as $O(r^{-1+\delta })$ with $\delta >0$, where $r=|z-p_{j}|$. This implies that
\begin{align*}
\int_{\Delta(p_{j},\varepsilon )} |\theta \sigma |^{2} & \leq K' \int \sum_{k=1}^{r_{j}}
|r^{-1+\delta}\sigma_{k}^{j}|^{2 }
+ K' \int \sum_{k=r_{j}+1}^{r} |r^{-1}\sigma_{k}^{j}|^{2} \\
& \leq K'' \int \sum_{k=1}^{r_{j}} |r^{-1+\delta}|^{2 } + K'' \int \sum_{k=r_{j}+1}^{r}
|r^{-1+\alpha_{k}^{j}}|^{2}.
\end{align*}
By Hypothesis \ref{main}, $\alpha_{k}^{j} > 0$ for all $j \in \{ r_{j}+1,\ldots,r \}$.
It then follows that this last expression is finite, which proves that
any holomorphic section of $E$ is in $\tilde{L}^{2}$. On the other hand,
if a section $\sigma = \sum_{k}\phi_{k}^{j}\sigma_{k}^{j}$ of $E$ is meromorphic in
$p_{j}$, then there is at least one $k \in \{ 1,\ldots,r\}$ such that $\phi_{k}^{j}$
has a pole in $p_{j}$. Suppose $k \in \{ 1,\ldots,r_{j}\}$: then
$|\phi_{k}^{j}\sigma_{k}^{j}|\sim 1/r$, and $\sigma $ is clearly not in $L^{2}$. Suppose
now $k \in \{ r_{j}+1,\ldots,r \}$: then again by Hypothesis \ref{main} we have
$\lambda_{k}^{j} \neq 0$, and therefore $|\theta \land \phi_{k}^{j}\sigma_{k}^{j}| \sim r^{-2+\delta }$, and
so $\theta \land \sigma \notin L^{2}$. Hence, the sections of $\tilde{L}^{2}(\Delta(p_{j},\varepsilon ),E)$ in the kernel
of $\dbar^{\E}$ are exactly the local holomorphic sections of $E$, in other
words the local sections of $\E$. This shows local exactness in $\tilde{L}^{2}(E)$.
The next thing we show is that in $\Delta(p_{j},\varepsilon )$ the complex
(\ref{resole}) is exact at $\tilde{L}^{2}(\Omega^{0,1} \otimes E)$:
let $v \d \bar{z} \in \tilde{L}^{2}(\Delta(p_{j},\varepsilon ), \Omega^{0,1} \otimes E)$ be
an arbitrary section; for $\varepsilon >0$ sufficiently small we wish to find
$u \in \tilde{L}^{2}(\Delta(p_{j},\varepsilon ),E)$ such that
\begin{equation} \label{dbareq}
\dbar^{\E}u=v \d\bar{z}
\end{equation}
We can suppose without restricting generality that $v= f \sigma_{k}^{j}$, with
$f $ a function defined in $\Delta(p_{j},\varepsilon )$. Since $\sigma_{k}^{j}$ is a
holomorphic section of $E$, solving (\ref{dbareq}) boils down to solving
the usual Cauchy-Riemann equation on the disk
\begin{equation} \label{dbareq2}
\frac{\partial g}{\partial \bar{z} } = f
\end{equation}
with $u= g \sigma_{k}^{j} \in \tilde{L}^{2}(\Delta(p_{j},\varepsilon ), E)$. Exactness
near a singularity at a finite point is given by the following claim:
\begin{clm} \label{crsol}
For $f \in L^{2}$ the equation (\ref{dbareq2}) has a
solution $g$ such that $g r^{-1+\delta } \in L^{2}$ for any $\delta >0$.
For $f$ such that $fr^{\alpha } \in L^{2}$ with $0<\alpha <1$, (\ref{dbareq2}) has a solution $g$
such that $g r^{-1+\alpha } \in L^{2}$.
\end{clm}
\begin{proof}
The first statement is established combining the usual resolution of the
Cauchy-Riemann equation for $f \in L^{2}$ by an $L^{2,1}$-function $g$ and the
estimation (\ref{regineq}).
The second one is a direct consequence of Proposition I.3 of \cite{Biq91}. One
might also prove it by direct estimations on the solution given by the
Cauchy kernel, as in \cite{At-Pat-Sin}.
\end{proof}
Now let us come back to exactness at a singularity in a finite point:
for the regular case $k\in \{ 1,\ldots,r_{j} \}$ we have $f\in L^{2}$ and
$|\theta\land g \sigma_{k}^{j}| \leq |g|r^{-1+\delta }$, so we can apply directly the first
statement of the claim;
for the singular case $k\in \{ r_{j}+1,\ldots,r \}$ by definition $|\theta\land f
\sigma_{k}^{j} \d \bar{z}| \sim |f|r^{-1+\alpha }$ is in $L^{2}$ with $\alpha >0$ by
Hypothesis \ref{main}, therefore we can apply the second statement of
the claim. Remark that in this case even a stronger condition then the
assumption $fr^{\alpha } \in L^{2}$ of the claim holds. However, we will need the
claim in its full generality to show exactness at infinity.
We now come to exactness at infinity. Recall that $\xi \notin \hat{P}$ implies
$\theta_{\xi } $ is an isomorphism
$L^{2}(\Omega^{0,b}) \ra L^{2}(\Omega^{1,b})$ for $b \in \{ 0,1 \}$. Therefore, the sections at
infinity of the sheaves $\tilde{L}^{2}(\Omega^{0,b})$ and $L^{2}(\Omega^{0,b})$
coincide. First, we consider exactness in
$\tilde{r} \tilde{L}^{2}(E)= \tilde{r} {L}^{2}(E)$: by the definition of $\E$, its local sections are the
holomorphic linear combinations $\sigma = \sum_{k} \phi_{k}^{\infty }\sigma_{k}^{\infty } $. First we
check that these sections verify $r^{-1}\sigma \in L^{2}$: since $|\phi_{k}^{\infty }| \leq K$
and $|\sigma_{k}^{\infty }| \sim r^{-\alpha_{k}^{\infty}}$ with $\alpha_{k}^{\infty} >0$ by Hypothesis \ref{main},
we see that $r^{-1}\phi_{k}^{\infty }\sigma_{k}^{\infty } \in L^{2}$. On the other hand, if we have a
section $\sigma = \sum_{k} \phi_{k}^{\infty }\sigma_{k}^{\infty } $ in the kernel of
$\dbar^{\E}$, then for all $k$ the function $\phi_{k}^{\infty }$ is either
holomorphic or meromorphic; but if $r^{-1}\sigma \in L^{2}$, then it implies that
$\phi_{k}^{\infty }$ is holomorphic for all $k$. This proves exactness in the first term.
Next we come to the term $L^{2}(\Omega^{0,1} \otimes E )$: for a section $v \d \bar{z} \in
L^{2}(\C \setminus \Delta(R), \Omega^{0,1}\otimes E)$ we search $u \in rL^{2}(\C \setminus
\Delta(R), E)$ such that $ \dbar^{\E}u=v$. Suppose $v=f \sigma_{k}^{\infty }$ and
$u=g \sigma_{k}^{\infty }$ again. In the coordinate $w=1/z=\rho e^{-\theta }$ on $\Delta(0,1/R)$
we find (for simplicity we took $R=1$ and wrote $\Delta= \Delta(0,1/R)$ ):
\begin{align*}
\int_{\Delta} | f |^{2} \rho^{2\alpha -4} |\d w |^{2} & = \int_{\C \setminus \Delta} | f |^{2}
r^{-2\alpha } |\d z |^{2} < \infty \\
\int_{\Delta} | g |^{2} \rho^{2\alpha -2} |\d w |^{2} & = \int_{\C \setminus \Delta} | g |^{2}
r^{-2-2\alpha } |\d z |^{2} < \infty .
\end{align*}
On the other hand, the Cauchy-Riemann equation
$$
\frac{\partial g}{\partial \bar{z} }=f
$$
transforms into
$$
\frac{\partial g}{\partial \bar{w} }= -\frac{f}{\bar{w}^{2}}.
$$
and we conclude applying Claim \ref{crsol} to $-f/\bar{w}^{2}$.
\end{proof}
We can also show the counterpart of Lemma \ref{lemrese} for $\F$:
\begin{lem} \label{lemresf} The complex
\begin{equation} \label{resolf}
\F \inc \tilde{r} \tilde{L}^{2}(\Omega^{1,0} \otimes E) \xrightarrow{\dbar^{\E}} L^{2}(\Omega^{1,1} \otimes E)
\end{equation}
is a resolution of $\F$.
\end{lem}
\begin{proof}
Away from the singularities this is also given by classical elliptic
theory, therefore we focus our attention on a neighborhood of a
singular point.
Let us first treat the case of a singularity at a finite point $p_{j} \in P$. A
local section of $\F$ is then by definition a section
$\sigma = \sum_{k}\phi_{k}^{j}\sigma_{k}^{j} \d z$ such that $\phi_{k}^{j}$ is holomorphic for
$k \in \{ 1,\ldots r_{j} \}$ and has a pole of order at most one in $p_{j}$ for
$k \in\{r_{j}+1,\ldots r\}$.
From the form of the parabolic structure, it follows that
$|\phi_{k}^{j}\sigma_{k}^{j}| \sim O(1)$ for $k \in \{ 1,\ldots r_{j} \}$ and
$|\phi_{k}^{j}\sigma_{k}^{j}| \sim O(r^{-1+\alpha_{k}^{j}})$ for $k \in\{r_{j}+1,\ldots r\}$.
By Hypothesis \ref{main} we have $\alpha_{k}^{j}>0$, thus $\sigma \in L^{2}(\Omega^{1,0}\otimes
E)$. On the other hand, if a section $\sigma = \sum_{k}\phi_{k}^{j}\sigma_{k}^{j} \d z$
of $\Omega^{1,0}\otimes E$ satisfies $\dbar^{\E}\sigma=0 $, but $ \sigma \notin L^{2}(\Omega^{1,0}\otimes E)$ then
either $\phi_{k}^{j}$ has a pole for some $k \in \{ 1,\ldots r_{j} \}$ or $\phi_{k}^{j}$
has an at least double pole for some $k \in\{r_{j}+1,\ldots r\}$, and therefore $\sigma $ is not a
local section of $\F$. This shows exactness in the first term.
Consider now exactness at the second term in $\Delta(p_{j},\varepsilon )$: here
we need to solve (\ref{dbareq2}), for $f \in L^{2}$ with the solution $g$ in $L^{2}$
in the regular case; and for $f$ such that $fr^{\alpha } \in L^{2}$ with the solution $g$ such
that $gr^{\alpha } \in L^{2}$ in the singular case. Both follow from Claim \ref{crsol}.
There now remains to show exactness at infinity: this is done similarly to
the case of $\E$.
\end{proof}
\subsection{Hypercohomology and $L^{2}$-cohomology}
We can use the results of the last section in order to deduce the following:
\begin{prop}\label{algl2}
The first $L^{2}$-cohomology $\hat{V}_{\xi } = L^{2}H^{1 }(D_{\xi }'')$ of
(\ref{higgsellcompl}) is isomorphic to the hypercohomology $\H^{1 }(\E \xrightarrow{\theta_{\xi } \land }\F)$.
\end{prop}
\begin{proof} By Lemmas \ref{lemrese} and \ref{lemresf}, $\theta_{\xi }$ defines a
morphism of resolutions
\begin{equation} \xymatrix{ \tilde{L}^{2}(\Omega^{0,1}\otimes E) \ar[r]^{\theta_{\xi } \land } & {L}^{2}(\Omega^{1,1}\otimes E) \\
\tilde{r} \tilde{L}^{2}(E) \ar[r]^(.4){\theta_{\xi } \land} \ar[u]^{\bar{\partial}^{\E} } &
\tilde{r} \tilde{L}^{2}(\Omega^{1,0}\otimes E) \ar[u]_{\bar{\partial}^{\E} } \\
\mathcal{E} \ar[r]^{\theta_{\xi } \land} \ar@{^{(}->}[u] & \mathcal{F} \ar@{^{(}->}[u] .
} \label{mr}
\end{equation}
Therefore, by general theory, the hypercohomology of the sheaf map $\E
\xrightarrow{\theta_{\xi } \land }\F$ identifies to the cohomology of the single
complex formed by the double complex $\mathcal{D}^{r}_{\xi }$:
\begin{eqnarray} \label{drcomplex}
\xymatrix{ \tilde{L}^{2}(\Omega^{0,1}\otimes E) \ar[r]^{\theta_{\xi } \land} & {L}^{2}(\Omega^{1,1}\otimes E) \\
\tilde{r} \tilde{L}^{2}(E) \ar[r]^(.4){\theta_{\xi } \land} \ar[u]^{\bar{\partial}^{\E} } &
\tilde{r} \tilde{L}^{2}(\Omega^{1,0}\otimes E) \ar[u]_{\bar{\partial}^{\E} }.
}
\end{eqnarray}
We show that the first cohomology of the single complex of this double
complex is isomorphic to the first cohomology of the single complex associated
to the double complex $\mathcal{D}_{\xi }$:
\begin{eqnarray} \label{dcomplex}
\xymatrix{ \tilde{L}^{2}(\Omega^{0,1}\otimes E) \ar[r]^{\theta_{\xi } \land} & {L}^{2}(\Omega^{1,1}\otimes E) \\
\tilde{L}^{2}(E) \ar[r]^(.4){\theta_{\xi } \land} \ar[u]^{\bar{\partial}^{\E} } &
\tilde{L}^{2}(\Omega^{1,0}\otimes E) \ar[u]_{\bar{\partial}^{\E} }.
}
\end{eqnarray}
We define a map
$$
\iota : H^{1}(\mathcal{D}_{\xi }) \lra H^{1}(\mathcal{D}^{r}_{\xi })
$$
as follows: represent a cohomology class of $H^{1}(\mathcal{D}_{\xi })$ by a
couple
$$
(\kappa\d \bar{z},\nu \d z )\in \tilde{L}^{2}(\Omega^{0,1}\otimes E) \oplus \tilde{L}^{2}(\Omega^{1,0}\otimes E),
$$
and use the inclusion (\ref{inclvsp}) to map it into the cohomology class represented by the same couple
$(\kappa,\nu )$ in $H^{1}(\mathcal{D}^{r}_{\xi })$. This is well defined, since if
$(\kappa \d \bar{z}+\dbar^{\E}\lambda ,\nu \d z + \theta_{\xi }\lambda )$
is a couple in $H^{1}(\mathcal{D}_{\xi })$ representing the same class
as $(\kappa \d \bar{z},\nu \d z)$, for $\lambda \in \tilde{L}^{2}(E)$, then in particular
$\lambda \in\tilde{r} \tilde{L}^{2}(E)$, and so the two couples are cohomologuous in
$H^{1}(\mathcal{D}^{r}_{\xi })$ as well. This also shows that $\iota $ is injective.
We only need to prove surjectivity: suppose we have a couple
$(\kappa\d \bar{z},\nu \d z)\in \tilde{L}^{2}(\Omega^{0,1}\otimes E) \oplus \tilde{r} \tilde{L}^{2}(\Omega^{1,0}\otimes E)$
representing a class in $H^{1}(\mathcal{D}^{r}_{\xi })$. It is clearly sufficient
to prove that this class can be represented by a couple vanishing in a
neighborhood of infinity. Since $\theta_{\xi }$ is an isomorphism at infinity, we
can put (restricting to a smaller neighborhood of infinity if necessary)
$\lambda = \theta_{\xi }^{-1}(\nu \d z)$. This is then a section in $\tilde{r} \tilde{L}^{2}(E)$,
and the couple $(\kappa\d \bar{z} - \dbar^{\E} \lambda ,\nu \d z - \theta_{\xi }\lambda) $ is
cohomologuous to $(\kappa\d \bar{z},\nu \d z)$ in $H^{1}(\mathcal{D}^{r}_{\xi })$. By
definition, the $(1,0)$-term of this couple vanishes at infinity. The same
thing is true for the $(0,1)$-part, because $\theta_{\xi } (\kappa\d \bar{z} -
\dbar^{\E} \lambda) = -\dbar^{\E}(\nu \d z - \theta_{\xi }\lambda) =0$ near infinity and $\theta_{\xi }$ is an
isomorphism there. This finishes the proof of the proposition, for the
$L^{2}$-cohomology of (\ref{higgsellcompl}) is by definition the cohomology
of the single complex associated to $\mathcal{D}_{\xi }$.
\end{proof}
\subsection{The spectral curve} \label{specsect}
In the explicit identification of the hypercohomology, the following notions will be of much importance.
Recall that (up to wedge product by $\d z$) $\theta_{\xi }$ is a meromorphic section of $End(E)$ over $\CP$.
\begin{defn} \label{specsetdef}
For $\xi \in \hat{\C} \setminus \hat{P}$, the set of zeros of $det{(\theta_{\xi })}$ is called the \emph{spectral set}
corresponding to $\xi$. We denote it by $\Sigma_{\xi }$.
\end{defn}
\begin{lem} \label{efdiv}
For each $\xi \in \hat{\C} \setminus \hat{P}$, the spectral set is an effective divisor of $\CP$,
in other words a finite set of points with multiplicities in $\N$.
\end{lem}
\begin{proof}
The section $\det{(\theta_{\xi })}$ of $End(V)$ is holomorphic with respect to $\dbar^{\E}$. We only
need to check it does not vanish identically for any $\xi$. Suppose there exists $\xi$ such that
$$
det{(\theta_{\xi}(q))} = 0
$$
for all $q \in \C \setminus P$. In different terms, $\theta$ has a constant eigenvalue over $\C\setminus P$;
in particular, the residue of this eigenvalue at infinity is $0$. This contradicts $\lambda^{\infty}_{k}\neq 0$
for all $k \in \{ 1,\ldots,n \}$ (see (\ref{mainii}) of Hypothesis \ref{main}).
\end{proof}
A basic property is the following.
\begin{clm} \label{merospec}
The points of $\Sigma_{\xi }$ define a multi-valued meromorphic function of $\xi \in \hat{\C}$.
\end{clm}
\begin{proof}
By assumption, $det(\theta_{\xi }(z))$ depends holomorphically on
$\xi \in \hat{\C}$ and meromorphically on $z$. We conclude using the implicit function theorem,
namely that the solutions of a meromorphic equation depending holomorphically on a variable are
meromorphic in this variable.
\end{proof}
\begin{defn}
The graph of the multi-valued meromorphic function
\begin{align*}
\hat{\C} \setminus \hat{P} & \lra \CP \\
\xi & \mapsto \Sigma_{\xi }
\end{align*}
is called the \emph{spectral curve} of the Higgs bundle. It is denoted by $\Sigma $.
\end{defn}
This object was first studied by N. Hitchin in \cite{Hit2}.
By Claim \ref{merospec} the spectral curve is an analytic subvariety
$$
\Sigma \xrightarrow{\j} (\hat{\C } \setminus \hat{P}) \times \CP ,
$$
of (complex) dimension one. (Here $\j$ stands for inclusion.)
Moreover, by construction it is naturally a branched cover of $\hat{\C }$ via projection to the first factor.
Here is an important property.
\begin{prop} \label{red}
The spectral curve $\Sigma$ is reduced; in other words, $det(\theta_{\xi })$ vanishes only up to the first order except
for a finite set of points of $\Sigma$.
\end{prop}
\begin{proof} Suppose $\Sigma$ has infinitely many points $(q,\xi)$ where $det(\theta_{\xi })$ vanishes up to order higher
than one. Since $\Sigma$ has a natural extension into a compact curve in $\CP \times \CPt$ (see Section \ref{ext}),
this means that for any $\xi$ some zero $q(\xi) \in\Sigma_{\xi}$ of $\theta_{\xi}$ has multiplicity higher than one; in different
terms, some irreducible component of $\Sigma$ has multiplicity higher than one. In particular, as $\xi \ra \infty$, at least two
of the $q_{k}(\xi)$ must have the same Laurent expansions. This is impossible by (\ref{asymbehinf})
and the assumption $\lambda^{j}_{k}\neq \lambda^{j}_{k'}$ for $k \neq k'$ made in (\ref{maini}) of Hypothesis \ref{main}.
\end{proof}
\subsection{Explicit computation of the hypercohomology}
Let us now compute the hypercohomology of
\begin{equation} \label{sheafmap}
\E \xrightarrow{\theta_{\xi } \land }\F
\end{equation}
Consider arbitrary algebraic resolutions of the sheaves $\E$ and $\F$ such
that $\theta_{\xi } \land $ induce a morphism of resolutions
\begin{equation} \xymatrix{ K^{0,1} \ar[r]^{\theta_{\xi } \land } & K^{1,1} \\
K^{0,0} \ar[r]^{\theta_{\xi } \land} \ar[u]^{\delta } &
K^{1,0} \ar[u]_{\delta } \\
\mathcal{E} \ar[r]^{\theta_{\xi } \land} \ar@{^{(}->}[u] & \mathcal{F.} \ar@{^{(}->}[u]
} \label{algmr}
\end{equation}
For example, one might take resolutions by \v{C}ech cochains.
By definition, the first filtration $K_{p}$ of the single complex associated to
(\ref{algmr}) is given by
\begin{align*}
K_{0} & = (K^{0,1} \oplus K^{0,0} ) \oplus ( K^{1,1} \oplus K^{1,0} ) \\
K_{1} & = K^{1,1} \oplus K^{1,0} .
\end{align*}
The first page of the spectral sequence corresponding to this filtration is
given by
\begin{equation} \xymatrix{ (\Cs^{0})^{[1]}(\CP) & (\Cs^{1})^{[1]}(\CP) \\
(\Cs^{0})^{[0]} (\CP) \ar[u]^{\delta } &
(\Cs^{1})^{[0]}(\CP) \ar[u]_{\delta }
} \label{ss1}
\end{equation}
where $\Cs^{j}$ is the $j$-th cohomology sheaf of the map (\ref{sheafmap}),
and the vertical sequences come from resolutions
\begin{align*}
\Cs^{0} & \inc (\Cs^{0})^{[0]} \xrightarrow{\delta } (\Cs^{0})^{[1]} \\
\Cs^{1} & \inc (\Cs^{1})^{[0]} \xrightarrow{\delta } (\Cs^{1})^{[1]}
\end{align*}
by taking global sections. Let us now describe explicitly the cohomology sheaves.
Recall from definition \ref{specsetdef} that $q \in \Sigma_{\xi}$ are exactly the points where
the map $\theta_{\xi }(q): E(q) \ra E(q)$ is not surjective.
After all this preparation, we have the following characterization:
\begin{lem}
The cohomology sheaf $\Cs^{0}$ of order $0$ of the sheaf map
(\ref{sheafmap}) is $0$. If $\det{(\theta_{\xi })}$ has a zero of order $1$
in all points of $q \in \Sigma_{\xi }$, then the first cohomology sheaf $\Cs^{1}$ is the
sky-scraper sheaf $\Sky_{\xi }$ whose stalk over a point $q \in \Sigma_{\xi }$ is the
finite-dimensional subspace $coKer(\theta_{\xi}(q)) \subset E(q)$, and all other stalks are $0$.
\end{lem}
\begin{rk}
The cokernel of $\theta_{\xi}(q)$ is naturally identified with the orthogonal of the image with respect
to the fiber metric, or, which is the same thing, with the kernel of $\theta_{\xi}^{\ast}(q)$.
This allows us to think of $coKer(\theta_{\xi}(q))$ as a subspace of $E(q)$.
\end{rk}
\begin{proof}
Let us start with $\Cs^{0}$: suppose we have a section $\phi \in \E|_{U}$ on
an open set $U \subset \CP$ such that $\theta_{\xi } \phi =0$. Since on the open subset
$U \setminus \Sigma_{\xi }$ the map $\theta_{\xi }: E(q) \ra E(q) $ is an isomorphism,
we deduce that $\phi =0$ on this set. But a holomorphic section vanishing on
an open set vanishes everywhere, thus $\phi =0$ on all of $U$. This gives the
first statement of the lemma.
We now come to $\Cs^{1}$: let $U \subset \CP$ be an open subset. If $U \cap \Sigma_{\xi }
= \emptyset $ then $\theta_{\xi }$ is an invertible holomorphic endomorphism of $\E$ on $U$, therefore
$\Cs^{1}|_{U} =0$. Suppose now $U$ contains exactly one point $q \in \Sigma_{\xi}$.
Then, for any section $\phi \in \E|_{U}$ the vector $(\theta_{\xi } \phi)(q) $ lies by
definition in the image of $\theta_{\xi }(q)$, which is just the orthogonal of
$coKer( \theta_{\xi }(q))$. Therefore, this latter is contained in
$\Cs^{1}|_{U}$. On the other hand, the condition that $\theta_{\xi }$ has a zero
of order $1$ in $q$ means that any section $\psi \in \E|_{U}$ such that
$\psi(q) \bot coKer(\theta_{\xi }(q))$ is in $Im(\theta_{\xi })$. This proves the second statement.
\end{proof}
\begin{rk} By Proposition \ref{red}, the condition of $\det{(\theta_{\xi })}$ having a first-order zero in
all points of $ \Sigma_{\xi }$ is generic in $\xi$: it is verified for all $\xi$ except
for twice the eigenvalues of $\theta(q)$ for the finite number of points
$q$ of $\Sigma$ of multiplicity higher than one. For the discrete set of $\xi$
where there exists a $q \in \Sigma_{\xi}$ with a multiple zero, one introduces
the flag
$$
E(q) = F_{0}E(q) \supseteq coKer(\theta_{\xi }(q))=F_{1}E(q)\supseteq \ldot
\supset F_{r_{q}}E(q)=\{ 0\} ,
$$
the subscript of $F$ being the order of zero of $\theta_{\xi }^{\ast}(q)$ along the given subspace,
and proves that the cohomology sheaf $\Cs^{1}|_{U}$ over an open set containing $q$ as the only
element of $\Sigma_{\xi }$ is in this case equal to the jet space
$$
\bigoplus_{m=1}^{r_{q}-1}F_{m}E(q).
$$
The assumptions that for fixed $j \in \{ 1,\ldots,n \}$ all the $\lambda^{j}_{k} $ be different for $k \in \{ r_{j}+1,\ldots,r \} $
and for fixed $l \in \{ 1,\ldots,n' \}$ all the $\lambda^{\infty }_{k}$ be different for $k \in \{ 1+a_{l}, \ldots a_{l+1} \}$
(see (\ref{maini}) and (\ref{mainii}), Hypothesis \ref{main}), mean that in the punctures
of $\CPt$ the limit states have first-order zeros.
\end{rk}
Now since a resolution of the sky-scraper sheaf $\Sky_{\xi }$ is given by
$$
\Sky_{\xi } \inc \Sky_{\xi } \ra 0,
$$
the first page of the hypercohomology spectral sequence (\ref{ss1}) becomes
$$
\xymatrix{ 0 & 0 \\
0 \ar[u]^{\delta } &
\bigoplus_{q \in \Sigma_{\xi }} coKer(\theta_{\xi }(q)). \ar[u]_{\delta }
} \label{ss1b}
$$
All this implies the following:
\begin{prop} \label{hss} The hypercohomology spectral sequence corresponding to the
first filtration collapses in its first page, and we have a natural
isomorphism
$$
\H^{1 }(\E \xrightarrow{\theta_{\xi } \land }\F) \simeq \bigoplus_{q \in \Sigma_{\xi }} coKer(\theta_{\xi }(q)).
$$
\end{prop}
\begin{proof}
This is a consequence of the standard fact that a spectral sequence
collapses as soon as non-zero elements only appear in one of its rows.
Furthermore, an explicit isomorphism can be given as follows: fix a radially invariant
bump-function $\chi$ on the unit disk $\Delta \subset \C$, equal to $0$ on the boundary
of $\Delta$ and to $1$ in $0$, and such that $\d \chi $ is supported
on the annulus $ 1/3<r<2/3 $. For any complex number $a \neq 0$ set $\chi_{a }(z)=
\chi(z/a)$. Now choose $\varepsilon_{0}>0$ so that the distance in $\C $ between any two distinct points
of the finite set $P \cup \Sigma_{\xi }$ is at least $3\varepsilon_{0}$.
For any $(v_{q})_{q \in \Sigma_{\xi }} \in \oplus coKer(\theta_{\xi }(q))$ consider the section
$v_{\varepsilon_{0}} =\sum_{q \in \Sigma_{\xi } } v_{q} \chi_{\varepsilon }(z-q)$.
Because $\d \chi_{\varepsilon_{0}}$ is
supported on the annulus $\varepsilon_{0}/3 <r< 2\varepsilon_{0}/3 $, the section
$\dbar^{\E}(v_{\varepsilon_{0}} \d z)\in \Omega^{1,1}\otimes E$ is supported outside a neighborhood
of $\Sigma_{\xi }$. Since this latter is the zero set of $\det{(\theta_{\xi })}$, it then
follows that there exists a section $t_{\varepsilon_{0}} \d \bar{z} \in \Omega^{0,1}\otimes E$ such that
$\theta_{\xi }\land (t_{\varepsilon_{0}}\d \bar{z}) + \dbar^{\E}(v_{\varepsilon_{0}}\d z)=0$, and
$t_{\varepsilon_{0}}$ is supported on the support of $\dbar^{\E} v_{\varepsilon_{0}}$, that is
outside a neighborhood of $\Sigma_{\xi }$ and of infinity. The couple
$(v_{\varepsilon_{0}}\d z,t_{\varepsilon_{0}}\d \bar{z})$ therefore defines a cocycle in the
single complex associated to $ \mathcal{D}_{\xi }$, and using Proposition \ref{algl2} we can define a map
\begin{align} \label{psi}
\Psi_{\xi } : \bigoplus_{q \in \Sigma_{\xi }} coKer(\theta_{\xi }(q)) & \lra H^{1}(\mathcal{D}_{\xi })
= \H^{1 }(\E \xrightarrow{\theta_{\xi } \land }\F) \notag \\
(v_{q})_{q \in \Sigma_{\xi }} & \mapsto [(v_{\varepsilon_{0}}\d z,t_{\varepsilon_{0}}\d \bar{z})],
\end{align}
where $[(v_{\varepsilon_{0}}\d z,t_{\varepsilon_{0}}\d \bar{z})]$ stands for
the cohomology class in $H^{1}(\mathcal{D}_{\xi })$ of this couple.
We need to show that this map does not depend on $\varepsilon_{0} >0$ chosen,
provided that it is sufficiently small as explained above. Consider
therefore the section $v_{\varepsilon_{1}}$ for $\varepsilon_{1} < \varepsilon$.
Since in the union of the disks of radius $ \varepsilon_{1}/3 $ around the
elements of $\Sigma_{\xi }$ we have $v_{\varepsilon_{1}} = v_{\varepsilon_{0}}$, and
$\theta_{\xi }$ is invertible outside this set, there exists a section $u \in \Gamma(E)$
such that $\theta_{\xi }u + v_{\varepsilon_{1}} \d z=v_{\varepsilon_{0}}\d z$. Then, as in the
proof of Proposition \ref{algl2}, the couple $(v_{\varepsilon_{0}}\d z,t_{\varepsilon_{0}}\d \bar{z})$
is equal to $(v_{\varepsilon_{1}}\d z+ \theta_{\xi }u ,t_{\varepsilon_{1}}\d \bar{z}+\dbar^{\E}u)$, and
the two couples define the same cohomology class in $H^{1}(\mathcal{D})$.
This then allows us to fix $\varepsilon_{0}>0$ sufficiently small once and for all.
In a similar way, one can prove that $\Psi_{\xi }$ is independent of the actual cut-off function $\chi $ as well.
Finally, the inverse of $\Psi_{\xi }$ can be obtained as follows: let the cohomology class
$\eta \in H^{1}(\mathcal{D}_{\xi })$ be represented by a $1$-form $\eta^{1,0} \d z + \eta^{0,1} \d \bar{z}$, where
$\eta^{1,0}$ and $\eta^{0,1}$ are sections of $E$. Then we have
\begin{equation} \label{psiinv}
\Psi_{\xi }^{-1} \eta = (eval_{q } \eta^{1,0} )_{ q \in \Sigma_{\xi } },
\end{equation}
where $eval_{q } \eta^{1,0}$ stands for evaluation of the section $\eta^{1,0}$ in the point $q$.
\end{proof}
\begin{rk}
Notice that the formula (\ref{psiinv}) is independent of the $1$-form representative of $\eta$;
in particular, the $(1,0)$-part of the harmonic representative of a cohomology class
$\Psi_{\xi }(v_{q})_{q \in \Sigma_{\xi }}$ vanishes in the $q \in \Sigma_{\xi }$ where $v_{q}=0$.
\end{rk}
\section[Extension over the singularities]{Extension of the Higgs bundle over the singularities} \label{ext}
The interpretation of the holomorphic bundle underlying the transformed
Higgs bundle in terms of hypercohomology established in the
previous section allows us to extend it over the singular points $\hat{P} \cup
\{ \infty \}$ in the parameter space ${\CPt}$. At each puncture, we need to do two
things: first, define the fiber of the transformed vector bundle over it.
This then extends the holomorphic structure induced by $\dbar^{\hat{\E}}$ over the puncture
in a natural way: a holomorphic section through the singular point will be a
continuous section in a neighborhood of it, that is holomorphic in the punctured neighborhood.
(Continuity is defined at the same time as the exceptional fiber.)
The second thing to do then is to give an explicit basis of holomorphic sections with respect
to this extended holomorphic structure. It is important to note that the extensions $^{i}\hat{\E}$
we define here are \emph{not} the transformed extensions given in Definition \ref{transfext},
but rather ones induced by the original Higgs bundle, and for which computations are more comfortable.
This is why we will call $^{i}\hat{\E}$ the \emph{induced extension}.
We study the link between these two extensions in Section \ref{secttop}.
\subsection{Extension to logarithmic singularities} \label{logext}
First, we consider the case of points of the set $\hat{P}$.
We shall now describe the extension $^{i}\hat{\E}$ over such a point.
Notice first that as the deformation $\theta_{\xi }$ has a well-defined extension over these
points, its hypercohomology spaces are also well-defined there. In particular, in view of
Proposition \ref{algl2}, we may extend the transformed vector bundle $\hat{V}$ by putting
\begin{equation*}
\hat{V}_{\xi_{l}}=\H^{1}(\E \xrightarrow{\theta_{\xi_{l} } \land }\F)
\end{equation*}
This is the definition of the fiber over such a point.
In order to give explicit representatives of holomorphic sections,
let us examine what happens to the fiber $\hat{V}_{\xi }$
when $\xi $ approaches one of the points of $\hat{P}=\{ \xi_{1}, \ldots, \xi_{n'} \}$, say $\xi_{l}$.
First, let us find the spectral points.
\begin{clm}\label{nobranching}
As $\xi \ra \xi_{l}$, exactly $m_{l}=a_{l+1}-a_{l}$ branches of the meromorphic functions $q_{k}\in \Sigma_{\xi}$
converge to infinity, while all others remain in a bounded region of $\C$. Moreover,
labelling the spectral points converging to infinity by $q_{1+a_{l}}(\xi),\ldots,q_{a_{l+1}}(\xi)$,
they admit the asymptotic behavior
\begin{equation} \label{zerobeh}
{q}_{k}(\xi )=\frac{2\lambda_{k }^{\infty }}{(\xi-\xi_{l} )} + O(|\xi-\xi_{l}|^{-\delta}),
\end{equation}
where $\delta>0$ can be chosen arbitrarily small. In particular, the branches converging to $\infty\in\CP$ of the
spectral curve are not ramified over the point $\xi_{l}$.
\end{clm}
\begin{proof}
As it can be seen from (\ref{modelhiggsinf}), exactly $m_{l}$
of the eigenvalues of the leading order term near infinity of the Higgs
field $\theta_{\xi }$ converges to $0$. Recall from Definition \ref{specsetdef} that $\Sigma_{\xi }$ is
the vanishing set of $det(\theta_{\xi })$. This implies that (counted with multiplicities) exactly $m_{l}$ of the
points $q(\xi ) \in \Sigma_{\xi }$ converge to infinity; label these by $1+a_{l},\ldots,a_{l+1}$. All the other
spectral points remain therefore bounded. By assumption (see (\ref{modelhiggsinf}))
in a holomorphic trivialisation of the bundle $\E$ in a neighborhood of $\infty \in \CP$, ignoring the factor
$\d z$ the field $\theta_{\xi }$ is of the form
$$
\frac{1}{2} (A - \xi \Id) + \frac{C}{z} + O(z^{-2}),
$$
where $O(z^{-2})$ stands for holomorphic terms independent of $\xi$.
Suppose first that the field is exactly equal to the polar part in this formula,
in other words the $O(z^{-2})$ term is equal to $0$. Then the solutions
$\tilde{q}_{1}(\xi),\ldots,\tilde{q}_{r}(\xi)$ are clearly given by
\begin{equation*}
\tilde{q}_{k}(\xi )=\frac{2\lambda_{k }^{\infty }}{(\xi-\xi_{l} )}.
\end{equation*}
In general, since $det(\theta_{\xi })$ is holomorphic in $z$, we can apply Rouch\'e's theorem to compare
the position of the zeros of $det(\theta_{\xi })$ with those of the polar part studied above.
This yields that the solutions $q_{k}(\xi ) \in \C$ of $det(\theta_{\xi })(q(\xi ))=0$ near infinity are close to
$\tilde{q}_{k}(\xi )$; more precisely for any $\delta>0$, there exists $K>0$ such that for all $|\xi-\xi_{l}|$
sufficiently small we have
$$
\left\vert q_{k}(\xi )-\tilde{q}_{k}(\xi ) \right\vert < K |\xi-\xi_{l}|^{-\delta}.
$$
Remark here that as $\xi \ra \xi_{l}$ the behavior of $|\xi-\xi_{l}|^{-\delta}$ is small compared to
$|\tilde{q}_{k}(\xi)|=c|\xi-\xi_{l}|^{-1}$. In other words, we have the expansion (\ref{zerobeh})
so that $q_{k}(\xi )$ converges indeed to infinity asymptotically proportionally to $(\xi-\xi_{l} )^{-1}$ for
$a_{l} < k \leq a_{l+1}$, while all other holomorphic families of zeros of $\det(\theta_{\xi })$ remain bounded.
The condition that the $\lambda_{1+a_{l}}, \ldots, \lambda_{a_{l+1}}$ are all distinct (see (\ref{mainii}), Hypothesis
\ref{main}) now implies that there is no splitting of the solutions at infinity,
that is to say locally near $\xi=\xi_{l}$ any $q_{k}(\xi)$ with $a_{l} < k \leq a_{l+1}$ itself forms a meromorphic
function without branching. Indeed, the occurrence of a branching at infinity implies that the Puiseux series
of the corresponding solutions agree, which is not the case here because of the asymptotic behaviors
(\ref{zerobeh}) with different leading coefficients.
\end{proof}
Now, recall that for fixed $\xi \in \hat{\C} \setminus \hat{P}$, in the
explicit description of $\hat{V}_{\xi }$ given in the proof of Proposition
\ref{hss}, we considered the zeros $q_{k}(\xi )$ for $ k=1,\ldots,r$ of $det(\theta_{\xi })(q)$,
and for each $q_{k}(\xi )$ an element $v_{k}(\xi )$ of the subspace
$coKer(\theta_{\xi })_{q_{k}(\xi )} \subset E_{q_{k}(\xi )}$. Then we extended each
$v_{k}(\xi)$ holomorphically into a neighborhood of $q_{k}(\xi )$, and multiplied the section
we obtained by a bump-function equal to $1$ in a small disk around $q_{k}(\xi)$
and to $0$ on the boundary of a slightly larger disk.
This section of $\F$ constituted the $(1,0)$-part of the element
in $\H^{1}(\E \xrightarrow{\theta_{\xi }} \F) \simeq \hat{V}_{\xi }$, and we chose the $(0,1)$-part in
such a way that the couple be in $Ker(D_{\xi }'')$.
In what follows, we wish to do the same thing, but for all $\xi $ in a
neighborhood of $\xi_{l}$ at the same time.
Let us consider one meromorphic family of zeros $q_{k}(\xi )$ with $a_{l} < k \leq a_{l+1}$.
We have just seen that $q_{k}(\xi )$ converges to $\infty $ as $\xi \ra \xi_{l}$;
therefore, we need to take a holomorphic section of $\E$ at infinity, extending an element of the
cokernel of $\theta_{\xi_{l}} $. One can check from formula (\ref{modelhiggsinf})
that this cokernel is equal to the vector subspace of the fiber $\F_{\infty }= E_{\infty } \otimes \d z$ generated by
$\{ \sigma_{m}^{\infty } (\infty ) \d z \}_{m=1+a_{l}}^{a_{l+1}}$, where $\{ \sigma_{m}^{\infty } \}_{m=1}^{r}$
is the holomorphic trivialisation of $\E$ at infinity considered in (\ref{holtrivinf}).
Furthermore, since the metric $h$ is mutually bounded with the diagonal model
$$
diag(|z|^{-2\alpha^{\infty}_{k}}),
$$
the orthogonal of the image of $\theta_{\xi}$ in $E(q_{k}(\xi))$ converges to $\sigma_{k}^{\infty}(\infty)$ as $\xi \ra \xi_{l}$.
Let $\varsigma_{k}(z )$ be a holomorphic extension of $\sigma_{k}^{\infty }(\infty) $ to a neighborhood of infinity such that for any
$\xi \in \hat{\C}$ sufficiently close to $\xi_{l}$, the vector $\varsigma_{k}(q_{k}(\xi ) )\d z$ be in the cokernel of
$\theta_{\xi }(q_{k}(\xi ))$.
Such an extension exists because $\theta_{\xi }$ varies holomorphically with $\xi $ and by Claim \ref{nobranching}
$q_{k}(\xi )$ is a genuine (single-valued) meromorphic function of $\xi$.
A holomorphic section $\hat{\sigma}_{k}$ of $\hat{\E}$ around $\xi_{l}$ is then given by the section constructed as follows:
for $\xi $ sufficiently close to $\xi_{l}$ such that $\varsigma_{k} $ is defined in $q_{k}(\xi ) $, set
\begin{equation} \label{vlog}
v_{k } (z, \xi ) = \chi_{\varepsilon_{0} (\xi -\xi_{l})^{-1}} (z- q_{k}(\xi )) \varsigma_{k}(z ) ,
\end{equation}
where we recall from the proof of Proposition \ref{hss} that $\chi_{\varepsilon_{0} (\xi -\xi_{l})^{-1}}$ is
a bump-function on a disk centered at $0$ and of diameter $\varepsilon_{0} |\xi -\xi_{l}|^{-1}$ with
$\varepsilon_{0}$ sufficiently small only depending on the parameters of the initial connection,
fixed once and for all.
(The importance of this choice will become clear in Theorem \ref{parweightthmlog}.)
Also, let $t_{k } (z,\xi ) \d \bar{z} \in \Gamma (\C , E \otimes \Omega^{0,1})$ be the unique solution of the equation
\begin{equation} \label{tlog}
\dbar^{\E} v_{k }(z,\xi ) \d z = - \theta_{\xi } t_{k }(z,\xi ) \d \bar{z}.
\end{equation}
Then consider the cohomology class $\hat{\sigma}^{l}_{k}(\xi)$ in
$\H^{1 }(\E \xrightarrow{\theta_{\xi } \land }\F) \simeq \hat{V}_{\xi } $ of the couple
$(v_{k } (z, \xi ) \d z, t_{k } (z,\xi ) \d \bar{z})$ defined as above. Since the choice of $\varsigma_{k}$ is independent
of $\xi $ and moreover $\theta_{\xi }$ and $ q_{k}(\xi )$ depend holomorphically on $\xi $, it follows that
$\hat{\sigma}^{l}_{k}$ is $\dbar^{\hat{\E}}$-holomorphic in $\xi $ outside of $\xi_{l}$.
\begin{defn}
Let the extension $^{i}\hat{\E}$ of $\hat{\E}$ to $\xi_{l}$ be defined by the holomorphic trivialisation
given by the sections $\hat{\sigma}^{l}_{k}$ for all choice of $k \in \{ 1+a_{l}, \ldots, a_{l+1} \}$ and for some
holomorphic extension $\varsigma_{k}$ of $\sigma^{\infty }_{k}(\infty )$ such that for any $\xi \in \hat{\C}$ sufficiently close to
$\xi_{l}$, we have $\varsigma_{k}(q_{k}(\xi ) )\d z \in coKer(\theta_{\xi }(q_{k}(\xi ) ) )$.
\end{defn}
\subsection{Extension to infinity} \label{infext}
In order to define the fiber over infinity, we first rephrase what we have done until now to obtain the
holomorphic bundle $\hat{\E}=(\hat{V},\dbar^{\hat{\E}})$ underlying the
transformed Higgs bundle: we considered the sheaves $\E$ and $\F$ over $\CP$, we pulled them back to
$\CP \times \hat{\C}$ by the projection map $\pi_{1}$ on the first factor, and formed the sheaf map
$$
\pi_{1}^{\ast }\E \xrightarrow{\theta_{\bullet } } \pi_{1}^{\ast }\F
$$
equal to $\theta_{\xi }$ on the fiber $\CP \times \{ \xi \}$. We then defined the vector
bundle
$$
\hat{V}_{\bullet } = \H^{1}(\pi_{1}^{\ast }\E \xrightarrow{\theta_{\bullet } } \pi_{1}^{\ast }\F),
$$
over $\hat{\C} \setminus \hat{P}$ and we let $\dbar^{\hat{\E}}$ be the
partial connection induced by $\hat{\d}^{0,1}$. In what follows, we keep on
writing $\E$ and $\F $ for their pull-back to the product, whenever this
does not cause confusion. Notice that $\theta_{\bullet }$ is
holomorphic in both coordinates. We wish to extend the hypercohomology of
this sheaf map over infinity; we will be done if we can extend the map
$\theta_{\bullet}$ over infinity in a holomorphic manner. Indeed, the hypercohomology
of a holomorphic family of sheaf morphisms is a holomorphic vector bundle
over the base space of the deformations, in our case $\CPt$. Notice that by definition
$\theta_{\xi }= \theta - \xi/2 \d z \land $, so it becomes singular as we let $\xi $ converge to infinity. However, we can
slightly change the sheaf $\F $ in such a way that there exist a natural
extension of $\theta_{\bullet }$. Again, we follow \cite{Jardim}.
Consider the projections $\pi_{j}$ to the $j$-th coordinate in the product
manifold $\CP \times \widehat{\mathbf{CP}}^{1}$, and set
$\tilde{\F } = \pi_{2}^{\ast }\mathcal{O}_{\widehat{\mathbf{CP}}^1}(1) \otimes \F $.
Recall that $\mathcal{O}_{\widehat{\mathbf{CP}}^1}(1)$ admits two global
holomorphic sections $s_{0}$ and $s_{\infty }$, characterized by the
fact that if $\hat{U}_{0}$ and $\hat{U}_{\infty }$ are the standard neighborhoods of
$0\in \widehat{\mathbf{CP}}^1$ and $\infty \in \widehat{\mathbf{CP}}^1$ with coordinates $\xi $ and $\zeta=\xi^{-1} $
vanishing in $0$ and $\infty $ respectively, then we have
\begin{align}
s_{0}(\xi )= \xi && s_{\infty }(\xi )=1 && \mbox{in } \hat{U}_{0} \label{o1zero} \\
s_{0}(\zeta )= 1 && s_{\infty }(\zeta )=\zeta && \mbox{in } \hat{U}_{\infty }. \label{o1inf}
\end{align}
Notice that here $\xi $ is the standard coordinate of $\C$ we used to define
$\theta_{\xi }$. Therefore for $\eta \in \widehat{\mathbf{CP}}^1$ we put
\begin{align}
\tilde{\theta}_{\eta } & : \E \lra \tilde{\F } \\
\tilde{\theta}_{\eta } & = s_{\infty }(\eta ) \otimes \theta -\frac{1}{2} s_{0}(\eta ) \otimes \d z \land , \label{globdefhiggs}
\end{align}
We remark that by (\ref{o1zero}), on $\hat{U}_{0}=\C $ we have
$\tilde{\theta}_{\xi }= \theta - \xi/2 \d z \land =\theta_{\xi }$, so $\tilde{\theta}_{\bullet }$ is indeed an extension of the
deformation $\theta_{\bullet }$ to infinity.
Therefore, in what follows we keep on writing $\theta $ for $\tilde{\theta}$ whenever this does
not cause any confusion.
In the same manner, we see that
$$
{\theta }_{\infty }= - \frac{1}{2} s_{0}(\xi )_{\xi = \infty } \otimes \d z \land : \E \lra \F \otimes \mathcal{O}_{\CPt }(1)_{\xi = \infty }.
$$
From the definition of the sheaves $\E$ and $\F$ one can see
that the cohomology sheaves of
this map are $\Cs^{0}(\d z \land )=0$ and $\Cs^{1}(\d z \land )= \Sky_{\infty }$,
the sky-scraper sheaf supported in points of $P$ and
having stalk equal to $ s_{0}(\xi )_{\xi = \infty } \otimes coKer(Res(\theta,p)) $ in $p \in P$.
Therefore, as in Proposition \ref{hss}, we obtain that the first
hypercohomology space of this map equals $ s_{0}(\xi )_{\xi = \infty } \otimes \left( \oplus_{p\in P} coKer(Res(\theta,p)) \right)$,
and all its other hypercohomology spaces vanish. The extension of the vector bundle
$\hat{V}$ to infinity is then given by setting $\hat{V}_{\eta }
= \H^{1}(\E \xrightarrow{{\theta }_{\eta }} \tilde{\F} )$ for all $\eta \in \CPt \setminus \hat{P}$.
In particular, any local section at $\zeta = 0$ of $\hat{E}$ is a family of sections of
the sheaf $\tilde{\F}$, and therefore can be written
\begin{equation} \label{holsectinf}
s_{0}(\zeta ) \otimes \psi(z, \zeta ),
\end{equation}
where $\psi(z, \zeta )$ are sections of $\F $ depending on the parameter $\zeta $.
\begin{defn}
The extension $^{i}\hat{\E}$ of the holomorphic structure of $\hat{\E}$ to infinity is the extension
whose holomorphic sections at infinity can be written as in (\ref{holsectinf}), with $\psi(z, \zeta )$
holomorphic in $\zeta $.
\end{defn}
We come to the explicit description of a holomorphic section of $^{i}\hat{\E}$ at $\xi = \infty $
with respect to this extension. We make a similar construction as in the case of logarithmic singularities:
first, we make a basic remark.
\begin{clm} \label{convzeros}
As $\xi \ra \infty $, all zeros of $det(\theta_{\xi })$ converge to one of the points of $P$. Moreover, supposing
$q(\xi)\ra p_{j}$, we have the asymptotic behavior
\begin{equation} \label{asymbehinf}
q(\xi)
= p_{j}+2 \frac{\lambda_{k}^{j}}{\xi} + O(\xi^{-2+\delta}),
\end{equation}
where $\lambda_{k}^{j}$ is a non-vanishing eigenvalue of the residue of $\theta$ at $p_{j}$ and $\delta>0$ can be chosen
arbitrarily small. In particular, the spectral curve is not branched over the point $\xi=\infty$.
\end{clm}
\begin{proof}
Let us consider the deformation of the Higgs field in terms of the coordinate $\zeta=\xi^{-1 }$ in $\hat{U}_{\infty }$.
As we see from (\ref{o1inf}) and (\ref{globdefhiggs}), it is given by
$$
{\theta }_{\zeta }=\zeta \theta - \frac{1}{2} \d z \land .
$$
Notice that as $\zeta \ra 0$, the first term on the right-hand side in a fixed
point $z \in \CP \setminus P$ becomes insignificant, and $\theta_{\zeta }(z)$ converges to
$-1/2\d z \land $. Therefore, for $|\zeta |$
sufficiently small, all zeros of $det(\theta_{\xi })$ are in a neighborhood of $P$.
In order to determine the asymptotic of this convergence,
remember that in a holomorphic trivialisation of $E$ in some neighborhood of $p_{j}$ the Higgs field is equal to the
model (\ref{modelhiggs}) up to terms in $O(z-p_{j})$. As in the case $\xi \ra \xi_{l}$, the solutions are close
to those of the diagonal model $det(diag(\theta_{\zeta}(\tilde{q})))=0$ (see Claim \ref{nobranching}).
This equation is
$$
\Pi_{k=1}^{r} \left(\frac{\zeta \lambda_{k}^{j}}{\tilde{q}-p_{j}} - \frac{1}{2} \right) =0.
$$
The solutions $\tilde{q}_{k}^{j}(\zeta )$ are clearly given by
$$
\tilde{q}_{k}^{j}(\zeta ) = p_{j}+2 \zeta \lambda_{k}^{j} = p_{j}+2 \frac{ \lambda_{k}^{j}}{\xi }.
$$
Here the upper index of the solution stands for the point $p_{j}\in P$ it converges to, and
the lower index $k \in \{r_{j}+1,\ldots,r\}$ is determined by the extension of the cokernel of $\theta_{\zeta}$
at the point. An application of Rouch\'e's theorem gives again the claim.
Finally, $\Sigma$ is not ramified at $\xi=\infty$ because this would imply that at least two of the $q_{k}(\xi)$
admit the same Puiseux expansion, which is impossible because of (\ref{asymbehinf}) and (\ref{maini}) of
Hypothesis \ref{main}.
\end{proof}
Furthermore, by Claim \ref{merospec} the points of $\Sigma_{\xi }$ define a
multi-valued meromorphic function in the variable $\xi $ near infinity.
Let $q_{k}^{j}(\xi ) \in \Sigma_{\xi }$ be such a holomorphically varying zero of
$det(\theta_{\xi } )$, and suppose it converges to $p_{j} \in P$ as $\xi \ra \infty $.
We can let the index $k$ to vary from $r_{j}+1$ to $r$.
Consider the diagram
$$
\xymatrix@!=16pt{
& \hspace{3pt} \Sigma_{} \ar@{^{(}->}[d]^(.4){\j} & \\
& \CP \times \CPt \ar[dl]_(.55){\pi_{1}} \ar[dr]^{\pi_{2}} & \\
\CP & & \CPt }
$$
where $\j$ is inclusion and the two other arrows are canonical projections. In order to define a local
holomorphic section of the transformed bundle, we need to choose elements of $coKer(\theta_{\xi }(q_{k}^{j}(\xi )))$
for all $\xi $, such that they depend holomorphically with $\xi $. It is clear that this is equivalent
to choose a local holomorphic section $\psi$ of $\j^{\ast } \pi_{1}^{\ast } \F$ over the branch $(q_{k}^{j}(\xi ),\xi)$
near the point $(p_{j}, \infty )$ such that for all $\xi $, we have $\psi(q_{k}^{j}(\xi ), \xi) \in coKer(\theta_{\xi }(q_{k}^{j}))$.
Since any local section of $\F $ near $p_{j}$ multiplied by $(z-p_{j})$ is a local section of
the sheaf $\E \otimes \d z$, the section $(q_{k}^{j}(\xi )-p_{j}) \psi $ of $\j^{\ast } \pi_{1}^{\ast } \F$
near $(p_{j},\infty )$ is in fact a local holomorphic section of $\j^{\ast } \pi_{1}^{\ast } (\E \otimes \d z)$ on the branch
$( q_{k}^{j}(\xi ),\xi )$ of the spectral curve $\Sigma \subset \CP \times \CPt$. Furthermore, because of Claim \ref{convzeros},
$(q_{k}^{j}(\xi ), \xi ) \mapsto q_{k}^{j}(\xi )$ is a simple cover near $p_{j}$ without branching.
In particular, for all $q$ sufficiently close to $p_{j}$ there exists a unique $\xi (q)$
such that $q=q_{k}^{j}(\xi (q))$. Therefore, $(q_{k}^{j}(\xi )-p_{j}) \psi(q_{k}^{j}(\xi ), \xi)$ is the lift from $\CP$ of a section
$\varsigma_{k}^{j} (z) \d z$ of $\E \otimes \d z$ in a neighborhood of $p_{j} $, such that for all $q$ we have
\begin{equation} \label{varsigmacoker}
\varsigma_{k}^{j} (q) \d z \in coKer(\theta_{\xi (q)}(q)).
\end{equation}
In particular, $\varsigma_{k}^{j} (p_{j}) \d z \in coKer(\theta_{\infty } (p_{j}))=E_{sing} \otimes \d z$, as it can easily
be checked using formula (\ref{globdefhiggs}). Conversely, we may consider any section $\varsigma_{k}^{j}(z)$
satisfying (\ref{varsigmacoker}), lift $\varsigma_{k}^{j}(z) \d z$ to a section of $\j^{\ast } \pi_{1}^{\ast } (\E \otimes \d z)$,
and divide the result by $q-p_{j}$ to obtain $\psi$.
Fix now for all $k=\{ r_{j}+1,\ldots,r \}$ a section $\varsigma_{k}^{j} $ satisfying (\ref{varsigmacoker}).
All that we have said above motivates the definition:
\begin{equation} \label{vinf}
v_{k}^{j}(z, \xi )= \chi_{\varepsilon_{0} \xi^{-1}}(z-q_{k}^{j}(\xi )) \frac{ \varsigma_{k}^{j} (z) }{z-p_{j}} \otimes s_{0} (\xi ),
\end{equation}
where we recall again from the proof of Proposition \ref{hss} that $\chi_{\varepsilon_{0}\xi^{-1 }}$ is a
bump-function over the disk of radius $\varepsilon_{0}/|\xi |$. Remark that evaluation of
$v_{k}^{j}(z, \xi ) \d z$ in $z=q_{k}(\xi )$ is by definition in
the cokernel of $\theta_{\xi }$. Also, as in the case of logarithmic singularities, for all $\xi $ close to
infinity, let $t_{k}^{j}(z,\xi )$ be the unique section of $E $ satisfying the equation (\ref{tlog})
for all $z$, in other words such that $D_{\xi }''(v_{k}^{j}(z,\xi ) \d z, t_{k}^{j}(z,\xi ) \d \bar{z} )=0$.
A holomorphic trivialisation of $^{i}\hat{\E}$ at infinity is then given by the $D_{\xi }''$-harmonic representatives
$\hat{\sigma}_{k}^{\infty}(\xi )$ of the couples $(v_{k}^{j} (z,\xi ) \d z, t_{k}^{j}(z, \xi ) \d \bar{z} )$ for all
$k=\{ r_{j}+1,\ldots,r \}$ and all $j=\{1,\ldots,n\}$.
\section{Singularities of the transformed Higgs field} \label{sing}
In this part, we describe the eigenvalues of the singular parts of the transformed Higgs
field $\hat{\theta}^{H}$ at the singularities. This establishes points (\ref{iv}), (\ref{vi}) and (\ref{vii})
of Theorem \ref{mainthmhiggs}.
\subsection{The case of a logarithmic singularity}
Recall from (\ref{htrhiggs}) that the transformed Higgs field is defined as multiplication by the coordinate
$-z/2$ of a harmonic spinor, followed by projection onto harmonic forms.
\begin{lem} \label{treigen}
The set of eigenvalues of the transformed Higgs field $\hat{\theta }^{H}$ on the fiber $\hat{E}_{\xi }^{H}$
(with multiplicities) is equal to $-\Sigma_{\xi }/2$ (with multiplicities), where $\Sigma_{\xi }$ is the set of zeros
of $det(\theta_{\xi })$.
\end{lem}
\begin{proof}
Let a cohomology class in the space $ \hat{E}_{\xi }^{H}= H^{1}(\mathcal{D}_{\xi })$ (see \ref{dcomplex})
be represented by $1$-forms $(v(\xi ) \d z, t(\xi ) \d \bar{z}) \in (\Omega^{1,0} \oplus \Omega^{1,0}) \otimes E$.
Since this spinor is not harmonic, first of all we need a technical result:
\begin{clm} Let $(v(\xi )\d z, t(\xi )\d \bar{z}) \in (\Omega^{1,0} \oplus \Omega^{1,0}) \otimes E $ be annihilated by $D_{\xi }''$.
Then we have
$$
\hat{\pi}^{H}_{\xi }(z \hat{\pi}^{H}_{\xi }(v(\xi )\d z, t(\xi )\d \bar{z}) )
= \hat{\pi}^{H}_{\xi }(z (v(\xi )\d z, t(\xi )\d \bar{z}) ).
$$
In words, the action of the Higgs field can be computed on any representative section in
$Ker(D_{\xi }'')$.
\end{clm}
\begin{proof} This is straightforward: we need to show
$$
\hat{\pi}_{\xi }^{H}(z (\mbox{Id} -\hat{\pi}^{H}_{\xi })(v(\xi )\d z, t(\xi )\d \bar{z}) ) = 0,
$$
which is equivalent to
$$
z \Dir_{\xi }'' G_{\xi } (\Dir_{\xi }'')^{\ast } (v(\xi )\d z, t(\xi )\d \bar{z}) \bot \hat{E}_{\xi }^{H}.
$$
Now the only thing to remark is that if $(v(\xi )\d z, t(\xi )\d \bar{z}) \in Ker(D_{\xi }'')$, then this implies that
$$
(\Dir_{\xi }'')^{\ast } (v(\xi )\d z, t(\xi )\d \bar{z}) = (D_{\xi }'')^{\ast } (v(\xi )\d z,t(\xi )\d \bar{z}) \in \Omega^{0}\otimes E,
$$
and by diagonality of $G_{\xi }$ with respect to the decomposition $S^{+}\otimes E = (\Omega^{0}\otimes E) \oplus (\Omega^{2}\otimes E)$
(see Lemma \ref{diaglem}), also
$$
G_{\xi } (\Dir_{\xi }'')^{\ast } (v(\xi )\d z, t (\xi ) \d \bar{z})\in \Omega^{0} \otimes E.
$$
Therefore we have
$$
\Dir_{\xi }'' G_{\xi } (\Dir_{\xi }'')^{\ast } (v(\xi )\d z, t(\xi )\d \bar{z})
= D_{\xi }'' G_{\xi } (D_{\xi }'')^{\ast } (v(\xi )\d z,t(\xi )\d \bar{z}),
$$
and we conclude using the commutation relation
$$
[z, D_{\xi }'']=0
$$
combined with $Im(D_{\xi }'') \bot \hat{E}_{\xi }^{H}$.
\end{proof}
The proof of the lemma is now immediate: via the map (\ref{psiinv}),
\[
\Psi_{\xi }^{-1} (z ( v(\xi )\d z, t(\xi )\d \bar{z} )) = ( q \cdot eval_{q } v(\xi ) )_{ q \in \Sigma_{\xi } }
\]
multiplication by $z$ goes over to multiplication by $q$ in the point $q \in \Sigma_{\xi }$, and via (\ref{psi})
this is then re-transformed into multiplication by the constant $q$ on the component of $v(\xi )$
localized near $q$.
\end{proof}
\begin{thm} \label{eigenlog}
The eigenvalues of the transformed Higgs field $\hat{\theta }^{H}$ have first-order poles in the points of
$\hat{P}$. Furthermore, the non-vanishing eigenvalues of its residue in the puncture $\xi_{l}$ are
equal to $\{ -\lambda^{\infty }_{1+a_{l}}, \ldots , -\lambda^{\infty }_{a_{l+1}} \}$, where $\{ \lambda^{\infty }_{1+a_{l}}, \ldots, \lambda^{\infty }_{a_{l+1}} \}$
are the eigenvalues of the residue of the original Higgs field $\theta $ at infinity,
restricted to the eigenspace of $A$ corresponding to the eigenvalue $\xi_{l}$.
\end{thm}
\begin{proof}
As we have seen in (\ref{zerobeh}), the point $q_{k}(\xi ) \in \Sigma_{\xi }$ converges to infinity at
the first order with $2\lambda_{k}^{\infty }(\xi - \xi_{l})^{-1}$ as $\xi \ra \xi_{l }$, where $k \in \{ 1+a_{l},\ldots,a_{l+1} \}$
is an index such that the eigenvalue $\lambda_{k}^{\infty }$ of the residue term of $\theta $ at infinity appears in the
eigenspace of the second order term $A$ corresponding to the eigenvalue $\xi_{l }$. By Lemma \ref{treigen},
the transformed Higgs field has a logarithmic singularity at $\xi_{l}$, and
the corresponding residue is $-\lambda_{k}^{\infty }$.
\end{proof}
\subsection{The case of infinity}
We wish to show the following.
\begin{thm} \label{eigeninf}
The transformed Higgs field has a second order singularity at infinity. The set of eigenvalues of
its leading order term is $\{ -p_{1}/2, \ldots , -p_{n}/2 \}$, where $\{ p_{1}, \ldots , p_{n} \}=P$ is the
set of punctures of the original Higgs bundle. The multiplicity of the eigenvalue $-p_{j }/2$
is equal to $r-r_{j} = rk (Res(\theta , p_{j}))$. The set of eigenvalues of the residue of the transformed
Higgs field restricted to the eigenspace of the second-order term corresponding to the
eigenvalue $-p_{j }/2$ is $\{ -\lambda^{j}_{k} \}_{k \in \{ r_{j}+1, \ldots , r \}}$.
\end{thm}
\begin{proof}
In Claim \ref{convzeros} we have proved that as $\zeta \ra 0$, all zeros of $det(\theta_{\zeta } )$ must converge to
one of the points of $P$. Furthermore, the expansion of a spectral point $q_{k}$ converging to
$p_{j}$ is (\ref{asymbehinf}).
By Lemma \ref{treigen}, on the corresponding components $\hat{\theta }^{H}$ is just multiplication
by $- \Sigma_{\xi} \d \xi /2$. Hence, we see that the eigenvalues of the leading-order term of the transformed
Higgs field are equal to $\{ -p_{j}/2 \}_{ j=1,\ldots,n}$, while those of its first-order term are
$\{-\lambda_{k}^{j} \}_{j=1,\ldots,n;k=r_{j}+1,\ldots,r}$.
\end{proof}
\section{Parabolic weights} \label{parweightsect}
Here we compute the parabolic weights of the transformed Higgs bundle with respect to the induced
extension.
\subsection{The case of infinity}
\begin{thm} \label{parweightthminf}
The parabolic weight of the extension $^{i}\hat{\E}$ of the transformed Higgs bundle at infinity described in Subsection
\ref{infext}, restricted to the eigenspace of
$\hat{\theta }$ corresponding to the eigenvalue $-p_{j}/2$ of its second order term and the eigenvalue
$- \lambda^{j}_{k}$ of its residue is equal to $-1+\alpha^{j}_{k}$, where $\alpha^{j}_{k}$ is the parabolic weight on
the $\lambda^{j}_{k}$-eigenspace of the residue of the original Higgs bundle at $p_{j}$.
\end{thm}
\begin{proof}
We prove the statement in two steps. In the first one, we show that it is true supposing the
original Higgs bundle only has one logarithmic point of a precise form.
In the second one, we show how the case with only one
logarithmic point and the exponential decay results of Section \ref{harmspinsect} imply the general case.
\paragraph{Step 1.} Let us first suppose that the set of logarithmic singularities is reduced to
a single point $p_{1}$, that we may take to be $0$ without restricting generality. Furthermore,
we suppose that $E$ is a holomorphically trivial bundle over $\C$ and that in a global holomorphic
trivialisation $\{ \sigma_{k} \}$ the Higgs field is equal to
$$
\theta = diag\left( \frac{\lambda_{k}}{z } \right)_{k=1,\ldots,r} \d z
$$
and the metric is just
\begin{equation} \label{modeleucl}
h(\sigma_{k}, \sigma_{k}) = |z|^{2\alpha_{k}}.
\end{equation}
This defines a parabolic Higgs bundle with weights $\alpha_{k}$ at $0$ and $-\alpha_{k}$ at infinity, the
field having deformation
\begin{equation} \label{higgsfield}
\theta_{\xi } = diag\left( \frac{\lambda_{k}}{z } - \frac{\xi}{2} \right)_{k=1,\ldots,r} \d z
\end{equation}
and the $D''$-operator
\begin{equation}
D''_{\xi } = \dbar + diag\left( \lambda_{k} \frac{\d z}{z} - \frac{\xi}{2} \d z \right)_{k=1,\ldots,r} .
\end{equation}
Recall from Subsection \ref{infext} that a representative $(v_{\xi } \d z, t_{\xi } \d \bar{z})$
of any spinor $ \psi_{\xi }$ is supported in the finite collection of disks
$ \cup_{q(\xi) \in \Sigma_{\xi } } \Delta ( q(\xi) , \varepsilon_{0}|\xi |^{-1})$. By Claim \ref{convzeros}, the points $q(\xi )$ are
given by
\begin{equation} \label{vanish}
q_{k} (\xi ) = \frac{2\lambda_{k}}{\xi }.
\end{equation}
Define a family of homotheties indexed by $\xi \in \hat{\C } \setminus \hat{P}$
\begin{align} \label{homoth}
h_{\xi }: \C & \lra \C \notag \\
w & \mapsto z = \frac{w}{\xi };
\end{align}
in such a way that
\begin{align}
h_{\xi }^{-1}(0) & = 0 \notag \\
h_{\xi }^{-1}( q_{k}(\xi ) ) & = 2\lambda_{k} \hspace{1cm} \mbox{for} \hspace{5mm}
k=r_{1},\ldots,r. \label{confzero}
\end{align}
Therefore, this corresponds to a family of coordinate changes $z \leftrightarrow w$ in the plane, such that the position of
the zeros of the Higgs field $\theta_{\xi }$ after applying $h_{\xi }^{-1}$ is constant (the $2\lambda_{k}$ for $k=r_{1},\ldots,r$),
as well as that of the poles ($0$ and $\infty $). Moreover, $\d z = \xi^{-1} \d w$ implies
\begin{align} \label{confhiggs}
h_{\xi }^{\ast } \theta_{\xi } = diag\left[ \lambda_{k} \frac{\d w}{w} -
\frac{1}{2} \d w \right]_{k=1,\ldots,r} ,
\end{align}
and so
\begin{align} \label{confdsecond}
h_{\xi }^{\ast }D''_{\xi } & = \dbar + diag\left[ \lambda_{k} \frac{\d w}{w} -
\frac{1}{2} \d w \right]_{k=1,\ldots,r} ,
\end{align}
where $\dbar $ stands this time for the Dolbeault operator with respect to the $w$-coordinate.
The crucial observation is that this operator is independent of $\xi$.
On the other hand, remark that the Euclidean metric on the base space and the fiber metric (\ref{modeleucl})
behave under these coordinate changes as
\begin{align}
(h_{\xi })_{\ast } |\d w |^{2} & = |\xi |^{2} |\d z |^{2} \label{confeucl} \\
|\sigma_{k}(z)|^{2} & = |\xi |^{-2\alpha_{k} } |w|^{2\alpha_{k}} . \label{confherm}
\end{align}
In other words, if we denote by $h^{(w)}$ the model hermitian metric on $h_{\xi }^{\ast } E$
equal in the basis $h_{\xi }^{\ast } \sigma_{k}$ to
$$
h^{(w)} = diag(|w|^{2\alpha_{k}}),
$$
then the homotheties $h_{\xi}$ induce a family of tautological isomorphisms of Hermitian fiber bundles
\begin{align} \label{confisom}
( h_{\xi }^{\ast } E, h^{(w)}) & \lra (E, h) \\
(h_{\xi }^{\ast } \sigma_{k})(w ) & \mapsto |\xi |^{\alpha_{k} } \sigma_{k}(z) \notag .
\end{align}
We deduce from (\ref{confeucl}) that in the basis $h_{\xi }^{\ast } \sigma_{k}$ the pull-back $h_{\xi }^{\ast } \Delta_{\xi }$ of the
Laplacian of the Dirac operator $\Dir_{\xi }''$ has the form
\begin{equation} \label{conflapl}
|\xi |^{2} \left[\Delta + diag \absl \frac{\lambda_{k} }{w}-\frac{1}{2} \absr^{2}_{k=1,\ldots,r} \right] ,
\end{equation}
where $\Delta$ stands for the usual Laplace operator on functions with respect to the
metric $|\d w |^{2}$. The operator $\Delta^{(w)}$ between brackets in this formula
is a bounded operator from the weighted Sobolev space $H^{2}(S^{+} \otimes E, |\d w|^{2})$ to $L^{2}(S^{+} \otimes E, |\d w|^{2})$.
The weight at $0$ is determined by the condition that for a section $u \in H^{2}$ we have
$u/|w|^{2} \in L^{2}$, and this gives therefore exactly the maximal domain of $\Delta^{(w)}$
(see Theorem \ref{Hilbertisom}). We infer that the pull-back $h_{\xi }^{\ast } G_{\xi }$ of the Green's operator
of $\Delta_{\xi }$ is
\begin{equation} \label{confgreen}
|\xi |^{-2} G^{(w)},
\end{equation}
where $G^{(w)}$ is the inverse of $\Delta^{(w)}$. It also follows from Theorem \ref{Hilbertisom} that $G^{(w)}$
is a bounded linear operator from $L^{2}(S^{+} \otimes E, |\d w|^{2})$ to $H^{2}(S^{+} \otimes E, |\d w|^{2})$.
Because $\Delta^{(w)}$ is diagonal in the basis $\sigma_{k}$, the same is true for $G^{(w)}$.
Remark that the pull-backs $h_{\xi }^{\ast }\hat{\pi}_{\xi}$ of the orthogonal projections onto $\Delta_{\xi}$-harmonic spinors
are all equal to the orthogonal projection $\hat{\pi}^{(w)}$ onto $\Delta^{(w)}$-harmonic spinors: indeed, the
conformal factor $|\xi |^{2}$ in (\ref{conflapl}) changes neither the space of harmonic spinors nor the
orthogonal projection operator onto them. In particular, since $\Delta^{(w)},G^{(w)}$ and $h$ are diagonal in the
basis $\sigma_{k}$, the same thing is true for all $\hat{\pi}_{\xi}$.
Now notice that by the definition of the $\dbar^{\hat{\E}}$-holomorphic extension to infinity of the
transformed bundle given in (\ref{vinf}) and via the identification (\ref{confisom}), the sections
$|\xi|^{\alpha_{k}} h_{\xi}^{\ast } (v_{k}(z,\xi) \d z)$ (modulo the value of the section $s_{0}$ of $\mathcal{O}_{\CPt }(1)$)
coincide: indeed,
$$
|\xi|^{\alpha_{k}} \chi_{\varepsilon_{0}/\xi}(z-q_{k}(\xi )) \sigma_{k}(z) \frac{\d z}{z} =
\chi_{\varepsilon_{0}} \left(w - 2\lambda_{k} \right) (h_{\xi }^{\ast }\sigma_{k})(w) \frac{\d w}{w}.
$$
It then follows from formula (\ref{confdsecond}) together with the definition
(\ref{tlog}) that the coefficient of $s_{0}$ in $|\xi|^{\alpha_{k}} h_{\xi}^{\ast } t_{k}(z,\xi) \d \bar{z}$
is also independent of $\xi$. From the fact that the projections $\hat{\pi}_{\xi}$ are also constant, we deduce that
the coefficient of $s_{0}$ in the pull-back
\begin{equation}\label{confspinor}
(h_{\xi}^{\ast }\hat{\sigma}_{k})(w,\xi) = |\xi|^{\alpha_{k}} \hat{\sigma}^{\infty}_{k}(z,\xi)
\end{equation}
of the spinors $|\xi|^{\alpha_{k}} \hat{\sigma}^{\infty}_{k}(z,\xi)$ representing $ |\xi|^{\alpha_{k}}(v_{k}(z,\xi) \d z, t_{k}(z,\xi) \d \bar{z})$
does not depend on $\xi$.
Therefore, denoting by $f_{k}(z,\xi)$ the coefficient of $s_{0}$ in $\hat{\sigma}^{\infty}_{k}(z,\xi)$ and by
$(h_{\xi}^{\ast }f_{k})(w,\xi)$ the coefficient of $s_{0}$ in $(h_{\xi}^{\ast }\hat{\sigma}^{\infty}_{k})(w,\xi)$, we see
by invariance of the $L^{2}$-norm of $1$-forms by conformal coordinate change that
$$
\int_{\C} \vert f_{k}(z,\xi) \vert_{h,|d z|^{2}}^{2} |\d z|^{2} =
|\xi|^{-2\alpha_{k}}\int_{\C}\vert (h_{\xi}^{\ast }f_{k})(w,\xi) \vert_{h^{(w)},|d w|^{2}}^{2} |\d w|^{2},
$$
for all $\xi$, with the integral on the right-hand side a constant independent of $\xi$.
On the other hand, recall from (\ref{o1zero}) that on the affine chart $\hat{U}_{0}$ of $\CP$ we have
$s_{0}(\xi )= \xi $. Observe also that the transformed Hermitian metric $\hat{h}$ is defined in the chart
$\hat{U}_{0}$, and that for any harmonic spinor $f$ we have
$$
\hat{h}(\xi f,\xi f)=|\xi|^{2} \hat{h}(f,f) = |\zeta|^{-2} \hat{h}(f,f)
$$
with $\zeta=\xi^{-1}$ the local coordinate centered at $0$ of the singularity at infinity.
This means that the effect on the parabolic weights of multiplying by $s_{0}$ is adding $-1$.
On the other hand, the $-\lambda_{k}$-eigenspace of the residue of the transformed Higgs bundle at infinity is spanned by
$\hat{\sigma}^{\infty}_{k}$. From all that has been said above, we deduce
\begin{equation}\label{asymbehharmspin}
\hat{h}(\hat{\sigma}^{\infty}_{k},\hat{\sigma}^{\infty}_{k})=M|\zeta|^{-2+2\alpha_{k}},
\end{equation}
where $M$ is independent of $\xi$; in different terms, that the parabolic weight of the transformed
Higgs bundle at infinity on the $-\lambda_{k}$-eigenspace of the residue is equal to $-1+\alpha_{k}$.
\paragraph{Step 2.}
Starting from now, we drop the assumption that the set of logarithmic singularities is reduced to a point.
In this part, we patch together solutions to local problems provided by Step 1, and use the results of
Section \ref{harmspinsect} to estimate the defect of these patched sections to be solutions of the global
problem. We find that the interaction between solutions to local problems near different punctures is
small as $|\xi|$ gets large.
Let $( \dbar^{\E}, \theta )$ be a Higgs bundle with some logarithmic singularities $P= \{ p_{1}, \ldots,p_{n} \}$.
In a holomorphic trivialisation $\{ \sigma^{j}_{k} \}_{k=1}^{r} $ near each one of these points, up to terms
in $O(1) \d z$, the Higgs field has the form
\begin{equation}\label{dpjagain}
\theta^{j} = \frac{A^{j}}{z-p_{j}} \d z ,
\end{equation}
where the ${A}^{j}$ are some diagonal matrices as in (\ref{dpj}).
The deformation of these local models is
$$
\theta^{j}_{\xi } = \left[\frac{A^{j}}{z-p_{j}} - \frac{\xi}{2}\right] \d z,
$$
and similarly the deformation of the local $D''$-operators $(D'')^{j}$ is
\begin{align*}
(D''_{\xi })^{j} & = \dbar^{\E } + \theta_{\xi } \\
& = \dbar^{\E } + \left[ \frac{A^{j}}{z-p_{j}} - \frac{\xi}{2} \right] \d z,
\end{align*}
Finally, that of the Dirac operator $\Dir^{j}= (D'')^{j} - ((D'')^{j})^{\ast }$ is
$$
\Dir^{j}_{\xi } = (D''_{\xi })^{j} - ((D''_{\xi })^{j})^{\ast },
$$
adjoint being taken relative to the harmonic metric corresponding to $(D'')^{j}$.
Now for all $j$ we can consider the extension of $\theta^{j}$ to a trivial bundle $E^{j}$ over the whole plane
by keeping the same formula (\ref{dpjagain}) for it, endowed with the model metric
$$
h^{j}=diag(|z-p_{j}|^{2\alpha^{j}_{k}})_{k=1}^{r}.
$$
It is clear that this extension only has one regular
singularity (in $p_{j}$) and an irregular one at infinity, so all the results of Step 1 hold for them.
In particular, for representatives
$$
(v_{k}^{j}(z,\xi) \d z, t_{k}^{j}(z,\xi) \d \bar{z})
$$
as described in Subsection \ref{infext} we have a harmonic representative
$$
\hat{\sigma}^{\infty}_{k}(z,\xi) \in Ker(\Dir^{j}_{\xi })^{\ast } \subset H^{1}(\C, S^{-} \otimes E^{j})
$$
with
$$
\int_{\C} \absl \hat{\sigma}^{\infty}_{k}(z,\xi) \absr_{h^{j},|dz|^{2}}^{2} |\d z|^{2} = |\xi|^{2-2\alpha^{j}_{k}}.
$$
This growth is measured with respect to the diagonal model metric $h^{j}$;
however, since the spinor $\hat{\sigma}^{\infty}_{k}$ is exponentially concentrated near $p_{j}$ and here $h^{j}$
is mutually bounded with the harmonic metric $h$ of $(\E,\theta)$, this implies
\begin{equation}\label{diagharm}
c|\xi|^{2-2\alpha^{j}_{k}} \leq \int_{\C} \absl \hat{\sigma}^{\infty}_{k}(z,\xi) \absr_{h,|dz|^{2}}^{2} |\d z|^{2}\leq C|\xi|^{2-2\alpha^{j}_{k}}
\end{equation}
for some $0<c<C$. Let $\chi^{j} $ be a cut-off function supported
in a disk $\Delta(p_{j}, 3 \varepsilon_{0})$, equal to $1$ on $\Delta(p_{j}, 2\varepsilon_{0})$, such that
$| \nabla \chi^{j} | \leq K$. Then for $\varepsilon_{0}>0$ fixed sufficiently small, the global section of $S^{-} \otimes E$ defined by
$$
\hat{\sigma}(z,\xi) = \chi^{j}(z) \hat{\sigma}^{\infty}_{k}(z,\xi)
$$
has a meaning, for the holomorphic trivialisation $\{ \sigma^{j}_{k} \} $ is defined in
$\Delta(p_{j}, 3\varepsilon_{0})$ provided $\varepsilon_{0}$ is sufficiently small.
Now notice that if $q(\xi ) \ra p_{j}$ as $\xi \ra \infty$ and more precisely
$$
q(\xi ) = p_{j} + \frac{2\lambda^{j}_{k}}{\xi } + O(|\xi |^{-2}),
$$
in other words on the component of the transformed bundle with eigenvalue of the second-order part of
$\hat{\theta}$ at infinity equal to $-p_{j}/2$ and eigenvalue of the residue of $\hat{\theta}$ at infinity equal to $-\lambda^{j}_{k}$,
the holomorphic extension $\varsigma^{j}_{k}$ of the cokernel has as parabolic weight the $\alpha^{j}_{k}$ corresponding to
the eigenspace of the eigenvalue $\lambda^{j}_{k}$ of the residue of $\theta$.
Recall that the harmonic metric on the transformed side is just $L^{2}$-metric
of the $\Delta_{\xi}$-harmonic representative with respect to the harmonic metric $h$ of the original Higgs bundle.
The statement of the theorem will therefore follow once we prove that the harmonic representative of
$\hat{\sigma}(z,\xi)$ satisfies the inequality
\begin{equation} \label{asymbehharmspin2}
c|\xi|^{2-2\alpha^{j}_{k}} \leq \int_{\C} \absl \hat{\pi}_{\xi}\hat{\sigma}(z,\xi) \absr_{h,|dz|^{2}}^{2} |\d z|^{2}\leq C|\xi|^{2-2\alpha^{j}_{k}}.
\end{equation}
for some $0<c<C$. Our first aim is to prove the following.
\begin{lem} \label{lemstep4}
There exists $\delta>0$ and $K>0$ such that for $|\xi|$ sufficiently large the inequality
$$
\left\Vert \Dir_{\xi }^{\ast } \hat{\sigma}(\xi) \right\Vert_{L^{2}(\C )}^{2}
\leq K | \xi |^{2-2 \delta} \left\Vert \hat{\sigma}(\xi) \right\Vert_{L^{2}(\C )}^{2}
$$
holds.
\end{lem}
\begin{proof}
Covering the annulus centered at $p_{j}$ of radii $2\varepsilon_{0}$ and $2R_{0}$ by a finite number of disks
of radius $\varepsilon_{0}$, we deduce from Lemmas \ref{expdecord} and \ref{expdecinf} that the
$\Dir^{j}_{\xi }$-harmonic spinor $\hat{\sigma}^{\infty}_{k}(z,\xi)$ is concentrated in $H^{1}$-norm, up to a factor
decreasing exponentially with $|\xi |$, in the disk $\Delta(p_{j},2\varepsilon_{0})$.
In particular, it is concentrated up to an exponentially decreasing factor in the same disk in $L^{2}$-norm as well.
Denoting by $\cdot $ Clifford multiplication, we have the estimation
\begin{align*}
\int_{\C } \absl \Dir_{\xi }^{\ast } (\chi^{j}(z) \hat{\sigma}^{\infty}_{k}(z,\xi)) \absr^{2} | \d z |^{2} \leq &
\int_{\C } \absl \chi^{j}(z) \Dir_{\xi }^{\ast } \hat{\sigma}^{\infty}_{k}(z,\xi) \absr^{2} | \d z |^{2} \\
& + \int_{\C } \absl ( \nabla \chi^{j} )(z) \cdot \hat{\sigma}^{\infty}_{k}(z,\xi) \absr^{2} | \d z |^{2} \\
\leq & \int_{\Delta(p_{j}, 3 \varepsilon_{0}) } \absl \Dir_{\xi }^{\ast } \hat{\sigma}^{\infty}_{k}(z,\xi) \absr^{2} | \d z |^{2} \\
& + K \int_{\Delta(p_{j}, 3 \varepsilon_{0}) \setminus \Delta(p_{j}, 2 \varepsilon_{0})} \absl \hat{\sigma}^{\infty}_{k}(z,\xi) \absr^{2} |\d z|^{2}.
\end{align*}
Again, by Lemma \ref{expdecord} the second integral on the right-hand side is bounded by an exponentially decreasing
multiple of $\| \hat{\sigma}^{\infty}_{k}(z,\xi) \|^{2}_{L^{2}(\C)} $ as $|\xi| \ra \infty$. Therefore, we only need to treat
$$
\left\Vert \Dir_{\xi }^{\ast } \hat{\sigma}^{\infty}_{k}(z,\xi) \right\Vert_{L^{2}( \Delta(p_{j}, 3 \varepsilon_{0}) )}^{2}.
$$
Remark that by hypothesis,
$$
(\Dir^{j}_{\xi })^{\ast } \hat{\sigma}^{\infty}_{k}(z,\xi) = 0,
$$
so we have
$$
\Dir_{\xi }^{\ast } \hat{\sigma}^{\infty}_{k}(z,\xi) = \left[ \Dir_{\xi }^{\ast } - (\Dir^{j}_{\xi })^{\ast } \right] \hat{\sigma}^{\infty}_{k}(z,\xi).
$$
This is then bounded by
$$
\hat{\sigma}^{\infty}_{k}(z,\xi) O(|z-p_{j}|^{-1+\delta }),
$$
where $O(|z-p_{j}|^{-1+\delta })$ stands for a term bounded from above by a constant (independent of $\xi$)
times $|z-p_{j}|^{-1+\delta }$, because $\Dir_{\xi}^{\ast }$ and $(\Dir^{j}_{\xi })^{\ast }$ are Dirac operators having
the same local model at the puncture and their difference is clearly independent of $\xi$.
In order to study this quantity, we make use of the coordinate $w= \xi(z -p_{j}) $ analogously to that
introduced in (\ref{homoth}). Under this coordinate change, the disk $\Delta(p_{j},3 \varepsilon_{0})$ goes
into the (varying) disk $\Delta(0, 3 \varepsilon_{0}|\xi|)$. Hence, we need to prove
\begin{align*}
\int_{\Delta(0 , 3 \varepsilon_{0} |\xi |)} & |w |^{-2+2\delta } |\xi|^{2-2\delta }
\absl (h_{\xi}^{\ast}\hat{\sigma}^{\infty}_{k})(w,\xi)\absr^{2}_{|d z |^{2},h} |\xi |^{-2} |\d w|^{2} \\
& \leq K |\xi |^{2-2\delta} \int_{\C } \absl (h_{\xi}^{\ast}\hat{\sigma}^{\infty}_{k})(w,\xi)\absr^{2}_{|d z |^{2},h} |\xi|^{-2} |\d w|^{2}
\end{align*}
Recall from (\ref{confspinor}) that in the coordinate $w$ the spinors
$|\xi|^{-\alpha^{j}_{k}}h_{\xi}^{\ast}\hat{\sigma}^{\infty}_{k}$ are independent of $\xi$.
Therefore this boils down to
\begin{align} \label{perturbest}
\int_{\Delta(0 , 3 \varepsilon_{0} |\xi |)} & |w |^{-2+2\delta }
\absl (h_{\xi}^{\ast}\hat{\sigma}^{\infty}_{k})(w)\absr^{2}_{|d z|^{2}} |\d w |^{2}\notag \\
& \leq K \int_{\C } \absl (h_{\xi}^{\ast}\hat{\sigma}^{\infty}_{k})(w)\absr^{2}_{|d z |^{2}} |\d w |^{2}
\end{align}
for a suitable constant $K>0$.
Because
$$
(h_{\xi}^{\ast}\hat{\sigma}^{\infty}_{k})(w) \in H^{1}(\C),
$$
in particular we have
$$
(h_{\xi}^{\ast}\hat{\sigma}^{\infty}_{k})(w) \in L^{2}(\C),
$$
and also
$$
\frac{1}{w} (h_{\xi}^{\ast}\hat{\sigma}^{\infty}_{k})(w) \in L^{2}_{loc}.
$$
near the origin. This implies $|w|^{-1+\delta}(h_{\xi}^{\ast}\hat{\sigma}^{\infty}_{k})(w) \in L^{2}(\C)$. Therefore,
$$
K = 2\frac{\left\Vert |w|^{-1+\delta}(h_{\xi}^{\ast}\hat{\sigma}^{\infty}_{k})(w) \right\Vert_{L^{2}(\C)}^{2}}
{\left\Vert (h_{\xi}^{\ast}\hat{\sigma}^{\infty}_{k})(w) \right\Vert_{L^{2}(\C)}^{2}}
$$
has the desired property.
\end{proof}
The lemma has the following consequence.
\begin{lem} \label{asymptnormharm}
As $| \xi | \ra \infty $, we have the estimate
$$
\left\vert \left\Vert \hat{\sigma}(\xi) \right\Vert_{L^{2}}^{2} -
\left\Vert \hat{\pi}^{H}_{\xi} \hat{\sigma}(\xi) \right\Vert_{L^{2}}^{2} \right\vert \leq
K |\xi|^{-2\delta} \left\Vert \hat{\sigma}(\xi) \right\Vert_{L^{2}}^{2}
$$
with $K>0$ independent of $\xi$.
\end{lem}
\begin{proof}
It is sufficient to bound
$$
\left\Vert \hat{\sigma}(\xi) -\hat{\pi}^{H}_{\xi} \hat{\sigma}(\xi) \right\Vert_{L^{2}}^{2}
$$
as in the lemma. The $\Dir_{\xi }^{\ast }$-harmonic representative $\hat{\pi}^{H}_{\xi} \hat{\sigma}(\xi)$ of
$\hat{\sigma}(\xi)$ is given by the formula
$$
(\Id - \Dir_{\xi } G_{\xi }\Dir_{\xi }^{\ast }) \hat{\sigma}(\xi) ,
$$
so the difference with $\hat{\sigma}(\xi)$ itself is
$$
\Dir_{\xi } G_{\xi }\Dir_{\xi }^{\ast } \hat{\sigma}(\xi).
$$
Since for any positive spinor $\varphi $ the estimation
$$
\left\Vert \Dir_{\xi } \varphi \right\Vert_{L^{2}(\C )}^{2} \leq
K \left\Vert \varphi \right\Vert_{H^{1}(\C )}^{2} + K |\xi |^{2} \left\Vert \varphi \right\Vert_{L^{2}(\C )}^{2}
$$
holds, we deduce that
\begin{equation*}
\left\Vert \Dir_{\xi } G_{\xi }\Dir_{\xi }^{\ast } \hat{\sigma}(\xi) \right\Vert_{L^{2}(\C )}^{2} \leq
K \left\Vert G_{\xi }\Dir_{\xi }^{\ast } \hat{\sigma}(\xi) \right\Vert_{H^{1}(\C )}^{2} +
K |\xi |^{2} \left\Vert G_{\xi }\Dir_{\xi }^{\ast } \hat{\sigma}(\xi) \right\Vert_{L^{2}(\C )}^{2} .
\end{equation*}
Lemma \ref{estgreen} implies that both terms on the right-hand side are bounded from above by
\begin{equation*}
K |\xi |^{-2} \left\Vert \Dir_{\xi }^{\ast } \hat{\sigma}(\xi) \right\Vert_{L^{2}(\C )}^{2}.
\end{equation*}
We conclude by Lemma \ref{lemstep4}.
\end{proof}
We can now finish the proof of Theorem \ref{parweightthminf}: as $|\xi|$ goes to infinity,
by Lemma \ref{asymptnormharm}, we have
$$
\frac{\left\Vert \hat{\pi}^{H}_{\xi} \hat{\sigma}(\xi) \right\Vert_{L^{2}}^{2}}
{\left\Vert \hat{\sigma}(\xi) \right\Vert_{L^{2}}^{2}} \lra 1.
$$
In words, the norm of the harmonic representative of the spinor $\hat{\sigma}(z,\xi)$
is asymptotically equal to the norm of $\hat{\sigma}(z,\xi)$ itself. On the other hand, as it has already been
remarked in the proof of Lemma \ref{lemstep4}, we have
$$
\frac{\Vert \hat{\sigma}(z,\xi) \Vert_{L^{2}(\C,h)}^{2}}{\Vert \hat{\sigma}^{\infty}_{k}(z,\xi) \Vert_{L^{2}(\C,h)}^{2}}
\lra 1
$$
exponentially as $\xi \ra \infty$. Finally, by
(\ref{diagharm}) the $L^{2}$-norm of the spinors $\hat{\sigma}^{\infty}_{k}(z,\xi)$ as measured by the harmonic
metric $h$ satisfy
\begin{equation}
c|\xi |^{2-2\alpha^{j}_{k}} \leq \Vert \hat{\sigma}^{\infty}_{k}(z,\xi) \Vert_{L^{2}}^{2} \leq C |\xi |^{2-2\alpha^{j}_{k}}
\end{equation}
for some $0<c<C$, where $\alpha^{j}_{k}$ is a parabolic weight of the original Higgs bundle at the point $p_{j}$.
All this then implies (\ref{asymbehharmspin2}), so it follows that the parabolic weight of the transformed Higgs bundle
on the given component is equal to $\alpha^{j}_{k}-1$, as it was stated in the theorem.
\end{proof}
\subsection{The case of logarithmic singularities}
Next we compute the parabolic weights at a puncture $\xi_{l}$
corresponding to the extension of the holomorphic structure of $\hat{\E}$ given in Subsection \ref{logext}.
Explicitly, here is the result we wish to show.
\begin{thm} \label{parweightthmlog}
The parabolic weight of the extension $^{i}\hat{\E}$ of the transformed Higgs bundle at the puncture $\xi_{l}$,
restricted to the $-\lambda_{k}^{\infty }$-eigenspace of the residue of the transformed Higgs field (here
$k \in \{ 1+a_{l},\ldots,a_{l+1 } \}$) is equal to $-1+\alpha^{\infty }_{k}$, where $\alpha^{\infty }_{k}$ is the parabolic weight of
the original Higgs field at infinity, restricted to the $\xi_{l}$-eigenspace of the second-order term and the
$\lambda_{k}^{\infty }$-eigenspace of the first-order term of the polar part of the Higgs field.
\end{thm}
\begin{proof}
We follow the proof of Theorem \ref{parweightthminf}.
Again, we divide the proof into two steps according to the number of distinct eigenvalues $\xi_{l}$ of
the second order term of $D$ at infinity. Recall that some of the spectral points $q_{k}\in \Sigma_{\xi}$
converge to infinity as $\xi\ra \xi_{l}$, whereas others remain bounded.
\paragraph{Step 1.} First we suppose that $n'=1$, that is to say $A$ is a simple diagonal matrix,
and that in a global holomorphic basis $\{\sigma^{\infty}_{k}\}$ the Higgs field has is of the form
\[
\theta = \frac{\xi_{1}}{2} \d z + diag(\lambda^{\infty}_{k})\frac{\d z}{z}
\]
with one regular singularity in $0$ and an irregular one at infinity, and finally the harmonic metric is
\begin{equation}\label{modelherminf}
h^{\infty}=diag(|z|^{-2\alpha^{\infty}_{k}})_{k=1}^{r}.
\end{equation}
This induces a parabolic structure on $\E$ with weights $-2\alpha^{\infty}_{k}$ at $0$ and $2\alpha^{\infty}_{k}$ at infinity.
The deformed field is
\[
\theta_{\xi} = \frac{\xi_{1}-\xi}{2} \d z + diag(\lambda^{\infty}_{k})\frac{\d z}{z},
\]
and the spectral points are
\[
\frac{2\lambda_{k}}{\xi - \xi_{1}}.
\]
Making the coordinate change
\begin{align}\label{homothlog}
h_{\xi }: \C & \lra \C \notag \\
w & \mapsto z = \frac{w}{\xi-\xi_{1}}
\end{align}
the field writes
\begin{equation}\label{confhiggslog}
\theta_{\xi} = -\frac{1}{2} \d w + diag(\lambda^{\infty}_{k})\frac{\d w}{w}.
\end{equation}
The Euclidean metric $|\d z|^{2}$ on the base and the fiber metric $h^{\infty}$ are transformed into
\begin{align}
|\xi-\xi_{1}|^{-2}&|\d w|^{2} \label{confeucllog} \\
diag(|\xi - \xi_{1}|^{2\alpha^{\infty}_{k}}&|w|^{-2\alpha^{\infty}_{k}})_{k=1}^{r}\label{confhermlog}
\end{align}
and the position of the spectral points become simply
\[
2\lambda_{k},
\]
independent of $\xi$. As in the case of the singularity at infinity, writing $h^{(w)}$ for the diagonal
model metric
$$
diag(|w|^{-2\alpha^{\infty}_{k}})_{k=1}^{r}
$$
the coordinate changes induce tautological isomorphisms of Hermitian fiber bundles
\begin{align} \label{confisomlog}
( h_{\xi }^{\ast } E, h^{(w)}) & \lra (E, h^{\infty}) \\
(h_{\xi }^{\ast } \sigma_{k})(w ) & \mapsto |\xi - \xi_{1}|^{-\alpha_{k}} \sigma_{k}(z) \notag .
\end{align}
Via this isomorphism the representatives $v_{k}(z,\xi)$ given in (\ref{vlog}) behave as follows:
\[
|\xi - \xi_{1}|^{-\alpha_{k}}v_{k}(z,\xi)=v_{k}(w),
\]
which is independent of $\xi$, or equivalently
\[
|\xi - \xi_{1}|^{-\alpha_{k}}v_{k}(z,\xi)(\xi - \xi_{1})\d z=v_{k}(w)\d w,
\]
independent of $\xi$. By the equation (\ref{tlog}), this implies
\[
|\xi - \xi_{1}|^{-\alpha_{k}}t_{k}(z,\xi)(\bar{\xi} - \bar{\xi}_{1})\d \bar{z}=t_{k}(w)\d \bar{w},
\]
independently of $\xi$. Exactly as in the case of the singularity at infinity, the Laplacian and the Green's
operator of $\Dir_{\xi}^{\ast}$ in the coordinate $w$ only depend on $\xi$ through a conformal factor
$|\xi-\xi_{1}|^{-2}$ and $|\xi-\xi_{1}|^{2}$ respectively, so the pull-back $h_{\xi }^{\ast }\hat{\pi}_{\xi}$ of the projection
onto $\Dir_{\xi}^{\ast}$-harmonic spinors is independent of $\xi$. We deduce using invariance of the $L^{2}$-norm of
$1$-forms by conformal coordinate change that for the $\Dir_{\xi}^{\ast}$-harmonic spinor $\hat{\sigma}_{k}(z,\xi)$
representing the cohomology class of $(v_{k}(z,\xi)\d z,t_{k}(z,\xi)\d \bar{z})$ we have
\[
\int_{\C}\left\vert \hat{\sigma}_{k}(z,\xi) \right\vert_{h^{\infty},|dz|^{2}}^{2} |\d z|^{2} =
|\xi - \xi_{1}|^{2\alpha_{k}-2} \int_{\C}\left\vert \hat{\sigma}_{k}(w) \right\vert_{h^{(w)},|dw|^{2}}^{2} |\d w|^{2},
\]
where $\hat{\sigma}_{k}(w)$ is the harmonic spinor representing $(v_{k}(w)\d w,t_{k}(w)\d \bar{w})$. We see
also that the integral on the right-hand side is independent of $\xi$, hence we have the desired behavior
giving parabolic weight $-1+\alpha_{k}$ on this component.
\paragraph{Step 2.}
We drop the assumption that the second-order term $A$ of the original Higgs field is a simple matrix.
Let $\chi$ be a fixed cut-off function supported on the complementary $\C \setminus \Delta(0,1/ \varepsilon_{0})$ of a
large disk, equal to $1$ on $\C \setminus \Delta(0,2/ \varepsilon_{0})$. In $\C \setminus \Delta(0,1/ \varepsilon_{0})$,
the Higgs field is up to a perturbation
\[
\theta^{\infty}=\frac{1}{2}A\d z + C \frac{\d z}{z}
\]
with $A$ and $C$ diagonal matrices as in (\ref{modelhiggsinf}), therefore decomposes into a direct sum of
problems studied in Step 1.
In particular, for each such model problem with eigenvalue of the second-order term $\xi_{l}$ we have
harmonic spinors $\hat{\sigma}^{l}_{k}(z,\xi)$ where $k \in \{1+a_{l},\ldots,a_{l+1} \}$, such that
$$
\int_{\C}\left\vert\hat{\sigma}^{l}_{k}(z,\xi)\right\vert_{|dz|^{2},h^{\infty}} |\d z|^{2} = |\xi-\xi_{l}|^{-2+2\alpha^{\infty}_{k}}.
$$
Again, since the harmonic metric $h$ of the Higgs bundle $(\E,\theta)$ is mutually bounded
with $h^{\infty}$ in a neighborhood of infinity and $\hat{\sigma}^{l}_{k}$ is supported there, this implies
\begin{equation}\label{cKineqlog}
c|\xi-\xi_{l}|^{-2+2\alpha^{\infty}_{k}} \leq \int_{\C}\left\vert\hat{\sigma}^{l}_{k}(z,\xi)\right\vert_{|dz|^{2},h} |\d z|^{2}
\leq C |\xi-\xi_{l}|^{-2+2\alpha^{\infty}_{k}}
\end{equation}
for some $0<c<C$. The section
\[
\hat{\sigma}(z,\xi)= \chi(z) \hat{\sigma}^{l}_{k}(z,\xi)
\]
is well-defined because the local holomorphic trivialisation $\sigma^{\infty}_{k}$ of $\E$ is defined in
$\C \setminus \Delta(0,1/ \varepsilon_{0})$ for $\varepsilon_{0}>0$ sufficiently small.
The statement of the theorem will again follow if we prove
\begin{equation}\label{asymbehharmlog}
c|\xi-\xi_{l}|^{-2+2\alpha^{\infty}_{k}} \leq \int_{\C}\left\vert \hat{\pi}_{\xi^{H}}\hat{\sigma}(z,\xi)\right\vert_{|dz|^{2},h} |\d z|^{2}
\leq C |\xi-\xi_{l}|^{-2+2\alpha^{\infty}_{k}}
\end{equation}
where $\hat{\pi}_{\xi}^{H}\hat{\sigma}(z,\xi)$ is the harmonic representative of $\hat{\sigma}(z,\xi)$.
As a first step in this direction, we prove:
\begin{lem}\label{lemstep4log}
There exists $\delta>0$ and $K>0$ such that for $|\xi|$ sufficiently large the inequality
$$
\left\Vert \Dir_{\xi }^{\ast } \hat{\sigma}(z,\xi) \right\Vert_{L^{2}(\C )}^{2}
\leq K |\xi -\xi_{l}|^{2+2 \delta} \left\Vert \hat{\sigma}(z,\xi) \right\Vert_{L^{2}(\C )}^{2}
$$
holds.
\end{lem}
\begin{proof}
We follow the proof of Lemma \ref{lemstep4}. We set $(D''_{\xi})^{\infty}=\dbar^{\E}+\theta^{\infty}$ and let $\Dir_{\xi}^{\infty}$
(respectively $(\Dir_{\xi}^{\infty})^{\ast}$) stand for its Dirac operator (respectively its adjoint).
By Lemma \ref{expdecord}, $\hat{\sigma}^{l}_{k}$ is supported in $L^{2}$-norm up to an exponentially decreasing factor
in $\xi$ in $\C \setminus \Delta(0,1/ \varepsilon_{0})$. Therefore, the lemma reduces to the same estimation for
$\hat{\sigma}^{l}_{k}$. Moreover, by assumption we have
\[
(\Dir_{\xi}^{\infty})^{\ast}\hat{\sigma}^{l}_{k}(z,\xi)=0,
\]
so
\[
\Dir_{\xi }^{\ast } \hat{\sigma}^{l}_{k}(z,\xi) = [\Dir_{\xi }^{\ast } - (\Dir_{\xi}^{\infty})^{\ast}] \hat{\sigma}^{l}_{k}(z,\xi).
\]
The difference on the right-hand side of this formula is bounded above by $K|z|^{-1-\delta}$ for some $K>0$
independent of $\xi$, because the two Dirac operators depend on $\xi$ in the same way, hence their difference
does not depend on it at all. Introducing the coordinate $w=z(\xi -\xi_{l})$, this becomes
$K|w|^{-1-\delta}|\xi-\xi_{l}|^{1+\delta}$. Therefore, it is sufficient to prove
\begin{align*}
\int_{\C \setminus \Delta(0,|\xi -\xi_{l}|/ \varepsilon_{0})}&
|w|^{-2-2\delta}|\xi -\xi_{l}|^{2+2\delta} \absl \hat{\sigma}^{l}_{k}(z,\xi)\absr_{|dz|^{2},h}^{2} |\xi -\xi_{l}|^{-2} |\d w|^{2} \\ \leq
& K |\xi -\xi_{l}|^{2+2\delta} \int_{\C} \absl \hat{\sigma}^{l}_{k}(z,\xi)\absr_{|dz|^{2},h}^{2} |\xi -\xi_{l}|^{-2} |\d w|^{2},\notag
\end{align*}
for a suitable $K>0$, or more simply
\begin{align}\label{ineqxideltalog}
\int_{\C \setminus \Delta(0,|\xi -\xi_{l}|/ \varepsilon_{0})}|w|^{-2-2\delta} &
\absl \hat{\sigma}^{l}_{k}(z,\xi)\absr_{|dz|^{2},h}^{2} |\d w|^{2} \notag \\
& \leq K \int_{\C} \absl \hat{\sigma}^{l}_{k}(z,\xi)\absr_{|dz|^{2},h}^{2} |\d w|^{2}.
\end{align}
This goes similarly to (\ref{perturbest}): because in the coordinate $w=h_{\xi}^{-1}z$ the spinor
$|\xi -\xi_{l}|^{2-2\alpha^{\infty}_{k}}\hat{\sigma}^{l}_{k}(z,\xi)$ is independent of $\xi$ (see Step 1) and $h$ and $h^{\infty}$ are mutually
bounded, it boils down to
\begin{align*}
\int_{\C \setminus \Delta(0,|\xi -\xi_{l}|/ \varepsilon_{0})}|w|^{-2-2\delta} &
\absl (h_{\xi}^{\ast}\hat{\sigma}^{l}_{k})(w)\absr_{|dz|^{2},h^{\infty}}^{2} |\d w|^{2} \\
& \leq K \int_{\C} \absl (h_{\xi}^{\ast}\hat{\sigma}^{l}_{k})(w)\absr_{|dz|^{2},h^{\infty}}^{2} |\d w|^{2}.
\end{align*}
Now remark that $h_{\xi}^{\ast}\hat{\sigma}^{l}_{k}\in H^{1}(\C,|\d w|^{2},h^{\infty})$ implies in particular that
$h_{\xi}^{\ast}\hat{\sigma}^{l}_{k}\in L^{2}(\C,|\d w|^{2},h^{\infty})$. Furthermore, near the origin
$|w|^{-1-\delta} h_{\xi}^{\ast}\hat{\sigma}^{l}_{k}\in L^{2}_{loc}(|\d w|^{2},h^{\infty})$ provided that $\delta<\alpha^{\infty}_{k}$.
Hence $|w|^{-1-\delta} h_{\xi}^{\ast}\hat{\sigma}^{l}_{k}\in L^{2}(\C,|\d w|^{2},h^{\infty})$, and
$$
K = 2 \frac{\left\Vert|w|^{-1-\delta} h_{\xi}^{\ast}\hat{\sigma}^{l}_{k}\right\Vert_{L^{2}(\C,|d w|^{2},h^{\infty})}^{2}}
{\left\Vert h_{\xi}^{\ast}\hat{\sigma}^{l}_{k}\right\Vert_{L^{2}(\C,|d w|^{2},h^{\infty})}^{2}}
$$
has the desired property (\ref{ineqxideltalog}).
\end{proof}
This has the following consequence.
\begin{lem}\label{asymptnormharmlog}
As $\xi \ra \xi_{l}$, we have the estimate
\[
\left\vert \left\Vert \hat{\sigma}(z,\xi) \right\Vert_{L^{2}}^{2} -
\left\Vert \hat{\pi}^{H}_{\xi} \hat{\sigma}(z,\xi)\right\Vert_{L^{2}}^{2} \right\vert \leq
K |\xi-\xi_{l}|^{2\delta} \left\Vert \hat{\sigma}(z,\xi) \right\Vert_{L^{2}}^{2}
\]
for some $K>0$ independent of $\xi$.
\end{lem}
\begin{proof}
Again as in Lemma \ref{asymptnormharm}, it is sufficient to bound
$$
\left\Vert \hat{\sigma}(z,\xi) -\hat{\pi}^{H}_{\xi} \hat{\sigma}(z,\xi) \right\Vert_{L^{2}}^{2}
$$
as in the lemma, where
$$
\hat{\pi}^{H}_{\xi} \hat{\sigma}(z,\xi)=(\Id - \Dir_{\xi } G_{\xi }\Dir_{\xi }^{\ast }) \hat{\sigma}(z,\xi)
$$
is the $\Dir_{\xi }^{\ast }$-harmonic representative of $\hat{\sigma}(\xi)$. Thus by Lemma \ref{estgreenlog}
we have for the norm of the difference
$$
\left\Vert \Dir_{\xi} G_{\xi}\Dir_{\xi}^{\ast }\hat{\sigma}(z,\xi) \right\Vert_{L^{2}}^{2}
\leq K|\xi-\xi_{l}|^{-2} \left\Vert \Dir_{\xi }^{\ast }\hat{\sigma}(z,\xi) \right\Vert_{L^{2}}^{2}
$$
and we conclude using Lemma \ref{lemstep4log}.
\end{proof}
We are now ready to finish the proof of Theorem \ref{parweightthmlog}: by Lemma \ref{asymptnormharmlog}, as
$\xi\to\xi_{l}$ the norm of the harmonic representative of the spinor $\hat{\sigma}(z,\xi)$ verifies
$$
\frac{\left\Vert \hat{\pi}^{H}_{\xi} \hat{\sigma}(\xi) \right\Vert_{L^{2}}^{2}}
{\left\Vert \hat{\sigma}(\xi) \right\Vert_{L^{2}}^{2}} \lra 1.
$$
On the other hand, since the support of $\chi$ in the coordinate $w$ is
$\C \setminus \Delta(0,|\xi -\xi_{l}|/ \varepsilon_{0})$, and these sets exhaust $\C$ as $\xi\to\xi_{l}$, we have that
$$
\frac{\left\Vert \hat{\sigma}(\xi) \right\Vert_{L^{2}}^{2}}{\left\Vert \hat{\sigma}^{l}_{k}(\xi) \right\Vert_{L^{2}}^{2}}\lra 1.
$$
By (\ref{cKineqlog}) the $L^{2}$-norm of $\hat{\sigma}^{l}_{k}(z,\xi)$ as measured by
the harmonic metric $h$ satisfies
$$
c|\xi-\xi_{l}|^{-2+2\alpha^{\infty}_{k}} \leq \int_{\C}\left\vert\hat{\sigma}^{l}_{k}(z,\xi)\right\vert_{|dz|^{2},h}^{2} |\d z|^{2}
\leq C |\xi-\xi_{l}|^{-2+2\alpha^{\infty}_{k}}.
$$
Putting together all this, we obtain (\ref{asymbehharmlog}),
so that on the component of $\hat{E}$ near $\xi_{l}$ on which the transformed Higgs field has eigenvalue
$-\lambda^{\infty}_{k}$, the parabolic weight of the induced extension is $-1+\alpha^{\infty}_{k}$.
\end{proof}
\section{The topology of the transformed bundle} \label{secttop}
In this section, we compute the topology of the underlying holomorphic
bundle $^{i}\hat{\E}$ of the transformed Higgs bundle (see (\ref{defholtrafo})) relative to its
extension over the punctures given in Section \ref{ext}. We then deduce the topology
of the transformed Higgs bundle relative to its transformed extension given by Definition \ref{transfext}.
We recall that we have denoted
\begin{equation}
\hat{r} = \sum_{p \in P} rk(Res(\theta , p))).
\end{equation}
The result we wish to show is the following:
\begin{thm} \label{thmtop}
The rank of $^{i}\hat{\E}$ is equal to $\hat{r}$, whereas its degree is equal to
$\hat{r}+ deg(\E ) + r$, where $r$ and $deg(\E ) $ are the rank and degree of $\E$, respectively.
\end{thm}
Notice that it gives in particular (\ref{i}) of Theorem \ref{mainthmhiggs}.
\begin{proof}
Recall that we have denoted by $\E$ the sheaf of holomorphic
sections of the bundle $\E$ underlying the original Higgs bundle; $\F$ was defined as a sheaf of
meromorphic sections of $\E \otimes \Omega^{1,0}$ having singularities at $P \cup \{ \infty \}$
with singular parts in prescribed spaces (see Subsection \ref{sheaves});
and finally $\tilde{\F}=\pi_{1}^{\ast }\F \otimes \pi_{2}^{\ast } \mathcal{O}_{\CPt}(1)$.
By hypothesis, $\theta$ (and so $\theta_{\eta } $ for any $\eta $) is holomorphic with respect to the holomorphic
structure $\dbar^{\E}$. Thus we may consider the holomorphic chain complex
\[
\xymatrix{
\E \ar[r] \ar[d]_{\Id} & 0 \ar[d] \\
\E \ar[r]^{\theta_{\eta }} \ar[d] & \tilde{\F} \ar[d]^{\Id} \\
0 \ar[r] & \tilde{\F}
}
\]
in $\eta \in \widehat{\mathbf{CP}}^1 $. The hypercohomology long exact sequence
associated to it yields the exact sequence of cohomology spaces
\begin{align} \label{les}
0 \lra H^{0}({\CP} , \E) & \xrightarrow{\theta_{\eta}} H^{0}({\CP} , \tilde{\F})
\lra \H^{1}( \E \xrightarrow{\theta_{\eta }}\tilde{\F})\notag \\
& \lra H^{1}({\CP},\E) \xrightarrow{\theta_{\eta}} H^{1}({\CP} , \F) \lra 0,
\end{align}
since we have seen that $\H^{0}(\E \xrightarrow{\theta_{\eta }}\tilde{\F} ) = \H^{2}(\E
\xrightarrow{\theta_{\eta }}\tilde{\F} )=0$.
All of the spaces in this exact sequence come with a natural holomorphic
structure over $\CPt$:
\begin{itemize}
\item{the cohomology spaces of $\E$ because this latter is
trivial over $\CPt$}
\item{those of $\tilde{\F}$ because this latter is the tensor
product of a trivial vector bundle over $\CPt$ and $\mathcal{O}_{\CPt}(1)$}
\item{finally, $\H^{1}( \E \xrightarrow{\theta_{\bullet }}\tilde{\F} )=\hat{V}_{\bullet }$ has its
holomorphic structure $\dbar^{\hat{\E}}$ induced by $\hat{\d}^{0,1}$, extended to
the singularities in Section \ref{ext} by the induced extension $^{i}\hat{\E}$.}
\end{itemize}
Moreover, all of the maps in the
exact sequence (\ref{les}) vary holomorphically in $\eta \in \CPt$ with respect to
these structures and extensions: this follows from the definition of $\tilde{\F}$ and that of the induced
extension. Therefore, it induces an exact sequence of the sheaves over ${\CPt}$
of holomorphic sections of the corresponding cohomology spaces:
\begin{align*}
0 \lra \mathcal{O}( H^{0}( \E)) & \xrightarrow{\theta_{\eta }} \mathcal{O}(H^{0}( \tilde{\F})) \lra \mathcal{O}(^{i}\hat{\E}) \\
& \lra \mathcal{O}(H^{1}(\E)) \lra \mathcal{O}(H^{1}( \F)) \lra 0,
\end{align*}
where $\mathcal{O} $ stands to denote the sheaf of regular sections on $\CPt$ with respect
to the above mentioned holomorphic structures.
By additivity of the Chern character, we deduce the equality
\begin{align}
ch(^{i}\hat{\E})= & ch(\mathcal{O}(\widehat{\mathbf{CP}}^1, H^{0}( \tilde{\F}))) -
ch(\mathcal{O}(\widehat{\mathbf{CP}}^1, H^{1}( \tilde{\F}))) \label{chern1} \\
& - ch(\mathcal{O}(\widehat{\mathbf{CP}}^1, H^{0}( \E))) + ch(\mathcal{O}(\widehat{\mathbf{CP}}^1, H^{1}( \E)))
\label{chern2}
\end{align}
in $H^{\ast }(\widehat{\mathbf{CP}}^1)$. Put $\pi = \pi_{2}$, the projection onto the second
factor in $\CP \times \widehat{\mathbf{CP}}^1$. One has direct image sheaves $\pi_{\ast }\E$ and
$\pi_{\ast }\tilde{\F}$ on $\widehat{\mathbf{CP}}^1$ defined by
\begin{align*}
\pi_{\ast }\E |_{U} & = \mathcal{O}(U,H^{0}(\CP, \E )) \\
\pi_{\ast }\tilde{\F}|_{U} & = \mathcal{O}(U,H^{0}(\CP ,\tilde{\F})))=\mathcal{O}(U,H^{0}(\CP ,{\F})) \otimes
\mathcal{O}_{\widehat{\mathbf{CP}}^1}(1)(U),
\end{align*}
for any open set $U \in \CP$, and one can form the ``virtual'' sheaves
\begin{align*}
\pi_{!}\E|_{U} & = \mathcal{O}(U,H^{0}(\CP , \E)) - \mathcal{O}(U,H^{1}(\CP , \E)) \\
\pi_{!} \tilde{\F}|_{U} & = \mathcal{O}(U,H^{0}(\CP , \tilde{\F})) - \mathcal{O}(U,H^{1}(\CP , \tilde{\F})).
\end{align*}
Again by additivity of the Chern character, the right-hand-side of
(\ref{chern1}) is equal to $ch(\pi_{!} \tilde{\F})$, which is in turn equal
to
$$
\pi_{\ast }(ch(\tilde{\F}) \cup Td(T_{\pi }) ),
$$
by the Grothendieck-Riemann-Roch theorem, where
$$
T_{\pi } = T(\CP \times \widehat{\mathbf{CP}}^1) - \pi^{\ast }T\widehat{\mathbf{CP}}^1 = \pi_{1}^{\ast}T\CP
$$
is the relative tangent bundle of $\pi $, and $Td$ stands for its Todd class. Moreover, $\pi_{\ast} $ is just evaluation
on the fundamental cycle of $\CP$. Similarly, we see that (\ref{chern2}) is just
$$
- ch(\pi_{!} \E ) = - \pi_{\ast }(ch(\E) \cup Td(T_{\pi }) ),
$$
and thus we obtain
\begin{equation} \label{grr}
ch(^{i}\hat{\E})= [ ( ch(\tilde{\F})-ch(\E) ) \cup Td(\pi_{1}^{\ast} T \CP ) ]/[\CP ].
\end{equation}
Now we have
\begin{align*}
ch(\E ) & = r + c_{1}(\E) \\
ch(\tilde{\F}) & = \left[ r + c_{1}(\E ) + h \sum_{p \in P} rk(Res(\theta ,
p)) \right] (1+ \hat{h}) \\
Td(T\CP) & = Td(\mathcal{O}_{\CP}(2))= 1+ h,
\end{align*}
where $r$ is the rank of the bundle $\E$, $c_{1}(\E)$ is its first Chern
class, and $h$ and $\hat{h}$ are the hyper-plane classes of $\CP$ and
$\widehat{\mathbf{CP}}^1$ respectively. Putting all this together, we obtain
\[
ch(\tilde{\F}) - ch(\E ) = \hat{r} h + [r+ c_{1}(\E)+ \hat{r}] \hat{h},
\]
and plugging this into (\ref{grr}),
\begin{equation} \label{top}
ch(^{i}\hat{\E}) = \hat{r}+ [r+ deg(\E)+ \hat{r}] \hat{h},
\end{equation}
as we wished.
\end{proof}
We are now ready to pass back to the transformed extension of the Higgs bundle introduced in Definition
\ref{transfext}, hence establishing points (\ref{ii}), (\ref{v}) and (\ref{viii}) of Theorem
\ref{mainthmhiggs}.
\begin{cor} \label{pardegcor}
The parabolic weights of the transformed Higgs bundle endowed with its transformed extension are
$\alpha_{k}^{\infty }$ at the logarithmic punctures (on the same subspace as in Theorem \ref{parweightthmlog})
and $\alpha_{k}^{j }$ at infinity (on the subspace in Theorem \ref{parweightthminf}).
The degree of the transformed Higgs bundle $\hat{\E}$ with respect to its transformed extension is equal to
the degree of ${\E}$.
\end{cor}
\begin{proof}
Recall from Theorems \ref{parweightthmlog} and \ref{parweightthminf} that the parabolic weigths of
the transformed Higgs bundle relative to the induced extensions considered in Subsections \ref{logext} and \ref{infext}
are equal to $-1+\alpha_{k}^{\infty}$ at the logarithmic punctures and to $-1+\alpha_{k}^{j}$ at infinity.
On the other hand, by Definition \ref{transfext}, the parabolic weights of the transformed Higgs bundle
with respect to its transformed extension are required to have parabolic weights between $0$ and $1$.
This means that a local holomorphic trivialisation of the singular component of the transformed extension
$\hat{\E}$ near the puncture $\xi_{l}$ is
\[
(\xi -\xi_{l}) \hat{\sigma}^{l}_{k}(\xi ) ,
\]
where $\hat{\sigma}^{l}_{k}(\xi )$ is the local holomorphic section of the extension $^{i}\hat{\E}$ at $\xi_{l}$
defined in Subsection \ref{logext} and $k \in \{1+a_{l},\ldots,a_{l+1}\}$. On the regular component of
$\hat{\E}|_{\xi_{l}}$ the harmonic representatives have bounded norm, which gives $0$ parabolic weight.
Therefore on this component one does not need to change the trivialisation. Similarly, a local holomorphic
frame of $\hat{\E}$ near infinity can be expressed by
\[
\xi^{-1} \hat{\sigma}_{k }^{\infty}(\xi) ,
\]
where $\hat{\sigma}^{\infty}_{k}$ is the local holomorphic section of the extension $^{i}\hat{\E}$ at infinity
defined in Subsection \ref{infext} localized near $p_{j}$ for some $j\in \{ 1,\ldots,n\}$, and $k \in \{r_{j}+1,\ldots,r\}$.
Clearly, this way we increased all non-vanishing parabolic weights by $1$. On the other hand,
by Remark \ref{pardegvanish} even if the algebraic geometric degree of the bundle depends on the choice
of extensions, the parabolic degree with respect to a fixed metric is independent of them, because it is always $0$.
Recall from Definition \ref{parintcon} that
\[
deg_{par}(^{i}\hat{\E }) = deg(^{i}\hat{\E }) + \sum_{j \in \{1,\ldots,n, \infty \} } \sum_{k=r_{j}+1}^{r} (-1+ \alpha^{j}_{k}).
\]
This quantity is therefore equal to
\begin{equation} \label{pardegEtr}
deg_{par}(\hat{\E} ) = deg(\hat{\E} ) + \sum_{j \in \{1,\ldots,n, \infty \} } \sum_{k=r_{j}+1}^{r} \alpha^{j}_{k}.
\end{equation}
Putting these expressions together, we deduce that
\[
deg(\hat{\E} ) = deg(^{i}\hat{\E }) - \hat{r} -r,
\]
where we recall again that we have defined
\[
\hat{r}= \sum_{j=1}^{n} rk (Res(\theta, p_{j} )).
\]
Using formula (\ref{top}) we get
\begin{equation} \label{p-top}
deg(\hat{\E })= deg(\E ).
\end{equation}
\end{proof}
\chapter{The inverse transform} \label{chinv}
In this chapter we construct the inverse of the transform introduced in the
previous chapters. In line with the properties of the ordinary Fourier
transform and its algebraic counterparts, the inverse is defined by a
formula which only differs from the transform in a sign.
Recall from Section \ref{flattr} that the transformed flat connection on
$\hat{E}_{\bullet }=L^{2}H^{1}(D_{\bullet })$ is defined by the $L^{2}$-orthogonal
projection of $\hat{\d}-z\d \xi \land$.
For any parabolic vector bundle with integrable connection $ (F,D^{F},h^{F})$ on
$\hat{\C}$ satisfying the conditions of Section \ref{flatintro} (i.e. having
a finite number of simple poles in finite points and a second-order
pole at infinity, such that the eigenvalues and parabolic weights meet the conditions imposed
in Theorem \ref{mainthm}), one can define the inverse transformed bundle with
integrable connection $(\check{F},\check{D}^{F},\check{h}^F)$ on $\C$ by a procedure similar to the one
defining $(\hat{E},\hat{D},\hat{h})$ starting from $(E,D,h)$: namely, consider the deformation
\begin{equation} \label{invdeform}
D^{F}_{z}=D^{F} + z \d \xi \land
\end{equation}
of the connection parametrized by $z$ in $\C$ minus a finite set,
and let $\check{F}_{z}$ be the first $L^{2}$-cohomology of
$$
F \xrightarrow{D^{F}_{z}} \Omega^{1}_{\hat{C}} \otimes F \xrightarrow{D^{F}_{z}} \Omega^{2}_{\hat{C}} \otimes F.
$$
These vector spaces are of the same dimension and form a
smooth vector bundle over $\C$ minus a finite number of points. The
critical points are easily seen to be the opposites of the eigenvalues of the second-order
term of ${D}^{F}$ at infinity. The proof goes similarly to the case
of the direct transform. We also define the Hilbert bundle $\check{H}$ over
$\C $, the $L^{2}$-metric $\check{h}$ and the orthogonal projection
$\check{\pi}_{z}: \check{H}_{z} \ra \check{F}_{z} $ in an analogous manner as in Section
\ref{flattr}.
Next, let the inverse transformed integrable connection
$\check{D}^{F}$ be defined by the parallel sections
$\check{\pi}_{z}(e^{(z_{0}-z)\xi } \phi_{z_{0}} (\xi ))$ for any harmonic section
$\phi_{z_{0}} (\xi ) \in \check{F}_{z_{0}}$. Equivalently, denoting by $\check{\d}$
the trivial connection with respect to $w$ in the trivial Hilbert bundle
$\check{H}$, the inverse transformed flat connection can be given by the
formula
\begin{equation} \label{flinvexpl}
\check{D}^{F} = \check{\pi}_{z} (\check{\d} + z \d \xi ),
\end{equation}
as it can be seen by the argument given in Section \ref{flattr}, changing
signs. Finally, we define the inverse transformed metric $\check{h}^F$ on the fiber
$\check{F}_{z_{0}}$ again as the $L^{2}$-norm on $\hat{\C}$ of a ${D}^{F}_{z_{0}}$-harmonic representative.
We can now state the
\begin{thm} \label{invtr}
The inverse transform of $\Nahm : (E,D,h)\mapsto (\hat{E},\hat{D}, \hat{h})$ is
$\Nahm^{-1} :(F,D^{F},h^{F}) \mapsto (\check{F},\check{D}^{F},\check{h}^{F})$. In different terms, for any
bundle with integrable connection and harmonic metric $(E,D,h)$ satisfying
the conditions of Section \ref{flatintro} and the ones imposed in Theorem \ref{mainthm}, there exists a canonical
Hermitian bundle isomorphism $\omega $ between $\check{\hat{E}}$ and $E$ such
that $\omega^{\ast } D = \check{\hat{D}} $.
\end{thm}
\begin{rk}
As one can check using the transform on the level of singularity parameters described in Theorem
\ref{mainthm}, the assumptions (1) and (2) of that theorem are symmetric, in the sense that
if they are fulfilled by $(E,D)$ than the same is true for $(\hat{E},\hat{D})$. Therefore, the
transform $\check{}$ can be applied to this latter, so the affirmation of the theorem has
a meaning.
\end{rk}
\begin{proof}
The proof is done in four steps:
first, we prove that the fibers over $0 \in \C$ of $E$ and $\check{\hat{E}}$ are
canonically isomorphic. Next we show the same thing for the other fibers.
Then we prove that the integrable connections are the same, and finally
we establish equality of the harmonic metrics and parabolic structures.
\paragraph{Step 1.}
Consider the product manifold $\C \times \hat{\C}$, and let $\pi_{1}$ and $\pi_{2}$
be the projection to the first and second factor, respectively.
Denote by $E$ the pull-back vector bundle $\pi_{1}^{\ast }E$ on the product, and
define the connection $\D =\pi_{1}^{\ast }D - \xi \d z - z \d \xi $. Notice that on the
fiber $\C \times \{ \xi_{0} \}$ this just gives the deformation $D_{\xi_{0}}$. Now
form the double complex
$$
\Dou^{p,q} = \Omega_{{\C}}^{p} \otimes \Omega_{\hat{\C}}^{q}(E),
$$
where $\Omega_{{\C}}^{p}$ (respectively $\Omega_{\hat{\C}}^{q}$) denote smooth
$p$-forms (smooth $q$-forms) on $\C$ ($\hat{\C}$); and
with differentials $\d_{1}=D_{\xi }, \d_{2}=\hat{\d}-z \d \xi \land $.
Remark that these differentials commute (in the graded sense), and their
sum is just $\D$. The desired
isomorphism will result from the study of the spectral sequences
corresponding to the two different filtrations of this double complex.
Namely, consider the first filtration of $\Dou$: the first page of the
corresponding spectral sequence $\S_{1}^{ \bullet , \bullet }$ is
\begin{eqnarray} \label{D1}
\xymatrix{ 0 & \Omega_{\hat{\C}}^{2} \otimes \hat{E} & 0 \\
0 & \Omega_{\hat{\C}}^{1} \otimes \hat{E} \ar[u]^{\d_{2}^{\sharp }} & 0 \\
0 & \hat{E} \ar[u]^{\d_{2}^{\sharp }} & 0 }
\end{eqnarray}
where $\d_{2}^{\sharp }$ stands for the operator induced by $\d_{2}$. More
precisely, this operator is obtained as follows. Consider for example a
local section of $\hat{E}$: if $B(\xi_{0} )$ is an open ball in $\hat{\C}$,
it is given by cohomology classes $[\phi_{\xi }]$ in
$L^{2}H^{1}(D_{\xi })$ changing smoothly with $\xi \in B(\xi_{0} )$. Here $\phi_{\xi }=\phi_{\xi }(z)$ is a global
$L^{2}$-section of $E$ over $\C$, in the kernel of $\Dir_{\xi }^{\ast }$. In
particular, $D_{\xi } \phi_{\xi }=0$, and since the two differentials commute, we
then have $D_{\xi } \circ \d_{2} \phi_{\xi }=0$. In other words, $\d_{2}\phi_{\xi}$
is a $\d_{1}$-closed section of $\Dou^{1,1}$ on $\C \times B(\xi_{0} )$; hence we
may consider its cohomology class with respect to $\d_{1}$, and letting $\xi$
vary these give a section of $\Omega^{1} \otimes \hat{E}$ over $B(\xi_{0} )$, which is
by definition $\d_{2}^{\sharp } [\phi_{\xi }]$. Now remark that under the
isomorphism of the first $L^{2}$-cohomology of the elliptic complex (\ref{ltwocompl})
and the space of $\Dir_{\xi }$-harmonic sections given in Theorem
\ref{laplker}, this induced connection goes over to $\hat{D}$
defined in Section \ref{flattr};
in other words, under these identifications $\d_{2}^{\sharp }=\hat{D}$.
Moreover, the connection
$\hat{D}$ also satisfies the conditions of Section \ref{flatintro}.
Therefore, by Chapter \ref{chFr} and Section \ref{l2coh}
the $L^{2}$-cohomology of $\hat{D}=\hat{D}_{0}$ is non-trivial
only in degree $1$, and so when passing to the second page $\S_{2}^{ \bullet , \bullet }$ of the
spectral sequence, we obtain by definition
$\S_{2}^{1,1}=\check{\hat{E}}_{0}$ and all other terms equal to $0$. In
particular, the spectral sequence collapses at the second page, and the
total cohomology of the double complex is canonically isomorphic to $\check{\hat{E}}_{0}$ in degree $2$
and vanishes in all other degrees.
Consider now the second filtration of $\Dou$: in order to form the first
page $\tilde{\S}_{1}^{ \bullet , \bullet }$ of the corresponding spectral sequence, we first take
cohomology on each column of the double complex with respect to
$\d_{2}=\hat{\d} -z \d \xi $, and so it is equal to
\begin{eqnarray} \label{D2}
\xymatrix{ 0 & 0 & 0 \\
0 & 0 & 0 \\
L^{2}(\C,E) e^{z\xi } \ar[r]^(.42){\d_{1}^{\sharp }} &
L^{2}(\C,\Omega_{\C}^{1} \otimes E) e^{z\xi } \ar[r]^(.48){\d_{1}^{\sharp }} &
L^{2}(\C,\Omega_{\C}^{2} \otimes E) e^{z\xi }. }
\end{eqnarray}
In words: for example, the $(0,0)$-term consists of $L^{2}$-sections of $E$
on $\C \times \hat{\C}$ which are a product of an arbitrary section of $E$ on
$\C$ and the function $e^{z\xi }$. Now notice that the only
possibility for a non-zero section of this form to be in $L^{2}$ on $\{z \} \times \hat{\C}$ is for $z=0$.
Put another way, the cohomology along the slices $\{z\} \times \hat{\C}$ vanishes
for all $z \neq 0$. Hence we may replace the double complex $\Dou$ without changing the spectral
sequence associated with this filtration (and so the total cohomology),
by the double complex $\germ$ whose component of bidegree $(p,q)$ is the
space of $L^{2}$-forms with values in $E$ of bidegree $(p,q)$ defined on
$V_{0} \times \hat{\C}$ for any neighborhood $V_{0}$ of $0 \in \C$, and where we
identify such forms if they coincide on an arbitrary neighborhood of
$\{0\} \times \hat{\C}$. Of course, the differentials of this new double complex
are induced by those of $\Dou$ in a trivial way.
The idea now is to consider the spectral sequence $\Sg$ corresponding to the
first filtration of $\germ$: by the general theory of spectral sequences,
this will then abut to the total cohomology of $\germ$, which is, as we
saw in the previous paragraph, equal to that of $\Dou$, that is to
$\check{\hat{E}}_{0}$. First trivialize $E$ in $V_{0}$: this just means
that we identify the total space of the bundle with $V_{0} \times E_{0}$.
Since the vector bundle $E$ on $\C \times \hat{\C}$ is just the pull-back of $E$
on $\C$, this also gives an identification of $E \ra V_{0} \times \hat{\C}$ with
the trivial bundle $ (V_{0} \times \hat{\C}) \times E_{0}$.
Without loss of generality we may assume $0 \notin P$, so for $V_{0}$ sufficiently small
the connection $D$ can also be taken by a gauge transformation $\tilde{g}$ to the
trivial one. Thus in this trivialisation and gauge we have
$\d_{1}=\d -\xi \d z$ where $\d$ stands for the trivial connection in the $z$
direction. The first page $\Sg_{1}^{ \bullet , \bullet }$ is then equal to
the cohomology spaces with respect to this differential:
\begin{eqnarray}
\label{germD1} \xymatrix{ \Omega_{\hat{\C}}^{2} \otimes L^{2}(\hat{\C},E_{0}) e^{z\xi } & 0 & 0\\
\Omega_{\hat{\C}}^{1} \otimes L^{2}(\hat{\C},E_{0}) e^{z\xi } \ar[u]^{\d_{2}^{\sharp }} & 0 & 0 \\
L^{2}(\hat{\C},E_{0}) e^{z\xi } \ar[u]^{\d_{2}^{\sharp }} & 0 & 0, }
\end{eqnarray}
where, as before, $ L^{2}(\hat{\C},E_{0}) e^{z\xi }$ stands to denote
functions with values in $E_{0}$ of
the form $\gamma(\xi ) e^{z \xi }$ but this time on $V_{0} \times \hat{\C}$, and the
$L^{2}$ condition now only implies that $\gamma $ must be rapidly decreasing as $|\xi| \ra \infty $.
The next remark is that since we only have terms in degree $p=0$, the
differential induced by $\d_{2}$ is just itself: indeed, it is by
definition $\d_{2}$ modulo the image of $\d_{1}$, but this latter vanishes
for $p=0$. Thus, in order to obtain the second page $\Sg_{2}^{ \bullet , \bullet}$
of the spectral sequence, we take cohomology with respect to
$\d_{2}=\hat{\d} -z \d \xi \land$. Notice that via the gauge transformation
$e^{-z \xi }$ the whole picture can be rephrased as the de Rham cohomology of
rapidly decreasing sections $\sigma $ on $\hat{\C}$ with values in $E_{0}$,
which is similar to compactly supported de Rham cohomology.
Therefore in $\Sg_{2}^{ \bullet , \bullet }$ all elements except for the one
corresponding to bidegree $(0,2)$ vanish, and this latter is canonically
isomorphic to $E_{0}$ via mapping an element $\gamma_{0} \in E_{0}$ into the germ
$$
[\gamma_{0} \chi(\xi ) e^{z\xi } \d \xi \land \d \bar{\xi }],
$$
where $\chi$ is a fixed exponentially decreasing bump-function on $\hat{\C}$ with
integral (with respect to the volume form $|\d \xi |^{2}$) equal to $1$, and
$[.]$ stands to denote the de Rham cohomology class of exponentially decreasing
forms on $\hat{\C} $ with values in $E_{0}$.
Conversely, for an arbitrary class $ [\gamma_{0}( \xi ) e^{z \xi } \d \xi \land \d \bar{\xi }]$ where
$\gamma_{0}( \xi ) e^{z \xi }$ is a germ of exponentially decreasing functions on
$\hat{\C}$ with values in $E_{0}$ and in the kernel of $\d_{1}=(\d -\xi \d z)$, we may define
\begin{align}\label{explisom}
[\gamma_{0}( \xi ) e^{z \xi } \d \xi \land \d \bar{\xi }] \mapsto &
eval_{z=0}\int_{\hat{\C}} \gamma_{0}( \xi ) e^{z \xi } |\d \xi |^{2} \notag \\
& = \int_{\hat{\C}} \gamma_{0}(\xi ) |\d \xi |^{2} \in E_{0}
\end{align}
and verify readily that it is independent of the section representing a cohomology
class.
The fact that $E_{0}$ and $\check{\hat{E}}_{0}$ are canonically isomorphic
now follows from the fact that they are both canonically isomorphic to
(different gradings of) the total cohomology of the double complex $\Dou$.
\paragraph{Step 2.} The first thing to do is to describe explicitly the
isomorphism obtained above. Let $\left[\check{\hat{\delta}}_{0} \right]$ be an element in
$\check{\hat{E}}_{0}$: it is a class in the cohomology space
$\S_{2}^{1,1}$ in the spectral sequence corresponding to the first
filtration of $\Dou$. Hence it is represented by a $(1,1)$-form
$\check{\hat{\delta}}_{0}(z;\xi )$ over $\C \times \hat{\C}$ such that
\begin{enumerate}
\item{ $(D-\xi \d z \land) \check{\hat{\delta}}_{0}(z;\xi ) =0$
\item{ $(\hat{\d}-z \d \xi \land)^{\sharp }\check{\hat{\delta}}_{0}(z;\xi )=0$; in other words, there exists a
$(0,2)$-form $\gamma_{0}(z;\xi ) $ over $\C \times \hat{\C}$
satisfying
$$
D_{\xi } \gamma_{0}(z;\xi ) = (\hat{\d}-z \d \xi \land)\check{\hat{\delta}}_{0}.
$$}
\end{enumerate}
Concatenating the map
$$
\left[ \check{\hat{\delta}}_{0} \right] \mapsto \gamma_{0}(z;\xi )
$$
with an analog of (\ref{explisom}), namely
\begin{equation} \label{eval0}
[\gamma_{0}( z;\xi ) ] \mapsto eval_{z=0}
\int_{\hat{\C}} \gamma_{0}( z;\xi )
\end{equation}
we get the canonical isomorphism
$$
\omega_{0}: \left[ \check{\hat{\delta}}_{0} \right] \mapsto \delta_{0} = eval_{z=0}
\int_{\hat{\C}} \gamma_{0}( z;\xi )
$$
between $\check{\hat{E}}_{0}$ and $ E_{0}$ provided by the previous step.
Fix now an arbitrary $z_{0} \in \C$, and consider the double complex
$\Dou_{z_{0}}$ having the same $(p,q)$-components as $\Dou$, but with
differentials $\d_{1}=D_{\xi }, \d_{2}=\hat{\d}-(z-z_{0}) \d \xi \land $. In order
to obtain the components of the first page $(\S_{z_{0}})_{1}^{\bullet , \bullet}$ of the
spectral sequence corresponding to the first filtration of $\Dou_{z_{0}}$,
we need to take cohomology with respect to $\d_{1}$, hence these
will be the same as those of $\Dou$ in (\ref{D1}), and the
differentials will be induced by $\d_{2}$. Now since $z_{0}$ is a
constant, observe that for any local section
$\phi_{\xi }(z) \in Ker \Dir_{\xi}^{\ast}$ in $\xi $ of harmonic sections over $\C$ the relation
$$
\d_{2}^{\sharp } \phi_{\xi } = [(\hat{\d}-(z-z_{0}) \d \xi \land) \phi_{\xi }] =
[(\hat{\d}-z \d \xi \land) \phi_{\xi }] + z_{0} \d \xi \land \phi_{\xi }=
\hat{D}_{z_{0}}(\phi_{\xi }),
$$
holds, where $\hat{D}_{z_{0}}$ is the deformation of $\hat{D}$ introduced
in (\ref{invdeform}). To get the second page of the spectral sequence, we
take cohomology with respect to $\d_{2}^{\sharp } = \hat{D}_{z_{0}}$, and
therefore if $z_{0}$ does not belong to the set of opposites of eigenvalues of the
leading term of $\hat{D}$ then this is a finite-dimensional space, equal by
definition to $\check{\hat{E}}_{z_{0}}$. Notice that by the results of
Subsection \ref{sing}, the set of $z_{0}$ where this does not hold is
exactly $P$, the set of singularities (at finite points) of $E$.
Similarly, the second filtration of $\Dou_{z_{0}}$ gives rise to a spectral
sequence whose first page is (analogously to (\ref{D2}))
\begin{eqnarray*} \label{Dz2}
\xymatrix{ 0 & 0 & 0 \\
0 & 0 & 0 \\
L^{2}(\C,E) e^{(z-z_{0})\xi } \ar[r]^(.45){\d_{1}^{\sharp }} &
L^{2}(\C,\Omega_{\C}^{1} \otimes E) e^{(z-z_{0})\xi } \ar[r]^{\d_{1}^{\sharp }} &
L^{2}(\C,\Omega_{\C}^{2} \otimes E) e^{(z-z_{0})\xi }. }
\end{eqnarray*}
Hence the only fiber $\{ z \} \times \hat{\C}$ over which these spaces are
non-trivial is for $z=z_{0}$, so we may consider the double complex
$\germz$ whose components are germs of forms in a neighborhood
$V_{z_{0}}\times\hat{\C}$ of the fiber $\{ z_{0} \} \times \hat{\C}$, two such germs
being identified if they coincide in any such neighborhood, and with
differentials coming from those of $\Dou_{z_{0}}$. As before, the spectral
sequences corresponding to the second filtration of these double complexes
agree starting from the first page, so in particular their total
cohomologies are the same. Now, we pass back again to the first filtration
and compute the spectral sequence of $\germz$ with respect to it: in
a convenient trivialisation of $E$ in $V_{0}$ and gauge, the first page is
equal to
\begin{eqnarray}
\label{germDz1} \xymatrix{ \Omega_{\hat{\C}}^{2} \otimes L^{2}(\hat{\C},E_{z_{0}}) e^{z\xi } & 0 & 0\\
\Omega_{\hat{\C}}^{1} \otimes L^{2}(\hat{\C},E_{z_{0}}) e^{z\xi } \ar[u]^{\d_{2}^{\sharp }} & 0 & 0 \\
L^{2}(\hat{\C},E_{z_{0}}) e^{z\xi } \ar[u]^{\d_{2}^{\sharp }} & 0 & 0, }
\end{eqnarray}
with differentials given by $\d_{2}=\hat{\d}-(z-z_{0}) \d \xi \land $.
As in step 1, the second page therefore contains only one non-vanishing
component: the one corresponding to bidegree $(0,2)$, and it is canonically
isomorphic to the vector space $E_{z_{0}}$; this proves that the vector
spaces $E_{z_{0}}$ and $\check{\hat{E}}_{z_{0}}$ are canonically
isomorphic to each other. Again, an element
$\left[ \check{\hat{\delta}}_{z_{0}} \right]$ of
$\check{\hat{E}}_{z_{0}}$ is represented by a $(1,1)$-form
$\check{\hat{\delta}}_{z_{0}}(z;\xi )$ over $\C \times \hat{\C} $
satisfying
$(\hat{\d}-(z-z_{0}) \d \xi )^{\sharp }\check{\hat{\delta}}_{z_{0}}(z;\xi )=0$, i.e. there exists a
$(0,2)$-form $\gamma_{z_{0}}(z;\xi ) $ over $\C \times \hat{\C}$
with
$$
D_{\xi } (\gamma_{z_{0}}(z;\xi ) ) = (\hat{\d}- (z-z_{0})
\d \xi \land )\check{\hat{\delta}}_{z_{0}}(z;\xi ),
$$
and an explicit way of describing the obtained isomorphism is given by
\begin{equation} \label{evalz}
\omega_{z_{0}} : \left[\check{\hat{\delta}}_{z_{0}} \right] \mapsto \delta_{z_{0}} = eval_{z=z_{0}}
\int_{\hat{\C}} \gamma_{z_{0}}( z;\xi )
\end{equation}
\paragraph{Step 3.} By the previous points, we have that the bundle
$\check{\hat{E}}$ is isomorphic to $E$ via the isomorphisms $\omega_{\bullet }$.
Now we prove that the integrable connection
$\check{\hat{D}}$ on $\check{\hat{E}}$ is carried into $D$ on $E$ by this
bundle isomorphism: for this, it is clearly sufficient to prove that any
local parallel section for $\check{\hat{D}}$ is carried into a parallel
section for $D$.
For simplicity, we shall consider a local section near $w=0$, but we
will see that the proof does not use this.
For this purpose, we need to work on the product $\C \times \hat{\C} \times \C$,
parametrized by $(z, \xi , w)$; we keep on writing the variable $w$ in lower index.
We shall consider $E$ as being a bundle over this space by pull-back,
without writing it out explicitly. Let $\left[\check{\hat{\delta}}_{w} \right]$ be a
$\check{\hat{D}}$-parallel local section of $\check{\hat{E}}$.
As in Step 2, such a section is represented by giving a global
$(1,1)$-form $\check{\hat{\delta}}_{w}(z;\xi )$ of $E$ on $\C \times \hat{\C}$ for each $w$ in a neighborhood
$V_{0}$ of $0 \in \C$, verifying
\begin{enumerate}
\item{$D_{\xi_{0}}\check{\hat{\delta}}_{w}(z;\xi )=0$ for all fixed $w_{0} \in V_{0}$
and $\xi_{0} \in \hat{\C}$}
\item{$(\d_{2} -(z- w_{0}) \d \xi \land)^{\sharp} \check{\hat{\delta}}_{w}(z;\xi )=0$ for
all fixed $w_{0} \in V_{0}$
\item{the section in $w$ of the cohomology classes of the above elements
is $\check{\hat{D}}$-parallel.}
\end{enumerate}
By Hodge theory, we may suppose that $\check{\hat{\delta}}_{w_{0}}(z;\xi_{0} )$ is
the $D_{\xi_{0}}$-harmonic representative of
$\left[ \check{\hat{\delta}}_{w_{0}}|_{\C \times \{ \xi_{0} \}} \right]$
and also that $\check{\hat{\delta}}_{w_{0}}(z;\xi )$ is the
$\hat{D}_{w_{0}}$-harmonic representative of $\left[\check{\hat{\delta}}_{w_{0}} \right]$.
This way we rephrase the above conditions as
\begin{enumerate}
\item{for all fixed $w_{0} \in V_{0}$ and $\xi_{0} \in \hat{\C}$
its restriction to the fiber $\C \times \{ \xi_{0} \} \times \{ w_{0} \}$ is
in $\hat{E}_{\xi_{0}}$, that is
$\Dir_{\xi_{0}}^{\ast } \check{\hat{\delta}}_{w_{0}}(z;\xi_{0})=0$} \label{cond1}
\item{for all fixed $w_{0} \in V_{0}$ the global section in $\xi $ of the above
elements of $\hat{E}_{\xi}$ is in $\check{\hat{E}}_{w_{0}}$, in different
terms
$\hat{\Dir}_{w_{0}}^{\ast }\check{\hat{\delta}}_{w_{0}}(z;\xi )=0 $}\label{cond2}
\item{and for all $w_{0} \in V_{0}$,
$\check{\pi}_{w}\circ (\check{\d}+\xi \d w \land)\check{\hat{\delta}}_{w}(z; \xi)|_{w=w_{0}}=0$.} \label{cond3}
\end{enumerate}
As before, (\ref{cond2}) means that for all $w \in V_{0}$ there exists
$\gamma_{w}(z;\xi ) \in \Gamma(\C \times \hat{\C}, \Omega^{2,0} \otimes E)$ such that
\begin{equation} \label{gammadelta1}
D_{\xi } \gamma_{w}(z;\xi ) = (\hat{\d}-(z-w) \d \xi \land) \check{\hat{\delta}}_{w}(z;\xi );
\end{equation}
and by Hodge theory, such a section can be defined by the formula
\begin{equation} \label{gammadelta2}
\gamma_{w}(z;\xi ) = G_{\xi } D_{\xi }^{\ast } (\hat{\d}-(z-w) \d \xi \land) \check{\hat{\delta}}_{w}(z;\xi ),
\end{equation}
where $G_{\xi }$ is the Green's operator of $\Dir_{\xi }^{\ast } \Dir_{\xi }$. (Here
we used that $G_{\xi }$ is diagonal with respect to the
decomposition $\Omega_{\C}^{0} \oplus \Omega_{\C}^{2}$, a standard consequence of the fact
that $\Dir_{\xi }^{\ast } \Dir_{\xi }$ is diagonal with respect to the same
decomposition, which comes immediately from harmonicity of the metric.)
Now by (\ref{evalz}) and (\ref{gammadelta1}) we have
\begin{align*}
D \delta (w)|_{w=w_{0}} = & D \left( eval_{z=w} \int_{\hat{\C}} \gamma_{w}( z;\xi ) \right)_{|{w=w_{0}}}\\
= & \int_{\hat{\C}} D \gamma_{w_{0}}( z;\xi )|_{z=w_{0}} + \check{\d} \gamma_{w}( w_{0};\xi )|_{w=w_{0}} \\
= & \int_{\hat{\C}} \xi \d z \land \gamma_{w_{0}}( w_{0};\xi ) \\
& +(\hat{\d}-(w_{0}-w_{0}) \d \xi \land) \check{\hat{\delta}}_{w_{0}}(w_{0};\xi )
+ \check{\d} \gamma_{w}( {w_{0}};\xi )|_{w=w_{0}}
\end{align*}
(remember that $\check{\d}$ stands for
the trivial connection with respect to $w$ in the trivial Hilbert bundle
$\check{H}$, whereas $\hat{\d}$ is the trivial connection with respect to
$\xi $ in the trivial Hilbert bundle $\hat{H}$).
The integral of the middle term in this last formula vanishes by Stokes's
theorem. Furthermore, on the diagonal $z=w$ of $\C \times \C$ we have $\d z =\d w$,
so we are left with
$$
\int_{\hat{\C}} (\check{\d} + \xi \d w \land ) \gamma_{w_{0}}( {w_{0}};\xi ).
$$
Applying to this quantity (\ref{gammadelta2}) and the commutation relations
\begin{align} \label{comm1}
[ \check{\d} + \xi \d w \land , \hat{\d}-(z-w) \d \xi \land ] = 0 & &
[ \check{\d} + \xi \d w \land , D_{\xi }] = 0
\end{align}
we obtain
\begin{equation} \label{int1}
\int_{\hat{\C}} G_{\xi } D_{\xi }^{\ast } (\hat{\d}-(z-w) \d \xi \land) (\check{\d}+ \xi \d w \land)
\check{\hat{\delta}}_{w_{0}}(w_{0};\xi ).
\end{equation}
Consider now condition (\ref{cond3}) above: denoting by $\hat{\Dir}_{w}$
and $\hat{\Dir}_{w}^{\ast }$ the positive and negative Dirac operators of
the deformation $\hat{D} + w \d \xi $, moreover by $\hat{G}_{w}$ the Green's
operator of $\hat{\Dir}_{w}^{\ast } \hat{\Dir}_{w}$, it can be rewritten as
\begin{equation*}
(\Id - \hat{\Dir}_{w} \hat{G}_{w} \hat{\Dir}_{w}^{\ast }) (\check{\d}+
\xi \d w \land) \check{\hat{\delta}}_{w}(z;\xi ) = 0.
\end{equation*}
In order to finish the proof, it is sufficient to prove the commutation
relation
\begin{equation} \label{comm2}
[\check{\d}+ \xi \d w \land , \hat{\Dir}_{w}] = 0.
\end{equation}
Indeed, this then implies
\begin{align*}
[\check{\d}+ \xi \d w \land , \hat{\Dir}_{w}^{\ast }] = 0 & & [\check{\d}+ \xi
\d w \land , \hat{G}_{w}] = 0,
\end{align*}
and interchanging $\check{\d}+ \xi \d w \land $ turn by turn with
$\hat{\Dir}_{w_{0}}^{\ast }$, $\hat{G}_{w_{0}}$ and $\hat{\Dir}_{w_{0}}$
using each time condition (\ref{cond2}), we get
\begin{align*}
(\check{\d}+ \xi \d w \land ) \check{\hat{\delta}}_{w_{0}}(w_{0};\xi ) & =
(\check{\d}+ \xi \d w \land ) (\Id - \hat{\Dir}_{w_{0}} \hat{G}_{w_{0}}
\hat{\Dir}_{w_{0}}^{\ast }) \check{\hat{\delta}}_{w_{0}}(w_{0};\xi ) \\
& = (\Id - \hat{\Dir}_{w_{0}} \hat{G}_{w_{0}} \hat{\Dir}_{w_{0}}^{\ast })
(\check{\d}+ \xi \d w \land) \check{\hat{\delta}}_{w_{0}}(w_{0};\xi ) \\
& = 0,
\end{align*}
and so (\ref{int1}) is equal to $0$; but on the other hand it is just the
expression for $D \delta (w)|_{w=w_{0}}$, and this shows that $\delta (w)$ is parallel in
$w_{0}$. There remains to show (\ref{comm2}): recall that $\hat{\Dir}_{w} =
\hat{D}_{w}-\hat{D}_{w}^{\ast }$, with
$$
\hat{D}_{w} = \hat{\pi}_{\xi } (\hat{\d} - (z-w) \d \xi ).
$$
Now the first relation in (\ref{comm1}) and
$\hat{\pi}_{\xi } = (\Id - \Dir_{\xi} G_{\xi } \Dir_{\xi}^{\ast } ) $ combined with the
second relation in (\ref{comm1}) show that
$$
[\check{\d}+ \xi \d w \land , \hat{D}_{w}] = 0,
$$
and we conclude.
\paragraph{Step 4.} Here we wish to show that the double transformed metric
$\check{\hat{h}}$ is equal to $h$. In Step 3 we have already shown that the flat connections $D$ and
$\check{\hat{D}}$ agree. On the other hand, using the results of Section \ref{sectharm} twice, we see that
$\check{\hat{h}}$ is a harmonic metric for $\check{\hat{D}}=D$.
Therefore by uniqueness (up to a constant) of the harmonic metric corresponding to an
integrable connection, we get that $\check{\hat{h}}=h$.
An equivalent way of deducing the same assertion would be as follows: using again the already proved
equality $\check{\hat{D}}=D$ and uniqueness of the harmonic metric, we will be done if we can prove
that the unitary part $\check{\hat{D}}^{+}$ (with respect to $\check{\hat{h}}$)
of the double transformed flat connection $\check{\hat{D}}$ is equal to
$D^{+} $, the unitary part of $D$ with respect to $h$. This can be done in a completely analogous way to Steps
1-3. The changes we have to make are the following: consider the double complex $\Dou^{H}_{z_{0}}$ having the
same components as $\Dou_{z_{0}}$, but with differentials $\d_{1}=D_{\xi }^{H}$ and
$\d_{2}= \hat{\d} - z/2 \d \xi \land - \bar{z}/2 \d \bar{\xi} \land $. One establishes that these operators commute,
therefore $\Dou^{H}_{z_{0}}$ really forms a double complex. We then see from (\ref{hdecomp}) that the deformation
$$
\hat{D}^{H}_{w} = \hat{D}^{H} + \frac{1}{2} w \d \xi \land + \frac{1}{2} \bar{w} \d \bar{\xi} \land
$$
induced from the differential
$$
\hat{\d} - \frac{1}{2} (z-w) \d \xi \land - \frac{1}{2} (\bar{z} - \bar{w}) \d \bar{\xi} \land
$$
is the natural deformation of the Higgs-bundle structure induced by the deformation $\hat{D}_{w}$.
In concrete terms, they are related by the gauge transformation $g^{-1}$. Therefore the double transformed
bundle $\check{\hat{E}}^{H}$ is isomorphic to $g^{-1}gE=E$, and the unitary connection
$$
\check{\hat{D}}^{+} = \check{\pi}_{w} \circ \left( \check{\d} + \frac{\xi }{2} \d w \land +
\frac{\bar{\xi} }{2} \d \bar{w} \land \right)
$$
is identified to $D^{+}$ just as $\check{\hat{D}}$ with $D$, using the commutation relations
\begin{align*}
\left[\check{\d} + \frac{\xi }{2} \d w \land + \frac{\bar{\xi} }{2} \d \bar{w} \land ,
\hat{\d} - \frac{1}{2} (z-w) \d \xi \land - \frac{1}{2} (\bar{z} - \bar{z}) \d \bar{\xi} \land \right] =0,
\\ \left[\check{\d} + \frac{\xi }{2} \d w \land + \frac{\bar{\xi} }{2} \d \bar{w} \land ,
D_{\xi }^{H} \right] =0
\end{align*}
instead of (\ref{comm1}), which together imply the analog
$$
\left[\check{\d} + \frac{\xi }{2} \d w \land + \frac{\bar{\xi} }{2} \d \bar{w} \land ,
\hat{\Dir}_{w}^{H} \right] = 0
$$
of (\ref{comm2}) for the deformed Dirac operator
$$
\hat{\Dir}_{w}^{H} = D_{w}^{H} - (D_{w}^{H})^{\ast }.
$$
This then allows us to conclude equality of the unitary connections.
Since the Hermitian bundles $(\check{\hat{E}}, \check{\hat{h}})$ and $(E,h)$ coincide, so do the flags
of their parabolic structures in the singular points; as well as the
parabolic weights, because they are supposed to be between $0$ and $1$, and there is a unique way of choosing
holomorphic sections with such behaviors.
\end{proof}
| -325,925.806091 |
[
-2.87890625,
2.560546875
] | 18.067803 |
[
-2.53125,
0.50537109375,
-2.29296875,
-6.40625,
-1.15625,
9.359375
] |
[
2.814453125,
8.7109375,
1.46484375,
5.7734375
] | 1,330 | 37,183 |
[
-3.568359375,
4.09375
] | 38.611396 |
[
-5.25390625,
-3.962890625,
-5.7421875,
-2.419921875,
1.478515625,
13.3515625
] | 1.3014 | 5.772594 | 10.520937 | 1.801496 |
[
1.8991138935089111
] | -185,013.637478 | 5.073367 | -325,411.605985 | 0.489284 | 6.322696 |
[
-1.6015625,
-3.37109375,
-3.982421875,
-5.48046875,
1.806640625,
12.7890625
] |
[
-5.6328125,
-2.41015625,
-1.8125,
-0.90771484375,
3.87890625,
4.25390625
] | |
BkiUa_U4ubnjoqNEqKRH
|
\section{Introduction}\label{s:intro}
\noindent
The space $BV$ of functions of bounded variation, consisting of real-valued functions $u$ defined in a domain
of $\mathbb{R}^m$ whose distributional
derivative $Du$ is a finite Radon measure, may contain discontinuous functions and, precisely for this reason,
can be used to model a variety of phenomena, while on the PDE side it plays an important role in the theory
of conservation laws \cite{Daf,Bre}. In more recent times, De Giorgi and the first author introduced the distinguished
subspace $SBV$ of special functions of bounded variation, whose distributional derivative consists of an absolutely
continuous part and a singular part concentrated on a $(m-1)$-dimensional set, called (approximate) discontinuity
set $S_u$. See \cite{AmbFusPal} for a full account of the theory, whose applications include the minimization of the
Mumford-Shah functional \cite{MumSha89} and variational models in fracture mechanics.
In a vector-valued setting, also the spaces $BD$ and $SBD$ play an important role, in connection with
problems involving linearized elasticity and fracture (see also the recent work by Dal Maso on the
space $GSBD$ \cite{Dal11})
It is well know that $|Du|$ vanishes on $\Haus{n-1}$-negligible sets, hence $BV$ and all related spaces can't
be used to describe singularities of higher codimension. For this reason, having in mind application to the
Ginzburg-Landau theory (where typically singularities, e.g. line vortices in $\mathbb{R}^3$ have codimension 2) Jerrard and Soner
introduced in \cite{JerSon02} the space $B_nV$ of functions of bounded higher variation, where $n$ stands
for the codimension: roughly speaking
it consists of Sobolev maps $u:\Omega\to\mathbb{R}^n$ whose distributional Jacobian $Ju$ (well defined, at least as
a distribution, under appropriate integrability assumptions) is representable by a vector-valued measure: in this
case the natural vector space is the space $\Lambda_{m-n}\mathbb{R}^m$ of $(m-n)$-vectors. Remarkable extensions
of the $BV$ theory have been discovered in \cite{JerSon02}, as the counterpart of the coarea formula and of
De Giorgi's rectifiability theorem for sets of finite perimeter. Even before \cite{JerSon02}, the distributional jacobian has been studied
in many fundamental works as \cite{Mor,Bal77,GiaModSou,Sve88,MulSpe95} in connection with variational problems
in nonlinear elasticity (where typically $m=n$ and $u$ represents a deformation map), e.g. to model cavitation effects.
As a matter of fact, since $Ju$ can be equivalently described as a flat $(m-n)$-dimensional current,
an important tool in the study of $Ju$ is the well-developed machinery
of currents, both in the Euclidean and in metric spaces, see
\cite{Fed,Fle66,AmbKir00,Whi99rect}.
The fine structure of the measure $Ju$ has been investigates in subsequent papers: using
precisely tools from the theory of metric
currents \cite{AmbKir00}, De Lellis in \cite{Del00}
characterized $Ju$ in terms of slicing and proved rectifiability of the (measure theoretic)
support $S_u$ of the $(m-n)$-dimensional part of $Ju$, while
in \cite{DelGhi10} the second author and De Lellis characterized the absolutely continuous part of $Ju$ with respect to
$\Leb{m}$ in terms of the Sobolev gradient $\nabla u$.
Also, in \cite{Del00} the analog of the space $SBV$ has been introduced, denoted $SB_nV$: it consists
of all functions $u\in B_nV$ such that $Ju=R+T$, with $\|R\|\ll\Leb{m}$ and $\|T\|$ concentrated on a
$(m-n)$-dimensional set.
The main goal of this paper is to study the compactness properties of $SB_nV$.
Even in the standard $SBV$ theory, a uniform control on the energy of Mumford-Shah type
$$
\int (|u_h|^s+|\nabla u_h|^p) d\Leb{m}+\Haus{m-n}(S_{u_h})
$$
(with $s>0$, $p>1$)
along a sequence $(u_h)$ does not provide a control on $Du_h$. Indeed, only the $\Haus{m-1}$-dimensional
measure of $S_{u_h}$ does not provide a control on the width of the jump.
This difficulty leads \cite{DeGAmb89} to the space
$GSBV$ of \emph{generalized} special functions of bounded variation, i.e. the space of all
real-valued maps $u$ whose truncates $(-N)\lor u\land N$ are all $SBV$. Since both the approximate gradient $\nabla u$
and the approximate discontinuity set $S_u$ behave well under truncation, it turns out that also the
energy of $u_n^N:=(-N)\lor u_n\land N$ is uniformly controlled, and now also $|Du_n^N|$; this is the very first
step in the proof of the compactness-lower semicontinuity theorem in $GSBV$, which shows that
the sequence $(u_h)$ has limit points with respect to local convergence in measure, that any
limit point $u$ belongs to $GSBV$, and that
$$
\int (|u|^s+|\nabla u|^p) d\Leb{m}\leq\liminf_h\int (|u_h|^s+|\nabla u_h|^p) d\Leb{m},\qquad
\Haus{m-1}(S_u)\leq\liminf_h\Haus{m-1}(S_{u_h}).
$$
In the higher codimension case, if we look for energies of the form
$$
\int (|u|^s+|\nabla u|^p+|M(\nabla u)|^\gamma) d\Leb{m}+\Haus{m-n}(S_{u})
$$
(with $s^{-1}+(n-1)/p<1$, $\gamma>1$) now involving also the minors $M(\nabla u)$ of
$\nabla u$, the same difficulty exists, but the truncation argument does not work anymore. Indeed,
the absence of $S_u$, namely the absolute continuity of $Ju$, may be due to very precise cancellation
effects that tend to be destroyed by a left composition, thus causing the appearance of new singular
points (see Example~\ref{ex:monopole} and the subsequent observation). Also, unlike the codimension 1 theory, no ``pointwise'' description
of $S_u$ is presently available.
For these reasons, when looking for compactness properties in $SB_nV$, we have been led to
define the space $GSB_nV$ of \emph{generalized} special functions of bounded higher variation
as the space of functions $u$ such that $Ju$ is representable in the form $R+T$, with
$R$ absolutely continuous with respect to $\Leb{m}$ and $T$ having finite \emph{size}
in an appropriate sense, made rigorous by the slicing theory of flat currents
(in the same vein, one can also define $GB_nV$, but our main object of investigation will be
$GSB_nV$). In particular, for $u\in GSB_nV$ the distribution $Ju$ is not necessarily
representable by a measure. The similarity between $GSBV(\Omega)$ and $GSB_nV(\Omega)$
is not coincidental, and in fact we prove that in the scalar case $n=1$ these two spaces are essentially
the same; on the contrary for $n\geq 2$ their properties are substantially different.
In order to study the $T$ part of $Ju$ we use the notion of
size of flat current with possibly infinite mass developed, even in metric spaces, in \cite{AmbGhi12},
see also \cite{HarDeP12} for the case of currents with finite mass.
The paper is organized as follows: after posing
the proper definitions in the context of metric currents
we briefly review the space $B_nV$ studied in
\cite{JerSon02,Del00,Del01}. In section~\ref{s:size}
we present the notion of size of a flat current,
we relate it to the concept of distributional
jacobian and define our new space of functions $\GSB{m}{n}$.
The main result of the paper is presented
in section \ref{s:compactness}, where
with the help of the slicing theorem we will
generalize to our setting the compactness theorem
of $GSBV$, as well as the closure theorem
in $SB_nV$ due to De Lellis in \cite{Del00,Del01}.
We finally apply the compactness theorem to show the existence
of minimizers for a general class of energies that feature both a volume
and a size term. The model problem is a new functional of Mumford-Shah
type that we here introduce, in the spirit of \cite{DeGCarLea89}.
We analyse its minimization together with
suitable Dirichlet boundary conditions, both in the interior and in the
closure of $\Omega$. In particular we show that minimizers
must be nontrivial (i.e.: $S_u\neq\emptyset$), at least for suitable
boundary data; we also compare our choice of the energy
with the classical $p$-energy of sphere-valued maps, see
\cite{GiaModSou,HarLinWan98,BreCorLie86}.
Regarding the problem in $\overline{\Omega}$
the higher codimension of the singular set allows
concentration of the jacobian at the boundary,
providing some interesting examples that we briefly include
in subsection \ref{ss:traces}.
Similar variational problems in the framework
of cartesian currents have been considered in \cite{Muc10},
where the author proves existence of minimizers in the set of
maps whose graph is a normal current:
the boundaries of these graphs enjoy a decomposition
into vertical parts of integer dimension, inherited from general properties
of integral currents, which relates to the space $B_nV$, see \cite{Muc11}.
In a forthcoming paper \cite{Ghi12} we show how the Mumford-Shah energy
can be approximated, in the sense of $\Gamma$-convergence, by
a family of functionals defined on maps with absolutely continuous
jacobian:
$$
F_\varepsilon(u,v;\Omega):= \int_\Omega (|u|^s+|\nabla u|^p+(v + k_\varepsilon)|M(\nabla u)|^\gamma) d\Leb{m}+\beta\int_\Omega \varepsilon^{q-n}|\nabla v|^q + \frac{(1-v)^n}{\varepsilon^n}\,d\Leb{m},
$$
(here $\beta, \gamma, q, k_\varepsilon$ are suitable parameters, $v:\Omega\rightarrow [0,1]$ is Borel).
Following \cite{AmbTor90,AmbTor92,AlbBalOrl05,ModMor77}
the control variable $v$ dims the concentration of $M(\nabla u)$:
the price of the transition between $0$ and $1$ is captured
by the Modica-Mortola term which detects $(m-n)$-dimensional sets.
We occasionally appeal to the metric theory of currents because
the main tool in the definition of size and in the proof of
the rectifiability theorem is the slicing technique,
a basic ingredient of the metric theory. For instance the argument
in \cite{AmbGhi12} proving the rectifiability
of currents with finite size uses Lipschitz restrictions and maps of
metric bounded variation taking values into an appropriate space
of flat chains with a suitable hybrid metric. However, no significant
simplification comes from the Euclidean theory,
except the fact that suffices to consider linear instead of
Lipschitz maps.
\subsection{Acknowledgements}
The authors wish to thank Camillo De Lellis, Nicola Fusco,
Bernd Kirchheim, Domenico Mucci and Emanuele Spadaro
for many useful discussions and comments.
This work has been supported by the ERC grant ADG GeMeThNES.
\section{Distributional Jacobians}\label{s:jac}
\noindent
We begin by fixing some basic notions about currents
and recalling some properties of the distributional jacobian.
\subsection{Exterior algebra and projections}
Our ambient space will be $\mathbb{R}^m$ with the standard basis
$e_1,\dots,e_m$ and its dual $e^1,\dots,e^m$.
For every $1\leq k \leq m$ we let
$$
{\bf O}_{k}=\big\{\pi:\mathbb{R}^m\rightarrow\mathbb{R}^{k}:
\pi\circ\pi^* = I_k\big\}
$$
be the space of orthogonal projections of rank $k$.
We will also need to fix coordinates according to some
projection $\pi\in{\bf O}_{k}$: we agree that
$\mathbb{R}^m\ni z = (x,y)\in\mathbb{R}^{k}\times\mathbb{R}^{m-k}$ are orthogonal coordinates
with positive orientation such that $\pi(z)=x$.
In particular we let $A^x = A\cap\pi^{-1}(x)$ be the
restriction of any $A\subset\mathbb{R}^m$ to the fiber $\pi^{-1}(x)$
and $i^x=\mathbb{R}^{m-k}\rightarrow\mathbb{R}^m$ be the isometric
injection $i^x(y) = (x,y)$.
As customary the symbols $\Lambda_k \mathbb{R}^m$
and $\Lambda^k \mathbb{R}^m$ will respectively denote the spaces of
$k$-vectors and $k$-covectors in $\mathbb{R}^m$.
The contraction operation
$\res:\Lambda_{q}\mathbb{R}^m\times\Lambda^{p}\mathbb{R}^m\rightarrow \Lambda_{q-p}\mathbb{R}^m$
between a $q$-vector $\zeta$ and a $p$-covector $\alpha$, with $q\geq p$, is defined as:
\begin{equation}\label{restr}
\langle \zeta\res\alpha,\beta \rangle = \langle \zeta,\alpha\wedge\beta\rangle
\qquad\text{ whenever }
\beta\in\Lambda^{q-p}\mathbb{R}^m.
\end{equation}
If $L:\mathbb{R}^m\rightarrow\mathbb{R}^n$ is linear then
\begin{equation}\label{M_nL}
M_nL:=e_1\wedge\dots\wedge e_m\res L^1\wedge\dots\wedge L^n \in\Lambda_{m-n}\mathbb{R}^m
\end{equation}
represents the collection of determinants of $n\times n$
minors of $L$.
In fact, if $\underline{i}:\{1,\dots,m-n\}\rightarrow \{1,\dots,m\}$
is an increasing selection of indexes, and if
$\overline{i}:\{1,\dots,n\}\rightarrow \{1,\dots,m\}$
is the complementary increasing selection,
then the $e_{\underline{i}}$ component of $M_nL$ is
\begin{equation}\label{M_nLi}
\langle M_n L,e^{\underline{i}}\rangle =
\langle e_1\wedge\dots\wedge e_m, L^1\wedge\dots\wedge L^n\wedge e^{\underline{i}}\rangle
= (-1)^{\sigma}\det([L]_{\overline{i}}),
\end{equation}
where $[L]_{\overline{i}}$ is the $n\times n$ submatrix
$L^{\ell}_j$ with $j=\overline{i}(1),\dots,\overline{i}(n)$,
$\ell=1,\dots,n$ and $\sigma$ is the sign of the permutation
$$
(1,\dots,m)\mapsto (\overline{i}(1),\dots,\overline{i}(n),\underline{i}(1),\dots,\underline{i}(m-n)).
$$
When ${\rm rk}(L)=n$, choosing an orthonormal frame $(e_i)$ so
that $\ker(L)=<e_{n+1},\dots,e_m>$ we have $L = (A,{\bf 0})$ and
by \eqref{M_nLi} $M_nL = \det(A)e_{n+1}\wedge\dots\wedge e_m$.
In particular $M_nL$ is a simple $(m-n)$-vector.
Recall that the spaces $\Lambda_k\mathbb{R}^m$ and $\Lambda^k\mathbb{R}^m$
can be endowed with two different pairs of dual norms.
The first one is called norm, it is denoted by $|\cdot|$ and
it comes from the scalar product where the multivectors
\begin{equation}\label{frame}
\{e_{i(1)}\wedge\dots\wedge e_{i(k)}\}_{i} \quad\text{ and }
\quad \{e^{i(1)}\wedge\dots\wedge e^{i(k)}\}_{i}
\end{equation}
indexed by increasing maps $i:\{1,\dots,k\}\rightarrow \{1,\dots,m\}$
form a pair of dual orthonormal bases. The second one is
called mass, comass for the space of covectors, and it
is defined as follows: the comass of $\phi\in\Lambda^k\mathbb{R}^m$ is
$$
\|\phi\| := \sup \big\{ \langle \phi,v_1\wedge\dots\wedge v_k\rangle : v_i\in \mathbb{R}^m, |v_i|\leq 1 \big\};
$$
and the mass of $\xi\in\Lambda_k\mathbb{R}^m$ is defined, by duality, by
$$
\|\xi\|:=\sup\big\{\langle \xi,\phi\rangle : \|\phi\|\leq 1\big\}.
$$
As described in \cite[1.8.1]{Fed}, in general $\|\xi\|\leq|\xi|$
and equality holds if and only if $\xi$ is simple.
Therefore $|M_nL| = \|M_nL\|$. Moreover using the Pitagora's Theorem
for the norm and Binet's formula we have the following relation:
\begin{align}\label{normsup}
\sup_{\pi\in{\bf O}_{m-n}}|M_nL\res d\pi| & = \sup_{\pi\in{\bf O}_{m-n}}|d\pi(M_nL)| =
\sup_{\pi\in{\bf O}_{m-n}}\big|\sum_{\underline{i}}M_nL^{\underline{i}}\,d\pi(e_{\underline{i}})\big| \notag\\
& \leq \sup_{\pi\in{\bf O}_{m-n}}|M_nL|\big(\sum_{\underline{i}}|d\pi(e_{\underline{i}})|^2\big)^{\frac 12} = |M_nL|\sup_{\pi\in{\bf O}_{m-n}}|\det(\pi\circ\pi^*)| = |M_nL|
\end{align}
where $d\pi$ stands for $d\pi^1\wedge\dots\wedge d\pi^{m-n}$, and
the equality is realized by the orthogonal projection onto $\ker(L)$.
We adopt the convention of choosing the mass and comass norms
to measure the length of $k$-vectors and $k$-covectors respectively.
\subsection{Currents in $\Omega\subset\mathbb{R}^m$}
We briefly recall the basic definitions and properties
of classical currents in $\mathbb{R}^m$. This theory was introduced
by De Rham in \cite{DeR}, along the lines of the previous
work on distributions by Schwartz \cite{Sch}, and the subsequently
put forward by Federer and Fleming in \cite{FedFle60};
we refer to \cite{Fed} for a complete account of it.
The classical framework is best for treating the concepts of
distributional jacobian in a subset of $\mathbb{R}^m$; however we
will need to use the metric theory of Ambrosio and Kirchheim
\cite{AmbKir00} to define the concept of size and of concentration
measure. We will clearly outline the interplay between the two
approaches.
We give for granted the concepts of derivative, exterior differentiation, pull-back
and support of a test functions: they can all be defined
by expressing the form in the coordinates given by the
frame \eqref{frame}, see \cite[4.1.6]{Fed}.
We begin by defining the space of smooth, compactly supported test forms:
\begin{definition}[Smooth test forms]\label{d:smoothforms}
We let $\mathscr{D}^k(\Omega)$ be the space of smooth, compactly supported
$k$-differential forms:
\begin{equation}
\mathscr{D}^k(\Omega)= \bigcup_{K\Subset\Omega}\mathscr{D}_K^k(\Omega),
\qquad \mathscr{D}^k_K(\Omega)=\big\{\omega\in
C^{\infty}(\Omega,\Lambda^k\mathbb{R}^m),\,{\rm spt}(\omega)\subset K\big\}.
\end{equation}
Each space $\mathscr{D}_K^k(\Omega)$ is endowed with the topology
given by the seminorms
$$p_{K,j}(\omega)=\sup\{\|D^{\alpha}\omega(x)\|, x\in K,
|\alpha|\leq j\},$$ and $\mathscr{D}^k(\Omega)$ is endowed with the finest topology
making the inclusions $\mathscr{D}^k_K(\Omega)\hookrightarrow\mathscr{D}^k(\Omega)$
are continuous.
\end{definition}
This topology is locally convex, translation invariant and Hausdorff;
moreover a sequence $\omega_j\rightarrow\omega$ in $\mathscr{D}^k(\Omega)$ if
and only if there exists $K\Subset\Omega$ such that ${\rm spt}(\omega_j)\subset K$ and
$p_{K,j}(\omega_j-\omega)\rightarrow 0$ for every $j\geq 0$.
\begin{definition}[Classical currents and weak* convergence]\label{d:current}
A current $T$ is a continuous linear functional on $\mathscr{D}^k(\Omega)$. The space
of $k$-currents is denoted by $\mathscr{D}_k(\Omega)$.
We say that a sequence $(T_h)$ weak* converges to $T$, $T_h\overset{*}{\rightharpoonup} T$,
whenever
\begin{equation}\label{wstar}
T_h(\omega)\rightarrow T(\omega)\qquad \forall\omega\in\mathscr{D}^k(\Omega).
\end{equation}
\end{definition}
The support of a current is
$$
{\rm spt}(T):= \bigcap\{C: T(\omega)=0\,\,\forall\omega\in\mathscr{D}^k(\Omega),\,{\rm spt}(\omega)\cap C = \emptyset\}.
$$
The boundary operator is the adjoint of exterior differentiation:
$$
\partial T(\omega):=T(d \omega);
$$
let also $\phi:\Omega\rightarrow\mathbb{R}^{p}$
be a proper Lipschitz map: we let the push-forward of $T\in\mathscr{D}_k(\Omega)$
via $\Phi$ by duality:
$$
(\Phi_{\#}T)(\omega) := T(\Phi^{\#}\omega)\qquad \forall \omega\in\mathscr{D}^{k}(\mathbb{R}^p).
$$
According to \eqref{restr} given $T\in\mathscr{D}_k(\Omega)$ and
$\tau\in\mathscr{D}^{\ell}(\Omega)$ with $\ell\leq k$ we set the restriction
$$
(T\res\tau)(\eta) = T(\tau\wedge\eta) \qquad \forall \eta\in\mathscr{D}^{k-\ell}(\Omega).
$$
\begin{definition}[Finite mass and Normal currents]\label{d:mass}
We say that $T\in\mathscr{D}_k(\Omega)$ is a current of finite
mass if there exists a finite Borel measure $\mu$ in $\Omega$
such that
\begin{equation}\label{mass}
|T(\omega)|\leq \int_{\Omega}\|\omega(x)\|d\mu(x)\qquad \forall \omega\in\mathscr{D}^k(\Omega).
\end{equation}
The total variation of $T$ is the minimal $\mu$ satisfying
\eqref{mass} and is denoted by $\|T\|$, and the mass
$\mathbf{M}(T):=\|T\|(\Omega)$.
As customary we let $\mathbf{M}_k(\Omega)$ be the space of
finite mass $k$-dimensional currents
and $\N_k(\Omega)$ be the subspace of normal currents:
$$
\N_k(\Omega)=\mathbf{M}_k(\Omega)\cap\{T:\partial T\in\mathbf{M}_{k-1}(\Omega)\}.
$$
\end{definition}
Therefore every finite mass current can be represented as
$T= \overset{\rightarrow}{T}\wedge \|T\|$, and admits
and extension to $k$-forms with bounded Borel coefficients.
In particular we can restrict every finite mass current $T$ to
every open set $A$.
We denote
$$
{\bf E}^m = e_1\wedge\dots\wedge e_m\wedge\Leb{m}
$$
the top dimensional $m$-current representing the Lebesgue
integration on $\mathbb{R}^m$ with the standard orientation.
As explained in \cite[4.1.7, 4.1.18]{Fed}, \cite[3.2]{AmbKir00},
every function $f\in L^1(\Omega,\Lambda_{m-k}\mathbb{R}^m)$
induces a $k$-current of finite mass via the action
\begin{equation}\label{Eresf}
({\bf E}^m\res f)(\omega)=\int_{\Omega} \langle f\wedge\omega, e_1\wedge\dots\wedge e_m\rangle \,d\Leb{m} \qquad \forall \omega\in\mathscr{D}^{k}(\Omega).
\end{equation}
Note that $f_h\rightharpoonup f$ weakly in $L^1$ entails ${\bf E}^m\res f_h\overset{*}{\rightharpoonup} {\bf E}^m\res f$.
\subsection{Flat currents}
In order to treat objects with possibly infinite mass, the
right subspace of $\mathscr{D}^k(\Omega)$ retaining some useful properties
such as slicing and restrictions is the space of flat currents.
\begin{definition}[Flat norm and flat currents, {\cite[4.1.12]{Fed}}]\label{d:flat}
For every $\omega\in\mathscr{D}^k(\Omega)$ we let
$$
\mathbf{F}(\omega)=\max\big\{\sup_{x\in\Omega}\|\omega(x)\|,\sup_{x\in\Omega}\|d\omega(x)\|\big\}.
$$
The flat norm is defined as
\begin{align}
\mathbf{F}(T) &= \inf\{\mathbf{M}(T - \partial Y) + \mathbf{M}(Y):\, Y\in \mathbf{M}_{k+1}(\Omega)\}\label{flatinf}\\
&=\sup\{T(\omega):\,\omega\in\mathscr{D}^k(\Omega),\,
\mathbf{F}(\omega)\leq 1\}. \label{flatsup}
\end{align}
The space $\mathbf{F}_k(\Omega)$ of flat $k$-dimensional
currents in an open subset $\Omega \subset\mathbb{R}^m$
is the $\mathbf{F}$-completion of $\N_k(\Omega)$ (see \cite{Fle66}
and \cite[4.1.12]{Fed} for the equivalence of \eqref{flatinf} and \eqref{flatsup}).
\end{definition}
It is straightforward to prove that $\mathbf{F}$ is a norm;
furthermore if $T$ is flat, so is $\partial T$ and
$$
\mathbf{F}(\partial T)\leq \mathbf{F}(T) \leq \mathbf{M}(T).
$$
Throughout all the paper we will deal with three notions of convergence:
\begin{itemize}
\item the convergence w.r.t. the flat norm $\mathbf{F}$ defined in \eqref{flatinf} above;
\item the weak convergence in $L^p$, $1\leq p <\infty$, denoted
by $\rightharpoonup$;
\item the weak* convergence of currents \eqref{wstar}.
\end{itemize}
The map \eqref{Eresf} $f\mapsto {\bf E}^m\res f$ embeds $L^1(\Omega,\Lambda_{m-k}\mathbb{R}^m)$
into $\mathbf{F}_k(\Omega)$ by \cite[4.1.18]{Fed}, and the three aforementioned
topologies are ordered from the strongest to the weakest.
\subsection{Slicing}
As explained in \cite[4.2]{Fed} and \cite{AmbGhi12}, every
$T\in\mathbf{F}_k(\Omega)$ can be sliced via a Lipschitz map
$\pi\in{\rm Lip}(\Omega,\mathbb{R}^{\ell})$, $1\leq\ell\leq k$:
the result is a collection of currents
$$
\langle T,\pi,x\rangle \in\mathbf{F}_{k-\ell}(\Omega) \qquad
\mbox{uniquely determined up to $\Leb{\ell}$ negligible sets}
$$
expressing the action of $T$ against tensor product
forms $(\phi\circ\pi) d\pi\wedge\psi $, for
$\phi\in\mathscr{D}^{0}(\mathbb{R}^{\ell})$ and $\psi\in\mathscr{D}^{k-\ell}(\Omega)$:
$$
T\big( (\phi\circ\pi) d\pi \wedge\psi\big) =
\int_{\mathbb{R}^{\ell}} \phi(x) \langle T,\pi,x\rangle(\psi)d\Leb{\ell}(x).
$$
The slices satisfy several properties: amongst them we recall
\begin{equation}\label{slicecontroimm}
\text{$\langle T,\pi,x\rangle$ is concentrated on $\pi^{-1}(x)$ for
$\Leb{\ell}$-a.e. $x\in\pi(\Omega)$,}
\end{equation}
\begin{equation}
\int_{\pi(\Omega)}\mathbf{F}(\langle T,\pi,x\rangle)\,d\Leb{\ell}(x) \leq {\rm Lip}(\pi)^{\ell}\mathbf{F}(T), \label{fubinislices}
\end{equation}
and we refer to \cite[4.2.1]{Fed} and to \cite{AmbKir00} for a general account
in the Euclidean and general metric settings.
We stress the following fact, which is a key tool to extend
many properties like restrictions and slicing from normal
to flat currents, and that will be used
later on. Suppose $(T_h)\subset\mathbf{F}_k(\Omega)$ satisfy
$$
\sum_h \mathbf{F}(T_{h+1}-T_h)<+\infty
$$
and let $\pi\in{\rm Lip}(\Omega,\mathbb{R}^{\ell})$ fixed.
Then
$$
\mathbf{F}\big(\langle T_h,\pi,x\rangle - \langle T,\pi,x\rangle\big)\rightarrow 0
$$
for $\Leb{\ell}$-almost every $x\in\mathbb{R}^{\ell}$.
Recall that for the finite mass current $R = \rho\Leb{m}$ with
$\rho\in L^1(\Omega,\Lambda_{k}\mathbb{R}^m)$ Federer's coarea formula implies that
at almost every $x\in\mathbb{R}^{\ell}$ it holds:
\begin{equation}\label{sliceac}
\langle R,\pi,x\rangle = (\rho(x,\cdot)\res d\pi)\,\Haus{m-\ell} \res\pi^{-1}(x).
\end{equation}
\subsection{Distributional jacobian}
We will assume throughout all
the paper that $m\geq n$ are positive integers
and that $p$ and $s$ are positive exponents satisfying
\begin{equation}\label{exp}
\quad \frac{1}{s}+\frac{n-1}{p} \leq 1.
\end{equation}
The definition of distributional jacobian takes
advantage of the divergence structure of jacobians
$$
d(u^1 du^2\wedge\dots\wedge du^n) = du^1 \wedge\dots\wedge du^n \quad \forall u\in C^1(\mathbb{R}^n,\mathbb{R}^n),
$$
which allows to pass the exterior derivative to the test
form and hence weakens the minimal regularity assumptions
on the map $u$.
\begin{definition}[Distributional Jacobian]\label{d:djac}
Let $u\in W^{1,p} (\Omega, \mathbb{R}^n) \cap L^{s}(\Omega, \mathbb{R}^n)$.
We denote by $j(u)$ the $(m-n+1)$-dimensional flat current
\begin{equation}\label{prejac}
\langle j(u),\omega \rangle :=(-1)^n \int_{\Omega} u^1 du^2\wedge\dots\wedge du^n\wedge \omega\qquad
\forall \omega \in \mathscr{D}^{m-n+1}(\Omega);
\end{equation}
we define the distributional Jacobian of $u$ as the $(m-n)$-dimensional
flat current
\begin{equation}\label{jac}
Ju := \partial j(u)\in\mathbf{F}_{m-n}(\Omega).
\end{equation}
\end{definition}
A few observations are in order: first of all the integrability
assumption $u\in W^{1,p}\cap L^{s}$ and the exponent bound \eqref{exp}
ensure that $j(u)$ is a well-defined flat current of finite mass,
since it acts on test forms as the
integration against an $L^1(\Omega,\Lambda_{m-n+1}\mathbb{R}^m)$
function: $j(u) = (-1)^n{\bf E}^m\res u^1du^2\wedge\dots\wedge du^n$.
As a consequence $Ju\in\mathbf{F}_{m-n}(\Omega)$ as
declared in \eqref{jac}. Furthermore for $p\geq \frac{mn}{m+1}$
the constraint \eqref{exp} is satisfied with the Sobolev
exponent $p^*$ in place of $s$, hence definition \ref{d:djac}
makes sense for $u\in W^{1,p}$ in this range
of summability.
In \cite{BreNgu11} the authors showed that $Ju$ can
be defined in the space $W^{1-\frac{1}{m},m}(\Omega)$,
which contains $L^s\cap W^{1,p}$ for every $s,p$ as in \eqref{exp}.
This extension exploits the trace space nature of $W^{1-\frac{1}{m},m}$,
expressing $Ju$ as a boundary integral in $\mathbb{R}^m_+$.
Finally in the special situation $n=1$ the minimal requirement to
give meaning to \eqref{prejac} is $u\in L^1(\Omega)$, and the Jacobian
reduces to the distributional derivative $Ju = -\partial({\bf E}^m\res u)$:
\begin{equation}\label{scalar}
\big\langle Ju,\sum_i (-1)^{i-1}\omega_i \widehat{dx^i} \big\rangle =
-\sum_i\int_{\Omega}u\frac{\partial \omega_i}{\partial x^i}dx =
\sum_i \langle D_i u,\omega_i\rangle .
\end{equation}
Regarding the convergence properties of these currents,
we note the following:
\begin{proposition}\label{p:conv}
Let
$u_h,u \in W^{1,p}(\Omega,\mathbb{R}^n)\cap L^{s}(\Omega,\mathbb{R}^n)$
satisfy
\begin{itemize}
\item $u_h\rightarrow u$ in $L^s(\Omega,\mathbb{R}^n)$,
\item $\nabla u_h \rightharpoonup \nabla u$ weakly in $L^p(\Omega,\mathbb{R}^{n\times m})$.
\end{itemize}
Then $\mathbf{F}(Ju_h -Ju)\rightarrow 0$.
\end{proposition}
\begin{proof}
Let us rewrite the difference $u_h^1 du_h^2\wedge\dots\wedge du_h^n - u^1 du^2\wedge\dots\wedge du^n$
in the following way:
\begin{multline*}
u_h^1 du_h^2 \wedge\dots\wedge du_h^n - u^1 du^2\wedge\dots\wedge du^n =\\
= (u_h^1 - u^1) du_h^2\wedge\dots\wedge du_h^n +
u^1\sum_{k=2}^n du_h^2\wedge du_h^{k-1}\wedge d(u^k_h - u^k) \wedge du^{k+1}\wedge\dots\wedge du^n.
\end{multline*}
We can actually write each addendum in the last
summation as
\begin{equation*}
-(u^k_h - u^k)du_h^2\wedge du_h^{k-1}\wedge du^1 \wedge du^{k+1}\wedge\dots\wedge du^n + d\zeta^k_h,
\end{equation*}
where we set
\begin{equation}\label{chain}
\zeta^k_h = (-1)^{k-2} u^1(u^k_h - u^k) du_h^2
\wedge\dots\wedge du_h^{k-1}\wedge du^{k+1}\wedge\dots\wedge du^n
\in L^1(\Omega,\Lambda^{n-2}\mathbb{R}^m).
\end{equation}
Notice that we can always assume $s\geq p$, hence
$\zeta^k_h\in L^1$.
To show \eqref{chain} it is sufficient to
approximate both $u$ and $u_h$ in the strong
topology with regular functions and apply the
Leibniz rule; the same approximation shows that
$d\zeta^k_h\in L^1$ and hence
$\int_{\Omega} d\zeta^k_h\wedge d\omega =0$ for each
$\omega \in \mathscr{D}^{m-n+1}(\Omega)$. By the calculations
above we can estimate
\begin{align*}
|\langle Ju_h - Ju,\omega\rangle| & =
|\langle j(u_h) - j(u),d\omega\rangle| \\
&\leq \|d\omega\|_{L^{\infty}}
\sum_{k=1}^n \|u^k_h - u^k\|_{L^s}\|du^1_h\|_{L^p}\cdots
\|du^{k-1}_h\|_{L^p}\|du^{k+1}_h\|_{L^p}\cdots\|du^n_h\|_{L^p} \\
& \leq C\,\mathbf{F}(\omega) \left(\sup_h \|\nabla u_h\|_{L^p}\right)^{n-1} \|u_h-u\|_{L^s}.
\end{align*}
Taking the supremum on test functions $\omega$ with
$\mathbf{F}(\omega) \leq 1$
we immediately obtain the asserted convergence.
\end{proof}
A natural question is the relation between the
summability exponent $p$ and the regularity
of the distribution $Ju$. There is a main
difference between $p\geq n$ and $p<n$: if the
gradient $\nabla u$ has a sufficiently high summability,
then $Ju$ is an absolutely continuous measure.
In fact let $u_h=u*\rho_h$,
where $\rho_h$ is a standard approximation of the identity:
since $p\geq n$ the continuous embedding
$W^{1,p} \hookrightarrow W^{1,n}_{{\rm loc}}$
implies that $u_h\rightarrow u$ both in
$W^{1,p}\cap L^{s}$ and $W^{1,n}_{{\rm loc}}$.
Taking a test form
$\psi$ with compact support
we can use Proposition \ref{p:conv} to pass to the
limit in the integration by parts formula
$$
\langle Ju_h,\psi\rangle =
(-1)^n \int_{\Omega} u_h^1 du_h^2\wedge\dots\wedge du_h^n\wedge d\psi =
\int_{\Omega} du_h^1\wedge du_h^2\wedge\dots\wedge du_h^n\wedge \psi,
$$
yielding
$Ju = {\bf E}^m \res du^1\wedge\dots\wedge du^n$.
On the other hand when $p<n$ there are several examples
of functions whose jacobian is not in $L^1$: for instance when $m=n$
the ``monopole'' function $u(x):= \frac{x}{|x|}$ satisfies
$Ju = \Leb{n}(B_1) \llbracket 0\rrbracket$, where
$\llbracket 0\rrbracket$ is the Dirac's mass in the origin.
More generally:
\begin{example}[Zero homogeneous functions, $m=n$, {\cite[3.2]{JerSon02}}]\label{ex:monopole}
Let $\gamma:S^{n-1}\rightarrow \mathbb{R}^n$ be smooth and let $u(x):=\gamma(\tfrac{x}{|x|})$.
Then
\begin{equation}\label{curve}
Ju =Area(\gamma) \llbracket 0 \rrbracket
\end{equation}
where $Area(\gamma)$ is the signed area enclosed by $\gamma$.
\end{example}
\begin{proof}
Outside the origin $u$ is smooth and takes values into
the $(n-1)$-dimensional submanifold $\gamma(S^{n-1})$,
hence ${\rm spt}(Ju)\subset\{0\}$.
Set $t=|x|$ and $y=\tfrac{x}{|x|}$:
then $d\gamma = \sum_i \frac{\partial u }{\partial x^k} tdy^k$, so
$$t^{n-1}du^2\wedge \dots \wedge du^n = d\gamma^2\wedge \dots\wedge d\gamma^n
\in \Lambda^{n-1}{\rm Tan\,} S^{n-1}.$$
Hence the only term of $d\omega$ surviving in the wedge product
is $\tfrac{\partial \omega}{\partial t}dt$. Therefore
\begin{align}
(-1)^n\int_{\mathbb{R}^n}u^1 &du^2\wedge\dots\wedge du^n\wedge d\omega =
(-1)^n\int_{\mathbb{R}^n}\frac{\partial \omega}{\partial t} \gamma^1(y)
du^2\wedge\dots\wedge du^n\wedge dt\notag\\
&= -\int_{\mathbb{R}^n}\frac{\partial \omega}{\partial t} u^1(y) dt\wedge du^2\wedge\dots\wedge du^n \notag\\
&= -\int_{\partial B_t}\left( \int_0^{+\infty} \frac{\partial \omega}{\partial t} dt \right)u^1(x) du^2\wedge\dots\wedge du^n \notag \\
&= -\int_{\partial B_1}\left( \int_0^{+\infty} \frac{\partial \omega}{\partial t} dt \right)\gamma^1(y) d\gamma^2\wedge\dots\wedge d\gamma^n \notag \\
& =\omega(0) \int_{S^{n-1}}\gamma^1(y) d\gamma^2\wedge\dots\wedge d\gamma^n. \label{lastintegral}
\end{align}
Setting $\Upsilon(t,y):=t\gamma(y)$ the Lipschitz extension
to the unit ball $B_1\subset \mathbb{R}^n$, by Stokes' Theorem \eqref{lastintegral} equals to
\begin{align}
\omega(0)\int_{\partial B_1}\Upsilon^1(1,y) d\Upsilon^2\wedge\dots\wedge d\Upsilon^n
&= \omega(0) \int_{B_1}d\Upsilon^1\wedge\dots\wedge d\Upsilon^n \notag\\
&= \omega(0)\int_{B_1}\det(\nabla\Upsilon) dx
=\omega(0) \int_{\mathbb{R}^n}\deg(\Upsilon,w,B_1)dw. \label{signedarea}
\end{align}
It is well known that \eqref{signedarea} represents the
signed area enclosed by the surface $\gamma(S^{n-1})$.
\end{proof}
This example immediately outlines one of the biggest differences
with the scalar case.
Consider as in \cite{JerSon02} the ``eight-shaped" loop in $\mathbb{R}^2$ :
\begin{equation}\label{ottosemplice}
\gamma(\theta)=
\left\{
\begin{array}{ll}
& (\cos(2\theta)-1,\sin(2\theta))\qquad\mbox{ for }\theta\in[0,\pi],\\
&(1-\cos(2\theta),\sin(2\theta))\qquad\mbox{ for }\theta\in[\pi,2\pi].
\end{array}
\right.
\end{equation}
and let $u$ be the zero homogeneous extension.
$\gamma$ encloses the union $B_1(-e_1)\cup B_1(e_1)$ with
degree $+1$ and $-1$ respectively: in light of \eqref{curve} $Ju=0$.
However a left composition with a smooth map $F:\mathbb{R}^2\rightarrow\mathbb{R}^2$
easily destroys the cancellation, causing the appearance
of a Dirac's mass in $0$. Hence the estimate
\begin{equation}\label{Hadamard}
\|J(F\circ u)\|\leq {\rm Lip}(F)^2\|Ju\|
\end{equation}
doesn't hold anymore if $u$ is not regular. Note that this
phenomenon does not appear for $n=1$ and $u\in BV(\Omega)$,
as Vol'pert chain rule provides exactly the estimate \eqref{Hadamard}
(see \cite[Theorem 3.96]{AmbFusPal}).
The failure of \eqref{Hadamard} is related to the validity
of a strong coarea formula for jacobians of vector valued maps,
namely equation (1.7) in \cite{JerSon02}.
The (weak) coarea amounts instead to decompose the current $Ju$ of
a $B_nV$ map (see next paragraph for the definition) into the
superposition of integral currents corresponding to the level
sets of $u$: letting $u_{y}(x):=\tfrac{u(x)-y}{|u(x)-y|}$, it
is proved in \cite[Theorem 1.2]{JerSon02} that
$$
Ju=\frac{1}{\Leb{n}(B_1)}\int_{\mathbb{R}^n} Ju_y\,dy
$$
as currents. However, because of some cancellation phenomena
like in \eqref{ottosemplice}, \eqref{Hadamard}, the strong version
of the coarea formula
\begin{equation}\label{strongcoarea}
\|Ju\|=\frac{1}{\Leb{n}(B_1)}\int_{\mathbb{R}^n} \|Ju_y\|\,dy
\end{equation}
might well fail. Once again observe that for $n=1$ the
equality \eqref{strongcoarea} has been proved by Fleming and
Rishel to holds for every $u\in BV$, see \cite[Theorem 3.40]{AmbFusPal}.
For a more detailed analysis we refer to
\cite{JerSon02,Del01,MulSpe95,DeP12}.
For later purposes we report the dipole construction,
introduced by Brezis, Coron and Lieb in \cite{BreCorLie86}:
it consists of a map taking values into a sphere which is constant
outside a prescribed compact set, its jacobian is the difference of two Dirac's masses
and satisfies suitable $W^{1,p}$ estimate. We write $(y,z)\in\mathbb{R}^{n-1}\times\mathbb{R}$ and denote by
$\mathcal{N} = (0,1) \in S^{n-1}$ the north pole.
\begin{example}[Dipole, {\cite[2.2]{BreNgu11}}]\label{ex:dipole}
Let $n\geq 2$, $\nu\in \Z$, $\rho>0$: there exists a map $f_{\nu,\rho}:\mathbb{R}^n\rightarrow S^{n-1}$
with the following properties:
\begin{itemize}
\item $f_{\nu,\rho}\equiv\mathcal{N}$ outside $\{|y|+|z|<\rho\}$;
\item $f_{\nu,\rho} - \mathcal{N} \in W^{1,p}(\mathbb{R}^n,\mathbb{R}^n)$ for every $p<n$ with estimates
$$
\|\nabla f_{\nu,\rho}\|^p_{L^p} \leq C_p\, \nu^{\frac{p}{n-1}}\rho^{n-p};
$$
\item $Jf_{\nu,\rho} = \nu\Leb{n}(B^n_1)\big(\llbracket (0,-\rho) \rrbracket - \llbracket (0,\rho) \rrbracket\big)$.
\end{itemize}
\end{example}
The locality of the dipole construction allows
to glue several copies of dipoles to produce interesting examples.
\begin{example}[Finiteness of $\mathbf{F}(Jg)$ does not imply finiteness of $M(Jg)$]\label{ex:dipole1}
We build a map $g$ such that $\mathbf{F}(Jg)$ is finite and $Jg$ has an infinite mass.
The construction starts from $f_{\nu,\rho}(\cdot,0):\mathbb{R}^{n-1}\rightarrow S^{n-1}$,
which is a smooth map equal to $\mathcal{N}$ outside $B^{n-1}_{\rho}$ and such that
$\deg(f_{\nu,\rho}(\cdot,0)) = \nu$. For $|z|<\rho$
we extend by $f_{\nu,\rho}(y,z) = f_{\nu,\rho}(\tfrac{\rho y}{\rho- |z|},0)$
and we set $f_{\nu,\rho}\equiv\mathcal{N}$ at points $|z|\geq\rho$.
Choosing a sequence of positive radii $(\rho_k)$
we can glue an infinite number of dipoles along the $z$ axis:
\begin{equation*}
g(y,z) = f_{1,\rho_k}(y, z -z_k)
\qquad\mbox{ for }\quad |z - z_k|\leq 2\rho_k,
\end{equation*}
where $z_0=0$ and $z_k = 2\sum_{j=0}^{k}\rho_j$.
The function $g$ belongs to $L^{\infty}\cap W^{1,p}$ provided $\sum_k\rho_k^{n-p}<\infty$:
in this case note that
$$
Jg = \Leb{n}(B_1)\sum_k \llbracket (0,z_k-\rho_k)\rrbracket - \llbracket(0,z_k+\rho_k)\rrbracket,
$$
hence $\mathbf{F}(Jg)\leq 2\Leb{n}(B_1)\sum_k\rho_k<\infty$ but $\mathbf{M}(Jg)=+\infty$.
\end{example}
More complicated examples, including maps such that
$Ju$ is not even a Radon measure, are presented in
\cite{JerSon02, MulSpe95, AlbBalOrl05}.
\subsection{Functions of bounded $n$-variation}\label{ss:bnv}
The space of functions of bounded $n$-variation has
been introduced by Jerrard and Soner in the fundamental
paper \cite{JerSon02}.
\begin{definition}\label{d:B_nV}
$B_nV(\Omega,\mathbb{R}^n)$ is the space of functions
$u\in W^{1,p}(\Omega,\mathbb{R}^n)\cap L^s(\Omega,\mathbb{R}^n)$
such that $Ju$ is a current of finite mass.
\end{definition}
Notice that in this case the action of $Ju$ can be
represented as the integration against a $\Lambda_{m-n}\mathbb{R}^m$-valued
Radon measure. Clearly the following statement is an easy
improvement of \ref{p:conv}, since every continuous function
with compact support can be uniformly
approximated by a Lipschitz function
with the same $L^{\infty}$ bound.
\begin{corollary}
Assume the same hypotheses of Proposition \ref{p:conv}.
If in addition
$$
(u_h)\subset B_nV(\Omega,\mathbb{R}^n) \qquad\text{and}\qquad \|Ju_h\|(\Omega) \leq C <\infty
$$
then
$u\in B_nV(\Omega,\mathbb{R}^n)$ and $Ju_h\overset{*}{\rightharpoonup} Ju$
in the sense of measures.
\end{corollary}
Furthermore it is a general result
on normal currents, contained for example in \cite[4.1.21]{Fed} and
in \cite[Theorem 3.9]{AmbKir00} for the metric spaces statement,
that if $T$ is a normal $k$-current then $\|T\|\ll\Haus{k}$.
In light of the Example \ref{ex:cantor}
(with a trivial extension in case $m>n$) $\|Ju\|\ll\Haus{m-n}$
is the only possible bound on the Hausdorff dimension of
$Ju$.
As in the theory of $BV$ functions $Ju$ satisfies a canonical
decomposition in three mutually singular parts according
to the dimensions (see \cite{Del00,AmbFusPal,JerSon02}):
\begin{equation}\label{dec}
Ju = \nu \cdot \Leb{m} + J^c u + \theta\cdot \Haus{m-n}\res S_u
\end{equation}
where the decomposition is uniquely determined by these three properties:
\begin{itemize}
\item $\nu =\frac{dJu}{d\Leb{m}} \in L^1(\Omega, \Lambda_{m-n}\mathbb{R}^m)$
is the Radon-Nikodym derivative of $Ju$ with respect to $\Leb{m}$;
\medskip
\item $\|J^cu\|(F)=0$ whenever $\Haus{m-n}(F)<\infty$;
\medskip
\item $\theta \in L^1(\Omega,\Lambda_{m-n}\mathbb{R}^m,\Haus{m-n})$
is a $\Haus{m-n}$-measurable function and $S_u\subset\Omega$ is $\sigma$-finite w.r.t. $\Haus{m-n}$.
\end{itemize}
The intermediate measure $J^c u$ is known as the Cantor part of $Ju$.
\begin{example}[Summability exponent $p$ versus $\dim_{\mathscr{H}}{\rm spt}(Ju)$,
{\cite[Theorem 5.1]{Mul93}}]\label{ex:cantor}
For every $\alpha\in [0,n]$
there exists a continuous $B_nV$ map
$$
u_{\alpha}\in C^0(\mathbb{R}^n,\mathbb{R}^n)\cap\bigcap_{p<n}W^{1,p}_{{\rm loc}}(\mathbb{R}^n,\mathbb{R}^n)
$$
such that $Ju_{\alpha}$ is a nonnegative Cantor measure
satisfying
$$
c\Haus{\alpha}\res {\rm spt}(Ju_{\alpha}) \leq Ju_{\alpha} \leq C \Haus{\alpha}\res {\rm spt}(Ju_{\alpha})
$$
for some $c,C>0$. In particular ${\rm spt}(Ju)$ has Hausdorff dimension $\alpha$.
\end{example}
Hence no bound on $p$
is sufficient to constrain the singularity of $Ju$.
Adding $m-n$ dummy variables to the domain the same
examples show $\alpha$ can range in the interval $[m-n,m]$
regardless how close $p$ is to $n$.
It has been proved in \cite{Mul90} and \cite{DelGhi10} that
\begin{equation}\label{eq:ambr2}
\nu(x) = M_n\nabla u(x)= e_1\wedge\dots\wedge e_m \res du^1(x)\wedge\dots\wedge du^n(x) \in\Lambda_{m-n}\mathbb{R}^m
\end{equation}
at $\Leb{m}$-almost every point $x\in\Omega$.
The set $S_u$ is unique up to $\Haus{m-n}$-negligible sets,
and can be characterized by
$$
S_u :=\left\{x\in\Omega : \limsup_{\rho\downarrow 0}\frac{\|Ju\|(B_{\rho}(x))}{\rho^{m-n}} >0\right\}.
$$
Moreover $S_u$ it has been shown in \cite{Del00} using
some general properties of normal and flat currents that $S_u$
is countably $\Haus{m-n}$-rectifiable and that for $\Haus{m-n}$-a.e.
$x\in S_u$ the multivector $\theta(x)$ is simple and it orients
the approximate tangent space $\Tan{m-n}(S_u,x)$.
\begin{definition}\label{d:SBnV}
We denote, in analogy with the $SBV$ theory, by $SB_nV$ the set of $B_nV$ functions such that $J^c u = 0$.
\end{definition}
The space $SB_nV$ enjoys a closure property proved in \cite{Del00}:
\begin{theorem}[Closure Theorem for $SB_nV$]\label{t:chiusura}
Let us consider $u,\,u_h\in B_nV(\Omega,\mathbb{R}^n)$
and suppose that
\begin{itemize}
\item[(a)]
$u_h\rightarrow u \mbox{ strongly in }L^s(\Omega,\mathbb{R}^n)$ and
$\nabla u_h \rightharpoonup \nabla u\mbox{ weakly in }L^p(\Omega,\mathbb{R}^{n\times m})$,
\item[(b)] if we write
\begin{equation*}
Ju_h = \nu_h \cdot \Leb{m} + \theta \cdot \Haus{m-n}\res S_{u_h}
\end{equation*}
then $|\nu_h|$ are equiintegrable in $\Omega$ and $\Haus{m-n}(S_{u_h}) \leq C <\infty$.
\end{itemize}
Then $u\in SB_nV(\Omega, \mathbb{R}^n)$ and
\begin{equation*}
\nu_h \rightharpoonup \nu \mbox{ weakly in }L^1(\Omega, \Lambda_{m-n}\mathbb{R}^m),\quad
\Haus{m-n}(S_{u}) \leq \liminf_h \Haus{m-n}(S_{u_h}).
\end{equation*}
\end{theorem}
\subsection{Slicing Theorem}
We aim to apply the slicing operation to $Ju \in\mathbf{F}_{m-n}(\Omega)$
in the special case $\ell = m-n$, thus reducing ourselves to
$0$-dimensional slices; moreover we want to relate these
slices to the Jacobian of the restriction $J(u|_{\pi^{-1}(x)})$.
In \cite{Del00}, the author extended a classical result
on restriction of $BV$ functions (see \cite[Section 3.11]{AmbFusPal})
to Jacobians:
\begin{theorem}[Slicing]\label{t:slicing}
Let $u\in W^{1,p}\cap L^s(\Omega,\mathbb{R}^n)$ be a function,
and let $\pi\in {\bf O}_{m-n}$.
Then for $\Leb{m-n}$-almost every $x\in\mathbb{R}^{m-n}$
\begin{equation}
\langle Ju, \pi, x\rangle = (-1)^{(m-n)n}i^x_{\#}(Ju^x),
\end{equation}
where $u^x = u\circ i^x$. Moreover $u\in B_nV(\Omega,\mathbb{R}^n)$
if and only if for every
$\pi\in{\bf O}_{m-n}$
the following two conditions hold:
\begin{align*}
\mbox{{\upshape (i)}}& \qquad u^x\in B_nV(\Omega^x,\mathbb{R}^n)\quad\mbox{ for }
\quad\Leb{m-n}\mbox{-almost every }x\in\mathbb{R}^{m-n}, \\
\mbox{{\upshape (ii)}}& \qquad \int_{\pi(\Omega)} |Ju^x|(\Omega^x) \,d\Leb{m-n}(x) <\infty.
\end{align*}
In this case the Distributional Jacobian of the restriction $u^x$ is equal
(up to sign) to the slice of $Ju$ at $x$:
\begin{equation}
\langle Ju, \pi, x\rangle = (-1)^{(m-n)n}i^x_{\#}(Ju^x),
\end{equation}
and this slicing property holds separately for the absolutely continuous part,
the Cantor part and the Jump part of $Ju$, namely:
\begin{itemize}
\item $\langle J^a u, \pi, x\rangle = (-1)^{(m-n)n}i^x_{\#}(J^a u^x),$
\item $\langle J^c u, \pi, x\rangle = (-1)^{(m-n)n}i^x_{\#}(J^c u^x),$
\item $\langle J^s u, \pi, x\rangle = (-1)^{(m-n)n}i^x_{\#}(J^s u^x).$
\end{itemize}
\end{theorem}
\section{Size of a current and a new class of maps}\label{s:size}
\noindent
As anticipated in the abstract, we are interested in broadening the class $B_nV$ to include
vector valued maps satisfying a weaker control than the mass bound: this lack of control on
$\mathbf{M}(Ju)$ already appears in Theorem \ref{t:chiusura} when we require \emph{a priori} the limit $u$ to be in $B_nV$.
We relax our energy by considering a mixed control
of $Ju$, where we bound part of the current $Ju$
with its size. In general it is
possible to define $\mathbf{S}(T)$
for every flat current $T\in\mathbf{F}_k(\Omega)$, even if $T$ has
infinite mass: this size quantity was introduced in \cite{AmbGhi12},
borrowing some ideas already used by Hardt
and Rivi\`ere in \cite{HarRiv03}, Almgren \cite{Alm86},
Federer \cite{Fed86}, and
agrees with the classical notion of size for finite mass
currents. For example a polyhedral
chain
$$
P = \sum_{i=1}^n a_i \llbracket Q_i \rrbracket
$$
where $a_i\in\mathbb{R}$ and $\llbracket Q_i \rrbracket$ are
the integration currents over some
pairwise disjoint $k$-polygons
$Q_i$, has mass
$\mathbf{M}(P) = \sum_i |a_i|\Haus{k}(Q_i)$
and size $\mathbf{S}(P) = \sum_i\Haus{k}(Q_i)$.
The main idea behind the definition is to
detect the support of the $0$-dimensional slices of $T$
via some $\pi\in{\bf O}_k$ and then to optimize
the choice of projection $\pi$.
\begin{definition}[Size of a flat current, {\cite[Definition 3.1]{AmbGhi12}}]\label{d:size}
We say that $T\in\mathbf{F}_k(\Omega)$ has finite size if
there exists a positive Borel measure $\mu$ such that
\begin{align}\label{mu}
& \Haus{0}\res{\rm spt}(T) \leq \mu &\mbox{ in the case }k=0,\notag \\
& \mu_{T,\pi} := \int_{\mathbb{R}^k}\Haus{0}\res {\rm spt}\langle
T,\pi,x\rangle\,d\Leb{k}(x)\leq \mu \,\,\,\quad\forall \pi
\in{\bf O}_k &\mbox{ in the case }k\geq 1.
\end{align}
The choice of $\mu$ can be optimized by choosing the least
upper bound of the family $\{\mu_{T,\pi}\}$ in the lattice of
nonnegative measures:
\begin{equation}\label{supmu}\mu_{T} := \bigvee_{\pi\in{\bf O}_k}\mu_{T,\pi}= \bigvee_{\pi\in{\bf O}_k}\int_{\mathbb{R}^k}\Haus{0}\res {\rm spt}\langle
T,\pi,x\rangle\,d\Leb{k}(x) .
\end{equation}
We set $\mathbf{S}(T):=\mu_T (\Omega)$.
\end{definition}
It can be proved (see \cite{AmbGhi12}) that for every flat
current with finite size there exists
unique (up to null sets) countably $\Haus{m-n}$-rectifiable set,
denoted $set(T)$, such that
$$
\mu_T= \Haus{m-n}\res set(T),
$$
so that in particular $\Haus{m-n}(set(T)) = \mathbf{S}(T)$.
A pointwise constructions of $set(T)$ can also be given as follows
$$
set(T) :=\left\{x\in\Omega : \limsup_{\rho\downarrow 0}\frac{\mu(B_{\rho}(x))}{\rho^{m-n}} >0\right\}.
$$
The following result, which we will not use, holds for a fairly
general class of metric spaces and fits
naturally in the context of calculus of variations:
\begin{theorem}[Lower semicontinuity of size, {\cite[Theorem 3.4]{AmbGhi12}}]\label{tlsc}
Let $(T_h)\subset \mathbf{F}_k(\Omega)$ be a sequence of currents with
equibounded sizes and converging to $T$ in the flat norm:
$$
\mathbf{S}(T_h)\leq C<\infty,\qquad \lim_{h}\mathbf{F}(T_h-T)= 0.
$$
Then $T$ has finite size and
\begin{equation}
\mathbf{S}(T)\leq \liminf_h \mathbf{S}(T_h).
\end{equation}
\end{theorem}
We remark that the definition of size in the metric space
contest of \cite{AmbGhi12} is slightly different, since supremum \eqref{supmu}
was taken among all $1$-Lipschitz maps $\pi\in{\rm Lip}_1(\Omega,\mathbb{R}^{k})$.
However, when the ambient space is Euclidean, the rectifiability and lower semicontinuity results obtained there,
as well as the characterization of $\mu_T$ in terms
of $set(T)$ can be readily proved using only the subset of orthogonal projections.
The space of generalized functions of bounded higher variation
is described in terms of the decomposition \eqref{dec}:
we relax the requirement on the addendum of lower dimension
and require only a size bound, retaining the mass bound
on the diffuse part.
Following the previous definitions
we consider the Sobolev functions $u$ whose jacobian
can be split in the sum of two parts, $R$ and $T$, such that:
\begin{itemize}
\item $R$ has finite mass and $\|R\|(F)=0$ whenever $\Haus{m-n}(F)<\infty$;
\medskip
\item $T$ is a flat chain of finite size.
\end{itemize}
In formulas:
\begin{definition}[Special functions of bounded higher variation]\label{d:GSBnV}
The space of generalized functions of bounded higher variation is defined by
\begin{multline}
\GB{m}{n}(\Omega) = \big\{
u\in W^{1,p}(\Omega,\mathbb{R}^n) \cap L^s(\Omega,\mathbb{R}^n):
Ju = R + T,\, \mathbf{M}(R)+ \mathbf{S}(T)<\infty,\, \\
\|R\|(F)=0\, \forall F :\Haus{m-n}(F)<\infty
\big\}.
\end{multline}
Analogously, the space of generalized special functions of bounded higher variation
is defined by
\begin{multline}\label{GSBnV}
\GSB{m}{n}(\Omega) = \big\{
u\in W^{1,p}(\Omega,\mathbb{R}^n) \cap L^s(\Omega,\mathbb{R}^n):
Ju = R + T,\, \mathbf{M}(R)+ \mathbf{S}(T)<\infty,\,
\|R\|\ll\Leb{m}\big\}.
\end{multline}
In accordance with the classical $BV$ theory we denote $S_u := set(T_u)$.
\end{definition}
This space is clearly meant to mimic the
aforementioned $SB_nV$ class.
In particular, thanks to the slicing properties of flat currents and the definition of size,
the slicing theorem for $\GSB{m}{m}({\Omega})$ can be stated in the following way:
\begin{align}\label{slicingGSBnV}
u\in\GSB{m}{n}(\Omega)\Longleftrightarrow\forall\pi\in{\bf O}_{m-n}\quad
\left\{
\begin{array}{ll}
& u^x\in\GSB{n}{n}(\Omega^x),\\
& \\
&\int_{\pi(\Omega)} \mathbf{M}(R_{u^x}) + \mathbf{S}(T_{u^x}) \,d\Leb{m-n}(x) <\infty.
\end{array}
\right.
\end{align}
In the following propositions we describe some
useful properties of the class $\GSB{m}{n}(\Omega)$.
\begin{lemma}\label{l:sjbvn}
If $m=n$ then $\GSB{n}{n}(\Omega) = SB_nV(\Omega)$.
\end{lemma}
\begin{proof}
The statement relies on the fact that a flat $0$-current
of finite size coincides with a finite sum of Dirac masses,
and in particular it has finite mass.
This property, reminiscent of Schwartz lemma for
distributions, has been proved in \cite{AmbGhi12}, Theorem 3.3.
Therefore the current
$$
T = Ju - R
$$
has finite mass, hence $\mathbf{M}(Ju) \leq \mathbf{M}(R) + \mathbf{M}(T) <\infty$ which
means $u\in B_nV(\Omega)$.
\end{proof}
Since the Radon-Nikodym decomposition of a measure
into the sum of an absolutely continuous and
a singular part is unique, by slicing also $R$ and $T$ are
uniquely determined in the decomposition. Therefore
we can write $Ju = R_u + T_u$, so that $S_u$ is a well
defined set.
A very well known space of functions implemented in the
calculus of variations is $GSBV$. The main idea behind this space,
introduced in \cite{DeGAmb89} (see also \cite[Section 4.5]{AmbFusPal}),
is to consider functions $u$ whose derivative $Du$ loses any kind
of local integrability, but nevertheless retains some
of the structure of $SBV$ functions. Setting $u^N:=(-N)\lor u\land N$
for every $N>0$ we define
$$
GSBV(\Omega)=\{u:\Omega\rightarrow\mathbb{R}\,\mbox{ Borel}:
\, u^N\in SBV(\Omega)\,\mbox{ for all integers }N>0\}.
$$
The countable set of truncation given by $N\in\N$ is enough to
provide the existence of an approximate differential $\nabla^*u$
and of a countably $\Haus{m-1}$-rectifiable singular set $S^*_u$ such
that for every $N$
$$
\|Du^N\|\leq |\nabla^* u|\chi_{\{|u|\leq N\}}\Leb{m} + 2N\Haus{m-1}\res S^*_u.
$$
Moreover an analog of the slicing theorem for $BV$ function
is available also in $GSBV$, see \cite[Proposition 4.35]{AmbFusPal}.
\begin{proposition}[Comparison between $GSB_1V$ and $GSBV$]\label{GSBV}
A function $u\in GSB_1V(\Omega)$ if and
only if $u\in GSBV(\Omega)$, $u\in L^1(\Omega)$,
$\nabla^*u\in L^1(\Omega,\mathbb{R}^n)$ and
$\Haus{m-1}(S^*_u)<\infty$.
\end{proposition}
\begin{proof}
With abuse of notation, motivated by \eqref{scalar}
we identify for scalar functions the action of $Ju$ on
$\mathscr{D}^{m-1}(\Omega)$ with the action of the distributional
derivative $Du$ on $C^{\infty}_c(\Omega,\mathbb{R}^n)$
(see the map ${\mathbf D}^{m-1}$ in \cite[1.5.2]{Fed}).
Consider first the case $m=1$. Let $u\in GSB_1V(\Omega)$:
writing $R_u = \rho\Leb{1}$ and
$T_u = \sum_{k=1}^{\mathbf{S}(T_u)}a_k\llbracket x_k\rrbracket$,
thanks to \eqref{scalar} we know that for $\omega\in\mathscr{D}^0(\Omega)$
$$
\langle D u,\omega\rangle =
\int_{\Omega}\rho\omega\,dx + \sum_{k=1}^{\mathbf{S}(T_u)}a_k\omega(x_k).
$$
This proves that $u\in SBV(\Omega)$, $S_u\subset set(T_u)$ and
$ u'(x)=\rho(x)$ almost everywhere. In particular for $N>0$ fixed
\begin{equation}\label{boh}
\|Du^N\|\leq |\rho|\Leb{1} + 2N\Haus{m-1}\res set(T_u).
\end{equation}
For $m\geq 2$ the slicing Theorem \ref{t:slicing} applied to a coordinate
projection onto a hyperspace implies that
almost every slice $u^x$ is in $\GSB_1(\Omega^x)$, hence for every
$N>0$ the estimate \eqref{boh} holds for $u^x$. Integrating back we have
$\|Du^N\|(\Omega)<\infty$, hence $u\in GSBV(\Omega)$.
On the other if $u\in GSBV(\Omega)\cap L^1(\Omega)$
we know that $Du^N\overset{*}{\rightharpoonup} Du$ in the sense of distributions, and
also in the flat norm, since the weak derivative is a distribution of
order $1$. Moreover $\nabla u^N\rightarrow \nabla^*u$ strongly in $L^1$,
hence also in the flat norm. Therefore the jump parts also converge
to some flat $T_u$:
$$
D^j u^N \overset{\mathbf{F}}{\rightarrow} T_u\in\mathbf{F}_{m-1}(\Omega).
$$
Recall that for $v\in BV$ the jump part of the derivative $Dv$ can
be expressed in terms of the approximate upper and lower limits $v_{\pm}$
and of the approximate tangent $(m-1)$-vector $\tau$ in the following way:
\begin{equation}\label{eq:jump}
D^j v = (v_+ - v_-)\tau\,\Haus{m-1}\res S_{v}.
\end{equation}
Hence if $m=1$ then $\Haus{0}({\rm spt} T_u)\leq \lim_N\Haus{0}(S_{u^N})\leq \Haus{0}(S^*_u)$;
in the general case can be achieved using the slicing Theorem \ref{t:slicing}
and Proposition \cite[4.35]{AmbFusPal}.
\end{proof}
\subsection{Some examples}
The following observation shows that when $n\geq 2$
it is hopeless to rely on truncation to get mass bounds
for $Ju$.
\begin{example}[$L^{\infty}$ bound for $n\geq 2$]\label{ex:Linfinity}
For $n\geq 2$ let $\gamma_k:S^{n-1}\rightarrow S^{n-1}$
be a smooth map with degree $k$, and call $u_k$ its zero
homogeneous extension to $\mathbb{R}^n$. Then $\|u_k\|_{L^{\infty}}\leq 1$
but by Example \ref{ex:monopole} $Ju = k\Leb{n}(B_1)\llbracket 0 \rrbracket$.
\end{example}
On the contrary for $n=1$ and $u\in BV(\Omega)$ the approximate upper and
lower limits $u_{\pm}$ of $u$ characterize the singular set:
$S_u=\{ x\in\Omega: u_+(x)>u_-(x)\}$. Equation
\eqref{eq:jump} implies that an
$L^{\infty}$ bound on $u$ together with a size bound
$\Haus{m-1}(S_u)<\infty$ gives a mass bound on $Du$.
We now adapt the construction in \ref{ex:dipole1}, building
a map whose jacobian has infinite mass but finite size:
\begin{example}[$\mathbf{S}(Ju)<\infty$ but $\mathbf{M}(Ju)=\infty$]\label{ex:finitesizejac}
Set $m=n+1$ and let us write $(x,y,z)$ the coordinates of $\mathbb{R}\times\mathbb{R}^{n-1}\times \mathbb{R}$.
Besides $\nu\in\Z$ and $\rho>0$ fix
an extra parameter $R\geq\rho$. We extend the function
$f_{\nu,\rho}(y,z)$ of Example \ref{ex:dipole} to $\mathbb{R}^{n+1}$ by
\begin{equation*}
h_{\nu,\rho,R}(x,y,z)=\left\{
\begin{array}{ll}
f_{\nu,\rho}\Big(\frac{Ry}{R-|x|},\frac{Rz}{R-|x|}\Big)
\qquad &\mbox{ for }\quad |x|<R,\\
\mathcal{N}\qquad &\mbox{ for }\quad |x|\geq R.
\end{array}
\right.
\end{equation*}
Clearly $h_{\nu,\rho,R}\neq\mathcal{N}$ in the set $\{|x|/R + |y|/\rho +|z|/\rho<1 \}$;
by simmetry we can do the computations in $\{x<0\}$.
Let us estimate the partial derivatives:
\begin{align*}
&\Big|\tfrac{\partial h_{\nu,\rho,R}}{\partial x}(x,y,z)\Big| \leq
\tfrac{R(|y|+|z|)}{(R+x)^2}|\nabla f_{\nu,\rho}|\Big(\tfrac{Ry}{x+R},\tfrac{Rz}{x+R}\Big)
\leq \tfrac{\rho }{x+R} |\nabla f_{\nu,\rho}|\Big(\tfrac{Ry}{x+R},\tfrac{Rz}{x+R}\Big),\\
& |\nabla_{y,z} h_{\nu,\rho,R}| \leq \tfrac{R}{x+R}|\nabla f_{\nu,\rho}| \Big(\tfrac{Ry}{x+R},\tfrac{Rz}{x+R}\Big).
\end{align*}
Since $\rho\leq R $
\begin{align}
\int_{\mathbb{R}^{n+1}}|\nabla h_{\nu,\rho,R}|^p\,dxdydz &\leq
2(n+1)\int_{-R}^0 \int_{\{|y|/\rho +|z|/\rho<(x+R)/R\}} \Big(\tfrac{R}{x+R}\Big)^p
|\nabla f_{\nu,\rho}|^p \Big(\tfrac{Ry}{x+R},\tfrac{Rz}{x+R}\Big)dydz \,dx \notag \\
&\leq 2(n+1)\int_{0}^R \Big(\tfrac{R}{x}\Big)^{p-n} \int_{\{|y| +|z|<\rho\}}
|\nabla f_{\nu,\rho}(y,z)|^p\,dydz\,dx \notag\\
&\leq C_p\, \nu^{\frac{p}{n-1}}\rho^{n-p}R. \label{pnorm}
\end{align}
Moreover $Jh_{\nu,\rho,R}$ is the integral cycle $\nu\cdot \zeta_{\#}\llbracket [0,1]\rrbracket$,
where $\zeta:[0,1]\rightarrow \mathbb{R}^{n+1}$ is the following closed curve:
\begin{eqnarray*}
\zeta(t) =\left\{
\begin{array}{ll}
( 4Rt-R,0,-4\rho t) \quad &\mbox{ for }\quad t\in [0,\tfrac 14],\\
(4Rt-R,0,4\rho t-2\rho) \quad &\mbox{ for }\quad t\in [\tfrac 14,\tfrac 12],\\
(3R - 4Rt,0,4\rho t-2\rho) \quad &\mbox{ for }\quad t\in [\tfrac 12,\tfrac 34],\\
(3R - 4Rt,0,4\rho-4\rho t) \quad &\mbox{ for }\quad t\in [\tfrac 34,1].
\end{array}
\right.
\end{eqnarray*}
Since ${\rm Lip}(\zeta)\leq CR$ we have $\mathbf{M}(Jh_{\nu,\rho,R})\leq C \nu R $
and $\mathbf{S}(Jh_{\nu,\rho,R}) \leq CR$.
Like in \ref{ex:dipole} we glue infinite copies of $h_{\nu_k,\rho_k,R_k}$
along the $z$ axis and obtain a map $g$: the Sobolev
norm of $g$ can be estimated by \eqref{pnorm}:
\begin{equation*}\label{eq:gnorm}
\|\nabla g\|_{L^p}^p \leq C \sum_k \nu_k^{\frac{p}{n-1}}\rho_k^{n-p}R_k
\end{equation*}
and
\begin{eqnarray*}
&\mathbf{M}(Jg) \leq C \sum_k \nu_kR_k,\\
&\mathbf{S}(Jg) \leq C \sum_k R_k.
\end{eqnarray*}
Choosing $\nu_k = k$, $R_k=\tfrac{1}{k^2}$ and $\rho_k=e^{-k}$
we obtain a $S^{n-1}$-valued $W^{1,p}$ function constant outside a compact
set and whose Jacobian has infinite mass but finite size.
\end{example}
\subsection{$Ju$ and approximate differentiability} We now extend
to $\GSB_n$ the pointwise characterization of the absolutely continuous
part of $Ju$.
\begin{proposition}[$Det = det$ in the $GSB_nV$ class]\label{p:det}
Let $u\in \GSB{m}{n}(\Omega)$
and write $Ju = R + T$ as in Definition \ref{d:GSBnV}.
Let $\nabla u$ be the approximate differential
of $u$.
Then
\begin{equation}\label{z}
\frac{dR}{d\Leb{m}} = M_n\nabla u
\quad\text{$\Leb{m}$-almost everywhere in $\Omega$.}
\end{equation}
\end{proposition}
\begin{proof}
For the ease of notation let $\nu:=\frac{dR}{d\Leb{m}}$.
Fix a projection $\pi\in{\bf O}_{m-n}$ and let us write
the coordinates $z= (x,y)$ accordingly.
For a fixed $x\in\mathbb{R}^{m-n}$ we note that the injection $i^x$ and
the complementary projection $\pi^{\perp}\sbarretta _{\pi^{-1}(x)}$ are
one the inverse of the other.
Recall the slicing Theorem for general Sobolev functions gives
\begin{equation}\label{eq:ambr1}
\langle Ju, \pi, x\rangle = (-1)^{(m-n)n}i^x_{\#}(Ju^x).
\end{equation}
Taking Lemma \ref{l:sjbvn} into account,
for almost every $x\in \mathbb{R}^{m-n}$ it holds $u^x \in B_nV(\Omega^x)$
and $\mathbf{M}(\langle R,\pi,x\rangle)) + \mathbf{S}(\langle T,\pi,x\rangle) <\infty$,
hence
\eqref{sliceac} gives
\begin{equation}\label{eq:slicexesima}
\langle Ju, \pi, x\rangle = \nu(x,\cdot)\res d\pi \Haus{n}\res\pi^{-1}(x) + \langle T,\pi,x\rangle.
\end{equation}
Pushing forward \eqref{eq:slicexesima} via $\pi^{\perp}$ by \eqref{eq:ambr1} it follows that
$(-1)^{(m-n)n}Ju^x = \nu(x,\cdot)\res d\pi \Leb{n} + \tilde{T}^x$,
with $\tilde{T}^x= \pi^{\perp}_{\#}\langle T,\pi,x\rangle$.
But the finiteness of the size of $\langle T,\pi,x\rangle$
implies that $\tilde{T}^x$ is a
sum of $\mathbf{S}(\langle T,\pi,x\rangle)$ Dirac masses.
In particular by equation \eqref{eq:ambr2} in the case $m=n$
we know that
$$
(-1)^{(m-n)n}\nu(x,\cdot)\res d\pi = \det \nabla_y u(x,\cdot).
$$
Using \eqref{M_nLi} we obtain $\nu(x,\cdot)\res d\pi = M_n\nabla u (x,\cdot)\res d\pi$
for almost every $x$.
We recover the equality \eqref{z} by taking orthogonal
projections $\pi$ onto every $(m-n)$-dimensional coordinate
subspace.
\end{proof}
It will be useful to extend the result of Proposition~\ref{p:det} to
the lower order determinants: let $u\in\GSB{n}{n}(\Omega)$
and $w\in{\rm Lip}(\Omega,\mathbb{R}^n)$. We denote by $\Gamma(u,w)$
the sum of the jacobians of the functions obtained by
replacing at least one component of $u$ with the
respective component of $w$, but not all of them.
More precisely for every $I\subset\{1,\dots,n\}$
such that $0<|I|<n$ we construct the function
$u_I$ whose components are
\begin{eqnarray*}
u_I^k=\left\{
\begin{array}{ll}
u^k \text{ if }k\not\in I,\\
w^k \text{ if }k \in I.
\end{array}
\right.
\end{eqnarray*}
Then we let $\Gamma(u,w)=\sum_{0<|I|<n}Ju_I$.
By the multilinearity of jacobians, it is easy to check that if $u$ is Lipschitz the identity
\begin{equation}\label{eq:ambr3}
J(u+w) =Ju+\Gamma(u,w)+Jw
\end{equation}
holds pointwise $\Leb{m}$-a.e. in $\Omega$.
\begin{corollary}\label{c:detgeneral}
Let $\Omega\subset\mathbb{R}^m$, $w\in{\rm Lip}(\Omega,\mathbb{R}^n)$ and $u\in\GSB{m}{n}(\Omega)$.
Then, in the sense of distributions, it holds
\begin{equation}\label{eq:ambr4}
J(u+w) =Ju+ \Gamma(u,w)+Jw.
\end{equation}
\end{corollary}
\begin{proof} The proof uses the following observation: if $u_h\rightarrow u$ in $L^s$
and $\nabla u_h\rightharpoonup \nabla u$ in $L^p$, then
by Reshetnyak's Theorem and the inequality $p>n-1$ every minor of $\nabla u$ of order
$k<n$ is weakly continuous in $L^{\frac pk}$.
It follows that $\Gamma(u_h,w)\to\Gamma(u,w)$, so that we can pass to the limit in \eqref{eq:ambr3}
to obtain \eqref{eq:ambr4}.
\end{proof}
\section{Compactness}\label{s:compactness}
\noindent
\begin{theorem}[Compactness for the class $\GSB{m}{n}$]\label{t:compactness}
Let $s>0,\,p>1$ be exponents with $\tfrac 1s+ \tfrac{n-1}{p}\leq 1$ and let
$\Psi:[0,\infty)\rightarrow [0,\infty)$ be a
convex increasing function satisfying
$\lim\limits_{t\rightarrow\infty}{\Psi(t)}/{t} = \infty$.\\
Let $(u_h)\subset \GSB{m}{n}(\Omega)$ be such that
$u_h\rightarrow u$ in $L^s(\Omega,\mathbb{R}^n)$ and
$\nabla u_h\rightharpoonup \nabla u$
weakly in $L^p(\Omega,\mathbb{R}^{n\times m})$.
Suppose that $Ju_h = R_{u_h}+T_{u_h}$ fulfil
\begin{equation}\label{bound}
K:= \sup_h\int_{\Omega}\Psi\left(\big|\frac{dR_{u_h}}{d\Leb{m}}\big|\right) d\Leb{m} + \mathbf{S}(T_{u_h}) <\infty.
\end{equation}
Then $u\in\GSB{m}{n}(\Omega)$ and, writing $Ju = R_{u}+ T_u$,
\begin{align}
& \frac{dR_{u_h}}{d\Leb{m}}\rightharpoonup \frac{dR_{u}}{d\Leb{m}}
\qquad\text{weakly in }L^1(\Omega,\Lambda_{m-n}\mathbb{R}^m),\label{wregpart} \\
\medskip
& \mathbf{S}(T_u)\leq \liminf_{h}\mathbf{S}(T_{u_h})\label{lscsize}.
\end{align}
\end{theorem}
\begin{proof} Without loss of generality we can assume $\Psi$ to have
at most a polynomial growth at infinity, for otherwise
it is sufficient to take $\tilde{\Psi}(t) := \min\{\Psi(t), t^2\}$.
In particular we will use the inequality
\begin{equation}\label{Delta_2}
\Psi(2t) \leq C\Psi(t) \qquad\forall t>0
\end{equation}
(this inequality is known as $\Delta_2$ condition in the literature, see for
instance \cite[8.6]{AdaFou}).
We shorten $T_h, R_h$ in place of $T_{u_h}$ and $R_{u_h}$ respectively and
denote by $\rho_h=M_n\nabla u_h$ the densities of $R_h$ with respect to $\Leb{m}$.
We know from Proposition \ref{p:conv}
that $Ju_h\rightarrow Ju$ in the flat norm. Possibly extracting a subsequence
we can assume with no loss of generality that:
\begin{itemize}
\item [(a)] the limit $\lim_h \,\mathbf{S}(T_h)$ exists,
\item [(b)] $\rho_h\rightharpoonup \rho$
weakly in $L^1(\Omega,\Lambda_{m-n}\mathbb{R}^m)$,
\item [(c)] $(u_h)$ rapidly converges to $u$ in $L^s$:
as a consequence of Proposition \ref{p:conv} we have also
\begin{equation}\label{fastS}
\sum_h \mathbf{F}(Ju_{h}-Ju) < \infty.
\end{equation}
\end{itemize}
Indeed, if we prove the result under these additional assumptions, then we can use the
weak compactness of $\rho_h$ in $L^1$ provided by
the Dunford-Pettis Theorem,
and the fact that any subsequence admits a further
subsequence satisfying (a), (b), (c) to obtain the general statement.
We shall let $R:= \rho\Leb{m}$ be the limit current:
since the flat and weak convergences in (b) and
(c) are stronger than the weak* convergence
for currents, putting them together we obtain a flat current
$T:=Ju-R$ such that
\begin{equation}\label{wconv}
T_h \overset{*}{\rightharpoonup} T.
\end{equation}
The proof is divided in three steps: we first address
the special case $m=n$, then we use this case and the
slicing Theorem \eqref{slicingGSBnV} to show the lower
semicontinuity of size in the second step.
The main difficulty is in the third step, where we prove
\eqref{wregpart}, because weak convergence behaves badly
under the slicing operation.
\\ {\bf Step 1: $m=n$.}
We can apply a very particular case of Blaschke's compactness
Theorem \cite[4.4.15]{AmbTil} to the sets ${\rm spt}(T_h)$, which
have equibounded cardinality, to obtain a finite set $N\subset\overline{\Omega}$
and a subsequence $(T_{h'})$ such that
${\rm spt}(T_{h'})\longrightarrow N$ in the sense of Hausdorff convergence.
By \eqref{wconv} we immediately obtain that ${\rm spt}(T)\subset N\cap\Omega$, hence
$\mathbf{S}(T)<\infty$ and $u\in\GSB{n}{n}(\Omega)$. In addition, since any point
in ${\rm spt} T$ is the limit of points in ${\rm spt} T_{h'}$ it follows that
\begin{equation*}
\mathbf{S}(T)\leq\liminf_{h'}\mathbf{S}(T_{h'})=
\lim_{h}\mathbf{S}(T_{h}).
\end{equation*}
Finally, since $Ju = R + T$ it must be $T=T_u$, which yields \eqref{lscsize},
and $R=R_u$, which together with (b) yields \eqref{wregpart}.\\
{\bf Step 2: $m\geq n$.} Let us fix $A\subset\Omega$ open, $\pi\in{\bf O}_{m-n}$ and $\varepsilon\in (0,1)$:
the bound \eqref{bound}, \eqref{sliceac} and Fatou's lemma imply that
\begin{align}\label{liminf}
+\infty>K & \geq \liminf_h\left\{ \mu_{T_h}(A) +
\varepsilon\int_{A}\Psi(|\rho_h|) d\Leb{m}\right\} \\
& \geq \liminf_h\left\{ \mu_{T_h,\pi}(A) +
\varepsilon\int_{A}\Psi(|\rho_h\res d\pi |) d\Leb{m} \right\} \label{eq:ambr5} \\
& = \int_{\mathbb{R}^{m-n}}\liminf_h \left[ \Haus{0}(A^x\cap {\rm spt}(\langle T_h,\pi,x\rangle))
+\varepsilon \int_{A^x}\Psi(|\rho_h\res d\pi |)dy \right]dx \notag\\
& = \int_{\mathbb{R}^{m-n}}\liminf_h \left[ \Haus{0}(A^x\cap {\rm spt}( T_{u^x_h}))
+\varepsilon \int_{A^x}\Psi(|\rho_h^x|)dy \right]dx,\label{finalfatou}
\end{align}
with $\rho^x_h:=M\nabla u_h(x,\cdot)\res d\pi$. By \eqref{liminf} we can choose for almost every $x\in\mathbb{R}^{m-n}$
a subsequence $h' = h'(x,A)$, possibly depending
on $x$ and on the set $A$, realizing the finite
lower limit:
\begin{equation*}
\liminf_h \, \Haus{0}(A^x\cap {\rm spt}( T_{u^x_h}))
+\varepsilon \int_{A^x}\Psi(|\rho_h^x|)dy.
\end{equation*}
Recall that thanks to (c) $Ju^x_h\overset{\mathbf{F}}{\rightarrow}Ju^x$ for almost every $x$.
We can therefore apply step 1 to the sequence $u^x_h\in\GSB{n}{n}(\Omega^x)$,
which converges rapidly to $u^x$, to conclude that
$u^x\in\GSB{n}{n}(\Omega^x)$ and that
\begin{align}\label{fatouchain}
\Haus{0}(A^x\cap {\rm spt}( T_{u^x}))& \leq\liminf_{h'}\Haus{0}(A^x\cap {\rm spt}( T_{u^x_{h'}})) \notag\\
& \leq \liminf_{h'}\Haus{0}(A^x\cap {\rm spt}( T_{u^x_{h'}}))
+\varepsilon \int_{A^x}\Psi(|\rho_{h'}^x|) dy \notag\\
& = \liminf_{h}\Haus{0}(A^x\cap {\rm spt}( T_{u^x_{h}}))
+\varepsilon \int_{A^x}\Psi(|\rho_h^x|)dy.
\end{align}
Integrating in $x$ and applying \eqref{eq:ambr5} as well as the monotonicity
of $\Psi$ we entail
\begin{equation}
\mu_{T,\pi}(A)\leq \liminf_h \left\{ \mu_{T_h,\pi}(A) +
\varepsilon\int_{A}\Psi(|\rho_h|) d\Leb{m} \right\} =:\eta_\varepsilon(A).
\end{equation}
The map $A\mapsto \eta_\varepsilon(A)$ is a finitely superadditive set-function,
with $\eta_\varepsilon(\Omega)\leq\liminf_h\mathbf{S}(T_{u_h})+K\varepsilon$.
Therefore if $B_1,\dots,B_N$ are pairwise disjoint
Borel sets and $K_i\subset B_i$ are compact,
we can find pairwise disjoint open sets $A_i$
containing $K_i$ and apply the superadditivity to get
$$
\sum_{i=1}^N \mu_{T,\pi_i}(K_i)\leq \sum_{i=1}^N \eta_\varepsilon(A_i)\leq\eta_\varepsilon(\Omega).
$$
Since $K_i$ are arbitrary, the same inequality holds
with $B_i$ in place of $K_i$; since also $B_i$, $\pi_i$ and
$N$ are arbitrary, it follows that $\mu_T$ is a finite
Borel measure and $\mu_T(\Omega)\leq\eta_\varepsilon(\Omega)$.
Hence $u\in \GSB{n}{n}(\Omega)$ because
$Ju = R + T$, $\mathbf{S}(T)<\infty$ and $R$ is an
absolutely continuous measure.
Letting $\varepsilon\downarrow 0$ we also prove \eqref{lscsize}.
For later purposes we notice that we proved
\begin{equation}\label{eq:lscA}
\mu_{T_u}(A)\leq \liminf_h \mu_{T_{u_h}}(A).
\end{equation}
{\bf Step 3: proof of \eqref{wregpart}.}
In order to prove \eqref{wregpart}, since the space $\Lambda_{m-n}\mathbb{R}^m$
is finite dimensional, we will prove that
\begin{equation}\label{wregpartpi}
\rho_h\res d\pi\rightharpoonup M_n\nabla u\res d\pi
\qquad\text{weakly in }L^1(\Omega,\Lambda_{m-n}\mathbb{R}^m)\end{equation}
for every orthogonal projection $\pi$ onto a coordinate
subspace.
We fix an open $A\subset\Omega$ and $a\in\mathbb{R}$.
From now on $w:A\rightarrow\mathbb{R}^n$ will be an affine map such that
$$
\nabla_x w=0,\qquad \det(\nabla_y w) = a.
$$
Let us compute $J(u_h+w)$:
thanks to Corollary~\ref{c:detgeneral} we get
$$
J(u_h+w) = Ju_h + \Gamma(u_h,w) + a{\bf E}^m\res d\pi.
$$
We are now ready to prove the last part of the Theorem.
We argue as in step 2, but this time we change
the form of the energy and we analyse the convergence
of a perturbed sequence of maps.
First of all we note that the sequence
\begin{equation}\label{pertenergy}
\int_{A}\Psi \left( \big| \rho_h\res d\pi + a \big| \right) d\Leb{m}
+ \varepsilon\mu_{T_h,\pi}(A) + \varepsilon\int_A|\nabla u_h|^p d\Leb{m}
\end{equation}
is still bounded from above, because
$|\alpha+\beta|^p\leq 2^{p-1}(|\alpha|^p+|\beta|^p)$, \eqref{Delta_2} and
the convexity of $\Psi$ imply that
\begin{equation*}\label{Deltabound}
\int_{A}\Psi \left( \big| \rho_h\res d\pi + a \big| \right) d\Leb{m} \leq
\frac C2 \int_{A}\Psi \left( \big| \rho_h \big| \right) d\Leb{m}
+\frac C2 \Psi(|a|)\Leb{m}(A) \leq \frac C2 \big(K + \Psi(|a|)\Leb{m}(A)\big).
\end{equation*}
We consider the sequence $(u_h + w)\subset\GSB{m}{n}(A)$ and
the perturbed energy \eqref{pertenergy}:
arguing as in the chain of inequalities \eqref{liminf}-\eqref{finalfatou}
for almost every $x$ we can find a suitable
subsequence $h'=h'(x,A)$ realizing the finite lower limit of the sliced energies
\begin{equation}\label{eq:energyx}
\int_{A^x}\Psi \left( \big| \rho^x_h+ a \big| \right) dy
+ \varepsilon\Haus{0}(A^x\cap {\rm spt}( T_{u^x_h})) + \varepsilon\int_{A^x}|\nabla u^x_h|^p dy.
\end{equation}
Since $\Psi$ is superlinear at infinity, up to subsequences
the densities $\rho^x_{h'} + a$ weakly converge to
some function $r^x$ in $L^1(A^x)$: in particular the associated
currents weak* converge
\begin{equation}\label{eq:rx}
(\rho^x_{h'} + a){\bf E}^{n}\res A^x \overset{*}{\rightharpoonup} r^x {\bf E}^{n}\res A^x.
\end{equation}
Thanks to the fast convergence (c) we also know that
$u^x\rightarrow u$ in $L^s(A^x)$; moreover the
boundedness of the Dirichlet term in \eqref{eq:energyx} implies also that
$\nabla_y u^x_{h'}\rightharpoonup \nabla u^x$ in $L^p(A^x,\mathbb{R}^n)$,
hence by step 2 we get
$$
u^x\in\GSB_n(A^x)\qquad\mbox{ and }\qquad T_{u^x_{h'}}\overset{*}{\rightharpoonup} T_{u^x}.
$$
The weak convergence of the gradients in $L^p$ also allows
to use the continuity property of $\Gamma(\cdot,w^x)$
along the sequence of restrictions $(u^x_{h'})$ and deduce that
$$
(\rho^x_{h'} + a){\bf E}^{n}\res A^x = J(u^x_{h'} + w^x) -\Gamma(u^x_{h'},w^x) - T_{u^x_{h'}} \overset{*}{\rightharpoonup}
J(u^x + w^x) -\Gamma(u^x,w^x) - T_{u^x}
$$
in the sense of distributions.
By Corollary \ref{c:detgeneral} and Proposition
\ref{p:det} we are able to identify the weak limit in \eqref{eq:rx}
\begin{equation}\label{eq:identification}
r^x = \det\nabla_y u^x +a = M_n\nabla u(x,\cdot)\res d\pi + a.
\end{equation}
We fix a a convex increasing function
with superlinear growth $\varphi$ satisfying
\begin{equation}\label{eq:ambr6}
\lim_{t\rightarrow+\infty}\frac{\Psi(t)}{\varphi(t)}=+\infty.
\end{equation}
Using the previous convergence \eqref{eq:rx}, \eqref{eq:identification} on almost
every slice and integrating with respect to $x$ we
deduce by the convexity of $\varphi$ that
$$
\int_A \varphi\left( \big|M_n\nabla u\res d\pi + a \big| \right) d\Leb{m} \leq \liminf_h
\int_A \varphi(|\rho_h\res d\pi + a|) d\Leb{m}
+\varepsilon\mu_{T_h,\pi}(A)+ \varepsilon\int_A|\nabla u_h|^p d\Leb{m}.
$$
Adding this inequality on a finite number of disjoint
open subsets $A_j$, with arbitrary choices of $a_i\in\mathbb{R}$, we obtain
$$
\int_{\Omega} \varphi\left( \big| M_n\nabla u\res d\pi + \xi \big| \right) d\Leb{m}
\leq \liminf_h
\int_{\Omega} \varphi (|\rho_h\res d\pi+\xi|) d\Leb{m}
+\varepsilon\mathbf{S}(T_h) +\varepsilon\int_{\Omega}|\nabla u_h|^p d\Leb{m},
$$
where $\xi:=\sum_j a_j\chi_{A_j}$. Letting $\varepsilon\downarrow 0$ we
can disregard the size and Dirichlet terms in the last inequality to
get
\begin{equation}\label{Psiineq}
\int_{\Omega} \varphi\left( \big| M_n\nabla u\res d\pi + \xi \big| \right) d\Leb{m}
\leq \liminf_h
\int_{\Omega} \varphi (|\rho_h\res d\pi+\xi|) d\Leb{m}.
\end{equation}
Taking $\varphi_n(t):= \tfrac{\varphi(t)}{n}\vee t$, we have that
$\varphi_n$ are still convex, increasing, superlinear
at infinity and satisfy \eqref{eq:ambr6}, therefore \eqref{Psiineq} is applicable with $\varphi=\varphi_n$.
Given $\delta>0$ fix $C_{\delta}$ such that $\varphi_1(t)\leq\delta\Psi(t)$ for $t>C_{\delta}$;
we also let $\Omega_{h,\delta} = \big\{| \rho_h\res d\pi + \xi | >C_{\delta}\big\}$.
By applying \eqref{Psiineq} with $\varphi=\varphi_n$ we have therefore
\begin{align*}
\int_{\Omega}\big| & M_n\nabla u\res d\pi + \xi \big| d\Leb{m}\leq
\int_{\Omega}\varphi_n\left(\big| M_n\nabla u\res d\pi + \xi \big| \right) d\Leb{m} \leq
\liminf_h \int_{\Omega}\varphi_n(|\rho_h\res d\pi + \xi|) d\Leb{m} \\
& \leq \liminf_h \int_{\Omega}\big| \rho_h\res d\pi + \xi \big|
+ \limsup_h \int_{\Omega_{h,\delta}}\varphi_1\left(\big| \rho_h\res d\pi + \xi \big| \right)
+ \sup_{0\leq t\leq C_\delta}\{\varphi_n(t) - t \}\Leb{m}(\Omega_{h,\delta}^c) \\
& \leq \liminf_h \int_{\Omega}\big| \rho_h\res d\pi + \xi \big|
+ \limsup_h \delta \int_{\Omega_{h,\delta}}\Psi\left(\big| \rho_h\res d\pi + \xi \big| \right)
+ \sup_{0\leq t\leq C_\delta}\{\varphi_n(t) - t \}\Leb{m}(\Omega) .
\end{align*}
Letting $n\rightarrow\infty$ the third term vanishes because
$\varphi_n(t)\downarrow t$ uniformly
on compact sets. Eventually,
sending $\delta\downarrow 0$ we obtain
\begin{equation}\label{dalmaso}
\int_{\Omega}\big| M_n\nabla u\res d\pi + \xi \big| \leq \liminf_h
\int_{\Omega}\big| \rho_h\res d\pi+ \xi\big|.
\end{equation}
Inequality \eqref{dalmaso} is actually valid for every $\xi\in L^1(\Omega)$
by approximation, since the set of functions of type $\sum_j a_j\chi_{A_j}$
is dense in $L^1$.
We therefore address the last point \eqref{wregpartpi}
thanks to Lemma \ref{l:dalmaso} below: the weak limit $\rho$ must
be the equal to $M_n\nabla u$, the density of $R_u$ with respect to $\Leb{m}$.
\end{proof}
\begin{lemma}\label{l:dalmaso}
Let $(z_h)\subset L^1(\Omega)$ be a weakly compact sequence
and suppose that, for some $z\in L^1(\Omega)$, it holds
$$
\int_{\Omega}|z+\xi|\, d\Leb{m}\leq
\liminf_h\int_{\Omega}|z_h+\xi|\, d\Leb{m}\qquad\forall \xi\in L^1(\Omega).
$$
Then $z_h\rightharpoonup z$ weakly in $L^1(\Omega)$.
\end{lemma}
We refer to \cite{Amb89} for the proof.
\section{Applications}
\noindent
We now present an application of Theorem \ref{t:compactness}
to a minimization problem. The choice of our Lagrangian is motivated
by the introduction of a new functional of the calculus of variation,
presented in \ref{ss:MS},
aiming to generalize the classical Mumford-Shah energy
\cite{MumSha89,DeGCarLea89,AmbFusPal} to
vector valued maps with singular set of codimension at least 2.
The discussion in the introduction already mentioned the central
role of the distributional jacobian in relation to low dimensional singularities:
in this model we replace the singularities of the derivative by the singularities
of the jacobian and we measure them with the size functional of section~\ref{s:size}.
\subsection{Existence result for general Lagrangians}
We fix an open, regular and bounded subset $\Omega$ of $\mathbb{R}^m$.
For approximately differentiable maps $u:\Omega\rightarrow\mathbb{R}^n$
we let $M\nabla u$ be the vector of all minors of $k\times k$
submatrices of $\nabla u$, with $k$ ranging from $1$ to $n$ and we let
$\kappa=\sum_{k=1}^n\binom{m}{k}\binom{n}{k}$ be its dimension.
Given $w\in\mathbb{R}^\kappa$ we let $w_{\ell}$ the variables relative to
the $\ell\times\ell$ minors. We also denote $\mathscr{L}_m$ the
$\sigma$-algebra of Lebesgue measurable subsets of $\mathbb{R}^m$ and
$\mathscr{B}(\mathbb{R}^{n+\kappa})$ the $\sigma$-algebra of Borel subsets of $\mathbb{R}^{n+\kappa}$.
For the bulk part of the energy it is natural to treat polyconvex Lagrangians:
the lower semicontinuity properties of such energies with respect
to the weak $W^{1,p}$ convergence for $p<n$ has been thoroughly
studied, see \cite{CelDal94,FusHut95,FonLeoMal05,Mar86}.
\begin{theorem}[Existence of minimizers for polyconvex Lagrangians]\label{t:existlagr}
Assume $r,p$ satisfy $r<\infty$, $\tfrac 1r + \tfrac{n-1}{p}<1$ and
let $c>0$ be a positive constant.
Let $f:\Omega\times\mathbb{R}^n\times\mathbb{R}^\kappa\rightarrow[0,+\infty)$ satisfy
the following hypotheses:
\begin{itemize}
\item[(a)] $f$ is $\mathscr{L}_m\times\mathscr{B}(\mathbb{R}^{n+\kappa})$-measurable;
\item[(b)] for $\Leb{m}$-a.e. $x\in\Omega$, $(u,w)\mapsto f(x,u,w)$ is lower semicontinuous;
\item[(c)] for $\Leb{m}$-a.e. $x\in\Omega$, $w\mapsto f(x,u,w)$ is convex in $\mathbb{R}^\kappa$ for every $u\in\mathbb{R}^n$;
\item[(d)]
$
f(x,u,w)\geq c\big(|u|^r + |w_1|^p + \Psi(|w_n|)\big)
$
for some function $\Psi$ satisfying the hypotheses of Theorem \ref{t:compactness}.
\end{itemize}
Let also $g:\Omega\rightarrow [c,+\infty)$ be a lower semicontinuous function.\\
Then, for every $u_0\in W^{1-\frac 1p,p}(\partial \Omega,\mathbb{R}^n)$ there exists
a solution to the problem
\begin{equation*}\label{eq:lagrangian}
\min_{u\in\GSB_n(\Omega),\,u=u_0\mbox{ on }\partial\Omega}\left\{ \int_{\Omega} f\big(x,u(x),M\nabla u(x)\big)\,dx +
\int_{\Omega\cap S_u}g(x)\,d\Haus{m-n}(x)\right\}. \tag{P}
\end{equation*}
\end{theorem}
\begin{proof}
Suppose the energy \eqref{eq:lagrangian} is finite for some function in $\GSB_n(\Omega)$ with
trace $u_0$, otherwise there is nothing to prove.
Pick a minimizing sequence $(u_h)$: by the growth assumption
(d) there exist $u \in L^r\cap W^{1,p}$ and a subsequence
(not relabeled) such that
\begin{equation}\label{eq:strongL1}
u_h\rightarrow u \mbox{ in }L^1
\end{equation}
and $\nabla u_h\rightharpoonup\nabla u$ in $L^p$.
Since $\tfrac 1r + \tfrac{n-1}{p}<1$ we can choose $s<r$
such that $\tfrac 1s + \tfrac{n-1}{p}\leq 1$: by Chebycheff's
inequality we know that $u_h\rightarrow u$ in $L^s$.
Moreover we know that the absolutely continuous parts of $Ju_h$
satisfy
\begin{equation*}
\int_{\Omega}\Psi\left(\big| M_n\nabla u_h\big|\right)\,dx \leq C,
\end{equation*}
and that by the lower bound $g\geq c$ we also have:
\begin{equation*}
\sup_h\Haus{m-n}(S_{u_h}\cap \Omega)<\infty.
\end{equation*}
Hence by the compactness Theorem \ref{t:compactness}, together
with the classical Reshetnyak's Theorem for the minors of order less
than $n$, we know that
\begin{equation}\label{eq:weakL1}
M\nabla u_h\rightharpoonup M\nabla u\quad\mbox{ weakly in }L^1.
\end{equation}
By \eqref{eq:strongL1} and \eqref{eq:weakL1}
the lower semicontinuity result of Ioffe
\cite{Iof77I,Iof77II} (see also \cite[Theorem 5.8]{AmbFusPal})
implies
$$
\liminf_h \int_{\Omega} f\big(x,u_h(x),M\nabla u_h(x)\big)\,dx
\geq \int_{\Omega} f\big(x,u(x),M\nabla u(x)\big)\,dx.
$$
Finally, $g$ being lower semicontinuous,
the superlevel sets $\{g>t\}$ are open, hence
\begin{align*}
\liminf_h \int_{\Omega\cap S_{u_h}}g(x)\,d\Haus{m-n}(x) &=
\liminf_h\int_0^{+\infty} \Haus{m-n}(S_{u_h}\cap \{g>t\})\,dt \\
&\geq \int_0^{+\infty} \liminf_h \Haus{m-n}(S_{u_h}\cap \{g>t\})\,dt \\
&\geq \int_0^{+\infty} \Haus{m-n}(S_{u}\cap \{g>t\})\,dt \\
&= \int_{\Omega\cap S_{u}}g(x)\,d\Haus{m-n}(x),
\end{align*}
because the size is lower semicontinuous on open sets, see \eqref{eq:lscA}.
\end{proof}
Recall that by the Sobolev embedding we can drop the growth
condition on $u$ provided $p>\tfrac{mn}{m+1}$.
Notice also that we can formulate problem \eqref{eq:lagrangian}
and the corresponding boundary value condition
in a slightly different way, in order to include in the energy the possible
appearance of singularities at the boundary.
Let $U\Supset\Omega$ be a bounded open subset of $\mathbb{R}^m$:
we formulate the minimization problem in the following way:
\begin{equation*}\label{eq:lagrclosure}
\min_{u\in\GSB_n(U), \,u=u_0\mbox{ in }U\setminus\Omega}
\left\{ \int_{U} f\big(x,u(x),M\nabla u(x)\big)\,dx +
\int_{U\cap S_u}g(x)\,d\Haus{m-n}(x)\right\}\tag{P'}
\end{equation*}
Every competitor being equal to $u_0$ in $U\setminus\overline{\Omega}$,
problem \eqref{eq:lagrclosure} accounts for variations of $Ju$ in the
closure $\overline{\Omega}$. Moreover Theorem \ref{t:existlagr} readily
applies to this case, as the condition $u=u_0$ in $U\setminus\Omega$ is
closed for the strong $L^1$ convergence. To explicit the dependence on the energy
on the datum $u_0$ and on the domain $U$
we adopt in the sequel the notation
$$
F(u,\Omega;u_0,U)
$$
for the energy in \eqref{eq:lagrclosure}.
\subsection{A Mumford-Shah functional of codimension higher than one}\label{ss:MS}
As anticipated in the beginning of the section the study of general
functionals of the form \eqref{eq:lagrangian} was modeled on the
Mumford-Shah type functional
\begin{equation}\label{eq:MS}
MS(u,\Omega):= \int_{\Omega} |u|^r + |\nabla u|^p + |M_n\nabla u|^\gamma\,dx + \Haus{m-n}(S_u\cap\Omega)
\end{equation}
defined on $\GSB_n(\Omega)$,
with $r,p$ satisfying \eqref{exp}, $\gamma>1$,
together with suitable boundary data. Theorem \ref{t:existlagr}
shows the existence of minimizers of \eqref{eq:MS} for both Dirichlet
problems \eqref{eq:lagrangian} and \eqref{eq:lagrclosure}:
it is however desirable that at least for some boundary datum
$u_0$ the minimizer presents some singularity.
In the next proposition we show that this is the case:
\begin{proposition}[Nontrivial minimizers for $MS$, formulation \eqref{eq:lagrclosure}]\label{p:nontrivial}
Let $m=n$ and $u_0:B_2\rightarrow \mathbb{R}^n$ be the identity:
$u_0(x)= x$. Then for $\varepsilon$ sufficiently small every minimizer
$u\in\GSB_n(B_2)$ of
$$
MS_\varepsilon(u,B_1;x,B_2):=\int_{B_2} \varepsilon\big(|u|^r + |\nabla u|^p \big) +|\det\nabla u|^\gamma\,dx + \varepsilon\Haus{0}(S_u\cap B_2)
$$
such that $u(x)= x$ in $B_2\setminus B_1$ must satisfy
$$
S_u \cap\overline{B_1}\neq \emptyset.
$$
\end{proposition}
\begin{proof}
We show that for every competitor $v$ with $\|Jv\|\ll\Leb{n}$
and for $\varepsilon$ small enough it holds:
$$
MS_\varepsilon(v,B_1;x,B_2)>MS_\varepsilon(w,B_1;x,B_2),
$$
where
\begin{eqnarray*}
w(x)=
\left\{
\begin{array}{ll}
\tfrac{x}{|x|} & \mbox{ in }B_1,\\
x &\mbox{ in }B_2\setminus B_1.
\end{array}
\right.
\end{eqnarray*}
For the rest of the proof $c$ will denote a generic positive constant
we do not keep track of. Let us compute the energy of $\tfrac{x}{|x|}$:
the Dirichlet and $L^r$ parts are simply constants. Moreover
\begin{equation*}
\det \nabla w = \chi_{B_2\setminus B_1} \quad\mbox{ and }\quad S_{ w} =\{0\}.
\end{equation*}
Hence $MS_\varepsilon( w,B_1;x,B_2) = c\varepsilon + \Leb{n}(B_2\setminus B_1)$.
On the contrary for almost every radius $\rho$ it holds:
$$
\int_{B_\rho} \det\nabla v\,dx = \int_{\partial B_\rho} v^1 dv^2\wedge\dots\wedge dv^n
$$
(see \cite[Lemma 2.1]{DelGhi10} for a simple proof of this fact).
Since $u(x) = x$ outside $B_1$ for almost every
$\rho \in (1,2)$ we have $\int_{B_\rho} \det\nabla v\,dx = \Leb{n}(B_{\rho})$,
hence by Jensen's inequality
$$
\int_{B_\rho} |\det\nabla v|^\gamma\,dx \geq \Leb{n}(B_{\rho}).
$$
Summing up:
$$
MS_\varepsilon(v,B_1;x,B_2) \geq \int_{B_\rho} |\det\nabla v|^\gamma\,dx
\geq \Leb{n}(B_{\rho}) > c\varepsilon + \Leb{n}(B_2\setminus B_1) =
MS_\varepsilon(w,B_1;x,B_2)
$$
choosing first $\rho$ sufficiently close to 2 and then $\varepsilon$ sufficiently small.
Therefore the minimizer $u$ must have a nonempty
singular set $S_u$,
and since $u$ is linear in the open set $B_2 \setminus B_1$, the singularity must
be in $\overline{B_1}$.
\end{proof}
It is easy to generalize the same proposition to the case
$m\geq n$, by simply taking the trivial extension in the extra
variables and showing that every minimizer has a nontrivial singular
set. In analogy with \cite{HarLinWan98}, we expect however the
singularities to appear in the interior.
The argument in Proposition \ref{p:nontrivial} essentially
exploits the presence of the jacobian term: this is not
coincidental, as the next proposition shows. Recall that
the sum of a $GSB_nV$ function and a $C^1$ function is
again in $GSB_nV$.
\begin{proposition}
Every local minimizer of
$$
u\in GSB_nV(\Omega)\mapsto \int_{\Omega}|\nabla u|^p\,dx + \Haus{m-n}(S_u\cap\Omega)
$$
is locally of class $C^{1,\alpha}$ in $\Omega$.
\end{proposition}
\begin{proof}
It is sufficient to perform an outer variation of the minimizer $u$
along a $\phi\in C^1_c(\Omega,\mathbb{R}^n)$ map: $\varepsilon\mapsto u +\varepsilon\phi$
and apply Corollary \ref{c:detgeneral} to obtain that
$$
S_{u + \varepsilon\phi} = S_u.
$$
Hence the size term is constant and $u$ satisfies:
$$
\int_{\Omega} |\nabla u|^{p-2}\nabla u \nabla \phi\,dx = 0 \quad \forall \phi\in C^1_c(\Omega,\mathbb{R}^n).
$$
Therefore $u$ is a $p$-harmonic $W^{1,p}$ function,
hence $u\in C^{1,\alpha}_{\rm loc}$ by \cite{Uhl77, DiB83}.
\end{proof}
\subsection{Traces}\label{ss:traces}
In the spirit of solving \eqref{eq:lagrclosure},
the nonuniqueness Example \ref{ex:cerchi} below raises the problem
of the dependence of the energy on the extension $u_0:U\setminus\Omega\rightarrow\mathbb{R}^n$
to a given Sobolev trace $u\sbarretta_{\partial\Omega}$.
The example was communicated to us by C. De Lellis, see
also Examples 1 and 2 in Section 3.2.5 of \cite{GiaModSou},
and the discussion on weak and strong anchorage condition therein.
It shows that if we want to detect the presence of singularities
of $Ju$ at the boundary of $\Omega$, the Sobolev trace is not
sufficient to characterize it.
\begin{example}[Singularity at the boundary]\label{ex:cerchi}
Let $u:\mathbb{R}^2\rightarrow S^1$ be defined by
\begin{equation}\label{eq:cerchi}
u(x,y)=\left( \frac{y^2- (x-1)^2}{(x-1)^2+y^2},\frac{2(1-x)y}{(x-1)^2+y^2}\right).
\end{equation}
This map represents the normal unit vectorfield
of the family of circles centered on the real axis and tangent
to $S^1$ in the point $(1,0)$. If $\theta$ is the angle that
the vector $(x-1,y)$ makes with the real axis, we can write
$u(x,y)=(-\cos(2\theta),-\sin(2\theta))$, hence by Example
\eqref{ex:monopole} $Ju=2\pi\llbracket (1,0)\rrbracket$. Note that $u$ is the identity
map when restricted to $S^1$.
Nonetheless we can construct another map $\tilde{u}$
\begin{equation}\label{eq:cerchitilde}
\tilde{u}(x,y)=\left\{
\begin{array}{ll}
u(x,y) \qquad &\mbox{ for }\quad |x|<1,\\
\Big(\frac{x}{\sqrt{x^2+y^2}},\frac{y}{\sqrt{x^2+y^2}}\Big)\qquad &\mbox{ for }\quad |x|\geq 1.
\end{array}
\right.
\end{equation}
In this case, by Example~\ref{ex:monopole}, $J\tilde{u}=\pi\llbracket (1,0)\rrbracket$.
Hence $u\sbarretta_{B_1}$ admits two different Sobolev extensions
$u$ and $\tilde{u}$ sharing the same trace at the boundary
but whose jacobians are different in $\overline{\Omega}$: the
trace of a Sobolev function does not characterize the jacobian
$Jv\res{\partial\Omega}$ of all the possible extensions $v$.
\end{example}
It is interesting to know when part of the distributional jacobian
can be represented as a boundary integral. Recall that
the slicing Theorem \ref{t:slicing} already provides an answer
to this question,
because if $u:\Omega\rightarrow\mathbb{R}^n$ then
$\partial (j(u)\res\{\pi>t\}) = Ju\res\{\pi>t\} + \langle j(u),\pi,t\rangle$,
where $\pi$ is the distance from $\partial\Omega$.
However, as Example \ref{ex:cerchi} shows, this statement holds only
for $\Leb{1}$-a.e. $t$. The following
proposition improves the general result by slicing, under additional
hypotheses on the summability of $u$ and of its trace.
This result is already present in the literature,
see \cite[Vol. I, p. 274]{GiaModSou} and \cite[Lemma 6.1]{AlbBalOrl05}:
we report the proof for the reader's convenience.
Denote for simplicity $g(u):=u^1 du^2\wedge\dots\wedge du^n$.
\begin{proposition}[Stokes' Theorem]\label{p:stokes}
If $u\in W^{1,n}(\Omega,\mathbb{R}^n)$ and $u\sbarretta_{\partial\Omega}\in W^{1,n-1}(\partial\Omega, \mathbb{R}^n) \cap L^{\infty}(\partial\Omega,\mathbb{R}^n)$ then
Stokes' theorem holds:
$$
\partial (j(u)\res\Omega) = Ju\res\Omega + \langle j(u),\partial\Omega\rangle
$$
with the representation
$$
\langle j(u),\partial\Omega\rangle(\omega) = \int_{\partial\Omega} \langle g(u),\tau_{\partial\Omega}\rangle\omega\,d\Haus{n-1}
$$
where $\tau_{\partial\Omega}$ orients $\partial\Omega$ as the boundary of $\Omega$.
In particular $\langle j(u),\partial\Omega\rangle$ depends
only on the trace $u\sbarretta_{\partial\Omega}$.
\end{proposition}
\begin{proof}
Suppose for simplicity that $\Omega = \mathbb{R}^{n}_+ = \mathbb{R}^n\cap\{x^n>0\}$, ${\rm spt}(u)\subset B_1$ and let $\phi:\mathbb{R}^{n-1}\rightarrow \mathbb{R}$ be a positive convolution kernel with compact support in $\mathbb{R}^{n-1}$. Set
$$
u_\varepsilon(x',x^n) = \frac{1}{\varepsilon^{n-1}}\int_{\mathbb{R}^{n-1}} u(x'-y', x^n)\phi\left(\frac{x'-y'}{\varepsilon}\right)\,dy':
$$
since the convolution in the $x'$ variables commutes with the trace operator
we still have $u_\varepsilon\sbarretta_{\mathbb{R}^{n-1}}(x') = u_\varepsilon(x',0)$; moreover $u_\varepsilon(\cdot,0)\in C^1(\mathbb{R}^{n-1},\mathbb{R}^n)$ and the following estimates hold:
\begin{equation}\label{eq:convol}
\|u_\varepsilon\|_{W^{1,n}(\mathbb{R}^n_+,\mathbb{R}^n)} \leq \|u\|_{W^{1,n}(\mathbb{R}^n_+,\mathbb{R}^n)},\qquad \|u_\varepsilon(\cdot,0)\|_{W^{1,n-1}(\mathbb{R}^{n-1},\mathbb{R}^n)} \leq \|u(\cdot,0)\|_{W^{1,n-1}(\mathbb{R}^{n-1},\mathbb{R}^n)}
\end{equation}
and since the translations are strongly continuous in $L^p$,
\begin{equation}\label{eq:convolest}
\|u_\varepsilon-u\|_{W^{1,n}(\mathbb{R}^n_+,\mathbb{R}^n)} + \|u_\varepsilon(\cdot,0)-u(\cdot,0)\|_{W^{1,n-1}(\mathbb{R}^{n-1},\mathbb{R}^n)}\rightarrow 0.
\end{equation}
We claim that Stokes' Theorem holds for $u_\varepsilon$: for every
$\omega\in \mathscr{D}^0(\mathbb{R}^n)$
\begin{equation}\label{eq:stokes}
\partial(j(u_\varepsilon)\res\mathbb{R}^n_+)(\omega) = \int_{\mathbb{R}^n_+}\omega\det\nabla u_\varepsilon\,dx +
\int_{\mathbb{R}^{n-1}\times\{0\}} \omega g(u_\varepsilon(\cdot,0)).
\end{equation}
In fact extending $u_\varepsilon(x',x^n):=u_\varepsilon(x',0)$ for $x^n \in [-1,0]$ and then convolving
with a smooth kernel $\rho_\delta$ supported in $B_\delta(0)$ we obtain a smooth
$u_{\varepsilon,\delta}\in C^{\infty}(\mathbb{R}^n,\mathbb{R}^n)$ such that ${\rm spt}(u_{\varepsilon,\delta})\subset B_2\times [-2,2]$,
\begin{eqnarray}
& u_{\varepsilon,\delta}(x',x^n) \rightarrow u_\varepsilon(x',0) \qquad &\mbox{ in } C^1_{\rm loc}(\mathbb{R}^{n}_-,\mathbb{R}^n), \notag\\
& u_{\varepsilon,\delta}\rightarrow u_\varepsilon\qquad&\mbox{ in }W^{1,n}_{\rm loc}(\mathbb{R}^{n-1}\times(-1,+\infty),\mathbb{R}^n).\label{eq:convdelta}
\end{eqnarray}
More precisely it holds: $u_{\varepsilon,\delta}(x',-\delta) \rightarrow u_\varepsilon(x',0)$
in $C^1(\mathbb{R}^{n-1},\mathbb{R}^n)$.
Hence
$$
\partial(j(u_{\varepsilon,\delta})\res\{x^n>-\delta\})(\omega) =
\int_{\{x^n>-\delta\}}\omega\det\nabla u_{\varepsilon,\delta}\,dx +
\int_{\mathbb{R}^{n-1}\times\{-\delta\}} \omega g(u_{\varepsilon,\delta}(\cdot,-\delta)):
$$
letting $\delta\downarrow 0$ the left hand side converges to $\partial(j(u_\varepsilon)\res\mathbb{R}^n_+)(\omega)$
by \eqref{eq:convdelta}. The boundary term in right hand side tends to
$$
\int_{\mathbb{R}^{n-1}\times\{0\}} \omega(\cdot,0) g(u_\varepsilon(\cdot,0))
$$
because the convergence is $C^1$ and $\omega$ is smooth. Regarding the volume integral
we can estimate
$$
|\nabla u_{\varepsilon,\delta}(x)| =| (\rho_\delta*\nabla u_\varepsilon)(x)| \leq
\|u_\varepsilon(\cdot,0)\|_{C^1} + \fint_{B_\delta(x)\cap\{y^n>0\}}|\nabla u_\varepsilon(y)|\,dy
$$
hence
\begin{align*}
\left| \int_{ \{|x^n|<\delta\}}\omega\det\nabla u_{\varepsilon,\delta}\,dx \right|
&\leq \|\omega\|_{C^0} \int_{ \{|x^n|<\delta\}}|\nabla u_{\varepsilon,\delta}|^n\,dx \\
&\leq c_n\|\omega\|_{C^0} \left( \|u_\varepsilon(\cdot,0)\|_{C^1}^n\delta+
\int_{ \{|x^n|<\delta\}}\fint_{B_\delta(x)\cap\{y^n>0\}}|\nabla u_\varepsilon(y)|^n\,dy\,dx \right) \\
&\leq c_n\|\omega\|_{C^0} \left( \|u_\varepsilon(\cdot,0)\|_{C^1}^n\delta+
\int_{ \{0<x^n<2\delta\}}|\nabla u_\varepsilon(x)|^n\,dx \right) \rightarrow 0.
\end{align*}
Clearly $ \int_{ \{ x^n>\delta\}}\omega\det\nabla u_{\varepsilon,\delta}\,dx \rightarrow
\int_{ \{ x^n>0\}}\omega\det\nabla u_{\varepsilon}\,dx $, therefore \eqref{eq:stokes} is true.
We now want to pass to the limit for $\varepsilon\downarrow 0$ in \eqref{eq:stokes}.
The left hand side goes to $\partial(j(u)\res\mathbb{R}^n_+)(\omega) $ because of \eqref{eq:convolest};
similarly for the volume term. Regarding the boundary term the convergence of
the minors on the slice needs to be improved.
The estimates \eqref{eq:convol} and the classical result \cite{CoiLioMeySem93} gives
a uniform bound of the Hardy norm \cite[Chapter IV]{Ste} of the minors of order $n-1$:
$$
\|du^2_\varepsilon(\cdot,0)\wedge\dots\wedge du^n_\varepsilon(\cdot,0)\|_{\mathcal{H}^1(\mathbb{R}^{n-1},\mathbb{R}^n)}\leq \|u(\cdot,0)\|^{n-1}_{W^{1,n-1}(\mathbb{R}^{n-1},\mathbb{R}^n)}.
$$
We already know from Reshetnyak's Theorem that
$
du^2_\varepsilon(\cdot,0)\wedge\dots\wedge du^n_\varepsilon(\cdot,0)\overset{*}{\rightharpoonup} du^2(\cdot,0)\wedge\dots\wedge du^n(\cdot,0)$
in the sense of distributions; moreover
smooth functions are dense in $VMO(\mathbb{R}^{n-1},\mathbb{R}^n)$ and $VMO^* = \mathcal{H}^1$,
so
$$
du^2_\varepsilon(\cdot,0)\wedge\dots\wedge du^n_\varepsilon(\cdot,0)\overset{*}{\rightharpoonup}
du^2(\cdot,0)\wedge\dots\wedge du^n(\cdot,0)\qquad\mbox{ in}\quad\sigma(\mathcal{H}^1,VMO).
$$
Finally the trace $u_\varepsilon(\cdot,0)$ belongs to
$$W^{1-\frac{1}{n},n}(\mathbb{R}^{n-1},\mathbb{R}^n)\subset VMO(\mathbb{R}^{n-1},\mathbb{R}^n)$$ (see \cite[Theorem 7.58]{Ada}, \cite[Example 2]{BreNir95} for the inclusions). Hence $\|u_\varepsilon(\cdot,0) - u(\cdot,0)\|_{VMO}\rightarrow 0$ strongly and we can pass to the limit in
\eqref{eq:stokes}
$$
\int_{\mathbb{R}^{n-1}} \omega g(u_\varepsilon(\cdot,0)) \rightarrow \int_{\mathbb{R}^{n-1}} \omega g(u(\cdot,0)).
$$
By \eqref{eq:convolest} also the left hand side of \eqref{eq:stokes} converges to $\int_{\mathbb{R}^n_+}\omega\det\nabla u\,dx$.
\end{proof}
In the example
above the smooth extension $\tilde{u}$ is certainly preferable to $u$,
where an ``extra" singularity comes from the outside.
A partial answer
to this problem can be given if we assume a better differentiability of the
outer extension, up to $\partial \Omega$:
\begin{proposition}[See {\cite[Vol. I, p.266]{GiaModSou}}]\label{p:uniqext}
Let $v,w\in L^s\cap W^{1,p}(U,\mathbb{R}^n)$ satisfy the following conditions:
\begin{itemize}
\item $v\sbarretta_{\,\Omega} = w\sbarretta_{\,\Omega}$;
\item $v\sbarretta_{\,U\setminus \Omega}, w\sbarretta_{\,U\setminus \Omega} \in W^{1,n}(U\setminus\Omega,\mathbb{R}^n)$;
\item $v\sbarretta_{\,\partial \Omega}=
w\sbarretta_{\,\partial \Omega} \in W^{1,n-1}(\partial \Omega,\mathbb{R}^n)$.
\end{itemize}
Then:
$$
Jv-Jw = \big( \det \nabla v -\det\nabla w\big){\bf E}^n\res (U\setminus\Omega).
$$
\end{proposition}
\begin{proof}
We can write
\begin{align}
Jv &= \partial j(v) = \partial (j(v)\res\Omega) + \partial (j(v)\res(U\setminus \Omega))
= \partial (j(v)\res\Omega) + Jv\res (U\setminus \Omega ) - \langle j(v),\partial\Omega\rangle.
\end{align}
Subtracting the analogous expression for $Jw$ we obtain
$$
Jv-Jw = (Jv - Jw)\res (U\setminus \Omega) - \langle j(v) -j(w) ,\partial\Omega\rangle=
\big( \det \nabla v -\det\nabla w\big){\bf E}^n\res (U\setminus\Omega)
$$
because Proposition \ref{p:stokes}
applied to the open set $U\setminus\Omega$ implies that $v\sbarretta_{\partial\Omega} = w\sbarretta_{\,\partial\Omega}$, hence $\langle g(v) - g(w),\tau_{\,\partial\Omega}\rangle = 0$.
\end{proof}
Therefore, if we aim at formulating problem \eqref{eq:lagrclosure}
in a local way, that is depending only on the values of $u$ in
$\overline{\Omega}$, at least when the trace is sufficiently ``nice",
we can proceed as follows.
If $u\sbarretta_{\partial\Omega}$ belongs to $W^{1,n-1}$
and admits a $W^{1,n}$ extensions outside $\Omega$,
we can conventionally agree to pick one of such extensions
to $U\setminus\Omega$:
the result of Proposition \ref{p:uniqext}
implies that the jacobian in $\overline{\Omega}$ of every competitor
does not depend on the particular choice we made.
Note however that the smoothness of the trace
does not imply membership of the extension to $\GSB_n(U)$. In fact,
it is sufficient to place the infinite dipoles of the function $g$ in
Example \ref{ex:dipole} so that the singularities lie on $\partial B_1$
and do not overlap. The constant extension outside the ball
provides a map whose jacobian has both infinite mass and size.
In conclusion, in order to solve Problem \eqref{eq:lagrclosure}
it seems necessary to impose membership of the competitors
to $GSB_nV(U)$, while for a fairly broad class of boundary data
the energy in $\overline{\Omega}$ shall not depend on the particular extension.
| -99,472.506053 |
[
-2.654296875,
2.4140625
] | 20.455671 |
[
-2.51171875,
0.69970703125,
-1.8896484375,
-6.23828125,
-1.251953125,
8.875
] |
[
3.888671875,
9.546875,
0.38916015625,
5.19140625
] | 485 | 10,918 |
[
-3.248046875,
3.77734375
] | 33.008779 |
[
-5.3984375,
-4.45703125,
-6.01953125,
-2.7890625,
1.8515625,
14.234375
] | 0.583998 | 6.773724 | 25.709837 | 1.216889 |
[
1.6942793130874634
] | -55,275.913497 | 6.116596 | -99,918.676547 | 0.20964 | 6.533647 |
[
-1.5341796875,
-3.45703125,
-4.16796875,
-5.42578125,
1.8828125,
12.828125
] |
[
-5.6328125,
-1.9375,
-1.83984375,
-0.9287109375,
3.53125,
3.71875
] | |
BkiUdW85qsJBjnJ89-mW
|
\section{Introduction and Motivation}
Perhaps the hottest topic in recent years regarding LHSPs is the discussion around physical unclonable functions (PUFs). The basic notion of a PUF involves a unique and reliable set of challenge-response pairs (CRPs), or inputs and outputs to a physical function, that can be used for security purposes including device identification, authentication~\cite{yan2015novelway}, and key generation. For a specific challenge, a PUF returns an associated response by way of a one-way function, meaning that the challenge cannot be re-obtained from the response. The size of the CRP set characterizes a PUF as being either strong (having a large enough number of CRPs that recording them all would be infeasible) or weak (having a small enough number of CRPs that recording them all would be feasible). Depending on which category the PUF falls into, it may or may not be suited for a specific use case. A number of PUF implementations exist, and a shared benefit of these LHSPs is that a PUF can provide device security without the need for additional hardware components intended solely for a security purpose. This is because PUFs take advantage of the inherent differences between silicon chips on the micro or nano scale that arise in the manufacturing process. They use these differences to provide a unique set of seemingly random yet reliable responses to corresponding challenges. This method of taking advantage of inherent uniqueness make PUFs especially desirable in situations where small size is a necessary constraints, such as in the case of Internet of Things
(IoT) technology~\cite{hassan2017internet}~\cite{tehranipoor2017invest}.
As discussed in~\cite{tehranipoor2017exploring}, employing secure methods of IoT device authentication and data protection is critical, yet poor security design is a common issue facing the IoT world~\cite{wortman2017proposing}, including devices on the consumer electronic market, in the medical and healthcare domains, and in other areas. In~\cite{tehranipoor2018low}, Tehranipoor et al. discuss consumer electronic device authentication and propose a low-cost PUF verification solution that uses keys generated from biometric signals~\cite{karimian2016evolving}. Another consumer electronics authentication scheme, termed P2M-Sec, is proposed in~\cite{wortman2018p2m} that uses PUF responses as seeds for a pseudorandom number generator. Multiple PUFs are investigated for use in the scheme, including an arbiter, ring oscillator~\cite{aguirre2020systematic}, bistable ring oscillator, butterfly, and DRAM-based PUF~\cite{commercial2017}. Similarly in~\cite{karimian2018secure}, Karimian et al. propose a PUF-based authentication framework with particular emphasis on ECG biometrics. The framework combines PUFs with hardware obfuscation, and makes non-volatile storage of keys (which can leave keys vulnerable to attack) unnecessary. The authors detail several case studies with their framework, and report that using the framework with an ECG provides the best trade-off between reliability, key length, entropy, and cost. In~\cite{karimian2017noise}, the authors discuss key generation techniques with ECGs based on different feature extraction techniques with varying noise sources. They propose the NA-IOMBA (noise aware interval optimized mapping bit allocation) protocol for improved key reliability, building off of their previously proposed IOMBA technique~\cite{karimian2016highly}. In~\cite{karimian2019heartid}, using ECG-based authentication is explored in the context of autonomous cars (HeartID), with heart signals being used to authenticate interaction between a person and a car (such as allowing entry to the vehicle).
One type of module often inherent in IoT devices that has been proposed for use as a PUF (as well as a TRNG) is dynamic random-access memory~\cite{anagnostopoulos2018overview}. Fig.~\ref{fig:noisydram} shows a key generation scheme proposed in~\cite{karimian2019generate} that can be used for device identification with DRAM-based PUFs, such as that proposed in~\cite{tehranipoor2016dram}~\cite{tehranipoor2015dram} by Tehranipoor et al. The 128-bit PUF relies on the random startup values of cell capacitors in an on-board DRAM module in order to generate a unique response when the device is turned on. In this case, the power-up sequence would be regarded as the challenge, and the values of the cells of interest make up the corresponding response. As seen in Fig.~\ref{fig:dramnet}, a similar use case of the entropic behavior observed in DRAM has been proposed in~\cite{karimian2019dramnet}~\cite{dramnet2} for an authentication scheme called DRAMNet. It uses a convolutional neural network (CNN) to extract and classify unique features from a DRAM image structure (created by converting the power-up sequence values into a 2-dimensional array) in order to authenticate a device with greater than 98\% accuracy and precision.
\begin{figure}
\centering
\includegraphics[width=0.8\linewidth]{NoisyDRAM2019.png}
\caption{DRAM-based PUF key generation~\cite{karimian2019generate}.}
\label{fig:noisydram}
\end{figure}
Several quality metrics are used to evaluate PUFs, including uniqueness, uniformity, and reliability. Uniqueness can be determined by calculating the inter-chip Hamming distance, which determines how different a PUF's responses will typically be when the same PUF is implemented on different chips and given the same challenges. Ideally, the inter-chip Hamming distance will be 50\%, meaning half of the bits in a PUF's response on one chip will be flipped when it's given the same challenge on another chip. Uniformity refers to the ratio of ones to zeros in PUF responses. Ideally, a PUF should exhibit 50\% ones and 50\% zeros on average in its responses in order to keep bit predictability as low as possible. Determining a PUF's reliability involves examining how well its CRP set holds up under varying environmental conditions. Ideally, a response will be exactly the same each time it's given the same challenge under all operating conditions, such as at different temperatures or after aging. In~\cite{anagnostopoulos2018securing}, DRAM-based PUFs, namely retention-based and row hammer PUFs, are evaluated in the context of temperature and voltage variations, and are shown to be highly dependent on temperature. The effect of temperature variation on PUFs is further investigated in~\cite{anagnostopoulos2018addressing}, and a method is proposed to account for the negative impacts. Specifically, temperature sensors, which are often present in DRAM modules, are used with a DRAM-based PUF in order to create an authentication protocol suitable for varying temperatures that restricts access to PUF responses when it's determined that the operation could be compromised by the environment's temperature.
Much research is put into ensuring PUF reliability. For example, in a paper by Tehranipoor et al.~\cite{tehranipoor2017investigation}, a report is given on the reliability of three DRAM-based PUFs aged 18 months. The authors show that the PUFs underwent only slight losses in their reliability over this period of time - the best of the three dropping from 88.9\% reliability to 87.5\%, and the worst from 89.7\% to 81.9\% over the period of accelerated aging. Another example of a PUF proposed specifically with reliability in mind is a phase calibrated ring oscillator PUF that can be implemented on a field-programmable gate array (FPGA)~\cite{yan2017phase}. The PUF is shown to have better response reliability than that of a traditional ring oscillator PUF, which relies on inherent wire and inverter delay to generate responses. Other PUF metrics including power requirements and footprint are also very important (especially in the realm of IoT), but uniqueness, uniformity, and reliability are among the most crucial in regards to a PUF's intended operation.
True random number generators (TRNGs) are another form of LHSP. Unlike a pseudo-random number generator (PRNG) that is ultimately deterministic in generating seemingly random outputs, a TRNG is non-deterministic, meaning it does not use an algorithm to generate outputs (which would reveal its secrets if known). Rather, a TRNG relies on a physical process to generate random bit streams. A few examples that a TRNG could utilize for this purpose include phase noise, thermal noise, jitter, random telegraph noise, and the photoelectric effect~\cite{tehranipoor2016robust}. TRNGs are used in a wide variety of security applications, with cryptographic algorithms often relying on them for key generation. As an example of a TRNG, one proposed by Tehranipoor et al.~\cite{tehranipoor2016robust} utilizes the remanence effect of DRAM (where some information remains in the memory cells after powering down) in order to generate random cryptographic keys. DRAM is also proposed for use as a TRNG in~\cite{eckert2017drng}, but the startup values are used instead. Another TRNG proposed in~\cite{tehranipoor2017study} is based on power supply noise, and based on results from a NIST statistical test suite, the authors conclude that the variations in voltage exhibit enough entropy that they can be used to generate random bits characteristic of a TRNG. Similarly in~\cite{tehranipoor2018dvft}, a dynamic feedback voltage tuning (DVFT) power supply noise-based TRNG is proposed. Six different power supplies are evaluated, and Gaussian distributions are observed, demonstrating non-cyclostationary behavior that can be utilized in the context of true random number generation~\cite{tehranipoor2018towards}.
\section{Literature Review}
While the concept of a PUF or TRNG may be nice, actually implementing one in practice that is fully resistant to attack is another story. In many cases regarding PUFs, these supposed "unclonable" functions have in fact been shown to be clonable through modeling. Equivalently, numerous successful attacks have been mounted against TRNGs. These attacks can come in various forms - many of which will be discussed in this section. In particular, the following literature review for PUFs is primarily focused on recent reports on PUF modeling attacks via machine learning (ML) and PUFs designed with ML-resistance in mind. The following subsection discusses recent attacks on TRNGs.
\subsection{Attacks on PUFs}
In this subsection, we provide a summary of some of the most recent developments in machine learning-based attacks on PUFs, as well as proposed ML-resistant PUFs. In all cases, the proposed/attacked PUFs are thought to meet at least a minimum set of requirements for uniqueness, uniformity, and reliability.
Beginning with a paper by Nguyen et al.~\cite{nguyen2019interpose}, the authors present the Interpose PUF and evaluate it against ML attacks. The Interpose PUF is a strong PUF consisting of two XOR arbiters, and the authors conclude that it is not vulnerable to logistic regression (LR) and covariance matrix adaptation evolution strategies (CMA-ES). In~\cite{wisiol2020splitting} however, it is shown that the Interpose PUF, which was thought to be more secure than existing XOR arbiter PUFs due to its resistance to LR and CMA-ES, is vulnerable to deep learning. The attack times recorded are comparable to those for other XOR arbiter PUFs, and the authors present results showing that they were able to achieve an attack success rate of at least 88\% on several variations of the Interpose PUF with a multilayer perceptron (MLP) neural network.
In~\cite{delvaux2019machine}, an ML-based side-channel impersonator attack is proposed and used to evaluate five different arbiter PUFs, specifically the PolyPUF by Konigsmark et al., the OB-PUF by Gao et al., the RPUF by Ye et al., the LHS-PUF by Idriss et al., and the PUF-FSM by Gao et al. All PUFs are successfully modeled, and it is noted that often the physical security of PUF protocols is not given enough attention in the design phase as opposed to mathematical security, which can result in unseen vulnerabilities.
In a paper by Wisiol et al., the authors present an attack on the Lightweight Secure PUF that focuses on attacking the input transformations (that result in local minima) that can be used to model the PUF. The Lightweight Secure PUF was previously thought to be significantly more resistant to ML than traditional XOR arbiter PUFs, with the input transformations being introduced for this purpose. The authors of this paper show that unfortunately these input transformations can actually be exploited to achieve successful modeling times comparable to those recorded when attacking classic XOR arbiter PUFs. LR and a correlation attack are used in the evaluation. The authors also propose a potential low-overhead countermeasure using fix-point-free permutations as an input transformation alternative that does not result in local minima. They find that this technique is able to significantly increase the PUF's resistance to their proposed attack.
In~\cite{laguduva2019machine} Laguduva et al. introduce a non-invasive architecture-independent ML attack targeting PUF-based IoT nodes. The three PUFs evaluated are of arbiter, XOR, and Lightweight architectures, and the attack involves the characterization of a given PUF via an optimal function in order to determine whether an LR, random forest (RF), or a neural network ML algorithm should be used to model it. With this attack strategy they find that they're able to achieve a modeling accuracy of ~95.3\%. The authors, however, also propose a countermeasure (that they call "discriminator") to their attack which is able distinguish cloned PUFs from the original for secure authentication from a cloud server with an accuracy of ~96\%.
\begin{figure}[htp]
\centering
\includegraphics[width=0.8\linewidth]{ICEM2020.png}
\caption{Authentication with DRAMNet~\cite{karimian2019dramnet}.}
\label{fig:dramnet}
\end{figure}
In~\cite{wang2019adversarial}, a CRP mechanism is suggested as a countermeasure to ML modeling attacks. Traditionally, a response for a given challenge is fixed (assuming proper reliability), but the authors propose adding a feature to strong PUFs that is capable of inverting responses with a chosen probability in order to throw off an attacker's modeling scheme. LR and CMA-ES are used to test the effectiveness of the proposed mechanism with a 64-bit arbiter PUF, and the authors find that the PUF's resistance does indeed improve with the countermeasure active. The prediction accuracy by LR without any response inversion is ~99\%, and when an inversion probability of 50\% is active, the accuracy drops to ~55\%. Similarly for CMA-ES, the prediction accuracy drops from ~99\% without any inversion to ~78\% with a 50\% probability of inversion.
In~\cite{zhuang2019strong}, a subthreshold current array PUF is proposed that is resistant to ML attacks. The PUF consists of two transistor arrays that exhibit nonlinear behavior due to the nonlinearity of the I-V characteristics for the subthreshold region of a MOSFET. Support vector machine (SVM) and LR algorithms along with neural networks are used to evaluate the PUF, which is shown to outperform the arbiter and 3-XOR PUFs it's compared against in terms of attack resistance. In~\cite{mahmoodi2019chipsecure}, a reconfigurable flash-based PUF called ChipSecure is proposed that takes advantage of random I-V characteristics in flash memory, consisting of two layers of primitive blocks and a shift register. The authors attack their proposed design with an MLP and report that achieved prediction accuracy is close to an ideal 50\%.
In~\cite{kroeger2020effect}, the authors present results on the effect of PUF aging on ML attacks. Specifically, they report that ML models trained with a dataset of CRPs recorded at a certain PUF age lose their effectiveness when attempting to predict on an attack dataset of CRPs collected after the PUF has aged.
\begin{figure}
\centering
\includegraphics[width=0.7\linewidth]{ARCHINDEP.png}
\caption{Architecture-independent PUF attack model~\cite{laguduva2019machine}.}
\label{fig:archindep}
\end{figure}
In a paper by Wang et al., the authors proposed a Lattice strong PUF that's shown to be resistant to ML with both classical and quantum computing. The PUF is made with a physically obfuscated key, a learning-with-errors decryption function block, and a linear-feedback shift register (used as a pseudo-random number generator). They attack their PUF with SVM, LR, and deep learning neural networks, and find that the prediction accuracy is ~50\% after 1 million training CRPs in all cases. In~\cite{wu2020ct}, the authors propose a configurable tri-state strong PUF that combines an arbiter, ring oscillator, and bistable ring PUF for improved ML resistance over the configurable ring oscillator and dual-mode PUFs it's compared against. They attack all three PUFs with LR and neural networks, and report that while the dual-mode PUF's resistance to LR is similar to that of their tri-state PUF, they were able to achieve greater than 90\% modeling accuracy for the dual-mode PUF with neural networks, but only 50-60\% for their tri-state PUF. The configurable ring oscillator PUF is shown to be easily modeled by both LR and neural networks.
In a paper by Ge et al., the authors propose a challenge preprocessing structure for an arbiter PUF that's shown to improve ML resistance by hiding the linear correlation seen between a challenge and its corresponding response in a traditional arbiter PUF. Flip-flops are used to preprocess the original challenge before it's sent to the arbiter PUF, throwing off the direct relationship between the original challenge and the response that follows. The highest modeling accuracy achieved by the authors with linear regression, LR, SVM, and backpropogation neural network (BPNN) ML algorithms is ~61\% (with the BPNN). In~\cite{pugazhenthi2019dla}, a DRAM-based PUF (DLA-PUF) that uses utilizes DRAM startup values is attacked with Naive Bayes (NB), LR, SVM, and CNN ML algorithms. It is shown to be resistant to these attack strategies when 3 DRAM startup values are used in its scheme.
In~\cite{ma2018machine}, the authors propose an arbiter-based multi-PUF that uses a group of weak PUFs to obfuscate challenges given to a strong PUF by taking the XOR of the weak PUF response bits and the challenge, and feeding the result into the strong PUF. The intention is to improve resistance to ML attacks. The authors attack their design with LR and CMA-ES, and find the prediction rate is ~50\% with LR (as opposed to 100\% when attacking a traditional arbiter PUF) and ~80\% with CMA-ES when the challenge size is at least 32 bits. In~\cite{wang2018machine}, a dual-mode PUF that works with both an odd number of inverters like a reconfigurable ring oscillator PUF and an even number of inverters like a bistable ring PUF is proposed, with the assumption that no ML attack can model both a ring oscillator and bistable ring PUF. Thus, as long as the current working mode can be hidden from attackers, its ML resistance should increase. They attack their PUF with LR and a neural network, and find that the LR accuracy is ~50\% and that the neural network accuracy is ~60\%, which is a significant improvement over the ML resistance of a ring oscillator PUF on its own (for which ~99\% modeling accuracy can be achieved).
In~\cite{xi2017strong}, the authors propose a subthreshold current array strong PUF that they show to be resistant to ML. The PUF boasts $2^{65}$ CRPs, and consists of two-dimensional transistor arrays with transistors operating in the subthreshold region with high voltage variability. The PUF is attacked with SVM, LR, and neural network algorithms, and from the authors' results is estimated to be 100 times more resistant to these attacks than a traditional arbiter PUF. In~\cite{tanaka2018coin}, a coin-flipping PUF is proposed that utilizes nonlinearity found in the convergence time of a bistable ring PUF with regard to threshold voltage variations. Responses are generated based on a ring oscillator's value at the convergence time of its associated bistable ring. The PUF is attacked with SVM, ES, RF, bagging, and boosting ML algorithms, and it is shown that the prediction accuracy is ~50\% across the board. In~\cite{pang2017novel}, the authors propose a double-layer 16Mb RRAM (resistive random-access memory) array PUF that uses the resistance values of cells to generate responses. The authors attack their double-layer PUF along with a single-layer version and a traditional arbiter PUF with a neural network, and are able to achieve a ~90\% success rate on the traditional arbiter, ~70\% on the single-layer version of their PUF, and only ~55\% on the double-layer version.
\subsection{Attacks on TRNGs}
This subsection discusses some of the most recent developments in TRNGs and their vulnerabilities. In a paper by Osuka et al., the authors investigate injecting sinusoidal waves through a power cable in order to non-invasively attack a TRNG by lowering its entropy and estimating the internal state of a ring oscillator through side-channel analysis. They show that when a signal at a particular frequency is injected, the TRNG's oscillator becomes locked, leading to a predictable generation of bits. The authors also propose a countermeasure to the attack, suggesting that including ferrite cores in the power line could help suppress the frequencies that are capable of locking the ring oscillator.
In a paper by Koyen et al., three forms of attack on a ring oscillator TRNG implemented on an FPGA are investigated. Of the three attack methods: voltage manipulation, ring oscillator locking, and replica observation, the voltage manipulation attack is reported to be the most effective. In a paper by Bonny et al., an attack on an FPGA-based non-autonomous chaotic oscillator TRNG is demonstrated. Their attack consists of applying clock glitches to the function clock in order to compromise the oscillator, and they show that with properly tuned attack parameters, they are able to disrupt the randomness of the generated bits and cause predictable behavior. In~\cite{govindan2018hardware}, a hardware Trojan-based attack on an FPGA-based TRNG is presented, as shown in Fig.~\ref{fig:tht}. The Trojan causes the TRNG to generate previously used keys with higher probability when active, while remaining undetectable otherwise. Specifically, the Trojan becomes active when the die temperature exceeds a set threshold. The authors use a transition effect ring oscillator-based Trojan detection technique along with a NIST statistical randomness testing suite to evaluate their proposed Trojan in terms of detectability, and find that both are unreliable and thus ineffective at detecting the Trojan.
\begin{figure}
\centering
\includegraphics[width=0.7\linewidth]{HTH.png}
\caption{Hardware Trojan structure and impact~\cite{govindan2018hardware}.}
\label{fig:tht}
\end{figure}
Another hardware Trojan for ring oscillator-based TRNGs has been proposed by Ghandali et al. that generates predictable outputs via a stochastic Markov Chain model when triggered. The TRNG's source of entropy is also disabled at high temperatures, while exhibiting proper operation under normal operating conditions in order for the Trojan to go unnoticed. In~\cite{karimian2015genetic}, the authors propose a hardware Trojan detection scheme that uses ring oscillators to collect information from ICs in order to determine whether or not the IC is Trojan-free. The feature selection approach is based on a genetic algorithm that's combined with SVM, and is shown to have a ~30\% improvement in accuracy and ~97\% improvement in equal error rate over principle component analysis, which is one of the most commonly used methods of Trojan detection.
In a paper by Bahadur et al., the authors propose a side-channel resistant reconfigurable Galois ring oscillator-based TRNG implemented on an FPGA. Unlike traditional ring oscillator-based TRNGs, it is not dominated by any single frequency, and as a result is less vulnerable to frequency locking by electromagnetic and frequency injection attacks. In a paper by Japa et al., a Tunnel FET (field effect transistor) TRNG is proposed that relies on delays in an ambipolarity-based ring oscillator for its source of entropy. The authors deduce that their design is more resistant to reverse engineering attacks than traditional ring oscillator-based TRNGs because their design's FET transmission gates never switch off (which would cause the FETs to enter the tri-state region and stop oscillation), hindering an effective attack strategy for traditional ring oscillator-based TRNGs.
In~\cite{prada2020auto}, the authors propose a two-oscillator, two-counter auto-calibrated ring oscillator TRNG implemented on an FPGA that relies on accumulated oscillator jitter as its source of entropy as opposed to instantaneous jitter, which can make traditional ring oscillator TRNGs prone to power supply injection attacks. In~\cite{luo2020high}, a TRNG that combines a ring oscillator with a chaotic cellular automata topology is proposed. The design is tested through HSpice simulations, on an FPGA, and on an ASIC, and evaluated against three different attacks, namely frequency injection, power, and thermal attacks. From their experiments, the authors show that their TRNG is resistant to all three, suffering little entropy loss in all cases. In~\cite{yang2016total}, the authors propose an attack and TRNG failure detection scheme called TRNG On-the-fly Tests. The method is centered around monitoring the entropy of a TRNG after determining important statistical features, and they test their method on a traditional ring oscillator-based TRNG along with a carry-chain TRNG to show that their method is effective.
\section{Conclusions}
Maintaining awareness of recently discovered vulnerabilities in existing methods of data protection is vital for ensuring that we stay on top of what is secure, and what will leave information at risk. This encourages the continued development of improved security primitives and protocols, leading to more effective protection of users' data in scenarios where sensitive information is handled. We hope to have provided the reader with a brief yet informative overview of where PUF and TRNG LHSPs stand today, and in what direction the field of lightweight hardware-based security is moving~\cite{tehranipoor2017design}.
\section{Introduction and Motivation}
Perhaps the hottest topic in recent years regarding LHSPs is the discussion around physical unclonable functions (PUFs). The basic notion of a PUF involves a unique and reliable set of challenge-response pairs (CRPs), or inputs and outputs to a physical function, that can be used for security purposes including device identification, authentication~\cite{yan2015novelway}, and key generation. For a specific challenge, a PUF returns an associated response by way of a one-way function, meaning that the challenge cannot be re-obtained from the response. The size of the CRP set characterizes a PUF as being either strong (having a large enough number of CRPs that recording them all would be infeasible) or weak (having a small enough number of CRPs that recording them all would be feasible). Depending on which category the PUF falls into, it may or may not be suited for a specific use case. A number of PUF implementations exist, and a shared benefit of these LHSPs is that a PUF can provide device security without the need for additional hardware components intended solely for a security purpose. This is because PUFs take advantage of the inherent differences between silicon chips on the micro or nano scale that arise in the manufacturing process. They use these differences to provide a unique set of seemingly random yet reliable responses to corresponding challenges. This method of taking advantage of inherent uniqueness make PUFs especially desirable in situations where small size is a necessary constraints, such as in the case of Internet of Things
(IoT) technology~\cite{hassan2017internet}~\cite{tehranipoor2017invest}.
As discussed in~\cite{tehranipoor2017exploring}, employing secure methods of IoT device authentication and data protection is critical, yet poor security design is a common issue facing the IoT world~\cite{wortman2017proposing}, including devices on the consumer electronic market, in the medical and healthcare domains, and in other areas. In~\cite{tehranipoor2018low}, Tehranipoor et al. discuss consumer electronic device authentication and propose a low-cost PUF verification solution that uses keys generated from biometric signals~\cite{karimian2016evolving}. Another consumer electronics authentication scheme, termed P2M-Sec, is proposed in~\cite{wortman2018p2m} that uses PUF responses as seeds for a pseudorandom number generator. Multiple PUFs are investigated for use in the scheme, including an arbiter, ring oscillator~\cite{aguirre2020systematic}, bistable ring oscillator, butterfly, and DRAM-based PUF~\cite{commercial2017}. Similarly in~\cite{karimian2018secure}, Karimian et al. propose a PUF-based authentication framework with particular emphasis on ECG biometrics. The framework combines PUFs with hardware obfuscation, and makes non-volatile storage of keys (which can leave keys vulnerable to attack) unnecessary. The authors detail several case studies with their framework, and report that using the framework with an ECG provides the best trade-off between reliability, key length, entropy, and cost. In~\cite{karimian2017noise}, the authors discuss key generation techniques with ECGs based on different feature extraction techniques with varying noise sources. They propose the NA-IOMBA (noise aware interval optimized mapping bit allocation) protocol for improved key reliability, building off of their previously proposed IOMBA technique~\cite{karimian2016highly}. In~\cite{karimian2019heartid}, using ECG-based authentication is explored in the context of autonomous cars (HeartID), with heart signals being used to authenticate interaction between a person and a car (such as allowing entry to the vehicle).
One type of module often inherent in IoT devices that has been proposed for use as a PUF (as well as a TRNG) is dynamic random-access memory~\cite{anagnostopoulos2018overview}. Fig.~\ref{fig:noisydram} shows a key generation scheme proposed in~\cite{karimian2019generate} that can be used for device identification with DRAM-based PUFs, such as that proposed in~\cite{tehranipoor2016dram}~\cite{tehranipoor2015dram} by Tehranipoor et al. The 128-bit PUF relies on the random startup values of cell capacitors in an on-board DRAM module in order to generate a unique response when the device is turned on. In this case, the power-up sequence would be regarded as the challenge, and the values of the cells of interest make up the corresponding response. As seen in Fig.~\ref{fig:dramnet}, a similar use case of the entropic behavior observed in DRAM has been proposed in~\cite{karimian2019dramnet}~\cite{dramnet2} for an authentication scheme called DRAMNet. It uses a convolutional neural network (CNN) to extract and classify unique features from a DRAM image structure (created by converting the power-up sequence values into a 2-dimensional array) in order to authenticate a device with greater than 98\% accuracy and precision.
\begin{figure}
\centering
\includegraphics[width=0.8\linewidth]{NoisyDRAM2019.png}
\caption{DRAM-based PUF key generation~\cite{karimian2019generate}.}
\label{fig:noisydram}
\end{figure}
Several quality metrics are used to evaluate PUFs, including uniqueness, uniformity, and reliability. Uniqueness can be determined by calculating the inter-chip Hamming distance, which determines how different a PUF's responses will typically be when the same PUF is implemented on different chips and given the same challenges. Ideally, the inter-chip Hamming distance will be 50\%, meaning half of the bits in a PUF's response on one chip will be flipped when it's given the same challenge on another chip. Uniformity refers to the ratio of ones to zeros in PUF responses. Ideally, a PUF should exhibit 50\% ones and 50\% zeros on average in its responses in order to keep bit predictability as low as possible. Determining a PUF's reliability involves examining how well its CRP set holds up under varying environmental conditions. Ideally, a response will be exactly the same each time it's given the same challenge under all operating conditions, such as at different temperatures or after aging. In~\cite{anagnostopoulos2018securing}, DRAM-based PUFs, namely retention-based and row hammer PUFs, are evaluated in the context of temperature and voltage variations, and are shown to be highly dependent on temperature. The effect of temperature variation on PUFs is further investigated in~\cite{anagnostopoulos2018addressing}, and a method is proposed to account for the negative impacts. Specifically, temperature sensors, which are often present in DRAM modules, are used with a DRAM-based PUF in order to create an authentication protocol suitable for varying temperatures that restricts access to PUF responses when it's determined that the operation could be compromised by the environment's temperature.
Much research is put into ensuring PUF reliability. For example, in a paper by Tehranipoor et al.~\cite{tehranipoor2017investigation}, a report is given on the reliability of three DRAM-based PUFs aged 18 months. The authors show that the PUFs underwent only slight losses in their reliability over this period of time - the best of the three dropping from 88.9\% reliability to 87.5\%, and the worst from 89.7\% to 81.9\% over the period of accelerated aging. Another example of a PUF proposed specifically with reliability in mind is a phase calibrated ring oscillator PUF that can be implemented on a field-programmable gate array (FPGA)~\cite{yan2017phase}. The PUF is shown to have better response reliability than that of a traditional ring oscillator PUF, which relies on inherent wire and inverter delay to generate responses. Other PUF metrics including power requirements and footprint are also very important (especially in the realm of IoT), but uniqueness, uniformity, and reliability are among the most crucial in regards to a PUF's intended operation.
True random number generators (TRNGs) are another form of LHSP. Unlike a pseudo-random number generator (PRNG) that is ultimately deterministic in generating seemingly random outputs, a TRNG is non-deterministic, meaning it does not use an algorithm to generate outputs (which would reveal its secrets if known). Rather, a TRNG relies on a physical process to generate random bit streams. A few examples that a TRNG could utilize for this purpose include phase noise, thermal noise, jitter, random telegraph noise, and the photoelectric effect~\cite{tehranipoor2016robust}. TRNGs are used in a wide variety of security applications, with cryptographic algorithms often relying on them for key generation. As an example of a TRNG, one proposed by Tehranipoor et al.~\cite{tehranipoor2016robust} utilizes the remanence effect of DRAM (where some information remains in the memory cells after powering down) in order to generate random cryptographic keys. DRAM is also proposed for use as a TRNG in~\cite{eckert2017drng}, but the startup values are used instead. Another TRNG proposed in~\cite{tehranipoor2017study} is based on power supply noise, and based on results from a NIST statistical test suite, the authors conclude that the variations in voltage exhibit enough entropy that they can be used to generate random bits characteristic of a TRNG. Similarly in~\cite{tehranipoor2018dvft}, a dynamic feedback voltage tuning (DVFT) power supply noise-based TRNG is proposed. Six different power supplies are evaluated, and Gaussian distributions are observed, demonstrating non-cyclostationary behavior that can be utilized in the context of true random number generation~\cite{tehranipoor2018towards}.
\section{Literature Review}
While the concept of a PUF or TRNG may be nice, actually implementing one in practice that is fully resistant to attack is another story. In many cases regarding PUFs, these supposed "unclonable" functions have in fact been shown to be clonable through modeling. Equivalently, numerous successful attacks have been mounted against TRNGs. These attacks can come in various forms - many of which will be discussed in this section. In particular, the following literature review for PUFs is primarily focused on recent reports on PUF modeling attacks via machine learning (ML) and PUFs designed with ML-resistance in mind. The following subsection discusses recent attacks on TRNGs.
\subsection{Attacks on PUFs}
In this subsection, we provide a summary of some of the most recent developments in machine learning-based attacks on PUFs, as well as proposed ML-resistant PUFs. In all cases, the proposed/attacked PUFs are thought to meet at least a minimum set of requirements for uniqueness, uniformity, and reliability.
Beginning with a paper by Nguyen et al.~\cite{nguyen2019interpose}, the authors present the Interpose PUF and evaluate it against ML attacks. The Interpose PUF is a strong PUF consisting of two XOR arbiters, and the authors conclude that it is not vulnerable to logistic regression (LR) and covariance matrix adaptation evolution strategies (CMA-ES). In~\cite{wisiol2020splitting} however, it is shown that the Interpose PUF, which was thought to be more secure than existing XOR arbiter PUFs due to its resistance to LR and CMA-ES, is vulnerable to deep learning. The attack times recorded are comparable to those for other XOR arbiter PUFs, and the authors present results showing that they were able to achieve an attack success rate of at least 88\% on several variations of the Interpose PUF with a multilayer perceptron (MLP) neural network.
In~\cite{delvaux2019machine}, an ML-based side-channel impersonator attack is proposed and used to evaluate five different arbiter PUFs, specifically the PolyPUF by Konigsmark et al., the OB-PUF by Gao et al., the RPUF by Ye et al., the LHS-PUF by Idriss et al., and the PUF-FSM by Gao et al. All PUFs are successfully modeled, and it is noted that often the physical security of PUF protocols is not given enough attention in the design phase as opposed to mathematical security, which can result in unseen vulnerabilities.
In a paper by Wisiol et al., the authors present an attack on the Lightweight Secure PUF that focuses on attacking the input transformations (that result in local minima) that can be used to model the PUF. The Lightweight Secure PUF was previously thought to be significantly more resistant to ML than traditional XOR arbiter PUFs, with the input transformations being introduced for this purpose. The authors of this paper show that unfortunately these input transformations can actually be exploited to achieve successful modeling times comparable to those recorded when attacking classic XOR arbiter PUFs. LR and a correlation attack are used in the evaluation. The authors also propose a potential low-overhead countermeasure using fix-point-free permutations as an input transformation alternative that does not result in local minima. They find that this technique is able to significantly increase the PUF's resistance to their proposed attack.
In~\cite{laguduva2019machine} Laguduva et al. introduce a non-invasive architecture-independent ML attack targeting PUF-based IoT nodes. The three PUFs evaluated are of arbiter, XOR, and Lightweight architectures, and the attack involves the characterization of a given PUF via an optimal function in order to determine whether an LR, random forest (RF), or a neural network ML algorithm should be used to model it. With this attack strategy they find that they're able to achieve a modeling accuracy of ~95.3\%. The authors, however, also propose a countermeasure (that they call "discriminator") to their attack which is able distinguish cloned PUFs from the original for secure authentication from a cloud server with an accuracy of ~96\%.
\begin{figure}[htp]
\centering
\includegraphics[width=0.8\linewidth]{ICEM2020.png}
\caption{Authentication with DRAMNet~\cite{karimian2019dramnet}.}
\label{fig:dramnet}
\end{figure}
In~\cite{wang2019adversarial}, a CRP mechanism is suggested as a countermeasure to ML modeling attacks. Traditionally, a response for a given challenge is fixed (assuming proper reliability), but the authors propose adding a feature to strong PUFs that is capable of inverting responses with a chosen probability in order to throw off an attacker's modeling scheme. LR and CMA-ES are used to test the effectiveness of the proposed mechanism with a 64-bit arbiter PUF, and the authors find that the PUF's resistance does indeed improve with the countermeasure active. The prediction accuracy by LR without any response inversion is ~99\%, and when an inversion probability of 50\% is active, the accuracy drops to ~55\%. Similarly for CMA-ES, the prediction accuracy drops from ~99\% without any inversion to ~78\% with a 50\% probability of inversion.
In~\cite{zhuang2019strong}, a subthreshold current array PUF is proposed that is resistant to ML attacks. The PUF consists of two transistor arrays that exhibit nonlinear behavior due to the nonlinearity of the I-V characteristics for the subthreshold region of a MOSFET. Support vector machine (SVM) and LR algorithms along with neural networks are used to evaluate the PUF, which is shown to outperform the arbiter and 3-XOR PUFs it's compared against in terms of attack resistance. In~\cite{mahmoodi2019chipsecure}, a reconfigurable flash-based PUF called ChipSecure is proposed that takes advantage of random I-V characteristics in flash memory, consisting of two layers of primitive blocks and a shift register. The authors attack their proposed design with an MLP and report that achieved prediction accuracy is close to an ideal 50\%.
In~\cite{kroeger2020effect}, the authors present results on the effect of PUF aging on ML attacks. Specifically, they report that ML models trained with a dataset of CRPs recorded at a certain PUF age lose their effectiveness when attempting to predict on an attack dataset of CRPs collected after the PUF has aged.
\begin{figure}
\centering
\includegraphics[width=0.7\linewidth]{ARCHINDEP.png}
\caption{Architecture-independent PUF attack model~\cite{laguduva2019machine}.}
\label{fig:archindep}
\end{figure}
In a paper by Wang et al., the authors proposed a Lattice strong PUF that's shown to be resistant to ML with both classical and quantum computing. The PUF is made with a physically obfuscated key, a learning-with-errors decryption function block, and a linear-feedback shift register (used as a pseudo-random number generator). They attack their PUF with SVM, LR, and deep learning neural networks, and find that the prediction accuracy is ~50\% after 1 million training CRPs in all cases. In~\cite{wu2020ct}, the authors propose a configurable tri-state strong PUF that combines an arbiter, ring oscillator, and bistable ring PUF for improved ML resistance over the configurable ring oscillator and dual-mode PUFs it's compared against. They attack all three PUFs with LR and neural networks, and report that while the dual-mode PUF's resistance to LR is similar to that of their tri-state PUF, they were able to achieve greater than 90\% modeling accuracy for the dual-mode PUF with neural networks, but only 50-60\% for their tri-state PUF. The configurable ring oscillator PUF is shown to be easily modeled by both LR and neural networks.
In a paper by Ge et al., the authors propose a challenge preprocessing structure for an arbiter PUF that's shown to improve ML resistance by hiding the linear correlation seen between a challenge and its corresponding response in a traditional arbiter PUF. Flip-flops are used to preprocess the original challenge before it's sent to the arbiter PUF, throwing off the direct relationship between the original challenge and the response that follows. The highest modeling accuracy achieved by the authors with linear regression, LR, SVM, and backpropogation neural network (BPNN) ML algorithms is ~61\% (with the BPNN). In~\cite{pugazhenthi2019dla}, a DRAM-based PUF (DLA-PUF) that uses utilizes DRAM startup values is attacked with Naive Bayes (NB), LR, SVM, and CNN ML algorithms. It is shown to be resistant to these attack strategies when 3 DRAM startup values are used in its scheme.
In~\cite{ma2018machine}, the authors propose an arbiter-based multi-PUF that uses a group of weak PUFs to obfuscate challenges given to a strong PUF by taking the XOR of the weak PUF response bits and the challenge, and feeding the result into the strong PUF. The intention is to improve resistance to ML attacks. The authors attack their design with LR and CMA-ES, and find the prediction rate is ~50\% with LR (as opposed to 100\% when attacking a traditional arbiter PUF) and ~80\% with CMA-ES when the challenge size is at least 32 bits. In~\cite{wang2018machine}, a dual-mode PUF that works with both an odd number of inverters like a reconfigurable ring oscillator PUF and an even number of inverters like a bistable ring PUF is proposed, with the assumption that no ML attack can model both a ring oscillator and bistable ring PUF. Thus, as long as the current working mode can be hidden from attackers, its ML resistance should increase. They attack their PUF with LR and a neural network, and find that the LR accuracy is ~50\% and that the neural network accuracy is ~60\%, which is a significant improvement over the ML resistance of a ring oscillator PUF on its own (for which ~99\% modeling accuracy can be achieved).
In~\cite{xi2017strong}, the authors propose a subthreshold current array strong PUF that they show to be resistant to ML. The PUF boasts $2^{65}$ CRPs, and consists of two-dimensional transistor arrays with transistors operating in the subthreshold region with high voltage variability. The PUF is attacked with SVM, LR, and neural network algorithms, and from the authors' results is estimated to be 100 times more resistant to these attacks than a traditional arbiter PUF. In~\cite{tanaka2018coin}, a coin-flipping PUF is proposed that utilizes nonlinearity found in the convergence time of a bistable ring PUF with regard to threshold voltage variations. Responses are generated based on a ring oscillator's value at the convergence time of its associated bistable ring. The PUF is attacked with SVM, ES, RF, bagging, and boosting ML algorithms, and it is shown that the prediction accuracy is ~50\% across the board. In~\cite{pang2017novel}, the authors propose a double-layer 16Mb RRAM (resistive random-access memory) array PUF that uses the resistance values of cells to generate responses. The authors attack their double-layer PUF along with a single-layer version and a traditional arbiter PUF with a neural network, and are able to achieve a ~90\% success rate on the traditional arbiter, ~70\% on the single-layer version of their PUF, and only ~55\% on the double-layer version.
\subsection{Attacks on TRNGs}
This subsection discusses some of the most recent developments in TRNGs and their vulnerabilities. In a paper by Osuka et al., the authors investigate injecting sinusoidal waves through a power cable in order to non-invasively attack a TRNG by lowering its entropy and estimating the internal state of a ring oscillator through side-channel analysis. They show that when a signal at a particular frequency is injected, the TRNG's oscillator becomes locked, leading to a predictable generation of bits. The authors also propose a countermeasure to the attack, suggesting that including ferrite cores in the power line could help suppress the frequencies that are capable of locking the ring oscillator.
In a paper by Koyen et al., three forms of attack on a ring oscillator TRNG implemented on an FPGA are investigated. Of the three attack methods: voltage manipulation, ring oscillator locking, and replica observation, the voltage manipulation attack is reported to be the most effective. In a paper by Bonny et al., an attack on an FPGA-based non-autonomous chaotic oscillator TRNG is demonstrated. Their attack consists of applying clock glitches to the function clock in order to compromise the oscillator, and they show that with properly tuned attack parameters, they are able to disrupt the randomness of the generated bits and cause predictable behavior. In~\cite{govindan2018hardware}, a hardware Trojan-based attack on an FPGA-based TRNG is presented, as shown in Fig.~\ref{fig:tht}. The Trojan causes the TRNG to generate previously used keys with higher probability when active, while remaining undetectable otherwise. Specifically, the Trojan becomes active when the die temperature exceeds a set threshold. The authors use a transition effect ring oscillator-based Trojan detection technique along with a NIST statistical randomness testing suite to evaluate their proposed Trojan in terms of detectability, and find that both are unreliable and thus ineffective at detecting the Trojan.
\begin{figure}
\centering
\includegraphics[width=0.7\linewidth]{HTH.png}
\caption{Hardware Trojan structure and impact~\cite{govindan2018hardware}.}
\label{fig:tht}
\end{figure}
Another hardware Trojan for ring oscillator-based TRNGs has been proposed by Ghandali et al. that generates predictable outputs via a stochastic Markov Chain model when triggered. The TRNG's source of entropy is also disabled at high temperatures, while exhibiting proper operation under normal operating conditions in order for the Trojan to go unnoticed. In~\cite{karimian2015genetic}, the authors propose a hardware Trojan detection scheme that uses ring oscillators to collect information from ICs in order to determine whether or not the IC is Trojan-free. The feature selection approach is based on a genetic algorithm that's combined with SVM, and is shown to have a ~30\% improvement in accuracy and ~97\% improvement in equal error rate over principle component analysis, which is one of the most commonly used methods of Trojan detection.
In a paper by Bahadur et al., the authors propose a side-channel resistant reconfigurable Galois ring oscillator-based TRNG implemented on an FPGA. Unlike traditional ring oscillator-based TRNGs, it is not dominated by any single frequency, and as a result is less vulnerable to frequency locking by electromagnetic and frequency injection attacks. In a paper by Japa et al., a Tunnel FET (field effect transistor) TRNG is proposed that relies on delays in an ambipolarity-based ring oscillator for its source of entropy. The authors deduce that their design is more resistant to reverse engineering attacks than traditional ring oscillator-based TRNGs because their design's FET transmission gates never switch off (which would cause the FETs to enter the tri-state region and stop oscillation), hindering an effective attack strategy for traditional ring oscillator-based TRNGs.
In~\cite{prada2020auto}, the authors propose a two-oscillator, two-counter auto-calibrated ring oscillator TRNG implemented on an FPGA that relies on accumulated oscillator jitter as its source of entropy as opposed to instantaneous jitter, which can make traditional ring oscillator TRNGs prone to power supply injection attacks. In~\cite{luo2020high}, a TRNG that combines a ring oscillator with a chaotic cellular automata topology is proposed. The design is tested through HSpice simulations, on an FPGA, and on an ASIC, and evaluated against three different attacks, namely frequency injection, power, and thermal attacks. From their experiments, the authors show that their TRNG is resistant to all three, suffering little entropy loss in all cases. In~\cite{yang2016total}, the authors propose an attack and TRNG failure detection scheme called TRNG On-the-fly Tests. The method is centered around monitoring the entropy of a TRNG after determining important statistical features, and they test their method on a traditional ring oscillator-based TRNG along with a carry-chain TRNG to show that their method is effective.
\section{Conclusions}
Maintaining awareness of recently discovered vulnerabilities in existing methods of data protection is vital for ensuring that we stay on top of what is secure, and what will leave information at risk. This encourages the continued development of improved security primitives and protocols, leading to more effective protection of users' data in scenarios where sensitive information is handled. We hope to have provided the reader with a brief yet informative overview of where PUF and TRNG LHSPs stand today, and in what direction the field of lightweight hardware-based security is moving~\cite{tehranipoor2017design}.
| -24,970.958449 |
[
-3.275390625,
3.0234375
] | 65.909091 |
[
-2.990234375,
1.7666015625,
-1.88671875,
-4.82421875,
-0.85791015625,
6.4765625
] |
[
3.66015625,
7.32421875,
2.595703125,
7.05078125
] | 360 | 7,774 |
[
-1.87890625,
1.6201171875
] | 20.112079 |
[
-6.1015625,
-3.16796875,
-3.89453125,
-2.083984375,
2.126953125,
11.84375
] | 1.576044 | 40.134913 | 13.750965 | 1.86123 |
[
2.338493585586548
] | -23,681.231926 | 5.550039 | -24,416.583628 | 0.417188 | 5.82085 |
[
-3.55078125,
-3.7109375,
-3.306640625,
-4.0390625,
2.875,
10.765625
] |
[
-6.16796875,
-2.625,
-2.421875,
-2.521484375,
3.916015625,
6.56640625
] | |
BkiUdlg4eILhQJztnknv
|
\section{Introduction}\label{sec:intro}
One of the most challenging problem in condensed matter theory is to obtain sufficiently accurate approximations of the ground state and low-energy excitations of
generic many-body Hamiltonians. For this reason, several numerical approaches have been devised in the last 30 years, including density-matrix renormalization group
(DMRG)~\cite{white1992}, dynamical mean-field theory~\cite{georges1992}, and quantum Monte Carlo (QMC) techniques~\cite{white1989}. Among them, the natural extensions
of DMRG, such as matrix-product states~\cite{ostlund1995,perez2007} in one dimension and projected entangled-pair states (PEPS)~\cite{verstraete2004a,verstraete2004b}
in two dimensions, represent a promising computational framework to get excellent approximations of strongly-correlated lattice models. Within this approach, the
ground-state wave function is represented by means of local tensors, typically associated to the sites of the underlying lattice. These tensors have two kinds of
indices: a single {\it physical} index specifying the local physical configuration (e.g., $S^z=\pm 1/2$ in a spin-$1/2$ model) and a collection of {\it bond} indices
(whose number, usually, equals the the coordination number of the lattice), each one having dimension $D$. This latter quantity controls the accuracy of the
{\it Ansatz}, the exact ground-state wave function being achieved for $D \to \infty$. The bond indices on nearest-neighbor sites are contracted together, thus
defining a {\it tensor network}. Furthermore, infinite systems may be considered by embedding the tensor network within a suitable environment. Such tensor networks,
dubbed iPEPS, are the main focus of this work.
In the last decade, several algorithms to optimize iPEPS {\it Ans\"atze} have been developed and used to address challenging problems in different correlated systems,
ranging from the characterization of unconventional states of matter (e.g., spin liquids in highly-frustrated spin model) to the competition between stripes and
superconductivity in Hubbard or $t{-}J$ models. For example, the ground state of the spin-$1/2$ Heisenberg model has been analysed on the kagome~\cite{liao2017}
and Shastry-Sutherland~\cite{corboz2013} lattices. Hamiltonians with an enlarged $SU(N)$ ``spin'' symmetry have been also considered, to assess both magnetically
ordered and disordered phases~\cite{corboz2011,corboz2012}. In this regard, a particular attention has been devoted to the possibility to stabilize chiral spin
liquids~\cite{chen2020,chen2021}. As far as electronic models are concerned, the evidence in favor of stripes has been pushed for $t{-}J$~\cite{corboz2014} and
Hubbard~\cite{zheng2017} models, in the vicinity of the hole doping $\delta=1/8$.
By construction, the iPEPS {\it Ansatz} has been devised to describe both gapped and gapless phases, satisfying the area law~\cite{eisert2010}; however, within the
optimization procedure in generic systems, gapless states are not easily obtained. Similarly, it is also not easy to correctly describe states in the vicinity of
continuous quantum phase transitions. This is in a stark contrast with what happens in one spatial dimension, where tensor-network states, equipped with an appropriate
scaling theory, may capture very well gapless phases and critical phenomena~\cite{pollmann2009,pirvu2012,rams2018,vanhecke2019}. Recently, a progress in the iPEPS
analysis of critical and/or gapless systems has been made thanks to two key developments. The first one is the introduction of optimization techniques, based either
on diagram summations~\cite{corboz2016,vanderstraeten2016} or the so-called automatic differentiation~\cite{liao2019}, which substantially improve the accuracy with
respect to the commonly used imaginary time evolution methods. The second one is the development of finite-correlation-length scaling (FCLS)~\cite{rader2018,corboz2018}
that allowed to leverage the well-established finite-size scaling approach to iPEPS states. In fact, the accuracy of thermodynamic estimates based on finite-$D$
iPEPS calculations can be considerably improved. The efficiency of these advances was recently demonstrated on a paradigmatic problem of the spin-$1/2$ Heisenberg
$J_1-J_2$ model on a square lattice by estimating the magnetization curve within the N\'eel phase, even in the vicinity of the transition to the quantum spin-liquid
state~\cite{hasik2021}.
In this work, we pursue the idea of describing a continuous phase transition within the iPEPS formalism. In particular, as for the $J_1-J_2$ model, we focus on a
case where the N\'eel order is melted by quantum fluctuations, namely a two-dimensional system of coupled spin-$1/2$ Heisenberg ladders defined by
\begin{equation}\label{eq:ham}
{\cal H} = J \sum_{R} {\bf S}_{R} \cdot {\bf S}_{R+\hat{x}} + \sum_{R} J_{R} {\bf S}_{R} \cdot {\bf S}_{R+\hat{y}},
\end{equation}
where ${\bf S}_{R}=(S^x_{R},S^y_{R},S^z_{R})$ is the $S=1/2$ operator on the site $R=(x,y)$ of a square lattice, $\hat{x}$ and ${\hat{y}}$ are unit vectors in $x$
and $y$ direction, and $J_{R}=J$ or $J_{R}=\alpha J$, depending on the parity of $y$. By varying $\alpha$, this model interpolates between the Heisenberg model on
the square lattice at $\alpha=1$ and a system of decoupled two-leg ladders at $\alpha=0$. In the following, we will take $J$ as the energy scale. Owing to the absence
of the sign problem, this system can be studied by unbiased QMC techniques and, therefore, offers an excellent benchmark for the accuracy of iPEPS to describe a
non-trivial quantum phase transition, e.g., beyond the simplest case of the Ising model in transverse field. Indeed, the Hamiltonian~\eqref{eq:ham} displays a quantum
critical point at $\alpha_c=0.31407(5)$, separating a gapless antiferromagnet and a gapped paramagnet~\cite{matsumoto2001}. The critical exponents are compatible with
the ones of the classical three-dimensional $O(3)$ Heisenberg model, as expected.
A recent iPEPS investigation of this Heisenberg model~\cite{hasik2019} highlighted the difficulties faced by the unrestricted optimizations of iPEPS tensors across
the phase transition. In fact, due to the unbroken $SU(2)$ symmetry in the paramagnetic side, strong finite-$D$ effects are present, thus impeding a systematic
analysis. Moreover, the optimization procedure is problematic also within the antiferromagnetic phase, where the expected $U(1)$ symmetry around the direction of
the staggered magnetization is usually broken. In this work, we want to constrain the iPEPS {\it Ansatz} by imposing this $U(1)$ symmetry, thus limiting finite-$D$
effect, and optimize such symmetric tensors with the automatic differentiation~\cite{liao2019}. In addition, we extend the FCLS analysis~\cite{rader2018,corboz2018}
in situations with a spatial anisotropy, due to the presence of two length scales, since the correlations along the spatial $x$ and $y$ directions will be generically
different in the ground state of Eq.~\eqref{eq:ham}. These two improvements allow us to get accurate results for the antiferromagnetic order parameter up to the
critical point, as confirmed by a direct comparison with QMC calculations. Finally, we included an external staggered magnetic field in the Hamiltonian~\eqref{eq:ham},
which allows us to inspect the response in the order parameter, even very close to the critical point.
\section{Methods}\label{sec:methods}
Here, we briefly describe the structure of the iPEPS {\it Ansatz} that is used to investigate the Hamiltonian~\eqref{eq:ham}. In order to account for both the
antiferromagnetic order in the gapless regime and the short-range valence-bond correlations in the gapped one, we consider four rank-5 tensors $t=\{a,b,c,d\}$,
each one with a physical index $s$ (with dimension $2$, suitable for $S=1/2$ degrees of freedom) and four auxilliary indices $u,l,d,r$ (with dimension $D$). These
tensors are arranged in a $2\times2$ unit cell to tile the square lattice:
\begin{equation}\label{eq:ansatz}
|\textrm{iPEPS}(a,b,c,d)\rangle = \vcenter{\hbox{\includegraphics[scale=0.25]{fig1.pdf}}}.
\end{equation}
Observables are obtained by computing effective environments within the corner-transfer matrix method, generalized for extended unit cells~\cite{corboz2014}.
The gapped phase retains all the symmetries of the Hamiltonian and, in particular it has the full $SU(2)$ spin symmetry; by contrast, the gapless phase breaks the
spin-rotational symmetry, leading to a finite collinear magnetization and a residual $U(1)$ symmetry about it. Therefore, we impose an explicit $U(1)$ spin symmetry
on the tensors. This can be achieved by associating integer ``charges'' to the two physical spin-$1/2$ components and the $D$ virtual degrees of freedom (on each
of the bond indices)~\cite{hasik2019}. To preserve point-group symmetries of the underlying lattice, a straightforward choice is to take the same charges on all
the bond indices that are equivalent under the symmetries (e.g., the $C_{4v}$ symmetry used in Ref.~\cite{hasik2019} led to such choice). Here, we do not want to
impose any symmetry within the $2\times 2$ unit cell, which allows the possibility that different charges are defined on different indices. Thus, each of the four
tensors will possess a set of charges ${\vec v}^{\gamma}_j=(v_0,\dots,v_{D-1})$ with $\gamma=a,b,c,d$ and $j=u,l,d,r$. Instead, the same charges
${\vec u}=(u^{\uparrow},u^{\downarrow})$ for the physical indices are taken for all tensors. Therefore, the $U(1)$ symmetry is realized by enforcing a selection
rule for the non-zero elements of the tensors $t^s_{uldr}$:
\begin{equation}
u^{s} + v^{\gamma}_u + v^{\gamma}_l + v^{\gamma}_d + v^{\gamma}_r = N,
\end{equation}
where, without loss of generality, we chose the case with $N=0$ to work with invariant tensors.
The definition of these charges allows us to improve the numerical efficiency, since tensors may arranged in blocks with definite values of them, allowing a
block-sparse representation. For a detailed treatment of tensor networks with $U(1)$ symmetry and linear algebra with block-sparse tensors, see Ref.~\cite{singh2011}.
Importantly, once the blocks are defined, the gradient optimization changes the tensors elements without mixing different blocks. The implementation of linear
tensor algebra for abelian-symmetric tensors is provided by open-source library {\it YAST}~\cite{yast} and the iPEPS algorithms built on top of it are available
in the {\it peps-torch} library~\cite{pepstorch}.
From a practical point of view, one possibility to identifying $U(1)$ charges (for the best variational states) is to perform imaginary-time evolution at different
values of $\alpha$ by using a two-site simple-update scheme~\cite{jiang2008} starting from product states with $U(1)$ symmetry. Remarkably, this approach does not
lead to substantial improvements (in terms of physical quantities, as energy or magnetization) with respect to the charges obtained from the unrestricted optimizations
of single-site iPEPS {\it Ansatz} with the $C_{4v}$ lattice symmetry in the Heisenberg model with $\alpha=1$~\cite{hasik2021}, see Appendix~\ref{sec:app-u1classes}.
\section{Results}\label{sec:results}
Let us now show the results obtained within the optimization of the iPEPS {\it Ans\"atze}. In order to perform a careful comparison with QMC calculations (which are
numerically exact, since the spin model does not suffer from the sign problem), we apply the stochastic series expansion method~\cite{sandvik1999} and perform
extrapolations by decreasing the temperature (to reach ground-state properties) and increasing the size of the cluster (to reach the thermodynamic limit). Most of
the QMC calculations were done on $L\times L$ lattices with periodic boundary conditions, at temperature $T=1/(2L)$, by using the ALPS libraries~\cite{ALPS13,ALPS2}.
\subsection{Phase diagram}
In Fig.~\ref{fig:phasediag}, we report the energy per site $e$ and the staggered magnetization $m^2$ (averaged over the $2\times 2$ unit cell), for different
values of $\alpha$ across the quantum critical point. The outcome shows that, for small bond dimension, i.e., $D \le 3$, the accuracy is quite poor in both the
antiferromagnetic and paramagnetic phases. However, once the bond dimension becomes large enough to capture the entanglement structure of the ground state, the
tensor network provides an excellent variational description, not only in the paramagnetic phase, where the iPEPS parametrization is particularly suitable, but
also within the magnetically ordered phase. The accuracy of the ground-state energy is exemplified by showing the relative error with respect to the value $e_{QMC}$,
obtained within the QMC approach (extrapolated in the thermodynamic limit). Indeed, the quantity $\Delta e/|e_{QMC}|$, with $\Delta e=|e-e_{QMC}|$ is vanishing
within the statistical errors, see Fig.~\ref{fig:phasediag}. Most importantly, the magnetization curves for finite values of the bond dimension $D$ follow the QMC
data faithfully in the bulk of each phase, still overestimating the order parameter close to the critical point. We would like to mention that the present QMC
results allow us to locate the critical point at $\alpha_c=0.31467(1)$, which is slightly different from the previous estimate $\alpha_c=0.31407(5)$~\cite{matsumoto2001}
(see Appendix~\ref{sec:app-qmc-alphac}).
\begin{figure}
\includegraphics[width=\columnwidth]{fig2.pdf}
\caption{\label{fig:phasediag}
Energy per site (upper panel) and staggered magnetization (lower panel) for optimized iPEPS with $U(1)$ symmetry and $2\times2$ unit cell. The results for $D=3,\ldots,6$
are reported, as well as the ones obained with the FCLS extrapolation, see text. In addition, QMC results are also shown for the thermodynamic extrapolation. The inset
in the upper panel shows the relative error of the iPEPS energies (with FCLS extrapolations) with respect to the QMC ones (with thermodynamic extrapolations). The inset
in the bottom panel gives a sketch of the super-exchanges in Hamiltonian~\eqref{eq:ham}, where intra-ladder $J$ (inter-ladder $\alpha J$) couplings are denoted by
black solid (gray dashed) lines.}
\end{figure}
\subsection{Finite-correlation-length scaling}
Obtaining accurate and reliable estimates of the observables (e.g., the energy per site or the staggered magnetization) within iPEPS {\it Ans\"atze} represents an
important issue that has been addressed since their definition. Recently, an interesting step forward has been obtained for magnetic states that break a continuous
symmetry (e.g., the antiferromagnetic one in the Heisenberg model). In this case, the presence of gapless modes induces a diverging correlation length. When working
on finite clusters, as in standard QMC approaches, this fact translates into well-defined size-scaling laws of the physical quantities~\cite{neuberger1989,fisher1989}.
Instead, within iPEPS, we directly work in the thermodynamic limit (that is mimicked by the embedding procedure) and, therefore, it is not possible to straightforwardly
apply the same scaling laws; still, the correlation length $\xi$ is finite for any finite value of the bond dimension $D$ and diverges with $D\to \infty$. As a result,
it has been suggested~\cite{rader2018,corboz2018} that the FCLS analysis can be defined in terms of $\xi$ instead of the cluster size as
\begin{eqnarray}
\label{eq:fclse}
e(\xi) &=& e_0 + \frac{a}{\xi^3} + O\left(\frac{1}{\xi^4}\right), \label{eqn:FCLS_erg}\\
m^2(\xi) &=& m^2_0 + \frac{b}{\xi} + O\left(\frac{1}{\xi^2}\right), \label{eq:fclsm}
\end{eqnarray}
where $a$ and $b$ are suitable constants. The system we are interested in, i.e., the Heisenberg model on coupled two-leg ladders, presents a complication, as it
breaks the $\pi/2$-rotational symmetry for $\alpha \neq 1$, leading to two distinct length scales. Within finite-$D$ iPEPS calculations, we have access to both of
them by looking at the spectrum of the transfer matrices along $x$ (i.e., within ladders) and $y$ (i.e., across ladders). Thus, we obtain two correlation lengths
($\xi_x$ and $\xi_y$) and perform FCLS independently for both of them. The extrapolated values $e_0$ and $m^2$ are independent of choice of the correlation length
direction, while the parameters $a$ and $b$ depend on the choice. It must be emphasized that these FCLS relations are expected to be valid exclusively for a Goldstone
phase with a spontaneously broken continuous symmetry, where the correlation length diverges with increasing $D$. Instead, within the paramagnetic gapped phase,
$\xi$ remains finite in the thermodynamic limit, thus implying that Eqs.~\eqref{eq:fclse} and~\eqref{eq:fclsm} cannot be applied. The impossibility to fit the
numerical data for a given $\alpha$ within this scheme is then taken as the evidence that the state is not gapless (antiferromagnetic). At the quantum critical point,
we still expect Eq.~\eqref{eqn:FCLS_erg} to hold, while the order parameter scaling is replaced with a quantum critical form $m(\xi) \approx \xi^{-\Delta_m}$, with
$\Delta_m$ the scaling dimension of the order parameter at the critical point~\cite{rader2018}.
The results for the energy per site $e$ are reported for selected values of $\alpha$ in Fig.~\ref{fig:fcls-e}. Within the magnetically ordered phase (but also close
to the quantum critical point), the extrapolations of the FCLS analysis using either $\xi_x$ or $\xi_y$ give consistent values, providing an internal verification of
the approach. Most importantly, the staggered magnetization $m^2$ can be also extracted for infinite correlation length, see Fig.~\ref{fig:fcls-m2}. The extrapolated
results are in very good agreement with QMC estimates, up to values of the inter-ladder couplings that are very close to the critical point. Beyond the critical point,
the system develops a gap and thus the FCLS fails to provide the correct scaling, where correlation lengths remain finite. Nevertheless, within the paramagnetic regime,
finite-$D$ iPEPS calculations are able to recover accurate results. Indeed, for $\alpha \lesssim 0.28$, a vanishing magnetization is already obtained with $D=5$.
However, upon approaching criticality, i.e., at $\alpha=0.3$, finite-$D$ iPEPS retain a small residual magnetization (at least up to $D=6$). In fact, close to the
critical point, the correlation length far surpasses the values of $\xi_x$ and $\xi_y$ that our finite-$D$ iPEPS can support. Hence, the FCLS analysis can be used to
improve the estimates of the energy. Still, some important deviations in the scaling laws are clearly visible, see Fig.~\ref{fig:fcls-m2}. Finally, let us remark that
the iPEPS optimization in the narrow region near criticality is technically difficult due to presence of instabilities (see Appendix~\ref{sec:opt-instability}).
\begin{figure}
\includegraphics[width=\columnwidth]{fig3.pdf}
\caption{\label{fig:fcls-e}
FCLS extrapolations for the energy $e$, by using both horizontal ($\xi_x$, full symbols and lines) and vertical ($\xi_y$, empty symbols and dashed lines) correlation
lengths, for a few values of the inter-ladder couplings $\alpha$. The symbols correspond to iPEPS with $D=3$ (diamonds), $D=4$ (squares), $D=5$ (triangles), and $D=6$
(pentagons). The colors follow the $\alpha$ color scheme of Fig.~\ref{fig:fcls-m2}.}
\end{figure}
\begin{figure}
\includegraphics[width=\columnwidth]{fig4.pdf}
\caption{\label{fig:fcls-m2}
FCLS extrapolations for staggered magnetization $m^2$, by using both vertical ($\xi_y$, left part) and horizontal ($\xi_x$, right part) correlation lengths for
$0.29 \le \alpha \le 1$. The symbols correspond to iPEPS with $D=3$ (diamonds), $D=4$ (squares), $D=5$ (triangles), and $D=6$ (pentagons); thermodynamic extrapolations
from QMC data are also reported (stars).}
\end{figure}
\begin{figure*}
\includegraphics[width=\textwidth]{fig5.pdf}
\caption{\label{fig:hz-alpha}
$U(1)$-symmetric iPEPS with $D=3,\ldots,6$ in the presence of external staggered field $h$. Upper panels: Running exponent as a function of $h$ for the gapped phase
with $\alpha=0.2$ (left), at criticality $\alpha=0.31407$ (middle), and in the N\'eel phase $\alpha=0.4$ (right). The value of the critical exponent $1/\delta$ is
reported at the critical point. QMC results are also reported for comparison, with faded colors marking the data affected by finite lattice size. Bottom panels:
horizontal correlation length $\xi_x$ as a function of $h$.}
\end{figure*}
\subsection{The effect of a staggered magnetic field}
In the previous section, we focused on estimating observables of the ground state of the $SU(2)$-symmetric Hamiltonian~\eqref{eq:ham}. Here, we explicitly break this
symmetry to investigate the phase transition with a complementary non-symmetric approach~\cite{binder1984,demidio2021,rader2018}. In particular, we follow the approach
of Refs.~\cite{rader2018,demidio2021}, which recently used tensor networks to compute running exponents in critical quantum systems.
To explicitly break the $SU(2)$, we supplement the Hamiltonian~\eqref{eq:ham} with an external {\it staggered} field $h$, which directly couples to the order parameter:
\begin{equation}\label{eq:ham-with-hstag}
{\cal H}_h = {\cal H} - h\sum_{R} (-1)^{x+y}S^z_{R}\,,
\end{equation}
where $R=(x,y)$. Then, for any $h \ne 0$, the ground state has a finite correlation length and develops a finite staggered magnetization $m=m(h,\alpha)$ in response
to the staggered field $h$. The response for $h \to 0^+$ is different within magnetically ordered or disordered phases. In both cases, the magnetization is an analytic
function of $h$, with two distinct regimes:
\begin{eqnarray}
m(\alpha,h) &=& a(\alpha) h +O(h^2) \qquad\mathrm{for}\;\alpha<\alpha_c,\, h\to0^+, \\
m(\alpha,h) &=& m(\alpha,0) +O(h) \qquad\mathrm{for}\;\alpha>\alpha_c,\, h\to0^+,
\end{eqnarray}
where $a(\alpha)$ is a suitable constant. By contrast, at criticality (i.e., for $\alpha=\alpha_c$), the magnetization is not analytic and the response follows a
power-law behavior:
\begin{equation}
m(\alpha_c,h) \propto h^{1/\delta}\;, \;\; \mathrm{for}\; h\to0^+\,,
\end{equation}
where $\delta$ is a critical exponent, which only depends on the universality class of the phase transition. The best estimate of $\delta$ (within the expected
universality class of the classical three-dimensional $O(3)$ Heisenberg model) $1/\delta=0.20916$~\cite{chester2021}.
To probe the system's response at fixed $\alpha$, we define the logarithmic derivative
\begin{equation}
[1/\delta](\alpha,h) = \frac{\partial \log m(\alpha,h)}{\partial \log h}\,,
\end{equation}
which is usually referred to as {\it running exponent}. For each value of $h$, we optimize the $U(1)$-symmetric iPEPS {\it Ansatz} and compute the average staggered
magnetization $m(\alpha,h)$. Then, we estimate the logarithmic derivative by using finite differences. The QMC results are based on a direct improved estimator for
the running exponent, see Ref.~\cite{demidio2021}. The outcomes are shown in Fig.~\ref{fig:hz-alpha}, where the running exponent and the correlation length $\xi_x$
of iPEPS are reported for the three different regimes (with $D=2,\ldots,6$).
In the gapped phase (for $\alpha=0.2$), the magnetization is linear in $h$ (linear response) and, therefore, the running exponent saturates at $1$ for $h \to 0$,
as also obtained numerically from both QMC and iPEPS data (with $D \geq 4$). The iPEPS correlation length saturates for $h\lesssim 0.005$, with small finite-$D$
corrections. Indeed, in the gapped phase, even finite-$D$ iPEPS provide precise description of the system as the area law is upheld, which is supported by a direct
comparison between iPEPS (with $D \geq 4$) and QMC (for $L \geq 32$). The $D=3$ iPEPS cannot faithfully capture gapped phase at $\alpha=0.2$, since it retains finite
$m$ and thus responds more akin to the N\'eel phase.
Within the magnetically ordered phase (for $\alpha=0.4$), the magnetization saturates to a finite value and the running exponent goes to zero for $h \to 0$. Indeed,
the logarithmic derivative $[1/\delta](\alpha,h)$ has a non-monotonic behavior, with a broad peak at intermediate values of the staggered field $h$. This feature is
correctly captured by iPEPS. QMC calculations also reproduce the broad peak, even though at small values of $h$ a huge upturn is present due to size effects;
therefore, for large enough system size $L$, the two methods converge to the same curve for sufficiently large values of the staggered field $h$. Here, it seems as
though QMC simulations provide an upper bound of the thermodynamic limit, while iPEPS provide a lower bound of it. Still the presence of the broad peak is robust and
appears in the region of $h$ where both $D \geq 5$ iPEPS and $L \geq 32$ QMC data are in excellent agreement.
At the critical point $\alpha=\alpha_c$, the running exponent is expected to converge towards $1/\delta=0.20916$, as mentioned before. The available iPEPS and QMC
calculations (for $D \leq 6$ and $L \leq 128$, respectively) are still somewhat away from the saturation regime, however the numerical data monotonously increase,
without the broad peak in the magnetically orderd phase. While the validity of the QMC results is limited by the finite size $L$, for iPEPS it is limited by the
finite correlation length induced by the bond dimension $D$. Within iPEPS, as the field $h$ is decreased, the correlation lengths initially grow, but then flatten
out (with large finite-$D$ corrections), in contrast to the expected diverging behavior of the gapless regime. From the direct comparison between QMC and iPEPS data,
we can estimate that the iPEPS response for $D=6$ gives a faithful estimation of the exact result down to $h \approx 10^{-2}$. Then, for smaller fields, the running
exponent data for iPEPS and large-$L$ QMC simulations start to deviate as iPEPS becomes increasingly biased by the induced finite correlation length.
\section{Conclusions and outlook}\label{sec:conclusions}
The analysis of quantum phase transitions by variational methods relies on a few aspects: first, a flexible {\it Ansatz} for the ground-state wave fuction, allowing
a description of different kinds of phases by tuning its parameters; second, a practical optimization scheme of such parameters to get the best approximate ground
state; finally, a way to analyze the results, possibly extrapolating to the thermodynamic limit. Here, we have shown that symmetric tensor networks, when implemented
with a variational optimization, allow for accurate description of the transition between a gapped paramagnet and gapless N\'eel antiferromagnet, thus overcoming the
issues arising within the imaginary-time evolution approach that has been emphasized in a recent work~\cite{hasik2019}. The selection of the correct symmetry structure
for tensors by the choice of their charges is crucial, as they determine the physical properties of the wave function. Most importantly, once the tensors have been
variationally optimized, our simulations show that the physical observables, such as the energy or the order parameter, are robust to small variations of these charges.
Although the region close to the critical point remains a challenge, it is in principle tractable by increasing the bond dimension $D$. Already with data up to $D=6$
and FCSL scaling, we could locate the critical point with an accuracy of about $5\%$, demonstrating the applicability of this analysis even in the case of two
different length scales $\xi_x$ and $\xi_y$. Finally, inspired by the recent works of Refs.~\cite{rader2018,demidio2021}, we have included an external staggered field
$h$ in the Hamiltonian and compared the iPEPS results with the ones obtained by QMC. Away from the critical point $\alpha>\alpha_c$ (i.e., within the N\'eel phase)
the running exponent $[1/\delta](\alpha,h)$ shows a broad peak at finite field $h$, indicative of magnetic order~\cite{demidio2021}. The robustness of the peak can
be established by locating the inflection point in the growing iPEPS correlation length as the external field $h$ is decreased. Peaks at fields larger than inflection
point represent genuine features of the system, while finite-$D$ effects may generate spurious peaks at fields smaller than the inflection point. Our findings
corroborate the fact that the analysis of the running exponent is a useful diagnosis within the antiferromagnetic regime, away from the critical point. This proof of
concept analysis shows the potential of the method put forward in Ref.~\cite{demidio2021} for future application of iPEPS, e.g., on questions of the stability of
quantum spin liquids.
\begin{acknowledgments}
J.H. thanks D. Poilblanc and P. Corboz for valuable discussions. This project has received funding from the European Research Council (ERC) under the European Union's
Horizon 2020 research and innovation programme (grant agreement No 101001604), TNSTRONG ANR-16-CE30-0025, TNTOP ANR-18-CE30-0026-01 grants awarded from the French
Research Council, the Austrian Science Fund FWF project I-4548, and the SFB BeyondC Project No. F7108-N38. This work was also granted access to the HPC resources of CALMIP supercomputing center under the
allocations 2017-P1231 and 2021-P0677.
\end{acknowledgments}
| -16,838.328309 |
[
-3.181640625,
2.88671875
] | 22.097378 |
[
-2.400390625,
0.427001953125,
-2.322265625,
-6.3359375,
-1.138671875,
9.625
] |
[
4.484375,
9.2265625,
3.02734375,
7.80078125
] | 221 | 4,028 |
[
-3.359375,
3.736328125
] | 24.323958 |
[
-6.4765625,
-4.8515625,
-5.10546875,
-2.794921875,
2.296875,
14.359375
] | 1.316194 | 14.375987 | 28.152929 | 1.894709 |
[
1.9358265399932861
] | -12,234.396636 | 5.65864 | -16,346.660302 | 0.912561 | 5.862961 |
[
-2.482421875,
-4.1953125,
-4.18359375,
-5.17578125,
2.3046875,
13.25
] |
[
-5.8828125,
-2.646484375,
-2.4609375,
-1.6806640625,
4.0078125,
5.96875
] | |
BkiUfQPxK0zjCrsOBupv
|
\section{Introduction}
Cooperative control of multi-agent systems has traditionally focused on designing distributed control laws in order to achieve global tasks such as consensus, formation control, network connectivity and collision avoidance. Over the last few years, the field of control of multi-agent systems under high-level task specifications has been gaining attention. In this work, we aim to impose specific probability bounds to each robot in order to satisfy one specification task formula. Such formulas can be ``The probability of a robot to periodically survey regions A, B, C, avoid region D is always more than 0.9", or ``The probability of a robot to visit regions A, B, C, in this order is more that 0.8".
Temporal properties for multi-agent plan synthesis under Linear Temporal Logic (LTL) formulas has been considered e.g., in \citep{guo_203_reconfiguration1, guo_2015_reconfiguration, belta_optimality, zavlanos_2016_multi-agent_LTL}. Timed temporal properties given in Metric Temporal Logic (MTL) and Metric Interval Temporal Logic (MITL) have been addressed in \citep{quottrup_timed_automata, frazzoli_MTL, alex_2016_acc, alex_acc_2017, alex_sofie_ifac_2017} respectively.
Most of the existing formal synthesis frameworks are based on the discretization of the agent's motion in a partitioned environment to a finite Transition System (TS) (the process is called abstraction) under the following assumptions. First, the measurements of the current state are accurate. Second, the transition system is either purely deterministic (namely, each control action enables a unique transition) or purely nondeterministic (namely, each control action enable multiple transitions). However, in realistic applications of robotic systems, noisy sensors and actuators can cause both of the aforementioned assumptions to fail. Motivated by this, we aim to model the multi-agent system in a probabilistic way such that the above issues are taken into consideration.
Some recent works model the system in a probabilistic way and imposes high-level specifications, given in Linear Temporal Logic (see e.g., \citep{belta_ifac_2011_undertain_environment, murray_2012_mdp_ltl, belta_2011_mdp_ltl_tac, topcu_2015_mdp_ltl, belta_2015_mdp_ltl_ijrr}). In \citep{liu_2016_mdp_mitl}, the authors modeled the system with an MDP and computed policies such that the satisfaction of a formula given in MITL, is maximized. Other works model the system in a probabilistic way with Markov Decision Processes (MDPs) and introduce high-level tasks in PCTL (see e.g., \citep{belta_2011_modp, belta_2012_tro_pctlalgorithms, lin_2015_counterexample_single, lin_2015_counterexample_multi}). However, all these works are restricted to single agent planning and they are not expendable to multi-agent systems in a straightforward way since in the multi-agent case potential couplings may occur among the agents.
By extending these works to multi-agent systems, in order to develop an approach in which the system noise, model errors and external disturbances is explicitly considered, in this work we consider that each agent is modeled as a MDP and the task specifications are given in PCTL formulas. Motivated by the fact that in real applications, the agents (robots) are required to collaborate with each other to perform a task, we assume that there are agents in the system that are dependent to each other. They need to communicate, collaborate through sharing a common action in order to achieve a desired task.
The main contribution of the paper is to develop a strategy for controlling a general framework of $N$ individual MDPs with respect to individual agents' specifications given in PCTL formulas. The proposed solution can handle the dependencies between the agents by considering agents clustering and MDP product construction and has provably better complexity than the centralized approach. When applied to robotic systems, our approach provides a framework for multi-robot control from temporal logic specifications with probabilistic guarantees. To the best of the authors' knowledge this is the first work that addresses the cooperative task planning for multi-agent systems under probabilistic temporal logic specifications in the presence of dependencies between the agents.
The paper is divided into five parts. In Section \ref{sec: preliminaries}, notation and preliminaries are given. Section \ref{sec: prob_formulation} provides the modeling of the system and the problem statement. Section \ref{sec:problem_solution} provides the technical details of the solution. Finally, conclusions and future work directions are discussed in Section \ref{sec: conclusions}.
\section{Notation and Preliminaries} \label{sec: preliminaries}
In this section, the mathematical notation and preliminary definitions from probabilistic model checking that are necessary for this paper are introduced.
Denote by $\mathbb{N}$ the set of natural numbers. Given a set $S$, denote by $|S|$ its cardinality and by $2^S$ the set of all its subsets. Denote by $\underset{i=1}{\overset{N}{\bigtimes}} S_i = S_1 \times \ldots \times S_N$ the $n$-th fold Cartesian product of the sets $S_1, \dots, S_N$.
An \textit{atomic proposition} $\chi$ is a statement that is either True $(\top)$ or False $(\bot)$.
\subsection{Markov Decision Processes} \label{sec:mdp_definitions}
MDPs offer a mathematical framework for modeling systems with stochastic dynamics. These models provide an effective way for describing processes in which sequential decision making is required for a system.
\begin{definition}
A \textit{probability distribution} over a countable set $S$ is a function $\sigma: S \to [0, 1]$ satisfying $\sum_{s \in S}^{}\sigma(s) = 1$. Define by $\Sigma(S)$ the set of all probability distributions over the set $S$.
\end{definition}
\begin{definition} \label{def:dtmc}
A Discrete Time Markov Chain (DTMC) $\mathcal{D}$ is a tuple $(S, s_0, P)$ where: $S$ is a finite set of states; $s_{0} \in S$ is the initial state; $P : S \times S \to [0,1]$ is the transition probability matrix where for all $s \in S$ it holds that $\sum_{s' \in S}^{} P(s,s') = 1$.
\end{definition}
\begin{definition} \label{def:mdp}
A Markov Decision Process (MDP) $\mathcal{M}$ is a tuple $(S, s_0, Act, T)$ where: $S$ is a finite set of states; $s_{0} \in S$ is the initial state; $Act$ is a finite set of actions (controls); $T: S \to 2^{Act \times \Sigma(S)}$ is the transition probability function.
\end{definition}
Denote by $\mathcal{A}(s)$ the set of all actions that are available at the state $s \in S$ and let $\delta(s, \alpha, s') \in [0, 1]$ be the probability of transitioning from the state $s$ to the state $s'$ under the action $\alpha \in \mathcal{A}(s)$. For a state $s \in S$ and an action $\alpha \in \mathcal{A}(s)$, define the set $\text{Post}(s, \alpha) = \{s' \in S: \delta(s,\alpha,s')>0\}$.
The transition probability function $T$ can be represented as a matrix with $\sum_{i = 0}^{|S|-1} |\mathcal{A}(s_i)|$ rows and $|S|$ columns.
An execution of an MPD is represented by a \textit{path}. Formally, an \textit{infinite path} $r$ is a sequence of states of the form: $r = s_0 \overset{\alpha_0}{\longrightarrow} s_1 \overset{\alpha_1}{\longrightarrow} \dots \overset{\alpha_{k-1}}{\longrightarrow} s_k \overset{\alpha_k}{\longrightarrow} s_{k+1} \dots,$
such that $s_k \in S_k, \alpha_k \in \mathcal{A}(s_k)$ and $\delta(s_k, \alpha_{k+1}, s_{k+1}) > 0, \forall k \geq 0$. A \textit{finite path} $\rho = s_0 \overset{\alpha_0}{\longrightarrow} s_1 \overset{\alpha_1}{\longrightarrow} \dots \overset{\alpha_{n-1}}{\longrightarrow} s_n$ is a prefix of an infinite path ending in a state. In case of the actions are not taken into consideration, the infinite and finite run can be written as $r = s_1s_2 \dots s_n \dots$ and $\rho = s_1s_2 \dots s_n$ respectively. Denote by $|\rho| = n$ the length of the finite path and by $r(k), \rho(k)$ the $k$-th element of the paths $r, \rho$ respectively. The set of all finite and infinite paths are defined by $FPath$ and $IPath$ respectively.
A control policy at each state of an MDP and is formally defined as follows:
\begin{definition} (Control Policy) A \textit{control policy} $\mu: FPath \\ \to Act$ of an MDP model $\mathcal{M}$ is a function mapping a finite path $\rho = s_0 \overset{\alpha_0}{\longrightarrow} s_1 \overset{\alpha_1}{\longrightarrow} \dots \overset{\alpha_{n-1}}{\longrightarrow} s_n$, of $\mathcal{M}$ onto an action in $\mathcal{A}(s_n)$ and specifies for every finite path, the next action to be enabled. If a control policy depends only on the last state of the finite path $\rho$, then it is called a \textit{stationary policy}.
\end{definition}
Denote by $M$ the set of all control policies. Under a control policy $\mu \in M$, an MDP becomes a DTMC $\mathcal{D}_\mu$ (see Def. \ref{def:dtmc}). Let $IPath_\mu \subseteq IPath$ and $FPath_\mu \subseteq FPath$ denote the set of infinite and finite paths that can be produced under the control policy $\mu$. For each policy $\mu \in M$, a probability measure $Prob_\mu$ over the set of all paths (under the control policy $\mu$) $IPath_\mu$ is induced. A probability measure $Prob_\mu^{fin}$ over the set of paths $FPath_\mu$ for a finite path $\rho$, is defined as: $Prob_\mu^{fin} (\rho) = 1, \text{if} \ |\rho| = 0$ and $Prob_\mu^{fin} (\rho) = P(s_0,s_1)P(s_1,s_2) \ldots P(s_{n-1},s_n), \text{otherwise}$, where $P(s_k, s_{k+1}), k \geq 0$ are the corresponding transition probabilities in $\mathcal{D}_\mu$.
Define also the set of all infinite paths with prefix $\rho$ as:
\begin{equation*}
C(\rho) = \{r \in IPath_\mu : \rho \ \text{is a prefix of r} \}.
\end{equation*}
By invoking theorems from probability theory \citep{marta_2004_book}, we have that:
\begin{equation*}
Prob_\mu(C(\rho)) = Prob_\mu^{fin}(\rho), \forall \rho \in FPath_\mu.
\end{equation*}
where $Prob_\mu^{fin}(\rho)$ as is defined previously.
\subsection{Probabilistic Computational Tree Logic (PCTL)} \label{sec:PCTL}
Probabilistic Computational Tree Logic (PCTL) \citep{hansson_1994_pctl} is used to express properties of MDPs. PCTL formulas can be recursively defined as follows:
\begin{align}
\varphi &:= \top \ | \ \chi \ | \ \neg \varphi \ | \ \varphi_1 \wedge \varphi_2 \ | \ \mathcal{P}_{\bowtie p} [\psi], \hspace{5mm} (state \ formulas) \notag \\
\psi &:= \bigcirc \varphi \ | \ \varphi_1 \ \mathcal{U}^{\leq k} \ \varphi_2, \hspace{25mm} (path \ formulas) \notag
\end{align}
where $\chi \in Act$ is an action, $\bowtie = \{<, >, \leq, \geq \}$, $p \in [0, 1]$ and $k \in \mathbb{N} \cup \{\infty\}$. In the syntax above, we distinguish between state formulas $\varphi$ and path formulas $\psi$, which are evaluated over states and paths, respectively. A property of a model will always be expressed as a state formula; path formulas only occur as the parameter of the probabilistic path operator $\mathcal{P}_{\bowtie p} [\psi]$. Intuitively, a state $s$ satisfies $\mathcal{P}_{\bowtie p} [\psi]$ (we write $s \models \mathcal{P}_{\bowtie p} [\psi]$) if there exists a control policy $\mu$ under which the probability of all paths starting from $s$ is in the range of the interval $\bowtie p$.
For path formulas, we allow the ``next" ($\bigcirc$) operator which is true if the state formula $\varphi$ is satisfied in the next state and the ``bounded until" ($\mathcal{U}^{\leq k}$) which is true if $\varphi_2$ is satisfied within $k$ steps and $\varphi_1$ holds up until that point. The unbounded ``until" operator $\mathcal{U}$ is the same as $\mathcal{U}^{\leq k}$ as $k \to \infty$.
\begin{definition} \citep{hansson_1994_pctl} (Semantics of PCTL) For any state $s \in S$, the satisfaction relation $\models$ is defined inductively as follows:
\begin{align*}
s &\models \top \Leftrightarrow \forall s \in S, \\
s &\models \chi \Leftrightarrow \chi \in \mathcal{A}(s), \label{eq:modified_sem} \\
s &\models \neg \varphi \Leftrightarrow s \not \models \varphi, \\
s&\models \varphi_1 \wedge \varphi_2 \Leftrightarrow s \models \varphi_1 \ \text{and} \ s \models \varphi_2, \\
s &\models \mathcal{P}_{\bowtie p} [\psi] \Leftrightarrow Prob_\mu(s, \psi) \bowtie p,
\end{align*}
where $Prob_\mu(s, \psi)$ denotes the probability that a path starting from the state $s$ satisfies the path formula $\psi$ under the specific control policy $\mu$. Moreover, for any path $r \in IPath$ we have that:
\begin{align*}
r &\models \bigcirc \varphi \Leftrightarrow r(1) \models \varphi, \\
r &\models \varphi_1 \ \mathcal{U}^{\leq k} \ \varphi_2 \Leftrightarrow \exists \ i \leq k, \notag \\
&\hspace{25mm} r(i) \models \varphi_2 \wedge r(j) \models \varphi_1 \ \forall j < i.
\end{align*}
For the operators $\square$ (always) and $\Diamond$ (eventually) it holds that:
\begin{align}
\mathcal{P}_{\bowtie p} \left[ \Diamond^{\leq k} \varphi \right] &= \mathcal{P}_{\bowtie p} \left[ \top \ \mathcal{U}^{\leq k} \varphi \right], \notag \\
\mathcal{P}_{\bowtie p} \left[ \square^{\leq k} \varphi \right] &= \mathcal{P}_{\overline{\bowtie} p} \left[ \Diamond^{\leq k} \neg \varphi \right], \notag
\end{align}
where $\bowtie = \{<, >, \leq, \geq \}$ and $\overline{\bowtie} = \{>, <, \geq, \leq\}$.
\end{definition}
\begin{remark}
Traditionally, the PCTL semantics are defined over a set of atomic propositions. However, in this paper, we aim to introduce dependencies over the actions among the agents. Therefore, the PCTL semantics are defined over a set of actions.
\end{remark}
\subsection{Probabilistic Verification} \label{sec:probabilistic_model_checking}
There are three problems that have generally been considered in probabilistic model checking of stochastic systems:
\begin{itemize}
\item (Model Checking) Given an MDP $\mathcal{M}$ and a property $\varphi$, check which of the states of the MDP $\mathcal{M}$ satisfy $\varphi$.
\item (Controller Synthesis): Given an MDP $\mathcal{M}$ and a property $\varphi$, find all the control policies under which the formula is satisfied.
\item (Existence) Given an MDP $\mathcal{M}$ and a property $\varphi$, find, if it exists, a control policy $\mu$ such that the MDP $\mathcal{M}$ satisfies the property $\varphi$ under $\mu$.
\end{itemize}
In this paper, we are mainly interested in the Controller Synthesis problem. The motivation for that is the following: if one control policy fails to guarantee the satisfaction of a formula, it may exists another policy under which the formula is satisfied. We refer the reader to \citep{marta_2004_book, alvaro_1995_probabilistic_model_checking} for more details about probabilistic model checking.
\section{Problem Formulation} \label{sec: prob_formulation}
\subsection{System Model and Abstraction}
Consider a multi-agent team with $N \geq 2$ agents operating in the bounded workspace $\mathcal{W} \subseteq \mathbb{R}^n$. Let $\mathcal{V} = \left\{ 1,\ldots, N\right\}$ denote the index set of the agents. The workspace $\mathcal{W} = \mathop{\bigcup} \limits_{\ell \in \mathcal{W}}^{} \gamma_{\ell}$ is partitioned using a finite number (assume $W$) of regions of interest $\gamma_1, \ldots, \gamma_W$. Denote by $\gamma_\ell^i$ the fact that the agent $i$ is occupying the region $\gamma_\ell$, where $i \in \mathcal{V}, \ell \in \mathcal{W}$.
We assume that each agent is programmed with a small set of feedback control primitives, which are not completely reliable, allowing it to move inside each region and from a region to an adjacent regions. It is also assumed that the probabilities of these transitions are known.
\begin{assumption} \label{assumption:basic_assumption}
We assume here that an abstraction of the dynamics of each robot into a MDP is given, and that a low level continuous time controller that allows each robot to transit from one region $\gamma_\ell$ to an adjacent region $\gamma_l$ with $\ell,l \in \mathcal{W}$, can be designed.
\end{assumption}
This modeling has been also considered in \citep{belta_2012_tro_pctlalgorithms}.
\begin{definition} \label{def:agent_mdp}
The motion of each agent $i \in \mathcal{V}$ in the workspace can be described by a Markov Decision Process (MDP) $\mathcal{M}_i =(S_i, s_0^i, Act_i, T_i)$ where:
\begin{itemize}
\item $S_i = \left\{ \gamma_1^i, \gamma_2^i, \ldots, \gamma_W^i \right\}$ is the set of states of agent $i$. The number of states for each agent is $|S_i| = W$, meaning that $S_i$ includes all regions within $\mathcal{W}$.
\item $s_0^i \in S_i$ is the initial state of agent $i$ (the initial region where agent $i$ may start). Note that the initial state is known and deterministic i.e. we know exactly the region from which each agent starts its motion.
\item $Act_i$ is a finite set of actions (controls).
\item $T_i: S_i \to 2^{Act_i \times \Sigma(S_i)}$ is the transition probability function.
\end{itemize}
\end{definition}
\begin{remark} \label{remark:agents_synch}
We investigate here, under which conditions two or more agents are visiting simultaneously a specific region of the workspace. Consider the $\{i_1,\dots,i_c\} \subseteq \mathcal{V}, c \geq 2$ agents of the multi-agent system. Let $r_{i_1} = s_0^{i_1} s_1^{i_1} s_2^{i_1} \dots s_k^{i_1} \dots s^{i_1}_n, \dots, r_{i_c} = s_0^{i_c} s_1^{i_c} s_2^{i_c} \dots s_k^{i_c} \dots s^{i_c}_n$, be the finite paths of length $n$ that are executed by the corresponding MDPs $\mathcal{M}_1, \dots, \mathcal{M}_c$, respectively, where $s_z^{j} \in S_z, \forall z \geq 0, j \in \{i_1,\dots,i_c\}$. Then, if there exists $k \geq 1$ such that for all the $k$-th elements of the above runs (in the same position at every path) it holds that $s_k^{i_1}=\dots=s_k^{i_c} = s^{\text{meet}}_k$, then we say that the agents $\{i_1,\dots,i_c\}$ are visiting simultaneously the same region $s_k$. If there does not exist such region $s^{\text{meet}}_k$, then the agents cannot meet simultaneously to one region.
\end{remark}
\subsection{Handshaking Actions}
The motivation for introducing dependencies in the multi-agent system comes from real applications where more than one agents (robots) need to collaborate with each other in oder to perform a desired task. For example, imagine two aerial manipulators that are required to meet and grasp an object simultaneously and deliver it to a specific location in a warehouse.
In order to be able to introduce dependencies in the actions between the agents, we write the action set of each agent as: $Act_i = \{\Pi_i, \hat{\Pi}_i\}, i \in \mathcal{V}$, where $\Pi_i$ is a finite set of actions that the agent $i$ need to execute in collaboration with other agents (\textit{handshaking} actions) and $\hat{\Pi}_i$ is a finite set of actions that the agent $i$ executes independently of the other agents (\textit{independent} actions). For the independent actions it holds that: $\hat \Pi_i \cap \hat \Pi_j = \emptyset, \forall i \neq j, i,j \in \mathcal{V}$.
The independent actions can always be executed without any constraints. On the other hand, for the handshaking actions, we have the following requirements:
\begin{itemize}
\item First, the agents are required to meet and occupy the same region of the workspace (not necessarily a specific region).
\item Once they meet, they need to execute \text{simultaneously} the same action.
\item All the agents that share an action are required to execute it in order for the task to be completed properly.
\end{itemize}
Formally, the handshaking actions are defined as follows:
\begin{definition} \label{def:handshaking_actions}
(Handshaking Actions) Let $\{i_1, \dots, i_{c}\} \subseteq \mathcal{V}, c \geq 2$ be a set of agents that need to collaborate in order to execute simultaneously a task under the action $\alpha$. The following two properties should hold in order for $\alpha$ to be well-posed handshaking action:
\begin{enumerate}
\item $\alpha \in \bigcap_{i \in \{i_1,\dots,i_c\}} \Pi_{i}$.
\item Let the following finite paths of length $n$:
\begin{align*}
r_{i_1} &= s_0^{i_1} \overset{\alpha_0^{i_1}}{\longrightarrow} \ldots \overset{\alpha_{k-1}^{i_1}}{\longrightarrow} s_{k}^{i_1} \overset{\alpha}{\longrightarrow} s_{k'}^{i_1} \dots \overset{\alpha_{n-1}^{i_1}}{\longrightarrow} s_n^{i_1}, \\
&\qquad \qquad \vdots \\
r_{i_c} &= s_0^{i_c} \overset{\alpha_0^{i_c}}{\longrightarrow} \ldots \overset{\alpha_{k-1}^{i_c}}{\longrightarrow} s_{k}^{i_c} \overset{\alpha}{\longrightarrow} s_{k'}^{i_c} \dots \overset{\alpha_{n-1}^{i_c}}{\longrightarrow} s_n^{i_c},
\end{align*}
be executed by the MDPs $\mathcal{M}_{i_1}, \dots, \\ \mathcal{M}_{i_c}$ respectively. Here, $s_{k}^{i_1}, \dots, s_{k}^{i_c}$ are the regions that the $i_1, \dots, i_c$ should occupy in order to execute the handshaking action $\alpha$ simultaneously. Then, there should exist at least one $k \geq 0$ such that $s_{k}^{i_1} = \dots = s_{k}^{i_c} = s^{\text{meet}}_k$ and $\delta(s_{k}^{j}, \alpha, s_{k'}^{j}) > 0$ for at least one $s_{k'}^j \in \text{Post}(s_k^j,\alpha)$ for every $j \in \{i_1,\dots,i_c\}$.
\end{enumerate}
\end{definition}
Notice that the same condition for a state $s^{\text{meet}}_k$ as in condition (2) was mentioned in Remark \ref{remark:agents_synch}, but here the existence of a common action $\alpha$ is also required. It should be also noted that every region of the workspace in which the agents can potentially meet, can serve as a region that a handshaking action can be executed (if such an action exists).
\subsection{Dependencies} \label{sec:dependencies}
Suppose that one agent $i$ receives a cooperative task that involves other agent's $j \in \mathcal{V} \backslash \{i\}$ participation. This means that both agents need to execute the same action
at the same region so as for the task to be performed. The dependencies are formally defined as:
\begin{definition} \label{def:dependency_relation}
The agents $i,j \in \mathcal{V}$ are called \textit{dependent} if one the following statements holds:
\begin{enumerate}
\item Agent $i$ depends on agent $j$ if $\Pi_{i} \cap \Pi_j \neq \emptyset$.
\item Agent $i$ depends on agent $j$ if $\Pi_{i} \cap \Pi_j \neq \emptyset$.
\item There exist at least one region $s^{\text{meet}}_k, k \geq 0$ of the workspace such that the second condition of Def. \ref{def:handshaking_actions} holds.
\end{enumerate}
\end{definition}
Conditions (1),(2) can be checked by comparing all the elements of the sets $\Pi_{i}, \Pi_j, \forall i,j \in \mathcal{V}$ one by one. Condition (3) can be checked by using graph search algorithms.
\begin{remark}
It should be noticed from the above definitions that all the agents that share an action, they are required to meet and execute it simultaneously.
\end{remark}
\begin{remark}
Due to fact that the control policies are defined over finite paths, the handshaking actions are defined with respect to finite paths as well. Therefore, the graph search algorithm for condition (3) is searching into a finite graph.
\end{remark}
\begin{assumption} \label{assumption:dependency_assumption}
It is assumed that there exists at least $2$ agents that are dependent. Otherwise, there exist no dependencies between the agents and the problem that is later defined can be straightforwardly solved by solving the controller synthesis methodology of Section \ref{sec:problem_solution} for each agent independently.
\end{assumption}
\subsection{Problem Statement}
We define here the problem that we aim to solve in this paper:
\begin{problem} \label{problem:basic_problem}
Given $N$ agents performing in the workspace $\mathcal{W}$, under the Assumptions \ref{assumption:basic_assumption}, \ref{assumption:dependency_assumption}, individual task specification formulas $\varphi_1, \ldots, \varphi_N$ over the actions $\Pi_i \cup \hat{\Pi}_i, i \in \mathcal{V}$ given in PCTL with semantics as in Sec. \ref{sec:PCTL}, synthesize individual control policies $\mu_1, \ldots, \mu_N$ (if there exists one) which guarantee the satisfaction of the formulas $\varphi_1, \ldots, \varphi_N$ respectively.
\end{problem}
\section{Problem Solution} \label{sec:problem_solution}
\subsection{Overview}
An overview of the proposed solution is given as follows:
\begin{itemize}
\item Step 1: First, the dependencies among the agents are modeled as a dependency graph (see Section \ref{section:dependency_graph}). The agents are split into clusters and each cluster contains the agents that are dependent according to Def. \ref{def:dependency_relation}.
\item Step 2: For each cluster of agents, the mutual specification $\varphi_m$ and the product MDP $\widetilde{M}$ are defined (see Section \ref{sec:product_MDP}).
\item Step 3: By utilizing the controller synthesis algorithms of Section \ref{sec:algorithms}, we design a control policy $\widetilde{\mu}$ of each cluster that guarantees the satisfaction of $\varphi_m$ (if such a control policy exists). We provide in Sec. \ref{sec:path_policy_projection} the definition of successful control policies, which project onto local control policies $\mu_1, \dots, \mu_N$ for each agent, which finally are a solution to Problem 1.
\end{itemize}
An algorithm describing all the steps of the proposed procedure is given in Section \ref{sec:algorithm}. Probabilistic model checking algorithms, which can compute all the control policies under which a PCTL formula $\varphi_{m}$ is satisfied, are presented in Section \ref{sec:algorithms}. The computational complexity of the proposed framework is discussed in Section \ref{section:complexity}.
Problem \ref{problem:basic_problem} can be solved in a centralized way by computing the product of all individual MDPs $\mathcal{M}_i, i \in \mathcal{V}$ (see Definition \ref{def:product_MDP}) and perform the proposed methodology of this paper to the centralized system without any clustering among the agents. However, this solution leads to a high computational burden and state explosion of the product MDP $\mathcal{M}$. A comparison of the computational complexity of the proposed framework that exploits the potential sparsity of dependencies with the centralized approach is discussed in Section \ref{section:complexity}.
\subsection{Modeling the Dependencies} \label{section:dependency_graph}
Based on the dependency relation of the Def. \ref{def:dependency_relation}, the dependency graph associated with the handshaking actions $\Pi_i, i \in \mathcal{V}$ is defined as follows:
\begin{definition} \label{def:dependency_graph}
The \textit{dependency graph} $\mathcal{G} = (\mathcal{V}, \mathcal{E})$, is an undirected graph that consists of the set of vertices $\mathcal{V}$ in which each of the agents is node for the graph and the edge set $\mathcal{E}$ which is defined as follows:
\begin{equation*}
\mathcal{E} = \{\{i, j\} : i \ \text{is dependent to} \ j \ \text{and} \ i,j \in \mathcal{V}, i \neq j\}.
\end{equation*}
\end{definition}
In order to proceed, the following definition is required:
\begin{definition} \citep{mesbahi_2010_graph_theory}
Let $\mathcal{G} = (\mathcal{V}, \mathcal{E})$ be an undirected graph. Then every graph $\mathcal{G}' = (\mathcal{V}', \mathcal{E}')$ with $\mathcal{V}' \subseteq \mathcal{V}$ and $\mathcal{E}' \subseteq \mathcal{E}$ if called a \textit{subgraph} of the graph $\mathcal{G}$.
\end{definition}
\begin{definition} \label{def:dependency_cluster}
The set $\mathcal{C} = \{C_\ell : \ell \in \mathbb{M}\} \subseteq \mathcal{V}$, where $\mathbb{M} = \{1,\dots,m\}$, forms a \textit{dependency cluster} if and only if $\forall i,j \in \mathcal{C}$ there is a path from node $i$ to node $j$ in the dependency graph $\mathcal{G}$.
\end{definition}
Define the function $f: \mathcal{V} \to \mathbb{M}$ which maps each agent to the index of the cluster that it belongs to. It can be observed that $\bigcup_{\ell \in \mathbb{M}} C_\ell = \mathcal{V}$ and $\sum_{\ell \in \mathbb{M}}^{} |C_\ell| = |\mathcal{V}| = N$.
Each agent $i \in \mathcal{V}$ for which there exist no $j \in \mathcal{V}$ such that $j \in C_{f(i)}$ will be called an \textit{independent agent}. For an independent agent it holds that $|C_{f(i)}| = 1$.
From Definition \ref{def:dependency_cluster}, it follows that every dependency cluster $C_\ell \in C, \ell \in \mathbb{M}$ is the vertex set of the subgraphs $G^{(\ell)} = (\mathcal{C}_\ell, \mathcal{E}_\ell), \mathcal{E}_\ell \subseteq \mathcal{E}, \ell \in \mathbb{M}$ of the system graph $\mathcal{G}$.
Loosely speaking, two agents belong to the same cluster when they are directly dependent or transitively dependent by a dependency chain. An example of a dependency graph and dependency clusters is given as follows:
\begin{example} \label{example:dependency_graph_example}
Consider $N = 6$ agents with $\mathcal{V} = \{1,\ldots,6\}$, $\mathcal{E} = \{\{1, 2\}, \{3, 4\}, \{4, 5\}\}$. The $m = 3$ clusters are given as: $C_1 = \{1,2\}, C_2 = \{3,4,5\}$ and $C_3 = \{6\}$ and the corresponding subgraphs $\mathcal{G}^{(1)} = (C_1, \mathcal{E}_1 = \{1,2\}), \mathcal{G}^{(2)} = (C_2, \mathcal{E}_2 = \{\{3,4\}, \{4,5\}\})$ and $\mathcal{G}^{(3)} = (C_3, \mathcal{E}_3 = \emptyset)$. Moreover, $f(1) = f(2) = 1, f(3) = f(4) = f(5) = 2, f(6) = 3$. The dependency graph is depicted in Fig. \ref{fig:dependency_graph_example}.
\begin{figure}[ht!]
\vspace{2mm}
\centering
\begin{tikzpicture} [scale = 1.2]
\node(1) [line width = 0.85] at (-3,0)[shape=circle,draw][fill=green!40] {$ $};
\node(2) [line width = 0.85] at (-2,1)[shape=circle,draw][fill=green!40] {$ $};
\node(3) [line width = 0.85] at (0,0)[shape=circle,draw][fill=red!40] {$ $};
\node(4) [line width = 0.85] at (2,0)[shape=circle,draw][fill=red!40] {$ $};
\node(5) [line width = 0.85] at (1,1)[shape=circle,draw][fill=red!40] {$ $};
\node(6) [line width = 0.85] at (-1,-2)[shape=circle,draw][fill=blue!40] {$ $};
\path [-] [line width = 0.85]
(1) edge node [above] {} (2)
(3) edge node [above] {} (5)
(4) edge node [above] {} (5);
\draw[black,thick,dashed] (-2.5,0.5) circle (1.2cm);
\draw[black,thick,dashed] (1,0.5) circle (1.5cm);
\draw[black,thick,dashed] (-1,-2) circle (0.7cm);
\node at (-3.2, 0.35) {$1$};
\node at (-2.2, 1.3) {$2$};
\node at (0, 0.35) {$3$};
\node at (1, 1.35) {$4$};
\node at (2, 0.35) {$5$};
\node at (-1, -1.65) {$6$};
\node at (-4, 1.0) {$C_1$};
\node at (2.8, 1.0) {$C_2$};
\node at (-2.1, -1.9) {$C_3$};
\end{tikzpicture}
\caption{An example of a dependency graph $\mathcal{G} = (\mathcal{V}, \mathcal{E})$ and its subgraphs for $N = 6$ agents.}
\label{fig:dependency_graph_example}
\end{figure}
\end{example}
According to the mathematical derivation above, Assumption \ref{assumption:dependency_assumption} is modified as follows:
\begin{assumption}
There exists at least one dependency cluster $C_\ell, \ell \in \mathbb{M}$ (as was defined in Definition \ref{def:dependency_cluster}) of the dependency graph $\mathcal{G}$ of the under consideration multi-agent system, which contains at least two dependent agents. I.e., there exists $\ell \in \mathbb{M}$ such that: $|C_\ell| = 2, \text{if} \ N = 2$ and $|C_\ell| \in [2, N-1], \text{if} \ N > 2$.
\end{assumption}
By employing the above computation, the initial multi-agent system is modeled as $m$ subgraphs $\mathcal{G}^{\ell}, \ell \in \mathbb{M}$ which capture the dependencies between the agents, as they are defined in Def. \ref{def:dependency_relation}. This forms a convenient modeling of the system's dependencies in order to compute the product MDP of every subsystem $\ell \in \mathbb{M}$ in the next Section.
\subsection{Product Markov Decision Process} \label{sec:product_MDP}
Define here the \emph{mutual specification} of a cluster of agents $C_\ell$ as:
\begin{align}
\varphi_m^{\ell} = \bigwedge_{i \in C_\ell}^{} \varphi_i, \ell \in \mathbb{M}, \label{eq:formula_mut}
\end{align}
over the set of actions $\bigcup_{i \in C_\ell}^{} \left(\Pi_i \cup \hat{\Pi}_i\right)$. If the satisfaction of $\varphi_m^\ell$ for each cluster $C_\ell$ is guaranteed, it holds by definition that the satisfaction of all the individual formulas $\varphi_i, i \in \mathcal{V}$ is guaranteed as well. Thus, a method for finding a team control policy that guarantees the satisfaction of $\varphi_m^\ell, \ell \in \mathbb{M}$ should be provided.
In the sequel, we construct a product MDP that captures the collaborative behavior of all the agents within a cluster. Having $\widetilde{M}_\ell$, allow us to synthesize a control policy $\widetilde{\mu}$ for $C_\ell$, which guarantees the satisfaction of the collaborative formula $\varphi_m^\ell$. Subsequently, the team control policy $\widetilde{\mu}_\ell$ can be projected onto the local agents' control policies $\mu_1, \dots, \mu_N$ which are a solution to Problem 1.
\begin{definition}\label{def:product_MDP} (Product MDP) The \textit{product MDP} $\widetilde{\mathcal{M}}_\ell$ for the cluster of agents $C_\ell$ is a tuple $(\widetilde{S}_\ell, \widetilde{s}_{0}^{\ell}, \widetilde{Act}_\ell, \widetilde{T}_\ell)$ where:
\begin{itemize}
\item $\widetilde{S}_\ell = \underset{i \in C_\ell}{\overset{}{\bigtimes}} S_i$ is the set of states.
\item $\widetilde{s}_{0}^{\ell} = \underset{i \in C_\ell}{\overset{}{\bigtimes}} s_0^i$ is the initial state.
\item $\widetilde{Act}_\ell = \bigcup_{i \in C_\ell}^{} Act_i = \bigcup_{i \in C_\ell}^{} \left\{\Pi_i, \hat{\Pi}_i \right\}$ is the set of actions.
\item $\widetilde{T}_\ell : \widetilde{S}_\ell \to 2^{\widetilde{Act}_\ell \times \Sigma(\widetilde{S}_\ell)}$ is the transition probability function for the product system. Similar to $\delta$ of Def. \ref{def:mdp}, we define $\widetilde{\delta}(\widetilde{s}, \widetilde{\alpha}, \widetilde{s}') \in [0,1]$ the probability of transitioning from the state $\widetilde{s}$ to the state $\widetilde{s}'$ under the action $\widetilde{\alpha}$. Let $C_\ell = \{i_1, \dots, i_{|C_\ell|}\}$ be an enumeration of the agents of the cluster $C_\ell$. Then, $\widetilde{\delta}_\ell$ is defined as follows:
\begin{enumerate}
\item $\widetilde{\delta}((s_{i_1}, \ldots, s_{i_{|C_\ell|}}), \alpha, (s'_{i_1}, \ldots, s'_{i_{|C_\ell|}})) = \\ \prod_{j \in C_\ell}^{} \delta_{j}(s_{j}, \alpha, s'_{j})$, if $\alpha \in \bigcap_{j \in C_\ell}^{} \mathcal{A}(s_{j})$.
\item $\widetilde{\delta}( (\widetilde{s}_{i_1}, \ldots, \widetilde{s}_{k_1}, \ldots, \widetilde{s}_{k_\nu}, \ldots, \widetilde{s}_{i_{|C_\ell|}}), \alpha, (\widetilde{s}_{i_1}, \\ \ldots, \widetilde{s}'_{k_1}, \ldots, \widetilde{s}'_{k_\nu}, \ldots, \widetilde{s}_{i_{|C_\ell|}})) = \prod_{j = 1}^{\nu} \delta_{k_j}(s_{k_j}, \alpha, \\ s'_{k_j})$ if $$\alpha \in \left[\bigcap_{j=1}^{\nu} \mathcal{A}(s_{k_j}) \right] \mathbin{\Bigg\backslash} \left[\bigcup_{z \in C_\ell \backslash \{k_1, \ldots, k_\nu\}}^{} \mathcal{A}(s_z) \right],$$ for $k_j \in C_\ell, j \in \{1, \ldots, \nu\}$.
\end{enumerate}
\end{itemize}
\end{definition}
Intuitively, (1) denotes that all the agents $i_1, \dots, i_{|C_\ell|}$ of the cluster $|C_\ell|$ are located in the states $s_{i_1}, \ldots, s_{i_{|C_\ell|}}$ respectively, and they are simultaneously transiting to the states $s'_{i_1}, \ldots, s'_{i_{|C_\ell|}}$ with action $\alpha$. (2) denotes that among all the agents of the cluster $C_\ell$, only the agent $\{k_1, \dots, k_\nu\} \subsetneq C_\ell$ are transiting simultaneously to the states $s'_{k_1}, \dots, s'_{k_\nu}$ respectively. (2) can not be handshaking action since for the handshaking action all the agents of the cluster should transit simultaneously to the next state. In order for (1) to be a handshaking transition according to Def. \ref{def:handshaking_actions} it is required also that $s_{i_1} = \ldots = s_{i_{|C_\ell|}}$.
\begin{remark}
In the case of a cluster $\ell \in \mathbb{M}$ that contains an independent agent $i \in \mathcal{V}$ with the property $|C_\ell| = |C_{f(i)}| = 1$, the product MDP $\widetilde{\mathcal{M}}_\ell$ coincides with the individual MDP $\mathcal{M}_i$ of Def. \ref{def:agent_mdp} ($\widetilde{\mathcal{M}}_\ell \equiv \mathcal{M}_i$).
\end{remark}
The infinite path $\widetilde{r}$, the finite path $\widetilde{\rho}$, the control policy $\widetilde{\mu}$ and the set of all infinite and finite paths $\widetilde{FPath}$ and $\widetilde{IPath}$, are defined similarly to Sec. \ref{sec:mdp_definitions}.
\subsection{Designing the Control Policies $\mathcal{\widetilde{\mu}}$} \label{sec:path_policy_projection}
The product MDP $\widetilde{M}_\ell, \ell \in \mathbb{M}$ of each cluster captures the paths and the control policies of the agents that belong to the same cluster and they are required to collaborate for achieving a task or acting independently.
By employing the controller synthesis algorithms (see Section \ref{sec:algorithms}), the control policies $\widetilde{\mu}_\ell$ for the team of agents in each cluster can be designed. The algorithms can compute all the control policies $\widetilde{\mu}_\ell$ that guarantees the satisfaction of formula $\varphi^\ell_m$ from \eqref{eq:formula_mut}.
What remains is to project these policies onto the individual control policies of the agents of each cluster in such a way that they serve as a solution to Problem 1.
Consider a cluster of agents $C_\ell = \{i_1, \dots, i_{|C_\ell|}\}$. A control policy $\widetilde{\mu}_\ell(\widetilde \rho) = \widetilde{\mu}_\ell(\widetilde{s}_0\widetilde{s}_1 \dots \widetilde{s}_n) \subseteq \widetilde{Act}$ for the finite path $\widetilde{\rho} = \widetilde{s}_0\widetilde{s}_1 \dots \widetilde{s}_n$ of length $n$, where $\widetilde{s}_k = (s^k_{i_1}, \dots, s^k_{i_{|C_\ell|}}), k \in \{1,\dots,n\}$, projects onto the local individual control policies $\mu_{j}(s^1_{j}, \dots, s^n_{j}), j \in \{i_1, \dots, i_{|C_\ell|}\}$, of the agents $\{i_1, \dots, i_{|C_\ell|}\}$ of the cluster $C_\ell, \ell \in \mathbb{M}$. Note that: $\mu_{j} \subseteq \widetilde{Act} \ \bigg\lvert_{j} \subseteq Act_{j} , j \in \{i_1, \dots, i_{|C_\ell|}\}$, and $\widetilde{Act} \ \bigg\lvert_{j}$ is the set of actions of the agent $j$ that are appearing in the s et $\widetilde{Act}$.
The set $\widetilde{\mu}(\widetilde{\rho})$ contains control policies that are either handshaking or independent. Let us also define the following set of handshaking actions: $\text{Succ}(\alpha, \ell) = \{\alpha \in \Pi_{i_1} \cap \dots \cap \Pi_{i_{|C_\ell|}}: \alpha \in \widetilde{\mu}_\ell(\widetilde{\rho})\}$, which is the subset of $\widetilde{\mu}(\widetilde{\rho})$ that contains the handshaking actions. We need to search now if all the projections $\mu_{i_j}, \forall j \in \{1,\dots,|C_\ell|\}$ follow the handshaking rules of Def. \ref{def:handshaking_actions}.
\begin{definition} \label{def:control_policy_solution}(Successful Control Policy)
Let $\widetilde{\mu}_\ell(\widetilde \rho) = \widetilde{\mu}_\ell(\widetilde{s}_0\widetilde{s}_1 \dots \widetilde{s}_n) \subseteq \widetilde{Act}$ be a control policy of a cluster $C_\ell$. The control policy $\widetilde{\mu}_\ell(\widetilde{\rho})$ is called \textit{successful} if for all $\alpha \in \text{Succ}(\alpha,\ell)$ it holds that $s^n_{i_1} = \dots = s^n_{i_{|C_\ell|}}$ and $\delta(s_j^n, \alpha, (s_j^n)') > 0$
for at least one $(s_j^n)' \in \text{Post}(s_j^n,\alpha), j \in \{i_i, \dots, i_{|C_\ell|}\}$.
\end{definition}
Let $SP(\ell) = \left\{\widetilde{\mu}_\ell(\widetilde{\rho}) \subseteq \widetilde{Act}: \widetilde{M}_\ell \models \varphi_m^\ell \right\}, \ell \in \mathbb{M},$ denotes the set of all the control policies that guarantee the satisfaction of the formula $\varphi_m^\ell$. All the control policies of $SP$ needs to be checked if they are successful. If $SP(\ell) = \emptyset$ for at least one $\ell \in \mathbb{M}$, then the Problem \ref{problem:basic_problem} has no solution. The set $SP(\ell)$ is computed by employing the algorithms of Section \ref{sec:algorithms}.
\subsection{Proposed Algorithm} \label{sec:algorithm}
The proposed procedure of solving Problem \ref{problem:basic_problem} can be shown in Algorithm \ref{alg:basic_algorithm}. The function $\text{checkDepend}()$ determines the dependent agents according to Def. \ref{def:dependency_relation}. The product and projection that were introduced in Sec. \ref{sec:product_MDP}, \ref{sec:path_policy_projection}, are computed by the functions $\text{product}(), \text{projection}()$ respectively. The algorithms of Sec. \ref{sec:algorithms} are incorporated in the function $\text{controlSynthesis}()$. By employing Def. \ref{def:control_policy_solution}, the function $\text{succPolicy}()$ determines if a sequence of control policies are successful.
\begin{remark}
Even though our proposed solution is centralized in each cluster, it is partially decentralized in terms of the whole multi-agent system.
\end{remark}
\begin{algorithm}
\caption{- SolveProblem1($\cdot$)}
\begin{algorithmic}[1]
\State \textbf{Input:} MDPs: $\mathcal{M}_1,\dots,\mathcal{M}_N$; \\
\hspace{11mm} PCTL Formulas: $\varphi_1,\dots,\varphi_N$
\State \textbf{Output:} $\mu_1, \dots, \mu_N$ \\
\State $\mathcal{C} = \{C_\ell, \ell \in \mathbb{M}\}=\text{checkDepend}(Act_1, \dots, Act_N)$
\State $\varphi_m^{\ell} = \bigwedge_{i \in C_\ell}^{} \varphi_i$
\For {$z \in C_\ell =\{i_1, \dots, i_{|C_\ell}|\}$}
\State $\widetilde{M}_\ell = \text{product}(\{\mathcal{M}_j, j \in C_\ell\})$
\State $SP(\ell) = \text{controlSynthesis}(\widetilde{M}, \varphi_m^\ell)$
\For {$\widetilde{\mu}_\ell \in SP(\ell)$}
\State $\{\mu_1,\dots,\mu_N\} = \text{projection}(\widetilde{M}_\ell)$
\If {$\text{succPolicy}(\{\mu_1,\dots,\mu_N\}) = \top$}
\State $\text{solFound} = 1$
\State $\text{return} \ \{\mu_1,\dots,\mu_N\}$ \Comment{Solution found}
\Else
\State $\text{go to} \ 12$ \Comment{Search other control policies}
\EndIf
\EndFor
\If {$\text{solFound} \neq 1$}
\State Problem 1 has no solution
\EndIf
\EndFor
\end{algorithmic}
\label{alg:basic_algorithm}
\end{algorithm}
\subsection{Algorithms for Probabilistic Control Synthesis} \label{sec:algorithms}
We are investigating here algorithms of computing all the control policies $\widetilde{\mu}_\ell \in SP(\ell)$. Once these control policies are found, then by following Algorithm \ref{alg:basic_algorithm}, the individual policies $\mu_{j}, j \in \{i_1, \dots, i_{|C_\ell|}\}$ can be designed and the Problem 1 is solved (if there exists a solution). For more details about the algorithms we refer to \citep{marta_2004_book, marta_2007_stochastic, marta_paper_2_2011, alvaro_1995_probabilistic_model_checking, belta_2012_tro_pctlalgorithms}
First, define $\text{Sat}(\varphi^\ell_{m}) = \{s \in S: s \models \varphi^\ell_m\}$ as the set of states that satisfy $\varphi_m^{\ell}$. Then we have: $\text{Sat}(\top) = S, \text{Sat}(\pi) = \{s \in S : \pi \in \mathcal{A}(s)\}, \text{Sat}(\neg \varphi^\ell_{m}) = S \backslash \text{Sat}(\varphi^\ell_{m}), \text{Sat}(\varphi^{\ell}_{m,1} \wedge \varphi^{\ell}_{m,2}) = \text{Sat}(\varphi^{\ell}_{m,1}) \cap \text{Sat}(\varphi^{\ell}_{m,2})$ for two PCTL formulas $\varphi^{\ell}_{m,1}, \varphi^{\ell}_{m,2}$. Define also the minimum and the maximum probabilities of satisfying the formula under the control policy $\mu$ for a starting state $s$:
\begin{subequations}
\begin{align}
Prob_{\max}(s, \psi) &= \sup_{\mu \in M} \{Prob_\mu(s,\psi)\}, \label{eq:prob_max} \\
Prob_{\min}(s, \psi) &= \inf_{\mu \in M} \{Prob_\mu(s,\psi)\}. \label{eq:prob_min}
\end{align}
\end{subequations}
where $M$ is set of all control policies. It has been proved in \citep{alvaro_1995_probabilistic_model_checking}, that the model checking problem problem of the operator $\mathcal{P}_{\bowtie p}[\psi]$ can be reduced to the computation of \eqref{eq:prob_max}, \eqref{eq:prob_min} according to the following:
\begin{align}
\bullet \ \text{If} \ \bowtie &= \{\geq, >\} \ \text{then} \notag \\
&\hspace{15mm} s \models \mathcal{P}_{\bowtie p}[\psi] \Leftrightarrow Prob_{\min}(s, \psi) \bowtie p. \label{eq:condition_p_min} \\
\bullet \ \text{If} \ \bowtie &= \{\leq, <\} \ \text{then} \notag \\
&\hspace{15mm} s \models \mathcal{P}_{\bowtie p}[\psi] \Leftrightarrow Prob_{\max}(s, \psi) \bowtie p. \label{eq:condition_p_max}
\end{align}
For the controller synthesis (as was defined in Section \ref{sec:probabilistic_model_checking}) of the path operators $\mathcal{P}_{\bowtie p}[\bigcirc \varphi^{\ell}_{m}]$, $\mathcal{P}_{\bowtie p}[\varphi^{\ell}_{m} \mathcal{U}^{\leq k} \varphi^{\ell}_{m}]$, we utilize the Algorithms 2,3 respectively as follows:
\subsubsection{Algorithm 2}
If the formula is $\varphi^{\ell}_{m} = \mathcal{P}_{\bowtie p}[\bigcirc \varphi^{\ell}_{m,1}]$, initially the maximum probability of satisfying $\varphi^{\ell}_{m}$ at the state $s\in S$:
\begin{equation}
Prob_{\max}(s, \varphi^{\ell}_{m}) = \underset{\alpha \in \mathcal{A}(s)}{\text{max}} \left(\sum_{s' \in Sat(\varphi^{\ell}_{m,1})}^{} \delta(s, \alpha, s')\right), \label{eq:alg_1}
\end{equation}
is computed for every $s \in S$. By replacing $\max$ with $\min$ in \eqref{eq:alg_1}, $Prob_{\min}(s, \bigcirc \varphi^{\ell}_{m,1})$ can be computed. Define the vector $\Phi(s) = 1, \text{if} \ s \in Sat(\varphi^{\ell}_{m, 1})$ or $\Phi(s) = 0, \text{otherwise}$. Perform now the matrix multiplication $X = T \cdot \Phi$. $X$ is a vector whose entries are the probabilities of satisfying $\bigcirc \varphi_1$, where each row corresponds to a state-action pair. After obtaining the vector $X$, eliminate the state-actions pairs whose probabilities are not in the range of $\bowtie p$ by taking into consideration the conditions \eqref{eq:condition_p_min}, \eqref{eq:condition_p_max}. This operation determines all the states $s \in S$ and all the actions $\mu \in M$ that satisfy the formula $\varphi^{\ell}_{m}$.
\subsubsection{Algorithm 3}
For the formula of the form $\phi_m^{\ell} = \mathcal{P}_{\bowtie p} \left[\varphi^{\ell}_{m,1} \ \mathcal{U}^{\leq k} \ \varphi^{\ell}_{m,2}\right]$, define by $S^{\text{yes}} = \text{Sat}(\varphi^{\ell}_{m,2}), S^{no} = S \backslash \left[\text{Sat}(\varphi^{\ell}_{m,1}) \cup \text{Sat}(\varphi^{\ell}_{m,2})\right],$ and $S^{rem} = S \backslash (S^{yes} \cup S^{\text{no}})$ the states that always satisfy the specification, the states that never satisfy the specification and the remaining states, respectively. Compute the maximum probability of satisfying $\varphi^{\ell}_{m}$ at the state $s\in S$ as: $Prob_{\max}(s, \varphi^{\ell}_{m}) = 1 \ \text{or} \ 0, \text{if} \ s \in S^{\text{yes}} \ \text{or} \ s \in S^{\text{no}}$ respectively. For $s \in s \in S^{\text{rem}}$ and $k \ge 0$ compute recursively the following:
\begin{align}
&Prob_{\max}(s, \varphi^{\ell}_{m}, k) \notag \\
&= \underset{\alpha \in \mathcal{A}(s)}{\text{max}} \left( \sum_{s' \in S^{rem}}^{} \delta(s, \alpha, s') Prob_{\max}(s, \varphi^{\ell}_{m}, k-1)\right. \notag \\
&\hspace{40mm}\left. +\sum_{s' \in S^{yes}}^{} \delta(s, \alpha, s')\right), \label{eq:prob_k}
\end{align}
with $Prob_{\max}(s, \varphi^{\ell}_{m}, 0) = 0$. The computation can be carried out in $k$ iterations, each similar to the process of Algorithm 2. By replacing $\max$ with $\min$ in \eqref{eq:prob_k}, $Prob_{\min}(s, \varphi^{\ell}_{m}, k)$ can be computed.
\subsubsection{Algorithm 4}
The form $\phi_m^{\ell} = \mathcal{P}_{\bowtie p} \left[\varphi^{\ell}_{m,1} \ \mathcal{U}^{} \ \varphi^{\ell}_{m,2}\right]$ is in fact the same as $\phi_m^{\ell} = \mathcal{P}_{\bowtie p} \left[\varphi^{\ell}_{m,1} \ \mathcal{U}^{\leq k} \ \varphi^{\ell}_{m,2}\right]$ as $k \to \infty$. With this approach, Algorithm 3 can be used to solve this problem.
\begin{remark}
The resulting control strategies of the aforementioned algorithms are stationary. Therefore, the control policies $\widetilde{\mu}_\ell(\widetilde{s}_1\widetilde{s}_2\dots\widetilde{s}_n)$ depend only to the state $\widetilde{s}_n$.
\end{remark}
\subsection{Computational Complexity} \label{section:complexity}
According to \citep{katoen}, the model checking of an MDP $\mathcal{M}$ is polynomial in the number of states of $\mathcal{M}$ and linear in the length of the formula $\varphi$. Denote by $|\varphi|$ the length of the formula $\varphi$ in terms of the number of the operator it has e.g., $|\mathcal{P}_{\geq 0.5} [\bigcirc \{red\} ]| = 2$ . The complexity can be expressed mathematically as $$\mathcal{O}(\text{poly}(|\mathcal{M}|) |\varphi| \kappa(\varphi)),$$ where $\kappa(\varphi) = \max \{k: \phi_1 \mathcal{U}^{\leq k} \varphi_2\}$, $\varphi_1, \varphi_2$ are subformulas of $\varphi$ and $\varphi_1 \mathcal{U}^{\leq k} \varphi_2$ are possible until operators involving in $\varphi$. Define also $$\text{poly}(n) = 2^{\mathcal{O}(\log(n))}.$$ If $\varphi$ does not contain a bounded until operator then $\kappa(\varphi) = 1$.
The number of states of the the product MDP in the centralized solution is $|\widetilde{S}| = \prod_{i \in \mathcal{V}} |S_i| = W^N$ and the corresponding complexity is in the class of $$O = \mathcal{O}\left( 2^{\mathcal{O}(N \log(W))} \cdot |\varphi^{\ell}_m| \cdot \kappa(\varphi_{m}^\ell) \right),$$ where $\varphi^\ell_{m}$ as it is defined in \eqref{eq:formula_mut} for $|C_\ell| = N$.
The worst case complexity of the proposed framework is when $1$ agent is independent and the other $N-1$ agents are dependent to each other. Then, there are two clusters $\ell \in \{1,2\}$: the first contains the independent agent and the other one contains the remaining agents. The corresponding MDPs have $|\widetilde{S}_\ell| = |\underset{i \in C_\ell}{\overset{}{\bigtimes}} S_i| = W^{|C_\ell|}, \ell \in \{1,2\}$ states i.e., $W, W^{N-1}$ states respectively. Thus, the worst case complexity of our framework is:
\begin{align}
&\widetilde{O} = \mathcal{O}\bigg(2^{\mathcal{O}(\log(W))}\cdot |\varphi_{m}^1| \cdot \kappa(\varphi_{m}^1) \notag \\
&\hspace{20mm}+ 2^{\mathcal{O}((N-1) \log(W))} \cdot |\varphi_{m}^2| \cdot \kappa(\varphi_{m}^2)\bigg).
\end{align}
The best case complexity of the proposed framework is when every agent is dependent to at most one other agent. Formally, if $N$ is odd number then $|C_{\ell'}| = 1$ for only one $\ell' \in \mathbb{M}$ and $|C_\ell| = 2, \forall \ell \in \mathbb{M} \backslash \{\ell'\}$. In this case, the best case complexity is in the class:
\begin{align}
&\bar{O} = \mathcal{O}\bigg( \sum_{\ell \in \mathbb{M} \backslash \{\ell'\}}^{} [2^{\mathcal{O}((N-1) \log(W))} \cdot |\varphi_{m}^\ell| \cdot \kappa(\varphi_{m}^\ell)] \notag \\
&\hspace{30mm}+ 2^{\mathcal{O}(\log(W))} \cdot |\varphi_{m}^{\ell'}| \cdot \kappa(\varphi_{m}^{\ell'}) \bigg). \notag
\end{align}
If $N$ is even number, then previous summation in performed in all the elements $\ell \in \mathbb{M}$. In total, it holds that: $\bar{O} < \widetilde{O} < O$ which verifies that our proposed framework achieves significantly better computational complexity than the centralized one.
\section{Conclusions and Future Work} \label{sec: conclusions}
We have proposed a systematic method for designing control policies for multi-agent systems. We assume that the system is under the presence of model uncertainties and actuation failures, thus the modeling is performed through MDPs. The agents are divided into dependency clusters which indicate the team of agents that they need to share an action in order to achieve a desired task. With the proposed framework, each agent is guaranteed to perform a task given in PCTL formulas. The computational complexity of the proposed framework is significantly better than the complexity of the centralized framework. Future efforts will be devoted towards performing the abstraction of the stochastic system which is given according to Assumption \ref{assumption:basic_assumption}.
\bibliographystyle{plainnat}
| -48,980.173885 |
[
-3.26171875,
3
] | 46.292135 |
[
-3.92578125,
-0.12078857421875,
-2.087890625,
-6.63671875,
-0.361328125,
8.734375
] |
[
3.068359375,
8.203125,
1.662109375,
7.265625
] | 311 | 6,424 |
[
-3.54296875,
4.18359375
] | 30.919069 |
[
-6.56640625,
-4.671875,
-5.07421875,
-2.2265625,
2.630859375,
13.015625
] | 0.990889 | 23.61444 | 20.392279 | 2.220706 |
[
1.6136587858200073
] | -30,170.582709 | 5.655511 | -48,604.315862 | 0.418375 | 5.944524 |
[
-2.857421875,
-3.7265625,
-4.1171875,
-5.33203125,
2.630859375,
12.6484375
] |
[
-6.60546875,
-2.654296875,
-2.578125,
-1.900390625,
4.3671875,
5.4453125
] | |
BkiUfufxK3YB9i3RN54o
|
\section{Introduction}
It is well-known that Dzyaloshinskii-Moriya interaction~\cite{dzyaloshinsky1958, moriya1960} (DMI) can lead to non-collinear magnetic structures~\cite{dzyaloshinsky1958, dzyaloshinskii1964}. Despite many years passed since the first observation of helical structures in non-centrosymmetric magnets~\cite{ludgren1970}, this type of compounds still attracts significant attention. In particular, it is stimulated by their omnifarious phase diagrams, which include regions hosting topologically nontrivial phases~\cite{bogdanov1989,bogdanov1994,muhlbauer2009}. Moreover, isolated skyrmions and their ordered arrays (skyrmion lattices, SkLs) have several promising technological applications~\cite{fert2013}. These magnetic structures can be also stabilized in layered nanostructures with interfacial DMI (iDMI)~\cite{fert1980, fert2017}.
In this context, developing of characterization methods of skyrmion-hosting materials becomes more and more demanded by material science and magnetism communities. Recently, the inelastic small-angle neutron scattering (SANS) on spin waves~\cite{toperverg1983,okorokov1986} (called SWSANS below) was shown to be fruitful for various chiral cubic helimagnets (e.g., MnSi) studies in fully-polarized by external field phase~\cite{grig2015,siegfried2017,grig2018jmmm,grig2018prb,ukleev2022}. Using this method, important information about spin-wave dynamics can be obtained. The latter is based on the theoretical prediction by Kataoka~\cite{kataoka1987}. In simple words, the SWSANS intensity is distributed on a three-dimensional sphere, which results in a circular spot when projected on a detector plane in the experiment. The radius of the spot is related to spin-wave stiffness and energy gap in the spectrum, whereas the center corresponds to the spiral vector. Moreover, even magnon damping can be studied using this technique, see Refs.~\cite{deriglazov1992, ukleev2022}.
In another type of non-centrosymmetric systems, so-called polar magnets, DMI favors cycloidal magnetic structures and N\'{e}el skyrmions and SkLs, which were indeed observed in Refs.~\cite{kezsmarki2015neel,kurumaji2017}. The same is also true for thin films with iDMI. Symmetry of such systems is $C_{nv}$ which is lower than the cubic one. So, we can expect different inelastic SANS maps with conventional B20 helimagnets picture.
In the present research, we show that the SWSANS technique can be also used for characterization of the compounds with interfacial-like DMI. Combining different experimental geometries gives a possibility for a comprehensive description of the system properties, such as spin-wave stiffness constants for in- and out-of-plane directions, DMI constant and single-ion anisotropy. These parameters are analytically connected with the SWSANS cutoff curves in different azimuthal directions (not usually circles, in contrast to the cubic helimagnets) in four various cases.
The rest of the paper is organised as follows. In Sec.~\ref{SecModel} we present the model under investigation. Its properties are briefly summarized in Sec.~\ref{SecLow}. Sec.~\ref{SecFP} addresses the magnon spectrum in the fully polarized by external field phase for the two cases of in-plane and perpendicular field. Small-angle neutron scattering in the fully polarized phase is discussed in Sec.~\ref{SecSANS}, where four particular geometries of the experiment are proposed and analyzed theoretically. Sec.~\ref{SecConc} contains discussion related to thin films and our conclusions.
\section{Model}
\label{SecModel}
We consider a spin Hamiltonian, which includes exchange coupling, Dzyaloshinskii-Moriya interaction, single-ion anisotropy, and Zeeman energy:
\begin{eqnarray} \label{ham1}
\mathcal{H} &=& \mathcal{H}_{\textrm{EX}} + \mathcal{H}_{\textrm{DMI}} + \mathcal{H}_{\textrm{AN}} + \mathcal{H}_\textrm{Z}, \nonumber \\
\mathcal{H}_{\textrm{EX}} &=& -\frac{1}{2} \sum_{\mathbf{R},\mathbf{R}^\prime} J_{\mathbf{R}-\mathbf{R}^\prime} \mathbf{S}_\mathbf{R} \cdot \mathbf{S}_{\mathbf{R}^\prime}, \nonumber \\
\mathcal{H}_{\textrm{DMI}} &=& \frac12 \sum_{\mathbf{R},\mathbf{R}^\prime} \mathbf{D}_{\mathbf{R} - \mathbf{R}^\prime} \cdot \left[ \mathbf{S}_\mathbf{R} \times \mathbf{S}_{\mathbf{R}^\prime} \right], \\
\mathcal{H}_{\textrm{AN}} &=& - K \sum_{\mathbf{R}} \left(S^z_\mathbf{R}\right)^2, \nonumber \\
\mathcal{H}_\textrm{Z} &=& - \mathbf{h} \cdot \left( \sum_\mathbf{R} \mathbf{S}_\mathbf{R} \right). \nonumber
\end{eqnarray}
Here $\mathbf{R}$ enumerates all spins, which are arranged for definiteness in a simple tetragonal lattice with $C_{4v}$ symmetry. We choose $z$ axis to be a high-symmetry direction. According to $\mathcal{H}_{\textrm{AN}}$, it is the easy axis for $K>0$ and the hard one for $K<0$ (the easy plane case). In Zeeman term, we use standard description of the magnetic field in energy units, so $\mathbf{h} = - g \mu_B \mathbf{H}$. DMI in the present system deserves more detailed discussion. In general, $C_{nv}$ symmetry allows for the following Lifshitz invariant in energy density~\cite{bogdanov1989} ($\mathbf{M}$ is the magnetization):
\begin{eqnarray}
W = \gamma \left[ M_z \frac{\partial M_x}{\partial x} - M_x \frac{\partial M_z}{\partial x} + M_z \frac{\partial M_y}{\partial y} - M_y \frac{\partial M_z}{\partial y}\right]
\end{eqnarray}
Microscopically, this form can be reproduced by the scheme shown in Fig.~\ref{figDMI}. Note, that this type of DMI is relevant to interfacial DMI, which arises due to symmetry breaking on the boundary between magnetic material and another material~\cite{fert1980} (usually heavy metal with large spin-orbit coupling). We also would like to point out that in our model we neglect magnetodipolar interaction, which can be important in multilayered systems with iDMI and can lead to a hybrid Bloch/N\'{e}el-type helicoids~\cite{legrand2018}. However, we focus on the effect of the iDMI on SWSANS spectra, where the dipolar forces are expected to play minor role.
\begin{figure}
\centering
\includegraphics[width=4cm]{DMI1.pdf}\\
\caption{Scheme of Dzyaloshinskii-Moriya interaction considered in the present study. DMI is allowed in $xy$ plane perpendicular to the high-symmetry $z$ axis. For the central magnetic ion vectors of DMI interactions with nearest neighbors are shown by the arrows. In the thin film case, $xy$ plane is parallel to interfaces, which break the inversion symmetry. }\label{figDMI}
\end{figure}
Subsequent discussion is much easier in the reciprocal space, so we introduce the Fourier transform on the lattice
\begin{eqnarray} \label{fourier1}
\mathbf{S}_\mathbf{R} = \frac{1}{\sqrt{N}} \sum_{\mathbf{q}} \mathbf{S}_\mathbf{q} e^{i \mathbf{q} \cdot \mathbf{R}},
\end{eqnarray}
where $N$ is a number of the lattice sites. Then, the counterparts of various interaction in the Hamiltonian~\eqref{ham1} read
\begin{eqnarray} \label{ham2}
\mathcal{H}_{\textrm{EX}} &=& - \frac12 \sum_{\mathbf{q}} J_{\mathbf{q}} \mathbf{S}_\mathbf{q} \cdot \mathbf{S}_{-\mathbf{q}}, \nonumber \\
\mathcal{H}_{\textrm{DMI}} &=& \frac12 \sum_{\mathbf{q}} \mathbf{D}_{\mathbf{q}} \cdot [\mathbf{S}_\mathbf{q} \times \mathbf{S}_{-\mathbf{q}}], \nonumber \\
\mathcal{H}_{\textrm{AN}} &=& - K \sum_{\mathbf{q}} S^z_\mathbf{q} S^z_{-\mathbf{q}}, \\
\mathcal{H}_{\textrm{Z}} &=& -\sqrt{N} \mathbf{h} \cdot \mathbf{S}_\mathbf{0}. \nonumber
\end{eqnarray}
Below we assume, that the exchange is ferromagnetic and for small $q$ it can be expanded as follows:
\begin{eqnarray} \label{stiff}
J_\mathbf{q} \approx J_\mathbf{0} - \frac{A_\perp (q^2_x + q^2_y)}{S} - \frac{A_\parallel q^2_z}{S}.
\end{eqnarray}
Here $A_\perp$ and $A_\parallel$ are corresponding spin-waves stiffnesses. Note also that in long-wavelength limit symmetry-allowed anisotropic exchange terms with sufficient accuracy can be mimicked by the single-ion anisotropy. However, anisotropic exchange can be responsible for some fine effects, e.g., the modulation vector modulus dependence on the spin structure orientation. It was directly shown in FeGe in Ref.~\cite{ukleev2021}.
Fourier transform of DMI in the nearest-neighbors approximation is given by
\begin{eqnarray}
\mathbf{D}_\mathbf{q} = i D \sum_{b} \sin{\mathbf{q} \cdot \mathbf{b}} [\hat{z} \times \mathbf{b}] \approx i D \sum_{b} \mathbf{q} \cdot \mathbf{b} [\hat{z} \times \mathbf{b}],
\label{dq1}
\end{eqnarray}
where $b$ stands for all the bonds with nearest-neighbors of a certain spin, see Fig.~\ref{figDMI}. This formula can be further simplified using the property ($\alpha$ and $\beta$ vector components $x,y,z$)
\begin{eqnarray}
\sum_b b_\alpha b_\beta = 2 \delta_{\alpha\beta}.
\end{eqnarray}
So, in the case of tetragonal lattice, we have Eq.~\eqref{dq1} in the form
\begin{eqnarray} \label{dq2}
\mathbf{D}_\mathbf{q} = 2 i D [\hat{z} \times \mathbf{q}].
\end{eqnarray}
Note that for hexagonal lattice it also holds but with factor $3$ instead of $2$, so the discussion above can be easily applied in this case too. Evidently, the Fourier transform of DMI is lying in $xy$ plane and is perpendicular to the corresponding momentum.
\section{Moderate fields}
\label{SecLow}
We start our analysis of the above model from the relatively small fields case, where at $T=0$ non-collinear spin structures (cycloids or N\'{e}el skyrmions) are possible. We also neglect the single-ion anisotropy term in Hamiltonian~\eqref{ham1}. For a single-modulated structures description we use the Kaplan helix representation~\cite{Kaplan1961}:
\begin{eqnarray}
\label{kaplan}
\mathbf{S}_\mathbf{R} = S \left( \mathbf{A} e^{i \mathbf{k} \cdot \mathbf{R}} + \mathbf{A}^* e^{- i \mathbf{k} \cdot \mathbf{R}} \right) \cos{\alpha} + S \hat{c} \sin{\alpha}.
\end{eqnarray}
Here $\alpha$ is the cone angle, $\mathbf{A} = (\hat{a} - i \hat{b})/2, \, \mathbf{A}^* = (\hat{a} + i \hat{b})/2$, $\hat{a},\hat{b},\hat{c}$ is some orthogonal basis. Well-known particular cases are a screw spiral with modulation vector $\mathbf{k} \parallel \hat{c}$ and a cycloid with $\mathbf{k} \perp \hat{c}$. Useful properties for calculations are the following: $\mathbf{A}^2=0, \, \mathbf{A} \cdot \mathbf{A}^* = 1/2, \, \hat{c} \times \mathbf{A} = i \mathbf{A}, \, \mathbf{A} \times \mathbf{A}^* = i \hat{c}/2$. From Eq.~\eqref{kaplan} one can deduce
\begin{eqnarray}
\mathbf{S}_\mathbf{k} &=& \sqrt{N} S \mathbf{A} \cos{\alpha} , \nonumber \\ \mathbf{S}_{-\mathbf{k}} &=& \sqrt{N} S \mathbf{A}^* \cos{\alpha}, \\ \mathbf{S}_\mathbf{0} &=& \sqrt{N} S \hat{c} \sin{\alpha}. \nonumber
\end{eqnarray}
All the harmonics with other $\mathbf{q}$ are zero. Plugging these formulas into Hamiltonian~\eqref{ham2}, one can calculate the spin structure energy per one spin
\begin{eqnarray} \label{ekaplan1}
E &=& - \frac{S^2}{2} \left( J_\mathbf{k} \cos^2{\alpha} + J_\mathbf{0} \sin^2{\alpha} \right) - \frac{i S^2}{2} \mathbf{D}_\mathbf{k} \cdot \hat{c} \cos^2{\alpha} \nonumber \\ &&- S \mathbf{h}\cdot \hat{c} \sin{\alpha}.
\end{eqnarray}
For small $k$ we can rewrite it using Eqs.~\eqref{stiff} and~\eqref{dq2} as follows:
\begin{eqnarray} \label{ekaplan2}
E &=& - \frac{S^2}{2} J_\mathbf{0} + \frac{S (A_\perp k^2_\perp + A_\parallel k^2_\parallel) \cos^2{\alpha}}{2} \nonumber \\
&& - S^2 D \hat{c} \cdot [\mathbf{k} \times \hat{z}] \cos^2{\alpha} - S \mathbf{h}\cdot \hat{c} \sin{\alpha}.
\end{eqnarray}
Here we divide modulation vector onto two parts, $k_\parallel$ is along $z$ axis and $k_\perp$ lies in $xy$ plane. Evidently, minimal $E$ requires $k_\parallel = 0$. For in-plane magnetic fields we have $\hat{c}$ along $\mathbf{h}$ and $\mathbf{k}$ along $D \hat{z} \times \mathbf{h}$ (its direction is dependent on the sign of $D$). So, in general, the solution is the conical cycloid, which energy~\eqref{ekaplan2} can be rewritten as
\begin{eqnarray} \label{ekaplan3}
E &=& - \frac{S^2}{2} J_\mathbf{0} + \frac{S \cos^2{\alpha}}{2}(A k^2_\perp - 2 D k_\perp) - S h \sin{\alpha}. \nonumber \\
\end{eqnarray}
Its minimization with respect to $\alpha$ and $k_\perp$ yields
\begin{eqnarray} \label{sp1}
k_\perp &=& \frac{S D}{A_\perp} \equiv k, \\ \label{alpha1}
\sin{\alpha} &=& \frac{h}{h_{C2}}, \, h < h_{C2}, \\ \label{hc21}
h_{C2} &=& A_\perp k^2.
\end{eqnarray}
At fields $h \geq h_{C2}$ the system is in fully polarized phase. Notation $C2$ is related to the fact, that at very small fields in-plane anisotropy and higher order terms in Eq.~\eqref{dq1} play important role in the cycloid orientation determination~\cite{maleyev2019}, and the regime $\hat{c} \parallel \mathbf{h}$ is correct for moderate fields \mbox{$h>h_{C1}$} only.
In the case of iDMI (e.g., thin film of a ferromagnet with neighboring nonmagnetic materials), the equations above should be modified. Lets consider system with M layers of magnetic material and two interfaces with DMI constants $D_1$ and $D_2$. Due to opposite directions of perpendicular to interfaces vectors, the effective DMI reads
\begin{eqnarray}
D_{eff} = \frac{D_1-D_2}{M}.
\end{eqnarray}
which should be used in the energy function~\eqref{ekaplan1}.
All the layers feel the exchange stiffness and the external field, but the DMI is only on the interfaces. Then, the parameters of the cycloid solution are altered according to the following formulas:
\begin{eqnarray} \label{sp2}
k &=& \frac{S D_{eff}}{A_\perp}, \\
\sin{\alpha} &=& \frac{h}{h_{C2}}, \, h < h_{C2}, \\
h_{C2} &=& A_\perp k^2 = \frac{S^2 D^2_{eff}}{A_\perp}.
\end{eqnarray}
We see, that they are identical to the bulk ones~\eqref{sp1},~\eqref{alpha1}, and~\eqref{hc21} upon the substitution $D \rightarrow D_{eff}$.
\section{Fully polarized phase}
\label{SecFP}
\subsection{In-plane field}
Here we consider spin waves spectrum in relatively large in-plane external field $h \geq h_{C2}$. We neglect small in-plane anisotropic terms, so the result is independent on the magnetic field orientation in $xy$ plane. We choose $x$ axis along $\mathbf{h}$ and use the following approximate Holstein-Primakoff spin operators representation~\cite{Holstein1940}:
\begin{eqnarray} \label{spinrep}
S^x_\mathbf{R} &=& S - a^\dagger_\mathbf{R} a_\mathbf{R}, \nonumber \\
S^y_\mathbf{R} &=& \sqrt{\frac{S}{2}} \left(a^\dagger_\mathbf{R} + a_\mathbf{R}\right), \\
S^z_\mathbf{R} &=& i \sqrt{\frac{S}{2}} \left(a^\dagger_\mathbf{R} - a_\mathbf{R}\right).
\end{eqnarray}
The Fourier transform of magnon creation-annihilation operators reads [cf. Eq.~\eqref{fourier1}]
\begin{eqnarray} \label{fourier2}
a_\mathbf{R} = \frac{1}{\sqrt{N}} \sum_{\mathbf{q}} a_\mathbf{q} e^{i \mathbf{q} \cdot \mathbf{R}}, \quad
a^\dagger_\mathbf{R} = \frac{1}{\sqrt{N}} \sum_{\mathbf{q}} a^\dagger_\mathbf{q} e^{-i \mathbf{q} \cdot \mathbf{R}}.
\end{eqnarray}
Using this formulas, we obtain spin operators components in reciprocal space in the form
\begin{eqnarray} \label{spinrep2}
S^x_\mathbf{q} &=& \sqrt{N} S \delta_{\mathbf{q},\mathbf{0}} - \frac{1}{\sqrt{N}} \sum_\mathbf{p} a^\dagger_\mathbf{p} a_{\mathbf{p} + \mathbf{q}}, \nonumber \\
S^y_\mathbf{q} &=& \sqrt{\frac{S}{2}} \left(a^\dagger_{-\mathbf{q}} + a_\mathbf{q}\right), \\
S^z_\mathbf{q} &=& i \sqrt{\frac{S}{2}} \left(a^\dagger_{-\mathbf{q}}- a_\mathbf{q}\right). \nonumber
\end{eqnarray}
Next, one can calculate bilinear in Bose-operators part of Hamiltonian~\eqref{ham2}, which reads
\begin{eqnarray} \label{hamb1}
\mathcal{H}^{(2)}_{\textrm{EX}} &=& S \sum_{\mathbf{q}} \left( J_\mathbf{0} - J_\mathbf{q} \right) a^\dagger_\mathbf{q} a_{\mathbf{q}}, \nonumber \\
\mathcal{H}^{(2)}_{\textrm{DMI}} &=& - 2 S D\sum_{\mathbf{q}} q_y a^\dagger_\mathbf{q} a_{\mathbf{q}}, \\
\mathcal{H}^{(2)}_{\textrm{AN}} &=& - S K \sum_{\mathbf{q}} \left( a^\dagger_\mathbf{q} a_{\mathbf{q}} + \frac{a_{\mathbf{q}} a_{-\mathbf{q}} + a^\dagger_{\mathbf{q}} a^\dagger_{-\mathbf{q}}}{2} \right), \nonumber \\
\mathcal{H}^{(2)}_{\textrm{Z}} &=& h \sum_{\mathbf{q}} a^\dagger_\mathbf{q} a_{\mathbf{q}}. \nonumber
\end{eqnarray}
So, without contribution from the anisotropy, the result is very simple:
\begin{eqnarray} \label{hamb2}
\mathcal{H}^{(2)} &=& \sum_\mathbf{q} \left[ S\left( J_\mathbf{0} - J_\mathbf{q} \right) - 2 S D q_y + h \right] a^\dagger_\mathbf{q} a_{\mathbf{q}} \nonumber \\ &\equiv& \sum_\mathbf{q} \varepsilon_\mathbf{q} a^\dagger_\mathbf{q} a_{\mathbf{q}},
\end{eqnarray}
where $\varepsilon_\mathbf{q}$ is the magnon energy. For small $q$ it can be further simplified,
\begin{eqnarray} \label{specin}
\varepsilon_\mathbf{q} &=& A_\perp \left[q^2_x + (q_y - k)^2 \right] + A_\parallel q^2_z + h - h_{C2} \Longleftrightarrow \nonumber \\
\varepsilon_\mathbf{q} &=& A_\perp (\mathbf{q}_\perp - \mathbf{k})^2 + A_\parallel q^2_z + \Delta.
\end{eqnarray}
This result can be compared with the one by Kataoka for B20 helimagnets~\cite{kataoka1987}. The similarity is that the gap $\Delta = h - h_{C2}$ and the spectrum is non-reciprocal ($\varepsilon_\mathbf{q} \neq \varepsilon_{-\mathbf{q}}$) due to DMI. However, the spectrum minimum lies in perpendicular to $\mathbf{h}$ direction (along $\mathbf{k}$ -- the cycloid modulation vector).
\subsection{The role of magnetic anisotropy in spin wave spectrum}
Taking into account $\mathcal{H}^{(2)}_{\textrm{AN}}$ term in Eq.~\eqref{hamb1}, we obtain the bilinear part of the Hamiltonian for small $q$ in the following form [cf. Eq.~\eqref{hamb2}]:
\begin{eqnarray}
\mathcal{H}^{(2)} &=& \sum_\mathbf{q} \left( E_\mathbf{q} a^\dagger_\mathbf{q} a_{\mathbf{q}} + B_\mathbf{q} \frac{a_{\mathbf{q}} a_{-\mathbf{q}} + a^\dagger_{\mathbf{q}} a^\dagger_{-\mathbf{q}}}{2} \right), \nonumber \\
E_\mathbf{q} &=& A_\perp \left[q^2_x + (q_y - k)^2 \right] + A_\parallel q^2_z + h - h_{C2} - S K , \nonumber \\
B_\mathbf{q} &=& - S K.
\end{eqnarray}
Then, the Bogoliubov transformation~\cite{Holstein1940} can be used to obtain the magnon spectrum:
\begin{eqnarray} \label{specinan}
\varepsilon^\prime_\mathbf{q} &=& \sqrt{E^2_\mathbf{q} - B^2_\mathbf{q}} = \sqrt{(\varepsilon_\mathbf{q} - 2 S K)\varepsilon_\mathbf{q}},
\end{eqnarray}
where $\varepsilon_\mathbf{q}$ is the spectrum without the anisotropy~\eqref{specin}. First, we see, that the anisotropy renormalizes the critical field which is now given by \mbox{$h^\prime_{C2} = A_\perp k^2 + 2 S K$}. Second, for $\varepsilon_\mathbf{q} \gg S |K|$ ($\mathbf{q}$ not very close to $\mathbf{k}$ or magnetic field not very close to $h_{C2}$) we obtain
\begin{eqnarray} \label{specinan2}
\varepsilon^\prime_\mathbf{q} &\approx& \varepsilon_\mathbf{q} -S K,
\end{eqnarray}
which can be considered as an effective gap renormalization $\Delta \rightarrow \Delta - S K$.
\subsection{Perpendicular field $\mathbf{h} \parallel \hat{z}$}
It is also useful to consider the spectrum of the fully polarized phase in magnetic field along $\hat{z}$. In this case, in spin operators quantization rules~\eqref{spinrep2} one should make the following substitutions $x \rightarrow z, \, y \rightarrow x, \, z \rightarrow y $. It is easy to see that after this substitutions, DMI includes only terms with odd number of Bose-operators, because cross product always includes $S^z$ component [see Eqs.~\eqref{ham2},~\eqref{dq2}, and~\eqref{spinrep2}]. Moreover, linear terms vanish because $\mathbf{D}_\mathbf{0}=\mathbf{0}$. So, in the linear spin-wave theory DMI does not influence magnon spectrum. However, in contrast with usual ferromagnets there will be some quantum corrections to the spectrum even at $T=0$ due to DMI.
Next, the contribution from the single-ion anisotropy reads
\begin{eqnarray}
\mathcal{H}^{(2)}_\textrm{AN} = 2 S K \sum_\mathbf{q} a^\dagger_\mathbf{q} a_{\mathbf{q}},
\end{eqnarray}
which along with terms from exchange coupling and Zeeman term [see Eq.~\eqref{hamb1}] gives the magnon spectrum
\begin{eqnarray} \label{specperp}
\varepsilon_\mathbf{q} &=& A_\perp q^2_\perp + A_\parallel q^2_\parallel + h + 2 S K \nonumber \\ &\equiv& A_\perp q^2_\perp + A_\parallel q^2_\parallel +\Delta^\prime.
\end{eqnarray}
It indicates that even at $h=0$ perpendicular collinear phase can be observed as metastable for the easy axis anisotropy, whereas in the easy plane case one needs $h \geq 2 S |K|$. We denote the gap here as $\Delta^\prime$ to avoid confusion with the different one for the in-plane field ($\Delta$), see Eq.~\eqref{specin}.
\section{Small-angle neutron scattering in fully polarized phase}
\label{SecSANS}
Here we discuss how inelastic small-angle neutron scattering on magnons can be used to obtain the corresponding material parameters from the experiment. In particular, we derive expressions for the so-called cutoff angle in four different experimental geometries. Note that the equations below are written for unpolarized neutrons, however, generalization to the polarized case is straightforward~\cite{maleev2002}. The latter can be important to pinpoint magnetic scattering~\cite{toperverg1983,okorokov1986}. We also would like to point out, that the sign of $D$ can be only determined with the polarized neutrons~\cite{maleyev1995,grigoriev2009}.
So, in our case, we use the following equation for neutron scattering cross-section (see, e.g., Ref.~\cite{maleev2002}):
\begin{eqnarray} \label{sigma1}
\sigma(\o,\mathbf{Q}) &=& \frac{1}{\pi} \frac{k_f}{k_i} \left[ 1 - \exp{\left( -\frac{\o}{T} \right)}\right]^{-1} r^2 |F_m|^2 \nonumber \\ && \mathrm{Im} \, \chi^{(S)}_{\alpha \beta}(\o,\mathbf{Q}) (\delta_{\alpha\beta} - \hat{Q}_\alpha \hat{Q}_\beta).
\end{eqnarray}
Here $k_i(k_f)$ is incident (scattered) neutron momentum, $\o$ and $\mathbf{Q}$ are transferred energy and momentum, respectively, $r$ is the classical electron radius, $F_m$ is the magnetic form factor of the ions and $\chi^{(S)}$ is the symmetric part of magnetic susceptibility. The latter in our case reads (diagonal components of the transverse susceptibility)
\begin{eqnarray}
\mathrm{Im}\,\chi^\perp(\o,\mathbf{Q}) = \frac{\pi \langle S \rangle}{2} \left[ \delta(\o-\varepsilon_\mathbf{Q}) - \delta(\o+\varepsilon_{-\mathbf{Q}}) \right],
\end{eqnarray}
where the terms in brackets correspond to emission and absorbtion of magnons and $\langle S \rangle$ is thermodynamical average of the spin.
Next, as in Ref.~\cite{grig2015}, we consider the case $\o \ll T$ and replace $\left[ 1 - \exp{\left( -\o/T \right)}\right]^{-1}$ by $T/\o$ in Eq.~\eqref{sigma1}. Moreover, $k_f \approx k_i$ because the momentum transfer is small. Finally, in SWSANS experiments one should average the cross-section over $\omega$, so we obtain
\begin{eqnarray} \label{sigma2}
\sigma(\mathbf{Q}) &\propto& \langle S \rangle T \int \frac{d \o}{\o} \left( 1 + \frac{\hat{\mathbf{Q}} \cdot \mathbf{h}}{h} \right) \\ \nonumber &&\left[ \delta(\o-\varepsilon_\mathbf{Q}) - \delta(\o+\varepsilon_{-\mathbf{Q}}) \right].
\end{eqnarray}
Here we explicitly use the fact that the sample is magnetized along the field. Particular scattering and magnetic field geometries are considered separately below. In the obtained results, three dimensionless parameters are important:
\begin{eqnarray} \label{Aanis}
a &=& \frac{A_\parallel}{A_\perp}, \\
\t_0 &=& \frac{E_i}{A_\parallel k^2_i}, \\
\t_B &=& \frac{k}{k_i}.
\end{eqnarray}
The first one measures the stiffness anisotropy, the second is the ratio of the incident neutron energy to the certain characteristic magnetic energy and the third one is so-called Bragg angle, indicating position of the scattering peak from an incommensurate order in the detector plane.
The findings of subsections below are summarized in Fig.~\eqref{figmaps}.
\begin{figure}
\centering
\includegraphics[width=4.2cm]{ScatXYa.png}
\hfill
\includegraphics[width=4.2cm]{ScatYZb.png}
\vspace{1cm}
\includegraphics[width=4.2cm]{ScatXYc.png}
\hfill
\includegraphics[width=4.2cm]{ScatYZd.png}
\caption{Sketches of SWSANS maps in four different geometries. Stiffness anisotropy parameter $a=0.64$ [see Eq.~\eqref{Aanis}] is used. In the case of $\mathbf{H} \parallel \hat{z}$ [(a) and (b), perpendicular field] the magnon spectrum is reciprocal and the scattering is ferromagnetic-like. Moreover, in panel (b) ellipticity of the cutoff angle related to $a$ is pronounced. When $\mathbf{H} \parallel \hat{x}$ (in-plane field) the magnon spectrum is non-reciprocal, and the SANS signals are centered at the Bragg angles [(c) and (d)]. For the scattering in $yz$ plane the ellipticity is also visible. In the case of polarized neutrons, in panels (c) and (d) contributions centered at $\pm \t_B$ has unequal intensities; the difference being $\propto Q^2_x/\mathbf{Q}^2$, which can be used to determine the sign of $D$. }\label{figmaps}
\end{figure}
\subsection{Scattering in $xy$ plane, $\mathbf{h} \parallel \hat{z}$}
In this case, the transferred momentum can be written in the following way:
\begin{eqnarray} \label{Q1}
\mathbf{Q} = k_i \left(\t_x,\t_y,\frac{\o}{2 E_i} \right),
\end{eqnarray}
where $E_i$ is energy of the incident neutron. Under conditions of SANS experiment, $Q \ll k_i$ and $\t_x, \t_y$ measure a point in a detector (scattering ``angles'').
Next, delta-functions in Eq.~\eqref{sigma2} determine so-called cutoff in the detector plane at which the cross-section diverges (in integrable way) and after which the signal is zero (however, it lasts beyond the cutoff due to finite magnon lifetime, see Ref.~\cite{ukleev2022} for the corresponding theory). The cutoff can be obtained by considering solutions of equation $\o = \varepsilon_{\mathbf{Q}}$ with respect to $\o$ (equation $\o + \varepsilon_{-\mathbf{Q}} =0$ yields the same physics). It is convenient to rewrite this equation using the variable $t = \o/2 E_i$, which after some transformations [we use the spectrum~\eqref{specperp}] reads
\begin{eqnarray}
t^2 - 2 \t_0 t + \frac{\Delta^\prime}{A_\parallel k^2_i} + \frac{\t^2_x + \t^2_y}{a}=0.
\end{eqnarray}
So, in this case the standard ferromagnetic-like picture arises: the scattering is located in a circle centered at $(\t_x,\t_y)=0$ and bounded by the cutoff satisfying the condition:
\begin{eqnarray}
\frac{\t^2_C}{a} = \t^2_0 - \frac{\Delta^\prime}{A_\parallel k^2_i}.
\end{eqnarray}
Under the magnetic field growth $\t^2_C$ linearly decreases (general property for all geometries considered in the present study), which can be used for the experimental data interpretation.
In Eq.~\eqref{sigma2} the following substitution should be done
\begin{eqnarray}
1 + \frac{\hat{\mathbf{Q}} \cdot \mathbf{h}}{h} \rightarrow 1+ \frac{(\o/2E_i)^2 }{\t^2_x + \t^2_y + (\o/2E_i)^2 }.
\end{eqnarray}
Importantly, one can see that it is isotropic in $\t_x \t_y$ plane.
\subsection{Scattering in $xz$ plane, $\mathbf{h} \parallel \hat{z}$}
In this case we have [cf. Eq.~\eqref{Q1}]
\begin{eqnarray} \label{Q2}
\mathbf{Q} = k_i \left(\t_x, \frac{\o}{2 E_i}, \t_z \right),
\end{eqnarray}
Equation $\o = \varepsilon_{\mathbf{Q}}$ is equivalent to
\begin{eqnarray}
t^2 - 2 a \t_0 t + \frac{\Delta^\prime}{A_\perp k^2_i} + \t^2_x + a \t^2_z=0.
\end{eqnarray}
So, the cutoff line in $\t_x\t_z$ plane here is an ellipse. Explicitly:
\begin{eqnarray}
\t^2_x + a \t^2_z = a^2 \t^2_0 - \frac{\Delta^\prime}{A_\perp k^2_i}.
\end{eqnarray}
Importantly, the ratio of this ellipse semi axes can be directly used for the parameter $a$ determination. Furthermore, the cross-section~\eqref{sigma2} acquires weak angular dependence,
\begin{eqnarray}
1 + \frac{\hat{\mathbf{Q}} \cdot \mathbf{h}}{h} \rightarrow 1+ \frac{\t^2_z}{\t^2_x + (\o/2E_i)^2 + \t^2_z }.
\end{eqnarray}
\subsection{Scattering in $xy$ plane, $\mathbf{h} \parallel \hat{x}$}
In this case, the magnon spectrum becomes non-reciprocal [see Eq.~\eqref{specin}]. So, the scattering patterns from $\o = \varepsilon_{\mathbf{Q}}$ and from $\o + \varepsilon_{-\mathbf{Q}} =0$ are centered in $(0, \t_B)$ and $(0, -\t_B)$, respectively, for magnetic field along $x$ axis. The result is a superposition of these two contributions, which can overlap. For brevity, below we consider contribution from $\o = \varepsilon_{\mathbf{Q}}$ only; its counterpart can be obtained in a straightforward way.
In this geometry we have
\begin{eqnarray} \label{Q3}
\mathbf{Q} = k_i \left(\t_x, \t_y, \frac{\o}{2 E_i} \right),
\end{eqnarray}
however, the spectrum is given by Eq.~\eqref{specin} or by Eq.~\eqref{specinan2} if the single-ion anisotropy contribution is taken into account. It is convenient to use
\begin{eqnarray}
\t_{rel} = \sqrt{\t^2_x + (\t_y - \t_B)^2},
\end{eqnarray}
which is the distance with respect to the Bragg angle in the detector plane. After some calculation we have the following equation for $t$:
\begin{eqnarray}
t^2 - 2 \t_0 t + \frac{\Delta}{A_\parallel k^2_i} + \frac{\t^2_{rel}}{a}=0.
\end{eqnarray}
Hence, the cutoff is given by
\begin{eqnarray}
\frac{\t^2_{relC}}{a} = \t^2_0 - \frac{\Delta}{A_\parallel k^2_i}.
\end{eqnarray}
In the cross-section~\eqref{sigma2} angle-dependent factor emerges from $1 + \hat{\mathbf{Q}} \cdot \mathbf{h}/h $ under the integration:
\begin{eqnarray}
1+ \frac{\t^2_x}{\t^2_x + \t^2_y + (\o/2E_i)^2 }.
\end{eqnarray}
\subsection{Scattering in $yz$ plane, $\mathbf{h} \parallel \hat{x}$}
\label{SANSd}
Here the transferred momentum is
\begin{eqnarray} \label{Q4}
\mathbf{Q} = k_i \left(\frac{\o}{2 E_i}, \t_y, \t_z \right).
\end{eqnarray}
Energy conservation law leads to
\begin{eqnarray}
t^2 - 2 a \t_0 t + \frac{\Delta}{A_\perp k^2_i} + (\t_y - \t_B)^2 + a \t^2_z=0.
\end{eqnarray}
So, the cutoff line is the ellipse centered in $(0,\t_B)$ and satisfying the following equation:
\begin{eqnarray}
(\t_y - \t_B)^2 + a \t^2_z = a^2 \t^2_0 - \frac{\Delta}{A_\perp k^2_i}.
\end{eqnarray}
In this case, in Eq.~\eqref{sigma2} the following substitution is in order
\begin{eqnarray}
1 + \frac{\hat{\mathbf{Q}} \cdot \mathbf{h}}{h} \rightarrow 1+ \frac{(\o/2E_i)^2 }{\t^2_y + \t^2_z + (\o/2E_i)^2 }.
\end{eqnarray}
\section{Discussion and conclusions}
\label{SecConc}
For thin films characterization various methods are usually used (see Ref.~\cite{kuepferling2020} and references therein). As it follows from the theoretical considerations, their properties in the fully polarized phase should be almost the same with bulk systems with the difference that the effective DMI should be used. However, the problem of weak signal in SWSANS measurements is expected to be crucial. Presumably, it can be overcome by stacking these layers with metallic spacers to increase scattering volume or using some off-specular scattering in reflective geometry (see, e.g., Refs.~\cite{lauter2006,zabel2007,toperverg2015}).
Another obstacle in the thin film case can be a possible breakdown of the parallel to high symmetry axis spin wave stiffness $A_\parallel$ and the corresponding momentum $q_z$ notation. For example, in a single film case magnon states size-quantization effect can become important. In this case, one would have (cf. Subsec.~\ref{SANSd}) transferred momentum in the form of Eq.~\eqref{Q4} and the following energy conservation law:
\begin{eqnarray}
t^2 - 2 \t^\prime_0 t + \frac{\Delta}{A_\perp k^2_i} + (\t_y - \t_B)^2=0,
\end{eqnarray}
where $\t^\prime_0 = E_i/ A_\perp k^2_i$ and the gap $\Delta$ can possible acquire some additional contribution due to the size quantization effect. This equation determines the cutoff along the $\t_y$ axis in the detector plane
\begin{eqnarray}
|\t_y - \t_B|\Bigr|_C = (\t^\prime_0)^2 - \frac{\Delta}{A_\perp k^2_i}.
\end{eqnarray}
At the same time, the signal $\t_z$ dependence is expected to have power-law decaying tails with the characteristic scale $\t_{zC} \sim 1/(k_i d)^2$, where $d$ is the magnetic film thickness.
To conclude, inelastic small-angle neutron scattering on magnons in the fully polarized phase (SWSANS) is proposed as a tool for determination of various parameters of ferromagnets with interfacial-like DMI, namely, spin wave stiffness along and perpendicular to high-symmetry axis, constant in DMI, and single-ion anisotropy. The method relies on the cutoff feature of SWSANS maps. The latter is theoretically connected with the system parameters in four various experimental geometries. We show that elliptical scattering patterns can be observed and the ratio between the corresponding semi axes yields the ratio between spin wave stiffnesses. Furthermore, for in-plane field, scattering patterns are centered at the so-called Bragg angle, which is straightly related to DMI magnitude. Finally, variation of the external magnetic field allows to quantify all the parameters listed above as well as the single-ion anisotropy.
\begin{acknowledgments}
The work is dedicated to blessed memory of S.V.\ Maleev who inspired this research. We are grateful to V.U.\ Ukleev for valuable discussions. The reported study was funded by the Russian Federation President Grant No. MK-1366.2021.1.2.
\end{acknowledgments}
| -37,052.854126 |
[
-2.5,
2.30078125
] | 33.994334 |
[
-3.630859375,
0.03631591796875,
-2.330078125,
-5.89453125,
-0.429443359375,
8.609375
] |
[
3.44140625,
8.359375,
3.9765625,
5.18359375
] | 237 | 3,828 |
[
-3.029296875,
3.474609375
] | 30.201301 |
[
-6.3984375,
-3.923828125,
-3.943359375,
-2.3203125,
2.171875,
11.8125
] | 0.724823 | 19.070794 | 29.022989 | 2.122711 |
[
1.6998871564865112
] | -23,190.515059 | 6.126959 | -36,617.802625 | 0.341093 | 6.035777 |
[
-2.763671875,
-3.892578125,
-4.03125,
-5.1171875,
2.611328125,
12.8125
] |
[
-5.6953125,
-2.34765625,
-2.6328125,
-1.8544921875,
3.859375,
5.07421875
] | |
BkiUdX85qdmDH5kXI0R3
|
\section{Introduction}
\label{sec:intro}
A general and consistent notion of energy or mass poses a difficulty in the context of general relativity. Due to the weak equivalence principle, the energy momentum distribution of the gravitational field is locally vanishing for a freely-falling observer moving along a geodesic. Because of these local considerations, quasilocal constructions were brought forward, see for instance \cite{Szabados:2004vb} and references therein. Amongst the candidates is a construction by Hawking \cite{Hawking:1968qt}, which will be referred to as Hawking energy in the following. Based on a closed spacelike 2-surface $S$ in spacetime, it phenomenologically aims to relate the energy/matter content enclosed by $S$ to the amount of light bending on $S$.
A sensible energy definition should match previously established concepts of energy in highly symmetric or asymptotic settings, such as the ADM- or Bondi-mass, but still be general enough to also apply to more general set-ups. A central challenge for many quasilocal energy definitions is to provide a sensible meaning to the concepts of positivity and monotonicity in a physically realistic and general enough context. For asymptotically flat spacetimes filled with matter obeying the dominant energy condition (DEC), several positive mass theorems for the ADM-mass \cite{Schon:1979rg,schoen1981,witten1981} as well as for the Bondi-mass \cite{Ludvigsen_1981,Horowitz:1981uw,PhysRevLett.48.369} were established. Later, positivity of mass was extended also to asymptotically AdS-spacetimes and to Einstein-Maxwell theory \cite{gibbons1983}. Closely linked to the question of positivity is the Penrose conjecture \cite{Penrose:1973um,Mars:2009cj}, relating the total mass of a spacetime to the area of the outermost apparent horizon. The corresponding Riemannian version, the Riemann-Penrose inequality, was proven by Huisken \& Ilmanen \cite{huisken2001} using the observation by Geroch that the Hawking energy behaves monotonously under the inverse mean curvature flow \cite{1973NYASA.224..108G}; it was also independently proven by Bray \cite{bray2001}. More generally, in \cite{Bray:2006pz} the authors examined under which flows the Hawking energy is monotonous.
The context in which the present work studies energy in a spacetime is observational cosmology \cite{Ellis1985315,ellis_maartens_maccallum_2012}, whose fundamental objective is to infer properties of the universe solely based on local observations. In particular, we would like to find a meaningful notion of energy for the observable universe, that is, the part of spacetime causally connected to and in the past of the observation event. The natural geometric object directly related to the causal boundary, but also to cosmological observations, is the past lightcone of an observer.
We stress that the lightcone is a geometric object associated to the spacetime at a given point without any further specifications. Therefore, a natural question is whether a well-defined notion of energy on the past lightcone exists. Positivity of energy for null-geodesically complete, globally smooth lightcones was shown in \cite{Chrusciel:2014gja}. Specifically for the Hawking energy, positivity and monotonicity results were established in \cite{Christodoulou:1988,Eardley:1979dra}. However, due to strong gravitational fields, potentially sourced by local inhomogeneities, the lightcone might develop singularities and caustics \cite{Ellis:1998ha,Friedrich:1983vi,Hasse_1996,arnold1985singularities}, see also \cite{Perlick2004} for a general overview about lightcones in the context of gravitational lensing. The aim of this paper is to study the properties of the Hawking energy in such a cosmological set-up, even admitting certain types of singularities which generically appear in lightcones.\\
This work is structured as follows. The Hawking energy and its main properties are discussed in section \ref{sec:Hawkingenergy}, before the cosmological set-up together with the slicing construction of the lightcone is explained in section \ref{sec:cosmolsetup}. The weak lensing regime in absence of self-intersections is studied in section \ref{sec:weaklensing}. The effect of self-intersections on the lightcone geometry is addressed in section \ref{sec:stronglensing}. Section \ref{sec:singularities} establishes the well-definedness of the Hawking energy and its derivative in the presence of swallow-tail type singularities. The rescaling freedom of the null generators of the lightcone is discussed in section \ref{sec:rescaling}, before moving to a discussion on monotonicity and possible extensions in sections \ref{sec:monotonicity} \& \ref{sec:extensions}. We conclude in section \ref{sec:conclusions}.
\section{Hawking Energy}
\label{sec:Hawkingenergy}
Unless stated otherwise, we assume a globally hyperbolic Lorentzian spacetime $(M,g)$ satisfying the Einstein field equations (EFEs):
\begin{equation}
R_{ab}-\frac{1}{2}R g_{ab}=8\pi T_{ab}\quad,
\end{equation}
with Ricci tensor $R_{ab}$, Ricci scalar $R$, and the energy-momentum tensor $T_{ab}$ satisfying the DEC. A potential cosmological constant can be accommodated in $T_{ab}$ in the following discussions. The signature convention is $(-+++)$ and we use units in which $c=G=1$.\\
A spacelike 2-surface $S$ in $M$ uniquely defines two distinct orthogonal null congruences, both of which are either future or past directed, represented by two null vector fields $l^a$ and $n^a$. These congruences are often referred to as outgoing and ingoing, their expansion scalars are denoted by $\theta_+=\nabla_al^a$ and $\theta_-=\nabla_an^a$ respectively, where $\nabla_a$ is the covariant derivative associated with the spacetime metric $g$. Hawking's original definition \cite{Hawking:1968qt} of the energy $E(S)$ associated with a spacelike surface $S$ of spherical topology reads:
\begin{equation}
E(S):= \frac{\sqrt{A(S)}}{(4\pi)^{3/2}}\left(2\pi+\frac{1}{4}\int_S\theta_+\theta_-\,\mathrm{d}S\right)\quad,
\label{eq:energy}
\end{equation}
where $A(S)=\int_S \mathrm{d}S$ denotes the area of the surface $S$ given in terms of the pullback $\mathrm{d}S$ of the canonical spacetime volume form onto $S$. This definition satisfies several important limits briefly reviewed here, see also \cite{Szabados:2004vb,Eardley:1979dra}:
\begin{itemize}
\item[(i)] The Hawking energy of any point in spacetime should vanish, hence $E(S)\rightarrow 0$ for $S$ degenerating to a point.
\item[(ii)] For a small sphere of (area) radius $r\rightarrow 0$ about point $p$, one finds for the leading order in $r$ \cite{Horowitz:1982}:
\begin{align}
E(S)&\sim r^5\, B_{abcd}t^at^bt^ct^d\ge 0\quad\text{in vacuum}\\
E(S)&\sim r^3\,T_{ab}t^at^b\quad\text{in non-vacuum}
\end{align}
with the Bel-Robinson tensor $B_{abcd}$ \footnote{The Bel-Robinson tensor is defined as\\ $B_{abcd}:=C_{aecf}\,C_{b\;d}^{\;e\;f}-\frac{3}{2} g_{a[b}C_{jk]cf} C^{jk\;f}_{\;\;\;d}$ .}, $t^a\in T_pM$ a unit timelike vector orthogonal to $S$, and the energy-momentum tensor $T_{ab}$. If the DEC holds, then $E(S)\ge0$ also in the non-vacuum case.
\item[(iii)] For large spheres near null infinity $\mathcal{I}^\pm$, the Bondi-Sachs energy is recovered \cite{Hawking:1968qt}: $E(S)\rightarrow E_{\text{Bondi-Sachs}}$.
\item[(iv)] For large spheres near spatial infinity $i^0$, the ADM-mass is recovered \cite{Szabados:2004vb}: $E(S)\rightarrow E_{\text{ADM}}$ .
\item[(v)] In a spherically symmetric spacetime, the Hawking energy coincides with the Misner-Sharp energy, e.g.\ \cite{Carrera:2008pi}.
\item[(vi)] If $S$ is a metric sphere in Minkowski spacetime: $E(S)=0$ \cite{Szabados:2004vb}.
\item[(vii)] Given a null hypersurface with $\theta_+=0$, for instance a non-expanding horizon or a Killing horizon. For any spacelike spherical cross section $S$, one finds:
\begin{equation}
E(S)=\sqrt{\frac{A(S)}{16\pi}}\quad.
\end{equation}
In particular, for a cross section of the event horizon of a Kerr-Newman black hole, the irreducible mass $M_{\text{irr}}$ is recovered, see e.g.\ \cite{Eardley:1979dra}.
\end{itemize}
Two other properties one would expect from an energy definition are positivity and monotonicity. However, it appears that this is not given in the general case. Concerning positivity, it is worth pointing out that (vi) only holds for metric spheres and not for arbitrary topological spheres on Minkowski spacetime. In fact, the Hawking energy might become negative for suitably shaped spheres\footnote{In general, the Hawking energy turns negative if according to (\ref{eq:ww}) the mean curvature $H$ of $S$ within the spacelike hypersurface $\Sigma$ is large enough compared to the mean curvature $\tau$ of $\Sigma$ in $M$.}. In order to maintain a vanishing energy for any spacelike topological sphere in Minkowski space, Hayward proposed a modification by including shear and twist terms \cite{Hayward:1993ph}. However, it is negative for small spheres in vacuum \cite{Bergqvist_1994}. A general positivity result for maximal slices was obtained in \cite{Christodoulou:1988}. Furthermore, one would naturally expect the energy to increase if the domain, i.e.\ the surface $S$, is enlarged. Since in general there are many ways to enlarge $S$, one would have to specify a particular construction to give a more precise meaning to the statement. Eardley was able to construct a special family of surfaces along which the Hawking energy increases monotonously \cite{Eardley:1979dra}. This result is essential in order to establish monotonicity in the weak lensing case and will be discussed in greater detail in section \ref{sec:weaklensing}.
In the light of these results, a natural question is whether positivity and monotonicity of the Hawking energy can be established in particular, physically relevant set-ups, such as the past lightcone of an observer in cosmology.
\section{Cosmological Set-up}
\label{sec:cosmolsetup}
The cosmological context in which we aim to answer this question is provided by the observational approach by Ellis and others \cite{Ellis1985315,ellis_maartens_maccallum_2012}. Based solely on data on the past lightcone of an observer, it aims at deducing the spacetime geometry in the vicinity of the lightcone without further model assumptions. Mathematically, it constitues a characteristic final value problem, see e.g.\ \cite{ChoquetBruhat:2010ih,Chrusciel:2012ap} and references therein, with final data given on the past lightcone and a solution in the chronological past of the event is constructed by propagating the data on the lightcone into its interior via the EFEs. The observer is assumed to be a point $p$ in spacetime $M$ and a future-pointing normalised timelike vector $u^a\in T_pM$. This is a good approximation as long as the duration of observation is negligible compared to the dynamical timescale of the universe. Almost all cosmologically relevant information, such as light and gravitational waves, travels with the speed of light, hence the central geometric object of interest is the past lightcone $C^-(p)$ of the observer at $p\in M$. It is a null hypersurface and can be uniquely constructed once the point $p\in M$ is specified. In Minkowski spacetime, it is an undistorted cone with topology $\mathbb{R}\times S^2$. However, the presence of matter or other inhomogeneities will in general deform the lightcone. Two regimes can be distinguished:
\begin{itemize}
\item \textit{Weak Lensing Regime}: the lightcone remains an embedded surface, but is weakly deformed, preserving the $\mathbb{R}\times S^2$ topology. Hence, no multiple images of the same source appear.
\item \textit{Strong Lensing Regime}: the lightcone is strongly deformed and intersects itself. Changes in topology cause multiple imaging.
\end{itemize}
More formally, the past lightcone $C^-(p)$ of a cosmological observer $(p,u^a)$ in a globally hyperbolic spacetime $M$ is the image of the exponential map $\exp_p$ along past-pointing null vectors $\in T_pM$ on its maximal domain of definition. Sufficiently close to $p$, the exponential map is always injective. At self-intersections, the exponential map fails to be injective, i.e.\ points may be reached along multiple null geodesics starting at $p$. Another crucial observation is that past null geodesics issued at $p$ are initially part of the boundary $\dot{I}^-(p)$ of the chronological past $I^-(p)$ of $p$, but might leave the boundary into the interior. Thus, they are not exclusively confined to $\dot{I}^-(p)$ but rather to $\dot{I}^-(p)\cup I^-(p)$. The last point along a null generator $\gamma(\tau)$ still in $\dot{I}^-(p)$ is called cut point of $\gamma$. The union of all cut points of all past-pointing null generators is then referred to as cut locus $L^-(p)$ of the past lightcone $C^-(p)$. Any point of a generator beyond the cut point lies in the chronological past of $p$ and therefore can also be reached along a timelike curve from $p$. At a cut point, multiple null generators intersect, either infinitesimally close generators resulting in a conjugate point, or globally different generators, see Fig.\ \ref{fig:spherlens}. Furthermore, since $\dot{I}^-(p)$ is an achronal boundary and therefore a Lipschitz continuous submanifold \cite{Hawking:1973uf}, the same holds for the part of the lightcone contained in the boundary, $C^-(p)\cap \dot{I}^-(p)$. Additionally, the cut locus has measure zero in $\dot{I}^-(p)$, thus, $C^-(p)\cap \dot{I}^-(p)$ is differentiable everywhere except at $p$ and the cut locus \cite{Perlick2004}.
\begin{center}
\begin{figure}
\includegraphics[width=0.4\textwidth]{./figures/lightcone.pdf}
\caption{Lightcone $C^-(p)$ of point $p$ going through a gravitational lensing event causing $C^-(p)$ to intersect itself at the cut locus $L^-(p)$ (blue), which is part of the exterior $C^-(p)\cap\dot{I}^-(p)$. Two different null generators (red) intersect at the cut locus, after which they turn into the interior $I^-(p)$ (dashed). The conjugate point $q$, where infinitesimally close generators intersect, is of swallow-tail type. Two cusp ridges originating from $q$ remain in $I^-(p)$.}
\label{fig:spherlens}
\end{figure}
\end{center}
In this cosmological set-up, we study the properties of the Hawking energy on the past lightcone $C^-(p)$ of a cosmological observer. In principle, the Hawking energy can be inferred directly from observations, at least within the ideal observational cosmology framework described in \cite{Ellis1985315}. For example, the energy associated with a round sphere of comoving radius $r$ in a FLRW-universe is $E=4/3\pi a^3r^3\rho$, where $\rho(t)$ is the matter density of the cosmic fluid and $a(t)$ the scale factor.
In particular, we are interested in monotonicity properties of $E$ along a family of two dimensional slices $(S_t)$ down the lightcone. Before turning to more formal and rigorous statements in sections \ref{sec:weaklensing} \& \ref{sec:monotonicity}, we first provide an intuitive argument in favour of monotonicity.
The past lightcone in Fig.\ \ref{fig:spherlens} can be sliced into two dimensional spacelike surfaces, for instance by a one-parameter family of (partial) Cauchy surfaces $\Sigma_t$. The part of such a lightcone slice contained in the past causal boundary $\dot{I}^-(p)$ is denoted by $S_t$: $S_t:=C^-(p)\cap \dot{I}^-(p)\cap\Sigma_t$. Since $\dot{I}^-(p)$ is the past causal boundary of $I^-(p)$, any matter respecting the DEC can only leave $I^-(p)$ to the future, in particular, nothing can enter $I^-(p)$ from outside. Therefore, taking two different slices $S_t$ and $S_{t'}$ with $t<t'$ as depicted in Fig.\ \ref{fig:monotonicity}, matter may only leave $I^-(p)$ between $t$ and $t'$. Turning the argument around, the surfaces $S_t$ should enclose more and more matter towards the past. Each $S_t$ is typically a closed, spacelike surface and thus has an associated Hawking energy $E(S_t)$. By the above argument, the Hawking energy should then be monotonously increasing along the family $(S_t)$ down the lightcone. Though, this naive argument only holds for $\theta_+>0$ everywhere on $C^-(p)$ as we shall see later.
\begin{center}
\begin{figure}
\includegraphics[width=0.45\textwidth]{./figures/monotonicity_plot.pdf}
\caption{Causal matter can only leave $I^-(p)$ to the future, but nothing can enter from the outside. Thus, the Hawking energy should monotonously increase from $S_{t'}$ to $S_t$.}
\label{fig:monotonicity}
\end{figure}
\end{center}
It is crucial to note that this argument only holds for surfaces which are part of the causal boundary. As mentioned above, the lightcone generators leave the boundary after self-intersections and the interior parts of $C^-(p)$, that is the part contained in the chronological past, can be penetrated by timelike curves. Therefore, we have to exclude the interior parts of $C^-(p)$ and restrict our monotonicity discussion to the part of the lightcone contained in the causal boundary $C^-(p)\cap \dot{I}^-(p)$. Also from a geometric point of view, $C^-(p)\cap \dot{I}^-(p)$ is a much better-behaved hypersurface than $C^-(p)$ since it is a Lipschitz manifold, whereas $C^-(p)$ might in general fail to be a manifold due to complicated self-intersections in the interior of $\dot{I}^-(p)$.\\
In the subsequent sections, we do not use Cauchy surfaces, but rather adopt the following construction in order to generate the 1-parameter family of lightcone slices. We start with an initial lightcone cut $S$ sufficiently close to $p$, guaranteeing that $S$ is a topological sphere and that the Hawking energy is positive due to the small sphere limit \cite{Horowitz:1982}. The part of the lightcone in the past of $S$ is generated by the past-pointing null geodesics associated with the null generators $l^a$. The lightcone can then be sliced into constant affine parameter distance slices $S_\lambda$, with $\lambda\geq 0$ and $S_{\lambda=0}=S$. However, since $l^a$ is a null vector field, we have a pointwise rescaling freedom $l^a\rightarrow\alpha l^a$, with $\alpha>0$ a function on $S$. After fixing the rescaling freedom in a suitable manner, as is done in the subsequent sections, we walk down a unit distance along the generators and arrive at a new spacelike cut, where the rescaling procedure is repeated etc. The function $\alpha$ extends to a function on $C^-(p)\cap \dot{I}^-(p)$ and encodes the particular choice of the lightcone foliation $(S_\lambda)$. Changing from one foliation with corresponding affine parameter $\lambda$ to another one with $\tilde{\lambda}$ relates the change of $E$ for each foliation via
\begin{equation}
\frac{\partial E}{\partial \tilde{\lambda}}=\frac{\partial E}{\partial \lambda}\frac{\partial\lambda}{\partial\tilde{\lambda}}\quad.
\label{eq:ooo}
\end{equation}
Thus, if $E$ is monotonously increasing along the $\lambda$-foliation, it increases along the new foliation if $\frac{\partial\lambda}{\partial\tilde{\lambda}}>0$, that is, if $\lambda$ is an increasing function of $\tilde{\lambda}$.
\section{Weak Lensing Regime}
\label{sec:weaklensing}
This section addresses monotonicity in scenarios excluding caustics, a discussion including these can be found in the next sections. The following results can be understood as an application of Eardley's findings \cite{Eardley:1979dra} to lightcones. Each lightcone slice on the boundary, $S_\lambda\subset C^-(p)\cap \dot{I}^-(p)$, comes with an associated Hawking energy $E(S_\lambda)$. In the following, it is assumed that each slice $S_\lambda$ is topologically a sphere, a brief discussion of different topologies can be found in section \ref{sec:extensions}. The change of the Hawking energy (\ref{eq:energy}) assigned to $S_\lambda$ along the outgoing null direction $l^a$ is given by $\partial_\lambda E(S_\lambda)\equiv \dot{E}(S_\lambda)$:
\begin{align}
\dot{E}(S_\lambda)&= \frac{E(S_\lambda)}{2A(S_\lambda)}\int_{S_\lambda} \theta_+\,\mathrm{d}S_\lambda+\nonumber\\
&+\frac{\sqrt{A(S_\lambda)}}{(4\pi)^{3/2}}\int_{S_\lambda}\left[ \dot{\theta}_+\theta_-+\theta_+\dot{\theta}_-+\theta_+^2\theta_-\right]\mathrm{d}S_\lambda\quad,
\label{eq:aa}
\end{align}
where we used $A(S)=\int_S\mathrm{d}S$ and $\dot{(\mathrm{d}S)}=\theta_+\mathrm{d}S$. At the same time, the vector field $l^a$ will be taken to be identical to the null generators of the past lightcone $C^-(p)$. Next, we will make use of the Sachs equation for the evolution of $\theta_+$ (see e.g.\ \cite{Hawking:1973uf}):
\begin{equation}
\dot{\theta}_+=-\frac{1}{2}\theta_+^2-\sigma_{ab}\sigma^{ab}-R_{ab}l^al^b\quad.\label{eq:sachseq}
\end{equation}
Since $l^a$ generates a null hypersurface, the vorticity term in the general Sachs equation is vanishing and thus absent in (\ref{eq:sachseq}). $\sigma_{ab}$ denotes the shear tensor of the congruence $l^a$. Using the EFEs, the right hand side of (\ref{eq:sachseq}) is non-positive if the DEC holds.
The evolution equation of $\theta_-$ along $l^a$ can be derived from \cite{Gourgoulhon:2005ng}:
\begin{equation}
\dot{\theta}_-=D_a\Omega^a+\Omega_a\Omega^a-\frac{1}{2}{}^2R+\frac{1}{2}h^{ab}R_{ab}-\theta_+\theta_-\quad.\label{eq:-evolution}
\end{equation}
It describes the change of the expansion of the ingoing null congruence $n^a$ along the outgoing one. $\Omega_a=\nabla_ln_a$ denotes the change of the ingoing null vector along the outgoing one. $h_{ab}$ denotes the two dimensional Riemannian metric on $S_\lambda$ defined by the pullback of the spacetime metric $g_{ab}$ onto $S_\lambda$. They are related via
\begin{equation}
h_{ab}=g_{ab}+l_an_b+n_al_b\quad.
\label{eq:hab}
\end{equation}
$D_a$ is the covariant derivative on $S_\lambda$ compatible with its induced metric $h_{ab}$ and related to the spacetime covariant derivative $\nabla$ via the projection operator onto $TS_\lambda$, $\Pi_a^{\;b}=\delta_a^{\;b}+l_an^b+n_al^b$. For instance, $D_aX^b=\Pi_a^{\;c} \Pi_d^{\;b}\nabla_cX^d$ for any $X^a\in TS_\lambda$. The two dimensional Ricci scalar of $S_\lambda$ is denoted by ${}^2R$. Using the EFE, we find
\begin{equation}
h^{ab}R_{ab}=R+2R_{ab}l^an^b=16\pi T_{ab}l^a n^b\ge0\quad,
\label{eq:DEC}
\end{equation}
if the DEC is satisfied. Inserting (\ref{eq:sachseq}), (\ref{eq:-evolution}), and (\ref{eq:DEC}) into (\ref{eq:aa}) yields:
\begin{align}
\dot{E}(S_\lambda)&= \frac{E(S_\lambda)}{2A(S_\lambda)}\int_{S_\lambda} \theta_+\,\mathrm{d}S_\lambda\nonumber\\
&+\frac{\sqrt{A(S_\lambda)}}{(4\pi)^{3/2}}\int_{S_\lambda}\bigg\{ -\theta_-\left(\frac{1}{2}\theta_+^2+\sigma_{ab}\sigma^{ab}+R_{ab}l^al^b\right)\nonumber\\
&+\theta_+\left(D_a\Omega^a+\Omega_a\Omega^a-\frac{1}{2}{}^2R+8\pi T_{ab}l^an^b \right) \bigg\}\,\mathrm{d}S_\lambda\quad.
\label{eq:Edot}
\end{align}
Eardley \cite{Eardley:1979dra} established a monotonicity results for a particular family of surfaces $(S_r)$. Starting off with a surface $S$ with $\theta_+> 0$ \& $\theta_-\le 0$ almost everywhere. One can define a constant $r$ on $S$ by $A(S)=:4\pi r^2$. Although $n^a$ is normalised such that $n^al_a=-1$, there is still a pointwise rescaling freedom of $l^a$ left: $l^a\rightarrow \alpha l^a$ with $\alpha>0$. It is used to rescale $l^a$ such that $\theta_+=\frac{2}{r}$. Since $\partial_\lambda r=1$, $r$ is also a parameter along the congruence. In fact, $r$ corresponds to an area distance function
\begin{equation}
r=\sqrt{\frac{A}{4\pi}}\quad,
\end{equation}
which is related to the luminosity distance via Etherington's reciprocity theorem \cite{Etherington}.
Starting with the initial surface $S$ being a lightcone section arbitrarily close to the tip $p$, the remaining lightcone is foliated by level surfaces $S_r$ of constant $r$. Along this special family of surfaces $S_r$, (\ref{eq:Edot}) can be further simplified by inserting the explicit expressions for $A$ and $\theta_+$:
\begin{align}
\dot{E}(S_r)=\frac{1}{4\pi}\int_{S_r}\bigg\{ -\frac{r}{4}\theta_-\left(\sigma_{ab}\sigma^{ab}+R_{ab}l^al^b\right)\nonumber\\
+\frac{1}{2}\left(\Omega_a\Omega^a+\frac{1}{2}R+R_{ab}l^an^b\right) \bigg\}\,\mathrm{d}S_r\quad,
\label{eq:ss}
\end{align}
where the Gauss-Bonnet theorem for a sphere $\int_S{}^2R\,\mathrm{d}S=8\pi$ was used as well as $\int_S D_a\Omega^a\,\mathrm{d}S=0$, because $S$ is a closed surface. Extending the assumption $\theta_+>0$ and $\theta_-\le 0$ to all $S_r$, and further assuming the DEC, we find the right hand side of (\ref{eq:ss}) to be non-negative, because $\sigma_{ab}\sigma^{ab}\ge 0$ and $\Omega_a\Omega^a\ge 0$, which can be verified by direct calculation. Using the EFEs, the curvature terms are shown to be non-negative because of the DEC. Thus, the Hawking energy increases monotonously along the particular foliation $(S_r)$ of $C^-(p)$ given the above assumptions. In fact, monotonicity can be established for a whole class of foliations, namely those with $\partial_{\tilde{\lambda}}\lambda>0$, cf. (\ref{eq:ooo}). Eardley's precise expression in Newman-Penrose variables \cite{Eardley:1979dra} is recovered after making use of the identities $\mu=\frac{\theta_-}{2}$, $\frac{1}{2}\sigma_{ab}\sigma^{ab}=|\sigma|^2$, $\phi_{00}=\frac{1}{2}R_{ab}l^al^b$, $\Omega_a\Omega^a=2\pi\bar{\pi}=2|\alpha+\bar{\beta}|^2$ and $\frac{1}{4}R+\frac{1}{2}R_{ab}l^an^b=3\Lambda+\phi_{11}$.
Additionally, if the initial sphere $S$ is sufficiently close to the lightcone tip $p$, the Hawking energy is positive due to the small sphere limit of \cite{Horowitz:1982}. Summarising, we found that the Hawking energy on the past lightcone $C^-(p)$ of an observer $p$ is positive and monotonously increasing to the past, provided that $\theta_+>0$ \& $\theta_-\le 0$ almost everywhere, and matter obeys the DEC.
The central assumption is the strict positivity of the expansion $\theta_+$ of the outgoing null congruence $l^a$ generating the lightcone. Firstly, this excludes spacetimes with certain global properties, such as the existence of past apparent horizons, beyond which $\theta_+$ turns negative \cite{Ellis:2015928}. This is the case in many cosmological settings, in particular FLRW dust universes with a positive cosmological constant, the Einstein static universe or other recollapsing models. In any case, the monotonicity results remain true even in such spacetimes in a suitably close neighbourhood of the observer $p$. Secondly, this also excludes local regions of $C^-(p)$ with negative expansion, as is the case in the presence of caustics due to local inhomogeneities causing strong gravitational lensing. A discussion on the inclusion of caustics can be found in the next chapter, the results so far only hold in the case of an empty cut locus of $C^-(p)$, $L^-(p)=\emptyset$. In particular, this includes the weak lensing regime.
\section{Geometry of $C^-(p)$ in the presence of strong lenses}
\label{sec:stronglensing}
In more realistic cosmological set-ups, the past lightcone will typically display self-intersections. Considering that gravitational lenses, such as galaxy clusters, galaxies, or individual stars, exist on different scales, caustics are expected to from hierarchical patterns on the past lightcone. Ellis et al.\ \cite{Ellis:1998ha} estimated the total number of caustics on our past lightcone due to inhomogeneities to be of the order $10^{22}$. The presence of caustics was neglected in the original observational cosmology programme of reconstructing the spacetime metric and energy momentum tensor from observables \cite{Ellis1985315}. However, their presence affects cosmological distances in such a way that observed area distances to objects are increased \cite{Ellis:1998ha}.
In general, these self-intersections may be arbitrarily complicated. Yet, it was shown that the multitude of these self-intersections can be divided into stable and unstable ones in the following sense. The set of points in spacetime $M$ that can be reached by the outgoing, respectively ingoing, null geodesic congruence emanating from an orientable, spacelike, smooth surface $S$ in $M$ is called wavefront, see e.g.\ \cite{Perlick2004}. The caustic of a wavefront is defined to be the set of points where the wavefront fails to be an immersed submanifold of $M$. In particular, the past lightcone $C^-(p)$ of $p$ is a wavefront if $S$ is chosen suitably close to $p$. Stability refers to arbitrarily small perturbations of the initial surface $S$, see e.g.\ \cite{Hasse_1996} for more details. A classification of stable caustics of wavefronts was established by \cite{Friedrich:1983vi,Low:1993,Hasse_1996,Low:1998}, using Arnol'd's singularity theory of Lagrangian and Legendrian maps \cite{arnold1985singularities,Ehlers2}. Of particular relevance for the present work, Low showed \cite{Low:1993,Low:1998} that only two types of stable caustics appear in the intersection of a lightcone with a spacelike hypersurface, referred to as cusp and swallow-tail singularities, cf.\ Fig.\ref{fig:swallowtail}. For a detailed account on the different types of caustics in the context of gravitational lensing, we refer the reader to \cite{Petters2001}.
In particular, we are interested the simple lensing configuration displayed in Fig.\ref{fig:spherlens}, consisting of a single swallow-tail singular point and two cusp ridges present in $C^-(p)\cap\Sigma_t$. The presence of self-intersections renders $C^-(p)$ Lipschitz continuous on the measure zero set of self-intersections, in other words, the light cone remains smooth almost everywhere. The following analysis also extends to more general lensing configurations, in particular, the results may be applied to multi-lens configurations in which the single lens in Fig.\ref{fig:spherlens} occurs multiple times; see also the discussion in section \ref{sec:monotonicity}. More formally, the analysis given below includes the subclass of configurations in which the singular points as well as the cut loci in $S$ are isolated and have measure zero. Physically, these configurations may be thought of as multiple local overdensities leading to isolated gravitational lenses of the type shown in Fig.\ref{fig:spherlens}.
Since singular points are conjugate points along the null generators with respect to $p$, the expansion of the lightcone generators $\theta_+=-\infty$ at a singular point. Hence, $\theta_+$ is a smooth function almost everywhere on $S$, apart from the singular points. Large regions of $S$ will display a positive $\theta_+$, and by continuity, any singular point is surrounded by a neighbourhood with negative $\theta_+$.
Also, at least for the lens configuration in Fig.\ref{fig:spherlens} and multi-lens set-ups thereof, only swallow-tail singular points can be found in the exterior part $C^-(p)\cap \dot{I}^-(p)$, whereas cusp singular points exclusively appear at self-intersections in the interior $I^-(p)$. Hence, for the configurations considered in this paper, it suffices to take only swallow-tail singular points on $S$ into account. Moreover, these observations seem to suggest that of all stable singular points according to Arnol'd, only swallow-tail ones appear in exterior part $C^-(p)\cap \dot{I}^-(p)\cap \Sigma_t$, however, it would be desireable to make a more rigorous statement on this matter.
\section{Hawking energy in the presence of singularities}
\label{sec:singularities}
Given the above set-up of a light cone including caustics, it is a natural question whether or not the Hawking energy for a surface containing these types of singularities is well-defined. In particular, since at singular points $\theta_+=-\infty$, one might wonder whether the integral $\int_S \theta_+\theta_-\,\mathrm{d}S$ in (\ref{eq:energy}) is well-defined, that is finite. In the following, we show that this is indeed the case for swallow-tail singularities only, whereas the integral is divergent if $S$ contains cusp points. Therefore, a finite Hawking energy can only be assigned to the exterior part of the lightcone, since the presence of cusp singularities in the interior causes the energy to diverge when integrated over the whole lightcone slice.
The past-pointing null vectors $l^a$ and $n^a$ orthogonal to the spacelike codimension-2 surface $S$ can be decomposed into a timelike (future-pointing) unit normal $t^a$ as well as an orthogonal spacelike unit normal $v^a$ via
\begin{equation}
l^a=\frac{1}{\sqrt{2}}\left(-t^a+v^a\right)\quad\&\quad
n^a=\frac{1}{\sqrt{2}}\left(-t^a-v^a\right)\quad.
\label{eq:ln}
\end{equation}
Following \cite{Szabados:2004vb}, the tangent bundle $TM$ of the spacetime $M$ can be decomposed into the sum of
the tangent bundle $TS$ of $S$ and the normal bundle $NS$ of $S$ by using the corresponding projectors
\begin{align}
\Pi^a_b&:=\delta^a_b+t^at_b-v^av_b=\delta^a_b+l^an_b+n^al_b\quad\text{and}\nonumber\\
O^a_b&:=\delta^a_b-\Pi^a_b
\end{align}
respectively. For example, the spacetime metric $g$ can be decomposed into the intrinsic metric $h$ on $S$ and an orthogonal part, cf.\ (\ref{eq:hab}):
\begin{equation}
g_{ab}=h_{ab}-t_at_b+r_ar_b=h_{ab}-l_an_b-n_al_b\quad.
\label{eq:metricdecomp}
\end{equation}
Corresponding to each normal, there is an associated extrinsic curvature $\tau_{ab}$ and $H_{ab}$:
\begin{equation}
\tau_{ab}=\Pi^c_a\Pi^d_b\nabla_ct_d \quad\&\quad H_{ab}=\Pi^c_a\Pi^d_b\nabla_cv_d\quad.
\label{eq:extrinsiscurv}
\end{equation}
$\tau_{ab}$ is the extrinsic curvature (or second fundamental form) of the spacelike hypersurface $\Sigma$ with timelike normal $t^a$ embedded in spacetime. $\Sigma$ creates the lightcone slices: $S=C^-(p)\cap\dot{I}^-(p)\cap \Sigma$. $H_{ab}$ is the extrinsic curvature of the spacelike 2-surface $S$ with spacelike normal $v^a$ within the spacelike hypersurface $\Sigma$. Taking the trace results in the mean curvatures $\tau$ and $H$ of $S$ in each direction. Using the definitions $\theta_+=\nabla_al^a$ and $\theta_-=\nabla_an^a$, together with (\ref{eq:ln}) \& (\ref{eq:extrinsiscurv}), yields the following relation between the null expansions and mean curvatures:
\begin{equation}
\theta_\pm= \frac{1}{\sqrt{2}}\left(-\tau\pm H\right).
\label{eq:expansions}
\end{equation}
Because $\theta_+=-\infty$ at singular points, (\ref{eq:expansions}) implies that one of the mean curvatures has to diverge. Since $\tau$ is the mean curvature of the smooth spacelike hypersurface $\Sigma$, it is finite, and hence, $H\rightarrow-\infty$ at singular points. The product of expansions can be written as the norm of the main curvature vector $Q^a$ of $S$:
\begin{align}
Q^a:&=-\theta_-l^a-\theta_+ n^a=\tau t^a-H v^a\quad\text{thus}\nonumber\\
-Q^aQ_a&=2\theta_+\theta_-=\tau^2-H^2\quad.\label{eq:meancurv}
\end{align}
In the generic case where $\theta_+>0$ and $\theta_-<0$, $Q^a$ is spacelike, it is null if one of the expansions is zero, for example on horizons, and becomes timelike if the expansions have the same sign, for instance for trapped surfaces. These results imply that every singular point on $S$ is surrounded by a "trapped ring", where $\theta_+\theta_->0$ (see Fig.\ref{fig:trapped ring}).
\begin{figure}
\includegraphics[width=0.4\textwidth]{./figures/trappedrings.pdf}
\caption{A singular point $q$ on $S$ is surrounded by a trapped, ring-like region (grey), where the product of the null expansions $\theta_+\theta_-$ is positive.}
\label{fig:trapped ring}
\end{figure}
Using (\ref{eq:meancurv}), the integral expression appearing in (\ref{eq:energy}) becomes
\begin{equation}
E(S)=\frac{\sqrt{A(S)}}{(4\pi)^{3/2}}\left(2\pi+\frac{1}{8}\int_S \left(\tau^2-H^2\right)\,\mathrm{d}S\right)\quad.
\label{eq:ww}
\end{equation}
The fact that $H$ diverges at singular points was also more formally established in \cite{geometryoffronts} (c.f.\ corollary 3.5), where the authors proved that
the mean curvature of a hypersurface in a Riemannian manifold diverges at swallow-tail or cusp singular points. Although $H$ is divergent, one might still hope that $\int_S H^2\;\mathrm{d}S$ in (\ref{eq:ww}) is finite. In the following, we show that this is the case only for swallow-tail singularities, whereas the integral diverges for cusp singularities. Therefore, the Hawking energy is well-defined, i.e.\ finite-valued, for Lipschitz surfaces only containing swallow-tail singularities.
It suffices to study the integral in a neighbourhood $Q\subset S$ of a singular point, because the mean curvature is always finite-valued at non-singular points of $S$. Below, we arrive at explicit expressions for the mean curvature near a cusp and swallow-tail singular point, if $S$ is embedded in Euclidean space, and find that $\int_Q H^2\;\mathrm{d}S$ diverges for a cusp, but is finite for a swallow-tail point. This result can be immediately extended to the Riemannian case by replacing the Euclidean metric $g$ in the calculation below with its Riemannian counterpart, altering the result only by finite factors.
The following calculation and notation follows \cite{geometryoffronts}. Given a smooth map $f: M\rightarrow N$ from an oriented 2-manifold $M$ into an oriented Riemannian 3-manifold $N$ with metric $g$. $f$ is called an instantaneous wavefront if there exists a unit vector field $\nu^a\in N$ along $f$ such that $g(f_*X,\nu)=0\quad \forall X\in TM$. $\nu^a$ is called the normal vector of the instantaneous wavefront $f$. An instantaneous wavefront is the intersection of a wavefront with a spacelike hypersurface \cite{Perlick2004}. $q\in M$ is called a singular point of the front $f$, if $f$ is not an immersion at $q$. A singular point is called cusp point or swallow-tail point respectively, if it is locally diffeomorphic to
\begin{align}
f_C(u,v)&:=(u^2,u^3,v)\quad\text{or}\nonumber\\ f_S(u,v)&:=(3u^4+u^2v,4u^3+2uv,v)\label{eq:singularities}
\end{align}
at $(u,v)=(0,0)$. The mean curvature $H$ of the front $f$ with normal vector $\nu^a$ is
\begin{equation}
H:=\frac{EN-2FM+GL}{4\lambda^2}\quad,
\end{equation}
with $f_u=\partial_uf$, $f_v=\partial_vf$, $E=g(f_u,f_u)$, $F=g(f_u,f_v)$, $G=g(f_v,f_v)$, $|\lambda|=\sqrt{EG-F^2}$, $L=-g(f_u,\nu_u)$, $M=-g(f_v,\nu_u)=-g(f_u,\nu_v)$, $N=-g(f_v,\nu_v)$.
Computing the mean curvature for cusp and swallow-tail singularities near the singular point $(0,0)$ yields:
\begin{align}
H_C&=-\frac{3}{2u(9u^2+4)^{3/2}}\quad\text{and}\\ H_S&=\frac{u^4+4u^2+1}{8(6u^2+v)(u^4+u^2+1)^{3/2}}\quad.
\label{eq:meancurvatures}
\end{align}
Inserting these into the integral expression in (\ref{eq:ww}) using $\mathrm{d}S=|\lambda|\,\mathrm{d}u\,\mathrm{d}v$ and setting the integration range\\ $Q_C=\left\{v\in[b_1,b_2],u\in[-a,a]\right\}$ yields
\begin{align}
&\int_{Q_C} H_C^2\,\mathrm{d}S= \frac{9}{4}v|_{b_1}^{b_2}\int_{-a}^a\frac{\mathrm{d}u}{|u|\left(9u^2+4\right)^{5/2}}\nonumber\\&\quad\overset{u\rightarrow 0}{\approx} \frac{9}{128}v|_{b_1}^{b_2}\int_{-a}^a\frac{\mathrm{d}u}{|u|}
=\frac{9}{128}(b_2-b_1)\cdot 2\left[\ln(|u|)\right]_0^a=+\infty
\end{align}
for the cusp case. In the case of the swallow-tail, we must be careful to only integrate over the outer part of the surface, i.e.\ the part contained in $C^-(p)\cap\dot{I}^-(p)$, see Fig.\ref{fig:swallowtail}. This is done by restricting the integration range to $v\geq-2u^2$ in the above parametrization, \\$Q_S=\left\{ v\in[-2u^2,a], u\in[-b,b] \right\}$, yielding
\begin{align}
\int_{Q_S} H_S^2\,\mathrm{d}S&=\int_{-b}^{b}\mathrm{d}u\int_{-2u^2}^{a}\mathrm{d}v\, \frac{\left(u^4+4u^2+1\right)^2}{32\left( u^4+u^2+1\right)^{5/2}(6u^2+v)}\nonumber\\
&= \int_{-b}^b\mathrm{d}u\, \frac{\left(u^4+4u^2+1\right)^2}{32\left( u^4+u^2+1\right)^{5/2}}\ln\left(\frac{3}{2}+\frac{a}{4u^2}\right)\nonumber\\
&\overset{u\rightarrow 0}{\approx}\frac{1}{32} \int_{-b}^{b}\ln\left(\frac{a}{4u^2}\right)\,\mathrm{d}u\nonumber\\
&=\frac{1}{16}\left[2u+u\ln\left(\frac{a}{4u^2}\right)\right]_0^b<\infty\quad.
\end{align}
Summarising, we showed that the integrals in (\ref{eq:ww}) are finite, thus the Hawking energy for a topological sphere $S$ containing swallow-tail singularities is well-defined.
\begin{figure}
\includegraphics[width=0.49\textwidth]{./figures/Swallowtail.pdf}
\caption{Front containing a swallow-tail singularity. The outer part $\subset \dot{I}^-(p)$ is coloured in red and satisfies $v\geq-2u^2$. It is a zoom-in of Fig.\ref{fig:spherlens} near the swallow-tail point $q$.}
\label{fig:swallowtail}
\end{figure}
Next, we address the derivative of $E$ along the null generators (\ref{eq:Edot}), which can be further simplified by making use of the contracted Gauss equation, cf. \cite{Gourgoulhon:2005ng}:
\begin{equation}
{}^2R=h^{ac}h^{bd}R_{abcd}-\theta_+\theta_-+2\sigma^+_{ab}\sigma_-^{ab}\quad.
\end{equation}
Applying the Ricci decomposition of the Riemann tensor (see also \cite{Faraoni:2015cnd}) and using the metric decomposition (\ref{eq:metricdecomp}) together with the EFE yields:
\begin{align}
h^{ac}h^{bd}R_{abcd}&=h^{ac}h^{bd}C_{abcd}+16\pi T_{ab}l^an^b+\frac{8\pi}{3}T\quad\text{with}\nonumber\\
h^{ac}h^{bd}C_{abcd}&=2C_{lnnl}\quad,
\end{align}
after using the symmetries of the Weyl tensor. The Weyl tensor term vanishes if $l^a$ belongs to a null geodesic congruence. This can be seen by taking the shear evolution equation along the null congruence, contracting it with $n^an^b$ and noting that $\sigma^+_{ab}n^a=0$:
\begin{equation}
C_{nlnl}=-n^an^b\nabla_l\sigma^+_{ab}-\theta_+\sigma^+_{ab}n^an^b=0\quad.
\end{equation}
Summarising, we find
\begin{equation}
{}^2R=16\pi T_{ln}+\frac{8\pi}{3}T-\theta_+\theta_-+2\sigma^+_{ab}\sigma_-^{ab}\quad.
\label{eq:2dricci}
\end{equation}
Next, the term $\Omega_a\Omega^a$ appearing in (\ref{eq:Edot}) can be expressed in terms of the energy momentum tensor. \\Recalling that $\Omega_a=\nabla_ln_a$ as well as taking into account that $\Omega_al^a=0=\Omega_an^a$, we are left with
\begin{align}
\nabla_l\left( n^\alpha\nabla_ln_\alpha \right)&=\nabla_l\left(\frac{1}{2}\nabla_l(n_\alpha n^\alpha)\right)=0\qquad\Leftrightarrow\nonumber\\
\Omega_a\Omega^a&=-n^a\nabla_l\Omega_a\quad.
\end{align}
Using the evolution equation for $\Omega_a$ along the null \\generators (cf.\ \cite{Gourgoulhon:2005ng}),
\begin{equation}
\nabla_l\Omega_\nu=-\Theta_\nu^{\,\alpha} \Omega_\alpha-\theta_+\Omega_\nu+8\pi T_{\mu\nu}l^\mu+\frac{1}{2}D_\nu\theta_+-D_\alpha\sigma_{+\;\nu}^{\,\alpha}\quad,
\end{equation}
and contracting it with $n^a$, we end up with
\begin{equation}
\Omega_a\Omega^a=-8\pi T_{ln}\quad.
\label{eq:omega}
\end{equation}
Thus, inserting (\ref{eq:2dricci}) and (\ref{eq:omega}) into (\ref{eq:Edot}), we find
\begin{widetext}
\begin{equation}
\dot{E}=\frac{E(S)}{2A(S)}\int_S \theta_+\,\mathrm{d}S
+\frac{\sqrt{A(S)}}{(4\pi)^{3/2}}
\int_S \left\{-\left( \theta_-\sigma^+_{ab}\sigma_+^{ab}+\theta_+\sigma^+_{ab}\sigma_-^{ab} \right)
-8\pi\left(\theta_-T_{ll}+\theta_+\left[T_{ln}+\frac{1}{6}T\right]\right)+\theta_+D_\alpha\Omega^\alpha\right\}\,\mathrm{d}S\quad.
\label{eq:XXX}
\end{equation}
\end{widetext}
This expression describes how the Hawking energy changes along constant affine parameter slices of spherical topology of the past lightcone $C^-(p)$. The shear and matter effects separate into two different contributions.
Before discussing the terms and addressing monotonicity, we first comment on whether or not the first derivative $\dot{E}$, in addition to the energy itself, is well-defined. One can check with (\ref{eq:meancurvatures}) that $\int_S H_S\,\mathrm{d}S$ as well as $\int_S H_S^2\,\mathrm{d}S$ are finite, however, $\int_S H_S^3\,\mathrm{d}S$ diverges. Since the first as well as the terms involving the energy momentum tensor are proportional to $H$, they are finite. Because $S$ is a manifold without boundary, $\int_S \theta_+D_\alpha\Omega^\alpha=-\int_S \Omega^\alpha D_\alpha\theta_+$, and $D_\alpha\theta_+$ is proportional to $\partial_uH$ and $\partial_vH$. Again, one can check explicitly with the help of (\ref{eq:meancurvatures}) that these derivatives are finite. Turning to the shear terms, we note that the shear tensors $\sigma^\pm_{ab}$ are also diverging at singular points, in particular in the same way as $\theta_\pm$ for non-degenerate singular points such as cusp or swallow-tail \cite{Seitz:1994xf}. Therefore, $\theta_-\sigma^+_{ab}\sigma_+^{ab}$ and $\theta_+\sigma^+_{ab}\sigma_-^{ab}$ are of order $H^3$, hence their integrals over $S$ diverge. However, close to a singular point, both terms have opposite signs and cancel each other. This can be seen by rewriting the expression with the help of $Q_{ab}^c:=h^d_a h^e_b\nabla_d h^c_e$ and noting that $\Theta_{ab}=-l_cQ_{ab}^c$, $\Xi_{ab}=-n_c Q_{ab}^c$:
\begin{equation}
\theta_-\sigma^+_{ab}\sigma_+^{ab}+\theta_+\sigma^+_{ab}\sigma_-^{ab}=\sigma_+^{ab}Q_cQ_{ab}^c\quad.
\end{equation}
Next, expressing $Q^a$ and $Q_{ab}^C$ in terms of the timelike and spacelike unit normals $t^a$ and $v^a$ yields:
\begin{align}
\sigma_+^{ab}Q_c&Q_{ab}^c= \frac{1}{\sqrt{2}}\tau\tau_{ab}\left(\tau^{ab}+H^{ab}\right)+\frac{1}{2}\tau^2\theta_+\nonumber
\\&-\frac{1}{\sqrt{2}}HH_{ab}\tau^{ab}
+\frac{1}{\sqrt{2}}HH_{ab}H^{ab}-\frac{1}{2}\theta_+H^2\quad.
\end{align}
All terms apart from the last two are at most of the order $H^2$ and thus integrable. The last two terms are diverging as $H^3$ near a singular point, but being of opposite sign, they precisely cancel each other. Hence, the integral of the shear terms is also finite.
Summarising, we found that the Hawking energy as well as its first derivative along the null generators of the past lightcone are well-defined even for surfaces including swallow-tail type singularities. Knowing that (\ref{eq:XXX}) is a well-defined quantity, we now address the rescaling freedom of $l^a$ before studying the monotonicity of (\ref{eq:XXX}).
\section{Choice of rescaling}
\label{sec:rescaling}
As mentioned earlier, once a scaling function $\alpha$ is chosen, the Hawking energy will monotonously increase and be positive along the family of constant (affine) parameter surfaces $(S_\lambda)$ associated with this rescaling, if and only if (\ref{eq:XXX}) is positive. Hayward \cite{Hayward:1993ma} pointed out that the sign of $\int_S \theta_\pm\,\mathrm{d}S$ is not an invariant under rescaling $l^a$. In particular, if $\theta_\pm$ changes its sign on $S$, $\int_S \theta_\pm\,\mathrm{d}S$ can take any sign and value by constructing an appropriate rescaling function $\alpha$ on $S$. In fact, since all terms appearing in (\ref{eq:XXX}) are not invariant under rescaling, one can use the rescaling freedom to simplify its right-hand-side. As in the weak lensing case, the term $\int_S\theta_+D_a\Omega^a$ can be eliminated even if $\theta_+$ is not strictly positive anymore. Under rescaling $l^a\rightarrow\alpha l^a$, $\alpha>0$, $\Omega_a$ transforms as
\begin{equation}
\Omega_a\rightarrow\Omega+D_a\ln\alpha\quad\Rightarrow\quad D_a\Omega^a\rightarrow D_a\Omega^a+D_aD^a\ln\alpha\quad,
\end{equation}
leading to the following Poisson equation for $\alpha$ on $S$, if the rescaling is used to eliminate $D_a\Omega^a$:
\begin{equation}
D_aD^a\ln\alpha=-D_a\Omega^a\quad.
\label{eq:poisson}
\end{equation}
We have two cases to consider depending on whether $S$ is smooth or Lipschitz.
\subsection{Poisson equation on smooth Riemannian manifold}
For smooth $S$, we have the following existence theorem for the Poisson equation:\\
\textit{Theorem:} On a closed Riemannian manifold $M$, if $\rho$ is a smooth function satisfying $\int_M \rho=0$: $\exists$ smooth solution to $\Delta\Phi=\rho$, unique up to the addition of a constant.\\
Since $S$ is a closed manifold, $\int_SD_a\Omega^a=0$ and therefore we can find an $\alpha$ such that $D_a\Omega^a=0$ after rescaling.
\subsection{Poisson equation on a Lipschitz manifold}
If $S$ is only Lipschitz continuous, we would still like to eliminate $\int_S\theta_+D_a\Omega^a$. Because the Poisson equation (\ref{eq:poisson}) contains second derivatives, it is ill-defined on a Lipschitz manifold. However, one can adapt a weak (i.e.\ distributional) formulation in the following way. Recall that a Riemannian Lipschitz manifold $(M,q)$ is a manifold $M$ equipped with a positive-definite metric $q$, for which all transition maps are locally Lipschitz functions. By Rademacher's theorem, a locally Lipschitz function $f:\mathbb{R}^n\rightarrow\mathbb{R}^m$ is differentiable almost everywhere (w.r.t\ to the n-dim. Lebesque measure). We denote the linear space of all Lipschitz continuous functions $\phi: M\rightarrow\mathbb{R}$ for which the norm
\begin{equation}
||\phi||^2:=\int_M (\phi^2+|\nabla\phi|^2)d\mu\;<\infty
\end{equation}
is finite by $\text{Lip}^{1,2}(M)$. This norm is well-defined because Rademacher's theorem ensures the existence of the gradient almost everywhere. We then define the Sobolev space $W^{1,2}(M)$ as the Cauchy completion of $\text{Lip}^{1,2}(M)$ with respect to the above norm $||\,.\,||$ . If $M$ is a compact, connected, oriented, Lipschitz manifold without boundary, the weak version of the Poisson equation reads
\begin{equation}
-\int_M\braket{\nabla\phi,\nabla f}d\mu=\int_M g\, \phi\,d\mu\quad,
\end{equation}
$\forall\, \phi\in W^{1,2}(M)$, given that $f\in W^{1,2}(M)$ and $g\in L^2(M)$ (see theorem 1.3 in \cite{laplacebeltrami}). Thus, provided that $\theta_+\in W^{1,2}(M)$ and $D_a\Omega^a\in L^2(M)$, we can find a function $\alpha$ such that
\begin{equation}
\int_S \theta_+\left(D_a\Omega^a+D_aD^a\ln\alpha\right)\mathrm{d}S=0\quad.
\end{equation}
Hence, one way to use the rescaling freedom is to eliminate the term $\int_S\theta_+ D_a\Omega^a\,\mathrm{d}S$ in (\ref{eq:XXX}), provided that $\theta_+\in W^{1,2}(S)$. $D_a\Omega^a$ is in $L^2(M)$ because of $\int_S D_a\Omega^a\,\mathrm{d}S=0$. If $\theta_+\notin W^{1,2}(S)$, we can use the rescaling of $l^a$ to achieve $\int_S\theta_+\,\mathrm{d}S>0$, but then assumptions on $D_a\Omega^a$ have to be made.
\section{Monotonicity}
\label{sec:monotonicity}
The crucial difference to the weak lensing case is that there now exists a region of negative $\theta_+$ on $S$ connected to the singular point. This implies that locally, the area decreases along $l^a$ although the total area of $S$ can still increase if $\dot{A}(S)=\int_S\theta_+\,\mathrm{d}S>0$. It is precisely this region in which energy can now be injected into the interior of $\dot{I}^-(p)$ from the exterior along causal curves. Hence, the naive monotonicity argument related to Fig.\ref{fig:monotonicity} holds only for regions with positive $\theta_+$ and fails in regions of negative $\theta_+$. In general, two different effects concerning monotonicity have to be taken into account:
\begin{itemize}
\item[(i)] A variation in the area $A$ leads to a change in the energy, because the amount of matter enclosed by $S$ changes. This effect is manifested in the first term in (\ref{eq:XXX}), describing nothing other than the change of $A$ along the null generators $l^a$.
\item[(ii)] Energy may leave $I^-(p)$ only in regions with $\theta_+>0$, and enter $I^-(p)$ only where $\theta_+<0$. This is accounted for by the second integral in (\ref{eq:XXX}), stating two contributions: shear and matter. The first corresponds to energy transported by the pure gravitational field in the form of gravitational waves, and is even present in vacuum. The latter contribution is due to matter encoded in the energy momentum tensor satisfying the DEC. A potential cosmological constant can be accommodated in the energy momentum tensor.
\end{itemize}
Assume in the following that $A$ is increasing along the family of surfaces, i.e. $\int_S \theta_+\,\mathrm{d}S>0$. In the case of a vacuum spacetime, the matter terms vanish and one only has to deal with the net flux of in- and outgoing shear contributions. Furthermore, by the Goldberg-Sachs theorem \cite{Goldberg2009}, the geodesic congruence $l^a$ in a vacuum spacetime $M$ is shear free, i.e.\ $\sigma^+_{ab}=0$, if and only if $M$ is algebraically special, that is $l^a$ is a repeated principle null direction, see also \cite{Ellis:2011pi} and references therein for generalisations. Demanding vanishing shear within the class of non-vacuum spacetimes imposes a strong constraint, see \cite{Adamo2012,Ellis:2011pi}.\\
So far, the studied configuration contained only one strong gravitational lens, see Fig. \ref{fig:spherlens}. Nevertheless, the obtained results can easily be generalised to configurations with multiple isolated strong lensing events taking place, that is, the swallow-tail singular points on $S$ have to be isolated. The more lensing events happen, the larger the fraction of $S$ with negative $\theta_+$. Having more and more lensing events present will ultimately turn $\int_S\theta_+\,\mathrm{d}S$ negative and therefore the whole lightcone will refocus. This indicates that enough energy is concentrated in the interior to cause the shrinking of $S$.
\section{Extensions}
\label{sec:extensions}
Until now, the discussion was restricted to a family of topological 2-spheres. In the following, we briefly review how a change of topology affects the results. One could imagine that more complicated lensing configurations may cause $C^-(p)\cap\dot{I}^-(p)$ to still consist of one connected component, but to be topologically different from $\mathbb{R}\times S^2$. The Hawking energy can be generalised to arbitrary closed orientable surfaces $\tilde{S}$, characterised by their genus $g$, by using the Gauss Bonnet theorem $\int_{\tilde{S}} {}^2R\,\mathrm{d}\tilde{S}=8\pi(1-g)$, see for instance \cite{Hayward:1993ph}. Then, (\ref{eq:energy}) reads instead
\begin{equation}
E(\tilde{S}):= \frac{\sqrt{A(\tilde{S})}}{32\pi^{3/2}}\left(8\pi(1-g(\tilde{S}))+\int_{\tilde{S}}\theta_+\theta_-\,\mathrm{d}\tilde{S}\right)\quad.
\end{equation}
However, if $g(\tilde{S})\geq 1$, and the surface is non-trapped on average in the sense of \cite{Hayward:1993ma}, i.e.\ $\int_{\tilde{S}} \theta_+\theta_-\,\mathrm{d}\tilde{S}<0$, the Hawking energy is negative.
In principle, it is also possible that $C^-(p)\cap\dot{I}^-(p)$ splits into $n$ disconnected components of genus $g_i$ :
\begin{equation}
C^-(p)\cap\dot{I}^-(p)=\mathbb{R}\times S_{g_1}\times\dots \times S_{g_n}\quad.
\end{equation}
This can happen for instance in situations topologically similar to a Schwarzschild black hole (see e.g.\ \cite{Perlick2004}). Hayward \cite{Hayward:1993ph} observed that the Hawking energy for $n$ disconnected surfaces is superadditive. If we denote $S_1\cup\dots\cup S_n=S_{\cup}$, then
\begin{equation}
E_\cup=\sqrt{\frac{A_\cup}{A_1}}E_1+\dots+\sqrt{\frac{A_\cup}{A_n}} E_n>E_1+\dots+E_n\quad.
\end{equation}
This property is in contrast to the expected subadditivity of gravitational systems.
\section{Conclusions}
\label{sec:conclusions}
The Hawking energy provides a reasonable definition of energy in the setting of cosmology. The past lightcone of a point $p$ in spacetime is closely linked to cosmological observations and therefore provides the ideal geometric structure to study the properties of the Hawking energy in a physical set-up. The part of it within the causal boundary, $C^-(p)\cap\dot{I}^-(p)$, provides the mathematical arena in which the Hawking energy is studied. It is a Lipschitz continuous manifold, potentially containing points where $\theta_+$ is singular, and is assumed to have spherical topology: $C^-(p)\cap\dot{I}^-(p)\simeq\mathbb{R}\times S^2$. This seems to be a natural case, but it would be interesting to get a better understanding whether, and under what circumstances other topologies may arise.
Assuming that the universe is described by a globally hyperbolic spacetime in which all matter obeys the DEC, strong gravitational fields may cause the lightcone to intersect itself locally at singular points, or globally. Since these singular points are conjugate to $p$, the presence of singularities indicates the existence of regions on $S$ where the expansion $\theta_+$ of the null generators is negative. The only two stable types of singularities appearing in lightcone slices are cusp and swallow-tail singularities \cite{Low:1993}. As a main result, we find that the Hawking energy of surfaces containing cusp singular points is divergent, but finite for swallow-tail singularities.
For a lightcone, the presence or absence of self-intersections gives rise to two natural regimes. The weak lensing regime, in which self-intersections are absent, exhibits a positive expansion parameter $\theta_+>0$ everywhere on smooth surfaces $S$. The Hawking energy is positive and monotonously increases along the null generators of the past lightcone, following directly from Eardley's results \cite{Eardley:1979dra} and the small sphere limit \cite{Horowitz:1982}. Studying a lightcone going through multiple isolated strong gravitational lenses, for instance caused by isolated local matter overdensities, we find that the Hawking energy associated with the exterior part of a lightcone slice, exclusively containing swallow-tail singular points, is finite. The energy of the total lightcone slice, however, is infinite due to the presence of cusp singularities in the interior part. Monotonicity (\ref{eq:XXX}) depends upon two effects. Firstly, the area of $S$ changes along the null generators. Secondly, and in contrast to the weak lensing case, matter may enter the interior of $\dot{I}^-(p)\cap C^-(p)$ through regions where $\theta_+<0$. Hence in general, the Hawking energy is not monotonous along the past lightcone anymore and monotonicity depends on the balance of in- and outgoing energy flux.
More generally, since the lightcone construction differs from other instantaneous wavefronts only by the small sphere limit,
the results concerning well-definedness and monotonicity of the Hawking energy extend to all instantaneous wavefronts. In particular, wavefronts containing swallow-tail singular points have a finite associated Hawking energy, whereas the energy of wavefronts containing cusp singularities diverges.
With the above results at hand, it might be interesting to study the energy of general wavefronts, such as light signals or gravitational waves, propagating through an (inhomogeneous) universe. In the particular context of inhomogeneous cosmology, it might be a useful tool to compare an inhomogeneous universe with a FLRW-reference universe and thereby address the so-called fitting problem in inhomogeneous cosmology \cite{ellis_maartens_maccallum_2012} by comparing domains of equal mass or energy, as it was advocated in \cite{Ellis:1998ha}.
\begin{acknowledgments}
The author would like to thank L. Brunswic, T. Buchert, M. Carfora, D. Giulini, V. Perlick, and R. Tanzi for fruitful discussions and comments. This work is supported by the DFG Research Training Group "Models of Gravity" as well as by the ERC Advanced Grant 740021--ARTHUS, PI: T. Buchert.
\end{acknowledgments}
\interlinepenalty=10000
\enlargethispage{3\baselineskip}
| -41,832.539656 |
[
-3.166015625,
2.826171875
] | 36.81592 |
[
-2.65234375,
0.921875,
-1.8037109375,
-5.99609375,
-0.93701171875,
8.2890625
] |
[
3.8046875,
8.921875,
1.8564453125,
5.94140625
] | 407 | 7,321 |
[
-3.64453125,
4.21875
] | 26.560079 |
[
-5.68359375,
-4.296875,
-5.16015625,
-2.5078125,
1.78125,
13.4453125
] | 0.911494 | 19.387641 | 22.63352 | 1.807643 |
[
2.329263925552368
] | -26,273.602264 | 5.994263 | -41,101.066791 | 1.385471 | 6.051427 |
[
-2.248046875,
-3.5546875,
-3.650390625,
-5.01953125,
1.9609375,
12.2421875
] |
[
-5.64453125,
-2.453125,
-2.05859375,
-1.5576171875,
4.1484375,
5.1640625
] | |
BkiUbjE5qoaAwj4XNMHR
|
\part{Introduction}
\label{part:intro}
\section{Motivation}
A fundamental idea in classifying Lie algebras is not only to list the possible algebras, but to find each algebra's subalgebras. The traditional approach to this problem follows the work of Eugene Dynkin \cite{dyn2}, who considered complex Lie algebras. Many properties of an algebra are more easily seen when it is considered as a complex algebra, and it is possible to list the complex algebras which exist within, and which contain, a given complex Lie algebra.
It is also possible to determine the real subalgebras of a complex Lie algebra. However, for applications, such as to physics, it is important not only to know the subaglebras of a real algebra but also to describe them in terms of an explicit representation. Dynkin's methods for classifying complex algebras are not helpful in this situation.
There are four infinite families of complex Lie algebras and exactly five algebras, called the {\it exceptional Lie algebras}, which do not fit into those families \cite{killing, cartan}. The infinite families of Lie algebras correspond to certain well-known groups over the three division algebras $\mathbb{R}$, $\mathbb{C}$, and $\mathbb{H}$ \cite{baez}. It is known that the exceptional Lie algebras, named $G_2$, $F_4$, $E_6$, $E_7$, and $E_8$, are related to the octonions $\mathbb{O}$, the largest division algebra \cite{baez}. The smallest algebra, $G_2$, is the automorphism group of the octonions. The other four exceptional Lie algebras appear as the bottom row and right column of the Magic Square of Lie algebras, constructed first by Freudenthal and Tits in 1964 and 1966, respectively \cite{freudenthal_magic_square, tits_magic_square}. Each division algebra $\mathbb{F} = \mathbb{R}, \mathbb{C}, \mathbb{H}, \mathbb{O}$ labels both a column and row in the $4 \times 4$ Magic Square, and each entry is a Lie algebra built over $\mathbb{F}_1 \times \mathbb{F}_2$. The Magic Square idea naturally yields several nested chains of subalgebras associated with the division algebras. For example, in \cite{gursey_magic_square}, G\"ursey showed that
$$ SU(2) \times U(1) \subset SU(2) \times U(1) \times SU(2) \times U(1) $$
$$ \hspace{3cm} \subset SU(4) \times SU(2) \times U(1) \subset SO(10) \times SO(2) \subset E_6$$
We will construct additional subalgebra chains contained within $E_6$ in this thesis.
As discussed in more detail below, the exceptional Lie group $E_6$ is related to the exceptional Jordan algebra of $3 \times 3$ hermitian octonionic matrices. The corresponding Dynkin diagram for $E_6$ shows that it contains the group $SO(8)$, indicating that $E_6$ may share some of the triality characteristics exhibited by $SO(8)$. While the Dynkin diagram can be used to reconstruct the commutation relations for $E_6$, and while these commutation relations can in turn be used to find subalgebras of $E_6$, we note that this is done entirely in the complex setting. The subalgebras identified by these constructions are complex Lie algebras, and a real form of the algebra might not contain a real form of the complex subalgebra.
The division algebras also seem to play an important role in the physics of string theory and supersymmetry. In 1983, Kugo and Townsend \cite{kugo_townsend} related spinors in spacetimes of dimension $3, 4,$ and $6$ with the sequence of Lorentz groups $SL(2,\mathbb{R})$, $SL(2,\mathbb{C})$, and $SL(2,\mathbb{H})$. In \cite{manogue_fairlie_composite_string}, Fairlie and Manogue extended this to $SL(2,\mathbb{O})$ and constructed a parametrization of the bosonic string coordinates in space-times of dimension $3$, $4$, $6$, and $10$. This parametrization exists in those dimensions due to the identity $|z||z^\prime| = |zz^\prime|$ which holds for the four division algebras. In \cite{manogue_fairlie_covariant_superstring}, Fairlie and Manogue gave a parametrization of the covariant superstring that both encodes the bosonic and fermionic components of the superstring and satisfies three of the four equations of motion. Division algebras again played an integral part in this construction, leading Fairlie and Manogue to assert that the special dimensionality associated with sypersymmetric gauge theories is just a reflection of the properties of division algebras.
A number of results specifically relate the octonions to the superstring and supersymmetry. In \cite{manogue_sudbery_gen_sol}, Manogue and Sudbery use an octonionic formalism for ten-dimensional vectors and spinors to solve the equations of motion for the Green-Schwarz Lagrangian for the superstring. By representing a vector in $(9+1)$-dimensional Minkowski space with a $2 \times 2$ hermitian octonionic matrix, they construct a lightlike ten-vector by {\it squaring} a column of two octonions, which is a spinor. In \cite{schray}, Schray combines the spinor and vector into a $3 \times 3$ Grassmann, octonionic Jordan matrix and expresses the Lorentz and supersymmetry transformations of fermionic and bosonic variables in terms of Jordan products. This connection between the Jordan algebra and superstrings has been pursued by others \cite{corrigan1989}. In fact, the octonionic Jordan algebra has another nice connection to physics. In \cite{gunaydin_1993}, G\"unaydin notes that the reduced structure group of the Jordan algebra of Hermitian octonionic $3 \times 3$ matrices is a real form of the complex Lie algebra $E_6$ with signature $(52,26)$. Other approaches also relate $E_6$ to the octonionic Jordan algebra, see \cite{toppan, lin}. The exceptional Lie group $E_6$ seems to be the right size to contain the Lie group $SU(3,\mathbb{C}) \times SU(2,\mathbb{C}) \times U(1,\mathbb{C})$, which describes the fundamental forces in the Standard Model of particle physics, and the Lorentz group $SO(3,1,\mathbb{R})$ of special relativity \cite{e6_and_decays, e6_and_physics}. In the Standard Model, the strong force is associated with the Lie group $SU(3,\mathbb{C})$, the weak force is tied to $SU(2,\mathbb{C})$, and the electromagnetic force is represented by $U(1,\mathbb{C})$.
\section{Summary}
Groups are used in quantum mechanics and particle physics in a fundamental way. In quantum mechanics, operators describe the energy and time evolution of a particle. The transformations which leave the Hamiltonian operator invariant in the Schr\"odinger equation form a group. The particle is only measured with certain values, and these operator eigenvalues may be labeled with a representation of the group. In traditional quantum mechanics, the representations are affiliated with complex Lie algebras. This complexification process involves a square root of -1, usually labeled $i$, and causes some difficulty. Myrheim notes in \cite{myrheim} that the standard complex notation identifies the role played by several different square roots of -1 in the field equation and Hilbert space of quantum mechanics. When working with the octonions, it is even more difficult to identify the object or objects which should play the various roles of $\sqrt{-1}$. If $\sqrt{-1}$ lies within the octonions, which one is 'special' and why was it chosen? If $\sqrt{-1}$ lies outside the octonions, where does it come from and how does it interact with the octonions? By avoiding issues related to the complexification of Lie algebras, we can leave these questions unanswered while still learning a great deal about the real structure of $sl(3,\mathbb{O})$, a real form of $E_6$.
Given the connection between the Lie group $E_6$, Jordan matrices, and superstrings, several authors have discussed the mathematical properties of the exceptional Lie algebra $E_6$ and its subalgebra $F_4$ \cite{schafer,lin,allenAndFerrar}. However, these discussions typically involve complex Lie algebras instead of real Lie algebras.
In \cite{manogue_dray}, Manogue and Dray construct the Lie group $SL(3,\mathbb{O})$, and the Lie algebra $sl(3,\mathbb{O})$ corresponding to this group is a real form of the Lie algebra $E_6$. The present work builds upon their results to provide the algebraic subalgebra structure of $sl(3,\mathbb{O})$. We provide the multiplication table for the real Lie algebra $sl(3,\mathbb{O})$ and give towers of subalgebras in $sl(3,\mathbb{O})$. We use our knowledge of the multiplication table of $sl(3,\mathbb{O})$ to label the towers with the appropriate Casimir operators and give explicit bases for the various subalgebras. We use the Casimir operators and explicit bases to distinguish between the physically interesting subalgebras of $sl(3,\mathbb{O})$ and other physicially uninteresting, but isomorphic, subalgebras.
Even though the material presented in Chapters $2$ and $3$ summarizes known results, the geometric treatment of root and weight diagrams in Chapter $2$ is not readily available in the literature. While the interpretation of the group $E_6$ as $SL(3,\mathbb{O})$ is due to Manogue and Dray \cite{manogue_dray}, the detailed analysis of the Lie algebra $sl(3,\mathbb{O})$ presented here is new. In particular, the notion of continuous type transformations in Section \ref{ch:E6_basic_structure.Type_Transformation}, the use of involutive automorphisms in Chapter \ref{ch:E6_further_structure}, and the detailed towers of subalgebras with nested Casimir operators in Section \ref{ch:E6_further_structure.chains_of_subalgebras} are due to the author.
\subsection{Thesis Organization}
This thesis is arranged into seven chapters. The second chapter provides a general introduction to Lie groups and Lie algebras. Section \ref{ch:Lie_groups_Lie_algebras.Basic_Definitions} gives the basic definitions we use for the study of Lie groups and Lie algebras. We introduce an example in Section \ref{ch:Lie_groups_Lie_algebras.Basic_Definitions.An_Example} and use this example to illustrate the definitions in the further sections. The definitions needed for Lie groups are given in Section \ref{ch:Lie_groups_Lie_algebras.Basic_Definitions.Lie_Groups}, while definitions for Lie algebras are contained within Section \ref{ch:Lie_groups_Lie_algebras.Basic_Definitions.Lie_Algebras}.
Properties of a Lie group can often be studied using its associated Lie algebra (and vice versa) using the tools of differentiation, exponentiation, and commutation; these tools are defined in Section \ref{ch:Lie_groups_Lie_algebras.Basic_Definitions.Diff_Exp_Comm}. The regular representation of a Lie algebra and the algebra's Killing form can be used to identify the signature of a simple Lie algebra; these tools are described in Section \ref{ch:Lie_groups_Lie_algebras.Basic_Definitions.regular_rep_Killing_form}. We finish Section \ref{ch:Lie_groups_Lie_algebras.Basic_Definitions} with a discussion in Section \ref{ch:Lie_groups_Lie_algebras.Basic_Definitions.conventions} of the different conventions employed by mathematicians and physicists when defining the Lie algebras of the classical matrix groups; These groups and algebras are listed in Section \ref{ch:Lie_groups_Lie_algebras.Basic_Definitions.classical_matrix_groups}.
Section \ref{ch:Joma_Paper} includes verbatim a non-technical paper \cite{joma_paper} which covers the classification of the simple complex Lie algebras using their root and weight diagrams, and concludes with the complex Lie subalgebra structure of $E_6$. This paper is intended for a non-expert audience and the definitions are kept deliberately informal; we therefore precede the paper here with precise definitions in Section \ref{ch:Joma_Paper.definitions}.
The paper itself begins in Section \ref{ch:Joma_Paper.Introduction} with a brief introduction.
In Section \ref{ch:Joma_Paper.Root_Weight_Construction}, we show how to construct the root and weight diagrams of any simple complex algebra from its Dynkin diagram. This section concludes with the root and weight diagrams of the rank $3$ complex simple Lie algebras. In Section \ref{ch:Joma_Paper.subalgebras} we show how root and weight diagrams of algebras $g$ and $h$ can be used to show $g \subset h$. In particular, we develop two methods, called {\it slicings} and {\it projections}, which we use to identify the complex subalgebras of algebras whose root and weight diagrams are of any dimension. In Section \ref{ch:Joma_Paper.subalgebras_greater_than_3}, we apply these two methods to the four-dimensional root diagram of $F_4$. We discuss how these methods may be applied to the six-dimensional diagrams of $E_6$, and produce a diagram showing the nesting of complex subalgebras contained within the complex Lie algebra $E_6$. This diagram will be utilized and expanded upon in Chapters \ref{ch:E6_basic_structure} and \ref{ch:E6_further_structure}, where we produce the nesting of the real subalgebras contained within the real Lie algebra $sl(3,\mathbb{O})$, which is a real form of the complex Lie algebra $E_6$. The conclusion of this paper, in Section \ref{ch:Joma_Paper.conclusion}, is also incorporated in the conclusion (Chapter \ref{ch:Conclusion}) of this thesis.
Chapter \ref{ch:Division_Algebras_Applications} discusses the division algebras and their application to structures pertinent to geometry and physics. The first two sections of this chapter summarize the work of Manogue and Schray, given in \cite{manogue_schray}. Section \ref{ch:Division_Algebras_Applications.Quaternions_Octonions} discusses the algebraic and geometric properties of the four normed division algebras. Their algebraic properties are covered in Section \ref{ch:Division_Algebras_Applications.Quaternions_Octonions.Division_Algebras}, and in Section \ref{ch:Division_Algebras_Applications.Quaternions_Octonions.Conjugation_Reflecitons_Rotations}, we construct rotations and reflections in $\mathbb{R}^4$ and $\mathbb{R}^8$ using conjugation maps. These maps will play a vital role in the construction of the generalized Lorentz transformations of $SO(9,1,\mathbb{R})$, which is discussed in Section \ref{ch:Division_Algebras_Applications.Lorentz_Transformations.Lorentz_Transformations_for_Spacetimes_involving_Division_Algebras}, and in our construction of $SL(3,\mathbb{O})$ in Chapter \ref{ch:E6_basic_structure}. We discuss triality as it pertains to the octonions in Section \ref{ch:Division_Algebras_Applications.Quaternions_Octonions.Triality}. Section \ref{ch:Division_Algebras_Applications.Lorentz_Transformations} discusses Lorentz transformations and spacetime. In particular, Section \ref{ch:Division_Algebras_Applications.Lorentz_Transformations.General_Lorentz_Transformations} discusses~$(k+1)$-dimensional spacetime and Lorentz transformations for any finite~$k$, Section \ref{ch:Division_Algebras_Applications.Lorentz_Transformations.Division_Algebras_and_Spacetimes} shows how spacetimes where~$k = 2, 3, 5, 9$ can be identified with vectors and spinors by using the division algebras, and Lorentz transformations utilizing the division alagebras are constructed for these particular spacetimes in Section \ref{ch:Division_Algebras_Applications.Lorentz_Transformations.Lorentz_Transformations_for_Spacetimes_involving_Division_Algebras}. We conclude Chapter \ref{ch:Division_Algebras_Applications} by constructing the exceptional Jordan Algebra $M_3(\mathbb{O})$, which is also known as the {\it Albert Algebra}. We describe the Lie group $SL(3,\mathbb{O})$ as the group which preserves the determinant of $M_3(\mathbb{O})$. By letting $SL(3,\mathbb{O})$ act on the Albert Algebra in Chapter \ref{ch:E6_basic_structure}, we are able to construct the commutation table of the Lie algebra $sl(3,\mathbb{O})$.
Chapter \ref{ch:E6_basic_structure} incorporates the work of Manogue and Dray \cite{manogue_dray} to construct the Lie group $SL(3,\mathbb{O})$ and explicitly extends their results to the Lie algebra $sl(3,\mathbb{O})$, which is a real form of $E_6$. We extend the construction of $SL(2,\mathbb{O}) = SO(9,1,\mathbb{R})$ which was given by Manogue and Schray in \cite{manogue_schray} and reviewed in Section \ref{ch:Division_Algebras_Applications.Lorentz_Transformations} from the $2 \times 2$ case to the $3 \times 3$ case. This work uses the exceptional Jordan Algebra $M_3(\mathbb{O})$ and a discrete {\it type} map, as described in \cite{manogue_dray}, and builds upon the work of Manogue and Dray who showed that $SL(3,\mathbb{O})$ contains three $SO(9,1,\mathbb{R})$ subgroups. We expand the results of Manogue and Dray by using the explicit representation of the Lie group $SL(3,\mathbb{O})$ to construct its Lie algebra $sl(3,\mathbb{O})$ in Section \ref{ch:E6_basic_structure.ConstructingE6algebra}. In particular, we find an explicit basis for the underlying vector space of the algebra and use the group transformations to define the product in the algebra. These computations were done using Maple and {\sc Perl}. We also extend the work of Manogue and Dray by giving explicit bases for the three subalgebra chains of $so(9,1,\mathbb{R})$ within $sl(3,\mathbb{O})$ and identifying six Casimir operators for the exceptional Lie algebra $E_6$. We discuss triality as it relates to the Lie groups $G_2$ and $SO(8,\mathbb{R})$ in Section \ref{ch:E6_basic_structure.Triality}, and introduce the notion of {\it strong} triality. We also relate triality to our notion of {\it type} in this section. In Section \ref{ch:E6_basic_structure.Type_Transformation}, we generalize our discrete notion of type to a continuous transformation in the group $SL(3,\mathbb{O})$. We identify this transformation in a particular subgroup $SO(3,\mathbb{R})$ of $SL(3,\mathbb{O})$, and, using its corresponding algebra $so(3,\mathbb{R})$, we find that any algebra containing this algebra $so(3,\mathbb{R})$ must be {\it type independent}. We use this idea to separate the algebra $sl(3,\mathbb{O})$ into chains of subalgebra of the form $g_1 \oplus g_2$. In Section \ref{ch:E6_basic_structure.Reduction_of_O} we give an explicit basis for each of the subaglebras $su(n,\mathbb{F})$ and $sl(n,\mathbb{F})$ for $\mathbb{F} = \mathbb{C}, \mathbb{H}, \mathbb{O}$ and $n = 2,3$. We again create a chain of subalgebras $g_1 \oplus g_2$ of $sl(3,\mathbb{O})$ where $g_1$ is either $su(n,\mathbb{F})$ or $sl(n,\mathbb{F})$ and $g_2$ is the maximal subalgebra of $sl(3,\mathbb{O})$ which commutes with $g_1$. In Section \ref{ch:E6_basic_structure.fix_l} we identify a subalgebra $F_\l$ of $sl(3,\mathbb{O})$ which is the stabilizer of the type $1$ octonionic unit $\l$ in $M_3(\mathbb{O})$. This subalgebra is neither simple nor semi-simple, and consists of $so(8,1,\mathbb{R})$ together with an abelian ideal of dimension $16$. Finally, in Section \ref{ch:E6_basic_structure.Gell-Mann}, we give an isomorphism between a basis for our preferred octonionic representation of the subalgebra $su(3,\mathbb{C})^C \subset G_2$ and the $3 \times 3$ Gell-Mann matrices.
In Chapter \ref{ch:E6_further_structure}, we find additional subalgebras of $sl(3,\mathbb{O})$ by adapting methods involving automorphisms of Lie algebras. We review the properties of automorphisms and involutive automorphisms of complex Lie algebras in \ref{ch:E6_further_structure.Automorphisms_of_Lie_Algebras}. The Killing form is used to show that an involutive automorphism of a complex Lie algebra $g^\mathbb{C}$ will map one real form of $g^\mathbb{C}$ to a possibly different real form of $g^\mathbb{C}$. In Section \ref{ch:E6_further_structure.Automorphisms_of_Lie_Algebras.id_algebras} we provide standard theorems and lemmas from \cite{cornwell, gilmore} that help us establish the identity and ``$\mathbb{R}$-simpleness'' of a real subalgebra of $sl(3,\mathbb{O})$. One such technique which we repeatedly employ identifies a real subalgebra of $sl(3,\mathbb{O})$ using only the algebra's dimension, rank, and signature. We construct three involutive automorphisms of $E_6$ in Section \ref{ch:E6_further_structure.three_automorphisms} and identify new subalgebras of $sl(3,\mathbb{O})$ using these automorphisms. In particular, if $\phi$ is such an involutive automorphism, then both $sl(3,\mathbb{O}) \cap \phi\left(sl\left(3,\mathbb{O}\right)\right)$ and the pre-image of the compact part of $\phi\left(sl\left(3,\mathbb{O}\right)\right)$ are subalgebras of $sl(3,\mathbb{O})$. We use the methods from Section \ref{ch:E6_further_structure.Automorphisms_of_Lie_Algebras.id_algebras} to determine the $\mathbb{R}$-semi-simple decompositions of certain real subalgebras of $sl(3,\mathbb{O})$. We expand this approach in Section \ref{ch:E6_further_structure.more_applications_of_three_automorphisms} to find additional subalgebras of $sl(3,\mathbb{O})$ by using the fact that the set of involutive automorphisms form a group. We use the action of the individual automoprhisms to partition the basis of $sl(3,\mathbb{O})$ into sets and combine various sets to form subalgebras of $sl(3,\mathbb{O})$. This approach yields a number of subalgebras of $sl(3,\mathbb{O})$ which could not be readily found using the methods of Chapter \ref{ch:E6_basic_structure}. We find a real form of $F_4$ with signature $(36,16)$ and multiple real forms of $A_5$, $C_4$ and $C_3$. We provide explicit bases for these real forms of these subalgebras and identify their Casimir operators. Finally, in Section \ref{ch:E6_further_structure.chains_of_subalgebras}, we produce chains of subalgebras of $sl(3,\mathbb{O})$ in which each larger algebra is obtained by extending the basis of a subalgebra. In particular, the subalgebras in these chains use the same choice of Casimir operators. We find this structure to be much more intricate than the structure indicated for the complex Lie algebra $E_6$ indicated in Chapter \ref{ch:Joma_Paper.subalgebras_greater_than_3}.
We summarize this work in Chapter \ref{ch:Conclusion} after identifying some open questions in Chapter \ref{ch:Open_Questions}.
\newpage{}
\part{Lie Groups and Lie Algebras}
\label{ch:Lie_groups_Lie_algebras}
This chapter begins with a brief introduction to the principle ideas needed for the study of Lie groups and Lie algebras. In Section 2.1.1, we begin with an example of a Lie group. We will develop this example throughout Section 2.1 to help illustrate the definitions of a group, algebra, differentiation and exponentiation, as found in \cite{boothby}. Although we give the general definitions of differentiation and exponentiation, we shall also give equivalent definitions that are helpful for the study of the classical matrix groups, which is of particular interest to this thesis. After discussing the adjoint representation of a Lie algebra and the Killing form in Section 2.1.7, we include a paper in Section 2.2 that discusses the classification of Lie algebras. This paper is to appear in the Journal of Online Mathematics, and is included without modification. It concludes with a tower of complex Lie algebras contained within $E_6$. Additional information about Lie groups and Lie algebras can be found in \cite{gilmore, cornwell, jacobsen}.
\section{Basic Definitions}
\label{ch:Lie_groups_Lie_algebras.Basic_Definitions}
\subsection{An Example}
\label{ch:Lie_groups_Lie_algebras.Basic_Definitions.An_Example}
We begin with a simple example. The solid ball of radius $r = \pi$, in three dimensions, is an example of a Lie group of rotations. Each point $p$ in the ball can be used to define an axis of rotation extending from the origin through the point $p$, while the distance $|p|$ from the origin to $p$ can be used to specify the amount of rotation about that axis. Hence, the point at the origin represents a rotation about any axis through $0$ radians, or the identity transformation, while the point $p = (x = \pi, y = 0, z = 0)$ represents a rotation of $\pi$ radians about the $x$-axis. Every rotation $p$ has an inverse rotation $-p$, and we note that antipodal points on the boundary of the ball may be identified since they are the same rotation. If we compose one rotation $p$ about one axis with another rotation $q$ about a second axis, the result is a rotation $s = q \circ p$ about a third axis. It is important to note that the resulting axis and amount of rotation depends continuously on both $p$ and $q$. Thus, we have given the ball of radius $r = \pi$ in three dimensions a group structure.
\subsection{Lie Groups}
\label{ch:Lie_groups_Lie_algebras.Basic_Definitions.Lie_Groups}
Informally, a Lie group is both a group and a manifold. Before we give a precise definition, we must first define the concepts of a topological and differentiable manifold as well as a group. We follow the treatment of Lie groups given in \cite{boothby}, where additional information may be found.
A {\it topological manifold} $M$ of dimension $n$ is a topological space which is Hausdorff, locally Euclidean of dimension $n$, and has a countable basis of open sets. A {\it differentiable manifold} is a topological manifold $M$ with a $C^\infty$ (smooth) structure, which is a family $\mathcal{U} = \lbrace \left( u_\alpha, \phi_\alpha \right) \rbrace$ of coordinate neighborhoods such that
\begin{enumerate}
\item $\cup\ u_\alpha$ covers $M$
\item $u_\alpha \cap u_\beta \ne \emptyset \implies \phi_\alpha \circ \phi_\beta^{-1} $ and $\phi_\beta \circ \phi_\alpha^{-1}$ are diffeomorphisms
\item Any coordinate neighborhood $\left( u_\gamma, \phi_\gamma \right)$ compatible with every $ \left( u_\alpha, \phi_\alpha \right) \in \mathcal{U}$ is in $\mathcal{U}$
\end{enumerate}
In general, a differentiable manifold may require more than one coordinate neighborhood. The first condition above guarantees that there is a coordinate neighborhood for each part of the manifold, while the second condition states that we can smoothly change from one coordinate neighborhood to another whenever their intersection is non-empty. The third condition guarantees that our atlas $\mathcal{U}$ of coordinate neighborhoods is complete.
A {\it group} is a set $G$ together with a map $\cdot : G \times G \to G$, referred to as multiplication, such that
\begin{enumerate}
\item $\exists \, e \in G$ such that $x \cdot e = x = e \cdot x \; \forall \, x \in G$
\item $\forall \, x \in G, \; \exists \, x^{-1} \in G$ such that $x \cdot x^{-1} = e = x^{-1} \cdot x$
\item $(x \cdot y) \cdot z = x \cdot (y \cdot z) \; \forall \, x,y,z \in G$
\end{enumerate}
That is, there is an identity element in the group, there is an inverse for every element in the group, and the multiplication is associative.
A {\it Lie group} is a differentiable manifold $G$ which is also a group whose composition map $G \times G \to G$ defined by $(x,y) = x\cdot y$ and inverse map $G \to G$ defined by $x \to x^{-1}$ are both $C^\infty$ mappings.
In the example from Section \ref{ch:Lie_groups_Lie_algebras.Basic_Definitions.An_Example}, the ball of radius $r = \pi$ is a $3$-dimensional topological manifold. To treat this solid ball as a differentiable manifold, we must be able to put coordinates on the ball by mapping it to $\mathbb{R}^3$. We already saw that we could interpret the solid ball as a group in Section \ref{ch:Lie_groups_Lie_algebras.Basic_Definitions.An_Example}. The solid ball became a Lie group once it was assigned a smooth map giving the inverse of each rotation and a smooth map composing two rotations. We note that the identity $e$ of this Lie group is the origin of the ball. We stress that while we are primarily interested in thinking of Lie groups as rotation groups for this thesis, there are many other examples of Lie groups.
\subsection{Lie Algebras}
\label{ch:Lie_groups_Lie_algebras.Basic_Definitions.Lie_Algebras}
We review here the definition of a Lie algebra in its general formalism. This treatment again is based upon \cite{boothby}. Additional information about Lie algebras may also be found in \cite{cornwell, gilmore, jacobsen, onishchik, bourbaki}.
A {\it real Lie algebra} $g$ is a vector space $V$ over $\mathbb{R}$ along with a product $\left[ \hspace{.15cm} , \hspace{.15cm} \right] : g \times g \to g$, called a commutator or bracket, which is bilinear,
$$ \left[ax + by, z \right] = a\left[x, z\right] + b\left[ y, z\right]$$
anti-commutative,
$$ \left[ x, y\right] = - \left[ y, x \right]$$
and satisfies the Jacobi identity
$$\left[ \left[ x,y \right], z\right] + \left[ \left[ y,z \right], x \right] + \left[ \left[ z, x \right], y \right] = 0$$
for any $x,y,z \in g$ and $a,b \in \mathbb{R}$.
A {\it complex Lie algebra} $g$ is a vector space $V$ over $\mathbb{C}$ with a bilinear product which satisfies the three properties above, but now with $a,b \in \mathbb{C}$.
{\bf Example 1:} Given a Lie group $G$, let $L_p : G \to G$ be left translation by $p$ (more formally a diffeomorphism from $G$ to itself given by $L_p(r) = pr$ for $p,r \in G$). A vector field $X$ on G is {\it left-invariant} if, for any $p, r \in G$, it satisfies $(dL_{rp^{-1}})(X_p) = X_r$ where $X_p$ is the value of $X$ at $p$.
Let $\mathfrak{X}(G)$ denote the set of all $C^\infty$ left-invariant vector fields defined on $G$.
This is a vector space over $\mathbb{R}$, and it may be equipped with a multiplication
$\left[ \hspace{.15cm}, \hspace{.15cm} \right] : \mathfrak{X}(G) \times \mathfrak{X}(G) \to \mathfrak{X}(G)$ given by
$$ \left[ X, Y \right]_p f = (XY - YX)_p f = X_p(Yf) - Y_p(Xf), \hspace{.75cm} f \in C^\infty(p)$$
called the commutator of $X$ and $Y$. As shown in \cite{boothby}, the commutator of left-invariant vector fields is again left-invariant. Hence, the vector space $\mathfrak{X}(G)$ along with the product $\left[ X, Y \right]$ is a real Lie algebra whose dimension is the dimension of the Lie group $G$.
{\bf Example 2:} The Lie algebra of a classical matrix group $G$ is the tangent space $T_eG$ at the identity $e$ of the group with the product $\left[ \hspace{.15cm}, \hspace{.15cm} \right] : T_eG \times T_eG \to T_eG$ defined by
$$ \left[ X, Y \right] = X Y - Y X$$
for any $X, Y \in T_eG$ where $X Y$ denotes ordinary matrix multiplication, as shown in \cite{cornwell}. This concept of a Lie algebra of a matrix Lie group will be useful in Chapter \ref{ch:E6_basic_structure}.
There is a $1-1$ correspondence between left-invariant vector fields on a Lie group $G$ and the vectors of $T_eG$, as any vector from $T_eG$ may be used to generate a left-invariant vector field on $G$ and each left-invariant vector field on $G$ is completely determined by its value at the identity element $e$ of $G$.
In our example using the solid ball of rotations as the Lie group, the corresponding Lie algebra consists of the vector space $\mathbb{R}^3$. The tangent vector to each rotation at the identity (origin) in the group is parallel to the axis of rotation. The commutator of two tangent vectors $x$ and $y$ can be given by their cross product, $\left[ x, y \right] = x \times y$ in $\mathbb{R}^3$.
Finally, we introduce some standard terminology for Lie algebras:
\begin{itemize}
\item A {\it basis} for the Lie algebra $g$ is a basis of the underlying vector space.
\item Given subsets $g_1, g_2$ of a Lie algebra $g$, the notation $\left[ g_1, g_2 \right]$ denotes the set
$$ \{ \left[ z_1, z_2 \right] | z_1 \in g_1, z_2 \in g_2 \} $$
\item A {\it Lie subalgebra} $g_1$ of a Lie algebra $g$ is a subset of $g$ that forms a Lie algebra with the same commutator and field as that of $g$.
\item Given a basis $B = \lbrace b_1, \cdots, b_n \rbrace$ of a real (complex) Lie algebra $g$, the {\it structure constants of $g$ with respect to the basis $B$ } are the $n^3$ real (complex) numbers $c^r_{pq}$ satisfying
$$ [ b_p, b_q] = \sum_{r=1}^{n} c^r_{pq} b_r \hspace{.75cm} p,q = 1,2,\cdots, n$$
\item The {\it complexification $V^\mathbb{C}$ of a real vector space $V$} is the complex vector space
$$V \otimes_\mathbb{R} \mathbb{C} = V \oplus i V$$
The {\it complexification $g^\mathbb{C}$ of a real Lie algebra $g$} is a complexification of the vector space $g$ with the commutator
$$ \left[ x_1 + i y_1, x_2 + i y_2 \right] = \left[ x_1, x_2 \right] - \left[ y_1, y_2 \right] + i\left( \left[ x_1, y_2 \right] + \left[ y_1, x_2 \right] \right), \hspace{.75cm} x_1, x_2, y_1, y_2 \in g$$
extending the commutator of $g$.\footnote{Care must be taken when considering real Lie algebras whose elements are complex matrices, see \cite{big_cornwell}.} Further information may be found in \cite{onishchik}.
\item A {\it real form} of a complex Lie algebra $g$ is a real Lie algebra $h$ whose complexification $h^\mathbb{C}$ is isomorphic to $g$ as a complex Lie algebra.
\item A Lie algebra $g$ is {\it abelian} if its commutator is identically zero, i.e. $\left[ g, g \right] = 0$.
\item A subalgebra $g_1$ of $g$ is called {\it invariant} if if $[a, b] \in g_1$ for all $a \in g_1$ and $b \in g$. Equivalently, we have $\left[ g_1, g \right] \subset g_1$.
\item A real or complex Lie algebra $g$ is called {\it simple} if its complexification $g^\mathbb{C}$ is not abelian and does not possess a proper invariant Lie subalgebra.
\item A real or complex Lie algebra $g$ is called {\it semi-simple} if its complexification $g^\mathbb{C}$ does not possess an abelian invariant subalgebra. Equivalently, a semi-simple Lie algebra is a direct sum of simple Lie algebras.
\end{itemize}
Throughout this thesis, we often write $C = R$ where $C$ is a complex Lie algebra and $R$ is a real Lie algebra whose complexification is isomorphic to $C$. Given our primary interest in the structure of the real subalgebras of a real form of $E_6$, we define two non-standard definitions for real Lie algebras corresponding to simple and semi-simple Lie algebras:
\begin{enumerate}
\item If $g$ is a real Lie algebra that is not abelian and does not possess a proper invariant Lie subalgebra, then we say $g$ is {\it $\mathbb{R}$-simple}.
\item If the real Lie algebra $g$ does not possess an abelian invariant subalgebra, then it is called {\it $\mathbb{R}$-semi-simple}. An $\mathbb{R}$-semi-simple Lie algebra $g$ is the direct sum of $\mathbb{R}$-simple real Lie algebras.
\end{enumerate}
These terms are intended to comment only on the real structure of a real Lie algebra. With this terminology, the real Lie algebra $so(3,1,\mathbb{R})$ is $\mathbb{R}$-simple and not decomposable into the form $g_1 \oplus g_2$, even though $so(3,1,\mathbb{R})$ and its complexification, $D_2 = A_1 \oplus A_1$, are not simple.
We will encouter the difference between simple, semi-simple, and abelian Lie algebras again in Section \ref{ch:Lie_groups_Lie_algebras.Basic_Definitions.regular_rep_Killing_form}.
Finally, we note that derivations may be used to construct a Lie algebra from any algebra $A$ over a field $\mathbb{F} = \mathbb{R}$ or $\mathbb{C}$. If $A$ is an algebra over the field $\mathbb{F}$, then an endomorphism\footnote{An {\it endomorphism} is an $\mathbb{F}$-linear map from a vector space $V$ to itself, where $V$ is a vector space over the field $\mathbb{F}$. We use $End_\mathbb{F} V$ to denote the associative algebra of all $\mathbb{F}$-linear maps from $V$ to $V$ where $V$ is a vector space over $\mathbb{F}$. }
$D$ of the vector space $A$ is called a {\it derivation} if $A(xy) = A(x)y + xA(y)$ for all $x,y \in A$. If $D_1$ and $D_2$ are derivations of $A$, then
$$[D_1, D_2] = D_1 D_2 - D_2 D_1$$
is also a derivation of $A$. The set of derivations of $A$ is a subalgebra of $gl(A)$. This construction will be useful in Section \ref{ch:Lie_groups_Lie_algebras.Basic_Definitions.regular_rep_Killing_form} in the particular case that $A$ is a Lie algebra.
\subsection{Correspondence between Lie groups and Lie algebras}
\label{ch:Lie_groups_Lie_algebras.Basic_Definitions.Diff_Exp_Comm}
There is a very nice correspondence between elements of a Lie group $G$ and elements of its corresponding Lie algebra $g$.
Roughly speaking, algebra elements are produced from group elements by differentiation, while exponentiation produces a group element from an element in the Lie algebra. As seen in the two examples above, elements of the Lie algebra of a Lie group $G$ may be described either as the set of left-invariant vector fields of $G$ or as tangent vectors at the identity of $G$.
Using these two processes, the commutator of Lie algebra elements can be described using the Lie group. The following treatment again follows \cite{boothby}.
We begin with {\it differentiation}, the process used to pass from the Lie group to the associated Lie algebra. Given a Lie group $G$ and the Lie group $\mathbb{R}$, the additive group of real numbers, if $\alpha : \mathbb{R} \to G$ is a homomorphism of $\mathbb{R}$ into $G$, then the image $\alpha(\mathbb{R})$ is called a {\it one-parameter subgroup of $G$}. Note that the homomorphism property requires both that $\alpha(0) = e$, the identity element in $G$, and that
$$ \alpha(s_1) \alpha(s_2) = \alpha(s_1 + s_2) \hspace{.75cm} s_1,s_2 \in \mathbb{R}$$
The corresponding left-invariant vector field $X$ is determined by its value $X_e$ at the identity of $G$, which is given by
$$ X_e = \dot \alpha = \frac{d \alpha(s)}{d s}|_{s = 0}$$
It is an element of the Lie algebra $g$ associated to the Lie group $G$.
We will often use the notation $\dot \alpha$ to denote the tangent vector at the identity of the one-parameter subgroup $\alpha : \mathbb{R} \to G$.
Having created an element of the Lie algebra from a one-parameter subgroup of a Lie group $G$, we now describe the inverse process, called {\it exponentiation}. Let $G$ be a Lie group and $X$ a $C^\infty$ vector field on $G$.
For each $p \in G$, there is a neighborhood $V \subset G$, a real number $\delta > 0$, and a $C^\infty$ mapping
$$ \gamma^v : I_\delta \times V \to G$$
which satisfies
$$ \frac{\partial \gamma^v(s,q)}{\partial s} = X_{\gamma^v(s,q)}$$
and
$$ \gamma^v(0,q) = q $$
for all $q \in V$, where $I_\delta = \left( -\delta, \delta \right)$ is a real open interval.
This theorem is the manifold version of the existence theorem for ordinary differential equations.
We use this theorem in the special case where $X$ is a left-invariant vector field of $G$, in which case the interval $I_\delta$ may be extended to all of $\mathbb{R}$ and we may take $V = G$. Let $p = e \in G$.
As noted in Section \ref{ch:Lie_groups_Lie_algebras.Basic_Definitions.Lie_Algebras}, each left-invariant vector field $X$ on $G$ may be identified by its value $X_e$ at the point $e$ in $G$, that is, with a tangent vector $v = X_e$ in $T_eG$. Then the mapping $\gamma^v: \mathbb{R} \times G \to G$ given above is a global action of $\mathbb{R}$ on $G$, and the map $\gamma^v_e : \mathbb{R} \to G$ given by
$$ \gamma^v_e(s) = \gamma^v(s,e) \hspace{.75cm} s \in \mathbb{R}$$
is a one-parameter subgroup of $G$. We note that $\gamma^v_e$ is defined for each particular $v \in T_eG$. Indeed, the map $\gamma_e : \mathbb{R} \times T_eG \to G$ given by
$$\gamma_e(s,v) = \gamma^v_e(s)$$
is a $C^\infty$ map. This map satisfies $\gamma_e(s,s_1 v) = \gamma_e(s_1 s, v)$ for all $s, s_1 \in \mathbb{R}$. The {\it exponential map}, $\exp{}: T_eG \to G$, is defined by the formula
$$\exp{(v)} = \gamma_e(1, v)$$.
The exponential map derives its name from the case of its use with the classical matrix groups. Consider the Lie group $G = GL(n,\mathbb{F})$, with $\mathbb{F} = \mathbb{R} $ or $\mathbb{C}$, which consists of all non-singular $n\times n$ matrices over $\mathbb{F}$. The Lie algebra of this Lie group is $g = gl_n(\mathbb{F})$, which is isomorphic to $M_n(\mathbb{F})$, the set of all $n \times n$ matrices over $\mathbb{F}$. Let $A \in gl_n(\mathbb{F})$. The flow of the left-invariant vector field defined by $A$ is obtained by integrating $\frac{d X}{d s} = X \cdot A$ with $s \in \mathbb{R}$ and $X(0) = Id$. If $\mathbb{F}^n$ is equipped with a norm and $M_n(\mathbb{F})$ with the associated endomorphism norm, then
$$ || A^n || \le ||A||^n$$
Hence, the series
$$ \sum_{r = 0}^\infty \left( \frac{s^r}{r!} \right) A^r$$
converges, as do all of its derivatives. Thus,
$$ \exp{(sA)} = \sum_{r = 0}^\infty \left( \frac{s^r}{r!} \right) A^r$$
giving the exponential map its name.
We can illustrate differentiation and exponentiation using our example of the solid ball of rotations from Section \ref{ch:Lie_groups_Lie_algebras.Basic_Definitions.An_Example}. We begin with differentiation. Let $G$ be the Lie group under consideration. Choose a point $p$ in the solid ball, and consider the straight line path $\alpha: \mathbb{R} \to G$ extending through $-p$ and $p$, which is given by
$$ \alpha(s) = s\vec{p} \hspace{.75cm} s \in \mathbb{R}$$
where $\vec{p}$ is the vector pointing from the origin to $p$.
This map is a homomorphism, and the line traced out by $\alpha$ is a one-parameter subgroup of $G$.
We note that $\alpha(1) = \vec{p}$ and $\alpha(-1) = -\vec{p}$ are rotations about the same axis but in opposite directions. Then
$$ \dot \alpha \equiv \frac{d \alpha(s)}{d s}|_{s=0} = \vec{p}$$
is the tangent vector to the path $\alpha$ at the origin. It is an element of the Lie algebra $g$ associated to the Lie group $G$.
Our example also illustrates the exponential of a Lie algebra element. The Lie group $G$ is of dimension three, implying that its Lie algebra $g = T_eG$ is also of dimension three. Let $\vec{v} \in T_eG$ be any vector in $\mathbb{R}^3$ and define the path $\gamma^{\vec{v}}_e : \mathbb{R} \to G$ in $G$ by
$$ \gamma^{\vec{v}}_e(s) = s\vec{v} \hspace{.75cm} s \in \mathbb{R}$$
The image of this map is a line extending through the origin in $G$. Hence, it is a one-parameter subgroup of $G$. The line also depends continuously on our choice of $\vec{v} \in \mathbb{R}^3$, so we may define $ \gamma_e : \mathbb{R} \times T_eG \to G$ by
$$\gamma_e(s,\vec{v}) = \gamma^{\vec{v}}_e(s) = s\vec{v}$$
The exponential map $\exp{} : T_eG \to G$ is then given by the formula
$$\exp{(\vec{v})} = \gamma_e(1,\vec{v}) = \vec{v} = v$$
where we identify $\vec{v}$ with the point $v \in G$. We emphasize that the image of
$$\exp{(s\vec{v})} = \gamma_e(1,s\vec{v}) = s\vec{v} = \gamma^{\vec{v}}_e(s) \hspace{.75cm} s \in \mathbb{R}$$
is not only a line through the origin containing $v$, but it is a one-parameter subgroup of $G$ whose tangent vector at the identity is $\vec{v} \in T_eG$.
It will be useful in Section \ref{ch:E6_basic_structure.ConstructingE6algebra} to have an expression for the commutator of two Lie algebra elements in terms of the corresponding group elements. Given a Lie algebra $g$ of a Lie group $G$, we identify $g$ with the tangent space $T_eG$ of $G$. Let $u,v \in T_eG$, and let $\exp{(su)}$ and $\exp{(sv)}$ denote their corresponding one-parameter subgroups in $G$. We define the curve $\sigma : \mathbb{R} \to G$ in $G$ by
$$ \sigma{(s)} = \exp{(-\frac{s}{2}u)} \exp{(-\frac{s}{2}v)} \exp{(\frac{s}{2}u)} \exp{(\frac{s}{2}v)} \hspace{.75cm} s \in \mathbb{R}$$
We note that $\sigma{(0)} = e$ is the identity in $G$, but that the image of $\sigma{(\mathbb{R})}$ is not in general a one-parameter subgroup of $G$. Nevertheless, its tangent vector is the commutator $\left[ u, v \right]$ in the algebra.\footnote{ The first-order derivative of $\sigma{(s)}$ vanishes at $s = 0$. However, the second-order derivative $\frac{d^2 \sigma{(s)}}{ds^2}|_{s=0}$ will be tangent to the curve $\sigma{(s)}$ at $s = 0$. Further information may be found in \cite{varadarajan}.}
We finish this section with a note on the direct products of Lie groups and the direct sum of Lie algebras. In particular, we show that if $g_1$ and $g_2$ are the Lie algebras of the Lie groups $G_1$ and $G_2$, then the Lie algebra of the direct product $G_1 \times G_2$ is $g_1 \oplus g_2$. Further information regarding this fact may be found in \cite{cornwell}.
Let $G_1$, $G_2$ be two Lie groups with identities $e_1$ and $e_2$, respectively. Consider the direct product of the manifolds $G_1$ and $G_2$, denoted $G_1\times G_2$, with the product map
$$ (x_1, y_1) (x_2, y_2) \to (x_1 x_2, y_1 y_2) \textrm{ for } x_1, x_2 \in G_1, y_1, y_2 \in G_2$$
and inverse map
$$ (x_1, y_1) \to ((x_1)^{-1}, (y_1)^{-1}) \textrm{ for } x_1, \in G_1, y_1 \in G_2 $$
where $(x_1)^{-1}$ and $(y_1)^{-1}$ is the inverse of $x_1$ and $y_1$ in $G_1$ and $G_2$, respectively. Since $G_1$ and $G_2$ are Lie groups, the product defined in this way also makes $G_1\times G_2$ into a Lie group. Its identity element is $(e_1,e_2)$. As a manifold, its dimension is $|G_1|+|G_2|$.
The direct sum of Lie algebras $g_1$ and $g_2$ over $\mathbb{R}$ is denoted $g_1\oplus g_2$. It is a direct sum of the underlying vector spaces with the product
$$ [(x_1, y_1), (x_2, y_2) ] = [ [x_1,x_2], [y_1,y_2]] \textrm{ for } x_1, x_2 \in g_1, y_1, y_2 \in g_2 $$
which is anti-commutative, satisfies the Jacobi identity, and is bilinear. The dimension of $g_1 \oplus g_2$ is $|g_1|+|g_2|$.
Cornwell \cite{cornwell} uses the fact that every finite-dimensional Lie algebra has a finite-dimensional faithful linear representation to prove that if $G_1$ and $G_2$ are two matrix Lie groups of dimensions $|G_1|$ and $|G_2|$, then $G_1 \times G_2$ is a matrix Lie group of dimension $|G_1|+|G_2|$, and that if $g_1$ and $g_2$ are the real Lie algebras of $G_1$ and $G_2$, respectively, then the real Lie algebra of $G_1 \times G_2$ is isomorphic to $g_1 \oplus g_2$.
Noting that the Lie algebra of $G_1\times G_2$ will have dimension $|G_1|+|G_2|$, we shall prove this theorem for general finite-dimensional Lie groups $G_1$ and $G_2$ by constructing a basis for $g_1 \oplus g_2$ from $G_1 \times G_2$. Let $\mathbb{B}$ be a basis for $G_1$. For each $x_i \in \mathbb{B}$, the image of $(\exp{(sx_i)}, e_2)$ with $s \in \mathbb{R}$ is a one-parameter subgroup of $G_1 \times G_2$. Hence, we obtain $|G_1|$ one-parameter subgroups of $G_1 \times G_2$. Similarly, we construct $|G_2|$ one-parameter subgroups in $G_1 \times G_2$ of the form $(e_1, \exp{(sy_i)})$ for each $y_i$ in a basis of $G_2$. Hence, we have $|G_1| + |G_2|$ subgroups of $G_1 \times G_2$. Differentiating, we obtain a set of $|G_1| + |G_2|$ linearly independent elements of $g_1 \oplus g_2$, each of the form $(x_i, 0)$ or $(0, y_i)$. Hence, we have constructed a basis of $g_1 \oplus g_2$ from $G_1 \times G_2$. Since their dimension are the same, we have shown that the Lie algebra of $G_1 \times G_2$ is $g_1 \oplus g_2$.
\subsection{Regular Representation and Killing Form}
\label{ch:Lie_groups_Lie_algebras.Basic_Definitions.regular_rep_Killing_form}
In this section, we describe the adjoint representation of a Lie algebra and the Killing form. We begin with the adjoint representation of the Lie group, and show how it produces the adjoint representation of the Lie algebra associated with the Lie group. We then discuss the Killing form of the Lie algebra, a tool which is fundamental in classifying real and complex Lie algebras. Finally, we include a note on general representations of Lie algebras and include definitions which will be useful in Section \ref{ch:Joma_Paper.Root_Weight_Construction}. The treatment of the adjoint representation of a Lie group, representations of a Lie algebra, and the Killing form may be found in a number of references, including \cite{cornwell, boothby, knapp}.
We first define a representation of a Lie group $G$ on a finite-dimensional vector space $V$ before considering the case where $V = g$, the Lie algebra of the Lie group.
A {\it finite-dimensional representation of a topological group $G$} is a continuous homomorphism $\Psi : G \to GL_{\mathbb{C}}(V)$ from $G$ into the group of invertible linear transformations on a finite-dimensional complex vector space $V$.
In the case that $G$ is a Lie group, it is often useful to let $V = g$, the Lie algebra of $G$.
Identify $g$ with the tangent space $T_eG$ to $G$ at the identity. Then, for any element $p \in G$, the map $\Psi_p : G \to G$ given by $\Psi_p(r) = prp^{-1}$ is a smooth isomorphism.
If $\alpha : \mathbb{R} \to G$ is a smooth curve in $G$ such that $\alpha(0) = e \in G$ and $ \dot \alpha \in g$, then $\Psi_p(\alpha(t)) = p \alpha(t) p^{-1}$ is also a smooth curve in $G$ which passes through $e$ at $t = 0$. Hence, the tangent vector to this curve, $p \dot \alpha p^{-1}$, is in $g$. Thus, the corresponding isomorphism $Ad(p) := d\Psi_p : T_eG \to T_eG$ is a representation $Ad : G \to Aut(T_eG)$ of $G$ on its tangent space $T_eG$. Identifying $T_eG$ with $g$, we call this isomorphism the {\it adjoint representation of the group on its algebra}. Further information on representations may be found in \cite{knapp}.
Having defined a representation of a Lie group, we now discuss the related concept for a Lie algebra.
A {\it representation of a Lie algebra $g$ on a vector space $V$} is a homomorphism of Lie algebras $\rho : g \to (End_{\mathbb{F}}(V))^\mathbb{K}$ where $V$ is a vector space over $\mathbb{F} = \mathbb{R} \textrm{ or } \mathbb{C}$.
The map $\rho$ must be $\mathbb{K}$-linear and satisfy
$$ \rho([x,y]) = \rho(x)\rho(y) - \rho(y)\rho(x) \textrm{ for all } x,y \in g$$
We note that $\mathbb{K} = \mathbb{R}$ if $g$ is a real Lie algebra, and $\mathbb{K} = \mathbb{C}$ if $g$ is a complex Lie algebra.
\footnote{The dimension of the space of endomorphisms $(End_{\mathbb{R}}(V))^\mathbb{R}$ is twice that of $(End_{\mathbb{C}}(V))^\mathbb{R}$.}
Ado has proven that every finite-dimensional Lie algebra has a finite-dimensional faithful linear representation \cite{bourbaki}.
As shown in \cite{knapp}, if $\Psi$ is a representation of a Lie group on a vector space $V$, then the differential at $e \in G$ gives a representation of the real Lie algebra $g$ of $G$ on the space $V$. We illustrate this using the adjoint representation of a matrix Lie group $G$ on its Lie algebra $g$.
Let $\alpha: \mathbb{R} \to G$ be a smooth curve in $G$ such that $\alpha(0) = e \in G$ and $\dot \alpha = y \in g$.
The map $t \to Ad_{\alpha(t)}(x)$, with $x \in g$, is a smooth map into $g$. Computing its derivative, we find that
$$
\begin{array}{ccl}
\frac{d}{dt} Ad_{\alpha(t)}(x) & = & \frac{d}{dt} [ \alpha(t) x \alpha^{-1}(t) ] \\
& = & \frac{d \alpha(t)}{dt} x \alpha^{-1}(t) + \alpha(t) x \frac{d \alpha^{-1}(t)}{dt} \\
& = & \frac{d \alpha(t)}{dt} x \alpha^{-1}(t) - \alpha(t) x \alpha^{-1}(t) \frac{d \alpha(t)}{dt} \alpha^{-1}(t) \\
\end{array}
$$
which, when evaluated at $t = 0$, gives
$$ \begin{array}{ccl}
\frac{d}{dt} Ad_{\alpha(t)}(x)|_{t = 0} & = & \dot \alpha x - x \dot \alpha \\
& = & yx - xy
\end{array}
$$
which is also in $g$. The homomorphism $Ad$ induces a homomorphism of Lie algebras, denoted by $ad: g \to End(g)$, given by $ad_x(y) = [x,y]$. This homomorphism is known as the {\it adjoint representation of a Lie algebra $g$}.
The adjoint representation of a Lie algebra may also be defined without any reference to its Lie group.
It is important to note that, for any Lie algebra $g$, the linear map $ad : g \to End(g)$ given by
$$(ad\; x)(y) = [x,y]$$
is a representation of the Lie algebra $g$. This homomorphism satisfies
$$ ad\; [x,y] = (ad\; x) (ad\; y) - (ad\; y) (ad\; x)$$
which can be seen by rewriting the Jacobi identity for $g$ as
$$ [ [x,y], z] = [x, [y,z]] - [y, [x,z]]$$
However, the Jacobi identity of $g$ may also be rewritten as
$$ [ x, [y,z]] = [[x,y],z] + [y,[x,z]]$$
or
$$ (ad\; x)([y,z]) = [(ad\; x)(y), z] + [y, (ad\; x)(z)]$$
showing that $ad$ is a derivation of $g$.
Hence, as noted in Section \ref{ch:Lie_groups_Lie_algebras.Basic_Definitions.Lie_Algebras}, the image $ad\; g$ of $g$ under this representation is a Lie algebra, and its commutator is given by
$$ [ad\; x, ad\; y] = (ad\; x) (ad\; y) - (ad\; y) (ad\; x)$$
where the product on the right hand side is the the product in $End(g)$. Further information may be found in \cite{knapp}.
We find that the adjoint representation of a Lie algebra $g$ is particularly useful. The representation is fully defined by its action on a basis of $g$. Hence, given a basis $\lbrace b_1, \cdots, b_n \rbrace$ of an $n$-dimensional Lie algebra $g$, we may express the linear transformation $ad_{b_i}$ as an $n \times n$ matrix $\ad{b_i}$ whose components are given by
$$ [ b_i, b_j ] = \sum_{k = 1}^n \ad{b_i}_{kj}b_k$$
We note that the components of $\ad{b_i}$ are the structure constants of $g$. The coefficients in column $j$ of $\ad{a}$ are the coefficients in the expansion of $ad_a(b_j) = [a, b_j] $ using the basis previously chosen. Hence, the components of the matrices $\ad{b_i}$, $i = 1, \cdots, n$, are the structure constants of $g$. Even though the individual structure constants are basis dependent, the entire collection describes the entire algebra, regardless of the chosen basis, in the following way: Given a real (complex) Lie algebra $g$, we define the {\it Killing form} to be a symmetric bilinear $B$ form over $\mathbb{R}$ ($\mathbb{C}$) on $g$ whose value is given by $B(x,y) = \tr{\ad{x} \ad{y}}$ for any $x,y \in g$. The Killing form is invariant under all automorphisms of $g$. Further information on the Killing form may be found in \cite{knapp}.
We now list some important results related to the Killing form which will be particularly useful in Chapters \ref{ch:E6_basic_structure} and \ref{ch:E6_further_structure}. Consult \cite{cornwell, gilmore} for a further discussion of these facts.
The Killing form provides useful criteria for proving that a Lie algebra $g$ is semi-simple, known as {\it Cartan's criterion for semi-simplicity}. Given a Lie algebra $g$ with basis $\lbrace b_1, \cdots, b_n \rbrace$, we may use the Killing form to define an $n \times n$ matrix $\mathbf{B}$ whose elements are defined by $B_{ij} = B(b_i, b_j)$. The Lie algebra $g$ is semi-simple if and only if the Killing form is non-degenerate, that is, if $\det{ \mathbf{B}} \ne 0$.
According to a theorem in \cite{cornwell}, it is possible to choose a basis $\lbrace b_1, \cdots, b_n \rbrace$ of $g$ such that $B(b_i, b_i) \in \lbrace -1, 0 +1 \rbrace$ and $B(b_i, b_j) = 0 $ for $i \ne j$. That is, $\mathbf{B}$ is a diagonal matrix with diagonal entries equal to $+1$, $-1$, or $0$. Let $d_{+}, d_{-},$ and $d_{0}$ denote the number of $+1$, $-1$, and $0$ diagonal entries in $\mathbf{B}$ with such a basis chosen. While there are any number of ways to choose the basis of $g$, a theorem \cite{cornwell} on bilinear forms states that if a basis of $g$ is chosen so that $\mathbb{B}$ is diagonal with entries $+1, -1$ and $0$, then the values $d_{+}$, $d_{-}$, and $d_{0}$ are invariant.
If $d_{0} > 0$, then $\mathbf{B}$ is degenerate and $g$ is not semi-simple.
On the other hand, if $d_{0} = 0$, then $g$ is semi-simple and the Killing form provides a further description of the Lie algebra $g$.
The elements $b_i$ such that $B(b_i, b_i) = -1$ correspond to {\it compact generators} or {\it rotations} in the Lie group, while the elements for which $B(b_i,b_i) = +1$ correspond to {\it non-compact generators} or {\it boosts}. In this case, we say that $(d_{-}, d_{+})$ is the {\it signature} of $g$. We note that a Lie algebra is compact if and only if its Killing form is negative definite. Indeed, as noted in \cite{knapp}, a Lie algebra $g$ of a Lie group $G$ is compact if and only if $G$ is compact.
There are two other aspects of the Killing form which we will find useful in Chapters \ref{ch:E6_basic_structure} and \ref{ch:E6_further_structure}. First, we say $x \in g$ is a {\it null rotation} if $B(x,x) = 0$. We note that this definition neither implies that $\mathbf{B}$ is a diagonal matrix nor that $g$ has an abelian invariant subalgebra. We will encounter null rotations in Section \ref{ch:E6_basic_structure.fix_l}. Second, if $\phi$ is any automorphism of $g$, then
$$B(\phi(x), \phi(y)) = B(x,y) \textrm{ for } x,y \in g$$
We will use this important property of the Killing form extensively in Chapter 5,
Further information about the properties of the Killing form can be found in \cite{cornwell}.
In my work below, I often represent the adjoint representation of a Lie algebra in a {\it commutation table $C$}. Having chosen a basis $\lbrace b_1, \cdots, b_n \rbrace$, the $ij$ entry in the table $C$ is given by $C_{ij} = [ b_i, b_j]$. This entry contains the expansion of the $j$-th column of $\ad{b_i}$ with the basis $\lbrace b_1, \cdots, b_n \rbrace$. Thus, each column of $C$ represents a complete $n\times n$ matrix in the adjoint representation. The Killing form of $b_i$ and $b_j$ is computed in part by expanding the $i$-th and $j$-th columns of $C$ into their equivalent $n\times n$ adjoint representations of $b_i$ and $b_j$. If $g$ is abelian, then $C$ will be the zero matrix. If $g$ has an invariant subalgebra $g^\prime \subset g$, we may expand a basis $\lbrace b_1, \cdots, b_k \rbrace$ for $g^\prime$ to a basis of $g$. Then each of the entries in the first $k$ rows and first $k$ columns of $C$ will be linear combinations of $\lbrace b_1, \cdots, b_k \rbrace$ only. Finally, if $g$ is semi-simple but not simple, we may choose a basis of $g$ over $\mathbb{C}$ such that $C$ is block diagonal. This block-diagonal structure may not be possible if $g$ is a real Lie algebra and the basis must be chosen over $\mathbb{R}$. However, if $g$ is a real Lie algebra which is $\mathbb{R}$-semi-simple, then we may choose a basis of $g$ over $\mathbb{R}$ such that $C$ is block diagonal.
\subsection{Different Conventions used by Physicists and Mathematicians}
\label{ch:Lie_groups_Lie_algebras.Basic_Definitions.conventions}
Mathematicians and physicists have slightly different conventions for the differentiation and exponentiation processes related to the classical matrix groups. The conventions above are typically used by mathematicians. For the classical matrix groups, these conventions lead to Lie algebra elements which are anti-hermitian. In physics, one of the fundamental ideas in quantum mechanics is that observables are associated with hermitian matrices. Hence, physicists insert a factor of $-i$ in the definition of differentiation, using $\sigma = -i \dot \alpha = -i \frac{d \alpha(s)}{d s}|_{s=0}$. This factor of $-i$ is removed in the exponentiation process, using the definition $\exp(s\sigma) = \sum_{m=0}^\infty \frac{{i s \sigma}^m}{m!} = \sum_{m=0}^\infty \frac{( s \dot \alpha)^m}{m!}$, with $s \in \mathbb{R}$, for the classical matrix groups.
It is important to note that these differences do not affect the content of the commutator in the algebra. If $\lbrace \dot \alpha_x, \dot \alpha_y, \dot \alpha_z \rbrace$ are the mathematicians' Lie algebra elements, the corresponding physicists' conventions are $\lbrace -i \dot \alpha_x, -i\dot \alpha_y, -i\dot \alpha_z \rbrace$. The mathematicians' commutator $[ \dot \alpha_x, \dot \alpha_y ] = \dot \alpha_z$ and physicists' commutator $[ -i \dot \alpha_x, -i \dot \alpha_y] = -i(-i \dot \alpha_z)$ are consistent, even if the physicists' commutator initially appears to have an extra factor of~$i$.
\subsection{Classical Matrix Groups and their Algebras}
\label{ch:Lie_groups_Lie_algebras.Basic_Definitions.classical_matrix_groups}
We list here the classical matrix groups, using their standard definitions as found in \cite{curtis}. We use $GL(n,\mathbb{F})$ with $\mathbb{F} = \mathbb{R}$ or $\mathbb{C}$ to denote the general linear group composed of $n\times n$ invertible matrices over $\mathbb{F}$ under the operation of matrix multiplication. The notation $A^T$ is used to denote the transpose of the matrix $A$. The hermitian conjugate of the matrix $A$ is denoted by $A^\dagger = \left( \overline{A} \right)^T$. That is, if $A = \left(a_{ij}\right)$, then the hermitian conjugate of $A$ is given by $A^\dagger = \left( \overline{a_{ji}}\right)$. Finally, the $n \times n$ identity matrix whose off-diagonal components are zero and whose diagonal components are one is denoted by $\mathbb{I}_n$, and $J_n = \left( \begin{array}{cc} 0 & \mathbb{I}_n \\ -\mathbb{I}_n & 0 \end{array} \right)$.
\begin{enumerate}
\item Real special linear group of degree $n$: $$ SL(n,\mathbb{R}) = \lbrace g \in GL(n,\mathbb{R}) | \mydet{g} = 1 \rbrace$$
\item Complex special linear group of degree $n$: $$SL(n,\mathbb{C}) = \lbrace g\in GL(n,\mathbb{C}) |\mydet{g} = 1 \rbrace$$
\item Real orthogonal group of degree $n$: $$O(n,\mathbb{R}) = \lbrace g \in GL(n,\mathbb{R}) | g^{T} g = \mathbb{I}_n \rbrace$$
\item Complex orthogonal group of degree $n$: $$O(n,\mathbb{C}) = \lbrace g \in GL(n,\mathbb{C}) | g^\dagger g = \mathbb{I}_n \rbrace$$
\item Real (Complex) special orthogonal group of degree $n$: $$SO(n,\mathbb{R}) = O(n,\mathbb{R}) \cap SL(n,\mathbb{R})$$
$$SO(n,\mathbb{C}) = O(n,\mathbb{C}) \cap SL(n,\mathbb{C})$$
\item Real symplectic group $$Sp(n,\mathbb{R}) = \lbrace g \in GL(2n,\mathbb{R}) | g^T J_n g = J_n \rbrace$$
\item Complex symplectic group $$Sp(n,\mathbb{C}) = \lbrace g \in GL(2n,\mathbb{C}) | g^\dagger J_n g = J_n \rbrace$$
\item Unitary group of degree $n$: $$U(n, \mathbb{C}) = \lbrace g \in GL(n,\mathbb{C}) | g^\dagger g = \mathbb{I}_n \rbrace$$
\item Special unitary group of degree $n$: $$SU(n,\mathbb{C}) = U(n) \cap SL(n,\mathbb{C})$$
\item Unitary symplecitic group of degree $n$: $$Sp(n) = U(2n,\mathbb{C})\cap Sp(n,\mathbb{C})$$
\end{enumerate}
For each of the classical matrix groups just given, we list the corresponding Lie algebras:
\begin{enumerate}
\item $$sl(n,\mathbb{R}) = \lbrace x \in M(n,\mathbb{R}) | \tr{X} = 0 \rbrace$$
\item $$sl(n,\mathbb{C}) = \lbrace x \in M(n,\mathbb{C}) | \tr{x} = 0 \rbrace$$
\item $$o(n,\mathbb{R}) = \lbrace x \in M(n,\mathbb{R}) | x^T = -x \rbrace $$
\item $$o(n,\mathbb{C}) = \lbrace x \in M(n,\mathbb{C}) | x^T = -x \rbrace $$
\item $$so(n,\mathbb{R}) = o(n,\mathbb{R}) \cap sl(n,\mathbb{R})$$
$$so(n,\mathbb{C}) = o(n,\mathbb{C}) \cap sl(n,\mathbb{C})$$
\item $$sp(n,\mathbb{R}) = \lbrace x \in M(2n,\mathbb{R}) | x^T J_n + J_n x = 0 \rbrace$$
\item $$sp(n,\mathbb{C}) = \lbrace x \in M(2n,\mathbb{C}) | x^T J_n + J_n x = 0 \rbrace$$
\item $$u(n,\mathbb{C}) = \lbrace x \in M(n,\mathbb{C}) | X^\dagger = -x \rbrace$$
\item $$su(n,\mathbb{C}) = \lbrace x \in M(n,\mathbb{C}) | x^\dagger = -x, \tr{x} = 0 \rbrace = u(n) \cap sl(n,\mathbb{C})$$
\item $$sp(n,\mathbb{C}) = \lbrace x \in M(2n,\mathbb{C}) | x^\dagger J_n + J_n x = 0, x^\dagger = -x \rbrace = u(2n,\mathbb{C}) \cap sp(n,\mathbb{C})$$
\end{enumerate}
\section{Classifying Complex Lie Algebras}
\label{ch:Joma_Paper}
Lie algebras can be represented by various diagrams.
The remaining part of Chapter \ref{ch:Lie_groups_Lie_algebras} is a paper, to appear in the {\it Journal of Online Mathematics and its Applicataions}, which discusses the classification of complex Lie algebras. It is included without modification, and includes some minor duplication of material previously covered in Chapter \ref{ch:Lie_groups_Lie_algebras}. It is intended for a non-expert audience, and therefore many definitions have been kept deliberately informal in this non-technical paper. Some of the terms with informal definitions have been defined previously in Chapter \ref{ch:Lie_groups_Lie_algebras}, and we have included the precise definitions of the other terms in Section \ref{ch:Joma_Paper.definitions}. The actual text of the paper begins in Section \ref{ch:Joma_Paper.Introduction}.
The paper is organized as follows. Section \ref{ch:Joma_Paper.Root_Weight_Construction} discusses how the root and weight diagrams of a complex Lie algebra my be constructed from the algebra's Dynkin diagram. This section includes a brief introduction to Lie algebras and other concepts covered previously in this thesis. An algebra's root and weight diagrams may be used to identify its subalgebras. This is shown for the three-dimensional case in Section \ref{ch:Joma_Paper.subalgebras} using the algebra $B_3 = so(7)$. In particular, two methods are described which find all the sublagebras of the rank $3$ algebras. These methods are generalized to higher dimensional diagrams in Section \ref{ch:Joma_Paper.subalgebras_greater_than_3}, including a detailed treatment of $F_4$. This paper concludes with a tower of the complex Lie algebras contained within $E_6$. The goal of the rest of this thesis is to expand upon the structure of this tower by considering the real subalgebras of $sl(3,\mathbb{O})$, our real form of $E_6$.
\subsection{Precise Definitions}
\label{ch:Joma_Paper.definitions}
We list here precise definitions for the terms which we left inadequately defined in the following paper.
\begin{enumerate}
\item In the following paper, we often write $C = R$ where
$$C = A_n, B_n, C_n, D_n, E_6, E_7, E_8, F_4, G_2$$
is the Dynkin classification of the complex Lie algebra and $R$ is a real algebra whose complexification is $C$. The equal sign is a short notation for the statement {\it The complexification of $R$ is equal as a complex Lie algebra to $C$.}
The {\it Cartan subalgebra} $h$ of a Lie algebra $g$ is a maximal nilpotent subalgebra of $g$. Equivalently, it can be shown that the Cartan subalgebra $h$ of a Lie algebra $g$ is the maximal abelian subalgebra $h$ of a simple Lie algebra $g$. See \cite{varadarajan} for additional information.
\item A {\it representation} $p$ of a real (resp. complex) Lie algebra $g$ on a real (resp. complex) vector space $V$ is a mapping which assigns to each $x \in g$ a linear transformation $p(x)$ on $V$, and which is such that
$$ p(a x + by) = ap(x) + bp(y) \textrm{ for all } x,y \in g \textrm{ and scalars } a,b; $$
$$ p\left( [x,y] \right) = p(x)p(y) - p(y)p(x)$$
For linear transformations on a vector space, the expression $[X,Y]$ will mean $XY-YX$. Further information about these definitions may be found in \cite{Hausner_and_schwartz}.
Equivalently, a representation of a real (resp. complex) Lie algebra $g$ is a homomorphism $p : g \to gl_d(\mathbb{F})$ for some $d$. This $d$, called the {\it degree} of the representation, is unrelated to the dimension $n$ of the Lie algebra $g$. Two representations $p, p^\prime$ of a Lie algebra $g$ of degree $d$ are called {\it equivalent} if there exists a non-singular matrix $T$ over $\mathbb{F}$ such that $p^\prime(x) = T^{-1}p(x)T$ for all $x \in g$. The degree of the adjoint representation of a Lie algebra $g$ is equal to the dimension of $g$. Further information may be found in \cite{carter_macdonald_segal_taylor}.
We tacitly require that $p$ is not the trivial representation, i.e. $p(x) = 0$ for all $x \in g$.
\item If $p$ is a representation of a complex Lie algebra $g$ on a vector space $V$, then the linear functional $\lambda$ on $g$ is called a {\it weight} of $p$ if there exists a non-zero vector $v \in V$ such that $p(x)v = \lambda(x)v$ for each $x \in g$.
The vector $v$ is called a {\it weight vector} belonging to the weight $\lambda$.
A weight is called a {\it zero weight} if the linear functional is identially zero, and a {\it non-zero weight} if the linear functional is not identically zero.
Further information about these definitions may be found in \cite{Hausner_and_schwartz}.
We once refer to the adjoint representation as the $n \times n$ representation, in Section \ref{ch:Joma_Paper.Root_Weight_Construction}.
\item In the case that $p$ is the adjoint representation of a complex Lie algebra $g$ on a vector space $V$, the weights are called {\it roots}. We reserve the term {\it root vector} for another use in this paper, but refer to the weight vectors in the adjoint representation as {\it operators}. The {\it zero roots} and {\it non-zero roots} are defined analogously. We use the symbol $\Delta$ to denote the set of non-zero roots.
Further information about these definitions may be found in \cite{Hausner_and_schwartz}.
\item A {\it simple set of roots} is a set $\lbrace r_1, \cdots, r_k \rbrace$ of linearly independent roots of $\Delta$ such that every root $r \in \Delta$ may be written as $$r = \sum_{i = 1}^k c_i r_i$$ where the coefficients $c_i$ are unique and are integers which either satisfy $c_i \ge 0$ or $c_i \le 0$ for all $i$ for each root $r$. The ordered basis $\lbrace r_1, \cdots, r_k \rbrace$ of simple roots can be used to provide an ordering for the roots of $\Delta$, by writing $r > r^\prime$ if $r = \sum_{i}^{k} c_i r_i$ and $r^\prime = \sum_{i}^{k} c_i^\prime r_i$ and $c_i > c_i^\prime$ at the first $i$ for which $c_i \ne c_i^\prime$. The operator $v$ is called a {\it raising operator} if its associated root $\lambda$ satisfies $\lambda > 0$, where $0$ is the zero linear functional. It is called a {\it lowering operator} if its associated root $\lambda$ satisfies $\lambda < 0$. Further information about these definitions may be found in \cite{Hausner_and_schwartz}.
\item Each weight $\lambda$ of a representation of a Lie algebra $g$ may be written uniquely as a real linear combination of the simple roots $\lbrace r_1, \cdots, r_k \rbrace$ of the adjoint representation of $g$. The ordered basis $\lbrace r_1, \cdots, r_k \rbrace$ of simple roots provides an ordering for the weights, by writing $\lambda > \lambda^\prime$ if $\lambda = \sum_{i}^k c_i r_i$ and $\lambda^\prime = \sum_{i}^k c^\prime_i r_i$ and $c_i > c_i^\prime$ at the first $i$ for which $c_i \ne c_i^\prime$. The {\it highest weight} $\lambda_0$ of a representation of a Lie algebra $g$ satisfies $\lambda_0 > \lambda$ for every weight $\lambda$. Further information about these definitions may be found in \cite{Hausner_and_schwartz}.
\item A {\it root diagram} is a diagram in $\mathbb{R}^d$, where $d$ is the rank of the Lie algebra $g$, in which every root of the adjoint representation is plotted as a point $\lambda$ in $\mathbb{R}^d$, vectors are drawn extending from the origin to each $\lambda$, and a vector is drawn between two roots $\lambda_1$ and $\lambda_2$ precisely when their difference is another root in $\Delta$. Further information about this definition may be found in \cite{kass}.
\item We reserve the term {\it root vector} for either the arrow in space which points from the origin to the root $\lambda$ in a root or weight diagram of $g$ or a translation of that arrow which does not reverse its orientation. The endpoint of a root vector is referred to as a {\it state}, to help differentiate the root from the vector pointing towards it.
\item A {\it weight diagram} is a diagram in $\mathbb{R}^d$, where $d$ is the rank of the Lie algebra $g$, in which every weight of a non-trivial representation is plotted as a point $\lambda$ in $\mathbb{R}^d$ and a vector is drawn between two weights $\lambda_1$ and $\lambda_2$ precisely when their difference is a root in $\Delta$. Further information about this definitions may be found in \cite{kass}.
\item A {\it minimal weight diagram} of a Lie algebra $g$ is a diagram with the least number of weights which still contains a vector for every root in $\Delta$. There may be more than one minimal weight diagram in the sense that each arises from a different representation of the Lie algebra $g$. However, all minimal weight diagrams are congruent.
\end{enumerate}
In the following paper, we often refer to the image of the representation $\Gamma$ of a Lie algebra as a representation of the Lie algebra. We also refer to the weight and root diagrams as a representation of the Lie algebra. We use this abuse of terminology not only because it is prevalent in the literature but also because the image of $\Gamma$ may be used to construct the weight or root diagrams, which in turn may be used to recreate the complexified algebra's structure constants.
The root and weight diagrams in this paper are best viewed in color. In particular, we often refer to roots which have been colored red, blue, etc. We have strived to include other descriptions which may be useful if this paper is printed in black and white only.
We describe in the paper how we may identify $g_1$ as a subalgebra of $g_2$ using root and weight diagrams, and simply state that $g_1$ is a subalgebra of $g_2$ if a root or weight diagram of $g_1$ may be identified as a subdiagram of $g_2$. We also comment that both diagrams must use the same highest weight. We list here the precise criteria that allows us to state that the complex Lie algebra $g_1$ is a subalgebra of the complex Lie algebra $g_2$:
We say $D_1$ may be {\it embedded} in $D_2$ if there is an isometry $r: \mathbb{R}^n \to \mathbb{R}^n$ such that the image of each vertex in $D_1$ is a vertex in $D_2$. Each root or weight diagram $D_i$ of an algebra $g_i$ has a {\it highest shell of weights } $\overline{W_i}$, which consists of the weights which are the furthest from the center of the diagram $D_i$. The diagram $D_1$ is a {\it subdiagram} of $D_2$ if $D_1$ may be embedded in $D_2$ and the image of $\overline{W_1}$ is contained within $\overline{W_2}$. In this case, the algebra represented by the image of $D_1$ in $D_2$ will close, allowing us to say that the algebra $g_1$ is a subalgebra of $g_2$.
Now, let $p: \mathbb{R}^n \to \mathbb{R}^{n-k}$ denote a projection with $k < n$ which satisfies $|p(x) - p(y)| \le |x - y|$ for all $x,y \in \mathbb{R}^n$, and write $pD_2$ for the result of projecting the diagram $D_2$ from $\mathbb{R}^n $ to $\mathbb{R}^{n-k}$. Considered as a diagram itself, $pD_2$ will contain a weight $pW^0_2$ such that $|pW^0_2| \ge |pW^i_2|$ for every other weight $pW^i_2$ of $pD_2$, where $|pW^i_2|$ denotes the distance from the origin to the weight $pW^i_2$. Let $\overline{pW_2}$ denote the set of weights which are a distance $|pW^0_2|$ away from the center of $pD_2$. Then the algebra $g_1$ is a subalgebra of $g_2$ if $D_1$ may be embedded in $pD_2$.
\subsection{Introduction}
\label{ch:Joma_Paper.Introduction}
Lie algebras are classified using Dynkin diagrams, which encode the geometric structure of root and weight diagrams associated with an algebra. This paper begins with an introduction to Lie algebras, roots, and Dynkin diagrams. We then show how Dynkin diagrams define an algebra's root and weight diagrams, and provide examples showing this construction. In Section~\ref{ch:Joma_Paper.subalgebras}, we develop two methods to analyze subdiagrams. We then apply these methods to the exceptional Lie algebra ~$F_4$, and describe the slight modifications needed in order to apply them to ~$E_6$. We conclude by listing all Lie subalgebras of ~$E_6$.
\subsection{Root and Weight Diagrams of Lie Algebras}
\label{ch:Joma_Paper.Root_Weight_Construction}
We summarize here some basic properties of root and weight diagrams. Further information can be found in ~\cite{jacobsen}, ~\cite{cornwell}, and ~\cite{wiki_lie_algebra}. A description of how root and weight diagrams are applied to particle physics is also given in ~\cite{cornwell}.
\subsubsection{Lie Algebras}
A Lie algebra $g$ of dimension $n$ is an $n$-dimensional vector space along with a product $[\ ,\ ]$:$ g \times g \to g$, called a commutator, which is anti-commutative ($[x,y] = -[y,x]$) and satisfies the Jacobi Identity
$$[x,[y,z]] + [y,[z,x]] + [z,[x,y]] = 0$$
for all $x,y,z \in g$. A Lie algebra is called simple if it is non-abelian and contains no non-trivial ideals. All complex semi-simple Lie algebras are the direct sum of simple Lie algebras. Thus, we follow the standard practice of studying the simple algebras, which are the building blocks of the semi-simple algebras.
There are four infinite families of Lie algebras as well as five exceptional Lie algebras. The algebras in the four infinite families correspond to special unitary matrices (or their generalizations) over different division algebras. The algebras ~$B_n$ and ~$D_n$ correspond to real special orthogonal groups in odd and even dimensions, ~$SO(2n+1)$ and ~$SO(2n)$, respectively. The algebras ~$A_n$ are the complex special unitary groups ~$SU(n+1)$, and the algebras ~$C_n$ correspond to unitary groups ~$SU(n,\mathbb{H})$ over the quaternions.
While Lie algebras are usually classified using their complex representations, there are particular real representations, based upon the division algebras, which are of interest in particle physics. Manogue and Schray ~\cite{manogue_schray} describe the use of quaternions ~$\mathbb{H}$ to construct ~$su(2,\mathbb{H})$ and ~$sl(2, \mathbb{H})$, which are real representations of ~$B_2=so(5)$ and ~$D_3 = so(5,1)$, respectively. As they discuss, their construction naturally generalizes to the octonions ~$\mathbb{O}$, yielding the real representations ~$su(2,\mathbb{O})$ and ~$sl(2,\mathbb{O})$ of ~$B_4 = so(9)$ and ~$D_5 = so(9,1)$, respectively. This can be further generalized to the ~$3 \times 3$ case, resulting in ~$su(3, \mathbb{O})$ and ~$sl(3, \mathbb{O})$, which preserve the trace and determinant, respectively, of a ~$3 \times 3$ octonionic hermitian matrix, and which are real representations of two of the exceptional Lie algebras, namely ~$F_4$ and ~$E_6$, respectively ~\cite{corrigan}. The remaining three exceptional Lie algebras are also related to the octonions ~\cite{baez, okubo}. The smallest, ~$G_2$, preserves the octonionic multiplication table and is ~$14$-dimensional, while ~$E_7$ and ~$E_8$ have dimensions ~$133$ and ~$248$, respectively. A major step in describing all possible representations of ~$E_8$ was recently completed by the Atlas Project \cite{atlas_E8}.
In Sections \ref{ch:Joma_Paper.subalgebras} and \ref{ch:Joma_Paper.subalgebras_greater_than_3}, we label the Lie algebras using the standard complex Lie algebra label (e.g. ~$A_n, B_n$) and also with the name of a particular real form (e.g. ~$su(3)$, ~$so(7)$) of the algebra. In section ~$4$, when discussing the subalgebras of ~$E_6$, we also give particular choices of real forms.
\subsubsection{Roots and Root Diagrams}
Every simple Lie algebra ~$g$ contains a maximal abelian subalgebra ~$h \subset g$, called a Cartan subalgebra, whose dimension is called the rank of ~$g$. There exists a suitably normalized basis ~$\{h_{1}, \cdots, h_{l}\}$ of ~$h$ which can be extended to a basis
$$\{ h_{1}, \cdots, h_{l}, g_1, g_{-1}, g_2, g_{-2}, \cdots, g_{\frac{n-l}{2}}, g_{-\frac{n-l}{2}} \} $$
of ~$g$ satisfying:
\begin{enumerate}
\item $[h_{i}, g_{j}] = \lambda_i^j g_{j}$ (no sum), $\lambda_i^j \in \mathbb{R}$
\item $[h_{i}, h_{j}] = 0 $
\item $[g_{j}, g_{-j}] \in h$
\end{enumerate}
The basis elements ~$g_j$ and ~$g_{-j}$ are referred to as raising and lowering operators. Property ~$1$ associates every ~$g_{j}$ with an ~$l$-tuple of real numbers ~$r^{j} = \langle\lambda_1^j, \cdots, \lambda_l^j\rangle$, called roots of the algebra, and this association is one-to-one. Further, if ~$r^{j}$ is a root, then so is ~$-r^{j} = r^{-j}$, and these are the only two real multiples of ~$r^{j}$ which are roots. According to Property ~$2$, each ~$h_i$ is associated with the ~$l$-tuple ~$\langle0, \cdots, 0\rangle$. Because this association holds for every ~$h_i \in h$, these ~$l$-tuples are sometimes referred to as zero roots. For raising and lowering operators ~$g_j$ and ~$g_{-j}$, Property ~$3$ states that ~$r^{j} + r^{-j} = \langle0, \cdots, 0\rangle$.
Let ~$\Delta$ denote the collection of non-zero roots. For roots ~$r^{i}$ and ~$r^{j} \ne -r^i$, if there exists ~$r^{k} \in \Delta$ such that ~$r^i + r^j = r^k$, then the associated operators for ~$r^i$ and ~$r^j$ do not commute, that is, $[ g_i, g_j ] \ne 0$. In this case, ~$[g_i, g_j] = C^{k}_{ij}g_{k}$ (no sum), with ~$C^k_{ij} \in \mathbb{C}, C^i_{ij} \ne 0$. If ~$r^i + r^j \not\in \Delta$, then ~$[g_i, g_j] = 0$.
When plotted in ~$\mathbb{R}^l$, the set of roots provide a geometric description of the algebra. Each root is associated with a vector in ~$\mathbb{R}^l$. We draw ~$l$ zero vectors at the origin for the ~$l$ zero roots corresponding to the basis ~$h_1, \cdots, h_l$ of the Cartan subalgebra. For the time being, we then plot each non-zero root ~$r^i = \langle\lambda_1^i, \cdots, \lambda_l^i\rangle$ as a vector extending from the origin to the point ~$\langle\lambda_1^i, \cdots, \lambda_l^i\rangle$. The terminal point of each root vector is called a state. As is commonly done, we use ~$r^i$ to refer to both the root vector and the state. In addition, we allow translations of the root vectors to start at any state, and connect two states ~$r^i$ and ~$r^j$ by the root vector ~$r^k$ when ~$r^k + r^i = r^j$ in the root system. The resulting diagram is called a root diagram.
As an example, consider the algebra ~$su(2)$, which is classified as ~$A_1$. The algebra ~$su(2)$ is the set of ~$2 \times 2$ complex traceless Hermitian matrices. Setting
\[
\sigma_1 = \left[ \begin{array}{cc}
0 & 1 \\
1 & 0 \\
\end{array}
\right]
\hspace{.5cm}
\sigma_2 = \left[ \begin{array}{cc}
0 & -i \\
i & 0 \\
\end{array}
\right]
\hspace{.5cm}
\sigma_3 = \left[ \begin{array}{cc}
1 & 0 \\
0 & -1 \\
\end{array}
\right]
\] we choose the basis ~$h_1 = \frac{1}{2}\sigma_3$ for the Cartan subalgebra ~$h$, and use~$g_1 = \frac{1}{2}(\sigma_1+i\sigma_2)$ and~$g_{-1} = \frac{1}{2}(\sigma_1-i\sigma_2)$ to extend this basis for all of ~$su(2)$. Then
\begin{enumerate}
\item $[h_1, h_1] = 0$
\item $[h_1, g_1] = 1 g_1$
\item $[h_1, g_{-1}] = -1 g_{-1}$
\item $[g_{1}, g_{-1}] = h_1$
\end{enumerate}
By Properties ~$2$ and ~$3$, we associate the root vector ~$r^1 = \langle1\rangle$ with the raising operator ~$g_1$ and the root vector ~$r^{-1} = \langle-1\rangle$ with the lowering operator ~$g_{-1}$. Using the zero root ~$\langle0\rangle$ associated with ~$h_1$, we plot the corresponding three points $(1)$, $(-1)$, and $(0)$ for the states ~$r^1$, ~$r^{-1}$, and ~$h_1$. We then connect the states using the root vectors. Instead of displaying both root vectors ~$r^1$ and ~$r^{-1}$ extending from the origin, we have chosen to use only the root vector ~$r^{-1}$, as ~$r^{-1} = - r^{1}$, to connect the states ~$(1)$ and ~$(0)$ to the states ~$(0)$ and ~$(-1)$, respectively. The resulting root diagram is illustrated in Figure ~\ref{fig:a1_root_diagram}.
\begin{figure}[htbp]
\begin{minipage}[t]{6in}
\begin{center}
\leavevmode
\resizebox{3in}{!}{\includegraphics*[50,210][531,262]{A1_root_diagram}}
\end{center}
\end{minipage}
\caption{Root diagram of $A_1 = su(2)$}
\label{fig:a1_root_diagram}
\end{figure}
\subsubsection{Weights and Weight Diagrams}
An algebra ~$g$ can also be represented using a collection of ~$d \times d$ matrices, with ~$d$ unrelated to the dimension of ~$g$. The matrices corresponding to the basis ~$h_1, \cdots, h_l$ of the Cartan subalgebra can be simultaneously diagonalized, providing ~$d$ eigenvectors. Then, a list ~$w^m$ of ~$l$ eigenvalues, called a {\it weight}, is associated with each eigenvector. Thus, the diagonalization process provides ~$d$ weights for the algebra ~$g$. The roots of an ~$n$-dimensional algebra can be viewed as the non-zero weights of its ~$n \times n$ representation.
Weight diagrams are created in a manner comparable to root diagrams. First, each weight ~$w^i$ is plotted as a point in ~$\mathbb{R}^l$. Recalling the correspondence between a root ~$r^i$ and the operator ~$g^i$, we draw the root ~$r^k$ from the weight ~$w^i$ to the weight ~$w^j$ precisely when ~$r^k + w^i = w^j$, which at the algebra level occurs when the operator ~$g^k$ raises (or lowers) the state ~$w^i$ to the state ~$w^j$.
The root and minimal non-trivial weight diagrams of the algebra ~$A_2 = su(3)$ are shown in Figure ~\ref{fig:root_and_weight_diagrams_A2}. The algebra has three pairs of root vectors, which are oriented east-west (colored blue), roughly northeast-southwest (colored red), and roughly northwest-southeast (colored green).
The algebra's rank is the dimension of the underlying Euclidean space, which in this case is~$l=2$, and the number of non-zero root vectors is the number of raising and lowering operators. The minimal representation contains three independent arrows, which provide six different roots (although we've only indicated one of each raising and lowering root pair). Using the root diagram, the dimension of the algebra can now be determined either from the number of non-zero roots or from the number of root vectors extending in different directions from the origin. Both diagrams indicate that the dimension of ~$A_2 = su(3)$ is ~$8 = 6+2$. Root and (non-trivial) weight diagrams indicate the underlying algebra's dimension and rank, which is (almost) enough information to identify the algebra.
\footnote{For algebras of rank~$6$ and lower, the exceptions are that ~$B_n = so(2n+1)$ and ~$C_n=sp(2\cdot n)$ have the same dimension for each rank ~$n$, and that ~$B_6$, ~$C_6$, and ~$E_6$ all have dimension~$78$.}
\begin{figure}[htbp]
\begin{center}
\begin{minipage}[t]{5cm}
\begin{center}
\leavevmode
\resizebox{3cm}{!}{\includegraphics*[120,80][484,400]{adjoint_rep_A2}}
\end{center}
\end{minipage}
\hspace{1cm}
\begin{minipage}[t]{5cm}
\begin{center}
\leavevmode
\resizebox{3cm}{!}{\includegraphics*[200,100][380,260]{weight_3_rep_A2}}
\end{center}
\end{minipage}
\end{center}
\caption{Root and minimal weight diagrams of ~$A_2=su(3)$}
\label{fig:root_and_weight_diagrams_A2}
\end{figure}
\subsubsection{Constructing Root Diagrams from Dynkin Diagrams}
In 1947, Eugene Dynkin simplified the process of classifying complex semi-simple Lie algebras by using what became known as Dynkin diagrams ~\cite{dyn2}. He realized that every root in a rank ~$l$ algebra can be expressed as an integer sum or difference of ~$l$ simple roots. Further, the relative lengths and interior angle between pairs of simple roots fits one of four cases. A Dynkin diagram records the configuration of an algebra's simple roots.
Each node in a Dynkin diagram represents one of the algebra's simple roots. Two nodes are connected by zero, one, two, or three lines when the interior angle between the roots is ~$\frac{\pi}{2}$, ~$\frac{2\pi}{3}$, ~$\frac{3\pi}{4}$, or ~$\frac{5\pi}{6}$, respectively. If two nodes are connected by ~$n$ lines, then the magnitudes of the corresponding roots satisfy the ratio ~$1:\sqrt{n}$. An arrow is used in the Dynkin diagram to point toward the node for the smaller root. If two roots are orthogonal, no direct information is known about their relative lengths.
We give the Dynkin diagrams for the rank ~$2$ algebras in Figure ~\ref{fig:rank_2_dynkin_diagrams} and the corresponding simple root configurations in Figure ~\ref{fig:rank_2_simple_roots}. For each algebra, the left node in the Dynkin diagram corresponds to the root ~$r^1$ of length ~$1$, colored red and lying along the horizontal axis, and the right node corresponds to the other root ~$r^2$, colored blue.
\begin{figure}[htbp]
\begin{minipage}[t]{6in}
\begin{center}
\leavevmode
\xymatrixcolsep{10pt}
\xymatrix@M=0pt@W=0pt{
*+=[o][F-]{\hspace{.35cm} } & & *+=[o][F-]{\hspace{.35cm} } & \hspace{.65cm} & *+=[o][F-]{\hspace{.35cm} } \ar@{-}[rr]& & *+=[o][F-]{\hspace{.35cm} } & \hspace{.65cm} & *+=[o][F-]{\hspace{.35cm} } \ar@2{-}[rr] & \rangle & *+=[o][F-]{\hspace{.35cm} } & \hspace{.65cm} & *+=[o][F-]{\hspace{.35cm} } & \langle & *+=[o][F-]{\hspace{.35cm} } \ar@2{-}[ll] & \hspace{.65cm} & *+=[o][F-]{\hspace{.35cm} } \ar@3{-}[rr] & \rangle & *+=[o][F-]{\hspace{.35cm} } \\
}\\
\vspace{1em}
\hspace{-.00in} $D_2 = so(4)$ \hspace{.29in} $A_2 = su(3)$ \hspace{.30in} $B_2 = so(5)$ \hspace{.35in} $C_2 = sp(2\cdot 2)$ \hspace{.25in} $G_2 = Aut(\mathbb{O})$
\caption{Rank 2 Dynkin diagrams}
\label{fig:rank_2_dynkin_diagrams}
\end{center}
\end{minipage}
\end{figure}
\begin{figure}[htbp]
\begin{minipage}[t]{6in}
\begin{center}
\leavevmode
\resizebox{5.0in}{!}{\includegraphics*[20,50][553,120]{step0_simple_roots_to_full_roots_no_title}}
\\
\vspace{1em}
\hspace{-.00in} $D_2 = so(4)$ \hspace{.29in} $A_2 = su(3)$ \hspace{.30in} $B_2 = so(5)$ \hspace{.35in} $C_2 = sp(2\cdot 2)$ \hspace{.25in} $G_2=Aut(\mathbb{O})$
\caption{Rank 2 simple roots}
\label{fig:rank_2_simple_roots}
\end{center}
\end{minipage}
\end{figure}
In ~$\mathbb{R}^l$, each root ~$r^i$ defines an ~$(l-1)$-dimensional hyperplane which contains the origin and is orthogonal to ~$r^i$. A {\it Weyl reflection} for ~$r^i$ reflects each of the other roots ~$r^j$ across this hyperplane, producing the root ~$r^k$ defined by
$$r^k = r^j - 2(\frac{r^j \textrm{ {\tiny $\bullet$} } r^i}{|r^i|})\frac{r^i}{|r^i|} $$
According to Jacobsen ~\cite{jacobsen}, the full set of roots can be generated from the set of simple roots and their associated Weyl reflections.
We illustrate how the full set of roots can be obtained from the simple roots using Weyl reflections in Figure ~\ref{fig:rank_2_weyl_reflections_animation}. We start with the two simple roots for each algebra, as given in Figure ~\ref{fig:rank_2_simple_roots}. For each algebra, we refer to the horizontal simple root, colored red, as ~$r^1$, and the other simple root, colored blue, as ~$r^2$. Step ~$1$ shows the result of reflecting the simple roots using the Weyl reflection associated with ~$r^1$. In this diagram, the black thin line represents the hyperplane orthogonal to ~$r^1$, and the new resulting roots are colored green. Step ~$2$ shows the result of reflecting this new set of roots using the Weyl reflection associated with ~$r^2$. At this stage, both ~$D_2 = so(4)$ and~$A_2 = su(3)$ have their full set of roots. We repeat this process again in steps 3 and 4, using the Weyl reflections associated first with ~$r^1$ and then with ~$r^2$. The full root systems for ~$B_2 = so(5)$ and ~$C_2 = sp(2\cdot 2)$ are obtained after the first three Weyl reflections. Only ~$G_2$ requires all four Weyl reflections.
\begin{figure}[htbp]
\begin{center}
\begin{minipage}[t]{5in}
\begin{center}
\leavevmode
Step ~$1$: Weyl reflection using root $r^1$\\
\resizebox{5in}{!}{\includegraphics*[8,22][567,123]{step1_simple_roots_to_full_roots_correct}}
\begin{minipage}[t]{5in}
\mbox{\hspace{-.05in} $D_2 = so(4)$ \hspace{.20in} $A_2 = su(3)$ \hspace{.24in} $B_2 = so(5)$ \hspace{.20in} $C_2 = sp(2\cdot 2)$ \hspace{.10in} $G_2 = Aut(\mathbb{O})$}
\end{minipage}
\end{center}
\vspace{.6cm}
\end{minipage}
\begin{minipage}[t]{5in}
\begin{center}
\leavevmode
Step ~$2$: Weyl reflection using root $r^2$\\
\resizebox{5in}{!}{\includegraphics*[8,22][567,141]{step2_simple_roots_to_full_roots_correct}}
\begin{minipage}[t]{5in}
\mbox{\hspace{-.05in} $D_2 = so(4)$ \hspace{.20in} $A_2 = su(3)$ \hspace{.24in} $B_2 = so(5)$ \hspace{.20in} $C_2 = sp(2\cdot 2)$ \hspace{.10in} $G_2 = Aut(\mathbb{O})$}
\end{minipage}
\end{center}
\vspace{.6cm}
\end{minipage}
\begin{minipage}[t]{5in}
\begin{center}
\leavevmode
Step ~$3$: Weyl reflection using root $r^1$\\
\resizebox{5in}{!}{\includegraphics*[8,22][567,141]{step3_simple_roots_to_full_roots_correct}}
\begin{minipage}[t]{5in}
\mbox{\hspace{-.05in} $D_2 = so(4)$ \hspace{.20in} $A_2 = su(3)$ \hspace{.24in} $B_2 = so(5)$ \hspace{.20in} $C_2 = sp(2\cdot 2)$ \hspace{.10in} $G_2 = Aut(\mathbb{O})$}
\end{minipage}
\end{center}
\vspace{.6cm}
\end{minipage}
\begin{minipage}[t]{5in}
\begin{center}
\leavevmode
Step ~$4$: Weyl reflection using root $r^2$\\
\resizebox{5in}{!}{\includegraphics*[8,4][567,141]{step4_simple_roots_to_full_roots_correct}}
\begin{minipage}[t]{5in}
\mbox{\hspace{-.05in} $D_2 = so(4)$ \hspace{.20in} $A_2 = su(3)$ \hspace{.24in} $B_2 = so(5)$ \hspace{.20in} $C_2 = sp(2\cdot 2)$ \hspace{.10in} $G_2 = Aut(\mathbb{O})$}
\end{minipage}
\end{center}
\caption{Generating an algebra's full root system using Weyl reflections}
\label{fig:rank_2_weyl_reflections_animation}
\end{minipage}
\end{center}
\end{figure}
\begin{figure}[htbp]
\begin{center}
\begin{minipage}[t]{2.4cm}
\begin{center}
\leavevmode
\resizebox{2.1cm}{!}{\includegraphics*[202,144][389,333]{D2_root_diagram}}\\
$D_2 = so(4)$
\end{center}
\end{minipage}
\begin{minipage}[t]{2.4cm}
\begin{center}
\leavevmode
\resizebox{2.1cm}{!}{\includegraphics*[120,70][500,400]{adjoint_rep_A2}}\\
$A_2 = su(3)$
\end{center}
\end{minipage}
\begin{minipage}[t]{2.4cm}
\begin{center}
\leavevmode
\resizebox{2.1cm}{!}{\includegraphics*[120,60][480,420]{B2_root_diagram}}\\
$B_2=so(5)$
\end{center}
\end{minipage}
\begin{minipage}[t]{2.4cm}
\begin{center}
\leavevmode
\resizebox{2.1cm}{!}{\includegraphics*[230,170][375,315]{C2_root_diagram}}\\
$C_2=sp(2\cdot 2)$
\end{center}
\end{minipage}
\begin{minipage}[t]{2.4cm}
\begin{center}
\leavevmode
\resizebox{2.1cm}{!}{\includegraphics*[152,79][440,401]{G2_root_diagram}}\\
$G_2 = Aut(\mathbb{O})$
\end{center}
\end{minipage}
\caption{Root diagrams of simple rank ~$2$ algebras}
\label{fig:rank_2_root_diagrams}
\end{center}
\end{figure}
The full set of roots have been produced once the Weyl reflections fail to produce additional roots. The root diagrams are completed by connecting the tips of the roots ~$r^i$ and ~$r^j$ via the root ~$r^k$ precisely when ~$r^k + r^i = r^j$. From the root diagrams in Figure ~\ref{fig:rank_2_root_diagrams}, it is clear that the dimension of~$A_2=su(3) \textrm{ is }8$ and the dimension of ~$G_2 \textrm{ is }14$, while both ~$B_2=so(5)$ and ~$C_2=sp(2\cdot 2)$ have dimension ~$10$. Further, since the diagram of ~$B_2$ can be obtained via a rotation and rescaling of the root diagram for ~$C_2$, it is clear that ~$B_2$ and ~$C_2$ are isomorphic.
\subsubsection{Constructing Weight Diagrams from Dynkin Diagrams}
Root diagrams are a specific type of weight diagram. While the states and roots can be identified with each other in a root diagram, this does not happen for general weight diagrams. Weight diagrams are a collection of states, called weights, and the roots are used to move from one weight to another. Although it has only one root diagram, an algebra has an infinite number of weight diagrams.
Like a root diagram, any one of an algebra's weight diagrams can be constructed entirely from its Dynkin diagram. The Dynkin diagram records the relationship between the ~$l$ simple roots of a rank ~$l$ algebra. Each simple root ~$r^1, \cdots, r^l$ is associated with a weight ~$w^1, \cdots, w^l$. When all integer linear combinations of these weights are plotted in ~$\mathbb{R}^l$, they form an infinite lattice of possible weights. This lattice contains every weight from every ~$d \times d$ ~$(d \ge 0)$ representation of the algebra. A particular weight diagram is chosen by specifying a highest weight ~$W$ in this lattice. Weyl reflections related to the weights ~$w^1, \cdots, w^l$ then define the boundary of weights ~$\overline{W}$ for that particular weight diagram, and root vectors are used to connect pairs of valid weights in ~$\overline{W}$ and within ~$\overline{W}$.\footnote{The distance between adjacent weights in the infinite lattice is less than or equal to the length of each root vector. Root vectors do not always connect adjacent weights, but often skip over them.} Choosing a new highest weight ~$W$ outside the original weight diagram's boundary ~$\overline{W}$ selects a new, larger, weight diagram, while choosing a new highest weight inside of ~$\overline{W}$ selects a new, smaller, weight diagram.
An equivalent method ~\cite{kass} constructs weight diagrams while avoiding the explicit use of Weyl reflections. First, the ~$l$ simple roots ~$r^1, \cdots, r^l$ are used to define the Cartan matrix ~$A$, whose components are ~$a_{ij} = 2 \frac{r^i \textrm{ {\tiny $\bullet$} } r^j}{r^i \textrm{ {\tiny $\bullet$} } r^i}$. Equivalently, the components are:
\[
a_{ij} =\begin{cases} 2, &\text{if $i = j$;}\\
0, &\text{if $r^i$ and $r^j$ are orthogonal;}\\
-3, &\text{if the interior angle between $r^i$ and $r^j$ is $\frac{5\pi}{6}$, and $\sqrt{3}|r^i| = |r^j|$;}\\
-2, &\text{if the interior angle between roots $r^i$ and $r^j$ is $\frac{3\pi}{4}$, and $\sqrt{2}|r^i| = |r^j|$;}\\
-1, &\text{in all other cases;}\\
\end{cases}
\]
Cartan matrices are invertible. Thus, the {\it fundamental weights} ~$w^1, \cdots, w^l$ defined by \[w^i = \sum_{k=1}^{l}(A^{-1})_{ki}r^k\] are linearly independent. We create an infinite lattice ~$\mathbb{W} = \{ m_i w^i | m_i \in \mathbb{Z}\}$ of possible weights in ~$\mathbb{R}^l$, and then label each weight ~$W^i = m^i_j w^j \in \mathbb{W}$ using the ~$l$-tuple ~$M^i = \left[m^i_1, \cdots, m^i_l\right]$, called a {\it mark}. We choose one of the infinite number of weight diagrams by specifying ~$l$ non-negative integers ~$m^0_1, \cdots, m^0_l$ for the mark of the highest weight ~$W^0$.
While the components of the Cartan matrix record the geometry of the simple roots, the components of each mark record the geometric configuration of the weights within the lattice. Each weight ~$W^i$ is part of a shell of weights ~$\overline{W}$ which are equidistant from the origin.
As discussed earlier, Weyl reflections associated with the fundamental weights can be used to find the weights in ~$\overline{W}$. The diagram's entire set of valid weights can also be determined using the mark ~$M^i$ of each weight. The positive integers ~$m^i_j \in M^i$ list the maximum number of times that the simple root ~$r^j$ can be subtracted from ~$W^i$ while keeping the new weights on or within the boundary ~$\overline{W}$.
Thus, weights ~$W^i - 1 r^j, \cdots, W^i - (m^i_j) r^j \textrm{ (no sum)}$ are valid weights occurring on or inside ~$\overline{W}$. These new weights have marks ~$M^i - A_i^{\mathsf{T}}, M^i - 2 A_i^{\mathsf{T}}, \cdots, M^i - m^i_j A_i^{\mathsf{T}}$, where ~$A_i^{\mathsf{T}}$ is the transpose of the ~$i$th column of the Cartan matrix ~$A$. Thus, given a weight diagram's highest weight ~$W^0$, this procedure selects all of the diagram's weights from the infinite lattice. The diagram is completed by connecting any two weights ~$W^i$ and ~$W^k$ by the root ~$r^j$ whenever ~$W^i + r^j = W^k$.
We carry out this procedure for the algebra $B_3 = so(7)$, whose simple roots are
\[
\begin{array}{ccc}
r^1 = \langle\sqrt{2},0,0\rangle, & r^2 = \langle-\sqrt{\frac{1}{2}}, -\sqrt{\frac{3}{2}}, 0\rangle, & r^3 = \langle0, \sqrt{\frac{2}{3}}, \sqrt{\frac{1}{3}}\rangle
\end{array}
\]
\noindent
We produce the Cartan matrix ~$A$, find ~$A^{-1}$, and list the fundamental weights.
\hspace{-.5cm}
\begin{minipage}[t]{4.05cm}
\begin{center}
Cartan Matrix
\end{center}
$$ A = \left( \begin{array}{ccc}
2 & -1 & 0 \\
-1 & 2 & -1 \\
0 & -2 & 2
\end{array}
\right)
$$
\end{minipage}
\hfill
\begin{minipage}[t]{4.05cm}
\begin{center}
Inverse Cartan Matrix
\end{center}
$$ A^{-1} = \left( \begin{array}{ccc}
1 & 1 & \frac{1}{2} \\
1 & 2 & 1 \\
1 & 2 & \frac{3}{2}
\end{array}
\right)
$$
\end{minipage}
\hfill
\begin{minipage}[t]{4.05cm}
\begin{center}
Fundamental Weights
\end{center}
$$
\begin{array}{c}
w^1 = 1r^1 + 1r^2 + 1r^3 \\
w^2 = 1r^1 + 2r^2 + 2r^3 \\
w^3 = \frac{1}{2}r^1 + 1r^2 + \frac{3}{2}r^3
\end{array}
$$
\end{minipage}
\noindent
Starting with the highest weight ~$W^0 = w^1$, whose mark is $\left[1,0,0\right]$, the above procedure generates the following weights:
\noindent
\begin{minipage}[htbp]{5in}
\begin{center}
\xymatrixcolsep{6pt}
\xymatrixrowsep{1pt}
\xymatrix@W=0pt{
& \textrm{Marks} & \hfill & \textrm{Weights} \\
& \left[1, 0, 0\right]\ar[ddl]_{-r^1} & & W^0 = \left[1, 0, 0\right] \textrm{ {\tiny $\bullet$} } \langle w^1, w^2, w^3 \rangle = 1r^1 + 1r^2 + 1r^3\\
& & & \\
\left[-1, 1, 0\right]\ar[ddd]_{-r^2} & & & W^1 = \left[-1, 1, 0\right] \textrm{ {\tiny $\bullet$} } \langle w^1, w^2, w^3 \rangle = 0r^1 + 1r^2 + 1r^3\\
& & & \\
& & & \\
\left[0, -1, 2\right]\ar[ddr]^{-r^3}& & & W^2 = \left[0, -1, 2\right] \textrm{ {\tiny $\bullet$} } \langle w^1, w^2, w^3 \rangle = 0r^1 + 0r^2 + 1r^3\\
& & & \\
& \left[0, 0, 0\right]\ar[ddr]^{-r^3} & & W^3 = \left[0, 0, 0\right] \textrm{ {\tiny $\bullet$} } \langle w^1, w^2, w^3 \rangle = 0r^1 + 0r^2 + 0r^3\\
& & & \\
& & \left[0, 1, -2\right]\ar[ddd]_{-r^2} & W^4 = \left[0, 1, -2\right] \textrm{ {\tiny $\bullet$} } \langle w^1, w^2, w^3 \rangle = 0r^1 + 0r^2 -1r^3\\
& & & \\
& & & \\
& & \left[1, -1, 0\right]\ar[ddl]_{-r^1} & W^5 = \left[1, -1, 0\right] \textrm{ {\tiny $\bullet$} } \langle w^1, w^2, w^3\rangle = 0r^1 - 1r^2 -1r^3\\
& & & \\
& \left[-1, 0, 0\right] & & W^6 = \left[-1, 1, 0\right] \textrm{ {\tiny $\bullet$} } \langle w^1, w^2, w^3\rangle = -1r^1 - 1r^2 -1r^3
}
\end{center}
\end{minipage}
We plot the weight ~$W^0, \cdots, W^6$ and the appropriate lowering roots ~$-r^1, \cdots, -r^3$ at each weight for this weight diagram of ~$B_3 = so(7)$ in Figure ~\ref{fig:weight_skeleton}. The full set of roots are used to connect pairs of weights, giving the complete weight diagram of ~$B_3 = so(7)$ in Figure ~\ref{fig:weight_diagram}.
\begin{figure}[hbtp]
\begin{minipage}[hb]{7cm}
\begin{center}
\leavevmode
\resizebox{3.5cm}{!}{\includegraphics*[100,90][338,320]{weight_B3_skeleton}}
\caption{\label{fig:weight_skeleton} \hbox{$B_3 = so(7)$ weight skeleton}}
\end{center}
\end{minipage}
\hfill
\begin{minipage}[hb]{7cm}
\begin{center}
\leavevmode
\resizebox{3.5cm}{!}{\includegraphics*[100,90][338,320]{weight_B3_filled_in_skeleton}}
\caption{\label{fig:weight_diagram} \hbox{$B_3=so(7)$ weight diagram}}
\end{center}
\end{minipage}
\end{figure}
\subsubsection{Rank 3 Root and Weight Diagrams}
Using the procedures outlined above, we catalog the root and minimal weight diagrams for the rank ~$3$ algebras. We did this by writing a Perl program to generate the weight and root diagram structures and return executable Maple code. We then used Maple and Javaview to display the pictures.
The ~$3$-dimensional root diagrams of the rank ~$3$ algebras are given in Figure ~\ref{fig:Rank_3_root_diagrams}. The root diagrams of ~$A_3=su(4)$ and $D_3=so(6)$ are identical. Hence, $A_3 = D_3$. These algebras have dimension ~$15$, as their root diagram contains ~$12$ non-zero states. The ~$B_3=so(7)$ and ~$C_3=sp(2\cdot 3)$ algebras both have dimension ~$21$, and their root diagrams each contain ~$18$ non-zero states. Each of these algebras contain the ~$A_2=su(3)$ root diagram, which is a hexagon. This can be seen at the center of each rank ~$3$ algebra by turning its root diagram in various orientations. The modeling kit {\it ZOME} ~\cite{zome} can be used to construct the rank ~$3$ root diagrams. The kit contains connectors of the right length and nodes with the correct configuration of connection angles to construct most of the rank ~$3$ root diagrams.\footnote{The root diagram for the algebra ~$C_3 = sp(2\cdot 3)$ can not be built to scale because it requires an angle not in the existing {\it ZOME} tools.}
\begin{figure}[htbp]
\begin{minipage}[t]{4.1cm}
\begin{center}
\leavevmode
\resizebox{4.0cm}{!}{\includegraphics*[185,155][352,304]{A3_root_diagram}}\\
$su(4) = A_3 = D_3 = so(6)$
\end{center}
\end{minipage}
\hfill
\begin{minipage}[t]{4.1cm}
\begin{center}
\leavevmode
\resizebox{4.0cm}{!}{\includegraphics*[204,160][370,308]{B3_has_A2}}\\
$B_3 = so(7)$
\end{center}
\end{minipage}
\hfill
\begin{minipage}[t]{4.1cm}
\begin{center}
\leavevmode
\resizebox{4.0cm}{!}{\includegraphics*[159,109][440,362]{C3_root_diagram}}\\
$C_3 = sp(2\cdot 3)$
\end{center}
\end{minipage}
\caption{Rank 3 root diagrams}
\label{fig:Rank_3_root_diagrams}
\end{figure}
An algebra's minimal weight diagram has the fewest number of weights while still containing all of the roots. Figure ~\ref{fig:rank_3_minimal_weight_diagrams} shows the minimal weight diagrams for ~$A_3=D_3$, ~$B_3$, and ~$C_3$. Each root occurs once in the diagram for ~$A_3=D_3$, while in the diagram for ~$B_3$ every root is used twice. The minimal weight diagram for ~$C_3$ is centered about the origin, and the roots passing through the origin (colored red, blue, and brown) occur once, while the other roots occur twice.
\begin{figure}[htbp]
\begin{minipage}{4.1cm}
\begin{center}
\leavevmode
\resizebox{3.4cm}{!}{\includegraphics*[242,130][370,240]{A3_min_weight}}\\
$su(4) = A_3 = D_3 = so(6)$
\end{center}
\end{minipage}
\hfill
\begin{minipage}{4.1cm}
\begin{center}
\leavevmode
\resizebox{3.4cm}{!}{\includegraphics*[225,145][350,268]{B3_min_weight}}\\
$B_3 = so(7)$
\end{center}
\end{minipage}
\hfill
\begin{minipage}{4.1cm}
\begin{center}
\leavevmode
\resizebox{3.4cm}{!}{\includegraphics*[158,115][431,365]{C3_min_weight}}\\
$C_3=sp(2\cdot 3)$
\end{center}
\end{minipage}
\caption{Rank 3 minimal weight diagrams}
\label{fig:rank_3_minimal_weight_diagrams}
\end{figure}
\subsection{Subalgebras of Complex Lie Algebras}
\label{ch:Joma_Paper.subalgebras}
We have already noted that we can recognize $A_2 = su(3)$ as a subalgebra of $A_3 = su(4)$, $B_3 = so(7)$, and $C_3 = sp(2\cdot 3)$ by identifying its hexagonal root diagram in each of the larger root diagrams. This section develops two further methods which help us identify subalgebras.
\subsubsection{Subalgebras of ~$B_3=so(7)$ using Root Diagrams}
Root and weight diagrams can be used to identify subalgebras. For algebras of rank ~$l \le 3$, we can recognize subdiagrams corresponding to subalgebras by building the algebra's root and weight diagrams and rotating them in ~$\mathbb{R}^3$.
We illustrate the process of finding all subalgebras of ~$B_3=so(7)$ by using its root diagram. When rotated to the position in Figure ~\ref{fig:B3_slice_to_B2}, the large horizontal ~$2$-dimensional rectangle through the origin (containing roots colored green and magenta) contains eight non-zero nodes. This subdiagram is the root diagram of the ~$10$-dimensional algebra ~$B_2 = C_2$. The smaller rectangular diagrams lying in parallel planes above and below this large rectangle contain all of the roots vectors of ~$B_2=C_2$. These diagrams are minimal weight diagrams of ~$B_2 = C_2$.
Two additional rotations of the root diagram of ~$B_3=so(7)$ produce subalgebras. In Figure ~\ref{fig:B3_slice_to_D2}, the horizontal plane through the origin contains only two orthogonal roots, which are colored blue and fuchsia. These roots comprise the root diagram of ~$D_2=su(2)+su(2)$. We have already identified the hexagonal ~$A_2=su(3)$ root diagram in the horizontal plane containing the origin in Figure ~\ref{fig:B3_slice_to_D3_and_A2}. Each horizontal triangle above and below this plane is a minimal weight representation of ~$A_2$. In addition, the subdiagram containing the bottom triangle, middle hexagon, and top triangle is the ~$15$-dimensional, rank ~$3$ algebra ~$A_3 = D_3$. Thus, it is possible to identify both rank ~$l-1$ and rank ~$l$ subalgebras of a rank ~$l$ algebra.
\begin{figure}[htbp]
\begin{center}
\begin{minipage}[t]{4.1cm}
\begin{center}
\leavevmode
\resizebox{4.0cm}{!}{\includegraphics*[150,175][320,310]{B3_slice_to_B2}}
\caption{ \hbox{$B_2 \subset B_3$}}
\label{fig:B3_slice_to_B2}
\end{center}
\end{minipage}
\hfill
\begin{minipage}[t]{4.1cm}
\begin{center}
\leavevmode
\resizebox{3.2cm}{!}{\includegraphics*[175,190][325,360]{B3_slice_to_D2}}
\caption{ \hbox{$D_2 \subset B_3$}}
\label{fig:B3_slice_to_D2}
\end{center}
\end{minipage}
\hfill
\begin{minipage}[t]{4.9cm}
\begin{center}
\leavevmode
\resizebox{4.0cm}{!}{\includegraphics*[152,167][322,323]{B3_slice_to_D3_and_A2}}
\caption{ \hbox{$A_2 \subset D_3 \subset B_3$}}
\label{fig:B3_slice_to_D3_and_A2}
\end{center}
\end{minipage}
\\
\vspace{.5cm}
\begin{center}Identified Subalgebras of $B_3=so(7)$ using the Root Diagram\end{center}
\end{center}
\end{figure}
Not all subalgebras of a rank ~$l$ algebra can be identified as subdiagrams of its root or weight diagram. When extending a rank ~$l$ algebra to a rank ~$l+1$ algebra, each root ~$r^i = \langle \lambda_1^i, \cdots, \lambda_l^i \rangle$ is extended in ~$\mathbb{R}^{l+1}$ to the roots ~$r^{i_1}, \cdots, r^{i_m}$, where ~$r^{i_j} = \langle \lambda_1^i, \cdots, \lambda_l^i, \lambda_{l+1}^{i_j} \rangle$. Here, ~$\lambda_{l+1}^{i_j}$ is one of ~$m \ge 1$ different eigenvalues values defined by the extension of the algebra. Hence, although roots ~$r^{i_1}, \cdots, r^{i_m}$ are all distinct in ~$\mathbb{R}^{l+1}$, they are the same root when restricted to their first ~$l$ coordinates. Thus, we can identify subalgebras of a rank ~$l+1$ algebra by projecting its root and weight diagrams along any direction.
Just as our eyes ``see'' objects in ~$\mathbb{R}^3$ by projecting them into ~$\mathbb{R}^2$, we can identify subalgebras of ~$B_3=so(7)$ by projecting its root and weight diagram into ~$\mathbb{R}^2$. In Figures ~\ref{fig:B3_project_to_B2} and ~\ref{fig:B3_project_to_G2}, we have rotated the root diagram of ~$B_3$ so that our eyes project one of the root vectors onto another root vector. This method allows us to identify ~$B_2=so(5)$ and ~$G_2$ as subalgebras of ~$B_3$ by recognizing their root diagrams in Figures ~\ref{fig:B3_project_to_B2} and ~\ref{fig:B3_project_to_G2}, respectively. Although we previously identified ~$B_2 \subset B_3$ by using a subdiagram confined to a plane, we could not identify ~$G_2$ as a subalgebra of ~$B_3$ using that method.
\begin{figure}[hbtp]
\begin{center}
\begin{minipage}[t]{7cm}
\begin{center}
\leavevmode
\resizebox{4cm}{!}{\includegraphics*[190,125][360,300]{B3_project_to_B2}}
\caption{\hbox{$B_2=so(5) \subset B_3=so(7)$}}
\label{fig:B3_project_to_B2}
\end{center}
\end{minipage}
\hfill
\hspace{-.5cm}
\begin{minipage}[t]{7cm}
\begin{center}
\leavevmode
\resizebox{3.5cm}{!}{\includegraphics*[250,180][400,350]{B3_project_to_G2}}
\caption{$G_2 \subset B_3=so(7)$}
\label{fig:B3_project_to_G2}
\end{center}
\end{minipage}
\\
\vspace{.5cm}
\begin{center}Identify Subalgebras of ~$B_3=so(7)$ using Projections of its Root Diagram\end{center}
\end{center}
\end{figure}
\subsubsection{Geometry of Slices and Projections}
The procedures in subsections ~$2.4$ and ~$2.5$ allow us to create root and weight diagrams for algebras of rank greater than ~$3$. We now give precise definitions for the two processes we used in subsection ~$3.1$ to identify subalgebras of ~$B_3 = so(7)$.
We first generalize the method that allowed us to find subalgebras of ~$B_3=so(7)$ by identifying subdiagrams in specific cross-sections of its root diagram. We refer to this method as ``slicing'', based on an analogy to slicing a loaf of bread. A knife, making parallel cuts through the bread, creates several independent slices of the bread. We use the same idea to slice an algebra's root or weight diagram. A set of ~$l-1$ linearly independent vectors ~$V = \{v^1, \cdots, v^{l-1}\}$ defines an ~$(l-1)$-dimensional hyperplane in ~$\mathbb{R}^l$. Given ~$V$, a {\it slice}~$D_{\alpha}$ of an algebra's ~$l$-dimensional diagram ~$D$ is the subdiagram consisting of the vertex ~$\alpha$ and all root vectors and vertices in the hyperplane spanned by ~$V$ containing ~$\alpha$. A {\it slicing} of a diagram~$D$ using ~$V$ separates ~$D$ into a finite set of disjoint slices. The root vectors which connect vertices from two different slices are called {\it struts}. We are interested in slices of ~$D$ which contain diagrams corresponding to the algebra's subalgebras. Hence, the slices must contain root vectors of ~$D$, and in practice we choose ~$V$ to consist of integer linear combinations of simple roots.
A diagram's slices can tell us about its original structure. When dealing with bread, we can obviously stack the slices on top of each other, in order, to recreate an image of the pre-cut loaf of bread - our mind removes the cuts made by the knife. When dealing with root and weight diagrams, we do not have the benefit of using the shape of an ``outer crust'' to guide the stacking of the slices of the diagram.
Instead of severing the struts, we color them grey to make them less prominent. This allows us to use the slices and struts to recreate the structure of the original root or weight diagram.
When the dimension of the diagram is greater than ~$3$, stacking slices on top of each other is not an effective means of recreating the root or weight diagram. Instead, we lay the slices out along one direction, much as slices of bread are laid in order along a countertop to make sandwiches. This allows us to display a~$3$-dimensional diagram in two dimensions, as we have shown for ~$B_3=so(7)$ in Figure ~\ref{fig:B3_slice_using_roots_1_and_2}, or a~$4$-dimensional diagram in three dimensions, as we have shown for the root diagram of ~$B_4 =so(9)$ in Figure ~\ref{fig:B4_slice_showing_B3}. Of course, a~$5$-dimensional diagram can be displayed in three dimensions by first laying~$4$-dimensional slices along the ~$x$ axis, and then slicing each of these diagrams and spreading them along directions parallel to the ~$y$ axis. This procedure generalizes easily to diagrams for rank six algebras, and can be modified to allow any compact ~$n$-dimensional diagram to be displayed in three dimensions.
\begin{figure}[htbp]
\begin{center}
\begin{minipage}[t]{5in}
\begin{center}
\leavevmode
\resizebox{8.5cm}{!}{\includegraphics*[40,160][555,310]{B3_slice_roots_1_2}}
\caption{ \label{fig:B3_slice_using_roots_1_and_2} Slicing of $B_3=so(7)$ using root $r^1$, colored red, and root $r^2$, colored blue.}
\end{center}
\end{minipage}
\hfill
\begin{minipage}[t]{5in}
\begin{center}
\leavevmode
\resizebox{11cm}{!}{\includegraphics*[86,188][505,290]{B4_adj36_slice_2_3_4}}
\caption{\label{fig:B4_slice_showing_B3} Slicing of ~$B_4=so(9)$ using roots ~$r^1$ (red), ~$r^2$ (green), and ~$r^3$ (blue).}
\end{center}
\end{minipage}
\end{center}
\end{figure}
For the Lie algebras of rank ~$l = 6$ or less, we implement a slicing as follows. We first build the algebra's root or weight diagram as described in subsection ~$2.5$. Given three vectors ~$v^1$, ~$v^2$, and ~$v^3$ to define the slicing, we apply an orthonormal transformation so that the slices are contained within the first ~$3$~coordinates of ~$\mathbb{R}^l$. We then use the projection (given here for ~$l=6$) ~$\mathbb{R}^6 \to \mathbb{R}^3: (x,y,z,u_1,u_2, u_3) \to (x + s_1\textrm{ {\tiny $\bullet$} } u_1, y + s_2\textrm{ {\tiny $\bullet$} } u_2, z + s_3 \textrm{ {\tiny $\bullet$} } u_3)$, where ~$s_1$, ~$s_2$, and ~$s_3$ are separation factors used to separate the slices from each other when placed on our~$3$-dimensional countertop.\footnote{An equivalent projection ~$\mathbb{R}^6 \to \mathbb{R}^3: (x,y,z,u_1,u_2,u_3) \to (x+s_1 u_1 + s_2 u_2 + s_3 u_3, y, z)$ with~$s_1 > \eta_2 s_2 > \eta_3 s_3$ for sufficiently large ~$\eta_2$ and ~$\eta_3$ will string a ~$6$-dimensional diagram along one axis. The ~$u_1$ coordinate will separate the different ~$5$-dimensional slices, with ~$s_1 u_1$ moving these different slices very far apart. The ~$u_2$ and ~$u_3$ coordinates will locally separate the ~$4$-dimensional and ~$3$-dimensional slices along the ~$x$ axis, but sufficiently small ~$s_2$ and ~$s_3$ will keep the subslices of one ~$5$-dimensional slice from interfering with another ~$5$-dimensional slice. This method generalizes to~$n$-dimensional diagrams, but creates a very long string of the resulting diagrams.}
When helpful, we keep the grey colored struts in the sliced diagrams.\footnote{The large number of grey lines in a diagram can hide the important roots within each slice in addition to causing computational overload.} While slicing preserves the length and direction of any root vector within a slice, laying the slices out along one direction obviously changes these characteristics for the grey struts.
The second method used to find subalgebras of ~$B_3=so(7)$ in subsection~$3.1$ involved projecting the ~$3$-dimensional diagram into a ~$2$-dimensional diagram. A projection of an ~$l$-dimensional diagram to an ~$(l-1)$-dimensional diagram is accomplished by projecting along a direction specified by ~$p$. To be useful, this projection must preserve the lengths of the roots, and the angles between them, when those roots are orthogonal to ~$p$.
Given a direction specified by a vector ~$p$, we create a linear transformation to change the basis from the standard basis ~$e^1, \cdots, e^l$ to a new orthonormal basis whose first basis vector is ~$\frac{p}{|p|}$. This is accomplished by applying Gram-Schmidt orthonormalization to the ordered set of vectors ~$P = \{p, e^1, e^2, \cdots, e^l\}$, which is linearly dependent, and keeping the first ~$l$ non-zero vectors. We then use a linear transformation to convert the standard basis to this new basis and apply it to the simple roots. Finally, we throw away the first coordinate in the expression for each simple root. This allows us to build the root or weight diagram following the procedure in subsection ~$2.5$, using the original simple roots to define the weights ~$W^0\textrm{,} \cdots \textrm{,} W^n$, which are then constructed using our projected simple roots. It is faster computationally to apply the projection to the simple roots before building the diagram than to apply the projection to the entire diagram after it has been built. As we can only display diagrams in three dimensions, when ~$l \ge 4$, we repeat this procedure ~$l-3$ times using ~$l-3$ projection directions ~$p^1, \cdots, p^{l-3}$.
The Gram-Schmidt process smoothly transforms a set of linearly independent vectors into a set of orthonormal vectors, and we choose our vectors in ~$P$ in a smooth way. However, as ~$P$ is linearly dependent, our resulting change of basis transformation will not smoothly depend on ~$p$ if ~$p \in span(e^1, \cdots, e^{l-1})$. Thus, we place the restriction that ~$|\frac{p}{|p|} \cdot e^l| > \epsilon$, for some small ~$\epsilon$. In practice, we are usually interested in directions ~$p$ which are integer or half-integer linear combinations of the simple roots, and change ~$p$ to ~$p + 0.015e^1 + 0.015e^2 + \cdots + 0.015e^l$. This assures that our projection smoothly depends upon ~$p$, or in the case ~$l \ge 4$, on ~$p^1, \cdots, p^{l-3}$.
By setting the separation factors ~$s_i = 0$, the slicing method can be used to produce another projection of the diagram ~$D$. This choice for ~$s_i$ collapses the separate slices ~$D_{\alpha}$ onto one another, and centers them about the origin. Given a hyperplane ~$V$, this {\it slice and collapse} method projects the slices along a direction perpendicular to ~$V$. This provides less flexibility that the true projection method, which allows a projection along any direction ~$p$ when ~$\{p\} \cap V = 0$. Nevertheless, by setting some ~$s_i = 0$, the slice and collapse method can produce useful projections of ~$5$-dimensional and ~$6$-dimensional diagram.
The slight difference between the projection method and slice and collapse method is illustrated in Figure ~\ref{fig:C4_project_along_root_1} and Figure~\ref{fig:C4_slice_using_roots_2_3_4}. Figure ~\ref{fig:C4_project_along_root_1} projects the root diagram of ~$C_4=sp(2\cdot 4)$ along the simple root ~$r^1$. The result is the root diagram of ~$C_3 = sp(2\cdot 3)$. Figure ~\ref{fig:C4_slice_using_roots_2_3_4} collapses the slices of the root diagram of ~$C_4 = sp(2\cdot 4)$, defined using the simple roots ~$r^2, r^3$, and ~$r^4$, onto the origin. While this diagram contains the ~$C_3$ root diagram, consisting of two large triangles on either side of a large hexagon, it also contains smaller triangles on either side of the hexagon. These small triangles are part of an octahedron, which is one of the original slices of ~$C_4$. In Figure ~\ref{fig:C4_slice_using_roots_2_3_4}, the octahedron is actually disjoint from the ~$C_3$ root diagram.\footnote{While the root vectors in the octahedron and the ~$C_3$ root diagram appear to intersect, they do not terminate or start from any common vertex.} However, in the true projection, in Figure ~\ref{fig:C4_project_along_root_1}, the octahedron is placed by the projection either above or below the origin. Whenever the slice and project method produces overlapping disjoint diagrams, a better projection can be obtained by translating the collapsed slice away from the origin.
\begin{figure}[htbp]
\begin{center}
\begin{minipage}[t]{6cm}
\begin{center}
\leavevmode
\resizebox{!}{4cm}{\includegraphics*[124,87][375,369]{C4_projection_along_root_1}}
\caption{\label{fig:C4_project_along_root_1} Projecting the ~$C_4=sp(2\cdot 4)$ root diagram along simple root ~$r^1$}
\end{center}
\end{minipage}\hfill
\begin{minipage}[t]{6cm}
\begin{center}
\leavevmode
\resizebox{!}{4cm}{\includegraphics*[164,84][394,327]{C4_slice_uses_roots_2_3_4_not_same_as_proj_along_r1}}
\caption{\label{fig:C4_slice_using_roots_2_3_4} Collapsing the slices of ~$C_4=sp(2\cdot 4)$ defined by simple roots ~$r^2$, ~$r^3$, and ~$r^4$ onto the origin.}
\end{center}
\end{minipage}
\end{center}
\end{figure}
\subsubsection{Identifying Subalgebras using Slicings and Projections}
Given an algebra ~$g$, the slicing and projection techniques described above produce subdiagrams of the algebra's root and weight diagrams. We then compare these subdiagrams to a list of known Lie algebra diagrams. If the subdiagram's vertices and root configuration exactly matches the configuration of a diagram for the Lie algebra ~$g^\prime$, then ~$g^\prime$ is a subalgebra of ~$g$.
While a projection along one of the diagram's root vectors will only allow an identification of subalgebras of rank ~$l-1$, slicings of a rank ~$l$ algebra's root diagram allow identifications of subalgebras of rank ~$l$ and ~$l-1$. When successfully slicing root diagrams, the middle slice will be the root diagram of a rank ~$l-1$ subalgebra, and other slices will be different weight diagrams for that subalgebra~\cite{danielsson}. We use a subdiagram consisting of some slices to identify a rank ~$l$ subalgebra. In addition, that subdiagram must contain the original diagram's highest weight. For instance, the short roots in the diagram of ~$C_3 = sp(2\cdot 3)$ look like they form the ~$A_3 = D_3$ algebra in Figure ~\ref{fig:Rank_3_root_diagrams}. However, the highest weight of the ~$C_3$ diagram is the furthest away from the origin, at the tip of the longest root extending from the origin. This weight is outside any embedding of the ~$A_3 = D_3$ root diagram into the root diagram of ~$C_3$. In fact, ~$A_3 = D_3$ is not a subalgebra of ~$C_3$, as there are two operators $g_1$ and $g_2$ corresponding to the short roots whose commutator is an operator corresponding to one of the longer roots. These rules are sufficient to identify rank ~$l$ and rank ~$l-1$ subalgebras of a rank ~$l$ algebra.
These methods only allow us to find subalgebras of the complex simple Lie algebras. For the case of real subalgebras of real Lie algebras, these techniques can only indicate which containments are not possible (i.e. a real form of ~$C_3$ can not contain a real form of ~$A_3$). See ~\cite{aaron_thesis} for information regarding the real Lie subalgebras of ~$E_6$.
\subsection{Applications to Algebras of Dimension Greater than~$3$}
\label{ch:Joma_Paper.subalgebras_greater_than_3}
We now show how these two techniques can be applied to find subalgebras of rank ~$l$ algebras, for ~$l \ge 4$. We begin by using slices and projections to find subalgebras of the exceptional Lie algebra ~$F_4$. We then show how to apply these techniques to algebras of higher rank.
\subsubsection{Subalgebras of ~$F_4$ using Slices}
We apply the slice and projection techniques to the ~$52$-dimensional exceptional Lie algebra ~$F_4$, whose Dynkin diagram is shown in Figure ~\ref{fig:F4_dynkin_diagram}. We number the nodes ~$1$ through ~$4$, from left to right, and use this numbering to label the simple roots ~$r^1, \dotsc, r^4$. Thus, the magnitude of ~$r^1$ and ~$r^2$ is greater than the magnitude of ~$r^3$ and ~$r^4$. We color these simple roots magenta ($r^1$), red ($r^2$), blue ($r^3$), and green ($r^4$).
\begin{figure}[htbp]
\begin{center}
\leavevmode
\xymatrixcolsep{10pt}
\xymatrix@M=0pt@W=0pt{
*+=[o][F-]{\hspace{.25cm} } \ar@{-}[rr]& & *+=[o][F-]{\hspace{.25cm} } \ar@2{-}[rr] & \rangle & *+=[o][F-]{\hspace{.25cm} } \ar@{-}[rr] & \hspace{.25cm} & *+=[o][F-]{\hspace{.25cm} }
}
\caption{$F_4$ Dynkin diagram}
\label{fig:F4_dynkin_diagram}
\end{center}
\end{figure}
We consider the slicing of ~$F_4$ defined using roots ~$r^2$, ~$r^3$, and ~$r^4$. Laying the slices along the ~$x$ axis, the large number of grey struts in the resulting diagram, Figure ~\ref{fig:F4_slice_showing_C3_greylines}, makes it difficult to observe the underlying structure of each slice, and so they are removed from the diagram in Figure ~\ref{fig:F4_slice_showing_C3}. This diagram clearly contains three nontrivial rank ~$3$ root or weight diagrams. Comparing this diagram to the root diagrams in Figure ~\ref{fig:Rank_3_root_diagrams}, we identify the middle diagram, containing ~$18$ non-zero vertices, as the root diagram of ~$C_3=sp(2\cdot 3)$. The other two slices are identical non-minimal weight diagrams of ~$C_3$. Because there are~$46$~non-zero vertices visible in Figure ~\ref{fig:F4_slice_showing_C3}, it is clear that two single vertices are missing from this representation of the root diagram of ~$F_4$, which has dimension ~$52$.
\begin{figure}[htb]
\begin{center}
\begin{minipage}[t]{12cm}
\begin{center}
\leavevmode
\resizebox{11cm}{!}{\includegraphics*[0,0][411,145]{F4_C3_adj52_slice_1_2_3_grey}}
\caption{\label{fig:F4_slice_showing_C3_greylines} Slicing of ~$F_4$ using roots ~$r^2$ (red), ~$r^3$ (blue), and ~$r^4$ (green). Grey colored struts connect vertices from different slices.}
\end{center}
\end{minipage}\hfill
\begin{minipage}[t]{12cm}
\begin{center}
\leavevmode
\resizebox{11cm}{!}{\includegraphics*[40,160][545,320]{F4_C3_adj52_slice_1_2_3}}
\caption[Slicing of ~$F_4$ using roots ~$r^2$, ~$r^3$, and ~$r^4$. Eliminating the struts shows~$C_3=sp(2\cdot 3)~\subset~F_4$]{\label{fig:F4_slice_showing_C3} \hbox{Slicing of ~$F_4$ using roots ~$r^2$, ~$r^3$, and ~$r^4$.} \hbox{Eliminating the struts shows~$C_3=sp(2\cdot 3)~\subset~F_4$}}
\end{center}
\end{minipage}
\end{center}
\end{figure}
Figure ~\ref{fig:F4_slice_showing_B3} is the result of slicing the root diagram of ~$F_4$ using the simple roots ~$r^1$, ~$r^2$, and ~$r^3$. The center diagram again contains ~$18$ non-zero weights, which we identify as ~$B_3=so(7)$ using Figure ~\ref{fig:Rank_3_root_diagrams}. Hence, ~$B_3 \subset F_4$. Furthermore, as all ~$48$ non-zero vertices are present and there are ~$5$ nontrivial slices in the root diagram, we compare this sliced root diagram of ~$F_4$ with that of ~$B_4=so(9)$, which is shown in Figure ~\ref{fig:B4_slice_showing_B3}, and see that ~$B_3 \subset B_4 \subset F_4$. An additional slicing of ~$B_4$ shows ~$D_4 =so(8) \subset B_4 \subset F_4$.
\begin{figure}[htbp]
\begin{center}
\begin{minipage}[t]{12cm}
\begin{center}
\leavevmode
\resizebox{11cm}{!}{\includegraphics*[67,187][526,292]{F4_B3_adj52_slice_1_2_3}}
\caption{\label{fig:F4_slice_showing_B3} \hbox{Slice of ~$F_4$ showing ~$B_3=so(7) \subset B_4 =so(9) \subset F_4$}}
\end{center}
\end{minipage}
\end{center}
\end{figure}
\subsubsection{Subalgebras of ~$F_4$ using Projections} Given the ~$4$-dimensional root diagram of ~$F_4$, we can observe its ~$3$-dimensional shadow when projected along any one direction. However, as a single projection eliminates the information contained in one direction, it is not possible to understand the root diagram of ~$F_4$ using a single projection. We work around this problem by creating an animation of projections, in which the direction of the projection changes slightly from one frame to the next.
The Dynkin diagram of ~$F_4$ reduces to the Dynkin diagram of ~$C_3=sp(2\cdot 3)$ or ~$B_3=so(7)$ by eliminating either the first or fourth node. The simple roots ~$r^1$ and ~$r^4$ define a plane in ~$\mathbb{R}^4$, and we choose a projection vector ~$p_{\theta} = \cos \theta r^1 + \sin \theta r^4$ to vary discretely in steps of size ~$\frac{\pi}{18}$ from ~$\theta = 0$ to ~$\theta = \frac{\pi}{2}$ in this plane. Each value of ~$\theta$ produces a frame of the animation sequence using the projection procedures of section~$3.2$. The resulting animation is displayed in Figure ~\ref{animation_F4}.
The result of each projection of ~$F_4$ is a diagram in three dimensions. We create the animation using Maple, and the software package Javaview is used to make a live, interactive applet of the animation. The Javaview applet allows the animation to be rotated in ~$\mathbb{R}^3$ as it plays. In particular, when ~$\theta = 0$, we can rotate the diagram to show a weight diagram of ~$C_3=sp(2\cdot 3)$, which our eyes project down to the root diagram of ~$B_2 = C_2$. Without rotating the diagram, the animation continuously changes the projected diagram as ~$\theta$ increases. When ~$\theta = \frac{\pi}{2}$, our eyes project the root diagram of ~$B_3 = so(7)$ down to the root diagram of ~$G_2$. However, it is also possible to rotate the animation to see the root diagram of ~$G_2$ at various other values of ~$\theta$. The interactive animation makes it easier to explore the structure of ~$F_4$.
This interactive animation can also illustrate an obvious fact about planes in ~$\mathbb{R}^4$. As ~$p_{\theta}$ is confined to a plane, there is a plane ~$P^{\perp}$ which is orthogonal to each of the projection directions. Thus, the projection does not affect ~$P^{\perp}$, and it is possible to see this plane in ~$\mathbb{R}^3$ by rotating the animation to the view shown in the sixth diagram in Figure ~\ref{animation_F4}. In this configuration, the roots and vertices in this diagram do not change as the animation varies from ~$\theta = 0$ to ~$\theta = \frac{\pi}{2}$. While this is obvious from the standpoint of Euclidean space, it is still surprising this plane can be seen in ~$\mathbb{R}^3$ even as the projected diagram is continually changing.
\begin{figure}[htb]
\begin{center}
\begin{minipage}[t]{4.1cm}
\begin{center}
\leavevmode
\resizebox{3.8cm}{!}{\includegraphics*[193,128][400,347]{anim_F4_0}}\\
$\theta = 0$
\end{center}
\end{minipage}
\hfill
\begin{minipage}[t]{4.1cm}
\begin{center}
\leavevmode
\resizebox{3.8cm}{!}{\includegraphics*[193,128][400,347]{anim_F4_25}}\\
$\theta = \frac{\pi}{8}$
\end{center}
\end{minipage}
\hfill
\begin{minipage}[t]{4.1cm}
\begin{center}
\leavevmode
\resizebox{3.8cm}{!}{\includegraphics*[193,128][400,347]{anim_F4_50}}\\
$\theta = \frac{\pi}{4}$
\end{center}
\end{minipage}
\hfill
\begin{minipage}[t]{4.1cm}
\begin{center}
\leavevmode
\resizebox{3.8cm}{!}{\includegraphics*[193,128][400,347]{anim_F4_75}}\\
$\theta = \frac{3\pi}{8}$
\end{center}
\end{minipage}
\hfill
\begin{minipage}[t]{4.1cm}
\begin{center}
\leavevmode
\resizebox{3.8cm}{!}{\includegraphics*[193,128][400,347]{anim_F4_100}}\\
$\theta = \frac{\pi}{2}$
\end{center}
\end{minipage}
\hfill
\begin{minipage}[t]{4.1cm}
\begin{center}
\leavevmode
\resizebox{3.8cm}{!}{\includegraphics*[199,145][394,338]{anim_F4_const_view}}\\
Orthogonal View in $\mathbb{R}^4$
\end{center}
\end{minipage}
\hfill
\caption{Animation of ~$F_4$, projecting along the \hbox{direction~$p_{\theta} = \cos(\theta)r^1 + \sin(\theta)r^4$} which is confined to a plane in~$\mathbb{R}^4$ containing ~$r^1$ and ~$r^4$.}
\label{animation_F4}
\end{center}
\end{figure}
\subsubsection{Modifications of methods for ~$E_6$}
Of particular interest is the exceptional Lie algebra~$E_6$, which preserves the determinant of elements of the Cayley plane. As explained in Section~$2.1$, this allows us to write ~$E_6 = sl(3,\mathbb{O})$. It therefore naturally contains the subalgebras ~$sl(2,\mathbb{O})$ and ~$su(2, \mathbb{O})$, which are identified as real forms of ~$D_5$ and ~$B_4$, respectively ~\cite{aaron_thesis}.
Because the root diagram of ~$E_6$ is a ~$6$-dimensional diagram, we apply three consecutive slicings to determine subalgebras of ~$E_6$. The first slicing creates a string of ~$5$-dimensional diagrams along the ~$x$ axis. The second slicing, when applied to each of these diagrams, creates a string of ~$4\textrm{-dimensional}$ diagrams laid parallel to the ~$y$-axis. The third slicing turns each ~$4$-dimensional diagram into a string of ~$3$-dimensional diagrams laid out parallel to the ~$z$ axis. We apply the same process to rank ~$5$ algebras and use two slicings. To identify a rank ~$5$ subalgebra ~$g$ of ~$E_6$, we need to identify its sliced root diagram as a subdiagram of the sliced ~$E_6$ diagram. Of course, we must be careful to check that the highest weight of the ~$E_6$ root diagram is used in the subdiagram. A similar technique is applied for finding rank ~$4$ subalgebras of both rank ~$5$ algebras and of ~$E_6$.
The projection technique can also be used to identify subalgebras of rank ~$E_6$. In one version, we project the root diagram of ~$E_6$ along one direction, thereby creating a diagram that possibly corresponds to a rank ~$5$ algebra ~$g$. We then apply the same pair of projections to our projected ~$E_6$ diagram and to the candidate root diagram of ~$g$. If these two projections preserve the number of vertices in the ~$5$-dimensional diagrams, it is possible to compare the resulting diagrams in ~$\mathbb{R}^3$. If we have identified the correct subalgebra of ~$E_6$, the resulting two diagrams should match for every pair of projections applied to the ~$5\textrm{-dimensional}$ diagrams.
Projections of rank ~$5$ and ~$6$ algebras can also be simulated using slicings of their root diagrams. This is done using the slice and collapse technique, which collapses all the slices onto one another in a particular direction. When using this technique, we draw the grey struts, as we are now interested in the root diagram's structure after the projection. This technique provides clearer pictures compared to the pure projection method.
\subsubsection{Towers of Complex Lie Algebras}
We list in Figure ~\ref{fig:subalgebras_of_E6} the subalgebras of ~$E_6$ found using the slicing and projection techniques applied to an algebra's root diagram. As mentioned in Section ~$2.1$, we list certain real representations of the subalgebras of the ~$sl(3,\mathbb{O})$ representation of ~$E_6$ in the diagram. The particular real representations are listed below each algebra.
We use different notations to indicate the particular method that was used to identify subalgebras. The notation ~$\xymatrix@1{A\ar@{.>}[r]&B}$ indicates the slicing method was used to identify ~$A$ as a subalgebra of ~$B$. The notation ~$\xymatrix@1{A\ar@{->}[r]&B}$ indicates that ~$A$ was identified as a subalgebra of ~$B$ using the normal projection technique, while we indicate projections done by the slice and collapse method as ~$\xymatrix@1{A\ar@{^{}.>}[r]^{s+p}&B}$. If both dotted and solid arrows are present, then ~$A$ can be found as a subalgebra of ~$B$ using both slicing and projection methods. If~$A$ and ~$B$ have the same rank, only the slicing method allows us to identify the root diagram of ~$A$ as a subdiagram of ~$B$. This case is indicated in the diagram using the notation ~$\xymatrix@1{*+++\txt{$A$}\ar@{^{(}.>}[r]&B}$. Each of the subalgebra inclusions below can be verified online ~\cite{subalgebras_online}.
\begin{figure}[htbp]
\begin{center}
\begin{minipage}{6in}
\begin{center}
\xymatrix@M=3.5pt@H=1pt{
& & *++\txt{$E_6$\\ $sl(3,\mathbb{O})$} & \\
\\
& *++\txt{$F_4$\\ $su(3,\mathbb{O})$} \ar[uur]^(.4){s+p} & *++\txt{$D_5 = so(10)$\\$sl(2,\mathbb{O})$} \ar@{.>}[uu]{} & *++\txt{$A_5 = su(6, \mathbb{C})$\\$sl(3,\mathbb{H})$} \ar@{.>}[uul]{} \\
& & & \\
*++\txt{$C_4 = sp(2\cdot 4)$\\$su(3,1, \mathbb{H})$} \ar@/^4pc/[uuuurr]^{s+p} & *++\txt{$B_4 = so(9)$\\$su(2,\mathbb{O})$} \ar@{_{(}.>}[uu]^(.4){} \ar[uur]^(.4){s+p} & D_4 = so(8) \ar@{.>}[uu]_(.3){} \ar@{_{(}.>}[l]_(.4){} & A_4 = su(5, \mathbb{C}) \ar@{.>}[uul]<1ex>{} \ar[uul]_(.6){s+p} \ar@{.>}[uu]<1ex>{} \ar[uu]_(.4){} \\
& & & \\
*++\txt{$C_3=sp(2\cdot 3)$\\$su(3,\mathbb{H})$} \ar@/_4pc/[uuuurrr]^(.6){} \ar@{.>}[uu]<1ex>{} \ar[uu]^(.4){} \ar@{.>}[uuuur]{} & B_3 = so(7) \ar@{.>}[uu]{} \ar[uur]<.5ex>^(.6){} & *++\txt{$ A_3 = su(4,\mathbb{C})$\\$D_3=so(6)$\\$sl(2,\mathbb{H})$} \ar[uur]{} \ar@{_{(}.>}[l]^(.4){} \ar@{.>}[uur]<1ex>{} \ar@{.>}[uull]<-1ex>^(.4){} \ar@{.>}[uu]{} & \\
& & & \\
*++\txt{$B_2 = so(5)$\\$C_2 = sp(2\cdot 2)$\\$su(2,\mathbb{H})$} \ar@/_2.3pc/[uurr]{} \ar@{.>}[uu]{} \ar@{.>}[uur]<1ex> \ar[uur]^(.4){} & G_2 = Aut(\mathbb{O}) \ar[uu]^(.4){} & A_2 = su(3,\mathbb{C}) \ar@{.>}[uull]<0ex>{} \ar@{.>}[uu]<1ex>{} \ar[uu]_(.4){} \ar@{_{(}.>}[l]^(.4){} \\
& \save[]+<0cm,.1cm>*++\txt{$D_2 = su(2,\mathbb{C}) \oplus su(2,\mathbb{C})$} \ar@{_{(}.>}[ul]^(.4){} \ar@{_{(}.>}[u]^(.4){} \restore & & \\
& & A_1 = su(2,\mathbb{C}) \ar@{.>}[uu]<1ex>{} \ar[uu]_(.4){} \ar@{.>}[ul]<1ex>{} \ar[ul]^(.4){} &\\
}
$\xymatrix@1{A\ar@{.>}[r]&B}$ indicates $A$ is realized using a slice of $B$.
\\
$\xymatrix@1{A\ar@{->}[r]&B}$ indicates A is a projection of B.
\\
$\xymatrix@1{A\ar@{->}[r]^{s+p}&B}$ indicates B projects to A with the slice and project method.
\\
$\xymatrix@1{*+++\txt{$A$} \ar@{^(.>}[r]&B}$ signals A and B have the same rank, but A is a subdiagram of B.
\caption{Subalgebras of ~$E_6$ together with some important real representations}
\label{fig:subalgebras_of_E6}
\end{center}
\end{minipage}
\end{center}
\end{figure}
Lists of subalgebra inclusions are found in ~\cite{danielsson}, which applies subalgebras to particle physics, and in ~\cite{gilmore}, which recreates the subalgebra lists of ~\cite{van_der_waerden}. However, the list in ~\cite{gilmore} mistakenly has ~$C_4$ and ~$B_3$ as subalgebras of ~$F_4$, instead of ~$C_3$ and ~$B_4$. Further, the list omits the inclusions ~$G_2 \subset B_3$, ~$C_4 \subset E_6$, ~$F_4 \subset E_6$, and ~$D_5 \subset E_6$. The correct inclusions of ~$C_3 \subset F_4$ and ~$B_4 \subset F_4$ are listed in Section ~$8$ of ~\cite{van_der_waerden}, but the ~$B_n$ and ~$C_n$ chains are mislabeled in the final table of \cite{van_der_waerden} which was used by Gilmore in ~\cite{gilmore}. Although \cite{van_der_waerden} uses root systems to determine subalgebra inclusions, it mistakenly claims that ~$D_n \subset C_n$ as a subalgebra in his Section ~$21$ of \cite{van_der_waerden}, which is not true since their root diagrams are based upon inequivalent highest weights.
In ~\cite{dyn2}, Dynkin classified subalgebras depending upon the root structure. If the root system of a subalgebra can be a subset of the root system of the full algebra, the subalgebra is called a {\it regular} subalgebra. Otherwise, the subalgebra is {\it special}. A complete list of regular and special subalgebras is given in ~\cite{danielsson}. All of the regular embeddings of an algebra in a subalgebra of ~$E_6$ can be found using the slicing method. In many cases, the projection technique also identifies these regular embeddings of subalgebras, but there are regular embeddings which are not recognized as the result of projections. The special embeddings of an algebra in a subalgebra of ~$E_6$ can only be found using the projection technique. We conjecture that all the subalgebras of a complex Lie algebra may be found using our slicing and projection methods, as these methods find the same regular and special subalgebras of a complex Lie algebra as found by Dynkin.
\subsection{Conclusion}
\label{ch:Joma_Paper.conclusion}
We have presented here methods which illustrate how root and weight diagrams can be used to visually identify the subalgebras of a given Lie algebra. While the standard methods of determining subalgebras rely upon adding, removing, or folding along nodes in a Dynkin diagram, we show here how to construct any of a Lie algebra's root or weight diagrams from its Dynkin diagram, and how to use geometric transformations to visually identify subalgebras using those weight and root diagrams. In particular, we show how these methods can be applied to algebras whose root and weight diagrams have dimensions four or greater. In addition to pointing out the erroneous inclusion of ~$C_4 \subset F_4$ in ~\cite{gilmore, van_der_waerden}, we provide visual proof that ~$C_4 \subset E_6$ and list all the subalgebras of ~$E_6$. While we are primarily concerned with the subalgebras of ~$E_6$, these methods can be used to find subalgebras of any rank ~$l$ algebra.
\newpage{}
\part{Division Algebras and Applications}
\label{ch:Division_Algebras_Applications}
This chapter includes a review of the concepts we will use in the construction of the group~$E_6 = SL(3,\mathbb{O})$ in Chapter \ref{ch:E6_basic_structure}. Readers familiar with the octonions and Lorentz transformations may want to skip this chapter, although they may want to quickly review of notion of triality in Section \ref{ch:Division_Algebras_Applications.Quaternions_Octonions.Triality} and Lorentz groups involving division algebras in Section \ref{ch:Division_Algebras_Applications.Lorentz_Transformations}. In Section \ref{ch:Division_Algebras_Applications.Quaternions_Octonions}, we review the properties of the division algebras and point out that the complexes, quaternions, and octonions naturally describe Euclidean spaces of dimension $2$, $4$, and $8$.
In Section \ref{ch:Division_Algebras_Applications.Quaternions_Octonions.Conjugation_Reflecitons_Rotations}, we review the work of Manogue and Schray \cite{manogue_schray} who show how to construct rotations and reflections in~$\mathbb{R}^4$ and~$\mathbb{R}^8$ using certain conjugation maps in the division algebras. These maps will be used extensively in the construction of our real form of~$E_6$ in Chapter \ref{ch:E6_basic_structure}. We finish our review of the division algebras with a summary of triality in Section \ref{ch:Division_Algebras_Applications.Quaternions_Octonions.Triality}. This idea will be again visited in Section \ref{ch:E6_basic_structure.Triality}, where we find that~$sl(3,\mathbb{O})$, the Lie algebra corresponding to the Lie group~$SL(3,\mathbb{O})$, exhibits interesting characteristics due to triality.
We review the properties of Lorentz Transformations in Section \ref{ch:Division_Algebras_Applications.Lorentz_Transformations} and pay particular attention to the notion that the division algebras may be used to construct Lorentz Transformations in~$k+1$ dimensions, where \hbox{$k-1 = |\mathbb{K}|$} is the dimension of the division algebra~$\mathbb{K} = \mathbb{R}, \mathbb{C}, \mathbb{H}, \mathbb{O}$. We conclude Chapter \ref{ch:Division_Algebras_Applications} in Section \ref{ch:Division_Algebras_Applications.Jordan_Algebras} by describing how the three smallest division algebras may be used to construct Jordan algebras while the octonions may be used to construct the exceptional Jordan algebra $M_3(\mathbb{O})$. Octonionic Lorentz transformations and the exceptional Jordan algebra are utilized in Chapter \ref{ch:E6_basic_structure} to give a description of the Lie group $SL(3,\mathbb{O})$ and its associated Lie algebra $sl(3,\mathbb{O})$.
\section{Normed Division Algebras}
\label{ch:Division_Algebras_Applications.Quaternions_Octonions}
The complexes, quaternions and octonions are division algebras of dimension~$2$,~$4$, and~$8$ over the reals.
Section \ref{ch:Division_Algebras_Applications.Quaternions_Octonions.Division_Algebras} includes a review of the basic algebraic and geometric properties of the normed division algebras. These algebras provide a nice description of~$|\mathbb{K}|$-dimensional Euclidean space for~$\mathbb{K} = \mathbb{R}, \mathbb{C}, \mathbb{H},$ or~$\mathbb{O}$. Certain multiplication maps in the division algebra can result in rotations or reflections in the corresponding Euclidean space, and this discussion, based on the work of \cite{manogue_schray}, is included in Section \ref{fig:octonionic_multiplication}. Triality is a concept closely related to the multiplication properties of the division algebras as well as to representations of Lie algebras, and is discussed in Section \ref{ch:Division_Algebras_Applications.Quaternions_Octonions.Triality}. Particular attention is paid to the triality related to the octonions and representations of~$so(8,\mathbb{R})$.
\subsection{Reals, Complexes, Quaternions and Octonions}
\label{ch:Division_Algebras_Applications.Quaternions_Octonions.Division_Algebras}
We review here the properties of the four division algebras, using the construction provided using the Cayley-Dickson process. Additional information about this construction and the properties of the division algebras may be found in either \cite{dickson} or \cite{baez}.
\subsubsection{Real and Complex Numbers}
\label{ch:Division_Algebras_Applications.Quaternions_Octonions.Division_Algebras.Real_and_Complex_Numbers}
The Cayley-Dickson process can be used to create a~$2n$-dimensional algebra from an~$n$-dimensional associative algebra.
The real numbers,~$\mathbb{R}$, are associative, commutative, and have an identity, denoted~$1$. Applied to~$\mathbb{R}$, the Cayley-Dickson process creates a two-dimensional algebra~$\mathbb{C} = \mathbb{R} \oplus \mathbb{R}i$ over~$\mathbb{R}$, using~$i$ to denote a square root of~$-1$.
This algebra has multiplication
\[ (a, b) (c, d) = (ac-bd, ad+bc) \]
or equivalently,
\[ (a+bi) (c+di) = ac-bd + (ad+bc)i \]
In addition, this algebra has the property that ~$i^2 = -1$, or ~\hbox{$(0,1) (0,1) = (-1,0)$}. For any ~$x,y,z \in \mathbb{C}$, this algebra is associative
\[ (xy)z = x(yz) \]
and commutative
\[ x y = y x \]
There is a conjugation map taking~$a+bi$ to~$\overline{a+bi} = a-bi$. Using this map, we can define the norm
$$|a+bi|^2 = (a+bi) \overline{(a+bi)} = a^2 + b^2$$
of any complex number~$a+bi$. Notice that the only complex number of norm~$0$ is the complex number~$0$ and that ~$z^{-1} = \frac{\overline{z}}{|z|^2}$ is the inverse of any non-zero complex number~$z$. We call~$Re(z) = \frac{1}{2} \left( z + \overline{z} \right)$ and~$Im(z) = \frac{1}{2}\left( z - \overline{z}\right)$ the {\it real part} and {\it imaginary part} of~$z \in \mathbb{C}$.
One nice characteristic of the complex numbers is their ability to describe points in a~$2$-dimensional plane. Each complex number~$a+bi \in \mathbb{C}$ may be identified with the point~$(a,b) \in \mathbb{R}^2$. Euler's formula
$$ e^{i\alpha} = \cos \alpha + i \sin \alpha$$
may be used to write every complex number~$z = a+bi$ in the form~$z = |z|e^{i \alpha}$. The distance from the origin to~$(a,b)$ is denoted by~\hbox{$|z| = \sqrt{a^2 + b^2}$}, while~$\alpha$ is the angle between the positive real axis and the ray extending from the origin to~$(a,b)$. The quantity~$e^{i\alpha}$ is called a {\it phase}. We note that complete set of complex phases describes~$S^1$.
\subsubsection{Quaternion Numbers}
\label{ch:Division_Algebras_Applications.Quaternions_Octonions.Division_Algebras.Quaternion_Numbers}
As the complex numbers are associative, we can again use the Cayley-Dickson process and produce the quaternions~$\mathbb{H} = \mathbb{C} \oplus \mathbb{C}j$, a four-dimensional division algebra over~$\mathbb{R}$, by using~$j$, another square root of~$-1$. The multiplication is again given by the rule
\[ (a, b) (c, d) = (ac-bd, ad+bc) \]
where~$a, b, c, d \in \mathbb{C}$. Setting~$a = q_1 + q_2 i$ and~$b = q_3 + q_4 i$, we will often write the quaternionic number ~$(a,b) = a + b j = q_1 + q_2 i + q_3 j + q_4 k$, where~$k$ is yet another square root of~$-1$ and is the product~$k = i j$.
This multiplication rule leads to the products
$$
\begin{array}{ccc}
i\; j = k & j \; k = i & k \; i = j \\
j \; i = -k & k \; j = -i & i \; k = -j
\end{array}
$$
This multiplication is nicely summarized in Figure \ref{fig:quaternionic_mult}, where the arrow indicates whether the product of two imaginary unit quaternions will carry a plus (with the arrow) or minus sign. The quaternions are not commutative, but they are still associative.
\begin{figure}[htbp]
\vspace{1cm}
\begin{center}
\begin{minipage}{3in}
\begin{center}
\mbox{
\xymatrix@M=3pt@H=10pt{
*++\txt{$j$} \ar@/_1pc/[dr] & & *++\txt{$i$} \ar@/_2.3pc/[ll] \\
& *++\txt{$k$} \ar@/_1pc/[ur] & \\
}
}
\end{center}
\caption{Quaternionic multiplication}
\label{fig:quaternionic_mult}
\end{minipage}
\end{center}
\end{figure}
Although they are not commutative, the quaternions have similar norm properties to the complex numbers.
Again, there is a conjugation map~$q \to \overline{q}$ where
$$\overline{q_1 + q_2 i + q_3 j + q_4 k} = q_1 - q_2 i - q_3 j - q_4 k$$
allowing the definition of a norm~$|q|^2 = q_1^2 + q_2^2 + q_3^2 + q_4^2$ for any quaternion~$q \in \mathbb{H}$. The inverse of any non-zero~$q \in \mathbb{H}$ is given by~$q^{-1} = \frac{\overline{q}}{|q|^2}$. Again,~$Re(q) = \frac{1}{2} \left( q + \overline{q} \right)$ and~$Im(q) = \frac{1}{2}\left( q - \overline{q}\right)$ are called the {\it real part} and {\it imaginary part} of~$q \in \mathbb{H}$.
Just as the complex numbers can be used to describe a plane, the quaternions may be used to describe points in a~$4$-dimensional space. The quaternion~$q = q_1 + q_2 i + q_3 j + q_4 k$ may be identified with the point~$(q_1, q_2, q_3, q_4) \in \mathbb{R}^4$. We note that the imaginary quaternions may be identified with vectors in~$\mathbb{R}^3$, in which case the quaternionic multiplication corresponds to the ordinary cross-product. The set of imaginary unit quaternions form the sphere~$S^{2}$ in~$\mathbb{R}^3$. For any~$s \in S^{2}$, we may form the complex subalgebra with basis~$\lbrace 1, s \rbrace$. Hence, Euler's formula may be used to write any quaternion~$q \in \mathbb{H}$ in the form~$q = |q|e^{s \alpha}$, where~$s$ is the unit imaginary quaternion pointing toward~$q$.
\subsubsection{Octonion Numbers}
\label{ch:Division_Algebras_Applications.Quaternions_Octonions.Division_Algebras.Octonion_Numbers}
When applied to the quaternions, the Cayley-Dickson process produces the octonions,~$\mathbb{O} = \mathbb{H} + \mathbb{H}\l$, an eight-dimensional division algebra over~$\mathbb{R}$. Here,~$\l$ is yet another square root of~$-1$, which is orthogonal to~$\mathbb{H}$. The multiplication is again given by the rule
$$(a, b) (c, d) = (ac - bd, ad+bc)$$
with~$a,b,c,d \in \mathbb{H}$.
We will often write the octonion~$(a,b) = a+b\l$ as
$$q_1 + q_2 i + q_3 j + q_4 k + q_5 k\l + q_6 j\l + q_7 i\l + q_8 \l$$
where~$a = q_1 + q_2 i + q_3 j + q_4 k$,~$b = q_8 + q_7 i + q_6 j + q_5 k$ and~$k\l, j\l,$ and~$i\l$ are the products of~$k, j$, and~$i$ with~$\l$, respectively. We refer to~$\lbrace i, j, k, k\l, j\l, i\l, \l \rbrace$ as the standard basis of unit imaginary octonions. It is convenient to encode the multiplication of the octonions as shown in Figure \ref{fig:octonionic_multiplication}, which contains seven directed loops, six of which are shown as directed lines instead of loops. Each loop contains three octonions~$p$,~$q$ and~$r$. The product of any two of these octonions~$p$ and~$q$ is~$\pm r$, where the positive sign is chosen if the order of multiplication follows the arrow and the negative sign is chosen if the order goes against the arrow. Hence, we see that~$i \; \l = i\l$ and~$\l \; i\l = i$ but~$\l \; i = - i\l$. We note that the octonions are not commutative, since they are constructed from the quaternions, and not associative, as~$i \; (j \; \l) = i \; (j\l) = -k\l$ but~$(i \; j) \; \l = (k) \; \l = k\l$. However, the octonions are {\it alternative}, since
$$ x(xy) = (xx)y \hspace{2cm} (yx)x = y(xx)$$
for any $x,y \in \mathbb{O}$.
The conjugation map~$q \to \overline{q}$ changes the sign of every imaginary basis unit in~$q$. With this modification, the norm~$|q|^2 = q \overline{q}$, inverse~$q^{-1} = \frac{q}{|q|}$, real~$Re(q) = \frac{1}{2}(q + \overline{q})$ and imaginary~$Im(q) = \frac{1}{2}(q - \overline{q})$ parts of an octonion~$q \in \mathbb{O}$ are similar to those for the other division algebras.
\begin{figure}[htb]
\begin{center}
\hspace{-.25in}
\begin{minipage}{5.0in}
\begin{center}
\mbox{
\xymatrix@M=3pt@H=10pt{
& & & & *++\txt{$k\l$} \ar[ddll] & & & & \\
\\
& & *++\txt{$j$} \ar@/_2.7pc/[dddrr] \ar[dddll] \ar[drr] & & & & *++\txt{$i$} \ar@/_3pc/[llll] \ar[uull] \ar[dll] \\
& & & & *++\txt{$\l$} \ar[ddrrrr] \ar[ddllll] \ar[uuu] & & & & \\
\\
*++\txt{$i\l$} \ar[rrrr] & \hspace{1.0cm} & & & *++\txt{$k$} \ar@/_2.6pc/[uuurr] \ar[uu] \ar[rrrr] & & & \hspace{1.0cm} & *++\txt{$j\l$} \ar[uuull] \\
}
}
\end{center}
\caption{Octonionic multiplication}
\label{fig:octonionic_multiplication}
\end{minipage}
\end{center}
\end{figure}
Just as the complex numbers and quaternions could be used to describe~$\mathbb{R}^2$ and~$\mathbb{R}^4$, the octonions may be used to describe points in~$\mathbb{R}^8$ using the obvious identification. The imaginary octonions describe~$\mathbb{R}^7$, and the set of imaginary unit octonions form the six-sphere~$S^6$ in~$\mathbb{R}^7$. Of course, Euler's formula again allows us to write any octonion~$q \in \mathbb{O}$ in the form~$q = |q|e^{s \alpha}$ where now~$s \in S^6$ points in the same direction as~$Im(q)$.
According to a theorem by Artin, the subalgebra generated by any two elements in an alternative algebra is associative \cite{schafer_book}.
For the octonions, two imaginary orthogonal units~$s_1, s_2 \in S^6$ define a quaternionic subalgebra spanned by~$\langle 1, s_1, s_2, s_1 \; s_2 \rangle$. This result may be generalized to the case where~$s_1$ and~$s_2$ are not orthogonal (but not parallel) by finding the orthogonal projection of~$s_2$ onto~$s_1$ and re-normalizing the resulting octonions.
We note that each triple of imaginary unit octonions in the multiplication diagram in Figure \ref{fig:octonionic_multiplication} define a quaternionic subalgebra. When working with one or two octonions, it is advantageous to consider the complex or quaternionic subalgebra generated by those octonions.
\subsection{Conjugation, Reflections, and Rotations}
\label{ch:Division_Algebras_Applications.Quaternions_Octonions.Conjugation_Reflecitons_Rotations}
As mentioned in the previous section, each division algebra~$\mathbb{K} = \mathbb{R}, \mathbb{C}, \mathbb{H}, \mathbb{O}$ may be identified with~$\mathbb{R}^{|\mathbb{K}|}$. This identification allows us to produce a geometric transformation in~$\mathbb{R}^{|\mathbb{K}|}$ using multiplication in ~$\mathbb{K}$. In this section, we describe the transformation corresponding to the conjugation map~$f_x : \mathbb{K} \to \mathbb{K}$ given by~$f_x(y) = x y \overline{x}$ for each~$x \in \mathbb{K}$. Noting that~$\mathbb{R}$ and ~$\mathbb{C}$ are commutative, implying the conjugation map is the identity map. Hence, we only discuss the quaternionic or octonionic case. The material in this section is a summary of the treatment given in \cite{manogue_schray}, and will be fundamental to the construction of~$SL(2,\mathbb{O})$ in Section \ref{ch:Division_Algebras_Applications.Lorentz_Transformations} and the construction of ~$SL(3,\mathbb{O})$ in Section \ref{ch:E6_basic_structure.Lorentz_Transformations}.
Consider the case~$\mathbb{K} = \mathbb{H}$, and let~$x = e^{i \alpha/2} \in \mathbb{H}$,~$\alpha \in \mathbb{R}$, be a phase. We shall see that the conjugation map~$f_x : \mathbb{H} \to \mathbb{H}$ produces a rotation in one plane in~$\mathbb{R}^4$ through an angle~$\alpha$; this is most easily seen using an explicit example. Let~$q = q_1 + q_2 i + q_3 j + q_4 k \in \mathbb{H}$. Then~$f_x(q) = e^{i\alpha / 2} q e^{-i\alpha / 2}$. The phase~$e^{i\alpha / 2}$ commutes with the basis elements~$1$ and~$i$. However, for~$j$ or~$k$, we have
$$ j \; e^{-i\alpha / 2} = e^{i \alpha /2}\; j \hspace{1cm} k\; e^{-i\alpha / 2} = e^{i \alpha /2}\; k $$
Hence, the conjugation map~$f_x$ fixes the~$(1,i)$ plane and rotates the~$(j,k)$ plane through an angle of~$\alpha$ radians:
$$
\begin{array}{ccl}
e^{i\alpha / 2}(q)e^{-i\alpha / 2}& =& e^{i\alpha / 2}(q_1 + q_2 i + q_3 j + q_4 k)e^{-i \alpha / 2}\\
&= & e^{i\alpha / 2} e^{-i\alpha / 2}(q_1 + q_2 i) + e^{i\alpha / 2} e^{i \alpha / 2}(q_3 j + q_4 k) \\
&= & q_1 + q_2 i + e^{i\alpha}(q_3 j + q_4 k)
\end{array}
$$
This calculation may be generalized. For any imaginary unit $s \in \mathbb{H}$ and $\alpha \in \mathbb{R}$, the phase~$e^{s\alpha / 2} \in \mathbb{H}$ defines a unique plane~$P$ in~$\mathbb{R}^4$ perpendicular to~$1$ and~$s$ whenever $e^{s\alpha / 2} \ne \pm 1$.
Then, analogous to the example given above, the conjugation map~$f_{e^{s\alpha / 2}} : \mathbb{H} \to \mathbb{H}$ will fix the~$(1,s)$ plane and produce a rotation in the plane~$P$ through~$\alpha$ radians.
In the case~$\mathbb{K} = \mathbb{O}$, the conjugation map~$f_x : \mathbb{O} \to \mathbb{O}$ no longer rotates one plane in~$\mathbb{R}^8$. Indeed, if~$s$ is an imaginary unit octonion and~$\alpha \in \mathbb{R}$, then~$e^{s\alpha/2}$ is a phase in~$\mathbb{O}$. Assume that~$Im(e^{s\alpha}) \ne 0$, or equivalently, that~$\alpha$ is not an integer multiple of~$2\pi$. Then, orthogonal to the~$(1,s)$ plane in~$\mathbb{R}^8$, there are three planes~$P_1, P_2, P_3$ which are also pair-wise orthogonal. We will now show that the conjugation map produces a rotation in three planes, not one!
Given an imaginary unit octonion~$s \in \mathbb{O}$, pick an orthonormal basis~$\lbrace 1, s, p_1, s_1, p_2, s_2, p_3, s_3 \rbrace$ for~$\mathbb{O}$ over~$\mathbb{R}$ with the property that~$s = p_a \; s_a$ for each~$a = 1, 2, 3$. Let~$P_a$ denote the~$(p_a, s_a)$ plane. By the construction of the basis,~$P_1, P_2, P_3$ are pairwise orthogonal and perpendicular to the~$(1,s)$ plane.
Expand~$q \in \mathbb{O}$ in terms of the orthonormal basis. Conjugating~$q$ by the phase~$e^{s\alpha /2}$, we see that~$e^{s\alpha/2}$ will commute with~$1$ and~$e^{s\alpha/2}$, but
$$ p_a e^{- s \alpha/2} = e^{s \alpha/2} p_a \hspace{1cm} s_a e^{- s \alpha/2} = e^{s \alpha/2} s_a$$
for each~$a = 1,2,3$.
Hence, just as with the quaternions, the plane spanned by~$p_a$ and~$s_a$ is rotated by~$\alpha$ radians. We note, however, that in the case~$\mathbb{K} = \mathbb{O}$, the conjugation map rotates all three planes~$P_a$ through an angle of~$\alpha$ radians while fixing the~$(1,s)$ plane.
Manogue and Schray \cite{manogue_schray} note that a rotation of a single plane in~$\mathbb{R}^8$ may be constructed as the composition of two {\it flips}, which are reflections of~$\mathbb{R}^8$ across a two-dimensional plane containing the origin. In terms of octonions, a flip is accomplished with the conjugation map~$f_x : \mathbb{O} \to \mathbb{O}$ where again $s \in \mathbb{O}$ is a unit imaginary octonion but now the phase angle~$\alpha$ in~$x = e^{s \alpha / 2}$ is chosen to be~$\alpha = \pi$. This again fixes the~$(1,s)$ plane, but causes the three pairwise orthogonal planes~$P_a$ for~$a = 1, 2, 3$ perpendicular to the~$(1,s)$ plane in~$\mathbb{R}^8$ to rotate by~$\pi$ radians about the~$s$-axis. Each rotation is indeed a reflection across the~$(1,s)$ plane. Hence, in order to rotate the~$(r,s)$ plane through~$\alpha$ radians, where~$r,s \in \mathbb{O}$ are orthogonal imaginary unit octonions, we compose two reflections using the map~$f_{r,s,\alpha /2} : \mathbb{O} \to \mathbb{O}$
given by
$$f_{r,s,\alpha/2}(q) = \left( \cos \left(\alpha/2 \right) r + \sin \left(\alpha/2 \right) s \right) \left( r q \overline{r}\right) \left( \overline{ \cos \left(\alpha/2 \right) r + \sin \left(\alpha/2 \right) s } \right)$$
The first conjugation~$q \to r q \overline{r}$ in the map above reflects~$\mathbb{R}^8$ about the direction~$r$, while the second conjugation map reflects all of~$\mathbb{R}^8$ back across the direction~$\cos \left(\alpha/2 \right)r + \sin \left(\alpha/2\right) s$. Together, these two reflections cause the~$(r,s)$ plane to be rotated by~$ \alpha$ radians. In the directions orthogonal to this plane, the composition of the two reflections is the identity transformation. We shall use this map extensively in Chapter \ref{ch:E6_basic_structure} to construct a basis for~$SL(3,\mathbb{O})$.
Finally, notice that since the octonions~$r$ and~$\cos \left(\alpha /2 \right) r + \sin \left( \alpha /2 \right) s$ are imaginary, then ~$\overline{r} = -r$ and~$\overline{\cos \left(\alpha /2 \right) r + \sin \left( \alpha /2 \right) s} = -\cos \left(\alpha /2 \right) r - \sin \left( \alpha /2 \right) s$. The fact that a rotation in a plane spanned by two imaginary orthogonal unit octonions involves two sign changes will be significant in future discussions of triality in Section \ref{ch:E6_basic_structure.Triality}.
\subsection{Triality}
\label{ch:Division_Algebras_Applications.Quaternions_Octonions.Triality}
In this section, we discuss triality as it relates to octonions and three different representations of~$SO(8,\mathbb{R})$.
In 1925, Cartan \cite{cartanTriality} noted that the Dynkin diagram~$D$ of~$so(8,\mathbb{R})$, shown in Figure \ref{fig:dynkin_diagram_so8}, contained a three-fold symmetry~$\tau$ which satisfies~$\tau^2 = \tau^{-1}$ and~\hbox{$\tau^3 = Id$}. He referred to these symmetries as ``triality'', and noted that~$\tau$ was an outer automorphism, not an inner automorphism, of~$so(8,\mathbb{R})$.
In \cite{baez}, Baez mentions that each of the exterior nodes in the Dynkin diagram of~$so(8,\mathbb{R})$ may be identified with a spinor, dual-spinor, and vector representation of~$SO(8,\mathbb{R})$. Hence, the triality~$\tau$ should provide an automorphism between these three representations of~$SO(8,\mathbb{R})$.
Baez, however, discusses triality for the most part as a property of the division algebras. After looking at that version of triality, we spend the remainder of this section discussing the more specific version of triality related to octonions given by Conway in \cite{conway}. This idea of triality will be visited again in Chapter \ref{ch:E6_basic_structure.Triality} as we discuss our real form of~$E_6$.
Additional information regarding triality and octonions may be found in \cite{baez, conway, schafer_book, jacobsen}, although we note that Jacobsen \cite{jacobsen} and Schafer \cite{schafer_book} give an infinitesimal version of triality which is less helpful for this work and hence we do not discuss their treatments of triality.
\begin{figure}[htbp]
\begin{center}
\begin{picture}(200,80)
\put(100,50){\circle*{5}}
\put(100,50){\line(2,1){44.72}}
\put(144,72){\circle*{5}}
\put(100,50){\line(0,-1){50}}
\put(100,0){\circle*{5}}
\put(100,50){\line(-2,1){44.72}}
\put(56,72){\circle*{5}}
\end{picture}
\caption{Dynkin Diagram of~$so(8,\mathbb{R})$}
\label{fig:dynkin_diagram_so8}
\end{center}
\end{figure}
Given inner product spaces~$V_1, V_2,$ and~$V_3$, all of the same dimension, Baez \cite{baez} defines a {\it normed triality} as a trilinear map
$$ t: V_1 \times V_2 \times V_3 \to \mathbb{R}$$
where
$$ |t(v_1, v_2, v_3)| \le ||v_1|| ||v_2|| ||v_3|| $$
and such that for all~$v_1, v_2$ there exists~$v_3 \ne 0$ for which this bound is attained - and similarly for cyclic permutations of~$1, 2, 3$.
This map may be dualized, resulting in a bilinear map
$$ m : V_1 \times V_2 \to V_3^*$$
which Baez conveniently calls a ``multiplication''. Using this dualized normed triality map, left multiplication by~$v_1 \in V_1, v_1 \ne 0$ gives an isomorphism $V_2 \to V_3^*$ while right multiplication by~$v_2 \in V_2, v_2 \ne 0$ gives an isomorphism $V_1 \to V_3^*$. Similarly, appropriate results hold for cyclic permutations of~$1,2,3.$ Baez notes that with a normed triality, if unit vectors are picked from any two of the three spaces $V_1$, $V_2$, or $V_3$, then all three spaces may be identified giving a normed division algebra. Evidently, every normed division algebra has a normed triality.
We are especially interested in the triality associated with the octonions, and find the particular treatment given in Conway \cite{conway} to be useful. Conway discusses triality in relation to octonions and $SO(8,\mathbb{R})$, and so we must first indicate how octonions may be used to construct three different representations of~$SO(8,\mathbb{R})$.
For any unit~$a \in \mathbb{O}$, let~$L_a$,~$R_a$, and~$S_a$ denote the left, right, and symmetric multiplication maps from~$\mathbb{O}$ to~$\mathbb{O}$ given by
$$ L_a(x) = ax \hspace{1cm} R_a(x) = xa \hspace{1cm} S_a(x) = axa \hspace{1cm} (x \in \mathbb{O})$$
As~$\mathbb{O}$ may be identified with~$\mathbb{R}^8$, and these multiplications are invertible and preserve the norm of~$x$, it can be shown \cite{conway} that~$\left( \mathbb{O}, L \right)$,~$\left( \mathbb{O}, R \right)$, and~$\left( \mathbb{O}, S \right)$ are representations of~$SO(8,\mathbb{R})$. These representations are often called the ``spinor,'' ``dual-spinor,'' and ``vector'' representations due to the multiplication utilized in each space. Denote a general transformation on~$\mathbb{O}$ induced by a sequence of left, right, and symmetric multiplications of unit octonions by~$L_\alpha$,~$R_\beta$, and~$S_\gamma$, respectively. That is,
$$ L_\alpha = L_{a_1} \cdots L_{a_{m_1}} \hspace{1cm} R_\beta = R_{b_1} \cdots R_{b_{m_2}} \hspace{1cm} S_\gamma = S_{c_1} \cdots S_{c_{m_3}}~$$
for~$a_i, b_i, c_i \in \mathbb{O}$ with~$|a_i| = |b_i| = |c_i| = 1$.
We use~$L_{-\alpha}$ to denote the transformation given by~$L_{-a_1} \cdots L_{-a_{m_1}}$. Similarly,~$R_{-\beta} = R_{-b_1} \cdots R_{-b_{m_2}}$ and~$S_{-\gamma} = S_{-c_1} \cdots S_{-c_{m_3}}$, implying~$S_{-\gamma} = S_{\gamma}$.
We are now prepared to define the specific version of triality which is helpful in the octonionic case, as presented in \cite{conway}.
Suppose~$x,y,z \in \mathbb{O}$ satisfy~$xy = z$. If~$S_\gamma$ is any transformation in~$\left( \mathbb{O}, S \right)$,
then there exists
\hbox{$L_\alpha \in \left( \mathbb{O}, L \right)$} and~$R_\beta \in \left( \mathbb{O}, R \right)$
for which~$L_\alpha(x) R_\beta(y) = S_\gamma(z)$. Moreover,~$\left( L_\alpha, R_\beta \right)$ and~$\left(L_{-\alpha}, R_{-\beta} \right)$ are the only choices from~$(\mathbb{O},L)\times (\mathbb{O},R)$ which satisfy this equation. Hence, for each representation of~$SO(8,\mathbb{R})$, we instead consider~$PSO(8,\mathbb{R})$ by writing~$[L_\alpha] = \lbrace L_\alpha, L_{-\alpha} \rbrace$ to signify that~$L_\alpha$ is identified with ~$L_{-\alpha}$, and similarly for~$[R_\beta]$ and~$[S_\gamma]$.
In \cite{conway}, Conway states that $PSO(8, \mathbb{R})$ has an outer automorphism~$\tau$ of order 3 such that~$\tau \left( [L_\alpha], [R_\beta], [S_\gamma] \right) = \left( [R_\beta], [S_\gamma], [L_\alpha] \right)$. This means that the general transformation~$[L_\alpha] \in PSO(8, \mathbb{R})$ may be written as either a composition of symmetric multiplications~$[S_\gamma] \in PSO(8,\mathbb{R})$ or, using~$\tau^2$, as a composition of right multiplications~$[R_\beta] \in PSO(8,\mathbb{R})$. This triality map~$\tau$ on the three representations of~$PSO(8,\mathbb{R})$ induces a map
$$ SO(8,\mathbb{R}) \to SO(8,\mathbb{R}) \to SO(8,\mathbb{R})$$
between the spaces
$$ (\mathbb{O}, L) \to (\mathbb{O},R) \to (\mathbb{O},S) $$
generated by
$$ L_\alpha \to R_\beta \to S_{\gamma}$$
In the case of~$(\mathbb{O}, L), (\mathbb{O}, R)$, and~$(\mathbb{O},S)$, this not only implies that each left multiplication may be written as the composition of right multiplications, but Conway shows that the general left multiplication requires an expression involving exactly seven right right multiplications \cite{conway}.
This form of triality will prove useful in Chapter \ref{ch:E6_basic_structure}, and in particular in Section \ref{ch:E6_basic_structure.Triality}, as we analyze the structure of~$SL(3,\mathbb{O})$.
\section{Lorentz Transformations}
\label{ch:Division_Algebras_Applications.Lorentz_Transformations}
Informally, a~$(k+1)$-dimensional spacetime consists of one time-like direction and~$k$ space-like directions. Spacetimes for certain values of~$k$ naturally match physical phenomena, for instance,~$(3+1)$-dimensional spacetime, called {\it Minkowski spacetime}, is a natural place for working with Einstein's theory of special relativity. As noted in \cite{kugo_townsend, manogue_fairlie_composite_string, manogue_fairlie_covariant_superstring, manogue_sudbery_gen_sol}, each of the division algebras~$\mathbb{K} = \mathbb{R}, \mathbb{C}, \mathbb{H}, \mathbb{O}$ give rise to particular \hbox{$(k+1)$-dimensional} spacetimes applicable to physics, where~$k-1 = |\mathbb{K}|$. In this section, we first show how~$(k+1)$-dimensional spacetimes may be represented using~$2 \times 2$ Hermitian matrices over~$\mathbb{K}$ for~\hbox{$k-1 = |\mathbb{K}|$}. We then show how the division algebras may be used to construct isometries in these spacetimes. All of the concepts in Section \ref{ch:Division_Algebras_Applications.Lorentz_Transformations.General_Lorentz_Transformations} through Section \ref{ch:Division_Algebras_Applications.Lorentz_Transformations.Lorentz_Transformations_for_Spacetimes_involving_Division_Algebras}
follow the treatment given in \cite{manogue_schray}, which is fundamental to the construction of~$SL(3,\mathbb{O})$ in Chapter \ref{ch:E6_basic_structure}.
\subsection{Spacetime and Lorentz Transformations}
\label{ch:Division_Algebras_Applications.Lorentz_Transformations.General_Lorentz_Transformations}
Let~$V$ be a~$(k+1)$-dimensional real vector space, and let
~$\lbrace e_1, \cdots, e_{k+1} \rbrace$ denote the standard ordered orthonormal basis on~$V$.
A~$(k+1)$-dimensional {\it spacetime}~$(V,g)$ is a \hbox{$(k+1)$-dimensional} real vector space~$V$ equipped with a non-degenerate, symmetric, bilinear form~$g$ with signature~$(+, -, -, \cdots, -)$.
We use~$t$ to signify the light-light coordinate of~$e_1$, and write
$$\mathbf{x} = \left(\begin{array}{cccccc} t & x_2 & x_3 & \cdots & x_{k} & x_{k+1} \end{array} \right)^T \in V$$
Then the squared length of~$\mathbf{x}$ is given by
$$ |\mathbf{x} |^2 = x^{T} g x = t^2 - x_2^2 - x^3_2 - \cdots - x_{k+1}^2$$
We note that $\mathbf{x} \in V$ is called a {\it null vector} if $|\mathbf{x}| = 0$.
A {\it Lorentz Transformation} is a map~$M : V \to V$ which preserves the squared length of~$\mathbf{x}$, that is,
$$ |\mathbf{x} |^2 = |(M\mathbf{x}) |^2$$
Writing~$\mathbf{x}$ as a~$(k+1)$ column vector, it is convenient to use matrix multiplication to represent a Lorentz transformation. Indeed, if~$M_{e_a,e_b} : V \to V$ is a rotation through an angle~$\alpha$ in the plane spanned by~$e_a,e_b$,
then the components of~$M_{e_a,e_b}$ are given by either (assuming~$a < b$)
$$ (M_{e_1,e_b})_{(1,1)} = (M_{e_1,e_b})_{(b,b)} = \cosh \alpha \hspace{1cm}
(M_{e_1,e_b})_{(1,b)} = (M_{e_1,e_b})_{(b,1)} = \sinh \alpha~$$
$$(M_{e_1,e_b})_{(c,c)} = 1 \textrm{ if } c \ne 1 \textrm{ and } c \ne b$$
or, when~$a \ne 1$, by
$$ (M_{e_a,e_b})_{(a,a)} = (M_{e_a,e_b})_{(b,b)} = \cos \alpha \hspace{1cm}
(M_{e_a,e_b})_{(a,b)} = -(M_{e_a,e_b})_{(b,a)} = -\sin \alpha~$$
$$(M_{e_a,e_b})_{(c,c)} = 1 \textrm{ if } c \ne a \textrm{ and } c \ne b$$
where all other entries are~$0$.
Hence, we see that~$M_{e_1,e_3}$ and~$M_{e_2,e_{k+1}}$ are given by
$$
M_{e_1,e_3} = \left( \begin{array}{cccccc
\cosh \alpha & 0 & \sinh \alpha & 0 & \cdots & 0 \\
0 & 1 & 0 & 0 & \cdots & 0 \\
\sinh \alpha & 0 & \cosh \alpha & 0 & \cdots & 0 \\
0 & 0 & 0 & 1 & \cdots & 0 \\
\vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\
0 & 0 & 0 & 0 & \cdots & 1 \\
\end{array} \right)$$
$$ M_{e_2,e_{k+1}} = \left( \begin{array}{cccccc}
1 & 0 & 0 & \cdots & 0 & 0 \\
0 & \cos \alpha & 0 & \cdots & 0 & -\sin \alpha \\
0 & 0 & 1 & \cdots & 0 & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\
0 & 0 & 0 & \cdots & 1 & 0 \\
0 & \sin \alpha & 0 & \cdots & 0 & \cos \alpha \\
\end{array} \right)
$$
Any Lorentz transformation may be written as some composition of the matrices
$$M_{e_{1},e_{2}}, M_{e_{1},e_3}, \cdots, M_{e_1,e_{k+1}}, M_{e_2,e_3}, \cdots, M_{e_k,e_{k+1}}$$
using appropriate angles.
\subsection{Spacetimes related to Division Algebras}
\label{ch:Division_Algebras_Applications.Lorentz_Transformations.Division_Algebras_and_Spacetimes}
For reasons expressed above, we are particularly interested in~$(k+1)$-dimensional spacetimes where~$k = 2, 3, 5, 9$ is one more than the dimension of~\hbox{$\mathbb{K} = \mathbb{R}, \mathbb{C}, \mathbb{H}, \mathbb{O}$}. For such values of~$k$,
it is convenient to write an element of~$(k+1)$-dimensional spacetime using a {\it vector}, which is a~$2 \times 2$ hermitian matrix
$$ \mathbf{X} = \left( \begin{array}{cc} t+z & \overline{q} \\ q & t-z \end{array} \right)$$
with~$q \in \mathbb{R}, \mathbb{C}, \mathbb{H}, \mathbb{O}$, respectively.
In the complex case, we may identify the element
$$\mathbf{x} = \left( \begin{array}{cccc} t & x & y & z \end{array} \right)^T$$
of~$(3+1)$ spacetime dimensions with the vector
$$ \mathbf{X} = \left( \begin{array}{cc} t+z & x-iy \\ x+iy & t-z \end{array} \right)$$
by writing
$$\mathbf{X} = t \sigma_t + x \sigma_x + y \sigma_y + z \sigma_z$$
where~$\sigma_t$ is the~$2 \times 2$ identity matrix and~$\sigma_x$,~$\sigma_y$, and~$\sigma_z$ are the Pauli spin matrices
$$ \sigma_x = \left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right) \hspace{1cm}
\sigma_y = \left( \begin{array}{cc} 0 & -i \\ i & 0 \end{array} \right)\hspace{1cm}
\sigma_z = \left( \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right)$$
Equipped with a metric~$\langle \; , \; \rangle$ whose value is given by~$\langle \sigma_a, \sigma_b \rangle = \frac{1}{2}\tr{\sigma_a \sigma_b}$, it can be shown that the Pauli matrices along with~$\sigma_t$ form an orthonormal basis of a four-dimensional vector space over~$\mathbb{R}$.
This identification between~$(3+1)$-dimensional spacetime and complex vectors may be extended to the quaternionic or octonionic case, involving spacetimes of dimension~$(5+1)$ and~$(9+1)$, respectively, by
relabeling~$\sigma_y$ as~$\sigma_i$ and including ~$\sigma_q$, which is the matrix~$\sigma_i$ with~$i$ replaced with~$q = i,j,k$ or~$q = i, \cdots, i\l, \l$, respectively.
We now show that the squared length of an element in~$(k+1)$-dimensional spacetime may be expressed as the determinant of its corresponding~$2 \times 2$ hermitian matrix, where~$k-1$ is the dimension of one of the division algebras. We show this for the case involving octonions and~$(9+1)$-dimensional spacetime, and note that the other cases correspond to subalgebras and subspaces of this case.
Let~$q = q_x + q_i i + q_j j + q_k k + q_{k\l}k\l + q_{j\l}j\l + q_{i\l}i\l + q_\l \l \in \mathbb{O}$. The general octonionic vector
$$ \mathbf{X} = \left( \begin{array}{cc} t+z & \overline{q} \\ q & t-z \end{array} \right), q \in \mathbb{O}, t,z \in \mathbb{R}$$
satisfies its characteristic equation
$$ \mathbf{X}^2 - (\tr \mathbf{X}) \mathbf{X} + (\det{\mathbf{X}}) I = 0$$
where~$\tr \mathbf{X}$ denotes the trace of~$\mathbf{X}$. We may solve this equation, giving an expression for~$\det{\mathbf{X}}$ (See Section \ref{ch:Division_Algebras_Applications.Jordan_Algebras}). As~$\mathbf{X}$ is a $2 \times 2$ hermitian matrix and the components of~$\mathbf{X}$ lie in a complex subalgebra of~$\mathbb{O}$, the expression for the determinant of~$\mathbf{X}$ takes the familiar form
$$\det{\mathbf{X}} = (t+z)(t-z) - |q|^2$$
Hence, using the identification between an element
$$\left( \begin{array}{ccccccc} t & q_x & q_i & \cdots & q_{i\l} & q_\l & z \end{array} \right)^T$$
of~$(9+1)$-dimensional spacetime and the octonionic vector~$\mathbf{X}$ given above, we see that
$$\begin{array}{ccl} |( \begin{array}{ccccccc} t & q_x & q_i & \cdots & q_{i\l} & q_\l & z \end{array})^T|^2 &=& t^2 - q_x^2 - q_i^2 - \cdots - q_{i\l}^2 - q_\l^2 - q^2 \\
&=& \det{\mathbf{X}} \end{array}$$
We note that the time-like variable~$t$ is found using
$$t = \frac{1}{2}\tr{\mathbf{X}}$$
Similar results can be found for the real, complex, or quaternionic cases involving spacetimes of dimension~$(2+1)$,~$(3+1)$, and~$(5+1)$, respectively.
Finally, we can not discuss vectors for spacetimes associated to the division algebras without mentioning spinors and dual-spinors.
A {\it spinor} or {\it Weyl spinor} is an element of~$\mathbb{K} \oplus \mathbb{K}$, for~$\mathbb{K} = \mathbb{R}, \mathbb{C}, \mathbb{H}, \mathbb{O}$, which we conveniently write as a two-component column~$v = \left( \begin{array}{c} b \\ c \end{array} \right)$ with $b,c \in \mathbb{K}$. A {\it dual spinor} is an element of the vector space dual to~$\mathbb{K} \oplus \mathbb{K}$. It is convenient to write a dual spinor $v^\dagger = \left( \begin{array}{cc} \overline{b} & \overline{c} \end{array} \right)$ as the hermitian conjugate, or dagger, of a spinor.
Let~$\mathbb{K}$ be one of the four division algebras and let the vector~$v \in \mathbb{K} \oplus \mathbb{K}$ and spinor~$v^\dagger \in (\mathbb{K} \oplus \mathbb{K})^*$ be written as above. There are two meaningful products which relate spinors to vectors. The product~$v v^\dagger$ is the~$2 \times 2$ Hermitian matrix
$$ v v^\dagger = \left( \begin{array}{cc} |b|^2 & b\overline{c} \\ c\overline{b} & |c|^2 \end{array} \right)$$
called the {\it square of a spinor}. Using the correspondence previously described in this chapter, we see that~$v v^\dagger$ is an element of a~$(k+1)$-dimensional spacetime where~$k-1 = |\mathbb{K}|$. However, as
$$ \begin{array}{ccl}
\mydet{ v v^\dagger } &=& \det{ \left( \begin{array}{cc} |b|^2 & b\overline{c} \\ c\overline{b} & |c|^2 \end{array} \right)} \\
& = & |b|^2 |c|^2 - (b\overline{c})\overline{(b\overline{c})} \\
& = & |b|^2 |c|^2 - |bc|^2 \\
& = & 0\\
\end{array}$$
we see that a squared spinor corresponds to a null vector in~$(k+1)$-dimensional spacetime. The other product,~$v^\dagger v = |b|^2 + |c|^2$, gives twice the time-like coordinate of this null vector.
\subsection{Lorentz Transformations related to Division Algebras}
\label{ch:Division_Algebras_Applications.Lorentz_Transformations.Lorentz_Transformations_for_Spacetimes_involving_Division_Algebras}
For~$k=2,3,5,9$, the elements of~$(k+1)$-dimensional spacetime may be represented using real, complex, quaternionic, or octonionic vectors. The final step is to construct Lorentz transformations. As we shall see, such transformations will make use of the division algebras and the geometric transformations associated with its multiplication, as reviewed in Section \ref{ch:Division_Algebras_Applications.Quaternions_Octonions.Conjugation_Reflecitons_Rotations}. This treatment again follows the work of \cite{manogue_schray}. The constructions and notation in this section are fundamental to the construction of~$SL(3,\mathbb{O})$ in Chapter \ref{ch:E6_basic_structure}.
For the general case, let~$\mathbf{X}$ be an octonionic vector. Since vectors are hermitian matrices, a Lorentz transformation must preserve both the hermiticity of~$\mathbf{X}$ as well as~$\det{\mathbf{X}}$. For general~$M$, neither one-sided multiplication~\hbox{$\mathbf{X} \to M \mathbf{X}$} nor~\hbox{$\mathbf{X} \to \mathbf{X} M$} preserves the hermiticity of~$\mathbf{X}$. However, the result of a transformation using conjugation
$$\mathbf{X} \to M \mathbf{X} M^\dagger$$
is a hermitian matrix for any matrix~$M$, provided the result is well-defined.
As shown in \cite{manogue_schray}, the product $M \mathbf{X} M^\dagger := \left(M \mathbf{X} \right) M^\dagger = M \left( \mathbf{X} M^\dagger \right)$ is well-defined if either the components of $M$ lie in a complex subspace of $\mathbb{O}$ or the columns of the imaginary part of $M$ must be real multiples of each other.
In addition, the squared length of~$\mathbf{x}$ corresponding to~$\mathbf{X}$ will be preserved if~$\det{\mathbf{X}} = \mydet{M \mathbf{X} M^\dagger}$. For such~$M$, we also have
$$\mydet{M \mathbf{X} M^\dagger} = \mydet{M M^\dagger} \mydet{\mathbf{X}}$$
requiring that the matrices~$M$ also satisfy the condition\footnote{Over the complex numbers, we can multiply~$M$ by an overall phase~$e^{i\alpha}$ without changing~$|\det{M}|$ or~$\det{(M\mathbf{X} M^{\dagger})}$, and hence we require that~$\det{M} = 1$. Thus, the Lorentz group~$SO(3,1)$ is essentially~$SL(2,\mathbf{C})$.}
$$\mydet{M M^\dagger} = 1$$
In the case of complex vectors~$\mathbf{X} \in M_{2}(\mathbb{C})$, we choose basis transformations
$$R_{ab}: \mathbb{R} \times M_2(\mathbb{C}) \to M_2(\mathbb{C})$$
where
$$R_{ab}(\alpha)(\mathbf{X}) = M_{a,b}(\frac{\alpha}{2}) \; \mathbf{X} \; M_{a,b}^\dagger (\frac{\alpha}{2})$$
rotates the~$(a,b)$ plane through an angle~$\alpha$, for~$a,b \in \lbrace t,x,y,z \rbrace$. For rotations in the~$(x,y)$,~$(y,z)$, and~$(z,x)$ planes, the matrices~$M_{a,b}(\alpha)$ are given by:
\[
M_{x,y}(\alpha) = \left( \begin{array}{cc} e^{i\alpha} & 0 \\ 0 & e^{-i\alpha}\end{array}\right)
\hspace{1cm}
M_{y,z}(\alpha) = \left( \begin{array}{cc} \cos \alpha & i\sin \alpha \\ i \sin \alpha & \cos \alpha\end{array}\right)
\]
\[ M_{z,x}(\alpha) = \left( \begin{array}{cc} \cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha\end{array}\right) \]
\noindent while the~$M_{a,b}(\alpha)$ matrices with~$a=t$ are given by
\[
M_{t,z}(\alpha) = \left( \begin{array}{cc} e^{\alpha} & 0 \\ 0 & e^{-\alpha}\end{array}\right)
\hspace{1cm}
M_{t,x}(\alpha) = \left( \begin{array}{cc} \cosh \alpha & \sinh \alpha \\ \sinh \alpha & \cosh \alpha\end{array}\right)
\]
\[M_{t,y}(\alpha) = \left( \begin{array}{cc} \cosh \alpha & i\sinh \alpha \\ -i\sinh \alpha & \cosh \alpha\end{array}\right) \]
\noindent
We find it convenient to call the Lorentz transformations which change the time-like coordinate {\it boosts}. We label those transformations not with the label~$R_{t,b}(\alpha)$ but with the label~$B_{t,b}(\alpha)$.
These transformations also work in the quaternionic case~$\mathbf{X} \in M_2(\mathbb{H})$ with a slight modification. In this larger setting, we are now working with three imaginary units,~$i$,~$j$, and~$k$, instead of just one,~$i$. Hence, it becomes helpful to relabel the Lorentz transformations~$R_{xy}$,~$R_{yz}$, and~$R_{ty}$ as~$R_{xi}$,~$R_{iz}$, and~$R_{ti}$. We create six new Lorentz transformations by replacing the~$i$ in the subscripts for the three transformations~$R_{ab}(\alpha)$ and~$M_{a,b}(\alpha)$ with~$j$ and~$k$. This gives 12 Lorentz transformations, each of which rotates a plane containing the~$t$,~$x$, or~$z$ direction. Finally, it is only necessary to include the rotations which rotate a plane spanned by two imaginary quaternions in~$5+1$ dimensions. However, in Section \ref{ch:Division_Algebras_Applications.Quaternions_Octonions.Conjugation_Reflecitons_Rotations}, we saw that the conjugation map
$$q \to e^{i\alpha/2} q e^{-i\alpha/2}$$
fixes the~$(1,i)$ plane and produces a rotation through an angle~$\alpha$ in the~$(j,k)$ plane. Thus, for~$s = p \, q$ for~$p,q \in \lbrace i,j,k \rbrace, p\ne q$, the Lorentz transformation~\hbox{$R_{p,q}: \mathbb{R} \times M_2(\mathbb{C}) \to M_2(\mathbb{C})$}
given by
$$ R_{p,q}(\alpha)(\mathbf{X}) = \left( \begin{array}{cc} \exp({s \; \alpha/2}) & 0 \\ 0 & \exp({s \; \alpha/2}) \end{array} \right)
\mathbf{X} \left( \begin{array}{cc} \exp({s \; \alpha/2}) & 0 \\ 0 & \exp({r\; \alpha/2}) \end{array} \right)^\dagger$$
produces a rotation in the~$(p,q)$ plane through an angle of~$\alpha$. The transformations involving two imaginary units are called {\it transverse rotations}.
In the octonionic case, Manogue and Schray \cite{manogue_schray} provided an explicit description of vector Lorentz transformations in~$(9+1)$ spacetime dimensions by generalizing the above results. The boosts and non-transverse rotations generalize to this case in the obvious way, by extending the transformations~$B_{tq}(\alpha), R_{xq}(\alpha),$ and~$R_{zq}(\alpha)$ from~$q = i,j,k$ to now~$q = i,j,k,k\l,j\l,i\l,\l$. More care must be taken to generalize the transverse rotations, as we saw in Section \ref{ch:Division_Algebras_Applications.Quaternions_Octonions.Conjugation_Reflecitons_Rotations} that the conjugation map~$q \to e^{i\alpha} q e^{-i\alpha}$ rotates three planes simultaneously instead of just one plane in the octonionic setting. As discussed in Section \ref{ch:Division_Algebras_Applications.Quaternions_Octonions.Conjugation_Reflecitons_Rotations}, nested flips may be used to produce a rotation spanned by two orthogonal pure imaginary units~$p$ and~$q$. That is, the transformation~$R_{p,q}: \mathbb{R} \times M_2(\mathbb{O}) \to M_2(\mathbb{O})$ given by
$$ R_{p,q}(\alpha) = M_{2(p,q)}\left(\alpha/2\right) \left( M_{1(p)} \mathbf{X} M_{1(p)}^\dagger \right) M_{2(p,q)}\left(\alpha/2\right)^\dagger$$
where
$$\begin{array}{ccl} M_{1(p)} & = & \left( \begin{array}{cc} -p & 0 \\ 0 & -p \end{array} \right) \\
M_{2(p,q)}(\alpha/2) & = & \left( \begin{array}{cc} \cos(\alpha/2)\; p + \sin(\alpha/2) \;q & 0 \\
0 & \cos(\alpha/2)\; p + \sin(\alpha/2)\; q
\end{array} \right)
\end{array}
$$
is the transverse rotation in the plane defined by two orthogonal imaginary units~$p, q \in \mathbb{O}$.
There are~$21$ different ways to pick pairs of these units from the seven orthogonal imaginary units in~$\mathbb{O}$, adding~$21$ transverse rotations to the~$15$ general rotations and~$9$ boosts.
Manogue and Schray \cite{manogue_schray} identified these~$45$ transformations as generators of the group $SO(9,1)$ and found several important subgroups using their finite Lorentz transformations. The~$21$ transverse rotations produce all rotations among the various plains spanned by the any two of the seven imaginary orthogonal units of~$\mathbb{O}$, generating the group~$SO(7)$. The group~$SO(8)$ is obtained by adding the seven~$R_{x,q}$ transformations, where~$q = i,j,k,k\l,j\l,i\l,\l$, while adding the remaining (non-boost) rotations gives~$SO(9)$.
In Table \ref{table:finite_octonionic_lorentz_transformations}, we list the finite Lorentz transformations as categorized by Manogue and Schray. The Category 1 transformations are boosts, while the Category 2 transformations are rotations in a plane containing either the~$x$ or~$z$ axis. The Category 3 transformations are the transverse rotations in the plane defined by two imaginary units~$p, q \in \mathbb{O}$ where~$p \perp q$. We note that each of these basic transformations $R_{ab}$ only affect the~$(a,b)$ plane in~$9+1$-dimensional spacetime. Hence,~$R_{ab}(\alpha)R_{ab}(\gamma) = R_{ab}(\alpha + \gamma)$ for any~$\alpha, \gamma \in \mathbb{R}$. That is, each of the~$45$ basic transformations of~$SL(2,\mathbb{O})$ listed in Table \ref{table:finite_octonionic_lorentz_transformations} are generators of a one-parameter subgroup of~$SL(2,\mathbb{O})$.
\begin{table}[hbtp]
\begin{center}
\begin{minipage}{4.75in}
\[
\mathbf{X} \to \left\{ \begin{array}{ccl}
M\mathbf{X} M^\dagger, &\hspace{1cm} & \textrm{ for Categories 1 and 2} \\
M_2 \left( M_1 \mathbf{X} M_1^\dagger \right) M_2^\dagger, & & \textrm{ for Category 3}
\end{array}
\right.
\]
$$
\begin{array}[t]{|l|c|}
\hline
\begin{minipage}[t]{0.85in}\textrm{Category 1:\\ Boosts}\end{minipage} &
\begin{array}{ccl}
B_{tz} & t \longleftrightarrow z & M = \left(
\begin{array}{cc} \exp \left(\frac{\alpha}{2} \right) & 0 \\
0 & \exp \left(-\frac{\alpha}{2} \right) \\
\end{array}
\right) \\
B_{tx} & t \longleftrightarrow x & M = \left(
\begin{array}{cc} \cosh \left( \frac{\alpha}{2} \right) & \sinh \left( \frac{\alpha}{2} \right) \\
\sinh \left( \frac{\alpha}{2} \right) & \cosh \left( \frac{\alpha}{2} \right)
\end{array}
\right) \\
B_{tq} & t \longleftrightarrow q & M = \left(
\begin{array}{cc} \cosh \left( \frac{\alpha}{2} \right) & q \sinh \left( \frac{\alpha}{2} \right) \\
-q \sinh \left( \frac{\alpha}{2} \right) & \cosh \left( \frac{\alpha}{2} \right)
\end{array}
\right) \\
\end{array} \\
\hline
\hline
\begin{minipage}[b]{0.85in} Category~2:\\ Rotations \end{minipage} &
\begin{array}{ccl}
R_{xq} & x \longleftrightarrow q & M = \left(
\begin{array}{cc}
\exp \left( q \frac{\alpha}{2} \right) & 0 \\
0 & \exp \left( -q \frac{\alpha}{2} \right)
\end{array} \right) \\
R_{xz} & x \longleftrightarrow z & M = \left(
\begin{array}{cc}
\cos \left( \frac{\alpha}{2} \right) & -\sin \left(\frac{\alpha}{2} \right) \\
\sin \left( \frac{\alpha}{2}\right) & \cos \left(\frac{\alpha}{2} \right)
\end{array} \right) \\
R_{zq} & q \longleftrightarrow z & M = \left(
\begin{array}{cc}
\cos \left( \frac{\alpha}{2} \right) & q \sin \left(\frac{\alpha}{2} \right) \\
q \sin \left( \frac{\alpha}{2}\right) & \cos \left(\frac{\alpha}{2} \right)
\end{array} \right) \\
\end{array} \\
\hline
\hline
\begin{minipage}[c]{0.85in}Category 3:\\ Transverse\\[-.66em] Rotations \\ \end{minipage} &
\begin{array}{ccl}
R_{p,q} & p \longleftrightarrow q & M_1 = -p \; \mathbb{I}_2 \\
& & M_2 = \left( \; \cos \left(\frac{\alpha}{2}\right) p + \sin \left( \frac{\alpha}{2} \right) q \; \right) \mathbb{I}_2
\end{array} \\
\hline
\end{array}
$$
\end{minipage}
\caption{Finite octonionic Lorentz transformations}
\label{table:finite_octonionic_lorentz_transformations}
\end{center}
\end{table}
Finally, we note that Manogue and Schray carefully constructed their finite Lorentz transformations so that they were compatible, meaning that the parentheses may be re-arranged when it is applied to a squared spinor~$v v^\dagger$. More precisely, a transformation~$\mathbf{X} \to M \mathbf{X} M^\dagger$ is {\it compatible } if
$$M (v v^\dagger) M^\dagger = (M v) (v^\dagger M^\dagger) = (M v) (M v)^\dagger$$
This re-arrangement of parenthesis is possible for octonionic spinors~$v$ provided that the elements of~$M$ lie in a complex subalgebra \cite{manogue_schray}.
\noindent We use compatible vector finite Lorentz transformations to give spinor and dual spinor transformations simply by using the same matrices but applying only left-multiplication or right-multiplication, respectively, instead of conjugation. That is, if~$R: \mathbb{R} \times M_2(\mathbb{O}) \to M_2(\mathbb{O})$ is a transformation given by
$$R(\alpha)(\mathbf{X}) = M_n(\alpha)\left( \cdots \left(M_1(\alpha) \mathbf{X} M_1^\dagger(\alpha) \right) \cdots \right) M_n^\dagger(\alpha)$$
then the spinor and dual spinor versions of~$R$ are then $R: \mathbb{R} \times (\mathbb{O}\oplus \mathbb{O}) \to (\mathbb{O}\oplus \mathbb{O})$ and $R: \mathbb{R} \times (\mathbb{O}\oplus \mathbb{O})^* \to (\mathbb{O}\oplus \mathbb{O})^*$,
respectively, where
$$R(\alpha)(\theta) = M_n(\alpha)\left( \cdots \left(M_1(\alpha) \; \theta \right) \cdots \right) $$
and
$$R(\alpha)(\theta^\dagger) = \left( \cdots \left(\theta^\dagger \; M_1^\dagger(\alpha) \right) \cdots \right) M_n(\alpha)^\dagger$$
We deliberately give the spinor and dual spinor versions of the transformations the same name as the vector Lorentz transformation, as the context will make it clear which version is appropriate. We utilize the vector, spinor, and dual spinor versions of each of the finite Lorentz transformations in our construction of~$E_6 = SL(3,\mathbb{O})$ in Chapter \ref{ch:E6_basic_structure}.
\section{Jordan Algebras and Albert Algebras}
\label{ch:Division_Algebras_Applications.Jordan_Algebras}
We give here the definition of the Albert algebra, which is the exceptional Jordan algebra. One particularly nice characteristic of the Albert algebra is that it naturally contains both spinors and vectors. We develop the invariants for the Albert algebra corresponding to the trace and determinant of a matrix, and discuss how these invariants are tied to certain Lie groups. Additional information on Jordan and Albert algebras, including their role in physics, may be found in \cite{okubo, manogue_dray_exceptional_eigenvalue_problem}.
A {\it Jordan algebra } is a commutative algebra~$A$ with product denoted by~$\circ$ that satisfies the Jordan identity
$$ (x^2 \circ y) \circ x = x^2\circ (y\circ x)$$
for any~$x,y \in A$, where~$x^2 \equiv x \circ x$. Given an algebra~$A$ with a bi-linear product~$xy$, we can create a new algebra~$(A,\circ)$ by introducing the commutative product
$$ x \circ y = \frac{1}{2} \left(xy + yx \right)$$
The algebra~$(A,\circ)$ may or may not satisfy the Jordan identity. If~$A$ is an associative algebra, then~$(A,\circ)$ will always satisfy the Jordan identity and is called a {\it special Jordan algebra}. In 1934, Jordan, von Neumann, and Wigner \cite{jordan_von_neumann_wigner} proved that all finite-dimensional simple Jordan algebras are special Jordan algebras except for one case, known as the exceptional Jordan algebra.
The exceptional Jordan algebra~$\mathbf{M}_3(\mathbb{O})$, also known as the Albert algebra, consists of~$3 \times 3$ octonionic hermitian matrices with the product
$$ X \circ Y = \frac{1}{2}\left(XY + YX \right)$$
where~$XY$ is matrix multiplication,~$X,Y \in \mathbf{M}_3(\mathbb{O})$.
Although this product is commutative, it is not associative as
$$
\begin{array}{l}
(X \circ Y) \circ Z - X \circ (Y \circ Z) = \\
\hspace{.6cm} \frac{1}{4} \left( (XY)Z + (YX)Z + Z(XY) + Z(YX) \right. \\
\hspace{1.2cm} \left. -X(YZ) - X(ZY) - (YZ)X - (ZY)X
\right)
\end{array}
$$
Nevertheless, the exceptional Jordan algebra does satisfy the Jordan identity.
Now, for $X \in \mathbf{M}_3(\mathbb{O})$, define~$X^2 := (X \circ X)$ and~$X^3 := X^2 \circ X \equiv X \circ X^2$.
A general element of~$\chi = \mathbf{M}_3(\mathbb{O})$ may be written in components as
\[
\chi = \left( \begin{array}{ccc}
p & \overline{a} & c \\
a & m & \overline{b} \\
\overline{c} & b & n \\
\end{array} \right)
\]
\noindent
where~$p, m, n \in \mathbb{R}$ and~$a,b,c \in \mathbb{O}$.
There are three natural ways to identify a~$2 \times 2$ Hermitian octonionic vector~$\mathbf{X}$, 2-component octonionic spinor~$\theta$, and 2-component octonionic dual spinor~$\theta^\dagger$ in~$\chi$. We will find it convenient in Chapter \ref{ch:E6_basic_structure} to use the following identification of~$\mathbf{X}$,~$\theta$, and~$\theta^\dagger$ in~$\chi$:
\[
\chi = \left( \begin{array}{c|c}
\mathbf{X} & \theta \\
\hline
\theta^\dagger & n
\end{array} \right) =
\left( \begin{array}{cc|c}
p & \overline{a} & c \\
a & m & \overline{b} \\
\hline
\overline{c} & b & n
\end{array} \right)
\]
Standard expressions for the determinant of a matrix rely upon the associativity of the underlying division algebra, but a general element of the exceptional Jordan algebra contains three independent octonionic directions~$a$,~$b$, and~$c$. Hence, except for very particular choices of~$a, b, c \in \mathbb{O}$, standard expressions of the determinant of a matrix fail to hold for \hbox{$\chi \in M_3(\mathbb{O})$}.
Nevertheless, octonionic~$3 \times 3$ matrices satisfy their characteristic equation. We now produce an equivalent expression of the determinant for~$3 \times 3$ octonionic hermitian matrices.
The characteristic equation of a~$1 \times 1$ matrix~$A$ is
$$ A - \sigma_1(A) = 0$$
Solving for~$\sigma_1(A)$ and taking the trace, we see that~$\tr{A} = \sigma_1(A)$ and note that for~$1 \times 1$ matrices,~$\det{A} = \sigma_1(A)$.
For the case when~$A$ is a~$2 \times 2$ matrix, the characteristic equation is
$$ A^2 - A \; \tr{A} + \sigma_2(A) \; \mathbb{I}_2 = 0$$
Again, we take the trace giving
$$ \tr{A^2} - (\tr{A})^2 + 2\sigma_2(A) = 0$$
Solving for~$\sigma_2(A)$, we obtain an expression for the~$2 \times 2$ invariant~$\sigma$:
$$ \sigma(A) := \frac{1}{2}\left( \left( \tr{A}\right)^2 - \tr{A^2}\right)$$
We note that for~$2 \times 2$ matrices, the determinant of~$A$ is~$\sigma_2(A)$.
If~$A$ is a~$3 \times 3$ matrix, then the characteristic equation is given by
$$ A^3 - A^2 \; \tr{A} + A \; \sigma(A) - \det{A} \;\mathbb{I}_3 = 0$$
Again, taking the trace and solving for~$\det{A}$ finds
$$\det(A) = \frac{1}{3}\left( \tr{A^3} - \frac{3}{2}\tr{A^2}\tr{A} + \frac{1}{2}\left(\tr{A}\right)^3 \right)$$
where we have substituted in the expression for~$\sigma(A)$ previously found.
This expression for~$\det(A)$ gives the determinant of a~$3 \times 3$ matrix in the special case that~$A \in \mathbf{M}_3(\mathbb{O})$. \footnote{
The general pattern for the characteristic equation of an~$n \times n$ matrix is given by
$$ A^n - A^{n-1}\epse_1(A) + A^{n-2}\epse_2(A) - A^{n-3}\epse_3(A) + \cdots \pm A^{1}\epsilon_{n-1}(A) \mp \det(A)\mathbb{I}_n = 0$$
where~$\epse_n(A)$ is the expression found when solving the~$n\times n$ case for~$\det{A}$.}
Equivalently, we find that the determinant of
\[
\chi = \left( \begin{array}{ccc}
p & \overline{a} & c \\
a & m & \overline{b} \\
\overline{c} & b & n \\
\end{array} \right)
\in \mathbf{M}_3(\mathbb{O})
\]
is given by
$$ \det{\chi} = pmn - (p|b|^2 + m|c|^2 + n|a|^2) + 2Re(\overline{a} \overline{b} \overline{c})$$
This may be rewritten in the more convenient form
$$ \det{ \left( \chi \right) } = \det{ \left( \begin{array}{cc} p & \overline{a} \\ a & m \end{array} \right)} n + 2\left( \begin{array}{cc} p & \overline{a} \\ a & m \end{array} \right) \cdot \left[ \left( \begin{array}{c} c \\ \overline{b} \end{array} \right) \left( \begin{array}{cc} \overline{c} & b \end{array} \right) \right] $$
where $A \cdot B = \tr{A \circ B} - \tr{A} \tr{B}$ is the Lorentzian inner product of vectors $A$ and $B$.
Of course, the trace of~$\chi$ is~$\tr{\chi} = p + m + n$.
Just as in the case of~$n \times n$ real, complex, or quaternionic matrices, the expressions for the trace and determinant of a Jordan matrix will be helpful in defining certain Lie groups. In particular, we will see in the next chapter that the exceptional Jordan algebra is closely tied to the exceptional Lie groups~$G_2$,~$F_4$, and~$E_6$, as well as a property of octonions called triality.
\newpage{}
\part{The Basic Structure of~$E_6$ }
\label{ch:E6_basic_structure}
In this chapter, we explore the Lie group~$SL(3,\mathbb{O})$ and its associated Lie algebra~$sl(3,\mathbb{O})$. We provide our methods for producing and studying our representation of the~$78$-dimensional exceptional Lie group~$E_6$ and its associated Lie algebra. This group preserves the determinant of the exceptional Jordan Algebra~$\mathbf{M}_3(\mathbb{O})$, justifying the interpretation
$$E_6 \equiv SL(3,\mathbb{O})$$
We show how to realize this group by extending the description of
$$SL(2,\mathbb{O}) \equiv SO(9,1,\mathbb{R})$$
given in~\cite{manogue_schray} from the~$2 \times 2$ case to the~$3 \times 3$ case.
This process produces~$135$ group generators, of which only~$78$ are independent. We follow the common practice of using linear dependencies among the associated Lie algebra elements to reduce our list to the~$78$ independent group generators, giving us a preferred basis for~$E_6$.
After constructing the group~$SL(3,\mathbb{O})$ in Section~\ref{ch:E6_basic_structure.LieGroup} and discussing how to construct its algebra in Section~\ref{ch:E6_basic_structure.E6_Lie_Algebra}, we study the characteristics of the group and algebra, including the relevant subgroups and subalgebras, in the remainig sections of this chapter. In particular, we discuss the group~$SL(3,\mathbb{O})$ in Section~\ref{ch:E6_basic_structure.LieGroup}, presenting our construction of the group elements in Section~\ref{ch:E6_basic_structure.Lorentz_Transformations} and giving an improved basis of the~$3\times 3$ transformations generalizing the~$2 \times 2$ transverse rotations in Section~\ref{ch:E6_basic_structure.Lorentz_Transformations.BetterBasis}. We study the Lie algebra corresponding to this group in Section~\ref{ch:E6_basic_structure.E6_Lie_Algebra}. In particular, we discuss our methods which allow us to construct the algebra commutator for~$sl(3,\mathbb{O})$ in Section~\ref{ch:E6_basic_structure.ConstructingE6algebra}. In Section~\ref{ch:E6_basic_structure.ConstructingE6algebra.LinearDependencies}, we use linear dependencies in the algebra to choose a preferred basis, and provide a basis for each of the three natural~$SO(9,1,\mathbb{O})$ subgroups of~$SL(3,\mathbb{O})$ in Section~\ref{ch:E6_basic_structure.Lorentz_Transformations.BetterBasisSubgroups}. We discuss triality as it relates to~$E_6$ in Section~\ref{ch:E6_basic_structure.Triality}. In Section~\ref{ch:E6_basic_structure.Type_Transformation}, we give some continuous extensions to the discrete type map used in our construction of~$SL(3,\mathbb{O})$ and find subgroups which contain or are independent of this type map in Section~\ref{ch:E6_basic_structure.Type_Transformation.Subgroups}. Although our construction relies upon the octonions, we obtain other subgroups of~$SL(3,\mathbb{O})$ by restricting~$\mathbb{O}$ to the other division algebras. These findings are summarized in Section~\ref{ch:E6_basic_structure.Reduction_of_O}. Finally, we give an isomorphism between our preferred basis of~$su(3,\mathbb{C}) \subset G_2$ and the Gell-Mann matrices in Section~\ref{ch:E6_basic_structure.fix_l}.
\section{The Lie Group~$E_6 = SL(3,\mathbb{O})$ }
\label{ch:E6_basic_structure.LieGroup}
As mentioned in Section~\ref{ch:Joma_Paper.subalgebras_greater_than_3}, it is known that~$E_6$ is the group which preserves the determinant of the Jordan matrix~$\chi \in M_3(\mathbb{O})$. It is in this sense that~$E_6$ is~$SL(3,\mathbb{O})$. Dray and Manogue~\cite{manogue_dray} show~$E_6$ is the appropriate union of three copies of~\hbox{$SO(9,1,\mathbb{R}) = SL(2,\mathbb{O})$}. In Section~\ref{ch:E6_basic_structure.Lorentz_Transformations}, we explicitly construct three different~$3 \times 3$ transformations from each of the~$2 \times 2$ finite Lorentz transformations which, as Manogue and Schray showed \cite{manogue_schray}, form a basis for~$SL(2,\mathbb{O})$. These~$2 \times 2$ transformations were also discussed in Section~\ref{ch:Division_Algebras_Applications.Lorentz_Transformations.Lorentz_Transformations_for_Spacetimes_involving_Division_Algebras}. This construction provides~$135$ transformations in~$E_6$, which has dimension~$78$. In Section~\ref{ch:E6_basic_structure.Lorentz_Transformations.BetterBasis}, we choose an explicit basis for a preferred subgroup~$SO(7,\mathbb{R})$ of each~$SL(2,\mathbb{O})$. We find this new basis will be helpful when discussing triality and the subgroup~$G_2$ in relation to~$SL(3,\mathbb{O})$, and will use this basis for the remainder of this thesis after Section~\ref{ch:E6_basic_structure.E6_Lie_Algebra}.
\subsection{Lorentz Transformations in~$3 \times 3$ case}
\label{ch:E6_basic_structure.Lorentz_Transformations}
We first describe the three natural ways to identify a vector~$\mathbf{X} \in M_2(\mathbb{O})$ in an element~$\chi$ of the exceptional Jordan algebra~$M_3(\mathbb{O})$. We write
$$\chi = \left( \begin{array}{ccc} t+z & \overline{a} & c \\ a & t-z & \overline{b} \\ \overline{c} & b & n \end{array} \right) \in M_3(\mathbb{O})$$
$$ t,z,n \in \mathbb{R} \hspace{1cm} a, b, c \in \mathbb{O}$$
where we use
$$ a = a_x + a_i i + a_j j + a_k k + a_{k\l} k\l + a_{j\l} j\l + a_{i\l} i\l + a_\l \l \in \mathbb{O}$$
and similar expressions for~$b$ and~$c$ to denote a general element of~$\mathbb{O}$. Recall from Section~\ref{ch:Division_Algebras_Applications.Lorentz_Transformations.Division_Algebras_and_Spacetimes} that an octonionic vector~$\mathbf{X}$ is a~$2 \times 2$ hermitian matrix, an octonionic spinor~\hbox{$\theta \in (\mathbb{O} \oplus \mathbb{O})$} may be written as a two-component column matrix, and a dual spinor~\hbox{$\theta^\dagger \in (\mathbb{O} \oplus \mathbb{O})^*$} may be written as the hermitian conjugate, or dagger, of a spinor. There are three natural ways to identify a~$2 \times 2$ hermitian matrix in~$\chi$, as shown in Table~\ref{table:three_types_of_vector_locations}. In each case, this identification breaks~$\chi$ into a block structure leaving a~$1\times 2$ octonionic submatrix and a~$2\times 1$ octonionic submatrix which are hermitian conjugates of each other. We identify them with the octonionic spinor and dual spinor, respectively.
\begin{table}[tbp]
\begin{center}
\begin{tabular}{ccccc}
\textrm{Location 1} & & \textrm{Location 2} & & \textrm{Location 3}\\
$\left( \begin{array}{c|c} \mathbf{X} & \theta \\ \hline \theta^\dagger & \cdot \end{array}\right)$ & &
$\left( \begin{array}{c|c} \cdot & \theta^\dagger \\ \hline \theta & \mathbf{X} \end{array}\right)$ & &
$\left( \begin{array}{c|c|c} \mathbf{X}_{2,2} & \theta_2 & \mathbf{X}_{2,1} \\ \hline \overline{\theta_2} & \cdot & \overline{\theta_1} \\ \hline \mathbf{X}_{1,2} & \theta_{1} & \mathbf{X}_{1,1} \end{array}\right)$ \\
$\parallel$ & & $\parallel$ & & $\parallel$ \\
$\left( \begin{array}{cc|c} t+z & a & \overline{c} \\ \overline{a} & t-z & b \\ \hline c & \overline{b} & n \end{array} \right)$ & &
$\left( \begin{array}{c|cc} t+z & a & \overline{c} \\ \hline \overline{a} & t-z & b \\ c & \overline{b} & n \end{array} \right)$ & &
$\left( \begin{array}{c|c|c} t+z & a & \overline{c} \\ \hline \overline{a} & t-z & b \\ \hline c & \overline{b} & n \end{array} \right)$ \\
\end{tabular}
\caption{\noindent Three natural locations of~$\mathbf{X}$,~$\theta$, and~$\theta^\dagger$ in~$\chi$}
\label{table:three_types_of_vector_locations}
\end{center}
\end{table}
Each of the~$2 \times 2$ Lorentz transformations~$R \in SL(2,\mathbb{O})$ given in~\cite{manogue_schray} and used in Section~\ref{ch:Division_Algebras_Applications.Lorentz_Transformations.Lorentz_Transformations_for_Spacetimes_involving_Division_Algebras} are expressed using a nesting of~$2 \times 2$ matrices~$M_1, M_2, \cdots, M_n$. That is, if~\hbox{$\mathbf{X} \in M_2(\mathbb{O})$} is a vector, then~$R$ is given by
$$ R(\mathbf{X}) = M_n \left( \cdots \left( M_2 \left( M_1 \; \mathbf{X} M_1^\dagger \right) M_2^\dagger \right) \cdots \right) M_n^\dagger$$
As noted in Section~\ref{ch:Division_Algebras_Applications.Lorentz_Transformations.Lorentz_Transformations_for_Spacetimes_involving_Division_Algebras}, the spinor (dual-spinor) version of the Lorentz transformation is constructed from nesting matrices using left (right) multiplication by matrices:
$$ R(\theta) = M_n \left( \cdots \left( M_1 \; \theta \right) \cdots \right) $$
$$ R(\theta^\dagger) = \left( \cdots \left( \theta^\dagger M_1^\dagger \right) \cdots \right) M_n^\dagger$$
If we treat the upper left~$2 \times 2$ submatrix of~$\chi$ as the vector, as in Location 1 in Table~\ref{table:three_types_of_vector_locations}, and~$M$ is a~$2 \times 2$ matrix which we embed into the upper left~$2\times 2$ submatrix of a~$3 \times 3$ matrix~$\mathbb{M}$ which is block diagonal and~$\mathbb{M}_{3,3} = 1$, then we note that conjugating~$\chi$ by~$\mathbb{M}$ gives the following block structure
$$
\begin{array}{ccc}
\chi & \to & \mathbb{M} \chi \mathbb{M}^\dagger \\
\left( \begin{array}{c|c} \mathbf{X} & \theta \\ \hline \theta^\dagger & n \end{array} \right) & \to &
\left( \begin{array}{c|c} M & 0 \\ \hline 0 & 1 \end{array} \right)
\left( \begin{array}{c|c} \mathbf{X} & \theta \\ \hline \theta^\dagger & n \end{array} \right)
\left( \begin{array}{c|c} M & 0 \\ \hline 0 & 1 \end{array} \right)^\dagger \\
& & = \left( \begin{array}{c|c} M \mathbf{X} M^\dagger & M\theta \\ \hline (M \theta)^\dagger & n \end{array} \right)
\end{array}
$$
where we must worry about~$M \mathbf{X} M^\dagger$ being well-defined.
But each of the matrices~$M_a = M_1, \cdots, M_n$ involved in the expression for the Lorentz transformation~$R$ satisfy certain conditions, outlined in~\cite{manogue_schray}, which ensures~$M \mathbf{X} M^\dagger$ is well-defined.
In particular, constructing the~$3 \times 3$ matrix~$\mathbb{M}_a$ for each~$2 \times 2$ matrix~$M_a$ in this way and conjugating~$\chi$ successively with matrices~$\mathbb{M}_1, \cdots, \mathbb{M}_n$ gives
$$\chi \to \mathbb{M}_n \left( \cdots \left( \mathbb{M}_1 \chi \mathbb{M}_1^\dagger \right) \cdots \right) \mathbb{M}_n^\dagger$$
The resulting matrix has the block structure
$$
\begin{array}{c}
\left( \begin{array}{c|c} M_n \left( \cdots \left( M_1 \mathbf{X} M_1^\dagger\right) \cdots \right) M_n^\dagger & M_n \left( \cdots \left( M_1 \theta \right) \cdots \right) \\ \hline \left(M_n \left( \cdots \left( M_1 \theta \right) \cdots \right)\right)^\dagger & n \end{array} \right) \\
= \left( \begin{array}{c|c} R(\mathbf{X}) & R(\theta) \\ \hline R(\theta^\dagger) & n \end{array} \right)
\end{array}
$$
in which the vector, spinor, and dual spinor have all transformed according to their respective versions of the Lorentz transformation~$R$.
We now generalize this previous construction for all three natural locations of the vector~$\mathbf{X}$ in~$\chi$. Let~$M_1, \cdots, M_n$ be the matrices involved in the expression of the~$2 \times 2$ Lorentz transformation. Let~$T^{(a)}: M_2(\mathbb{O}) \to M_3(\mathbb{O})$ be an embedding of a~$2 \times 2$ matrix~$M$ into a~$3 \times 3$ matrix according to
$$T^{(1)}(M) = \left( \begin{array}{c|c} M & 0 \\ \hline 0 & 1 \end{array} \right) \hspace{2cm} T^{(2)}(M) = \left( \begin{array}{c|c} 1 & 0 \\ \hline 0 & M \end{array} \right) $$
$$T^{(3)}(M) = \left( \begin{array}{c|c|c} M_{2,2} & 0 & M_{2,1} \\ \hline 0 & 1 & 0 \\ \hline M_{1,2} & 0 & M_{1,1} \end{array} \right) $$
for~$a = 1,2,3$. We call this the {\it discrete}\footnote{This type map is expanded to a continuous map in Section~\ref{ch:E6_basic_structure.Type_Transformation}} {\it type map}. Given a~$2 \times 2$ matrix~$M$, we abbreviate~$T^{(a)}(M)$ as~$M^{(a)}$ and refer to~$M^{(1)}, M^{(2)},$ and~$M^{(3)}$ as type~$1, 2$, and~$3$ versions of~$M$. We note that
$$M^{(2)} = \mathcal{T} M^{(1)} \mathcal{T}^\dagger \hspace{1.5cm} M^{(3)} = \mathcal{T} M^{(2)} \mathcal{T}^\dagger \hspace{1.5cm} M^{(1)} = \mathcal{T} M^{(3)} \mathcal{T}^\dagger$$
is a cyclic permutation of the three types of~$M$, where
$$\mathcal{T} = \left( \begin{array}{ccc} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0\end{array} \right)$$
If~$R \in SL(2,\mathbb{O})$ is a vector Lorentz transformation which is constructed from the nesting of conjugations of~$2 \times 2$ matrices~$M_1, \cdots, M_n$, then it can be shown that
$$ \chi \to M_n^{(a)} \left( \cdots \left( M_1^{(a)} \chi M_1^{(a)^\dagger} \right) \cdots \right)M_n^{(a)\dagger}$$
respects the block structure of~$\chi$, in the sense that identifying a vector~$\mathbf{X}$, spinor~$\theta$, and dual spinor~$\theta^\dagger$ in~$\chi$ according to the Location~$a$ given in Table~\ref{table:three_types_of_vector_locations} results in the transformations
$$ \mathbf{X} \to R(\mathbf{X}) \hspace{1cm} \theta \to R(\theta) \hspace{1cm} \theta^\dagger \to R(\theta^\dagger)$$
for the appropriate vector, spinor, and dual-spinor versions~$R$ of the Lorentz transformation. We call~$R^{(a)}$ the transformation~$R$ of {\it Type~$a$}.
{\bf Lemma: } For any Lorentz transformation~$R \in SL(2,\mathbb{O})$, the corresponding~$3 \times 3$ transformations~$R^{(a)}: M_3(\mathbb{O}) \to M_3(\mathbb{O})$ preserves the determinant of elements of~$M_3(\mathbb{O})$ for~$a = 1,2,3$.
{\bf Proof: } We first prove this for the case~$a = 1$:
By definition, the finite Lorentz transformation~$R \in SL(2,\mathbb{O})$ preserves the determinant of the~$2 \times 2$ vector~$\mathbf{X} \in M_2(\mathbb{O})$. That is,~$\det{ \left( \mathbf{X} \right) } = \det{\left( R( \mathbf{X}) \right)}$.
Since~$R$ is a compatible transformation, then~\hbox{$R(\theta \theta^\dagger) = R(\theta) \; R(\theta^\dagger)$} for the appropriate vector, spinor, and dual-spinor versions of the Lorentz transformation, giving
$$\det{ \left(\theta \; \theta^\dagger \right)} = \det{ \left( R(\theta) \; R(\theta^\dagger) \right)}$$
As shown in Section~\ref{ch:Division_Algebras_Applications.Jordan_Algebras}, the determinant of~$\chi \in M_3(\mathbb{O})$ is given by
$$ \det{ \left( \chi \right) } = \det{\left( \begin{array}{cc}\mathbf{X} & \theta \\ \theta^\dagger & n \end{array} \right)} = \left( \det{ \mathbf{X} } \right) n + 2\mathbf{X} \cdot \theta \theta^\dagger$$
where~$A \cdot B = \tr{A \circ B} - \tr{A} \tr{B}$ is the Lorentzian inner product of~$A$ and~$B$.
But
Lorentz transformations preserve the Lorentzian inner product of~$\mathbf{X}$ and~$\theta \; \theta^\dagger$, so that
$$\begin{array}{ccc}
R(\mathbf{X}) \cdot R(\theta \; \theta^\dagger) & = & \mathbf{X} \cdot (\theta \; \theta^\dagger)
\end{array}
$$
Hence, we see that
$$ \begin{array}{ccl}
\det{ \left(R^{(1)}(\chi)\right) } & = & \det{\left( \begin{array}{c|c} R(\mathbf{X}) & R(\theta) \\ \hline R(\theta^\dagger) & n \end{array} \right)}\\
& = &\det{\left(R(\mathbf{X}) \right)} n + 2 R(\mathbf{X}) \cdot \left(R(\theta) R(\theta^\dagger) \right)\\
& = &\det{\left(R(\mathbf{X}) \right)} n + 2 R(\mathbf{X}) \cdot R(\theta \theta^\dagger)\\
& = &\det{ \left( \mathbf{X} \right) } n + 2\mathbf{X} \cdot (\theta \theta^\dagger) \\
& = & \det{(\chi)}
\end{array} $$
where the first two equalities are true by definition, the third equality is a result of compatibility, and the fourth equality results from the use of Lorentz transformations. Hence, the transformation~$R^{(1)}$ preserves the determinant of~$\chi \in M_3(\mathbb{O})$ for each~$R \in SL(2,\mathbb{O})$.
For the case~$a = 2$ and~$a = 3$, we note that the determinant of~$\chi \in M_3(\mathbb{O})$ may be written
$$ \det{\chi} = (t+z)(t-z)n - ((t+z)|b|^2 + (t-z)|z|^2 + n|a|^2) + 2Re(\overline{a} \overline{b} \overline{c})$$
as was shown in Section~\ref{ch:Division_Algebras_Applications.Jordan_Algebras} with~$t+z = p$ and~$t-z = m$. We see that it is cyclic in~$a,b,c$ and~$t+z,t-z,n$. That is,~$\det{( \mathcal{T} \chi \mathcal{T}^\dagger) } = \det{ \chi}$. Hence,
$$\det{ \chi} = \det{(R^{(1)}(\chi))} = \det{(R^{(2)}(\chi))} = \det{(R^{(3)}(\chi))}$$
showing that each of the three types of~$3 \times 3$ transformation constructed from \hbox{$R \in SL(2,\mathbb{O})$} are in~$SL(3,\mathbb{O})$.
\hspace{5.40in}$\square$
We conclude with a result showing that the~$3 \times 3$ transformations we just constructed will form a one-parameter subgroup of~$SL(3,\mathbb{O})$ provided that the original~$2 \times 2$ transformation is both compatible and forms a one-parameter subgroup of~$SL(2,\mathbb{O})$.
{\bf Lemma: } If the finite Lorentz transformation~$R: \mathbb{R} \times M_2(\mathbb{O}) \to M_2(\mathbb{O})$ is both compatible and a one-parameter subgroup of~$SL(2,\mathbb{O})$, then~$R^{(a)}: \mathbb{R}\times M_3(\mathbb{O}) \to M_3(\mathbb{O})$ is a one-parameter subgroup of~$SL(3,\mathbb{O})$ for~$a = 1,2,3$.
{\bf Proof: } Our construction of~$R^{(a)}$ fixes a block structure in~$\chi$ in which the vector, spinor, and dual-spinor version of the transformation~$R$ act on the appropriate blocks of~$\chi$. By hypothesis, the vector transformation~$R$ is a one-parameter subgroup. It remains to show that the spinor and dual-spinor transformations are one-parameter subgroups.
But if the vector version of~$R$ is compatible and a one-parameter subgroup, then for any~\hbox{$\alpha, \beta \in \mathbb{R}$}, we see that
$$\begin{array}{ccl}
\left( R(\alpha)\left(R(\beta)(\theta)\right) \right) \; \left (R(\alpha)\left(R(\beta)(\theta^\dagger)\right) \right) & = & R(\alpha)\left( R(\beta)(\theta) \; R(\beta)(\theta^\dagger) \right) \\
& = & R(\alpha)\left( R(\beta)(\theta \; \theta^\dagger) \right) \\
& = & R(\alpha + \beta)(\theta \; \theta^\dagger) \\
& = & \left( R(\alpha + \beta)(\theta) \right) \; \left( R(\alpha+\beta)(\theta^\dagger)\right) \\
\end{array}
$$
where~$\theta$ is a spinor and~$\theta^\dagger$ is a dual spinor, and we have used the appropriate vector, spinor, or dual spinor version of the vector Lorentz transformation~$R$. Hence, the products of spinor and dual-spinor transformations are one-parameter subgroups. But this requires the spinor and dual-spinor transformations themselves to be one-parameter subgroups.
Since each of the vector, spinor, and dual-spinor versions of the transformation~$R$ are one-parameter subgroups, then our construction produces a new transformation~$R^{(a)}$ which is also a one-parameter subgroup.
\hspace{5.40in}$\square$
This lemma shows that since each of the basis transformations~$R$ constructed by Manogue and Schray~\cite{manogue_schray} which were discussed in~\ref{ch:Division_Algebras_Applications.Lorentz_Transformations.Lorentz_Transformations_for_Spacetimes_involving_Division_Algebras} are compatible one-parameter subgroups of~$SL(2,\mathbb{O})$, then~$R^{(a)}$ is a one-parameter subgroup of~$SL(3,\mathbb{O})$ for~$a = 1,2,3$. This result will be useful in Section~\ref{ch:E6_basic_structure.ConstructingE6algebra}.
\subsection{A Basis for the Category 3 Transformations}
\label{ch:E6_basic_structure.Lorentz_Transformations.BetterBasis}
The type maps provide three different copies of~$SO(9,1)$ transformations which act on~$\chi$. Hence, there are at least three~$SO(9,1)$ groups, one of each type, as well as subgroups~$SO(n,\mathbb{R})$ and~$SO(n,1,\mathbb{R})$ for~$n \le 9$, in the group~$E_6 = SL(3,\mathbb{O})$. This gives~$3 \times 45 = 135$ generators in~$E_6$, which only has dimension~$78$. In the next section, we will use linear dependencies among the corresponding Lie algebra elements for these transformations to reduce the~$135$ generators down to a basis of~$78$ independent generators. In the remainder of this section, we introduce an alternate basis for the Category~$3$ transformations which were given by Manogue and Schray \cite{manogue_schray} and reviewed in Section \ref{ch:Division_Algebras_Applications.Lorentz_Transformations.Lorentz_Transformations_for_Spacetimes_involving_Division_Algebras}. This new basis will prove useful in determining the dependencies among the~$135$ algebra generators.
In the~$2 \times 2$ setting, each Category~$3$ transformation~$R_{p,q}$ rotates a single plane spanned by the imaginary octonions~$p$ and~$q \perp p$.
Rather than simply rotating one plane at a time, we choose a new basis of transformations, where each transformation simultaneously rotates two or three planes. For each basis octonion, say~$q = i$, there are three pairs of basis octonions~$\lbrace j, k \rbrace, \lbrace k\l, j\l \rbrace, \lbrace \l, i\l \rbrace$ which generate a quaternionic subalgebra containing~$q = i$. Note that we have chosen the ordering of the pairs in such a way that the product of the pair is~$q = i$, e.g. ~$(j) (k) = i$.
Hence, we have used the three triples (each pair along with~$q = i$) to create a right-handed, three-dimensional coordinate frame. The planes spanned by~$\lbrace j, k \rbrace$,~$\lbrace k\l, j\l \rbrace$, and~$\lbrace \l, i\l \rbrace$ are orthogonal to each other, and the Category~$3$ transformations~$R_{j,k}(\alpha)$,~$R_{k\l,j\l}(\alpha)$, and~$R_{\l,i\l}(\alpha)$ rotate the appropriate plane counter-clockwise about the~$q=i$ axis through the angle~$\alpha$. From these rotations, we create the new class of transformations~$A_q$,~$G_q$, and~$S_q$ where, for~$q = i$, we have \footnote{Because the rotations are in orthogonal planes, our definition does not depend on the order of the rotations, i.e.~$A_i(\alpha) = R_{j,k}(\alpha) \circ R_{k\l, j\l}(-\alpha) = R_{k\l, j\l}(-\alpha) \circ R_{j,k}(\alpha)$}
$$ A_i(\alpha) = R_{j,k}(\alpha) \circ R_{k\l, j\l}(-\alpha) \hspace{1.5cm} G_i(\alpha) = R_{j,k}(\alpha) \circ R_{k\l, j\l}(\alpha) \circ R_{\l,i\l}(-2\alpha)$$
$$ S_i(\alpha) = R_{j,k}(\alpha) \circ R_{k\l, j\l}(\alpha) \circ R_{\l,i\l}(\alpha)$$
where~$\circ$ is used to denote composition of the group transformations.
The~$A_i$ transformation rotates the~$(j,k)$ plane counter-clockwise and the~$(k\l, j\l)$ plane clockwise about the~$q = i$ axis. The~$G_i$ transformation rotates two planes counter-clockwise about the~$q=i$ axis through an angle of~$\alpha$ and one plane clockwise about the~$q=i$ axis through an angle of~$2\alpha$.
The~$S_i$ transformation rotates all three planes counter-clockwise about the~$q = i$ axis through the angle~$\alpha$.\footnote{ We choose the labels~$A_q$ since the seven~$A_q$ tranformations along with~$G_\l$ are isomorphic to an~$SU(3,\mathbb{C})$ normally represented by Gell-Mann matrices, which are typically labeled using the greek symbol~$\Lambda$, and the Roman symbol~$A$ closely resembles~$\Lambda$. The~$G_q$ label is chosen for the relationship between these transformations and the group~$G_2$. The~$S_q$ label is chosen to remind us that these are the symmetric transformations.}
Combined, the seven transformations of each class~$A_q$,~$G_q$, and~$S_q$ form a new basis for~$SO(7,\mathbb{R})$. Finally, we expand these~$2 \times 2$ transformations to the~$3 \times 3$ case by using the type map~$T^{(a)}, a = 1,2,3$ which is applied to each matrix used in the transformations~$A_q, G_q$, and~$S_q$ to produce the transformations~$A^{(a)}_q$,~$G^{(a)}_q$, and~$S^{(a)}_q$ for~$a = 1,2,3$.
{\bf Lemma: } The~$3 \times 3$ transformations~$A^{(a)}_q$,~$G^{(a)}_q$, and~$S^{(a)}_q$ built from the~$2 \times 2$ one-parameter transformations~$R_{p,q}$ are also one parameter transformations.
{\bf Proof: } The~$2 \times 2$ transformations~$R_{p,q}$ are one-parameter transformations, and rotate one plane spanned by~$p$ and~$q$. Our new transformations rotate two or three planes at once, but because these planes are pair-wise orthogonal, the effect of one transformation in the plane spanned by~$p$ and~$q$ does not affect the transformation in the plane spanned by~$u$ and~$v$. Hence, the rotations which comprise~$A_q, G_q,$ and~$S_q$ commute with each other. But then we can rearrange all of the rotations in~$A_q(\alpha) \circ A_q(\beta)$ to form~$A_q(\alpha+\beta)$, and similarly for the transformations~$G_q$ and~$S_q$. Hence, the~$2 \times 2$ transformations~$A_q$,~$G_q$, and~$S_q$ are one-parameter transformations. That their~$3 \times 3$ generalizations~$A^{(a)}_q, G^{(a)}_q$, and~$S^{(a)}_q$ are also one-parameter transformations follows from the last lemma of Section~\ref{ch:E6_basic_structure.Lorentz_Transformations}.
\hspace{5.40in}$\square$
We list here our conventions for our new basis for the~$SO(7)$ transformations. Each of the~$A_q$,~$G_q$, and~$S_q$ transformations are labeled by a unit octonion~$q \in \lbrace i,j,k,k\l,j\l, i\l, l \rbrace$ about which the planes rotate. For each~$q$, we have three pairs of unit octonions which multiply to~$q$.
Our conventions for the pairs are listed in Table~\ref{table:AGS_conventions}. We note that~$\l$ only appears in the third pair, if at all.
\begin{table}[htbp]
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
$q$ & First pair & Second pair & Third pair \\
\hline
\hline
$i$ & $(j , k )$ & $(kl , jl)$ & $(l , il)$ \\
$j$ & $(k , i )$ & $(il , kl)$ & $(l , jl)$ \\
$k$ & $(i , j )$ & $(jl , il)$ & $(l , kl)$ \\
$kl$ & $(jl , i)$ & $(j , il) $ & $(k , l) $ \\
$jl$ & $(i , kl)$ & $(il , k) $ & $(j , l) $ \\
$il$ & $(kl , j)$ & $(k , jl) $ & $(i , l) $ \\
$l$ & $(il , i)$ & $(jl , j) $ & $(kl , k)$ \\
\hline
\end{tabular}
\caption{\noindent Quaternionic subalgebras chosen for~$A_q$,~$G_q$, and~$S_q$}
\label{table:AGS_conventions}
\end{center}
\end{table}
In the~$2 \times 2$ case, we may reproduce the original Category~$3$ transformations~$R_{q,r}(\alpha)$ simply by combining our new transformations with appropriate angles. For instance,
$$R_{j,k}(6\alpha) = A_i(3\alpha) \circ G_i(\alpha) \circ S_i(2\alpha)$$
We use the anlges~$3 \alpha$,~$\alpha$, and~$2\alpha$ so that the clockwise and counter-clockwise rotations in the various planes cancel each other out, leaving only the rotation by~$6\alpha$ in the~$j,k$ plane. These dependencies also extend to the~$3 \times 3$ transformations.
\section{The Lie Algebra~$sl(3,\mathbb{O})$ of~$E_6 = SL(3,\mathbb{O})$}
\label{ch:E6_basic_structure.E6_Lie_Algebra}
We have expanded the~$45$ finite Lorentz transformations from the~$2 \times 2$ case to the~$3 \times 3$ case, giving us~$3 \times 45 = 135$ transformations. Each of these transformations preserves the determinant of the Jordan matrix~$\chi \in M_3(\mathbb{O})$. Hence, they are a subset of~$SL(3,\mathbb{O}) = E_6$, which is a~$78$-dimensional exceptional Lie group. The Lie algebra of this group is~$sl(3,\mathbb{O})$, which is the topic of this section.
In Section~\ref{ch:E6_basic_structure.ConstructingE6algebra}, we present the techniques which allow us to conclude that our transformations generate the entire group. Normally, we would do this by associating each transformation with its tangent vector in the Lie algebra~$sl(3,\mathbb{O})$. Then, using linear dependencies in the Lie algebra, we could reduce our~$135$ generators to~$78$ independent generators to show that we have the entire group~$SL(3,\mathbb{O})$. However, the nesting of matrices involved in our representation of the transformations makes this difficult. Instead, we use the local action of our representation of~$SL(3,\mathbb{O})$ on~$M_3(\mathbb{O})$ to give a homomorphism of~$sl(3,\mathbb{O})$ into the Lie algebra of all vector fields on~$M_3(\mathbb{O})$. The tangent vectors to the group orbits at a point~$\chi$ in~$M_3(\mathbb{O})$ constitute a homomorphic copy of the Lie algebra~$sl(3,\mathbb{O})$. We also use the tangent vectors to the group orbits to construct the complete commutator multiplication in the Lie algebra~$sl(3,\mathbb{O})$. In Section~\ref{ch:E6_basic_structure.ConstructingE6algebra.LinearDependencies}, we use the linear dependencies among these tangent vectors to find linear dependencies among the~$135$ group generators constructed in the previous section. These results are combined with our construction of~$SL(3,\mathbb{O})$ in Section~\ref{ch:E6_basic_structure.Lorentz_Transformations.BetterBasis} to give an explicit basis for subgroups~$SO(9,1,\mathbb{R})$ and~$SO(n,1,\mathbb{R})$ for~$n \le 9$, as well as for the exceptional groups~$G_2$ and~$F_4 = SU(3,\mathbb{O})$.
In what follows, we find it cumbersome to always refer to {\it tangent vectors to the group orbits} and {\it the homomorphic copy of the Lie algebra~$sl(3,\mathbb{O})$}. Hence, we often refer to the group orbit of a one-parameter subgroup of~$SL(3,\mathbb{O})$ as a one-parameter path in~$M_3(\mathbb{O})$ and frequently state that the tangent vector to this curve is the element of the Lie algebra~$sl(3,\mathbb{O})$. We hope this abuse of language makes the following methods more intuitive.
\subsection{Constructing the algebra for~$SL(3,\mathbb{O})$}
\label{ch:E6_basic_structure.ConstructingE6algebra}
We begin by associating each transformation in the group with a vector in the Lie algebra. Each of the~$135$ transformations are one-parameter curves in the group. Given a one-parameter curve~$R(\alpha)$ in a classical Lie group, the traditional method for associating it with the Lie algebra generator~$\dot R$ is to find its tangent vector~$\dot R = \frac{\partial R(\alpha)}{\partial \alpha } \mid_{_{\alpha = 0}}$ at the identity in the group. However, our Category~$3$ transformations use nested matrices
$$ \chi \to M_4^{(a)} \left( M_3^{(a)} \left( M_2^{(a)} \left( M_1^{(a)} \chi M_1^{(a)\dagger} \right) M_2^{(a)\dagger} \right) M_3^{(a)\dagger} \right) M_4^{(a)\dagger}$$
and it is not possible to apply this technique to the one-parameter transformations~$A^{(a)}_q$, $G^{(a)}_q$, and~$S^{(a)}_q$, with~$a = 1,2,3$. Instead, we let our one-parameter transformations~$R$ act on
the exceptional Jordan algebra~$M_3(\mathbb{O})$, as it is a~$27$-dimensional representation of~$E_6$~\cite{corrigan}.
When the transformation~$R$ acts on~$\chi \in M_3(\mathbb{O})$, it produces a curve~$R(\alpha)(\chi)$ in~$M_3(\mathbb{O})$. This curve~$R(\alpha)(\chi)$ is a one-parameter curve in~$M_3(\mathbb{O})$, since~$R$ is a one-parameter transformation that is the identity,~$R(0)(\chi) = \chi$, when~$\alpha = 0$. We use the tangent vector at the identity to this curve in~$M_3(\mathbb{O})$ to represent our tangent vector to the original one-parameter transformation~$R$. That is, we have the association indicated in Figure~\ref{figure:lie_group_to_lie_algebra_commutators} between the group transformations and the tangent vectors.
This procedure produces~$135$ tangent vectors, one for each of the curves~$R(\alpha)(\chi)$.
We also use group orbits to construct the commutator of two tangent vectors.
In the traditional approach to the classical matrix groups, the commutator of the tangent vectors~$\dot R_1$ and~$\dot R_2$ is defined as~\hbox{$\left[ \dot R_1, \dot R_2 \right] = \dot R_1 \dot R_2 - \dot R_2 \dot R_1$}. However, we are not using tangent vectors to the transformations themselves but tangent vectors to curves in~$M_3(\mathbb{O})$. To find the commutator of the tangent vectors to the curves~$R_1(\alpha)(\chi)$ and~$R_2(\alpha)(\chi)$, we create a new curve in~$M_3(\mathbb{O})$ defined by
$$[ R_2, R_1](\alpha)(\chi) = R_2(-\frac{\alpha}{2}) \circ R_1(-\frac{\alpha}{2}) \circ R_2(\frac{\alpha}{2}) \circ R_1(\frac{\alpha}{2})(\chi)$$
where~$\circ$ denotes composition.
This new path is not a one-parameter curve, and its first derivative is identically zero at~$\alpha = 0$, but~$\frac{\partial^2}{\partial \alpha^2}[R_2, R_1](\alpha)(\chi) |_{_{\alpha = 0}}$ is tangent to the curve~$[R_2, R_1](\alpha)(\chi)$ at~$\alpha = 0$. Therefore, we define the commutator of~$\dot R_1$ and~$\dot R_2$ to be
$$ \left[ \dot R_2, \dot R_1 \right] = \frac{\partial^2}{\partial \alpha^2}[R_2, R_1](\alpha)(\chi) |_{_{\alpha = 0}}$$
This formula agrees with the usual definition \cite{gilmore}.
We have a~$1 - 1$ correspondence between the group transformation~$R$, the orbit~$R(\alpha)(\chi)$ in~$M_3(\mathbb{O})$, and our tangent vector to the group orbit~$\dot R(\alpha)(\chi)$, allowing us to identify the group transformation given the tangent vector~$\dot R(\alpha)(\chi)$ to the group orbit~$R(\alpha)(\chi)$. In particular, we can use this correspondence to identify the algebra element corresponding to~$\left[ \dot R_2, \dot R_1 \right]$ for any~$\dot R_2$ and~$\dot R_1$. Our construction of the commutator, and this correspondence, is summarized in Figure~\ref{figure:lie_group_to_lie_algebra_commutators}.
\begin{figure}[htbp]
\begin{minipage}{6in}
\begin{center}
\xymatrix@M=2pt{
*++\txt{Lie Group~$G$} & *++\txt{$R(\alpha)$}\ar@/_3.5pc/[ddd] \ar[d] & \hspace{1.5cm} & *++\txt{$ R_2(\alpha),R_1(\alpha) $} \ar[d] \ar@/^5.0pc/[ddd]\\
*++\txt{$\textrm{Path in } M_3(\mathbb{O})$} & *++\txt{$R(\alpha)(\chi)$} \ar[d]^{\frac{\partial }{\partial \alpha} |_{_{\alpha = 0}}} & & *++\txt{$\left[R_2,R_1\right](\alpha)(\chi)$} \ar[d]^{\frac{\partial^2 }{\partial \alpha^2}|_{_{\alpha = 0}}} \\
*++\txt{$\textrm{Tangent vector}$\\$\textrm{to the path}$} & *++\txt{$\frac{\partial R(\alpha)(\chi)}{\partial \alpha}|_{_{\alpha = 0}}$} \ar[d] & & *++\txt{$\frac{\partial^2 \left[ R_2, R_1 \right](\alpha)(\chi)}{\partial \alpha^2}|_{_{\alpha = 0}}$} \ar[d] \\
*++\txt{Lie Algebra~$g$} & *++\txt{$\dot R$} & *++\txt{$\dot R_2, \dot R_1 $} \ar@/^2pc/[uuur]^{1-1} \ar@{.>}[r]^{\left[ \hspace{.15cm}, \hspace{.15cm} \right]} & *++\txt{$\left[ \dot R_2, \dot R_1 \right]$}\\
}
\caption{\noindent Calculating tangent vectors and their commutators}
\label{figure:lie_group_to_lie_algebra_commutators}
\end{center}
\end{minipage}
\end{figure}
We note that since we are using the local action of~$SL(3,\mathbb{O})$ on~$M_3(\mathbb{O})$ to give a homomorphic image of~$sl(3,\mathbb{O})$, our construction does not lead to a readily available exponential map giving the group element corresponding to~$\left[\dot R_1, \dot R_2\right]$. In particular, we are not always able to find the one-parameter curve whose tangent vector is~$\left[ \dot R_1, \dot R_2 \right]$.
To illustrate our techniques, we compute the tangent vector for the group transformations~$A^{(1)}_\l$. By our construction given in Section~\ref{ch:E6_basic_structure.Lorentz_Transformations.BetterBasis}, the finite Lorentz transformation~$A^{(1)}_\l(\alpha)$ conjugates the Jordan matrix~$\chi \in M_3(\mathbb{O})$ with the Category~$3$ matrices used in~$R^{(1)}_{i\l,i}(\alpha)$ and~$R^{(1)}_{j\l,j}(-\alpha)$. This gives
\[ \begin{array}{ccc}
A^{(1)}_{\l}(\alpha)(\chi) & = & R^{(1)}_{j\l,j}(-\alpha) \circ R^{(1)}_{i\l,i}(\alpha)(\chi) \\
\end{array}
\]
where
\[
\begin{array}{cl}
R^{(1)}_{i\l,i}(\alpha)(\chi) = & \left( \begin{array}{ccc} e^{\l \alpha/2 }i\l &0&0\\0&e^{\l \alpha/2 }i\l&0\\0&0&1\\ \end{array} \right) \hspace{6cm}\\
& \hspace{1cm} \left[
\left( \begin{array}{ccc} -i\l&0&0\\0&-i\l&0\\0&0&1 \end{array} \right)
\chi
\left( \begin{array}{ccc} -i\l&0&0\\0&-i\l&0\\0&0&1 \end{array} \right)^\dagger
\right] \hspace{2cm} \\
& \hspace{5cm}
\left( \begin{array}{ccc} e^{\l\alpha/2}i\l&0&0\\0&e^{\l\alpha/2}i\l&0\\0&0&1\\ \end{array} \right)^\dagger
\end{array}
\]
and
\[
\begin{array}{cl}
R^{(1)}_{j\l,j}(-\alpha)(\chi) = & \left( \begin{array}{ccc} e^{-\l \alpha/2 }j\l &0&0\\0&e^{-\l \alpha/2 }j\l&0\\0&0&1\\ \end{array} \right) \hspace{6cm} \\
& \hspace{1cm} \left[
\left( \begin{array}{ccc} -j\l&0&0\\0&-j\l&0\\0&0&1 \end{array} \right)
\chi
\left( \begin{array}{ccc} -j\l&0&0\\0&-j\l&0\\0&0&1 \end{array} \right)^\dagger
\right] \hspace{2cm}\\
& \hspace{4cm}
\left( \begin{array}{ccc} e^{-\l \alpha/2}j\l&0&0\\0&e^{-\l\alpha/2}j\l&0\\0&0&1\\ \end{array} \right)^\dagger\\
\end{array}
\]
\noindent It is useful to use a computer algebra software program to compute the result of conjugating~$\chi$ with the above matrices. Then, the tangent vector to this group orbit is found to be
$$\dot A^{(1)}_\l = \frac{\partial A^{(1)}_\l(\alpha)(\chi)}{\partial \alpha} =
\left[ \begin {array}{ccc} 0& \overline{a_1}& c_1\\
\noalign{\medskip}a_1&0&\overline{b_1}\\
\noalign{\medskip}\overline{c_1}&b_1&0
\end {array} \right]
$$
where
\begin{equation}
\label{eq:Al_tangent_coeff}
\begin{array}{c}
a_1 = -{\it a_{i\l}}\,{\it i}+{\it a_{j\l}}\,{\it j}-{\it a_j}\,{\it j\l}+{\it a_i}\,{\it i\l}\\
b_1 = -{\it b_{i\l}}\,{\it i}+{\it b_{j\l}}\,{\it j}-{\it b_j}\,{\it j\l}+{\it b_i}\,{\it i\l}\\
c_1 = -{\it c_{i\l}}\,{\it i}+{\it c_{j\l}}\,{\it j}-{\it c_j}\,{\it j\l}+{\it c_i}\,{\it i\l}
\end{array}
\end{equation}
The calculations involved in finding the commutator of two transformations are straight-forward, but time-consuming. We used a {\tt MAPLE} program to calculate all~$3003$ commutators of pairs of the~$135$ different group generators as well as identify its tangent vector as a linear combination of the tangent vectors to our preferred~$78$ transformations. Running this calculation on~$12$ computers running at~$1.5Ghz$, it finished in just under~6 hours.
To simplify notation, we will use the traditional notation~$\dot R$ in place of~$\frac{\partial R(\alpha)(\chi)}{\partial \alpha}|_{_{\alpha = 0}}$ for the tangent vector to~$R(\alpha)(\chi)$ at~$\alpha = 0$. Using the isomorphism above, we can regard these tangent vectors as our Lie algebra.
\subsection{Linear Dependencies}
\label{ch:E6_basic_structure.ConstructingE6algebra.LinearDependencies}
We shall now give the dependencies among the group transformations by using linear dependencies among the Lie algebra elements. In doing so, we will indicate which transformations can be eliminated leaving our preferred basis for the group~$SL(3,\mathbb{O})$ and the algebra~$sl(3,\mathbb{O})$. Since we are using a homomorphic image of the Lie algebra~$sl(3,\mathbb{O})$, we check that the indicated dependencies actually do provide dependencies among the group transformations.
We begin with the Category~$3$ transformations. Among the~$21$ transformations~$A_q$,~$G_q$, and~$S_q$ of each type, direct examination of the tangent vectors shows that
$$\dot A^{(1)}_q = \dot A^{(2)}_q = \dot A^{(3)}_q \hspace{2cm} \dot G^{(1)}_q = \dot G^{(2)}_q = \dot G^{(3)}_q$$
for each basis octonion~$q$. That is, the~$A_q$ and~$G_q$ transformations are type independent, allowing us to drop the type designation and simply write~$\dot A_q$ and~$\dot G_q$. These fourteen transformations generate~$G_2 = Aut(\mathbb{O})$, which is the smallest of the exceptional Lie groups. The type independence of these transformations is an effect of {\it triality}, which will be further discussed in Section 4.3.
When added to the fourteen~$G_2$ transformations, the seven~$S^{(a)}_q$ transformations of each type produce a basis for an~$SO(7)$ of type~$a$, with~$a = 1, 2, 3$. However, the~$S^{(a)}_q$ transformations are not independent, as their tangent vectors satisfy
$$\dot S^{(1)}_q + \dot S^{(2)}_q + \dot S^{(3)}_q = 0$$
Hence, the union of any two of the~$SO(7)$ groups contains the third. In particular, we may use the transformations~$S^{(a)}_q$ of type~$1$ and type~$2$ to generate the type~$3$ transformations~$S^{(3)}_q$. These linear dependences have reduced our~$63$ Category~$3$ transformations by~$35$, trimming our original~$135$ transformations down to~$100$.
Among the Category~$2$ transformations, we have the relation
$$ \dot R^{(1)}_{xq} + \dot R^{(2)}_{xq} + \dot R^{(3)}_{xq} = 0$$
This identity allows us to eliminate another seven transformations, and we choose to keep the type~$1$ and type~$2$ transformations. The seven group transformations~$R^{(a)}_{xq}$ of type~$a$ are added to the group~$SO(7)$ of type~$a$ to obtain~$SO(8)$. But the dependency among the types for the~$R^{(a)}_{xq}$ transformations indicates the group transformation~$R^{(3)}_{xq}$ can be obtained from the group transformations~$R^{(1)}_{xq}$ and~$R^{(2)}_{xq}$. That is, the union of the groups~$SO(8)$ of type~$1$ and type~$2$ contains the group~$SO(8)$ of type~$3$.
A final pair of relations exist among the Category~$2$ and Category~$3$ transformations, however, which will eliminate an additional~$14$ transformations and condense all of the different~$SO(8)$ groups into one! Having eliminated the~$S_q$ and~$R_{xq}$ transformations of type~$3$, these final relations allow us to write the type~$2$ transformations as linear combinations of the type~$1$ transformations.
The tangent vectors satisfy
$$ \dot R^{(2)}_{xq} = -\frac{1}{2}\dot R^{(1)}_{xq} - \frac{1}{2}\dot S^{(1)}_q \hspace{2cm} \dot S^{(2)}_q = \frac{3}{2} \dot R^{(1)}_{xq} - \frac{1}{2}\dot S^{(1)}_q $$
Hence, we may express any of the~$S^{(2)}_q$ and~$R^{(2)}_{xq}$ transformations in terms of~$S^{(1)}_q$ and~$R^{(1)}_{xq}$ transformations which are in the~$SO(8)$ of type~$1$. Hence, there is only one~$SO(8)$! This group contains three different copies of~$SO(7)$ built upon the unique group~$G_2$. Again, this is a result of triality, which will be further discussed in Section 4.2.
Having reduced the~$135$ transformations to~$100$ and then by another~$21$ to~$79$, we expect only one additional linear dependency among the transformations. Not surprisingly, we are left with~$52$ Category~$2$ and~$3$ transformations. These rotations preserve the trace of~$\chi \in M_3(\mathbb{O})$ and form the Lie group~$F_4$. Among the boost transformations in Category~$1$, we see that
$$ \dot B^{(1)}_{tz} + \dot B^{(2)}_{tz} + \dot B^{(3)}_{tz} = 0$$
From the three boosts~$B^{(1)}_{tz}$,~$B^{(2)}_{tz}$, and~$B^{(3)}_{tz}$, we choose to keep the boosts~$B^{(1)}_{tz}$ and~$B^{(2)}_{tz}$ in our preferred basis for~$SL(3,\mathbb{O})$. We will see in the algebra that there is some evidence that we should choose the boost~$\dot B^{(2)}_{tz} - \dot B^{(3)}_{tz}$ in place of~$\dot B^{(2)}_{tz}$, but this choice makes little difference given
the identity among~$\dot B^{(1)}_{tz}, \dot B^{(2)}_{tz}$, and~$\dot B^{(3)}_{tz}$.
It can be shown that the final~$78$ tangent vectors corresponding to the remaining boosts and rotations are linearly independent.
By looking at the tangent vectors to the group actions for the rotations, direct computation shows there are~$24$ hermitian and~$24$ anti-hermitian trace-free tangent vectors in addition to the fourteen~$G_2$ tangent vectors. The tangent vectors for the~$26$ boosts are non-trace-free and hermitian. It turns out that the~$64$ non-$G_2$ tangent vectors correspond to the~$64$ independent trace-free octonionic~$3 \times 3$ matrices. Taken with the~$G_2$ transformations, this gives~$78$ independent transformations. Hence, we do have the group~$SL(3,\mathbb{O}) = E_6$. The commutation table for~$sl(3,\mathbb{O})$ is located online at~\cite{commutation_table_online}, and using this table, we can further identify
$$\dot B^{(1)}_{tz}, \dot B^{(2)}_{tz}, \dot R^{1}_{x\l}, \dot A_\l, \dot G_\l, \textrm{ and }\dot S^{(1)}_{\l}$$
as the six elements corresponding to the Casimir operators in the complexified Lie algebra~$sl(3,\mathbb{O})$. Hence, we will refer to these six elements as Casimir operators, but note that this is not a choice of orthogonal Casimir operators.
\subsection{Triplets of Subgroup Chains}
\label{ch:E6_basic_structure.Lorentz_Transformations.BetterBasisSubgroups}
With our particular choice of basis for~$E_6$, we can identify two separate~$SO(n)$ subgroup structures within~$E_6$.
Figure~\ref{figure:type_1_2_3_so} shows the~$SO(n)$ subgroup chain of~$SO(9,1)$ of type~$1$ in~$SL(3,\mathbb{O})$, while Figure~\ref{figure:type_1_2_3_G2_so7} shows the three~$SO(9)$ subgroup chains of~$F_4$ within~$E_6$.
In both subgroup structures, there is only one~$SO(8)$. While~$G_2 \subset SO(7)$, it is not a subset of~$SO(6)$ in Figure~\ref{figure:type_1_2_3_so}. Hence, we omit~$G_2$ from Figure~\ref{figure:type_1_2_3_so}, but include it in Figure~\ref{figure:type_1_2_3_G2_so7} since the~$SO(7)$ basis here may be restricted to to the basis of~$G_2$. We also note that~$SU(3)$, consisting of~$A_i, \cdots, A_{i\l}, A_\l, G_\l$, is contained both within~$G_2$ and~$SO(6)$. G\"unaydin denotes this~$SU(3)$ subgroup of~$G_2$ as~$SU(3)^C$ in~\cite{gunaydin_1974}, and we adopt that convention. We use this same notation for the subgroup~$SU(2)^C \subset SU(3)^C$, which consists of~$A_k, A_{k\l}$ and~$A_\l$.
The figures indicate which Casimir operator is added to a group when it is expanded to a larger group. The additional Casimir operator and basis elements are also indicated in Table~\ref{table:basis_for_type_1_2_3_so} and Table~\ref{table:basis_type_1_2_3_G2_so7}.
\begin{figure}[htbp]
\begin{center}
\begin{minipage}{6in}
\begin{center}
\[
\xymatrix{
& *++\txt{$sl(3,\mathbb{O})$\\$[E_6]$} & \\
*++\txt{$so(9,1) = sl(2,\mathbb{O})$\\ $[D_5]$}\ar[ur]^{B^{(2)}_{tz}} & *++\txt{$so(9,1) = sl(2,\mathbb{O})$\\ $[D_5]$}\ar[u]_{B^{(1)}_{tz}} & *++\txt{$so(9,1) = sl(2,\mathbb{O})$\\ $[D_5]$}\ar[ul]^{B^{(3)}_{tz} \to B^{(1)}_{tz}}_{B^{(2)}_{tz}} \\
*++\txt{$so(9) = su(2,\mathbb{O})$\\ $[B_4]$}\ar[u]^{B^{(1)}_{tz}} & *++\txt{$so(9) = su(2,\mathbb{O})$\\ $[B_4]$}\ar[u]^{B^{(2)}_{tz}} & *++\txt{$so(9) = su(2,\mathbb{O})$\\ $[B_4]$}\ar[u]^{B^{(3)}_{tz}} \\
& *++\txt{$so(8)$\\$[D_4]$}\ar[u] \ar[ur] \ar[ul] \\
*++\txt{$so(7) = su(1,\mathbb{O})$\\ $[B_3]$}\ar[ur]^{R^{(1)}_{x\l}} & *++\txt{$so(7) = su(1,\mathbb{O})$\\ $[B_3]$}\ar[u]^{S^{(2)}_\l \to S^{(1)}_\l}_{R^{(1)}_{x\l}} & *++\txt{$so(7) = su(1,\mathbb{O})$\\ $[B_3]$}\ar[ul]^{S^{(3)}_\l \to S^{(1)}_\l}_{R^{(1)}_{x\l}} \\
*++\txt{$so(6)$\\ $[D_3]$}\ar[u] & *++\txt{$so(6)$\\ $[D_3]$}\ar[u] & *++\txt{$so(6)$\\ $[D_3]$}\ar[u] \\
*++\txt{$so(5)$\\ $[B_2]$}\ar[u]_{G_\l-2S^{(1)}_\l} & *++\txt{$so(5)$\\ $[B_2]$}\ar[u]_{G_\l-2S^{(2)}_\l} & *++\txt{$so(5)$\\ $[B_2]$}\ar[u]_{G_\l-2S^{(3)}_\l} \\
*++\txt{$so(4) = so(3)\oplus so(3)$\\ $[D_2]$}\ar[u] & *++\txt{$so(4) = so(3)\oplus so(3)$\\ $[D_2]$}\ar[u] & *++\txt{$so(4) = so(3)\oplus so(3)$\\ $[D_2]$}\ar[u] \\
& *++\txt{$so(3)$\\ $[B_1]$}\ar[u]_{G_\l+S^{(2)}_\l} \ar[ul]_{G_\l+2S^{(1)}_\l} \ar[ur]_{G_\l+2S^{(3)}_\l} & \\
& *++\txt{$u(1)$}\ar[u]_{A_\l} &
}
\]
\caption{Chain of subgroups~$SO(n) \subset SO(9,1) \subset SL(3,\mathbb{O})$}
\label{figure:type_1_2_3_so}
\end{center}
\end{minipage}
\end{center}
\end{figure}
\begin{figure}[htbp]
\begin{center}
\begin{minipage}{6in}
\[
\xymatrix{
& *++\txt{$sl(3,\mathbb{O})$\\$[E_6]$} & \\
& *++\txt{$su(3,\mathbb{O})$\\$[F_4]$}\ar[u]^{B^{(1)}_{tz}, B^{(2)}_{tz}} \\
*++\txt{$so(9) = su(2,\mathbb{O})$\\ $[B_4]$}\ar[ur] & *++\txt{$so(9) = su(2,\mathbb{O})$\\ $[B_4]$}\ar[u] & *++\txt{$so(9) = su(2,\mathbb{O})$\\ $[B_4]$}\ar[ul] \\
& *++\txt{$so(8)$\\$[D_4]$}\ar[u] \ar[ur] \ar[ul] \\
*++\txt{$so(7) = su(1,\mathbb{O})$\\ $[B_3]$}\ar[ur]^{R^{(1)}_{x\l}} & *++\txt{$so(7) = su(1,\mathbb{O})$\\ $[B_3]$}\ar[u]^{S^{(2)}_\l \to S^{(1)}_\l}_{R^{(1)}_{x\l}} & *++\txt{$so(7) = su(1,\mathbb{O})$\\ $[B_3]$}\ar[ul]^{S^{(3)}_\l \to S^{(1)}_\l}_{R^{(1)}_{x\l}} \\
& *++\txt{$Aut(\mathbb{O})$\\$[G_2]$}\ar[u]^{S^{(2)}_\l} \ar[ur]^{S^{(3)}_\l} \ar[ul]^{S^{(1)}_\l} \\
& *++\txt{$su(3,\mathbb{C})^C$\\$[A_2]$}\ar[u] \\
& *++\txt{$su(2,\mathbb{C})^C$\\$[A_1]$}\ar[u]^{G_\l} \\
& *++\txt{$u(1,\mathbb{C})$}\ar[u]^{A_\l}
}
\]
\caption{Chain of subgroups~$SU(3)^C \subset G_2 \subset SO(8) \subset F_4 \subset E_6$}
\label{figure:type_1_2_3_G2_so7}
\end{minipage}
\end{center}
\end{figure}
\begin{table}[htbp]
\begin{center}
\begin{tabular}{|l|l|l|l|}
\hline
Algebra & Casimir & Basis & Set for~$q$ \\
& Operator & & \\
\hline
\hline
$sl(3,\mathbb{O}) = E_6$ & $B_{tz}^{(2)}$ & $B^{(2)}_{tx}, B^{(3)}_{tx}$ & \\
& & $B^{(2)}_{tq}, B^{(3)}_{tq}$ & $i,j,k,k\l,j\l,i\l,\l$ \\
& & $R^{(2)}_{zq}, R^{(3)}_{zq}$ & $i,j,k,k\l,j\l,i\l,\l$ \\
& & $R^{(2)}_{xz}, R^{(3)}_{xz}$ & \\
\hline
$sl(2,\mathbb{O}) = so(9,1) = D_5$ & $B^{(1)}_{tz}$ & $B^{(1)}_{tx}, B^{(1)}_{tq}$ & $i,j,k,k\l,j\l,i\l,\l$ \\
& & $R^{(2)}_{xz}, R^{(3)}_{xz}$ & \\
& & $R^{(2)}_{zq}, R^{(3)}_{zq}$ & $k,k\l,\l$ \\
& & $B^{(1)}_{tq}$ & $i,j,i\l,j\l$ \\
\hline
$su(2,\mathbb{O}) = so(9) = B_4$ & & $R^{(1)}_{xz}, R^{(1)}_{zq}$ & $i,j,k,k\l,j\l,i\l,\l$ \\
\hline
$so(8) = D_4$ & $R^{(1)}_{x\l}$ & $R^{(1)}_{xq}$ & $i,j,k,k\l,j\l,i\l$ \\
\hline
$su(1,\mathbb{O}) = so(7) = B_3$ & & $G_q - S^{(1)}_q$ & $i,j,k,k\l,j\l,i\l$ \\
\hline
$so(6) = D_3$ & $G_\l-S^{(1)}_\l$ & $3A_q - G_q - 2S^{(1)}_q$ & $i,j$ \\
& & $3A_q + G_q + 2S^{(1)}_q$ & $i\l,j\l$ \\
\hline
$so(5) = B_2$ & & $3A_q + G_q + 2S^{(1)}_q$ & $i,j$ \\
& & $3A_q - G_q - 2S^{(1)}_q$ & $i\l,j\l$ \\
\hline
$so(4) = so(3)\oplus so(3) = D_2$ & $G_\l +2S^{(1)}_\l$ & $G_q+2S^{(1)}_q$ & $k,k\l$\\
\hline
$so(3)^C = B_1$ & &$A_q$ & $k,k\l$\\
\hline
$u(1)$ & $A_\l$ & & \\
\hline
\end{tabular}
\caption{\noindent Basis for~$SO(n) \subset SO(9,1)$ subgroup chain of Type~$1$}
\label{table:basis_for_type_1_2_3_so}
\end{center}
\end{table}
\begin{table}[htbp]
\begin{center}
\begin{tabular}{|l|l|l|l|}
\hline
Algebra & Casimir & Basis & Set for~$q$ \\
& Operator & & \\
\hline
\hline
$sl(3,\mathbb{O}) = E_6$ & $B^{(1)}_{tz}, B^{(2)}_{tz}$ & $B^{(1)}_{tx}, B^{(2)}_{tx}, B^{(3)}_{tx}$ & \\
& & $B^{(1)}_{tq}, B^{(2)}_{tq}, B^{(3)}_{tq}$ & $i,j,k,k\l,j\l,i\l,\l$ \\
\hline
$su(3,\mathbb{O}) = F_4 $ & & $R^{(2)}_{zq}, R^{(3)}_{zq}$ & $i,j,k,k\l,j\l,i\l,\l$ \\
& & $R^{(2)}_{xz}, R^{(3)}_{xz}$ & \\
\hline
$su(2,\mathbb{O}) = so(9) = B_4$ & & $R^{(1)}_{xz}, R^{(1)}_{zq}$ & $i,j,k,k\l,j\l,i\l,\l$ \\
\hline
$so(8) = D_4$ & $R^{(1)}_{x\l}$ & $R^{(1)}_{xq}$ & $i,j,k,k\l,j\l,i\l$ \\
\hline
$su(1,\mathbb{O}) = so(7) = B_3$ & $S^{(1)}_\l$ & $S^{(1)}_q$ & $i,j,k,k\l,j\l,i\l$ \\
\hline
$Aut(\mathbb{O}) = G_2$ & & $G_q$ & $i,j,k,k\l,j\l,i\l$ \\
\hline
$su(3,\mathbb{C})^C$ & $G_\l$ & $A_q$ & $i,j,j\l,i\l$ \\
\hline
$su(2,\mathbb{C})^C = A_1$ & &$A_q$ & $k,k\l$\\
\hline
$u(1)$ & $A_\l$ & & \\
\hline
\end{tabular}
\caption{\noindent Basis for~$G_2 \subset SO(7) \subset F_4$ subgroup chain}
\label{table:basis_type_1_2_3_G2_so7}
\end{center}
\end{table}
\section{Triality}
\label{ch:E6_basic_structure.Triality}
In this section, we expand upon the discussion of triality given in Section~\ref{ch:Division_Algebras_Applications.Quaternions_Octonions.Triality} as it relates to~$SL(3,\mathbb{O})$ and its Lie algebra. In particular, we note that each~$3 \times 3$ transformation in our basis of~$SL(3,\mathbb{O})$ naturally implements a~$2 \times 2$ Lorentz transformation and as well as a spinor and dual spinor version of this transformation. However, we may also view~$\chi$ as consisting of three octonions as well as the three real components on its diagonal. The~$SO(8,\mathbb{R})$ transformations, which fix the diagonal components, only act on the octonions via left, right, or symmetric multiplication.
In this section, we find an interesting connection between these different left, right, and symmetric transformations.
In particular, we discuss how the spinor, vector, and dual-spinor transformations embedded into our~$3 \times 3$ transformation use triality to simplify the subgroup structure of~$SO(8,\mathbb{R})$ within~$SL(3,\mathbb{O})$. We also discuss triality in relation to our type map, and introduce a strong formulation of triality.
We quickly review the two versions of triality most pertinent to this work. Baez~\cite{baez} mentions that each of the exterior nodes in the Dynkin diagram of~$so(8,\mathbb{R})$ may be identified with a spinor, dual-spinor, and vector representation of~$SO(8,\mathbb{R})$. Both Baez~\cite{baez} and Conway~\cite{conway} discuss a triality between transformations of~$\mathbb{O}$ via left, right, and symmetric multiplication. Conway further states that each left multiplication may be written as a product of at most seven right or seven symmetric multiplications. Further information regarding these claims may be found in~\cite{baez, conway} or Section~\ref{ch:Division_Algebras_Applications.Quaternions_Octonions.Triality}.
We first consider triality in relation to the transformations contained within~$G_2$.
Consider the one-parameter transformations~$A^{(a)}_{q}(\theta)$ and~$G^{(a)}_{q}(\theta)$, with $a = 1,2,3$, as constructed in Section~\ref{ch:E6_basic_structure.Lorentz_Transformations.BetterBasis}. In Section~\ref{ch:E6_basic_structure.ConstructingE6algebra}, we calculated~$\dot A^{(1)}_{\l}$, which was found to be
$$\dot A^{(1)}_\l(\theta)(\chi) = \frac{\partial A^{(1)}_\l(\theta)(\chi)}{\partial \alpha} =
\left[ \begin {array}{ccc} 0& a_1& \overline{c_1}\\
\noalign{\medskip}\overline{a_1}&0&b_1\\
\noalign{\medskip}c_1&\overline{b_1}&0
\end {array} \right]
$$
$$a_1 = -{\it a_{i\l}}\,{\it i}+{\it a_{j\l}}\,{\it j}-{\it a_j}\,{\it j\l}+{\it a_i}\,{\it i\l}$$
$$b_1 = -{\it b_{i\l}}\,{\it i}+{\it b_{j\l}}\,{\it j}-{\it b_j}\,{\it j\l}+{\it b_i}\,{\it i\l}$$
$$c_1 = -{\it c_{i\l}}\,{\it i}+{\it c_{j\l}}\,{\it j}-{\it c_j}\,{\it j\l}+{\it c_i}\,{\it i\l}$$
\noindent The above expressions for~$a_1, b_1$, and~$c_1$ are very similar; Direct computation of the tangent vectors for~$A^{(2)}_\l$ or~$A^{(3)}_\l$ yields the exact same expression for~$a_1, b_1$, and~$c_1$.
That is,
the one-parameter curves~$A^{(1)}_{\l}$,~$A^{(2)}_{\l}$, and~$A^{(3)}_{\l}$ all have this same tangent vector. Hence, we refer to this transformation as~$A_{\l}$ without ambiguity. Similarly, since the one-parameter curves~$G^{(1)}_\l, G^{(2)}_\l, G^{(3)}_\l$ have the same tangent vector, we also refer to this transformation as~$G_\l$ without a need to refer to type. Similar results hold for~$A_q$ and~$G_q$ for~$q = i,j,k,k\l,j\l,i\l$.
In terms of vector, spinor, and dual spinor transformations, the~$G_2$ transformations are type independent. This requires further explanation. Let~$q = i,j,k,k\l,j\l,i\l,\l$. We may choose to view~$\chi$ as containing the vector~$\mathbf{X}$, spinor~$\theta$, and dual spinor~$\theta^\dagger$ in any one of the three natural locations given in Table~\ref{table:three_types_of_vector_locations} of Section~\ref{ch:E6_basic_structure.Lorentz_Transformations}. Regardless of this choice, the type~$1$ transformation~$A^{(1)}_q$ (and similarly~$G^{(1)}_q$) produces the same transformation on~$\mathbf{X}, \theta, \theta^\dagger$. On the other hand, we may fix a location for~$\mathbf{X}$ in~$\chi$, and notice that the three types~$a$ of~$A^{(a)}_q$ (and similarly for~$G^{(a)}_q$) produce the same transformation on~$\mathbf{X}$. We say that the~$G_2$ transformations are type independent not only because the type~$1$ transformation may be written in terms of the type~$2$ or type~$3$ transformation, but because all three types are the same transformation!
Conway~\cite{conway} states that a general transformation of~$SO(8,\mathbb{R})$ via left-sided multiplication may be expressed in terms of at most seven right-sided multiplications or via at most seven symmetric multiplications. Our basis~$G_2$ transformations use symmetric, left, and right-sided matrix multiplication on the vector~$\mathbf{X}$, spinor~$\theta$, and dual spinor~$\theta^\dagger$ in~$\chi$ and are contained in~$SO(8,\mathbb{R})$. By looking explicitly at the matrices involved in the type~$1$ expression for~$A_\l$, we find
\[
\begin{array}{ccl}
A_{l}(\alpha)(\chi) & = & A_{l}(\alpha)\left( \begin{array}{cc} \mathbf{X} & \theta \\ \theta^\dagger & n \end{array} \right) \\
& = & R^{(1)}_{j\l,j}(-\alpha) \circ R^{(1)}_{i\l,i}(\alpha)\left( \begin{array}{cc} \mathbf{X} & \theta \\ \theta^\dagger & n \end{array} \right) \\
\end{array}
\]
where the vector~$\mathbf{X}$, spinor~$\theta$, and dual spinor~$\theta^\dagger$ transform according to
$$ \mathbf{X} \to \left( M_{2^\prime} \left( M_2 \left( M_{1^\prime} \left(M_1 \mathbf{X} M_1^\dagger \right) M_{1^\prime}^\dagger \right) M_2^\dagger \right) M_{2^\prime}^\dagger \right)
$$
$$ \theta \to \left( M_{2^\prime} \left( M_2 \left( M_{1^\prime} \left(M_1 \theta \right) \right) \right) \right) $$
$$ \theta^\dagger \to \left( \left( \left( \left( \theta^\dagger M_1^\dagger \right) M_{1^\prime} ^\dagger \right) M_2^\dagger \right) M_{2^\prime}^\dagger \right)$$
\noindent with the~$2 \times 2$ matrices~$M_1, M_{1^\prime}, M_2$, and~$M_{2^\prime}$ given by
$$ M_1 = -i\l I_2 \hspace{2cm} M_{1^\prime} = \left( \cos \left(\frac{\alpha}{2}\right) i\l + \sin \left(\frac{\alpha}{2}\right) i \right) I_2$$
$$ M_2 = -j\l I_2 \hspace{2cm} M_{2^\prime} = \left( \cos \left(-\frac{\alpha}{2}\right) j\l + \sin \left(-\frac{\alpha}{2}\right) j \right) I_2$$
\noindent Indeed, we notice that the matrices involved in this~$G_2$ transformation, and all other~$G_2$ transformations, are diagonal and imaginary. Hence, for these matrices,~$M^\dagger = -M$ and the vector transforms not only by conjugation but by symmetric multiplication! Indeed, the spinor and dual spinor transform via four multiplications from the left or right, respectively. We also note that our other~$G_2$ transformations of the form~$G_q$ have the vector, spinor, and dual spinor transform with expressions involving multiplication by six matrices. Hence, there are certain transformations within~$SO(8,\mathbb{R})$ which may be expressed via symmetric, left, or right matrix multiplication using the same matrices. We note that all fourteen of our basis transformations in~$G_2$ have this property.
Of course, we previously discussed (Section~\ref{ch:Division_Algebras_Applications.Quaternions_Octonions.Triality}) Conway's claim as it applied to transformations involving octonions. The matrix~$\chi$ contains three octonions \hbox{$a, b, c \in \mathbb{O}$}.
We find by examining the transformations on~$a,b,c$ for the~$A_l$ transformation of type~$1$
$$
A_\l(\alpha)(\chi) = A_\l(\alpha)\left( \begin{array}{cc|c} t+z & a & \overline{c} \\ \overline{a} & t-z & b \\\hline c & \overline{b} & n \end{array} \right) = \left( \begin{array}{cc|c} t+z & a_1 & \overline{c_1} \\ \overline{a_1} & t-z & b_1 \\\hline c_1 & \overline{b_1} & n \end{array} \right)$$
\noindent that the octonions~$a$,~$b$, and~$c$ transform in the vector, spinor, and dual-spinor pieces via
$$ a \to a_1 = e^{-l\frac{\alpha}{2}}j\l \left( -j\l \left( e^{\l\frac{\alpha}{2}}i\l \left((-i\l) a (-i\l)^\dagger \right) (e^{\l\frac{\alpha}{2}}i\l)^\dagger \right) (-j\l)^\dagger \right) (e^{-l\frac{\alpha}{2}}j\l)^\dagger $$
$$ b \to b_1 = e^{-l\frac{\alpha}{2}}j\l \left( -j\l \left( e^{\l\frac{\alpha}{2}}i\l \left(-i\l \left(b\right) \right) \right) \right) $$
$$ c \to c_1 = \left( \left( \left( c (-i\l)^\dagger \right) (e^{\l\frac{\alpha}{2}}i\l)^\dagger \right) (-j\l)^\dagger \right) (e^{-l\frac{\alpha}{2}}j\l)^\dagger $$
\noindent However, since all three types of the~$A_\l$ transformation are the same, we see that~$a$,~$b$, and~$c$ undergo the same transformation using conjugation, left-sided, and right-sided multiplication by octonions. These octonions transform the same way using different forms of multiplication by the same octonions. This is a stronger condition than Conway gives in~\cite{conway} for triality, and we call this property {\it strong triality}. Equivalently, a transformation in~$SL(3,\mathbb{O})$ which is type independent exhibits strong triality. Hence, every transformation in~$G_2$ gives octonionic transformations which have strong triality.
Finally, we note that~$SO(8,\mathbb{R}) \in SL(3,\mathbb{O})$ also exhibits triality.
There is an~$SO(7)$ group of each type~$a = 1,2,3$, which consists of~$G_2$ along with the seven transformations~$S^{(a)}_{q}$. We expand any one~$SO(7)$ group to~$SO(8)$ by including the~$R^{(a)}_{xq}$ transformations of type~$a$. The linear dependencies in the algebra allow us to express any type~$2$ or type~$3$ transformation of the form~$\dot R^{(a)}_{xq}$ or~$\dot S^{(a)}_{q}$ in terms of type~$1$ transformations \footnote{These expressions are cyclic in~$a = 1, 2, 3$, so that we may write any~$SO(8)$ transformation of type~$a$ in terms of transformations of another type.}
$$ \dot R^{(2)}_{xq} = -\frac{1}{2}\dot R^{(1)}_{xq} - \frac{1}{2}\dot S^{(1)}_q \hspace{2cm} \dot S^{(2)}_q = \frac{3}{2} \dot R^{(1)}_{xq} - \frac{1}{2}\dot S^{(1)}_q $$
$$ \dot R^{(3)}_{xq} = -\frac{1}{2}\dot R^{(1)}_{xq} + \frac{1}{2}\dot S^{(1)}_q \hspace{2cm} \dot S^{(3)}_q = -\frac{3}{2} \dot R^{(1)}_{xq} - \frac{1}{2}\dot S^{(1)}_q $$
\noindent
Since the tangent vectors for the transformations on the right-hand side of each equality commute, their corresponding group transformations commute. Hence, we may express the corresponding type~$2$ or type~$3$ group transformation on the left-hand side in terms of type~$1$ transformations.
For instance, the expression for~$\dot R^{(2)}_{x\l}$ given above leads to the group transformation identity
$$ R^{(2)}_{x\l}(\alpha)(\chi) = S^{(1)}_\l(-\frac{\alpha}{2})(\chi) \circ R^{(1)}_{x\l}(-\frac{\alpha}{2})$$
Given that it is possible in the algebra~$so(8)$ to express every transformation in terms of type~$1$ transformations or~$G_2$ transformations, we see that the group~$SO(8)$ exhibits triality. In particular, while there are three different~$SO(7)$ groups built from~$G_2$, there is only one~$SO(8)$.
We now show that transformations in~$SO(8,\mathbb{R}) - G_2$ exhibit the more traditional version of triality described by Baez~\cite{baez} and Conway~\cite{conway} which was described in Section~\ref{ch:Division_Algebras_Applications.Quaternions_Octonions.Triality}. Consider the two transformations~$R^{(2)}_{x\l}(\alpha)$ and~$S^{(1)}_{\l}(-\frac{\alpha}{2}) \circ R^{(1)}_{x\l}(-\frac{\alpha}{2})$, which were just shown to be equal, and their effect on the octonions~$a,b,c \in \mathbb{O}$ which results from them being applied to~$\chi$.
Following the construction given in Section~\ref{ch:E6_basic_structure.LieGroup}, explicit calculation shows that~$a$,~$b$, and~$c$ transform under~$R^{(2)}_{x\l}(\alpha)(\chi)$ according to
$$\begin{array}{ccccc} a \to a e^{-\l\frac{\alpha}{2}} & \hspace{1cm} & b \to e^{\l\frac{\alpha}{2}} b e^{-\l\frac{\alpha}{2}} & \hspace{1cm} & c \to e^{\l\frac{\alpha}{2}}c \end{array}$$
\noindent while under~$S^{(1)}_{\l}(-\frac{\alpha}{2}) \circ R^{(1)}_{x\l}(-\frac{\alpha}{2})(\chi)$, we see that~$b$ and~$c$ transform via left-sided and right-sided multiplication
$$\begin{array}{ccc}
b& \to& e^{-\l\frac{\alpha}{4}}k\l \left( -k\l \left( e^{-\l\frac{\alpha}{4}}j\l \left( -j\l \left( e^{-\l\frac{\alpha}{4}}i\l \left( -i\l \left( e^{-\l\frac{\alpha}{4}} (b) \right)\right)\right)\right)\right)\right)\\
c& \to& \left( \left( \left( \left( \left( \left( (c) e^{\l\frac{\alpha}{4}} \right) i\l \right) (-i\l e^{\l\frac{\alpha}{4}}) \right) j\l \right) (-j\l e^{\l\frac{\alpha}{4}}) \right) k\l \right) (-k\l e^{\l\frac{\alpha}{4}})
\end{array}$$
\noindent and~$a$ transforms via multiplication by conjugate octonions, although we have omitted the expression for this transformation due to space constraints. Due to the equalities of the group transformations, we see that these new expressons for~$a$,~$b$, and~$c$ are equal to the previous expressions for all values of~$\alpha$. In particular, we have
$$\begin{array}{ccc}
e^{\l\frac{\alpha}{2}} b e^{-\l\frac{\alpha}{2}} &=& e^{-\l\frac{\alpha}{4}}k\l \left( -k\l \left( e^{-\l\frac{\alpha}{4}}j\l \left( -j\l \left( e^{-\l\frac{\alpha}{4}}i\l \left( -i\l \left( e^{-\l\frac{\alpha}{4}} (b) \right)\right)\right)\right)\right)\right)\\
e^{\l\frac{\alpha}{2}}c &=& \left( \left( \left( \left( \left( \left( (c) e^{\l\frac{\alpha}{4}} \right) i\l \right) (-i\l e^{\l\frac{\alpha}{4}}) \right) j\l \right) (-j\l e^{\l\frac{\alpha}{4}}) \right) k\l \right) (-k\l e^{\l\frac{\alpha}{4}})
\end{array}
$$
\noindent That is, we may produce the~$b \to e^{\l \frac{\alpha}{2}} b e^{-\l \frac{\alpha}{2}}$ action by nesting seven left-sided multiplications, and we may also produce the action of multiplying the octonion~$c$ on the left by nesting seven right-sided multiplications. The expression for~$a$ would show that multiplying~$a$ on the right by an octonion produces the same action as conjugating~$a$ with a series of octonions. These identities hold for all values of~$\alpha \in \mathbb{R}$. Also, these expressions are not unique.\footnote{ Note, too, that each factor~$e^{-\l\frac{\alpha}{4}}$ and~$e^{\l\frac{\alpha}{4}}$ next to~$b$ and~$c$, respectively, on the right hand side of each equality generates a rotation in the~$1\leftrightarrow \l$ plane and is perpendicular to the other three planes being by the~$S^{(1)}_{\l}$ transformation. Hence, we are free to commute the phase~$e^{-\l\frac{\alpha}{4}}$ across any pair of factors of the form~$(e^{-\l\frac{\alpha}{4}}q)(-q)$ in the expression for~$b$ and similarly in the expressions for~$c$ and~$a$.}
All of the equivalent ways of writing type~$2$ or type~$3$ transformations in~$SO(8) - G_2$ in terms of type~$1$ transformations (and cyclic permutations) will produce similar identities involving seven multiplications.
\section{Type Transformation}
\label{ch:E6_basic_structure.Type_Transformation}
Our construction of~$3 \times 3$ Lorentz transformations produces equivalent~$SO(9,1)$ groups of type~$1$, type~$2$ and type~$3$ in the following sense: The discrete type transformation~\hbox{$\mathbf{T}: M_3(\mathbb{O}) \to M_3(\mathbb{O})$} given by
$$\mathbf{T}(M) = \mathcal{T} M \mathcal{T}^\dagger$$
with
$$\mathcal{T} = \left( \begin{array}{ccc} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0\end{array} \right)$$
cycles the matrices used in our ~$3 \times 3$ finite Lorentz transformations from type~$1$ to type~$2$ and type~$3$ matrices
$$ \mathbf{T}(M^{(1)}) = M^{(2)} \hspace{1.5cm} \mathbf{T}(M^{(2)}) = M^{(3)} \hspace{1.5cm} \mathbf{T}(M^{(3)}) = M^{(1)}$$
Hence, we have~$\mathbf{T}^3 = Id$ and~$\mathbf{T}^2 = \mathbf{T}^{-1}$. In addition, the type transformation is in~$SL(3,\mathbb{O})$ since
$$\det(\mathbf{T}(\chi)) = \det(\chi)$$
While it is {\bf not} one of our basis group transformations, we have the identities
$$ \mathbf{T} = R^{(3)}_{xz}(-\pi) \circ R^{(1)}_{xz}(-\pi) \hspace{1cm} \mathbf{T}^{-1} = R^{(2)}_{xz}(\pi) \circ R^{(1)}_{xz}(\pi) $$
We may expand~$\mathbf{T}^3 = Id$ for these expressions, showing that
$$\begin{array}{ccl}
\mathbf{T}^{-1} &=& R^{(3)}_{xz}(-\pi) \circ R^{(1)}_{xz}(-\pi) \circ R^{(3)}_{xz}(-\pi) \circ R^{(1)}_{xz}(-\pi)\\
\mathbf{T} &=& R^{(2)}_{xz}(\pi) \circ R^{(1)}_{xz}(\pi) \circ R^{(2)}_{xz}(\pi) \circ R^{(1)}_{xz}(\pi)
\end{array}
$$
We also note that
$$ Id = \mathbf{T} \circ \mathbf{T}^{-1} = R^{(3)}_{xz}(-\pi) \circ R^{(1)}_{xz}(-\pi) \circ R^{(2)}_{xz}(\pi) \circ R^{(1)}_{xz}(\pi) $$
and that
$$ Id = R^{(2)}_{xz}(\pi) \circ R^{(1)}_{xz}(\pi) \circ R^{(2)}_{xz}(\pi) \circ R^{(1)}_{xz}(\pi) \circ R^{(2)}_{xz}(\pi) \circ R^{(1)}_{xz}(\pi)$$
Combining these two expressions and solving for~$\mathbf{T}$, we obtain the expression
$$ \mathbf{T} = R^{(1)}_{xz}(\pi) \circ R^{(3)}_{xz}(\pi) \circ R^{(2)}_{xz}(\pi) \circ R^{(1)}_{xz}(\pi)$$
which involves all three types. Similarly, many additional expressions for~$\mathbf{T}$ and~$\mathbf{T}^{-1}$ may be obtained from these expressions.
We also note that each of these expressions may be expanded into a {\it continuous type transformation} from~$\mathbb{R} \to SL(3,\mathbb{O})$ by letting the fixed angle~$\pi$ or ~$-\pi$ become an arbitrary angle~$\alpha$ which varies over~$\mathbb{R}$. The resulting transformations are not one-parameter subgroups of~$SL(3,\mathbb{O})$, but they do connect a type~$1$ transformation with either a type~$2$ or type~$3$ transformation.
\subsection{Type Independent and Dependent Subgroups}
\label{ch:E6_basic_structure.Type_Transformation.Subgroups}
We list here some important groups which contain the type transformation.
The group~$\langle R^{(1)}_{xz}, R^{(2)}_{xz}, R^{(3)}_{xz} \rangle$ is the standard representation of~$SO(3,\mathbb{R})$, and we label this group~$SO(3,\mathbb{R})_s$ using the subscript~$s$ to designate it as the standard representation.
This group obviously contains~$\mathbf{T}$, as does the standard representation
$$ SL(3,\mathbb{R})_s = \langle R^{(1)}_{xz}, R^{(2)}_{xz}, R^{(3)}_{xz}, B^{(1)}_{tz}, B^{(2)}_{tz}, B^{(1)}_{tx}, B^{(2)}_{tx}, B^{(3)}_{tx} \rangle$$
Using~$\l$ as our preferred complex unit, we see that the standard representation
$$ SU(3,\mathbb{C})_s = \langle R^{(1)}_{xz}, R^{(2)}_{xz}, R^{(3)}_{xz}, R^{(1)}_{x\l}, R^{(2)}_{x\l}, R^{(1)}_{z\l}, R^{(3)}_{z\l}, R^{(3)}_{z\l}\rangle$$
of~$SU(3,\mathbb{C})$, which consists of~$3\times 3$ special unitary matrices, contains~$SO(3,\mathbb{R})_s$. Hence, $SU(3,\mathbb{C})_s$ also contains~$\mathbf{T}$, as does, of course, the group
$$ SL(3,\mathbb{C})_s = SU(3,\mathbb{C})_s \cup \langle B^{(1)}_{tz}, B^{(2)}_{tz}, B^{(1)}_{tx}, B^{(2)}_{tx}, B^{(3)}_{tx}, B^{(1)}_{t\l}, B^{(2)}_{t\l}, B^{(3)}_{t\l} \rangle$$
These four groups~$SO(3,\mathbb{R})_s$,~$SL(3,\mathbb{R})_s$,~$SU(3,\mathbb{C})_s$, and~$SL(3,\mathbb{C})_s$ are important because they contain the type transformation~$\mathbf{T}$. If, for instance, the transformation~$R^{(1)}$ is in a group~$G$ which has one of these groups as a subgroup, then the subgroup forces~$G$ to contain~$R^{(2)}$ and~$R^{(3)}$ as well. On the other hand, if we add one of those four groups to any type-specific group~$G$, then the new group must contain~$R^{(1)}$,~$R^{(2)}$, and~$R^{(3)}$ for each transformation~$R \in G$. The new group, in most cases, becomes much larger.
We are careful to point out that the standard representations~$SO(3,\mathbb{R})_s$ and~$SU(3,\mathbb{C})_s$ are {\bf not} our preferred representations for the groups~$SO(3,\mathbb{R})^C$ and~$SU(3,\mathbb{C})^C$, which are subgroups of~$G_2$. For any~$R \in G_2$, we have~$R^{(1)} = R^{(2)} = R^{(3)}$. Thus, we see that~$SU(3,\mathbb{C})_s$ and~$SU(3,\mathbb{C})^C$ are both type independent, but in~$SU(3,\mathbb{C})_s$ the transformations~$R^{(1)}, R^{(2)}$ and~$R^{(3)}$ are distinct while in in~$SU(3,\mathbb{C})^C$ the three transformations are equal.
We use the type transformation~$\mathbf{T}$ to provide insight into the structure of~$SL(3,\mathbb{O})$. The algebras~$g$ in the left column of Figure~\ref{figure:type_subalgebras} are subalgebras of the type~$1$ copy of~$sl(2,\mathbb{O})$, while each algebra~$g^\prime$ in the right column is the largest algebra of~$sl(3,\mathbb{O})$ such that~$g \oplus g^\prime$ is still~$\mathbb{R}$-simple. When we restrict~$g$ to a smaller subalgebra of~$sl(2,\mathbb{O})$, it is sometimes possible to exand the type independent subalgebra~$g^\prime$ to a larger subalgebra of~$sl(3,\mathbb{O})$. Each arrow in the diagram indicates containment.
\begin{figure}[htbp]
\[
\xymatrixcolsep{1pt}
\xymatrixrowsep{30pt}
\xymatrix@M=0pt{
*++\txt{$sl(2,\mathbb{O})$ \\ $[D_5 $} & \txt{$\oplus$ \\ $\oplus$} & *++\txt{$ \lbrace \dot B^{(2)}_{tz} - \dot B^{(3)}_{tz} \rbrace $ \\ $ su(1)]$} \ar[d]<1ex> \\
*++\txt{$su(2,\mathbb{O})$ \\ $[B_4 $} \ar@{.>}[u]<1ex> & \txt{$\oplus$ \\ $\oplus$} & *++\txt{$ \lbrace \dot B^{(2)}_{tz} - \dot B^{(3)}_{tz} \rbrace $\\ $ su(1)]$} \ar[d]<1ex> \\
*++\txt{$su(1,\mathbb{O})$ \\ $[B_3$} \ar@{.>}[u]<1ex> & \txt{$\oplus$ \\ $\oplus$} & *++\txt{$ (\lbrace B^{(1)}_{tz} \rbrace \oplus \lbrace \dot B^{(2)}_{tz} - \dot B^{(3)}_{tz} \rbrace)$ \\ $ (su(1) \oplus su(1))]$}\ar[d]<1ex> \\
*++\txt{$G_2 $ \\ $[G_2 $} \ar@{.>}[u]<1ex> & \txt{$\oplus$ \\ $\oplus$} & *++\txt{$ sl(3,\mathbb{R})_s$\\$(A_1 \oplus A_1)]$} \ar[d]<1ex> \\
*++\txt{$su(3,\mathbb{C})^C $ \\ $ [A_2$} \ar@{.>}[u]<1ex> & \txt{$\oplus$\\ $\oplus$} & *++\txt{$ sl(3,\mathbb{C})_s$\\$(A_2 \oplus A_2)]$} \ar[d]<1ex> \\
*++\txt{$so(3,\mathbb{R})^C $ \\ $[A_1 $} \ar@{.>}[u]<1ex> & \txt{$\oplus$ \\ $\oplus$} & *++\txt{$ sl(3,\mathbb{H})$\\ $ A_5]$} \\
}
\]
\caption{Type dependent and independent subalgebras of~$E_6$}
\label{figure:type_subalgebras}
\end{figure}
\section{Reduction of~$\mathbb{O}$ to~$\mathbb{H}$,~$\mathbb{C}$, and~$\mathbb{R}$}
\label{ch:E6_basic_structure.Reduction_of_O}
We find subalgebras of~$E_6$ by restricting our octonionic~$E_6$ generators to be quaternionic, complex, or real.
In the first case, we limit ourselves to the quaternionic case by using only the algebra generators~$\dot R_{ab}$ or~$\dot R_{aq}$ where~$a,b \in \lbrace t,x,z \rbrace$ and~$q \in \lbrace k,kl,l\rbrace$. Of course, we must also include those generators in~$SO(7)$ which mix up these imaginary quaternionic pieces. Given our definition of~$A_q, G_q,$ and~$S_q$, the particular combinations
$$ \dot G_k - \dot S^{(1)}_k \hspace{1.5cm} \dot G_{kl} - \dot S^{(1)}_{kl} \hspace{1.5cm} \dot G_l - \dot S^{(1)}_l$$
provide the permutations of~$\lbrace k,kl,l\rbrace$ while fixing~$\lbrace i,j,il,jl\rbrace$. This provides~$35$ transformations, of which~$14$ are boosts. The~$21$ compact generators form the algebra~$su(3,\mathbb{H})$, a real form of~$C_3 = sp(2\cdot 3)$, while all~$35$ together form~$sl(3,\mathbb{H})$, a real form of~\hbox{$A_5 = su(6,\mathbb{C})$}. Restricting only to type~$1$ transformations, we obtain~$10$ rotations and~$5$ boosts. This restriction reduces the algebra~$sl(3,\mathbb{H})$ to~$sl(2,\mathbb{H}) = so(5,1)$, a real form of~$D_3 = so(6)$. The algebra containing only the~$10$ rotations is~$su(2,\mathbb{H}) = so(5)$, a real form of~$C_2 = sp(2\cdot 2)$.
We note that~$\dot A_k, \dot A_{kl},$ and~$\dot A_l$ fix our preferred quaternions~$\lbrace k, kl, l\rbrace$ and permute the four {\it orthogonal quaternions}~$\lbrace i, j, jl, il \rbrace$ among themselves. Hence, we have~\hbox{$sl(3,\mathbb{H}) \oplus so(3)^C$}, where~$\lbrace \dot A_k, \dot A_kl, \dot A_l\rbrace$ form the~$so(3)^C$. This direct sum structure~$g \oplus so(3)^C$ holds for all quaternionic subalgebras~$g$ of~$sl(3,\mathbb{H})$.
Given the octonion~$\l$, there are at least two interesting ways to break the octonions into quaternionic subalgebras. As discussed above, we chose a preferred quaternion algebra containing~$k, k\l$ and~$\l$. However, another choice would be to leave~$\l$ alone, instead using~$i, j,$ and~$k$.
Using this quaternionic sublagebra instead of~$\lbrace 1, k,kl,l\rbrace$, direct calculation shows that
$$\dot A_i + \frac{1}{3} \dot G_i + \frac{2}{3} \dot S^{(1)}_i \hspace{1.5cm} \dot A_j + \frac{1}{3} \dot G_j + \frac{2}{3} \dot S^{(1)}_j \hspace{1.5cm} \dot A_k + \frac{1}{3} \dot G_k + \frac{2}{3} \dot S^{(1)}_k$$
permutes our imaginary quaternionic units~$\lbrace i, j, k\rbrace$ while
$$\dot G_i - \dot S^{(1)}_i \hspace{1.5cm}\dot G_j - \dot S^{(1)}_j \hspace{1.5cm} \dot G_k - \dot S^{(1)}_k $$
form the~$so(3)$ which fixes our quaternionic subalgebra and permutes the orthogonal quaternions~$\lbrace kl, jl, il, l \rbrace$. However, neither of these Lie algebras contain any of the preferred Casimir operators~$\lbrace \dot B^{(1)}_{tz}, \dot B^{(2)}_{tz}, \dot R^{(1)}_{xl}, \dot A_{l}, \dot G_l, \dot S^{(1)}_{l} \rbrace $.
As previously discussed, we obtain the classical Lie algebras~$su(3,\mathbb{C})_s$ and~$sl(3,\mathbb{C})_s$ by choosing those transformations~$\dot R_{ab}$ and~$\dot R_{aq}$ of all three types, where~$a,b \in \lbrace t, x, z \rbrace$ and~$q = l$. As there is only one octonionic unit used to form~$\mathbb{C}$, we do not need to use any of the transformations from~$SO(7)$. While using all~$16$ transformations gives~$sl(3,\mathbb{C})_s$, a real form of~$A_2 \oplus A_2 = su(3,\mathbb{C}) \oplus su(3,\mathbb{C})$ with~$8$ boosts, we obtain~$su(3,\mathbb{C})_s$ by using only the~$8$ compact generators. Restricting ourselves to the type~$1$ case reduces these two algebras to~$sl(2,\mathbb{C})_s = so(3,1)_s$ and~$su(3,\mathbb{C})_s$, which are real forms of~\hbox{$D_2 = su(2,\mathbb{C}) \oplus su(2,\mathbb{C})$} and~$A_1 = su(3,\mathbb{C})$.
We also note that when we restrict~$sl(3,\mathbb{C})_s$ to~$sl(2,\mathbb{C})_s$, the smaller algebra no longer contains the type transformation~$\mathbf{T}$ but it uses the octonionic direction~$\l$. Direct calculation shows that~$sl(2,\mathbb{C})_s \oplus so(6)$, where~$so(6)$ fixes~$\l$, is an~$\mathbb{R}$-simple subalgebra of~$sl(3,\mathbb{O})$.
Finally, by restricting to real transformations, we see that in the~$3 \times 3$ case, the nine elements~$\dot R^{(T)}_{ab}, a,b \in \lbrace t,x,z \rbrace, T = 1,2,3$ along with the relation~$\dot R^{(1)}_{tz}+\dot R^{(2)}_{tz} + \dot R^{(3)}_{tz} = 0$ give a real form of~$A_2 = su(3,\mathbb{C})$ with~$5$ non-compact elements. Note that this is {\bf not} our standard~$su(3,\mathbb{C})^C \subset G_2$, nor the standard~$su(3,\mathbb{C})_s$ as it is not complex. It may be restricted to either~$so(3,\mathbb{R})_s$, whose group contains the type transformation, or~$so(2,1)_s$, which is a type~$1$ non-compact form of~$A_1 = so(3,\mathbb{R})$.
The results of restricting~$sl(3,\mathbb{O})$ to~$sl(n,\mathbb{F})$ for~$n = 1,2,3$ and~$\mathbb{F} = \mathbb{R}, \mathbb{C}, \mathbb{H}, \mathbb{O}$ is given in Figure~\ref{fig:division_algebra_subalgebras_of_E6}.
For each algebra~$g$ in Figure~\ref{fig:division_algebra_subalgebras_of_E6}, we list in Figure~\ref{fig:division_algebra_subalgebras_and_perp_algebras_of_E6} the maximal subalgebra~$g^\prime$ of~$E_6$ such that we have the structure~$g \oplus g^\prime$. Here~$so(6)$ denotes the algebra which permutes~$\lbrace i,j,k,k\l,j\l,i\l\rbrace$ but fixes~$\l$. While~$so(6) \not \subset G_2$, we do have~$su(3,\mathbb{C})^C \subset so(6)$. The solid arrow indicates inclusion when the algebra~$g$ is expanded to a larger Lie algebra and the dotted arrow indicates the result of restricting the algebra~$g^\prime$ to a smaller algebra.
\begin{figure}[htbp]
\begin{center}
\begin{minipage}{6in}
\begin{center}
\xymatrixcolsep{8pt}
\xymatrix@M=1pt@H=0pt{
& & & *++\txt{$sl(3,\mathbb{O})$\\$[E_6]$} & & & \\
& & & *++\txt{$sl(3,\mathbb{H})$\\$[A_5]$}\ar[u] & & & \\
& & & *++\txt{$sl(3,\mathbb{C})_s$\\$[A_2\oplus A_2]$}\ar[u] & & & \\
*++\txt{$sl(2,\mathbb{O})$\\$[D_5]$} \ar[uuurrr] & *++\txt{$sl(2,\mathbb{H})$\\$[A_3 = D_3]$} \ar[uurr] \ar[l] & *++\txt{$sl(2,\mathbb{C})_s$\\$[A_1 \oplus A_1]$} \ar[ur] \ar[l] & & *++\txt{$su(3,\mathbb{C})_s$\\$[A_2]$} \ar[ul] \ar[r]& *++\txt{$su(3,\mathbb{H})$\\$[C_3]$} \ar[uull] \ar[r]& *++\txt{$su(3,\mathbb{O})$\\$[F_4]$} \ar[uuulll]\\
& & & *++\txt{$su(2,\mathbb{C})_s$\\$[A_1]$} \ar[d] \ar[ul] \ar[ur] & & & \\
& & & *++\txt{$su(2,\mathbb{H})$\\$[B_2 = C_2]$}\ar[d] \ar[uurr] \ar[uull] & & & \\
& & & *++\txt{$su(2,\mathbb{O})$\\$[B_4]$}\ar[uuurrr] \ar[uuulll] & & & \\
}
\caption{Subalgebras~$sl(n,\mathbb{F})$ and~$su(n,\mathbb{F})$ of ~$sl(3,\mathbb{O})$}
\label{fig:division_algebra_subalgebras_of_E6}
\end{center}
\end{minipage}
\end{center}
\end{figure}
\begin{landscape}
\begin{figure}[htbp]
\begin{center}
\xymatrixcolsep{12pt}
\xymatrixrowsep{12pt}
\xymatrix@M=2pt@H=2pt{
& & & *++\txt{$sl(3,\mathbb{O})\oplus 0 $\\$[E_6 \oplus 0]$} \ar@{.>}[d]<1ex> \ar@{.>}[dddrrr]<1ex> \ar@{.>}[dddlll]<1ex>& & & \\
& & & *++\txt{$sl(3,\mathbb{H}) \oplus so(3)^C$\\$[A_5 \oplus A_1]$}\ar[u]<1ex> \ar@{.>}[d]<1ex> \ar@{.>}[ddll]<1ex> \ar@{.>}[ddrr]<1ex>& & & \\
& & & *++\txt{$sl(3,\mathbb{C})_s \oplus su(3)^C$\\$[(A_2\oplus A_2) \oplus A_2 ]$}\ar[u]<1ex> \ar@{.>}[dl]<1ex> \ar@{.>}[dr]<1ex>& & & \\
*++\txt{$sl(2,\mathbb{O}) \oplus 0$\\$[D_5 \oplus 0]$} \ar@{.>}[r]<-1ex> \ar[uuurrr]<1ex> \ar@{.>}[rrrddd]<1ex>& *++\txt{$sl(2,\mathbb{H}) \oplus so(3)^C$\\$[(A_3 = D_3) \oplus A_1]$} \ar@{.>}[r]<-1ex> \ar[uurr]<1ex> \ar[l]<-1ex> \ar@{.>}[ddrr]<1ex> & *++\txt{$sl(2,\mathbb{C})_s \oplus so(6)$\\$[(A_1 \oplus A_1) \oplus D_3 ]$} \ar[ur]<1ex> \ar@{.>}[dr]<1ex> \ar[l]<-1ex> & & *++\txt{$su(3,\mathbb{C})_s \oplus su(3)^C$\\$[A_2 \oplus A_2]$} \ar[ul]<1ex> \ar@{.>}[dl]<1ex> \ar[r]<1ex>& *++\txt{$su(3,\mathbb{H}) \oplus so(3)^C$\\$[C_3\oplus A_1]$} \ar[uull]<1ex> \ar@{.>}[ddll]<1ex> \ar[r]<1ex> \ar@{.>}[l]<1ex>& *++\txt{$su(3,\mathbb{O}) \oplus 0$\\$[F_4\oplus 0]$} \ar[uuulll]<1ex> \ar@{.>}[dddlll]<1ex> \ar@{.>}[l]<1ex>\\
& & & *++\txt{$su(2,\mathbb{C})_s \oplus so(6)$\\$[A_1 \oplus D_3]$} \ar[d]<-1ex> \ar[ul]<1ex> \ar[ur]<1ex> & & & \\
& & & *++\txt{$su(2,\mathbb{H}) \oplus so(3)^C$\\$[(B_2 = C_2) \oplus A_1]$}\ar[d]<-1ex> \ar[uurr]<1ex> \ar[uull]<1ex> \ar@{.>}[u]<-1ex> & & & \\
& & & *++\txt{$su(2,\mathbb{O})\oplus 0$\\$[B_4 \oplus 0]$}\ar[uuurrr]<1ex> \ar[uuulll]<1ex> \ar@{.>}[u]<-1ex> & & & \\
}
\caption{Subalgebras~$sl(n,\mathbb{F}) \oplus g_1$ and~$su(n,\mathbb{F}) \oplus g_1$ of ~$sl(n,\mathbb{O})$}
\label{fig:division_algebra_subalgebras_and_perp_algebras_of_E6}
\end{center}
\end{figure}
\end{landscape}
\section{A Subalgebra fixing~$\l$}
\label{ch:E6_basic_structure.fix_l}
We examine here a subalgebra of~$sl(3,\mathbb{O})$ which fixes a preferred octonionic unit in~$\chi$. Given
$$\chi = \left( \begin{array}{ccc} t+z & \overline{a} & c \\ a & t-z & \overline{b} \\ \overline{c} & b & n \end{array} \right) \in M_3(\mathbb{O})$$
we choose to treat the upper~$2 \times 2$ submatrix as a vector. We call the unit octonion~$\l$ in this vector {\it special} and look for transformations which fix the coefficient~$a_\l$ of~$\l$ near the identity. That is, we find~$Stab(\l)$ for this special~$\l$. At the Lie algebra level, this implies that the coefficient of our special~$\l$ in the tangent vector must vanish, ensuring that~$\l$ is fixed by the corresponding one-parameter transformation in the group.
Examination of the~$78$ tangent vectors in our preferred basis for~$SL(3,\mathbb{O})$ results in~$52$ elements which fix our special~$\l$. Direct computation shows these elements form a~$52$-dimensional subalgebra~$Stab(\l)$ of~$sl(3,\mathbb{O})$.
However,~$Stab(\l)$ is not a simple algebra. It contains
$$su(3,\mathbb{C})^C = \langle A_i, A_j, A_k, A_{k\l}, A_{j\l}, A_{i\l}, A_{\l}, G_\l \rangle$$
which is a subalgebra of~$G_2$. The algebra~$su(3,\mathbb{C})^C$ can be expanded to the algebra~$so(6,\mathbb{R})$ of type~$1$ which fixes~$\l$ via
$$\begin{array}{ccl}
so(6,\mathbb{R}) &= & su(3,\mathbb{C})^C \cup \langle G_i + 2S^{(1)}_i, G_j + 2S^{(1)}_j, G_k + 2S^{(1)}_k, \\
& & \hspace{2cm} G_{k\l} + 2S^{(1)}_{k\l}, G_{j\l} + 2S^{(1)}_{j\l}, G_{i\l} + 2S^{(1)}_{i\l}, S^{(1)}_\l \rangle
\end{array}$$
We note that we may expand~$so(6,\mathbb{R})$ into~$so(8,1,\mathbb{R})$ which does {\bf not} contain our preferred~$so(8,\mathbb{R})$. In particular, this~$so(8,1,\mathbb{R})$ is an algebra of type~$1$ and does not contain~$G_2$. The basis of this~$so(8,1,\mathbb{R})$ is expanded from that of~$so(6,\mathbb{R})$ by including the transformations~$R^{(1)}_{xz}, B^{(1)}_{tx}, B^{(1)}_{tz}$, which form~$so(2,1,\mathbb{R})$, and
$$ B^{(1)}_{tq}, R^{(1)}_{xq}, R^{(1)}_{zq}$$
where~$q = i,j,k,k\l,j\l,i\l$, but not~$q = \l$.
We note that~$so(2,1,\mathbb{R})$ commutes with~$so(6,\mathbb{R})$ since it contains purely real transformations.
We note that each of the~$3 \times 3$ Lorentz transformations corresponding to the Lie algebra elements in~$so(8,1,\mathbb{R})$ are one-parameter transformations. Hence, then the group~$SO(8,1,\mathbb{R})$ also fixes the coefficient of~$\l$ in the vector.
At the identity, the tangent vectors of our type~$2$ and type~$3$ transformations do not fix~$\l$. However, the following real linear combinations of these tangent vectors do fix the~$\l$ of type~$1$:
$$ b_2 = \langle B^{(2)}_{tq} + R^{(2)}_{zq} | q = i,j,k,k\l, j\l, i\l \rangle$$
$$ b_3 = \langle B^{(3)}_{tq} - R^{(3)}_{zq} | q = i,j,k,k\l, j\l, i\l \rangle$$
$$ b_\l = \langle B^{(2)}_{tx}+R^{(2)}_{xz}, B^{(3)}_{tx}-R^{(3)}_{xz}, B^{(2)}_{t\l}+R^{(2)}_{x\l}, B^{(3)}_{t\l}-R^{(3)}_{x\l} \rangle$$
No other real linear combinations of tangent vectors fix the~$\l$ of type~$1$.
The algebra~$b = b_2 \oplus b_3 \oplus b_\l$ is abelian, and~$[ so(8,1,\mathbb{R}), b ] \subset b$. Hence,~$b$ is an ideal, implying~$Stab(\l)$ is not simple. We note that
$$ [su(3)^C,b_2 ] \subset b_2 \hspace{1.5cm} [su(3)^C,b_3 ] \subset b_3 \hspace{1.5cm} [su(3)^C,b_\l] = 0$$
and that~$ [so(6), b ] \subset b$ since~$[so(6), b_3] \subset b$.
The Killing form for~$E_6$ shows that the elements of~$b_2$, along with~$B^{(2)}_{tz} + R^{(2)}_{xz}$ and~$B^{(2)}_{t\l}+R^{(2)}_{x\l}$, are null transformations.
\section{Gell-Mann Matrices and~$su(3) \subset G_2$}
\label{ch:E6_basic_structure.Gell-Mann}
We list in Table~\ref{table:gell-mann_and_su3} the isomorphism between the~$3 \times 3$ Gell-Mann matrices and the subalgebra~\hbox{$su(3, \mathbb{C})^C \subset G_2$}. This isomorphism respects~$\left[ \hspace{.15cm}, \hspace{.15cm} \right]$ in the sense that the structure constants are the same for each algebra. We use~$\iota$ to denote a square root of~$-1$ which commutes with all octonions. This is required, since the Gell-Mann matrices are Hermitian and use physicists' conventions, while our~$su(3,\mathbb{C})^C$ uses mathematicians' conventions.
\begin{table}[htbp]
\begin{center}
\begin{tabular}{|cl|}
\hline
Octonionic~$su(3)$ & Gell-Mann \\
Transformation & Matrix \\
\hline
$\dot A_k $ & $\lambda_1= - \iota \left(\begin{array}{ccc}0 & 1 & 0 \\1 & 0 & 0 \\0 & 0 & 0 \\\end{array} \right) $ \\
$\dot A_{k\l}$ & $\lambda_2= - \iota \left( \begin{array}{ccc}0 & -i & 0 \\i & 0 & 0 \\0 & 0 & 0 \\\end{array} \right) $\\
$\dot A_\l$ & $ \lambda_3= \iota \left( \begin{array}{ccc}1 & 0 & 0 \\0 & -1 & 0 \\0 & 0 & 0 \\\end{array} \right)$\\
$\dot A_i$ & $\lambda_4= - \iota \left( \begin{array}{ccc} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \\ \end{array} \right)$ \\
$\dot A_{i\l}$ & $ \lambda_5= \iota \left( \begin{array}{ccc}0 & 0 & -i \\0 & 0 & 0 \\i & 0 & 0 \\ \end{array} \right)$ \\
$\dot A_{j\l}$ & $ \lambda_6= - \iota \left( \begin{array}{ccc}0 & 0 & 0 \\0 & 0 & 1 \\0 & 1 & 0 \\ \end{array} \right) $ \\
$\dot A_{j}$ & $ \lambda_7= - \iota \left( \begin{array}{ccc}0 & 0 & 0 \\0 & 0 & -i \\0 & i & 0 \\ \end{array} \right) $\\
$ \dot G_\l$ & $\lambda_8= - \iota \sqrt{3} \left( \begin{array}{ccc}\frac{1}{\sqrt{3}} & 0 & 0 \\0 & \frac{1}{\sqrt{3}} & 0 \\0 & 0 & -\frac{2}{\sqrt{3}} \end{array} \right)$ \\
\hline
\end{tabular}
\caption{\noindent Isomorphism between~$su(3,\mathbb{C})^C$ and Gell-Mann matrices}
\label{table:gell-mann_and_su3}
\end{center}
\end{table}
\newpage{}
\part{The Further Structure of~$E_6$}
\label{ch:E6_further_structure}
In this chapter, we identify additional subalgebras and subalgebra chains of~$sl(3,\mathbb{O})$, our preferred real form of~$E_6$. We identify real subalgebras of~$sl(3,\mathbb{O})$ by adapting techniques from the study of complex Lie algebras. In particular, if~$g$ is a real form of a complex Lie algebra~$g^\mathbb{C}$, we use a map~$\phi$ which sends the complex Lie algebra~$g^\mathbb{C}$ to itself while sending the particular real algebra~$g$ to another real form~$g^\prime$ of~$g^\mathbb{C}$. The algebra~$g \cap g^\prime$ is then an easily identifiable subalgebra of~$g$. In addition, the maximal compact subalgebra~$g^\prime_{c}$ of~$g^\prime$ can be easily identified, providing a simple way to identify the subalgebra~$\phi^{-1}\left(g^\prime_{c}\right)$ in the original algebra~$g$.
The chapter is organized as follows. We discuss the properties of automorphisms of Lie algebras, particularly involutive automorphisms, in Section~\ref{ch:E6_further_structure.Automorphisms_of_Lie_Algebras}. These automorphisms will later be used to find subalgebras of~$sl(3,\mathbb{O})$ in Sections~\ref{ch:E6_further_structure.three_automorphisms} and~\ref{ch:E6_further_structure.more_applications_of_three_automorphisms}. While the automorphisms may be used to find subalgebras of~$sl(3,\mathbb{O})$, we still must identify and classify those subalgebras. Section~\ref{ch:E6_further_structure.Automorphisms_of_Lie_Algebras.id_algebras} reviews the methods we utilize to determine whether a given subalgebra of~$sl(3,\mathbb{O})$ is semi-simple, simple, or neither, in addition to developing methods to classify it as a classical matrix algebra. We construct three involutive automorphisms in Section~\ref{ch:E6_further_structure.three_automorphisms}, which we then use to find some subalgebras of~$sl(3,\mathbb{O})$ which we did not find in Chapter~\ref{ch:E6_basic_structure}. These automorphisms are combined in Section~\ref{ch:E6_further_structure.more_applications_of_three_automorphisms} to give new involutive automorphisms and new subalgebras of~$sl(3,\mathbb{O})$. These subalgebras are used to fill out the chain of subalgebras contained within~$sl(3,\mathbb{O})$ in Section~\ref{ch:E6_further_structure.chains_of_subalgebras}, where we construct a tower of subalgebras in~$sl(3,\mathbb{O})$ based upon our preferred choice of Casimir operators.
\section{Automorphisms of Lie Algebras}
\label{ch:E6_further_structure.Automorphisms_of_Lie_Algebras}
We summarize here the standard definitions and treatment of automorphisms of Lie algebras, according to~\cite{gilmore}. These methods will be used in Sections~\ref{ch:E6_further_structure.three_automorphisms} and~\ref{ch:E6_further_structure.more_applications_of_three_automorphisms} to identify simple subalgebras of~$sl(3,\mathbb{O})$.
Let~$g$ be a real form of the complex Lie algebra~$g^\mathbb{C}$.
An {\it automorphism of~$g^\mathbb{C}$} is a map~\hbox{$\phi : g^\mathbb{C} \to g^\mathbb{C}$} which is an isomorphism of the underlying vector space over~$\mathbb{C}$ and satisfies
$$ \phi \left(\left[X,Y\right]\right) = \left[ \phi(X), \phi(Y) \right] \hspace{1cm} X,Y \in g$$
This condition means that~$\phi$ respects the algebraic structure of~$g^\mathbb{C}$. An {\it involutive automorphism of~$g^\mathbb{C}$} is an automorphism~$\phi$ such that
$$\phi^2 = \textrm{Id}$$
As discussed in~\cite{gilmore}, involutive automorphisms map simple Lie algebras to simple Lie algebras. Hence,
{\bf Lemma: } If~$\phi : g^\mathbb{C} \to g^\mathbb{C}$ is an involutive automorphism of the associated complex Lie algebra~$g^\mathbb{C}$ of a real semi-simple Lie algebra~$g$, then the subalgebra~$g_1 \subset g$ is simple if and only if~$\phi\left( g_1 \right)$ is simple.
Proof: Assume that~$g_1$ is not simple. That is, either~$g_1$ is abelian, or it possesses a proper invariant subalgebra~$h_1$. If~$g_1$ is abelian, then for every~$x,y \in g_1$, we have
$$\left[ \phi(x), \phi(y) \right] = \phi \left( \left[ x, y \right] \right) = 0$$
showing that~$\phi(g_1)$ is also abelian. On the other hand, if~$h_1 \subset g_1$ is a proper invariant subalgebra of~$g_1$, then for every~$x \in g_1, y \in h_1$, we have~$[x,y] \in h_1$.
This implies
$$\left[ \phi(x), \phi(y) \right] = \phi \left( \left[ x, y \right] \right) \in \phi(h_1)$$
Hence,~$\phi(h_1)$ is a proper invariant subalgebra of~$\phi(g_1)$. We conclude that if~$g_1$ is not simple, then~$\phi(g_1)$ is not simple. Since involutive automorphisms are invertible, we can reverse the arguments above to show that~$g_1$ is not simple if~$\phi(g_1)$ is not simple, thus proving our claim.
\hspace{5.40in}$\square$
The set of automorphisms form a group, as do the set of involutive automorphisms. This property of involutive automorphisms will be utilized in Section~\ref{ch:E6_further_structure.more_applications_of_three_automorphisms}. Finally, as mentioned in Section~\ref{ch:Lie_groups_Lie_algebras.Basic_Definitions.regular_rep_Killing_form}, we note that if~$\phi$ is an automorphism of a Lie algebra~$g$ with Killing form~$B$, then~$B(\phi(x), \phi(y)) = B(x,y)$ for all~$x,y \in g$. This interaction between the Killing form of a complex Lie algebra~$g^\mathbb{C}$ and one of its involutive automorphisms allows us to naturally separate~$g^\mathbb{C}$ into two orthogonal subspaces, which we now show. Further details may be found in~\cite{gilmore}.
Suppose~$g^\mathbb{C}$ is a semi-simple complex Lie algebra and~$\phi : g^\mathbb{C} \to g^\mathbb{C}$ is an involutive automorphism. The following argument from~\cite{gilmore} shows that we may use~$\phi$ to naturally separate~$g^\mathbb{C}$ into two useful subspaces. Since~$\phi^2 - \textrm{Id} = \left( \phi - \textrm{Id}\right)\left(\phi + \textrm{Id}\right) = 0$, the involutive automorphism~$\phi$ splits the vector space~$g^\mathbb{C}$ into subspaces~$\mathcal{P}$ and~$\mathcal{N}$ which are associated with the eigenvalues~$+1$ and~$-1$ of~$\phi$, respectively. Using these eigensubspaces, we write
$$g^\mathbb{C} = \mathcal{P} \oplus \mathcal{N}$$
where we note that this is not a direct sum of Lie algebras! However, as
$$
\begin{array}{ccl}
\phi\left(g^\mathbb{C}\right) &=& \phi\left(\mathcal{P}\right) \oplus \phi\left(\mathcal{N}\right)\\
&=& (+1)\mathcal{P} \oplus (-1)\mathcal{N}
\end{array}
$$
and the Killing form of the semi-simple Lie algebra~$g^\mathbb{C}$ is non-degenerate, then
$$B\left(\mathcal{P}, \mathcal{N}\right) = B\left(\phi\left(\mathcal{P}\right), \phi\left(\mathcal{N}\right)\right) = B\left(+\mathcal{P},-\mathcal{N}\right) =-B\left(\mathcal{P},\mathcal{N} \right) = 0$$
showing that~$\mathcal{P}$ and~$\mathcal{N}$ are orthogonal as vector spaces.
We use the Killing form and the involutive automorphism~$\phi$ to show that~$\mathcal{P}$ closes under commutation, that~$\left[ \mathcal{N}, \mathcal{N} \right] \subset \mathcal{P}$, and that~$\left[ \mathcal{P}, \mathcal{N} \right] \subset \mathcal{N}$.
For any~$p_1, p_2 \in \mathcal{P}$ and any~\hbox{$n \in \mathcal{N}$}, we have
$$\begin{array}{ccl}
B\left(\left[p_1,p_2\right], n \right) & = &B\left(\left[\phi(p_1),\phi(p_2)\right], \phi(n) \right) \\
& = &B\left(\left[+p_1, +p_2\right], -n \right) \\
& = &-B\left(\left[p_1, p_2\right], n \right) \\
& = & 0 \\
\end{array}
$$
Hence,~$\left[ \mathcal{P}, \mathcal{P} \right]$ is orthogonal to~$\mathcal{N}$, implying~$\left[ \mathcal{P}, \mathcal{P} \right] \subset \mathcal{P}$ since~$B$ is non-degenerate. Thus,~$\mathcal{P}$ closes under commutation and is a subalgebra of~$g$. Note that if we had chosen~$p_1, p_2 \in \mathcal{N}$ instead, then the two plus signs in the commutator in the second line would become negative signs, resulting in no overall sign change. Hence, the same calculation shows that~$\left[ \mathcal{N}, \mathcal{N} \right] \subset \mathcal{P}$. Finally,
for~$n_1 \in \mathcal{N}$ and~$p_1,p_2 \in \mathcal{P}$, we have
$$\begin{array}{ccl}
B\left(\left[n_1,p_1\right], p_2 \right) &=& B\left(\left[\phi(n_1), \phi(p_1)\right], \phi(p_2)\right) \\
&=& B\left(\left[-n_1, p_1\right], p_2\right) \\
&=& -B\left(\left[n_1, p_1\right], p_2\right) \\
&=& 0 \\
\end{array}
$$
This shows~$\left[\mathcal{N}, \mathcal{P}\right]$ is orthogonal to~$\mathcal{P}$, resulting in ~$\left[ \mathcal{N}, \mathcal{P} \right] \subset \mathcal{N}$.
This proves~$\mathcal{P}$ and~$\mathcal{N}$ are orthogonal complementary subspaces of~$g^\mathbb{C}$.
When restricted to~$g$, a slight modification of the involutive automorphism~$\phi$ can be used to produce another real form of~$g^\mathbb{C}$. We again use~$\phi$ to split~$g$ into subspaces~$\mathcal{P} \oplus \mathcal{N}$, and then introduce the involutive automorphism~$\phi^* : g^\mathbb{C} \to g^\mathbb{C}$ via
$$\phi^*(p+n) = \phi(p) + \xi \phi(n)$$
where~$p \in \mathcal{P}$,~$n \in \mathcal{N}$, and~$\xi$ is a square root of~$-1$ which commutes with all imaginary units used in the representation of~$g$.
As mentioned in Section~\ref{ch:Lie_groups_Lie_algebras.Basic_Definitions.Lie_Algebras}, the structure constants of a real form~$g$ of~$g^\mathbb{C}$ are real, and since
$$ \left[ \mathcal{P}, \mathcal{P} \right] \subset \mathcal{P} \hspace{1cm} \left[ \mathcal{P}, \xi \mathcal{N} \right] \subset \xi \mathcal{N} \hspace{1cm} \left[ \xi \mathcal{N}, \xi \mathcal{N}, \right] \subset (\xi)^2 \mathcal{P} = (-1)\mathcal{P}$$
then~$\mathcal{P} \oplus \xi \mathcal{N}$ also has real structure constants. It is a different real form of the complex Lie algebra~$g^\mathbb{C}$. Further information regarding the interplay between involutive automorphisms, the Killing form, and real forms of a complex Lie algebra may be found in~\cite{gilmore}.
\subsection{Identification of Algebras}
\label{ch:E6_further_structure.Automorphisms_of_Lie_Algebras.id_algebras}
In Sections~\ref{ch:E6_further_structure.three_automorphisms} and~\ref{ch:E6_further_structure.more_applications_of_three_automorphisms}, we will use involutive automorphisms to find many semi-simple real subalgebras of~$sl(3,\mathbb{O})$. Identifying these subalgebras can be fairly easy if we know they are simple. Establishing whether a subalgebra is simple or not, however, may be time consuming. In this section, we outline the various methods we use for identifying simple subalgebras of~$sl(3,\mathbb{O})$, and verifying that they are indeed simple.
In the following discussion,~$g$ and~$g^\prime$ are real Lie algebras. The complexification of~$g$ is denoted~$g^\mathbb{C}$. We are primarily interested in the real subalgebras of~$sl(3,\mathbb{O})$, and may utilize a procedure in our specific case even though it does not apply to more general Lie algebras. We also find it useful to consider the following example to illustrate how the various methods may be used:
{\bf Example 1: } Let~$\varrho$ be a subalgebra of~$sl(3,\mathbb{O})$ whose dimension is~$36$, and assume that~$\varrho$ contains precisely~$4$ Casimir operators. We note that~$\varrho$ has rank four if it is simple, but we do not assume that~$\varrho$ is simple or semi-simple.
\subsubsection{Traditional Methods }
\label{traditional_methods}
The classic approach to classifying the real Lie algebra~$g$ is to utilize its Killing form~$B$ and construct the Dynkin diagram for its associated complex Lie algebra~$g^\mathbb{C}$. As mentioned in Section~\ref{ch:Lie_groups_Lie_algebras.Basic_Definitions.regular_rep_Killing_form}, if the Killing form is non-degenerate, then~$g$ is semi-simple. According to~\cite{dyn2}, if~$g$ is semi-simple, then it is possible to construct the Dynkin diagram associated with~$g^\mathbb{C}$. This diagram will be connected if and only if~$g^\mathbb{C}$ is simple~\cite{dyn2}. In this case, the Dynkin diagram uniquely identifies the complex Lie algebra~$g^\mathbb{C}$ as one of the classical Lie algebras~$A_l, B_l, C_l, D_l$ for~$l \ge 1$ or as one of the exceptional Lie algebras~$G_2, F_4, E_6, E_7$, or~$E_8$. Conversely, if~$g^\mathbb{C}$ is semi-simple but the Dynkin diagram is not connected, then~$g^\mathbb{C}$ is a direct sum of simple complex Lie algebras which can be identified from the resulting diagram. Further information regarding these methods may be found in~\cite{cornwell, gilmore}.
We also note that if~$g$ is a simple Lie algebra, then the Killing form provides the signature of~$g$. As stated in~\cite{gilmore}, except in some cases involving real forms of~$A_l$ and~$D_l$ for various values of~$l$, if~$g$ is not compact, then the signature, rank, and dimension of~$g$ identifies~$g$ as a particular real form of~$g^\mathbb{C}$. We do not encounter any of the exceptions to this rule, due to the rank and signature of these algebras. Tables of the various real forms of the complex Lie algebras may be found in~\cite{gilmore}.
For the algebra~$\varrho$ in Example 1, if we know that~$\varrho$ is simple, then we may classify it as either a real form of~$B_4$ or~$C_4$. In this case, the signature of~$\varrho$ will usually help us identify the particular real form when we use the tables listed in~\cite{gilmore} by matching the rank, dimension, and signature. This will fail, however, if~$\varrho$ is compact. If it is not known that~$\varrho$ is simple, we note that it is a tedious process to calculate the Killing form of~$\varrho$ and to construct the Dynkin diagram of~$\varrho^\mathbb{C}$ for such a large algebra. Indeed, while the traditional methods described above are effective, we prefer to find other methods which may be more helpful in identifying subalgebras of~$sl(3,\mathbb{O})$ with less effort.
\subsubsection{Counting Methods for Complex Algebras}
According to~\cite{gilmore}, the complex simple Lie algebra~$g^\mathbb{C}$ may be identified by its dimension~$n$ and rank~$l$ in all but a few cases. The complex Lie algebra~$A_l$ has dimension~$n = l^2-1$ for~$l \ge 1$ and the dimension of~$D_l$ is~$n = l(2l-1)$ for~$l \ge 4$. The exceptional Lie algebras~$G_2$,~$F_4$,~$E_6$,~$E_7$, and~$E_8$ have dimensions~$14, 52, 78, 133$, and~$248$. The subscript indicates the rank of each algebra. These algebras have different dimensions and rank. The main exception is that both~$B_l$ and~$C_l$ have dimension~$l(2l+1)$, but that for~$l \ge 3$, these algebras are not isomorphic. We also note that~$B_6$ and~$C_6$ have the same dimension as~$E_6$.
This counting argument can help us classify the complexified algebra of a simple real Lie algebra~$g$. As noted in Section~\ref{traditional_methods}, we may then use the signature of~$g$ and tables found in~\cite{gilmore} to identify the particular real form of~$g$. However, this method fails if it is not known that~$g$ is semi-simple, as is the case with the algebra~$\varrho$ in Example 1.
For the algebra~$\varrho$ in Example 1, if we know that~$\varrho$ is simple but do not want to determine its Killing form~$B$, then there is still a method that may identify~$\varrho$ as either a real form of~$B_4$ or~$C_4$. As shown in Figure~\ref{fig:subalgebras_of_E6} in Section~\ref{ch:Joma_Paper.subalgebras_greater_than_3}, the Lie algebra~$B_4$ contains a subalgebra~$D_3$ of dimension~$28$, while the largest simple subalgebra contained within~$C_4$ is~$C_3$, which has dimension~$21$. Hence, if we are able to find a simple subalgebra of dimension~$28$ within~$\varrho$, then we know that~$\varrho$ must be a real form of~$B_4$. We note that the failure to find such a subalgebra does not mean~$\varrho$ must be a real form of~$C_4$, as not every real form of a complex algebra~$g^\mathbb{C}$ will contain real forms of the complex subalgebras of~$g^\mathbb{C}$.
\subsubsection{Involutive Automorphisms and Compact Subalgebras}
As shown in~\cite{gilmore} and listed in its tables of real forms of complex Lie algebras, each real form~$g$ and~$g^\prime \ne g$ of~$g^\mathbb{C}$ contains a maximal compact subalgebra~$g_1 \oplus g_2 \oplus \cdots \oplus g_k \subset g$ and~$g^\prime_1 \oplus \cdots \oplus g^\prime_j \subset g^\prime$, respectively. According to~\cite{cornwell}, there is an involutive automorphism~$\phi^* : g^\mathbb{C} \to g^\mathbb{C}$ such that~$\phi^*(g) = g^\prime$ and~$\phi^*(g) \cap g = g_1 \oplus g_2 \oplus \cdots \oplus g_k$ is the maximal compact subalgebra of~$g$. When~$g \subset sl(3,\mathbb{O})$, this method merely identifies the maximal compact subalgebra of~$g$. However, the pre-image of the maximal compact subalgebra of~$g^\prime = \phi^*(g)$ is also a subalgebra of~$g$ and the involutive automorphism may be used to determine the signature of this subalgebra in~$g$. Again,~\cite{gilmore} lists the semi-simple decomposition of this non-compact subalgebra in~$g$.
Given a subalgebra~$g \subset sl(3,\mathbb{O})$, assume we know~$g$ should have the semi-simple decomposition~$g = g_1 \oplus g_2 \oplus \cdots \oplus g_k$ where~$|g_1| \le |g_2| \le \cdots \le |g_k|$ and~$|g_i|$ is the dimension of~$g_i$. By identifying the smallest subalgebras~$g_1, \cdots, g_{k-1}$ in~$g$ which commute with each other and also which commute with~$g_k = g - g_1 - \cdots - g_{k-1}$, we automatically show that~$g_k$ is simple. This method is most effective when~$k = 2$ or~$k=3$ and the dimension of~$g_k$ is very large compared to~$g_1, \cdots, g_{k-1}$. Effectively, by finding some very small simple Lie algebras in~$g$, this method allows us to find a very large simple subalgebra of~$g$ with very little work.
We now show how this method may be used to show an algebra is simple by applying it the algebra~$\varrho$ given in Example 1. By assumption,~$\varrho$ is a~$36$-dimensional subalgebra~$\varrho$ of~$sl(3,\mathbb{O})$. First, we note that there is a real form~$g^\prime$ of~$E_6$ with signature~$(36,42)$, and that~\cite{gilmore} identifies its compact maximal subalgebra, denoted~$\mathcal{P}$, as~$C_4$. There are involutive automorphisms~$\phi^*: E_6 \to E_6$ such that~$\phi(sl(3,\mathbb{O})) = g^\prime$. We search for one such that the inverse image of~$\mathcal{P}$ is~\hbox{$\varrho \in sl(3,\mathbb{O})$}. If this succeeds, we have identified~$\varrho$ as a real form of~$C_4$. If it fails, however, all is not lost - we have still shown that the pre-image of~$\mathcal{P}$ in~$sl(3,\mathbb{O})$ is both simple and a real form of~$C_4$.
\subsubsection{Counting for Matrix Groups}
We now show how to classify the simple real classical Lie algebras, which are $su(p,q,\mathbb{F})$ and~$sl(n,\mathbb{F})$ for~$\mathbb{F} = \mathbb{R}, \mathbb{C}, \mathbb{H},$ or~$\mathbb{O}$. While the signature of a simple algebra~$g$ was used in the past to help identify a maximal compact subalgebra, we will now see that its particular real form is identified by the algebra's signature, rank, and dimension.
For~$\mathbb{F} = \mathbb{R}, \mathbb{C}, \mathbb{H}$, or~$\mathbb{O}$, and for~$n \ge 1$ (with~$n \le 3$ when~$\mathbb{F} = \mathbb{O}$), let~$|so(\textrm{im}(\mathbb{F}))|$ denote the dimension of the special orthogonal group acting on the imaginary basis units of the division algebra~$\mathbb{F}$. Hence, we have
$$
\begin{array}{ccc}
|so(\textrm{im}(\mathbb{R}))| = |so(0)| = 0 & \hspace{1cm} & |so(\textrm{im}(\mathbb{C}))| = |so(1)| = 0 \\
|so(\textrm{im}(\mathbb{H}))| = |so(3)| = 3 & & |so(\textrm{im}(\mathbb{O}))| = |so(7)| = 21
\end{array}
$$
\noindent We note that~$Aut(\mathbb{H}) = so(\textrm{im}(\mathbb{H}))$, as they are both~$so(3)$, and that
$$G_2 = Aut(\mathbb{O}) \subset so(\textrm{im}(\mathbb{O}))$$
We now derive a formula for the dimension of~$SU(n,\mathbb{F}), \mathbb{F} \ne \mathbb{O}$ by counting the dimension of its associated Lie algebra~$su(n,\mathbb{F})$. Let~$|\mathbb{F}|$ denote the dimension of the division algebra~$\mathbb{F}$.
An~$n \times n$ matrix~$m \in su(n,\mathbb{F})$ satisfies~$m^\dagger = -m$ and~$\tr{m} = 0$. For the diagonal entries of~$m$, the condition~$m^\dagger = -m$ guarantees the diagonal entries are pure imaginary. Combined with the condition~$\tr{m} = 0$, there are~$(|\mathbb{F}|-1)(n-1)$ real degrees of freedom on the main diagonal. The condition~$m^\dagger = -m$ eliminates half of the~$|\mathbb{F}|(n(n-1))$ real degrees of freedom for the off-diagonal entries of~$m$. Finally, we add~$|so(\textrm{im}(\mathbb{F}))|$ degrees of freedom for the transformations which mix up the imaginary units of the division algebra~$\mathbb{F}$. Thus, the Lie group~$SU(n,\mathbb{F})$ has dimension
$$|SU(n,\mathbb{F})| = |su(n,\mathbb{F})| = (|\mathbb{F}|-1)(n-1) + |\mathbb{F}|\frac{n(n-1)}{2} + |so(\textrm{im}(\mathbb{F}))|$$
which reduces to the following dimensions for the division algebras over~$\mathbb{F} = \mathbb{R}, \mathbb{C}$, and~$\mathbb{H}$:
$$
\begin{array}{|c|c|c|}
\hline
\mathbb{F} & SU(n,\mathbb{F}) & \textrm{Dimension} \\
\hline
\mathbb{R} & SO(n,\mathbb{R}) & \frac{n(n-1)}{2} \\
\mathbb{C} & SU(n,\mathbb{C}) & (n+1)(n-1) = n^2 - 1\\
\mathbb{H} & SU(n,\mathbb{H}) = SP(2\cdot n) & n(2n+1) \\
\hline
\end{array}
$$
Traditionally, the Lie algebra~$sp(n)$ consists of~$2n \times 2n$ complex matrices constructed in a way that resembles the representation of quaternions via~$2 \times 2$ complex matrices. We do not use this convention, and instead use a representation which truly uses quaternions. With our convention, the matrices in~$su(n,\mathbb{H}) = sp(n)$ are~$n \times n$ matrices.
In the case~$\mathbb{F} = \mathbb{O}$, the above formula produces~$|su(1,\mathbb{O})| = 21$ and~$|su(2,\mathbb{O})| = 36$, matching the dimensions of~$so(7)$ and~$so(9)$, respectively.
For the case~$n = 3$, we must modify the formula since seven of the~$so(\textrm{im}(\mathbb{F}))$ transformations (which are not in~$G_2$) can be generated from the type~$2$ and type~$3$ off-diagonal generators. This over-counting is a result of triality, since the three copies of~$so(7)$ (each of which contains~$G_2$) in~$so(8)$ are not independent (see Section~\ref{ch:E6_basic_structure.Triality}). Thus, we reduce the dimension of~$su(3,\mathbb{O})$ from~$59$, as given by the formula, to~$52$, which is the dimension of~$F_4$.
$$
\begin{array}{|c|c|c|}
\hline
\mathbb{F} = \mathbb{O} & SU(n,\mathbb{F}) & \textrm{Dimension} \\
\hline
& SU(1,\mathbb{O}) & 21 \\
& SU(2,\mathbb{O}) & 36 \\
& SU(3,\mathbb{O}) & 52 \\
\hline
\end{array}
$$
To produce a formula for~$SL(n,\mathbb{F})$, we again count the dimension of the associated Lie algebra~$sl(n,\mathbb{F})$. The~$n \times n$ matrix~$m \in sl(n,\mathbb{F})$ satisfies~$m^\dagger = \pm m$ and~$\tr{m} = 0$. When compared with the conditions for~$m \in su(n,\mathbb{F})$, there are~$n-1$ additional degrees of freedom for the diagonal entries and~$|\mathbb{F}|\frac{n(n-1)}{2}$ additional degrees of freedom for the off-diagonal entries of the~$sl(n,\mathbb{F})$ matrices than existed for the~$su(n,\mathbb{F})$ matrices.
This produces the dimension formula
$$|sl(n,\mathbb{F})| = |su(n,\mathbb{F})| + (n-1) + |\mathbb{F}|\frac{n(n-1)}{2}$$
\noindent This formula gives the following dimensions of~$SL(n,\mathbb{F})$ for~$\mathbb{F} = \mathbb{R}, \mathbb{C}$, and~$\mathbb{H}$:
$$
\begin{array}{|c|c|c|}
\hline
\mathbb{F} & SL(n,\mathbb{F}) & \textrm{Dimension} \\
\hline
\mathbb{R} & SL(n,\mathbb{R}) & (n+1)(n-1) \\
\mathbb{C} & SL(n,\mathbb{C}) & 2(n-1)(n+1) \\
\mathbb{H} & SL(n,\mathbb{H}) & 4n^2 - 1 \\
\hline
\end{array}
$$
In the case~$\mathbb{F} = \mathbb{O}$, when~$n = 1$, we find the dimension of~$sl(1,\mathbb{O})$ is~$21$. In fact, for all~$\mathbb{F}$, we have~$|sl(1,\mathbb{F})| = |su(1,\mathbb{F})|$,
since~$sl(n,\mathbb{F})$ adds Hermitian trace-free matrices to the anti-Hermitian trace-free matrices of~$su(n,\mathbb{F})$, and in the case~$n =1$, no such matrices may be added to~$su(1,\mathbb{F})$.
For~$\mathbb{F} = \mathbb{O}$ and~$n = 2, 3$, the formula gives the dimensions of~$sl(2,\mathbb{O})$ and~$sl(3,\mathbb{O})$ as~$45$ and~$78$, which are the dimensions of~$so(9,1)$ and~$E_6$, respectively.
$$
\begin{array}{|c|c|c|}
\hline
\mathbb{F} = \mathbb{O} & SU(n,\mathbb{F}) & \textrm{Dimension} \\
\hline
& SL(1,\mathbb{O}) & 21 \\
& SL(2,\mathbb{O}) & 45 \\
& SL(3,\mathbb{O}) & 78 \\
\hline
\end{array}
$$
Finally, we note that~$su(p,q,\mathbb{F})$ is another real form of the real algebra~$su(n,\mathbb{F})$ when~$p+q = n$. Thus, both~$su(p,q,\mathbb{F})$ and~$su(n,\mathbb{F})$ have the same dimension and rank. The algebra~$su(n,\mathbb{F})$ is compact, containing no boosts, while the algebra~$su(p,q,\mathbb{F})$ contains~$|\mathbb{F}|pq$ boosts. The algebra~$sl(p,q,\mathbb{F})$ is~$sl(n,\mathbb{F})$, since~$sl(n,\mathbb{F})$ already contains the anti-hermitian and hermitian matrices which correspond to the compact and non-compact generators needed for~$su(p,q,\mathbb{F})$. Nevertheless, we will use~$sl(p,q,\mathbb{F})$ in certain instances to differentiate the algebra from another (preferred) algebra~$sl(n,\mathbb{F})$.
We conclude that in all but a few cases (which are not relevant for subalgebras of~$sl(3,\mathbb{O})$), knowing the rank, dimension, and signature of a finite-dimensional simple Lie algebra~$g$ is enough information to identify~$g$ as~$su(p,q,\mathbb{F})$ or as~$sl(n,\mathbb{F})$, where~$\mathbb{F} = \mathbb{R}, \mathbb{C}, \mathbb{H}, \mathbb{O}$ and~$n = p + q$. The exceptions to this rule are listed in~\cite{gilmore}.
This conclusion is very useful for identifying the particular real form of a simple Lie algebra contained within~$sl(3,\mathbb{O})$. Consider the~$36$-dimensional algebra~$\varrho$ in Example 1, which was introduced as the start of Section~\ref{ch:E6_further_structure.Automorphisms_of_Lie_Algebras.id_algebras}. Assuming that we have shown~$\varrho$ is simple, then we know~$g$ is either a real form of~$B_4$ or~$C_4$ since it has dimension~$36$ and rank~$4$. If~$\varrho$ contains~$8$ boosts, then it must be~$so(8,1,\mathbb{R})$, while if it contains 12 boosts, then it must be~$su(3,1,\mathbb{H})$. If~$\varrho$ is compact, we may use an involutive automorphism~$\phi^* : \varrho^* \to \varrho^*$ sending~$\varrho$ to another real form of its complexification~$\varrho^*$. Now, depending on the number of boosts in ~$\phi^*(\varrho)$, we determine that~$g^*$ is either~$B_4$, in which case~$\varrho = so(9,\mathbb{R})$, or~$C_4$, in which case~$\varrho = sp(4)$.
\subsubsection{Summary}
We summarize here the methods which allow us to quickly find and identify subalgebras of~$sl(3,\mathbb{O})$. We use an involutive automorphism~$\phi^*: g^\mathbb{C} \to g^\mathbb{C}$ not only to identify different real forms of~$g^\mathbb{C}$, but also to identify real subalgebras of our particular real form~$g$. When applied to the compact real form~$g = \mathcal{P} \oplus \mathcal{N}$, the map~$\phi^*$ changes the signature from~$(|\mathcal{P}|+|\mathcal{N}|, 0)$ to~$(|\mathcal{P}|,|\mathcal{N}|)$. Similar counting arguments give the signature of~$\phi^*(g)$ when~$g$ is non-compact, and we note that~$\phi^*$ changes some compact generators into non-compact generators, and vice-versa. We then use the rank, dimension, and signature of~$\phi^*(g)$ to identify the particular real form of the algebra. Using tables, found in~\cite{gilmore}, of real forms of~$g^\mathbb{C}$ that list the algebra's maximal compact subalgebra, we then identify~$\mathcal{P} = g \cap \phi^*(g)$ as a subalgebra of our original real form~$g$. In addition, it is very easy to identify the maximal compact subalgebra~$g_{(c)}$ of~$\phi^*(g)$. We then identify the pre-image of~$g_{(c)}$ as a non-compact subalgebra of~$g$. The involutive automorphisms~$\phi$, and their complex extensions~$\phi^*$, provide significant insight into the structure of our real form~$g = sl(3,\mathbb{O})$ of the algebra~$E_6$.
\section{Three Important Involutive Automorphisms of~$sl(3,\mathbb{O})$}
\label{ch:E6_further_structure.three_automorphisms}
We first make some comments about our preferred basis for~$sl(3,\mathbb{O})$, which is listed in Table~\ref{table:our_basis}. Let~$B$ and~$R$ be the vector subspaces consisting of boosts and rotations, respectively. Our preferred choice of basis favors type~$1$ transformations, in the sense that we choose to use the linear dependencies given in Section~\ref{ch:E6_basic_structure.ConstructingE6algebra.LinearDependencies} for~$so(8)$
to express~$G_2$,~$so(7)$, and~$so(8)$ transformations in terms of type~$1$ transformations.
Let~$T_1$ be the subspace spanned by~$\lbrace \dot B^{(2)}_{tz} - \dot B^{(3)}_{tz} \rbrace~$ and all type~$1$ transformations, and~$T_2$ and~$T_3$ the subspaces spanned by the type~$2$ and type~$3$ transformations which are not in~$T_1$.\footnote{With the condition~$\dot B^{(1)}_{tz} + \dot B^{(2)}_{tz} + \dot B^{(3)}_{tz} = 0$, we note that both~$\dot B^{(2)}_{tz}$ and~$\dot B^{(3)}_{tz}$ are viewed as elements of~$T_1$ since~$T_1$ contains~$\dot B^{(1)}_{tz}$ and~$\dot B^{(2)}_{tz} - \dot B^{(3)}_{tz}$.}
Let~$T_{(2,3)} = T_2 + T_3$. Finally, let~$H$ be the subspace spanned by those transformations with no labels from~$\lbrace i,j,j\l,i\l \rbrace$ and~$H^\perp$ the subspace spanned by those transformations labeled with one label from~$\lbrace i, j, j\l, i\l \rbrace$. This subspace~$H$ contains {\it quaternionic} basis elements (using the quaternionic algebra~$\langle 1, k, k\l, \l \rangle$) such as~$\dot A_{k\l}, \dot R^{(1)}_{z\l}$, and~$\dot B^{(3)}_{tx}$, while~$H^\perp$ contains {\it orthogonal-quaternionic} basis elements such as~$\dot A_i, \dot R^{(1)}_{zi\l},$ and~$\dot B^{(3)}_{tj\l}$.
The results from Section~\ref{ch:E6_further_structure.Automorphisms_of_Lie_Algebras} show that as a vector space,~$sl(3,\mathbb{O})$ splits into the pairs of orthogonal spaces~$R \oplus B$,~$T_1 \oplus T_{(2,3)}$, and~$H \oplus H^\perp$. Indeed, the pairs of orthogonal spaces~$R \oplus B$,~$T_1 \oplus T_{(2,3)}$, and~$H \oplus H^\perp$ satisfy the commutation relations:
$$
\begin{array}{ccc}
\left[R, R\right] \subset R & \left[T_1, T_1 \right] \subset T_1 & \left[H, H\right] \subset H \\
\left[R, B\right] \subset B & \hspace{1cm} \left[T_1, T_{(2,3)} \right] \subset T_{(2,3)} \hspace{1cm} & \left[H, H^\perp\right] \subset H^\perp \\
\left[B, B\right] \subset R & \left[T_{(2,3)}, T_{(2,3)} \right] \subset T_1 & \left[H^\perp, H^\perp\right] \subset H \\
\end{array}
$$
We define the three maps~$\phi_{(t)}$,~$\phi_{(2,3)}$, and~$\phi_{H}$ from~$sl(3,\mathbb{O})$ to~$sl(3,\mathbb{O})$, according to the rules
$$
\begin{array}{ccc}
\phi_{(t)}(r + b) = r + \xi b & \hspace{1cm} & r \in R, b \in B \\
\phi_{(2,3)}(g_1 + g_{(2,3)}) = g_1 + \xi g_{(2,3)} & & g_1 \in T_1, g_{(2,3)} \in T_{(2,3)} \\
\phi_{(H^\perp)}(h + h^\prime) = h + \xi h^\prime & & h \in H, h^\prime \in H^\perp\\
\end{array}
$$
where~$\xi = -1$. These maps respect the algebraic structure of~$sl(3,\mathbb{O})$, and each clearly satisfies~$\phi^2 = Id$. Thus,~$\phi_{(t)}$,~$\phi_{(2,3)}$, and~$\phi_{(H^\perp)}$ are involutive automorphisms, inducing the structure~$sl(3,\mathbb{O}) = \mathcal{P} \oplus \mathcal{N}$ for~$\mathcal{P} \oplus \mathcal{N} = R \oplus B$,~$\mathcal{P} \oplus \mathcal{N} = T_1 \oplus T_{(2,3)}$, and~$\mathcal{P} \oplus \mathcal{N} = H \oplus H^\perp$, respectively.
{\bf Lemma:} The signatures of~$R$,~$T_1$, and~$H$ are~$(52,0)$,~$(36,10)$, and~$(24,14)$.
{\bf Proof:} The algebra~$R$ consists of the compact generators of~$sl(3,\mathbb{O})$, of which there are~$52$. The algebra~$T_1$ contains~$so(9,1)$, which has signature~$(36,9)$, and the subalgebra~$\langle \dot B^{(2)}_{tz} - \dot B^{(3)}_{tz} \rangle$. The signature of~$H$ is found by counting the number of compact and non-compact generators in its basis.
\hspace{5.40in}$\square$
We extend our basis of~$sl(3,\mathbb{O})$ to a basis of the complex Lie algebra~$E_6$ by considering our basis now as a vector space over the complex numbers. Writing~$\sqrt{-1}$ for~$\xi$, we extend our involutive automorphisms~$\phi: sl(3,\mathbb{O}) \to sl(3,\mathbb{O})$ to \hbox{$\phi^*: E_6 \to E_6$}, giving the maps
$$ \phi^*_{(t)}(R + B) = R + \sqrt{-1}B \hspace{1cm} \phi^*_{(2,3)}(T_1 + T_{(2,3)}) = T_1 + \sqrt{-1}T_{(2,3)}$$
$$\phi^*_{(H^\perp)}(H + H^\perp) = H + \sqrt{-1}H^\perp$$
These involutive automorphisms change some compact and non-compact generators of~$sl(3,\mathbb{O})$ into non-compact and compact generators, respectively, of~$\phi^*\left( sl(3,\mathbb{O})\right)$. For instance, each boost~$b$ satisfies~$B(b, b) > 0$, where~$B( \hspace{.22cm} , \hspace{.22cm} )$ is the Killing form. Under~$\phi^*_{(t)}$, the boosts~$b$ satisfy
$$B\left(\phi^*_{(t)}(b), \phi^*_{(t)}(b)\right) = B(\xi b, \xi b) = \xi^2B(b,b) < 0$$
Hence,~$\phi^*_{(t)}(b)$ is a compact generator of~$\phi^*_{(t)}\left( sl(3,\mathbb{O}) \right)$.
The involutive automorphism~$\phi^*_{(2,3)}$ changes the type~$2$ and type~$3$ rotations to non-compact generators, and the boosts in~$T_{(2,3)}$ are changed to compact generators. Similarly, all orthogonal-quaternionic rotations and boosts in~$H^\perp$ change signature.
The involutive automorphism~$\phi^*_{(t)}$ transforms~$sl(3,\mathbb{O})$, which has signature~$(52,26)$, into the compact real form~$\phi^*_{(t)}\left(sl(3,\mathbb{O})\right)$, which has signature~$(78,0)$. The subalgebra
$$\mathcal{P} = \phi^*_{(t)}\left(sl(3,\mathbb{O})\right) \cap sl(3,\mathbb{O})$$
has dimension~$52$ and is compact. Hence, according to the tables in~\cite{gilmore}, we see that~$\mathcal{P}$ is the compact real form~$su(3,\mathbb{O})$ of~$F_4$. Of course, the compact part of~$\phi^*_{(t)}\left(sl(3,\mathbb{O})\right)$ is the entire algebra, so that in this case, its pre-image is already known.
We obtain two interesting results when we apply the involutive automorphism~$\phi^*_{(2,3)}$ to~$sl(3,\mathbb{O})$. First, the signature of~$g^\prime = \phi^*_{(2,3)}\left(sl(3,\mathbb{O})\right)$ is~$(52,26)$, giving another real form of~$E_6$ whose maximal compact subalgebra~$g^\prime_{(c)}$ is~$F_4$. Hence, the pre-image of~$g^\prime_{(c)}$ is a real form of~$F_4$ in~$sl(3,\mathbb{O})$. It has signature~$(36,16)$, and the~$16$ non-compact generators identify this real form of~$F_4$ as~$su(2,1,\mathbb{O})$. The second result comes from looking at
$$ \mathcal{P} = \phi^*_{(2,3)}\left(sl(3,\mathbb{O})\right) \cap sl(3,\mathbb{O})$$
Because~$|\mathcal{P}| = 46$, we see from the table in~\cite{gilmore}
that~$\mathcal{P}$ is a real form of~$D_5 \oplus D_1$. Of course,~$\phi^*_{(2,3)}$ fixes the algebra~$T_1$, which has signature~$(36,9)\oplus(0,1)$. Hence, we identify~$T_1$ as the subalgebra~$so(9,1) \oplus \langle \dot B^{(2)}_{tz} - \dot B^{(3)}_{tz} \rangle$.
We also obtain two results by applying~$\phi^*_{(H^\perp)}$ to~$sl(3,\mathbb{O})$. We find that the signature of~$g^\prime = \phi^*_{(H^\perp)}\left(sl(3,\mathbb{O})\right)$ is~$(36,42)$. The maximal compact subalgebra~$g^\prime_{(c)}$ of~$g^\prime$ is~$su(4,\mathbb{H})$, a real form of~$C_4$. The pre-image of~$g^\prime_{(c)}$ has signature~$(24,12)$, and due to the~$12$ non-compact generators, must be identified as~$su(3,1,\mathbb{H})_1$. The invariant subalgebra
$$ \mathcal{P} = \phi^*_{(H^\perp)}\left(sl(3,\mathbb{O})\right) \cap sl(3,\mathbb{O})$$
has dimension~$|\mathcal{P}| = 38$ and signature~$(24,14)$. According to the table in~\cite{gilmore}
~$\mathcal{P}$ is a real form of~$A_5 \oplus A_1$ with signature~$(21,15) \oplus (3,0)$. The involutive automorphism~$\phi^*_{(H^\perp)}$ obviously fixes~$H$. Of the~$24$ rotations unaffected by~$\phi^*_{(H^\perp)}$, there are~$21$ which are quaternionic and form the subalgebra~$su(3,\mathbb{H})$. The remaining three rotations~${A_k, A_{k\l}, A_{\l}}$ are elements of~$G_2$, but are transformations involving pairs of~$i,j,i\l$, and~$j\l$ only and leave~$k,k\l$, and~$l$ fixed. The four elements~$i,j,i\l$ and~$j\l$ can be paired into two complex pairs (of which~$i + i\l, j + j\l$ is one choice), and the~$\dot A_k, \dot A_{k\l}, \dot A_\l$ transformations act as~$su(2,\mathbb{C})$ transformations producing the other pairs of complex numbers. Hence, the~$24$ compact elements form the algebra~$su(3,\mathbb{H})\oplus su(2,\mathbb{C})^C$, where we continue to use superscript~$C$ notation to indicate this~$su(2,\mathbb{C})$ is a subalgebra of~$G_2$ and {\bf not} the
canonical~$su(2,\mathbb{C})_s$. We identify~$\mathcal{P} = H$ as the subalgebra~$sl(3,\mathbb{H}) \oplus su(2,\mathbb{C})^C$.
The set of involutive automorphisms form a group, allowing us to consider the compositions of the involutive automorphisms~$\phi^*_{(t)}$,~$\phi^*_{(2,3)}$, and~$\phi^*_{(H^\perp)}$. After working through one example, we provide the subalgebra structure of~$sl(3,\mathbb{O})$ that can be found using these maps in Table~\ref{table:maximal_fixed_and_compact_subalgebras}.
Consider the involutive automorphism~$\phi^*_{(2,3)} \circ \phi^*_{(H^\perp)} : E_6 \to E_6$. We define the following subspaces~$H_1$,~$H_{(2,3)}$,~$H^\perp_1$, and~$H^\perp_{(2,3)}$ which are the intersections of pairs of spaces~$T_1, T_{(2,3)}, H$, and~$H^\perp$, as indicated in Table~\ref{table:int_of_T_H}. For example,~$H_1 = H \cap T_1$ and~$H^\perp_{(2,3)} = H^\perp \cap T_{(2,3)}$. The second table in Table~\ref{table:int_of_T_H} indicates the number of basis elements which are boosts and rotations (in~$sl(3,\mathbb{O})$) in each of these spaces. These spaces satisfy the commutation rules given in Table~\ref{table_comm_of_T_H}.
\begin{table}[htbp]
\begin{center}
\begin{tabular}{c|cccc|cc}
$\cap$ & $T_1$ & $T_{(2,3)}$ & \hspace{2cm} &$\cap$ & $T_1$ & $T_{(2,3)}$\\
\cline{1-3}\cline{5-7}
$H$ & $H_1$ & $H_{(2,3)}$ & & $H$ & $(16,6)$ & $(8,8)$ \\
$H^\perp$ & $H^\perp_1$ & $H^\perp_{(2,3)}$ & & $H^\perp$ & $(20,4)$ & $(8,8)$\\
\multicolumn{3}{c}{Subspace} & & \multicolumn{3}{c}{Signature}
\end{tabular}
\caption{Intersections of Subspaces~$T_1$,~$T_{(2,3)}$,~$H$, and~$H^\perp$}
\label{table:int_of_T_H}
\end{center}
\end{table}
\begin{table}[htbp]
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
$\left[ \hspace{.15cm} , \hspace{.15cm} \right]$& $H_1$ & $H_{(2,3)}$ & $H^\perp_1$ & $H^\perp_{(2,3)}$ \\
\hline
$H_1$ & $H_1$ & $H_{(2,3)}$ & $H^\perp_1$ & $H^\perp_{(2,3)}$ \\
\hline
$H_{(2,3)}$ & $H_{(2,3)}$ & $H_1$ & $H^\perp_{(2,3)}$ & $H^\perp_1$ \\
\hline
$H^\perp_1 $ & $H^\perp_1$ & $H^\perp_{(2,3)}$ & $H_1$ & $H_{(2,3)}$\\
\hline
$H^\perp_{(2,3)}$ & $H^\perp_{(2,3)}$ & $H^\perp_1$ & $H_{(2,3)}$ & $H_1$ \\
\hline
\end{tabular}
\caption{Commutation structure of~$H_1, H_{(2,3)}, H^\perp_1,$ and~$H^\perp_{(2,3)}$}
\label{table_comm_of_T_H}.
\end{center}
\end{table}
We see from Table~\ref{table_comm_of_T_H} that~$H_1$,~$H_1 + H^\perp_{(2,3)}$,~$H_1 + H_{(2,3)}$, and~$H_1 + H^\perp_1$ are subalgebras of~$sl(3,\mathbb{O})$
The involutive automorphism~$\phi^*_{(2,3)} \circ \phi^*_{(H^\perp)}$ fixes
$$
\begin{array}{ccl}
\mathcal{P} & = & \phi^*_{(2,3)} \circ \phi^*_{(H^\perp)}\left(sl(3,\mathbb{O})\right) \cap sl(3,\mathbb{O})\\
& = & H_1 + H^\perp_{(2,3)} \\
\end{array}
$$
as a subalgebra, since everything in~$H_1$ is fixed by both automorphisms, while everything in~$H^\perp_{(2,3)}$ is multiplied by~$\xi^2 = -1$. The subalgebra~$\mathcal{P}$ has dimension~$38$ and signature~$(24,14)$, and is thus a real form of~$A_5 \oplus A_1$. It has a fundamentally different basis from~$sl(3,\mathbb{H})\oplus su(2,\mathbb{C})^C$, however, as it uses a mixture of the quaternionic transformations of type~$1$ with the orthogonal-quaternions transformations of type~$2$ and type~$3$, while~$sl(3,\mathbb{H})$ is comprised of only the quaternionic transformations. We call this algebra~$sl(2,1,\mathbb{H}) \oplus su(2,\mathbb{C})_2$, where the notation~$sl(2,1,\mathbb{H})$ is used to indicate that this is {\bf not} our standard~$sl(3,\mathbb{H})$ and
$$su(2,\mathbb{C})_2 = \langle \dot G_k + 2\dot S^1_k, \dot G_{k\l} + 2\dot S^1_{k\l}, \dot G_\l + 2\dot S^1_\l, \rangle$$
again corresponds to permutations of ~$\lbrace i,j,j\l,i\l \rbrace$, fixes~$\lbrace k, k\l, \l \rbrace$, but is not in~$G_2$. We also note that the compact subalgebra~$g^\prime_{(c)}$ of~$\phi^*_{(2,3)} \circ \phi^*_{(H^\perp)} \left( sl(3,\mathbb{O})\right)$ has dimension~$36$, and its preimage in~$sl(3,\mathbb{O})$ has signature~$(24,12)$. We identify this as~$su(3,1,\mathbb{H})_2$, and include the subscript~$2$ since this real algebra has a different basis than our previously identified~$su(3,1,\mathbb{H})$, which we had called~$su(3,1,\mathbb{H})_1$.
We summarize in Table~\ref{table:maximal_fixed_and_compact_subalgebras} the subalgebras achieved from our three involutive automorphisms as well as compositions of these automorphisms. For each involutive automorphism~$\phi^*$, the second column lists the signature of~$\phi^*(sl(3,\mathbb{O})$. The third column identifies the fixed subalgebra~$\mathcal{P} = \phi^*\left(sl(3,\mathbb{O})\right) \cap sl(3,\mathbb{O})$ as well as its signature. In the fourth column, we identify the pre-image of the maximal compact subalgebra of~$g^\prime = \phi^*\left(sl(3,\mathbb{O})\right)$ and lists both the algebra and its signature.
\begin{table}[htbp]
\begin{center}
\begin{tabular}{|cccc|}
\hline
Involutive & Signature of & Signature of & Signature of \\
Automorphism & $g^\prime = \phi^*\left(sl(3,\mathbb{O})\right)$ & $\mathcal{P} = g^\prime \cap sl(3,\mathbb{O})$ & $(\phi^*)^{-1}(g^\prime_{(c)})$ \\
\hline
$1$ & $(52,26)$ & $(52,26)$ & $(52,0)$ \\ & & $sl(3,\mathbb{O})$ & $su(3,\mathbb{O})$ \\
\hline
$\phi^*_{(t)}$ & $(78,0)$ & $(52,0)$ & $(52,26)$ \\ & & $su(3,\mathbb{O})$ & $sl(3,\mathbb{O})$ \\
\hline
$\phi^*_{(2,3)}$ & $(52,26)$ & $(36,10)$ & $(36,16)$ \\ & & $sl(2,\mathbb{O}) \oplus u(1)$ & $su(2,1,\mathbb{O})$ \\
\hline
$\phi^*_{(H^\perp)}$& $(36,42)$ & $(24,14)$ & $(24,12)$ \\ & & $sl(3,\mathbb{H})\oplus su(2,\mathbb{C})^C$ & $su(3,1,\mathbb{H})_1$\\
\hline
$\phi^*_{(2,3)}\circ \phi^*_{(t)}$ & $(46,32)$ & $(36,16)$ & $(36,10)$\\ & & $su(2,1,\mathbb{O})$ & $so(9,1) \oplus u(1)$\\
\hline
$\phi^*_{(H^\perp)}\circ \phi^*_{(t)}$ & $(38,40)$ & $(24,12)$ & $(24,14)$ \\ & & $su(3,1,\mathbb{H})_1$ & $sl(3,\mathbb{H}) \oplus su(2,\mathbb{C})^C$ \\
\hline
$\phi^*_{(2,3)}\circ \phi^*_{(H^\perp)}$ & $(36,42)$ & $(24,14)$ & $(24,12)$ \\ & & $sl(2,1,\mathbb{H})\oplus su(2,\mathbb{C})_2$ & $su(3,1,\mathbb{H})_2$ \\
\hline
$\phi^*_{(2,3)}\circ \phi^*_{(H^\perp)}\circ \phi^*_{(t)} $ & $(38,40)$ & $(24,12)$ & $(24,14)$ \\ & & $su(3,1,\mathbb{H})_2$ & $sl(2,1,\mathbb{H})\oplus su(2,\mathbb{C})_2$ \\
\hline
\end{tabular}
\caption{Involutive automorphisms with maximal subalgebras of~$sl(3,\mathbb{O})$}
\label{table:maximal_fixed_and_compact_subalgebras}
\end{center}
\end{table}
Despite recognizing~$H_1$,~$H_1 + H^\perp_{(2,3)}$,~$H_1 + H_{(2,3)}$, and~$H_1 + H^\perp_1$ as subalgebras of~$sl(3,\mathbb{O})$ using the commutation relations in Table~\ref{table_comm_of_T_H}, we note that~$su(3,1,\mathbb{H})_2$ is not any of these subalgebras. In the next section, we use another technique involving involutive automorphisms to give a finer refinement of subspaces of~$sl(3,\mathbb{O})$, allowing us to provide a nice basis for~$su(3,1,\mathbb{H})_2$ and other subalgebras of~$sl(3,\mathbb{O})$.
\section{Additional Subalgebras of~$sl(3,\mathbb{O})$ from Automorphisms}
\label{ch:E6_further_structure.more_applications_of_three_automorphisms}
We have already used involutive automorphisms to identify the maximal subalgebras of~$sl(3,\mathbb{O})$. This was done by separating the algebra into two separate spaces, one of which was left invariant by the automorphism. In this section, we use the group characteristic of the involutive automorphisms to separate~$sl(3,\mathbb{O})$ into four or more subspaces spanned by either the compact or non-compact generators, with the condition that the involutive automorphism either preserves the entire subspace or it changes the character of all the basis elements in the subspace. We identify additional subalgebras of~$sl(3,\mathbb{O})$ by taking various combinations of these subspaces.
We continue to use the automorphisms~$\phi^*_{(t)}$,~$\phi^*_{(2,3)}$, and~$\phi^*_{(H^\perp)}$, as well as the subspaces~$R, H, T_1, T_{(2,3)}, H$, and~$H^\perp$ defined in the previous section.
We first consider the involutive automorphism~$\phi^*_{(H^\perp)} \circ \phi^*_{(t)}: E_6 \to E_6$. This map fixes the subspaces~$R_H = R \cap H$, consisting of quaternionic rotations, and~$B_{H^\perp} = B \cap H^\perp$, consisting of orthogonal-quaternionic boosts. Under~$\phi^*_{(H^\perp)} \circ \phi^*_{(t)}$, the two subspaces~$B_{H}$ and~\hbox{$R_{H^\perp} = R \cap H^\perp$} change signature. These spaces consist of orthogonal-quaternionic rotations and quaternionic boosts, respectively. The dimensions of these four spaces are displayed in Table~\ref{table:H_T_subspaces}.
\begin{table}[htbp]
\begin{center}
\begin{tabular}{l|c|c|}
\cline{2-3}
& \multicolumn{2}{c|}{$\phi^*_{(H^\perp)} \circ \phi^*_{(t)}$ Signature} \\
\cline{2-3}
& $(1,0)$ & $(0,1)$ \\
\hline
\textrm{Same Signature} & $|R_H| = 24$ & $|B_{H^\perp}| = 12$ \\
\hline
\textrm{Change Signature} & $|B_H| = 14$ & $|R_{H^\perp}| = 28$ \\
\hline
\end{tabular}
\caption{Splitting of $E_6$ basis under~$\phi^*_{(H^\perp)} \circ \phi^*_{(t)}$}
\label{table:H_T_subspaces}
\end{center}
\end{table}
We list the signature of these spaces under~$\phi^*_{(H^\perp)} \circ \phi^*_{(t)}$, and can thus identify subalgebras of~$\phi^*_{(H^\perp)} \circ \phi^*_{(t)}\left( sl(3,\mathbb{O}) \right)$. However, we are primarily interested in the pre-image of these subalgebras in our preferred algebra~$sl(3,\mathbb{O})$. The signature of these spaces in~$sl(3,\mathbb{O})$ is straight-forward, as the rotations are compact and the boosts are not compact.
We use the subspaces represented in Table~\ref{table:H_T_subspaces} to identify subalgebras of~$sl(3,\mathbb{O})$. As previously identified, the~$24$ rotations in~$R_H$ fixed by the automorphism form the subalgebra~$su(3,\mathbb{H}) \oplus su(2,\mathbb{H}^\perp)_1$. The entries in the first column of Table~\ref{table:H_T_subspaces} represent all quaternionic rotations and boosts, and form the subalgebra~$sl(3,\mathbb{H})_1 \oplus su(2,\mathbb{H}^\perp)_1$. Of course, the entries on the diagonal,~$R_H$ and~$R_{H^\perp}$, form~$F_4$. We finally consider the entries~$R_H$ and~$B_{H^\perp}$ in the top row of the table. Two orthogonal-quaternionic boosts commute to a quaternionic rotation, and an orthogonal-quaternionic boost commuted with a quaternionic rotation is again an orthogonal-quaternionic boosts. Hence, the subspace~$R_H + B_{H^\perp}$ closes under commutation and is a subalgebra with signature~$(24,12)$. While both~$so(9-n, n, \mathbb{R})$ and~$su(4-n, n, \mathbb{H})$ have dimension~$36$, only~$su(3,1,\mathbb{H})$ has~$12$ boosts. We realize~$R_H + B_{H^\perp}$ is the previously identified~$su(3,1,\mathbb{H})_1$.
The algebra~$R_H + B_{H^\perp} = su(3,1,\mathbb{H})_1$ is a real form of the complex Lie algebra~$C_4$. Since~$C_4$ contains~$C_3$ but not~$B_3$, we note that the~$21$-dimensional subalgebra contained within~$R_H$ is a real form of~$C_3$, not of~$B_3$. In addition, any simple~$21$-dimensional subalgebra of~$R_H + B_{H^\perp}$ is a real from of~$C_3$. Eliminating the boosts from~$su(3,1,\mathbb{H})_1$ leaves~$su(3,\mathbb{H}) \oplus su(2,\mathbb{H}^\perp)_1$.
We next consider the involutive automorphism~$\phi^*_{(2,3)} \circ \phi^*_{(t)}: E_6 \to E_6$. This automorphism separates the basis for~$sl(3,\mathbb{O})$ into the subspaces
$$
\begin{array}{ccc}
R_1 = R \cap T_1 &\hspace{2cm} & B_1 = B \cap T_{(2,3)} \\
B_{(2,3)} = B \cap T_{(2,3)} & \hspace{2cm}& R_{(2,3)} = R \cap T_{(2,3)}\\
\end{array}
$$
This automorphism leaves the signature of~$R_1$ and~$B_{(2,3)}$ alone, while it reverses the signature of the generators in~$R_{(2,3)}$ and~$B_1$. This information is represented in Table~\ref{table:T_23_subspaces}
\begin{table}[htbp]
\begin{center}
\begin{tabular}{l|c|c|}
\cline{2-3}
& \multicolumn{2}{c|}{$\phi^*_{(2,3)} \circ \phi^*_{(t)}$ Signature} \\
\cline{2-3}
& $(1,0)$ & $(0,1)$ \\
\hline
\textrm{Same Signature} & $|R_1| = 36$ & $|B_{(2,3)}| = 16$ \\
\hline
\textrm{Change Signature} & $|B_1| = 10$ & $|R_{(2,3)}| = 16$ \\
\hline
\end{tabular}
\caption{Splitting of $E_6$ basis under $\phi^*_{(2,3)} \circ \phi^*_{(t)}$}
\label{table:T_23_subspaces}
\end{center}
\end{table}
We again use the subspaces represented in Table~\ref{table:T_23_subspaces} to identify subalgebras of~$sl(3,\mathbb{O})$. The subspace~$R_1$ is the subalgebra~$so(9,\mathbb{R})$, and contains all subalgebras~$so(n,\mathbb{R})$ for~$n \le 9$. The subspace~$R_1 + B_1$ is, of course,~$so(9,1,\mathbb{R}) \oplus u(1)$, where~$u(1) = \langle \dot B^{(2)}_{tz} - \dot B^{(3)}_{tz} \rangle$. Again, the complete set of rotations,~$R_1 + R_{(2,3)}$ is the subalgebra~$su(3,\mathbb{O})$, which is a real form of~$F_4$. Interestingly, the subspace~$R_1 + B_{(2,3)}$ in the top row is another form of~$F_4$. The~$16$ boosts in~$R_1 + B_{(2,3)}$ identify this version of~$F_4$ as~$su(2,1,\mathbb{O})$.
We finally consider the involutive automorphism~$\phi^*_{(2,3)} \circ \phi^*_{(H^\perp)} : E_6 \to E_6$ as the composition of the maps~$\phi^*_{(2,3)} \circ \phi^*_{(t)}$ and~$\phi^*_{(H^\perp)} \circ \phi^*_{(t)}$. A pair of maps normally separates~$E_6$ into four different subspaces, but we can create a finer refinement of subspaces of~$sl(3,\mathbb{O})$ by using the two copies of~$\phi^*{(t)}$ to separate the boosts and rotations from each other. These subspaces are represented in Table~\ref{table:23_H_subspaces}. We continue with our previous conventions for designating intersections of subspaces - that is,~$R_{(2,3),H^\perp} = R \cap T_{(2,3)} \cap H^\perp$. We use a~$4\times4$ table, rather than a~$4\times 2$ table, to help us identify relevant subalgebras.
\begin{table}[htbp]
\begin{center}
\begin{tabular}{l|cc|cc|}
\cline{2-5}
& \multicolumn{4}{c|}{$\phi^*_{(2,3)} \circ \phi^*_{(H^\perp)}$ Signature} \\
\cline{2-5}
& $(1,0)$ & $(1,0)$ & $(0,1)$ & $(0,1)$ \\
\hline
$\phi^*_{(2,3)} = Id, \phi^*_{(H^\perp)} = Id$ & $|R_{1,H}|$ & & $|B_{1,H}|$ & \\
& $= 16$ & & $=6$ & \\
$\phi^*_{(2,3)} \ne Id, \phi^*_{(H^\perp)} \ne Id$ & & $|R_{(2,3),H^\perp}|$ & & $|B_{(2,3),H^\perp)}|$ \\
& & $= 8$ & & $=8$ \\
\hline
$\phi^*_{(2,3)} \ne Id, \phi^*_{(H^\perp)} = Id$ & $|B_{(2,3),H}|$ & & $|R_{(2,3),H}|$ & \\
& $= 8$ & & $=8$ & \\
$\phi^*_{(2,3)} = Id, \phi^*_{(H^\perp)} \ne Id$ & & $|B_{1,H^\perp}|$ & & $|R_{1,H^\perp}|$ \\
& & $= 4$ & & $=20$ \\
\hline
\end{tabular}
\caption{Splitting of~$E_6$ basis under~$\phi^*_{(2,3)} \circ \phi^*_{(H^\perp)}$}
\label{table:23_H_subspaces}
\end{center}
\end{table}
Using this division of~$sl(3,\mathbb{O})$, we find a large list of subalgebras of~$sl(3,\mathbb{O})$ simply by combining certain subspaces. The subspace description of these algebras, as well as their identity and signature in~$sl(3,\mathbb{O})$, is listed in Table~\ref{table:finest_refinement_subalgebras}. We note that this fine refinement of~$sl(3,\mathbb{O})$ provides a description of the basis for~$su(3,1,\mathbb{H})_1$ and~$su(3,1,\mathbb{H})_2$, as well as~$sl(3,\mathbb{H})$ and~$sl(2,1,\mathbb{H})$. The explicit basis elements for each algebra are listed in Appendix~\ref{appendix.basis_for_various_subalgebras_of_E6}.
\begin{table}[htbp]
\begin{center}
\begin{tabular}{|l|l|c|c|}
\hline
Basis & Subalgebra of $sl(3,\mathbb{O})$ & Our & $\phi^*_{(2,3)}\circ \phi^*_{(H^\perp)}$ \\
& (our signature)& Signature & Signature \\
\hline
$R_{1,H}$ & $su(2,\mathbb{H})\oplus su(2,\mathbb{C})^C \oplus su(2)$ & $(10+3+3,0)$ & $(16,0)$ \\
\hline
$R_{1,H}+R_{(2,3),H^\perp}$ & $su(3,\mathbb{H})_2\oplus su(2,\mathbb{C})^C$ & $(21+3,0)$ & $(21+3,0)$ \\
\hline
$R_{1,H}+B_{1,H}$ & $sl(2,\mathbb{H})\oplus su(2,\mathbb{C})^C $ & $(21+1,6)$& $(15+1, 6)$ \\
& \hspace{.5cm} $\oplus su(2) \oplus u(1)$ & & \\
\hline
$R_{1,H}+B_{(2,3),H^\perp}$ & $su(2,1,\mathbb{H})_1 \oplus su(2,\mathbb{C})^C$ & $(16,8)$ & $(16,8)$ \\
\hline
$R_{1,H}+B_{(2,3),H}$ & $su(2,1,\mathbb{H})_2\oplus su(2,\mathbb{C})^C$ & $(13+3,8)$ & $(21+3,0)$\\
\hline
$R_{1,H}+R_{(2,3),H}$ & $su(3,\mathbb{H})\oplus su(2,\mathbb{C})^C $ & $(21+3,0)$ & $(13+3,8)$ \\
\hline
$R_{1,H}+B_{1,H^\perp}$ & $so(5,\mathbb{R})\oplus so(4,1,\mathbb{R})$ & $(10+6,4)$ & $(10+10,0)$\\
\hline
$R_{1,H}+R_{1,H^\perp}$ & $so(9) = su(2,\mathbb{O})$ & $(36,0)$ & $(16,20)$\\
\hline
$R_{1,H}+B_{1,H}$ & $sl(3,\mathbb{H}) \oplus su(2,\mathbb{C})^C $ & $(21+3,14)$ & $(21+3,14)$\\
\hspace{.5cm} $+B_{(2,3),H}+R_{(2,3),H}$ & & & \\
\hline
$R_{1,H}+B_{1,H}$ & $sl(2,1,\mathbb{H})_1\oplus su(2,\mathbb{C})_2$ & $(21+3,14)$ & $(21+3,14)$\\
\hspace{.5cm} $+R_{(2,3),H^\perp}$ & & & \\
\hspace{.5cm} $+B_{(2,3),H^\perp}$ & & & \\
\hline
$R_{1,H}+B_{1,H}$ & $sl(2,\mathbb{O})\oplus u(1)$ & $(35+1,10)$ & $(20, 25+1)$\\
\hspace{.5cm} $ +B_{1,H^\perp}$ & & & \\
\hspace{.5cm} $ +R_{1,H^\perp}$ & & & \\
\hline
$R_{1,H}+B_{(2,3),H^\perp}$ & $su(3,1,\mathbb{H})_1$ & $(24,12)$ & $(20,16)$ \\
\hspace{.5cm}$+R_{(2,3),H}+B_{1,H^\perp}$ & & & \\
\hline
$R_{1,H}+R_{(2,3),H^\perp}$ & $su(3,1,\mathbb{H})_2$ & $(24,12)$ & $(36,0)$ \\
\hspace{.5cm} $+B_{(2,3),H}+B_{1,H^\perp}$ & & & \\
\hline
$R_{1,H}+R_{(2,3),H^\perp}$ & $su(3,\mathbb{O})$ & $(52,0)$ & $(24,28)$ \\
\hspace{.5cm} $+R_{(2,3),H}+R_{1,H^\perp}$ & & & \\
\hline
$R_{1,H}+B_{(2,3),H^\perp}$ & $su(2,1,\mathbb{O})$ & $(36,16)$ & $(24,28)$ \\
\hspace{.5cm} $+B_{(2,3),H}+R_{1,H^\perp}$ & & & \\
\hline
\end{tabular}
\caption{Subalgebras of~$sl(3,\mathbb{O})$ using~$\phi^*_{(2,3)} \circ \phi^*_{(H^\perp)}$}
\label{table:finest_refinement_subalgebras}
\end{center}
\end{table}
\section{Chains of Subalgebras of~$sl(3,\mathbb{O})$ with Compatible Bases and Casimir Operators}
\label{ch:E6_further_structure.chains_of_subalgebras}
We used the involutive automorphisms to produce large simple subalgebras of~$sl(3,\mathbb{O})$. These subalgebras range in dimension from~$52$, for~$F_4$, to~$21$, for~$C_3$. Each of these subalgebras have involutive automorphisms which we can also use to find even smaller subalgebras, thereby giving a catalog of subalgebras chains contained within~$sl(3,\mathbb{O})$. However, having identified the real form of the large subalgebras of~$sl(3,\mathbb{O})$, it is not too difficult a task to find smaller algebras simply by looking for simple subalgebras of smaller dimension and/or rank, using the tables of real forms listed in~\cite{gilmore} when needed. In addition, we choose our representatives of the smaller subalgebras so that they use a subset of the preferred Casimir operators~$\lbrace \dot B^{(1)}_{tz}, \dot B^{(2)}_{tz}, \dot R^{(1)}_{x\l}, \dot S^{(1)}_{\l}, \dot G_{\l}, \dot A_{\l} \rbrace$ which we have chosen for~$sl(3,\mathbb{O})$. We list the basis for these subalgebras in Appendix~\ref{appendix.basis_for_various_subalgebras_of_E6}.
We display the chains of subalgebras of~$sl(3,\mathbb{O})$ in the following tables. Each table is built from an~$u(1)$ algebra, which consists of merely a single Casimir operator. We extend each algebra~$g$ to a larger algebra~$g^\prime$ by adding elements to the basis for~$g$.
In particular, each algebra of higher rank must add compatible Casimir operators. Figure~\ref{fig:clean_chain_of_subalgebras_of_E6} is built from the single~$u(1) = \langle G_l - S^1_l \rangle$ algebra. This basis can be extended to
$$u(1) \subset su(1,\mathbb{H}) \subset su(2,\mathbb{H}) \subset su(3,\mathbb{H})_1 \subset sl(3,\mathbb{H})$$
However, additional subalgebras of~$sl(3,\mathbb{O})$ can be inserted into this chain of subalgebras. For instance, we can insert~$sl(2,\mathbb{H})$ between~$su(2,\mathbb{H})$ and~$sl(3,\mathbb{H})$, and extend~$sl(2,\mathbb{H})$ to~$sl(2,\mathbb{O})$. We can expand~$su(1,\mathbb{H})$ to~$su(1,\mathbb{O}) = so(7)$ and insert the~$so(n,\mathbb{R})$ chain for~$n \ge 7$ into the figure. However, we note that~$so(7)$ uses a basis in this chain that is not compatible with~$G_2 = aut(\mathbb{O})$. We do not list all of the possible~$u(1)$ algebras, but do add the~$u(1) = \langle R^1_{x\l} \rangle$ subalgebra into the chain and extend it to~$su(2,\mathbb{C})_s$ and~$sl(2,\mathbb{C})_s$ which use the {\it standard} definition of~$su(2,\mathbb{C})$ and~$sl(2,\mathbb{C})$.
Finally, we have identified four different real forms of~$C_3$ which all contain~$su(2,\mathbb{H})$. Space constraints limit us to listing only~$su(2,1,\mathbb{H})_1$ and~$su(3,\mathbb{H})_1$ in Figure~\ref{fig:clean_chain_of_subalgebras_of_E6}, but the algebras
$$su(2,1,\mathbb{H})_2 \hspace{1cm} su(3,\mathbb{H})_2 \hspace{1cm} su(3,1,\mathbb{H})_2$$
should also be in this table. We list the four~$C_3$ algebras which are build from~$su(2,\mathbb{H})$ in
Figure~\ref{fig:different_C3s}, and include all the algebras which are built from the~$C_3$ algebras. Figure~\ref{fig:different_C3s} can be incorporated into Figure~\ref{fig:clean_chain_of_subalgebras_of_E6} without having to adjust choice of the Casimir operators.
While the algebras in Figure~\ref{fig:clean_chain_of_subalgebras_of_E6} are built from its subalgebras by extending the subalgebra's basis, we do not allow a change of basis in any of the algebras. In Figure~\ref{fig:clean_change_of_basis_chain_of_subalgebras_of_E6}, we do allow a change of basis at each subalgebra stage.
The major difference this change allows is that the chain
$$su(2,\mathbb{C}) \subset su(3,\mathbb{C}) \subset su(3,\mathbb{H})$$
may be included with the Casimir operators~$S^1_\l$ and~$G_\l$ for each subalgebra extension.
\hspace{-1in}
\begin{figure}[htbp]
\begin{center}
\begin{minipage}{6in}
\begin{center}
\hspace{-1in}
\xymatrixcolsep{5pt}
\xymatrix@M=3pt@H=1pt{
& & & *++\txt{$sl(3,\mathbb{O})$ \\$[E_6]$} & \\
\save[]+<0cm,.4cm>*++\txt{$sl(2,1,\mathbb{H}) \oplus su(2,\mathbb{C})_2$\\$[A_5 \oplus A_1]$} \ar@{^{(}.>}[urrr]<2.75ex>_{} \restore & & & & \save[]+<0cm,.4cm>*++\txt{$sl(3,\mathbb{H}) \oplus su(2,\mathbb{C})^C$\\$[A_5 \oplus A_1]$}\ar@{_{(}.>}[ul]_{} \restore \\
*++\txt{$sl(2,1,\mathbb{H})$\\$[A_5]$} \ar@{.>}[u]^{G_\l+S^1_\l} & *++\txt{$su(2,1,\mathbb{O})$\\$[F_{4(36,16)}]$} \ar@/^3pc/[uurr]^(.3){B^1_{tz}}^(.25){B^2_{tz}} & *++\txt{$su(3,\mathbb{O})$\\$[F_4]$} \ar[uur]^(.4){B^1_{tz}}^(.32){B^2_{tz}} & *++\txt{$sl(2,\mathbb{O}) = so(9,1)$\\$[D_5]$} \ar@{.>}[uu]_{B^2_{tz}} & *++\txt{$sl(3,\mathbb{H})$\\$[A_5]$} \ar@{.>}[u]_{A_\l} \\
& & & & \\
& *++\txt{$su(3,1,\mathbb{H})_1$\\$[C_4]$} \ar@/^4pc/[uuuurr]^(.5){B^1_{tz}}^(.45){B^2_{tz}} & *++\txt{$su(2,\mathbb{O}) = so(9)$\\$[B_4]$} \ar@{_{(}.>}[uul]^(.4){} \ar@{_{(}.>}[uu]^(.4){} \ar[uur]^(.65){B^1_{tz}}& *++\txt{$so(8)$ \\$[D_4]$} \ar@{_{(}.>}[l]^(.4){} & \\
& & & & \\
*++\txt{$su(2,1,\mathbb{H})_1$\\$[C_3]$} \ar[uuuu]<1ex>^(.55){B^1_{tz}}^(.45){B^2_{tz}} \ar@{.>}[uuuur]^(.5){G_\l+S^1_\l} \ar@{.>}[uur]{} \ar[uur]<1ex>_(.5){G_\l+S^1_\l} & *++\txt{$su(3,\mathbb{H})_1$\\$[C_3]$} \ar `u[r] `[rrru] [uurrruu] ^(.5){B^1_{tz}} ^(.4){B^2_{tz}} \ar@{.>}[uu]<2ex>{} \ar[uu]<1ex>_(.5){A_\l} \ar@{.>}[uuuur]^(.5){A_\l} & *++\txt{$su(1,\mathbb{O}) = so(7)$\\$[B_3]$} \ar[uur]^(.5){R^1_{x\l}} \ar@{.>}[uu]_(.5){R^1_{x\l}} & *++\txt{$sl(2,\mathbb{H})$\\$[A_3 = D_3]$} \ar `l[uuu] `[uuull] `[uuulll]_(.9){A_\l, B^2_{tz}} [uuuulll] \ar@/_4pc/[uuuu]^(.5){G_\l+S^1_\l}^(.4){A_\l} \ar@/_8pc/[uuuur]^(.7){G_\l+S^1_\l}^(.6){B^2_{tz}} & \\
& & & & \\
& *++\txt{$su(2,\mathbb{H}) = sp(2) $\\$[B_2 = C_2]$} \ar[uuuur]^(.8){A_\l}^(.75){G_1+S^1_\l} \ar@/_1.8pc/[uurr]^(.3){B^1_{tz}} \ar@{.>}[uul]^(.5){A_\l} \ar@{.>}[uu]<1ex>_(.5){G_\l+S^1_\l} & *++\txt{$[G_2]$} & *++\txt{$su(3,\mathbb{C})$\\$[A_2]$} & *++\txt{$sl(2,\mathbb{C})_s$\\ $[A_1\oplus A_1]$ } \ar@{.>}[uul]<1ex>{} \ar[uul]_(.5){G_\l-S^1_\l}\\
& & *++\txt{$so(4,\mathbb{R})$\\$[D_2 = C_1 \oplus C_1]$} \ar@{_{(}.>}[ul]^(.4){} \ar[uuuuur]^(.6){A_\l}^(.55){G_\l+S^1_\l} & & \\
& & & *++\txt{$su(1,\mathbb{H})$\\$[C_1]$} \ar[uuuul]<-1ex>_(.25){A_\l}_(.2){G_\l+S^1_\l} \ar@{.>}[ul]<1ex>{} \ar[ul]_(.45){R^1_{x\l}} & *++\txt{$su(2,\mathbb{C})_s$\\$[A_1]$} \ar[uu]_(.5){B^1_{tz}} \ar@/_1pc/[uuuulll]_(.25){G_\l+S^1_\l}_(.2){G_\l-S^1_\l} \\
& & & u(1) \ar[u]_(.4){G_\l - S^1_\l} & u(1) \ar[u]_(.4){R^1_{x\l}}\\
}
\caption{Preferred subalgebra chains of~$E_6$ using the same basis}
\label{fig:clean_chain_of_subalgebras_of_E6}
\end{center}
\end{minipage}
\end{center}
\end{figure}
\begin{figure}[htbp]
\begin{center}
\begin{minipage}{6in}
\begin{center}
\xymatrixcolsep{5pt}
\xymatrix@M=3pt@H=10pt{
& & *++\txt{$sl(3,\mathbb{O})$\\$[E_6]$} & & \\
*++\txt{$sl(2,1,\mathbb{H}) \oplus su(2,\mathbb{C})_2$\\$[A_5 \oplus A_1]$} \ar@{^{(}.>}[urr]^(.5){} & & & & *++\txt{$sl(3,\mathbb{H}) \oplus su(2,\mathbb{C})^C$\\$[A_5 \oplus A_1]$} \ar@{_{(}.>}[ull]^(.5){} \\
*++\txt{$sl(2,1,\mathbb{H})$\\$[A_5]$} \ar@{.>}[u]^(.5){G_\l + S^1_\l} & & & & *++\txt{$sl(3,\mathbb{H})$\\$[A_5]$} \ar@{.>}[u]_(.5){A_\l} \\
& *++\txt{$su(2,1,\mathbb{O})$\\$[F_{4(36,16)}]$} \ar[uuur]^(.5){B^1_{tz}}^(.4){B^2_{tz}} & & *++\txt{$su(3,\mathbb{O})$\\$[F_4]$} \ar[uuul]_(.5){B^1_{tz}}_(.4){B^2_{tz}} & \\
*++\txt{$su(3,1,\mathbb{H})_2$\\$[C_4]$} \ar@/^12.5pc/[uuuurr]^(.75){B^1_{tz}}^(.65){B^2_{tz}} & & & & *++\txt{$su(3,1,\mathbb{H})_1$\\$[C_4]$} \ar@/_12pc/[uuuull]_(.75){B^1_{tz}}_(.65){B^2_{tz}} \\
& & & & & \\
*++\txt{$su(2,1,\mathbb{H})_2$\\$[C_3]$} \ar@{.>}[uu]<1ex>^(.4){A_\l} \ar[uu]^(.4){} \ar@{.>}[uuur]_(.8){A_\l} \ar@/_3.5pc/[uuuurrrr]_(.55){A_\l} & *++\txt{$su(3,\mathbb{H})_2$\\$[C_3]$} \ar@{.>}[uul]<1ex>^(.4){G_\l+S^1_\l} \ar[uul]_(.4){} \ar@{.>}[uuurr]^(.75){G_\l+S^1_\l} \ar@/^1pc/[uuuul]^(.82){B^1_{tz}}^(.72){B^2_{tz}} & & *++\txt{$su(2,1,\mathbb{H})_1$\\$[C_3]$} \ar@{.>}[uur]<1ex>_(.5){} \ar[uur]^(.5){G_\l+S^1_\l} \ar@{.>}[uuull]_(.75){G_\l+S^1_\l} \ar@/^3pc/[uuuulll]_(.85){B^1_{tz}}_(.75){B^2_{tz}} & *++\txt{$su(3,\mathbb{H})_1$\\$[C_3]$} \ar@{.>}[uu]<1ex>_(.5){} \ar[uu]_(.4){A_\l} \ar@{.>}[uuul]_(.7){A_\l} \ar@/_4pc/[uuuu]_(.4){B^1_{tz}}_(.3){B^2_{tz}} \\
& & *++\txt{$su(2,\mathbb{H})$\\$[C_2]$} \ar@{.>}[ull]<1ex>^(.5){G_\l + S^1_\l} \ar@{.>}[ul]<1ex>^(.5){A_\l} \ar@{.>}[ur]<1ex>_(.5){A_\l} \ar@{.>}[urr]<1ex>_(.5){G_\l + S^1_\l}
}
\caption{\noindent Four real forms of~$C_3$}
\label{fig:different_C3s}
\end{center}
\end{minipage}
\end{center}
\end{figure}
\clearpage{}
\hspace{-1in}
\begin{figure}[htbp]
\begin{minipage}{6in}
\begin{center}
\xymatrixcolsep{5pt}
\hspace{-1in}
\xymatrix@M=3pt@H=10pt{
& & & *++\txt{$sl(3,\mathbb{O})$ \\$[E_6]$} & \\
\save[]+<0cm,.4cm>*++\txt{$sl(2,1,\mathbb{H}) \oplus su(2,\mathbb{C})_2$\\$[A_5 \oplus A_1]$} \ar@{^{(}.>}[urrr]<2.75ex>_{} \restore& & & & \save[]+<0cm,.4cm>*++\txt{$sl(3,\mathbb{H}) \oplus su(2,\mathbb{C})^C$\\$[A_5 \oplus A_1]$}\ar@{_{(}.>}[ul]_{} \restore \\
*++\txt{$sl(2,1,\mathbb{H})$\\$[A_5]$} \ar@{.>}[u]^{G_\l+S^1_\l} & *++\txt{$su(2,1,\mathbb{O})$\\$[F_{4(36,16)}]$} \ar@/^3pc/[uurr]^(.3){B^1_{tz}}^(.25){B^2_{tz}} & *++\txt{$su(3,\mathbb{O})$\\$[F_4]$} \ar[uur]^(.4){B^1_{tz}}^(.32){B^2_{tz}} & *++\txt{$sl(2,\mathbb{O}) = so(9,1)$\\$[D_5]$} \ar@{.>}[uu]_{B^2_{tz}} & *++\txt{$sl(3,\mathbb{H})$\\$[A_5]$} \ar@{.>}[u]_{A_\l} \\
& & & & \\
& *++\txt{$su(3,1,\mathbb{H})_1$\\$[C_4]$} \ar@/^4pc/[uuuurr]^(.5){B^1_{tz}}^(.45){B^2_{tz}} & *++\txt{$su(2,\mathbb{O}) = so(9)$\\$[B_4]$} \ar@{_{(}.>}[uul]^(.4){} \ar@{_{(}.>}[uu]^(.4){} \ar[uur]^(.65){B^1_{tz}}& *++\txt{$so(8)$ \\$[D_4]$} \ar@{_{(}.>}[l]^(.4){} & \\
& & & & \\
*++\txt{$su(2,1,\mathbb{H})_1$\\$[C_3]$} \ar[uuuu]<1ex>^(.55){B^1_{tz}}^(.45){B^2_{tz}} \ar@{.>}[uuuur]^(.5){G_\l+S^1_\l} \ar@{.>}[uur]{} \ar[uur]<1ex>_(.5){G_\l+S^1_\l} & *++\txt{$su(3,\mathbb{H})_1$\\$[C_3]$} \ar `u[r] `[rrru] [uurrruu] ^(.5){B^1_{tz}} ^(.4){B^2_{tz}} \ar@{.>}[uu]<2ex>{} \ar[uu]<1ex>_(.5){A_\l} \ar@{.>}[uuuur]^(.5){A_\l} & *++\txt{$su(1,\mathbb{O}) = so(7)$\\$[B_3]$} \ar[uur]^(.5){R^1_{x\l}} \ar@{.>}[uu]_(.5){R^1_{x\l}} & *++\txt{$sl(2,\mathbb{H})$\\$[A_3 = D_3]$} \ar `l[uuu] `[uuull] `[uuulll]_(.9){A_\l, B^2_{tz}} [uuuulll] \ar@/_4pc/[uuuu]^(.5){G_\l+S^1_\l}^(.4){A_\l} \ar@/_8pc/[uuuur]^(.7){G_\l+S^1_\l}^(.6){B^2_{tz}} & \\
& & & & \\
& *++\txt{$su(2,\mathbb{H}) = sp(2) $\\$[B_2 = C_2]$} \ar[uuuur]^(.8){A_\l}^(.75){G_1+S^1_\l} \ar@/_1.8pc/[uurr]^(.3){B^1_{tz}} \ar@{.>}[uul]^(.5){A_\l} \ar@{.>}[uu]<1ex>_(.5){G_\l+S^1_\l} & *++\txt{$[G_2]$} & *++\txt{$su(3,\mathbb{C})_s$\\$[A_2]$} \ar@{.>}[uull]_(.6){G_\l} & *++\txt{$sl(2,\mathbb{C})_s$\\ $[A_1\oplus A_1]$ } \ar@{.>}[uul]<1ex>{} \ar[uul]_(.5){G_\l-S^1_\l}\\
& & *++\txt{$so(4,\mathbb{R})$\\$[D_2 = C_1 \oplus C_1]$} \ar@{_{(}.>}[ul]^(.4){} \ar[uuuuur]^(.6){A_\l}^(.55){G_\l+S^1_\l} & & \\
& & & *++\txt{$su(1,\mathbb{H})$\\$[C_1]$} \ar[uuuul]<-1ex>_(.25){A_\l}_(.2){G_\l+S^1_\l} \ar@{.>}[ul]<1ex>{} \ar[ul]_(.45){R^1_{x\l}} & *++\txt{$su(2,\mathbb{C})_s$\\$[A_1]$} \ar[uu]_(.5){B^1_{tz}} \ar@{.>}[uul]<1ex>^(.5){} \ar[uul]_(.5){R^2_{x\l} \to S^1_\l } \\
& & & u(1) \ar[u]_(.4){G_\l - S^1_\l} & u(1) \ar[u]_(.4){R^1_{x\l}}\\
}
\caption{Preferred subalgebra chains of~$E_6$ allowing a change of basis}
\label{fig:clean_change_of_basis_chain_of_subalgebras_of_E6}
\end{center}
\end{minipage}
\end{figure}
\newpage{}
\part{Open Questions}
\label{ch:Open_Questions}
This study of $sl(3,\mathbb{O})$ has been driven by an underlying desire to find subalgebras useful for physics
and to find a subalgebra structure which treats ``$\l$ as special.
Although we have not fully resolved these issues, this study of~$sl(3,\mathbb{O})$ has raised some additional interesting questions.
Ultimately, the interesting subalgebra structure we seek involves~$so(3,1,\mathbb{R})$ along with $su(3,\mathbb{C}) \oplus su(2,\mathbb{C}) \oplus u(1,\mathbb{C})$. While we did not carry out an exhaustive search for this structure, we did find structures which are the direct sum of subalgebras of $sl(3,\mathbb{O})$, as indicated in Appendix~\ref{direct_sums_in_E6}.
This search may be more fruitful if we knew exactly how the desired algebra structure contains the direct sum $su(3,\mathbb{C}) \oplus su(2,\mathbb{C}) \oplus u(1,\mathbb{C})$ and $so(3,1,\mathbb{R})$.
There are a number of ways to interpret the phrase ``$\l$ is special'', and each leads to interesting substructures of $E_6$. In one scenario, we choose $\l$ as our preferred imaginary complex unit, leaving us free to choose the quaternionic subalgebra $\mathbb{H}$ to either contain $\l$ or to be independent of $\l$. For instance, choosing $\mathbb{H}$ generated by $\langle 1, k, kl, l \rangle $ obviously creates the scenario $\mathbb{C} \subset \mathbb{H} \subset \mathbb{O}$, while choosing $\mathbb{H}$ to be generated by $\langle 1, i, j, k \rangle$ breaks $\mathbb{O}$ into division algebras~$\mathbb{C}, \mathbb{H}$ whose intersection is $\mathbb{R}$. An interesting question would be to find the subalgebras of $sl(3,\mathbb{O})$ which respect these divisions of the octonions into $\mathbb{C}$ and $\mathbb{H}$.
The questions above may also be extended to the group $SL(3,\mathbb{O})$ in various ways. Although we found in Section \ref{ch:E6_basic_structure.fix_l} the subalgebra of $sl(3,\mathbb{O})$ which fixes the $\l$ in the type $T=1$ vector of $sl(3,\mathbb{O})$, it is not clear how to give an expression for the one-parameter curves associated with the $8$ null rotations. In addition, there may be transformations in $SL(3,\mathbb{O})$ which fix $\l$ but are not connected to the identity. There could also be discrete transformations in $SL(3,\mathbb{O})$ which form a group and also fix $\l$.
We could use our automorphisms to study the subalgebra structures of the other real forms of $E_6$. As a real algebra, we did not find any real form of $A_4$ in $sl(3,\mathbb{O})$, even though we can find it in $\phi^*_{H^\perp}\left( sl(3,\mathbb{O}) \right)$. We therefore do expect that there will be slight differences in the subalgebra structures of these different real forms of $E_6$.
Finally, this work may also be extended to study the structure of the final two exceptional Lie algebras $E_7$ and $E_8$.
\newpage{}
\part{Conclusion}
\label{ch:Conclusion}
We presented here a study of the subalgebra structure of $sl(3,\mathbb{O})$, a real form of the complex Lie algebra $E_6$. We first examined the subalgebra structure of the complex Lie algebra~$E_6$.
We then expanded upon Dray and Manogue's work with $SL(3,\mathbb{O})$ to provide subalgebras in the $3 \times 3$ case corresponding to the $2 \times 2$ subalgebra structure of $sl(2,\mathbb{O})$. Finally, we used automorphisms of real algebras to provide some subalgebra structures in $sl(3,\mathbb{O})$ which are distinctly $3 \times 3$.
In Chapter \ref{ch:Lie_groups_Lie_algebras}, we presented methods which illustrated how root and weight diagrams could be used to visually identify the subalgebras of a given Lie algebra. While the standard methods of determining subalgebras rely upon adding, removing, or folding along nodes in a Dynkin diagram, we showed here how to construct any of a Lie algebra's root or weight diagrams from its Dynkin diagram, and how to use geometric transformations to visually identify subalgebras using those weight and root diagrams. In particular, we showed how these methods can be applied to algebras whose root and weight diagrams have dimensions four or greater. In addition to pointing out the erroneous inclusion of ~$C_4 \subset F_4$ in ~\cite{gilmore, van_der_waerden}, we provided visual proof that ~$C_4 \subset E_6$ and listed all the complex subalgebras of ~$E_6$. While we were primarily concerned with the subalgebras of ~$E_6$, these methods could be used to find subalgebras of any rank ~$l$ algebra.
In Chapter~\ref{ch:Division_Algebras_Applications}, we discussed the four division algebras. We repeated the findings of Manogue and Schray who showed how multiplication in $\mathbb{H}$ and $\mathbb{O}$ can produce rotations in specific planes in $\mathbb{R}^4$ and $\mathbb{R}^8$. Lorentz transformations in $(k+1)$ dimensions were related to determinant or trace preserving transformations of $2 \times 2$ hermitian matrices using the expressions developed in \cite{manogue_schray}. After discussing the Albert algebra, we showed how to construct an expression for the determinant of a $3 \times 3$ octonionic hermitian matrix. This material was needed for the construction of our real form of $E_6$.
We discussed our construction of the Lie group $SL(3,\mathbb{O})$ and its associated Lie algebra $sl(3,\mathbb{O})$ in Chapter \ref{ch:E6_basic_structure}. The $2 \times 2$ formalism for hermitian octonionic matrices given in \cite{manogue_schray} was generalized to the $3 \times 3$ case using our notion of {\it type}. Despite the use of nested matrices in transformations, we constructed an association between group and algebra transformations and produced the multiplication table in the Lie algebra. We repeated many of the results shown in \cite{manogue_dray} which were needed for this work. We identified $F_4$, $G_2$, and a preferred $su(3,\mathbb{C}) \subset G_2$. We showed how our notion of {\it type} is related to {\it triality}, and identified a {\it strong} notion of triality among our transformations. We also showed the existence of {\it continuous type transformations}, and identified subgroups of $SL(3,\mathbb{O})$ which contain these transformations. We also used this type transformation to study type dependent and type independent subalgebras of $sl(3,\mathbb{O})$. The subalgebra structures of $so(9,1)$, $sl(2,\mathbb{O})$, and $su(2,\mathbb{O})$, which were known for the $2 \times 2$ case, were generalized to the $3 \times 3$ case. Finally, we found a non-simple subalgebra of $sl(3,\mathbb{O})$ which fixes $\l$.
In Chapter \ref{ch:E6_further_structure}, we used automorphisms of $sl(3,\mathbb{O})$ to find subalgebras of $sl(3,\mathbb{O})$. We adapted theory related to automorphisms of real forms of complex algebras to provide methods which helped us find subalgebras of our specific real form of $E_6$. This search provided some surprising results. We found multiple real forms of $C_3$ and $C_4$ in $sl(3,\mathbb{O})$. These real algebras differ both in how they reduce the octonions $\mathbb{O}$ to the quaternions $\mathbb{H}$ and how they used quaternionic and {\it orthogonal-quaternionic} spinor and dual spinor transformations. We saw that each octonion $q$ defined both a quaternionic subalgebra which contained $q$ and a quaternionic subalgebra which was perpendicular to $q$. Further, each of those subalgebras were used to construct real forms of $A_5$ and $C_4$ in $sl(3,\mathbb{O})$. The automorphisms were also used to show that $sl(3,\mathbb{O})$ contains both a compact and non-compact form of $F_4$. With the results of the previous section, we were able to construct subalgebra maps of $sl(3,\mathbb{O})$, labeled with Casimir operators, which show how its subalgebras sit in relation to each other. When used with our basis of $sl(3,\mathbb{O})$, the automorphisms could also be used to construct these maps for the other real forms of $E_6$.
\bibliographystyle{unsrt}
| -285,310.191297 |
[
-3.005859375,
2.68359375
] | 33.615288 |
[
-2.109375,
1.8916015625,
-1.6064453125,
-4.69921875,
-1.02734375,
6.3984375
] |
[
2.271484375,
8.4609375,
2.744140625,
6.5390625
] | 2,455 | 42,467 |
[
-3.388671875,
3.8125
] | 35.260784 |
[
-5.3359375,
-3.1484375,
-4.53125,
-2.103515625,
1.443359375,
11.828125
] | 0.649195 | 20.96366 | 13.278546 | 5.084761 |
[
3.129856824874878
] | -160,269.719349 | 5.694728 | -283,663.316809 | 0.227425 | 6.635486 |
[
-1.8603515625,
-3.26171875,
-3.771484375,
-4.90625,
1.9208984375,
11.640625
] |
[
-5.05078125,
-1.640625,
-1.923828125,
-1.1572265625,
3.62109375,
4.01171875
] | |
BkiUbdM5qsMAIwbUZ5AS
|
\section{Molecular Clouds in the Galactic Center}
~~~~The cold phase of the interstellar medium condenses into dense and compact clouds, containing predominantly self-gravitating molecular gas shaped by supersonic turbulence \citep{2001RvMP...73.1031F,2004RvMP...76..125M,McKee2007,2014prpl.conf...77P}. Fragmentation of this gas into collapsing cores seeds pre-stellar objects, leading to star formation and its feedback on the parent environment \citep{2015MNRAS.450.4035F,2018PhT....71f..38F}. Thus, molecular clouds compose a key part of the ISM life-cycle since they facilitate an extremely important link between global ($\gtrsim100$ pc) gas flows and cradles of individual massive stars ($\lesssim0.01$ pc) \citep{2015MNRAS.454..238W}.
Molecular clouds in the Central Molecular Zone (CMZ) of our Galaxy are of particular interest in this regard, since they evolve in a very extreme environment of the Galactic Center, and also because their measured star formation efficiency appears to be an order of magnitude lower compared to the molecular clouds in the Galactic disk \citep{2017A&A...603A..89K}. Additionally, these clouds trace the flows of the cold gas that might be responsible for feeding of the Galaxy's central supermassive black hole, Sgr A*, and hence regulating its feedback.
The internal structure of molecular clouds is determined by a complex, scale-dependent interplay between several processes, which are far from being fully understood despite their fundamental importance for a range of astrophysical contexts \citep[][]{McKee2007,Heyer2015}. Namely, driving and decay of the supersonic turbulence, generation of magnetic fields and cosmic ray propagation are all likely to play a role in the mass and energy flows across a range of scales spanning many orders of magnitude. On top of this, molecular clouds are essentially dynamical structures evolving as their internal life-cycle proceeds as well as in response to changes in external conditions \citep[e.g.][]{2015MNRAS.447.1059K}. In that sense, molecular clouds in the CMZ are subject to the most extreme and dynamic environment in the Milky Way, so they are ideal laboratories to study the aforementioned physical processes in action \citep[][]{2017A&A...603A..89K}.
For a typical molecular cloud, quasi-isothermal supersonic turbulence is believed to shape the density field at $1-10$ pc scales, resulting in the log-normal distribution function \citep{1994ApJ...423..681V} with the width determined by the Mach number of the turbulent motions and the nature of forcing \citep{2008ApJ...688L..79F}. On sub-pc scales, self-gravity of individual dense cores starts to take over, marking the transition to the coherent (rather than turbulent) motions \citep[e.g.][]{Goodman1998}, and it is these scales that are believed to be crucial for determining a cloud's overall massive star-formation efficiency \citep[e.g.][]{Williams2000,Ward2007,2018MNRAS.479.1702O,2019arXiv190200934K}. Thus, probing a range of scales from 10 pc down to 0.1 pc is absolutely necessary to reconstruct the physical picture of the multi-scale gas dynamics by separating various effects and revealing their interconnections.
Physical conditions and hence emission properties of the gas vary noticeably from scale to scale, so it is very hard to construct a probe that would be unbiased and free of opacity and projection effects. In contrast to the nearby molecular complexes, any emission from the clouds in the CMZ is strongly affected by interstellar absorption and by the fact that the typical angular resolution of a single-dish antenna ($\sim$ 30 arcsec) allows resolving only scales above 1.5 pc at the distance to the Galactic Center. Single-dish observations however provide information about the total mass of the cloud (hence its average density) and velocity dispersion (hence the Mach number of the gas motions inside it). Interferometric observations (e.g. with ALMA) are capable of resolving structures down to $ \sim 1$ arcsec (and even smaller) resolution, so that individual filaments and dense cores can be efficiently detected \citep{2015ApJ...802..125R,2017IAUS..322..162U}. Reconstruction of the density PDF with such data is however problematic, due to leakage of the large-scale power \citep{2016ApJ...832..143F}.
Fortunately, in the CMZ the Nature offers us a unique diagnostic tool - illumination of the clouds with the Sgr~A* flare, that is short enough to remove all adverse projection or opacity effects in the "reflected" signal. This X-ray reflection technique is very much complementary to both single-dish and interferometric observations in molecular line emission, basically providing a link between them.
\begin{figure}
\includegraphics[trim= 5cm 2cm 5cm 1cm,width=0.28\textwidth]{par_v4.pdf}
\includegraphics[trim= 0cm -2.2cm 0.cm 0cm, width=0.8\textwidth,clip=t,angle=0.,scale=0.46]{ks1_chandra_0015_lin2_v3.png}
\includegraphics[trim= 0mm -1.4cm 0.5cm 5cm, width=1\textwidth,clip=t,angle=0.,scale=0.3]{bright_filament_y15_y16.png}
\caption{\small {\bf Left:} Sketch of molecular gas in the Galactic Center region exposed to a 4 years long flare
that happened 110 years ago (a view from above the Galactic Plane). The light propagates from Sgr~A* to the dense gas and then to the observer \citep{1998MNRAS.297.1279S}. The locus of illuminated gas is a space between the two ellipsoids. Only this thin slice should be visible in X-rays, eliminating all adverse effects of projection. {\bf Middle:} \Chandra image of the reflected component in the central region of the Galaxy. This component can be easily separated from other background and foreground components using its distinct spectral shape.
\textbf{Right}: Changes in the 4-8 keV flux maps on a time scale of one year in a small patch $\sim20$~pc from Sgr~A*.
\label{fig:sketch}
}
\end{figure}
\section{Illumination of molecular clouds by Sgr~A* X-ray flare}
~~~~The supermassive black hole Sgr A* at the center of the Milky
Way is currently very dim \citep{2010RvMP...82.3121G}, although it has
experienced powerful outbursts in the recent past,
as revealed by reflected X-ray emission coming from dense
molecular clouds with a light-travel-time delay of tens to hundreds of years \citep{1993ApJ...407..606S,1996PASJ...48..249K,2004A&A...425L..49R,2010ApJ...719..143T,2015ApJ...814...94M,2017MNRAS.468.2822K}. One recent outburst took place
$t_{age}\sim 110~{\rm yrs}$ ago, when Sgr A* was more than a million times brighter than today for a period of time
shorter than $\Delta t\sim$ 1.6 years \citep{2017MNRAS.465...45C,2017MNRAS.468..165C,2017MNRAS.471.3293C}.
The upper limit on the flare duration is set by variability of the observed emission \citep[e.g.][]{2007ApJ...656L..69M,2010ApJ...714..732P,2013A&A...558A..32C}. The total energy emitted during such a flare amounts to $\sim$ few $ 10^{47}$~erg.
The nature of such flare remains unclear (it might be a signature of tidal disruption of a planet followed by a short episode of debris accretion onto the supermassive black hole). Clues could be \textbf{found} by recovering the {\bf light curve of the flare}, which is possible if a very compact knot is \textbf{found} in the illuminated molecular gas. If the light-crossing time of the knot is shorter than the duration of the flare, then the variation of its reflected emission accurately reproduces the flare profile. A comparison of light curves for several compact knots at different locations would be useful to verify if there was a single flare or many flares separated by extended quiescent periods.
{\bf The geometry of the illuminated region} is uniquely set by the time delay due to propagation of scattered light from Sgr~A* to the cloud and then to the observer (see Fig.~\ref{fig:sketch}). Hence, knowing the age of the outburst gives an opportunity to reconstruct the {\bf 3D location of the reflecting molecular gas} with respect to Sgr~A* (see Fig.~\ref{fig:sketch} and also the left panel in Fig.~\ref{fig:pdf}). Moreover, knowing that the outburst was shorter than a few years allows one to turn X-ray reflection into an extremely powerful tool for probing characteristics of the illuminated molecular gas itself. Indeed, the line-of-sight thickness of the illuminated region is just $\Delta z \sim c\Delta t\sim0.2$~pc (and possibly even narrower), i.e. much smaller then the typical size of the whole cloud ($\sim$10 pc). Thus, the outburst effectively `cuts out' a {\bf thin 2D slice of molecular gas} at any given moment, and surface brightness of the X-ray emission is directly proportional to the local gas density (see Figs.~\ref{fig:sketch} and~\ref{fig:pdf}). The reflected X-ray emission has a very characteristic spectral shape (Fig.~\ref{fig:shoulder}), which facilitates its clean separation from other background and foreground components. As a result, the observed X-ray surface brightness is linked to the underlying density field of the molecular gas in a straightforward fashion, so it can readily be exploited to derive statistical properties of the latter.
The first attempt of such a study \citep{2017MNRAS.465...45C} used \Chandra data to reconstruct the {\bf gas density PDF}. Although the measured distribution function was consistent with the expectations for quasi-isothermal supersonic turbulence, i.e. having log-normal shape with the width possibly indicating mostly solenoidal driving \citep{2017MNRAS.465...45C}, it has been recognized that the PDF measured this way suffers from substantial statistical issues both at the low-surface brightness end {(Eddington bias)} and for the brightest individual pixels (shot noise and sampling variance). These problems can be cured with increased exposure \citep{2019K}. However, simply accumulating the data taken at different epochs causes another problem -- all sub-pc structures are expected to be highly variable (see Fig.\ref{fig:sketch}) and co-adding the images smears out the smallest scales. In principle, the \Chandra angular resolution ($1''$ corresponds $\sim$0.04~pc) potentially offers the \textbf{dynamic range of some 200} in resolved spatial scales. Useful constraints can still be achieved with a Ms-long exposure, (collected over the period shorter than $\sim$4 months), but the \Chandra effective area is simply too small to exploit this diagnostic fully, while the angular resolution of \XMM limits the dynamic range. A natural solution - a telescope with the effective area ten times larger than \Chandra and a comparable angular resolution.
\begin{figure}
\includegraphics[trim= 0cm 6cm 0.cm 4cm, width=0.65\textwidth,clip=t,angle=0.,scale=0.5]{slab_tv1.pdf}
\includegraphics[trim= 0cm 6cm 0.cm 4cm, width=0.65\textwidth,clip=t,angle=0.,scale=0.5]{spec_rega.pdf}
\includegraphics[trim= 0mm 6cm 0.cm 4cm, width=0.65\textwidth,clip=t,angle=0.,scale=0.5]{spec_blob.pdf}
\caption{\small {\bf Left:} Predicted spectrum arising from a slab of cold gas with the Thomson optical depth $\tau_T=0.2$ illuminated by a power-law X-ray continuum. Only a narrow region near the prominent iron 6.4 keV line is shown (black line). The blue line shows a doubly-scattered emission profile. This emission can be observed even after the primary photons have already left the cloud. {\bf Center:} Predicted reflected emission, which will be observed with future X-ray observatories having $\sim 1500\;{\rm cm^2}$ effective area and $\sim 3$~eV energy resolution from the bright region shown in Fig.~1 (middle panel) in 100~ks observation. The blue curve corresponds to a thermal plasma emission, schematically illustrating the contribution of unresolved compact sources. \textbf{Right}: The simulated spectrum from one of the knots shown in the right panel of Fig.~\ref{fig:sketch} for an instrument with 10 times larger effective area than \Chandra and an energy resolution of 3~eV.
\label{fig:shoulder}
}
\end{figure}
A further nontrivial diagnostic can be provided by measurements of the total flux and time variations of the {\bf fluorescent line Compton shoulder}. For a compact cloud entirely illuminated by the flare, the spectrum should be similar to the one shown in Fig.~\ref{fig:shoulder}, where the flux in the line shoulder $F_{s}$ is approximately $\tau_{c}$ times smaller than the flux in the narrow fluorescent lines $F_{l}$. However, for a larger cloud, whose light-crossing time is longer than the duration of the flare, the strength of the shoulder should grow steadily with time as the flare front propagates through the cloud, reaching maximum values of $\tau_{c}\times F_{l}$ by the time when the front is about to leave the cloud. Once the front leaves the cloud, the ratio $\frac{F_{s}}{F_{line}}$ jumps to a level of order unity - an unambiguous spectral signature of the second (and higher-order) scattering \citep{1998MNRAS.297.1279S,2016A&A...589A..88M}. The life-time of the second scattering component is set by the light crossing time of the cloud or of the scattering environment on larger scales and, therefore, probes the mean gas density of the entire region and, possibly, low-density extended envelopes of molecular clouds connecting them with the ambient ISM.
\begin{figure}
\href{https://wwwmpa.mpa-garching.mpg.de/~churazov/gc_rot_1.mp4}{\includegraphics[trim= 1cm 0.cm 1.cm 0cm, width=0.7\textwidth,clip=t,angle=0.,scale=0.4]{3d.png}}
\includegraphics[trim= 0cm 7cm 0.cm 5cm, width=0.72\textwidth,clip=t,angle=0.,scale=0.5]{tf_a1_n4_e5p_lynx1arcsec05yr.pdf}
\includegraphics[trim= 1cm 7cm 0.cm 7cm, width=0.73\textwidth,clip=t,angle=0.,scale=0.5]{efloffset_phys_nl.pdf}
\caption{\small {
{\bf Left:} Reconstructed \href{https://wwwmpa.mpa-garching.mpg.de/~churazov/gc_rot_1.mp4}{\color{blue}{3D distribution}} of the molecular gas density from the \XMM data accumulated over 15 years of observations. The width of the reconstructed layer is $\sim 3.5\;{\rm pc}$ and the radius is $\sim 35\;{\rm pc}$; adapted from \citep{2017MNRAS.465...45C}.
\bf Center:} Linearity of the surface brightness of reflected X-ray emission and gas density (and characteristic size) of a substructure inside a molecular cloud with power-law relation between density (bottom axis) and size (top axis) in a form $ \rho\propto l^{-\alpha} $ with $\alpha=1$, and $\rho_0=10^4$ cm$^{-3}$ at $L_0=10$ pc. Front propagation speed along the line-of-sight is set equal to $v_z=0.7c$, the flare's duration is $ \Delta t=0.5$ yr. The right axis shows the expected number of counts per 1''$\times$1'' pixel. The red dashed curve (arbitrary normalization) shows mass distribution between the scales for a cloud having log-normal density distribution with mean density $\rho_0=10^4$ cm$^{-3}$ and dispersion $\sigma_s=1$, while the solid red line shows how this PDF will be distorted by the Eddington bias corresponding to $\approx2$ counts per pixel at $\rho_0=10^4$ cm$^{-3}$. {\bf Right:} Simulated shift of the line 6.4 keV centroid for a shear velocity field resembling observed average densities and velocity gradients in the Brick cloud based on HNCO data of the MOPRA survey \cite{2015ApJ...802..125R,2016ApJ...832..143F}.
\label{fig:pdf}
}
\end{figure}
The next frontier is the {\bf mapping of the gas velocity field} using the centroid shift of the 6.4 keV line.
Typical turbulent velocities of the order of few ${\rm km\;s^{-1}}$ can hardly be measured even with the next generation of X-ray telescopes. However, the shear velocity in the Galactic Center region is much larger and can be mapped with cryogenic bolometers. This is true not only for clouds separated by tens of pc, but also for the velocity variations within individual clouds (see Fig.~\ref{fig:pdf}).
Having in hand the 3D positions and the line-of-sight velocities, the next natural step is to {\bf cross-match the X-ray data and velocity-resolved emission of various molecular species}. This would allow converting PPV molecular data into a much more powerful dataset, since the line-of-sight positions of clouds will be known. The main limitation of this approach is that only molecular clouds that are illuminated by the flare during the time span of observations can be identified.
We reiterate here that the reflected X-ray signal above 4~keV is essentially insensitive to variations of temperature of the ionization state of the gas (at the levels characteristic for typical warm or cold phases of the ISM). There are subtle and interesting effects associated with electrons bound in different species, e.g., helium \citep{1996AstL...22..648S}, but such effects are beyond the scope of this paper.
\section{Summary}
~~~~~~Presence of large quantities of molecular gas in the vicinity of Sgr~A* opens the possibility for synergetic studies of the supermassive black hole past outbursts reflected by the gas in clouds and the fundamental properties of the molecular clouds themselves, which are not easily accessible without illumination by these outbursts. Table~\ref{tab:sum} summarizes the properties that can be constrained and the required capabilities of the telescopes. The full potential of this diagnostic can only be unraveled with the next generation of high-sensitivity, high-energy and high-angular resolution X-ray observatories (see Table~\ref{tab:req}).
Finally, high sensitivity and spectral resolution would allow for a systematic study of the {\bf reflected/delayed emission in other galaxies} \citep{1998MNRAS.301L...1G,2017IAUS..322..253C,2019ApJ...870...69F}, effectively probing the outburst rates (in particular, TDEs) over the last few hundred years in a large sample of galaxies.
\begin{table*}[b!]
\begin{center}
\caption{\small Road-map for probing Sgr~A* flare and molecular clouds with X-ray data. Driving capabilities are coded as follows: {\bf A} - large effective Area; {\bf I} - excellent Imaging; {\bf S} - excellent Spectral resolution; {\bf T} - possibility to make observations, separated in Time by few months; {\bf P} - Polarimetry in X-ray band. "Now" stands for current generation of X-ray telescopes.} \vspace{0.5cm}
\label{tab:sum}
\begin{tabular}{l | l | l}
\hline
\textbf{Parameter} & \textbf{Methods} & \textbf{Driving capabilities} \\
\hline
& {\it Sgr~A* flare diagnostic} & \\
\hline
$t_{age}$ & Structure functions in time and space & Now, {\bf T, I, A, S}\\
& Equivalent width of the 6.4 keV line & Now, {\bf S, A, I} \\
(also source position) & Polarization & {\bf P, A, S, I}\\
\hline
$\Delta t$ & Structure functions in time and space & Now, {\bf T, I, A, S}\\
& Smallest clouds (short light crossing time) & {\bf I, A, S}\\
\hline
$L_{X,min}$ & Brightest clouds (maximal density) & {\bf I, A}\\
\hline
$L_X(t)$ & Smallest clouds (short light crossing time)& {\bf I, A, S}\\ \hline
$L_{X}\Delta t$ & X-ray images + molecular data & {\bf I, A} \\
\hline
& {\it Molecular gas diagnostic} & \\
\hline
$PDF(\rho)$ & Deep X-ray images & {\bf I, A, S}\\
\hline
$\rho_{3D}(l,b,z)$ & Multiple X-ray images & Now, {\bf I, A, S} \\
\hline
$v_z(l,b,z)$ & High resolution X-ray spectroscopy & {\bf S, A, I}\\
\hline
PPV $\rightarrow$3D & High resolution X-ray spectroscopy & {\bf S, A, I}\\
\hline
Other galaxies & Sample of quiescent galaxies & {\bf I, A, S}\\
\hline
\end{tabular}
\end{center}
\end{table*}
\begin{table*}[b!]
\begin{center}
\caption{Main requirements for future X-ray observatories} \vspace{0.5cm}
\label{tab:req}
\begin{tabular}{l | l}
\hline
\textbf{Goal} & \textbf{Requirements} \\
\hline
Resolve 0.05~pc @ GC distance & PSF$\sim$1" \\
\hline
Get $>$200 counts in 6.4 keV line at scales $\sim$0.1~pc (Fig.\ref{fig:pdf}) & Area$\sim 1500\;{\rm cm^2}$ @ 6.4 keV\\
\hline
Error in velocity $< 10\;{\rm km\;s^{-1}}$ & FWHM $\sim$3 eV \\
\hline
\end{tabular}
\end{center}
\end{table*}
\newpage
\pagebreak
| -12,628.169052 |
[
-2.76171875,
2.689453125
] | 38.888889 |
[
-2.748046875,
1.181640625,
-1.158203125,
-4.2890625,
-0.50341796875,
6.09375
] |
[
5.32421875,
8.8984375,
5.57421875,
7.3671875
] | 265 | 2,684 |
[
-2.552734375,
2.572265625
] | 27.973863 |
[
-5.96484375,
-3.265625,
-3.46484375,
-2.5625,
1.3330078125,
11.46875
] | 1.839443 | 24.119116 | 34.23994 | 3.125132 |
[
2.8596372604370117
] | -10,018.061555 | 5.671386 | -12,324.928684 | 0.137958 | 5.834619 |
[
-3.671875,
-3.84765625,
-2.947265625,
-3.83984375,
2.50390625,
11.09375
] |
[
-5.953125,
-2.3828125,
-2.23046875,
-1.8291015625,
3.619140625,
5.92578125
] | |
BkiUdTrxK02iP5rWKmQx
|
\section*{Introduction}
The interplay between quantum physics and machine learning gives rise to an emergent research frontier of quantum machine learning that has attracted tremendous attention recently \cite{Carleo2019Machine,Sarma2019Machine,Biamonte2017Quantum,Dunjko2018Machine}.
In particular, certain carefully-designed quantum algorithms for machine learning, or more broadly artificial intelligence, may exhibit exponential advantages compared to their best possible classical counterparts \cite{Sarma2019Machine,Biamonte2017Quantum,Dunjko2018Machine,gao2018quantum,harrow2019low,Lloyd2018Quantum,Havlicek2019}.
An intriguing example concerns quantum generative adversarial networks (QGANs) \cite{Lloyd2018Quantum},
where near-term quantum devices have the potential to showcase quantum supremacy \cite{arute2019quantum} with real-life practical applications.
The general framework of QGANs consists of a generator learning to generate statistics for data mimicking those of a true data set, and a discriminator trying to discriminate generated data from true data \cite{Lloyd2018Quantum}. The generator and discriminator follow an adversarial learning procedure to optimize their strategies alternatively and arrive at a Nash equilibrium point, where the generator learns the underlying statistics of the true data and the discriminator can no longer distinguish the difference between the true and generated data.
Different versions of QGANs have been proposed~\cite{Demers2018Quantum,Lloyd2018Quantum,Zeng2019Learning,Zoufal2019Quantum,hu2019quantum,regitti2019,regitti2020}.
Recently, a proof-of-principle experimental QGAN demonstration has been reported \cite{hu2019quantum}, showing that the generator can indeed be trained via the adversarial learning process to replicate the statistics of the single-qubit quantum data output from a quantum channel simulator. Yet, in this experiment the gradient, which is crucial for training the QGAN, was estimated numerically by the finite-difference method with a classical computer. This finite-difference approach induces an inaccuracy to the gradient and therefore may significantly retard the convergence to the equilibrium point \cite{harrow2019low}. In addition, the quantum states involved in this experiment were all single-qubit states, and entanglement, a characterizing feature of quantumness and a vital resource for quantum advantages, was absent during the learning process \cite{Biamonte2017Quantum}.
In this paper, we add these two crucial yet missing blocks by reporting an experiment realization of a QGAN based on a programmable superconducting processor with multiple qubits.
Superconducting qubits are a promising platform for realizing QGANs, owing to their flexible design, excellent scalability and remarkable controllability.
In our implementation, both the generator and discriminator are composed of multiqubit parameterized quantum circuits, also referred to as quantum neural networks in some contexts~\cite{Jarrod2018QNN, Cerezo2020Plateaus, Andrea2020QNN, Patrick2020}.
Here, we benchmark the functionality of the quantum gradient method by learning an arbitrary mixed state, where the state is replicated with a fidelity up to 0.999. We further utilize our QGAN to learn an classical XOR gate, and the generator is successfully trained to exhibit a truth table close to that of the XOR gate.
\begin{figure*}
\includegraphics[width=1.0\textwidth]{Figure1}
\caption{\textbf{QGAN algorithm and its implementation.} \textbf{a,} Overview of the QGAN logic.
\textbf{b,} Sketch of the superconducting processor used to implement the QGAN algorithm, where the five qubits, Q$_0$ to Q$_4$, are interconnected by the central bus resonator.
\textbf{c,}
An instance of the experimental sequences for fulfilling the QGAN algorithm to learn the classical XOR gate. Both \textbf{G} and \textbf{D} are parameterized quantum circuits consisting of layers of the multiqubit entangling gate $U_{\textrm{ENT}}$ and the single-qubit rotations $\theta_{j,l,m}^{x/z}$, where $l$ is the layer index, $m \in \{\textbf{G},\textbf{D}\}$, and the superscript (x or z) refers to the axis in the Bloch sphere around which the state of Q$_j$ is rotated by the angle $\theta$.
Shown is the training sequence on \textbf{G} to optimize its parameter $\theta_{3,1,G}^x$ based on the quantum gradient subroutine, which includes two Hadamard gates ($H$), two controlled rotation gates,
and a $\pi/2$ rotation around x-axis ($X/2$).
In this instance, both $Q_1$-$Q_2$ and $Q_3$-$Q_4$ store the input label,
and \textbf{D}'s output score is encoded in Q$_1$ by $S^{{\textrm{D}},\textrm{R/G}}_n = \langle\sigma^z_{1}\rangle/2+1/2$, which can be obtained by directly measuring Q$_1$.
$Q_0$'s $\langle\sigma_0^z\rangle$ gives the score derivative with respect to the rotational angle parameter right before the first controlled rotation gate, i.e., $\partial\langle\sigma_1^z\rangle / \partial \theta_{3,1,G}^x = -\langle\sigma_0^z\rangle$ for the sequence displayed here. The first controlled rotation gate can be either a controlled X (CNOT) or controlled Z (CZ) gate depending on the single-qubit rotation axis it follows as shown in the lower right box.
}
\label{fig:Fig1}
\end{figure*}
We first introduce a general recipe for our QGAN and then apply it to two typical scenarios: learning quantum states and the classical XOR gate.
The overall structure of the QGAN is outlined in Fig.~\ref{fig:Fig1} (a), which includes
an input label, a real source (\textbf{R}), a generator (\textbf{G}), a discriminator (\textbf{D}), and a quantum gradient subroutine.
The input label sorts the training data stored in \textbf{R}, and also instructs \textbf{G} to generate data samples mimicking \textbf{R}. \textbf{D} receives the label and corresponding data samples from either \textbf{G} or \textbf{R}, and then evaluates with appropriate scores,
based on which a loss function $V$ is constructed to differentiate between \textbf{R} and \textbf{G}.
The adversarial training procedure is repeated in conjugation with the quantum gradient subroutine, which yields the partial derivatives of $V$ with respect to the parameters constructing \textbf{D} or \textbf{G}, so as to maximize $V$ for the optimal configuration of \textbf{D} and to minimize $V$ for the optimal \textbf{G} in terms of the chosen values of the constructing parameters.
Specifically, denoting the sets of parameters constructing \textbf{G} and \textbf{D} as $\vec{\theta}_\textrm{G}$ and $\vec{\theta}_\textrm{D}$, respectively,
the loss function $V$ is written as
\begin{equation}
V=\frac{1}{N} \sum_{n=1}^{N} \left[S_{n}^{\textrm{D}, \textrm{R}}(\vec{\theta}_\textrm{D})-S_{n}^{\textrm{D},\textrm{G}}(\vec{\theta}_\textrm{D},\vec{\theta}_\textrm{G})\right], \nonumber
\label{loss}
\end{equation}
where $N$ is the total number of data samples selected for training and
$S_{n}^{\textrm{D}, \textrm{R}}$ ($S_{n}^{\textrm{D}, \textrm{G}}$) represents the score of the $n$th data sample from \textbf{R} (\textbf{G}) evaluated by \textbf{D}.
\textbf{G} and \textbf{D} are trained alternately with \textbf{D} being trained first.
In \textbf{D}'s turn, we maximize the loss function by iteratively optimizing $\vec{\theta}_\textrm{D}$ according to
$\vec{\theta}^{i+1}_\textrm{D}=\vec{\theta}^i_\textrm{D}+ \alpha_\textrm{D} \nabla_{\vec{\theta}_\textrm{D}}V$,
where $i$ is the iteration step index and $\nabla_{\vec{\theta}_\textrm{D}}V$ denotes the gradient vector of the loss function
at the $i$th step;
we train \textbf{G} by minimizing the square of the loss function with an iteration relation
$\vec{\theta}^{i+1}_\textrm{G}=\vec{\theta}^i_\textrm{G}-\alpha_\textrm{G} \nabla_{\vec{\theta}_\textrm{G}}V^2$.
The learning rates can be adjusted by tuning $\alpha_\textrm{D}$ and $\alpha_\textrm{G}$ which are typically on the order of unity.
An important quantity that plays a vital role in training our QGAN is the gradient of the loss function with respect to a given parameter. Interestingly, owing to the special structures of our quantum circuits for the generator and the discriminator, a common approach to obtain such a gradient is to shift the corresponding parameter by $\pm \pi/2$ and measure the loss function at the shifted values \cite{Schuld2019}. In our experiment,
we employ a even simpler method---the Hadamard test quantum algorithm \cite{Mitarai2019}---to obtain the gradient.
This algorithm can reduce half of the time in comparison with the common approach to finish the training,
at the expense of an additional auxiliary qubit and two controlled gates (see next) \cite{SM}.
At each round of the training process, the parameters of the discriminator (or generator) are updated simultaneously after the gradients for all parameters are obtained, and a quantum process tomography (QPT) is performed to characterize the overlap fidelity between the generated data based on the updated parameters and the true data.
\begin{figure}[htb]
\includegraphics[width=0.49\textwidth]{Figure2}
\caption{{\bf QGAN performance in learning a mixed state.} \textbf{a,} Tracking of the loss function $V$, the output scores $S^{\textrm{D}, \textrm{R/G}}$, and the state fidelity between \textbf{R}/\textbf{G}'s output states $F$ during the adversarial training procedure with learning rates $\alpha_\textrm{D} = 0.8$ and $\alpha_\textrm{G} = 0.6$. The alternate training stages of \textbf{D} and \textbf{G} are marked by red and white regions, respectively. In each stage, the maximum step number is limited to 50 for \textbf{D} and 100 for \textbf{G}. Inset: the quantum circuit for generating $\rho_\textrm{R}$ for Q$_3$. The same circuit is also used for \textbf{G}, with random initial guesses for the two rotational angles which are then optimized.
\textbf{b,} The real and imaginary parts of the output density matirces of \textbf{R} (black frames) and the final \textbf{G} (solid bars).
}
\label{fig:Fig2}
\end{figure}
The above mentioned QGAN is experimentally realized on a superconducting quantum processor using 5 frequency-tunable transmon qubits labeled as Q$_j$ for $j=0$ to 4, where
all qubits are interconnected by a central bus resonator as illustrated in Fig.~\ref{fig:Fig1} (b).
The role arrangement of Q$_0$ to Q$_4$ can be visualized by the exemplary experimental sequence shown in Fig~\ref{fig:Fig1} (c). Q$_0$ is to assist the quantum gradient subroutine. Q$_1$-Q$_2$ stores the label which is passed to \textbf{D}, with the output score encoded in Q$_1$ by $S^{{\textrm{D}},\textrm{R/G}}_n = \langle\sigma^z_{1}\rangle/2+1/2$. For the experimental instances with \textbf{G} inserted, Q$_3$-Q$_4$ also stores the label as the input to \textbf{G} and the output data sample from \textbf{G} is passed to \textbf{D} via Q$_3$; for the experimental instances with \textbf{R} in replacement of \textbf{G}, only Q$_3$ stores the data sample from \textbf{R} as designated by the label. Details of the device parameters can be found in supplementary materials (SM)~\cite{SM} and Ref.~\cite{Song2019Generation}.
By tuning the qubits on resonance but detuned from the bus resonator in frequency, these qubits can be all effectively connected, which enables the flexible realizations of the multiqubit entangling gates among arbitrarily selected qubits.
The all-to-all interactions are described in the dispersive regime by the effective Hamiltonian $H_{\textrm{I}}=\sum \lambda_{jk}(\sigma_j^+\sigma_k^-+\sigma_j^-\sigma_k^+)$~\cite{Song2019Generation}, where $\sigma_j^+$ ($\sigma_j^-$) denotes the raising (lowering) operator of Q$_j$, and $\lambda_{jk}$ is the effective coupling strength between Q$_j$ and Q$_k$ mediated by the bus resonator.
Evolution under this Hamiltonian for an interaction time $\tau$ leads to the entangling operator with the form of $U_{\textrm{ENT}}=e^{-iH_{\textrm{I}}\tau}$, which can steer the interacting qubits into highly entangled state.
In our QGAN, the parameterized quantum circuits that comprise \textbf{G} and \textbf{D} leverage the naturally available multiqubit $U_{\textrm{ENT}}$s, with
the interaction time of the two-qubit $U_{\textrm{ENT}}$ fixed at around 50~ns for \textbf{G} and the three-qubit one fixed at around 55~ns for \textbf{D}~\cite{SM}.
We note that, in this hardware-efficient realization of the QGAN, the entangling operators can be any device-tailored operations that generate sufficient entanglement.
As laid out in Fig.~\ref{fig:Fig1} (c), the entangling operators $U_{\textrm{ENT}}$ are interleaved with the single-qubit $X$ and $Z$ rotations,
which successively rotate Q$_j$ around x- and z-axis in the Bloch sphere by angles of $\theta_{j,l,m}^x$ and $\theta_{j,l,m}^z$, where
$l$ is the layer index and $m \in \{\textbf{G}, \textbf{D}\}$.
The lengths of the $X$ and $Z$ rotations are fixed at 30 and 20 ns, respectively.
Taking into account the experimental imperfections, we perform numerical simulations to decide the depths of the interleaved layers consisting of $U_{\textrm{ENT}}$ and the single-qubit rotations,
for a balance between the learning fidelity and efficiency.
For example, to learn the XOR gate, the circuit layer depths are set to be 2 and 3 for \textbf{G} and \textbf{D} respectively, as shown in Fig.\ref{fig:Fig1} (c).
The QGAN learning process is guided by the gradient of the loss function with respect to $\vec{\theta}_\textrm{G}$ and $\vec{\theta}_\textrm{D}$.
To obtain these gradients, we adopt a quantum method called Hadamard test~\cite{Demers2018Quantum, Mitarai2019},
which is illustrated in the sequence instance in Fig.~\ref{fig:Fig1} (c).
For the partial derivative with respect to the parameter $\theta_{j,l,m}^{x}$ ($\theta_{j,l,m}^{z}$),
we insert the first controlled-X (Z) gate right after the single-qubit X (Z) rotation containing this parameter, with Q$_j$ as the target.
The second controlled-Z gate is applied at the end of the training sequence with Q$_1$ as the target.
The partial derivative of Q$_1$'s $\langle\sigma^z_{1}\rangle$, which relates to \textbf{D}'s output score,
is given by $\partial\langle\sigma^z_{1}\rangle/\partial\theta_{j,l,m}^{x (z)}=-\langle\sigma^z_{0}\rangle$, which can be directly obtained by measuring Q$_0$ and used in computing the gradient of the loss function.
See SM~\cite{SM} for more
details about the experimental realizations of the controlled-X (Z) gates,
as well as the theoretical and experimental verifications of the quantum gradient method.
To benchmark the functionality of the quantum gradient method and the learning efficiency of our QGAN circuit,
we first train an arbitrary mixed state as data which is a simulation of quantum channel.
As shown in the inset of Fig.\ref{fig:Fig2} (a), the mixed state for Q$_3$ reads
$\rho_\textrm{R} = \left(\begin{smallmatrix} 0.7396 &0.0431 + 0.3501i\\0.0431 - 0.3501i&0.2604\protect \end{smallmatrix}\right)$,
which is generated by applying two single-qubit X rotations (with the rotation angles of 1.35 on Q$_3$ and 0.68 on Q$_4$) followed by the
$U_{\textrm{ENT}}$ gate on Q$_3$ and Q$_4$.
Correspondingly, \textbf{G} is set up with a single layer and 2 parameters describing the single-qubit X rotation angles during the training, while \textbf{D} remains the one shown in Fig.\ref{fig:Fig1} (c) with 3 layers and all 18 parameters being trained.
The trajectories of the loss function and scores of data from \textbf{R}/\textbf{G} during the training process are recorded and plotted in Fig.~\ref{fig:Fig2} (a).
We optimize \textbf{D} at the beginning of the training to enlarge the distance between $S^{\textrm{D}, \textrm{R}}$ and $S^{\textrm{D}, \textrm{G}}$. At the end of this turn, \textbf{D} can discriminate datasets from \textbf{R} and \textbf{G} with the maximum probability.
In \textbf{G}'s turn,
$S^{\textrm{D}, \textrm{G}}$ moves towards $S^{\textrm{D}, \textrm{R}}$ which means that \textbf{G} is learning the behavior of \textbf{R}. Each turn ends when the optimal point of the loss function is reached, or the iteration number
goes over a preset limit.
As the adversarial learning process goes on, the value of the loss function oscillates from turn to turn
and eventually converges to 0
indicating that the learning
arrives at a Nash equilibrium point, where \textbf{G} is able to produce a mixed state $\rho_\textrm{G}$ which resembles $\rho_\textrm{R}$
and \textbf{D} can no longer distinguish between them~\cite{Lloyd2018Quantum,Demers2018Quantum}.
The training process is characterized by the similarity between datasets generated by \textbf{G} and \textbf{R}, which is quantified by the state fidelity $F(\rho_{\textrm{R}}, \rho_{\textrm{G}}) = \textrm{Tr}(\sqrt{\rho_{\textrm{R}}}\rho_{\textrm{G}}\sqrt{\rho_{\textrm{R}}})$,
As shown in Fig~\ref{fig:Fig2} (a), $F$ increases rapidly with the iteration steps, indicating the effectiveness of the adversarial learning.
The density matrices $\rho_{\textrm{R}}$ and the final $\rho_{\textrm{G}}$ are plotted in Fig~\ref{fig:Fig2} (b), which yields a state fidelity of around 0.999.
Now we apply the recipe to train the QGAN to replicate the statistics of an XOR gate, which is a classical gate with input-output rules $00 \rightarrow 0$, $01 \rightarrow 1$, $10 \rightarrow 1$, and $11 \rightarrow 0$. We use the computational basis state $|0\rangle$ and $|1\rangle$ to encode the classical data $0$ and $1$, respectively.
We randomly initialize the parameters of both \textbf{D} and \textbf{G}, and then update them alternately following the rules outlined above.
The trajectories of the key parameters benchmarking the QGAN performance
during the training process are
plotted in Fig.~\ref{fig:Fig3} (a), with the evolutions of two representative parameters construcing \textbf{D} and \textbf{G} shown in Fig.~\ref{fig:Fig3} (b).
Again, the loss function exhibits a typical oscillation during the adversarial process, and the training reaches its equilibrium after about 190 steps with an average state fidelity of 0.927. In addition, after training \textbf{G} successfully exhibits a truth table close to that of an XOR gate, as shown in the inset of Fig.~\ref{fig:Fig3} (a).
\begin{figure}
\includegraphics[width=0.49\textwidth]{Figure3}
\caption{{\bf QGAN performance in learning the XOR gate}. \textbf{a,} Tracking of of the loss function $V$, the averaged output scores $\bar{S}^{\textrm{D}, \textrm{R/G}}$, and the averaged state fidelity between \textbf{R}/\textbf{G}'s output states $\bar{F}$ during the adversarial training process with learning rates $\alpha_\textrm{D} = 1.0$ and $\alpha_\textrm{G} = 1.5$. The maximum step number is limited to 50 for both $\textbf{G}$ and $\textbf{D}$. $\bar{S}^{\textrm{D}, \textrm{R/G}}$ and $\bar{F}$ are averaged over four possible inputs.
Inset: The truth table of \textbf{G} after training in comparison with that of the XOR gate.
\textbf{b,} Trajectories of two representative parameters constructing \textbf{D} and \textbf{G}, $\theta_{3,1,D}^x$ and $\theta_{3,1,G}^x$, during the QGAN training.
All parameters in \textbf{G} and \textbf{D} are initialized randomly between 0 and $\pi$ and optimized alternately during the training.
}
\label{fig:Fig3}
\end{figure}
In conclusion, we have experimentally implemented a multi-qubit QGAN equipped with a quantum gradient algorithm on a programmable superconducting processor.
The results clearly show
the feasibility of QGAN for learning data with both classical and quantum statistics.
The parameterized quantum circuits for constructing quantum generators and discriminators do not require accurate implementations of specific quantum logics and can be achieved on the near-term quantum devices across different physical platforms.
Our implementation paves the way to the much-anticipated computing paradigm with combined quantum-classical processors, and holds the intriguing potential to realize practical quantum supremacy \cite{arute2019quantum} with noisy intermediate-scale quantum devices
\cite{Preskill2018quantumcomputingin}.
\section*{Acknowledgments}
\noindent Devices were made at the Nanofabrication Facilities at
Institute of Physics in Beijing and National Center for Nanoscience
and Technology in Beijing. The experiment was performed on the quantum computing platform at Zhejiang University.
{\bf Funding:} Supported by
National Basic Research Program of China (Grants No. 2017YFA0304300, No. 2016YFA0302104 and No.
2016YFA0300600),
National Natural Science Foundation of China (Grants No. 11934018 and No. 11725419),
the Zhejiang Province Key Research and Development Program (Grant No. 2020C01019),
the start-up fund from Tsinghua University (Grant No. 53330300320), and
Strategic Priority Research Program of Chinese Academy of Sciences
(Grant No. XDB28000000).
{\bf Author contributions:} Z.A.W, D.L.D. and H.F. proposed the idea; supervised by Z.B.L, J.G.T and H.W.,
K.H., C.S., K.X. and Q.G. conducted the experiment;
supervised by H.F., K.H. and Z.A.W. performed the numerical simulation;
H.L. and D.Z. fabricated the device;
K.H., Z.A.W., C.S., Z.W., D.L.D., H.W. and H.F. cowrote the manuscript; and all authors contributed to the experimental setup, discussions of the results, and development of the manuscript.
{\bf Competing interests:} Authors declare no competing interests.
{\bf Data and materials availability:} All data needed to evaluate the
conclusions in the paper are present in the paper or the
supplementary materials.
| -17,241.052268 |
[
-1.7236328125,
1.8251953125
] | 61.42132 |
[
-2.8203125,
0.73974609375,
-1.46484375,
-4.859375,
-0.90576171875,
7.1171875
] |
[
4.71875,
8.4453125,
2.98828125,
7.6484375
] | 188 | 2,803 |
[
-2.20703125,
2.287109375
] | 25.446512 |
[
-5.8828125,
-3.873046875,
-4.01171875,
-2.076171875,
2.060546875,
11.4921875
] | 1.276984 | 17.681886 | 30.43168 | 3.141309 |
[
2.597820281982422
] | -12,828.543875 | 5.866928 | -16,910.530902 | 1.276984 | 5.640394 |
[
-2.556640625,
-3.5,
-3.876953125,
-4.75390625,
2.32421875,
11.7578125
] |
[
-5.0078125,
-1.9462890625,
-1.88671875,
-1.2998046875,
3.119140625,
4.44921875
] | |
BkiUdBQ5qYVBSqkygqvC
|
\section{Introduction}
{Black Holes are one of the most enigmatic and fascinating objects in physics. It has been a very active area of research, and due to }
the spectacular observations from the LIGO \cite{ligo} and the Event Horizon collaborations \cite{EH}, there has been
a resurgence in the study of theoretical puzzles in black hole physics.
Moreover, because there are black hole solutions in alternative theories of gravity, black holes are used as workhorses to probe fundamental aspects of these theories (i.e supergravity, noncommutative gravity, Hordenski gravity, etc.).
{The idea of a noncommutative space-time was revived at the beginning of the century. There have been several attempts to study the possible effects of noncommutativity in the cosmological scenario. In previous works\cite{tachy,yee,yee2}, it has been argued that there is a possible relationship between the cosmological constant and the noncommutative parameters. Moreover, the effects of noncommutativity during inflation were explored\cite{brand}, but noncommutativity was only incorporated in the matter Lagrangian, neglecting the gravitational sector. The difficulties of analyzing noncommutative gravitational models arise from the complicated structure of noncommutative gravity. Writing a
noncommutative theory of gravity which only depends on the
commutative fields and their derivatives has field equations that are
highly nonlinear \cite{sabidograv1,sabidograv2}. To avoid these difficulties, it has been
proposed to introduce the effects of noncommutativity at the quantum level\cite{ncqc},
namely quantum cosmology, by deforming the minisuperspace through a Moyal deformation of the Wheeler-DeWitt (WDW) equation. It is then possible to proceed in noncommutative quantum mechanics \cite{gamboa}.} Using this approach for noncommutative cosmology and using the diffeomorphism to transform the
Schwarzschild metric into the Kantowski-Sachs (KS) metric, one obtains the noncommutative Wheeler-DeWitt (NC-WDW) equation
for the Schwarzschild black hole\cite{LopezDominguez:2007zz}. From the NC-WDW equation and the Feynman-Hibbs method, they calculate the entropy of the noncommutative black hole. This method was originally used for the Schwarzschild black hole, to reproduce the known results \cite{paths}.
We know that supergravity is the supersymmetric generalization of general relativity (GR).
Supersymmetric black hole solutions of supergravity theories played a crucial role in important developments in string black hole physics.
{In supergravity, only black holes that satisfy the BPS constraints are well understood.\footnote{{The simplest case for a spherical symmetric solution that meets this criteria is the Reissner-Nordstr$\ddot{o}$m extremal black hole.}} Consequently non BPS black holes have not been extensively studied.}
{One approach that has been used to study SUSY black holes is to use the relationship between the KS and the Schwarzschild metrics, to introduce supersymmetry}. Along this line of reasoning, classical (and quantum) supersymmetric Schwarzschild and Schwarzschild-(anti) de Sitter black hole models were proposed \cite{julio1,julio2}. By supersymmetrizing the WDW equation associated with the standard Schwarzschild black hole. {The authors derive a modified (SUSY quantum) Hamiltonian and its corresponding classical equations that in this sense define a supersymmetric generalization of the Schwarzschild and Schwarzschild-(anti) de Sitter space-times.}
{Exploiting the relationship between the KS and Schwarzschild metric
we can introduce new physical ideas to black holes. It was effective to introduce noncommutative effects using the WDW equation allowing us to construct noncommutative Schwarzschild black hole. It is reasonable to assume that noncommutative constructions which deal with the supersymmetric versions of the WDW equation have a similar behaviour \cite{eri_andres}, this motivate us to study the effects of noncommutativity on SUSY black holes, using the NC-SUSY WDW equation, therefore the main objective of this paper is to explore black holes in the context of noncommutativity and supersymmetry.}
The paper is organized as follows. In section \ref{sec_1}, we review the proposal for SUSY black holes \cite{julio1}, we also discuss noncommutative black holes. In Section \ref{sec_3} we present our proposal for noncommutative SUSY black Holes. Section \ref{final} is devoted to discussion and final remarks.
\section{Modifying the Wheeler-DeWitt equation}\label{sec_1}
{In this section we discuss how to introduce the new physics to the Schwarzschild black hole. We will briefly discuss SUSY black holes from the WDW equation. We also discuss the introduction of noncommutativity to black holes by deforming the WDW equation.}
\subsection{{Wheeler-DeWitt equation and the SUSY black hole}}
{Let us start by recalling} the classical and quantum aspects of the supersymmetric cosmological KS model and the Schwarzschild black hole. We use the square root and operator method. Where the resulting Hamiltonian has terms that allow us to find a supersymmetric solution \cite{julio1}. From the Schwarzschild metric
\begin{equation}
ds^{2}=-\left(1-\frac{2M}{r}\right)dt^{2}+\left( 1-\frac{2 M}%
{r}\right)^{-1}dr^{2}
+r^{2}\left(d \theta^{2}+\sin^{2}\theta d\varphi^{2}\right),
\end{equation}
we can find the relationship between the cosmological Kantowski-Sachs metric and the Schwarzschild metric,
by doing the coordinate transformation $t\leftrightarrow r$. Also, $g_{tt}$ and $g_{rr}$ change their sign and $\partial_{t}$ becomes a space-like vector. {Finally, we compare the Kantowski-Sachs metric with the parametrization by Misner and identify}%
\begin{equation}
N^{2}=\left(\frac{2M}{t}-1\right)^{-1},\quad
e^{2\sqrt{3}\beta}=\frac{2M}{t}-1,\quad
e^{ -2\sqrt{3}\beta}e^{-2\sqrt{3}\Omega}=t^{2},
\label{dif}
\end{equation}
where we know that
\begin{equation}
ds^{2}=-N^{2}dt^{2}+e^{2\sqrt{3}\beta} dr^{2}+
e^{-2\sqrt{3}\beta}
e^{\left( -2\sqrt{3}\Omega\right)} \left(
d\theta^{2}+\sin^{2}\theta d\varphi^{2}\right).
\label{KSmetric}%
\end{equation}
{We canonically quantize this model and get the} Wheeler-DeWitt equation for the
Kantowski-Sachs metric. {Which, with some particular factor ordering, is given by}
\begin{equation}
\left[ -\frac{\partial^{2}}{\partial\Omega^{2}}+\frac{\partial^{2}}%
{\partial\beta^{2}}+48e^{ -2\sqrt{3}\Omega } \right]
\psi(\Omega,\beta)=0.\label{ks}
\end{equation}
{To construct the supersymmetric generalization for the WDW equation Eq.(\ref{ks}), we follow} the procedure for SUSY quantum cosmology\cite{julio1}. {First we need to find a diagonal Hamiltonian operator constructing the supercharges and get the supersymmetric KS WDW equation}
\begin{equation}
\left[-\frac{\partial^{2}}{\partial\Omega^{2}}+\frac{\partial^{2}}%
{\partial\beta^{2}}+12e^{-2\sqrt{3}\Omega}(4\pm e^{\sqrt{3}\Omega})\right]
\psi_{\pm}(\Omega,\beta)=0.\label{kssusy}
\end{equation}
{For the metric, we apply the WKB method and from the semiclassical equivalent to Eq.(\ref{kssusy})}, we find
\begin{align}
4e^{-2\sqrt{3}\Omega}\left(1+2t\sqrt{3}\dot{\Omega}\right)-t^2
\left(4\pm e^{\sqrt{3}\Omega}\right)=0,\label{clasusy}
\end{align}
for our purposes, the only physically relevant asymptotic region of Eq.(\ref{clasusy}) is $4\ll e^{\sqrt{3}\Omega}$.
From the solutions for the asymptotic regions
we get
\begin{align}
e^{-\sqrt{3}\Omega}=\left(\frac{3}{4}\right)^{1/3}r^{2/3}
\left(\pm1+\frac{C}{\sqrt{r}}\right)^{1/3},\label{solposneg}
\end{align}
where $C$ is a constant. Using these solution and Eq.(\ref{KSmetric}) we construct the metric
\begin{align}
ds^{2}=&-\left(\frac{3}{4}\right)^{2/3}r^{-2/3}
\left(\pm1+\frac{C}{\sqrt{r}}\right)^{2/3}dt^{2}+
\left(\frac{4}{3}\right)^{2/3}r^{2/3}\left(\pm1+\frac{C}{\sqrt{r}}\right)^{-2/3}
dr^{2}\nonumber\\
&+r^{2} \left(d \theta^{2}+\sin^{2} \theta
d\phi^{2}\right).\label{solucionposneg}
\end{align}
which is formally equivalent to Eq.(\ref{KSmetric}). {Note that there are two cases in Eq.(\ref{solucionposneg}), it was shown that the interesting and physical case is for $C>0$}. For this case, there are two singularities in $r=0$ and $r=C^2$.
{Moreover, it was suggested that because the semiclassical limit of Eq.(\ref{kssusy}) contains the ``fermionic" information, consequently, it is reasonable to expect the lack of a horizon. This was analyzed by solving the Dirac equation in the Schwarzschild and Kerr backgrounds and determining that the spinors destroy the horizon\cite{gibbons}. }
\subsection{Noncommutative Black Hole}\label{sec_2}
The starting point for the analysis is to consider the noncommutative proposal of quantum cosmology \cite{ncqc}. Where the Cartesian coordinates $\Omega$ and $\beta$ of the KS minisuperspace variables are introduced as a canonical deformation in the algebra of the minisuperspace operators,
\begin{equation}
[\hat{\Omega},\hat{\beta}]=\vartheta,\quad
[\hat{\Omega},\hat{P_\Omega}]=[\hat{\beta},\hat{P_\beta}]=1,\quad [\hat{P_\Omega},\hat{P_\beta}]=0. \label{ncc}
\end{equation}
Given the Weyl quantization procedure in the context of the above description, the realization of the commutation relation Eq.(\ref{ncc}) between the minisuperspace variables is made by a specific Moyal product
\begin{equation}
f(\Omega,\beta)\ast g(\Omega,\beta)=f(\Omega,\beta) e^{(\frac{i\vartheta}{2})(\stackrel{\leftarrow}{\partial}_{{\Omega}}\stackrel{\rightarrow}{\partial}_{\beta} -\stackrel{\leftarrow}{\partial}_{\beta}\stackrel{\rightarrow}{\partial}_{\Omega})} g(\Omega,\beta).
\end{equation}
This leads to a shift in the variables
\begin{equation}
\hat{\Omega}=\Omega-\frac{\vartheta}{2}P_\beta,\quad \hat{\beta}=\beta+\frac{\vartheta}{2}P_\Omega. \label{shift}
\end{equation}
The Moyal product functions will be applied to find the WDW equation, this gives a modified
WDW equation for the noncommutative model%
\begin{equation}
\left[-P_{\Omega}^{2}+P_{\beta}^{2}-48e^{-2\sqrt{3}\Omega}\right]
\ast\psi(\Omega,\beta)=0.
\end{equation}
As is known in noncommutative quantum mechanics, the original
phase-space is modified. It is possible to reformulate in terms
of the commutative variables and the ordinary product of functions, if
the new variables satisfy Eq.(\ref{shift}). Consequently, the
original WDW equation changes, with a modified
potential $V(\Omega,\beta)$,
\begin{equation}
V\left( \Omega,\beta\right) \ast\psi\left( \Omega,\beta\right) =V\left(
\Omega-\frac{\vartheta}{2} P_{\beta},\beta+\frac{\vartheta}{2}P_{\Omega}\right)\psi\left(\Omega,\beta\right),
\end{equation}
so the NC-WDW equation takes the form%
\begin{equation}
\left[ -\frac{\partial^{2}}{\partial\Omega^{2}}+\frac{\partial^{2}}%
{\partial\beta^{2}}+48e^{\left( -2\sqrt{3}\Omega+\sqrt{3}\vartheta
P_{\beta}\right) } \right] \psi(\Omega,\beta)=0. \label{ncwdw}
\end{equation}
{The consequences of the NC-WDW equation were originally analyzed in cosmology \cite{ncqc}. At the quantum level, it gives several new maxima on the probability density depending on the value of the noncommutative parameter $\vartheta$. The classical solutions have been obtained for the model \cite{bastos1}. Moreover, using the NC-WDW in Eq.(\ref{ncwdw}) with the Feynman-Gibbs approach to statistical mechanics, the thermodynamics of noncommutative black holes were studied \cite{LopezDominguez:2007zz}. Moreover, using Eq.(\ref{ncwdw}), the singularity of noncommutative quantum black holes was discussed\cite{bastos2}.}
\section{Noncommutative SUSY Black Hole}\label{sec_3}
As already stated in the previous sections, to construct the noncommutative SUSY black hole we will start with the SUSY WDW equation for the Kantowski-Sachs cosmological model Eq.(\ref{kssusy}).
This gives the noncommutative SUSY WDW equation. After introducing the deformed algebra in Eq.(\ref{ncc}) we get a SUSY generalization of the Schwarzschild black hole
\begin{equation}
\left[ -\frac{\partial^{2}}{\partial\Omega^{2}}+\frac{\partial^{2}}%
{\partial\beta^{2}}+12e^{-2\sqrt{3}(\Omega-\frac{\vartheta}{2}P_\beta)}(4\pm e^{\sqrt{3}(\Omega-\frac{\vartheta}{2}P_\beta)})\right] \psi(\Omega,\beta)=0. \label{susyncwdw}
\end{equation}
Then we apply the WKB method to the NC SUSY WDW equation. Assuming that the wave function has the form
\begin{equation}
\psi(\Omega,\beta)=e^{i(S_1(\Omega)+S_2(\beta))},
\end{equation}
we can construct the Einstein-Hamilton-Jacobi (EHJ) equation. Finally one can derive the equation of motion\cite{julio1}. This approach is equivalent to using a modified Hamiltonian that includes the noncommutative deformation as well as the SUSY generalization.
From the EHJ equation one can identify $\frac{dS_1(\Omega)}{d\Omega}\rightarrow P_\Omega$ and $\frac{dS_1(\beta)}{d\beta}\rightarrow P_\beta$,
where $P_\Omega$ and $P_\beta$ are the conjugate momentum to $\Omega$ and $\beta$, respectively.
We can also consider the noncommutative relations Eq.(\ref{ncc}) and the shift in the noncommutative variable Eq.(\ref{shift}) in Eq.(\ref{kssusy}). From the noncommutative SUSY WDW equation we can write the Hamiltonian as
\begin{equation}
H=P_\Omega^2-P_\beta^2+12e^{-2\sqrt{3}(\Omega-\frac{\vartheta}{2}P_\beta)}(4\pm e^{\sqrt{3}(\Omega-\frac{\vartheta}{2}P_\beta)}).\label{hnc2}
\end{equation}
This is the same Hamiltonian we obtain from Eq. (\ref{susyncwdw}).
{ We are interested in asymptotic region where supersymmetry can dominate \cite{julio1}. Therefore, we take the approximation} $e^{\sqrt{3}\Omega}\gg 4 $. {This can be considered the ``classical supersymmetric" limit}. In this limit, the equations of motion are
\begin{eqnarray}
\dot{\Omega}&=&-\frac{e^{2\sqrt{3}(\Omega-\frac{\vartheta}{2}P_\beta)}P_\Omega}{12},\quad
\dot{P}_\Omega=\mp\frac{\sqrt{3}}{2}e^{\sqrt{3}(\Omega-\frac{\vartheta}{2}P_\beta)}\\
\dot{\beta}&=&\mp\frac{\sqrt{3}}{4}\vartheta e^{\sqrt{3}(\Omega-\frac{\vartheta}{2}P_\beta)}+\frac{e^{2\sqrt{3}(\Omega-\frac{\vartheta}{2}P_\beta)}}{12}P_\beta,\quad
\dot{P_\beta}=0.\nonumber
\end{eqnarray}
{As one expects that the effects of noncommutativity are very small, and to simplify the calculations, we take the approximation $\vartheta\ll 1$. Moreover, one can obtain definitions for the momentum from the Hamiltonian}
\begin{eqnarray}
P_\Omega&=&-12e^{-2\sqrt{3}\Omega}\dot{\Omega}-144\sqrt{3}\vartheta e^{-4\sqrt{3}\Omega}\dot{\beta}\dot{\Omega},\label{poc2a}\\
P_\beta&=&\pm3\sqrt{3}\vartheta e^{-\sqrt{3}\Omega}+12e^{-2\sqrt{3}\Omega}\dot{\beta}+144\sqrt{3}\vartheta e^{-4\sqrt{3}\Omega}\dot{\beta}^2.\label{pbc2a}
\end{eqnarray}
From the mentioned approximations we get
\begin{eqnarray}
&-&3\dot{\Omega}^2-72\sqrt{3}\vartheta e^{-2\sqrt{3}\Omega}\dot{\beta}\dot{\Omega}^2\mp\frac{3\sqrt{3}}{2}\vartheta e^{\sqrt{3}\Omega}\dot{\beta}+3\dot{\beta}^2\nonumber\\
&+&72\sqrt{3}\vartheta e^{-2\sqrt{3}\Omega}\dot{\beta}^3\mp\frac{1}{4} e^{3\sqrt{3}\Omega}=0.\label{edbo2}
\end{eqnarray}
Setting $t^2=e^{-2\sqrt{3}\Omega-2\sqrt{3}\beta}$ and using the change of variable $u=e^{-3\sqrt{3}\Omega}$, we get
\begin{eqnarray}
\mp\frac{9}{4}\vartheta t^{-3/2} u^{2/3}\pm\frac{3}{4}\vartheta t^{-1/2} u^{-1/3}\dot{u}\pm\frac{3}{8}t^{-1/2}-\frac{3}{2}t^{-5/2}u+t^{-3/2}\dot{u}\nonumber\\
+36\vartheta t^{-7/2}u^{5/3}-36\vartheta t^{-5/2} u^{2/3}\dot{u}+8\vartheta t^{-3/2}u^{-1/3}\dot{u}^2=0.
\end{eqnarray}
To solve this equation, we use a perturbative method. We start by proposing that $u=\pm\frac{3}{4}t^2+ct^{3/2}+\vartheta u_1+O(\vartheta^2)$. The first and second terms are the same as for the SUSY black hole\cite{julio1}. Substituting, we obtain
\begin{equation}
t^{-3/2} \dot u_1-\frac{3}{2}t^{-5/2}u_1=\pm\frac{27 \left(2 c\pm \sqrt{t}\right)}{8 \sqrt[3]{8 c t^{3/2}\pm 6 t^2}}.
\end{equation}
After solving for $t$ and interchanging $r\leftrightarrow t$, we arrive at
\begin{align}
e^{-\sqrt{3}\Omega}=\left(\frac{3}{4}\right)^{1/3}r^{2/3}\left[\pm1+Cr^{-1/2}+\frac{9}{40}\vartheta r^{-1/2} \left(9C\pm 4\sqrt{r}\right) \left(6C \pm6 \sqrt{r}\right)^{2/3}\right]^{1/3}\\
e^{-\sqrt{3}\beta}=\left(\frac{4}{3}\right)^{1/3}r^{1/3}\left[\pm1+Cr^{-1/2}+\frac{9}{40}\vartheta r^{-1/2} \left(9C\pm 4\sqrt{r}\right) \left(6C \pm6 \sqrt{r}\right)^{2/3}\right]^{-1/3}\label{6}
\end{align}
Finally, substituting the solutions in Eq.(\ref{KSmetric}), we find the noncommutative SUSY generalization to the Schwarzschild metric
\begin{eqnarray}
ds^2&=&-\left(\frac{3}{4}\right)^{2/3}r^{-2/3}\left[\pm1+\frac{C}{\sqrt{r}}+\frac{9\vartheta\left(9C\pm 4\sqrt{r}\right) \left(6C \pm6 \sqrt{r}\right)^{2/3}}{40\sqrt{r}} \right]^{2/3}dt^2\nonumber\\
&+&\left(\frac{4}{3}\right)^{2/3}r^{2/3}\left[\pm1+\frac{C}{\sqrt{r}}+\frac{9\vartheta\left(9C\pm 4\sqrt{r}\right) \left(6C \pm6 \sqrt{r}\right)^{2/3}}{40\sqrt{r}} \right]^{-2/3}dr^2\nonumber\\
&+&r^2(d\theta^2+\sin{\theta}d\phi^2).
\end{eqnarray}
{These solutions are valid only in the asymptotic region $e^{\sqrt{3}\Omega}>>4$. The different solutions to reconstruct the metric give rise to some branches that are not physical as well as some singularities, these are summarized in table I.
\noindent We can see that for $\vartheta=0$ one recovers the results for the SUSY black Hole \cite{julio1}.}
\begin{table}[h!]
\caption{Singular and regular points for the noncommutative SUSY metric.}
{\begin{tabular}{|c | c | c | c|}
\hline
\textbf{$\pm$ 1} & $\mathbf{r}$ & $\mathbf{C}$&\textbf{Type of region}\\
\hline
\hline
Positive/Negative & $r=0$ & Any value & Singular point \\
\hline
\hline
Positive & $r>0$ & $C>0$ & {Regular region}\\
\hline
Positive & $r=C^2$ & $C<0$ & {Singular point}\\
\hline
\hline
Negative & $r>0$ & $C<0$ & {Invalid solution for the metric}\\
\hline
Negative & $r=C^2$ & $C>0$ & {Singular point}\\
\hline
\end{tabular} \label{table1}}
\end{table}
{As in the commutative case, for the noncommutative SUSY case, we have a singularity in $r=0$. This singularity is independent of the value of $C$. We start with the solution with the positive sign, there are two possibilities, $C>0$ and $C<0$. The first case is a regular for $r>0$, for the second case we have a singularity in $r=C^2$. Taking the solution with the negative sign, again we have two cases. For positive $C$ the time component of the metric vanishes. Therefore, there is a singularity in $r = C^2$. The case for negative $C$ is negative and therefore this case is discarded.
In order to exhibit the singularities, the Kretschmann invariant is calculated and it is given by}
\begin{eqnarray}
K&=&
\frac{1}{864 \sqrt[3]{6} \left(\pm 1+\frac{C}{\sqrt{r}} \right)^{8/3}r^{22/3}}\\
&&\times\left[
\begin{array}{c}
3888\ 6^{2/3} C^4+ 216 C^3 \sqrt{r} \left(\pm 63\ 6^{2/3} -48 \sqrt{r} \left(C\pm\sqrt{r} \right)^{1/3}\right)\\
+27 C^2 \left(659\ 6^{2/3} r \mp1152 r^{3/2} \left(C\pm\sqrt{r} \right)^{1/3}+128 \sqrt[3]{6} r^{2} \left(C\pm\sqrt{r} \right)^{2/3}\right)\\
\pm24 C r^{3/2}\left( 431\ 6^{2/3} \mp1296 r^{1/2} \left(C\pm\sqrt{r} \right)^{1/3}+288 \sqrt[3]{6} r \left(C\pm\sqrt{r} \right)^{2/3}\right)\\
+32 r^2\left(108 \sqrt[3]{6} r \left(C\pm\sqrt{r} \right)^{2/3}+71\ 6^{2/3} \mp324 r^{1/2} \left(C\pm\sqrt{r} \right)^{1/3}\right)
\end{array}
\right]\nonumber\\
&+&\frac{\vartheta}{960 \left(\pm 1+\frac{C}{\sqrt{r}}\right)^{3}r^{15/2}}\nonumber\\
&&\times\left[
\begin{array}{c}
69984 C^5+ 972 C^4 \sqrt{r}\left(\pm281-16 \sqrt[3]{6} r^{1/2} \left(C\pm\sqrt{r} \right)^{1/3}\right)\\
+27 C^3 r\left(15355\mp 1984 \sqrt[3]{6} r^{1/2} \left(C\pm\sqrt{r} \right)^{1/3}\right)\pm13952 r^{5/2}\\
+9 C^2 r^{3/2}\left(\pm33767-7488 \sqrt[3]{6} r^{1/2} \left(C\pm\sqrt{r} \right)^{1/3}\right)\\
+84 C r^2\left(1267 \mp 432 \sqrt[3]{6} r^{1/2} \left(C\pm\sqrt{r} \right)^{1/3}\right)-6912 \sqrt[3]{6} r^{3} \left(C\pm\sqrt{r} \right)^{1/3}
\end{array}
\right].\nonumber
\end{eqnarray}
{From this equation, we can see that there are singularities for $r=0$ and $r=C^2$.
Moreover, from the Kretschmann invariant we conclude (as in the commutative case) that the singularity in the SUSY region remains.}
\section{Concluding remarks}\label{final}
In this paper we have combined the ideas of noncommutative black holes \cite{julio1} and SUSY black holes \cite{LopezDominguez:2007zz} to give a proposal for a noncommutative SUSY black hole. The approach is based on using the diffeomorphism between the KS and Schwarzschild metrics and introducing noncommutativity and supersymmetry to the KS-WDW equation. Furthermore, by solving the semiclassical approximation, we can reconstruct the metric for the noncummutative SUSY metric for the Schwarzschild black hole. Moreover, we calculate the Kretschmann scalar from which we conclude that for $\vartheta=0$ we recover the result for the SUSY case\cite{julio1}. This gives us confidence that this is a noncommutative generalization of ``SUSY Schwarzschild metric" \cite{julio1}. We also see that the singularities for $r=0$ and $r=C^2$ are not removed by combining supersymmetry with noncommutativity \footnote{Even if in the metric it seems that for $r=C^2$ and taking the positive sign, the time component of the metric is non zero.}. Therefore, the behavior of this noncommutative SUSY black hole is qualitatively the same as for the proposed SUSY Schwarzschild black hole\cite{julio1}.
{Alternatively, we could have started with the NC-WDW and constructed the SUSY generalization. In particular, supersymmetric versions of noncommutative quantum mechanical models have been constructed \cite{gamboa} with the noncommutative deformation introduced using the Bopp shift. One characteristic feature of these models, is that angular momentum type interaction are introduced. Therefore, it can be expected that in the SUSY deformation of the NC-WDW equation, new angular momentum type interaction might appear.}
{Finally, the procedure used in this paper} can be applied to other black holes, in particular the (anti)de-Sitter black hole \cite{julio2}. These ideas are under research and will be reported elsewhere.
\section*{Acknowledgements}
This work is supported by CONACYT grant 258982. M. S. is supported by {CIIC-071/2022}, Mena-Barboza is partially supported by Adler.
| -26,123.882173 |
[
-2.70703125,
2.65234375
] | 22.727273 |
[
-3.0390625,
0.6337890625,
-1.84375,
-4.7578125,
-0.63330078125,
7.04296875
] |
[
3.19921875,
8.6328125,
2.302734375,
5.203125
] | 140 | 2,379 |
[
-2.765625,
3.05078125
] | 29.891697 |
[
-5.83984375,
-4.41015625,
-4.7421875,
-2.0703125,
2.13671875,
12.21875
] | 0.777298 | 14.625513 | 31.904161 | 4.704363 |
[
2.7184624671936035
] | -17,209.459536 | 6.813787 | -25,912.607497 | 0.950031 | 5.587612 |
[
-2.314453125,
-3.705078125,
-4,
-5,
2.224609375,
12.40625
] |
[
-5.39453125,
-1.578125,
-1.94921875,
-0.65966796875,
2.884765625,
3.3203125
] | |
BkiUdqk25V5i1dkg3xWc
|
\section{Supplemental Materials}
\subsection{The inverse KS procedure with the energy-level calibration}
Here we explain the procedure of generating the training data through the Kohn-Sham inversion with the energy level calibration.
\begin{enumerate}
\item Take the parameters $(A, B)$ in the external potential $V_{\rm ext}$ randomly.
\item Solve the two-body Schr\"odinger equation for Hamiltonian Eq.~(2) to get the total energy $E_{\rm tot}$ and the electron density $n(x)$.
\item Set $V_{\rm Hxc}^0(x_i)=0$ for all $i$.
\item Solve the KS equation with the one-body potential $V_{\rm ext}+V_{\rm Hxc}^{t}$ with the exact diagonalization to get the KS energy $\{\varepsilon_{j}\}$ and the density $n^t(x)$
\begin{eqnarray}
&&H=\Bigl(-\frac{\nabla^2}{2}+V_{\rm ext}(r)+V_{\rm Hxc}^{t}(r)\Bigr)\\
&&n^t(x_i)=\sum_{j=1}^N |\varphi_{j}(x_i)|^2
\end{eqnarray}
\item For each $x_{i}$, update the $V_{\rm Hxc}^{\rm t}(x_i)$ by
\begin{eqnarray}
V_{\rm Hxc}^{\rm t+1}(x_i)=V_{\rm Hxc}^{\rm t}(x_i)+\alpha(n^t(x_i)-n(x_i))
\label{eq:updatingV_Hxc}
\end{eqnarray}
The mixing parameter $\alpha$ was set to $450$.
\item Repeat the steps 4--5 until $\sqrt{\sum^{N_{r}}_{i}|n^{t+1} (x_i) - n^{t} (x_i)|^2/N} <s$ is achieved. The convergence threshold $s$ was set to $5\times10^{-8}$ in the present work.
\item Adjust the constant part of the converged $V_{\rm Hxc}(x_i)$
\begin{eqnarray}
V_{\rm Hxc}\rightarrow V_{\rm Hxc}+\frac{1}{N}(E_{\rm tot}-\sum_{j=1}^N\varepsilon_{j})
\end{eqnarray}
so as to make the sum of KS energy $\sum_{i=1}^N\varepsilon_i^{\rm new}$ solved under the adjusted $V_{\rm Hxc}$ equates to $E_{\rm tot}$ (adjust $c$ in Eq.~(2) so that the second and later terms cancel).
\begin{eqnarray}
\sum_{j=1}^N\varepsilon_{j}^{\rm new} &=&
\sum_{j=1}^N\{\varepsilon_{j}^{\rm old}+\frac{1}{N}(E_{\rm tot}-\sum_{j'=1}^N\varepsilon_{j'}^{\rm old})\}\nonumber\\
&=&E_{\rm tot}
\end{eqnarray}
\end{enumerate}
\section{Charge density distributions for the model}
The charge density realized by the Hamiltonian Eq.~(2) shows rapid change across the line in the A-B plane. As discussed in the main text, this reflects the change of the number of one-particle bound states in the non-interacting case. In Fig.~\ref{fig:solution-trap}, we here show a typical behavior of $n$ across the boundary for the interacting case. The boundary lines for the interacting case (Fig.~3) were drawn on the basis of this rapid change.
\begin{figure}[b!]
\begin{center}
\includegraphics[scale=0.25]{lines_ed_171227.pdf}
\caption{(a) Boundary line where the character of $n(x)$ for Hamiltonian Eq.~(3) changes. The line for the non-interacting case (without the interaction term) is rigorously defined as the line across which the number of the one-particle bound states changes from 1 to 2, whereas that for the interacting case was drawn by observing the rapidly changing behavior of $n(x)$ derived from the two-particle wave function. (b) Snapshot of $n(x)$ calculated at the point $(A, B)=(4.0, 0.45)$ indicated by circle in panel (a). (c) Snapshot of $n(x)$ calculated at the point $(A, B)=(4.0, 0.55)$ indicated by square in panel (a). }
\label{fig:solution-trap}
\end{center}
\end{figure}
\begin{figure}[b!]
\begin{center}
\includegraphics[scale=0.26]{learningline_hige_log.pdf}
\caption{Typical learning curve of the training of NN-$V_{\rm Hxc}$ for the training data set I (see the main text). Each box represents the distribution of the test error for every 5,000 epochs. Box indicates the interquartile range, whiskers maximum and minimum values, and bold line in the box median of each 5,000 epochs.
}
\label{fig:learningline}
\end{center}
\end{figure}
\section{Optimization of the neural network}
The initial step size of the AdaM was set to 0.001. Randomly selected 95\% of the generated data were referred to for training and the remaining 5\% were used to evaluate the test error. Figure \ref{fig:learningline} shows the typical behavior of the test error during the optimization. Since the test error does not reduce monotonically with the AdaM, the error distribution for every 5000 epoch is represented by the box plot. In this study we stopped the optimizations after 100,000 steps, though we have found that the stable convergence of the KS cycle and the trend of transferability shown in the main text are robust against at which step the optimization is stopped.
\end{document}
| -5,740.139234 |
[
-2.404296875,
2.123046875
] | 35.483871 |
[
-9.25,
-6.546875,
-1.4052734375,
-6.453125,
3.21875,
10.1953125
] |
[
0.03729248046875,
6.84375,
0.357666015625,
5.55859375
] | 42 | 577 |
[
-3.5546875,
4.0234375
] | 30.665468 |
[
-6.03515625,
-4.84375,
-2.146484375,
-0.08038330078125,
2.857421875,
7.17578125
] | 1.580778 | 21.029332 | 46.967071 | 1.825981 |
[
1.7870075702667236
] | -4,334.269668 | 5.481802 | -5,633.146547 | 1.517547 | 4.966488 |
[
-2.970703125,
-4.234375,
-4.49609375,
-4.17578125,
2.99609375,
10.984375
] |
[
-6.89453125,
-3.951171875,
-2.3671875,
-1.376953125,
4.59375,
5.58984375
] | |
BkiUbXXxK7Tt6CA5FTG7
|
\section{Introduction}
What is meant by the term {\it Critical Metallicity} in the context
of first- and second-generation stars? We define
$Z_{\rm crit}$ as the heavy-element abundance at which metal-line
cooling of the gas begins to dominate over cooling by H and He
(and molecules H$_2$ and HD). In order for the gas to collapse gravitationally
and continue to radiate away the heat produced by adiabatic compression,
the radiative cooling time,
$t_{\rm cool} \approx (3nkT/2 {\cal L})$, must be less than the
gravitational collapse time, $t_{\rm coll} \approx
(3 \pi/32 G \rho)^{1/2}$, where ${\cal L}$ is the cooling rate per
volume and $\rho$ is the gas mass density. Recent studies,
based on Jeans-mass arguments and thermodynamic histories of cloud
collapse, suggest that the mode of star formation may change,
shifting from higher-mass stars at low-$Z$ (Pop~III) to a
more normal (Pop~II) initial mass function (IMF) at $Z > Z_{\rm crit}$.
At zero metallicity, because of
the lack of CNO-burning and the inefficiency of the p-p chain, the
first stars are smaller, hotter, and shorter-lived than their current
counterparts (e.g., Tumlinson \& Shull 2000). Their efficient production
rates of ionizing radiation give them special importance for IGM
reionization (Venkatesan, Tumlinson, \& Shull 2003; Wyithe \& Loeb
2003; Shull \& Venkatesan 2007). The most massive stars have large yields
of heavy elements (Heger \& Woosley 2002), particularly $\alpha$-process
elements (O, Si) and the iron-group. Thus, it is astrophysically
important to understand the transition from first to second-generation
stars when $Z > Z_{\rm crit}$. However, this transition probably varies
spatially and temporally, owing to the inhomogenous nature of metal
production and transport into the IGM.
In cold dark matter (CDM) cosmologies, the first galaxies in the
universe are predicted (Ricotti, Gnedin, \& Shull 2002, 2007)
to be $10^6$ times smaller than
the Milky Way, with characteristic masses comparable to mass estimates
for the smaller dwarf spheroidal galaxies (dSph) observed around our
Galaxy and Andromeda (Mateo 1998; Belokurov et~al.\ 2006). The
gravitational potentials of these $10^{6-8}~M_{\odot}$ objects are so weak
that warm and hot ionized phases of their interstellar medium
are weakly bound. As a result, each episode of star formation
may produce powerful outflows that could temporarily inhibit further
star formation.
The first subgalactic structures form from the collapse
of rare dark matter density perturbations, with masses
$10^{5-6}\,M_{\odot}$ at $z \sim 30-40$. The initial gas cooling is
from collisionally excited H$_2$ rotational and vibrational
transitions (Lepp \& Shull 1984). A minimum H$_2$ abundance $x_{H_2}
\approx 10^{-4}$ is required to trigger star formation in a dark halo
in less than a Hubble time. In dust-free gas, H$_2$ formation is
catalyzed by the H$^-$ ion, that forms as a consequence of the shocks
that partially ionize and heat the gas during the virialization
process. At a given redshift, the mass of the smaller halo that can
form stars is determined by its virial temperature and therefore by
its mass. This analytical result has been confirmed by hydrodynamical
cosmological simulations.
Abel, Bryan, \& Norman (2002) carried out
such numerical simulations for a selected $10^6$ M$_\odot$ halo, using
adaptive-mesh refinement that resolves the collapse over a large range of
scales. In this selected halo, they find that only one star forms,
with mass 10--100 $M_{\odot}$. Bromm, Coppi, \& Larsen
(1999) found similar results with a variety of initial conditions for
the protogalaxies. These numerical results confirm longstanding
theoretical suggestions that the first stars should be
massive: their characteristic mass reflects the larger Jeans mass in
the inefficiently cooling metal-free gas. However, the cooling by
trace-metal fine-structure lines depends on the gas density
(Santoro \& Shull 2006, 2008) and radiative coupling to the CMB.
Thus, the Jeans mass and critical metallicity are sensitive to
local gas density.
\section{Previous Calculations of $Z_{\rm crit}$ }
The first static models of $Z_{\rm crit}$ (Bromm \& Loeb 2003) found
$Z_{\rm crit} \approx 10^{-3.5} Z_{\odot}$ for fine-structure cooling
by [C~II] 158~$\mu$m and [O~I] 63~$\mu$m.
Santoro \& Shull (2006) confirmed these results for C~II and O~I,
but suggested that fine-structure lines of [Si~II] and [Fe~II]
might also contribute, since IGM ``metal pollution" from massive-star
nucleosynthesis is weighted toward heavier elements. They further
noted that $Z_{\rm crit}$
depends on the gas density, $n$, owing to the change in cooling
rate, from ${\cal L} \propto n^2$ (at low-density) to
${\cal L} \propto n$ (high-density), as the fine-structure levels
reach Boltzmann (LTE) populations at ``critical density"
($n_{\rm cr}$). For H$^{\circ}$ excitation at 200~K, $n_{\rm cr}$ ranges
from $3000$ cm$^{-3}$ ([C~II]) to $(1-2) \times 10^6$ cm$^{-3}$ ([O~I]
and [Fe~II]). Santoro \& Shull (2006) found that $Z_{\rm crit}$
exceeds $0.01Z_{\odot}$ at the low gas densities,
$n \approx$ 1--100~cm$^{-3}$, present in virialized halos at $z > 20$.
\begin{figure}
\includegraphics[height=10cm]{f1-small.ps}
\caption{Minimum critical metallicities, $Z_{\rm crit}$,
vs.\ total gas density $n$ (Santoro \& Shull 2006)
for static cooling at $T = 200$~K by individual heavy elements.
Curves correspond to gas enriched by C~II, Si~II, O~I, and Fe~II
(see labels). Bottom envelope shows all four species together
in solar abundance ratios. Minimum values occur at high densities
(near $n_{\rm cr}$ for each coolant)
at log~$(Z_{\rm crit}/Z_{\odot}) = -3.48$ (C~II), $-3.54$ (Si~II),
$-3.78$ (O~I), $-3.52$ (Fe~II), and $-4.08$ (all elements). }
\end{figure}
Figure 1 shows values of $Z_{\rm crit}$ for C, O, Si, and Fe,
where metallicity is labelled by a single parameter $Z$. In fact,
there is no single ``metallicity", since the primary coolants
(C, O, Si, Fe) are rarely produced in solar abundance ratios.
Because of the density dependence of the cooling, Santoro \& Shull
(2008) examined the thermodynamic history of cloud collapse
and refined the definition of $Z_{\rm crit}$. In the new models
they included collisional coupling of the gas and level populations
(H$_2$, HD, fine-structure lines) and radiative coupling
to the cosmic microwave background (CMB). The latter effect can be
especially important at high redshifts, $z = 25-30$, where the CMB
temperature, $T_{\rm CMB} = (82~K)[(1+z)/30]$, sets a floor on gas
temperature sufficient to increase the Jeans mass. This, in turn,
may produce more massive stars at high redshift (Tumlinson 2007).
\section{Results}
\begin{figure}
\includegraphics[height=12cm]{f2-small.ps}
\caption{Temperature--density ($\log T, \log n)$ evolution of a collapsing
gas cloud (Santoro \& Shull 2008) starting with virial conditions of
mass, density, and temperature at $z = 30$. Curves are labelled
by metallicity $\log (Z/Z_{\odot})$ or [Fe/H].
The initial rise in temperature comes from adiabatic heating,
while the subsequent decreases in $T$ are driven by cooling from
H$^{\circ}$-excited H$_2$ rotational lines and fine-structure
lines of [O~I], [Si~II], [Fe~II], [C~II].
The broad temperature minima occur at $n \approx 10^{5.5}$
to $10^{6.5}$ cm$^{-3}$, near the critical densities for
H$^{\circ}$ de-excitation of [O~I], [Si~II], and [Fe~II]
fine-structure levels.
When level populations reach LTE, the volume cooling rate
${\cal L} \propto n$ rather than $n^2$, and $T$ begins to rise
at $n > n_{\rm cr}$. Radiation and collisions couple the gas and
fine-structure levels to the CMB ($T_{\rm CMB} \approx$ 70--80 at
$z = $25--30). The stellar IMF may be imprinted when
$T \leq 200$~K and $n \approx 10^6$ cm$^{-3}$. }
\end{figure}
\begin{figure}
\includegraphics[height=8.5cm]{f3-small.ps}
\caption{Critical metallicity (Santoro \& Shull 2008)
for collapsing clouds, allowed to cool
for various fractions (5\%, 10\%, 15\%) of the local Hubble time
at $z = $ 20--30. As shown by the lower two curves, the gas within
virialized halos may need
to reach metallicities as large as $\sim$1--2\% $Z_{\odot}$ in order
to cool in 10--15\% of the local Hubble time, $t_H$. }
\end{figure}
The results of our time-dependent models are shown in Figures 2 and 3,
with the astrophysical importance summarized in the captions.
Figure 2 shows the thermodynamic $(T,n)$ collapse history at $z=30$.
After an initial rise due to adiabatic heating, the
temperature turns downward (inflection point at [Fe/H] $\leq -2.0$).
Once the density exceeds $n \geq 10^6$ cm$^{-3}$ at
$Z > 10^{-3.5} Z_{\odot}$, $T$ is driven toward $T_{\rm CMB}$.
Figure 3 illustrates the dependence of $Z_{\rm crit}$ on the
time allowed to cool, expressed as a fraction of the local Hubble
time. Most of the cooling time is spent at low densities.
In order to cool in 10--15\% of $t_H$, the gas must reach
$Z_{\rm crit} \approx 0.01 Z_{\odot}$, until the density reaches
the point ($\sim 10^6$ cm$^{-3}$) of most efficient cooling.
These far-infrared fine-structure lines dominate the H$_2$
cooling once $Z > Z_{\rm crit}$, and their detection represents
a challenge for FIR and sub-mm astronomy. As shown by
Santoro \& Shull (2006), the strongest lines are [O~I] 63 $\mu$m,
[Si~II] 34.8 $\mu$m, and [Fe~II] 25.99 $\mu$m, which at $z=4$
redshift to 316 $\mu$m, 174 $\mu$m, and 130 $\mu$m. For
$(10^8~M_{\odot}) M_8$ of high-density (LTE) gas at 200~K and
$Z = (0.01 Z_{\odot})Z_{0.01}$, the line luminosities are
predicted to be
$L_{\rm line} \approx ([0.8-2.0] \times 10^{41}~{\rm erg~s}^{-1})
M_8 Z_{0.01}$. The luminosity distance at $z \approx 4$ is
$d_L \approx 10^{29}$ cm, and the expected line fluxes are
$\sim10^{-21}$ W~m$^{-2}$, within reach of facilities
such as ALMA, SPICA, or SAFIR. At higher redshifts, these lines
shift into the 350~$\mu$m (sub-mm) window. The fluxes
will probably be lower, owing to the larger $d_L$, lower $Z$,
and smaller masses of cooling gas.
\section{Observations \& Discussion}
The ``fossil record" of the first stars may be observed in gas-phase
(IGM), as well as in metal-poor halo stars.
Metallicities $Z \sim 10^{-3}$ Z$_\odot$ are measured from
absorption lines in the Ly$\alpha$ forest at $z \sim 2-5$
(Songaila 2001; Schaye et~al.\ 2003; Pettini et~al.\ 2003;
Simcoe et~al.\ 2004). This mean metallicity, typically inferred from
abundances of C~IV and Si~IV, shows little evolution from $z=5$ to $z=2$.
However, a possible redshift-dependent ionization correction may
conspire to mask any real metallicity evolution.
The origin of the metals observed in the low-density Ly$\alpha$ forest at
redshifts $z \sim 2-5$ is still under some debate. One view invokes a
nearly uniform pre-enrichment of the intergalactic medium (IGM) produced
by the first stars at high-redshift (Madau et~al.\ 2001). The other view
attributes the origin of the observed metal lines to hot, metal-enriched
superbubbles located around Lyman-break galaxies (Adelberger et~al.\ 2003).
Given the difficulties associated with both scenarios, it is important to
know the amount and volume filling factor of metal-enriched IGM produced
by primordial stars and galaxies.
As modeled by many groups (e.g., Ricotti, Gnedin, \& Shull 2002, 2007),
the first sources of metals may come from ``dwarf primordial" (dPri)
galaxies, with virial temperatures $T_{\rm vir} \leq 20,000$ K and
circular velocities $v_c \leq 20$ km~s$^{-1}$.
In contrast to more massive galaxies, the
dark matter (DM) halos of dPri galaxies are too shallow to contain
much photoionized gas, with temperatures 10,000--20,000 K. During their
formation, the gas is only heated to temperatures below $10^4$~K, where
it is unable to cool by atomic hydrogen (Ly$\alpha$) line emission.
The mass of these halos is $M_{\rm dm} \leq 2 \times 10^{8}$ M$_\odot$
at their typical redshifts of formation ($z \geq 10$). These
galaxies rely primarily on the formation of H$_2$ to cool and form
stars, because metal cooling is negligible as long as
the gas has almost primordial composition. This situation changes,
after the metallicity rises above $Z_{\rm crit}$,
which could be as large as $0.01 Z_{\odot}$ at halo gas densities
$n \approx$ 10--100 cm$^{-3}$. As the first stars form, these
requirements no longer hold, since some gas
is heated above 10,000 K and is polluted with heavy elements.
At low redshifts ($z < 0.4$), ultraviolet spectrographs aboard
{\it Hubble} and {\it FUSE} have made similar measurements of the
gas-phase baryon content and metallicity of the IGM,
using Ly$\alpha$, Ly$\beta$, and trace metal lines of
O~VI, C~III, C~IV, etc. (Danforth \& Shull 2005, 2007).
The current observational sensitivity to metallicity is
$\sim10^{-3.5} Z_{\odot}$ at high redshift (Songaila 2001;
Simcoe et~al.\ 2004) and $10^{-2} Z_{\odot}$ at low redshift
(Danforth \& Shull 2007; Stocke et al. 2007).
Figure 4 shows how these lines can be used to derive
the statistical metallicity ($\sim0.1 Z_{\odot}$) from ratios of
O~VI to H~I. However, the IGM appears to have widely varying
metal abundances, with differences between ``filaments" and ``voids"
in the ``Cosmic Web" of absorbers.
If underdense regions of IGM contain some metals, then star formation
in high redshift galaxies may have pre-enriched it (Madau et~al.\ 2001).
If, instead, the metals detected along a line of sight are associated with
nearby bright galaxies at $z \sim 2-5$, the metal distribution is
probably quite inhomogeneous. In this second case, observations are
not probing a minimum floor of metal enrichment produced by the first
galaxies.
The degree of inhomogeneity in the IGM metal distribution IGM is
not well characterized, although this should change with the installation
of the powerful {\it Cosmic Origins Spectrograph} (COS) on the
{\it Hubble Space Telescope} in August 2008.
The spectroscopic throughput of COS should be over $20\times$ that of
STIS, and the resulting ultraviolet spectra will be used for a wide
range of observations of the IGM and surrounding galaxies:
(1) surveys of the baryon content and metallicity of the low-$z$ IGM;
(2) searches for nucleosynthetic patterns among heavy elements;
(3) absorber-galaxy correlation studies, seeking connections between
the IGM and the galaxies responsible for metal enrichment;
(4) Surveys of baryons and metals residing in absorbers within voids.
Preliminary studies of several of these issues with HST and FUSE
spectrographs is now underway (Danforth \& Shull 2005, 2007;
Stocke et al.\ 2006; Stocke et al.\ 2007).
\begin{figure}
\includegraphics[height=15cm]{f4-small.ps}
\caption{The ``fossil record" of the first stars may be studied
in the gas phase at $z < 0.4$, through quasar absorption lines.
This plot shows the ``multiphase ratio" of O~VI and H~I absorbers,
measured by {\it Hubble Space Telescope} and the {\it Far Ultraviolet
Spectroscopic Explorer} (Danforth \& Shull 2005, 2007). The ratio,
N(H~I)/N(O~VI) varies from high column density filaments in the
Cosmic Web to low-column gas in voids, with
N$_{\rm HI} < 10^{14}$ cm$^{-2}$. Our
surveys and abundance analyses show that low-$z$ filaments have
mean metallicities $\sim0.1 Z_{\odot}$, whereas no metals have
been detected in voids, down to limits of $0.02Z_{\odot}$ (Stocke
et al.\ 2007). The major conclusions of this study are:
(1) the filaments have increased their metallicity by factors
of 30--100 from $z \geq 3$ to the present; and (2) the IGM in voids
may contain low-metallicity pockets at $Z < 0.01 Z_{\odot}$. }
\end{figure}
\begin{theacknowledgments}
This work has been supported by the Astrophysical Theory program
at the University of Colorado, through NASA grant NNX07-AG77G, and
through grants from the Space Telescope Science Institute and
from NASA for far-UV studies with the FUSE satellite.
\end{theacknowledgments}
\bibliographystyle{aipproc}
| -12,047.526591 |
[
-3.486328125,
3.126953125
] | 28.482972 |
[
-2.837890625,
0.260986328125,
-1.869140625,
-6.0859375,
-0.378662109375,
7.9921875
] |
[
4.1328125,
7.91796875,
3.009765625,
6.88671875
] | 143 | 2,346 |
[
-3.23828125,
3.7421875
] | 30.194163 |
[
-6.00390625,
-4.09765625,
-3.75390625,
-2.576171875,
1.7705078125,
11.7421875
] | 0.973504 | 18.83875 | 34.953112 | 4.081345 |
[
2.8154611587524414
] | -9,692.123287 | 5.035379 | -11,614.171344 | 0.66029 | 5.872592 |
[
-3.431640625,
-3.810546875,
-2.875,
-3.84765625,
2.564453125,
10.515625
] |
[
-5.37890625,
-1.9921875,
-2.083984375,
-1.6337890625,
3.0546875,
4.67578125
] | |
BkiUc_Y4c3aisMwilqSJ
|
\section{Introduction}\label{intro}\setcounter{equation}{0}
Fractional evolution equations involve the derivatives of the form $\frac{d^{\alpha}}{dt^{\alpha}}$, where $\alpha>0$ is not necessarily an integer, have received much attention from researchers. This rising interest is motivated both by important applications of the theory, and by the difficulties involved in the mathematical structure. Fractional evolution equations appear in many physical phenomena arising from various scientific fields including analysis of viscoelastic materials, electrical engineering, control theory of dynamical systems, electrodynamics with memory, quantum mechanics, heat conduction in materials with memory, signal processing, economics, and many other fields (cf. \cite{HlR,AAK,YAR,IPF,MST}, etc). For an extensive study of the fractional differential equations (FDEs), we refer the interested readers to \cite{AAK,IPF}, etc.
The real-world phenomena such as natural disasters, harvesting, shocks, etc, experience unpredictable changes in their states for negligible time duration. These sudden changes are relevant and interesting, and well approximated by instantaneous impulses. Such processes are conveniently described by instantaneous impulsive evolution equations. The theory of instantaneous impulsive evolution equations has found various applications such as optimal control models in economics, threshold, pharmacokinetics, bursting rhythm models in medicine and frequency modulated systems etc (cf. \cite{BM,LVB,NE,TY} etc). Moreover, the evolution processes in pharmacotherapy, such as the consequent absorption of the body, in hemodynamical steadiness of a person and the insulin distribution in the bloodstream are moderate and continuous. Hence, the dynamics of such phenomena cannot be interpreted by the instantaneous impulsive systems. Therefore, this situation can be appropriately estimated as an impulsive action that starts suddenly and continuously active over a finite time interval. Thus, the dynamics of such processes are suitably described by non-instantaneous impulsive systems.
Hern\'andez et. al. \cite{EHD}, first introduced a class of impulses that are suddenly active for an arbitrary fixed point and remains in action on a finite time interval. Such impulses are called non-instantaneous impulses. Then, they developed the existence of solutions for non-instantaneous impulsive differential equations. Later, Wang et. al. \cite{JRW,JRY} extended this model to two general classes of impulsive differential equations, which are very useful to characterize certain dynamics of evolutionary processes in pharmacotherapy. For the study of non-instantaneous impulsive systems, one can refer to \cite{JMF,YTJ}, etc, and the references therein.
On the other hand, there are several real-world phenomena in which the current state of a system depends on the past states. The dynamics of such types of processes are suitably analyzed by delay (functional) evolution system, which naturally occurs in logistic reaction-diffusion processes with delay, ecological models, neural network, inferred grinding models, etc (cf. \cite{FC,LA,NJ}, etc). Moreover, the evolution equations with state-dependent delays are more frequent and widely applicable, see for example \cite{AO,FC}, etc, and the references therein.
The notion of controllability is one of the fundamental concepts in mathematical control theory and play a significant role in many control problems such as, irreducibility of transition semigroups \cite{GDJZ}, the optimal control problems \cite{VB}, stabilization of unstable systems via feedback control \cite{VB1} etc. Controllability (exact and approximate) means that the solution of a considered dynamical control system can steer from an arbitrary initial state to a desired final state by using some suitable control function. For the infinite dimensional control systems, the problem of approximate controllability is more substantial and is having broad range of applications, see for instance, \cite{M,TRR,TR,EZ}, etc. In the past few decades, the problem of approximate controllability for different kinds of dynamical systems (in Hilbert and Banach spaces) such as fractional evolution equations, Sobolev type systems, delay (functional) differential equations, impulsive systems, stochastic differential equations, etc, has been widely studied by many researchers and they have produced excellent results, see for instance, \cite{SAM,SM,AGK,M,ER,Z} etc, and the references therein.
It is well known that the slow diffusion processes are described by first-order evolution equations while the fast diffusion processes are characterized by second-order evolution systems. Similarly, FDEs of order $\alpha\in(0,1)$ stand for slow anomalous diffusion processes (subdiffusion process), while the fast anomalous diffusion processes (superdiffusion process) are suitably modeled by FDEs of order $\alpha>1$. Therefore, by this reason, the theory for fractional evolution equations of order $\alpha\in(0,1)$ is discrete from that for fractional evolution equations of order $\alpha>1$, and it inspired the researchers to study both theories separately (see \cite{Jkr,Jkj}, etc for more details). In the past two decades, a good number of results came out on the existence of solutions and approximate controllability for fractional systems of order $\alpha\in(0,1)$ in Hilbert and Banach spaces, see for example, \cite{DNC,JA,GR,SNS,MIN,NMI,JWY,Z}, etc, and the references therein. Recently, many authors established the existence of mild solutions and approximate controllability for fractional evolution equations of order $\alpha\in(1,2)$ by invoking the theory of sectorial operators via suitable fixed point theorems, see for instance, \cite{Qh,Sr,Sbl,Sb} etc.
Recently, a few developments have been reported on the existence and controllability of fractional evolution equations of order $1<\alpha<2$ by applying the cosine and sine family of bounded linear operators (cf. \cite{Jh,MMr,Yz} etc). A new concept of mild solutions for fractional evolution systems with order $\alpha\in (1, 2)$ in Banach spaces by using the Laplace transform and Mainardi's Wright-type function is introduced in \cite{Yz}. {Henr\'iquez et.al \cite{JFA} investigated the existence of a classical solution for the fractional homogeneous and inhomogeneous abstract Cauchy problems of order $\alpha\in(1,2)$ by using the $\alpha$-resolvent family corresponding to the homogeneous Cauchy problem}. Raja et. al. \cite{MMv} formulated sufficient conditions for the approximate controllability of fractional differential equation of order $1<\alpha<2$ in Hilbert spaces by applying the concept of cosine and sine family of operators via Schauder's fixed point theorem. Raja and Vijaykumar \cite{Mvv} derived sufficient conditions for the approximate controllability of fractional integro-differential system of order $\alpha\in(1,2)$ in Hilbert spaces. Later, Vijaykumar et.al. in \cite{Vcr} demonstrated the approximate controllability of Sobolev-type Volterra-Fredholm integro-differential equations of order $1<\alpha<2$ in Hilbert spaces by invoking the Krasnoselskii fixed point theorem.
To the best of our knowledge, the approximate controllability for impulsive fractional differential equations of order $1<\alpha<2$ with state-dependent delay in Banach spaces by using the idea of cosine and sine family of operators is not investigated yet, and it motivated us to encounter this problem. In this paper, we consider semilinear impulsive fractional differential equations of order $1<\alpha<2$ with non-instantaneous impulses and state-dependent delay in separable reflexive Banach spaces. We derive sufficient conditions for the approximate controllability of the considered system by applying the concept of $\alpha/2$-resolvent family (defined in \eqref{2.3}, which is related to the strongly continuous consine family generated by the operator $\mathrm{A}$), resolvent operator condition and Schauder's fixed point theorem. Moreover, we discuss the construction of a feedback control and the approximate controllability of the linear fractional control system of order $1<\alpha<2$ in details (see, Section \ref{sec3}), which lacks in the available literature. Furthermore, due to the observation related to characterization of the phase space for impulsive systems discussed in \cite{GU}, we replace the uniform norm on the phase space by the integral norm (see Example \ref{exm2.8}),
Let us consider the following non-instantaneous impulsive fractional order system with state-dependent delay:
\begin{equation}\label{1.1}
\left\{
\begin{aligned}
^C\mathrm{D}_{0,t}^{\alpha}x(t)&=\mathrm{A}x(t)+\mathrm{B}u(t)+f(t,x_{\varrho(t, x_t)}), \ t\in\bigcup_{j=0}^{p} (s_j, \tau_{j+1}]\subset J=[0,T], \\
x(t)&=h_j(t, x(\tau_j^{-})),\ t\in(\tau_j, s_j], \ j=1,\dots,p, \\
x'(t)&=h'_j(t, x(\tau_j^{-})),\ t\in(\tau_j, s_j], \ j=1,\dots,p, \\
x_{0}&=\psi\in \mathfrak{B}, x'(0)=\eta,
\end{aligned}
\right.
\end{equation}
where \begin{itemize} \item $^C\mathrm{D}_{0,t}^{\alpha}$ is the Caputo fractional derivative of order $\alpha$ with $1<\alpha<2$, \item the operator $\mathrm{A}:\mathrm{D(A)}\subset \mathbb{X}\to\mathbb{X}$ is linear (not necessarily bounded), where $ \mathbb{X}$ is a separable reflexive Banach space having a strictly convex dual $\mathbb{X}^*$, \item the linear operator $\mathrm{B}:\mathbb{U}\to\mathbb{X}$ is bounded with $\left\|\mathrm{B}\right\|_{\mathcal{L}(\mathbb{U};\mathbb{X})}= \tilde{M}$ and the control function $u\in \mathrm{L}^{2}(J;\mathbb{U})$ (admissible control class), where $\mathbb{U}$ is a separable Hilbert space, \item the appropriate function $f:J\times \mathfrak{B}\rightarrow \mathbb{X},$ where $\mathfrak{B}$ is a phase space satisfying some axioms which will be provided in next section, \item the functions $h_j:[\tau_j, s_j]\times\mathbb{X}\to\mathbb{X}$ and their derivatives $h'_j:[\tau_j, s_j]\times\mathbb{X}\to\mathbb{X}$ for $j=1,\ldots,p$, denote the non-instantaneous impulses and the fixed points $s_j$ and $\tau_j$ satisfy $0=\tau_0=s_0<\tau_1\le s_1\le \tau_2<\ldots<\tau_p\le s_p\le \tau_{p+1}=T$, the values $x(\tau_j^+)$ and $x(\tau_j^-)$ stand for the right and left limit of $x(t)$ at the point $t=\tau_j,$ respectively and satisfy $x(\tau_j^-)=x(\tau_j),$ for $j=1,\ldots,p$, \item the function $x_{t}:(-\infty, 0]\rightarrow\mathbb{X}$ with $x_{t}(\theta)=x(t+\theta)$ and $x_t\in \mathfrak{B}$, for each $t\in J$. The function $\varrho: J \times \mathfrak{B} \rightarrow (-\infty, T]$ is continuous.
\end{itemize}
The rest of the article is structured as follows: In section \ref{sec2}, we recall some basic definitions of fractional calculus, property of cosine family and the phase space axioms. Moreover, we also provide the assumptions and important results which is useful in the subsequent sections. Section \ref{sec3} is reserved for the approximate controllability of the linear fractional control system of order $1<\alpha<2$. For this, first we formulate the linear regulator problem, show the existence of an optimal control (Theorem \ref{th1}), and then we derive an explicit form of the optimal control in the feedback form in Lemma \ref{lem3.1}. Further, we examine the approximate controllability of the linear fractional control system \eqref{3.2}, by using the optimal control (Lemma \ref{lem3.4}). In section \ref{semilinear}, we first prove the existence of a mild solution for the control system \eqref{1.1} by applying Schauder's fixed point theorem (Theorem \ref{thm4.3}), and then demonstrate the approximate controllability of the system \eqref{1.1} in Theorem \ref{thm4.4}. In the final section, we present an example to support the application of the developed theory.
\section{Preliminaries}\label{sec2}\setcounter{equation}{0}
In the present section, we introduce some standard notations, basic definitions and facts about fractional calculus. Moreover, we also recall some important properties of cosine and sine families, and also provide the axioms of phase space.
Let $\mathrm{AC}([a,b];\mathbb{R})$ denote the space of all mappings from $[a,b]$ to $\mathbb{R},$ which is absolutely continuous and $\mathrm{AC}^n([a,b];\mathbb{R}),$ for $n\in\mathbb{N},$ represent the space of all functions $f:[a,b]\to\mathbb{R}$, which have continuous derivatives up to order $n-1$ with $f^{(n-1)}\in\mathrm{AC}([a,b];\mathbb{R})$.
First, we recall some basic definitions from fractional calculus and the details can be found in \cite{AAK,IPF}, etc.
\begin{Def}
The \emph{fractional integral} of order $\beta>0$ for a function $f\in\mathrm{L}^1([a,b];\mathbb{R})$, $a,b\in\mathbb{R}$ with $a<b$, is given as
\begin{align*}
I_{a}^{\beta}f(t):=\frac{1}{\Gamma(\beta)}\int_{a}^{t}\frac{f(s)}{(t-s)^{1-\beta}}\mathrm{d}s,\ \mbox{ for a.e. } \ t\in[a,b],
\end{align*}
where $\Gamma(\alpha)=\int_{0}^{\infty}t^{\alpha-1}e^{-t}\mathrm{d}t$ is the Euler gamma function.
\end{Def}
\begin{Def}
The \emph{Riemann-Liouville fractional derivative} of order $\beta>0$ for a function $f\in\mathrm{AC}^n([a,b];\mathbb{R})$ is defined as
\begin{align*}
^L\mathrm{D}_{a,t}^{\beta}f(t):=\frac{1}{\Gamma(n-\beta)}\frac{d^n}{dt^n}\int_{a}^{t}(t-s)^{n-\beta-1}f(s)\mathrm{d}s,\ \mbox{ for a.e. }\ t\in[a,b],
\end{align*}
where $n-1< \beta<n$.
\end{Def}
\begin{Def}
The \emph{Caputo fractional derivative} of a function $f\in\mathrm{AC}^n([a,b];\mathbb{R})$ of order $\beta>0$ is defined as
\begin{align*}
^C\mathrm{D}_{a,t}^{\beta}f(t):=\ ^L\mathrm{D}_{a,t}^{\beta}\left[f(t)-\sum_{k=1}^{n-1}\frac{f^{(k)}(a)}{k!}(x-a)^k\right],\ \mbox{ for a.e. } \ t\in[a,b].
\end{align*}
\end{Def}
\begin{rem}[\cite{IPF}]
If a function $f\in\mathrm{AC}^n([a,b];\mathbb{R})$, then the Caputo fractional derivative of $f$ can be written as
\begin{align*}
^C\mathrm{D}_{a,t}^{\beta}f(t)=\frac{1}{\Gamma(n-\beta)}\int_{a}^{t}(t-s)^{n-\beta-1}f^{(n)}(s)\mathrm{d}s,\ \mbox{for a.e.}\ t \in[a,b], \ n-1< \beta<n.
\end{align*}
\end{rem}
\subsection{The strongly continuous cosine family:} In this subsection, we first construct a strongly continuous cosine family $\left\{\mathrm{C}\left(t \right) :t\in \mathbb{R}\right\}$ in $\mathbb{X}$ generated by $\mathrm{A}$, and then review some important properties of the cosine and sine families. In order to construct a strongly continuous cosine family $\left\{\mathrm{C}\left(t \right) :t\in \mathbb{R}\right\}$, we impose the following assumptions on the linear operator $\mathrm{A}:\mathrm{D}(\mathrm{A})\to\mathbb{X}$.
\begin{Ass}\label{ass2.1}
The operator $\mathrm{A}$ fulfills the following conditions:
\begin{itemize}
\item[(R1)]The linear operator $\mathrm{A}$ is closed with dense domain $\mathrm{D}(\mathrm{A} )$ in $\mathbb{X}$.
\item[(R2)] For real $\lambda$, $\lambda>0,$ $\lambda^2$ is in the resolvent set $\rho(\mathrm{A})$ of $\mathrm{A}$.
The resolvent $\mathrm{R}(\lambda^2; \mathrm{A})$ exists, strongly infinitely differentiable, and satisfies
$$\left\|\left(\frac{\mathrm{d}}{\mathrm{d}\lambda}\right )^k(\lambda \mathrm{R}(\lambda^2; \mathrm{A}))\right\|_{\mathcal{L}(\mathbb{X})}\!\!\!\le\frac{Mk!}{\lambda^{k+1}},$$
for $k=0,1,2,\dots. $
\item[(R3)] $\mathrm{R}(\lambda^2;\mathrm{A})$ is compact for some $\lambda$ with $\lambda^2\in \rho(\mathrm{A}).$
\end{itemize}
\end{Ass}
\begin{Def}[\cite{TCC1}]\label{cosine}
A family of bounded linear operators $\{\mathrm{C}(t):t\in \mathbb{R}\}$ in a Banach space $\mathbb{X}$ is called \emph{a strongly continuous cosine family} if
\begin{enumerate}
\item [(1)] $\mathrm{C}(s+t)+\mathrm{C}(s-t)=2\mathrm{C}(s)\mathrm{C}(t)$, for $ s, t \in \mathbb{R}$;
\item [(2)] $\mathrm{C}(0)=\mathrm{I}$, where $\mathrm{I}$ represents the identity operator;
\item[(3)] $\mathrm{C}(t)x$ is continuous in t on $\mathbb{R},$ for each fixed $x\in\mathbb{X}$.
\end{enumerate}
\end{Def}
The family $\{\mathrm{S}(t):t\in \mathbb{R}\}$ on $\mathbb{X}$ defined by
\begin{align}
\nonumber \mathrm{S}(t)x&=\int_{0}^{t}\mathrm{C}(s)x\mathrm{d}s,\ x\in \mathbb{X},\ t\in\mathbb{R}.
\end{align}
is called \emph{strongly continuous sine family}.
The infinitesimal generator $\mathrm{A}$ of a strongly continuous cosine family $\{\mathrm{C}(t):t\in \mathbb{R}\}$ is defined by
$$\mathrm{A}x=\frac{\mathrm{d}^2}{\mathrm{d}t^2}\mathrm{C}(t)x\Big|_{t=0},\ x\in \mathrm{D}(\mathrm{A}),$$ where
\begin{align*}
\mathrm{D}(\mathrm{A})&= \Big\{ x\in \mathbb{X}:\mathrm{C}(t)x \text{ is two times continuously differentiable function} \text{ with respect to } t\Big\},
\end{align*}
equipped with the graph norm $$\left\|x\right\|_{\mathrm{D}(\mathrm{A})}=\left\|x\right\|_\mathbb{X}+\left\|\mathrm{A}x\right\|_\mathbb{X},\ x\in \mathrm{D}(\mathrm{A}).$$ We also define the set
$$\mathrm{E}=\Big\{x: \mathrm{C}(t)x \text{ is continuously differentiable function with respect to}\ t\Big\},$$ with the norm
$$\left\|x\right\|_1=\left\|x\right\|_\mathbb{X}+\sup_{0\le t\le 1}\left\|\mathrm{AS}(t)x\right\|_{\mathbb{X}},\ x\in \mathrm{E},$$
forms a Banach space, see \cite{JKI}.
\begin{lem}[Proposition 2.7, \cite{TCC2}]\label{lem2.1}
If the assumptions (R1)-(R2) hold true. Then the operator $\mathrm{A}$ generates a strongly continuous cosine family $\{\mathrm{C}(t):t\in \mathbb{R}\}$, which is uniformly bounded, that is, there exists a constant $M\ge1$ such that $\left\|\mathrm{C}(t)\right\|_{\mathcal{L}(\mathbb{X})}\le M$.
\end{lem}
\begin{lem}[Proposition 5.2, \cite{TCC2}]\label{lem2.2}
Suppose that the operator $\mathrm{A}$ generates a strongly continuous cosine family $\{\mathrm{C}(t):t\in\mathbb{R}\}$ on $\mathbb{X}$. Let $\{\mathrm{S}(t):t\in \mathbb{R}\}$ be the associated sine family on $\mathbb{X}$. Then the following are equivalent:
\begin{itemize}
\item [(i)] The operator $\mathrm{S}(t)$ is compact for every $t\in\mathbb{R}$.
\item [(ii)] Assumption (R3) holds.
\end{itemize}
\end{lem}
\begin{lem}[Propositions 2.1 and 2.2, \cite{TCC1}]\label{lem2.3}
Let the operator $\mathrm{A}$ generates a strongly continuous cosine family $\{\mathrm{C}(t):t\in\mathbb{R}\}$, which is uniformly bounded in $\mathbb{X}$, and let $\{\mathrm{S}(t):t\in \mathbb{R}\}$ be the associated sine family on $\mathbb{X}$. Then the following are true:
\begin{itemize}
\item [(i)] $\left\|\mathrm{S}(t_2)-\mathrm{S}(t_1)\right\|_{\mathcal{L}(\mathbb{X})}\le M\left|t_2-t_1\right|,$ for all $t_1,t_2\in\mathbb{R}$;
\item [(ii)] If $x\in\mathrm{E}$, then $\mathrm{S}(t)x\in\mathrm{D}(\mathrm{A})$ and $\frac{d}{dt}\mathrm{C}(t)x=\mathrm{A}\mathrm{S}(t)x$.
\end{itemize}
\end{lem}
\subsection{Phase space:} Let us provide the definition of the phase space $\mathfrak{B}$ in axiomatic form (cf. \cite{HY}) and appropriately modify for the impulsive systems (cf. \cite{VOj}). Specifically, the phase space $\mathfrak{B}$ is a linear space of all mapping from $(-\infty, 0]$ into $\mathbb{X}$ equipped with the seminorm $\left\|\cdot\right\|_{\mathfrak{B}}$ and satisfying the following axioms:
\begin{enumerate}
\item [(A1)] If $x: (-\infty, \nu+\sigma)\rightarrow \mathbb{X},\ \sigma>0,$ such that $x_{\nu}\in \mathfrak{B}$ and $x|_{[\nu, \nu+\sigma]}\in \mathrm{PC}([\nu, \nu+\sigma];\mathbb{X}),$ then for every $t\in [\nu, \nu+\sigma],$ the following conditions hold true:
\begin{itemize}
\item [(i)] $x_{t}$ is in $\mathfrak{B}$;
\item [(ii)] $\left\|x_{t}\right\|_{\mathfrak{B}}\leq \mathcal{P}(t-\nu)\sup\{\left\|x(s)\right\|_{\mathbb{X}}: \nu
\leq s\leq t\}+\mathcal{Q}(t-\nu)\left\|x_{\nu}\right\|_{\mathfrak{B}},$ where $\mathcal{P},\ \mathcal{Q}:[0, \infty)\rightarrow [0, \infty)$ such that $\mathcal{P}$ is continuous, $\mathcal{Q}$ is locally bounded, and both $\mathcal{P},\mathcal{Q}$ are independent of $x(\cdot).$
\end{itemize}
\item [(A2)] The space $\mathfrak{B}$ is complete.
\end{enumerate}
For any $\psi\in \mathfrak{B},$ the function $\psi_{t}, \ t\leq 0,$ is defined by $\psi_{t}(\theta)=\psi(t+\theta),\ \theta \in (-\infty, 0].$ If the function $x(\cdot)$ fulfills the axiom (A1) with $x_{0}=\psi,$ then we can extend the mapping $t\mapsto x_{t}$ by taking $x_{t}=\psi_{t}, \ t\leq0$, to the whole interval $(-\infty, T].$ Moreover, let us define a set
\begin{align*}
\mathfrak{R}(\varrho^{-})&=\{\varrho(s, \psi):\ \varrho(s, \psi)\leq 0, \ \text{ for }\ (s, \psi)\in J \times \mathfrak{B}\},
\end{align*}
where the function $\varrho$ is the same as the one defined in section \ref{intro}. Further, we assume that the function $\psi_{t}:\mathfrak{R}(\varrho^{-})\to\mathfrak{B}$ is continuous in $t$, and there exists a continuous and bounded function $\varTheta^{\psi}: \mathfrak{R}(\varrho^{-})\rightarrow (0, \infty)$ such that
\begin{align*}
\left\|\psi_{t}\right\|_{\mathfrak{B}}&\leq \varTheta^{\psi}(t)\left\|\psi\right\|_{\mathfrak{B}}.
\end{align*}
Let us define the set
\begin{align*}
\mathrm{PC}(J;\mathbb{X})&:=\Big\{x:J \rightarrow \mathbb{X} : x\vert_{t\in I_i}\in\mathrm{C}(I_i;\mathbb{X}),\ I_i:=(\tau_i, \tau_{i+1}],\ i=0,1,\ldots,p,\\& \qquad x(\tau_i^+) \mbox{ and }\ x(\tau_i^-) \mbox{ exist for each }\ i=1,\ldots,p, \ \mbox{and satisfy}\ x(\tau_i)=x(\tau_i^-)\Big\},
\end{align*}
with the norm $\left\|x\right\|_{\mathrm{PC}(J;\mathbb{X})}:=\sup\limits_{t\in J}\left\|x(t)\right\|_{\mathbb{X}}$.
\begin{lem}[Lemma 2.3, \cite{ER}]\label{lema2.1}
Let $x:(-\infty,T]\rightarrow \mathbb{X}$ be a function such that $x_0=\psi$ and $x|_J\in \mathrm{PC}(J;\mathbb{X})$. Then
$$\left\|x_s\right\|_\mathfrak{B}\le K_1\left\|\psi\right\|_\mathfrak{B}+K_2\sup\{\left\|x(\theta)\right\|_\mathbb{X}:\theta \in [0,\max\{0,s\}]\},\ s\in\mathfrak{R}(\varrho^-)\cup J,$$
where $$K_1=\sup_{t \in \mathfrak{R}(\varrho-)}\varTheta^\psi(t)+\sup_{t\in J} \mathcal{Q}(t),\ K_2=\sup_{t\in J}\mathcal{P}(t). $$
\end{lem}
\begin{Ex}\label{exm2.8}
Let $\mathfrak{B}=\mathcal{PC}_{g}(\mathbb{X})$ be the space consisting of all mappings $\phi:(-\infty,0]\to\mathbb{X}$ such that $\phi$ is left continuous, $\phi\vert_{[-q,0]}\in \mathcal{PC}([-q,0];\mathbb{X}),$ for $q>0,$ and $\int_{-\infty}^{0}\frac{\left\|\phi(\theta)\right\|_{\mathbb{X}}}{g(\theta)}\mathrm{d}\theta<\infty.$ The norm $\left\|\cdot\right\|_{\mathfrak{B}}$ is defined as
\begin{align}
\label{Bnor}\left\|\phi\right\|_{\mathfrak{B}}:=\int_{-\infty}^{0}\frac{\left\|\phi(\theta)\right\|_{\mathbb{X}}}{g(\theta)}\mathrm{d}\theta,
\end{align} where the function $g:(-\infty,0]\to[1,\infty)$ satisfies the following conditions:
\begin{enumerate}
\item $g(\cdot)$ is continuous with $g(0)=1$, and $\lim\limits_{\theta\to-\infty} g(\theta)=\infty,$
\item the function $\mathcal{O}(t):=\sup\limits_{-\infty<\theta\leq -t}\frac{g(t+\theta)}{g(\theta)}$ is locally bounded for $t\geq 0.$
\end{enumerate}
\end{Ex}
\subsection{Mild solution, resolvent operator and assumptions}In order to define the mild solution of the system \eqref{1.1}, we first consider Mainardi's Wright-type function $\mathrm{M}_{\upsilon}$ (cf. \cite{Fm},\cite{Ip})
\begin{align*}
\mathrm{M}_{\upsilon}(z)=\sum_{k=0}^{\infty}\frac{(-z)^{k}}{k!\Gamma(1-\upsilon(n+1))},\ \upsilon\in(0,1),\ z\in\mathbb{C}.
\end{align*}
Note that, for any $t>0$, $\mathrm{M}_{\upsilon}(t)\ge 0$ and it satisfies the following: $$\int_{0}^{\infty} s^{c}\mathrm{M}_{\upsilon}(s)\mathrm{d}s=\frac{\Gamma(1+\delta)}{\Gamma(1+\upsilon\delta)},\ \mbox{ for }\ -1<c<\infty.$$
For any $x\in\mathbb{X}$ and $\gamma=\frac{\alpha}{2}$, we define
\begin{align}
\label{2.3}\mathcal{C}_{\gamma}(t)&:=\int_{0}^{\infty}\mathrm{M}_{\gamma}(\theta)\mathrm{C}(t^{\gamma}\theta)\mathrm{d}\theta,\\ \mathcal{T}_{\gamma}(t)&:=\int_{0}^{t}\mathcal{C}_{\gamma}(s)\mathrm{d}s, \\
\mathcal{S}_{\gamma}(t)&:=\int_{0}^{\infty} \gamma\theta\mathrm{M}_{\gamma}(\theta)\mathrm{S}(t^{\gamma}\theta)\mathrm{d}\theta.
\end{align}
Next, we discuss some important properties of the operators $\mathcal{C}_{\gamma}(t), \mathcal{T}_{\gamma}(t)$ and $\mathcal{S}_{\gamma}(t)$ for $t\ge0$ in the following lemma.
\begin{lem}(\cite{Yz})\label{lem2.11}
The operators $\mathcal{C}_{\gamma}(t), \mathcal{T}_{\gamma}(t)$ and $\mathcal{S}_{\gamma}(t)$ have the following properties:
\begin{itemize}
\item [(i)] The operators $\mathcal{C}_{\gamma}(t), \mathcal{T}_{\gamma}(t)$ and $\mathcal{S}_{\gamma}(t)$ are linear. Moreover, for all $t\geq 0$,
\begin{align*}
\left\|\mathcal{C}_{\gamma}(t)\right\|_{\mathcal{L}(\mathbb{X})}\le M, \ \ \left\|\mathcal{T}_{\gamma}(t)\right\|_{\mathcal{L}(\mathbb{X})}\le Mt,\ \ \left\|\mathcal{S}_{\gamma}(t)\right\|_{\mathcal{L}(\mathbb{X})}\le\frac{M}{\Gamma(2\gamma)}t^{\gamma}.
\end{align*}
\item [(ii)] For all $t\ge 0$, the operator $\mathcal{C}_{\gamma}(t)$ is strongly continuous and the operators $\mathcal{T}_{\gamma}(t)$, $\mathcal{S}_{\gamma}(t)$ and $t^{\gamma-1}\mathcal{S}_{\gamma}(t)$ are uniformly continuous.
\item [(iii)] If $\mathrm{S}(t)$ is compact for $t\ge0$, then the operator $\mathcal{S}_{\gamma}(t)$ is also compact for $t\ge0$.
\end{itemize}
\end{lem}
\begin{Def}
A function $x(\cdot;\psi,\eta,u):(-\infty, T]\to\mathbb{X}$ is called a \emph{mild solution} of \eqref{1.1}, if $x(t)=\psi(t)$ on $t\in(-\infty,0]$ (where $\psi\in\mathfrak{B}$) and $x(\cdot)$ satisfies the following:
\begin{equation}\label{2.2}
x(t)=
\begin{dcases}
\mathcal{C}_{\gamma}(t)\psi(0)+\mathcal{T}_{\gamma}(t)\eta+\int_{0}^{t}(t-s)^{\gamma-1}\mathcal{S}_{\gamma}(t-s)\left[\mathrm{B}u(s)+f(s,x_{\varrho(s, x_s)})\right]\mathrm{d}s,\\ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \ t\in[0, \tau_1],\\
h_j(t, x(\tau_j^-)),\qquad\qquad\qquad\qquad\qquad\qquad\quad t\in(\tau_j, s_j],\ j=1,\ldots,p,\\
\mathcal{C}_{\gamma}(t-s_j)h_j(s_j, x(\tau_j^-))+\mathcal{T}_{\gamma}(t-s_j)h'_j(s_j, x(\tau_j^-))\\ \quad -\int_{0}^{s_j}(s_j-s)^{\gamma-1}\mathcal{S}_{\gamma}(s_j-s)\left[\mathrm{B}u(s)+f(s,x_{\varrho(s, x_s)})\right]\mathrm{d}s \\\quad+\int_{0}^{t}(t-s)^{\gamma-1}\mathcal{S}_{\gamma}(t-s)\left[\mathrm{B}u(s)+f(s,x_{\varrho(s, x_s)})\right]\mathrm{d}s, \\\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \ t\in(s_j,\tau_{j+1}],\ j=1,\ldots,p.
\end{dcases}
\end{equation}
\end{Def}
\begin{Def}
The system \eqref{1.1} is said to be approximately controllable on $J$, if for any initial function $\psi \in \mathfrak{B}, x_T\in\mathbb{X}$ and given $\epsilon>0$, there exist a control function $u\in\mathrm{L}^{2}(J;\mathbb{X})$ such that $$\left\|x(T)-x_T\right\|_{\mathbb{X}}\le\epsilon,$$ where $x(\cdot)$ is the mild solution of the system \eqref{1.1} with the control $u(\cdot)$.
\end{Def}
To investigate the approximate controllability of the system \eqref{1.1}, we first introduce the following operators:
\begin{equation}\label{2.1}
\left\{
\begin{aligned}
L_0^Tu&:=\int^{T}_{0}(T-t)^{\gamma-1}\mathcal{S}_{\gamma}(T-t)\mathrm{B}u(t)\mathrm{d}t,\\
\Phi_{0}^{T}&:=\int^{T}_{0}(T-t)^{\gamma-1}\mathcal{S}_{\gamma}(T-t)\mathrm{B}\mathrm{B}^*\mathcal{S}_{\gamma}(T-t)^*\mathrm{d}t,\\
\mathcal{R}(\lambda,\Phi_{0}^{T})&:=(\lambda \mathrm{I}+\Phi_{0}^{T}\mathscr{J})^{-1},\ \lambda > 0,
\end{aligned}
\right.
\end{equation}
where $\mathrm{B}^{*}$ and $\mathcal{S}_{\gamma}(t)^*$ represent the adjoint operators of $\mathrm{B}$ and $\mathcal{S}_{\gamma}(t),$ respectively. It is straightforward that the operators $L_0^T$ and $\Phi_{0}^{T}$ are linear and bounded. Moreover, the mapping $\mathscr{J} : \mathbb{X} \rightarrow 2^{\mathbb{X}^*}$ is defined as
\begin{align*}
\mathscr{J}[x]&=\{x^* \in \mathbb{X}^* : \langle x, x^* \rangle=\left\|x\right\|_{\mathbb{X}}^2= \left\|x^*\right\|_{\mathbb{X}^*}^2 \}, \ \mbox{ for all }\ x\in \mathbb{X},
\end{align*}
which is known as the duality mapping. The notation $\langle \cdot, \cdot \rangle $ denotes the duality pairing between $\mathbb{X}$ and its dual $\mathbb{X}^*$. Since $\mathbb{X}$ is a reflexive Banach space, then $\mathbb{X}^*$ becomes strictly convex (see \cite{AA}), which ensures that the mapping $\mathscr{J}$ is strictly monotonic, bijective and demicontinuous, that is, $$x_k\to x\ \mbox{ in }\ \mathbb{X}\ \mbox{ implies }\ \mathscr{J}[x_k] \xrightharpoonup{w} \mathscr{J}[x]\ \mbox{ in } \ \mathbb{X}^*\ \mbox{ as }\ k\to\infty.$$
\begin{rem}\label{rem2.8}
If $\mathbb{X}$ is a separable Hilbert space, which is identified by its own dual, then the resolvent operator is defined as $\mathcal{R}(\lambda,\Phi_{0}^{T}):=(\lambda \mathrm{I}+\Phi_{0}^{T})^{-1},\ \lambda > 0$.
\end{rem}
\begin{lem}[Lemma 2.2 \cite{M}]\label{lem2.9}
For $\lambda>0$ and every $h\in\mathbb{X}$, the equation
\begin{align}\label{2.4}\lambda z_{\lambda}+\Phi_{0}^{T}\mathscr{J}[z_{\lambda}]=\lambda h,\end{align}
has a unique solution which is given as
$$z_{\lambda}(h)=\lambda(\lambda \mathrm{I}+\Phi_{0}^{T}\mathscr{J})^{-1}(h)=\lambda\mathcal{R}(\lambda,\Phi_{0}^{T})(h)$$ and satisfies the following \begin{align}\label{2.5}
\left\|z_{\lambda}(h)\right\|_{\mathbb{X}}=\left\|\mathscr{J}[z_{\lambda}(h)]\right\|_{\mathbb{X}^*}\leq\left\|h\right\|_{\mathbb{X}}.
\end{align}
\begin{proof}
Since the operator $\Phi_0^T$ is linear and bounded, then proceeding similar way as in the proof of Lemma 2.2 \cite{M}, one can complete the proof.
\end{proof}
\end{lem}
We need to impose the following assumptions to obtain the approximate controllability of the system \eqref{1.1}.
\begin{Ass}\label{as2.1}
\begin{enumerate}
\item [\textit{($H0$)}] For every $h\in\mathbb{X}$, $z_\lambda=z_{\lambda}(h)=\lambda\mathcal{R}(\lambda,\Phi_{0}^{T})(h) \rightarrow 0$ as $\lambda\downarrow 0$ in strong topology, where $z_{\lambda}(h)$ is a solution of the equation \eqref{2.4}.
\item [\textit{(H1)}]
\begin{enumerate}
\item [(i)] Let $x:(-\infty, T]\rightarrow \mathbb{X}$ be such that $x_0=\psi$ and $x|_{J}\in \mathrm{PC}(J;\mathbb{X}).$ The mapping $t\mapsto f(t, x_{\varrho(t,x_{t})}) $ is strongly measurable on $J$ and the function $f(t,\cdot): \mathfrak{B}\rightarrow \mathbb{X}$ is continuous for a.e. $t\in J$. Also, the map $t\mapsto f(s,x_{t})$ is continuous on $\mathfrak{R}(\varrho^-) \cup J,$ for every $s\in J$.
\item [(ii)] For each positive integer $r$, there exists a constant $\delta\in[0,\gamma]$ and a function $\phi_r\in \mathrm{L}^{\frac{1}{\delta}}(J;\mathbb{R^{+}})$, such that $$ \sup_{\left\| \psi\right\|_{\mathfrak{B}}\leq r} \left\|f(t, \psi)\right\|_{\mathbb{X}}\leq \phi_{r}(t), \mbox{ for a.e.} \ t \in J \mbox{ and } \psi\in \mathfrak{B},$$ with
$$ \liminf_{r \rightarrow \infty } \frac {\left\|\phi_r\right\|_{\mathrm{L}^{\frac{1}{\delta}}(J;\mathbb{R^+})}}{r} = \zeta< \infty. $$
\end{enumerate}
\item [\textit{(H2)}] The impulses $ h_j:[\tau_j,s_j]\times\mathbb{X}\to\mathbb{X}$, for $j=1,\dots,p$, are such that
\begin{itemize}
\item [(i)] the impulses $h_j(\cdot,x):[\tau_j,s_j]\to\mathbb{X}$ are continuously differentiable for each $x\in\mathbb{X}$,
\item [(ii)] for all $t\in[\tau_j,s_j]$, the impulses $h_j(t,\cdot):\mathbb{X}\to\mathbb{X}$ and their derivatives $h'_j(t,\cdot):\mathbb{X}\to\mathbb{X}$ are completely continuous,
\item[(iii)] $\left\|h_{j}(t,x)\right\|_{\mathbb{X}}\le \kappa_{j},\ \left\|h'_{j}(t,x)\right\|_{\mathbb{X}}\le \vartheta_{j}, \text{for all}\ x\in\mathbb{X},\ t\in [\tau_{j},s_{j}],\ j=1,\cdots,p$, where $\kappa_{j}$'s and $\vartheta_{j}$'s are positive constants.
\end{itemize}
\end{enumerate}
\end{Ass}
The following version of the discrete Gronwall-Bellman lemma (cf. \cite{Ch}) is used in the sequel.
\begin{lem}\label{lem2.13}
If $\{f_n\}_{n=0}^{\infty}, \{g_n\}_{n=0}^{\infty}$ and $\{w_n\}_{n=0}^{\infty}$ are non-negative sequences and $$f_n\le g_n+\sum_{0\le k<n}w_kf_k,\ \text{ for }\ n\geq 0,$$ then $$f_n\le g_n+\sum_{0\le k<n}g_kw_k\exp\left(\sum_{k<j<n}g_j\right),\text{ for }\ n\geq 0.$$
\end{lem}
\section{Fractional order linear control problem}\label{sec3}\setcounter{equation}{0} This section deals with the linear fractional control problem and investigate the approximate controllability of the linear system. To achieve this goal, first we consider the linear-regulator problem with the cost functional defined as
\begin{equation}\label{3.1}
\mathscr{G}(x,u)=\left\|x(T)-x_{T}\right\|^{2}_{\mathbb{X}}+\lambda\int^{T}_{0}(T-t)^{\gamma-1}\left\|u(t)\right\|^{2}_{\mathbb{U}}\mathrm{d}t,
\end{equation}
where $\gamma=\frac{\alpha}{2}$ with $\alpha\in(1,2)$, $\lambda >0$, $x_{T}\in \mathbb{X}$ and $x(\cdot)$ is the solution of the linear fractional control system:
\begin{equation}\label{3.2}
\left\{
\begin{aligned}
^CD_{0,t}^{\alpha}x(t)&= \mathrm{A}x(t)+\mathrm{B}u(t),\ t\in J,\\
x(0)&=v,\ x'(0)=w,
\end{aligned}
\right.
\end{equation}
with the control $u(\cdot)\in \mathbb{U}$. Since $\mathrm{B}u\in\mathrm{L}^1(J;\mathbb{X})$ for any $u \in\mathrm{L}^{2}(J;\mathbb{U})=\mathcal{U}_{\mathrm{ad}}$ (admissible control class), then, $(t-s)^{\gamma-1}\mathcal{S}_{\gamma}(t-s)\mathrm{B}u(s)$ is integrable. Hence, the above linear system possess a unique mild solution $x\in\mathrm{C}(J;\mathbb{X})$ (see section 4.2, Chapter 4, \cite{P}) given by
\begin{align}
x(t)&=\mathcal{C}_{\gamma}(t)v+\mathcal{T}_{\gamma}(t)w+\int_{0}^{t}(t-s)^{\gamma-1}\mathcal{S}_{\gamma}(t-s)\mathrm{B}u(s)\mathrm{d}s,\ t\in J.
\end{align}
Next, we define the {\em admissible class} $\mathcal{A}_{\mathrm{ad}}$ for the optimal control problem \eqref{3.1}-\eqref{3.2} as $$\mathcal{A}_{\mathrm{ad}}=\{(x,u):x \text{ is the unique mild solution of } \eqref{3.2} \text{ with the control }u\in \mathcal{U}_{\mathrm{ad}}\}.$$ For any $u \in\mathrm{L}^{2}(J;\mathbb{U}),$ the existence of a unique mild solution ensures that the set $\mathcal{A}_{\mathrm{ad}}$ is nonempty. Our next aim is to prove the existence of an optimal pair, which minimize the cost functional \eqref{3.1}.
\begin{theorem}\label{th1} For given $v,w\in\mathbb{X}$, there exists a unique optimal pair $(x^0,u^0)\in\mathcal{A}_{\mathrm{ad}}$ of the problem:
\begin{align}\label{F}
\min\limits_{(x,u)\in \mathcal{A}_{\mathrm{ad}}}\mathscr{G}(x, u).
\end{align}
\end{theorem}
\begin{proof}
Let us assume $$L := \inf \limits _{u \in \mathcal{U}_{\mathrm{ad}}}\mathscr{G}(x,u).$$ Since $0\leq L< +\infty$, there exists a minimizing sequence $\{u^n\}_{n=1}^{\infty} \in \mathcal{U}_{\mathrm{ad}}$ such that $$\lim_{n\to\infty}\mathscr{G}(x^n,u^n) = L,$$ where $(x^n, u^n)\in\mathcal{A}_{\mathrm{ad}}$, for each $n\in\mathbb{N}$. Moreover, $x^n(\cdot)$ satisfies
\begin{align}\label{3.4}
x^n(t)&=\mathcal{C}_{\gamma}(t)v+\mathcal{T}_{\gamma}(t)w+\int_{0}^{t}(t-s)^{\gamma-1}\mathcal{S}_{\gamma}(t-s)\mathrm{B}u^n(s)\mathrm{d}s,\ t\in J.
\end{align}
Since $0\in\mathcal{U}_{\mathrm{ad}}$, without loss of generality, we may assume that $\mathscr{G}(x^n,u^n) \leq \mathscr{G}(x,0)$, where $(x,0)\in\mathscr{A}_{\mathrm{ad}}$. Using the definition of $\mathscr{G}(\cdot,\cdot)$, we obtain
\begin{align}\label{35}
T^{\gamma-1}\lambda\int^{T}_{0}\left\|u^n(t)\right\|^{2}_{\mathbb{U}}\mathrm{d}t&\leq \left\|x^n(T)-x_{T}\right\|^{2}_{\mathbb{X}}+\lambda\int^{T}_{0}(T-t)^{\gamma-1}\left\|u^n(t)\right\|^{2}_{\mathbb{U}}\mathrm{d}t\nonumber\\&\leq \left\|x(T)-x_{T}\right\|^{2}_{\mathbb{X}}\leq 2\left(\|x(T)\|_{\mathbb{X}}^2+\|x_T\|_{\mathbb{X}}^2\right)<+\infty.
\end{align}
The above estimate guarantees the existence of a large $R>0$ (independent of $n$) such that
\begin{align}\label{3.5}\int_0^T \|u^n(t)\|^2_{\mathbb{U}} \mathrm{d} t \leq R < +\infty .\end{align}
Using the relation \eqref{3.4}, we estimate
\begin{align*}
\left\|x^n(t)\right\|_{\mathbb{X}}&\le\left\|\mathcal{C}_{\gamma}(t)v\right\|_{\mathbb{X}}+\left\|\mathcal{T}_{\gamma}(t)w\right\|_{\mathbb{X}}+\left\|\int_{0}^{t}(t-s)^{\gamma-1}\mathcal{S}_{\gamma}(t-s)\mathrm{B}u^n(s)\mathrm{d}s\right\|_{\mathbb{X}}\nonumber\\&\le M\left\|v\right\|_{\mathbb{X}}+Mt\left\|w\right\|_{\mathbb{X}}+\frac{M\tilde{M}}{\Gamma(2\gamma)}\int_{0}^{t}(t-s)^{2\gamma-1}\left\|u^n(s)\right\|_{\mathbb{U}}\mathrm{d}s
\nonumber\\&\le M\left\|v\right\|_{\mathbb{X}}+MT\left\|w\right\|_{\mathbb{X}}+\frac{M\tilde{M}}{\Gamma(2\gamma)}\left(\frac{T^{4\gamma-1}}{4\gamma-1}\right)^{\frac{1}{2}}\left(\int_{0}^{t}\left\|u^n(s)\right\|_{\mathbb{U}}^2\mathrm{d}s\right)^{\frac{1}{2}}\nonumber\\&\le M\left\|v\right\|_{\mathbb{X}}+MT\left\|w\right\|_{\mathbb{X}}+\frac{M\tilde{M}}{\Gamma(2\gamma)}\left(\frac{T^{4\gamma-1}R}{4\gamma-1}\right)^{\frac{1}{2}}<+\infty,
\end{align*}
for all $t\in J$. Since the space $\mathrm{L}^{2}(J;\mathbb{X})$ is reflexive, then the Banach-Alaoglu theorem yields the existence of a subsequence $\{x^{n_k}\}_{k=1}^{\infty}$ of $\{x^n\}_{n=1}^{\infty}$ such that
\begin{align}\label{3.6}
x^{n_k}\xrightharpoonup{w}x^0\ \mbox{ in }\ \mathrm{L}^{2}(J;\mathbb{X}), \ \mbox{ as }\ k\to\infty.
\end{align}
The estimate \eqref{3.5} implies that the sequence $\{u^n\}_{n=1}^{\infty}$ is uniformly bounded in the space $\mathrm{L}^2(J;\mathbb{U})$. Once again, by applying the Banach-Alaoglu theorem, we can find a subsequence, say, $\{u^{n_k}\}_{k=1}^{\infty}$ of $\{u^n\}_{n=1}^{\infty}$ such that
\begin{align*}
u^{n_k}\xrightharpoonup{w}u^0\ \mbox{ in }\ \mathrm{L}^2(J;\mathbb{U})=\mathcal{U}_{\mathrm{ad}}, \ \mbox{ as }\ k\to\infty.
\end{align*}
Since the linear operator $\mathrm{B}$ is bounded (continuous) from $\mathbb{U}$ to $\mathbb{X}$, then we have
\begin{align}\label{3.7}
\mathrm{B} u^{n_k}\xrightharpoonup{w}\mathrm{B}u^0\ \mbox{ in }\ \mathrm{L}^2(J;\mathbb{X}),\ \mbox{ as }\ k\to\infty.
\end{align}
Moreover, by using the above convergences together with the compactness of the operator $(\mathrm{Q}f)(\cdot) =\int_{0}^{\cdot}(\cdot-s)^{\gamma-1}\mathcal{S}_{\gamma}(\cdot-s)f(s)\mathrm{d}s:\mathrm{L}^2(J;\mathbb{X})\rightarrow \mathrm{C}(J;\mathbb{X}) $ (see Lemma \ref{lem2.12} below), we obtain
\begin{align*}
\left\|\int_{0}^{t}(t-s)^{\gamma-1}\mathcal{S}_{\gamma}(t-s)\mathrm{B}u^{n_k}(s)\mathrm{d}s-\int_{0}^{t}(t-s)^{\gamma-1}\mathcal{S}_{\gamma}(t-s)\mathrm{B}u(s)\mathrm{d}s\right\|_{\mathbb{X}}\to0,
\end{align*}
as $k\to\infty$, for all $t\in J$. We now estimate
\begin{align}\label{3.8}
&\left\|x^{n_k}(t)-x^*(t)\right\|_{\mathbb{X}}\nonumber\\&=\left\|\int_{0}^{t}(t-s)^{\gamma-1}\mathcal{S}_{\gamma}(t-s)\mathrm{B}u^{n_k}(s)\mathrm{d}s-\int_{0}^{t}(t-s)^{\gamma-1}\mathcal{S}_{\gamma}(t-s)\mathrm{B}u^{0}(s)\mathrm{d}s\right\|_{\mathbb{X}}\nonumber\\&\to 0\ \mbox{ as } \ k\to\infty, \ \mbox{for all }\ t\in J,
\end{align}
where
\begin{align*}
x^*(t)=\mathcal{C}_{\gamma}(t)v+\mathcal{T}_{\gamma}(t)w+\int_{0}^{t}(t-s)^{\gamma-1}\mathcal{S}_{\gamma}(t-s)\mathrm{B}u^{0}(s)\mathrm{d}s,\ t\in J.
\end{align*}
The above expression guarantees that the function $x^*\in \mathrm{C}(J;\mathbb{X})$ is the unique mild solution of the equation \eqref{3.2} with the control $u^{0}\in\mathcal{U}_{\mathrm{ad}}$. Using the convergences \eqref{3.6} and \eqref{3.8} and the fact that the weak limit is unique, we deduce that $x^*(t)=x^0(t),$ for all $t\in J$. Therefore, the function $x^0$ is the unique mild solution of the system \eqref{3.2} with the control $u^{0}\in\mathcal{U}_{\mathrm{ad}}$, so that the whole sequence $x^n\to x^0\in\mathrm{C}(J;\mathbb{X})$. Hence, we have $(x^0,u^0)\in\mathscr{A}_{\mathrm{ad}}$.
Next, we show that the functional $\mathscr{G}(\cdot,\cdot)$ attains its minimum at $(x^0,u^0)$, that is, \emph{$\mathscr{G}=\mathscr{G}(x^0,u^0)$}. Note that the cost functional $\mathscr{G}(\cdot,\cdot)$ given in \eqref{3.1} is continuous and convex (see Proposition III.1.6 and III.1.10, \cite{EI}) on $\mathrm{L}^2(J;\mathbb{X}) \times \mathrm{L}^2(J;\mathbb{U})$, it follows that $\mathscr{G}(\cdot,\cdot)$ is weakly lower semi-continuous (Proposition II.4.5, \cite{EI}). That is, for a sequence
$$(x^n,u^n)\xrightharpoonup{w}(x^0,u^0)\ \mbox{ in }\ \mathrm{L}^2(J;\mathbb{X}) \times \mathrm{L}^2(J;\mathbb{U}),\ \mbox{ as }\ n\to\infty,$$
we have
\begin{align*}
\mathscr{G}(x^0,u^0) \leq \liminf \limits _{n\rightarrow \infty} \mathscr{G}(x^n,u^n).
\end{align*}
Hence, we obtain
\begin{align*} L \leq \mathscr{G}(x^0,u^0) \leq \liminf \limits _{n\rightarrow \infty} \mathscr{G}(x^n,u^n)= \lim \limits _{n\rightarrow \infty} \mathscr{G}(x^n,u^n) =L,\end{align*}
and thus $(x^0,u^0)$ is a minimizer of the problem \eqref{F}. Since the cost functional given in \eqref{3.1} is convex, the constraint \eqref{3.2} is linear and the class $\mathcal{U}_{\mathrm{ad}}=\mathrm{L}^2(J;\mathbb{U})$ is convex, then the optimal control obtained above is unique.
\end{proof}
\begin{rem}
Since $\mathbb{X}$ is a separable reflexive Banach space with strictly convex dual $\mathbb{X}^*$, then the fact 8.12 in \cite{MFb} ensures that the norm $\|\cdot\|_{\mathbb{X}}$ is Gateaux differentiable. Moreover, the Gateaux derivative $\partial_x$ of the function $\varphi(x)=\frac{1}{2}\|x\|_{\mathbb{X}}^2$ is the duality map, that is, $$\langle\partial_x\varphi(x),y\rangle=\lim_{\epsilon \to 0}\frac{\varphi(x+\epsilon y)-\varphi(x)}{\epsilon}=\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}\epsilon}\|x+\epsilon y\|_{\mathbb{X}}^2\Big|_{\epsilon=0}=\langle\mathscr{J}[x],y\rangle,$$ for $y\in\mathbb{X}$. Furthermore, as we know that $\mathbb{U}$ is a separable Hilbert space (identified with its own dual), then the space $\mathbb{U}$ admits a Fr\'echet differentiable norm (cf. Theorem 8.24, \cite{MFb}) .
\end{rem}
The expression for the optimal control in the feedback form is obtained in the following lemma:
\begin{lem}\label{lem3.1}
Assume that $(x,u)$ is the optimal pair for the problem \eqref{F}. Then the optimal control $u$ is given by
\begin{align*}
u(t)=\mathrm{B}^*\mathcal{S}_{\gamma}(T-t)^*\mathscr{J}\left[\mathcal{R}(\lambda,\Phi_{0}^T)\ell(x(\cdot))\right],\ t\in [0, T),\ \lambda>0,\ \frac{1}{2}<\gamma<1,
\end{align*}
with
\begin{align*}
\ell(x(\cdot))=x_{T}-\mathcal{C}_{\gamma}(T)v-\mathcal{T}_{\gamma}(T)w.
\end{align*}
\end{lem}
\begin{proof}
Let us consider the functional
\begin{align*}
\mathscr{L}(\varepsilon)=\mathscr{G}(x_{u+\varepsilon z},u+\varepsilon z),
\end{align*}
where $(x,u)$ is the optimal solution of \eqref{F} and $z\in \mathrm{L}^{2}(J;\mathbb{U})$. Also the function $x_{u+\varepsilon z}$ is the unique mild solution of \eqref{3.2} with the control $u+\varepsilon z$. Then, it is immediate that
\begin{align*}
x_{u+\varepsilon z}(t)= \mathcal{C}_{\gamma}(t)v+\mathcal{T}_{\gamma}(t)w+\int_{0}^{t}(t-s)^{\gamma-1}\mathcal{S}_{\gamma}(t-s)\mathrm{B}(u+\varepsilon z)(s)\mathrm{d}s.
\end{align*}
It is clear that the functional $\mathscr{L}(\varepsilon)$ has a critical point at $\varepsilon=0$. We now calculate the first variation of the cost functional $\mathscr{G}$ (defined in \eqref{3.1}) as
\begin{align*}
\frac{\mathrm{d}}{\mathrm{d}\varepsilon}\mathscr{L}(\varepsilon)\Big|_{\varepsilon=0}&=\frac{\mathrm{d}}{\mathrm{d}\varepsilon}\bigg[\left\|x_{u+\varepsilon z}(T)-x_{T}\right\|^{2}_{\mathbb{X}}+\lambda\int^{T}_{0}(T-t)^{\gamma-1}\left\|u(t)+\varepsilon z(t)\right\|^{2}_{\mathbb{U}}\mathrm{d}t\bigg]_{\varepsilon=0}\nonumber\\
&=2\bigg[\langle \mathscr{J}(x_{u+\varepsilon z}(T)-x_{T}), \frac{\mathrm{d}}{\mathrm{d}\varepsilon}(x_{u+\varepsilon z}(T)-x_{T})\rangle\nonumber\\&\qquad +2\lambda\int^{T}_{0}(T-t)^{\gamma-1}(u(t)+\varepsilon z(t),\frac{\mathrm{d}}{\mathrm{d}\varepsilon}(u(t)+\varepsilon z(t)))\mathrm{d}t\bigg]_{\varepsilon=0}\nonumber\\
&=2\left\langle\mathscr{J}(x(T)-x_T),\int_0^T(T-t)^{\gamma-1}\mathcal{S}_{\gamma}(T-t)\mathrm{B}z(t)\mathrm{d}t\right\rangle\\&\quad+2\lambda\int_0^T(T-t)^{\gamma-1}(u(t),z(t))\mathrm{d} t.
\end{align*}
By taking the first variation zero, we deduce that
\begin{align}\label{3.9}
0&=\left\langle\mathscr{J}(x(T)-x_T),\int_0^T(T-t)^{\gamma-1}\mathcal{S}_{\gamma}(T-t)\mathrm{B}z(t)\mathrm{d}t\right\rangle+\lambda\int_0^T(T-t)^{\gamma-1}(u(t),z(t))\mathrm{d} t\nonumber\\&=\int_0^T(T-t)^{\gamma-1}\left\langle\mathscr{J}(x(T)-x_T),\mathcal{S}_{\gamma}(T-t)\mathrm{B}z(t) \right\rangle\mathrm{d}t +\lambda\int_0^T(T-t)^{\gamma-1}(u(t),z(t))\mathrm{d}t\nonumber\\&= \int_0^T(T-t)^{\gamma-1}\left(\mathrm{B}^*\mathcal{S}_{\gamma}(T-t)^*\mathscr{J}(x(T)-x_T)+\lambda u(t),z(t) \right)\mathrm{d}t,
\end{align}
where $(\cdot,\cdot)$ is the inner product in the Hilbert space $\mathbb{U}$. Note that the function $z\in \mathrm{L}^{2}(J;\mathbb{U})$ is an arbitrary (one can take $z$ to be $\mathrm{B}^*\mathcal{S}_{\gamma}(T-t)^*\mathscr{J}(x(T)-x_T)+\lambda u(t)$), it is immediate that
\begin{align}\label{3.10}
u(t)&= -\lambda^{-1}\mathrm{B}^*\mathcal{S}_{\gamma}(T-t)^*\mathscr{J}(x(T)-x_T),
\end{align}
for a.e. $t\in [0,T]$. Along with the relations \eqref{3.9} and \eqref{3.10}, it is clear that $u\in\mathrm{C}([0,T];\mathbb{U})$. Using the above expression of control, we find
\begin{align}\label{3.11}
x(T)&=\mathcal{C}_{\gamma}(T)v+\mathcal{T}_{\gamma}(T)w-\int^{T}_{0}\lambda^{-1}(T-s)^{\gamma-1}\mathcal{S}_{\gamma}(T-s)\mathrm{B}\mathrm{B}^*\mathcal{S}_{\gamma}(T-s)^*\mathscr{J}(x(T)-x_T)\mathrm{d}s\nonumber\\
&=\mathcal{C}_{\gamma}(T)v+\mathcal{T}_{\gamma}(T)w-\lambda^{-1}\Phi_{0}^T\mathscr{J}\left[x(T)-x_{T}\right].
\end{align}
Let us assume
\begin{align}\label{3.12}
\ell(x(\cdot)):=x_{T}-\mathcal{C}_{\gamma}(T)v-\mathcal{T}_{\gamma}(T)w.
\end{align}
Further, by \eqref{3.11} and \eqref{3.12}, we have
\begin{align}\label{3.13}
x(T)-x_{T}&=-\ell(x(\cdot))-\lambda^{-1}\Phi_{0}^T\mathscr{J}\left[x(T)-x_{T}\right].
\end{align}
From \eqref{3.13}, one can easily get
\begin{align}\label{3.15}
x(T)-x_T=-\lambda\mathrm{I}(\lambda\mathrm{I}+\Phi_0^T\mathscr{J})^{-1}\ell(x(\cdot))=-\lambda\mathcal{R}(\lambda,\Phi_0^T)\ell(x(\cdot)).
\end{align}
Finally, from \eqref{3.10}, we find the optimal control as
\begin{align*}
u(t)=\mathrm{B}^*\mathcal{S}_{\gamma}(T-t)^*\mathscr{J}\left[\mathcal{R}(\lambda,\Phi_{0}^T)\ell(x(\cdot))\right],\ \mbox{ for all } \ t\in [0,T],
\end{align*}
which completes the proof.
\end{proof}
In the next lemma, we discuss the compactness of the operator $(\mathrm{Q}f)(\cdot) =\int_{0}^{\cdot}(\cdot-s)^{\gamma-1}\mathcal{S}_{\gamma}(\cdot-s) f(s)\mathrm{d}s:\mathrm{L}^2(J;\mathbb{X})\rightarrow \mathrm{C}(J;\mathbb{X}) ,\ \mbox{where}\ \gamma\in(\frac{1}{2},1)$ and $\mathbb{X}$ is a general Banach space.
\begin{lem}\label{lem2.12}
Suppose that the operator $\mathcal{S}_{\gamma}(t)$ is compact for $t\ge 0$. Let the operator $\mathrm{Q}:\mathrm{L}^{2}(J;\mathbb{X})\rightarrow \mathrm{C}(J;\mathbb{X})$ be defined as
\begin{align}
(\mathrm{Q}f)(t)= \int^{t}_{0}(t-s)^{\gamma-1}\mathcal{S}_{\gamma}(t-s)f(s)\mathrm{d}s, \ t\in J,\ \frac{1}{2}<\gamma<1.
\end{align}
Then the operator $\mathrm{Q}$ is compact.
\end{lem}
A proof of the above lemma is a straightforward adaptation of the proof of Lemma 3.2, \cite{SMJ}.
\begin{lem}\label{lem3.4}
The linear control system \eqref{3.2} is approximately controllable on $J$ if and only if Assumption (H0) holds.
\end{lem}
A proof of the above lemma can be obtained by proceeding similarly as in the proof of Theorem 3.2, \cite{SM}.
\begin{rem}\label{rem3.4}
Note that the operator $\Phi_{0}^{T}$ is positive if and only if Assumption (\textit{H0}) holds (see, Theorem 2.3, \cite{M}). The positivity of $\Phi_{0}^{T}$ is equivalent to $$ \langle x^*, \Phi_{0}^{T}x^*\rangle=0\Rightarrow x^*=0.$$ Using the definition of $\Phi_{0}^{T}$, we have
\begin{align}
\langle x^*, \Phi_{0}^{T}x^*\rangle =\int_0^T(T-t)^{\gamma-1}\left\|\mathrm{B}^*\mathcal{S}_{\gamma}(T-t)^*x^*\right\|_{\mathbb{X}^*}^2\mathrm{d}t.
\end{align}
The above fact and Lemma \ref{lem3.4} guarantee that the approximate controllability of the linear system \eqref{3.2} is equivalent to the condition $$\mathrm{B}^*\mathcal{S}_{\gamma}(T-t)^*x^*=0,\ 0\le t<T \Rightarrow x^*=0.$$
\end{rem}
\section{Approximate Controllability of the Fractional order Impulsive System} \label{semilinear}\setcounter{equation}{0}
The present section is reserved for the approximate controllability of the semilinear impulsive fractional control system \eqref{1.1}. For this, we first verify the existence of a mild solution of the system \eqref{1.1} with the control defined as
\begin{align}\label{C}
u^{\gamma}_{\lambda}(t)&=\sum_{j=0}^{p}u^{\gamma}_{j,\lambda}(t)\chi_{[s_j, \tau_{j+1}]}(t), \ t\in J,\ \frac{1}{2}<\gamma<1,
\end{align}
where
\begin{align*}
u^{\gamma}_{j,\lambda}(t)&=\mathrm{B}^*\mathcal{S}_{\gamma}(\tau_{j+1}-t)^*\mathscr{J}\left[\mathcal{R}(\lambda,\Phi_{s_j}^{\tau_{j+1}})g_j(x(\cdot))\right],
\end{align*}
for $t\in [s_j, \tau_{j+1}],j=0,1,\ldots,p$, with
\begin{align*}
g_0(x(\cdot))&=\xi_{0}-\mathcal{C}_{\gamma}(\tau_1)\psi(0)-\mathcal{T}_{\gamma}(\tau_1)\eta-\int^{\tau_1}_{0}(\tau_1-s)^{\gamma-1}\mathcal{S}_{\gamma}(\tau_1-s)f(s,\tilde{x}_{\varrho(s,\tilde{x}_s)})\mathrm{d}s,\nonumber\\
g_j(x(\cdot))&=\xi_{j}-\mathcal{C}_{\gamma}(\tau_{j+1}-s_j)h_j(s_j,\tilde{x}(\tau_j^-))-\mathcal{T}_{\gamma}(\tau_{j+1}-s_j)h'_j(s_j,\tilde{x}(\tau_j^-))\\&\quad+\int_{0}^{s_j}(s_j-s)^{\gamma-1}\mathcal{S}_{\gamma}(s_j-s)\left[f(s,\tilde{x}_{\varrho(s,\tilde{x}_s)})+\mathrm{B}\sum_{k=0}^{j-1}u^{\gamma}_{k,\lambda}(s)\chi_{[s_k, \tau_{k+1})}(s)\right]\mathrm{d}s \\&\quad-\int_{0}^{\tau_{j+1}}(\tau_{j+1}-s)^{\gamma-1}\mathcal{S}_{\gamma}(\tau_{j+1}-s)f(s,\tilde{x}_{\varrho(s,\tilde{x}_s)})\mathrm{d}s \\&\quad-\int_{0}^{s_j}(\tau_{j+1}-s)^{\gamma-1}\mathcal{S}_{\gamma}(\tau_{j+1}-s)\mathrm{B}\sum_{k=0}^{j-1}u^{\gamma}_{k,\lambda}(s)\chi_{[s_k, \tau_{k+1})}(s)\mathrm{d}s, \ j=1,\ldots,p,
\end{align*}
the function $\tilde{x}:(-\infty,T]\rightarrow\mathbb{X}$ is given by $$\tilde{x}(t)=\psi(t), \ t\in(-\infty,0], \ \tilde{x}(t)=x(t),\ t\in J,$$ and for arbitrary $\xi_{j}\in \mathbb{X}$ for $j=0,1,\ldots,p$.
\begin{rem}
Since the operator $\Phi_{s_j}^{\tau_{j+1}},$ for each $j=0,\ldots,p,$ is linear, bounded and non-negative, Lemma \ref{lem2.9} is valid for each $\Phi_{s_j}^{\tau_{j+1}},$ for $j=0,\ldots,p$.
\end{rem}
In the following theorem, we obtain the existence of a mild solution of the system \eqref{1.1} with the control given in \eqref{C}.
\begin{theorem}\label{thm4.3}
If Assumptions (R1)-(R3) and (H1)-(H2) hold true. Then for fixed $\xi_{j}\in\mathbb{X},$ for $j=0,1,\ldots,p$ and every $\lambda>0$, the system \eqref{1.1} with the control \eqref{C} has at least one mild solution on $J$, provided
\begin{align}\label{cnd}
\frac{2MT^{2\gamma-\delta}K_{2}\zeta}{\Gamma(2\gamma)c^{1-\delta}}\left\{1+\frac{(p+1)(p+2)R}{2}+\frac{p(p+1)R^2}{2}\sum_{k=0}^{p-1}e^{\frac{(p+k)(p-k-1)R}{2}}\right\}<1,
\end{align}
where $R=\frac{2T^{3\gamma}}{3\gamma\lambda}\left(\frac{M\tilde{M}}{\Gamma(2\gamma)}\right)^2$ and $c=\frac{2\gamma-\delta}{1-\delta}$.
\end{theorem}
\begin{proof}
Let us take a set $\mathcal{Z}:=\{x\in\mathrm{PC}(J;\mathbb{X}) : x(0)=\psi(0)\}$ with the norm $\left\|\cdot\right\|_{\mathrm{PC}(J;\mathbb{X})}$. Next, we consider a set $\mathcal{E}_{r}=\{x\in\mathcal{Z} : \left\|x\right\|_{\mathrm{PC}(J;\mathbb{X})}\le r\}$, for each $r>0$.
For $\lambda>0$, we define an operator $\mathscr{F}_{\lambda}:\mathcal{Z}\to\mathcal{Z}$ such that
\begin{equation}\label{2}
(\mathscr{F}_{\lambda}x)(t)=\left\{
\begin{aligned}
&\mathcal{C}_{\gamma}(t)\psi(0)+\mathcal{T}_{\gamma}(t)\eta+\int_{0}^{t}(t-s)^{\gamma-1}\mathcal{S}_{\gamma}(t-s)\left[\mathrm{B}u^{\gamma}_{\lambda}(s)+f(s,\tilde{x}_{\varrho(s, \tilde{x}_s)})\right]\mathrm{d}s,\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad t\in[0, \tau_1],\\
&h_j(t, \tilde{x}(\tau_j^-)),\qquad\qquad\qquad\qquad\qquad\qquad\ \ t\in(\tau_j, s_j], j=1,\ldots,p,\\
&\mathcal{C}_{\gamma}(t-s_j)h_j(s_j, \tilde{x}(\tau_j^-))+\mathcal{T}_{\gamma}(t-s_j)h'_j(s_j, \tilde{x}(\tau_j^-))\\& \quad -\int_{0}^{s_j}(s_j-s)^{\gamma-1}\mathcal{S}_{\gamma}(s_j-s)\left[\mathrm{B}u^{\gamma}_{\lambda}(s)+f(s,\tilde{x}_{\varrho(s, \tilde{x}_s)})\right]\mathrm{d}s \\&\quad+\int_{0}^{t}(t-s)^{\gamma-1}\mathcal{S}_{\gamma}(t-s)\left[\mathrm{B}u^{\gamma}_{\lambda}(s)+f(s,\tilde{x}_{\varrho(s, \tilde{x}_s)})\right]\mathrm{d}s,\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad t\in(s_j, \tau_{j+1}],\ j=1,\ldots,p.
\end{aligned}
\right.
\end{equation}
where $u^{\gamma}_{\lambda}(\cdot)$ is given in \eqref{C}. It is clear form the definition of $ \mathscr{F}_{\lambda}$ that the system $\eqref{1.1}$ has a mild solution, if the operator $ \mathscr{F}_{\lambda}$ has a fixed point. The proof that the operator $\mathscr{F}_{\lambda}$ has a fixed point is divided into the following steps.
\vskip 0.1in
\noindent\textbf{Step (1): } \emph{$ F_{\lambda}(\mathcal{E}_r)\subset \mathcal{E}_r,$ for some $ r $}. We achieve this goal by contradiction. To prove this, we assume that for any $\lambda>0$ and every $r > 0$, there exists $x^r(\cdot) \in \mathcal{E}_r$ such that $\left\|(\mathscr{F}_\lambda x^r)(t)\right\|_{\mathbb{X}} > r$, for some $t\in J$ ($t$ may depend upon $r$). First, we compute
\begin{align}\label{4.4}
\left\|g_0(x(\cdot))\right\|_{\mathbb{X}}&\le\left\|\xi_0\right\|_{\mathbb{X}}+\left\|\mathcal{C}_{\gamma}(\tau_1)\psi(0)\right\|_{\mathbb{X}}+\left\|\mathcal{T}_{\gamma}(\tau_1)\eta\right\|_{\mathbb{X}}\nonumber\\&\quad+\int_{0}^{\tau_1}(\tau_1-s)^{\gamma-1}\left\|\mathcal{S}_{\gamma}(\tau_1-s)f(s,\tilde{x}_{\varrho(s,\tilde{x}_s)})\right\|_{\mathbb{X}}\mathrm{d}s\nonumber\\&\le \left\|\xi_0\right\|_{\mathbb{X}}+M\left\|\psi(0)\right\|_{\mathbb{X}}+Mt\left\|\eta\right\|_{\mathbb{X}}+\frac{M}{\Gamma(2\gamma)}\int_{0}^{\tau_1}(\tau_1-s)^{2\gamma-1}\phi_ {r'}(s)\mathrm{d}s\nonumber\\&\le\left\|\xi_0\right\|_{\mathbb{X}}+M\left\|\psi(0)\right\|_{\mathbb{X}}+MT\left\|\eta\right\|_{\mathbb{X}}\nonumber\\&\quad+\frac{M}{\Gamma(2\gamma)}\left(\int_{0}^{\tau}(\tau_1-s)^{\frac{2\gamma-1}{1-\delta}}\mathrm{d}s\right)^{1-\delta}\left(\int_{0}^{\tau_1}(\phi_{r'}(s))^{\frac{1}{\delta}}\mathrm{d}s\right)^{\delta}\nonumber\\&\le\left\|\xi_0\right\|_{\mathbb{X}}+M\left\|\psi(0)\right\|_{\mathbb{X}}+MT\left\|\eta\right\|_{\mathbb{X}}+\frac{M\tau_{1}^{2\gamma-\delta}}{\Gamma(2\gamma)c^{1-\delta}}\left\|\phi_{r'}\right\|_{\mathrm{L}^{\frac{1}{\delta}}([0,\tau_1];\mathbb{R}^+)}\nonumber\\&\le\left\|\xi_0\right\|_{\mathbb{X}}+M\left\|\psi(0)\right\|_{\mathbb{X}}+MT\left\|\eta\right\|_{\mathbb{X}}+\frac{2MT^{2\gamma-\delta}}{\Gamma(2\gamma)c^{1-\delta}}\left\|\phi_{r'}\right\|_{\mathrm{L}^{\frac{1}{\delta}}(J;\mathbb{R}^+)}=:N_0,
\end{align}
where $c=\frac{2\gamma-\delta}{1-\delta}$ and $r'=K_{1}\left\|\psi\right\|_{\mathfrak{B}}+K_{2}r$. Next, we estimate
\begin{align*}
\left\|g_j(x(\cdot))\right\|_{\mathbb{X}}&\le\left\|\xi_j\right\|_{\mathbb{X}}+\left\|\mathcal{C}_{\gamma}(t-s_j)h_j(s_j,\tilde{x}(\tau_j^-))\right\|_{\mathbb{X}}+\left\|\mathcal{T}_{\gamma}(t-s_j)h'_j(s_j,\tilde{x}(t_j^-))\right\|_{\mathbb{X}}\nonumber\\&\quad+\int_{0}^{s_j}(s_j-s)^{\gamma-1}\left\|\mathcal{S}_{\gamma}(s_j-s)f(s,\tilde{x}_{\varrho(s,\tilde{x}_s)})\right\|_{\mathbb{X}}\mathrm{d}s\nonumber\\&\quad+\int_{0}^{\tau_{j+1}}(\tau_{j+1}-s)^{\gamma-1}\left\|\mathcal{S}_{\gamma}(\tau_{j+1}-s)f(s,\tilde{x}_{\varrho(s,\tilde{x}_s)})\right\|_{\mathbb{X}}\mathrm{d}s\nonumber\\&\quad+\int_{0}^{s_j}(s_j-s)^{\gamma-1}\left\|\mathcal{S}_{\gamma}(s_j-s)\mathrm{B}\sum_{k=0}^{j-1}u^\gamma_{k,\lambda}(s)\chi_{[s_k,\tau_{k+1}]}(s)\right\|_{\mathbb{X}}\mathrm{d}s\nonumber\\&\quad+\int_{0}^{s_j}(\tau_{j+1}-s)^{\gamma-1}\left\|\mathcal{S}_{\gamma}(\tau_{j+1}-s)\mathrm{B}\sum_{k=0}^{j-1}u^\gamma_{k,\lambda}(s)\chi_{[s_k,\tau_{k+1}]}(s)\right\|_{\mathbb{X}}\mathrm{d}s\nonumber\\&\le\left\|\xi_j\right\|_{\mathbb{X}}+M\kappa_{j}+M(t-s_j)\vartheta_{j}+\frac{M}{\Gamma(2\gamma)}\int_{0}^{s_j}(s_j-s)^{2\gamma-1}\phi_{r'}(s)\mathrm{d}s\nonumber\\&\quad+\frac{M}{\Gamma(2\gamma)}\int_{0}^{\tau_{j+1}}(\tau_{j+1}-s)^{2\gamma-1}\phi_{r'}(s)\mathrm{d}s+\frac{1}{\lambda}\left(\frac{M\tilde{M}}{\Gamma(2\gamma)}\right)^2\sum_{k=0}^{j-1}\left\|g_k(x(\cdot))\right\|_{\mathbb{X}}\nonumber\\&\qquad\times\int_{s_k}^{\tau_{k+1}}\left[(s_j-s)^{2\gamma-1}+(\tau_{j+1}-s)^{2\gamma-1}\right](\tau_{k+1}-s)^{\gamma}\mathrm{d}s\nonumber\\&\le\left\|\xi_j\right\|_{\mathbb{X}}+M\kappa_{j}+MT\vartheta_{j}+\frac{2M}{\Gamma(2\gamma)}\int_{0}^{\tau_{j+1}}(\tau_{j+1}-s)^{2\gamma-1}\phi_{r'}(s)\mathrm{d}s\nonumber\\&\quad+\frac{2}{\lambda}\left(\frac{M\tilde{M}}{\Gamma(2\gamma)}\right)^2\sum_{k=0}^{j-1}\left\|g_k(x(\cdot))\right\|_{\mathbb{X}}\int_{s_k}^{\tau_{k+1}}(\tau_{j+1}-s)^{2\gamma-1}(\tau_{k+1}-s)^{\gamma}\mathrm{d}s\nonumber\\&\le\left\|\xi_j\right\|_{\mathbb{X}}+M\kappa_{j}+MT\vartheta_{j}+\frac{2M}{\Gamma(2\gamma)}\int_{0}^{\tau_{j+1}}(\tau_{j+1}-s)^{2\gamma-1}\phi_{r'}(s)\mathrm{d}s\nonumber\\&\quad+\frac{2}{\lambda}\left(\frac{M\tilde{M}}{\Gamma(2\gamma)}\right)^2\sum_{k=0}^{j-1}\left\|g_k(x(\cdot))\right\|_{\mathbb{X}}\int_{s_k}^{\tau_{k+1}}(\tau_{j+1}-s)^{3\gamma-1}\mathrm{d}s\nonumber\\&\le\left\|\xi_j\right\|_{\mathbb{X}}+M\kappa_{j}+MT\vartheta_{j}+\frac{2M\tau_{j+1}^{2\gamma-\delta}}{\Gamma(2\gamma)c^{1-\delta}}\left\|\phi_{r'}\right\|_{\mathrm{L}([0,\tau_{j+1}];\mathbb{R}^+)}\nonumber\\&\quad+\frac{2}{\lambda}\left(\frac{M\tilde{M}}{\Gamma(2\gamma)}\right)^2\sum_{k=0}^{j-1}\left\|g_k(x(\cdot))\right\|_{\mathbb{X}}\frac{(\tau_{j+1}-s_k)^{3\gamma}-(\tau_{j+1}-\tau_{k+1})^{3\gamma}}{3\gamma}\nonumber\\&\le\left\|\xi_j\right\|_{\mathbb{X}}+M\kappa_{j}+MT\vartheta_{j}+\frac{2MT^{2\gamma-\delta}}{\Gamma(2\gamma)c^{1-\delta}}\left\|\phi_{r'}\right\|_{\mathrm{L}(J;\mathbb{R}^+)}\nonumber\\&\quad+\frac{2T^{3\gamma}}{3\gamma\lambda}\left(\frac{M\tilde{M}}{\Gamma(2\gamma)}\right)^2\sum_{k=0}^{j-1}\left\|g_k(x(\cdot))\right\|_{\mathbb{X}}\nonumber\\&\le N_j+R\sum_{k=0}^{j-1}\left\|g_k(x(\cdot))\right\|_{\mathbb{X}},
\end{align*}
where $R=\frac{2T^{3\gamma}}{3\gamma\lambda}\left(\frac{M\tilde{M}}{\Gamma(2\gamma)}\right)^2$ and $N_j=\left\|\xi_j\right\|_{\mathbb{X}}+M\kappa_{j}+M T\vartheta_{j} +\frac{2M T^{2\gamma-\delta}}{\Gamma(2\gamma)c^{1-\delta}}\left\|\phi_{r'}\right\|_{\mathrm{L}(J;\mathbb{R}^+)},$ for $j=1,\ldots,p.$ Using the discrete Gronwall-Bellman lemma (Lemma \ref{lem2.13}), we obtain
\begin{align}\label{4.5}
\left\|g_j(x(\cdot))\right\|_{\mathbb{X}}\le&N_j+R\sum_{k=0}^{j-1}N_ke^{\frac{(j+k)(j-k-1)R}{2}}=:C_j, \ \mbox{ for }\ j=1,\ldots,p.
\end{align}
Taking $t\in[0,\tau_1]$ and using the estimates \eqref{2.5}, \eqref{4.4},\eqref{4.5}, Lemma \ref{lem2.11} and Assumption \ref{as2.1} \textit{(H1)}-\textit{(H2)}, we evaluate
\begin{align}\label{4.21}
r&<\left\|(\mathscr{F}_{\lambda}x^r)(t)\right\|_\mathbb{X}\nonumber\\&=\left\|\mathcal{C}_{\gamma}(t)\psi(0)+\mathcal{T}_{\gamma}(t)\eta+\int_{0}^{t}(t-s)^{\gamma-1}\mathcal{S}_{\gamma}(t-s)\left[\mathrm{B}u^{\gamma}_{\lambda}(s)+f(s,\tilde{x}_{\varrho(s, \tilde{x}_s)})\right]\mathrm{d}s\right\|_{\mathbb{X}}\nonumber\\&\le M\left\|\psi(0)\right\|_{\mathbb{X}}+Mt\left\|\eta\right\|_{\mathbb{X}}+\frac{M\tilde{M}}{\Gamma(2\gamma)}\int_{0}^{t}(t-s)^{2\gamma-1}\left\|u^{\gamma}_{\lambda}(s)\right\|_{\mathbb{U}}\mathrm{d}s\nonumber\\&\quad+\frac{M}{\Gamma(2\gamma)}\int_{0}^{t}(t-s)^{2\gamma-1}\left\|f(s,\tilde{x}_{\rho(s,\tilde{x}_s)})\right\|_{\mathbb{X}}\mathrm{d}s\nonumber\\&\le M\left\|\psi(0)\right\|_{\mathbb{X}}+M\tau_{1}\left\|\eta\right\|_{\mathbb{X}}+\frac{M\tilde{M}}{\Gamma(2\gamma)}\int_{0}^{\tau_1}(\tau_1-s)^{2\gamma-1}\left\|u^{\gamma}_{\lambda}(s)\right\|_{\mathbb{U}}\mathrm{d}s\nonumber\\&\quad+\frac{M}{\Gamma(2\gamma)}\int_{0}^{\tau_1}(\tau_1-s)^{2\gamma-1}\left\|f(s,\tilde{x}_{\rho(s,\tilde{x}_s)})\right\|_{\mathbb{X}}\mathrm{d}s\nonumber\\&\le M\left\|\psi(0)\right\|_{\mathbb{X}}+M\tau_{1}\left\|\eta\right\|_{\mathbb{X}}+\frac{M\tilde{M}}{\Gamma(2\gamma)}\int_{0}^{\tau_1}(\tau_1-s)^{2\gamma-1}\left\|u^{\gamma}_{1,\lambda}(s)\right\|_{\mathbb{U}}\mathrm{d}s\nonumber\\&\quad+\frac{M}{\Gamma(2\gamma)}\int_{0}^{\tau_1}(\tau_1-s)^{2\gamma-1}\phi_{r'}(s)\mathrm{d}s\nonumber\\&\le M\left\|\psi(0)\right\|_{\mathbb{X}}+M\tau_1\left\|\eta\right\|_{\mathbb{X}}+\frac{1}{\lambda}\left(\frac{M\tilde{M}}{\Gamma(2\gamma)}\right)^{2}\left\|g_0((\cdot))\right\|_{\mathbb{X}}\int_{0}^{\tau_1}(\tau_1-s)^{3\gamma-1}\mathrm{d}s\nonumber\\&\quad+\frac{M}{\Gamma(2\gamma)}\int_{0}^{\tau_1}(\tau_1-s)^{2\gamma-1}\phi_{r'}(s)\mathrm{d}s\nonumber\\&\le M\left\|\psi(0)\right\|_{\mathbb{X}}+M\tau_1\left\|\eta\right\|_{\mathbb{X}}+\frac{N_{0}}{\lambda}\left(\frac{M\tilde{M}} {\Gamma(2\gamma)}\right) ^{2} \int_{0}^{\tau_1}(\tau_1-s)^{3\gamma-1}\mathrm{d}s\nonumber\\&\quad+\frac{M}{\Gamma(2\gamma)}\left(\int_{0}^{\tau_1}(\tau_1-s)^{\frac{2\gamma-1}{1-\delta}}\right)^{1-\delta}\left(\int_{0}^{\tau_1}\phi_{r'}^{\frac{1}{\delta}}(s)\mathrm{d}s\right)^{\delta}\nonumber\\&= M\left\|\psi(0)\right\|_{\mathbb{X}}+M\tau_1\left\|\eta\right\|_{\mathbb{X}}+\frac{N_{0}\tau_1^{3\gamma}}{3\gamma\lambda}\left(\frac{M\tilde{M}}{\Gamma(2\gamma)}\right)^{2}+ \frac {M\tau_1^{2\gamma-\delta}} {\Gamma(2\gamma)c^{1-\delta}}\left\|\phi_{r'}\right\|_{\mathrm{L}^{\frac{1}{\delta}}([0,\tau_1];\mathbb{R}^+)}\nonumber\\&\le M\left\|\psi(0)\right\|_{\mathbb{X}}+MT\left\|\eta\right\|_{\mathbb{X}}+\frac{N_{0}T^{3\gamma}}{3\gamma\lambda}\left(\frac{M\tilde{M}} {\Gamma(2\gamma)}\right) ^{2}+ \frac {MT^{2\gamma-\delta}} {\Gamma(2\gamma)c^{1-\delta}}\left\|\phi_{r'}\right\|_{\mathrm{L}^{\frac{1}{\delta}}(J;\mathbb{R}^+)}\nonumber\\&<M\left\|\psi(0)\right\|_{\mathbb{X}}+MT \left\|\eta\right\|_{\mathbb{X}}+\frac{2T^{3\gamma}}{3\gamma\lambda}\left(\frac{M\tilde{M}}{\Gamma(2\gamma)}\right)^{2}\sum_{k=0}^{p}C_k+\frac {2MT^{2\gamma-\delta}} {\Gamma(2\gamma)c^{1-\delta}}\left\|\phi_{r'}\right\|_{\mathrm{L}^{\frac{1}{\delta}}(J;\mathbb{R}^+)}\nonumber\\&=M\left\|\psi(0)\right\|_{\mathbb{X}}+MT\left\|\eta\right\|_{\mathbb{X}}+R\sum_{k=0}^{p}C_k+\frac{2MT^{2\gamma-\delta}}{\Gamma(2\gamma)c^{1-\delta}}\left\|\phi_{r'}\right\|_{\mathrm{L}^{\frac{1}{\delta}}(J;\mathbb{R}^+)},
\end{align}
where $C_0=N_0$. For $t\in(\tau_j,s_j],\ j=1,\ldots,p$, we obtain
\begin{align}\label{4.22}
r&<\left\|(\mathscr{F}_{\lambda}x^r)(t)\right\|_\mathbb{X}\le\left\|h_j(t, \tilde{x}(\tau_j^-))\right\|_{\mathbb{X}}\nonumber\\&\le \kappa_{j}< \kappa_{j}+R\sum_{k=0}^{p}C_k+\frac{2MT^{2\gamma-\delta}}{\Gamma(2\gamma)c^{1-\delta}}\left\|\phi_{r'}\right\|_{\mathrm{L}^{\frac{1}{\delta}}}(J;\mathbb{R}^+).
\end{align}
Taking $t\in(s_j,\tau_{j+1}], \ j=1,\dots,p$, we compute
\begin{align}\label{4.23}
r&<\left\|(\mathscr{F}_{\lambda}x^r)(t)\right\|_\mathbb{X}\nonumber\\&=\bigg\|\mathcal{C}_{\gamma}(t-s_j)h_j(s_j,\tilde{x}(\tau_j^-))+\mathcal{T}_{\gamma}(t-s_j)h'_j(s_j,\tilde{x}(\tau_j^-))\nonumber\\&\quad-\int_{0}^{s_j}(s_j-s)^{\gamma-1}\mathcal{S}_{\gamma}(s_j-s)\left[\mathrm{B}u^{\gamma}_{\lambda}(s)+f(s,\tilde{x}_{\varrho(s,\tilde{x}_s)})\right]\mathrm{d}s\nonumber\\&\quad+\int_{0}^{t}(t-s)^{\gamma-1}\mathcal{S}_{\gamma}(t-s)\left[\mathrm{B}u^{\gamma}_{\lambda}(s)+f(s,\tilde{x}_{\varrho(s,\tilde{x}_s)})\right]\mathrm{d}s\bigg\|_{\mathbb{X}}\nonumber\\&\leq\left\|\mathcal{C}_{\gamma}(t-s_j)h_j(s_j,\tilde{x}(\tau_j^-))\right\|_{\mathbb{X}}+\left\|\mathcal{T}_{\gamma}(t-s_j)h'_j(s_j,\tilde{x}(t_j^-))\right\|_{\mathbb{X}}\nonumber\\&\quad+\int_{0}^{s_j}(s_j-s)^{\gamma-1}\left\|\mathcal{S}_{\gamma}(s_j-s)\mathrm{B}u^{\gamma}_{\lambda}(s)\right\|_{\mathbb{X}}\mathrm{d}s\nonumber\\&\quad+\int_{0}^{s_j}(s_j-s)^{\gamma-1}\left\|\mathcal{S}_{\gamma}(s_j-s)f(s,\tilde{x}_{\varrho(s,\tilde{x}_s)})\right\|_{\mathbb{X}}\mathrm{d}s\nonumber\\&\quad+\int_{0}^t(t-s)^{\gamma-1}\left\|\mathcal{S}_{\gamma}(t-s)\mathrm{B}u^{\gamma}_{\lambda}(s)\right\|_{\mathbb{X}}\mathrm{d}s\nonumber\\&\quad+\int_{0}^t(t-s)^{\gamma-1}\left\|\mathcal{S}_{\gamma}(t-s)f(s,\tilde{x}_{\varrho(s,\tilde{x}_s)})\right\|_{\mathbb{X}}\mathrm{d}s\nonumber\\&\le M\kappa_{j}+M(t-s_j)\vartheta_{j}+\frac{M\tilde{M}}{\Gamma(2\gamma)}\int_{0}^{s_j}(s_j-s)^{2\gamma-1}\left\|u_{\lambda}^{\gamma}( s)\right\|_{\mathbb{U}}\mathrm{d}s\nonumber\\&\quad+\frac{M}{\Gamma(2\gamma)}\int_{0}^{s_j}(s_j-s)^{2\gamma-1}\left\|f(s,\tilde{x}_{\varrho(s,\tilde{x}_s)})\right\|_{\mathbb{X}}+\frac{M\tilde{M}}{\Gamma(2\gamma)}\int_{0}^{t}(t-s)^{2\gamma-1}\left\|u_{\lambda}^{\gamma}(s)\right\|_{\mathbb{U}}\mathrm{d}s\nonumber\\&\quad+\frac{M}{\Gamma(2\gamma)}\int_{0}^{t}(t-s)^{2\gamma-1}\left\|f(s,\tilde{x}_{\varrho(s,\tilde{x}_s)})\right\|_{\mathbb{X}}\mathrm{d}s \nonumber\\&\leq M\kappa_{j}+MT\vartheta_{j}+\frac{M\tilde{M}}{\Gamma(2\gamma)}\int_{0}^{s_j}(s_j-s)^{2\gamma-1}\left\|\sum_{k=0}^{j-1}u_{k,\lambda}^{\gamma}(s)\chi_{[s_k,\tau_{k+1}]}(s)\right \|_{\mathbb{U}}\mathrm{d}s\nonumber\\&\quad+\frac{M}{\Gamma(2\gamma)}\int_{0}^{s_j}(s_j-s)^{2\gamma-1}\phi_{r'}(s)\mathrm{d}s\nonumber\\&\quad+\frac{M\tilde{M}}{\Gamma(2\gamma)}\int_{0}^{\tau_{j+1}}(\tau_{j+1}-s)^{2\gamma-1}\left\|\sum_{k=0}^{j}u_{k,\lambda}^{\gamma}(s)\chi_{[s_k,\tau_{k+1}]}(s)\right\|_{\mathbb{U}}\mathrm{d}s\nonumber\\&\quad+\frac{M}{\Gamma(2\gamma)}\int_{0}^{t_{j+1}}(t_{j+1}-s)^{2\gamma-1}\phi_{r'}(s)\mathrm{d}s\nonumber\\&\leq M\kappa_{j}+MT\vartheta_{j}+\frac{1}{\lambda}\left(\frac{M\tilde{M}}{\Gamma(2\gamma)}\right)^{2}\sum_{k=0}^{j-1}\left\|g_{k}(x(\cdot))\right\|_{\mathbb{X}}\int_{s_k}^{\tau_{k+1}}(s_j-s)^{2\gamma-1}(\tau_{k+1}-s)^{\gamma}\mathrm{d}s\nonumber\\&\quad+\frac{Ms_j^{2\gamma-\delta}}{\Gamma(2\gamma)c^{1-\delta}}\left\|\phi_{r'}\right\|_{\mathrm{L}^{\frac{1}{\delta}}([0,s_j];\mathbb{R}^+)}+\frac{Mt_{j+1}^{2\gamma-\delta}}{\Gamma(2\gamma)c^{1-\delta}}\left\|\phi_{r'}\right\|_{\mathrm{L}^{\frac{1}{\delta}}([0,\tau_{j+1}];\mathbb{R}^+)}\nonumber\\&\quad+\frac{1}{\lambda}\left(\frac{M\tilde{M}}{\Gamma(2\gamma)}\right)^{2}\sum_{k=0}^{j}\left\|g_{k}(x(\cdot))\right\|_{\mathbb{X}}\int_{s_k}^{\tau_{k+1}}(s_j-s)^{2\gamma-1}(\tau_{k+1}-s)^{\gamma}\mathrm{d}s\nonumber\\&\le M\kappa_{j}+MT\vartheta_{j}+\frac{T^{3\gamma}}{3\gamma\lambda}\left(\frac{M\tilde{M}}{\Gamma(2\gamma)}\right)^{2}\sum_{k=0}^{j-1}\left\|g_k(x(\cdot))\right\|_{\mathbb{X}}+\frac{2MT^{2\gamma-\delta}}{\Gamma(2\gamma)c^{1-\delta}}\left\|\phi_{r'}\right\|_{\mathrm{L}^{\frac{1}{\delta}}(J;\mathbb{R}^+)}\nonumber\\&\quad+\frac{T^{3\gamma}}{3\gamma\lambda}\left(\frac{M\tilde{M}}{\Gamma(2\gamma)}\right)^{2}\sum_{k=0}^{j}\left\|g_k(x(\cdot))\right\|_{\mathbb{X}}
\nonumber\\&\leq M\kappa_{j}+MT\vartheta_{j}+\frac{2T^{3\gamma}}{3\gamma\lambda}\left(\frac{M\tilde{M}}{\Gamma(2\gamma)}\right)^{2}\sum_{k=0}^{j}\left\|g_k(x(\cdot))\right\|_{\mathbb{X}}+\frac{2MT^{2\gamma-\delta}}{\Gamma(2\gamma)c^{1-\delta}}\left\|\phi_{r'}\right\|_{\mathrm{L}^{\frac{1}{\delta}}(J;\mathbb{R}^+)}\nonumber\\&\le M\kappa_{j}+MT\vartheta_{j}+\frac{2T^{3\gamma}}{3\gamma\lambda}\left(\frac{M\tilde{M}}{\Gamma(2\gamma)}\right)^{2}\sum_{k=0}^{p}\left\|g_k(x(\cdot))\right\|_{\mathbb{X}}+\frac{2MT^{2\gamma-\delta}}{\Gamma(2\gamma)c^{1-\delta}}\left\|\phi_{r'}\right\|_{\mathrm{L}^{\frac{1}{\delta}}(J;\mathbb{R}^+)}\nonumber\\&\le M\kappa_{j}+MT\vartheta_{j}+\frac{2T^{3\gamma}}{3\gamma\lambda}\left(\frac{M\tilde{M}}{\Gamma(2\gamma)}\right)^{2}\sum_{k=0}^{p}C_k+\frac{2MT^{2\gamma-\delta}}{\Gamma(2\gamma)c^{1-\delta}}\left\|\phi_{r'}\right\|_{\mathrm{L}^{\frac{1}{\delta}}(J;\mathbb{R}^+)}\nonumber\\&=M\kappa_{j}+MT\vartheta_{j}+R\sum_{k=0}^{p}C_k+\frac{2MT^{2\gamma-\delta}}{\Gamma(2\gamma)c^{1-\delta}}\left\|\phi_{r'}\right\|_{\mathrm{L}^{\frac{1}{\delta}}(J;\mathbb{R}^+)}.
\end{align}
Using Assumption \ref{as2.1} (\textit{H2})(ii), we easily obtain
\begin{align*}
\liminf_{r \rightarrow \infty }\frac {\left\|\phi_{r'}\right\|_{\mathrm{L}^{\frac{1}{\delta}}(J;\mathbb{R^+})}}{r}&=\liminf_{r\rightarrow\infty}\left (\frac {\left\|\gamma_{r'}\right\|_{\mathrm{L}^{\frac{1}{\alpha_1}}(J;\mathbb{R^+})}}{r'}\times\frac{r'}{r}\right)=K_2\zeta.
\end{align*}
Thus, dividing by $r$ in the expressions \eqref{4.21}, \eqref{4.22}, \eqref{4.23} and then passing $r\to\infty$, we obtain
\begin{align*}
\frac{2MT^{2\gamma-\delta}K_{2}\zeta}{\Gamma(2\gamma)c^{1-\delta}}\left\{1+\frac{(p+1)(p+2)R}{2}+\frac{p(p+1)R^2}{2}\sum_{k=0}^{p-1}e^{\frac{(p+k)(p-k-1)R}{2}}\right\}>1,
\end{align*}
which is a contradiction to \eqref{cnd}. Therefore, for some $r>0$, $\mathscr{F}_{\lambda}(\mathcal{E}_{r})\subset \mathcal{E}_{r}.$
\vskip 0.1in
\noindent\textbf{Step (2): } Next, we verify that the operator \emph{$ \mathscr{F}_{\lambda}$ is compact.} To prove this claim, we use the infinite-dimensional version of the Ascoli-Arzela theorem (see, Theorem 3.7, Chapter 2, \cite{JYONG}). According to this theorem, it is enough to verify the following:
\begin{itemize}
\item [(i)] the image of $\mathcal{E}_r$ under $\mathscr{F}_{\lambda}$ is uniformly bounded (which is proved in Step (1)),
\item [(ii)] the image of $\mathcal{E}_r$ under $\mathscr{F}_{\lambda}$ is equicontinuous,
\item [(iii)] for any $t\in J$, the set $\mathcal{V}(t)=\{(\mathscr{F}_\lambda x)(t):x\in \mathcal{E}_r\}$ is relatively compact.
\end{itemize}
First, we show that the map $\mathscr{F}_{\lambda}$ is equicontinuous, that is, the image of $\mathcal{E}_r$ under $\mathscr{F}_{\lambda}$ is equicontinuous. For $t_1,t_2\in[0,\tau_1]$ such that $t_1<t_2$ and $x\in\mathcal{E}_r$, we evaluate the following:
\begin{align}\label{4.31}
&\left\|(\mathscr{F}_{\lambda}x)(t_2)-(\mathscr{F}_{\lambda}x)(t_1)\right\|_{\mathbb{X}}\nonumber\\&\le\left\|\left[\mathcal{C}_{\gamma}(t_2)-\mathcal{C}_{\gamma}(t_1)\right]\psi(0)\right\|_{\mathbb{X}}+\left\|\left[\mathcal{T}_{\gamma}(t_2)-\mathcal{T}_{\gamma}(t_1)\right]\eta\right\|_{\mathbb{X}}\nonumber\\&\quad+\left\|\int_{t_1}^{t_2}(t_2-s)^{\gamma-1}\mathcal{S}_{\gamma}(t_2-s)f(s,\tilde{x}_{\varrho(s,\tilde{x}_s)})\mathrm{d}s\right\|_{\mathbb{X}}\nonumber\\&\quad+\left\|\int_{t_1}^{t_2}(t_2-s)^{\gamma-1}\mathcal{S}_{\gamma}(t_2-s)\mathrm{B}u^{\gamma}_{\lambda}(s)\mathrm{d}s\right\|_{\mathbb{X}}\nonumber\\&\quad+\left\|\int_{0}^{t_1}\left[(t_2-s)^{\gamma-1}-(t_1-s)^{\gamma-1}\right]\mathcal{S}_{\gamma}(t_2-s)\mathrm{B}u^{\gamma}_{\lambda}(s)\mathrm{d}s\right\|_{\mathbb{X}}\nonumber\\&\quad+\left\|\int_{0}^{t_1}(t_1-s)^{\gamma-1}\left[\mathcal{S}_{\gamma}(t_2-s)-\mathcal{S}_{\gamma}(t_1-s)\right]\mathrm{B}u^{\gamma}_{\lambda}(s)\mathrm{d}s\right\|_{\mathbb{X}}\nonumber\\&\quad+\left\|\int_{0}^{t_1}\left[(t_2-s)^{\gamma-1}-(t_1-s)^{\gamma-1}\right]\mathcal{S}_{\gamma}(t_2-s)f(s,\tilde{x}_{\varrho(s,\tilde{x}_s)})\mathrm{d}s\right\|_{\mathbb{X}}\nonumber\\&\quad+\left\|\int_{0}^{t_1}(t_1-s)^{\gamma-1}\left[\mathcal{S}_{\gamma}(t_2-s)-\mathcal{S}_{\gamma}(t_1-s)\right]f(s,\tilde{x}_{\varrho(s,\tilde{x}_s)})\mathrm{d}s\right\|_{\mathbb{X}}\nonumber\\&\leq\left\|\mathcal{C}_{\gamma}(t_2)\psi(0)-\mathcal{C}_{\gamma}(t_1)\psi(0)\right\|_{\mathbb{X}}+\left\|\mathcal{T}_{\gamma}(t_2)-\mathcal{T}_{\gamma}(t_1)\right\|_{\mathcal{L}(\mathbb{X})}\left\|\eta\right\|_{\mathbb{X}}\nonumber\\&\quad+\frac{M}{\Gamma(2\gamma)}\int_{t_1}^{t_2}(t_2-s)^{2\gamma-1}\phi_{r'}(s)\mathrm{d}s+\frac{N_0}{\lambda}\left(\frac{M\tilde{M}}{\Gamma(2\gamma)}\right)^{2}\int_{t_1}^{t_2}(t_2-s)^{2\gamma-1}(\tau_1-s)^{\gamma}\mathrm{d}s\nonumber\\&\quad+\frac{N_0}{\lambda}\left(\frac{M\tilde{M}}{\Gamma(2\gamma)}\right)^{2}\int_{0}^{t_1}\left|(t_2-s)^{\gamma-1}-(t_1-s)^{\gamma-1}\right|(t_2-s)^{\gamma-1}(\tau_1-s)^{\gamma}\mathrm{d}s\nonumber\\&\quad+\frac{M\tilde{M}^{2}N_{0}}{\lambda\Gamma(2\gamma)}\int_{0}^{t_1}(t_1-s)^{\gamma-1}\left\|\mathcal{S}_{\gamma}(t_2-s)-\mathcal{S}_{\gamma}(t_1-s)\right\|_{\mathcal{L}(\mathbb{X})}(\tau_1-s)^{\gamma}\mathrm{d}s\nonumber\\&\quad+\frac{M}{\Gamma(2\gamma)}\int_{0}^{t_1}\left|(t_2-s)^{\gamma-1}-(t_1-s)^{\gamma-1}\right|(t_2-s)^{\gamma}\phi_{r'}(s)\mathrm{d}s\nonumber\\&\quad+\int_{0}^{t_1}(t_1-s)^{\gamma-1}\left\|\mathcal{S}_{\gamma}(t_2-s)-\mathcal{S}_{\gamma}(t_1-s)\right\|_{\mathcal{L}(\mathbb{X})}\phi_{r'}(s)\mathrm{d}s\nonumber\\&\leq\left\|\mathcal{C}_{\gamma}(t_2)\psi(0)-\mathcal{C}_{\gamma}(t_1)\psi(0)\right\|_{\mathbb{X}}+\left\|\mathcal{T}_{\gamma}(t_2)-\mathcal{T}_{\gamma}(t_1)\right\|_{\mathcal{L}(\mathbb{X})}\left\|\eta\right\|_{\mathbb{X}}\nonumber\\&\quad+\frac{M}{\Gamma(2\gamma)}\frac{(t_2-t_1)^{2\gamma-\delta}}{c^{1-\delta}}\left(\int_{t_1}^{t_2}(\phi_{r'}(s))^{\frac{1}{\delta}}\mathrm{d}s\right)^{\delta}\nonumber\\&\quad+\frac{N_{0}}{\lambda}\left(\frac{M\tilde{M}}{\Gamma(2\gamma)}\right)^{2}\int_{t_1}^{t_2}(t_2-s)^{2\gamma-1}(\tau_1-s)^{\gamma}\mathrm{d}s\nonumber\\&\quad+\frac{N_{0}}{\lambda}\left(\frac{M\tilde{M}}{\Gamma(2\gamma)}\right)^{2}\int_{0}^{t_1}\left|(t_2-s)^{\gamma-1}-(t_1-s)^{\gamma-1}\right|(t_2-s)^{\gamma-1}(\tau_1-s)^{\gamma}\mathrm{d}s\nonumber\\&\quad+\frac{M\tilde{M}^{2}N_{0}}{\lambda\Gamma(2\gamma)}\sup_{s\in[0,t_1]}\left\|\mathcal{S}_{\gamma}(t_2-s)-\mathcal{S}_{\gamma}(t_1-s)\right\|_{\mathcal{L}(\mathbb{X})}\int_{0}^{t_1}(t_1-s)^{\gamma-1}(\tau_1-s)^{\gamma}\mathrm{d}s\nonumber\\&\quad+\frac{M}{\Gamma(2\gamma)}\int_{0}^{t_1}\left|(t_2-s)^{\gamma-1}-(t_1-s)^{\gamma-1}\right|(t_2-s)^{\gamma}\phi_{r'}(s)\mathrm{d}s\nonumber\\&\quad+\sup_{s\in[0,t_1]}\left\|\mathcal{S}_{\gamma}(t_2-s)-\mathcal{S}_{\gamma}(t_1-s)\right\|_{\mathcal{L}(\mathbb{X})}\int_{0}^{t_1}(t_1-s)^{\gamma-1}\phi_{r'}(s)\mathrm{d}s.
\end{align}
Similarly for $t_1,t_2\in(s_j,\tau_{j+1}], \ j=1,\ldots,p$ with $t_1<t_2$ and $x\in\mathcal{E}_r$, one can compute
\begin{align}\label{4.32}
&\left\|(\mathscr{F}_{\lambda}x)(t_2)-(\mathscr{F}_{\lambda}x)(t_1)\right\|_{\mathbb{X}}\nonumber\\&\le\left\|\mathcal{C}_{\gamma}(t_2-s_j)h(s_j,x(\tau_j^-))-\mathcal{C}_{\gamma}(t_1-s_j)h(s_j,x(\tau_j^-))\right\|_{\mathbb{X}}\nonumber\\&\quad+\left\|\mathcal{T}_{\gamma}(t_2-s_j)-\mathcal{T}_{\gamma}(t_1-s_j)\right\|_{\mathcal{L}(\mathbb{X})}\vartheta_{j}+\frac{M}{\Gamma(2\gamma)}\frac{(t_2-t_1)^{2\gamma-\delta}}{c^{1-\delta}}\left(\int_{t_1}^{t_2}(\phi_{r'}(s))^{\frac{1}{\delta}}\mathrm{d}s\right)^{\delta}\nonumber\\&\quad+\frac{C_j}{\lambda}\left(\frac{M\tilde{M}}{\Gamma(2\gamma)}\right)^{2}\int_{t_1}^{t_2}(t_2-s)^{2\gamma-1}(\tau_{j+1}-s)^{\gamma}\mathrm{d}s\nonumber\\&\quad+\frac{M\tilde{M}}{\Gamma(2\gamma)}\int_{0}^{t_1}\left|(t_2-s)^{\gamma-1}-(t_1-s)^{\gamma-1}\right|(t_2-s)^{\gamma}\left\|u_{\lambda}^{\gamma}(s)\right\|_{\mathbb{U}}\mathrm{d}s\nonumber\\&\quad+\tilde{M}\sup_{s\in[0,t_1]}\left\|\mathcal{S}_{\gamma}(t_2-s)-\mathcal{S}_{\gamma}(t_1-s)\right\|_{\mathcal{L}(\mathbb{X})}\int_{0}^{t_1}(t_1-s)^{\gamma-1}\left\|u_{\lambda}^{\gamma}(s)\right\|_{\mathbb{U}}\mathrm{d}s\nonumber\\&\quad+\frac{M}{\Gamma(2\gamma)}\int_{0}^{t_1}\left|(t_2-s)^{\gamma-1}-(t_1-s)^{\gamma-1}\right|(t_2-s)^{\gamma}\phi_{r'}(s)\mathrm{d}s\nonumber\\&\quad+\sup_{s\in[0,t_1]}\left\|\mathcal{S}_{\gamma}(t_2-s)-\mathcal{S}_{\gamma}(t_1-s)\right\|_{\mathcal{L}(\mathbb{X})}\int_{0}^{t_1}(t_1-s)^{\gamma-1}\phi_{r'}(s)\mathrm{d}s\nonumber\\&\leq\left\|\mathcal{C}_{\gamma}(t_2-s_j)h(s_j,x(\tau_j^-))-\mathcal{C}_{\gamma}(t_1-s_j)h(s_j,x(\tau_j^-))\right\|_{\mathbb{X}}\nonumber\\&\quad+\left\|\mathcal{T}_{\gamma}(t_2-s_j)-\mathcal{T}_{\gamma}(t_1-s_j)\right\|_{\mathcal{L}(\mathbb{X})}\vartheta_{j}+\frac{M}{\Gamma(2\gamma)}\frac{(t_2-t_1)^{2\gamma-\delta}}{c^{1-\delta}}\left(\int_{t_1}^{t_2}(\phi_{r'}(s))^{\frac{1}{\delta}}\mathrm{d}s\right)^{\delta}\nonumber\\&\quad+\frac{C_j}{\lambda}\left(\frac{M\tilde{M}}{\Gamma(2\gamma)}\right)^{2}\int_{t_1}^{t_2}(t_2-s)^{2\gamma-1}(\tau_{j+1}-s)^{\gamma}\mathrm{d}s\nonumber\\&\quad+\sum_{k=0}^{j-1}\frac{C_k}{\lambda}\left(\frac{M\tilde{M}}{\Gamma(2\gamma)}\right)^{2}\int_{s_k}^{\tau_{k+1}}\left|(t_2-s)^{\gamma-1}-(t_1-s)^{\gamma-1}\right|(t_2-s)^{\gamma}(\tau_{k+1}-s)^{\gamma}\mathrm{d}s\nonumber\\&\quad+\frac{C_j}{\lambda}\left(\frac{M\tilde{M}}{\Gamma(2\gamma)}\right)^{2}\int_{s_j}^{s_1}\left|(t_2-s)^{\gamma-1}-t_1-s)^{\gamma-1}\right|(t_2-s)^{\gamma}(\tau_{j+1}-s)^{\gamma}\mathrm{d}s\nonumber\\&\quad+\sup_{s\in[0,s_1]}\left\|\mathcal{S}_{\gamma}(t_2-s)-\mathcal{S}_{\gamma}(t_1-s)\right\|_{\mathcal{L}(\mathbb{X})}\sum_{k=0}^{j-1}\frac{C_k}{\lambda}\frac{M\tilde{M}^{2}}{\Gamma(2\gamma)}\int_{s_k}^{\tau_{k+1}}(t_1-s)^{\gamma-1}(\tau_{k+1}-s)^{\gamma}\mathrm{d}s\nonumber\\&\quad+\sup_{s\in[0,s_1]}\left\|\mathcal{S}_{\gamma}(t_2-s)-\mathcal{S}_{\gamma}(t_1-s)\right\|_{\mathcal{L}(\mathbb{X})}\frac{C_j}{\lambda}\frac{M\tilde{M}^{2}}{\Gamma(2\gamma)}\int_{s_j}^{t_1}(t_1-s)^{\gamma-1}(t_{j+1}-s)^{\gamma}\mathrm{d}s\nonumber\\&\quad+\frac{M}{\Gamma(2\gamma)}\int_{0}^{t_1}\left|(t_2-s)^{\gamma-1}-(t_1-s)^{\gamma-1}\right|(t_2-s)^{\gamma}\phi_{r'}(s)\mathrm{d}s\nonumber\\&\quad+\sup_{s\in[0,t_1]}\left\|\mathcal{S}_{\gamma}(t_2-s)-\mathcal{S}_{\gamma}(t_1-s)\right\|_{\mathcal{L}(\mathbb{X})}\int_{0}^{t_1}(t_1-s)^{\gamma-1}\phi_{r'}(s)\mathrm{d}s.
\end{align}
Moreover, for $t_1,t_2\in(\tau_j, s_j], \ j=1,\ldots,p$ with $t_1<t_2$ and $x\in\mathcal{E}_r$, we have
\begin{align}\label{4.33}
\left\|(\mathscr{F}_{\lambda}x)(t_2)-(\mathscr{F}_{\lambda}x)(t_1)\right\|_{\mathbb{X}}&\le\left\|h_j(t_2,\tilde{x}(\tau_j^-))-h_j(t_1,\tilde{x}(\tau_j^-))\right\|_{\mathbb{X}}.
\end{align}
From the facts that the operator $\mathcal{C}_{\gamma}(t)x,\ x\in\mathbb{X}$ is uniformly continuous for $t\in J$, the operators $\mathcal{T}_{\gamma}(t)$ and $\mathcal{S}_{\gamma}(t)$ are uniformly continuous for all $t\in J$ and the impulses $h_{j}(\cdot, x)$ are continuous for $j=1,\ldots,p$ and each $x\in\mathbb{X}$, we conclude that the right hand side of the expressions \eqref{4.31}, \eqref{4.32} and \eqref{4.33} converge to zero as $|t_2-t_1|\rightarrow 0$. Hence, the image of $\mathcal{E}_r$ under $\mathscr{F}_\lambda$ is equicontinuous.
The set $\mathcal{V}(t)=\{(\mathscr{F}_\lambda x)(t):x\in \mathcal{E}_r\},$ for each $t\in J$, is relatively compact in $\mathcal{E}_r$ follows from the facts that the operator $\mathcal{S}_{\gamma}(t)$ is compact for $t\ge 0$, the impulses $h_{j}(t,\cdot),\ h_{j}'(t,\cdot)$, for $t\in[\tau_j,s_j], j=1,\ldots,p,$ are completely continuous and also the operator $(\mathrm{Q}f)(\cdot) =\int_{0}^{\cdot}(\cdot-s)^{\gamma-1}\mathcal{S}_{\gamma}(\cdot-s)f(s)\mathrm{d}s$ is compact (see Lemma \ref{lem2.12}). Hence, for each $t\in J$, the set $\mathcal{V} (t) = \{(\mathscr{F}_{\lambda}x)(t) : x\in\mathcal{E}_r\},$ is relatively compact.
Thus, by applying the Arzela-Ascoli theorem, we obtain that the operator $\mathscr{F}_{\lambda}$ is compact.
\vskip 0.1in
\noindent\textbf{Step (3): } In the final step, we prove that the operator \emph{$ \mathscr{F}_{\lambda}$ is continuous}. In order to prove this, we consider a sequence $\{{x}^n\}^\infty_{n=1}\subseteq \mathcal{E}_r$ such that ${x}^n\rightarrow {x}\mbox{ in }{\mathcal{E}_r},$ that is,
$$\lim\limits_{n\rightarrow \infty}\left\|x^n-x\right\|_{\mathrm{PC}(J;\mathbb{X})}=0.$$
From Lemma \ref{lema2.1}, we have
\begin{align*}
\left\|\tilde{x_{s}^n}- \tilde{x_{s}}\right\|_{\mathfrak{B}}&\leq K_{2}\sup\limits_{\theta\in J}\left\|\tilde{x^{n}}(\theta)-\tilde{x}(\theta)\right\|_{\mathbb{X}}=K_{2}\left\|x^{n}-x\right\|_{\mathrm{PC}(J;\mathbb{X})}\rightarrow 0 \ \mbox{ as } \ n\rightarrow\infty,
\end{align*}
for all $s\in\mathfrak{R}(\varrho^{-})\cup J$. Since $\varrho(s, \tilde{x_s^k})\in\mathfrak{R}(\varrho^{-})\cup J,$ for all $k\in\mathbb{N}$, then we conclude that
\begin{align*}
\left\|\tilde{x^n}_{\varrho(s,\tilde{x_s^k})}-\tilde{x}_{\varrho(s,\tilde{x_s^k})}\right\|_{\mathfrak{B}}\rightarrow 0 \ \mbox{ as } \ n\rightarrow\infty, \ \mbox{ for all }\ s\in J\ \mbox{ and }\ k\in\mathbb{N}.
\end{align*}
In particular, we choose $k=n$ and use the above convergence together with Assumption \ref{as2.1} \textit{(H1)} to obtain
\begin{align}\label{4.25}
& \left\|f(s,\tilde{x^n}_{\varrho(s,\tilde{x_s^n})})-f(s,\tilde{x}_{\varrho(s,\tilde{x_s})})\right\|_{\mathbb{X}}\nonumber\\&\leq\left\|f(s,\tilde{x^n}_{\varrho(s,\tilde{x_s^n})})-f(s,\tilde{x}_{\varrho(s,\tilde{x_s^n})})\right\|_{\mathbb{X}}+\left\|f(s, \tilde{x}_{\varrho(s,\tilde{x_s^n})})-f(s,\tilde{x}_{\varrho(s,\tilde{x_s})})\right\|_{\mathbb{X}}\nonumber\\&\to 0\ \mbox{ as }\ n\to\infty, \mbox{ uniformly for } \ s\in J.
\end{align}
Using the convergence \eqref{4.25} and the dominated convergence theorem (DCT), we conclude
\begin{align}\label{4.26}
&\left\|g_0(x^n(\cdot))-g_0(x(\cdot))\right\|_{\mathbb{X}}\nonumber\\&\le\left\|\int^{\tau_1}_{0}(\tau_1-s)^{\gamma-1}\mathcal{S}_{\gamma}(\tau_1-s)\left[f(s,\tilde{x^n}_{\varrho(s,\tilde{x^n_s})})-f(s,\tilde{x}_{\varrho(s,\tilde{x_s})})\right]\mathrm{d}s\right\|_{\mathbb{X}}\nonumber\\&\le\frac{M}{\Gamma(2\gamma)}\int^{\tau_1}_{0}(\tau_1-s)^{2\gamma-1}\left\|f(s,\tilde{x^n}_{\varrho(s,\tilde{x^n_s})})-f(s,\tilde{x}_{\varrho(s,\tilde{x_s})})\right\|_{\mathbb{X}}\mathrm{d}s\nonumber\\&\to 0\ \mbox{ as }\ n\to\infty.
\end{align}
Invoking the above convergences and the relation \eqref{2.5}, we estimate
\begin{align*}
& \left\|\mathcal{R}(\lambda,\Phi_{0}^{\tau_{1}})g_0(x^{n}(\cdot))-\mathcal{R}(\lambda,\Phi_{0}^{\tau_{1}})g_0(x(\cdot))\right\|_{\mathbb{X}}\nonumber\\&=\frac{1}{\lambda}\left\|\lambda\mathcal{R}(\lambda,\Phi_{0}^{\tau_1})\left(g_0(x^{n}(\cdot))-g_0(x(\cdot))\right)\right\|_{\mathbb{X}}\nonumber\\&\leq\frac{1}{\lambda}\left\|g_0(x^{n}(\cdot))-g_0(x(\cdot))\right\|_{\mathbb{X}}\nonumber\\&\to 0 \ \mbox{ as } \ n \to \infty.
\end{align*}
By using the demicontinuous property of the duality mapping $\mathscr{J}:\mathbb{X}\to\mathbb{X}^{*}$, one can easily obtain
\begin{align}\label{4.2}
\mathscr{J}\left[\mathcal{R}(\lambda,\Phi_{0}^{\tau_1})g_0(x^{n}(\cdot))\right]\xrightharpoonup{w}\mathscr{J}\left[\mathcal{R}(\lambda,\Phi_{0}^{\tau_1})g_0(x(\cdot))\right] \ \mbox{ as } \ n\to\infty \ \mbox{ in }\ \mathbb{X}^{*}.
\end{align}
From Lemma \ref{lem2.11}, we infer that the operator $\mathcal{S}_{\gamma}(t)$ is compact for $t>0$. Consequently, the operator $\mathcal{S}_{\gamma}(t)^*$ is also compact for $t>0$. Thus, by using this compactness and the weak convergence \eqref{4.2}, one can easily conclude that
\begin{align}\label{4.1}
& \left\|u^{n,\gamma}_{0,\lambda}(t)-u_{0,\lambda}^{\gamma}(t)\right||_{\mathbb{U}}\nonumber\\&\le\left\|\mathrm{B}^*\mathcal{S}_{\gamma}(\tau_{1}-t)^*\left(\mathscr{J}\left[\mathcal{R}(\lambda,\Phi_{0}^{\tau_{1}})g_0(x^{n}(\cdot))\right]-\mathscr{J}\left[\mathcal{R}(\lambda,\Phi_{0}^{\tau_{1}})g_0(x(\cdot))\right]\right)\right\|_{\mathbb{U}}\nonumber\\&\le\tilde{M}\left\|\mathcal{S}_{\gamma}(\tau_{1}-t)^*\left(\mathscr{J}\left[\mathcal{R}(\lambda,\Phi_{0}^{\tau_{1}})g_0(x^{n}(\cdot))\right]-\mathscr{J}\left[\mathcal{R}(\lambda,\Phi_{0}^{\tau_{1}})g_0(x(\cdot))\right]\right)\right\|_{\mathbb{X}}\nonumber\\&\to 0\ \mbox{ as }\ n\to\infty, \ \mbox{uniformly for}\ t\in[0,\tau_{1}].
\end{align}
Similarly, for $j=1$, we evaluate
\begin{align}\label{4.27}
&\left\|g_1(x^n(\cdot))-g_1(x(\cdot))\right\|_{\mathbb{X}}\nonumber\\&\le\left\|\mathcal{C}_{\gamma}(\tau_{2}-s_1)\left[h_1(s_1,\tilde{x^n}(\tau_1^-))-h_1(s_1,\tilde{x}(\tau_1^-))\right]\right\|_{\mathbb{X}}\nonumber\\&\quad+\left\|\mathcal{T}_{\gamma}(\tau_{2}-s_1)\left[h'_1(s_1,\tilde{x^n}(\tau_1^-))-h'_1(s_1,\tilde{x}(\tau_1^-))\right]\right\|_{\mathbb{X}}\nonumber\\&\quad+\left\|\int_{0}^{s_{1}}(s_{1}-s)^{\gamma-1}\mathcal{S}_{\gamma}(s_{1}-s)\left[f(s,\tilde{x^n}_{\varrho(s,\tilde{x^n_s})})-f(s,\tilde{x}_{\varrho(s,\tilde{x}_s)})\right]\mathrm{d}s\right\|_{\mathbb{X}}\nonumber\\&\quad+\left\|\int_{0}^{\tau_{2}}(\tau_{2}-s)^{\gamma-1}\mathcal{S}_{\gamma}(\tau_{2}-s)\left[f(s,\tilde{x^n}_{\varrho(s,\tilde{x^n_s})})-f(s,\tilde{x}_{\varrho(s,\tilde{x}_s)})\right]\mathrm{d}s\right\|_{\mathbb{X}}\nonumber\\&\quad+\left\|\int_{0}^{s_1}(s_1-s)^{\gamma-1}\mathcal{S}_{\gamma}(s_1-s)\mathrm{B}\left[u^{n,\alpha}_{0,\lambda}(s)-u^{\alpha}_{0,\lambda}(s)\right]\mathrm{d}s\right\|_{\mathbb{X}}\nonumber\\&\quad+\left\|\int_{0}^{s_1}(\tau_2-s)^{\gamma-1}\mathcal{S}_{\gamma}(\tau_2-s)\mathrm{B}\left[u^{n,\alpha}_{0,\lambda}(s)-u^{\alpha}_{0,\lambda}(s)\right]\mathrm{d}s\right\|_{\mathbb{X}} \nonumber\\&\leq M\left\|h_1(s_1,\tilde{x^n}(\tau_1^-))-h_1(s_1,\tilde{x}(\tau_1^-))\right\|_{\mathbb{X}}\nonumber\\&\quad+M(\tau_2-s)\left\|h'_1(s_1,\tilde{x^n}(\tau_1^-))-h'_1(s_1,\tilde{x}(\tau_1^-))\right\|_{\mathbb{X}}\nonumber\\&\quad+\frac{M}{\Gamma(2\gamma)}\int_{0}^{s_{1}}(s_{1}-s)^{2\gamma-1}\left\|f(s,\tilde{x^n}_{\varrho(s,\tilde{x^n_s})})-f(s,\tilde{x}_{\varrho(s,\tilde{x}_s)})\right\|_{\mathbb{X}}\mathrm{d}s\nonumber\\&\quad+\frac{M}{\Gamma(2\gamma))}\int_{0}^{\tau_{2}}(\tau_{2}-s)^{2\gamma-1}\left\|f(s,\tilde{x^n}_{\varrho(s,\tilde{x^n_s})})-f(s,\tilde{x}_{\varrho(s,\tilde{x}_s)})\right\|_{\mathbb{X}}\mathrm{d}s\nonumber\\&\quad+\frac{M\tilde{M}}{\Gamma(2\gamma)}\int_{0}^{s_1}(s_1-s)^{2\gamma-1}\left\|u^{n,\gamma}_{0,\lambda}(s)-u^{\gamma}_{0,\lambda}(s)\right\|_{\mathbb{U}}\mathrm{d}s\nonumber\\&\quad+\frac{M\tilde{M}}{\Gamma(2\gamma)}\int_{0}^{s_1}(\tau_2-s)^{\alpha-1}\left\|u^{n,\alpha}_{0,\lambda}(s)-u^{\alpha}_{0,\lambda}(s)\right\|_{\mathbb{U}}\mathrm{d}s\nonumber\\&\to0\ \mbox{ as }\ n\to\infty,
\end{align}
where we have used the convergences \eqref{4.25}, \eqref{4.1}, Assumption \ref{as2.1} (\textit{H2}) and DCT. Moreover, similar to the convergence \eqref{4.1}, one can prove
\begin{align*}
&\left\|u^{n,\gamma}_{1,\lambda}(t)-u_{1,\lambda}^{\gamma}(t)\right||_{\mathbb{U}}\to 0\ \mbox{ as }\ n\to\infty, \ \mbox{uniformly for}\ t\in[s_1,\tau_{2}].
\end{align*}
Further, employing a similar analogy as above for $j=2,\ldots,p,$ one obtains
\begin{align*}
&\left\|u^{n,\gamma}_{j,\lambda}(t)-u_{j,\lambda}^{\gamma}(t)\right||_{\mathbb{U}}\to 0\ \mbox{ as }\ n\to\infty, \mbox{ uniformly for }\ t\in[s_j,\tau_{j+1}], \ j=2,\ldots,p.
\end{align*}
Hence, we have
\begin{align}\label{4.29}
\left\|u^{n,\alpha}_{\lambda}(t)-u_{\lambda}^{\alpha}(t)\right||_{\mathbb{U}}\to 0\ \mbox{ as }\ n\to\infty,\ \mbox{ uniformly for }\ t\in[s_j,\tau_{j+1}],\ j=0,1,\ldots,p.
\end{align}
Using the convergences \eqref{4.25}, \eqref{4.29} and DCT, we arrive at
\begin{align*}
& \left\|(\mathscr{F}_{\lambda}x^n)(t)-(\mathscr{F}_{\lambda}x)(t)\right\|_{\mathbb{X}}\nonumber\\&\leq\int_{0}^{t}(t-s)^{\gamma-1}\left\|\mathcal{S}_{\gamma}(t-s)\mathrm{B}\left[u^{n,\alpha}_{\lambda}(s)-u_{\lambda}^{\gamma}(s)\right]\right\|_{\mathbb{X}}\mathrm{d}s\nonumber\\&\quad+\int_{0}^{t}(t-s)^{\gamma-1}\left\|\mathcal{S}_{\gamma}(t-s)\left[f(s,\tilde{x^n}_{\varrho(s,\tilde{x^n_s})})-f(s,\tilde{x}_{\varrho(s,\tilde{x}_s)})\right]\right\|_{\mathbb{X}}\mathrm{d}s\nonumber\\&\le\frac{M\tilde{M}}{\Gamma(2\gamma)}\int_{0}^{t}(t-s)^{2\gamma-1}\left\|u^{n,\gamma}_{\lambda}(s)-u_{\lambda}^{\gamma}(s)\right\|_{\mathbb{U}}\mathrm{d}s\nonumber\\&\quad+\frac{M}{\Gamma(2\gamma))}\int_{0}^{t}(t-s)^{2\gamma-1}\left\|f(s,\tilde{x^n}_{\varrho(s,\tilde{x^n_s})})-f(s,\tilde{x}_{\varrho(s, \tilde{x}_s)})\right\|_{\mathbb{X}}\mathrm{d}s\nonumber\\&\to 0 \ \mbox{ as }\ n\to\infty, \ \mbox{ uniformly for }\ t\in[0,t_1].
\end{align*}
Similarly, for $t\in(s_j,\tau_{j+1}],\ j=1,\ldots,p$, we deduce that
\begin{align*}
& \left\|(\mathscr{F}_{\lambda}x^n)(t)-(\mathscr{F}_{\lambda}x)(t)\right\|_{\mathbb{X}}\nonumber\\&\le\left\|\mathcal{C}_{\gamma}(t-s_j)\left[h_j(s_j,\tilde{x^n}(\tau_j^-))-h_j(s_j,\tilde{x}(\tau_j^-))\right]\right\|_{\mathbb{X}}\nonumber\\&\quad+\left\|\mathcal{T}_{\gamma}(t-s_j)\left[h'_j(s_j,\tilde{x^n}(\tau_j^-))-h'_j(s_j,\tilde{x}(\tau_j^-))\right]\right\|_{\mathbb{X}}\nonumber\\&\quad+\int_{0}^{s_j}(s_j-s)^{\gamma-1}\left\|\mathcal{S}_{\gamma}(s_j-s)\left[f(s,\tilde{x^n}_{\varrho(s, \tilde{x^n_s})})-f(s,\tilde{x}_{\varrho(s, \tilde{x}_s)})\right]\right\|_{\mathbb{X}}\mathrm{d}s\nonumber\\&\quad+\int_{0}^{s_j}(s_j-s)^{\gamma-1}\left\|\mathcal{S}_{\gamma}(s_j-s)\mathrm{B}\left[u^{n,\gamma}_{\lambda}(s)-u_{\lambda}^{\gamma}(s)\right]\right\|_{\mathbb{X}}\mathrm{d}s\nonumber\\&\quad+\int_{0}^{t}(t-s)^{\gamma-1}\left\|\mathcal{S}_{\gamma}(t-s)\mathrm{B}\left[u^{n,\gamma}_{\lambda}(s)-u_{\lambda}^{\gamma}(s)\right]\right\|_{\mathbb{X}}\mathrm{d}s\nonumber\\&\quad+\int_{0}^{t}(t-s)^{\gamma-1}\left\|\mathcal{S}_{\gamma}(t-s)\left[f(s,\tilde{x^n}_{\varrho(s,\tilde{x^n_s})})-f(s,\tilde{x}_{\varrho(s,\tilde{x}_s)})\right]\right\|_{\mathbb{X}}\mathrm{d}s\nonumber\\&\le M\left\|h_j(s_j,\tilde{x^n}(\tau_j^-))-h_j(s_j,\tilde{x}(\tau_j^-))\right\|_{\mathbb{X}}\nonumber\\&\quad+M(t-s_j)\left\|h'_j(s_j,\tilde{x^n}(\tau_j^-))-h'_j(s_j,\tilde{x}(\tau_j^-))\right\|_{\mathbb{X}}\nonumber\\&\quad+\frac{M}{\Gamma(2\gamma)}\int_{0}^{s_j}(s_j-s)^{2\gamma-1}\left\|f(s,\tilde{x^n}_{\varrho(s,\tilde{x^n_s})})-f(s,\tilde{x}_{\varrho(s, \tilde{x}_s)})\right\|_{\mathbb{X}}\mathrm{d}s\nonumber\\&\quad+ \frac{M\tilde{M}}{\Gamma(2\gamma)}\int_{0}^{s_j}(s_j-s)^{2\gamma-1}\left\|u^{n,\gamma}_{\lambda}(s)-u_{\lambda}^{\gamma}(s)\right\|_{\mathbb{U}}\mathrm{d}s\nonumber\\&\quad+\frac{M\tilde{M}}{\Gamma(2\gamma)}\int_{0}^{t}(t-s)^{2\gamma-1}\left\|u^{n,\gamma}_{\lambda}(s)-u_{\lambda}^{\gamma}(s)\right\|_{\mathbb{U}}\mathrm{d}s\nonumber\\&\quad+\frac{M}{\Gamma(2\gamma)}\int_{0}^{t}(t-s)^{2\gamma-1}\left\|f(s,\tilde{x^n}_{\varrho(s,\tilde{x^n_s})})-f(s,\tilde{x}_{\varrho(s,\tilde{x}_s)})\right\|_{\mathbb{X}}\mathrm{d}s\nonumber\\&\to 0 \ \mbox{ as }\ n\to\infty, \ \mbox{uniformly for}\ t\in(s_j,\tau_{j+1}].
\end{align*}
Moreover, for $t\in(\tau_j, s_j],\ j=1,\ldots,p$, using Assumption \ref{as2.1} (\textit{H2}), we obtain
\begin{align*}
\left\|(\mathscr{F}_{\lambda}x^n)(t)-(\mathscr{F}_{\lambda}x)(t)\right\|_{\mathbb{X}}&\le\left\|h_j(t,\tilde{x^n}(\tau_j^-))-h_j(t,\tilde{x}(\tau_j^-))\right\|_{\mathbb{X}}\to 0 \ \mbox{ as }\ n\to\infty.
\end{align*}
Therefore, it follows that $\mathscr{F}_{\lambda}$ is continuous.
Hence, by the application of \emph{Schauder's fixed point theorem}, we conclude that the operator $\mathscr{F}_{\lambda}$ has a fixed point in $\mathcal{E}_{r}$, or the system \eqref{1.1} has a mild solution.
\end{proof}
In order to verify the approximate controllability of the system \eqref{1.1}, we replace the assumption (\textit{H2}) of the function $f(\cdot,\cdot)$ by the following assumption:
\begin{enumerate}\label{as}
\item [\textit{(H3)}] The function $ f: J \times \mathfrak{B} \rightarrow \mathbb{X} $ satisfies Assumption (\textit{H1})(i) and there exists a function $ \phi\in \mathrm{L}^{\frac{1}{\delta}}(J;\mathbb{R}^+)$ with $\delta\in[0,\gamma]$ such that $$ \|f(t,\psi)\|_{\mathbb{X}}\leq \phi(t),\ \text{ for all }\ (t,\psi) \in J \times \mathfrak{B}. $$
\end{enumerate}
\begin{theorem}\label{thm4.4}
Let Assumptions (R1)-(R3), (H0)-(H1), (H3) and the condition \eqref{cnd} of Theorem \ref{thm4.3} be fulfilled. Then the system \eqref{1.1} is approximately controllable.
\end{theorem}
\begin{proof}
Theorem \ref{thm4.3}, ensures that for every $\lambda>0$ and $\xi_j\in \mathbb{X},$ for $j=0,1,\ldots p$, the system \eqref{1.1} has a mild solution, say, $x^{\lambda}\in\mathcal{E}_r$ with the control $u_{\lambda}^{\gamma}$ same as given in \eqref{C}. Then, it is immediate that
\begin{equation}\label{M}
x^{\lambda}(t)=\begin{dcases}
\mathcal{C}_{\gamma}(t)\psi(0)+\mathcal{T}_{\gamma}(t)\eta+\int_{0}^{t}(t-s)^{\gamma-1}\mathcal{S}_{\gamma}(t-s)\left[\mathrm{B}u_{\lambda}^{\gamma}(s)+f(s,\tilde{x^\lambda}_{\varrho(s, \tilde{x^\lambda_s})})\right]\mathrm{d}s,\\ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad t\in[0, \tau_1],\\
h_j(t, \tilde{x^\lambda}(\tau_j^-)),\qquad\qquad\qquad\qquad\qquad\qquad\qquad t\in(\tau_j, s_j],\ j=1,\ldots,p,\\
\mathcal{C}_{\gamma}(t-s_j)h_j(s_j, \tilde{x^\lambda}(\tau_j^-))+\mathcal{T}_{\gamma}(t-s_j)h'_j(s_j, \tilde{x^\lambda}(\tau_j^-))\\ \quad -\int_{0}^{s_j}(s_j-s)^{\gamma-1}\mathcal{S}_{\gamma}(s_j-s)\left[\mathrm{B}u_{\lambda}^{\gamma}(s)+f(s,\tilde{x^\lambda}_{\varrho(s, \tilde{x^\lambda_s})})\right]\mathrm{d}s \\\qquad+\int_{0}^{t}(t-s)^{\gamma-1}\mathcal{S}_{\gamma}(t-s)\left[\mathrm{B}u_{\lambda}^{\gamma}(s)+f(s,\tilde{x^\lambda}_{\varrho(s, \tilde{x^\lambda_s})})\right]\mathrm{d}s,\\ \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad t\in(s_j,\tau_{j+1}],\ j=1,\ldots,p.
\end{dcases}
\end{equation}
Next, we estimate
\begin{align}\label{4.35}
x^{\lambda}(T)&=\mathcal{C}_{\gamma}(T-s_p)h_p(s_p,\tilde{x^\lambda}(\tau_p^-))+\mathcal{T}_{\gamma}(T-s_p)h_{p}(s_p,\tilde{x^\lambda}(\tau_p^-))\nonumber\\&\quad-\int_{0}^{s_p}(s_p-s)^{\gamma-1}\mathcal{S}_{\gamma}(s_p-s)\left[\mathrm{B}u^{\gamma}_{\lambda}(s)+f(s,\tilde{x^\lambda}_{\varrho(s,\tilde{x^\lambda_s})})\right]\mathrm{d}s\nonumber\\&\quad+\int_{0}^{T}(T-s)^{\gamma-1}\mathcal{S}_{\gamma}(T-s)\left[\mathrm{B}u^{\gamma}_{\lambda}(s)+f(s,\tilde{x^\lambda}_{\varrho(s,\tilde{x^\lambda_s})})\right]\mathrm{d}s\nonumber\\&=\mathcal{C}_{\gamma}(T-s_p)h_p(s_p,\tilde{x^\lambda}(\tau_p^-))+\mathcal{T}_{\gamma}(T-s_p)h_{p}(s_p,\tilde{x^\lambda}(\tau_p^-))\nonumber\\&\quad-\int_{0}^{s_p}(s_p-s)^{\gamma-1}\mathcal{S}_{\gamma}(s_p-s)\left[\mathrm{B}u^{\gamma}_{\lambda}(s)+f(s,\tilde{x^\lambda}_{\varrho(s,\tilde{x^\lambda_s})})\right]\mathrm{d}s\nonumber\\&\quad+\int_{0}^{T}(T-s)^{\gamma-1}\mathcal{S}_{\gamma}(T-s)f(s,\tilde{x^\lambda}_{\varrho(s,\tilde{x^\lambda_s})})+\int_{0}^{s_p}(T-s)^{\gamma-1}\mathcal{S}_{\gamma}(T-s)\mathrm{B}u^{\gamma}_{\lambda}(s)\mathrm{d}s\nonumber\\&\quad+\int_{s_p}^{T}(T-s)^{\gamma-1}\mathcal{S}_{\gamma}(T-s)\mathrm{B}\mathrm{B}^*\mathcal{S}_{\gamma}(T-s)^*\mathscr{J}\left[\mathcal{R}(\lambda,\Phi_{s_p}^{T})g_p(x^\lambda(\cdot))\right]\mathrm{d}s\nonumber\\&=\mathcal{C}_{\gamma}(T-s_p)h_p(s_p,\tilde{x^\lambda}(\tau_p^-))+\mathcal{T}_{\gamma}(T-s_p)h_{p}(s_p,\tilde{x^\lambda}(\tau_p^-))\nonumber\\&\quad-\int_{0}^{s_p}(s_p-s)^{\gamma-1}\mathcal{S}_{\gamma}(s_p-s)\left[\mathrm{B}u^{\gamma}_{\lambda}(s)+f(s,\tilde{x^\lambda}_{\varrho(s,\tilde{x^\lambda_s})})\right]\mathrm{d}s\nonumber\\&\quad+\int_{0}^{T}(T-s)^{\gamma-1}\mathcal{S}_{\gamma}(T-s)f(s,\tilde{x^\lambda}_{\varrho(s,\tilde{x^\lambda_s})})+\int_{0}^{s_p}(T-s)^{\gamma-1}\mathcal{S}_{\gamma}(T-s)\mathrm{B}u^{\gamma}_{\lambda}(s)\mathrm{d}s\nonumber\\&\quad+\Phi_{s_p}^{T}\mathscr{J}\left[\mathcal{R}(\lambda,\Phi_{s_p}^{T})g_p(x^\lambda(\cdot))\right]\nonumber\\&=\xi_p-\lambda\mathcal{R}(\lambda,\Phi_{s_p}^{T})g_p(x^\lambda(\cdot)).
\end{align}
The sequence $x^{\lambda}(t)$ is bounded in $\mathbb{X},$ for each $t\in J$, follows from the argument $ x^{\lambda}\in\mathcal{E}_r $. Then by the Banach-Alaoglu theorem, we can find a subsequence, still labeled as $ x^{\lambda}$ such that
\begin{align*}
x^\lambda(t)\xrightharpoonup{w}z(t) \ \mbox{ in }\ \mathbb{X} \ \ \mbox{ as }\ \ \lambda\to0^+,\ t\in J.
\end{align*}
Using the condition (\textit{H2}) of Assumption \ref{as2.1}, we have the following convergence:
\begin{align}
h_p(t,x^\lambda(\tau_p^-))\to h_p(t,z(\tau_p^-)) \ \mbox{ in }\ \mathbb{X} \ \mbox{ as }\ \lambda\to0^+, \ \mbox{ for all }\ t\in J,\label{4.19}\\
h'_p(t,x^\lambda(\tau_p^-))\to h'_p(t,z(\tau_p^-)) \ \mbox{ in }\ \mathbb{X} \ \mbox{ as }\ \lambda\to0^+, \ \mbox{ for all }\ t\in J\label{4.20}.
\end{align}
Moreover, by using Assumption \textit{(H3)}, we obtain
\begin{align}
\int_{t_1}^{t_2}\left\|f(s,\tilde{x^{\lambda}}_{\varrho(s,\tilde{x^{\lambda}_s})})\right\|_{\mathbb{X}}^{2}\mathrm{d}s&\le \int_{t_1}^{t_2}\phi^2(s)\mathrm{d} s\leq \left(\int_{t_1}^{t_2}\phi^{\frac{1}{\delta}}(s)\mathrm{d}s\right)^{2\delta}(t_1-t_2)^{1-2\delta}<+\infty, \nonumber
\end{align}
for any $ t_1,t_2\in[0,T]$ with $t_1<t_2$. The above estimate guarantees that the sequence $ \{f(\cdot, \tilde{x^{\lambda}}_{\varrho(s,\tilde{x^{\lambda}_s})}): \lambda >0\}$ in $ \mathrm{L}^2([t_1,t_2]; \mathbb{X})$ is bounded. Once again by applying the Banach-Alaoglu theorem, we can extract a subsequence still denoted by$ \{f(\cdot, \tilde{x^{\lambda}}_{\rho(s,\tilde{x^{\lambda}}_s)}): \lambda > 0 \}$ such that
\begin{align}\label{4.36}
f(\cdot, \tilde{x^{\lambda}}_{\rho(s,\tilde{x^{\lambda}}_s)})\xrightharpoonup{w}f(\cdot) \ \mbox{ in }\ \mathrm{L}^2([t_1,t_2];\mathbb{X}) \ \mbox{ as }\ \lambda\to0^+.
\end{align}
Next, we compute
\begin{align*}
&\int_{0}^{s_p}\left\|u^{\gamma}_{\lambda}(s)\right\|^2_{\mathbb{U}}\mathrm{d}s\nonumber\\&=\int_{0}^{\tau_1}\left\|u^{\gamma}_{\lambda}(s)\right\|^2_{\mathbb{U}}\mathrm{d}s+\int_{\tau_1}^{s_1}\left\|u^{\gamma}_{\lambda}(s)\right\|^2_{\mathbb{U}}\mathrm{d}s+\int_{s_1}^{\tau_2}\left\|u^{\gamma}_{\lambda}(s)\right\|^2_{\mathbb{U}}\mathrm{d}s+\cdots+\int_{\tau_p}^{s_p}\left\|u^{\gamma}_{\lambda}(s)\right\|^2_{\mathbb{U}}\mathrm{d}s\nonumber\\&=\int_{0}^{\tau_1}\left\|u^{\gamma}_{0,\lambda}(s)\right\|^2_{\mathbb{U}}\mathrm{d}s+\int_{s_1}^{\tau_2}\left\|u^{\gamma}_{1,\lambda}(s)\right\|^2_{\mathbb{U}}\mathrm{d}s+\cdots+\int_{s_{p-1}}^{\tau_p}\left\|u^{\gamma}_{p-1,\lambda}(s)\right\|^2_{\mathbb{U}}\mathrm{d}s\nonumber\\&\le\left(\frac{M\tilde{M}}{\lambda\Gamma(2\gamma)}\right)^{2}\frac{T^{2\gamma+1}}{2\gamma+1}\sum_{j=0}^{p-1}C_j^2=C,
\end{align*}
where $C_j,$ for $j=1,\ldots,p-1$ are the same as given in \eqref{4.5} and $C_0=N_0$ given in \eqref{4.4}. Moreover, the above fact implies that the sequence $\{u^\gamma_{\lambda}(\cdot): \lambda >0\}$ in $ \mathrm{L}^2([0,s_p]; \mathbb{U})$ is bounded. Further, by an application of the Banach-Alaoglu theorem, we obtain a subsequence, still denoted by $\{u^\gamma_{\lambda}(\cdot): \lambda >0\}$ such that
\begin{align}\label{4}
u^\gamma_{\lambda}(\cdot)\xrightharpoonup{w}u^\gamma(\cdot) \ \mbox{ in }\ \mathrm{L}^2([0,s_p];\mathbb{U}) \ \mbox{ as }\ \lambda\to0^+.
\end{align}
Next, we compute
\begin{align}\label{4.37}
& \left\|g_{p}(x^{\lambda}(\cdot))-\omega\right\|_{\mathbb{X}}\nonumber\\&\le\left\|\mathcal{C}_{\gamma}(T-s_p)(h_p(s_p,\tilde{x^\lambda}(\tau_p^-))-h_p(s_p,z(\tau_p^-)))\right\|_{\mathbb{X}}\nonumber\\&\quad+\left\|\mathcal{T}_{\gamma}(T-s_p)(h'_p(s_p,\tilde{x^\lambda}(\tau_p^-))-h'_p(s_p,z(\tau_p^-)))\right\|_{\mathbb{X}}\nonumber\\&\quad+\left\|\int_{0}^{s_p}(s_p-s)^{\gamma-1}\mathcal{S}_{\gamma}(s_p-s)\mathrm{B}\left[u^\gamma_{\lambda}(s)-u^{\gamma}(s)\right]\mathrm{d}s\right\|_{\mathbb{X}}\nonumber\\&\quad+\left\|\int_{0}^{s_p}(s_p-s)^{\gamma-1}\mathcal{S}_{\gamma}(s_p-s)\left[f(s,\tilde{x^\lambda}_{\varrho(s,\tilde{x^\lambda_s})})-f(s)\right]\mathrm{d}s\right\|_{\mathbb{X}}\nonumber\\&\quad+\left\|\int_{0}^{s_p}(T-s)^{\gamma-1}\mathcal{S}_{\gamma}(T-s)\mathrm{B}\left[u^\gamma_{\lambda}(s)-u^{\gamma}(s)\right]\mathrm{d}s\right\|_{\mathbb{X}}\nonumber\\&\quad+\left\|\int_{0}^{T}(T-s)^{\gamma-1}\mathcal{S}_{\gamma}(T-s)\left[f(s,\tilde{x^\lambda}_{\varrho(s,\tilde{x^\lambda_s})})-f(s)\right]\mathrm{d}s\right\|_{\mathbb{X}}\nonumber\\&\le\left\|\mathcal{C}_{\gamma}(T-s_p)(h_p(s_p,x^\lambda(\tau_p^-))-h_p(s_p,z(\tau_p^-)))\right\|_{\mathbb{X}}\nonumber\\&\quad+\left\|\mathcal{T}_{\gamma}(T-s_p)(h'_p(s_p,x^\lambda(\tau_p^-))-h'_p(s_p,z(\tau_p^-)))\right\|_{\mathbb{X}}\nonumber\\&\quad+\left\|\int_{0}^{s_p}(s_p-s)^{\gamma-1}\mathcal{S}_{\gamma}(s_p-s)\mathrm{B}\left[u^\gamma_{\lambda}(s)-u^{\gamma}(s)\right]\mathrm{d}s\right\|_{\mathbb{X}}\nonumber\\&\quad+\left\|\int_{0}^{s_p}(s_p-s)^{\gamma-1}\mathcal{S}_{\gamma}(s_p-s)\left[f(s,\tilde{x^\lambda}_{\varrho(s,\tilde{x^\lambda_s})})-f(s)\right]\mathrm{d}s\right\|_{\mathbb{X}}\nonumber\\&\quad+\frac{T^{2\gamma-1}-(T-s_p)^{2\gamma-1}}{2\gamma-1}\left(\int_{0}^{s_p}\left\|\mathcal{S}_{\gamma}(T-s)\mathrm{B}\left[u^\gamma_{\lambda}(s)-u^{\gamma}(s)\right]\right\|^2_{\mathbb{X}}\mathrm{d}s\right)^{\frac{1}{2}}\nonumber\\&\quad+\left\|\int_{0}^{T}(T-s)^{\gamma-1}\mathcal{S}_{\gamma}(T-s)\left[f(s,\tilde{x^\lambda}_{\varrho(s,\tilde{x^\lambda_s})})-f(s)\right]\mathrm{d}s\right\|_{\mathbb{X}}\nonumber\\&\to 0\ \mbox{ as }\ \lambda\to0^+,
\end{align}
where
\begin{align*}
\omega &=\xi_p-\mathcal{C}_{\gamma}(T-s_p)h_p(s_p,z(\tau_p^-))-\mathcal{T}_{\gamma}(T-s_p)h'_p(s_p,z(\tau_p^-))\nonumber\\&\quad-\int_{0}^{T}(T-s)^{\gamma-1}\mathcal{S}_{\gamma}(T-s)f(s)\mathrm{d}s+\int_{0}^{s_p}(s_p-s)^{\gamma-1}\mathcal{S}_{\gamma}(s_p-s)\left[\mathrm{B}u^{\gamma}(s)+f(s)\right]\mathrm{d}s\nonumber\\&\quad-\int_{0}^{s_p}(T-s)^{\gamma-1}\mathcal{S}_{\gamma}(T-s)\mathrm{B}u^{\gamma}(s)\mathrm{d}s.
\end{align*}
In \eqref{4.37}, we have used the convergences \eqref{4.19}, \eqref{4.20}, \eqref{4.36}, \eqref{4}, DCT and the compactness of the operator $(\mathrm{Q}f)(\cdot) =\int_{0}^{\cdot}(\cdot-s)^{\gamma-1}\mathcal{S}_{\gamma}(\cdot-s)f(s)\mathrm{d}s:\mathrm{L}^2(J;\mathbb{X})\rightarrow \mathrm{C}(J;\mathbb{X}) $ (see Lemma \ref{lem2.12}).
Finally, by using the equality \eqref{4.35}, we estimate
\begin{align}
\left\|x^{\lambda}(T)-\xi_p\right\|_{\mathbb{X}}&\le\left\|\lambda\mathcal{R}(\lambda,\Phi_{s_p}^{T})g_p(x(\cdot))\right\|_{\mathbb{X}}\nonumber\\&\le\left\|\lambda\mathcal{R}(\lambda,\Phi_{s_p}^{T})(g_p(x(\cdot))-\omega)\right\|_{\mathbb{X}}+\left\|\lambda\mathcal{R}(\lambda,\Phi_{s_p}^{T})\omega\right\|_{\mathbb{X}}\nonumber\\&\le\left\|\lambda\mathcal{R}(\lambda,\Phi_{s_p}^{T})\right\|_{\mathcal{L}(\mathbb{X})}\left\|g_p(x(\cdot))-\omega\right\|_{\mathbb{X}}+\left\|\lambda\mathcal{R}(\lambda,\Phi_{s_p}^{T})\omega\right\|_{\mathbb{X}}.
\end{align}
Using the above inequality, \eqref{4.37} and Assumption \ref{as2.1} (\textit{H0}), we obtain
\begin{align*}
\left\|x^{\lambda}(T)-\xi_p\right\|_{\mathbb{X}}\to0\ \mbox{ as }\ \lambda\to0^+.
\end{align*}
Hence, the system \eqref{1.1} is approximately controllable on $J$.
\end{proof}
\section{Application}\label{app}\setcounter{equation}{0}
In this section, we investigate the approximate controllability of the fractional order wave equation with non-instantaneous impulses and delay:
\begin{Ex}\label{ex1} Let us consider the following fractional system:
\begin{equation}\label{ex}
\left\{
\begin{aligned}
\frac{\partial^\alpha v(t,\xi)}{\partial t^\alpha}&=\frac{\partial^2v(t,\xi)}{\partial \xi^2}+w(t,\xi)+\int_{-\infty}^{t}b(t-s)v(s-\beta(\|v(t)\|),\xi)\mathrm{d}s, \\&\qquad \qquad \qquad\qquad \ t\in\bigcup_{j=0}^{m} (s_j, \tau_{j+1}]\subset J=[0,T], \ \xi\in[0,\pi], \\
v(t,\xi)&=h_j(t,v(t_j^-,\xi)),\ \ \ t\in(\tau_j,s_j],\ j=1,\ldots, m,\ \xi\in[0,\pi],\\
\frac{\partial v(t,\xi)}{\partial t}&=\frac{\partial h_j(t,v(\tau_j^-,\xi))}{\partial t},\ t\in(\tau_j,s_j],\ j=1\ldots,m,\ \xi\in[0,\pi],\\
v(t,0)&=0=v(t,\pi), \qquad \ t\in J, \\
v(\theta,\xi)&=\psi(\theta,\xi), \frac{\partial v(0,\xi)}{\partial t}=\zeta_0(\xi),\ \xi\in[0,\pi], \ \theta\leq0,
\end{aligned}
\right.
\end{equation}
where $\alpha\in(1,2)$ and the function $w:J\times[0,\pi]\to[0,\pi]$ is square integrable in $t$ and $\xi$, and the functions $\beta:[0,\infty)\to[0,\infty)$ and $b:[0, \infty)\rightarrow\mathbb{R}$ are also continuous. The functions $\psi(\cdot,\cdot)$ and $h_j$ for $j=1,\ldots,p$ satisfy suitable conditions, which will be specified later.
\end{Ex}
\vskip 0.1 in
\noindent\textbf{Step 1:} \emph{Strongly continuous families and phase space:} Let $\mathbb{X}_p= \mathrm{L}^{p}([0,\pi];\mathbb{R})$, for $p\in[2,\infty)$, and $\mathbb{U}=\mathrm{L}^{2}([0,\pi];\mathbb{R})$. Note that $\mathbb{X}_p$ is a separable reflexive Banach space with strictly convex dual $\mathbb{X}_p^*=\mathrm{L}^{\frac{p}{p-1}}([0,\pi];\mathbb{R})$ and $\mathbb{U}$ is separable. We define the operator $\mathrm{A}_p:\mathrm{D}(\mathrm{A}_p)\subset\mathbb{X}_p\rightarrow \mathbb{X}_p$ as
\begin{align}\label{5.9}
\mathrm{A}_pf(\xi)=f''(\xi), \ \text{ where }\ \mathrm{D}(\mathrm{A}_p)= \mathrm{W}^{2,p}([0,\pi];\mathbb{R})\cap\mathrm{W}_0^{1,p}([0,\pi];\mathbb{R}).\end{align}
Moreover, the spectrum of the operator $\mathrm{A}_p$ is given by $\sigma(\mathrm{A}_p)=\{-n^2:n\in\mathbb{N}\}$. Then, for every $f\in\mathrm{D}(\mathrm{A}_p)$, the operator $\mathrm{A}_p$ can be written as
\begin{align*}
\mathrm{A}_pf&= \sum_{n=1}^{\infty}-n^{2}\langle f, w_{n} \rangle w_{n},\ \langle f,w_n\rangle :=\int_0^{\pi}f(\xi)w_n(\xi)\mathrm{d}\xi,
\end{align*}
where $w_n(\xi)=\sqrt{\frac{2}{\pi}}\sin(n\xi)$ are the normalized eigenfunctions (with respect to the $\mathbb{X}_2$ norm) of the operator $\mathrm{A}_p$ corresponding to the eigenvalues $-n^2$, $n\in\mathbb{N}$. The operator $\mathrm{A}_p$ satisfies all the conditions (\textit{R1})-(\textit{R3}) of Assumption \ref{ass2.1} (see application section of \cite{SS}). By applying Lemma \ref{lem2.1}, we deduce the existence of a strongly continuous cosine family $\mathrm{C}_p(t),\ t \in \mathbb{R}$ in $\mathbb{X}_p$. The compactness of the associated strongly continuous sine family $\mathrm{S}_p(t), t\in \mathbb{R},$ follows by Lemma \ref{lem2.2}. The strongly continuous families $\{\mathrm{C}_p(t):t\in\mathbb{R}\}$ and $\{\mathrm{S}_p(t):t\in\mathbb{R}\}$ can be written as
\begin{align*}
\mathrm{C}_p(t)f&=\sum_{n=1}^{\infty}\cos(nt)\langle f, w_{n} \rangle w_{n},\ f\in\mathbb{X}_p,\nonumber\\
\mathrm{S}_p(t)f&=\sum_{n=1}^{\infty}\frac{1}{n}\sin(nt)\langle f, w_{n} \rangle w_{n},\ f\in\mathbb{X}_p.\nonumber
\end{align*}
Next, we define the operator
\begin{align*}
\mathcal{S}_{\gamma,p}(t)f&=\int_{0}^{\infty} \gamma\theta\mathrm{M}_{\gamma}(\theta)\sum_{n=1}^{\infty}\frac{1}{n}\sin(nt^{\gamma}\theta)\langle f, w_{n} \rangle w_{n}\mathrm{d}\theta,\ f\in\mathbb{X}_p,
\end{align*}
where $\gamma=\frac{\alpha}{2}$.
\vskip 0.2 cm
\noindent \emph{Phase space:} The space $\mathfrak{B}=\mathcal{PC}_{g}(\mathbb{X}_p)$ (see, Example \ref{exm2.8}) is a phase space satisfying the axioms (A1) and (A2) with $\mathcal{P}(t) =t$ and $\mathcal{Q}(t)= \mathcal{O}(t)$, (see, section 5 of \cite{SS}). Specifically, one can choose the function $g(\theta)=e^{a\theta}$, for $a<0$. In this case, we assume the following condition:
\begin{itemize}
\item [$(C1)$] The function $\psi\in\mathcal{PC}_{g}(\mathbb{X})$, we assume $L:=\esssup\limits_{\theta\in(-\infty,0]}|b(-\theta)|g(\theta)$.
\end{itemize}
\vskip 0.1 cm
\noindent\textbf{Step 2:} \emph{Approximate controllability.}
Let us define $$x(t)(\xi):=v(t,\xi),\ \mbox{ for }\ t\in J\ \mbox{ and }\ \xi\in[0,\pi],$$ and the operator $\mathrm{B}:\mathbb{U}\to\mathbb{X}_p$ as $$\mathrm{B}u(t)(\xi):=w(t,\xi)=\int_{0}^{\pi}K(\zeta,\xi)u(t)(\zeta)\mathrm{d}\zeta, \ t\in J,\ \xi\in [0,\pi],$$ where $K\in\mathrm{C}([0,\pi]\times[0,\pi];\mathbb{R})$ is a symmetric kernel, that is, $K(\zeta,\xi)=K(\xi,\zeta),$ for all $\zeta,\xi\in [0,\pi]$. We assume that the operator $\mathrm{B}$ is one-one. Hence, the operator $\mathrm{B}$ is bounded (see application section of \cite{SMJ}). The symmetry of the kernel ensures that the operator $\mathrm{B}$ is self-adjoint, that is, $\mathrm{B}=\mathrm{B}^*$ . For example, one can take $K(\xi,\zeta)=e^{(\xi+\zeta)},\ \mbox{for all}\ \xi, \zeta\in [0,\pi]$.
The function $\psi:(-\infty,0]\rightarrow\mathbb{X}$ is given as
\begin{align}
\nonumber \psi(t)(\xi)=\psi(t,\xi),\ \xi\in[0,\pi].
\end{align}
Next, the functions $f, \varrho:J\times \mathfrak{B}\to\mathbb{X}$ are defined as
\begin{align}
\nonumber f(t,\psi)\xi&:=\int_{-\infty}^{0}b(-\theta)\psi(\theta,\xi)\mathrm{d}\theta,\\
\nonumber\varrho(t,\psi)&:=t-\beta(\|\psi(0)\|_{\mathbb{X}}),
\end{align}
for $\xi\in[0,\pi]$. It is easy to verify that the function $f$ is continuous and uniformly bounded by $L$. Hence, the function $f$ fulfills the condition \textit{$(H1)$} of Assumption \ref{as2.1} and the condition \textit{$(H3)$}.
Moreover, the impulse functions $h_{j}:[\tau_j,t_j]\times\mathbb{X}_p\to\mathbb{X}_p,$ for $j=1,\ldots,m,$ are defined as
\begin{align*}
h_{j}(t,x)\xi:=\int_{0}^{\pi}\rho_j(t,\xi,z)\cos^2(x(\tau_j^-)z)\mathrm{d}z, \ \mbox{ for }\ t\in(\tau_j,s_j],
\end{align*}
where, $\rho_j\in\mathrm{C}^{1}(J\times[0,\pi]^2;\mathbb{R})$. Clearly, the impulses $h_{j}$ for $j=1,\ldots,m,$ satisfy the condition \textit{$(H2)$} of Assumption \ref{as2.1}.
Using the above transformations, the system \eqref{ex} can be rewritten in the form \eqref{1.1} which satisfies Assumption \ref{as2.1} \textit{(H1)-(H2)} and Assumption (\textit{H3}). Next, we show that the corresponding linear fractional control system of the equation \eqref{1.1} is approximately controllable. To achieve this, we assume that
$$\mathrm{B}^*\mathcal{S}_{\gamma,p}(T-t)^*x^*=0,\ \mbox{ for any}\ x^*\in\mathbb{X}_p^*,\ 0\le t<T.$$ Since the operator $\mathrm{B}^*$ is one-one, we get that
\begin{align}\label{5.3}
\mathcal{S}_{\gamma,p}(T-t)^*x^*=0,\ \mbox{ for all } \ t\in [0,T).
\end{align}
Let us compute
\begin{align}\label{5.4}
\lim_{t\downarrow T}\frac{\mathcal{S}_{\gamma,p}(T-t)^*x^*}{(T-t)^\gamma}&= \lim_{t\downarrow T}\frac{\int_{0}^{\infty}\gamma\theta\mathrm{M}_{\gamma}(\theta)\sum_{n=1}^{\infty}\frac{1}{n}\sin(n(T-t)^{\gamma}\theta)\langle x^*, w_{n} \rangle w_{n}\mathrm{d}\theta}{(T-t)^{\gamma}}\nonumber\\&=\lim_{t\downarrow T}\sum_{n=1}^{\infty}\frac{1}{n}\gamma\int_{0}^{\infty}\theta\mathrm{M}_{\gamma}(\theta)\left(\frac{\sin(n(T-t)^{\gamma}\theta)}{(T-t)^\gamma}\right)\mathrm{d}\theta\langle x^*, w_{n} \rangle w_n \nonumber\\&=\sum_{n=1}^{\infty}\frac{1}{n}\gamma\int_{0}^{\infty}\theta\mathrm{M}_{\gamma}(\theta)\left(\lim_{t\downarrow T}\frac{\sin(n(T-t)^{\gamma}\theta)}{(T-t)^\gamma}\right)\mathrm{d}\theta\langle x^*, w_{n} \rangle w_n\nonumber\\&=\sum_{n=1}^{\infty} \gamma \int_{0}^{\infty} \theta^2\mathrm{M}_{\gamma}\mathrm{d}\theta\langle x^*, w_{n} \rangle w_n\nonumber\\&=\frac{2\gamma}{\Gamma(1+2\gamma)}\sum_{n=1}^{\infty}\langle x^*, w_{n} \rangle w_n,
\end{align}
where $\gamma\in(0,1)$. Combining the estimates \eqref{5.3} and \eqref{5.4}, we obtain
\begin{align*}
\sum_{n=1}^{\infty}\langle x^*, w_{n} \rangle w_n=0,
\end{align*}
which implies that $x^*=0$. Thus, by applying Lemma \ref{lem3.4} and Remark \ref{rem3.4}, we deduce that the corresponding linear fractional control system of \eqref{1.1} is approximately controllable. Finally, by Theorem \ref{thm4.4}, we conclude that the semilinear fractional control system \eqref{1.1} (equivalent to the system \eqref{ex}) is approximately controllable.
\section{Conclusions} In this manuscript, we considered the semilinear fractional control system \eqref{1.1} of order $1<\alpha<2$, where the operator $\mathrm{A}$ generates a strongly continuous cosine family $\{\mathrm{C}(t):t\in\mathbb{R}\}$. We first formulated the linear regulator problem and obtained the optimal control in the feedback form. With the help of this optimal control, we discussed the approximate controllability of the linear control system \eqref{3.2}. Then, we proved the existence of a mild solution of the semilinear fractional control system \eqref{1.1} for the suitable control function defined in \eqref{C} via the resolvent operator and the Schauder fixed point theorem. Then, we derived sufficient conditions for the approximate controllability of the system \eqref{1.1}, whenever the corresponding linear control system is approximately controllable. Moreover, we modified the axioms of phase space to deal with impulsive functional differential equations. Note that, in this work, we have used the idea of $\alpha/2$-resolvent family related to cosine family generated by the operator $\mathrm{A}$.
The existence of solutions and the approximate controllability results of the semilinear fractional abstract Cauchy problem of order $\alpha\in(1,2)$ by using the idea related to the $\alpha$-resolvent family introduced in \cite{JFA} is not investigated so far, and it will be addressed in a future work.
\medskip\noindent
{\bf Acknowledgments:} S. Arora would like first to thank the Council of Scientific and Industrial Research, New Delhi, Government of India (File No. 09/143(0931)/2013 EMR-I), for financial support to carry out his research work and also thank the Department of Mathematics, Indian Institute of Technology Roorkee (IIT Roorkee), for providing stimulating scientific environment and resources. M. T. Mohan would like to thank the Department of Science and Technology (DST), Govt of India for Innovation in Science Pursuit for Inspired Research (INSPIRE) Faculty Award (IFA17-MA110). J. Dabas would like to thank the Council of Scientific and Industrial Research, New Delhi, Government of India, project (ref. no. 25(0315)/20/EMR-II).
| -179,460.574142 |
[
-2.93359375,
2.677734375
] | 25.648022 |
[
-3.232421875,
0.958984375,
-1.8828125,
-5.890625,
-0.75830078125,
8.4765625
] |
[
2.650390625,
7.3203125,
0.69140625,
5.30078125
] | 539 | 7,045 |
[
-2.25390625,
2.05078125
] | 39.107297 |
[
-6.19140625,
-4.2890625,
-5.14453125,
-2.509765625,
2.3203125,
13.3671875
] | 0.409425 | 2.407467 | 25.209368 | 2.77894 |
[
2.36824893951416
] | -113,334.372828 | 10.054081 | -180,719.044088 | 0.777908 | 6.138536 |
[
-2.267578125,
-3.607421875,
-4.1171875,
-5.28515625,
2.23828125,
13
] |
[
-6.12109375,
-1.8564453125,
-1.95703125,
-0.916015625,
3.74609375,
3.96875
] | |
BkiUdlE5qoYAo52Vqk4-
|
\section{Introduction}
In recent years \ac{vi} sensors, which consist of one or more cameras and an \ac{imu}, have become increasingly popular for robust high frequency motion estimation~\citep{corke2007introduction,8633393,leutenegger2015keyframe,bloesch2017iterated,8421746}.
Before use, VI sensors need to be calibrated, which implies obtaining the parameters for the camera intrinsics, the camera-IMU extrinsics, and the time offset between the different sensors~\citep{tschopp2020versavis,nikolic2014synchronized}.
The performance of VI systems is highly dependent on the quality and accuracy of the calculated calibration parameters~\citep{kelly2011visual}.
Precise calibrations are usually obtained offline in controlled environments, following sophisticated motion routines ensuring observability~\citep{furgale2013unified,furgale2012continuous}.
This makes the entire process non-trivial for an inexperienced operator~\citep{kelly2011visual,nobre2019learning}.
Additionally, the motion primitives that most effectively render the best calibration results are unknown.
So instead of performing this task by hand or with a manually programmed operator, we propose the use of model-based deep \ac{rl} to learn the best motion primitives and perform them on a robotic arm.
Previous works in automatic calibration use trajectory optimization and reinforcement learning to address this problem.
For the class of optimization methods that maximize the observability Gramian of the trajectories on the calibration parameters~\citep{preiss2017trajectory,hausman2017observability}, it remains challenging to model and include other practical optimization objectives, such as maximizing target point coverage for camera calibration and trajectory length minimization for efficiency.
A learning-based approach was proposed by Nobre \textit{et al}. ~\citep{nobre2019learning}, where a set of trajectories were pre-defined empirically and Q-learning was applied to choose a sequence of those trajectories that renders sufficient observability of the calibration problem.
A disadvantage of this approach is that it remains restricted to the collection of pre-defined trajectories and therefore the possible range of movements is not fully explored.
We conclude that there are two aspects that could be further studied based on existing previous works:
\begin{itemize}
\item \textbf{Multiple objectives:} It is not obvious whether the trajectory that renders sufficient observability of states also provides the best calibration. Besides, other empirical requirements such as path length and camera coverage could also be included in the optimized objectives.
\item \textbf{Learn the trajectory:} A more general learning problem could be posed, where the predefined trajectories and their selection are learned jointly in a single optimization problem.
\end{itemize}
We aim to address both points by designing a \ac{rl}-based method to obtain the best sequence of \ac{vi} calibration trajectories that fulfill multiple objectives, including both observability and practical requirements.
The approach will be applied separately to calibrate both camera intrinsics and camera-IMU extrinsics, as these two steps have very different motion requirements.
The trajectories are executed by a robot arm during training in a realistic simulation for both dynamics and photorealism, thus guaranteeing the feasibility of execution by a real robot arm.
Additionally, we simulate different \ac{vi} sensor configurations to model the variability that also exists in the real world.
To solve the proposed problem, we model calibration as a \ac{mdp} and use model-based \ac{rl}~\citep{nagabandi2018neural} to establish the sequence of motion trajectories that optimizes sensor calibration accuracy.
Compared with other solutions the action space of our model has fewer constraints, making it possible to learn a more general policy for calibration.
In addition, we do not only consider different information-theoretic metrics of the trajectories, but also take camera coverage and path length into account, in the reward of our \ac{mdp} model.
For the learning algorithm, we propose a sample efficient model-based \ac{rl} method using the \ac{pso}~\citep{kennedy1995particle} algorithm to search for the optimal open-loop control sequence.
We adapt the \ac{pso} algorithm to \ac{rl} by utilizing gradient information and memorizing previous optimization results for initialization.
This method addresses difficulties in searching for action sequences in a high dimensional space and the high time cost of performing a calibration at each step.
The main contributions of this work are as follows:
\begin{itemize}
\setlength\itemsep{0em}
\item Our approach models the entire calibration process as a \ac{mdp} and to the best of our knowledge, we are the first to solve it using model-based deep reinforcement learning.
\item Our proposed model-based heuristic \ac{rl} algorithm with adapted \ac{pso} satisfies different practical requirements such as the capability of solving problems with high dimensional action space with high sample efficiency.
\item The evaluation shows that the learned trajectories deliver more accurate calibrations compared to handcrafted or random ones. We enable easy transferability to real scenarios by also simulating the robot arm together with different sensor configurations.
\end{itemize}
\section{Related Work}
\label{sec:rel-work}
The most popular method for camera calibration during the last decades is to use a known calibration pattern and apply nonlinear regression to obtain the parameters.
This method has been successfully used both for camera intrinsic~\citep{sturm1999plane} and extrinsic calibration~\citep{zhang2004extrinsic}.
For VI sensor calibration the most reliable and precise approaches are also based on the use of a calibration board.
A method presented in~\citep{mirzaei2008kalman} applies an Extended Kalman Filter to estimate the relative poses between sensors jointly.
A parametric method proposed in~\citep{furgale2013unified,furgale2012continuous} represents the pose and bias trajectories using B-splines and introduces a batch estimator in continuous-time.
While those methods are relatively efficient, they still require expert knowledge to obtain the required sensor data for good accuracy.
In addition, the optimal calibration movements remain unknown, especially when aiming to increase time efficiency and calibration accuracy.
The problem of designing the best motion primitive was first addressed by methods based on trajectory optimization to find the trajectory that renders the highest observability of calibration parameters.
A method proposed in~\citep{hausman2017observability} optimizes the expanded empirical local observability Gramian of unknown parameters based on the measurement model, to solve for the best state and input trajectories.
Preiss \textit{et al}. ~\citep{preiss2017trajectory} further extended the method to be obstacle-free and more balanced among multiple objectives.
Both of the approaches only optimize observability by using the observability Gramian of the trajectories, where other empirical and general requirements are difficult to model and cannot be easily included.
Our framework combines different evaluation metrics of information gain for variables in the reward design and learns to obtain the highest reward using model-based \ac{rl}.
Another class of methods applies \ac{rl} to get the best sequence of trajectories selected from a library of pre-designed trajectories.
Nobre \textit{et al}. ~\citep{nobre2019learning} modeled the calibration process as an \ac{mdp}, where the states are estimated parameters and the actions are choices of trajectories from the library at each step.
The \ac{mdp} planning problem is solved using Q-learning.
However, the method only yields suggestions on predefined motions to choose from, rather than exploring new possible motion primitives.
In contrast, our method includes the trajectory parameters in the action design and solves the entire calibration problem in an end-to-end fashion.
For such complex sequential decision-making problems, recent works have shown that deep \ac{rl} algorithms are capable of learning policies that render high performance~\citep{mnih2015human,lillicrap2015continuous,schulman2017proximal,haarnoja2018soft,schulman2015trust}.
Model-based and model-free deep \ac{rl} are two classes of deep \ac{rl} algorithms.
Model-free algorithms are capable of learning a wide range of sequential decision problems, but they require a large number of samples to achieve good performance~\citep{schulman2015trust,feinberg2018model,buckman2018sample}.
Nagabandi \textit{et al}. ~\citep{nagabandi2018neural} combined neural network model learning with sample-based \ac{mpc} to improve sample efficiency, and the policy is further fine-tuned with model-free algorithms.
Because of the high time cost to perform a calibration at each training step, model-based algorithms are suitable to reduce the number of required episodes to learn a good action sequence.
However, sample-based \ac{mpc} performs relatively poorly in high dimensional action space, therefore we substitute random sampling in~\citep{nagabandi2018neural} with a metaheuristic algorithm.
\ac{pso}~\citep{kennedy1995particle} is such a metaheuristic algorithm that has been widely used in the last two decades due to its good performance in complex high-dimensional problems, which cannot be solved using traditional deterministic algorithms.
Hein \textit{et al}. ~\citep{hein2016reinforcement} reformulated \ac{rl} problems as optimization tasks and applied \ac{pso} to search for optimal solutions.
In this paper, we combine \ac{mpc} with a modified \ac{pso} in a model-based framework to search for optimal action sequences.
\section{Method}
\label{sec:approach}
\begin{figure}[t]
\centering
\setlength{\belowcaptionskip}{-0.5cm}
\includegraphics[width=0.8\linewidth]{framework.png}
\caption{Overview of our calibration framework. The trajectories (actions) the agent chooses to take and their simulated calibration performance are recorded in real-time. This data is then used to train the agent with model-based \ac{rl}.}
\label{fig:framework}
\end{figure}
An overview of the proposed learning framework is shown in Figure~\ref{fig:framework}.
Given a set of trajectory parameters, the simulation platform executes the trajectory and records the resulting sensor data.
The estimation client then computes the calibration parameters based on the sensor data and transforms all the results into training data according to the \ac{mdp} formulation of the problem.
Finally, the agent samples from the recorded training data to learn the optimal trajectories using model-based heuristic \ac{rl} and chooses an action to execute.
\subsection{Visual-Inertial Calibration}
Our estimator follows the Kalibr framework~\citep{furgale2013unified,furgale2012continuous}, where the Levenberg-Marquardt algorithm is applied to minimize the loss between the obtained and predicted measurements to maximize the likelihood of the unknown parameters $Pr(X,\theta|D,L)$.
Here, $X$ is the estimated pose trajectory, $\theta$ depicts the calibration parameters, $D$ are the measurements of a \ac{vi} sensor consisting of images and inertial measurements and $L$ is the known position of landmarks on the calibration target.
As explained in detail in~\citep{schneider2019observability}, the covariance matrix of the known parameters $\Sigma_{X\theta}$ can be obtained from the Jacobian of all error terms and the stacked error covariances.
The covariance of the calibration parameters $\Sigma_\theta$ can be extracted from $\Sigma_{X\theta}$ and further normalized to $\overline{\Sigma}_\theta$.
The information gain can then be evaluated with the following metrics:
\begin{itemize}
\item \textbf{A-Optimality:} $H_{Aopt} = trace(\overline{\Sigma}_\theta)$
\item \textbf{D-Optimality:} $H_{Dopt} = \det(\overline{\Sigma}_\theta)$
\item \textbf{E-Optimality:} $H_{Eopt} = \max(eig(\overline{\Sigma}_\theta))$
\end{itemize}
Minimizing these $H$ metrics leads to the maximization of information gain and furthermore can be used for the reward design of our proposed method.
\subsection{MDP Model for Learning to Calibrate}
\label{subsec:POMDP}
The whole calibration process is modeled as an \ac{mdp}.
The process description includes a state $S_t$, action $A_t$, transition model $S_{t+1} = f(S_t,A_t)$, and reward $R_t$ at each time step $t$.
We define the action $A_t$ at each time as a looped parameterized trajectory with the same start and terminal pose.
Each action represents a trajectory subsequence, and these subsequences are concatenated to form the final calibration trajectory. At each step, the calibration process needs to be rerun over the entire sequence and cannot be run only over the newly acquired measurements.
The trajectory poses \{$[x_j, y_j, z_j,\alpha_j, \beta_j, \gamma_j]^\top\}_{j=1:J}$ are parameterized by $\{\sum_{q=1,2,4}\bm{a}_q(1-\cos \frac{2q\pi j}{J})+\bm{b}_q\sin \frac{2q\pi j}{J}\}_{j=1:J}$, where $J$ is the number of waypoints inside one action. $\bm{a}_q$ and $\bm{b}_q$ are $6\times1$ vectors.
Thus, for one trajectory 36 parameters are needed, which is the action space dimension of the \ac{mdp}.
The reason to choose $\sin$ and $\cos$ as basis functions is that they are capable of representing many good empirical trajectories for calibration.
Let $D_t$ be the measurements acquired with action $A_t$, and $Y_t$ be the vector that stacks all the calibration parameters and their information gain status, then $Y_t = [\theta_t^*, O_t]^T = Cal(\bigcup_{i=0}^{t-1} D_i)$, where $Cal$ represents the calibration process that returns the predicted calibration parameters $\theta^*$ and their respective information gain status $O_t$.
For camera calibration $O_t$ contains the progress of the coverage of the horizontal axis, the vertical axis, size and skew.
For camera-IMU calibration, $O_t$ is composed of the eigenvalues of the covariance matrix for extrinsics, used in the optimization process of the calibration tool.
Ignoring the sensor noise, we assume the action history sequence together with the calibration status determine the measurement sequence: $\bigcup_{i=0}^{t-1} D_i = h(A_{0:t-1}, Y_{0:t-1})$, where $h$ represents an unknown mapping.
In this way, the state $S_t$ is defined as the concatenation of all actions and calibration results of previous time steps: $S_t = [A_{0:t-1}, Y_{0:t}]$.
The state transition satisfies the Markov property and the transition model becomes
\begin{align}
S_{t+1} = A_{0:t}\cup Y_{0:t+1} = A_{0:t-1}\cup Y_{0:t}\cup Y_{t+1} \cup A_t = S_t\cup Cal(h(A_{0:t}, Y_{0:t}))\cup A_t
\label{eq:trans}
\end{align}
where the right hand side of \eqref{eq:trans} only depends on $S_t$ and $A_t$.
Finally, the reward at each step is composed of four parts: empirical reward $e_t$, information gain for the calibration parameters $o_t$, trajectory length $l_t$ and relative calibration error $d_t$.
The empirical reward encodes intuitive requirements such as image view coverage with target observations.
The information gain includes different evaluation metrics such as the determinant, trace, and eigenvalues of the aforementioned covariance matrix for the extrinsics.
The trajectory length $l$ is computed by summing up the position and Euler angle distances between each two neighboring waypoints
\begin{align}
l = & \sum_{j=1}^J(\|x_j-x_{j-1},y_j-y_{j-1},z_j-z_{j-1}\|_2+C\|\alpha_j-\alpha_{j-1},\beta_j-\beta_{j-1},\gamma_j-\gamma_{j-1}\|_2),
\label{eq:path_length}
\end{align}
where $C$ is a tuneable weighting factor to balance the importance between rotation and translation.
The calibration errors are defined as the Euclidean distance between the calibration result $\theta^*$ and ground truth $\theta$.
The relative calibration error is further normalized by the norm of the ground truth.
In a real world scenario, where the ground truth calibration parameters are not available, this reward term can instead be computed using the reprojection error of the calibration process, as it directly measures how close to the ground truth calibration we currently are.
Finally, the reward is the weighted sum of increments of each term at each time step
\begin{align}
R_t = \eta_1\Delta e_t + \eta_2\Delta o_t - \eta_3\Delta d_t - \eta_4\Delta l_t,
\end{align}
where $\eta_1,\cdots,\eta_4$ are positive weights that can be manually tuned separately.
\subsection{Model-based Heuristic Reinforcement Learning with PSO}
\textbf{Neural network dynamics and reward functions}:
We parameterize both the dynamics model and reward model as neural networks.
The network structures are shown in Figure~\ref{fig:network}.
Each model contains a recurrent neural network~\citep{chung2014empirical} to encode a hidden state to compress the information acquired so far, given the action history sequence $A_{0:t-1}$ and the calibration status sequence $Y_{0:t}$ as inputs.
The reward model then feeds the hidden state and current action input $A_t$ to a fully-connected network to predict the current reward $\hat{R}_t = g_\psi(A_{0:t-1},Y_{0:t},A_t)$.
With the same input, the dynamics model predicts the next calibration status $\hat{Y}_{t+1} = f_\phi(A_{0:t-1},Y_{0:t},A_t)$, where $\psi$ and $\phi$ are the weights of the two neural networks.
Given a batch of training data $\{Y_{0:t}^i, A_{0:t-1}^i, A_t^i, Y_{t+1}^i, R_t^i\}_{i=1}^N$, both models are trained using \ac{sgd} to minimizing the mean squared dynamics and reward errors
\begin{align}
\epsilon_{dyn} &= \frac{1}{N}\sum_{i=1}^N\|Y_{t+1}^i-\hat{Y}^i_{t+1}\|^2 \label{eq:dyn_f},\\
\epsilon_{reward} &= \frac{1}{N}\sum_{i=1}^N\|R_{t}^i-\hat{R}^i_{t}\|^2 \label{eq:rew_f}
\end{align}
respectively.
A higher model prediction accuracy is fundamental to the performance of the learned action sequence.
\begin{figure}[t]
\centering
\setlength{\belowcaptionskip}{-0.4cm}
\includegraphics[width=0.75\linewidth]{network_architecture.png}
\caption{Network architecture of the dynamics model (top) and reward model (bottom).}
\label{fig:network}
\end{figure}
\textbf{Model-based open-loop optimization}:
With the learned reward and dynamics model, we use a \acf{mpc} to control the agent.
In the \ac{mpc} framework, an open-loop action sequence $A^*_{t:T}$ from the current time step $t$ to the end time $T$ is first optimized to maximize the predicted future sum of rewards.
Then only the first action $A_t$ is executed and the corresponding new states and rewards are obtained.
The optimal action sequence until the end $A^*_{t+1:T}$ is then recalculated and the next action is executed.
For our problem, the open-loop optimization to solve the optimal future action sequence $A^*_{t:T}$ at each time step $t$ is formalized as
\begin{align}
A^*_{t:T} = \argmax_{A_{t:T}} \sum_{\tau=t}^T g_\psi(A_{0:\tau-1},Y_{0:t},\hat{Y}_{t+1:\tau},A_\tau),
\label{eq:optimization}
\end{align}
where $\hat{Y}_{t+1:\tau}$ is predicted using the dynamics model $f_\phi$ for each time step.
This non-linear optimization problem is solved using a modified version of the \ac{pso} algorithm.
For a particle swarm with $M$ particles in total, each particle position $P^t_{i\in\{1,2,\cdots,M\}}$ represents a potential solution of actions $A^i_{t:T}$.
For the position update, the velocities $v$ of particles of the original \ac{pso} algorithm include 3 components: social component, cognitive component, and inertia~\citep{kennedy1995particle}.
In our method, the cognitive component is modified to be the gradient of the optimization function, rather than the direction towards the local best position the particle has ever visited.
This is because the gradients are more informative to indicate the local optimal position.
Furthermore, the gradients can be obtained from the learned model as $\mu_i = \nabla_{A_{t:T}} \sum_{\tau=t}^T g_\psi(A_{0:\tau-1},Y_{0:t},\hat{Y}_{t+1:\tau},A_\tau)|_{A_{t:\tau}=P^t_i}$.
In this way, the particles move according to the following rules at each optimization iteration $\iota$
\begin{align}
\prescript{\iota}{}v_i &= \omega_0\prescript{\iota-1}{}v_i + c_1 (G_{best}-\prescript{\iota-1}{}P^t_i) + c_2 \prescript{\iota-1}{}\mu_i \label{eq:PSO1}\\
\prescript{\iota}{}P^t_i &= \prescript{\iota-1}{}P^t_i + \prescript{\iota}{}v_i,
\label{eq:PSO2}
\end{align}
where $G_{best}$ represents the global best position all the particles have covered. The first two terms in Equation~\eqref{eq:PSO1} represent the inertia and social components, the last one is the modified cognitive component.
The parameters $\omega_0,c_1, c_2$ are coefficients that can be tuned to trade-off between exploration and exploitation.
\begin{figure}[t]
\centering
\setlength{\belowcaptionskip}{-0.5cm}
\includegraphics[width=0.85\linewidth]{particle_saving.png}
\caption{Particle initialization and action selection in subsequent episodes.}
\label{fig:particles}
\end{figure}
Due to high computational costs, at each time step, only a small number $I$ of iterations are used for \ac{pso}.
A good solution cannot be guaranteed with a limited number of iterations and random initialization, but the performance can be improved by using previous optimization results to initialize new particles.
Although the optimization problems are different between different time steps in the same episode (the lengths of open-loop action sequences to optimize are different), they are similar between the same time step of different episodes.
Therefore, we save the positions of the top $K$ particles after each time step as $P^t_\text{best}$, for initialization of new particles of the same time step in subsequent episodes.
The other $M-K$ new particles are randomly initialized to enable exploration and avoid local minima.
\textbf{Model predictive control}:
After $I$ iterations and obtaining the final position of all particles $P^t_{1:M}= \{A^i_{t:T}\}_{i=1}^M$ at each time step, a particle $A^*_{t:T}$ is selected as the open-loop control solution.
Different policies are applied to select the particle for training and testing in our method.
For training episodes, in order to encourage exploration, the action sequence is randomly selected from the top $W$ particles with the the highest predicted reward sum, rather than always selecting the globally best particles.
To prevent the top $W$ particles from converging to the same point, $W>K$ should be satisfied.
In this way, the $W$ particles contain not only the $K$ particles that have been optimized continuously among episodes but also some new particles that are initialized randomly in the current episode.
For testing, only the particle with the highest predicted sum of rewards is selected.
Following the \ac{mpc} rule, we only extract the first action $A^*_t$ from $A^*_{t:T}$ for execution.
An overview of the particle initialization and action selection framework is shown in Figure~\ref{fig:particles}.
The entire algorithm is summarized in Algorithm~\ref{algo:MBHRL}.
\begin{algorithm}
\small
\caption{Model-based Heuristic Reinforcement Learning}\label{euclid}
\begin{algorithmic}[1]
\State set $T$, $M$, $K$, $W$ and $I$, where $K,W<M$
\State initialize $g_\psi$, $f_\phi$ and randomly initialize $\{P_\text{best}^t\}_{t=0}^{T}$
\State gather dataset $D_{RL}$ with random trajectories
\For{each episode}
\State reinitialize the \ac{mdp}
\For{t=0 \textbf{to} $T$}
\State train $g_\psi$ and $f_\phi$ by applying SGD to \eqref{eq:dyn_f}-\eqref{eq:rew_f} with $D_{RL}$
\State update $Y_{0:t}$, $A_{0:t-1}$
\State initialize $P^t_{1:M}$ by assigning $P^t_{1:M} = \{P^t_\text{best}$, randomly sampling $P^t_{K+1:M}$\}
\For{$\iota$=0 \textbf{to} $I$}
\State update $P^t_{1:M}$ using \eqref{eq:PSO1}-\eqref{eq:PSO2}
\EndFor
\State sort $P^t_{1:M}$ descendingly by the predicted reward sum
\State update the best K particles $P^t_\text{best}$ = $P^t_{1:K}$
\State randomly select $A^*_{t:T}$ from $P^t_{1:W}$
\State execute $A^*_t$ and obtain $Y_{t+1}$, $R_t$
\State add $\{Y_{0:t},A_{0:t-1},A_t,Y_{t+1},R_t\}$ to $D_{RL}$
\EndFor
\EndFor
\State \textbf{return} $g_\psi$ and $f_\phi$
\end{algorithmic}
\label{algo:MBHRL}
\end{algorithm}
\section{Implementation}
\label{sec:implement}
We evaluate our method by performing experiments in a simulation platform based on Gazebo~\citep{koenig2004design}.
This includes a checkerboard, and a \ac{vi} sensor consisting of a pinhole camera and an \ac{imu} mounted on the end-effector of a FRANKA EMIKA Panda robot arm.
To increase robustness and generalization, both the camera intrinsics and camera-IMU extrinsics are sampled from Gaussian distributions in every episode.
Each interaction episode is an independent calibration process without sharing measurement with other episodes.
However, the training process is off-policy so that all recorded interaction data is utilized to update the model to improve sample efficiency.
For the camera intrinsic calibration, we limit the maximum time step size $T$ to be 4 (run at most 4 looped trajectories) in each training episode.
The empirical reward is the coverage of the image view with target observations.
The accuracy reward is computed by dividing the decrease in Euclidean distance from the ground truth by the norm of the ground truth.
The reward and dynamic models for intrinsic calibration are trained in total for 1000 episodes.
For the camera-IMU extrinsic calibration, each training episode contains 3 time steps.
The empirical reward includes the entropy of IMU measurement data in 6 dimensions and number of target observations captured per image.
The model for extrinsic calibration is first trained for 900 episodes.
Subsequently, we fine-tune the agent for another 650 episodes by integrating Kalibr~\citep{furgale2013unified,furgale2012continuous} into the framework.
Two items are added to the reward.
One is the information gain reward, which is the negative A-optimality $H_{Aopt}$ computed based on the covariance matrix.
The other one is the calibration accuracy itself.
For \ac{pso}, we use $M=15$, $K=5$, $W=5$ and $I=5$.
Finally, it should be noted that when training on real systems, the ground truths are not available.
In this case calibration error could be replaced by the reprojection error given from the calibration process (see Appendix~\ref{apx:add_exp_results}).
\section{Experiments}
\label{sec:result}
In this section, we conduct two groups of experiments to evaluate the learned trajectories for camera intrinsic calibration and camera-IMU extrinsic calibration.
The learned trajectories are extracted from the result actions using an \ac{mpc} framework on the final reward model and dynamics model.
We compare our learned trajectories with empirically handcrafted trajectories and ones with random parameters as baselines.
The path lengths and calibration errors are computed as stated in Section~\ref{sec:approach}.
In addition, we also verified that a favorable policy can be learned when calibration error is substituted by the reprojection error.
These results are shown in Appendix~\ref{apx:add_exp_results}.
\subsection{Camera Intrinsic Calibration}
\begin{table}[htbp]
\footnotesize
\centering
\vspace{-0.2cm}
\setlength{\belowcaptionskip}{-0.2cm}
\setlength{\tabcolsep}{1.5mm}{
\begin{tabular}{c|cc|ccc}
& \multicolumn{2}{c|}{\textit{Camera Intrinsic Calibration}} & \multicolumn{3}{c}{\textit{Camera-IMU Extrinsic Calibration}} \\ \hline
& \textbf{Mean error} & \textbf{Path length} & \textbf{Mean error} & \textbf{Path length} & \textbf{Mean A-optimality} \\ \hline
Random trajectory & $\unit[0.560]{\%}$ & $\unit[9.627]{m}$ & $\unit[0.396]{\%}$ & $\unit[7.331]{m}$ & $\unit[4.16\cdot 10^{-08}]{}$ \\
Handcrafted trajectory & $\unit[0.196]{\%}$ & $\unit[11.116]{m}$ & $\unit[0.340]{\%}$ & $\unit[3.306]{m}$ & $\unit[1.97\cdot 10^{-07}]{}$ \\
\textbf{Learned trajectory} & $\mathbf{\unit[0.159]{\%}}$ & $\unit[11.037]{m}$ & $\unit[0.265]{\%}$ & $\unit[4.367]{m}$ & $\unit[5.83\cdot 10^{-08}]{}$ \\
\textbf{Fine-tuned trajectory} & --- & --- & $\mathbf{\unit[0.214]{\%}}$ & $\unit[3.241]{m}$ & $\unit[9.83\cdot 10^{-08}]{}$ \\ \hline
\end{tabular}}
\vspace{0.2em}
\caption{Comparison of mean relative calibration error, path length, and A-optimality between random, handcrafted, learned, and fine-tuned trajectories. Note that the handcrafted and learned trajectories are different for intrinsic and extrinsic calibration. The means are averaged over all intrinsics and extrinsics settings for evaluation. The path lengths are computed by Equation \ref{eq:path_length} where $\unit[C=1]{m/rad}$.}
\label{tab:average}
\end{table}
\begin{figure}[htbp]
\centering
\setlength{\belowcaptionskip}{-0.2cm}
\includegraphics[width=0.85\linewidth]{intrinsic_error_bar_chart2.png}
\caption{Relative errors of the intrinsic calibration results w.r.t. the ground truth for cameras with different horizontal \acp{fov}. For each pair of camera setting and policy, we perform the calibration experiments 5 times and calculate the average relative calibration errors w.r.t the ground truth.}
\label{fig:intrinsic_error}
\end{figure}
We evaluate our policy on cameras with the same image size ($\unit[640]{px}\times\unit[480]{px}$) but different \acp{fov}.
The relative errors of the intrinsic calibration results w.r.t. the ground truth in different settings of horizontal \acp{fov} are reported in Table~\ref{tab:average} and Figure~\ref{fig:intrinsic_error}.
For smaller \acp{fov}, the handcrafted trajectory renders slightly higher calibration accuracy than learned trajectories.
This may be because, with smaller \acp{fov}, the target appears larger in the image view, which makes it easier for both the learned and handcrafted trajectories to achieve high calibration accuracy.
However, with larger \acp{fov} which make the target smaller in the image view and harder to achieve a high image coverage with target observations, the learned trajectory shows better performance.
Table~\ref{tab:average} shows that under similar path lengths, our framework could learn how to perform favorable motion trajectories and collect enough measurements efficiently that yield the desired camera intrinsic parameters.
\subsection{Camera-IMU Extrinsic Calibration}
In this experiment, we train an agent to learn a policy for camera-\ac{imu} extrinsics calibration. As shown in Figure~\ref{fig:extrinsic_error} and Table~\ref{tab:average}, the trajectories generated by our learned policy without fine-tuning can achieve higher calibration accuracies compared with the random and handcrafted trajectories. The low mean A-optimality of learned trajectories can be interpreted as a confidence of Kalibr about the calibration results.
The fine-tuned trajectories perform the best as they achieve the lowest mean error and the shortest path length. The higher mean A-optimality of the fine-tuned trajectory compared with the learned trajectory is possibly caused by the lower path length. A lower path length provides less data, which results in Kalibr being less confident about the calibrations.
\begin{figure}[htbp]
\centering
\vspace{-0.2cm}
\setlength{\belowcaptionskip}{-0.2cm}
\includegraphics[width=0.85\linewidth]{extrinsic_error_histogram.png}
\caption{The distribution of the relative errors of extrinsic calibration results w.r.t. the ground truth. Each trajectory is tested with 9 different pairs of camera intrinsic and \ac{vi} extrinsic configurations.}
\label{fig:extrinsic_error}
\end{figure}
\section{Conclusion}
\label{sec:conclusion}
In this work, we introduced a novel framework that uses model-based deep \ac{rl} with an adapted version of \ac{pso} for sampling, to generate trajectories for efficiently collecting measurements to calibrate both camera intrinsic and \ac{vi} extrinsic parameters.
Our experiments show that under similar or even shorter path lengths, the trajectories generated by our learned policy can lead to more accurate results and a higher calibration confidence.
While the proposed model-based \ac{rl} framework is able to achieve state of the art performance in our \ac{mdp} model for \ac{vi} calibration, an interesting avenue for further improvement is to integrate our approach with model-free learners.
Additionally, model-free \ac{rl} can also be applied for further fine-tuning the learned policies.
\clearpage
\acknowledgments{This paper was partially supported by ABB Corporate Research, Siemens Mobility GmbH, ETH Mobility Initiative under the project \textit{PROMPT}, and the Luxembourg National Research Fund (FNR) 12571953.}
| -20,944.06145 |
[
-1.9677734375,
1.951171875
] | 62.273902 |
[
-3.0703125,
0.59521484375,
-1.935546875,
-5.2578125,
-0.5146484375,
7.38671875
] |
[
3.060546875,
7.49609375,
2.345703125,
7.83203125
] | 230 | 4,113 |
[
-2.724609375,
3.240234375
] | 25.009748 |
[
-6.5859375,
-5.41015625,
-5.2421875,
-2.19921875,
3.087890625,
14.0390625
] | 0.726402 | 23.161246 | 27.546803 | 1.935045 |
[
1.6693251132965088
] | -14,785.116128 | 6.192074 | -20,649.215936 | 0.494738 | 5.905912 |
[
-2.697265625,
-3.85546875,
-3.80078125,
-4.56640625,
2.60546875,
11.765625
] |
[
-5.79296875,
-2.35546875,
-2.662109375,
-1.763671875,
3.62890625,
5.43359375
] | |
BkiUed04eIOjR4Mj-00Z
|
\section{Introduction}
In recent years, the
homogeneous Fermi gas with attractive interactions has been studied extensively both theoretically and experimentally
due to the success in cooling atoms into ultracold dilute condensates
\cite{Ketterle02, Ketterle08, Giorgini08}. By
tuning the interaction strength through the
Feshbach resonance,\cite{Fano61, Feshbach62, Ketterle98, Verhaar98} the system can cross from the
Bardeen-Cooper-Schrieffer (BCS) superfluid phase, where the s-wave scattering length $a_s$ is
negative, to the Bose-Einstein condensate (BEC),
where $a_s$ is positive. Since there is no symmetry change of the
quantum state involved,
the system exhibits the well-known BCS-BEC crossover.
In the special case corresponding to the diverging scattering length,
$a_s \to \infty$, the system is in a strongly interacting regime called the unitary limit.
In this regime the interparticle spacing $r_s$ is the
only relevant scale, and the rest of the quantities are universal and system independent.
The total energy of this system can be conveniently written
as $E= \xi E_{\rm free}$, where $E_{\rm free}$ is the energy of the non-interacting
atomic gas and $\xi$ is a system independent parameter. Experimental measurements of $\xi$ have
been performed using $^6$Li and $^{40}$K atoms by investigating the
expansion rate of the atomic cloud and the sound propagation in it\cite{Bartenstein04,
Bourdel04, Thomas05, Stewart06, Thomas07}.
Simultaneously, a number of theoretical and numerical estimations of $\xi$ have been reported,
including diffusion Monte Carlo (DMC)
\cite{Carlson03,Carlson04,Astrak04,Astrak05,Pandharipande05,Needs10} as well as path
integral Monte Carlo, lattice simulations and analytical methods\cite{Lee06, Lee08, Trivedi07,
Burovski06, Gurarie07, Bulgac06, Bulgac08, Abe09a, Abe09b, Needs10}. The resulting
estimates fall between $\approx$ 0.25--0.45
showing that the actual value has not been settled yet and is still of significant
interest due to the universal nature of the unitary limit.
One of the most interesting properties of the unitary gas is the robust
presence of the pairing condensate which involves a large fraction of the
system.
The study of pairing effects is thus much more straightforward
than, say, in superconducting materials, where only a sliver of the fermions around
the Fermi level forms the condensate and the attractive interaction is much more complicated.
The quantum Monte Carlo (QMC)
methods have the advantage that the condensate can be detected directly,
by evaluating the off-diagonal two-particle density matrix and by monitoring
its behavior at large distances
\cite{Astrak05, Needs10}.
The goal of our study is twofold. First, any actual simulation involves
only a finite system, and the quantities relevant for the
thermodynamic limit have to be obtained from appropriate
extrapolations. The unitary
system is not trivial in this respect, since pairing with infinite
scattering length is described by a function with a slow fall-off
at large distances. It is necessary to analyze the finite-size
scaling of the quantities of interest and to test whether the actual limit of
infinite dilution, or, equivalently, of point-like character of the
interaction, has indeed been reached. Second, the impact of
the fixed-node approximation in the quantum Monte Carlo method
is not very well understood for this system since there is nothing
to compare with: so far the fixed-node formulation
of the QMC methods appears to be the only approach
that is able to provide an upper bound for the total energy.
This has motivated us to probe the accuracy of the nodes
by released-node QMC simulations and by improvements in the variational
flexibility of the employed wave functions.
We have carried out calculations
of the ground-state properties of the dilute unitary Fermi gas by the
fixed-node DMC (FN-DMC)\cite{Mitas01} method for 4, 14, 38 and 66
atoms. By using a more versatile
construction of the pair orbital in the BCS wave function
and by extrapolating the effective range of the two-particle
interactions $R_{\rm eff}$ to zero, we were able to obtain lower $\xi$
than, for example, the previously reported DMC calculations in Ref. \cite{Carlson03}
based on BCS trial functions.
These results suggest that
the extrapolation of $R_{\rm eff}$ is important,
especially for smaller systems.
In order to test the quality
of the nodal surface of the BCS wave function,
we have performed released-node DMC (RN-DMC)\cite{Ceperley80,Ceperley84} calculations
for 4 and 14 atoms. This procedure has been carried out starting from two types of
nodal constraints: from the BCS nodes and from the Hartree-Fock (HF) nodes.
Our RN-DMC results indicate that the nodal corrections
are driven mainly by long-range correlations which are difficult to sample
in the released-node framework due to the rapid growth of the bosonic noise.
We have calculated also the two-body density matrix and the condensate fraction
for the 66 atom system, and we
have estimated the corrections from the effective-range extrapolation on these quantities.
\section{method}
\subsection{Hamiltonian}
We consider a two-component Fermi gas with Hamiltonian
\begin{equation}
H=-\frac{1}{2}\sum_{i=1}^{N/2}\nabla_i^2 -
\frac{1}{2}\sum_{i'=1}^{N/2}\nabla_{i'}^2
+ \sum_{i,i'}V(r_{ii'})\,,
\end{equation}
where $N$ is the total number of atoms, $i$ and $i'$ correspond to
the spin-up and spin-down atoms, and $r_{ii'}$ denotes the distance $|\mathbf{r}_i-\mathbf{r}_{i'}|$.
The atoms are located in a cubic box
with the side $L$ and we impose the periodic boundary conditions. The
two-particle potential $V(r_{ii'})$ is taken in the P\"oschl-Teller
form
\begin{equation}
\label{eq:pot_PT}
V(r_{ii'})=-\frac{2\mu^2}{\cosh^2(\mu r_{ii'})}\,,
\end{equation}
whose effective range is $R_{\rm eff}=2/\mu$. The s-wave scattering
length $a_s$ is infinite for all values of $\mu\neq 0$.
\subsection{Trial wave functions}
In the majority of our calculations
we employ trial wave functions of the BCS
form multiplied with the Jastrow factor (BCS-Jastrow) as given by
\begin{equation}
\Psi_T(\textbf{R})=\Psi_{BCS}(\textbf{R})e^{J(\textbf{R})}\,,
\end{equation}
where
\begin{equation}
\Psi_{BCS}(\textbf{R})= \mathcal{A}\biggl[\prod_{i,i'=1}^{N/2}\phi(\mathbf{r}_i,\mathbf{r}_{i'})\biggr]
=\mathop{\rm det} [\phi(\mathbf{r}_i,\mathbf{r}_{i'})]\,.
\end{equation}
Here $\mathcal{A}$ represents the antisymmetrization operator and
$\phi(\mathbf{r}_i,\mathbf{r}_{i'})$ is the pair orbital. The vector
$\mathbf{R}$ encompasses all atomic coordinates $\mathbf{r}_i$ and
$\mathbf{r}_{i'}$. Additionally,
we have carried out a subset of calculations also
with the Hartree-Fock-Jastrow (HF-Jastrow) trial
functions, in which $\Psi_{BCS}$ is replaced with a product
of two Slater determinants of one-particle orbitals (simple plane
waves). The HF-Jastrow wave
function reads as
\begin{equation}
\Psi_{SJ}(\textbf{R})= \mathop{\rm det}[\varphi_a(\mathbf{r}_i)] \mathop{\rm det}[\varphi_a(\mathbf{r}_{i'})] e^{J(\textbf{R})}.
\end{equation}
The pair orbital $\phi(\mathbf{r}_i,\mathbf{r}_{i'})$ in $\Psi_{BCS}$ is written as a linear combination of
Gaussian functions
\begin{multline}
\phi (\mathbf{r}_i,\mathbf{r}_{i'})=\sum_{l,m,n=-1}^{1}\sum_{k} d_{k}
e^{-\alpha_{k}(x_i-x_i'+lL)^2}\\ \times e^{-\alpha_{k}(y_i-y_i'+mL)^2}e^{-\alpha_{k}(z_i-z_i'+nL)^2},
\end{multline}
where $d_{k}$ are expansion coefficients, and $\mathbf{r}_i=(x_i,y_i,z_i)$
and $\mathbf{r}_{i'}=(x_{i'},y_{i'},z_{i'})$ are coordinates of $i$ and $i'$ atoms inside
the simulation box of linear size $L$.
We choose sufficiently large exponents $\alpha_{k}$
so that only the first neighbor shell of periodic images
contributes to the sum, that is, the Gaussian functions are negligible at distances
larger than $3L/2$. The pair orbital is smooth with zero derivative at
the boundary of the simulation cell.
The Jastrow factor $J(\textbf{R})$ is constructed in a similar way as the pair orbital $\phi (\mathbf{r}_i,\mathbf{r}_{i'})$
for both different spin atoms and same spin atoms.
A typical trial wave function includes around 30 to 40 variational parameters that
are optimized by minimizing a linear combination of the total energy and its variance\cite{Umrigar05}.
Although the Jastrow factor does not change
the nodal surface, accurate description of the pair correlations
makes the variational optimization much more efficient and robust.
When the effective range of the potential approaches zero,
more Gaussian functions with larger exponents $\alpha_{k}$ are included in
the Jastrow factor in order to keep the accuracy of the trial function consistently high.
On the other hand, and somewhat surprisingly, we
find that similar adjustment of the pair orbital with the changing effective
range is relatively minor.
\subsection{Fixed-node and released-node DMC methods}
The DMC method projects out the ground state from a given trial function
$\Psi_T$ by means of an auxiliary evolution in the imaginary time,
$\Phi(\tau)\sim\exp(-\tau H)\Psi_T$.
By introducing importance sampling\cite{Mitas01} with the aid of a guiding function
$\Psi_G$, we can write an integral equation for
$\Phi(\textbf{R},\tau)$ in the form
\begin{multline}
\label{eq:DMC_integral}
\Psi_G(\textbf{R})\Phi(\textbf{R},\tau+\Delta\tau)= \\
\int d\textbf{R}' \frac{\Psi_G(\textbf{R})}{\Psi_G(\textbf{R}')} G(\textbf{R},\textbf{R}',\Delta\tau)
\Psi_G(\textbf{R}')\Phi(\textbf{R}', \tau)\,.
\end{multline}
For small $\Delta\tau$, the propagator $G(\textbf{R},\textbf{R}',\Delta\tau)$ can
be approximated using the Trotter-Suzuki formula as
\begin{multline}
\label{eq:Trotter}
\frac{\Psi_G(\textbf{R})}{\Psi_G(\textbf{R}')} G(\textbf{R},\textbf{R}',\Delta\tau)
\approx G_0(\textbf{R}, \textbf{R}'+\Delta\tau\mathbf{v}(\mathbf{R}'),\Delta\tau) \\
\times e^{-\Delta\tau[E_L(\textbf{R})+E_L(\textbf{R}')-2E_T]/2} ,
\end{multline}
where $\mathbf{v}(\mathbf{R}')\equiv\nabla\ln|\Psi_G(\textbf{R}')|$ and
$G_0(\textbf{R},\textbf{R}',\Delta\tau)$ is the Green's function for
non-interacting atoms that takes the form of the diffusion kernel. The
so-called local energy $E_L$ is given by
\begin{equation}
E_L(\textbf{R})=\frac{H\Psi_G(\textbf{R})}{\Psi_G(\textbf{R})}\,.
\end{equation}
The product $\Psi_G\Phi$ is represented by a set of samples (also
referred to as walkers) and this ensemble is evolved with the aid of a
stochastic process simulating Eqs.~\eqref{eq:DMC_integral}
and~\eqref{eq:Trotter}. In the fixed-node method we set $\Psi_G(\textbf{R})=\Psi_T(\textbf{R})$ and the fixed-node
condition is imposed by enforcing the sampling points to obey
\begin{equation}
\Psi_G(\textbf{R})\Phi(\textbf{R},\tau) \geq 0
\end{equation}
at all times. In the limit of long $\tau$ the solution
converges towards the lowest-energy state consistent with the boundary
conditions given by the fixed nodes.
\begin{figure}
\begin{center}
\includegraphics[height=3.00in,width=3.25in]{tot_extr.pdf}
\end{center}
\caption{The fixed-node energy for unpolarized unitary Fermi gas as a function
of
the interaction range $R_{\rm eff}/r_s$ with linear extrapolation to $R_{\rm eff}/r_s=0$.
The system sizes are 4, 14, 38
and 66 atoms from the top left to the bottom right.
The statistical error bars are smaller than the symbol size.}
\label{tot_extr}%
\end{figure}
In the RN-DMC method the guiding function has bosonic symmetry and
its square should be close to the square of the fermionic ground state.
We have used guiding functions in the form\cite{Ceperley84, Casulleras00}
\begin{equation}
\label{eq:RN_quide}
\Psi_G(\textbf{R})= \sqrt{\Psi_T^2(\textbf{R})+\alpha \left\langle \Psi_{T}^{2} \right\rangle}\,,
\end{equation}
where $\left\langle \Psi_{T}^{2}\right\rangle$ is the average value of $\Psi_{T}^2(\textbf{R}_0)$ over all configurations, and
$\textbf{R}_0$ are the walker positions right after the nodal
release.
The tunable parameter $\alpha$ controls the rate of walkers passing through
the nodal region.
The guiding function is non-negative everywhere and therefore
the stochastic process propagates
a mix of bosonic and fermionic states. The fermionic component
is filtered out
by reweighting with the factor
$\Psi_T/\Psi_G$ so that the fermionic-state
energy is given by
\begin{equation}
\langle\Phi_0|H|\Psi_T\rangle
=\frac{ \int d\textbf{R}\Phi_{0}(\textbf{R})\Psi_G(\textbf{R})\frac{\Psi_T(\textbf{R})}{\Psi_G(\textbf{R})}\frac{H\Psi_T(\textbf{R})}{\Psi_T(\textbf{R})} }
{ \int d\textbf{R}\Phi_{0}(\textbf{R})
\Psi_G(\textbf{R})\frac{\Psi_T(\textbf{R})}{\Psi_G(\textbf{R})} } \,,
\end{equation}
where $\Phi_0(\textbf{R})$ denotes the exact fermionic ground state.
Since this method is exponentially demanding both in the projection time
and in the number of atoms, it is important
to choose $\alpha$ so that the statistical information is recovered as quickly as
possible. If $\alpha$ is too large the fluctuations from the poor importance sampling
overwhelm any useful signal very rapidly. On the other hand, a too small value
can bias the results. Since it is difficult to reach reliable error bars
in this type of calculations, we have used the method mostly to identify the onset
and the amplitude of the energy decrease during the projection
period when the stochastic noise was acceptably small.
In the RN-DMC process, we can also pick up the statistical signal from the walkers
that have never crossed the nodal surface, and in essence this provides
the FN-DMC estimator.
By monitoring these paths as well, we can assess
the consistency of the estimators and somewhat better
tune the parameter $\alpha$ for providing better RN-DMC signal.
\begin{figure*}
[htb]
\includegraphics[
height=4in,
width=6.2500in
]%
{unpol4_compare_wn.pdf}%
\caption{The pair orbitals and FN-DMC and RN-DMC energies of the 4-atom unitary system
with $R_{\rm eff}/r_s=0.06397$.
The upper row shows
the pair orbitals with the lowest (left), intermediate (middle) and optimal
(right) accuracy with regard to the variational optimization.
The lower row shows the corresponding DMC energies
as functions of the projection time starting from the variational estimate.
Note that the resolution of the left and right
panels differs by an order of magnitude. The vertical dotted lines
indicate the instant of the nodal release.
}
\label{4p_wn}%
\end{figure*}
\section{Results}
For benchmark purposes we first
calculate perhaps the smallest nontrivial system---four atoms.
Our result is shown in Fig.~\ref{tot_extr}, upper left panel.
There is approximately $10\%$ energy drop when $R_{\rm eff}/r_s$ is
reduced from $0.1279$ to $0.003998$.
We extrapolate $R_{\rm eff}$
to zero using a linear fit and obtain $\xi_{2,2}=0.212(2)$. Here
and in the rest of the paper, the denominator $E_{\rm free}$ in the ratio
$\xi=E/E_{\rm free}$ is evaluated in the same finite volume subject to
the same boundary conditions as the nominator $E$.
The ground-state energy of this small and relatively simple system was
obtained also by two other numerical methods using a lattice
formulation of the unitary Fermi gas model: the iterative Lanczos
diagonalization and the auxiliary-field projection Monte Carlo
method\cite{Lee11}. The agreement between our fixed-node DMC results
and the outcome of these entirely different methods strongly suggests
that our range-extrapolated total energy of the four-atom system is
very accurate. This conclusion is further corroborated by our
released-node DMC results discussed below.
Our calculations with 14, 38 and 66 atoms are carried out analogously to
the 4-atom case,
and the extrapolated values of $\xi$ are 0.407(2), 0.409(3) and 0.398(3),
respectively, as
plotted in Fig.~\ref{tot_extr}.
In these calculations, the smallest effective range is $R_{\rm eff}/r_s=0.003125$.
It should be noted that in the previous DMC calculations,
the range of $R_{\rm eff}/r_s$ was between 0.1 to 0.2, and within this range our
DMC results agree well with the previously obtained values
\cite{Carlson04,Astrak04,Needs10}.
Reduction of $R_{\rm eff}$ decreases the energy in all cases
although the decrease per atom is smaller in larger systems.
To test the quality of the nodal surfaces, we have carried out
released-node calculations for 4-atom and 14-atom systems.
The RN-DMC calculations for 4 atoms were done with $R_{\rm eff}/r_s=0.06397$. In a typical released-node
run the number of walkers was about two million so that the error bars were initially
very small.
In Fig.~\ref{4p_wn}, the upper row shows the pair orbital along three
distinct directions (100, 110 and 111)
of the interparticle distance vector ${\bf r}_i-{\bf r}_j$.
The lower row shows the FN-DMC and RN-DMC energies as they evolve with the projection
time. The plots show convergence of the FN-DMC energy followed by the nodal release.
This is accomplished by switching
the guiding wave function from $\Psi_T(\textbf{R})$ to the bosonic
function $\Psi_G(\textbf{R})$ defined in Eq.~\eqref{eq:RN_quide}.
The released-node signal reflects the quality of the nodal surface of
the trial wave function employed in the FN-DMC simulation.
We have tested wave functions with intentionally
varied accuracy by employing suboptimal pair orbitals.
The plot of the energy evolution in the left panel of Fig.~\ref{4p_wn} shows
a clear and pronounced drop after the nodal release.
As the quality of the
pair orbital improves,
this drop shrinks. For the fully optimized BCS-Jastrow
wave function (the right panel in Fig.~\ref{4p_wn})
the energy is reduced by less than 0.002 within the longest projection time we have tried.
This fact as well as comparison with other methods\cite{Lee11}
indicate that our BCS wave functions are very accurate in this small
system and that the fixed-node error is marginal.
We observe an unexpectedly high sensitivity of the nodal quality to the details of the
pair orbital at large distances. This suggests an explanation for the
relatively slow convergence of the released-node energy: the
long-range tails of the pair orbital affect the nodal hypersurfaces,
although their contribution to the total energy is relatively small.
One can further deduce that this makes
the released-node method quite challenging to apply since
it requires sampling of long distances and the corresponding
correlations. This is, however, difficult to achieve because the
diffusive motion of walkers is slow, proportional to $t^{1/2}$, while
the growth of the noise is fast, proportional to $\exp(\Delta_{BF}t)$ where
$\Delta_{BF}$ is the difference between the bosonic and fermionic
ground-state energies.
The time step $\Delta\tau$ was set to $4\times10^{-5}r_s^2$ in all runs and
we have verified that the time-step bias of the RN-DMC results is negligible.
The converged RN-DMC energy should not depend on the parameter
$\alpha$ in the bosonic guiding function
$\Psi_G$ provided one would be able to evolve the stochastic process
with the error bars under control until
the full convergence. In the same time, the parameter $\alpha$ crucially affects the growth
of the fluctuations with the projection time
as illustrated in Fig.~\ref{4p_wn}.
\begin{figure
\begin{center}
\includegraphics[
height=2in,
width=2.800in
]%
{14p_rs5_u2.pdf}%
\caption{Evolution of the DMC energies for
the 14-atom system with the best optimized BCS-Jastrow wave function.
The runs are for $R_{\rm eff}/r_s=0.2$.
No statistically significant energy drop is observed after the nodal
release that is indicated with the vertical dotted line.}
\label{14p_rn}%
\end{center}
\end{figure}
The RN-DMC energy for 14 atoms with $R_{\rm eff}/r_s=0.2$ is shown in Fig.~\ref{14p_rn}. The error
bars are estimated from eight independent runs with two million walkers each.
In the interval of $E_Ft\leq 0.2$ after the nodal release the RN-DMC energy gain appears
to be very small and the error bars preclude to make any statistically
sound estimation for longer projection times.
The rapid loss of resolution is expected since
the difference between the bosonic and fermionic ground states grows with
the number of atoms.
Again, the RN-DMC signal exhibits little
dependence on $\alpha$ we choose.
In order to make a comparison with a case displaying a clear fixed-node bias, we have
carried out RN-DMC runs using the Slater-Jastrow trial wave function, see Fig.~\ref{14p_rn_hf}.
Since this wave function has the nodal surface of
the non-interacting Fermi gas,
the nodal surface is strongly distorted.
As a result, we see a very pronounced released-node signal.
However,
within the projection time interval of $E_Ft\leq0.2$, the energy drops by
only $\approx 0.015$ for the largest $\alpha$ we tested. This is very small considering
that the true ground-state energy is at least an order of magnitude lower. This again illustrates
the challenges of efficient application of the released-node method, at least for
present cases.
Comparison of the Slater and BCS wave functions shows
a significant effect of pairing as was demonstrated in earlier studies\cite{Carlson03}.
\begin{figure
\begin{center}
\includegraphics[
height=2.in,
width=2.800in
]%
{14p_rs5_u2hf.pdf}%
\caption{DMC energies for 14 atoms obtained with the Slater-Jastrow wave function.
The runs are for $R_{\rm eff}/r_s=0.2$. The RN-DMC energy drops are significant
when compared
to the RN-DMC signal from the BCS-Jastrow wave function. The parameter $\xi_{7,7}$
drops by $\approx$ 0.015 within $E_Ft\approx0.2$ after the nodal release.}
\label{14p_rn_hf}%
\end{center}
\end{figure}
In order to quantify the pairing effects
we calculate the two-body density matrix which enable us to evaluate the
condensate fraction.
The projected two-body density matrix for spin-up and spin-down atoms is defined as
\begin{equation}
\rho^{(2)}(\textbf{r})=\frac{N^2}{4V^2}
\frac{\int d\textbf{R} \Phi(\textbf{R})\Psi_T(\textbf{R}) \frac{\Psi_T(\textbf{r}_1+\textbf{r},\textbf{r}_2+\textbf{r})} {\Psi_T(\textbf{r}_1,\textbf{r}_2)}}
{\int d\textbf{R}\Phi(\textbf{R})\Psi_T(\textbf{R})}\,,
\end{equation}
where $N$ is the total number of atoms and $V$ is the volume of the simulation cell.
The density matrices have been calculated for the fixed-node
wave functions and hence they correspond to the mixed estimators
\cite{Mitas01}.
Nevertheless, the mixed-estimator bias is negligible since the
variational Monte Carlo and DMC estimates of $\rho^{(2)}$ coincide within
error bars. This is a further evidence of the high accuracy of our trial
wave functions.
The condensate fraction can be extracted from the two-body density matrix as
\begin{equation}c=\frac{2V^2}{N}\lim_{r \to \infty}\rho^{(2)}(r)\,.\end{equation}
The calculated density matrices are shown in Fig.~\ref{66p_rs10_dm} with
the condensate fraction estimated from the long-range limit.
The condensate fraction saturates
for $R_{\rm eff}\leq 0.5$ at $c=0.56(1)$.
This value is not too far from the results obtained previously \cite{Astrak05,Needs10}.
\begin{figure}
[ptb]
\begin{center}
\includegraphics[
height=2in,
width=2.800in
]%
{66p_rs10_dm.pdf}%
\caption{The two-body density matrix for 66 atoms calculated from the
FN-DMC mixed estimator. The condensate fraction converges
to 0.56(1) for $R_{\rm eff}/r_s\leq 0.05$.}
\label{66p_rs10_dm}%
\end{center}
\end{figure}
\begin{figure
\centering
\mbox{
{\resizebox{1.1in}{!}{\includegraphics{hf1}}
{\resizebox{1.1in}{!}{\includegraphics{unitary1}}}
{\resizebox{1.1in}{!}{\includegraphics{BEC1}}}
}
\mbox{
{\resizebox{1.1in}{!}{\includegraphics{hf2}}
{\resizebox{1.1in}{!}{\includegraphics{unitary2}}}
{\resizebox{1.1in}{!}{\includegraphics{BEC2}}}
}
\caption{Three-dimensional subsets of the nodal hypersurfaces for three types
of wave functions and corresponding phases in the 14-atom system.
The node is obtained by scanning the simulation cell with a pair of
spin-up and spin-down atoms sitting on the top of each other while keeping the rest
of the atoms at fixed positions (tiny spheres). From the left to the right,
the columns show the nodal surfaces of the wave functions corresponding
to the free Fermi gas,
the unitary limit and the BEC side of the crossover.
The lower row displays the same surfaces rotated by 45 degrees around the $z$-axis.
}
\label{node}
\end{figure}
To illustrate the character of the nodal surfaces in
the BCS-BEC systems, we present three-dimensional scans of the nodes for three wave functions corresponding
to the following scattering regimes:
the free atomic gas with no pairing, our best unitary-limit wave function, and the wave
function with enhanced pairing from the BEC side of the BEC-BCS phase
diagram ($a_sk_F=0.6592$).
The left column of Fig.~\ref{node} displays the nodal surface of the free atomic Fermi gas.
The delocalized nature of the system is apparent.
At the unitary limit, shown in the middle column of Fig.~\ref{node},
the shape of the nodal surface is significantly different as the pairing effects
clearly dominate and lead to a localized character of the nodes
from the perspective of a pair of up and down spin atoms.
The nodes on the BEC side (the right column) do not differ much from the unitary
limit, except for a slightly more pronounced localization.
\section{Conclusions}
We have carried
out QMC calculations of the zero-temperature, spin-unpolarized atomic Fermi gas
in the unitary limit.
We show that the interaction range impacts the resulting total energies significantly.
By extrapolating the interaction
range to zero we obtain the ratio $E/E_{\rm free}$
for 4, 14, 38 and 66 atoms to be 0.212(2), 0.407(2), 0.409(3) and 0.398(3), respectively.
Our results agree well with the previous fixed-node DMC
calculations when we employ similar simulation parameters such
as the atom density, the interaction range and the number of atoms.
From the released-node DMC calculations for 4 and 14 atoms
we have found that the convergence
to the correct and asymptotically exact ground-state energies is unfavorably slow compared to the
growth of the statistical noise.
We were able to identify only
small energy gains within the simulation times that allowed for
acceptable signal to noise ratio.
We have calculated the two-body density matrix and the condensate
fraction in the limit of zero interaction range,
and we have found only small changes in these quantities when compared with
the previous calculations. Our condensate
fraction from the fixed-node DMC simulations is 0.56(1).
During the preparation of our manuscript we became aware of a similar
study where an interaction-range extrapolation was also
performed\cite{Gandolfi10}. The final result for the 66-atom system was
$\xi_{33,33}=0.383(2)$, which is approximately 4\% lower than ours. We
believe that a large portion of this difference can be attributed to differences
in the functional forms of the pairing orbital. Some influence could come also due to the
differences between the extrapolation methods employed in this work and in
Ref.~\cite{Gandolfi10}.
\begin{acknowledgments}
This work is supported by ARO and by the NSF grants
DMR-0804549 and OCI-0904794.
\end{acknowledgments}
\bibliographystyle{apsrev4-1}
| -18,460.430813 |
[
-3.0390625,
2.783203125
] | 26.868327 |
[
-2.884765625,
0.261474609375,
-1.8720703125,
-5.67578125,
-0.89599609375,
7.89453125
] |
[
3.6015625,
8.125,
3.228515625,
5.88671875
] | 225 | 3,622 |
[
-3.60546875,
4.19140625
] | 25.521427 |
[
-6.2109375,
-4.3671875,
-4.48046875,
-2.5078125,
2.041015625,
12.6875
] | 1.834597 | 11.251632 | 28.989509 | 2.680375 |
[
1.6733005046844482
] | -13,364.533452 | 5.718664 | -18,160.686909 | 0.675904 | 5.742696 |
[
-2.810546875,
-4.02734375,
-3.59375,
-4.4765625,
2.484375,
12
] |
[
-5.421875,
-2.14453125,
-2.458984375,
-1.4580078125,
3.388671875,
4.5859375
] | |
BkiUdWg5qg5A59MXG8c0
|
\section{LE TRIPTYQUE DES TESTS CLASSIQUES DANS LE SYST\`EME SOLAIRE}
D\`es 1845, Le Verrier \`a l'observatoire de Paris (un an avant sa
d\'ecouverte de Neptune par le calcul \`a partir des perturbations
engendr\'ees sur Uranus), avait remarqu\'e que le demi grand-axe de
l'orbite de Mercure pr\'ecesse \`a chaque rotation avec un angle qui est
l\'eg\`erement en avance par rapport \`a la pr\'ediction th\'eorique
$\Delta_\mathrm{N}$. Son calcul de $\Delta_\mathrm{N}$, en th\'eorie de
Newton, \'etait fond\'e sur les perturbations induites par les autres
plan\`etes, principalement V\'enus qui est la plus proche de Mercure, et
Jupiter qui est la plus massive du syst\`eme solaire. L'avance anormale
du p\'erih\'elie \'etait rest\'ee inexpliqu\'ee et avait aliment\'e de
nombreuses sp\'eculations, parmi lesquelles l'existence d'une nouvelle
plan\`ete int\'erieure \`a l'orbite de Mercure (d\'enomm\'ee Vulcain par
Le Verrier), la pr\'esence possible d'un anneau de mati\`ere zodiacale
dans le plan de l'\'ecliptique, et m\^eme une modification de la loi
newtonienne en $1/r^2$. D\`es l'obtention des \'equations du champ
gravitationnel en novembre 1915, Einstein prouvera que les corrections
purement relativistes au mouvement d'une plan\`ete sur une ellipse
keplerienne impliquent une rotation suppl\'ementaire du grand axe de
l'ellipse donn\'ee par
$$\Delta_\mathrm{R} = \frac{6\pi GM_\odot}{c^2 a (1-e^2)}\,,$$ o\`u $a$
et $e$ sont le demi grand-axe et l'excentricit\'e de l'orbite, $M_\odot$
est la masse du Soleil, et $G$ et $c$ sont la constante de la
gravitation et la vitesse de la lumi\`ere. Num\'eriquement on trouve
43'' d'arc par si\`ecle, qui s'ajoutent donc \`a la pr\'ecession
newtonienne $\Delta_\mathrm{N}$ pour \^etre en parfait accord avec
l'observation~! C'est certainement ce succ\`es remarquable qui a
convaincu Einstein de la justesse de la th\'eorie naissante (c'\'etait
d'ailleurs \`a l'\'epoque la seule confrontation possible de la
th\'eorie \`a des observations r\'eelles).
Le deuxi\`eme ``test classique'', encore plus c\'el\`ebre, est celui de
l'angle de d\'eviation de la lumi\`ere en provenance d'une source lointaine
(un quasar dans les mesures r\'ecentes), par le champ de gravitation du
Soleil. Il est donn\'e en relativit\'e g\'en\'erale, dans le cas d'un rayon
rasant la surface du Soleil (rayon $R_\odot$), par
$$\alpha_\odot = \frac{4GM_\odot}{c^2 R_\odot}\,.$$ Cet angle vaut
\textit{deux} fois la valeur estim\'ee en th\'eorie de Newton, car en
effet si on consid\`ere la lumi\`ere comme faite de corpuscules de
vitesse $c$ (et de masse arbitraire, car la masse n'intervient pas), il
y a bien une d\'eviation de la lumi\`ere chez Newton~! En fait on peut
montrer que le facteur 4 dans l'expression de $\alpha_\odot$ se
d\'ecompose en ``2+2'', avec le premier 2 qui provient du principe
d'\'equivalence, $m_i=m_g$, qui est vrai en relativit\'e g\'en\'erale
comme en th\'eorie de Newton, et le second 2 qui est un effet
suppl\'ementaire d\^u \`a la courbure de l'espace en relativit\'e
g\'en\'erale. L'angle $\alpha_\odot$ vaut 1.75'' d'arc, et fut mesur\'e
lors d'une \'eclipse du Soleil par Eddington en 1919, qui put d'ores et
d\'ej\`a conclure que la th\'eorie de Newton \'etait exclue
exp\'erimentalement. (En cette ann\'ee du trait\'e de Versailles un
anglais mettait \`a mal la th\'eorie d'un autre anglais, et confirmait
exp\'erimentalement celle d'un allemand.)
L'effet Shapiro compl\`ete notre triptyque des tests classiques de la
relativit\'e g\'en\'erale dans le syst\`eme solaire. C'est un retard
d\^u au champ de gravitation dans les temps d'arriv\'ee de photons ayant
ras\'e la surface du Soleil. Non seulement la trajectoire de la
lumi\`ere est d\'evi\'ee de l'angle $\alpha_\odot$, mais les photons sur
leur trajectoire sont \textit{ralentis} par le champ du Soleil. L'effet
n'est pas du tout n\'egligeable, et il a \'et\'e calcul\'e et observ\'e
pour la premi\`ere fois par Shapiro en 1964. Son exp\'erience a
consist\'e \`a mesurer le temps d'aller-retour de photons radio \'emis
sur Terre vers Mercure, r\'efl\'echis sur le sol de Mercure et
renvoy\'es vers la Terre, lorsque la trajectoire des photons passe \`a
proximit\'e de la surface du Soleil. L'effet principal du ralentissement
de la lumi\`ere est donn\'e par
$$\Delta T =
\frac{4GM_\odot}{c^3}\log\left(\frac{4\,r_\oplus\,r_\otimes}{{R_\odot}^{\!2}}
\right)\,,$$ o\`u $r_\oplus$ et $r_\otimes$ sont les distances de la
Terre et de Mercure au Soleil. Contrairement aux autres tests, l'effet
Shapiro ne date pas de l'enfance de la relativit\'e g\'en\'erale.
Curieusement, Einstein n'a jamais pens\'e \`a calculer cet effet. Ayant
obtenu la trajectoire des photons au voisinage du Soleil et leur angle
de d\'eviation $\alpha_\odot$, il n'a apparemment jamais cherch\'e \`a
conna\^itre le mouvement ``horaire'' des photons sur leur trajectoire,
ce qui lui aurait donn\'e leur retard gravitationnel $\Delta T$ -- un
nouvel effet tout \`a fait int\'eressant.
La relativit\'e g\'en\'erale est maintenant v\'erifi\'ee dans le
syst\`eme solaire \`a mieux que 1/1000$^{\text{\`eme}}$ pr\`es. Des
mesures tr\`es pr\'ecises d'astrom\'etrie, telles que celles du futur
satellite GAIA qui sera lanc\'e par l'agence spatiale europ\'eenne,
devraient encore am\'eliorer la pr\'ecision sur la d\'eviation de la
lumi\`ere. Des th\'eories alternatives de la gravitation, comme par
exemple la th\'eorie de Brans et Dicke o\`u l'on rajoute un champ
scalaire au champ gravitationnel de la relativit\'e g\'en\'erale, et qui
fut une th\'eorie fameuse en son temps (1961), ont \'et\'e \'elimin\'ees
par ces observations. Cependant, dans le syst\`eme solaire, les vitesses
des corps sont tr\`es petites par rapport \`a la vitesse de la
lumi\`ere, $v\lesssim 10^{-4}\,c$, et le champ de gravitation est
faible, car le potentiel newtonien $U$ est tout petit en unit\'es
relativistes, $U\lesssim 10^{-6}\,c^2$. Les tests classiques n'ont donc
v\'erifi\'e qu'un r\'egime assez restreint de la th\'eorie, celui de sa
limite ``quasi-newtonienne''.
\section{LA RELATIVIT\'E G\'EN\'ERALE, UN OUTIL POUR
L'ASTROPHYSIQUE}
Apr\`es son enfance brillante, la vieille dame a connu une adolescence
difficile. Elle fut longtemps consid\'er\'ee comme un ``paradis pour le
th\'eoricien'', mais un ``d\'esert pour l'exp\'erimentateur''. En fait,
elle est rest\'ee \`a l'\'ecart du courant principal de la physique,
domin\'e par la m\'ecanique quantique et la th\'eorie quantique des
champs, jusqu'au d\'ebut des ann\'ees 1960 (notre th\'eorie est alors
quadrag\'enaire), \'epoque \`a laquelle elle a subi un renouveau et un
essor remarquables.
Du point de vue th\'eorique, cette \'epoque a vu l'\'elucidation du
concept de trou noir, et la magnifique d\'ecouverte par Kerr du trou
noir en rotation (1963). Le trou noir de Schwarzschild, sans rotation,
date des premiers mois de la relativit\'e g\'en\'erale, mais \`a
l'\'epoque on consid\'erait cette solution comme valable uniquement \`a
l'ext\'erieur d'une \'etoile, et ce n'est que dans les ann\'ees 60 que
l'on analysera les propri\'et\'es du trou noir au voisinage de
l'horizon, pour comprendre que ces objets peuvent r\'eellement exister
dans la nature. De m\^eme pour le rayonnement gravitationnel, dont on a
vraiment compris les caract\'eristiques pendant cette p\'eriode;
auparavant une controverse faisait rage sur l'existence r\'eelle des
ondes gravitationnelles~!
On peut dire que les exp\'eriences modernes de gravitation ont
commenc\'e avec la v\'erification pr\'ecise en laboratoire du d\'ecalage
gravitationnel vers le rouge ou effet Einstein, par Pound et Rebka en
1960. Souvent consid\'er\'ee comme le 4$^{\text{\`eme}}$ test classique
de la th\'eorie, cette v\'erification est en fait un test du principe
d'\'equivalence qui est donc plus g\'en\'eral. \`A la m\^eme \'epoque on
tentait la d\'etection du rayonnement gravitationnel \`a l'aide d'un
cylindre m\'etallique r\'esonnant appel\'e maintenant barre de Weber.
La relativit\'e g\'en\'erale \'emerge alors enfin en tant que th\'eorie
\textit{physique}, qui fait des pr\'edictions et voit ses pr\'edictions
r\'ealis\'ees. La d\'ecouverte en 1974 du pulsar binaire PSR~1913+16, et
la preuve exp\'erimentale de l'existence du rayonnement gravitationnel
tel qu'il est pr\'evu par la relativit\'e g\'en\'erale, illustre
merveilleusement la capacit\'e de pr\'ediction de notre th\'eorie (voir
la section suivante).
Aujourd'hui la relativit\'e g\'en\'erale est un ``outil'' permettant
d'explorer l'existence et de comprendre les observations de nouveaux
objets ou de nouveaux ph\'enom\`enes en astrophysique. Par exemple les
propri\'et\'es particuli\`eres du trou noir de Kerr sont utilis\'ees par
les astrophysiciens travaillant sur les objets compacts et les disques
d'accr\'etion autour de trous noirs. La relativit\'e g\'en\'erale va
probablement permettre d'ouvrir une nouvelle ``fen\^etre'' en
astronomie, celle des ondes gravitationnelles, car ce rayonnement a des
propri\'et\'es sp\'ecifiques tr\`es diff\'erentes des ondes
\'electromagn\'etiques.
Il faut pourtant garder \`a l'esprit que le domaine o\`u s'exerce la
relativit\'e g\'en\'erale est le \textit{macrocosme}. Cette th\'eorie
n'incorpore pas les lois de la m\'ecanique quantique, et il est probable
qu'elle doive \^etre consid\'er\'ee comme une th\'eorie ``effective''
valable uniquement \`a grande \'echelle. Assez \'etrangement, la force
gravitationnelle n'a pu \^etre test\'ee en laboratoire que jusqu'\`a une
\'echelle de l'ordre du millim\`etre. \`A une \'echelle microscopique,
inf\'erieure ou tr\`es inf\'erieure au millim\`etre, on ne conna\^it
exp\'erimentalement rien de la loi gravitationnelle et il est
vraisemblable que la relativit\'e g\'en\'erale \textit{stricto sensu} ne
s'applique plus.
\section{LE PULSAR BINAIRE PSR~1913+16}
L'ann\'ee 1974 fut faste pour les ``relativistes'' avec la d\'ecouverte
par Hulse et Taylor d'un syst\`eme extr\^emement int\'eressant~: le
pulsar binaire PSR~1913+16, qui valut \`a ses d\'ecouvreurs le prix
Nobel en 1993. C'est un pulsar, c'est-\`a-dire une \'etoile \`a neutrons
en rotation rapide sur elle-m\^eme (avec une p\'eriode de
$56\,\mathrm{ms}$), qui envoie \`a chaque rotation, tel un phare, du
rayonnement \'electromagn\'etique radio en direction de la Terre.
L'analyse des instants d'arriv\'ee des pulses radio montre (gr\^ace \`a
leur d\'ecalage Doppler) que PSR~1913+16 est en orbite autour d'une
\'etoile compagnon, probablement une autre \'etoile \`a neutrons.
L'orbite est une ellipse quasi-keplerienne de p\'eriode orbitale
$P\simeq 7^\mathrm{h}40^\mathrm{mn}$, d'excentricit\'e $e\simeq 0.617$
et de demi grand-axe $a\simeq 10^6\,\mathrm{km}$. Les masses du pulsar
et de son compagnon ($m_p$ et $m_c$) sont toutes deux environ \'egales
\`a $1.4\,M_\odot$ (qui est la masse des \'etoiles \`a neutrons).
PSR~1913+16 est un syst\`eme passionnant car les effets relativistes
jouent un r\^ole important dans sa dynamique. Par exemple, la
pr\'ecession relativiste $\Delta_\mathrm{R}$ du p\'eriastre de l'orbite
est de l'ordre de 4 degr\'es par an, \`a comparer avec les 43'' arc par
si\`ecle du p\'erih\'elie de Mercure.
Le syst\`eme double form\'e par le pulsar et son compagnon \'emet du
rayonnement gravitationnel, ce qui se traduit par une perte d'\'energie
orbitale, et donc par le rapprochement des deux \'etoiles l'une de
l'autre, et une lente d\'erive de la p\'eriode orbitale du mouvement
($\dot{P}<0$). On sait qu'en premi\`ere approximation le rayonnement
gravitationnel est quadrupolaire et le flux du rayonnement est donn\'e
par la formule dite du quadrupole d'Einstein. Si l'on applique cette
formule \`a un syst\`eme de deux masses ponctuelles en mouvement sur une
ellipse keplerienne on trouve un r\'esultat d\^u \`a Peters et Mathews
(1963),
$$\dot{P}=-\frac{192\pi}{5c^5} \left(\frac{2\pi
G}{P}\right)^{5/3}\!\!\!\!
\frac{m_p\,m_c}{(m_p+m_c)^{1/3}}\frac{1+\frac{73}{24}e^2+\frac{37}{96}
e^4}{(1-e^2)^{7/2}}\,.$$ Dans le cas du pulsar binaire cette formule
donne $\dot{P}=-2.4~10^{-12}\,\mathrm{s/s}$, qui repr\'esente donc la
d\'ecroissance de la p\'eriode orbitale mesur\'ee en secondes \`a chaque
seconde. Cette pr\'ediction purement th\'eorique est en excellent accord
(\`a mieux que $0.5\%$ pr\`es) avec les observations effectu\'ees par
Taylor et ses collaborateurs. C'est une v\'erification remarquable,
l'une des confirmations les plus importantes de la relativit\'e
g\'en\'erale, et l'une des mesures les plus pr\'ecises effectu\'ees en
astronomie. Mais pour nous elle repr\'esente une validation
observationnelle de l'ordre newtonien sur lequel est fond\'e nos
investigations post newtoniennes pour les binaires compactes spiralantes
(voir la derni\`ere section)~!
\section{LA D\'ETECTION INTERF\'EROM\`ETRIQUE DU RAYONNEMENT
GRAVITATIONNEL}
Une onde gravitationnelle est engendr\'ee par le mouvement
acc\'el\'er\'e des corps massifs. Dans notre th\'eorie, le champ de
gravitation est repr\'esent\'e par la m\'etrique de l'espace-temps, et
l'on peut montrer que les composantes de la m\'etrique ob\'eissent en
premi\`ere approximation, dans un syst\`eme de coordonn\'ees
particulier, \`a une \'equation de d'Alembert ou \'equation des ondes.
On a donc affaire \`a une onde gravitationnelle, qui doit \^etre vue
comme une perturbation de la surface de l'espace-temps se propageant \`a
la vitesse de la lumi\`ere. L'action de l'onde gravitationnelle sur la
mati\`ere se traduit par des d\'eformations analogues \`a celles
produites par un champ de mar\'ee. La variation relative de la taille
$L$ d'un d\'etecteur au passage de l'onde est donn\'ee par
$$\frac{\delta L}{L}\simeq\frac{h}{2}\,,$$ o\`u $h$ repr\'esente l'amplitude
de l'onde gravitationnelle, c'est-\`a-dire la modification de la m\'etrique de
l'espace-temps par rapport \`a la m\'etrique ``plate'' de Minkowski en
l'absence du champ gravitationnel.
De nouvelles exp\'eriences vont tenter pour la premi\`ere fois de
d\'etecter le rayonnement gravitationnel produit par des sources
cosmiques. Ces exp\'eriences sont fond\'ees sur l'interf\'erom\`etrie
\`a laser, et sont constitu\'ees de gigantesques interf\'erom\`etres de
Michelson avec cavit\'es Fabry-Perot. Elles ambitionnent de former un
r\'eseau international, comprenant des interf\'erom\`etres de grande
taille, LIGO aux \'Etats-Unis dont les bras ont une longueur de
$4\,\mathrm{km}$, et VIRGO qui est construit pr\`es de Pise avec des
bras de $3\,\mathrm{km}$ dans le cadre d'une collaboration
franco-italienne (voir la figure 1). Le r\'eseau comprend aussi des
d\'etecteurs de taille plus modeste, GEO \`a Hanovre et TAMA au Japon.
Le grand int\'er\^et de l'interf\'erom\'etrie \`a laser pour la
d\'etection des ondes gravitationnelles, est la bande de fr\'equence
tr\`es large du d\'etecteur, typiquement de $\sim 10\,\mathrm{Hz}$ \`a
$1000\,\mathrm{Hz}$ pour VIRGO. Les barres de Weber en revanche ne
peuvent d\'etecter qu'au voisinage de la fr\'equence de r\'esonance de
la barre.
\begin{figure}
\begin{center}
\epsfig{file=fig1.eps,width=3.4in}
\caption{Vue du d\'etecteur d'ondes gravitationnelles VIRGO
\`a Cascina, pr\`es de Pise.}
\end{center}
\end{figure}
L'amplitude attendue pour une onde gravitationnelle en provenance de
syst\`emes binaires \`a une distance de 100 Mpc est $h\sim 10^{-23}$, ce
qui, d'apr\`es l'estimation pr\'ec\'edente sur la variation de longueur
du d\'etecteur, donne $\delta L\sim 10^{-20}\,\mathrm{m}$ dans le cas de
VIRGO soit $10^{-5}\,\mathrm{fermi}$~! Comment est-il possible de
mesurer un tel d\'eplacement~? La r\'eponse est qu'en r\'ealit\'e on
mesure le d\'eplacement \textit{collectif} de $N$ atomes des miroirs en
entr\'ee et en bout de bras de l'interf\'erom\`etre. La mesure contient
donc un effet de moyenne sur les atomes ce qui permet de gagner un
facteur $\sqrt{N}$. Avec $N\sim 10^{18}$ ce qui correspond \`a une
couche atomique en surface du miroir on constate que la mesure effective
\`a r\'ealiser est beaucoup plus raisonnable, $\delta
L_\mathrm{eff}=\sqrt{N}\,\delta L\sim 10^{-11}\,\mathrm{m}$.
Il existe de nombreuses sources astrophysiques potentielles dont le
rayonnement gravitationnel pourrait \^etre d\'etect\'e par VIRGO et
LIGO. Les supernov\ae, qui sont des explosions d'\'etoiles massives en
fin de vie lorsqu'elles ont \'epuis\'e tout leur ``combustible''
nucl\'eaire, ont longtemps \'et\'e consid\'er\'ees comme des sources
d'ondes gravitationnelles int\'eressantes, mais on sait maintenant
qu'elles engendrent en fait peu de rayonnement. En effet l'effondrement
des couches internes de la supernova, qui devrait \^etre responsable de
la production du rayonnement gravitationnel, est essentiellement
sph\'erique, et d'apr\`es un th\'eor\`eme fameux de relativit\'e
g\'en\'erale, le champ ext\'erieur \`a une distribution sph\'erique de
mati\`ere est donn\'e par la solution de Schwarzschild qui est
\textit{statique} -- il n'y a donc pas de rayonnement \'emis. Beaucoup
plus int\'eressants pour VIRGO et LIGO sont les syst\`emes binaires, car
leur dynamique est fortement asym\'etrique et ils engendrent beaucoup de
rayonnement gravitationnel.
\section{SPIRALE ET MORT DES SYST\`EMES BINAIRES D'\'ETOILES
COMPACTES}
Une binaire compacte spiralante est ce que deviendra le pulsar binaire
PSR~1913+16 dans quelques centaines de millions d'ann\'ees, lorsqu'il
finira par fusionner avec son compagnon. Pendant toute sa vie il aura
\'emis son \'energie de liaison gravitationnelle sous forme de
rayonnement gravitationnel, jusqu'\`a ce que les deux \'etoiles \`a
neutrons tombent l'une sur l'autre. (Rappelons qu'une \'etoile \`a
neutrons est un astre compact, form\'e essentiellement de neutrons avec
une densit\'e comparable \`a celle de la mati\`ere nucl\'eaire, et dont
la taille est \`a peu pr\`es celle de l'agglom\'eration parisienne pour
une masse de $\sim 1.4\,M_\odot$.)
Dans les derniers instants avant la fusion finale, les deux objets
compacts (\'etoiles \`a neutrons ou trous noirs) d\'ecrivent une orbite
rapproch\'ee qui a la forme d'une spirale circulaire rentrante \`a cause
de la perte d'\'energie li\'ee \`a l'\'emission du rayonnement
gravitationnel. C'est ce rayonnement que l'on observera sur Terre o\`u
il d\'eformera l'espace-temps avec une amplitude relative de l'ordre de
$10^{-23}$ (voir la figure 2). Au cours de la phase spiralante, la
distance entre les deux \'etoiles diminue au cours du temps, et la
fr\'equence orbitale du mouvement, $\omega=2\pi/P$ o\`u $P$ est la
p\'eriode, augmente. On peut montrer que l'\'evolution de l'orbite est
adiabatique, dans le sens o\`u le changement relatif de fr\'equence
pendant une p\'eriode correspondante reste faible,
$\dot{\omega}/\omega^2 \lesssim 0.1$. Nous allons voir comment cette
propri\'et\'e d'adiabaticit\'e permet de d\'efinir un sch\'ema
d'approximation tr\`es puissant en relativit\'e g\'en\'erale, capable de
d\'ecrire la spirale avec grande pr\'ecision.
\begin{figure}
\begin{center}
\epsfig{file=fig2.eps,width=3.4in}
\caption{Onde gravitationnelle $h(t)$ \'emise par une binaire compacte
spiralante. La fr\'equence et l'amplitude de l'onde augmentent
adiabatiquement au cours du temps.}
\end{center}
\end{figure}
\`A la fin de la phase spiralante, le syst\`eme binaire atteint ce qu'on
appelle la derni\`ere orbite circulaire (dite aussi ``innermost circular
orbit'' ou ICO), \`a partir de laquelle l'\'evolution de l'orbite cesse
d'\^etre adiabatique. Les deux corps plongent alors l'un sur l'autre et
fusionnent tr\`es rapidement pour former un trou noir unique. \`A cause
de la dynamique violente qui conduit \`a la formation de ce trou noir,
celui-ci est initialement tr\`es d\'eform\'e et soumis \`a d'importantes
vibrations, mais il finira par atteindre, avec l'\'emission de ses modes
de vibration intrins\`eques en ondes gravitationnelles, un r\'egime
stationnaire d\'ecrit par la solution de Kerr pour le trou noir en
rotation.
Les binaires compactes spiralantes sont des syst\`emes parmi les plus
relativistes que l'on puisse imaginer, \`a la fois du point de vue de la
relativit\'e restreinte, car la vitesse orbitale atteint $\sim 0.5\,c$
au moment du passage \`a la derni\`ere orbite circulaire, et de la
relativit\'e g\'en\'erale, car les masses en jeu sont importantes,
$1.4\,M_\odot$ pour les \'etoiles \`a neutrons et peut-\^etre jusqu'\`a
$20\,M_\odot$ pour des trous noirs, donc les champs gravitationnels sont
intenses. Des milliers de cycles orbitaux sont parcourus en quelques
secondes avant la fusion finale, dans ce qu'on peut d\'ecrire de
fa\c{c}on imag\'ee comme une ``spirale infernale'', constituant les
derniers spasmes de l'agonie du syst\`eme binaire. La spirale et la mort
des binaires compactes repr\'esente donc un \'ev\'enement tout \`a fait
impressionnant du point de vue de l'astrophysique. C'est principalement
pendant la phase spiral\'ee pr\'ec\'edant imm\'ediatement la fusion que
l'onde gravitationnelle qui sera d\'etect\'ee par VIRGO et LIGO est
produite.
De tels \'ev\'enements cataclysmiques de fusion d'objets compacts se
produisent dans l'univers. \`A partir des syst\`emes binaires
d'\'etoiles \`a neutrons connus dans notre Galaxie (le pulsar binaire
PSR~1913+16 en est l'exemple le plus c\'el\`ebre), on peut en d\'eduire
que quelques fusions d'\'etoiles \`a neutrons devraient survenir par an
dans un volume de environ $100\,\mathrm{Mpc}$ de rayon centr\'e sur
notre Galaxie. Un tel volume contient des milliers de galaxies, dont
toutes celles de l'amas de la Vierge (qui a donn\'e son nom \`a VIRGO)
situ\'e au centre du super-amas de galaxies dans lequel nous vivons. \`A
cette distance le signal gravitationnel sera assez puissant pour \^etre
observ\'e par le r\'eseau actuel des d\'etecteurs ou par une
g\'en\'eration de d\'etecteurs de sensibilit\'e am\'elior\'ee. Il
rentrera dans leur bande de fr\'equence quelques minutes avant la
fusion, lorsque la fr\'equence du signal gravitationnel atteindra $f\sim
10\,\mathrm{Hz}$ (d'apr\`es les propri\'et\'es des ondes
gravitationnelles la fr\'equence de l'harmonique principale du signal
est double de la fr\'equence orbitale du mouvement, $f=\omega/\pi=2/P$).
VIRGO, qui a de tr\`es bonnes performances \`a basse fr\'equence gr\^ace
\`a ses tours d'isolation du bruit sismique terrestre, pourra notamment
capter tr\`es t\^ot le signal des binaires spiralantes d'\'etoiles \`a
neutrons, et augmenter ainsi le rapport signal-sur-bruit par une longue
int\'egration du signal \`a partir de la basse fr\'equence.
L'existence des binaires spiralantes de trous noirs est plus incertaine,
car malheureusement on ne conna\^it pas de syst\`emes de deux trous
noirs dans notre Galaxie. N\'eanmoins, gr\^ace \`a la simulation
num\'erique des phases successives d'\'evolution des syst\`emes
binaires, on peut estimer que le taux de fusion de deux trous noirs
pourrait \^etre comparable ou sup\'erieur \`a celui des \'etoiles \`a
neutrons. Les astrophysiciens pensent souvent que la premi\`ere
d\'etection du rayonnement gravitationnel par VIRGO et LIGO sera celle
d'une binaire spiralante de trous noirs.
\section{LE PROBL\`EME DES DEUX CORPS EN RELATIVIT\'E
G\'EN\'ERALE}
Pour le th\'eoricien ``relativiste'' l'int\'er\^et principal des
binaires compactes spiralantes r\'eside dans le fait que l'onde
gravitationnelle qu'elles \'emettent est \textit{calculable} avec grande
pr\'ecision, ind\'ependamment des d\'etails de la structure interne des
\'etoiles compactes, et de la pr\'esence possible d'un environnement
astrophysique ``sale''. Par exemple les effets non gravitationnels, qui
compliquent habituellement l'astrophysique des syst\`emes binaires
(champs magn\'etiques, pr\'esence d'un milieu interstellaire,
\textit{etc.}), sont n\'egligeables. La dynamique des binaires
spiralantes est domin\'ee par les effets gravitationnels orbitaux (les
effets de mar\'ees jouent tr\`es peu de r\^ole dans le cas de corps
compacts). On peut donc mod\'eliser le syst\`eme par deux particules
ponctuelles, sans structure interne, caract\'eris\'ees uniquement par
leurs masses $m_1$ et $m_2$, et aussi \'eventuellement par leurs spins.
Ainsi le probl\`eme th\'eorique des binaires spiralantes est un pur
probl\`eme de m\'ecanique c\'eleste~: le probl\`eme des deux corps
(consid\'er\'es comme ponctuels) en relativit\'e g\'en\'erale. On sait
qu'en th\'eorie de Newton le probl\`eme des 2 corps est
``int\'egrable'', mais qu'\`a partir de 3 corps les \'equations du
mouvement ne peuvent pas \^etre r\'esolues dans le cas g\'en\'eral. En
relativit\'e g\'en\'erale m\^eme pour 2 corps on ne peut pas \'ecrire de
fa\c{c}on exacte les \'equations du mouvement et encore moins les
r\'esoudre~! Le probl\`eme \`a un corps en revanche admet une solution
exacte qui est donn\'ee par la m\'etrique du trou noir de Schwarzschild.
N'admettant pas de solution exacte le probl\`eme relativiste des deux
corps doit \^etre trait\'e par des m\'ethodes d'approximations. Ce n'est
pas un drame car pratiquement tous les grands succ\`es de la
relativit\'e g\'en\'erale dans sa confrontation avec l'exp\'erience et
l'observation ont \'et\'es obtenus gr\^ace \`a de telles m\'ethodes.
Notre sch\'ema d'approximation fait intervenir ce que nous avons dit \`a
propos du spiralement adiabatique des binaires compactes. Il se trouve
en effet que le petit param\`etre adiabatique est en fait petit dans le
sens \textit{post newtonien}, car on a $\dot{\omega}/\omega^2 =
\mathcal{O}\left[(v/c)^5\right]$, o\`u $v$ est la vitesse orbitale des
corps et $c$ la vitesse de la lumi\`ere. L'approximation post
newtonienne consiste \`a d\'evelopper la relativit\'e g\'en\'erale
autour de la th\'eorie de Newton, sous la forme d'un d\'eveloppement en
puissances de $v/c$ lorsque $v/c\rightarrow 0$, ce qui peut \^etre vu
plus formellement comme un d\'eveloppement quand $c\rightarrow +\infty$.
L'ordre d'approximation $(v/c)^5$ correspond au premier effet de ce
qu'on appelle la \textit{r\'eaction de rayonnement}, c'est-\`a-dire
l'influence de l'\'emission du rayonnement gravitationnel sur le
mouvement du syst\`eme binaire, qui se traduit par un petit effet de
``freinage'' des corps sur leur orbite, et donc par une d\'ecroissance
de la p\'eriode orbitale $P$.
L'approximation post newtonienne est la seule technique connue qui
permet de d\'ecrire la phase spiralante des binaires compactes (pendant
laquelle on a $v/c\lesssim 0.5$), et elle est valable jusqu'\`a la
derni\`ere orbite circulaire. Pass\'ee cette orbite le d\'eveloppement
post newtonien devrait en principe \^etre remplac\'e par un calcul
d'int\'egration \textit{num\'erique} des \'equations d'Einstein. Un tel
calcul est indispensable pour d\'ecrire en d\'etail le m\'ecanisme de
fusion des deux horizons des trous noirs, et obtenir la forme d'onde
gravitationnelle produite lors de cette phase. Malheureusement la
relativit\'e num\'erique n'a pas encore r\'eussi \`a r\'esoudre ce
probl\`eme extr\^emement difficile, bien qu'il ait \'et\'e l'objet de ce
qu'on a appel\'e le ``binary black hole Grand challenge'', qui a
mobilis\'e de nombreux instituts am\'ericains mais n'a pas apport\'e les
r\'esultats escompt\'es. Il se trouve que vouloir calculer
num\'eriquement la fusion de deux trous noirs en utilisant la ``force
brute'' d'un ordinateur n'est pas r\'ealisable actuellement, malgr\'e
certaines perc\'ees remarquables ces derni\`eres ann\'ees. Heureusement,
dans le cas d'\'etoiles \`a neutrons ou de trous noirs peu massifs, la
plus grande partie du rapport signal-sur-bruit dans VIRGO et LIGO
proviendra de la phase spiralante pr\'ec\'edant la fusion, qui est
tr\`es bien d\'ecrite par la th\'eorie post newtonienne.
\section{PATRONS D'ONDES GRAVITATIONNELLES POUR VIRGO ET LIGO}
Le d\'eveloppement post newtonien va s'av\'erer l'outil id\'eal pour le
calcul de la radiation gravitationnelle d'une binaire compacte
spiralante. Et comme l'approximation post newtonienne va devoir \^etre
d\'evelopp\'ee jusqu'\`a un ordre tr\`es \'elev\'e ce probl\`eme va
devenir un vrai paradis pour le th\'eoricien~! Des \'etudes d'analyse du
signal dans les d\'etecteurs VIRGO et LIGO ont en effet montr\'e qu'une
pr\'ediction tr\`es pr\'ecise de la relativit\'e g\'en\'erale est
n\'ecessaire pour tirer parti de toute l'information potentielle
contenue dans le signal des binaires spiralantes. Pour d\'etecter le
signal gravitationnel (et l'analyser ult\'erieurement) on utilise la
pr\'ediction th\'eorique, que l'on appelle pour l'occasion le ``patron''
d'onde gravitationnelle, et on effectue sa corr\'elation avec le signal
de sortie du d\'etecteur. Si le patron est une copie fid\`ele du signal
r\'eel (c'est-\`a-dire si la pr\'ediction de la relativit\'e
g\'en\'erale est correcte), alors la corr\'elation est importante, et
l'on aura d\'etect\'e une onde gravitationnelle.
Il a \'et\'e montr\'e que les patrons d'ondes doivent prendre en compte
toutes les corrections relativistes dans le champ d'ondes
gravitationnelles jusqu'\`a la troisi\`eme approximation post
newtonienne, qui correspond \`a une pr\'ecision relative incluant tous
les termes jusqu'\`a l'ordre $(v/c)^6$ par rapport \`a la formule du
quadrupole d'Einstein pour le rayonnement gravitationnel. Dans le jargon
cette approximation s'appelle 3PN, et plus g\'en\'eralement les termes
post newtoniens $\sim (v/c)^{2n}$ sont dits d'ordre $n$PN.
Pour mieux comprendre ce que signifie la pr\'ecision 3PN, rappelons-nous
que la formule du quadrupole d'Einstein d\'ecrit l'\'emission des ondes
gravitationnelles \`a l'ordre dominant (quadrupolaire en relativit\'e
g\'en\'erale), qui est newtonien dans le sens o\`u \`a cet ordre
d'approximation le quadrupole peut \^etre calcul\'e avec la loi
newtonienne de la gravitation. C'est la formule ``newtonienne'' du
quadrupole qui permet d'expliquer le ph\'enom\`ene de r\'eaction de
rayonnement dans le pulsar binaire PSR~1913+16, dont on a d\'ej\`a vu
qu'il correspond lui-m\^eme \`a une correction dans l'\'equation du
mouvement \`a l'ordre $(v/c)^5$ soit 2.5PN. La pr\'ecision demand\'ee
pour les binaires spiralantes correspond donc en fait \`a une
contribution d'ordre 3PN+2.5PN c'est-\`a-dire $\sim (v/c)^{11}$ dans les
\'equations du mouvement de la binaire~!
C'est la premi\`ere fois dans l'histoire de la relativit\'e g\'en\'erale
que la r\'ealisation d'exp\'eriences nouvelles suscite des
d\'eveloppements th\'eoriques nouveaux. M\^eme pendant la p\'eriode de
son renouveau, notre vieille dame, confront\'ee \`a des observations et
des tests jamais effectu\'es auparavant, s'\'etait d\'edaigneusement
content\'ee de voir ses pr\'edictions d\'ej\`a ``sur \'etag\`ere''
confirm\'ees. Avant le d\'emarrage de la construction de LIGO et VIRGO
on pensait que les corrections relativistes \`a la formule du quadrupole
d'Einstein n'avaient qu'un int\'er\^et purement acad\'emique (par
exemple ces corrections sont compl\`etement n\'egligeables dans le cas
du pulsar binaire). La pr\'ediction th\'eorique ad\'equate pour les
binaires spiralantes n'existait pas, et il a fallu la d\'evelopper
sp\'ecialement dans le but de fournir les patrons d'ondes n\'ecessaires
\`a l'analyse du signal dans LIGO/VIRGO. Nous avons employ\'e des
m\'ethodes perturbatives analytiques, permettant d'it\'erer les
\'equations d'Einstein sous la forme d'une s\'erie post newtonienne,
d'abord pour des syst\`emes isol\'es g\'en\'eraux, puis dans
l'application \`a des syst\`emes binaires compacts. La pr\'ediction
th\'eorique de la relativit\'e g\'en\'erale, \`a la pr\'ecision 3PN et
m\^eme \`a 3.5PN, est maintenant sur \'etag\`ere (voir la section
suivante). La vieille dame est heureuse.
Un aspect int\'eressant de l'analyse du signal dans VIRGO et LIGO, qui
r\'esulte directement de l'application du d\'eveloppement post
newtonien, est la possibilit\'e d'effectuer des tests nouveaux de la
relativit\'e g\'en\'erale. Dans les patrons d'ondes des binaires
compactes spiralantes, d\'evelopp\'es \`a 3.5PN, existent en effet
plusieurs signatures caract\'eristiques de ce qu'on appelle les
``sillages'' (ou ``tails'') d'ondes gravitationnelles, produits par la
diffusion du rayonnement gravitationnel se propageant dans un
espace-temps distordu par la pr\'esence de la source elle-m\^eme. Cet
effet purement non-lin\'eaire dans la propagation du rayonnement
gravitationnel de sa source vers le d\'etecteur pourra \^etre observ\'e
pour la premi\`ere fois par comparaison des patrons d'ondes avec le
signal r\'eel dans VIRGO/LIGO. On ne conna\^it pas d'autres syst\`emes
que la spirale des binaires compactes, pour lesquels la d\'etection d'un
effet aussi fin que le sillage d'onde gravitationnelle soit possible.
\section{LA FASCINANTE APPROXIMATION 3PN}
Lors du calcul des patrons d'ondes de binaires compactes spiralantes,
l'approximation $\mathrm{3PN}$, \textit{i.e.} \`a l'ordre $\sim (v/c)^6$
au-del\`a de la formule du quadrupole, s'est av\'er\'ee ``fascinante''
par la complexit\'e des calculs en jeu et la richesse de la th\'eorie
\`a cet ordre. En effet \`a l'ordre $\mathrm{3PN}$ interviennent \`a la
fois les corrections relativistes dans les \'equations du mouvement et
les moments multipolaires de la source, et des effets non-lin\'eaires
associ\'es aux sillages d'ondes gravitationnelles -- une partie de
l'onde se propage \`a une vitesse inf\'erieure \`a $c$ (en moyenne) \`a
cause des diffusions sur la courbure de l'espace-temps induite par la
source. La difficult\'e technique principale est de mettre correctement
en \oe uvre le concept de masse ponctuelle mod\'elisant les corps
compacts. On a recours \`a des m\'ethodes de r\'egularisation du champ
propre de particules ponctuelles.
Le patron d'onde gravitationnelle fournit essentiellement l'\'evolution
temporelle de la phase orbitale $\phi$ de la binaire, par effet de
r\'eaction de rayonnement. La phase est \'ecrite sous la forme d'une
s\'erie post newtonienne, $\phi=\phi_\mathrm{N}\left[1+\sum_n
a_{(n\mathrm{PN})} \,x^n\right]$, o\`u le param\`etre post newtonien
$x\equiv(G m\,\omega/c^3)^{2/3}$ est fonction de la masse totale
$m=m_1+m_2$ et de la fr\'equence orbitale $\omega={\dot\phi}$.
L'approximation newtonienne $\phi_\mathrm{N}$ correspond au ${\dot P}$
du pulsar binaire, \`a ceci pr\`es que pour les binaires spiralantes
l'excentricit\'e de l'orbite est n\'egligeable ($e\simeq 0$). Tous les
coefficients post newtoniens jusqu'\`a l'ordre $\mathrm{3.5PN}$ inclus
sont maintenant connus. \`A titre d'exemple, le coefficient \`a
$\mathrm{3PN}$, qui a \'et\'e le plus difficile \`a calculer, est
donn\'e en fonction du rapport de masse $\nu\equiv m_1m_2/(m_1+m_2)^2$
par l'expression impressionnante
\begin{eqnarray}a_{(\mathrm{3PN})}&=&\frac{12348611926451}{18776862720}
\nonumber\\ &-& \frac{856}{21}\log\,(16\,x) - \frac{160}{3}\pi^2 -
\frac{1712}{21}C \nonumber\\ &+&
\left(-\frac{15737765635}{12192768}+\frac{2255}{48}\pi^2\right)\nu \nonumber\\
&+& \frac{76055}{6912}\nu^2 -\frac{127825}{5184}\nu^3\,.\nonumber
\end{eqnarray}
Toutes les fractions rationnelles qui apparaissent dans cette formule,
de m\^eme que les nombres irrationnels $\pi$ et la constante d'Euler
$C=\lim_{K\rightarrow\infty}[\sum_{k=1}^{K-1}\frac{1}{k}-\log K]$
(num\'eriquement $C\simeq 0.577$), sont d\'eduites de la relativit\'e
g\'en\'erale et de sa structure non-lin\'eaire. Les expressions comme
celles de $a_{(\mathrm{3PN})}$ sont utilis\'ees pour calculer la
corr\'elation entre le patron d'onde des binaires spiralantes et le
signal de sortie des d\'etecteurs LIGO et VIRGO.
Bien s\^ur il faudrait (id\'ealement) s'assurer que l'approximation
$\mathrm{3PN}$ est proche de la valeur ``exacte''. Cela ne peut pas se
\textit{prouver}, mais n\'eanmoins on peut avoir une id\'ee de la
``convergence'' de la s\'erie post newtonienne en \'etudiant le cas de
l'\'energie du syst\`eme binaire au passage \`a la derni\`ere orbite
circulaire dite ICO. La figure qui suit montre que les ordres post
newtoniens successifs semblent en effet converger vers une valeur
``exacte''. En effet $\mathrm{3PN}$ est tr\`es proche de $\mathrm{2PN}$;
par contre $\mathrm{1PN}$ est clairement insuffisant car l'ICO est une
orbite tr\`es relativiste ($v/c\sim\,0.5$). De plus, la figure montre un
tr\`es bon accord entre $\mathrm{3PN}$ et la valeur calcul\'ee par la
relativit\'e num\'erique dans le cadre d'un mod\`ele approch\'e dit de
``sym\'etrie h\'elico\"idale'' (Gourgoulhon, Grandcl\'ement et
Bonazzola, 2001).
\begin{figure}
\begin{center}
\epsfig{file=fig3.eps,width=3.4in}
\caption{\'Energie d'un syst\`eme binaire de masses \'egales \`a la
derni\`ere orbite circulaire (ICO) en fonction de la fr\'equence
orbitale. On montre les approximations post newtoniennes et le point
calcul\'e par la relativit\'e num\'erique.}
\end{center}
\end{figure}
En outre, on peut faire une estimation de la pr\'ecision du patron
d'onde en comptant le nombre de cycles orbitaux, dans la bande de
fr\'equence de VIRGO, d\^u \`a chacun des ordres post newtoniens. On
trouve qu'\`a l'approximation $\mathrm{3.5PN}$ l'erreur relative sur le
nombre de cycles dans le cas de deux \'etoiles \`a neutrons est
(probablement) inf\'erieure \`a un $1/10\,000^{\text{\`eme}}$.
\bigskip
{\bf Acknowledgements} L'auteur remercie Thibault Damour, Gilles
Esposito-Far\`ese, Guillaume Faye, \'Eric Gourgoulhon et Bala Iyer pour
les nombreuses interactions et/ou collaborations.
| -32,850.505068 |
[
-0.82421875,
0.431640625
] | 14.696486 |
[
-2.859375,
-0.01412200927734375,
-1.775390625,
-3.572265625,
0.744140625,
5.9609375
] |
[
0.139404296875,
5.75,
-0.352783203125,
1.0126953125
] | 196 | 5,068 |
[
-3.140625,
3.568359375
] | 26.076343 |
[
-3.890625,
-1.275390625,
-1.248046875,
-0.62939453125,
0.95556640625,
5.765625
] | 1.10114 | 10.218244 | 27.959747 | 1.693443 |
[
1.5792202949523926
] | -33,003.362509 | 5.51914 | -35,519.028566 | 1.501555 | 6.022852 |
[
-3.61328125,
-2.3828125,
-1.57421875,
-3.0625,
2.6796875,
8.1015625
] |
[
-5.39453125,
-0.80029296875,
-0.95458984375,
0.51025390625,
3.1484375,
2.271484375
] | |
BkiUfsA5qhLBrnqN0sQf
|
\section{Introduction}
\def\theequation{1.\arabic{equation}}\makeatother
\setcounter{equation}{0}
The goal of the present paper is to introduce and study a family of operators \ap \`a la Kantorovich'' for functions acting on locally compact topological groups. These operators include, in particular, the family of Kantorovich sampling type operators, which play a crucial role in the theory of Signal and Image Processing.
Kantorovich sampling type operators were introduced in \cite{BBSV3} and studied in \cite{BM1,BM2,VZ1,VZ2,CV1,CV2}; see also \cite{KA,BBSV1}. In \cite{BBSV3} the authors considered operators acting on functions defined on the real line with the idea of introducing a Kantorovich version of the generalized sampling operators. Here, we extend the theory to the general case when the underlying space is a locally compact topological group.
There are various reasons for which it is worth considering Kantorovich sampling operators in the setting of topological groups. First of all, one has the possibility to retrieve, from our operators, several families of Kantorovich type operators; second, one has the possibility to treat simultaneously both discrete and integral operators and both one and multidimensional operators. In particular, we will be able to define and study Kantorovich versions of sampling, convolution and Mellin operators all in one and multidimensional setting.
The starting point to understand the significance of sampling operators of Kantorovich type is the following (see \cite{BBSV3}): for a locally integrable function $f$ defined on $\mathbb R$, we define
\begin{equation}\label{1i} (S_wf)(x)=\sum_{-\infty}^\infty\chi(wx-k)w\int_{\frac{k}{w}}^{\frac{k+1}{w}}f(s)ds,\;x\in\mathbb R\s(w>0),\end{equation} where $\chi$ is a suitable kernel and $f$ is chosen in such a way that the series \eqref{1i} converges.
One of the main differences between the series \eqref{1i} and the generalized sampling series lies in the use of the mean values $\displaystyle{w\int_{\frac{k}{W}}^{\frac{k+1}{w}}f(s)ds}$ instead of the sampling values $f(k/w)$. This is because, in practical situations, more information is usually known around a point than exactly at that point, and therefore Kantorovich sampling series \eqref{1i} arises as a natural modification of the generalized sampling series in order to reduce time jitter errors; concerning sampling type series, the reader can see the book \cite{BMV} (Chapt. 8 and 9) and \cite{STEN1,BU,RS,BRS,BSPS,BH,BS1,BS2,BFS,BV1,BV2,BL1,STEN2,BL2,BSS,VI1,BV3,AV1,AV2,VI2,BBSV2,
BBSV4}). Moreover, by using the series \eqref{1i}, one can deal with integrable (e.g. discontinuous) functions as well, and in fact with functions lying in a general Orlicz space. This is another important difference between \eqref{1i} and the generalized sampling series, since the infinite sum of the latter is not suitable for integrable functions because it depends on function values $f(k/w).$ In addition, the generalized sampling operators are not continuous in $L^p(\mathbb R),$ while operators \eqref{1i} are instead continuous in $L^p(\mathbb R)$.
In this paper, we define operators which are analogous to \eqref{1i}, but where both the function $f$ and the sample values $k/w$ are defined in locally compact topological groups with regular Haar measures. In order to deal with this general setting, we have to introduce a whole framework in which a series like \eqref{1i} makes sense and can be studied in detail.
Namely, we introduce the family $(S_w)_{w>0}$ of integral operators defined as follows:
\begin{equation}\label{2i} S_wf(z)=\int_H\chi_w(z-h_w(t))\left[\dfrac{1}{\mu_G(B_w(t))}\int_{B_w(t)}f(u)d\mu_G(u)\right]d\mu_H(t),
\end{equation}
where $f:G \rightarrow \mathbb R$ is a measurable function for which the above integrals are well defined.
\\
The above operators \eqref{2i} contains, as particular cases, the Kantorovich sampling, convolution and Mellin type operators, as we will show along the paper.
One of the main results obtained reads as follows:
\begin{equation*}
\lim_{w\rightarrow\infty}||S_wf-f||_\infty=0 \end{equation*}
for $f\in C(G)$, where $C(G)$ is the space of uniformly continuous and bounded functions.
In the case of not necessarily continuous functions, we obtain a modular convergence result in a suitable subset $\mathcal Y$ of an Orlicz space, i.e. we prove that there exists a constant $\lambda\ma0$ such that $$\lim_{w\rightarrow\infty}I^G_\varphi[\lambda(S_wf-f)]=0,$$
for $f\in\mathcal Y$ and where $\varphi$ is a convex $\varphi$-function.
\\
As pointed out, in the particular cases of operators of Kantorovich type, $\mathcal Y$ coincides with the whole Orlicz space $L^{\varphi}.$
Moreover, we show how the general setting of Orlicz spaces allows us to apply our theory to well known examples of function spaces, as the $L^p-$spaces, the interpolations spaces (or Zygmund spaces) and the exponential spaces, the last two being very useful for PDE's and for embedding theorems. Moreover, the last example concerning exponential spaces is of particular interest since, in this context, the norm convergence is not equivalent to the modular one.
Finally we construct concrete examples of operators, by using kernels both with compact and without compact support, and we show with some graphical examples how the considered operators reconstruct the function in various situations.
\section{Preliminaries and notations}
\def\theequation{2.\arabic{equation}}\makeatother
\setcounter{equation}{0}
\noindent This section provides the background material which is needed throughout the paper.
First we review some notions concerning topological groups. In the following, we will deal with locally compact Hausdorff topological groups equipped with regular measures. If $H$ is such a topological group, we will denote by $\theta_H$ its neutral element. For simplicity, we will denote the group operation in $H$ by the symbol $+$. It is well-known (see, e.g. \cite{M,PON,BREN}) that there exists a unique (up to multiplication by a constant) left (resp. right) translation-invariant regular Borel measure $\mu_H$ (resp. $\nu_H$). In general, $\mu_H$ and $\nu_H$ are different, and indeed, if $A\subset H$ is a Borel set and if $-A$ denotes the set $-A:=\{-a\;|\;a\in A\}$, there exists a constant $\kappa\ma0$ such that $\mu(-A)=\kappa \nu(A).$ The measures $\mu_H$ and $\nu_H$ coincide if and only if $H$ is an \emph{unimodular} group.
Examples of unimodular groups are abelian groups, compact groups and discrete groups. In unimodular groups hence the right and the left invariant measures coincide, and we will denote simply by $\mu_H$ this measure. Note that, for unimodular groups, one has $\mu_H(A)=\mu_H(-A)$ for every Borel set $A\subset H$.
Let us now consider a locally compact abelian topological group $G$ with neutral element $\theta_G$. Then $G$ is an unimodular group. It can be shown (see \cite{PON}) that a countable symmetric local base $\mathcal B$ of the neutral element $\theta_G$ can be chosen in such a way that
\begin{itemize}
\item[($ \ast$)] if $U\in\mathcal B$ then there exists $V\in\mathcal B$ such that $V+V\subset U$ (and $V-V\subset U$).
\end{itemize}
From now on, when we will make use of a local base $\mathcal B$ of the neutral element $\theta_G$ of an abelian topological group $G$, we will agree that $\mathcal B$ satisfies the condition ($\ast$).
We now move our attention to some basic results on Orlicz spaces.
Let $\varphi:\mathbb R_0^+\rightarrow\mathbb R_0^+$ be a continuous function. We say that $\varphi$ is a \emph{$\varphi$-function} if moreover:
\begin{itemize}
\item[(a)] $\varphi(0)=0$ and $\varphi(u)\ma0$ for all $u\ma0$;
\item[(b)] $\varphi$ is non-decreasing on $\mathbb R_0^+$;
\item[(c)] $\displaystyle{\lim_{u\rightarrow\infty}\varphi(u)=+\infty}$.
\end{itemize}
Let $G$ be a locally compact abelian topological group with regular Haar measure $\mu_G$.
Let us denote by $M(G)$ the set of measurable bounded functions $f:G\rightarrow\mathbb R$.
Further, by $C(G)$ (resp. $C_c(G)$) we denote the set of functions $f:G\rightarrow\mathbb R$ which are
uniformly continuous and bounded (resp. continuous and with compact support), equipped with the standard $||\cdot||_\infty$ norm, where uniform continuity on $G$ means that:
for every $\ep\ma0$, there exists a compact set $B_\ep\in\mathcal B$ such that for every $s,z \in G$ with $s-z \in B_\ep,$ then $|f(s)-f(z)|\mi\ep.$
If a $\varphi$-function $\varphi$ is chosen, one can define a functional $I_\varphi^G:M(G)\rightarrow[0,\infty]$ by $$I^G_\varphi(f):=\int_G\varphi(|f(t)|)d\mu_G(t).$$ $I^G_\varphi$ is a \emph{modular functional} on $M(G)$: it generates the Orlicz space $$L^\varphi(G):=\{f\in M(G)\;|\;I^G_\varphi(\lambda f)\mi\infty\;\mbox{for some}\;\lambda\ma0\}.$$ The subset of $L^\varphi(G)$ consisting of those $f\in M(G)$ for which $I^G_\varphi(\lambda f)\mi\infty$ \emph{for every} $\lambda\ma0$ is denoted by $E^\varphi(G)$. In general one has $E^\varphi(G)\subset L^\varphi(G)$, and they coincide if and only if $\varphi$ satisfies the so called \emph{$\Delta_2$-condition}, i.e.,
\begin{center}\it there exists a number $M\ma0$ such that $\dfrac{\varphi(2u)}{\varphi(u)}\leq M$ for every $u\ma0.$.\end{center}
There are two different kinds of convergence which are usually used in the context of Orlicz spaces. The first one is determined by a norm on $L^\varphi(G),$ called the \emph{Luxemburg norm} and defined as $$||f||_\varphi:=\inf\{\lambda\ma0\;|\;I^G_\varphi(f/\lambda)\leq \lambda\}.$$ The second is a weaker kind of convergence, called \emph{modular convergence}:
\it a sequence $(f_n)\subset L^\varphi(G)$ converges modularly to $f\in L^\varphi(G)$ if $$\lim_{n\rightarrow\infty}I^G_\varphi[\lambda(f_n-f)]=0,$$ for some $\lambda\ma0$. \rm
Clearly if $(f_n)_n\subset L^\varphi(G)$ converges in the Luxemburg norm to $f$, then it converges modularly to $f$ as well. The converse is true if and only if the $\Delta_2$-condition is satisfied by $\varphi$.
Orlicz spaces are natural generalizations of $L^p$ spaces, and in fact if $1\leq p\mi\infty$ and $\varphi(u)=u^p$, the Orlicz space generated by $I^G_\varphi$ is exactly the Lebesgue space $L^p(G)$. The function $\varphi(u)=u^p$ satisfies the $\Delta_2$-condition, hence Luxemburg and modular convergences are the same in $L^p(G)$ and coincide with the convergence with respect to the standard norm.
There are other examples of Orlicz spaces which play an important role in functional analysis and PDEs. For instance, if we set $\varphi_{\alpha,\beta}(u)=u^\alpha\ln^\beta(e+u)$ ($\alpha\geq 1,\;\beta\ma0$), we obtain the \emph{interpolation space} (also called \emph{$L^\alpha\log^\beta L$-space}) $L^{\varphi_{\alpha,\beta}}(G);$ (see e.g \cite{STEI1,STEI2}.
As another example,
if $\alpha\ma0$, we can take $\varphi_\alpha(u)=\exp\left(u^\alpha\right)-1 $ ($u\in\mathbb R_0^+$). The Orlicz space obtained via $\varphi_\alpha$ is called the \emph{exponential space} $L^{\varphi_\alpha}(G)$ (see \cite{HEN}). This last example is particularly interesting because the function $\varphi_{\alpha,\beta}$ does not satisfy the $\Delta_2$-condition, hence in the space $L^{\varphi_{\alpha,\beta}}(G)$ Luxemburg and modular convergence are different.
For more information on Orlicz spaces and related topics, the reader can be addressed to \cite{KR,MU,RR1,RR2,BMV}.
\section{Approximation results}
\def\theequation{3.\arabic{equation}}\makeatother
\setcounter{equation}{0}
Let $H$ and $G$ be locally compact Hausdorff topological groups with regular Haar measures $\mu_H$ and $\mu_G$ respectively. Let us denote by $\theta_H$ (resp. $\theta_G$) the neutral element of $H$ (resp. $G$). We further assume that $G$ is abelian. Let $\mathcal B\subset G$ be a countable local base of the neutral element $\theta_G$ (which satisfies condition ($\ast$) of the previous section), ordered by inclusion.
For every $w\ma0$, let $h_w:H\rightarrow G$ be a map which restricts to a homeomorphism from $H$ to $h_w(H)$.
Let us further assume that for every $w\ma0$, there exists a family $\mathcal B_w=(B_w(t))_{t\in H}$ $\subset G$ of open nonempty subsets of $G$
such that
\begin{itemize}
\item[(i)] $0\mi\mu_G(B_w(t))\mi\infty$ for every $t\in H$ and $w\ma0$;
\item[(ii)] for every $w\ma0$ and $t\in H$, $h_w(t)\in B_w(t)$.
\item[(iii)] if $B\in\mathcal B$, there exists a number $\overline w\ma0$ such that for every $w\ma\overline w$ we have $h_w(t)-B_w(t)\subset B$, for every $t\in H$.
\end{itemize}
Let $(\chi_w)_{w\ma0}$ be a family of measurable kernel functionals; i.e., $\chi_w:G\rightarrow\mathbb R$, $\chi_w\in L^1(G)$ and is bounded in a neighborhood of $\theta_G$ ($w\ma0$). We assume that
\begin{itemize}
\item[($\chi_1$)] the map $t\mapsto \chi_w(z-h_w(t))\in L^1(H)$ for every $z\in G$;
\item[($\chi_2$)] for every $w\ma0$ and $z\in G$, $$\int_H\chi_w(z-h_w(t))d\mu_H(t)=1;$$
\item[($\chi_3$)] for every $w\ma 0$, $$m_{0.\pi}(\chi_w):=\sup_{z\in G}\int_H\left|\chi_w(z-h_w(t))\right|d\mu_H(t)\mi M \mi+\infty;$$
\item[($\chi_4$)] if $w\ma0$, $z\in G$ and $B\in\mathcal B$, set $B_{z,w}=\{t\in H\;|\;z-h_w(t)\in B\}\subset H$. Then
$$\lim_{w\rightarrow\infty}\int_{H\setminus B_{z,w}}\left|\chi_w(z-h_w(t))\right|d\mu_H(t)=0$$ uniformly with respect to $z\in G$;
\item[($\chi_5$)] for every $\ep\ma0$ and compact set $K\subset G$, there exists a symmetric compact set $C\subset G$ containing $\theta_G$ with $\mu_G(C)\mi\infty$ and such that $$\int_{z\notin C}\Upsilon_w(K)\left|\chi_w(z-h_w(t))\right|d\mu_G(z)\mi\ep$$ for every sufficiently large $w\ma 0$ and $h_w(t)\in K$, where $$\Upsilon_w(K):=\mu_H\{t\in H\;|\;h_w(t)\in K\}\s(w\ma0).$$
\end{itemize}
If $w\ma0$, we study the family of operators $\{S_w:M(G)\rightarrow \mathbb R\}_w$ defined as
\begin{equation}\label{1} S_wf(z)=\int_H\chi_w(z-h_w(t))\left[\dfrac{1}{\mu_G(B_w(t))}\int_{B_w(t)}f(u)d\mu_G(u)\right]d\mu_H(t),
\end{equation}
where $f:G \rightarrow \mathbb R$ is a measurable function such that the above integrals are well defined.
We make some concrete examples of operators of the kind \eqref{1}. In Section 4, we will study in detail these examples.
\\
{\bf Kantorovich Sampling Type Operators}. If $H=\mathbb Z$ and $G=\mathbb R$, we can choose $h_w:\mathbb Z\rightarrow\mathbb R:k\mapsto t_k/w$, where $\{t_k\}_k$ is a sequence of real numbers such that: $(i)$ $t_k\mi t_{k+1}$ ($k\in\mathbb Z$); $(ii)$ there exist numbers $0\mi\delta\mi\Delta$ such that $\delta\mi t_{k+1}-t_k\mi \Delta$ and $B_w(k)=[t_k/w,t_{k+1}/w]$ . We obtain $$S^{(1)}_wf(z)=\sum_{k\in\mathbb Z}\chi_w(z-t_k/w)\left(\dfrac{w}{t_{k+1}-t_k}\int_{t_k/w}^{t_{k+1}/w}f(u)du\right).$$ See e.g. \cite{BBSV3,VZ1,VZ2,CV1,CV2} for a detailed study of these operators.
\\
{\bf Kantorovich Convolution Type Operators}. If $H=G=\mathbb R$ and $h_w(t)=t/w$, we may choose $B_w(t)=[(t-1)/w,(t+1)/w]$ ($w\ma0$). Then we obtain $$S_w^{(2)}f(z)=\int_{-\infty}^{\infty}
\chi_w(z-t/w)\left(\dfrac{w}{2}\int_{(t-1)/w}^{(t+1)/w}f(u)du\right)dt.$$
\\
Moreover, if we choose $h_w(t)=t$ and $B_w(t)=[t-1/w,t+1/w]$ ($w\ma0$), then we have
$$S_w^{(3)}f(z)=\int_{-\infty}^\infty\chi_w(z-t)\left(\dfrac{w}{2}\int_{t-1/w}^{t+1/w}f(u)du\right)dt.$$
For the theory of classical convolution operators, see e.g. \cite{BN}.
\\
{\bf Kantorovich Mellin Type Operators}. If $H=G=\mathbb R^+$, then $\mu_H=\mu_G$ is the logarithmic measure, and the group operation is the product.
If $h_w(t)=t$, we take $B_w(t)=\left[t\dfrac{w}{w+1},t\dfrac{w+1}{w}\right]$ ($w\ma0$) and we have $$S_w^{(4)}f(z)=\int_0^\infty\chi_w(z/t)\left(\dfrac{1}{2\ln(1+1/w)}\int_{t\frac{w}{w+1}}^{t\frac{w+1}{w}}f(u)\dfrac{du}{u}\right)
\dfrac{dt}{t}.$$ We address the reader to the book \cite{M} and \cite{BJ1,BJ2,BM0,BM3} for the theory and results concerning Mellin type operators.\finedim
\bigskip
We move our attention to some preliminary results concerning the structure and the properties of the operators \eqref{1}.
\\
First of all, we observe that the operators $S_w$ map $L^\infty(G)$ into $L^\infty(G)$. In fact, for every $f \in L^\infty(G),$
\begin{equation*}\begin{split}
|S_wf(z)|\leq &\int_H\Big|\chi_w(z-h_w(t))\Big|\left[\dfrac{1}{\mu_G(B_w(t))}\int_{B_w(t)}|f(u)|d\mu_G(u)\right]d\mu_H(t)\leq\\&\leq ||f||_{\infty}\int_H|\chi_w(z-h_w(t))d\mu_H(t)\leq ||f||_\infty m_{0,\pi}(\chi_w)\end{split}\end{equation*} for $z\in G$, hence $||S_wf||_\infty\leq ||f||_\infty m_{0,\pi}(\chi_w)$ for every $w\ma0$.
From now on, if $A\subseteq G$ is a measurable set with $\mu_G(A)\mi\infty$ and $f:A\rightarrow\mathbb R$ is an integrable function, we will write $$\dashint_A f(u)d\mu_G(u):=\dfrac{1}{\mu_G(A)}\int_Af(u)d\mu_G(u).$$
Next we prove a first result of convergence.
\begin{theorem}\label{p1} Let $f\in C(G)$. Then
\begin{equation*}
\lim_{w\rightarrow\infty}||S_wf-f||_\infty=0. \end{equation*}
\end{theorem}
Clearly, Theorem \ref{p1} implies that if $f$ is continuous at a point $z\in G$, then $S_wf(z)$ converges to $f(z)$.
\noindent
\proof By the uniform continuity of $f$, for every $\ep\ma0$, there exists a compact set $B_\ep\in\mathcal B$ such that $|f(z)-f(u)|\mi\ep$ whenever $z-u\in B_\ep$. We can choose an open set $B^{(1)}\subset\mathcal B$ such that $B^{(1)}+B^{(1)}\subset B_\ep$. By (iii), there exists $\overline w\ma0$ such that if $w\ma\overline w$ then $h_w(t)-B_w(t)\subset B^{(1)}$, for every $t\in H$. We can write
\begin{equation*}\begin{split}|S_wf(z)-&f(z)|=\left|
\int_H\chi_w(z-h_w(t))\left[
\dashint_{B_w(t)}(f(u)-f(z))d\mu_G(u)\right]
d\mu_H(t)\right|\leq\\&\leq\int_H|\chi_w(z-h_w(t))|\left[
\dashint_{B_w(t)}|f(u)-f(z)|d\mu_G(u)\right]d\mu_H(t)\leq\\&\leq\int_{B_{z,w}^{(1)}}|\chi_w(z-h_w(t))|\left[
\dashint_{B_w(t)}|f(u)-f(z)|d\mu_G(u)\right]d\mu_H(t)+\\&+
\int_{H\setminus B_{z,w}^{(1)}}|\chi_w(z-h_w(t))|\left[
\dashint_{B_w(t)}|f(u)-f(z)|d\mu_G(u)\right]d\mu_H(t)=\\&=:I_1+I_2.
\end{split}\end{equation*}
We estimate $I_1$. If $z-h_w(t)\in B^{(1)}$, then, since $h_w(t)-B_w(t)\in B^{(1)}$ for $w\ma\overline w,$ we have $z-u=z-h_w(t)+h_w(t)-u\in B^{(1)}+B^{(1)}\subset B_\ep$ whenever $u\in B_w(t),$ for $w\ma\overline w.$ It follows that, for $w\ma\overline w,$
$$I_1\leq \ep \int_{B_{z,w}^{(1)}}|\chi_w(z-h_w(t))|d\mu_H(t)\leq \ep m_{0,\pi}(\chi_w)\mi \ep M.$$
For $I_2$, we have
$$I_2\leq ||f||_\infty\int_{H\setminus B_{z,w}^{(1)}}|\chi_w(z-h_w(t))|d\mu_H(t).$$ By the assumption ($\chi_4$), $I_2\rightarrow 0$ as $w\rightarrow\infty$, uniformly with respect to $z\in G$. Combining the estimates for $I_1$ and $I_2$ the proof follows at once.\finedim
\noindent
We now turn to a first result of convergence in Orlicz spaces.
\begin{theorem} \label{t2}Let $\varphi:\mathbb R_0^+\rightarrow\mathbb R_0^+$ be a convex $\varphi$-function and let $f\in C_c(G)$. Then $$\lim_{w\rightarrow\infty}||S_wf-f||_\varphi=0.$$\end{theorem}
\noindent
\proof We must show that
$$\lim_{w\rightarrow\infty}\int_G\left[\varphi(\lambda|S_wf(z)-f(z)|)\right]d\mu_G(z)=0,$$ for every $\lambda\ma0$.
Let us consider the family of functions $g_w(z):=\varphi(\lambda|S_wf(z)-f(z)|)$. Then $g_w:G\rightarrow\mathbb R$ is nonnegative for every $w\ma0$. Moreover, $\displaystyle{\lim_{w\rightarrow\infty}g_w(z)=0}$ uniformly with respect to $z\in G$. This is due to the the properties of $\varphi$ and the fact that $||S_wf-f||_\infty\rightarrow 0$ as $w\rightarrow\infty$ (see Theorem \ref{p1}). Now we show that it is possible to apply the Vitali convergence Theorem to the family $(g_w)_w$.
Let $K_1=Supp(f).$ Choose a symmetric compact set $K\subset G$ satisfying $K_1\varsubsetneq K$. By (iii), for sufficiently large $w\ma0$, $B_w(t)\subseteq h_w(t)+B$ ($B\in\mathcal B$). Then $B_w(t)\cap K_1=\emptyset$ for sufficiently large $w\ma0$ and therefore if $h_w(t)\notin K$ we have $$\int_{B_w(t)}f(u)du=0.$$
Now, fix $\lambda\ma0$, let $\ep\ma0$ and let $C\subset G$ with $\mu_G(C)\mi\infty$ be such that ($\chi_5$) is valid for $K$. Let us estimate $$I:=\int_{G\setminus C}g_w(z)d\mu_G(z).$$ We use the notation $K_{t,w}=\{t\in H\;|\;h_w(t)\in K\}$ ($t\in H,\;w\ma0$). We have
\begin{equation*}\begin{split}I&=\int_{G\setminus C}\varphi\left[\lambda|S_wf(z)-f(z)|\right]d\mu_G(z)\leq\\&\leq\int_{G\setminus C}\varphi\left(2\lambda||f||_\infty\int_{K_{t,w}}\left|\chi_w(z-h_w(t))\right|d\mu_H(t)\right)d\mu_G(z)\leq\\&\int_{G\setminus C}\left(\dfrac{1}{\Upsilon_w(K)M}\int_{K_{t,w}}\varphi\left(2\lambda M||f||_\infty\right)\Upsilon_w(K)|\chi_w(z-h_w(t))|d\mu_H(t)\right)d\mu_G(z)\leq\\&
\dfrac{1}{\Upsilon_w(K)M}\int_{K_{t,w}}\left(\varphi(2\lambda M||f||_\infty )\int_{G\setminus C}\Upsilon_w(K)|\chi_w(z-h_w(t))|d\mu_G(z)\right)d\mu_H(t)\leq\\&\dfrac{\ep}{\Upsilon_w(K)M}\int_{K_{t,w}}\varphi(2\lambda M||f||_\infty)d\mu_H(t)=\dfrac{\ep\varphi(2\lambda M||f||_\infty)}{M}\mi\infty,
\end{split}\end{equation*}
where we used the Jensen's inequality, the Fubini-Tonelli Theorem and ($\chi_5$).
Moreover, it is easy to see that, for every measurable set $A\subset G$ with $\mu_G(A)\mi\infty$, we have
$$\int_A\varphi(\lambda|S_wf(z)-f(z)|)d\mu_G(z)\leq\int_A\varphi(2\lambda M||f||_\infty)d\mu_H(t)=\varphi(2\lambda M||f||_\infty)\mu_G(A).$$ So, for fixed $\ep\ma0$, it suffices to take $\delta\mi\dfrac{\ep}{\varphi(2\lambda M||f||_\infty)}$, to obtain $$\int_A \varphi(\lambda|S_wf(z)-f(z)|)d\mu_G(z)\leq\ep,$$ for every measurable set $A\subset G$ with $\mu_G(A)\mi\delta.$ The Vitali convergence theorem can be applied, and the theorem is therefore proved. \finedim
\noindent
It is a matter of fact that, except for the standard operators defined on $\mathbb R$, when one has to face the problem of the convergence in the space $L^\varphi(G)$ for a generic function $f\in L^\varphi(G)$, then one has to make one additional assumption, which allows to compare the value of an integral over $H$ of the function $\varphi$ which involves $f$ and the sets $B_w(t)$ (in a sense specified below) with the value $I^G_\varphi(\lambda f)$. We formulate this assumption:
\begin{itemize}
\item[($\chi_6$)] We assume that there exists a vector subspace $\mathcal Y\subset L^\varphi(G)$ with $C_c^\infty(G)\subset \mathcal Y$ and such that, for every $g\in\mathcal Y$, there holds
\begin{equation}\label{cond}\limsup_{w\rightarrow\infty}||\chi_w||_{L^1(G)} I^H_\varphi \left(\dashint_{B_w(\cdot)}g(z)d\mu_G(z)\right)\leq CI^G_\varphi( g),\end{equation} for some $C\ma0$.\end{itemize}
We remark that condition \eqref{cond} is not assured in general: however,
in the cases when $H$ and $G$ are subgroups of $\mathbb R$, as we will see below, the \ap Kantorovich'' nature of the operators considered here allows us to discharge ($\chi_6$) on the kernels $\chi_w$.
Under this additional assumption we can prove the following
\begin{theorem}\label{t3}Let $\varphi$ be a convex $\varphi$-function and let $f\in\mathcal Y$, where $\mathcal Y$ is defined in ($\chi_6$). Then, if $\lambda\ma0$,
\begin{equation*}
I^G_\varphi(\lambda S_wf)\leq\dfrac{C}{M}I^\varphi_G(\lambda Mf)
\end{equation*}
for sufficiently large $w\ma0$.
In particular $S_w:\mathcal Y\rightarrow L^\varphi(G)$ is well defined for every $w\ma0$.\end{theorem}
\noindent
\proof Let $\lambda\ma 0$ be such that the quantity $I^G_\varphi[\lambda Mf]\mi\infty$. Then, using \eqref{cond} with $g=\lambda Mf$, \begin{equation*}\begin{split}
&I^G_\varphi[\lambda S_wf]=\int_G\varphi(\lambda |S_wf(z)|)d\mu_G(z)\leq \\&\leq
\int_G\varphi\left[\lambda\int_H|\chi_w(z-h_w(t))|
\left(\left|\dashint_{B_w(t)}f(u)d\mu_G(u)\right|\right)d\mu_H(t)\right]d\mu_G(z)
\leq\\&\leq\dfrac{1}{M}\int_G\left[\int_H\varphi\left(\lambda M\left|\dashint_{B_w(t)}f(u)
d\mu_G(u)\right|\right)|\chi_w(z-h_w(t))|d\mu_H(t)\right]d\mu_G(z)
\leq\\&\leq\dfrac{1}{M}\left[\int_H\varphi\left(\lambda M\left|\dashint_{B_w(t)}f(u)d\mu_G(u)\right|\right)\left(\int_G|\chi_w(z-h_w(t))|d\mu_G(z)\right)
\right]d\mu_H(t)\leq\\&\leq \dfrac{1}{M}||\chi_w||_{L^1(G)} I^H_\varphi\left(\lambda M\dashint_{B_w(\cdot)}f(u)d\mu_G(u)\right)\leq \dfrac{C}{M}I^G_\varphi[\lambda Mf],
\end{split}\end{equation*}
for sufficiently large $w\ma0$. \finedim
\noindent
Our next result concerns the convergence in the Orlicz space $L^\varphi(G)$ of $S_wf$ to $f$ as $w\rightarrow\infty$. To prove it, we need the following lemma (see \cite{BM00,BV1,BMV}).
\begin{lemma}The set $C_c^\infty(G)$ is dense in $L^\varphi(G)$ with respect to the modular convergence.\end{lemma}
Now, the main convergence result can be stated as follows:
\begin{theorem}\label{t6} Let $\varphi$ be a convex $\varphi$-function and let $f\in\mathcal Y$. Then there exists a constant $\lambda\ma0$ such that $$\lim_{w\rightarrow\infty}I^G_\varphi[\lambda(S_wf-f)]=0.$$\end{theorem}
\noindent
\proof Let $\ep\ma0$. By the above Lemma, we can find a function $g\in C_c(G)$ and a constant $\eta\ma0$ such that
$$I^G_\varphi[\eta(f-g)]\mi\ep.$$
Choose $\lambda\ma0$ such that $3\lambda(1+M)\mi\eta.$
We can write
\begin{equation*}\begin{split}I^G_\varphi&[\lambda(S_wf-f)]\leq
I^G_\varphi[3\lambda(S_wf-S_wg)]+I^G_\varphi[3\lambda(S_wg-g)]+
I^G_\varphi[3\lambda(f-g)]\leq\\&\leq \dfrac{C}{M}I^G_\varphi[\eta (f-g)]+I^G_\varphi[3\lambda(S_wg-g)]+I^G_\varphi[\eta(f-g)]\leq \\&\leq \left(\dfrac{C}{M}+1\right)\ep+I^G_\varphi[3\lambda(S_wg-g)].
\end{split}\end{equation*}
From Theorem \ref{t2} the proof follows easily since $\ep$ is arbitrarily chosen. \finedim
\noindent
\section{Applications}
\def\theequation{4.\arabic{equation}}\makeatother
\setcounter{equation}{0}
In this section, we will give concrete examples of applications of the theory developed in the previous section.
Some of these examples are known in the literature, whereas others are generalizations of well known operators to the \ap Kantorovich'' setting.
\begin{itemize}
\item[\bf(1)] We begin with a kind of operators discussed in \cite{BBSV3}. Let $H=(\mathbb Z,+)$ and $G=(\mathbb R,+)$ provided with the counting and Lebesgue measures respectively. Let us define $h_w:\mathbb Z\rightarrow\mathbb R:k\mapsto t_k/w$ ($w\ma0$), where $t_k\mi t_{k+1}$ for every $k\in\mathbb Z$, and $\delta\mi t_{k+1}-t_k\mi\Delta$ for some numbers $0\mi\delta\mi\Delta\mi\infty$. Set $\Delta_k=t_{k+1}-t_k.$ ($k\in\mathbb Z$), and let $B_w(k)=[t_k/w,t_{k+1}/w].$ If $f:\mathbb R\rightarrow\mathbb R$ is a measurable function, the corresponding family of operators is defined as
$$S_w^{(1)}f(z)=\sum_{k\in\mathbb Z}\chi_w(z-t_k/w)\dfrac{w}{\Delta_k}\int_{t_k/w}^{t_{k+1}/w}f(u)du.$$
For the family $(S_w^{(1)})_{w\ma0},$ a theory has already been introduced in \cite{BBSV3}. We show that the assumptions ($\chi_1$)--($\chi_6$) reduce to those in the above paper.
Assumptions from ($\chi_1$) to ($\chi_3$) can be easily rewritten with $\displaystyle{\sum_{k\in\mathbb Z}}$ instead of $\displaystyle{\int_H\dots d\mu(t)}$.
($\chi_4$) can be rewritten in a more familiar form as follows:
\ap for every $\gamma\ma0$
$$\lim_{w\rightarrow\infty}\sum_{|z-t_k/w|\ma\gamma}|\chi_w(z-t_k/w)|=0,$$ uniformly with respect to $z\in\mathbb R$'';
($\chi_5$) is equivalent to the following:
\ap for every $\ep\ma0$ and $\gamma\ma0$ there exists a number $M\ma0$ such that
$$\int_{|z|\ma M}w|\chi_w(z-t_k/w)|dz\mi\ep$$ for sufficiently large $w\ma0$ and $t_k/w\in[-\gamma,\gamma]$''. Indeed, in this case the compact set $K$ can be taken as $[-\gamma,\gamma]$ ($\gamma\ma0$), while the symmetric compact set $C$ is given by $[-M,M]$. Now, if we compute the quantity $\Upsilon_w([-\gamma,\gamma])$ we have
\begin{equation*}\begin{split}\dfrac{2\gamma w}{\Delta}-2\leq\Upsilon_w([-\gamma,\gamma])\leq 2\left(\dfrac{\gamma w}{\delta}+2\right)=4+2\dfrac{\gamma}{\delta}w.\end{split}\end{equation*}
It follows that
\begin{equation*}\begin{split}\int_{|z|\ma M}\left(\dfrac{2\gamma w}{\Delta}-2\right)|\chi_w(z-t_k/w)|dz&\leq\int_{|z|\ma M}\Upsilon_w([-\gamma,\gamma])|\chi_w(z-t_k/w)|dz\\ &\leq \int_{|z|\ma M}\left(4
+2\dfrac{\gamma w}{\delta} \right)|\chi_w(z-t_k/w)|dz.\end{split}\end{equation*}
So, since $X_w\in L^1(\mathbb R)$ for every $w\ma0$, ($\chi_5$) is equivalent to $$\int_{|z|\ma M}w|\chi_w(z-t_k/w)|dz\mi \ep$$ for sufficiently large $w\ma0$ and $t_k/w\in[-\gamma,\gamma]$;
($\chi_6$) translates to a condition on the kernels: indeed \eqref{cond} becomes
\begin{equation*}\begin{split}&||\chi_w||_1 I^\mathbb Z_\varphi\left[\alpha \left(\dfrac{w}{\Delta_k}\int_{t_k/w}^{t_{k+1}/w}f(z)dz\right)\right]\leq\\&\leq ||\chi_w||_1\dfrac{w}{\delta}\left(\sum_{k\in\mathbb Z}\int_{t_k/w}^{t_{k+1}/w}\varphi[\alpha|f(z)|]dz\right)=\\&=||\chi_w||_1\dfrac{w}{\delta} I_\varphi^\mathbb R[\alpha f] .\end{split}\end{equation*}
So it suffices to have $$\limsup_{w\rightarrow\infty}w||\chi_w||_1\mi\infty.$$ This happens, for example, if $\chi_w(s)$ is given by $\chi_w(s)=\chi(ws)$ where $\chi\in L^1(\mathbb R)$. Actually, in \cite{BBSV3}, the kernels under consideration have exactly this form.
Therefore, in this case, we have $\mathcal Y=L^\varphi(\mathbb R)$.
We now slightly modify the operator $S_w^{(1)}$ to show an interesting consequence of the Kantorovich frame. So, let $H=(\mathbb Z,+)$ and $G=(\mathbb R,+)$ provided with the counting and Lebesgue measures respectively as before, and consider the map $h_w:\mathbb Z\rightarrow\mathbb R: k\mapsto t_k,$ where the sequence $(t_k)_k\subset\mathbb R$ is chosen as above. Now, set $B_w(k):=[t_k-1/w,t_k+1/w]$ ($w\ma0$).
With these choices, we obtain the operators $$S_w^{(1,1)}f(z)=\sum_{k\in\mathbb Z}\chi_w(z-t_k)\dfrac{w}{2}\int_{t_k-1/w}^{t_{k}+1/w}f(u)du.$$ For the family $(S_w^{(1,1)})_w$, we have the same results as for the family $(S_w^{(1)})_w$ above. In particular, ($\chi_6$) translates to a condition on the family of kernels $(\chi_w)_w$ which is satisfied if $\chi_w(s)=\chi(ws)$ where $\chi\in L^1(\mathbb R)$, and so $\mathcal Y=L^\varphi(\mathbb R)$ (see the discussion above). However, this examples has an interesting feature. For fixed $k\in\mathbb Z$, if $f\in L^1_{loc}(\mathbb R)$, the quantity $$\dfrac{w}{2}\int_{t_k-1/w}^{t_{k}+1/w}f(u)du$$ converges, as $w\rightarrow\infty$, to the value $f(t_k)$, for a.a. $t_k$. This is the Lebesgue-Besicovich Differentiation Theorem (see \cite{EG}). Roughly speaking, this fact tells us that, for large $w\ma0$, the operators $S_w^{(1,1)}f(\cdot)$ can be asymptotically compared with the classical sampling operators $$S_wf(z)= \sum_{k\in\mathbb Z}\chi_w(z-t_k)f(t_k).$$
To retrieve the exact formula of the classical sampling operators, it suffices to take $t_k = s_k/w$ and $\chi_w(z)=\chi(wz).$ Hence we obtain
$$S_wf(z)=\sum_{k\in\mathbb Z}\chi(wz-s_k)f\left(\dfrac{s_k}{w}\right).$$
However, it is well known (see \cite{BMV}) that the sampling operators $S_wf(\cdot)$ do not converge to $f(\cdot)$ for an arbitrary function $f\in L^\varphi(\mathbb R)$. In fact, one can show that the convergence is assured in a proper subspace $\mathcal Y$ of $L^\varphi(\mathbb R)$, namely $\mathcal Y=BV^\varphi(\mathbb R)\cap E^\varphi(\mathbb R)$ (where $BV^\varphi(\mathbb R)$ is the subset of $M(\mathbb R)$ consisting of those $f\in M(\mathbb R)$ satisfying $\varphi(\lambda|f|)\in BV(\mathbb R)$ for every $\lambda\ma0$). This shows the importance of the regularizing property of the mean value in the Kantorovich type operators.
\item[\bf(2)] We take $H=G=\mathbb R$, $h_w:\mathbb R\rightarrow\mathbb R:t\mapsto t/w$ and $B_w(t)=[(t-1)/w,(t+1)/w]$ ($w\ma0$, $t\in\mathbb R$). In this case we have
$$S_w^{(2)}f(z)=\int_{-\infty}^\infty \chi_w(z-t/w)\left(\dfrac{w}{2}\int_{(t-1)/w}^{(t+1)/w}f(s)ds\right)dt,$$
which is a Kantorovich version of a convolution operator.
We rewrite the assumptions ($\chi_1$)--($\chi_6$).
Again, assumptions from ($\chi_1$) to ($\chi_3$) may be rewritten with $H=\mathbb R$ and the standard Lebesgue measure on $\mathbb R$ in place of $d\mu(t)$ and $d\mu(z)$.
($\chi_4$) can be easily written as $$\lim_{w\rightarrow\infty}\int_{|z-t/w|\ma\gamma}|\chi_w(z-t/w)|dt=0,$$ for every $\gamma\ma0$ and uniformly with respect to $z\in\mathbb R$;
($\chi_5$) assumes an interesting form: namely, (we can take $K=[-\gamma,\gamma]$ and $C=[-M,M]$) we require that for every $\ep\ma0$ and $\gamma\ma0$ there exists a number $M\ma0$ such that $$\int_{|z|\ma M} w|\chi_w(z-t/w)|dz\mi\ep,$$ for sufficiently large $w\ma0$ and $t/w\in[-\gamma,\gamma]$. Indeed, in this case he have
$\Upsilon_w([-\gamma.\gamma])=2\gamma w.$
($\chi_6$) can be rewritten by observing that
\begin{equation*}\begin{split}&\s\s||\chi_w||_1\int_\mathbb R\varphi\left[\dfrac{w}{2}\alpha\left|\int_{(t-1)/w}^{(t+1)/w}f(u)du\right|\right]dt\leq \\&\leq ||\chi_w||_1\dfrac{w}{2}\int_0^{2/w}\left(\int_\mathbb R\varphi\left(\alpha|f(s+(t-1)/w)|\right)dt\right)ds,
\end{split}
\end{equation*}
by using the Jensen's inequality, the change of variable $s=u-(t-1)/w$ and the Fubini-Tonelli theorem.
A further change of variable $\rho=s+(t-1)/w$ gives \begin{equation*}\begin{split}&||\chi_w||_1\int_\mathbb R\varphi\left[\dfrac{w}{2}\alpha\left|\int_{(t-1)/w}^{(t+1)/w}f(u)du\right|\right]dt\leq\\ &\leq ||\chi_w||_1\dfrac{w}{2}\int_0^{w/2}wI^\mathbb R_\varphi(\alpha f)ds=w||\chi_w||_1 I^\mathbb R_\varphi(\alpha f).\end{split}\end{equation*}
Again, the condition ($\chi_6$) is satisfied if $$\limsup_{w\rightarrow\infty}w||\chi_w||_1\mi\infty,$$ which is true if, for instance, $\chi_w(s)=\chi(ws)$ for every $s\in\mathbb R$, where $\chi\in L^1(\mathbb R)$. Again, in this case we have $\mathcal Y=L^\varphi(\mathbb R)$.
\item[\bf(3)] A similar example can be obtained by setting $h_w(t)=t$ for every $w\ma0$ and $t\in\mathbb R.$ In this case, we can take $B_w(t)=[t-1/w,t+1/w],$ and the operators have the form
$$S_w^{(3)}f(z)=\int_\mathbb R\chi_w(z-t)\left(\dfrac{w}{2}\int_{t-1/w}^{t+1/w}f(u)du\right)dt.$$
In this case, however, the statements of assumptions ($\chi_5$) and ($\chi_6$) slightly differ from those mentioned above, namely:
($\chi_5$) $\rightarrow$ for every $\ep\ma0$ and $\gamma\ma0$ there exists a number $M\ma0$ such that $$\int_{|z|\ma M}|\chi_w(z-t)|dz\mi\ep$$ for sufficiently large $w\ma0$ and $t\in[-\gamma,\gamma]$ (indeed, one has $\Upsilon_w([-\gamma,\gamma])=2\gamma$). The above assumption is clearly satisfied if, for example, the family $(\chi_w)_w$ is uniformly bounded by a function $\chi\in L^1(\mathbb R)$.
($\chi_6$) $\rightarrow$ $\displaystyle{\limsup_{w\rightarrow\infty}||\chi_w||_1\mi\infty,}$ arguing as before, and $\mathcal Y=L^\varphi(\mathbb R)$.
Note that, if $f\in L^1_{loc}(\mathbb R)$, in $S_w^{(3)}f(z)$ the factor $\displaystyle{\dfrac{w}{2}\int_{t-1/w}^{t+1/w}f(u)du}$ converges, as $w\rightarrow\infty$, to the value $f(t)$ for a.e. $t\in\mathbb R$ (this is the Lebesgue-Besicovich Differentiation Theorem again). Roughly speaking, this fact tells us that, for large values of $w\ma0$, $S_w^{(3)}f(z)$ can be related to the standard convolution operator
$$C_wf(z)=\int_\mathbb R\chi_w(z-t)f(t)dt.$$ Compare our results with the very well-known theorem which states that $C_wf(z)\rightarrow f(z)$ as $w\rightarrow\infty,$ when $f\in L^p(\mathbb R^N)$ and $\chi_w$ is a family of mollifiers (any sequence of mollifiers satisfies the assumptions ($\chi_1$)--($\chi_6$)).
\item[\bf(4)] Let $H=G=\mathbb R^+$. The group operation in $\mathbb R^+$ is the product, hence $\theta_G=1$. Set $h_w(t)=t$ and $B_w(t)=\left[t\dfrac{w}{w+1},t\dfrac{w+1}{w}\right]$ for every $w\ma0$ and $t\ma0$. The only regular Haar measure on $\mathbb R^+$ (up to a multiplicative constant) is the logarithmic measure $\mu(\mathbb R^+)=\displaystyle{\int_{\mathbb R^+} \dfrac{dt}{t}.} $ The family $\mathcal B$ can be taken as $\mathcal B=\{[1/\alpha,\alpha],\;\alpha\ma1\}.$ The Haar measure of $B_w(t)$ is $2\ln\dfrac{w+1}{w}.$ If $f\in M(\mathbb R^+)$, then we obtain
$$S_w^{(4)}f(z)=\int_0^\infty \chi_w\left(\dfrac{z}{t}\right)\dfrac{1}{2\ln(1+1/w)}\left(\int_{t\frac{w}{w+1}}^{t\frac{w+1}{w}} f(u) \dfrac{du}{u}\right) \dfrac{dt}{t}.$$
Assumptions ($\chi_1$)--($\chi_4$) can be easily adapted with $H=G=\mathbb R^+$ and $d\mu(t)=\dfrac{dt}{t}.$
As in the above example, ($\chi_5$) assumes the following form: \ap for every $\ep\ma0$ and $\gamma\ma1$, there exists a number $M\ma1$ such that $$\int_{z\notin[1/M,M]}|\chi_w(z/t)|\dfrac{dz}{z}\mi\ep$$ for sufficiently large $w\ma0$ and $t\in[1/\gamma,\gamma]$'' (indeed, in this case, $\Upsilon([1/\gamma,\gamma])=2\ln\gamma$).
It is possible to prove that ($\chi_6$) is satisfied if $$\limsup_{w\rightarrow\infty}||\chi_w||_1\mi\infty;$$ one has to keep in mind, however, that the $L^1$-norm is with respect to the measure $\dfrac{dt}{t};$ again, $\mathcal Y=L^\varphi(\mathbb R^+)$.
It is worth to spend again a word concerning the form of $S_w^{(4)}.$ If $w\rightarrow\infty$ and if $f\in L^1_{loc}(\mathbb R^+,dt/t)$, then the value $I:=\displaystyle{\dfrac{1}{2\ln(1+1/w)}\int_{t\frac{w}{w+1}}^{t\frac{w+1}{w}} f(u) \dfrac{du}{u}}$ converges to $f(t)$ for a.e. $t\in\mathbb R^+$. Indeed, by the change of variables $ts=u$, one gets $$I=\dfrac{1/w}{\ln(1+1/w)}\dfrac{w}{2}\int_{1-\frac{1}{w+1}}^{1+\frac{1}{w}} f(ts) \dfrac{ds}{s}:=\dfrac{1/w}{\ln(1+1/w)}I_1,$$ and $I_1$ converges to $f(t)$ as $w\rightarrow\infty$ for a.e. $t\in\mathbb R^+$ by the Lebesgue-Besicovich theorem again, since the term
$\dfrac{1/w}{\ln(1+1/w)}\dfrac{w}{2}$ is the reciprocal of the logarithmic (Haar) measure of the integration set and in fact it is an average.
Now, since the factor $\dfrac{1/w}{\ln(1+1/w)}$ converges to 1 as $w\rightarrow\infty$, then
the operator $S_w^{(4)}f(z)$ can be asymptotically compared, as $w\rightarrow\infty$, to the standard Mellin operator $$M_wf(z)=\int_0^\infty\chi_w\left(\dfrac{z}{t}\right)f(t)\dfrac{dt}{t}.$$
\item[\bf(5)] Our last application concerns operators which approximate functions $f:\mathbb R^N\rightarrow\mathbb R$. Let $(t_k)_{k\in\mathbb Z}$ be a sequence of real numbers such that:
$(i)$ $-\infty\mi t_{k}\mi t_{k+1}\mi\infty$; $(ii)$ $\displaystyle{\lim_{k\rightarrow\pm\infty}t_{k}=\pm\infty}$;
$(iii)$ $\delta\mi t_{k+1}-t_{k}\mi\Delta$ for some fixed numbers $0\mi\delta\mi\Delta\mi\infty.$ Let $\Delta_{k}=t_{k+1}-t_{k}.$
Let $H=\mathbb Z^N$ and $G=\mathbb R^N$. Denote points in $\mathbb Z^N$ by ${\bf k}=(k_1,\dots,k_N).$ For every $w\ma0$, let us define $h_w:\mathbb Z^N\rightarrow\mathbb R^N$ by $h_w({\bf k})=\dfrac{1}{w}{\bf t_{\bf k}}=(t_{k_1}/w,\dots,t_{k_N}/w)\in\mathbb R^N$. Let us consider the grid in $\mathbb R^N$ determined by the point ${\bf t_k}$. Then $R^N$ is divided into parallelepipeds $C_{\bf{k}}=[t_{k_1}/w,t_{k_1+1}/w]\times\dots\times [t_{k_{N}}/w,t_{k_N+1}/w]$. Each parallelepiped $C_{{\bf k}}$ has $N$-dimensional Lebesgue measure equal to $|C_{{\bf k}}|=\dfrac{\Delta_{k_1}\cdots\Delta_{k_N}}{w^N}.$
If $f\in M(\mathbb R^N)$, we are left with the family
$$S_w^{(5)}f({\bf z})= \sum_{{\bf k}\in\mathbb Z^N}\chi_w({\bf z}-{\bf t_k}/w)\dfrac{1}{|C_{{\bf k}}|}\int_{C_{{\bf k}}}f({\bf u})du.$$
Our theory applies to the family above (see \cite{CV1,CV2,BM1} for details concerning the series $S_w^{(5)}f$). Moreover, one can define operators analogous to $S^{(2)}_wf,\;S^{(3)}_wf$ and $S^{(4)}_wf$ in a multidimensional setting by adapting the construction we made above to those concrete cases.
\end{itemize}
\begin{remark}\rm As we mentioned before, in all our examples assumption ($\chi_6$) translates in a condition on the kernels $\chi_w$. This is due to the behaviour of the Kantoro\-vich-type operators in $L^\varphi(\mathbb R^N)$. \end{remark}
\section{Concrete results in Orlicz spaces}
\def\theequation{5.\arabic{equation}}\makeatother
\setcounter{equation}{0}
In this section we rewrite the results obtained before in the general setting to some important cases.
The first case we examine are the standard $L^p$-spaces. Let $\varphi(u)=u^p$ ($p\geq 1$), $u\in\mathbb R_0^+$. The Orlicz space $L^\varphi(G)$ coincides with the standard space $L^p(G)$. Theorems \ref{t3} and \ref{t6} can be summarized as follows:
\begin{theorem}\label{lp} Let $f\in\mathcal Y.$ Then $$||S_wf||_p\leq \left(CM^{p-1}\right)^{1/p}||f||_p.$$ Moreover
$$\lim_{w\rightarrow\infty}||S_wf-f||_p=0.$$
\end{theorem}
Other important examples of Orlicz spaces are furnished by the so-called \ap interpolation spaces'' ($L^\alpha\ln^\beta L$): these spaces are defined as the Orlicz spaces $L^{\alpha,\beta}(G)$, where $\alpha\geq1$ and $\beta\ma 0$. The generating function is given by $\varphi_{\alpha,\beta}(u)=u^\alpha\ln^\beta(e+u)$. Hence $L^{\alpha,\beta}(G)$ is made up of all the functions $f\in M(G)$ such that there exists a constant $\lambda \ma0$ satisfying $$I^{\alpha,\beta}(\lambda f):=\int_\mathbb G\left(\lambda |f(x)|\right)^\alpha \ln^\beta (e+\lambda|f(x)|)dx\mi+\infty.$$
The interested reader can be addressed to \cite{STEI1,STEI2} for more information on interpolation spaces. The function $\varphi_{\alpha,\beta}$ satisfies the $\Delta_2$-condition, hence modular and Luxemburg convergences are the same in $L^{\alpha,\beta}(G)$. Again, we can restate theorems \ref{t3} and \ref{t6} as follows:
\begin{theorem}\label{llog} Let $f\in\mathcal Y.$
Then \begin{equation*}\begin{split}&\int_G|S_wf(x)|^\alpha\ln^\beta\left(e+\lambda|S_wf(x)|\right)dx\leq \\&\leq CM^{\alpha-1}\int_G\left(|f(x)|^\alpha\ln^\beta(e+\lambda M|f(x)|)\right)dx,\end{split}\end{equation*} for every $\lambda\ma0$. Moreover, $$\lim_{w\rightarrow\infty}||S_wf-f||_{\varphi_{\alpha,\beta}}=0.$$
\end{theorem}
The last example of Orlicz space we consider is a space where the $\Delta_2$-condition is not fulfilled, hence modular and Luxemburg convergence are distinct. This space is the so called \ap exponential space''. To define an exponential space, let us fix $\alpha\ma0$, and consider the function $\varphi_\alpha:\mathbb R_0^+\rightarrow\mathbb R_0^+:u\mapsto\exp(u^\alpha)-1.$ The space $L^{\varphi_\alpha}(G)$ consists of those functions $f\in M(G)$ for which there exists a constant $\lambda\ma0$ such that $$I^\alpha(\lambda f):=\int_\mathbb G\left(\exp[(\lambda|f(x)|)^\alpha]-1\right)dx\mi +\infty;$$ (for more information on exponential spaces, see, e.g. \cite{HEN}).
Theorems \ref{t3} and \ref{t6} can be stated as
\begin{theorem}\label{exp} Let $f\in \mathcal Y.$
Then for some $\lambda\ma0$ we have
$$\int_G\left(\exp[(\lambda|S_wf(x)|)^\alpha]-1\right)dz\leq \dfrac{C}{M}\int_G
\left(\exp[(\lambda M|f(x)|)^\alpha]-1\right).
$$
Moreover, there exists a number $\lambda\ma0$ such that $$\lim_{w\rightarrow\infty}\int_G\left(\exp[(\lambda|S_wf(x)-f(x)|)^\alpha]-1\right)dx=0.$$
\end{theorem}
\section{Some graphical representations}
\def\theequation{6.\arabic{equation}}\makeatother
\setcounter{equation}{0}
This section provides some graphical representations of the convergence of the operators we have studied in the previous sections. In all the examples below the convergence must be interpreted as to be in the $L^p$ setting.
We will concentrate on the examples (2), (3) and (4) of Section 4,
since graphical examples of operators (1) can be found in \cite{BBSV3} and for operators (5) one can see \cite{CV1,CV2}.
Although the prototypical example of kernel is obtained from the Fejer's kernel function $$F(x)=\dfrac{1}{2}\sinc^2\left(\dfrac{x}{2}\right),$$ where $$\sinc(x)=\begin{cases}\dfrac{\sin\pi x}{\pi x},&x\in\mathbb R\setminus\{0\},\\1,&x=0\end{cases},$$
it will be convenient for computational purposes to take a kernel with compact support over $\mathbb R$. Well known examples of such kernels are those arising from linear combinations the so-called $B$-splines functions of order $n\in\mathbb N$, namely
$$M_n(x)=\dfrac{1}{(n-1)!}\sum_{j=0}^n(-1)^j\begin{pmatrix}n\\j\end{pmatrix}\left(\dfrac{n}{2}+x-1\right)_+^{n-1},$$ where the symbol $(\cdot)_+$ denotes the positive part. Below we represent the graphs of the functions $M_3(x),M_4(x)$ (Figure 1), and $M(x)=4M_3(x)-3M_4(x)$ (Figure 2).
\vspace{-.3cm}
\begin{figure}[htbp]
\begin{center}\includegraphics[width=.55\textwidth,height=.34\textwidth]
{M3}\includegraphics[width=.55\textwidth,height=.34\textwidth]
{M4}\end{center}
\vspace{-.5cm}
\caption{\small{\it The graphs of $M_3(u)$ and of $M_4(u)$ for $-5\leq u\leq5$.}}
\end{figure}
\vspace{-.7cm}
\begin{figure}[htbp]
\begin{center}\includegraphics[width=.7\textwidth,height=.47\textwidth]
{M}\end{center}
\vspace{-1.3cm}
\caption{\small{\it The graph of $M(u)$ for $-5\leq u\leq5$.}}
\end{figure}
In the following examples we will use the function $M(u)$ to define kernels which satisfy ($\chi_1$)--($\chi_6$). We start from the example considered in (2) of Section 4. We choose the kernels $\chi_w(u)$ defined as $\chi_w(u)=M(wu).$ We have $$S_w^{(2)}f(x)=\int_{-\infty}^\infty M(wx-t)\left(\dfrac{w}{2}\int_{(t-1)/w}^{(t+1)/w}f(s)ds\right)dt.$$
We take a discontinuous function $f(x)$, as follows
$$f(x)=\begin{cases}3e^x,&x\mi-1,\\-1,&-1\leq x\mi0,\\2,&0\leq x\mi 1,\\ x,&1\leq x\mi 2,\\-2e^{-x},&x\geq 2.\end{cases}.$$ The graphs below (Figure 3) show the behaviour of the operator $S_w^{(2)}$ for $w=5,10$ and 15 respectively.
\begin{figure}[htbp]
\includegraphics[width=.49\textwidth,height=.49\textwidth]
{g15}
\end{figure}
\begin{figure}[htbp]
\includegraphics[width=.49\textwidth,height=.49\textwidth]
{g110}
\includegraphics[width=.49\textwidth,height=.49\textwidth]
{g115}
\caption{\small{\it The graphs of the functions $S_5^{(2)}f(x),S_{10}^{(2)}f(x)$, $S_{15}^{(2)}f(x)$ compared with the graph of $f(x)$}}
\end{figure}
The next example represents the approximation of the operator $S_w^{(3)}$ of Section 4. In order to satisfy the assumptions ($\chi_1$)--($\chi_6$), we take the kernels $\chi_w(u)$ defined as $\chi_w(u)=wM(wu).$ So we are left with $$S_w^{(3)}f(x)=\int_{\mathbb R}wM(wx-wt)\left(\dfrac{w}{2}\int_{t-\frac{1}{w}}^{t+\frac{1}{w}}f(s)ds\right)dt.$$ We take the function $$f(x)=\begin{cases}3e^x,&x\mi-1,\\-1,&-1\leq x\mi0,\\2,&0\leq x\mi 2,\\-2e^{-x},&x\geq 2.\end{cases}$$
\vspace{-.4cm}
Below we represent the graphs of $S_5^{(3)}f(x), S_{10}^{(3)}f(x)$ and $S_{15}^{(3)}f(x)$ respectively (Fi\-gure 4).
\vspace{-.3cm}
\begin{figure}[htbp]
\includegraphics[width=.49\textwidth,height=.49\textwidth]
{c5}
\end{figure}
\vspace{-1cm}
\begin{figure}[htbp]
\includegraphics[width=.49\textwidth,height=.49\textwidth]
{c10}
\includegraphics[width=.49\textwidth,height=.49\textwidth]
{c15}
\caption{\small{\it The graphs of the functions $S_5^{(3)}f(x),S_{10}^{(3)}f(x)$, $S_{15}^{(3)}f(x)$ compared with the graph of $f(x)$}}
\end{figure}
\vspace{3cm}
Our last graphical representation takes into account the example (4) in Section 4. In this case, however, we cannot take kernels based on the function $M(u)$ as before, because of the base space $\mathbb R^+$ and of the measure $d\mu(t)=\dfrac{dt}{t}$. Suitable kernel functions in this case are familiar to those working with Mellin operators, namely we consider the kernels
\vspace{-.2cm}
$$\mathcal M_w(u)=\begin{cases} wu^w,&0<u<1,\\0,& \mbox{otherwise}\end{cases}.$$
\vspace{-.2cm}
Next, we consider the operators
\vspace{-.2cm}
$$S_w^{(4)}f(x)=\int_0^\infty\mathcal M_w\left(\dfrac{x}{t}\right)\dfrac{1}{2\ln(1+1/w)}\left(\int_{t\frac{w}{w+1}}^{t\frac{w+1}{w}}f(u)\dfrac{du}{u}\right)\dfrac{dt}{t}.$$
\vspace{-.2cm}
We consider the function $$f(x)=\begin{cases}2x,&0\leq x<2,\\1,&2\leq x<4,\\-25/x^3,&x\geq 4\end{cases}.$$
\vspace{-.3cm}
Below (Figure 5) we represent the approximation of the operators $S_w^{(4)}$ for $w=5,20$ and 30 respectively.
\begin{figure}[htbp]
\includegraphics[width=.49\textwidth,height=.49\textwidth]
{m5}
\end{figure}
\vspace{-.8cm}
\begin{figure}[htbp]
\includegraphics[width=.49\textwidth,height=.49\textwidth]
{m20}
\includegraphics[width=.49\textwidth,height=.49\textwidth]
{m30}
\caption{\small{\it The graphs of the functions $S_5^{(4)}f(x),S_{20}^{(4)}f(x)$, $S_{30}^{(4)}f(x)$ compared with the graph of $f(x)$}}
\end{figure}
\newpage
| -53,839.838921 |
[
-2.947265625,
2.611328125
] | 42.411642 |
[
-3.271484375,
0.580078125,
-1.8994140625,
-6.26171875,
-0.775390625,
8.9296875
] |
[
1.986328125,
7.48828125,
-0.2890625,
4.73828125
] | 351 | 5,114 |
[
-3.44140625,
3.96484375
] | 34.350733 |
[
-5.54296875,
-4.30859375,
-5.3828125,
-2.419921875,
2.013671875,
13.203125
] | 0.963855 | 28.12104 | 24.912006 | 4.126653 |
[
1.4194631576538086
] | -33,817.386051 | 6.329683 | -53,232.809718 | 0.667285 | 5.985079 |
[
-1.796875,
-3.158203125,
-3.8359375,
-5.3984375,
1.9306640625,
12.4921875
] |
[
-5.74609375,
-1.84765625,
-2.130859375,
-0.890625,
3.609375,
4.06640625
] | |
BkiUbwnxK4sA-7sDdyQM
|
\section{Conclusion}
We present the first large-scale densely-annotated material segmentation dataset which can train or evaluate indoor and outdoor scene parsing models.
\footnote{Our data is available at https://github.com/apple/ml-dms-dataset.}
We propose a benchmark on 46 kinds of materials. Our data can be a foundation for algorithms which utilize material type, make use of physical properties for simulations or functional properties for planning and human-computer interactions. We look forward to expanding the number of materials, finding new methods to reach even better full-scene material segmentation, and combining the point-wise annotations of MINC~\cite{minc} with our data in future work.
{\bf Acknowledgements.}
We thank Allison Vanderby, Hillary Strickland, Laura Snarr, Mya Exum, Subhash Sudan, Sneha Deshpande, and Doris Guo for their help with acquiring data; Richard Gass, Daniel Kurz and Selim Ben Himane for their support.
\section{Data Collection}
\label{sec:dataset}
\text{DMS}\xspace is a set of dense polygon annotations of 52 material classes across 44,560 images, which are a subset of OpenImages~\cite{openimages2}. We followed a four step process. First, a set of labels was defined. Next, a large set of images was studied by people and algorithms to select images for annotation. Next, the selected images were fully segmented and labeled by a human annotator. Finally, each segmented image was relabeled by multiple people and a final label map was created by fusing all labels. The last three steps were followed multiple times.
\subsection{Material Labels}
\label{sec:dataset1}
We choose to predefine a label set which is the approach of COCO-Stuff~\cite{cocostuff}. This encourages annotators to create consistent labels suitable for machine learning. We instructed annotators to assign \matlabel{not on list} to recognized materials which do not fit in any category and \matlabel{I cannot tell} to unknown and unrecognizable surfaces (\emph{e.g.}} \def\Eg{\emph{E.g.}, watermarks and under-/over-saturated pixels).
We defined a label set based on appearance, which is the approach of OpenSurfaces~\cite{opensurfaces}. A label can represent a solid substance (\emph{e.g.}} \def\Eg{\emph{E.g.}, wood), a distinctive arrangement of substances (\emph{e.g.}} \def\Eg{\emph{E.g.}, brickwork), a liquid (\emph{e.g.}} \def\Eg{\emph{E.g.}, water) or a useful non-material (\emph{e.g.}} \def\Eg{\emph{E.g.}, sky).
We used 35 labels from OpenSurfaces and \matlabel{asphalt} from~\cite{schwartz2019recognizing}.
We added 2 fine-grained people and animal categories (\matlabel{bone} and \matlabel{animal skin}). We introduced 3 labels for workplaces (\matlabel{ceiling tile}, \matlabel{whiteboard} and \matlabel{fiberglass wool}), 6 for indoor scenes (\matlabel{artwork}, \matlabel{clutter}, \matlabel{non-water liquid}, \matlabel{soap}, \matlabel{pearl} and \matlabel{gemstone}) and 4 for outdoors (\matlabel{sand}, \matlabel{snow}, \matlabel{ice} and \matlabel{tree wood}). \matlabel{Artwork} identifies an imitative surface which is photographic or fine art---affording further analysis by Materials In Paintings~\cite{van2021materials}. \matlabel{Clutter} is a region of visually indistinguishable manufactured stuff (typically a mixture of metal, plastic and paper) which occurs in trash piles. Lastly, we defined a label called \matlabel{engineered stone} for artificial surfaces which imitate stone, which includes untextured and laminated solid surfaces.
See Figure~\ref{fig:matlabels} for an example of each label.
\subsection{Image Selection}
\label{sec:dataset2}
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{figures/all_occurrences.pdf}
\caption{{\bf Image diversity.} We plot number of categories ($y$-axis) vs. occurrence in images (log-scale $x$-axis) of Places365 scene type (a), COCO objects (b), and SUN attributes (c). Our dataset ({\it blue}) is larger, more diverse and more balanced across categories (higher slope) compared to the largest segmentation dataset ({\it orange}).}
\label{fig:occurrences}
\end{figure}
Bell \emph{et al.}~\cite{minc} found that a balanced set of material labels can achieve nearly the same performance as a 9x larger imbalanced set. Since we collect dense annotations we cannot directly balance classes. Instead, we searched 191k images for rare materials and assumed common materials would co-occur. Furthermore, we ran Detectron~\cite{detectron} to detect COCO~\cite{coco} objects, and Places365~\cite{places365} to classify scenes and recognize SUN~\cite{sunattributes} attributes. EXIF information was used to infer country. These detections were used to select images of underrepresented scenes, objects and countries.
Figure~\ref{fig:occurrences} compares the diversity of the 45k images in \text{DMS}\xspace to the 19k images in OpenSurfaces by a plot of the number of categories, $y$, which have at least $x$ occurrences. Occurrences of scene type, object and SUN attribute are plotted. Note that the $x$-axis is logarithmic scale. We find our dataset is more diverse having more classes present in greater amounts (more than can be explained by the 2.24x difference in image count).
We balance the distribution of skin appearance in \text{DMS}\xspace so that algorithms trained with our data perform well on all kinds of skin~\cite{gendershades}. We use Fitzpatrick~\cite{fitzpatrick} skin type to categorize skin into 3 groups, inspired by an approach used by~\cite{towardsfairerdatasets}. We ran the DLIB~\cite{dlib} face detector and labeled a subset of the faces. Our 157 manual annotations were used to calibrate a preexisting face attribute predictor (trained on a different dataset) which was then used to predict skin types for the rest of \text{DMS}\xspace. We found that the ratio of the largest group to the smallest was 9.4. Next, we selected images which would increase the most underrepresented skin type group and found this reduced the ratio to 2.2. We calibrated the same detector for OpenSurfaces faces and measured its ratio as 10.4. According to the findings of~\cite{gendershades}, we expect skin classifiers trained on OpenSurfaces would underperform on dark skin. Table~\ref{tab:skintypes} shows the distribution of skin types.
We used Places365 scene type detection to select outdoor images but we found this did not lead to outdoor materials. We took two steps to address this. First, we annotated our images with one of nine \emph{photographic types} which distinguish outdoor from indoor from unreal images. Table~\ref{tab:photographic_type} shows the annotated types. Next, we used these labels to select outdoor scenes and underrepresented viewpoints. This was effective---growing the dataset by 17\% more than doubled 9 kinds of outdoor materials: \matlabel{ice} (3x), \matlabel{sand} (4.4x), \matlabel{sky} (8x), \matlabel{snow} (9.5x), \matlabel{soil} (3x), \matlabel{natural stone} (2.4x), \matlabel{water} (2.5x), \matlabel{tree wood} (2.3x) and \matlabel{asphalt} (9.2x).
\begin{table}[t]
\centering
\caption{{\bf Skin types.} We report estimated occurrences. Our dataset has 12x more occurrences of the smallest group and 4.8x more fair representation by ratio.}
\label{tab:skintypes}
\begin{tabular}{@{}lrr@{}}\toprule
& OpenSurfaces & \text{DMS}\xspace (Ours)\\\midrule
Type I-II (light) & 2,332 & 4,535\\
Type III-IV (medium) & 3,889 & 9,776\\
Type V-VI (dark) & 375 & 5,899\\\midrule
Ratio of largest to smallest group & 10.37\,:\,1 & 2.16\,:\,1\\
\bottomrule
\end{tabular}
\end{table}
\begin{table}[t]
\centering
\caption{{\bf Photographic types.} Our data contains indoor views ({\it top}), outdoor views ({\it middle}), and close-up and unusual views ({\it bottom}).}
\label{tab:photographic_type}
\begin{minipage}[c]{0.69\linewidth}
\begin{tabular}{@{}lp{3mm}r@{}}\toprule
Photographic Type & & Images \\\midrule
An area with visible enclosure & & 16,013 \\
A collection of indoor things & & 6,064 \\
A tightly cropped indoor thing & & 2,634 \\\midrule
A ground-level view of reachable outdoor things & & 3,127 \\
A tightly cropped outdoor thing & & 1,196 \\
Distant unreachable outdoor things & & 971 \\\midrule
A real surface without context & & 847 \\
Not a real photo & & 805 \\
An obstructed or distorted view & & 204 \\
\bottomrule
\end{tabular}
\end{minipage}
\begin{minipage}[c]{0.275\linewidth}
\includegraphics[height=10.9ex]{figures/example_photographic_types/8_22591_crop.jpg}\hfill
\includegraphics[height=10.9ex]{figures/example_photographic_types/7_22711_crop.jpg}\hfill
\includegraphics[height=10.9ex]{figures/example_photographic_types/6_228861_crop.jpg}
\includegraphics[height=10.9ex]{figures/example_photographic_types/9_232645_crop.jpg}\hfill
\includegraphics[height=10.9ex]{figures/example_photographic_types/5_228134_crop.jpg}\hfill
\includegraphics[height=10.9ex]{figures/example_photographic_types/4_193852_crop.jpg}
\includegraphics[height=10.9ex]{figures/example_photographic_types/2_230376_crop.jpg}\hfill
\includegraphics[height=10.9ex]{figures/example_photographic_types/1_46126_crop.jpg}\hfill
\includegraphics[height=10.9ex]{figures/example_photographic_types/3_57208_crop.jpg}
\end{minipage}
\end{table}
\subsection{Segmentation and Instances}
\label{sec:dataset3}
Images were given to annotators for polygon segmentation of the entire image.
We instructed annotators to segment parts larger than a fingertip, ignore gaps smaller than a finger, and to follow material boundaries tightly while ignoring geometry and shadow boundaries. Following~\cite{opensurfaces}, annotators were instructed to segment glass and mirror surfaces rather than the covered or reflected surfaces. Unreal elements such as borders and watermarks were segmented separately. Images with objectionable content (\emph{e.g.}} \def\Eg{\emph{E.g.}, violence) were not annotated.
Annotators segmented resized images, with median longest edge of 1024 pixels, creating over 3.2 million segments (counting only those larger than 100 pixels) with a mean of 72 segments per image. The created segments are detailed---wires, jewelry, teeth, eyebrows, shoe soles, wheel rims, door hinges, clasps, buttons and latches are some of the small and thin materials segmented separately. See Figure~\ref{fig:teaser} and Figure~\ref{fig:fusedlabels} for examples of detailed segmentations.
We defined a material instance as materials of the same type from the same manufacturing source. For example a wooden cabinet should be segmented separately from a wood floor but the planks making up a single-source floor would be one instance. \text{DMS}\xspace is the first large-scale densely segmented dataset to have detailed material instances.
\begin{table}[t]
\centering
\caption{{\bf Annotator agreement rates.} High rates indicate consistent label assignment. Low rates indicate disagreement, confusion or unstructured error.}
\label{tab:agreement}
\begin{tabular}{@{}llp{3mm}llp{3mm}llp{3mm}ll@{}}\toprule
Hair & 0.95 & & Glass & 0.80 & & Wood & 0.67 & & Non-clear plastic & 0.60\\
Skin & 0.93 & & Paper & 0.76 & & Tree wood & 0.66 & & Leather & 0.53\\
Foliage & 0.86 & & Carpet/rug & 0.73 & & Tile & 0.66 & & Cardboard & 0.53\\
Sky & 0.86 & & Nat. stone & 0.72 & & Metal & 0.65 & & Artwork & 0.51\\
Food & 0.84 & & Ceramic & 0.70 & & Paint/plaster & 0.62 & & Clear plastic & 0.50\\
Fabric/cloth & 0.82 & & Mirror & 0.68 & & Rubber & 0.61 & & Concrete & 0.45\\
\bottomrule
\end{tabular}
\end{table}
\subsection{Labeling}
The annotator who segmented an image also assigned labels based on their judgment and our instruction. We found that surfaces coated with another material or colored by absorbing ink required clarification. Appearance-changing coatings were labeled \matlabel{paint} while clear or appearance-enhancing coatings (\emph{e.g.}} \def\Eg{\emph{E.g.}, varnish, cosmetics, sheer hosiery) were labeled as the underlying material. Small amounts of ink (\emph{e.g.}} \def\Eg{\emph{E.g.}, printed text) are disregarded. Some surfaces imitate the appearance of other materials (\emph{e.g.}} \def\Eg{\emph{E.g.}, laminate). High-quality imitations were labeled as the imitated material and low-quality imitations as the real material.
Our instructions were refined in each iteration and incorrect labels from early iterations were corrected. Some cases needed special instruction. We instructed annotators to label electronic displays as \matlabel{glass} and vinyl projection screens as \matlabel{not on list}. Uncovered artwork or photographs were to be labeled \matlabel{artwork} while glass-covered art should be labeled \matlabel{glass}. In ambiguous cases, we assume framed artwork has a glass cover.
\matlabel{Sky} includes day sky, night sky and aerial phenomenon (\emph{e.g.}} \def\Eg{\emph{E.g.}, clouds, stars, moon, and sun).
We collected more opinions by presenting a segmentation, after removing labels, to a different annotator who relabeled the segments. The relabeling annotator could fix bad segments by adjusting polygons or assign special labels to indicate a segment does not follow boundaries or is made of multiple material types. We collected 98,526 opinions across 44,560 images consisting of 8.2 million segment labels (counting only segments larger than 100 pixels).
We studied label agreement by counting occurrences of a segment label and matching pixel-wise dominant label by a different annotator.
We found an agreement rate of 0.675. In cases of agreement, 8.9\% were unrecognizable (\matlabel{I cannot tell}) and 0.6\% were \matlabel{not on list}.
Table~\ref{tab:agreement} shows the agreement rate for classes larger than the median number of segments per class. Among the largest classes the most agreed-upon labels are \matlabel{hair}, \matlabel{skin}, \matlabel{foliage}, \matlabel{sky}, and \matlabel{food}.
We only analyze the largest classes since unstructured error (\emph{e.g.}} \def\Eg{\emph{E.g.}, misclicks) can overwhelm the statistics of small classes, which are up to 2,720 times smaller.
\begin{figure}[t]
\includegraphics[height=12.1ex]{figures/fusedlabel_examples/fused_202582_crop.png}\hfill
\includegraphics[height=12.1ex]{figures/fusedlabel_examples/fused_138031.png}\hfill
\includegraphics[height=12.1ex]{figures/fusedlabel_examples/fused_219628_crop.png}\hfill
\includegraphics[height=12.1ex]{figures/fusedlabel_examples/fused_41296.png}\hfill
\includegraphics[height=12.1ex]{figures/fusedlabel_examples/fused_53567.png}\hfill
\includegraphics[height=12.1ex]{figures/fusedlabel_examples/fused_56373_crop.png}
\caption{{\bf Fused labels.} We show segmentation quality and variety of scenes, activities and materials ({\it left to right:} building exterior, workplace, road, swimming pool, shop, dining room). See Table~\ref{tab:fusedcount} for color legend. Black pixels are unlabeled (no consensus).}
\label{fig:fusedlabels}
\end{figure}
\begin{table}[t]
\centering
\caption{{\bf Material occurrence in images.} We report the number of images in which a label occurs. The colors are used for visualizations.}
\label{tab:fusedcount}
\begin{tabular}{@{}p{3.1mm}lrp{3mm}p{3.1mm}lrp{3mm}p{3.1mm}lr@{}}\toprule
\cellcolor{matcolor29} & Paint/plaster & 39,323 & & \cellcolor{matcolor38} & Sky & 3,306 & & \cellcolor{matcolor8} & Chalkboard & 668 \\
\cellcolor{matcolor13} & Fabric/cloth & 31,489 & & \cellcolor{matcolor27} & Mirror & 3,242 & & \cellcolor{matcolor56} & Asphalt & 474 \\
\cellcolor{matcolor34} & Non-clear plas& 30,506 & & \cellcolor{matcolor4} & Cardboard & 3,150 & & \cellcolor{matcolor15} & Fire & 412 \\
\cellcolor{matcolor26} & Metal & 30,504 & & \cellcolor{matcolor17} & Food & 2,908 & & \cellcolor{matcolor19} & Gemstone & 369 \\
\cellcolor{matcolor20} & Glass & 28,934 & & \cellcolor{matcolor10} & Concrete & 2,853 & & \cellcolor{matcolor42} & Sponge & 326 \\
\cellcolor{matcolor52} & Wood & 24,248 & & \cellcolor{matcolor6} & Ceiling tile & 2,524 & & \cellcolor{matcolor12} & Eng. stone & 299 \\
\cellcolor{matcolor30} & Paper & 20,763 & & \cellcolor{matcolor43} & Natural stone & 2,076 & & \cellcolor{matcolor25} & Liquid & 294 \\
\cellcolor{matcolor37} & Skin & 18,524 & & \cellcolor{matcolor48} & Water & 2,063 & & \cellcolor{matcolor31} & Pearl & 282 \\
\cellcolor{matcolor21} & Hair & 17,766 & & \cellcolor{matcolor53} & Tree wood & 2,026 & & \cellcolor{matcolor11} & Cork & 273 \\
\cellcolor{matcolor16} & Foliage & 11,384 & & \cellcolor{matcolor51} & Wicker & 1,895 & & \cellcolor{matcolor36} & Sand & 272 \\
\cellcolor{matcolor46} & Tile & 10,173 & & \cellcolor{matcolor41} & Soil/mud & 1,855 & & \cellcolor{matcolor39} & Snow & 191 \\
\cellcolor{matcolor5} & Carpet/rug & 9,516 & & \cellcolor{matcolor44} & Pol. stone & 1,831 & & \cellcolor{matcolor40} & Soap & 154 \\
\cellcolor{matcolor7} & Ceramic & 8,314 & & \cellcolor{matcolor3} & Brickwork & 1,654 & & \cellcolor{matcolor9} & Clutter & 128 \\
\cellcolor{matcolor35} & Rubber & 7,811 & & \cellcolor{matcolor18} & Fur & 1,567 & & \cellcolor{matcolor23} & Ice & 96 \\
\cellcolor{matcolor24} & Leather & 7,354 & & \cellcolor{matcolor50} & Whiteboard & 1,171 & & \cellcolor{matcolor45} & Styrofoam & 88 \\
\cellcolor{matcolor33} & Clear plastic & 6,431 & & \cellcolor{matcolor49} & Wax & 1,107 & & \cellcolor{matcolor14} & Fiberglass wool & 33 \\
\cellcolor{matcolor32} & Artwork & 4,344 & & \cellcolor{matcolor47} & Wallpaper & 1,076 \\
\cellcolor{matcolor2} & Bone/horn & 3,751 & & \cellcolor{matcolor1} & Animal skin & 1,007 \\
\bottomrule
\end{tabular}
\end{table}
\subsection{Label Fusion}
Each annotator's segments are rendered to create a label map. Label maps were inspected for correctness and we fixed incorrect labels in 1,803 images.
Next, we create a single \emph{fused label map} for each image. First, we combined label maps pixel-wise by taking the strict majority label. Next, we overlaid manual corrections and reassigned non-semantic labels (\emph{e.g.}} \def\Eg{\emph{E.g.}, \matlabel{I cannot tell}) to \matlabel{no label}.
The fused maps have a mean labeled area fraction of 0.784. For comparison, we created fused label maps for OpenSurfaces and found its density is 0.210. \text{DMS}\xspace is 2.3x larger and 3.7x denser, which is 8.4x more labeled area.
Compared to the 3M points in MINC~\cite{minc}, DMS has 3.2M fused segments which carry more information about shape, boundary and co-occurrences. While MINC annotations span 10x more images, point annotations cannot evaluate segmentation boundaries for scene parsing tasks.
Example fused maps and class occurrences are shown in Figure~\ref{fig:fusedlabels} and Table~\ref{tab:fusedcount}.
The smallest class appears in 33 images whereas the largest class, \matlabel{paint}, appears in 39,323 images, which is 88\% of the images.
\captionsetup[subfigure]{labelformat=empty}
\begin{figure}[t]
\subfloat[Asphalt]{\includegraphics[height=7.5ex]{figures/class_examples/56_asphalt_small.jpg}}\hfill
\subfloat[Bone]{\includegraphics[height=7.5ex]{figures/class_examples/2_bone.jpg}}\hfill
\subfloat[Brick]{\includegraphics[height=7.5ex]{figures/class_examples/3_brick_small.jpg}}\hfill
\subfloat[Eng. stone]{\includegraphics[height=7.5ex]{figures/class_examples/12_engineeredstone_crop.jpg}}\hfill
\subfloat[Fabric]{\includegraphics[height=7.5ex]{figures/class_examples/13_fabric_small.jpg}}\hfill
\subfloat[Carpet]{\includegraphics[height=7.5ex]{figures/class_examples/5_carpet_small.jpg}}\hfill
\subfloat[Ceiling tile]{\includegraphics[height=7.5ex]{figures/class_examples/6_ceilingtile.jpg}}\hfill
\subfloat[Ceramic]{\includegraphics[height=7.5ex]{figures/class_examples/7_ceramic.jpg}}\hfill
\subfloat[Wax]{\includegraphics[height=7.5ex]{figures/class_examples/49_wax_crop.jpg}}
\subfloat[Wallpaper]{\includegraphics[height=7.5ex]{figures/class_examples/47_wallpaper_small.jpg}}\hfill
\subfloat[Clear plastic]{\includegraphics[height=7.5ex]{figures/class_examples/33_clearplastic_small.jpg}}\hfill
\subfloat[Plastic]{\includegraphics[height=7.5ex]{figures/class_examples/34_plastic_small.jpg}}\hfill
\subfloat[Concrete]{\includegraphics[height=7.5ex]{figures/class_examples/10_concrete_small.jpg}}\hfill
\subfloat[Artwork]{\includegraphics[height=7.5ex]{figures/class_examples/32_photograph.jpg}}\hfill
\subfloat[Cardboard]{\includegraphics[height=7.5ex]{figures/class_examples/4_cardboard_small.jpg}}\hfill
\subfloat[Chalkboard]{\includegraphics[height=7.5ex]{figures/class_examples/8_chalkboard.jpg}}\hfill
\subfloat[Fiberglass]{\includegraphics[height=7.5ex]{figures/class_examples/14_fiberglass.jpg}}\hfill
\subfloat[Rubber]{\includegraphics[height=7.5ex]{figures/class_examples/35_rubber_masked_small.jpg}}
\subfloat[Fur]{\includegraphics[height=7.5ex]{figures/class_examples/18_fur_small.jpg}}\hfill
\subfloat[Foliage]{\includegraphics[height=7.5ex]{figures/class_examples/16_foliage_small.jpg}}\hfill
\subfloat[Food]{\includegraphics[height=7.5ex]{figures/class_examples/17_food_small.jpg}}\hfill
\subfloat[Hair]{\includegraphics[height=7.5ex]{figures/class_examples/21_hair.jpg}}\hfill
\subfloat[Cork]{\includegraphics[height=7.5ex]{figures/class_examples/11_cork.jpg}}\hfill
\subfloat[Fire]{\includegraphics[height=7.5ex]{figures/class_examples/15_fire_small.jpg}}\hfill
\subfloat[Gemstone]{\includegraphics[height=7.5ex]{figures/class_examples/19_gemstone.jpg}}\hfill
\subfloat[Glass]{\includegraphics[height=7.5ex]{figures/class_examples/20_glass_small.jpg}}\hfill
\subfloat[Ice]{\includegraphics[height=7.5ex]{figures/class_examples/23_ice_small.jpg}}
\subfloat[Paper]{\includegraphics[height=7.3ex]{figures/class_examples/30_paper_small.jpg}}\hfill
\subfloat[Leather]{\includegraphics[height=7.3ex]{figures/class_examples/24_leather_small.jpg}}\hfill
\subfloat[Liquid]{\includegraphics[height=7.3ex]{figures/class_examples/25_liquid.jpg}}\hfill
\subfloat[Metal]{\includegraphics[height=7.3ex]{figures/class_examples/26_metal_small.jpg}}\hfill
\subfloat[Mirror]{\includegraphics[height=7.3ex]{figures/class_examples/27_mirror_small.jpg}}\hfill
\subfloat[Paint]{\includegraphics[height=7.3ex]{figures/class_examples/29_paint_masked_small.jpg}}\hfill
\subfloat[Pearl]{\includegraphics[height=7.3ex]{figures/class_examples/31_pearl_crop.jpg}}\hfill
\subfloat[Sponge]{\includegraphics[height=7.3ex]{figures/class_examples/42_sponge.jpg}}
\subfloat[Soap]{\includegraphics[height=7.5ex]{figures/class_examples/40_soap_crop.jpg}}\hfill
\subfloat[Clutter]{\includegraphics[height=7.5ex]{figures/class_examples/9_clutter_crop.jpg}}\hfill
\subfloat[Wicker]{\includegraphics[height=7.5ex]{figures/class_examples/51_wicker_small.jpg}}\hfill
\subfloat[Snow]{\includegraphics[height=7.5ex]{figures/class_examples/39_snow_small.jpg}}\hfill
\subfloat[Sand]{\includegraphics[height=7.5ex]{figures/class_examples/36_sand_small.jpg}}\hfill
\subfloat[Skin]{\includegraphics[height=7.5ex]{figures/class_examples/37_skin.jpg}}\hfill
\subfloat[Sky]{\includegraphics[height=7.5ex]{figures/class_examples/38_sky_small.jpg}}\hfill
\subfloat[Soil]{\includegraphics[height=7.5ex]{figures/class_examples/41_soil_small.jpg}}
\subfloat[Nat. stone]{\includegraphics[height=7.5ex]{figures/class_examples/43_stone_small.jpg}}\hfill
\subfloat[Pol. stone]{\includegraphics[height=7.5ex]{figures/class_examples/44_polishedstone_crop.jpg}}\hfill
\subfloat[Styrofoam]{\includegraphics[height=7.5ex]{figures/class_examples/45_styrofoam_crop.jpg}}\hfill
\subfloat[Tile]{\includegraphics[height=7.5ex]{figures/class_examples/46_tile_crop_small.jpg}}\hfill
\subfloat[Water]{\includegraphics[height=7.5ex]{figures/class_examples/48_water_crop_small.jpg}}\hfill
\subfloat[Whiteboard]{\includegraphics[height=7.5ex]{figures/class_examples/50_whiteboard_small.jpg}}\hfill
\subfloat[Wood]{\includegraphics[height=7.5ex]{figures/class_examples/52_wood_small.jpg}}\hfill
\subfloat[Tree wood]{\includegraphics[height=7.5ex]{figures/class_examples/53_treewood_small.jpg}}\hfill
\subfloat[Animal skin]{\includegraphics[height=7.5ex]{figures/class_examples/1_hide_small.jpg}}
\caption{{\bf Material labels.} For each label we show a cut-out example.}
\label{fig:matlabels}
\end{figure}
\section{Discussion and Conclusion}
\label{sec:conclusion}
{\bf Dense Annotation.}
Prior works~\cite{opensurfaces,minc,schwartz2019recognizing} instruct annotators to locate and segment regions made of a given material. Our approach is different. We instruct annotators to segment and label the entire image.
This approach collects different data because annotators address all surfaces---not just those which are readily recognized. We hypothesize this creates a more difficult dataset, and propose this approach is necessary for evaluation of scene parsing, which predicts all pixels.
{\bf Real vs. Synthetic.}
Synthetic data has achieved high levels of realism (\emph{e.g.}} \def\Eg{\emph{E.g.}, Hypersim~\cite{hypersim}) and may be a valuable generator of training data. We opted to label real photos because models trained on synthetic data need a real evaluation dataset to confirm the domain gap from synthetic to real has been bridged.
{\bf Privacy.}
Material predictions can be personal. Knowing a limb is not made of skin reveals a prosthetic. The amount of body hair reveals one aspect of appearance. Precious materials in a home reveals socio-economic status. Clothing material indicates degree of nakedness.
Care is needed if material segmentation is tied to identity.
Limiting predicted materials to only those needed by an application or separating personal materials from identity are two ways, among many possible ways, to strengthen privacy and protect personal information.
\section{Experiments}
\label{sec:experiments}
First, we investigate the impact of our data on training deep learning models with a cross-dataset comparison (Section~\ref{sec:crossdataset}). Then, we compare the impact of skin type distributions on fairness of skin recognition (Section~\ref{sec:skin}). Next, we establish a material segmentation benchmark for 46 kinds of materials (Section~\ref{sec:baseline}). Finally, we show predictions on real world images (Section~\ref{sec:real}).
{\bf Splits.} We created train, validation and test splits for our data by assigning images according to material occurrence. The smallest classes are assigned a ratio of 1\,:\,1\,:\,1, which increases to 2.5\,:\,1\,:\,1 for the largest. An image assignment impacts the ratio of multiple classes so small classes are assigned first. There are 24,255 training images, 10,139 validation images and 10,166 test images.
\subsection{Cross-Dataset Comparison}
\label{sec:crossdataset}
Does training with our data lead to a better model?
This experiment compares a model fit to our data against two baselines fit to OpenSurfaces data---the strongest published model~\cite{upernet} and a model with the same architecture as ours.
There are two sources of data. The first is OpenSurfaces data with the splits and 25 labels proposed by~\cite{upernet}. The second is comparable DMS training and validation data (\cite{upernet} does not define a test split) created by translating our labels to match~\cite{upernet}. The evaluation set, which we call Avg-Val, is made of both parts---the validation sets of OpenSurfaces and \text{DMS}\xspace, called OS-Val and \text{DMS}\xspace-Val, respectively---weighted equally.
For evaluation of our data we fit models to DMS training data and choose the model that performs best on \text{DMS}\xspace-Val. This model, which we call \text{DMS}\xspace-25, is a ResNet-50 architecture~\cite{he2016deep} with dilated convolutions~\cite{chen2017deeplab,yu2015multi} as the encoder, and Pyramid Pooling Module from PSPNet~\cite{zhao2017pyramid} as the decoder.
The first baseline (Table~\ref{tab:results1}, row 2) is UPerNet~\cite{upernet}, a multitask scene parsing model which uses cross-domain knowledge to boost material segmentation performance. The second baseline (Table~\ref{tab:results1}, row 3), called OS-25, has the same architecture as \text{DMS}\xspace-25 but is fit to OpenSurfaces training data.
Table~\ref{tab:results1} shows the results. We report per-pixel accuracy (Acc), mean class accuracy (mAcc), mean intersection-over-union (mIoU) and $\Delta$, the absolute difference in a metric across \text{DMS}\xspace-Val and OS-Val. A low $\Delta$ indicates a model is more consistent across datasets.
We find that fitting a model to \text{DMS}\xspace training data leads to higher performance and lower $\Delta$ on all metrics.
We also report the metrics on each validation set and find that both baselines underperform on \text{DMS}\xspace-Val.
We find that DMS-25 performs 0.01 lower on OS-Val mAcc compared to a model trained on OpenSurfaces data. This may be due to differences in annotation and image variety. We use our photographic type labels to investigate the larger performance gaps on \text{DMS}\xspace-Val.
Why do models trained with OpenSurfaces underperform on our validation images?
In Table~\ref{tab:results2} we report per-pixel accuracy of \text{DMS}\xspace-25, UPerNet, and OS-25 across nine categories. We find that \text{DMS}\xspace-25 performs consistently across categories with the lowest performing category (unreal images) 0.071 below the highest performing category (images of enclosed areas). UPerNet shows lower performance across all categories with a drop of 0.426 from images of enclosed areas to images of distant outdoor things. And OS-25 shows similar performance with a drop of 0.407. We observe that both UPerNet and OS-25 have low performance on outdoor images and images without any context.
This study shows that photographic types can improve our understanding of how material segmentation models perform in different settings. And, these results justify our decision to collect outdoor images and images of different photographic types.
\begin{table}[t]
\centering
\caption{{\bf Training data evaluation.} We compare segmentation of 25 materials with our training data ({\it row 1}) to OpenSurfaces data with two kinds of models ({\it rows 2 and 3}). Avg-Val is the equally-weighted validation sets of each dataset, \text{DMS}\xspace-Val and OS-Val. $\Delta$ is the difference in a metric across datasets. A convnet fit to our data achieves higher performance and is more consistent across datasets.}
\label{tab:results1}
\begin{tabular}{@{}lrlrlrcccccc@{}}\toprule
Training data & & Model & & Metric & & {Avg-Val }$\uparrow$ & $\Delta\downarrow$ & {\text{DMS}\xspace-Val }$\uparrow$ & {OS-Val }$\uparrow$\\\midrule
& & & & Acc & & {\bf 0.777} & {\bf 0.047} & 0.753 & 0.800\\
\text{DMS}\xspace (Ours) & & DMS-25 & & mAcc & & {\bf 0.689} & {\bf 0.006} & 0.686 & 0.692\\
& & & & mIoU & & {\bf 0.500} & {\bf 0.014} & 0.507 & 0.493\\\midrule
& & & & Acc & & 0.682 & 0.310 & 0.527 & 0.837\\
OpenSurfaces~\cite{opensurfaces} & & UPerNet~\cite{upernet} & & mAcc & & 0.486 & 0.274 & 0.349 & 0.623\\
& & & & mIoU & & 0.379 & 0.298 & 0.230 & 0.528\\\midrule
& & & & Acc & & 0.705 & 0.231 & 0.589 & 0.820\\
OpenSurfaces~\cite{opensurfaces} & & OS-25 & & mAcc & & 0.606 & 0.193 & 0.509 & 0.702\\
& & & & mIoU & & 0.416 & 0.199 & 0.316 & 0.515\\
\bottomrule
\end{tabular}
\end{table}
\begin{table}[t]
\centering
\caption{{\bf Performance analysis with photographic types.} A model fit to our data, \text{DMS}\xspace-25 ({\it Table~\ref{tab:results1}, row 1}), performs well on all photographic types whereas two models fit to OpenSurfaces, UPerNet and OS-25 ({\it Table~\ref{tab:results1}, rows 2-3}) have low performance outdoors ({\it middle}) and on surfaces without any context ({\it row 7}).}
\label{tab:results2}
\begin{tabular}{@{}lccccc@{}}\toprule
\multicolumn{1}{l}{Photographic Type} & \multicolumn{5}{c}{Per-Pixel Accuracy}\\
\cmidrule{2-6}
& \text{DMS}\xspace-25 (Ours) & & UPerNet~\cite{upernet} & & OS-25\\\midrule
An area with visible enclosure & 0.756 & & 0.615 & & 0.632\\
A collection of indoor things & 0.752 & & 0.546 & & 0.622\\
A tightly cropped indoor thing & 0.710 & & 0.441 & & 0.561\\\midrule
A view of reachable outdoor things & 0.750 & & 0.265 & & 0.388\\
A tightly cropped outdoor thing & 0.731 & & 0.221 & & 0.359\\
Distant unreachable outdoor things & 0.736 & & 0.189 & & 0.225\\\midrule
A real surface without context & 0.691 & & 0.222 & & 0.348\\
Not a real photo & 0.685 & & 0.528 & & 0.551\\
An obstructed or distorted view & 0.729 & & 0.370 & & 0.496\\
\bottomrule
\end{tabular}
\end{table}
\begin{table}[t]
\centering
\caption{{\bf Test set results.} We report metrics for our model, \text{DMS}\xspace-46. 17 materials, in italics, are new---not predicted by prior general-purpose models~\cite{minc,upernet,schwartz2019recognizing}.}
\label{tab:OM46_per_class_testset}
\begin{tabular}{@{}lccp{3mm}lccp{3mm}lcc@{}}\toprule
Category & Acc & IoU & & Category & Acc & IoU & & Category & Acc & IoU\\\midrule
Sky & 0.962 & 0.892 & & \textit{Chalkboard} & 0.712 & 0.548 & & \textit{Artwork} & 0.454 & 0.301 \\
Fur & 0.910 & 0.707 & & Paint/plaster & 0.694 & 0.632 & & Mirror & 0.452 & 0.278 \\
Foliage & 0.902 & 0.761 & & Wicker & 0.674 & 0.460 & & \textit{Sand} & 0.444 & 0.340 \\
Skin & 0.886 & 0.640 & & Natural stone & 0.665 & 0.436 & & \textit{Ice} & 0.440 & 0.362 \\
Hair & 0.881 & 0.673 & & Glass & 0.653 & 0.483 & & \textit{Tree wood} & 0.428 & 0.261 \\
Food & 0.868 & 0.668 & & Asphalt & 0.628 & 0.442 & & Pol. stone & 0.379 & 0.236 \\
\textit{Ceiling tile} & 0.867 & 0.611 & & Leather & 0.615 & 0.373 & & \textit{Clear plastic} & 0.360 & 0.222 \\
Water & 0.866 & 0.712 & & \textit{Snow} & 0.610 & 0.465 & & Rubber & 0.255 & 0.163 \\
Carpet/rug & 0.849 & 0.592 & & Concrete & 0.603 & 0.304 & & \textit{Clutter} & 0.182 & 0.152 \\
\textit{Whiteboard} & 0.838 & 0.506 & & Metal & 0.575 & 0.303 & & \textit{Fire} & 0.176 & 0.147 \\
Fabric/cloth & 0.801 & 0.692 & & \textit{Wax} & 0.573 & 0.371 & & \textit{Gemstone} & 0.116 & 0.096 \\
Wood & 0.797 & 0.635 & & Cardboard & 0.570 & 0.363 & & Eng. stone & 0.088 & 0.071 \\
Ceramic & 0.757 & 0.427 & & Wallpaper & 0.544 & 0.329 & & \textit{Cork} & 0.082 & 0.066 \\
Brickwork & 0.746 & 0.491 & & \textit{Non-clear plastic} & 0.519 & 0.321 & & \textit{Bone/horn} & 0.074 & 0.070 \\
Paper & 0.729 & 0.508 & & Soil/mud & 0.511 & 0.332 \\
Tile & 0.722 & 0.550 & & \textit{Animal skin} & 0.472 & 0.308 \\
\bottomrule
\end{tabular}
\end{table}
\subsection{Recognition of Different Skin Types}
\label{sec:skin}
Models trained on face datasets composed of unbalanced skin types exhibit classification disparities~\cite{gendershades}. Does this impact skin recognition? Without any corrections for skin type imbalance we find
that \text{DMS}\xspace-25 has a 3\% accuracy gap among different skin types on \text{DMS}\xspace-val (Type I-II: 0.933, Type III-IV: 0.924, Type V-VI: 0.903) while OS-25 has a larger gap of 13.3\% (Type I-II: 0.627, Type III-IV: 0.571, Type V-VI: 0.494). This confirms that skin type imbalance impacts skin recognition. Our contribution lies in providing more data for all skin types (Table~\ref{tab:skintypes}), which makes it easier for practitioners to create fair models.
\subsection{A Material Segmentation Benchmark}
\label{sec:baseline}
It is common practice to select large categories and combine smaller ones (our smallest occurs in only 12 training images) for a benchmark. Yet, we cannot know {\it a priori} how much training data is sufficient to learn a category. We choose to be guided by the validation data. We fit many models to all 52 categories then inspect the results to determine which categories can be reliably learned.
We select ResNet50~\cite{he2016deep} with dilated convolutions~\cite{chen2017deeplab,yu2015multi} as the encoder, and Pyramid Pooling Module from PSPNet~\cite{zhao2017pyramid} as the decoder. We choose this architecture because it has been shown to be effective for scene parsing~\cite{zhao2017pyramid,zhou2019semantic}.
Our best model, which we call \text{DMS}\xspace-52, predicts 52 materials with per-pixel accuracy 0.735, mean class accuracy 0.535 and mIoU 0.392 on \text{DMS}\xspace-val.
We inspected a few strongest \text{DMS}\xspace-52 fitted models and found that 6 categories consistently stood out as underperforming---having 0 accuracy in some cases and, at best, not much higher than chance. Those categories are \matlabel{non-water liquid}, \matlabel{fiberglass}, \matlabel{sponge}, \matlabel{pearl}, \matlabel{soap} and \matlabel{styrofoam}, which occur in 129, 12, 149, 129, 58 and 33 training images, respectively. Guided by this discovery we select the other 46 material labels for a benchmark.
We train a model, called \text{DMS}\xspace-46, to predict the selected categories, with the same architecture as DMS-52. We use a batch size of 64 and stochastic gradient descent optimizer with 1e-3 base learning rate and 1e-4 weight decay. We use ImageNet pretraining~\cite{zhou2017scene,zhou2019semantic} to initialize the encoder weights, and scale the learning rate for the encoder by 0.25. We update the learning rate with a cosine annealing schedule with warm restart~\cite{loshchilov2016sgdr} every 30 epochs for 60 epochs. Because the classes are imbalanced we use weighted symmetric cross entropy~\cite{wang2019symmetric}, computed across \text{DMS}\xspace training images, as the loss function, which gives more weight to classes with fewer ground truth pixels. We apply stochastic transformations for data augmentation (scale, horizontal and vertical flips, color jitter, Gaussian noise, Gaussian blur, rotation and crop), scale inputs into [0, 1], and normalize with mean = [0.485, 0.456, 0.406] and std = [0.229, 0.224, 0.225] from ImageNet~\cite{deng2009imagenet}. The training tensor has height and width of 512.
\text{DMS}\xspace-46 predicts 46 materials with per-pixel accuracy 0.731/0.729, mean class accuracy 0.598/0.585 and mIoU 0.435/0.420 on \text{DMS}\xspace-val/\text{DMS}\xspace-test respectively. We report the test set per-class accuracy and IoU in Table~\ref{tab:OM46_per_class_testset}. We find that \matlabel{sky}, \matlabel{fur}, \matlabel{foliage}, \matlabel{skin} and \matlabel{hair} have the highest recognition rates, similar to the findings of~\cite{minc}. 17 materials do not appear in any prior large-scale material benchmarks. Among these new materials we report high recognition rates for \matlabel{ceiling tile}, \matlabel{whiteboard} and \matlabel{chalkboard}.
To our knowledge, \text{DMS}\xspace-46 is the first material segmentation model evaluated on large-scale dense segmentations and predicts more classes than any general-purpose model.
\begin{figure}[t]
\includegraphics[height=13.0ex]{figures/predictions_om46/44.jpg}\hfill
\includegraphics[height=13.0ex]{figures/predictions_om46/56.jpg}\hfill
\includegraphics[height=13.0ex]{figures/predictions_om46/12.jpg}
\includegraphics[height=14.4ex]{figures/predictions_om46/21.jpg}\hfill
\includegraphics[height=14.4ex]{figures/predictions_om46/49.jpg}\hfill
\includegraphics[height=14.4ex]{figures/predictions_om46/5.jpg}\hfill
\includegraphics[height=14.4ex]{figures/predictions_om46/0.jpg}
\caption{{\bf Real-world examples.} Our model, \text{DMS}\xspace-46, predicts 46 kinds of indoor and outdoor materials. See Table~\ref{tab:fusedcount} for color legend.}
\label{fig:examples}
\end{figure}
\subsection{Real-World Examples}
\label{sec:real}
In Figure~\ref{fig:examples} we demonstrate \text{DMS}\xspace-46 on indoor and outdoor photos from daily life. Our model recognizes and localizes \matlabel{food} on \matlabel{ceramic} plates, workplace materials (\matlabel{whiteboard} and \matlabel{ceiling tile}), ground cover materials (\matlabel{soil}, \matlabel{stone}, \matlabel{foliage} and \matlabel{snow}), unprocessed \matlabel{tree wood}, and \matlabel{fire} on a \matlabel{wax} candle.
{\bf A Failure Case.}
The last image is a failure case where our model is confused by decorative tile artwork. We also see opportunities for further improving boundaries and localizing small surfaces.
\section{Introduction}
A goal of computer vision is to develop the cognitive ability to plan manipulation of something and predict how it will respond to stimuli. This is informed by the properties of what something is made of. Those properties can be discovered by segmenting a photograph into recognized materials. Material recognition can be understood through the science of material perception starting with Adelson's~\cite{thingsstuff} proposal to divide the world into \emph{things} (countable objects) and \emph{stuff} (materials). Adelson argued stuff is important because of its ubiquity in everyday life. Ritchie \emph{et al.}~\cite{ritchie2021material} describe material perception in two parts. The first part is categorical recognition of what something is made of. The second part is recognizing material properties (\emph{e.g.}} \def\Eg{\emph{E.g.}, glossy, flexible, sound absorbent, sticky) which tells us how something will feel or how it will interact with other objects. While Schwartz \emph{et al.}~\cite{schwartz2019recognizing} proposed to recognize properties from local image patches we follow Bell \emph{et al.}~\cite{minc} who segmented images by recognizing material classes.
Deep learning-based material recognition builds on some key developments. Sharan \emph{et al.}~\cite{sharan2013recognizing} showed that people can recognize 10 kinds of materials in the wild~\cite{fmd} with 85\% accuracy. Bell \emph{et al.}~\cite{opensurfaces}, following~\cite{labelme}, built an efficient annotation tool to create a large-scale material database from crowds and Internet photos. Next, Bell \emph{et al.}~\cite{minc} introduced large-scale training data and a deep learning approach leading to material segmentation as a building-block for haptics, material assignment, robotic navigation, acoustic simulation and context-aware mixed reality~\cite{gao2016deep,park2018photoshape,schissler2017acoustic,zhao2017fully,brandao2016material,chen2020context}. Xiao \emph{et al.}~\cite{upernet} introduced a multi-task scene parsing model which endows a photograph with a rich prediction of scene type, objects, object parts, materials and textures.
Despite widespread adoption of material segmentation, a lack of large-scale data means evaluation rests on the only large-scale segmentation dataset, OpenSurfaces~\cite{opensurfaces}. We find there is room for improvement and propose the Dense Material Segmentation dataset (DMS) which has 3.2 million segments across 44k densely annotated images, and
show empirically that our data leads to models which further close the gap between computer vision and human perception.
\begin{figure}[t]
\includegraphics[width=\linewidth]{figures/teaser3.pdf}
\caption{{\bf Densely annotated materials.} Our annotations are full-scene, highly detailed and enable prediction of 46 kinds of materials.}
\label{fig:teaser}
\end{figure}
There are goals to consider for a material dataset.
First, we need a general-purpose set of material labels. We want to mimic human perception so we choose distinguishable materials even if they are of the same type. For example, we separate clear from opaque plastic rather than have a single label for all plastics. We define fine-grained labels which have useful properties, physical or otherwise. For example, a painted whiteboard surface has utility not found in a \matlabel{paint} label---it is appropriate for writing, cleaning and virtual content display. These functional properties come from how the material is applied rather than its physical structure. Ultimately we choose a set of 52 labels based on prior work and useful materials we found in photographs (details in Section~\ref{sec:dataset1}).
Following~\cite{schwartz2019recognizing}, we also want indoor and outdoor scenes. Counter-intuitively, this could be unnecessary. Material is recognizable regardless of where it occurs in the world, and deep learning methods aim to create a model which generalizes to unseen cases. Thus, an indoor residential dataset~\cite{opensurfaces} could be sufficient. We find this is not the case. In Section~\ref{sec:crossdataset} we show that a model trained on~\cite{opensurfaces} performs worse on outdoor scenes. This is a key finding which impacts all algorithms which use~\cite{opensurfaces} for training. We also show that a model trained on our dataset is consistent across indoor and outdoor scenes.
We want our database to support many scene parsing tasks so we need broad coverage of objects and scene attributes (which include activities, \emph{e.g.}} \def\Eg{\emph{E.g.}, eating). In Section~\ref{sec:dataset2} we show that we achieve better coverage compared to~\cite{opensurfaces}.
We propose nine kinds of photographic types which distinguish different viewpoints and circumstances. Our motivation was to quantitatively evaluate cases where we had observed poor performance. This data can reveal new insights on how a model performs. We find that a state-of-the-art model underperforms in some settings whereas a model fit to our data performs well on all nine types.
Our final goal is to have diversity in skin types. Skin is associated with race and ethnicity so it is crucial to have fair representation across different types of skin. We compare our skin type data to OpenSurfaces~\cite{opensurfaces} in Section~\ref{sec:dataset2} and show our data has practical benefits for training in Section~\ref{sec:skin}.
The paper is organized as follows. In Section~\ref{sec:related} we review datasets. In Section~\ref{sec:dataset} we describe how we collected data to achieve our goals. In Section~\ref{sec:experiments} we compare our dataset to state-of-the-art data and a state-of-the-art model, study the impact of skin types on training, propose a material segmentation benchmark, and demonstrate material segmentation on real world photos.
In summary, our contributions are:
\begin{itemize}[topsep=3pt,itemsep=0pt]
\item We introduce \text{DMS}\xspace, a large-scale densely-annotated material segmentation dataset and show it is diverse with extensive analysis (Section~\ref{sec:dataset}).
\item We advance fairness toward skin types in material datasets (Section~\ref{sec:dataset2}).
\item We introduce photographic types which reveal new insights on prior work and show that a model fit to our data performs better across datasets and viewpoints compared to the state-of-the-art (Section~\ref{sec:crossdataset}).
\item We propose a new large-scale indoor and outdoor material segmentation benchmark of 46 materials and present a baseline result (Section~\ref{sec:baseline}).
\end{itemize}
\section{Related Work}
\label{sec:related}
{\bf Material Segmentation Datasets.} The largest dataset is OpenSurfaces~\cite{opensurfaces} which collected richly annotated polygons of residential indoor surfaces on 19k images, including 37 kinds of materials.
The largest material recognition dataset is the Materials in Context Database~\cite{minc} which is 3M point annotations of 23 kinds of materials across 437k images. This data enables material segmentation by CNN and a dense CRF tuned on OpenSurfaces segments.
The Local Materials Database~\cite{schwartz2019recognizing} collected segmentations, with the goal of studying materials using only local patches, of 16 kinds of materials across 5,845 images sourced from existing datasets.
The Light-Field Material Dataset~\cite{wang20164d} is 1,200 4D indoor and outdoor images of 12 kinds of materials.
The Multi-Illumination dataset~\cite{murmann2019dataset} captured 1,016 indoor scenes under 25 lighting conditions and annotated the images with 35 kinds of materials.
Table~\ref{tab:datasets} lists the largest datasets.
\begin{table}[t]
\centering
\caption{{\bf Large-scale datasets.} We propose a dataset with 23x more segments, more classes and 2.3x more images as the largest segment-annotated dataset.}
\label{tab:datasets}
\begin{tabular}{@{}lrlcrrl@{}}\toprule
Dataset & & Annotation & Classes & Images & & Scenes\\\midrule
OpenSurfaces~\cite{opensurfaces} & & 137k segments & 37 & 19,447 & & Indoor residential\\
Materials in Context~\cite{minc} & & 3M points & 23 & 436,749 & & Home interior \& exterior\\
Local Materials~\cite{schwartz2019recognizing} & & 9.4k segments & 16 & 5,845 & & Indoor \& outdoor\\
\text{DMS}\xspace (Ours) & & 3.2M segments & 52 & 44,560 & & Indoor \& outdoor\\
\bottomrule
\end{tabular}
\end{table}
Materials have appeared in purpose-built datasets. The Ground Terrain in Outdoor Scenes (GTOS) database~\cite{gtos} and GTOS-mobile~\cite{gtos2} are 30k images of hundreds of instances of 40 kinds of ground materials and 81 videos of 31 kinds of ground materials, respectively. The Materials in Paintings dataset~\cite{van2021materials} is bounding box annotations and extracted segmentations on 19k paintings of 15 kinds of materials depicted by artists, partly distinguished into 50 fine-grained categories.
COCO-Stuff~\cite{cocostuff} is segmentations of 91 kinds of stuff on 164k COCO~\cite{coco} images. While this is a source of material data, it is not a general-purpose material dataset because important surfaces (\emph{e.g.}} \def\Eg{\emph{E.g.}, objects labeled in COCO) are not assigned material labels.
ClearGrasp~\cite{cleargrasp} is a dataset of 50k synthetic and 286 real RGB-D images of glass objects built for robotic manipulation of transparent objects.
The Glass Detection Dataset~\cite{mei2020don} is 3,916 indoor and outdoor images of segmented glass surfaces.
The Mirror Segmentation Dataset~\cite{yang2019my} is 4,018 images with segmented mirror surfaces across indoor and outdoor scenes.
Fashionpedia~\cite{fashionpedia} is a database of segmented clothing images of which 10k are annotated with fashion attributes which include fine-grained clothing materials.
Figaro~\cite{figaro} is 840 images of people with segmented hair distinguished into 7 kinds of hairstyles.
{\bf Categorical Material Names.}
Bell \emph{et al.}~\cite{opensurfaces} created a set of names by asking annotators to enter free-form labels which were merged into a list of material names.
This approach is based on the appearance of surfaces as perceived by the annotators. Schwartz \emph{et al.}~\cite{schwartz2019recognizing} created a three-level hierarchy of material names where materials are organized by their physical properties. Some categories were added for materials which could not be placed in the hierarchy. In practice, both approaches resulted in a similar set of entry-level~\cite{ordonez2013large} names which also closely agree with prior studies of categorical materials in Internet images~\cite{fmd,hu2011toward}.
\section{Dataset Details}
In this section we supplement Section~\ref{sec:dataset} of the main paper.
In Table~\ref{tab:matarea} we list names used in annotation tools. For brevity, names in the main paper are shortened and ``Photograph/painting'' is called \matlabel{artwork}. We also report the number of images in which a material occurs and total area, the sum over all images of the fraction of pixels covered by a material.
In Table~\ref{tab:fusedpixelcount} we show the number of annotated pixels for each class. This count is according to the resized images which are smaller than the original images.
{
\begin{longtable}{@{}lrrrrcrrrr@{}}
\caption{{\bf Material occurrence.} We report the number of images and total area (in units of image proportion, rounded).}
\label{tab:matarea}\\
\toprule
\endfirsthead
\caption{continued from previous page}\\
\endhead
& \multicolumn{4}{c}{Image Count} & \phantom{\;} & \multicolumn{4}{c}{Total Area}\\
\cmidrule{2-5} \cmidrule{7-10}
& All & Train & Val & Test & & All & Train & Val & Test\\\midrule
Animal skin & 1,007 & 479 & 260 & 268 & & 34 & 14 & 8 & 11\\
Bone/teeth/horn & 3,751 & 2,084 & 858 & 809 & & 4 & 2 & 1 & 2\\
Brickwork & 1,654 & 862 & 388 & 404 & & 204 & 113 & 46 & 44\\
Cardboard & 3,150 & 1,773 & 681 & 696 & & 133 & 73 & 30 & 30\\
Carpet/rug & 9,516 & 5,470 & 2,073 & 1,973 & & 985 & 567 & 208 & 209\\
Ceiling tile & 2,524 & 1,460 & 529 & 535 & & 299 & 173 & 65 & 61\\
Ceramic & 8,314 & 4,608 & 1,854 & 1,852 & & 260 & 135 & 69 & 56\\
Chalkboard/blackboard\;\; & 668 & 332 & 166 & 170 & & 68 & 34 & 16 & 19\\
Clutter & 128 & 41 & 43 & 44 & & 12 & 3 & 5 & 5\\
Concrete & 2,853 & 1,381 & 731 & 741 & & 400 & 186 & 109 & 105\\
Cork/corkboard & 273 & 122 & 78 & 73 & & 9 & 4 & 2 & 3\\
Engineered stone & 299 & 134 & 81 & 84 & & 18 & 8 & 5 & 5\\
Fabric/cloth & 31,489 & 17,727 & 6,875 & 6,887 & & 4,799 & 2,732 & 1,038 & 1,030\\
Fiberglass wool & 33 & 12 & 9 & 12 & & 3 & 1 & 1 & 1\\
Fire & 412 & 184 & 110 & 118 & & 12 & 5 & 4 & 3\\
Foliage & 11,384 & 5,902 & 2,714 & 2,768 & & 1,377 & 640 & 372 & 364\\
Food & 2,908 & 1,553 & 687 & 668 & & 287 & 126 & 82 & 79\\
Fur & 1,567 & 761 & 398 & 408 & & 206 & 95 & 55 & 55\\
Gemstone/quartz & 369 & 165 & 99 & 105 & & 10 & 5 & 2 & 3\\
Glass & 28,934 & 16,142 & 6,378 & 6,414 & & 2,159 & 1,192 & 488 & 479\\
Hair & 17,766 & 10,076 & 3,823 & 3,867 & & 336 & 190 & 74 & 72\\
Ice & 96 & 31 & 32 & 33 & & 27 & 10 & 8 & 8\\
Leather & 7,354 & 4,146 & 1,609 & 1,599 & & 210 & 118 & 50 & 42\\
Liquid, non-water & 294 & 129 & 83 & 82 & & 9 & 2 & 4 & 3\\
Metal & 30,504 & 16,917 & 6,801 & 6,786 & & 805 & 427 & 187 & 190\\
Mirror & 3,242 & 1,871 & 684 & 687 & & 315 & 176 & 67 & 72\\
Paint/plaster/enamel & 39,323 & 21,765 & 8,773 & 8,785 & & 10,965 & 6,073 & 2,434 & 2,458\\
Paper & 20,763 & 11,692 & 4,592 & 4,479 & & 883 & 485 & 200 & 199\\
Pearl & 282 & 129 & 77 & 76 & & 0 & 0 & 0 & 0\\
Photograph/painting & 4,344 & 2,435 & 976 & 933 & & 174 & 90 & 41 & 43\\
Plastic, clear & 6,431 & 3,583 & 1,425 & 1,423 & & 129 & 69 & 28 & 31\\
Plastic, non-clear & 30,506 & 17,154 & 6,662 & 6,690 & & 1,278 & 708 & 282 & 288\\
Rubber/latex & 7,811 & 4,244 & 1,788 & 1,779 & & 65 & 32 & 17 & 16\\
Sand & 272 & 110 & 76 & 86 & & 70 & 24 & 20 & 26\\
Skin/lips & 18,524 & 10,444 & 4,014 & 4,066 & & 509 & 287 & 113 & 108\\
Sky & 3,306 & 1,447 & 911 & 948 & & 1,020 & 435 & 286 & 298\\
Snow & 191 & 70 & 60 & 61 & & 57 & 19 & 20 & 18\\
Soap & 154 & 58 & 50 & 46 & & 0 & 0 & 0 & 0\\
Soil/mud & 1,855 & 860 & 495 & 500 & & 165 & 73 & 42 & 51\\
Sponge & 326 & 149 & 89 & 88 & & 1 & 1 & 0 & 0\\
Stone, natural & 2,076 & 962 & 569 & 545 & & 355 & 156 & 102 & 98\\
Stone, polished & 1,831 & 993 & 435 & 403 & & 187 & 97 & 46 & 44\\
Styrofoam & 88 & 33 & 27 & 28 & & 2 & 1 & 0 & 1\\
Tile & 10,173 & 5,722 & 2,206 & 2,245 & & 1,490 & 845 & 321 & 323\\
Wallpaper & 1,076 & 577 & 252 & 247 & & 233 & 127 & 56 & 49\\
Water & 2,063 & 959 & 552 & 552 & & 564 & 260 & 156 & 149\\
Wax & 1,107 & 578 & 260 & 269 & & 7 & 3 & 2 & 2\\
Whiteboard & 1,171 & 642 & 265 & 264 & & 111 & 60 & 24 & 27\\
Wicker & 1,895 & 1,031 & 438 & 426 & & 75 & 35 & 22 & 18\\
Wood & 24,248 & 13,496 & 5,309 & 5,443 & & 3,608 & 2,006 & 802 & 800\\
Wood, tree & 2,026 & 929 & 561 & 536 & & 72 & 30 & 19 & 22\\
Asphalt & 474 & 211 & 132 & 131 & & 73 & 35 & 17 & 22\\
\bottomrule
\end{longtable}
}
\begin{table}[t]
\centering
\caption{{\bf Material occurrence in pixels.} We report the number of pixels covered by each label according to the resized images used by annotation tools.}
\label{tab:fusedpixelcount}
\begin{tabular}{@{}lrp{3mm}lr@{}}\toprule
Animal skin & 22,995,883 & & Paint/plaster/enamel & 7,796,144,397\\
Bone/teeth/horn & 3,050,548 & & Paper & 628,009,751\\
Brickwork & 145,410,237 & & Pearl & 411,455\\
Cardboard & 93,881,191 & & Photograph/painting & 123,296,052\\
Carpet/rug & 707,147,207 & & Plastic, clear & 93,002,805\\
Ceiling tile & 216,289,692 & & Plastic, non-clear & 906,618,216\\
Ceramic & 185,191,692 & & Rubber/latex & 45,644,757\\
Chalkboard/blackboard\;\; & 48,346,203 & & Sand & 47,860,125\\
Clutter & 8,845,550 & & Skin/lips & 359,727,474\\
Concrete & 283,303,562 & & Sky & 702,864,398\\
Cork/corkboard & 6,468,131 & & Snow & 40,936,881\\
Engineered stone & 13,140,139 & & Soap & 265,782\\
Fabric/cloth & 3,408,488,743 & & Soil/mud & 114,322,155\\
Fiberglass wool & 1,874,005 & & Sponge & 1,075,671\\
Fire & 7,965,989 & & Stone, natural & 253,271,347\\
Foliage & 961,103,715 & & Stone, polished & 134,425,626\\
Food & 192,755,372 & & Styrofoam & 1,552,343\\
Fur & 145,359,760 & & Tile & 1,068,909,615\\
Gemstone/quartz & 7,273,649 & & Wallpaper & 168,289,772\\
Glass & 1,535,538,311 & & Water & 390,040,955\\
Hair & 238,600,730 & & Wax & 4,791,692\\
Ice & 18,308,742 & & Whiteboard & 80,692,711\\
Leather & 149,122,712 & & Wicker & 50,066,493\\
Liquid, non-water & 5,861,652 & & Wood & 2,584,799,129\\
Metal & 573,827,793 & & Wood, tree & 50,922,547\\
Mirror & 224,631,105 & & Asphalt & 51,218,822\\
\bottomrule
\end{tabular}
\end{table}
\begin{table}[t]
\centering
\caption{{\bf Case resolution.} For some cases we provided additional instruction, which we summarize here.}
\label{tab:labeldesc}
\begin{tabular}{@{}lrl@{}}\toprule
Case & & Resolution\\\midrule
Skin with sparse hair & & \matlabel{Skin} for people; \matlabel{animal skin} for animals.\\
Coat of hair (\emph{e.g.}} \def\Eg{\emph{E.g.}, horse) & & \matlabel{Fur}.\\
Smoothed stone & & \matlabel{Polished stone}.\\
Laminated paper & & \matlabel{Clear plastic}.\\
Sauces & & \matlabel{Food} on food; \matlabel{non-water liquid} during preparation.\\
Chandelier prisms & & \matlabel{Gemstone} or \matlabel{glass} based on appearance.\\
Seasoned or blued metal & & \matlabel{Metal}.\\
Metal patina & & \matlabel{Metal}.\\
Printed text & & The underlying material.\\
Mirror-like finishes & & \matlabel{Mirror} if sole purpose is to reflect; the material otherwise.\\
Wrapped items & & The material of the wrap.\\
Electronic display & & \matlabel{Glass}.\\
Glass-top surface & & \matlabel{Glass}.\\
Thatch & & \matlabel{Wicker}.\\
Stained wood & & \matlabel{Wood}.\\
Projection screen & & \matlabel{Not on list}.\\
Vinyl & & The closest of \matlabel{non-clear plastic}, \matlabel{rubber} or \matlabel{leather}.\\
\bottomrule
\end{tabular}
\end{table}
\begin{table}[t]
\centering
\caption{{\bf Objects and functional spaces.} We report the number of images for the largest classes of detected objects ({\it top}) and estimated scene functions ({\it bottom}).}
\label{tab:topk}
\begin{tabular}{@{}lrrrrp{3mm}lrrrr@{}}\toprule
& All & Train & Val & Test & & & All & Train & Val & Test\\\midrule
person & 19,966 & 11,219 & 4,303 & 4,426 & & tie & 1,398 & 802 & 280 & 314\\
chair & 17,617 & 9,987 & 3,826 & 3,780 & & bench & 1,196 & 671 & 244 & 277\\
dining table & 8,086 & 4,511 & 1,765 & 1,806 & & keyboard & 1,192 & 648 & 272 & 272\\
bottle & 5,964 & 3,320 & 1,313 & 1,325 & & cell phone & 1,121 & 629 & 269 & 222\\
cup & 5,656 & 3,136 & 1,248 & 1,265 & & mouse & 939 & 516 & 199 & 224\\
potted plant & 5,078 & 2,762 & 1,122 & 1,191 & & refrigerator & 834 & 504 & 161 & 168\\
book & 4,384 & 2,465 & 976 & 939 & & backpack & 739 & 420 & 154 & 165\\
tv & 4,303 & 2,411 & 947 & 942 & & oven & 737 & 399 & 173 & 165\\
laptop & 3,076 & 1,737 & 664 & 675 & & remote & 718 & 403 & 166 & 148\\
bowl & 2,900 & 1,579 & 636 & 682 & & dog & 692 & 369 & 162 & 160\\
couch & 2,846 & 1,614 & 628 & 602 & & cat & 685 & 344 & 162 & 178\\
vase & 2,790 & 1,551 & 626 & 609 & & toilet & 677 & 383 & 144 & 149\\
bed & 2,357 & 1,348 & 524 & 482 & & knife & 579 & 335 & 123 & 120\\
sink & 1,747 & 949 & 395 & 402 & & car & 542 & 292 & 128 & 121\\
handbag & 1,617 & 906 & 366 & 345 & & boat & 524 & 227 & 136 & 161\\
wine glass & 1,473 & 797 & 332 & 343 & & suitcase & 510 & 310 & 94 & 106\\
clock & 1,452 & 814 & 294 & 343 & & spoon & 477 & 258 & 106 & 112\\\midrule
working & 14,343 & 8,032 & 3,124 & 3,166 & & swimming & 868 & 397 & 240 & 230\\
reading & 14,039 & 7,931 & 3,118 & 2,970 & & sports & 824 & 442 & 181 & 198\\
socializing & 8,545 & 4,869 & 1,794 & 1,873 & & using tools & 686 & 369 & 149 & 167\\
congregating & 7,317 & 4,129 & 1,559 & 1,620 & & praying & 649 & 363 & 144 & 138\\
eating & 5,862 & 3,217 & 1,294 & 1,345 & & touring & 626 & 283 & 159 & 180\\
shopping & 2,419 & 1,325 & 563 & 526 & & waiting in line & 593 & 362 & 118 & 113\\
studying & 2,070 & 1,147 & 459 & 463 & & exercise & 574 & 329 & 106 & 137\\
competing & 1,960 & 1,085 & 410 & 458 & & diving & 556 & 275 & 163 & 117\\
spectating & 1,489 & 845 & 305 & 335 & & bathing & 524 & 288 & 120 & 115\\
training & 1,335 & 744 & 295 & 295 & & research & 451 & 251 & 92 & 108\\
transporting & 1,153 & 587 & 268 & 297 & & cleaning & 445 & 247 & 94 & 104\\
boating & 876 & 371 & 235 & 267 & & driving & 404 & 199 & 92 & 113\\
\bottomrule
\end{tabular}
\end{table}
\begin{table}[t]
\centering
\caption{{\bf Judgments.} We report the number of unique opinions (\emph{i.e.}} \def\Ie{\emph{I.e.}, label maps) collected for images.}
\label{tab:judgments}
\begin{tabular}{@{}ccr@{}}\toprule
Label Map Count & & Images\\\midrule
1 & & 1,245\\
2 & & 35,039\\
3 & & 7,459\\
4 & & 122\\
5 & & 867\\
\bottomrule
\end{tabular}
\end{table}
\begin{figure}[t]
\includegraphics[height=13.5ex]{figures/fusedlabel_examples/fused_231048.png}\hfill
\includegraphics[height=13.5ex]{figures/fusedlabel_examples/fused_197290.png}\hfill
\includegraphics[height=13.5ex]{figures/fusedlabel_examples/fused_197794.png}\hfill
\includegraphics[height=13.5ex]{figures/fusedlabel_examples/fused_57588.png}\hfill
\includegraphics[height=13.5ex]{figures/density_representative_openmaterials.png}\hfill
\crule{0.15ex}{13.5ex}\hfill
\includegraphics[height=13.5ex]{figures/density_representative_opensurfaces.png}
\caption{{\bf Fused material labels.} {\it Left to right:} van, sports, aerial photo, conference and dining area. The 5th image has a label density close to the mean density of \text{DMS}\xspace. The rightmost image is a fused label map from OpenSurfaces with a label density close to the mean density of OpenSurfaces. See Table~\ref{tab:fusedcount} for color legend.}
\label{fig:morefusedlabels}
\end{figure}
We found that asking annotators to label all surfaces required extensive instruction. Our training document grew to include clarifications for rare and uncommon cases. In Table~\ref{tab:labeldesc} we summarize how we choose to resolve cases.
In Table~\ref{tab:topk} we report the number of images in which an object class is detected by~\cite{detectron}, and the number of images which are predicted by~\cite{places365} to have scene elements for an activity. There are 80 object classes and 30 functional scene attributes. For brevity, we report only the largest classes.
For most images we collected two unique opinions for labels. In Table~\ref{tab:judgments} we report the number of images with a given number of opinions.
In Figure~\ref{fig:morefusedlabels} we expand on Figure~\ref{fig:fusedlabels} by showing more fused label maps and we show a fused label map from \text{DMS}\xspace and OpenSurfaces which are representative of the mean density of the respective datasets.
\section{Skin Type Experiment}
In Section~\ref{sec:skin}, we compared skin accuracies for three skin groups, Type I-II, Type III-IV, and Type V-VI. In order to compute accuracy we have to assign ground truth pixels to a group. We do this for images which contain detections of only one skin group. However, there are images where multiple skin groups co-occur and where no skin groups were detected. We do not evaluate on these two scenarios to avoid assigning groups incorrectly.
\section{Benchmark Experiment Details}
In this section we include more details on training our material segmentation benchmark model, DMS-46, from Section~\ref{sec:baseline} of the main paper. All the models are trained on NVIDIA Tesla V100 GPUs with 32 GB of memory.
\subsection{Data Augmentation}
In this section we show details on how we apply different data augmentation in training. We apply the following data transformation in order:
{\bf Scale.} We first scale the input image so that the shortest dimension is 512 given that the training image size has height 512 and width 512. Then we randomly scale the input dimension with a ratio in [1, 2, 3, 4] uniformly.
{\bf Horizontal Flip.} We apply random horizontal flip with probability 0.5.
{\bf Vertical Flip.} We apply random vertical flip with probability 0.5.
{\bf Color Jitter.} We apply color jitter with probability 0.9, using torchvision\footnote{https://pytorch.org/vision/} ColorJitter with brightness 0.4, contrast 0.4, saturation 0.4, and hue 0.1.
{\bf Gaussian Blur or Gaussian Noise.} We apply this transformation with probability 0.5. Gaussian blur or Gaussian noise is selected with equal chance. We use a kernel size of 3 for Gaussian blur with uniformly chosen standard deviation in [0.1, 2.0]. Gaussian noise has mean of 0 and standard deviation 3 across all the pixels.
{\bf Rotation.} We apply random rotation in [-45, 45] degrees with probability 0.5. We fill 0 for the area outside the rotated color image and an ignore value for the rotated segmentation map. The loss calculation ignores those pixels.
{\bf Crop.} Finally, we randomly crop a subregion, height 512 and width 512, to feed into the neural network.
\subsection{Loss Function}
We use weighted symmetric cross entropy~\cite{wang2019symmetric} as the loss function for DMS-46. The weight ${W_{i}}$ for each class is calculated as a function of frequency of pixel count, ${F_{i}}$, for each material class ${i \in N}$~\cite{paszke2016enet}, in Equation~\ref{eq:weighted}.\begin{equation} \label{eq:weighted}
W_{i} = \frac{1} {\log \left(1.02 + \frac{F_{i}} {\sum_{i=1}^{N}F_{i}}\right)}
\end{equation}
The number 1.02 is introduced in~\cite{paszke2016enet} to restrict the class weights in [1, 50] as the probability approaches 0. The weights we are using for DMS-46 are presented in Table~\ref{tab:allnames}.
Symmetric cross entropy (SCE)~\cite{wang2019symmetric} is composed of a regular cross entropy (CE) and a reverse cross entropy (RCE) to avoid overfitting to noisy labels. Given the target distribution P and the predicted distribution Q, Equation~\ref{eq:sce} shows each part of the loss function for SCE. We choose $\alpha=1$ and $\beta=0.5$ for the weighting coefficients.
\begin{equation} \label{eq:sce}
L_{SCE} = \alpha L_{CE} + \beta L_{RCE} = \alpha (- \sum P \log Q \\) + \beta (- \sum Q \log P \\)
\end{equation}
\begin{table}[t]
\centering
\caption{{\bf Class weights.} We show the class weights we applied in the loss function for DMS-46.}
\label{tab:allnames}
\begin{tabular}{@{}lrp{3mm}lrp{3mm}lr@{}}\toprule
Label & Weight & & Label & Weight & & Label & Weight \\\midrule
Bone & 50.259 & & Whiteboard & 43.585 & & Hair & 33.870 \\
Wax & 50.140 & & Clear plastic & 42.709 & & Water & 30.402 \\
Clutter & 50.136 & & Soil & 42.585 & & Skin & 29.049 \\
Cork & 49.995 & & Cardboard & 42.482 & & Sky & 24.133 \\
Fire & 49.945 & & Artwork & 40.905 & & Metal & 23.981 \\
Gemstone & 49.826 & & Fur & 40.427 & & Paper & 22.447 \\
Engineered stone & 49.459 & & Pol. stone & 40.226 & & Carpet & 20.422 \\
Ice & 49.163 & & Brickwork & 38.979 & & Foliage & 19.325 \\
Animal skin & 48.646 & & Leather & 38.715 & & Non-clear plastic & 17.986 \\
Snow & 47.972 & & Food & 38.368 & & Tile & 15.895 \\
Sand & 47.603 & & Wallpaper & 37.854 & & Glass & 12.555 \\
Tree wood & 46.759 & & Ceramic & 37.201 & & Wood & 8.388 \\
Rubber & 46.672 & & Nat. stone & 35.919 & & Fabric & 6.596 \\
Wicker & 46.465 & & Mirror & 34.651 & & Paint & 3.415 \\
Chalkboard & 46.462 & & Ceiling tile & 34.617 \\
Asphalt & 46.447 & & Concrete & 34.095 \\
\bottomrule
\end{tabular}
\end{table}
\subsection{Model Architecture Implementation}
We select ResNet50~\cite{he2016deep} with dilated convolutions~\cite{chen2017deeplab,yu2015multi} as the encoder, and Pyramid Pooling Module from PSPNet~\cite{zhao2017pyramid} as the decoder. We choose this architecture because it has been shown to be effective for scene parsing~\cite{zhao2017pyramid,zhou2019semantic}. We use a publicly-available implementation of ResNet50dilated architecture with pre-trained weights (on an ImageNet task) from~\cite{zhou2017scene,zhou2019semantic}\footnote{https://github.com/CSAILVision/semantic-segmentation-pytorch}, under a BSD 3-Clause License.
\subsection{Material Class Selection For Benchmark}
In Section~\ref{sec:baseline} we reported empirically finding that six material categories (\matlabel{non-water liquid}, \matlabel{fiberglass}, \matlabel{sponge}, \matlabel{pearl}, \matlabel{soap} and \matlabel{styrofoam}) fail consistently across models. We present the three top candidates of DMS-52 which led us to this conclusion. Each one is the best fitted model, according to DMS-val, from a comprehensive hyper-parameter search on learning rate, learning rate scheduler, and optimizer.
The first model, called DMS-52, is the best model across all models, is introduced in the main paper, and we report the per-class performance in Table~\ref{tab:first}.
The second model, called DMS-52 variant A, has the same architecture as DMS-52 and uses all of OpenSurfaces data as additional training data. We report the per-class performance of DMS-52A in Table~\ref{tab:second}.
The third model, called DMS-52 variant B, has a ResNet101 architecture and uses OpenSurfaces data as additional training data. We report the per-class performance of DMS-52B in Table~\ref{tab:third}.
Across DMS-52, DMS-52A and DMS-52B the same six material classes are the worst-performing categories. Based on these findings we selected the other 46 categories for a benchmark and leave these six to future work.
\begin{table}[t]
\centering
\caption{{\bf DMS-Val results for DMS-52.} Results are sorted by accuracy.}
\label{tab:first}
\begin{tabular}{@{}lccp{3mm}lccp{3mm}lcc@{}}\toprule
& Acc & IoU & & & Acc & IoU & & & Acc & IoU\\\midrule
Sky & 0.937 & 0.891 & & Glass & 0.703 & 0.489 & & Animal skin & 0.396 & 0.268 \\
Fur & 0.913 & 0.694 & & Paper & 0.686 & 0.496 & & Rubber & 0.345 & 0.240 \\
Foliage & 0.897 & 0.769 & & Leather & 0.676 & 0.397 & & Pol. stone & 0.332 & 0.236 \\
Ceiling tile & 0.890 & 0.679 & & Nat. stone & 0.634 & 0.447 & & Tree wood & 0.327 & 0.224 \\
Hair & 0.885 & 0.673 & & Wax & 0.626 & 0.430 & & Ice & 0.320 & 0.284 \\
Food & 0.882 & 0.689 & & Wicker & 0.622 & 0.432 & & Bone & 0.213 & 0.178 \\
Water & 0.881 & 0.695 & & Wallpaper & 0.603 & 0.397 & & Clutter & 0.209 & 0.186 \\
Skin & 0.876 & 0.647 & & Concrete & 0.579 & 0.333 & & Gemstone & 0.127 & 0.077 \\
Carpet & 0.855 & 0.582 & & Soil & 0.578 & 0.376 & & Cork & 0.115 & 0.102 \\
Fire & 0.821 & 0.621 & & Cardboard & 0.571 & 0.340 & & Eng. stone & 0.096 & 0.069 \\
Wood & 0.801 & 0.657 & & Non-clear plastic & 0.562 & 0.322 & & {\bf Sponge} & 0.051 & 0.050 \\
Fabric & 0.787 & 0.690 & & Asphalt & 0.560 & 0.386 & & {\bf Liquid} & 0.048 & 0.044 \\
Brickwork & 0.785 & 0.514 & & Metal & 0.548 & 0.305 & & {\bf Fiberglass} & 0.034 & 0.034 \\
Whiteboard & 0.771 & 0.508 & & Sand & 0.548 & 0.407 & & {\bf Styrofoam} & 0.003 & 0.003 \\
Tile & 0.752 & 0.564 & & Snow & 0.495 & 0.414 & & {\bf Pearl} & 0.000 & 0.000 \\
Chalkboard & 0.747 & 0.616 & & Clear plastic & 0.441 & 0.254 & & {\bf Soap} & 0.000 & 0.000 \\
Ceramic & 0.746 & 0.482 & & Mirror & 0.423 & 0.297 \\
Paint & 0.707 & 0.640 & & Artwork & 0.407 & 0.271 \\
\bottomrule
\end{tabular}
\end{table}
\begin{table}[t]
\centering
\caption{{\bf DMS-Val results for DMS-52A.} Results are sorted by accuracy.}
\label{tab:second}
\begin{tabular}{@{}lccp{3mm}lccp{3mm}lcc@{}}\toprule
& Acc & IoU & & & Acc & IoU & & & Acc & IoU\\\midrule
Sky & 0.946 & 0.889 & & Leather & 0.695 & 0.407 & & Clear plastic & 0.405 & 0.255 \\
Fur & 0.921 & 0.692 & & Paint & 0.680 & 0.625 & & Rubber & 0.367 & 0.240 \\
Foliage & 0.912 & 0.768 & & Wicker & 0.670 & 0.436 & & Tree wood & 0.358 & 0.221 \\
Ceiling tile & 0.886 & 0.686 & & Concrete & 0.646 & 0.347 & & Wax & 0.327 & 0.246 \\
Hair & 0.883 & 0.677 & & Soil & 0.635 & 0.385 & & Ice & 0.230 & 0.228 \\
Water & 0.883 & 0.707 & & Fire & 0.626 & 0.570 & & Eng. stone & 0.207 & 0.108 \\
Skin & 0.877 & 0.636 & & Nat. stone & 0.620 & 0.439 & & Clutter & 0.204 & 0.185 \\
Food & 0.875 & 0.688 & & Wallpaper & 0.600 & 0.417 & & Bone & 0.167 & 0.139 \\
Carpet & 0.830 & 0.614 & & Asphalt & 0.599 & 0.401 & & Cork & 0.126 & 0.112 \\
Wood & 0.821 & 0.654 & & Cardboard & 0.586 & 0.362 & & Gemstone & 0.087 & 0.057 \\
Fabric & 0.801 & 0.700 & & Snow & 0.584 & 0.484 & & {\bf Sponge} & 0.066 & 0.060 \\
Whiteboard & 0.801 & 0.515 & & Non-clear plastic & 0.555 & 0.319 & & {\bf Fiberglass} & 0.029 & 0.029 \\
Brickwork & 0.789 & 0.496 & & Metal & 0.548 & 0.289 & & {\bf Liquid} & 0.009 & 0.009 \\
Ceramic & 0.772 & 0.471 & & Animal skin & 0.517 & 0.272 & & {\bf Pearl} & 0.000 & 0.000 \\
Tile & 0.745 & 0.576 & & Pol. stone & 0.489 & 0.254 & & {\bf Soap} & 0.000 & 0.000 \\
Chalkboard & 0.744 & 0.593 & & Sand & 0.463 & 0.389 & & {\bf Styrofoam} & 0.000 & 0.000 \\
Paper & 0.718 & 0.509 & & Artwork & 0.445 & 0.294 \\
Glass & 0.696 & 0.502 & & Mirror & 0.434 & 0.308 \\
\bottomrule
\end{tabular}
\end{table}
\begin{table}[t]
\centering
\caption{{\bf DMS-Val results for DMS-52B.} Results are sorted by accuracy.}
\label{tab:third}
\begin{tabular}{@{}lccp{3mm}lccp{3mm}lcc@{}}\toprule
& Acc & IoU & & & Acc & IoU & & & Acc & IoU\\\midrule
Sky & 0.943 & 0.865 & & Glass & 0.690 & 0.488 & & Tree wood & 0.352 & 0.257 \\
Foliage & 0.905 & 0.776 & & Nat. stone & 0.685 & 0.402 & & Rubber & 0.310 & 0.265 \\
Hair & 0.891 & 0.687 & & Wicker & 0.684 & 0.454 & & Animal skin & 0.301 & 0.254 \\
Water & 0.889 & 0.655 & & Paper & 0.681 & 0.510 & & Ice & 0.239 & 0.232 \\
Food & 0.862 & 0.687 & & Wallpaper & 0.651 & 0.384 & & Bone & 0.206 & 0.177 \\
Skin & 0.861 & 0.675 & & Leather & 0.603 & 0.431 & & Wax & 0.202 & 0.166 \\
Ceiling tile & 0.858 & 0.673 & & Snow & 0.593 & 0.507 & & Eng. stone & 0.198 & 0.106 \\
Carpet & 0.847 & 0.566 & & Concrete & 0.587 & 0.316 & & Cork & 0.192 & 0.134 \\
Fur & 0.829 & 0.720 & & Metal & 0.553 & 0.300 & & Clutter & 0.131 & 0.113 \\
Wood & 0.820 & 0.642 & & Soil & 0.542 & 0.337 & & Gemstone & 0.095 & 0.082 \\
Fabric & 0.789 & 0.701 & & Non-clear plastic & 0.540 & 0.344 & & {\bf Liquid} & 0.029 & 0.022 \\
Whiteboard & 0.752 & 0.539 & & Asphalt & 0.536 & 0.369 & & {\bf Fiberglass} & 0.017 & 0.016 \\
Fire & 0.739 & 0.654 & & Cardboard & 0.529 & 0.367 & & {\bf Sponge} & 0.003 & 0.003 \\
Ceramic & 0.737 & 0.499 & & Sand & 0.498 & 0.407 & & {\bf Pearl} & 0.000 & 0.000 \\
Brickwork & 0.734 & 0.501 & & Pol. stone & 0.459 & 0.238 & & {\bf Soap} & 0.000 & 0.000 \\
Chalkboard & 0.733 & 0.634 & & Artwork & 0.438 & 0.276 & & {\bf Styrofoam} & 0.000 & 0.000 \\
Paint & 0.705 & 0.633 & & Clear plastic & 0.392 & 0.251 \\
Tile & 0.704 & 0.535 & & Mirror & 0.358 & 0.265 \\
\bottomrule
\end{tabular}
\end{table}
\subsection{More Real-World Examples}
We show more DMS-46 predictions on real world images in Figure~\ref{fig:examples2}.
\begin{figure}[t]
\includegraphics[height=14.7ex]{figures/predictions_om46/1.jpg}\hfill
\includegraphics[height=14.7ex]{figures/predictions_om46/14.jpg}\hfill
\includegraphics[height=14.7ex]{figures/predictions_om46/41.jpg}\hfill
\includegraphics[height=14.7ex]{figures/predictions_om46/10.jpg}
\includegraphics[height=13.0ex]{figures/predictions_om46/40.jpg}\hfill
\includegraphics[height=13.0ex]{figures/predictions_om46/2.jpg}\hfill
\includegraphics[height=13.0ex]{figures/predictions_om46/3.jpg}
\includegraphics[height=13.55ex]{figures/predictions_om46/24.jpg}\hfill
\includegraphics[height=13.55ex]{figures/predictions_om46/32.jpg}\hfill
\includegraphics[height=13.55ex]{figures/predictions_om46/15_masked.jpg}\hfill
\includegraphics[height=13.55ex]{figures/predictions_om46/51.jpg}
\caption{{\bf Real-world examples.} Our model, \text{DMS}\xspace-46, predicts 46 kinds of indoor and outdoor materials. See Table~\ref{tab:fusedcount} for color legend.}
\label{fig:examples2}
\end{figure}
\section{Image Credits}
Photos in the paper and supplemental are used with permission. We thank the following Flickr users for sharing their photos with a CC-BY-2.0\footnote{https://creativecommons.org/licenses/by/2.0/} license. Some photos in the main paper were changed to remove logos or faces, scale, mask, or crop.
Image credits:
Random Retail,
Ross Harmes,
Amazing Almonds,
Jonathan Hetzel,
Patrick Lentz,
Colleen Benelli,
Jannes Pockele,
FaceMePLS,
Michael Button,
samuelrodgers752,
Ron Cogswell,
David Costa,
Janet McKnight,
Jennifer,
Adam Bartlett,
www.toprq.com/iphone,
Seth Goodman,
Municipalidad Antofagasta,
Tom Hughes-Croucher,
Travis Grathwell,
Associated Fabrication,
Tjeerd Wiersma,
mike.benedetti,
Frédéric BISSON,
Wendy Cutler,
with wind,
Barry Badcock,
Joel Kramer,
Gwydion M. Williams,
Andreas Kontokanis,
Jim Winstead,
Mike Mozart,
Keith Cooper,
Kurman Communications, Inc.,
Paragon Apartments,
Pedro Ribeiro Simões,
jojo nicdao,
Gobierno Cholula,
David Becker,
Emmanuel DYAN,
Ewen Roberts,
Supermac1961,
fugzu,
Erik (HASH) Hersman,
Eugene Kim,
Bernt Rostad,
andrechinn,
Geología Valdivia,
peapod labs,
Alex Indigo,
Turol Jones, un artista de cojones,
Blake Patterson,
cavenderamy,
tapetenpics,
DLSimaging,
Andy / Andrew Fogg,
Scott,
Justin Ruckman,
espring4224,
objectivised,
Li-Ji,
Bruno Kussler Marques,
and BurnAway.
| -61,747.389836 |
[
-1.9345703125,
2.119140625
] | 28.237792 |
[
-3.056640625,
1.1015625,
-1.208984375,
-3.732421875,
-0.57568359375,
5.70703125
] |
[
1.7412109375,
3.98828125,
1.87890625,
5.73046875
] | 1,313 | 8,588 |
[
-1.5546875,
1.8271484375
] | 41.727619 |
[
-5.546875,
-1.6533203125,
-2.138671875,
-1.5205078125,
0.7197265625,
8.2890625
] | 0.248942 | 18.021773 | 35.265363 | 15.961842 |
[
1.834907054901123
] | -45,442.012553 | 6.54844 | -59,825.818972 | 0.825065 | 6.972845 |
[
-3.6328125,
-3.064453125,
-2.51171875,
-3.26953125,
2.681640625,
9.1171875
] |
[
-6.328125,
-2.583984375,
-2.853515625,
-2.029296875,
4.3203125,
6.65234375
] | |
BkiUd9Y4eILhP_ZFB6pa
|
\section{Introduction}
Atomic structures are sensitive to the presence of external fields. For instance, light shifts may significantly alter the frequencies obtained in optical atomic clock~\cite{katori2003ultrastable}. Spectral line widths are also sensitive to applied fields. Recently, it has been shown that the lifetime of hydrogen metastable state can be influenced not only by the field intensity, but also by the chosen geometry of the electromagnetic fields~\cite{trappe2016geometric}. This paper focuses on the use of external fields to filter specific atomic states.
The atomic polarizer, which filters specific atomic states, is an essential component of matter-wave interferometers based on a multiple-state atomic source~\cite{miniatura1992,Kouchi2019entangled}
such as hydrogen Stern-Gerlach interferometers~\cite{miniaturaJPhysII91,robertJPhysII92,Impens17}. In the present work, we are interested in exploring the filtering of two specific metastable hyperfine structure states of the hydrogen atom. An implementation of this particular atomic polarizer can be obtained with of a region of static and uniform magnetic field. The role of the polarizer's magnetic field is twofold: it is used to tune the energy levels of the metastable hydrogen and also to generate the commoving electric field responsible for the decay, as shown in the classical references~\cite{Lamb50,Lamb52a}. The couplings between $2S_{1/2}$ and $2P_{1/2}$ atomic states dressed by the magnetic field filter out two specific $2S_{1/2}$ hyperfine structure states. We analyse the Lyman-$\alpha$ radiation rate associated with these decays and show that a specific pattern is obtained. This pattern by itself can be seen as an indication of a successful polarization process. However, a more robust pattern can also be observed if, after the desired polarization, we add an external electric field and observe the resulting Lyman-$\alpha$ emission, this time coming from the states transmitted by the initial polarization stage.
The paper is organized as follows. Section~\ref{Td} reviews the influence of the magnetic field on the $2S_{1/2}$ and $2P_{1/2}$ hyperfine structure states of the hydrogen atom. We then derive the time-dependent probabilities to find the hydrogen atom in a specific $2S_{1/2}$ or $2P_{1/2}$ hyperfine structure state. These expressions are used in Section~\ref{sec:fast} to propose a polarizing device consisting in a region of uniform and constant magnetic field. Finally, we discuss a criterion of effectiveness for the polarizer based on the Lyman-$\alpha$ radiation patterns produced by the decay of the atomic states of interest.
\section{Theoretical description}
\label{Td}
\subsection{Zeeman effect in hyperfine $H$ states}
We review here the Zeeman effect in the hyperfine structure states $2S_{1/2}$ and $2P_{1/2}$ of the hydrogen atom. As usual, we use decomposition on a basis of orbital angular momentum, electron spin, and nucleus spin eigenstates (Appendix~\ref{apendA}) to analyse the Zeeman effect. For this purpose, we use a decomposition on the hyperfine eigenstates $\ket{2L_{J},F,M_F}$, where $\vec{F}=\vec{J}+\vec{I}=(\vec{L}+\vec{S})+\vec{I},$ with $\vec{L}$ standing for the orbital angular momentum, $\vec{S}$ the electron spin and $\vec{I}$ the spin of the nucleus. We consider the following hyperfine structure states:
\begin{eqnarray}
&\ket{2S_{\frac{1}{2}},0,0},\ket{2S_{\frac{1}{2}},1,-1},\ket{2S_{\frac{1}{2}},1,0},\ket{2S_{\frac{1}{2}},1,1}, \nonumber \\
&\ket{2P_{\frac{1}{2}},0,0},\ket{2P_{\frac{1}{2}},1,-1},\ket{2P_{\frac{1}{2}},1,0},\ket{2P_{\frac{1}{2}},1,1}. \nonumber
\end{eqnarray}
The $2P_{3/2}$ states were not considered in this study for being about ten times further from the $2S_{1/2}$ states, regarding energy intervals. For that reason, the influence of the $\ket{2P_{3/2},F,M_F}$ states in the $\ket{2S_{1/2},F,M_F}$ states lifetimes is much less significant than the influence of the $\ket{2P_{1/2},F,M_F}$ states.
We will consider the total Hamiltonian operator
$
\hat{H}=\hat{H}_0+\hat{H}_B.
$
$\hat{H}_0$ is the non-perturbed Hamiltonian expressed in terms of hyperfine structure states~\cite{Gasenzer12metastable}. $\hat{H}_B=(\hat{L}_z+g_e\hat{S}_{ez}-\tilde{g}_p\hat{S}_{pz})\mu_BB/\hbar$ describes the atom-magnetic field interaction, where $\mu_{B}$ is the Bohr magneton, $g_e$ is the gyromagnetic factor of the electron, $\tilde{g}_p= g_p \mu_N /\mu_B=g_p m_e / m_p$ is the gyromagnetic factor of the nucleus multiplied by the ratio between the electron mass and the proton mass.
The eigenvalues of the Hamiltonian $\hat{H}$ in the subspace spanned by the $\ket{2S_{1/2},F,M_F}$ states can be easily determined. The same can be done for the $\ket{2P_{1/2},F,M_F}$ states. The expressions for the mentioned Hamiltonian eigenvalues are displayed in Appendix~\ref{apendB} and Fig.~\ref{Fig1} shows them as a function of the magnetic field intensity. We take as reference the energy of the fine state $2P_{1/2}$ in the absence of magnetic field.
\begin{figure}[h]
\centering
\textit{}\includegraphics[scale=0.38]{Fig01.jpg}
\caption {Zeeman splitting of the $2S_{1/2}$ and $2P_{1/2}$ hyperfine structure states. The energy eigenvalues represented are $h_1$ to $h_8$ for increasing energy at low magnetic field intensities.}
\label{Fig1}
\end{figure}
As expected, the energy eigenvalues associated to the angular momentum projections $M_F= \pm 1$, Eqs. (\ref{h2},\ref{h4},\ref{h6},\ref{h8}), behave linearly with the magnetic field intensity, while the energy eigenvalues associated to the angular momentum projections $M_F=0$, Eqs. (\ref{h1},\ref{h3},\ref{h5},\ref{h7}), behave quadratically with the magnetic field intensity when $(g_e+g_p)^{2} \mu_B ^2 B^2 \ll A_1 ^2$ and $(4-g_e+3g_p)^2\mu_B^2B^2 \ll 9A_2^2$, respectively, or $B \ll 63 $G. This behaviour is related to the mixture between the states $\ket{2S_{1/2},0,0}$ and $\ket{2S_{1/2},1,0}$ and between the states $\ket{2P_{1/2},0,0}$ and $\ket{2P_{1/2},1,0}$ caused by the external magnetic field. The new states can be written with respect to the $\ket{2S_{1/2},F,M_F}$ and $\ket{2P_{1/2},F,M_F}$ states as follows:
\begin{equation}\label{1}
\ket{1}=m_1(B) \ket{2P_{\frac{1}{2}},0,0} +n_1(B) \ket{2P_{\frac{1}{2}},1,0}
\end{equation}
\begin{equation}\label{2}
\ket{2}=\ket{2P_{\frac{1}{2}},1,-1}
\end{equation}
\begin{equation}\label{3}
\ket{3}=m_3(B) \ket{2P_{\frac{1}{2}},0,0} +n_3(B) \ket{2P_{\frac{1}{2}},1,0}
\end{equation}
\begin{equation}\label{4}
\ket{4}=\ket{2P_{\frac{1}{2}},1,1}
\end{equation}
\begin{equation}\label{5}
\ket{5}=m_5(B) \ket{2S_{\frac{1}{2}},0,0} +n_5(B) \ket{2S_{\frac{1}{2}},1,0}
\end{equation}
\begin{equation}\label{6}
\ket{6}=\ket{2S_{\frac{1}{2}},1,-1}
\end{equation}
\begin{equation}\label{7}
\ket{7}=m_7(B) \ket{2S_{\frac{1}{2}},0,0} +n_7(B) \ket{2S_{\frac{1}{2}},1,0}
\end{equation}
\begin{equation}\label{8}
\ket{8}=\ket{2S_{\frac{1}{2}},1,1}
\end{equation}
In the absence of an external magnetic field, we have $n_1(0)=m_3(0)=n_5(0)=m_7(0)=0$ while $m_1(0)=n_3(0)=m_5(0)=n_7(0)=1$. That means that all states are given in terms of only one state $\ket{2S_{1/2},F,M_F}$ or $\ket{2P_{1/2},F,M_F}$. For strong magnetic fields, on the other hand, we have $m_1(B)=n_1(B)=n_3(B)=m_5(B)=n_5(B)=n_7(B)\rightarrow 1/\sqrt{2}$, while $m_3(B)=m_7(B)\rightarrow -1/\sqrt{2}$. This means that, for high values of the magnetic field intensity, the states $\ket{5}$ and $\ket{7}$ are perfect mixtures of the states $\ket{2S_{1/2},0,0}$ and $\ket{2S_{1/2},1,0}$ and the states $\ket{1}$ and $\ket{3}$ are perfect mixtures of the states $\ket{2P_{1/2},0,0}$ and $\ket{2P_{1/2},1,0}$.
Figure~\ref{Fig1} reveals crossings between some of the $2S_{1/2}$ and $2P_{1/2}$ levels for well-defined values of the magnetic field. A strong coupling between certain states can be achieved by the motional electric field in the vicinity of the crossing regions. Selection rules impose $\Delta m_l=\pm 1$ with the motional electric field perpendicular to the quantization direction (See Appendix~\ref{apendC}).
Since the $2P_{1/2}$ states have short lifetime, of the order of $10^{-9}$s, it is possible to induce the decay of the $2S_{1/2}$ metastable states by creating a channel to the ground state through the unstable $2P_{1/2}$ states. Even a small external magnetic field may significantly enhance the decay probability of a moving atom in a $2S_{1/2}$ state. We study below how external fields affect the lifetimes of hyperfine states.
\subsection{Time-dependent evolution of the atomic states}
The quenching process will be described using time dependent perturbation theory taking $\gamma_{2s}\approx 7 s^{-1}$ and $\gamma_{2p}\approx 6\times 10^{8} s^{-1}$ as the $2S_{1/2}$ and $2P_{1/2}$ decay constants, noting that $\gamma_{2s} \ll \gamma_{2p}$. The expressions below give the temporal variation of the probability amplitude of the atom to be observed in a certain state considering external perturbation and spontaneous decay
\begin{equation}\label{ddtcsdet}
\frac{d}{dt}c_j(t)=-\frac{1}{2}\gamma_{2s}c_j(t) + \sum_{k \in P} \frac{V_{jk}}{i\hbar}e^{i\omega_{jk} t}c_k(t)
\end{equation}
\begin{equation}\label{ddtcpdet}
\frac{d}{dt}c_k(t)=-\frac{1}{2}\gamma_{2p}c_k(t) + \sum_{j \in S} \frac{V_{kj}}{i\hbar}e^{i\omega_{kj} t}c_j(t)
\end{equation}
where the sets $P = \{1,2,3,4 \} $ and $S = \{ 5,6,7,8 \} $ corresponds to the $2P_{1/2}$ and $2S_{1/2}$ states, respectively. The indices $j$ and $k$ denote $2S_{1/2}$ and $2P_{1/2}$ states, respectively. $c_j(t)$ and $c_k(t)$ are the corresponding time-dependent probability amplitudes. As discussed later, each state of the set $S$($P$) is coupled to two states of the set $P$($S$). However, as discussed in Appendix~\ref{apendC}, only one of these two couplings will be relevant for the quantum evolution. The relevant coupling depends on the electric field induced in the atomic frame as well as on the magnetic field.
Since these states are separated by an energy $\Delta \varepsilon_{jk} = h_j-h_k = \hbar \omega_{jk} $, being those states coupled by an electric field $\vec{E}$, the matrix elements $V_{kj}$ are:
\begin{equation}
V_{kj}=\bra{k}e\vec{E} \cdot \vec{r} \ket{j}=V_{jk}^*.
\end{equation}
See Appendix~\ref{apendC}.
The solutions $c_j(t)$ and $c_k(t)$ can be found by decoupling (\ref{ddtcsdet}) and (\ref{ddtcpdet}) through differentiation.
We take the atom initially in the $2S_{1/2}$ level so that $ c_j(0)=1 $ and $ c_k(0)=0 $.
The solutions can then be written as:
{\fontsize{9.5}{13}\selectfont
\begin{equation}\label{csdet}
c_j(t)=\Big( \frac{\lambda_{2}^{+}+\frac{1}{2}\gamma_{2s}}{\lambda_{2}^{+}-\lambda_{1}^{+}} \Big) e^{\lambda_{1}^{+} t}-\Big( \frac{\lambda_{1}^{+}+\frac{1}{2}\gamma_{2s}}{\lambda_{2}^{+}-\lambda_{1}^{+}} \Big)e^{\lambda_{2}^{+} t}
\end{equation}
\begin{equation}\label{cpdet}
c_k(t)=-\Big( \frac{(i\hbar)^{-1} V_{kj}}{\lambda_{2}^{-}-\lambda_{1}^{-}} \Big)e^{\lambda_{1}^{-} t}+\Big( \frac{(i\hbar)^{-1} V_{kj}}{\lambda_{2}^{-}-\lambda_{1}^{-}} \Big)e^{\lambda_{2}^{-} t}
\end{equation}
}
with $\lambda_{1,2}^{+}= -\gamma_{+}/2 \pm \sqrt{\gamma_{+}^2/4-\omega_{0_+}^2}$ and $\lambda_{1,2}^{-}= -\gamma_{-}/2 \pm \sqrt{\gamma_{-}^2/4
-\omega_{0_-}^2}$
and where $\gamma_{\pm} = (\gamma_{2s}+\gamma_{2p})/2 \pm i\omega_{kj}$, $\omega_{0_+}^2= |V_{kj}|^2/\hbar^2+\gamma_{2s}\gamma_{2p}/4 + i\gamma_{2s}\omega_{kj}/2$ and $\omega_{0_-}^2= |V_{kj}|^2/\hbar^2+\gamma_{2s}\gamma_{2p}/4 - i\gamma_{2p}\omega_{kj}/2$.
$\gamma_{\pm}$ and $\omega_{0_\pm}$ are dependent of the energy difference $\Delta \varepsilon_{jk}$, but that dependence is not explicitly expressed for the sake of simplicity.
The considered probabilities are then:
\begin{equation}\label{probtrabalhada}
\begin{aligned}
&|c_j(t)|^2 = \hspace{0.13cm} \frac{e^{-\Re(\gamma_+)t}}{\abs{2\beta_+}^2} \hspace{0.13cm} \Bigg\{ {} \bigg \{ \frac{\gamma_{2s}^2}{4}-\gamma_{2s} \Re\Big( \frac{\gamma_{+}}{2}+ \beta_+ \Big) \\
&+ \abs{ \frac{\gamma_{+}}{2}+ \beta_+ } ^2 \bigg \}\hspace{-0.03cm} e^{2\Re ( \beta_+ ) t} \hspace{-0.07cm} + \hspace{-0.08cm} \bigg \{\hspace{-0.07cm} \frac{\gamma_{2s}^2}{4}\hspace{-0.07cm} -\hspace{-0.05cm} \gamma_{2s} \hspace{-0.04cm} \Re\hspace{-0.03cm}\Big(\hspace{-0.03cm} \frac{\gamma_{+}}{2}\hspace{-0.07cm}-\hspace{-0.07cm} \beta_+ \hspace{-0.05cm} \Big) \\
& +\abs{ \frac{\gamma_{+}}{2}- \beta_+ } ^2 \bigg \} e^{-2\Re ( \beta_+ ) t} - \hspace{-0.02cm} \bigg \{ \frac{\gamma_{2s}^2}{4}-\gamma_{2s}\Re\Big( \frac{\gamma_{+}}{2} \Big) \\
& + \abs{\frac{\gamma_{+}}{2}}^2 -\abs{\beta_+}^2 \bigg \} \Big( e^{2i\Im ( \beta_+ ) t}+e^{-2i\Im ( \beta_+ ) t} \Big)\\
& - 2i {} \bigg \{ \Re\Big( \frac{\gamma_{+}}{2} \Big)\Im ( \beta_+ ) -\Im\Big( \frac{\gamma_{+}}{2} \Big)\Re\big( \beta_+ \big) \\
& -\frac{1}{2}\gamma_{2s}\Im\big( \beta_+ \big)\bigg \} \Big( e^{2i\Im ( \beta_+ ) t} - e^{-2i\Im ( \beta_+ ) t} \Big) \Bigg \},
\end{aligned}
\end{equation}
and
\begin{equation}\label{modck2}
\begin{aligned}
|c_k(t)|^2 & = \frac{|V_{kj}|^2 e^{-\Re(\gamma_-)t}}{\hbar^2 \abs{2\beta_-}^2} \hspace{0.13cm} \Bigg\{ {} e^{2\Re ( \beta_- ) t} + \\
& + e^{-2\Re ( \beta_- ) t} -2 \cos \big( 2 \Im ( \beta_- ) t \big) \Bigg \}
\end{aligned}
\end{equation}
where $\beta_{\pm}=\sqrt{\gamma^2_{\pm}/4-\omega_{0\pm
}^2}$.
Eq.~(\ref{probtrabalhada}), although seemingly complicated, falls for small values of the electric field, $|V_{kj}|^2 \ll \hbar^2 (\gamma_{2p}^2+4 \omega_{kj}^2)$, into the well-known expression \cite{Lamb50}
\begin{equation}\label{probapp1}
|c_j(t)|^2 \approx e^{-\gamma_{E}t},
\end{equation}
where $\gamma_{E}=\gamma_{2p}|V_{kj}|^2/\big(\hbar^2 (\gamma_{2p}^2/4 + \omega_{kj}^2 )\big)$ is the resulting decay rate of the state $2S_{1/2}$ due to the action of an external electric field.
\subsection{Decay for a resonant coupling}
When the energy difference between the two coupled states cancels, for a specific magnetic field intensity, a simpler interpretation of Eq.~(\ref{probtrabalhada}) is possible. With this condition, the parameters $\omega_{0+}$ and $\gamma_+$ turn real and can be interpreted as a frequency of oscillation and the damping constant, respectively, of a damped harmonic oscillator. Thus,
$\gamma_{\pm}=\gamma=(\gamma_{2p}+\gamma_{2s})/2$
and
$\omega_{0\pm}^2=\omega_0^2=|V_{kj}|^2/\hbar^2+(\gamma_{2s}\gamma_{2p})/4.$
Besides, Eq.~(\ref{probtrabalhada}) presents three different regimes depending on whether $\gamma ^2/4<\omega_0^2$, $\gamma ^2/4>\omega_0^2$ or $\gamma ^2/4=\omega_0^2$, corresponding to the under-damped, over-damped and critically damped regimes, respectively.
In the under-damped regime, we can write Eq.~(\ref{probtrabalhada}) as:
\begin{equation}\label{sub}
\begin{aligned}
|c_j(t)|^2= \bigg \{ & \frac{\bar{\gamma}^2}{4} + \omega^2 - \bigg ( \frac{\bar{\gamma}^2}{4} - \omega^2 \bigg )\cos(2\omega t)\\
& +\bar{\gamma} \omega \sin(2\omega t) \bigg \} \frac{e^{-\gamma t}}{2\omega^2},
\end{aligned}
\end{equation}
where $\bar{\gamma}=(\gamma_{2p}-\gamma_{2s})/2$ and $\omega=\sqrt{\omega_{0
}^2-\gamma^2/4}$.
For a sufficiently strong damping, meeting the condition $\gamma ^2/4>\omega_0^2$, condition in which the decay happens without oscillations, we have that:
\begin{equation}\label{super}
\begin{aligned}
|c_j(t)|^2= \bigg \{ \Big( & \beta + \frac{\bar{\gamma}}{2}\Big )^2 e^{2\beta t} + \Big(\beta - \frac{\bar{\gamma}}{2}\Big )^2 e^{-2\beta t} \\
& + 2\beta^2 -\frac{1}{2} \bar{\gamma}^2 \bigg \} \frac{e^{-\gamma t}}{4\beta^2},
\end{aligned}
\end{equation}
where $\beta=\sqrt{\gamma^2/4-\omega_{0
}^2}$.
In the critically damped regime, we can write:
\begin{equation}\label{critico}
|c_j(t)|^2= \bigg \{ 1 + \bar{\gamma} + \frac{1}{4}\bar{\gamma}^2 t^2 \bigg \} e^{-\gamma t}.
\end{equation}
The three different regimes of decay can be achieved by varying the atom's speed or with the addition of an external electric field.
\section{Results and Discussion}
\label{sec:fast}
In this Section, we apply the previous analysis of the Zeeman energy levels to discuss the working principle of an atomic polarizer. We also analyse the Lyman radiation rate emitted by the selected atomic fragments. This rate exhibits a characteristic pattern.
\subsection{The atomic polarizer}
We consider the decay rate of $2 S_{1/2}$ atoms (state $|6 \rangle =\ket{2S_{1/2},1,-1}$) propagating in a region of constant magnetic field with a velocity orthogonal to the field. In particular, we investigate the behaviour of this decay in the vicinity of the magnetic field $B_{3,6}=597 \: {\rm G}$ corresponding to the crossing between states $|3\rangle$ and $|6\rangle$. The probability of the atom being in the corresponding $2 S_{1/2}$ state as a function of time (given by Eq.~(\ref{probtrabalhada})) is presented on Figure~\ref{Fig2}. The relevant coupling in this region occurs between the states $|3\rangle$ and $|6\rangle$. Indeed, the couplings between $|6\rangle$ and $|4\rangle$ as well as $|6\rangle$ and $|2\rangle$ are forbidden by the selection rules. On the other hand, the coupling between $|6\rangle$ and $|1\rangle$ is strongly non-resonant and also suppressed by the asymptotic form of the eigenstate $|1 \rangle$ for strong magnetic field values (as detailed in Appendix~\ref{apendC}).
\begin{figure}[h]
\includegraphics[scale=.34]{Fig02.jpg}
\caption{a) Probability, $|c_{6}(t)|^2$, for a magnetic field intensity of $300 \: {\rm G}$ (dotted line), $400 \: {\rm G}$ (dashed line), $500 \: {\rm G}$ (dot-dashed line) and $B_{3,6}= 597 \: {\rm G}$ (solid line) for an atomic speed $v \: = \: 100 \: {\rm km/s}$.
b) Probability $|c_{6}(t)|^2$ for the magnetic field $B_{3,6}$ and with the atomic velocities: 100 km/s (solid line) and 200 km/s (dashed line), corresponding to the underdamped regime; 17.7 km/s (dot-dashed), corresponding to the critically damped regime; and 8 km/s (dot-dot-dashed), corresponding to the overdamped regime.
The dotted line shows the envelope $e^{-\frac{1}{2}(\gamma_{2s}+\gamma_{2p})t}$.}
\label{Fig2}
\end{figure}
Fig.~\ref{Fig2}a) shows that the decay depends strongly on the magnetic field intensity and is enhanced in the vicinity of the crossing value $B_{3,6}$. Fig.~\ref{Fig2}b) displays the time-dependent probability in the different regimes, Eqs.(~\ref{sub}-\ref{critico}), at the crossing magnetic field $B_{3,6}$ for four atomic velocities, with $\vec{v} \perp \vec{B}$. In the underdamped regime, the probability of decay has an exponential envelope $e^{-\frac{1}{2}(\gamma_{2s}+\gamma_{2p})t}$. This can be interpreted as follows: the atom undergoes Rabi oscillations between states $\ket{6}$ and $\ket{3}$ and spends on average an equal time in both states. Since one of these states ($2S_{1/2}$) has a much smaller decay rate, the lifetime of the moving atom is approximately twice that of the $2P_{1/2}$ state. The maximum decay is obtained at the crossing magnetic field value $B_{3,6}$.
The discussion above can be extended to the state $\ket{5}$ (coupled to state $\ket{4}$). The remaining $2 S_{1/2}$ states $(\ket{7},\ket{8})$ do not exhibit energy crossings and are thus not resonantly coupled. Indeed, their lifetimes (less affected by the presence of the magnetic field) are orders of magnitude larger than that of the states $\ket{5}$ and $\ket{6}$. This fact enables one to build a device which quenches two of the four $2S_{1/2}$ Zeeman states, i.e., which filters out the states $\ket{5}$ and $\ket{6}$ and preserves the states $\ket{7}$ and $\ket{8}$, consisting of a region of uniform and constant magnetic field with appropriate length through which the atomic beam must travel, as further discussed in Appendix~\ref{apendD}.
\subsection{Detection signal with Lyman-$\alpha$ radiation and Criterion for testing a polarizing device}
Detection can be done by sensing the Lyman-$\alpha$ radiation emitted by the propagating fragments. Since the probability of emission of Lyman-$\alpha$ radiation is $10^8$ times higher when the atom is in a $2P_{1/2}$ state when compared to a $2S_{1/2}$ state, the radiation rate may be taken as proportional to the probability of the atom being in one of the $2P_{1/2}$ states:
\begin{equation}\label{Plyman2P}
R_{Lyman-\alpha} \propto \sum_{k=1}^{4} |c_k(t)|^2,
\end{equation}
where the probabilities $|c_k(t)|^2$ are given by Eq.(\ref{modck2}).
Let us consider a non-polarized H(2S) beam (with four equally populated $2S_{1/2}$ Zeeman states), propagating in the x-direction through a region with an orthogonal magnetic field yielding near-resonant energy levels $\ket{3},\ket{4},\ket{5},\ket{6}$.
\begin{figure}[h!]
\includegraphics[scale=.49]{Fig03.jpg}
\caption{a) Joint Lyman-$\alpha$ emission originated by the quenching of the four $2S_{1/2}$ states when a H(2S) beam of velocity $\vec{v} = v \hat{x}$ with
$v \: = \: 100 \: {\rm km/s}$ travels through a region with a magnetic field of $\vec{B}= B \hat{z}$ ($B= 565 G$). b) Joint Lyman-$\alpha$ emission originated by the quenching of states $\ket{7}$ and $\ket{8}$ in presence of an identical magnetic field and an additional external electric field $\vec{E}_{ext}=E_{ext} \hat{y}$ with $E_{ext}=600 \: \rm{V/cm}$.}
\label{Fig3}
\end{figure}
Figure~\ref{Fig3}a) shows the profile of a joint Lyman-$\alpha$ emission rate produced by the quenching of the four $2S_{1/2}$ Zeeman states.
The dominant contribution to the Lyman-$\alpha$ radiation pattern comes from the coupling between the states $\ket{5}$ and $\ket{4}$, and $\ket{6}$ and $\ket{3}$, respectively, while the contribution from states $\ket{7}$ and $\ket{8}$ is negligible. For a magnetic field of 565 G we have an optimal value for a clear pattern, since the oscillations in the Lyman-$\alpha$ emission rate for the states that contribute the most get in phase, but significant variations in the magnetic field intensity still result in a similar pattern.
After roughly $t \simeq 20 \: {\rm ns}$ (corresponding to a path of a few mm at the considered velocity $v \: = \: 100 \: {\rm km/s}$), most atoms that were initially in the states $\ket{5}$ and $\ket{6}$ are in the ground state, and the resultant beam is almost completely composed by atoms in states $\ket{7}$ and $\ket{8}$, since the latter present a very low Lyman-$\alpha$ emission rate when compared to the former.
Although states $\ket{7}$ and $\ket{8}$ have a similar behaviour in the presence of external fields, regarding Lyman-$\alpha$ radiation emission, their simultaneous presence in the resulting beam drastically changes the radiation pattern in presence of an additional external electric field (fundamental to increase their decay rate). This can be seen in Figure~\ref{Fig3}b), which shows the Lyman radiation in presence of an additional external electric field for an atomic beam containing initially only equally populated
$\ket{7},\ket{8}$ states. One notes an oscillatory pattern with a beat frequency $\Delta \omega =\big( \sqrt{\Delta \varepsilon_{81}^2+4|V_{81}|^2}-\sqrt{\Delta \varepsilon_{72}^2+4|V_{72}|^2} \big) / \hbar$, that can be calculated from $\vec{v}$, $\vec{B}$ and the external electric field $\vec{E}_{ext}$.
\begin{figure}[h!]
\hspace{-0.06cm}\includegraphics[scale=.335]{Fig04.jpg}
\caption{a) Proposed setup for testing polarization with a fully working polarizer. A non-polarized H(2S) beam enters a region of constant magnetic field. In this regime, states $\ket{5}$ and $\ket{6}$ suffer quenching and the resulting Lyman-$\alpha$ radiation is detected by detector A. Upon leaving this first region, the beam is composed exclusively of states $\ket{7}$ and $\ket{8}$. It then travels to a second region with a constant magnetic field and an additional external electric field that quenches the remaining states. The Lyman-$\alpha$ radiation is then sensed by detector B. b) Testing setup with no polarizer. A non-polarized H(2S) beam travels through the apparatus until it gets to a region of constant electric and magnetic fields. Those fields cause intense quenching of the four hyperfine structure states and the resulting Lyman-$\alpha$ radiation is detected by detector B.
The magnetic field is only present in two specific regions for illustrative purposes, it may remain constant throughout the setup.}
\label{Fig4}
\end{figure}
In theory, a polarizer consisting of a region of approximately constant magnetic field with adequate intensity and extension should be able to filter two of the four states of hyperfine structure, as discussed. However, inherent limitations in the apparatus construction, such as resulting inhomogeneities of the external magnetic field, can compromise the efficiency of the device. The study presented here leads to a simple and robust criterion for testing a polarizer after its construction and verify if the desired polarization has been achieved. This technique could be used prior to inserting the polarizing device into a more complex apparatus.
Figure~\ref{Fig4} shows a sketch of a setup for that verification. Fig.~\ref{Fig4}a) shows an initially non-polarized beam containing equal parts of the four hyperfine structure states. The beam then enters a region of constant magnetic field that causes quenching of two of the four states preferentially. The two surviving states then enter a region with an additional external electric field strong enough to cause a decay within a few millimetres of propagation, corresponding to a few nanoseconds. Detectors A and B should capture spectrums like the ones indicated in Fig.~\ref{Fig4}a). In the absence of a polarizer, the detection patterns would be as shown in Fig.~\ref{Fig4}b).
The pattern obtained by detector A in Fig.~\ref{Fig4}a) can be seen as an indication of a successful polarization process. However, the pattern from detector B is much more robust, as it is little affected by the atomic speed. See Appendix~\ref{apendD}. Indeed, the quenching of the two remaining states relies mostly on the external electric field, and not on the motional electric field as previously. Nevertheless, the speed still influences the detection time of the radiation, but this effect can be minimized by shortening the path and by using an atomic beam of narrow velocity distribution. The beating pattern does not depend on the specific polarizing process, as long as it delivers a beam with similar proportions of states $\ket{7}$ and $\ket{8}$.
\section{Conclusion}
\label{Conc}
In this work, we have discussed how a tuning of the metastable hydrogen energy levels by a magnetic field can enable the realization of a two-state polarizer. This is apparent from the expressions of the time-dependent probability amplitudes for the $2S_{1/2}$ and $2P_{1/2}$ states. We have particularly investigated the effects of near-resonant coupling of the metastable states dressed by the magnetic field. We have also analysed the Lyman-$\alpha$ radiation rate, used in most detection schemes, for typical quenching conditions. The simultaneous presence of two $2S_{1/2}$ states after the polarization leads to a radiation pattern with beat notes that can be seen as an indication of successful polarization. We have established a simple and robust criterion for checking the effectiveness of a magnetic polarizer in atom-interferometers setup. Perspectives for this work include the design of specific electric and magnetic field geometries enabling the hydrogen polarization along a single atomic state instead of a two-state multiplicity.
\section{acknowledgments}
This work was partially supported by the Brazilian agencies CNPq, CAPES, and FAPERJ. It is part of the INCT-IQ from CNPq.
| -28,075.899528 |
[
-2.640625,
2.513671875
] | 41.975309 |
[
-3.669921875,
-0.023162841796875,
-2.32421875,
-6.171875,
-1.0068359375,
9.203125
] |
[
4.7890625,
8.1484375,
4.09765625,
7.51171875
] | 184 | 3,377 |
[
-2.767578125,
3.001953125
] | 32.114277 |
[
-6.3671875,
-4.26171875,
-4.078125,
-2.390625,
2.171875,
11.8359375
] | 1.502802 | 24.450289 | 28.901392 | 6.747828 |
[
2.0083279609680176
] | -18,425.59245 | 5.812852 | -27,331.905952 | 1.222618 | 5.709479 |
[
-2.73828125,
-3.884765625,
-4.19921875,
-5.33984375,
2.48046875,
12.8359375
] |
[
-5.3515625,
-2.056640625,
-2.45703125,
-1.7275390625,
3.52734375,
4.875
] | |
BkiUeTzxK6mkyCfODSv-
|
\section{Introduction}
The task of autonomous driving has received much deserved attention in recent years.
Current progress in computer vision helps in providing reliable scene understanding from visual and geometrical data.
In particular, autonomous agents must detect vehicles, pedestrians, cyclists, and other objects on the road to ensure a safe navigation.
Pipelines for autonomous driving leverage 3D computer vision for tasks such as
object detection~\cite{chen2017multi,ku2018joint},
tracking~\cite{Giancola_2019_CVPR,Hu_2019_ICCV} and
trajectory prediction~\cite{Chandra_2019_CVPR,Chang_2019_CVPR}.
Current state-of-the-art methods in 3D vehicle detection prefer LiDAR point clouds over images.
Performances of monocular~\cite{chen2016monocular,mousavian20173d,wang2019pseudo} and stereo-based~\cite{chen20173d,xu2018multi,Li_2019_CVPR} 3D object detection techniques are not yet comparable with LiDAR-based techniques~\cite{Yang2019std,liang2019multi}.
Recent image-based methods for 3D vehicle detection even relies on LiDAR supervision~\cite{wang2019pseudo} to generate surrogate point clouds from images.
Still, defining which point cloud representation fits the most with deep learning is not straightforward and remains an open problem~\cite{qi2017pointnet,riegler2017octnet,li2019deepgcns_journal}.
Recent advances in graph convolution networks~\cite{li2019deepgcns_journal} suggest that graph representations could provide better features for point cloud processing.
Such a representation already outperforms the state-of-the-art in many other computer vision tasks~\cite{Wang_2019_CVPR,Nguyen_2019_ICCV,Shu_2019_ICCV,wang2019deep}.
Thus, we investigate the use of a graph representation for LiDAR point cloud in the task of 3D vehicle detection.
\begin{figure}[t]
\begin{subfigure}{\linewidth}
\centering
\begin{overpic}[height=3.6cm,cfbox=Green 3pt 0pt,trim={8cm 4cm 8cm 3cm},clip]{imgs/RGCN_ReceptiveField.png}
\put (5,78)
{\Large\color{Green}\textbf{R-GCN}}
\end{overpic}
\begin{overpic}[height=3.6cm,cfbox=RoyalBlue 3pt 0pt,trim={9cm 6cm 11cm 4cm},clip]{imgs/CGCN_ReceptiveField.png}
\put (5,84)
{\Large\color{RoyalBlue}\textbf{C-GCN}}
\end{overpic}
\end{subfigure}%
\caption{
\textbf{Proposed GCN modules.}
Insights on the \textbf{\color{Orange}Receptive field} for both our proposed modules.
The \emph{per-proposal} \textbf{\color{Green}R-GCN} gathers meaningful features information between points on each proposal.
The \emph{per-frame} \textbf{\color{RoyalBlue}C-GCN} gathers contextual information between proposals on each frame.
Note that this non-contractual representation does not consider the dynamic graph on features for neighbors.
}
\label{fig:teaser}
\end{figure}
In our work, we propose two modules exploiting convolution operations on point cloud graph representations.
We tackle vehicle detection in a common two-stage fashion by generating a set of high-recall proposals from a LIDAR scene point clouds which we successively fine-tune to obtain refined detections.
In particular, we introduce PointRGCN: a novel pipeline consisting of two refinement modules.
R-GCN (\Figure{teaser}~(top)) provides a \emph{per-proposal} feature aggregation based on a GCN that utilizes all points contained a proposal.
C-GCN (\Figure{teaser}~(bottom)) gathers \emph{per-frame} information from all proposals, looking for a contextual consistency in the road scene.
We will release our code after the review process.
\mysection{Contributions.} We summarize our contributions as follows.
\textbf{(i)}
We are the first, to the best of our knowledge, to leverage graph representation and GCNs for the task of 3D vehicle detection.
\textbf{(ii)}
We propose R-GCN and C-GCN, two GCN-based modules for \emph{per-proposal} and \emph{per-frame} feature aggregation, designed to gather all features within and between proposals.
\textbf{(iii)}
We show competitive performances for our novel pipeline PointRGCN in the challenging KITTI~\cite{Geiger2012KITTI}\xspace 3D vehicle detection task and exhibit a $+2\%$ AP$_{BEV}$ boost on the easy subset.
\section{Related Work}
Our work relates to methods delving with 3D point cloud representations, graph convolutional networks, and 3D object detection architectures.
We present previous works on these topics, and highlight the novelty of our pipeline with respect to each of these works.
\mysection{3D Point Cloud Representation.}
Finding an efficient representation for sparse LiDAR point clouds is paramount for 3D computer vision tasks.
Current works leverage projection, voxelization or per-point features.
Image projections allow for efficient convolution operations on the image space, and are commonplace for 3D object classification and retrieval tasks~\cite{su2015multi,qi2016volumetric,tatarchenko2018tangent}.
Projecting provides coarse but structured inputs for further deep convolution network.
Voxelization is a volumetric representation that discretizes 3D space into a structured grid.
Voxelizing 3D space allows for 3D CNN processing~\cite{wu20153d,maturana2015voxnet,riegler2017octnet}, but are not memory efficient due to the sparsity of point clouds.
PointNet~\cite{qi2017pointnet} and PointNet++~\cite{qi2017pointnet++} solve the sparsity problem by learning a deep feature representation for each point, with all points contributing to a more complete representation.
However, the context shared between points is limited to point relations in 3D space~\cite{qi2017pointnet++}, and shapes still lack structure.
In order to overcome these issues, we use GCNs as the backbone for both refinement modules in our network.
\mysection{Graph Convolutional Network.}
To cope with current limitations of point cloud representation methods, Wang~\etal~\cite{wang2019dynamic} propose a graph representation for point clouds and show improved performances in classification and segmentation tasks.
Li~\etal~\cite{li2019deepgcns_journal} show that GCNs can run as deep as 112 layers by introducing residual skip connections, similar to ResNet.
GCNs have an increased receptive field with respect to PointNet++\cite{qi2017pointnet++} since the edge connections are dynamically assigned for each intermediate layer.
With such features, GCNs have shown success in tasks such as
segmentation~\cite{Wang_2019_CVPR},
point cloud deformation~\cite{Nguyen_2019_ICCV},
adversarial generation~\cite{Shu_2019_ICCV} and
point cloud registration~\cite{wang2019deep}.
However, To the best of our knowledge, we introduce the first 3D vehicle detection module using a graph representation on LiDAR input.
\mysection{Single-Stage 3D Object Detection.}
Single-stage detectors train a single network end-to-end to generate a small set of detection boxes with high precision.
VoxelNet~\cite{zhou2018voxelnet} directly generates detections from a Region Proposal Network using a voxelized point cloud.
SECOND~\cite{yan2018second} improves upon VoxelNet by leveraging sparse convolution operations on the voxelized input.
PointPillars~\cite{lang2019pointpillars} uses a pillars representation based on SECOND voxelisation with infinite height.
Voxel-FPN~\cite{wang2019voxel} introduces a multi-scale voxel feature pyramid network based on VoxelNet~\cite{zhou2018voxelnet}.
Qi~\etal~\cite{Qi_2019_ICCV} recently introduced a Deep Hough Voting scheme for indoor detection, adopting PointNet++\cite{qi2017pointnet++} for the feature backbone.
We preferred a two-stage detector since single-stage methods tend to attain lower performances, and GCNs are computationally expensive to compute on complete point clouds.
\mysection{Two-Stage 3D Object Detection.}
On the other hand, two-stage detection methods divide the detection pipeline into two sub-modules.
One module is trained to generate a large number of high-recall proposals, and a second one to refine those proposals.
Frustum PointNet~\cite{qi2018frustum} uses 2D proposals from RGB images and localizes 3D bounding boxes using a PointNet~\cite{qi2017pointnet} representation on the 2D proposal's frustum.
Liang \cite{liang2019multi} propose a multi sensor multi view aggregation taking into account both LiDAR and RGB camera information.
Alternatively, STD~\cite{Yang2019std} defines spherical anchors to obtain higher recall in the proposal stage.
Chen~\etal~\cite{Chen2019FastPointRCNN} use a voxelization for their Region Proposal Network (RPN), and use PointNet features for proposal refinement.
Also, the concurrent work of Lehner~\etal~\cite{lehner2019patch} proposes a coarse RPN on the top view projection which allows for a denser voxelization of proposals during refinement.
PointRCNN~\cite{shi2019pointrcnn} creates object proposals by first performing point segmentation using a PointNet++ backbone, regressing a proposal for each foreground point, and finally refining the proposal using PointNet++.
Part-$A^2$ Net~\cite{shi2019part} improves upon PointRCNN by adding a part aware loss for intra-proposal point locations, and performing proposal refinement by using a Voxel Feature Encoding.
In our work, we build upon PointRCNN~\cite{shi2019pointrcnn}\xspace by using its proposal scheme, but leverage GCNs to refine proposals.
Due to the computational and memory complexity of GCNs, we limit their usage to the refinement stage.
In particular, we combine point features for each proposal with R-GCN, and aggregate proposal context information for each frame with C-CGN.
\section{Methodology}
In this section, we present our 3D object detection pipeline, PointRGCN, which takes advantage of recent advances in GCNs.
Our model takes as an input a LIDAR scan of a scene and generates tight 3D bounding boxes for cars found in the LIDAR point cloud.
Similar to common practice, we split object detection into two stages: generating high recall proposals, and refining the set of proposals to get detections with high precision.
The main focus of our network is on the second stage of object detection: refining proposals.
Therefore, we take the region proposal network from an existing method, PointRCNN~\cite{shi2019pointrcnn}, to generate proposals.
During refinement, we take advantage of geometric information contained in point clouds by leveraging a graph representation for points within proposals.
Furthermore, we aggregate the contextual information across proposals in a second refinement step.
In the following section, we present an introduction to GCNs and how they are applied in the context of object detection.
We then present the details of our pipeline PointRGCN, including the two main components of our refinement network: \textbf{R-GCN}, and \textbf{C-GCN}.
\begin{figure*}[t]
\centering
\begin{overpic}[height=2.13cm,cfbox=Black 2pt 0pt,
trim={12cm 5cm 7cm 8cm},clip]{imgs/Pipeline_PC.png}
\put (4,5) {\large\color{Black}\textbf{Point Cloud}}
\end{overpic}
\begin{overpic}[height=2.13cm,cfbox=Orange 2pt 0pt,
trim={12cm 5cm 7cm 8cm},clip]
{imgs/Pipeline_Proposals.png}
\put (4,5) {\large\color{Orange}\textbf{Proposals}}
\end{overpic}
\begin{overpic}[height=2.13cm,cfbox=Green 2pt 0pt,
trim={8.5cm 3cm 8cm 4cm},clip]
{imgs/Pipeline_RGCN.png}
\put (5,6) {\large\color{Green}\textbf{R-GCN}}
\end{overpic}
\begin{overpic}[height=2.13cm,cfbox=RoyalBlue 2pt 0pt,
trim={12cm 5cm 7cm 8cm},clip]
{imgs/Pipeline_CGCN.png}
\put (4,5) {\large\color{RoyalBlue}\textbf{C-GCN}}
\end{overpic}
\begin{overpic}[height=2.13cm,cfbox=Black 2pt 0pt,
trim={12cm 5cm 7cm 8cm},clip]
{imgs/Pipeline_Final.png}
\put (4,5) {\large\color{Black}\textbf{Detection}}
\end{overpic}
\caption{
\textbf{PointRGCN:}
We propose a novel object detection pipeline that introduce Graph Convolution Networks (GCNs) in the refinement module.
{\color{orange}\textbf{Proposals}}: We generate proposals by regressing a bounding box per each foreground vehicle points, similar than PointRCNN~\cite{shi2019pointrcnn}.
{\color{ForestGreen}\textbf{R-GCN}}: We improve the per-proposal feature extraction used to classify and regress the 3D object proposals by introducing residual GCNs.
{\color{blue}\textbf{C-GCN}}: We share the features between proposals to embed a contexual consistency between the objects to detect.
}
\label{fig:pipeline}
\end{figure*}
\subsection{Graph Embedding and GCNs.}
\mysection{Definitions.}
A graph \ensuremath{\mathcal{G}}\xspace is defined as a tuple consisting of a set of nodes and a set of edges, $(\ensuremath{\mathcal{V}}\xspace, \ensuremath{\mathcal{E}}\xspace)$, in which an edge $e_{i,j}\in\ensuremath{\mathcal{E}}\xspace$, between nodes $v_i\in\ensuremath{\mathcal{V}}\xspace$ and $v_j\in\ensuremath{\mathcal{V}}\xspace$ takes a binary value signifying whether node $v_i$ is related to node $v_j$ or not.
Connections between nodes are directed, \ie an edge $e_{i,j}$ may differ from its reciprocal edge $e_{j,i}$.
Each node is represented by a $c$-dimensional feature vector, \ie $v_i \in \mathbb{R}^c$.
\mysection{Graph Convolutional Networks.}
GCNs are an extension of CNNs which operate on graph representations instead of regular image grids.
Given an input graph \ensuremath{\mathcal{G}}\xspace, a graph convolution operation \ensuremath{\mathcal{F}}\xspace aggregates features from $k$ nodes in a neighborhood $\ensuremath{\mathcal{N}}\xspace^{(k)}(v)$ of a given node $v$.
A convolution operation \ensuremath{\mathcal{F}}\xspace updates the value of the given node by aggregating features amongst its neighbor nodes, as presented in \Equation{GCN}.
This aggregates features from nearby nodes, mirroring the way convolutional filters aggregate nearby pixels.
Note that unlike CNNs, GCNs do not apply different weights to different neighbors.
\begin{equation}
\begin{aligned}
\ensuremath{\mathcal{F}}\xspace (\ensuremath{\mathcal{G}}\xspace_l,\ensuremath{\mathcal{W}}\xspace_l )& = \text{Update} (\text{Aggregate} ( \ensuremath{\mathcal{G}}\xspace_l , \ensuremath{\mathcal{W}}\xspace_l^{a} ) , \ensuremath{\mathcal{W}}\xspace_l^{u} ) \\
\ensuremath{\mathcal{G}}\xspace_{l+1} & = \ensuremath{\mathcal{F}}\xspace ( \ensuremath{\mathcal{G}}\xspace_l , \ensuremath{\mathcal{W}}\xspace_l ) + \ensuremath{\mathcal{G}}\xspace_l
\label{eq:GCN}
\end{aligned}
\end{equation}
\mysection{EdgeConv.}
Node features are aggregated around neighbors $\ensuremath{\mathcal{N}}\xspace^{(k)}(v_l)$ of nodes $v_l$ at layer $l$.
In particular, the feature difference $\mathbf{h}_{u_l}-\mathbf{h}_{v_l})$, for given neighbor features $u_l\in\ensuremath{\mathcal{N}}\xspace^{(k)}(v_l)$ is concatenated to the current node's features $\mathbf{h}_{v_l}$, and processed with a shared multi-layered perceptron (MLP).
This makes use of both a local geometry encoded by the feature difference as well as a global geometry obtained from the features of each node center $\mathbf{h}_{v_l}$.
Then, a max operation pools features between the neighbors and updates the current node's feature $\mathbf{h}_{v_{l+1}}$ for next layer $l+1$.
\Equation{EdgeConv} illustrates the aggregation and update steps introduced by EdgeConv~\cite{wang2019dynamic}\xspace.
\begin{equation}
\begin{aligned}
\mathbf{h}_{v_{l+1}} &
=\max(\{\mathbf{h}(u_l)|u_l\in\ensuremath{\mathcal{N}}\xspace^{(k)}(v_l)\})\\
\mathbf{h}(u_l) &
=\text{MLP}(\text{concat}(\mathbf{h}_{v_l},\mathbf{h}_{u_l}-\mathbf{h}_{v_l}))
\label{eq:EdgeConv}
\end{aligned}
\end{equation}
\mysection{MRGCN.}
Rather than performing the MLP operation on all neighbors before the maxpool operation in \Equation{EdgeConv}, MRGCN~\cite{li2019deepgcns_journal}\xspace performs the maxpool first, and applies the MLP only once.
In particular, the difference features $\mathbf{h}_{u_l}-\mathbf{h}_{v_l})$ for all neighbor features $u_l\in\ensuremath{\mathcal{N}}\xspace^{(k)}(v_l)$ are maxpooled into an intermediate representation $\mathbf{h}_{\ensuremath{\mathcal{N}}\xspace^{(k)}\left(v_l\right)}$.
Then, that representation is concatenated with the current node feature and processed once with an MLP.
This reduces the memory budget since a single intermediate feature vector is stored for each neighbor, rather than $k$ of them.
Also, the MLP is performed only once, allowing for faster inference.
\begin{equation}
\begin{aligned}
\mathbf{h}_{v_{l+1}}&
=\text{MLP}\left(\text{concat}\left(\mathbf{h}_{v_l},\mathbf{h}_{\ensuremath{\mathcal{N}}\xspace}\left(v_l\right)\right)\right)\\
\mathbf{h}_{\ensuremath{\mathcal{N}}\xspace^{(k)}}\left(v_l\right)&
=\max(\{\mathbf{h}_{u_l}-\mathbf{h}_{v_l}|u_l\in\ensuremath{\mathcal{N}}\xspace^{(k)}\left(v_l)\}\right)
\label{eq:MRGCN}
\end{aligned}
\end{equation}
\mysection{Dynamic Graphs.}
Dynamic GCNs recompute a new graph per layer using the feature space produced by each intermediate representation~\cite{wang2019deep}.
In particular, edges $e_{i,j}$ are recomputed in each layer by taking $k$ closest nodes using a Euclidean distance in the current layer's feature space $\mathbf{h}$.
By using dynamic graph updates, the receptive field becomes dynamic.
This allows relations to be drawn between far apart nodes, given they have similarities, and thus provides additional contextual information.
\mysection{Dilated GCNs.}
Convolutional kernel dilations showed improvements in CNNs~\cite{yu2015multi}, and have recently been extended into graph convolutions as well.
In the convolution domain, dilation increases a convolutional filter's receptive field without increasing the number of model parameters.
Dilation is achieved by using a larger convolution kernel, but fixing every other kernel coefficient to $0$.
Similarly, dilation on GCNs is performed by skipping neighbors to increase the receptive field~\cite{Li2019deepgcn}.
In this manner, further neighbors can be reached without increasing the total number of neighbors aggregated in each graph convolution.
A dilation of $d$ implies that every $d$-th neighbor is used for aggregation, which effectively increases the receptive field to $d \times k$ neighbors at a given layer.
We use a linearly increasing dilation with respect to the number of layers, \ie the $l$-th layer uses a dilation of $l$.
\mysection{Residual Skip Connections.}
In a similar manner, residual skip connections have been adopted from CNNs.
We apply residual skip connections by adding the previous graph nodes features to the next layer, providing a better gradient flow and faster convergence.
\subsection{PointRGCN Vehicle Detection Pipeline}
Our pipeline is composed of $3$ main modules: a region proposal network, a graph-based proposal feature extraction network, and a graph-based proposal context aggregation network.
The input to our network is a scene point cloud, and the output is a set of refined boxes \ensuremath{\mathcal{R}}\xspace (\Figure{pipeline}).
\mysection{RPN: Proposal Generation.}
This module takes as input the scene point cloud, and generates a set of proposal bounding boxes \ensuremath{\mathcal{R}}\xspace.
Proposal boxes $b^i$ are defined by a $7$-dimensional vectors consisting of a center, dimensions, and a rotation along the vertical axis: $b^i = (x, y, z, h, w, l, \theta)$.
The main requirement for this module is to have a high recall for the set of proposals \ensuremath{\mathcal{R}}\xspace in the input scene.
In our work, we leverage proposals from~\cite{shi2019pointrcnn}.
We do not alter on the proposal generation as the average recall of their method is relatively high already.
This proposal framework first learns to segment foreground and background points, and extrapolates a proposal bounding box $b^i$ for each foreground point.
Both proposals and segmentation features are used further down the pipeline during proposal refinement.
Current GCN-based feature extraction methods are memory intensive hence not suitable for feature extraction from large point clouds such as an outdoor scene captured with LIDAR.
As such, we rely on PointNet++~\cite{qi2017pointnet++} to extract meaningful per-point features for the segmentation performed in this stage.
\mysection{R-GCN: Proposal Feature Extraction.}
The aim of this network is to take a set of proposal boxes \ensuremath{\mathcal{R}}\xspace with a high recall, and extract a set of features from each proposal.
These features are later on used to regress better bounding boxes.
To achieve this, proposal boxes are first expanded to include extra context from objects surrounding the proposal.
LIDAR points lying within each of the expanded proposal boxes are cropped, and a total of $P$ points are kept to form the set of points $\ensuremath{\mathcal{P}}\xspace^i$ for any proposal $i$.
The isolated proposal point clouds are transformed to a canonical reference frame where the proposal's center lies at the origin, and where the proposal axes are aligned with the canonical frame's axes.
For each point, we project its canonical coordinates into a larger space to match the dimensionality of the point's RPN features.
We then concatenate the projected canonical coordinates with the corresponding RPN features, and reduce the concatenated vector's dimensionality in order to obtain a per-point feature vector.
Our R-GCN module processes these per-point feature vectors using $N_r$ layers of MRGCN~\cite{li2019deepgcns_journal}\xspace.
Each layer has a fixed number $F_r$ of filters, and residual skip connections are used between consecutive layers.
We then create a local multi-layer feature vector for each point by concatenating the output features from every graph convolution layer.
For each proposal, we learn a global point feature by projecting the local multi-layer feature into a $C_r$-dimensional space, and performing a max pool across all points.
The global point feature is then concatenated to every point in each proposal.
Finally, a max pool is performed across all points for each proposal box.
This generates an $(N_r \times F_r + C_r)$-dimensional output feature vector that captures both global and local information within proposals.
\mysection{C-GCN: Context Aggregation.}
We would like to exploit the fact that most proposals will be related by physical (\eg ground plane) and road (\eg lanes) constraints.
For this purpose, we leverage the information encapsulated in each proposal features to further refine all proposals.
We gather the set of $\vert \ensuremath{\mathcal{R}}\xspace \vert$ proposal feature outputs from our R-GCN module into a graph representation of the current frame.
In particular, each node on the graph represents a proposal with its R-GCN feature representation.
We then process the proposal graph with $N_c$ layers of EdgeConv~\cite{wang2019dynamic}\xspace, each with $F_c$ filters, and residual skip connections between consecutive layers.
We compute a global feature like for R-GCN and concatenate it to each proposal's local feature vector, producing an output of dimension $(N_c \times F_c + C_c)$ with aggregated proposal information.
\subsection{Detection Prediction}
In the last module, the feature vector generated for each proposal is used to regress a refined detection box along with a classification score for the given proposal.
Two sets of fully connected layers are used for this purpose: one for classification, and one for regression.
The purpose of classification is to predict whether a given proposal is good enough.
Regression is performed in two different manners specified below: binned regression, and residual regression as is done in~\cite{shi2019pointrcnn}.
Binned regression is performed for $f \in \{x,z,\theta\}$, while residual regression is performed for box features $f \in \{y,h,w,l\}$.
\mysection{Classification.}
Proposal classification is performed by predicting a confidence score of whether a proposal is good enough to predict a refined box or not.
It is assumed that a proposal which has a large IoU with a ground truth bounding box should lead to an even better refined detection.
Therefore, a proposal is considered positive if its IoU with the closest ground truth bounding box is larger than a set threshold, and negative if its IoU is lower than a different threshold.
Proposals in the gray area between the positive and negative thresholds are not used during training.
This classification score is used to determine which detection should be kept during inference.
\mysection{Binned Regression}
Binned regression is done by performing a classification among a set $V_f$ of different bins for each box feature $f$ using binned regression.
Alongside this bin classification, an intra-bin regression is also performed to allow for finer box prediction.
Bin centers $v^i_f$ for bin indices $i \in [0,\vert V_f \vert)$ are calculated as shown in \Equation{refine}, for a given feature $f \in \{x,z,\theta\}$.
Here, $S_f$ is the search space which is being discretized into bins, $\delta_f$ is the bin resolution, and $v_f$ is the proposal box feature.
The bin center with the highest classification score, $\hat{v}_f$, is used to generate the refined box.
Refined box features $\hat{r}_f$ are calculated as shown in \Equation{refine}, where $\widehat{breg}_f$ is the network regression output corresponding to the bin with the highest score.
The bin size $\delta_f$ is used to normalize regression outputs in order have a more stable training.
\begin{equation}
\begin{split}
v^i_f &= b_f + (i + 0.5) \delta_f - S_f\\
\hat{r}_f &= \hat{v}_f + \delta_f \widehat{breg}_f\\
\label{eq:refine}
\end{split}
\end{equation}
\mysection{Residual Regression.}
Residual regression can be thought of as a binned regression with a single bin except that bin classification is no longer necessary since there is a single bin.
Thus, only one regression value $\widehat{breg}_f$ is calculated by the network for each box feature $f$ that is being regressed in this manner.
Refined box features are obtained in the same manner as with binned regression, as shown in \Equation{refine}.
However, as bins are no longer being calculated, $\hat{v}_f$ and $\delta_f$ take new meanings with some subtle differences between the case when $f \in \{y\}$, and the case when $f \in \{h, w, l\}$.
When $f \in \{y\}$, $\hat{v}_f = b_f$, and $\delta_f = 1$.
This means the single bin center is located at the proposal's $y$ location, and there is no normalization of the calculated regression value.
If $f \in \{h, w, l\}$, both $\hat{v}_f$ and $\delta_f$ are taken to be the mean of the box feature among all training samples \ie $\hat{v}_f = \delta_f = \{1.53, 1.63, 3.88\}$, for $f \in \{h, w, l\}$.
Therefore, both the single bin's center and size are fixed to the mean of the corresponding box dimension.
\subsection{Losses}
Our final loss is composed of two main components, proposal classification $ \mathcal{L}_{cls} $ and proposal regression $ \mathcal{L}_{reg} $, which we minimize jointly.
The proposal regression loss can be further decomposed into a binned classification loss, and a binned regression loss.
The loss equations used for both classification and regression are shown in \Equation{Losses}.
A binary cross entropy loss is used for both proposal and bin classifications while a
smooth L1 loss is used for the regression tasks, \ie $\mathcal{F}_{cls}$ is the binary cross entropy loss, and $\mathcal{F}_{reg}$ is the smooth L1 loss.
$\mathcal{B}_{reg}$ is the subset of proposals with an IoU large enough to be used for regression.
\begin{equation}
\begin{split}
\mathcal{L}_{cls} = &\frac{1}{\vert\mathcal{B}\vert} \sum_b{\mathcal{F}_{cls}(\widehat{cls}^b, \widebar{cls}^b)} \\
\mathcal{L}_{reg} = &\frac{1}{\vert\mathcal{B}\vert} \sum_{b \in \mathcal{B}_{reg}} \sum_{f \in \{x,z,\theta\}}\mathcal{F}_{cls}(\widehat{bcls}^b_f, \widebar{bcls}^b_f)\\
+ &\frac{1}{\vert\mathcal{B}_{reg}\vert} \sum_{b \in \mathcal{B}_{reg}} \sum_f\mathcal{F}_{reg}(\widehat{breg}^b_f, \widebar{breg}^b_f)
\end{split}
\label{eq:Losses}
\end{equation}
\mysection{Regression Targets}
We compute the targets for regression as follows.
The ground truth bin for a given proposal $b$ and box feature $f$ can be computed as shown in Equation~\eqref{eq:gtBin}, where $\bar{b}_f$ is the ground truth bounding box with the highest IoU for proposal $b$.
\begin{equation}
\begin{split}
\widebar{bin}^b_f &= \floor*{\frac{\bar{b}_f - b_f + S_f}{\delta_f}} \\
\end{split}
\label{eq:gtBin}
\end{equation}
Binned regression classification targets are the one-hot encoded vector of ground truth bins $\widebar{bin}^b_f$ amongst all $\| \mathcal{V_f} \|$ possible bins, \ie $\widebar{bcls}^b_f = \text{onehot}(\widebar{bin}^b_f)$.
Once the ground truth bin has been computed for a given feature, its regression target can be computed as shown in \Equation{regressionTargets}, where $\bar{v}_f$ is the ground truth bin center.
In the cases where there is a single bin, the ground truth bin center is equal to the single bin center \ie $\bar{v}_f = \hat{v}_f$.
It is worth noting that regression is only performed for the single ground truth bin for each proposal.
\begin{equation}
\begin{split}
\widebar{breg}^b_f &= \frac{\bar{b}_f - \bar{v}_f}{\delta_f} \\
\end{split}
\label{eq:regressionTargets}
\end{equation}
\section{Discussions}
\section{Conclusion}
Current advances in graph convolutional networks have led to better performances in varied 3D computer vision tasks.
This has motivated us to leverage GCNs for the task of 3D vehicle detection, and demonstrate their effectiveness for vehicle detection.
We introduced two novel GCN-based modules for point feature and context aggregation both within and between proposals.
Making use of both modules, we presented a novel pipeline PointRGCN: a two-stage 3D vehicle detection pipeline that introduces graph representations for proposal boxes refinement.
We showed comparable results with recent works, and report a new state-of-the-art on the easy subset in BEV settings for the KITTI dataset.
Still, much work remains to be done in the field of GCNs, from which this work and others stemming from it will benefit.
\section{Supplementary Material.}
We present in \Figure{KITTIleaderboard} the Precision-Recall curves from the KITTI leaderboard, established on the \emph{testing} set.
In particular, we compare the results of PointRCNN, our R-GCN module alone, and our PointRGCN pipeline (from left to right).
We refer to the [email protected], [email protected], [email protected] and AOS metrics (from top to bottom).
Note how our model improves the [email protected] and AOS metrics on the Easy difficulty setting, reaching a recall of 100\%.
This shows that we are able to find \textbf{\emph{all easy vehicle samples}} in the 2D RGB frame with a precision of 60\%.
Such performances were previously unseen with PointRCNN.
To enable further comparison, we provide the ablation study of our method on BEV, 2D Bounding Boxes and AOS.
We complement the ablation study on R-GCN from \Table{AblationRGCN} in \Table{AblationRGCN_complement}
and on C-GCN from \Table{AblationCGCN} in \Table{AblationCGCN_complement}.
\begin{table*}[t]
\centering
\caption{
\textbf{Ablation for R-GCN} on KITTI \emph{validation}\xspace set.
We validate here the setup of our R-GCN network.
Our setup and best results in \textbf{bold}.
Time in ms.
}
\label{tab:AblationRGCN_complement}
\begin{tabular}{l||c|c|c||c|c|c||c|c|c||c}
& \multicolumn{3}{c||}{BEV @ 0.7 IoU}& \multicolumn{3}{c||}{2D @ 0.7 IoU}& \multicolumn{3}{c||}{AOS} & Time \\ \hline
\textbf{R-GCN} & Easy & Mode. & Hard & Easy & Mode. & Hard & Easy & Mode. & Hard & (ms) \\ \hline
\hline PointRGCN & 89.68 & 87.71 & 85.66 & 96.19 & 89.71 & 88.58 & 96.18 & 89.58 & 88.35 & 262 \\\hline
R-GCN alone & 89.76 & 87.80 & 86.11 & 96.22 & 89.59 & 88.94 & 96.21 & 89.45 & 88.73 & 239 \\\hline
\hline 1 layers & 89.40 & 87.59 & 85.86 & 90.36 & 89.38 & 88.65 & 90.35 & 89.23 & 88.43 & 135 \\\hline
3 layers & 89.67 & 87.77 &\D86.25 &\D96.67 & 89.46 & 88.84 &\D96.66 & 89.32 & 88.63 & 188 \\\hline
\D5 layers &\D89.76 &\D87.80 & 86.11 & 96.22 &\D89.59 &\D88.94 & 96.21 &\D89.45 &\D88.73 & 239 \\\hline
10 layers & 89.41 & 87.57 & 86.07 & 90.31 & 89.19 & 88.45 & 90.30 & 89.05 & 88.23 & 447 \\\hline
\hline w/ EdgeConv~\cite{wang2019dynamic}\xspace & 89.41 & 87.69 & 85.97 & 96.29 & 89.61 & 88.91 & 96.28 & 89.44 & 88.67 & 282 \\\hline
w/o residual & 87.04 & 84.53 & 78.03 & 89.09 & 87.52 & 87.11 & 89.08 & 87.33 & 86.82 & 237 \\\hline
w/o dilation & 87.84 & 84.70 & 78.31 & 89.60 & 87.56 & 87.11 & 89.59 & 87.36 & 86.81 & 232 \\\hline
w/o RPN feat. & 89.41 & 87.39 & 85.51 & 90.31 & 88.96 & 88.07 & 90.30 & 88.77 & 87.83 & 238 \\\hline
\hline
\end{tabular}
\end{table*}
\begin{table*}[t]
\centering
\caption{
\textbf{Ablation for C-GCN} on KITTI \emph{validation}\xspace set.
We validate here the setup of our C-GCN network.
Our setup and best results in \textbf{bold}.
Time in ms.
}
\label{tab:AblationCGCN_complement}
\begin{tabular}{l||c|c|c||c|c|c||c|c|c||c}
& \multicolumn{3}{c||}{BEV @ 0.7 IoU}& \multicolumn{3}{c||}{2D @ 0.7 IoU}& \multicolumn{3}{c||}{AOS} & Time \\ \hline
\textbf{C-GCN} & Easy & Mode. & Hard & Easy & Mode. & Hard & Easy & Mode. & Hard & (ms)\\ \hline\hline
PointRGCN & 89.68 & 87.71 & 85.66 & 96.19 & 89.71 & 88.58 & 96.18 & 89.58 & 88.35 & 262 \\ \hline
C-GCN alone & 89.57 & 86.86 & 84.90 & 96.16 & 89.36 & 87.67 & 96.15 & 89.21 & 87.42 & 147 \\\hline
\hline 1 layers &\D89.61 &\D87.35 &\D85.43 & 90.66 &\D89.48 &\D88.43 & 90.65 &\D89.35 &\D88.19 & 145 \\\hline
\D3 layers & 89.57 & 86.86 & 84.90 &\D96.16 & 89.36 & 87.67 &\D96.15 & 89.21 & 87.42 & 147 \\\hline
5 layers & 89.35 & 86.69 & 84.90 & 95.17 & 89.23 & 87.60 & 95.16 & 89.05 & 87.33 & 151 \\\hline
10 layers & 89.30 & 86.33 & 85.21 & 90.57 & 89.25 & 87.38 & 90.56 & 89.12 & 87.16 & 160 \\\hline
20 layers & 88.95 & 85.80 & 84.75 & 90.64 & 89.17 & 86.97 & 90.63 & 89.02 & 86.72 & 173 \\\hline
30 layers & 87.57 & 85.11 & 83.95 & 89.90 & 88.29 & 86.21 & 89.89 & 88.11 & 85.91 & 192 \\\hline
40 layers & 89.47 & 86.47 & 85.15 & 90.41 & 88.64 & 86.15 & 90.41 & 88.48 & 85.91 & 206 \\\hline
\hline w/ MRGCN~\cite{li2019deepgcns_journal}\xspace & 89.19 & 86.83 & 85.21 & 90.59 & 89.30 & 87.35 & 90.59 & 89.13 & 87.08 & 152 \\\hline
w/o residual & 87.29 & 84.46 & 78.22 & 89.39 & 87.70 & 87.03 & 89.38 & 87.47 & 86.71 & 150 \\\hline
w/ dilation & 89.32 & 86.92 & 85.21 & 95.72 & 89.47 & 87.97 & 95.71 & 89.32 & 87.73 & 150 \\\hline
w/ RPN feat. & 87.81 & 84.96 & 78.73 & 89.75 & 87.96 & 87.50 & 89.75 & 87.72 & 87.15 & 150 \\\hline
\hline
\end{tabular}
\end{table*}
\section{Experiments}
\label{sec:Experiments}
In this section, we detail the experiments performed to validate our pipeline.
We first present the experimental setup, including the dataset, network, and training details.
Afterwards, we show our main results on KITTI testing set and compare our pipeline with other state-of-the-art in both 3D object detection and BEV object detection.
Finally, we present ablation studies performed on different model components to validate our pipeline design choices.
\begin{table*}[t]
\centering
\caption{
\textbf{Main Results on KITTI \emph{testing} set.}
We report metrics published in papers.
The first $4$ methods leverage LiDAR and RGB information,
while the next $7$ LiDAR only.
For each columns, we highlight the \A{first}, \bfseries{second} and \C{third} best published method using LiDAR only.
Our method performs best on the easy difficulty for AP$_{BEV}$.
}
\label{tab:MainResults}
\begin{tabular}{l|c||c|c|c||c|c|c||c}
& & \multicolumn{3}{c||}{3D @ 0.7 IoU}& \multicolumn{3}{c||}{BEV @ 0.7 IoU} & Time \\ \hline
Method & Modality & Easy & Mode. & Hard & Easy & Mode. & Hard & (ms) \\ \hline\hline
MV3D~\cite{chen2017multi}\xspace & L+I & 66.77 & 52.73 & 51.31 & 85.82 & 77.00 & 68.94 & 240 \\ \hline
AVOD~\cite{ku2018joint}\xspace & L+I & 73.59 & 65.78 & 58.38 & 86.80 & 85.44 & 77.73 & 100 \\ \hline
AVOD-FPN~\cite{ku2018joint}\xspace & L+I & 81.94 & 71.88 & 66.38 & 88.53 & 83.79 & 77.90 & 100 \\ \hline
F-PointNet~\cite{qi2018frustum}\xspace& L+I & 81.20 & 70.39 & 62.19 & 88.70 & 84.00 & 75.33 & 170 \\ \hline
UberATG-MMF~\cite{liang2019multi}\xspace & L+I & 86.81 & 76.75 & 68.41 & 89.49 & 87.47 & 79.10 & 80 \\ \hline\hline
VoxelNet~\cite{zhou2018voxelnet}\xspace & L & 77.49 & 65.11 & 57.73 & 89.35 & 79.26 & 77.39 & 220 \\ \hline
PIXOR~\cite{yang2018pixor}\xspace & L & - & - & - & 84.44 & 80.04 & 74.31 & 100 \\ \hline
SECOND~\cite{yan2018second}\xspace & L & 83.13 & 73.66 & 66.20 & 88.07 & 79.37 & 77.95 & 50 \\ \hline
PointPillars~\cite{lang2019pointpillars}\xspace & L & 79.05 & 74.99 & 68.30 & 88.35 &\C{86.10}& 79.83 & 16 \\ \hline
PointRCNN~\cite{shi2019pointrcnn}\xspace & L &\C{85.94}&\bfseries{75.76}& 68.32 & 89.47 & 85.68 & 79.10 & 100 \\ \hline
Fast Point R-CNN~\cite{Chen2019FastPointRCNN}\xspace & L & 84.28 &\C{75.73}& 67.39 & 88.03 &\C{86.10}& 78.17 & 65 \\ \hline
STD~\cite{Yang2019std}\xspace & L &\A{86.61}&\A{77.63}&\A{76.06}&\C{89.66}&\A{87.76}&\A{86.89}& 80 \\ \hline\hline
R-GCN only (ours)& L & 83.42 & 75.26 &\C{68.73}&\A{91.91}& 86.05 &\bfseries{81.05}& 239 \\ \hline
PointRGCN (ours) & L &\bfseries{85.97}&\C{75.73}&\bfseries{70.60}&\bfseries{91.63}&\bfseries{87.49}&\C{80.73}& 262 \\ \hline \hline
\end{tabular}
\end{table*}
\begin{figure*}[!htbp]
\centering
\frame{\begin{overpic}[width=0.33\linewidth,trim={4cm 23cm 4cm 10cm},clip]{imgs/DetectionVisualizations/det_gt_rcnn_visualization_8.png}
\put (5,88) {\LARGE\color{RoyalBlue}\textbf{(a)}}
\end{overpic}}
\frame{\begin{overpic}[width=0.33\linewidth,trim={4cm 23cm 4cm 10cm},clip]{imgs/DetectionVisualizations/det_gt_rcnn_visualization_15.png}
\put (5,88) {\LARGE\color{RoyalBlue}\textbf{(b)}}
\end{overpic}}
\frame{\begin{overpic}[width=0.33\linewidth,trim={4cm 23cm 4cm 10cm},clip]{imgs/DetectionVisualizations/det_gt_rcnn_visualization_6154.png}
\put (5,88) {\LARGE\color{RoyalBlue}\textbf{(c)}}
\end{overpic}}
\caption{
\textbf{Qualitative Results:}
\textbf{Top:}
\textbf{\color{RoyalBlue}Ground truth},
\textbf{\color{Green}PointRCNN}~\cite{shi2019pointrcnn} and
\textbf{\color{Red}PointRGCN [ours]} detection results on BEV projections.
\textbf{Bottom:}
We project the detections from \textbf{\color{Red}PointRGCN [ours]} onto the RGB image plane for visualization. \textbf{(a)} showcases where our pipeline is able to detect unlabeled vehicles from the dataset, \textbf{(b)} showcases where our pipeline is able to avoid false positives compared with PointRCNN, and \textbf{(c)} shows failure cases for our pipeline.
}
\label{fig:Qualitative_Detection}
\end{figure*}
\subsection{Experimental setup}
\mysection{Dataset.}
We perform our experiments on the KITTI dataset~\cite{Geiger2012KITTI} for 3D object detection, where detections are divided into $3$ difficulty levels: easy, moderate, and hard, based on occlusion and distance with respect to the vehicle's camera.
There are three major labeled object classes: cars, pedestrians, and cyclists.
We report results only on the car category since it has the highest number of training samples.
Note that the same pipeline is applicable to other categories without further modifications.
The publicly available training set is divided into training and validation subset with $3712$ and $3769$ samples each as common practice, and all results reported are taken from the validation subset except for the final testing results.
\mysection{R-GCN.}
We refine proposals by considering their canonical representation, centered on the bounding box reference system.
Proposal bounding boxes are extended by $1$ meter in every direction before being cropped, and $512$ points are randomly sampled for each proposal to allow for batch training.
We use $N_r=5$ residual MRGCN~\cite{li2019deepgcns_journal}\xspace layers with $F_r=64$ filters each and a linearly increasing dilation.
We look for $k=16$ neighbors per node and use $C_r=1024$ for the global feature size.
We use an aggregation of both canonical point coordinates and RPN features as input for each node.
We report ablation studies on the type of convolution layer (\ie EdgeConv~\cite{wang2019dynamic}\xspace vs MRGCN~\cite{li2019deepgcns_journal}\xspace), network depth, the use of residual skip connections, the use of dilation and the importance of input RPN features in \Table{AblationRGCN}.
\mysection{C-GCN.}
We gather contextual information across proposals in order to detect more consistent and coherent bounding boxes.
For each proposal, we maxpool the R-GCN point features to define each node features.
We define the edges of the graph by dynamically looking for the closest $k=16$ neighbors in the feature space.
We use $N_c = 3$ residual EdgeConv~\cite{wang2019dynamic}\xspace layers with $F_c=64$ filters without any dilation.
We compute a global feature of size $C_c=1024$.
In our complete PointRGCN pipeline, we use the features from R-GCN, but also experiment with the the RCNN module introduced in PointRCNN~\cite{shi2019pointrcnn}\xspace, to highlight the contribution of our sole additional C-GCN.
We experiment with the type of layer (EdgeConv~\cite{wang2019dynamic}\xspace vs MRGCN~\cite{li2019deepgcns_journal}\xspace), the depth of the network, the residual skip connections, dilation and additional RPN features aggregation.
We report these ablation studies in \Table{AblationCGCN}.
\mysection{Bin Parameters.}
For location coordinates $f \in {x,z}$, we set the search space $S_f = 1.5m$, and the bin resolution $\delta_f = 0.5m$.
This leads to a total of $6$ bins for $f \in {x,z}$.
In the case of $f = \theta$, we use a search space $S_f = 22.5^\degree$, and bin resolution $\delta_f = 5^\degree $, for a total of $9$ bins.
\mysection{Training Details.}
We use the offline training scheme from \cite{shi2019pointrcnn} where RPN proposals and features are pre-computed and saved.
Data augmentation is performed by artificially adding vehicles to the scene along with random global rotations and translations.
During training, $300$ RPN proposals are saved for each frame, but we only consider $64$ proposals for the losses.
Proposals are considered positive if they have an IoU greater than $0.6$, and negative if they have an IoU lower than $0.45$.
Only proposals with an IoU of at least $0.55$ are considered for the regression loss.
The network is trained for $50$ epochs on KITTI \emph{train}\xspace only, using Adam optimizer, with a batch size of $256$ proposals (\ie $4$ frames), and a learning rate of $0.0002$.
We run all our training on v100 GPUs with 32GB of memory, and our validation and testing on GTX1080Ti GPUs with 12GB of memory.
\subsection{Main Results}
\Table{MainResults} reports the average precision at an IoU of $0.7$ for 3D and BEV object detection on KITTI \emph{test}\xspace.
We obtain results that are comparable against other published performances.
PointRGCN is on average $2^{nd}$, after the recently proposed method STD~\cite{Yang2019std}\xspace.
We consistently outperform our baseline PointRCNN~\cite{shi2019pointrcnn}\xspace, and achieve state-of-the-art on the easy difficulty on BEV detection task with an impressive $2.13\%$ improvement.
We believe the fine-grained information gathered by the graph representation helps in localizing vehicles in the BEV space, but struggles to improve further the height parameters (\ie $r_y$, $r_h$) with respect to PointRCNN~\cite{shi2019pointrcnn}\xspace.
\subsection{Ablation Study}
We compare our method with the state-of-the-art on KITTI \emph{validation}\xspace and provide extensive ablation studies for both our R-GCN and C-GCN modules.
In order to demonstrate the effectiveness of our R-GCN and C-GCN modules, we compare each of our modules against PointRCNN~\cite{shi2019pointrcnn}\xspace in \Table{AblationModule}.
An improvement in performance is obtained by using R-GCN over RCNN on KITTI \emph{validation}\xspace in hard settings.
By combining both modules we obtain a performance increase in the easy and moderate categories in KITTI \emph{validation}\xspace.
Additionally, our full pipeline generalize better than PointRCNN~\cite{shi2019pointrcnn}\xspace as shown from the overall increase in performance on KITTI \emph{test}\xspace.
This aligns with the better generalization capability of current GCN~\cite{li2019deepgcns_journal}.
\begin{table}[htbp]
\centering
\caption{
\textbf{Ablation for PointRGCN.}
We evaluate our modules R-GCN and C-GCN singularly and compared with PointRGCN, on KITTI \emph{validation}\xspace.
Best results shown in \textbf{bold}. Time in ms.
}
\label{tab:AblationModule}
\begin{tabular}{l||c|c|c||c}
Method & Easy & Mode. & Hard & Time \\ \hline\hline
PointRCNN~\cite{shi2019pointrcnn}\xspace &\D88.45 & 77.67 & 76.30 & 100 \\
\hline\hline
PointRGCN & 88.37 &\D78.54 & 77.60 & 262 \\ \hline
R-GCN & 88.08 & 78.45 &\D77.81 & 239 \\ \hline
\cite{shi2019pointrcnn} + C-GCN
& 87.56 & 77.58 & 75.94 & 147 \\ \hline
\hline
\end{tabular}
\end{table}
\subsection{Further Ablation on R-GCN}
We further investigate the contribution of R-GCN in \Table{AblationRGCN}, by ablating it using different depths, layer types, residual skip connections, dilations, and input features.
\begin{table}[htbp]
\centering
\caption{
\textbf{Ablation for R-GCN} on KITTI \emph{validation}\xspace set.
We validate here the setup of our R-GCN network.
Our setup and best results in \textbf{bold}.
Time in ms.
}
\label{tab:AblationRGCN}
\begin{tabular}{l||c|c|c||c}
\textbf{R-GCN} ([email protected])& Easy & Mode. & Hard & Time \\ \hline\hline
PointRGCN & 88.37 & 78.54 & 77.60 & 262 \\\hline
R-GCN alone & 88.08 & 78.45 & 77.81 & 239 \\\hline
\hline 1 layers & 87.29 & 78.09 & 77.34 & 135 \\\hline
3 layers & 87.96 &\D78.48 &\D77.94 & 188 \\\hline
\D5 layers &\D88.08 & 78.45 & 77.81 & 239 \\\hline
10 layers & 83.33 & 76.56 & 75.47 & 447 \\\hline
\hline w/ EdgeConv~\cite{wang2019dynamic}\xspace & 87.75 & 78.33 & 77.68 & 282 \\\hline
w/o residual & 82.05 & 73.12 & 73.04 & 237 \\\hline
w/o dilation & 82.84 & 73.34 & 73.31 & 232 \\\hline
w/o RPN feat. & 83.18 & 74.36 & 72.83 & 238 \\\hline
\hline
\end{tabular}
\end{table}
\mysection{Depth.}
We test our R-GCN module using $1$, $3$, $5$ and $10$ layers.
As shown in \Table{AblationRGCN}, the R-GCN module is fairly insensitive across different network depths.
The best results are achieved from $3$ to $5$ layers.
\mysection{Network Design.}
We show that dilation and residual connections are paramount.
Without residual skip connections, the performance drops $5.36\%$.
Li~\etal~\cite{li2019deepgcns_journal} showed that residual skip connection allows the gradient to flow better between layers, improving the smoothness of the GCN convergence.
Without dilation, the performance drops $5.14\%$ on the moderate validation subset.
Dilation provides the necessary receptive field for our GCN to gather enough feature information.
On the other hand, MRGCN~\cite{li2019deepgcns_journal}\xspace and EdgeConv~\cite{wang2019dynamic}\xspace perform fairly similarly, with the biggest difference being in memory and speed requirements.
We have measured EdgeConv to be approximately $40$ms slower than MRGCN when using a $5$-layer network.
Additionally, the $5$-layer network using MRGCN only consumes a maximum of $1840$MB of memory while EdgeConv consumes a maximumum of $2165$MB of memory.
\mysection{Input data.}
We exhibit a drop of $4.16\%$ on the moderate subset by using only point coordinates as features for the nodes of the graph.
RPN features are an important source of semantic information when processing proposal points since they have been trained for semantic segmentation.
\subsection{Further Ablation on C-GCN}
We further investigate the contribution of C-GCN in \Table{AblationCGCN}, by ablating it using different depths, layer types, residual skip connections, dilation, and input features.
\begin{table}[htbp]
\centering
\caption{
\textbf{Ablation for C-GCN} on KITTI \emph{validation}\xspace set.
We validate here the setup of our C-GCN network.
Our setup and best results in \textbf{bold}.
Time in ms.
}
\label{tab:AblationCGCN}
\begin{tabular}{l||c|c|c||c}
\textbf{C-GCN} ([email protected]) & Easy & Mode. & Hard & Time\\ \hline\hline
PointRGCN & 88.37 & 78.54 & 77.60 & 262 \\
C-GCN alone & 87.56 & 77.58 & 75.94 & 147 \\\hline
\hline 1 layers & 86.11 & 77.30 & 75.12 & 145 \\\hline
\D3 layers &\D87.56 &\D77.58 &\D75.94 & 147 \\\hline
5 layers & 86.45 & 76.95 & 75.88 & 151 \\\hline
10 layers & 85.67 & 76.56 & 75.08 & 160 \\\hline
20 layers & 82.69 & 75.45 & 72.96 & 173 \\\hline
30 layers & 83.12 & 74.15 & 71.98 & 192 \\\hline
40 layers & 81.44 & 72.22 & 67.48 & 206 \\\hline
\hline w/ MRGCN~\cite{li2019deepgcns_journal}\xspace & 85.02 & 76.09 & 73.51 & 152 \\\hline
w/o residual & 82.84 & 73.74 & 73.17 & 150 \\\hline
w/ dilation & 87.03 & 77.37 & 76.06 & 150 \\\hline
w/ RPN feat. & 82.89 & 74.05 & 73.74 & 150 \\\hline
\hline
\end{tabular}
\end{table}
\mysection{Depth.}
We tested our C-GCN modules with up to $40$ layers.
Since the number of nodes is lower than for the R-GCN, adding layers increases the inference time without providing improved performances.
We have thus chosen to use a total of $3$ layers for the C-GCN network in our pipeline.
\mysection{Network Design.}
We show on \Table{AblationCGCN} that MRGCN~\cite{li2019deepgcns_journal}\xspace performs slightly less than EdgeConv~\cite{wang2019dynamic}\xspace, as expected, since MRGCN is a simplification over EdgeConv.
Like for our R-GCN, residual skip connection are necessary to make the GCN converge smoothly, accounting for $3.84\%$ in the moderate difficulty.
Using dilation slightly improves performances on the hard difficulty, but prefer not using dilation since dilation may overflow neighbors for deeper networks.
Using a global feature derived from the RPN point features does not provide further improvements.
\subsection{Qualitative Results}
We show qualitative results of our detection pipeline in \Figure{Qualitative_Detection}.
It can be observed that in some cases our method is able to detect vehicles which are not labeled in the dataset due to their partial visibility, as in \Figure{Qualitative_Detection}(a).
Additionally, we show an improvement over PointRCNN in cases such as \Figure{Qualitative_Detection}(b), where false positives are avoided due to our C-GCN.
Failure cases can be observed in images \Figure{Qualitative_Detection}(c), where false positives occur due to large occlusion of vehicles in heavy traffic and due to vegetation clutter at large distances.
Overall, we are able to provide tight bounding boxes with high precision.
| -35,207.687972 |
[
-2.591796875,
2.4140625
] | 64.92622 |
[
-2.85546875,
1.4111328125,
-1.3427734375,
-3.84375,
-1.0458984375,
5.58203125
] |
[
2.630859375,
6.42578125,
2.37890625,
6.37890625
] | 749 | 6,234 |
[
-1.478515625,
1.5546875
] | 32.558714 |
[
-6.125,
-3.962890625,
-3.97265625,
-1.595703125,
2.53515625,
11.171875
] | 0.315773 | 33.017871 | 26.435675 | 6.967561 |
[
2.455155372619629
] | -25,537.952561 | 6.095926 | -34,616.739616 | 0.34735 | 6.297201 |
[
-2.74609375,
-3.57421875,
-3.83984375,
-4.3828125,
2.51953125,
11.5078125
] |
[
-5.43359375,
-1.5087890625,
-2.0859375,
-1.572265625,
3.3046875,
4.98828125
] | |
BkiUfbI25V5jXvdT3FCG
|
\section{Introduction}
Let $L/{\mathbb Q_p}$ be a finite extension with ring of integers ${\mathfrak o} = {\mathfrak o}_L$.
In \cite{PSS4} we have introduced certain sheaves of differential operators ${\widetilde{\sD}}^\dagger_{n,k,{\mathbb Q}}$ on a family of semistable models ${\mathfrak X}_n$ of the projective line and have shown that ${\mathfrak X}_n$ is ${\widetilde{\sD}}^\dagger_{n,k,{\mathbb Q}}$-affine.
In this paper we generalize this approach to (not necessarily semistable) formal models of general flag varieties of split reductive groups.
So let ${\mathbb G}_0$ be a reductive group scheme over ${\mathfrak o}$ with generic fibre ${\mathbb G}$. Denote by $X^{\rm rig}$ the rigid analytic space attached to the flag variety of ${\mathbb G}$. We consider a formal model ${\mathfrak X}$ of $X^{\rm rig}$ which we assume to be an admissible blow-up of a smooth formal model ${\mathfrak X}_0$. On such a model ${\mathfrak X}$ we introduce certain sheaves of differential operators ${\widetilde{\sD}}^\dagger_{{\mathfrak X},k,{\mathbb Q}}$, for $k \ge k_{\mathfrak X}$, where $k_{\mathfrak X}$ is a non-negative integer depending on ${\mathfrak X}$. Our first main result is then
\vskip8pt
{\bf Theorem 1.} {\it For all $k \ge k_{\mathfrak X}$ the formal scheme ${\mathfrak X}$ is ${\widetilde{\sD}}^\dagger_{{\mathfrak X},k,{\mathbb Q}}$-affine.}
\vskip8pt
This means that the global sections functor furnishes an equivalence of categories between coherent modules over ${\widetilde{\sD}}^\dagger_{{\mathfrak X},k,{\mathbb Q}}$ and finitely presented modules over the ring $H^0({\mathfrak X},{\widetilde{\sD}}^\dagger_{{\mathfrak X},k,{\mathbb Q}})$ (cf. \cite{BB81},\cite{BK80}, \cite{BK81} for the classical setting).
It is shown that $H^0({\mathfrak X},{\widetilde{\sD}}^\dagger_{{\mathfrak X},k,{\mathbb Q}})$ can be identified with a central reduction of Emerton's analytic distribution algebra ${\mathcal D}^{\rm an}({\mathbb G}(k)^\circ)$ of the wide open rigid analytic $k^{\rm th}$ congruence subgroup of ${\mathbb G}_0$, cf. \cite{EmertonA}.
\vskip8pt
As in \cite{PSS4} our main motivation for this result concerns locally analytic representations.
The category of admissible locally analytic representations of the $p$-adic group $G:={\mathbb G}(L)$\footnote{considered as a locally $L$-analytic group} with trivial infinitesimal character $\theta_0$ is anti-equivalent to the category of coadmissible modules over $D(G)_{\theta_0}$, the
central reduction of the locally $L$-analytic distribution algebra of $G$ at $\theta_0$. On the geometric side, we consider a projective system ${\mathcal F}$ of formal models ${\mathfrak X}$. We assume that each model in this system is an admissible blow-up of some smooth formal model of $X^{\rm rig}$, and that the system ${\mathcal F}$ is equipped with an action of $G$. Then we can form the limit of the sheaves ${\widetilde{\sD}}^\dagger_{{\mathfrak X},k,{\mathbb Q}}$ and obtain a sheaf of (infinite order) differential operators ${\widetilde{\sD}}^\dagger_{\infty,{\mathbb Q}}$
on ${\mathfrak X}_\infty = {\mathfrak X}_\infty({\mathcal F}) = \varprojlim_{\mathcal F} {\mathfrak X}$. In this situation, the localization functors for the various ${\mathfrak X}$ assemble to a functor ${\mathscr Loc}^\dagger_\infty$ from the coadmissible modules over $D(G)_{\theta_0}$ into the $G$-equivariant
${\widetilde{\sD}}^\dagger_{\infty,{\mathbb Q}}$-modules on ${\mathfrak X}_\infty$.
As in \cite{PSS4} we tentatively call its essential image the coadmissible (equivariant) ${\widetilde{\sD}}^\dagger_{\infty,{\mathbb Q}}$-modules. We then obtain
\vskip8pt
{\bf Theorem 2.} {\it The functors ${\mathscr Loc}^\dagger_\infty$ and $H^0({\mathfrak X}_\infty,\cdot)$ are quasi-inverse equivalences between the categories of coadmissible $D(G)_{\theta_0}$-modules and coadmissible equivariant ${\widetilde{\sD}}^\dagger_{\infty,{\mathbb Q}}$-modules.}
\vskip8pt
When the projective system ${\mathcal F}$ is cofinal in the system of all formal models of $X^{\rm rig}$, then ${\mathfrak X}_\infty = {\mathfrak X}_\infty({\mathcal F})$ is the Zariski-Riemann space attached to $X^{\rm rig}$. The latter space is in turn isomorphic (as a ringed space, after inverting $p$ on the structure sheaf) to the adic space attached to $X^{\rm rig}$, cf. \cite{SchnVPut}.
\vskip8pt
With Theorem 1 as key ingredient, the proof of Theorem 2 is a straightforward generalization of the ${\rm GL}_2$-case treated in \cite{PSS4}, but we give all the details.
\vskip8pt
In this paper we only treat the case of the central character $\theta_0$, but there is an extension of this theorem available for characters more general than $\theta_0$ by using twisted versions of the sheaves ${\widetilde{\sD}}^\dagger_{{\mathfrak X},k,{\mathbb Q}}$. Moreover, the construction of the sheaf ${\widetilde{\sD}}^\dagger_{\infty,{\mathbb Q}}$ carries over to general smooth rigid (or adic) spaces over $L$. We will discuss these addenda in a sequel paper.
\vskip8pt
We would also like to mention that K. Ardakov and S. Wadsley are developing a theory of ${\mathcal D}$-modules on general rigid spaces, cf. \cite{ArdakovICM}, \cite{AW_Dcap_I}. In their work they consider deformations of the sheaves of crystalline differential differential operators (as in \cite{AW}), whereas we take as a starting point deformations of Berthelot's rings of arithmetic differential operators. Insight from prior studies of logarithmic arithmetic differential operators (cf. \cite{PSS2}, \cite{PSS3}) suggested to consider these latter deformations, because their global sections turned out to be (central reductions of) the analytic distribution algebras mentioned above.
\vskip8pt
{\it Notation.} $L$ denotes a finite extension of ${\mathbb Q_p}$, with ring of integers ${\mathfrak o}$ and uniformizer ${\varpi}$. We put
${\mathfrak S} = {\rm Spf}({\mathfrak o})$ and $S = {\rm Spec}({\mathfrak o})$. Let $q$ denote the cardinality of the residue field ${\mathfrak o}/({\varpi})$ which we also denote by ${\mathbb F_q}$. ${\mathbb G}_0$ denotes a split reductive group scheme over ${\mathfrak o}$ and ${\mathbb B}_0 \subset {\mathbb G}_0$ a Borel subgroup scheme. The Lie algebra of ${\mathbb G}_0$ is denoted by ${\mathfrak g}_{{\mathfrak o}}$.
For any scheme $X$ over ${\mathfrak o}$, we denote by ${\mathcal T}_{X}$ its relative tangent sheaf. A coherent sheaf of ideals ${\mathcal I}\subset {\mathcal O}_{X}$ is called {\it open} if it contains a power $\varpi^N{\mathcal O}_{X}$. A scheme which arises from blowing up an
open ideal sheaf on $X$ will be called {\it an admissible blow-up} of $X$. We always denote by ${\mathfrak X}$ the completion of $X$ along its special fiber.
\vskip12pt
\section{The sheaves ${\widetilde{\cD}}^{(m)}_{X,k}$ and ${\widetilde{\sD}}^{(m)}_{{\mathfrak X},k}$}\label{new_sheaves}
\subsection{Definitions}
In this section, $X_0$ denotes a smooth $S$-scheme, and ${\mathfrak X}_0$ its formal completion.
Let ${\rm pr}:X\rightarrow X_0$ be an admissible blow-up of the scheme $X_0$,
defined by a sheaf of ideals
${\mathcal I}$, containing ${\varpi}^N$ and ${\mathfrak X}$ be the formal completion of $X$. For any integer $k\geq N$, and $m$ a fixed positive integer, we will define
a $p$-adically complete sheaf of arithmetic differential operators ${\widetilde{\sD}}^{(m)}_{{\mathfrak X},k}$ over the non-smooth formal scheme ${\mathfrak X}$.
To achieve this construction, we will first define a sheaf of differential
operators ${\widetilde{\cD}}^{(m)}_{X,k}$ on $X$ for every $k\geq N$.
The scheme $X_{0}$ being a smooth $S$-scheme, one can use the sheaves of arithmetic differential operators of
Berthelot defined
in \cite{BerthelotDI}. In particular, for a fixed $m\in{\mathbb N}$,
${\mathcal D}^{(m)}_{X_{0}}$ will denote the sheaf of differential operators over $X_{0}$ of level $m$. The usual sheaf of differential operators
(EGA IV) over $X_{0}$ will be denoted by ${\mathcal D}_{X_{0}}$.
Let $U$ be a smooth affine open scheme of $X_{0}$ endowed with coordinates
$x_1,\ldots,x_N$ and ${\mathfrak U}$ its formal completion. One denotes $\partial_l$ the derivation relative to $x_l$,
$\partial_l^{[\nu_l]}\in {\mathcal D}_{U}$ defined by $\nu_i!\partial_l^{[\nu_l]}=\partial_l^{\nu_l}$, and finally
$$\partial_{l}^{\langle \nu_l \rangle}=q_{\nu_l}!\partial_{l}^{[\nu_l]}.$$
Here, $q_{\nu_l}$ denotes the quotient of the euclidean division
of $\nu_l$ by $p^m$. For $(\nu_1,\ldots,\nu_N)\in{\mathbb N}^N$, let us define
$\underline{\partial}^{\langle \underline{\nu} \rangle}=\prod_{l=1}^N \partial_{l}^{\langle \nu_l \rangle}$, and $\underline{\partial}^{[\underline{\nu} ]}=\prod_{l=1}^N \partial_{l}^{[\nu_l]}$.
Restricted to $U$, the sheaf ${\mathcal D}^{(m)}_{X_{0},k}$ is a sheaf of free ${\mathcal O}_U$-algebras, with basis the elements $\underline{\partial}^{\langle \underline{\nu} \rangle}$.
Following ~\cite{PSS4}, given two positive integers, $k,m$,
one introduces the subsheaves of algebras ${\widetilde{\cD}}^{(m)}_{{X_{0},k}}$ (resp. ${\widetilde{\sD}}^{(m)}_{{\mathfrak X},k}$)
of differential operators of congruence
level $k$, which are locally free ${\mathcal O}_{U_i}$-algebras (resp. ${\mathcal O}_{{\mathcal U}}$-algebras), with basis the elements ${\varpi}^{k|\underline{\nu}|}\underline{\partial}^{\langle \underline{\nu} \rangle}$, meaning
that
$${\widetilde{\cD}}^{(m)}_{{X_{0},k}}(U)=\left\{\sum_{\underline{\nu}}{\varpi}^{k|\underline{\nu}|}a_{\underline{\nu}}\underline{\partial}^{\langle \underline{\nu} \rangle}\,|\, a_{\underline{\nu}}\in {\mathcal O}_{X_0}(U)\right\}.$$
Let $m$ be a fixed non negative integer, if $\nu$ is
a non negative integer, one denotes by $q$ the quotient of the euclidean division
of $\nu$ by $p^m$. Let $\nu\geq \nu'$ be two nonnegative integers and $\nu'':=\nu-\nu'$, then we introduce for the corresponding numbers $q,q'$ and $q''$
the following notation
$$\accfrac{\nu}{\nu'}=\frac{q!}{q'!q''!}.$$
We also define
$$ {\widetilde{\sD}}_{{\mathfrak X}_0,k}^{(m)}=\varprojlim_i {\widetilde{\cD}}^{(m)}_{X_{0},k}/p^i \hskip5pt \textrm{ and }\hskip5pt
{\widetilde{\sD}}^{\dagger}_{{\mathfrak X}_0,k,{\mathbb Q}}=\varinjlim_m {\widetilde{\sD}}_{{\mathfrak X}_0,k}^{(m)}\otimes {\mathbb Q}.$$
As explained in ~\cite{PSS4}, these sheaves of rings have the same finiteness properties as the usual
sheaves of Berthelot (corresponding to $k=0$). In particular, over an affine smooth scheme $U_{0}$ there is an equivalence of
categories between the coherent ${\widetilde{\cD}}^{(m)}_{U_{0},k}$-modules and the finite type modules over the algebra
$\Gamma(U_{0},{\widetilde{\cD}}^{(m)}_{U_{0},k})$, for every $k\geq 0$.
Let us now explain how to construct the desired sheaves over $X$.
Imitating 2.3.5 of ~\cite{BerthelotDI}, if we can prove that
${\mathcal O}_{X}$ can be endowed with a structure of ${\rm pr}^{-1}{\widetilde{\cD}}^{(m)}_{X_{0},k}$-module, then the sheaf
of ${\mathcal O}_{X}$-modules, ${\rm pr}^{*}{\widetilde{\cD}}^{(m)}_{X_{0},k}$, will be a sheaf of rings over ${\mathcal O}_{X}$. In order to prove
this, we need the following proposition.
\begin{prop} Let $k\geq N$. Then the sheaf of rings ${\mathcal O}_{X}$ can be endowed with a structure of
${\rm pr}^{-1}{\widetilde{\cD}}^{(m)}_{X_0,k}$-module, such that the map ${\rm pr}^{-1}{\mathcal O}_{X_0}\rightarrow {\mathcal O}_{X}$
is ${\rm pr}^{-1}{\widetilde{\cD}}^{(m)}_{X_0,k}$-linear.
\end{prop}
\begin{proof} It is enough to prove the statement in the case where $X_0$ is affine, say $X_0={\rm Spec} A$. Denote then $I=\Gamma(X_0,{\mathcal I})$,
which contains ${\varpi}^N$,
and $B$ the $A$-graded algebra $B=Sym_{A}\;I$, $B=\bigoplus_n I_n$ (meaning that $I_n$ equals as an abelian group $I^n$),
so that $X=Proj\; B$. Let $t\in I_d$ an homogeneous element of
degree $d>0$ of $B$, and $B[1/t]_0$ the algebra of degree $0$-elements in the localization $B[1/t]$.
It is enough to prove that $B[1/t]_0$ can be endowed with a structure of ${\widetilde{\cD}}^{(m)}_{{X_{0},k}}(X_0)$-module. We can even assume that $X_0$ is affine endowed
with local coordinates $x_1,\ldots,x_N$. In this case, we let $D^{(m)}_{k}=\Gamma(X_0,{\widetilde{\cD}}^{(m)}_{{X_{0},k}})$
which can be described in the following way
$$D^{(m)}_{k}= \left\{\sum_{\underline{\nu}}{\varpi}^{k|\underline{\nu}|}a_{\underline{\nu}}\underline{\partial}^{\langle \underline{\nu} \rangle}\,|\, a_{\underline{\nu}}\in A\right\}.$$
Let us observe that the algebra $B$ is a subgraded algebra of $A[T]$. Indeed, there is a graded ring morphism
$$\xymatrix{B \ar@{->}[r]^(.4){\varphi}& A[T]\\
x_n \in I_n \ar@{->}[r]& x_n T^n. }$$
We can identify ${\rm Spec}\; A[T]$ with $Y=\mathbb{A}^1_{X_0}$ and
consider ${\rm pr}_2$: $Y \rightarrow X_0$. By copying the classical proof, one can check that ${\rm pr}_2^{-1}{\widetilde{\cD}}^{(m)}_{{X_{0},k}}$ is a subsheaf
of the sheaf ${\widetilde{\cD}}^{(m)}_{{Y,k}}$, as sheaf of rings. Let $U_t$ be the affine open set of $Y$ where $\varphi(t)$ is nonzero.
and $C_t=\Gamma(U_t, {\mathcal O}_Y)$. By flatness of localization, there is an injective map $B_0[1/t]\subset C_t$. Moreover,
$C_t$ and $A[T]$ are $\Gamma(Y,{\widetilde{\cD}}^{(m)}_{{Y,k}})$-modules and thus $D^{(m)}_{k}$-modules. Take $P\in D^{(m)}_{k}$ and
$a\in A$, then $P(aT^n)=P(a)T^n$. This shows that the action of $D^{(m)}_{k}$ over $A[T]$ is graded.
To prove that
$B_0[1/t]$ is a $D^{(m)}_{k}$-module, we will first prove that
$B$ is a $D^{(m)}_{k}$-submodule of $A[T]$ and then that $B_0[1/t]$ is a $D^{(m)}_{k}$-submodule
of $C_t$. The fact that the defined actions will be compatible with the action of $D^{(m)}_{k}$ over $A$, will come
from the fact that this is true for the action of $D^{(m)}_{k}$ over $A[T]$.
Let us prove now that $B$ is a $D^{(m)}_{k}$-submodule of $A[T]$. To avoid too heavy notations, we identify
$B$ (resp. $B_t$) with their image into $A[T]$ (resp. $C_t$) via $\varphi$.
The ideal $I$ is trivially stable by
the action of $A$ by left multiplication, and $A=I_0$ is certainly a $D^{(m)}_{k}$-module.
Recall that the algebra $D^{(m)}_{k}$
is generated by the operators ${\varpi}^{k\nu_i} \partial_{i}^{\langle \nu_i \rangle}$ for all $\nu_i \leq p^m$.
Since $k\geq N$, it is clear that if $\nu_i\geq 1$,
${\varpi}^{k\nu_i} \partial_{i}^{\langle \nu_i \rangle}(IT)\subset {\varpi}^N AT \subset IT $, since the action of $D^{(m)}_{k}$ over $A[T]$ is graded, so that we can define an action
of $D^{(m)}_{k}$ : $I_1 \rightarrow I_1$.
We next prove by induction on $d$ that $I_d$ is stable by the action of $D^{(m)}_{k}$. Let
$$ x= {\varpi}^{N(d-b)}y_1\ldots y_b\,\in I_d.$$ Let $\nu_i\geq 1$ and $\nu_i\leq p^m$, then one computes
\begin{align*} {\varpi}^{N\nu_i}\partial_{i}^{\langle \nu_i \rangle}(y_1\ldots y_b) &= \sum_{\mu_i\leq \nu_i} \left({\varpi}^{N\mu_i}\accfrac{\mu_i}{\nu_i}
\partial_{i}^{\langle \mu_i \rangle}(y_1)\right){\varpi}^{N(\nu_i-\mu_i)}\partial_{i}^{\langle \nu_i-\mu_i \rangle}(y_2\cdots y_b)
\end{align*}
by \cite[2.2.4(iv)]{BerthelotDI}. By induction, we know that ${\varpi}^{N(\nu_i-\mu_i)}\partial_{i}^{\langle \nu_i-\mu_i \rangle}(y_2\cdots y_b)\in I_{b-1}$, that implies
that $${\varpi}^{N\nu_i}\partial_{i}^{\langle \nu_i \rangle}(x)\in I_d.$$
It remains now to prove that $B[1/t]_0$ is a $D^{(m)}_{k}$-submodule
of $C_t$. We have to check the following statement
\begin{gather}\label{stat-inter}
\forall \nu_i\leq p^m, \forall c\in {\mathbb N},\forall g\in I_{dc},\, {\varpi}^{N\nu_i}\partial_{i}^{\langle \nu_i \rangle}\left(\frac{g}{t^c}\right)\in B_0[1/t].
\end{gather}
Let us first prove by induction on $\nu_i$ that
\begin{gather*} {\varpi}^{N\nu_i}\partial_{i}^{\langle \nu_i \rangle} \left(t^{-1}\right)\in \frac{I_{\nu_id}}{t^{\nu_i+1}}.
\end{gather*}
This is true for $\nu_i=0$. Consider then the formula
$$ {\varpi}^{N(\nu_i+1)}\partial_{i}^{\langle \nu_i +1\rangle}(t^{-1})=-\sum_{\mu =0}^{\nu_i}\accfrac{\nu_i+1}{\mu }t^{-1}
{\varpi}^{N(\nu_i+1-\mu )}\partial_{i}^{\langle \nu_i+1 -\mu \rangle}(t)
{\varpi}^{N\mu }\partial_{i}^{\langle \mu \rangle}(t^{-1}).$$ By induction on $\nu_i$, one knows that for any integer $\mu\leq \nu_{i}$,
$$ {\varpi}^{N(\nu_i+1)}t^{-1}\partial_{i}^{\langle \nu_i+1 -\mu \rangle}(t)\partial_{i}^{\langle \mu \rangle}(t^{-1})
\in \frac{1}{t^{\mu +2}}I_{d}I_{\mu d}\, \subset \frac{I_{(\nu_i+1) d}}{t^{\nu_i+2}},$$
which proves our claim. Applying this claim to $t^c$ gives for $\mu\leq p^m$
$${\varpi}^{N\mu }\partial_{i}^{\langle \mu \rangle}(t^{-c})\in \frac{I_{\mu dc}}{t^{c(\mu +1)}}.$$
Then, we have the following formula
$${\varpi}^{N\nu_i}\partial_{i}^{\langle \nu_i \rangle}\left(\frac{g}{t^c}\right)=\sum_{\mu =0}^{\nu_i}
\accfrac{\nu_i}{\mu }{\varpi}^{N(\nu_i- \mu)}\partial_{i}^{\langle \nu_i- \mu \rangle}(g){\varpi}^{N \mu}\partial_{i}^{\langle \mu \rangle}(t^{-c}),$$
whose right-hand terms are contained in $$I_{dc}\frac{I_{ \mu dc}}{t^{c(\mu +1)}}\subset B_0[1/t],$$
and this completes the proof of~\ref{stat-inter} and the proposition.
\end{proof}
From this, we deduce as in 2.3.5 of \cite{BerthelotDI} the following.
\begin{cor} Under the hypothesis of the section, for $k\geq N$, the sheaf ${\rm pr}^*{\widetilde{\cD}}^{(m)}_{{X_{0},k}}$ is a sheaf of rings
over $X$.
\end{cor}
\vskip8pt
This allows us to introduce the following sheaves of rings over the admissible blow-up ${\mathfrak X}$ of ${\mathfrak X}_0$
$$ {\widetilde{\sD}}_{{\mathfrak X},k}^{(m)}=\varprojlim_i {\rm pr}^*{\widetilde{\cD}}^{(m)}_{X_{0},k}/p^i \hskip5pt \textrm{ and }\hskip5pt
{\widetilde{\sD}}^{\dagger}_{{\mathfrak X},k,{\mathbb Q}}=\varinjlim_m {\widetilde{\sD}}_{{\mathfrak X},k}^{(m)}\otimes {\mathbb Q}.$$
We abbreviate ${\widetilde{\cD}}^{(m)}_{X,k}:={\rm pr}^*{\widetilde{\cD}}^{(m)}_{{X_{0},k}}$ in the following.
\subsection{Finiteness properties of the sheaves ${\widetilde{\cD}}^{(m)}_{{X,k}}$}
We keep here the hypothesis of the beginning of the section.
For a given natural number $k\geq 0$ we let $${\widetilde{\cT}}_{X,k} := {\varpi}^k ({\rm pr})^*({\mathcal T}_{X_0}).$$
\begin{lemma}\label{tcT_lemma1}
(i) ${\widetilde{\cT}}_{X,k}$ is a locally free ${\mathcal O}_{X}$-module of rank equal to the relative dimension of $X_0$ over $S$\;,
\vskip8pt
(ii) Suppose $\pi: X'\rightarrow X$ is a morphism of admissible blow-ups of $X_0$. Let $k',k\geq 0 $. One has
$${\widetilde{\cT}}_{X',k'} = {\varpi}^{k'-k} {\rm \pi}^*({\widetilde{\cT}}_{X,k}) $$
as subsheaves of ${\mathcal T}_{X'}\otimes L$.
\end{lemma}
\vskip8pt
{\it Proof. } The assertion (i) follows directly from the definition. The assertion (ii) follows since $\pi$ is a morphism over $X_0$.
\qed
\vskip8pt
We also have
\begin{prop}\label{finite_tDm}
\begin{enumerate} \item The sheaves ${\widetilde{\cD}}^{(m)}_{{X,k}}$ are filtered by the order of differential operators
and there is a canonical isomorphism of graded sheaves of algebras
$${\rm gr}\left({\widetilde{\cD}}^{(m)}_{X,k}\right) \simeq {\rm Sym}({\widetilde{\cT}}_{X,k})^{(m)} = \bigoplus_{d \ge 0} {\rm Sym}^d({\widetilde{\cT}}_{X,k})^{(m)} \;.$$
\item
There is a basis of the topology ${\mathscr B}$ of $X$, consisting of open affine subsets, such that for any $U \in {\mathscr B}$, the ring
${\widetilde{\cD}}^{(m)}_{X,k}(U)$ is noetherian.
In particular, the sheaf of rings ${\widetilde{\cD}}^{(m)}_{X,k}$ is coherent.
\item The sheaf ${\widetilde{\sD}}^{(m)}_{{{\mathfrak X},k}}$ is coherent.
\end{enumerate}
\end{prop}
{\it Proof. }
Denote by ${\widetilde{\cD}}^{(m)}_{X_0,k;d}$ the sheaf of differential operators of ${\widetilde{\cD}}^{(m)}_{{X_0,k}}$ of
order less than $d$.
It is straightforward that we have an exact sequence of ${\mathcal O}_{X_0}$-modules on $X_0$
\begin{numequation}\label{fil_gr_ex_seq_n0} 0 \longrightarrow {\widetilde{\cD}}^{(m)}_{X_0,k;d-1} \longrightarrow {\widetilde{\cD}}^{(m)}_{X_0,k;d} \longrightarrow {\rm Sym}^d({\widetilde{\cT}}_{X_0,k})^{(m)} \longrightarrow 0 \;.
\end{numequation}
Now we apply ${\rm pr}^*$ and get an exact sequence since
${\rm Sym}^d({\widetilde{\cT}}_{X_0,k})^{(m)}$ is a free ${\mathcal O}_{X_0}$-module of finite rank. This gives (i). As we work with quasi-coherent sheaves
of ${\mathcal O}_X$-modules, it is enough to show that ${\widetilde{\cD}}^{(m)}_{{X,k}}(U)$ is noetherian in the case where
$U={\rm pr}^{-1}(U_0)$, and $U_0\subset X_0$ has some coordinates $x_1,\ldots,x_N$. In this case one has the following
description
$${\widetilde{\cD}}^{(m)}_{{X},k}(U)=\left\{\sum_{\underline{\nu}}{\varpi}^{k|\underline{\nu}|}a_{\underline{\nu}}\underline{\partial}^{\langle \underline{\nu} \rangle}\,|\, a_{\underline{\nu}}\in {\mathcal O}_{X}(U)\right\}.$$
By (i), the graded algebra ${\rm gr}{\widetilde{\cD}}^{(m)}_{{X},k}(U)$ is isomorphic to
${\rm Sym}_{{\mathcal O}(U)}({\widetilde{\cT}}_{X,k}(U))^{(m)}$, which is known to be noetherian \cite[Prop. 1.3.6]{Huyghe97}. This gives the
noetherianity of the algebras ${\widetilde{\cD}}^{(m)}_{{X},k}(U)$. As ${\mathscr B}$ we may take the set of open subsets
of $X$ that are contained in some ${\rm pr}^{-1}(U_0)$, for some open $U_0\subset X_0$ endowed with global coordinates. We finally
note that, since the sheaf ${\widetilde{\cD}}^{(m)}_{{X,k}}$ is ${\mathcal O}_X$-quasicoherent and has noetherian sections of over open affines,
it is actually a sheaf of coherent rings \cite[3.1.3(i)]{BerthelotDI}. The last assertion is a direct consequence of (ii), as in \cite{BerthelotDI}. We do not redo the proof here.
\qed
\vskip8pt
From these considerations, we give theorems A and B for affine schemes $X$, whose proofs follows readily
by the proofs given by Berthelot in~\cite{BerthelotDI}.
\begin{prop} Let $U\subset X$ be an affine subscheme of $X$, ${\mathfrak U}$ its formal completion.
\begin{enumerate}
\item The algebra $\Gamma(U,{\widetilde{\cD}}^{(m)}_{{U,k}})$ is noetherian and the functor $\Gamma(U,.)$ establishes an equivalence of categories
between coherent ${\widetilde{\cD}}^{(m)}_{{U,k}}$-modules and finite $\Gamma(U,{\widetilde{\cD}}^{(m)}_{{X,k}})$-modules.
\item The algebra $\Gamma({\mathfrak U},{\widetilde{\sD}}^{(m)}_{{{\mathfrak U},k}})$ is noetherian and the functor $\Gamma({\mathfrak U},.)$ establishes an equivalence of categories
between coherent ${\widetilde{\sD}}^{(m)}_{{{\mathfrak U},k}}$-modules and finite $\Gamma({\mathfrak U},{\widetilde{\sD}}^{(m)}_{{{\mathfrak U},k}})$-modules.
\end{enumerate}
\end{prop}
\section{Formal models of flag varieties}\label{models}
From now on $X_0$ denotes the smooth flag scheme ${\mathbb G}_0/{\mathbb B}_0$ of ${\mathbb G}_0$ and ${\mathfrak X}_0$ denotes its $p$-adic completion.
\subsection{Congruence group schemes}\label{groups}
We let ${\mathbb G}(k)$ denote the $k$-th scheme-theoretic congruence subgroup of the group scheme ${\mathbb G}_0$ \cite[1.]{WW80}, \cite[2.8]{YuSmoothModels}. So ${\mathbb G}(0)={\mathbb G}_0$ and ${\mathbb G}(k+1)$ equals the dilatation (in the sense of \cite[3.2]{BLR}) of the trivial subgroup of ${\mathbb G}(k)\otimes_{{\mathfrak o}}{\mathbb F}_q$ on ${\mathbb G}(k).$ In particular,
if ${\mathbb G}(k) = {\rm Spec}\; {\mathfrak o}[t_1,...,t_N]$ with a set of parameters $t_i$ for the unit section of ${\mathbb G}(k)$, then ${\mathbb G}(k+1)={\rm Spec} \; {\mathfrak o}[\frac{t_1}{\varpi},...,\frac{t_N}{\varpi}]$. The ${\mathfrak o}$-group scheme ${\mathbb G}(k)$ is again smooth, has Lie algebra equal to $\varpi^k{\mathfrak g}_{{\mathfrak o}}$ and its generic fibre coincides with the generic fibre of ${\mathbb G}_0$.
\subsection{A very ample line bundle on $X$} Let ${\rm pr}: X \rightarrow X_0$ be an admissible blow-up, and let ${\mathcal I} \subset X_0$ be the ideal sheaf that is blown up. Since ${\mathcal I}$ is open, there is $N \in {\mathbb Z}_{>0}$ such that
\begin{numequation}\label{invert_p_1}
p^N {\mathcal O}_{X_0} \subset {\mathcal I} \subset {\mathcal O}_{X_0} \;.
\end{numequation}
Put ${\mathcal S} = \bigoplus_{s \ge 0} {\mathcal I}^s$, then $X$ is glued together from schemes ${\bf Proj}({\mathcal S}(U))$ for affine open subsets $U \subset X$. On each ${\bf Proj}({\mathcal S}(U))$ there is an invertible sheaf ${\mathcal O}(1)$, and these glue together to give an invertible sheaf ${\mathcal O}(1)$ on $X$ which we will denote ${\mathcal O}_{X/X_0}(1)$ (cf. the discussion in \cite[ch. II,\S 7]{HartshorneA}.) This invertible sheaf is in fact the inverse image ideal sheaf ${\rm pr}^{-1}({\mathcal I}) \cdot {\mathcal O}_{X}$, cf. \cite[ch. II, 7.13]{HartshorneA}. From \ref{invert_p_1} conclude that
\begin{numequation}\label{invert_p_2}
p^N {\mathcal O}_{X} \subset {\mathcal O}_{X/X_0}(1) \subset {\mathcal O}_{X} \hskip10pt \mbox{and} \hskip10pt {\mathcal O}_{X} \subset {\mathcal O}_{X/X_0}(-1) \subset p^{-N}{\mathcal O}_{X}\;.
\end{numequation}
And for any $r \ge 0$ we get
\begin{numequation}\label{invert_p_3}
p^{rN} {\mathcal O}_{X} \subset {\mathcal O}_{X/X_0}(1)^{\otimes r} \subset {\mathcal O}_{X} \hskip10pt \mbox{and} \hskip10pt {\mathcal O}_{X} \subset {\mathcal O}_{X/X_0}(-1)^{\otimes r} \subset p^{-rN}{\mathcal O}_{X}\;.
\end{numequation}
\begin{lemma}\label{direct_im_str_sh} Suppose $X$ is normal. Let $X'$ be the blow-up of a coherent open ideal on $X$ and
let ${\rm pr}: X' \rightarrow X$ be the blow-up morphism. Then $({\rm pr})_*({\mathcal O}_{X'}) = {\mathcal O}_{X}$.
\vskip8pt
\end{lemma}
{\it Proof. } The morphism ${\rm pr}: X' \rightarrow X$ is a birational projective morphism of noetherian integral schemes. The assertion then follows exactly as in the proof of Zariski's Main Theorem as given in \cite[ch. III, Cor. 11.4]{HartshorneA}. \qed
\vskip8pt
\begin{lemma}\label{v_ample_sh_lemma} There is $a_0\in {\mathbb Z}_{>0}$ such that the line bundle
\begin{numequation}\label{v_ample_sh_disp}
{\mathcal L}_{X} = {\mathcal O}_{X/X_0}(1) \otimes {\rm pr}^*\Big({\mathcal O}_{X_0}(a_0)\Big)
\end{numequation}
on $X$ is very ample over ${\rm Spec}({\mathfrak o})$, and it is very ample over $X_0$.
\end{lemma}
{\it Proof. } By \cite[ch. II, ex. 7.14 (b)]{HartshorneA}, the sheaf
\begin{numequation}\label{v_ample_sh}
{\mathcal L} = {\mathcal O}_{X/X_0}(1) \otimes {\rm pr}^*\Big({\mathcal O}_{X_0}(a_0)\Big)
\end{numequation}
is very ample on $X$ over ${\rm Spec}({\mathfrak o})$ for suitable $a_0 > 0$. We fix such an $a_0$. By \cite[4.4.10 (v)]{EGA_II} it is then also very ample over $X_0$. \qed
\vskip8pt
\begin{lemma}\label{tcT_lemma}
Suppose $\pi: X'\rightarrow X$ is a morphism of admissible blow-ups of $X_0$.
(i) In the case $\pi_*{\mathcal O}_{X'}={\mathcal O}_X$, one has
$$ {\widetilde{\cT}}_{X,k} = {\varpi}^{k-k'} {\rm \pi}_*({\widetilde{\cT}}_{X',k'}) $$
as subsheaves of ${\mathcal T}_{X}\otimes L$.
\vskip8pt
(ii) There is a ${\mathcal O}_{X}$-linear surjection ${\mathcal O}_{X}\otimes_{{\mathfrak o}} \varpi^k{\mathfrak g}_{{\mathfrak o}}\twoheadrightarrow{\widetilde{\cT}}_{X,k}$.
\end{lemma}
{\it Proof. }
The assertion (i) follows from (ii) of the previous lemma~\ref{tcT_lemma1} together with the projection formula. Finally, (ii) follows from \cite[1.6.1]{NootHuyghe09} by applying $\pi^*$.
\qed.
\vskip12pt
\vskip12pt
\begin{prop}\label{graded_tcD} In the case $\pi_*{\mathcal O}_{X'}={\mathcal O}_X$, one has $ \pi_*\left({\widetilde{\cD}}^{(m)}_{X',k}\right) = {\widetilde{\cD}}^{(m)}_{X,k}$.
\vskip8pt
\end{prop}
{\it Proof. } The sheaves ${\widetilde{\cD}}^{(m)}_{X_0,k;d}$ are locally free of finite rank, and so are the sheaves ${\widetilde{\cD}}^{(m)}_{X,k;d}$, by
construction. We can thus apply the projection formula and get
$$\pi_*\Big({\widetilde{\cD}}^{(m)}_{X',k;d}\Big) = {\widetilde{\cD}}^{(m)}_{X,k;d} \;.$$
\vskip8pt
The claim follows because the direct image commutes with inductive limits on a noetherian space.
\qed
\vskip8pt
\begin{para} {\it Twisting by ${\mathcal L}_X$.} Recall the very ample line bundle ${\mathcal L}_X$ from \ref{v_ample_sh_lemma}. In the following we will always use this line bundle to 'twist' ${\mathcal O}_{X}$-modules. If ${\mathcal F}$ is a ${\mathcal O}_{X}$-module and $r \in {\mathbb Z}$ we thus put
$${\mathcal F}(r) = {\mathcal F} \otimes_{{\mathcal O}_{X}} {\mathcal L}_X^{\otimes r} \;.$$
\vskip8pt
Some caveat is in order when we deal with sheaves which are equipped with both a left and a right ${\mathcal O}_{X}$-module structure (which may not coincide). For instance, if ${\mathcal F}_d = {\widetilde{\cD}}_{X,k;d}^{(m)}$ then we let
$${\mathcal F}_d(r) = {\widetilde{\cD}}_{X,k;d}^{(m)}(r) = {\widetilde{\cD}}_{X,k;d}^{(m)} \otimes_{{\mathcal O}_{X}} {\mathcal L}_X^{\otimes r} \;,$$
\vskip8pt
where we consider ${\mathcal F}_d = {\widetilde{\cD}}_{X,k;d}^{(m)}$ as a {\it right} ${\mathcal O}_{X}$-module. Similarly we put
$${\widetilde{\cD}}_{X,k}^{(m)} \otimes_{{\mathcal O}_{X}} {\mathcal L}_X^{\otimes r} \;,$$
\vskip8pt
where we consider ${\widetilde{\cD}}_{X,k}^{(m)}$ as a {\it right} ${\mathcal O}_{X}$-module. Then we have ${\widetilde{\cD}}_{X,k}^{(m)}(r) = \varinjlim_d {\mathcal F}_d(r)$. When we consider the associated graded sheaf of ${\widetilde{\cD}}_{X,k}^{(m)}(r)$, it is with respect to the filtration by the ${\mathcal F}_d(r)$. The sheaf ${\widetilde{\cD}}_{X,k}^{(m)}(r)$ is a coherent left ${\widetilde{\cD}}_{X,k}^{(m)}$-module according to part (iii) of \ref{graded_tcD}.
\end{para}
\vskip8pt
\vskip12pt
\subsection{Global sections of ${\widetilde{\cD}}^{(m)}_{X,k}$, ${\widetilde{\sD}}^{(m)}_{{\mathfrak X},k}$, and ${\widetilde{\sD}}^\dagger_{{\mathfrak X},k,{\mathbb Q}}$}\label{global_sec}
\begin{para} {\it Divided power enveloping algebras.}
We denote by $D^{(m)}({\mathbb G}(k))$ the distribution algebra of the smooth ${\mathfrak o}$-group scheme ${\mathbb G}(k)$ of level $m$ \cite{HS13}.
It is noetherian and admits the following explicit description. Let ${\mathfrak g}_{\mathfrak o}={\mathfrak n}^{-}_{\mathfrak o}\oplus{\mathfrak t}_{\mathfrak o}\oplus{\mathfrak n}_{\mathfrak o}$ be a triangular decomposition of ${\mathfrak g}_{\mathfrak o}$.
We fix basis elements $(f_i),(h_j)$ and $(e_i)$ of the ${\mathfrak o}$-modules ${\mathfrak n}^{-}_{\mathfrak o}, {\mathfrak t}_{\mathfrak o}$ and ${\mathfrak n}_{\mathfrak o}$ respectively.
Then $D^{(m)}({\mathbb G}(k))$ equals the ${\mathfrak o}$-subalgebra of $U({\mathfrak g}) = U({\mathfrak g}_{\mathfrak o}) \otimes_{\mathfrak o} L$ generated by the elements
\begin{numequation}\label{generators_n}
q^{(m)}_{\underline{\nu}}! \frac{({\varpi}^k \underline{e})^{\underline{\nu}}}{\underline{\nu}!} \cdot q^{(m)}_{\underline{\nu}'}! {\varpi}^{k |\underline{\nu}'|}{\underline{h} \choose \underline{\nu}'} \cdot q^{(m)}_{\underline{\nu}''}! \frac{({\varpi}^k \underline{f})^{\underline{\nu}''}}{\underline{\nu}''!} \;.
\end{numequation}
In the case of the group ${\rm GL}_2$ we considered the same algebra in \cite{PSS4}. We denote by $\widehat{D}^{(m)}({\mathbb G}(k))$ the $p$-adic completion of $D^{(m)}({\mathbb G}(k))$. It is noetherian.
\end{para}
\vskip8pt
\begin{prop}\label{global_sections_tcD} There is a canonical homomorphism of ${\mathfrak o}$-algebras
\begin{numequation}\label{xi_uncompl} Q^{(m)}_{X,k}:
D^{(m)}({\mathbb G}(k))\rightarrow H^0(X, {\widetilde{\cD}}^{(m)}_{X,k}) \;,
\end{numequation}
\end{prop}
{\it Proof. }
Recall that a filtered ${\mathfrak o}$-algebra (or sheaf of such algebras) $A$ with positive filtration $F_iA$ and ${\mathfrak o}\subseteq F_0A$ yields the graded subring $R(A):=\oplus_{i\geq 0}F_iAt^{i}\subseteq A[t]$ of the polynomial ring over $A$, its associated Rees ring. Specialising $R(A)$ in an element $\lambda\in{\mathfrak o}$ yields a filtered subring $A_{\lambda}$ of $A$. For fixed $\lambda$, the formation of $A_{\lambda}$ is functorial in $A$.
We apply this remark to the filtered homomorphism
$$D^{(m)}({\mathbb G}(0)) \rightarrow {\widetilde{\cD}}^{(m)}_{X_0,0}$$
appearing in \cite[4.4.4]{HS13} (and denoted by $Q_m$ in loc.cit.) which comes by functoriality from the ${\mathbb G}(0)$-action on $X_0$. Passing to Rees rings and specialising the parameter in $\varpi^k$ yields a filtered homomorphism
$$D^{(m)}({\mathbb G}(k)) \rightarrow {\widetilde{\cD}}^{(m)}_{X_0,k}.$$ Taking global sections and using $H^0(X, {\widetilde{\cD}}^{(m)}_{X,k}) = H^0(X_0, {\widetilde{\cD}}^{(m)}_{X_0,k})$ by \ref{graded_tcD} yields a homomorphism $$ Q_{X,k}^{(m)}: D^{(m)}({\mathbb G}(k)) \rightarrow H^{0}(X,{\widetilde{\cD}}^{(m)}_{X,k})$$
as claimed.
\qed
\vskip8pt
Let ${\mathcal A}^{(m)}_{X,k}={\mathcal O}_{X}\otimes_{{\mathfrak o}}D^{(m)}({\mathbb G}(k))$ with the twisted ring multiplication coming from the action of $D^{(m)}({\mathbb G}(k))$ on ${\mathcal O}_{X}$ via $Q^{(m)}_{X,k}$. It has a natural filtration whose associated graded equals the ${\mathcal O}_{X}$-algebra ${\mathcal O}_{X}\otimes_{{\mathfrak o}}{\rm Sym}^{(m)}({\rm{Lie}}({\mathbb G}(k)))$ \cite[Cor. 4.4.6]{HS13}. In particular, ${\mathcal A}^{(m)}_{X,k}$ has noetherian sections over open affines.
\begin{prop}\label{prop-auxiliaryI}
The homomorphism ${\mathcal A}^{(m)}_{X,k}\rightarrow {\widetilde{\cD}}^{(m)}_{X,k}$ induced by $\xi^{(m)}_{X,k}$ is surjective. In particular, ${\widetilde{\cD}}^{(m)}_{{\mathfrak X},k}$ has noetherian sections over open affines.
\end{prop}
{\it Proof. } We adapt the argument of \cite[4.4.7.2(ii)]{HS13}: The homomorphism is filtered. Applying ${\rm Sym}^{(m)}$ to the surjection in (iv) of \ref{tcT_lemma} we obtain a surjection ${\mathcal O}_{X}\otimes_{{\mathfrak o}}{\rm Sym}^{(m)}({\rm{Lie}}({\mathbb G}(k))\rightarrow{\rm Sym}^{(m)}({\widetilde{\cT}}_{X,k})$ which equals the associated graded homomorphism by \ref{graded_tcD}. Hence the homomorphism is surjective as claimed.
\qed
\vskip8pt
\begin{prop}\label{prop-auxiliaryII}
Let ${\mathcal M}$ be a coherent ${\mathcal A}^{(m)}_{X,k}$-module.
(i) $H^0(X,{\mathcal A}^{(m)}_{X,k})=D^{(m)}({\mathbb G}(k))$.
\vskip8pt
(ii) There is a surjection ${\mathcal A}_{X,k}^{(m)}(-r)^{\oplus s}\rightarrow{\mathcal M}$ of ${\mathcal A}_{X,k}^{(m)}$-modules for suitable $r,s\geq 0$.
\vskip8pt
(iii) For any $i\geq 0$ the group $H^{i}(X,{\mathcal M})$ is a finitely generated $D^{(m)}({\mathbb G}(k))$-module.
\vskip8pt
(iv) The ring $H^{0}(X,{\widetilde{\cD}}^{(m)}_{X,k})$ is a finitely generated $D^{(m)}({\mathbb G}(k))$-module and hence noetherian.
\end{prop}
{\it Proof. } The points (i)-(iii) follow exactly as in \cite[A.2.6.1]{HS13}. Statement (iv) is a special case of (iii) by \ref{prop-auxiliaryI}.\qed
\vskip8pt
Now let ${\widetilde{\sD}}^{(m)}_{{\mathfrak X},k}$ be the $p$-adic completion of ${\widetilde{\cD}}^{(m)}_{X,k}$, which we will always consider as a sheaf on the formal scheme ${\mathfrak X}$. We also put ${\widetilde{\sD}}^{(m)}_{{\mathfrak X},k,{\mathbb Q}} = {\widetilde{\sD}}^{(m)}_{{\mathfrak X},k} \otimes_{\mathbb Z} {\mathbb Q}$, and
\begin{numequation}\label{tsD_dagger}
{\widetilde{\sD}}^\dagger_{{\mathfrak X},k} = \varinjlim_m {\widetilde{\sD}}^{(m)}_{{\mathfrak X},k} \;,
\end{numequation}
and
\begin{numequation}\label{tsD_dagger_Q}
{\widetilde{\sD}}^\dagger_{{\mathfrak X},k,{\mathbb Q}} = {\widetilde{\sD}}^\dagger_{{\mathfrak X},k} \otimes_{\mathbb Z} {\mathbb Q} = \varinjlim_m {\widetilde{\sD}}^{(m)}_{{\mathfrak X},k,{\mathbb Q}} \;.
\end{numequation}
We denote by $\widehat{D}^{(m)}({\mathbb G}(k))_{{\mathbb Q},\theta_0}$ the quotient of $\widehat{D}^{(m)}({\mathbb G}(k))\otimes {\mathbb Q}$ modulo the ideal generated
by the center of the ring $U({\mathfrak g})$ \cite[A.2.1]{HS13}. This is the same central reduction considered in \cite{PSS4} for the group ${\rm GL}_2$.
\vskip8pt
In the proposition below, and in the remainder of this paper, certain rigid analytic `wide open' groups ${\mathbb G}(k)^\circ$ will be used repeatedly. To define them, consider first the formal completion ${\mathfrak G}(k)$ be of the group scheme ${\mathbb G}(k)$ along its special fiber, which is a formal group scheme (of topologically finite type) over ${\mathfrak S} = {\rm Spf}({\mathfrak o})$. Then let $\widehat{{\mathfrak G}}(k)^\circ$ be the completion of ${{\mathfrak G}}(k)$ along its unit section ${\mathfrak S} \rightarrow {{\mathfrak G}}(k)$, and denote by ${\mathbb G}(k)^\circ$ its associated rigid space.
\vskip8pt
\begin{prop}\label{global_sec_tsD} (i) There is a basis of the topology ${\mathfrak U}$ of $X$, consisting of open affine subsets, such that for any $U \in {\mathfrak U}$, the ring $H^0(U,{\widetilde{\sD}}^{(m)}_{{\mathfrak X},k})$ is noetherian.
\vskip8pt
(ii) The transition map ${\widetilde{\sD}}^{(m)}_{{\mathfrak X},k,{\mathbb Q}} \rightarrow {\widetilde{\sD}}^{(m+1)}_{{\mathfrak X},k,{\mathbb Q}}$ is flat.
\vskip8pt
(iii) The sheaf of rings ${\widetilde{\sD}}^\dagger_{{\mathfrak X},k,{\mathbb Q}}$ is coherent.
\vskip8pt
(iv) The homomorphism $Q^{(m)}_{X,k}$ induces an algebra isomorphism
$$\widehat{D}^{(m)}({\mathbb G}(k))_{{\mathbb Q},\theta_0}\stackrel{\simeq}{\longrightarrow} H^0({\mathfrak X},{\widetilde{\sD}}^{(m)}_{{\mathfrak X},k,{\mathbb Q}}).$$
\vskip8pt
(v) The ring $H^0({\mathfrak X},{\widetilde{\sD}}^\dagger_{{\mathfrak X},k,{\mathbb Q}})$ is canonically isomorphic to the coherent $L$-algebra \linebreak ${\mathcal D}^{\rm an}({\mathbb G}(k)^\circ)_{\theta_0}$, where ${\mathcal D}^{\rm an}({\mathbb G}(k)^\circ)$ is the analytic distribution algebra in the sense of Emerton, cf. \cite[ch. 5]{EmertonA}.
\end{prop}
{\it Proof. } (i) This follows from \cite[3.3.4(i)]{BerthelotDI} together with \ref{prop-auxiliaryI}.
\vskip8pt
(ii) This can be proved as in \cite[sec. 3.5]{BerthelotDI}.
\vskip8pt
(iii) Follows from (i) and (ii).
\vskip8pt
(iv) This follows from statement (iv) in \ref{prop-auxiliaryII} together with \cite[Lem. A.3]{HS13}.
\vskip8pt
(v) Follows from (iv) and the description of the analytic distribution algebra as given in \cite[ch. 5]{EmertonA}, compare also
\cite[Prop. 5.2.1]{HS13}. \qed
\vskip12pt
\section{Localization on ${\mathfrak X}$ via ${\widetilde{\sD}}^\dagger_{{\mathfrak X},k,{\mathbb Q}}$}\label{loc_n}
The general line of arguments follows fairly closely \cite{NootHuyghe09}.
As usual, ${\rm pr}: X \rightarrow X_0$ denotes an admissible blow-up of $X_0$.
The number $k\geq 0$ is fixed throughout this section and chosen large enough so that the sheaf of coherent rings
${\widetilde{\cD}}^{(m)}_{X,k}$ is defined for all $m$.
\vskip12pt
\subsection{Cohomology of coherent ${\widetilde{\cD}}^{(m)}_{X,k}$-modules}
\begin{lemma}\label{van_coh} Let ${\mathcal E}$ be an abelian sheaf on $X$. For all $i >\dim X$ one has $H^i(X,{\mathcal E}) =0$.
\end{lemma}
{\it Proof. } Since the space $X$ is noetherian
the result follows from Grothendieck's vanishing theorem \cite[Thm. 2.7]{HartshorneA}. \qed
\vskip8pt
\begin{prop}\label{vanishing_coh_gr_Dnk}
There is a natural number $r_0$ such that
for all $r \ge r_0$ and all $i \ge 1$ one has
\begin{numequation}\label{vanishing_coh_gr_Dnk_disp}
H^i\left(X,{\rm gr}\left({\widetilde{\cD}}_{X,k}^{(m)}\right)(r)\right) = 0 \;.
\end{numequation}
\end{prop}
{\it Proof. } Since ${\mathcal L}_X$ is very ample over ${\mathfrak o}$ by \ref{v_ample_sh_lemma},
the Serre theorems \cite[II.5.17/III.5.2]{HartshorneA} imply that there is a number $u_0$ such that for all $u\geq u_0$ the module ${\mathcal O}_X(u)$ is generated by global sections and has no higher cohomology. After this remark we prove the proposition along the lines of
\cite[Prop. 2.2.1]{NootHuyghe09}. There is an ${\mathcal O}_{X_0}$-linear surjection $({\mathcal O}_{X_0})^{\oplus a}\rightarrow{\mathcal T}_{X_0}$ for a suitable natural number $a$. Applying $({\rm pr})^*$ and multiplication by $\varpi^k$ gives an ${\mathcal O}_X$-linear surjection
$({\mathcal O}_X)^{\oplus a}\simeq \varpi^k ({\mathcal O}_X)^{\oplus a} \rightarrow {\widetilde{\cT}}_{X,k}$. By functoriality we get a surjective
morphism of algebras
$$ {\mathcal C}:={\rm Sym}(({\mathcal O}_{X})^{\oplus a})^{(m)}\longrightarrow {\rm Sym}({\widetilde{\cT}}_{X,k})^{(m)}.$$
The target of this map equals ${\rm gr}\left({\widetilde{\cD}}_{X,k}^{(m)}\right)$ according to \ref{graded_tcD}. It therefore suffices to prove the following: given a coherent ${\mathcal C}$-module ${\mathcal E}$, there is a number $r_0$
such that for all $r\geq r_0$ and $i\geq 1$, one has $H^{i}(X,{\mathcal E}(r))=0$. To start with, $X$ is noetherian, hence ${\mathcal E}$ is quasi-coherent and so equals the union over its ${\mathcal O}_X$-coherent submodules ${\mathcal E}_i$. Since ${\mathcal E}$ is ${\mathcal C}$-coherent and ${\mathcal C}$ has noetherian sections over open affines
\cite[1.3.6]{Huyghe97}, there is a ${\mathcal C}$-linear surjection ${\mathcal C}\otimes_{{\mathcal O}_{X}} {\mathcal E}_i\rightarrow {\mathcal E}$. Choose a number $s_0$ such that ${\mathcal E}_i(-s_0)$ is generated by global sections. We obtain a ${\mathcal O}_{X}$-linear surjection ${\mathcal O}_{X}(s_0)^{\oplus a_0}\rightarrow {\mathcal E}_i$ for a number $a_0$. This yields a ${\mathcal C}$-linear surjection
$${\mathcal C}_0:={\mathcal C}(s_0)^{\oplus a_0}\longrightarrow {\mathcal E}.$$
The ${\mathcal O}_X$-module ${\mathcal C}_0$ is graded and each homogeneous component equals a sum of copies of ${\mathcal O}_X(s_0)$. It follows that
$H^{i}(X,{\mathcal C}_0(r)=0$ for all $r\geq u_0-s_0$ and all $i\geq 1$. The rest of the argument proceeds now as in \cite[2.2.1]{NootHuyghe09}.
\qed
\vskip8pt
\begin{cor}\label{vanishing_coh_Dnk} Let $r_0$ be the number occuring in the preceding proposition. For all $r \ge r_0$ and all $i \ge 1$ one has
\begin{numequation}\label{vanishing_coh_Dnk_disp}
H^i\left(X,{\widetilde{\cD}}_{X,k}^{(m)}(r)\right) = 0 \;.
\end{numequation}
\end{cor}
{\it Proof. } For $d\geq 0$ we let ${\mathcal F}_d = {\widetilde{\cD}}_{X,k;d}^{(m)}$. We consider the exact sequence
\begin{numequation}\label{ex_fil_seq_1}
0 \rightarrow {\mathcal F}_{d-1} \rightarrow {\mathcal F}_d \rightarrow {\rm gr}_d\left({\widetilde{\cD}}_{X,k}^{(m)}\right) \rightarrow 0
\end{numequation}
\vskip8pt
(where ${\mathcal F}_{-1}:=0$) from which we deduce the exact sequence
\begin{numequation}\label{ex_fil_seq_2}
0 \rightarrow {\mathcal F}_{d-1}(r) \rightarrow {\mathcal F}_d(r) \rightarrow {\rm gr}_d\left({\widetilde{\cD}}_{X,k}^{(m)}\right)(r) \rightarrow 0
\end{numequation}
\vskip8pt
because tensoring with a line bundle is an exact functor. Since cohomology commutes with direct sums, we have for all $r\geq r_0$ and $i\geq 1$ that
$$H^{i}(X,{\rm gr}_d\left({\widetilde{\cD}}_{X,k}^{(m)}\right)(r))=0$$
\vskip8pt
according to the preceding proposition. Using the sequence \ref{ex_fil_seq_2} we can then deduce by induction on $d$ that for all $r\geq r_0$ and $i\geq 1$
$$H^i(X, {\mathcal F}_d(r)) = 0 \;.$$
\vskip8pt
Because cohomology commutes with inductive limits on a noetherian scheme we obtain the asserted vanishing result. \qed
\vskip8pt
\begin{prop}\label{surjection} Let ${\mathcal E}$ be a coherent ${\widetilde{\cD}}^{(m)}_{X,k}$-module.
\vskip8pt
(i) There is a number $r=r({\mathcal E}) \in {\mathbb Z}$ and $s \in {\mathbb Z}_{\ge 0}$ and an epimorphism of ${\widetilde{\cD}}^{(m)}_{X,k}$-modules
$$\Big({\widetilde{\cD}}^{(m)}_{X,k}(-r)\Big)^{\oplus s} \twoheadrightarrow {\mathcal E} \;.$$
\vskip8pt
(ii) There is $r_1({\mathcal E}) \in {\mathbb Z}$ such that for all $r \ge r_1({\mathcal E})$ and all $i >0$
$$H^i\Big(X, {\mathcal E}(r)\Big) = 0 \;.$$
\vskip8pt
\end{prop}
{\it Proof. } (i) As $X$ is a noetherian scheme, ${\mathcal E}$ is the inductive limit of its coherent subsheaves. There is thus a coherent ${\mathcal O}_{X}$-submodule ${\mathcal F} \subset {\mathcal E}$ which generates ${\mathcal E}$ as a ${\widetilde{\cD}}^{(m)}_{X,k}$-module, i.e., an epimorphism of sheaves
$${\widetilde{\cD}}^{(m)}_{X,k} \otimes_{{\mathcal O}_{X}} {\mathcal F} \stackrel{\alpha}{\longrightarrow} {\mathcal E} \;,$$
\vskip8pt
where ${\widetilde{\cD}}^{(m)}_{X,k}$ is considered with its right ${\mathcal O}_{X}$-module structure. Next, there is $r>0$ such that the sheaf
$${\mathcal F}(r) = {\mathcal F} \otimes_{{\mathcal O}_{X}} {\mathcal L}_X^{\otimes r}$$
\vskip8pt
is generated by its global sections. Hence there is $s > 0$ and an epimorphism ${\mathcal O}_{X}^{\oplus s} \twoheadrightarrow {\mathcal F}(r)$, and thus an epimorphism of ${\mathcal O}_{X}$-modules
$$\left({\mathcal O}_{X}(-r)\right)^{\oplus s} \twoheadrightarrow {\mathcal F} \;.$$
\vskip8pt
From this morphism we get an epimorphism of ${\widetilde{\cD}}^{(m)}_{X,k}$-modules
$$\left({\widetilde{\cD}}^{(m)}_{X,k}(-r)\right)^{\oplus s} = {\widetilde{\cD}}^{(m)}_{X,k} \otimes_{{\mathcal O}_{X_n}} \left({\mathcal O}_{X}(-r)\right)^{\oplus s} \twoheadrightarrow {\widetilde{\cD}}^{(m)}_{X,k} \otimes_{{\mathcal O}_{X}} {\mathcal F} \stackrel{\alpha}{\longrightarrow} {\mathcal E} \;.$$
\vskip8pt
(ii) Consider for $i\geq 1$ the following assertion $(a_i)$: for any coherent ${\widetilde{\cD}}^{(m)}_{X,k}$-module ${\mathcal E}$, there is a number $r_i({\mathcal E})$ such that for all $r\geq r_i({\mathcal E})$ and all $i\leq j$ one has $H^{j}(X,{\mathcal E}(r))=0$. For $i>\dim X$ the assertion holds, cf. \ref{van_coh}. Suppose the statement $(a_{i+1})$ holds.
Using (i) we find an epimorphism of ${\widetilde{\cD}}^{(m)}_{X,k}$-modules
$$\beta: {\mathcal C}_0:=\Big({\widetilde{\cD}}^{(m)}_{X,k}(s_0)\Big)^{\oplus s} \twoheadrightarrow {\mathcal E}$$
for numbers $s_0\in{\mathbb Z}$ and $s\geq 0$. By \ref{graded_tcD}, the kernel ${\mathcal R} = \ker(\beta)$ is a coherent ${\widetilde{\cD}}^{(m)}_{X,k}$-module.
Recall the number $r_0$ of the preceding corollary. For any $r\geq \max (r_0-s_0, r_{i+1}({\mathcal R}))$ we have the exact sequence
$$ 0=H^{i}(X,{\mathcal C}_0(r))\longrightarrow H^{i}(X,{\mathcal E}(r))\longrightarrow H^{i+1}(X, {\mathcal R}(r))=0$$
which shows $H^{i}(X,{\mathcal E}(r))=0$ for these $r$. So we may take as $r_i({\mathcal E})$ any of these $r$ which is larger than $r_{i+1}({\mathcal E})$ and obtain the statement $(a_i)$. In particular, $(a_1)$ holds which proves (ii).
\qed
\vskip8pt
\begin{prop}\label{finite_power}
\vskip8pt
\vskip8pt
(i) Fix $r \in {\mathbb Z}$. There is $c_2 = c_2(r) \in {\mathbb Z}_{\ge 0}$ such that for all $i>0$ the cohomology group $H^i(X,{\widetilde{\cD}}_{X,k}^{(m)}(r))$ is annihilated by $p^{c_2}$.
\vskip8pt
(ii) Let ${\mathcal E}$ be a coherent ${\widetilde{\cD}}_{X,k}^{(m)}$-module. There is $c_3 = c_3({\mathcal E}) \in {\mathbb Z}_{\ge 0}$ such that for all $i>0$ the cohomology group $H^i(X,{\mathcal E})$ is annihilated by $p^{c_3}$.
\end{prop}
{\it Proof. } (i) Since the blow-up morphism ${\rm pr}: X \rightarrow X_0$ becomes an isomorphism over $X_0\times_{{\mathfrak o}} L$ any coherent module over ${\widetilde{\cD}}^{(m)}_{X,k}\otimes {\mathbb Q}$ induces a coherent module over the sheaf of usual differential operators on $X_0\times_{\mathfrak o} L$. By \cite{BB81} we conclude that the global section functor on $X$ is exact for coherent ${\widetilde{\cD}}^{(m)}_{X,k}\otimes{\mathbb Q}$-modules. In particular, the cohomology group $H^i(X,{\widetilde{\cD}}_{X,k}^{(m)}(r))$ is $p$-torsion. To see that the torsion is bounded, we
deduce from \ref{prop-auxiliaryI} that ${\widetilde{\cD}}_{X,k}^{(m)}(r)$ is a coherent module over ${\mathcal A}^{(m)}_{X,k}$. According to \ref{prop-auxiliaryII}, $H^i(X,{\widetilde{\cD}}_{X,k}^{(m)}(r))$ is therefore finitely generated over $D^{(m)}({\mathbb G}(k))$. This implies the claim.
\vskip8pt
(ii) We consider for any $i\geq 1$ the following assertion $(a_i)$: for any coherent ${\widetilde{\cD}}_{X,k}^{(m)}$-module ${\mathcal E}$, there is a number $r_i({\mathcal E})$ such that the groups $H^{j}(X,{\mathcal E}), i\leq j$ are
all annihilated by $p^{r_i({\mathcal E})}$. For $i>\dim X$ the assertion holds, cf. \ref{van_coh}. Let us assume $(a_{i+1})$ holds
and consider an arbitrary coherent ${\widetilde{\cD}}_{X,k}^{(m)}$-module ${\mathcal E}$.
Acccording to \ref{surjection} we have a ${\widetilde{\cD}}_{X,k}^{(m)}$-linear surjection
$$ {\mathcal E}_0:={\widetilde{\cD}}_{X,k}^{(m)}(r)^{\oplus s}\longrightarrow {\mathcal E}$$
for numbers $r\in{\mathbb Z}$ and $s\geq 0$.
Let ${\mathcal E}'$ be the kernel. We have an exact sequence
$$ H^{i}(X,{\mathcal E}_0)\stackrel{\iota}{\rightarrow}H^{i}(X,{\mathcal E})\stackrel{\delta}{\rightarrow}H^{i+1}(X,{\mathcal E}').$$
Then $p^{c_2(r)}$ annihilates the image of $\iota$ according to (i) and $p^{r_{i+1}({\mathcal E}')}$ annihilates the image of $\delta$ according to $(a_{i+1})$. So we may take as $r_i({\mathcal E})$ any number greater than the maximum of $r_{i+1}({\mathcal E})$ and $c_2(r)+r_{i+1}({\mathcal E}')$ and obtain the statement $(a_i)$. In particuar, $(a_1)$ holds which proves (ii). \qed
\vskip8pt
\subsection{Cohomology of coherent ${\widetilde{\sD}}^{(m)}_{{\mathfrak X}, k,{\mathbb Q}}$-modules}\label{coh_coh_tsD_mod}
We denote by $X_{j}$ the reduction of $X$ modulo $p^{j+1}$.
\vskip8pt
\begin{prop}\label{completion} Let ${\mathcal E}$ be a coherent ${\widetilde{\cD}}^{(m)}_{X,k}$-module on $X$ and $\widehat{{\mathcal E}} = \varprojlim_j {\mathcal E}/p^{j+1}{\mathcal E}$ its $p$-adic completion, which we consider as a sheaf on ${\mathfrak X}$.
\vskip8pt
(i) For all $i \ge 0$ one has $H^i({\mathfrak X},\widehat{{\mathcal E}}) = \varprojlim_j H^i\left(X_{j},{\mathcal E}/p^{j+1}{\mathcal E} \right)$.
\vskip8pt
(ii) For all $i>0$ one has $H^i({\mathfrak X},\widehat{{\mathcal E}}) = H^i(X,{\mathcal E})$.
\vskip8pt
(iii) $H^0({\mathfrak X},\widehat{{\mathcal E}}) = \varprojlim_j H^0(X,{\mathcal E})/p^{j+1}H^0(X,{\mathcal E})$.
\end{prop}
{\it Proof. } Put ${\mathcal E}_j = {\mathcal E}/p^{j+1}{\mathcal E}$. Let ${\mathcal E}_t$ be the subsheaf defined by
$${\mathcal E}_t(U) = {\mathcal E}(U)_{\rm tor} \;,$$
\vskip8pt
where the right hand side denotes the group of torsion elements in ${\mathcal E}(U)$. This is indeed a sheaf (and not only a presheaf) because $X$ is a noetherian space. Furthermore, ${\mathcal E}_t$ is a ${\widetilde{\cD}}^{(m)}_{X,k}$-submodule of ${\mathcal E}$. Because the sheaf ${\widetilde{\cD}}^{(m)}_{X,k}$ has noetherian rings of sections over open affine subsets of $X$, cf. \ref{graded_tcD}, the submodule ${\mathcal E}_t$ is a coherent ${\widetilde{\cD}}^{(m)}_{X,k}$-module. ${\mathcal E}_t$ is thus generated by a coherent ${\mathcal O}_{X}$-submodule ${\mathcal F}$ of ${\mathcal E}_t$. The submodule ${\mathcal F}$ is annihilated by a fixed power $p^c$ of $p$, and so is ${\mathcal E}_t$. Put ${\mathcal G} = {\mathcal E}/{\mathcal E}_t$, which is again a coherent ${\widetilde{\cD}}^{(m)}_{X,k}$-module. Using \ref{finite_power}, we can then assume, after possibly replacing $c$ by a larger number, that
\vskip8pt
$$\begin{array}{cl}
(a) & p^c{\mathcal E}_t = 0 \;,\\
(b) & \mbox{for all } i>0: p^cH^i(X,{\mathcal E}) = 0 \;,\\
(c) & \mbox{for all } i>0: p^cH^i(X,{\mathcal G}) = 0 \;.\\
\end{array}$$
\vskip8pt
From here on the proof of the proposition is exactly as in \cite[4.2.1]{PSS4}. \qed
\vskip8pt
\begin{prop}\label{surj_compl} Let ${\mathscr E}$ be a coherent ${\widetilde{\sD}}^{(m)}_{{\mathfrak X},k}$-module.
\vskip8pt
(i) There is $r_1({\mathscr E}) \in {\mathbb Z}$ such that for all $r \ge r_1({\mathscr E})$ there is $s \in {\mathbb Z}_{\ge 0}$ and an epimorphism of ${\widetilde{\sD}}^{(m)}_{{\mathfrak X},k}$-modules
$$\Big({\widetilde{\sD}}^{(m)}_{{\mathfrak X},k}(-r)\Big)^{\oplus s} \twoheadrightarrow {\mathscr E} \;.$$
\vskip8pt
(ii) There is $r_2({\mathscr E}) \in {\mathbb Z}$ such that for all $r \ge r_2({\mathscr E})$ and all $i >0$
$$H^i\Big({\mathfrak X}, {\mathscr E}(r)\Big) = 0 \;.$$
\vskip8pt
\end{prop}
{\it Proof. } (i) Because ${\mathscr E}$ is a coherent ${\widetilde{\sD}}^{(m)}_{{\mathfrak X},k}$-module, and because $H^0(U,{\widetilde{\sD}}^{(m)}_{{\mathfrak X},k})$ is a noetherian ring for all open affine subsets $U \subset {\mathfrak X}$, cf. \ref{global_sec_tsD}, the torsion submodule ${\mathscr E}_t \subset {\mathscr E}$ is again a coherent ${\widetilde{\sD}}^{(m)}_{{\mathfrak X},k}$-module.
As ${\mathfrak X}$ is quasi-compact, there is $c \in {\mathbb Z}_{\ge 0}$ such that $p^c {\mathscr E}_t = 0$.
Put ${\mathscr G} = {\mathscr E}/{\mathscr E}_t$ and ${\mathscr G}_0 = {\mathscr G}/p{\mathscr G}$. For $j \ge c$ one has an exact sequence
$$0 \rightarrow {\mathscr G}_0 \stackrel{p^{j+1}}{\longrightarrow} {\mathscr E}_{j+1} \rightarrow {\mathscr E}_j \rightarrow 0 \;.$$
\vskip8pt
We note that the sheaf ${\mathscr G}_0$ is a coherent module over
${\widetilde{\sD}}^{(m)}_{{\mathfrak X},k}/p{\widetilde{\sD}}^{(m)}_{{\mathfrak X},k}$. We view ${\mathfrak X}$ as a closed subset of $X$ and denote the closed embedding temporarily by $i$. Because the canonical map of sheaves of rings
\begin{numequation}\label{iso_uncompl_compl}
{\widetilde{\cD}}^{(m)}_{X,k}/p{\widetilde{\cD}}^{(m)}_{X,k} \stackrel{\simeq}{\longrightarrow} i_*\Big({\widetilde{\sD}}^{(m)}_{{\mathfrak X},k}/p{\widetilde{\sD}}^{(m)}_{{\mathfrak X},k}\Big)
\end{numequation}
is an isomorphism, $i_*{\mathscr G}_0$ can be considered a coherent ${\widetilde{\cD}}^{(m)}_{X,k}$-module via this isomorphism. Hence we can apply \ref{surjection} to $i_*{\mathscr G}_0$ and deduce that there is $r_2({\mathscr G}_0)$ such that for all $r \ge r_2({\mathscr G}_0)$ one has $$ H^1({\mathfrak X},{\mathscr G}_0(r))= H^1(X,i_*{\mathscr G}_0(r))=0.$$
The canonical maps
\begin{numequation}\label{surj_H0}
H^0({\mathfrak X},{\mathscr E}_{j+1}(r)) \longrightarrow H^0({\mathfrak X},{\mathscr E}_j(r))
\end{numequation}
are thus surjective for $r \ge r_2({\mathscr G}_0)$ and $j \ge c$. Similarly, ${\mathscr E}_c$ is a coherent module over ${\widetilde{\cD}}^{(m)}_{X,k}/p^c{\widetilde{\cD}}^{(m)}_{X,k}$-module, in particular a coherent ${\widetilde{\cD}}^{(m)}_{X,k}$-module. By \ref{surjection} there is $r_1({\mathscr E}_c)$ such that for every $r \ge r_1({\mathscr E}_c)$ there is $s \in {\mathbb Z}_{\ge 0}$ and a surjection
$$\lambda: \Big({\widetilde{\cD}}^{(m)}_{X,k}/p^c{\widetilde{\cD}}^{(m)}_{X,k}\Big)^{\oplus s} \twoheadrightarrow {\mathscr E}_c(r) \;.$$
\vskip8pt
Let $r_1({\mathscr E}) = \max\{r_2({\mathscr G}_0),r_1({\mathscr E}_c)\}$, and assume from now on that $r \ge r_1({\mathscr E})$. Let $e_1, \ldots, e_s$ be the standard basis of the domain of $\lambda$, and use \ref{surj_H0} to lift each $\lambda(e_t)$, $1 \le t \le s$, to an element of
$$\varprojlim_j H^0({\mathfrak X},{\mathscr E}_j(r)) \simeq H^0({\mathfrak X},\widehat{{\mathscr E}(r)}) \;,$$
\vskip8pt
by \ref{completion} (i). But $\widehat{{\mathscr E}(r)} = \widehat{{\mathscr E}}(r)$, and $\widehat{{\mathscr E}} = {\mathscr E}$, as follows from \cite[3.2.3 (v)]{BerthelotDI}. This defines a morphism
$$\Big({\widetilde{\sD}}^{(m)}_{{\mathfrak X},k}\Big)^{\oplus s} \longrightarrow {\mathscr E}(r)$$
\vskip8pt
which is surjective because, modulo $p^c$, it is a surjective morphism of sheaves coming from coherent ${\widetilde{\sD}}^{(m)}_{X,k}$-modules by reduction modulo $p^c$, cf. \cite[3.2.2 (ii)]{BerthelotDI}.
\vskip8pt
(ii) We deduce from \ref{vanishing_coh_Dnk} and \ref{completion} that for all $i>0$
$$H^i\Big({\mathfrak X},{\widetilde{\sD}}^{(m)}_{{\mathfrak X},k}(r)\Big) = 0 \;,$$
\vskip8pt
whenever $r \ge r_0$, where $r_0$ is as in \ref{v_ample_sh_lemma}. Since the sheaf ${\widetilde{\sD}}^{(m)}_{{\mathfrak X},k}$ is coherent, cf. \ref{global_sec_tsD}, and ${\mathfrak X}$ is a noetherian space of finite dimension, the statement in (ii) can now be deduced by descending induction on $i$
exactly as in the proof of part (ii) of \ref{surjection}. \qed
\vskip8pt
\begin{prop}\label{compl_finite_power_torsion} Let ${\mathscr E}$ be a coherent ${\widetilde{\sD}}^{(m)}_{{\mathfrak X},k}$-module.
\vskip8pt
(i) There is $c=c({\mathscr E}) \in {\mathbb Z}_{\ge 0}$ such that for all $i>0$
the cohomology group $H^i({\mathfrak X},{\mathscr E})$ is annihilated by $p^c$.
\vskip8pt
(ii) $H^0({\mathfrak X},{\mathscr E}) = \varprojlim_j H^0({\mathfrak X},{\mathscr E})/p^jH^0({\mathfrak X},{\mathscr E})$.
\end{prop}
{\it Proof. } (i) Let $r\in{\mathbb Z}$. By \ref{completion} we have for $i>0$ that
$$H^{i}({\mathfrak X},{\widetilde{\sD}}^{(m)}_{{\mathfrak X},k}(-r)) = H^{i}(X,{\widetilde{\cD}}^{(m)}_{X,k}(-r)) \;,$$
\vskip8pt
and this is annihilated by a finite power of $p$, by \ref{finite_power}. The proof now proceeds by descending induction exactly as in the proof of
part (ii) of \ref{finite_power}.
\vskip8pt
(ii) Let ${\mathscr E}_t \subset {\mathscr E}$ be the subsheaf of torsion elements and ${\mathscr G} = {\mathscr E}/{\mathscr E}_t$. Then the discussion in the beginning of the proof of \ref{completion} shows that there is $c \in {\mathbb Z}_{\ge 0}$ such that $p^c{\mathscr E}_t = 0$. Part (i) gives that $p^cH^1({\mathfrak X},{\mathscr E}) = p^cH^1({\mathfrak X},{\mathscr G}) = 0$, after possibly increasing $c$. Now we can apply the same reasoning as in the proof of \ref{completion} (iii) to conclude that assertion (ii) is true. \qed
\vskip8pt
\vskip8pt
\begin{para} Let ${\rm Coh}({\widetilde{\sD}}^{(m)}_{{\mathfrak X},k})$ (resp. ${\rm Coh}({\widetilde{\sD}}^{(m)}_{{\mathfrak X},k,{\mathbb Q}})$) be the category of coherent ${\widetilde{\sD}}^{(m)}_{{\mathfrak X},k}$-modules (resp. ${\widetilde{\sD}}^{(m)}_{{\mathfrak X},k,{\mathbb Q}}$-modules). Let ${\rm Coh}({\widetilde{\sD}}^{(m)}_{{\mathfrak X},k})_{\mathbb Q}$ be the category of coherent ${\widetilde{\sD}}^{(m)}_{{\mathfrak X},k}$-modules up to isogeny. We recall that this means that ${\rm Coh}({\widetilde{\sD}}^{(m)}_{{\mathfrak X},k})_{\mathbb Q}$ has the same class of objects as ${\rm Coh}({\widetilde{\sD}}^{(m)}_{{\mathfrak X},k})$, and for any two objects ${\mathcal M}$ and ${\mathcal N}$ one has
$$Hom_{{\rm Coh}({\widetilde{\sD}}^{(m)}_{{\mathfrak X},k})_{\mathbb Q}}({\mathcal M},{\mathcal N}) = Hom_{{\rm Coh}({\widetilde{\sD}}^{(m)}_{{\mathfrak X},k})}({\mathcal M},{\mathcal N}) \otimes_{\mathbb Z} {\mathbb Q} \;.$$
\vskip8pt
\end{para}
\vskip8pt
\begin{prop}\label{integral_models} (i) The functor ${\mathcal M} \mapsto {\mathcal M}_{\mathbb Q} = {\mathcal M} \otimes_{\mathbb Z} {\mathbb Q}$ induces an equivalence between ${\rm Coh}({\widetilde{\sD}}^{(m)}_{{\mathfrak X},k})_{\mathbb Q}$ and ${\rm Coh}({\widetilde{\sD}}^{(m)}_{{\mathfrak X},k,{\mathbb Q}})$.
\vskip8pt
(ii) For every coherent ${\widetilde{\sD}}^\dagger_{{\mathfrak X},k,{\mathbb Q}}$-module ${\mathscr M}$ there is $m \ge 0$ and a coherent ${\widetilde{\sD}}^{(m)}_{{\mathfrak X},k,{\mathbb Q}}$-module ${\mathscr M}_m$ and an isomorphism of ${\widetilde{\sD}}^\dagger_{{\mathfrak X},k,{\mathbb Q}}$-modules
$${\varepsilon}: {\widetilde{\sD}}^\dagger_{{\mathfrak X},k,{\mathbb Q}} \otimes_{{\widetilde{\sD}}^{(m)}_{{\mathfrak X},k,{\mathbb Q}}} {\mathscr M}_m \stackrel{\simeq}{\longrightarrow} {\mathscr M} \;.$$
\vskip8pt
If $(m', {\mathscr M}_{m'},{\varepsilon}')$ is another such triple, then there is $\ell \ge \max\{m,m'\}$ and an isomorphism of ${\widetilde{\sD}}^{(\ell)}_{{\mathfrak X},k,{\mathbb Q}}$-modules
$${\varepsilon}_\ell: {\widetilde{\sD}}^{(\ell)}_{{\mathfrak X},k,{\mathbb Q}} \otimes_{{\widetilde{\cD}}^{(m)}_{{\mathfrak X},k,{\mathbb Q}}} {\mathscr M}_m \stackrel{\simeq}{\longrightarrow} {\widetilde{\sD}}^{(\ell)}_{{\mathfrak X},k,{\mathbb Q}} \otimes_{{\widetilde{\cD}}^{(m')}_{{\mathfrak X},k,{\mathbb Q}}} {\mathscr M}_{m'}$$
\vskip8pt
such that ${\varepsilon}' \circ \Big({\rm id}_{{\widetilde{\cD}}^\dagger_{{\mathfrak X},k,{\mathbb Q}}} \otimes {\varepsilon}_\ell\Big) = {\varepsilon}$.
\end{prop}
{\it Proof. } (i) This is \cite[3.4.5]{BerthelotDI}. Note that the sheaf ${\widetilde{\sD}}^{(m)}_{{\mathfrak X},k}$ satisfies the conditions in \cite[3.4.1]{BerthelotDI}, by \ref{global_sec_tsD}. We point out that the formal scheme ${\mathcal X}$ in \cite[sec. 3.4]{BerthelotDI} is not supposed to be smooth over a discrete valuation ring, but only locally noetherian, cf. \cite[sec. 3.3]{BerthelotDI}.
\vskip8pt
(ii) This is \cite[3.6.2]{BerthelotDI}. In this reference the formal scheme is supposed to be noetherian and quasi-separated, but not necessarily smooth over a discrete valuation ring. \qed
\vskip8pt
\begin{thm}\label{acycl_tsD} Let
${\mathscr E}$ be a coherent ${\widetilde{\sD}}^{(m)}_{{\mathfrak X},k,{\mathbb Q}}$-module (resp. ${\widetilde{\sD}}^\dagger_{{\mathfrak X},k,{\mathbb Q}}$-module).
\vskip8pt
(i) There is $r({\mathscr E}) \in {\mathbb Z}$ such that for all $r \ge r({\mathscr E})$ there is $s \in {\mathbb Z}_{\ge 0}$ and an epimorphism of ${\widetilde{\sD}}^{(m)}_{{\mathfrak X},k,{\mathbb Q}}$-modules (resp. ${\widetilde{\sD}}^\dagger_{{\mathfrak X},k,{\mathbb Q}}$-modules)
$$\Big({\widetilde{\sD}}^{(m)}_{{\mathfrak X},k,{\mathbb Q}}(-r)\Big)^{\oplus s} \twoheadrightarrow {\mathscr E} \; \hskip16pt (\; \mbox{resp.} \; \Big({\widetilde{\sD}}^\dagger_{{\mathfrak X},k,{\mathbb Q}}(-r)\Big)^{\oplus s} \twoheadrightarrow {\mathscr E} \;) \;.
$$
\vskip8pt
(ii) For all $i >0$ one has $H^i({\mathfrak X}, {\mathscr E}) = 0$.
\vskip8pt
\end{thm}
{\it Proof. } (a) We first show both assertions (i) and (ii) for a coherent ${\widetilde{\sD}}^{(m)}_{{\mathfrak X},k,{\mathbb Q}}$-module ${\mathscr E}$. By \ref{integral_models} (i) there is a coherent ${\widetilde{\sD}}^{(m)}_{{\mathfrak X},k}$-module ${\mathscr F}$ such that ${\mathscr F} \otimes_{\mathbb Z} {\mathbb Q} = {\mathscr E}$. We use \ref{surj_compl} to find for every $r \ge r_1({\mathscr F})$ a surjection
$$\Big({\widetilde{\sD}}^{(m)}_{{\mathfrak X},k}(-r)\Big)^{\oplus s} \twoheadrightarrow {\mathscr F} \;,$$
\vskip8pt
for some $s$ (depending on $r$). Tensoring with ${\mathbb Q}$ gives then the desired surjection onto ${\mathscr E}$. Hence assertion (i). Furthermore, for $i>0$
$$H^i({\mathfrak X},{\mathscr E}) = H^i({\mathfrak X},{\mathscr F}) \otimes_{\mathbb Z} {\mathbb Q} = 0 \;,$$
\vskip8pt
by \ref{compl_finite_power_torsion}, and this proves (ii).
\vskip8pt
(b) Now suppose ${\mathscr E}$ is a coherent ${\widetilde{\cD}}^\dagger_{{\mathfrak X},k,{\mathbb Q}}$-module.
By \ref{integral_models} (ii) there is $m \ge 0$ and a coherent module ${\mathscr E}_m$ over ${\widetilde{\sD}}^{(m)}_{{\mathfrak X},k,{\mathbb Q}}$ and an isomorphism of ${\widetilde{\sD}}^\dagger_{{\mathfrak X},k,{\mathbb Q}}$-modules
$${\widetilde{\sD}}^\dagger_{{\mathfrak X},k,{\mathbb Q}} \otimes_{{\widetilde{\sD}}^{(m)}_{{\mathfrak X},k,{\mathbb Q}}} {\mathscr E}_m \stackrel{\simeq}{\longrightarrow} {\mathscr E} \;.$$
\vskip8pt
Now use what we have just shown for ${\mathscr E}_m$ in (a) and get the sought for surjection after tensoring with ${\widetilde{\sD}}^\dagger_{{\mathfrak X},k,{\mathbb Q}}$. This proves the first assertion. We have
$${\mathscr E} = {\widetilde{\sD}}^\dagger_{{\mathfrak X},k,{\mathbb Q}} \otimes_{{\widetilde{\sD}}^{(m)}_{{\mathfrak X},k,{\mathbb Q}}} {\mathscr E}_m = \varinjlim_{\ell \ge m} {\widetilde{\sD}}^{(\ell)}_{{\mathfrak X},k,{\mathbb Q}} \otimes_{{\widetilde{\sD}}^{(m)}_{{\mathfrak X},k,{\mathbb Q}}} {\mathscr E}_m = \varinjlim_{\ell \ge m} {\mathscr E}_\ell$$
\vskip8pt
where ${\mathscr E}_\ell = {\widetilde{\sD}}^{(\ell)}_{{\mathfrak X},k,{\mathbb Q}} \otimes_{{\widetilde{\sD}}^{(m)}_{{\mathfrak X},k,{\mathbb Q}}} {\mathscr E}_m$ is a coherent ${\widetilde{\sD}}^{(\ell)}_{{\mathfrak X},k,{\mathbb Q}}$-module. Then we have for $i>0$
$$H^i({\mathfrak X},{\mathscr E}) = \varinjlim_{\ell \ge m} H^i({\mathfrak X},{\mathscr E}_\ell) = 0 \;,$$
\vskip8pt
by part (a). And this proves assertion (ii). \qed
\vskip12pt
\subsection{${\mathfrak X}$ is ${\widetilde{\sD}}^{(m)}_{{\mathfrak X},k,{\mathbb Q}}$-affine and ${\widetilde{\sD}}^\dagger_{{\mathfrak X},k,{\mathbb Q}}$-affine}
\begin{prop}\label{prop-genglobal} (i) Let ${\mathscr E}$ be a coherent ${\widetilde{\sD}}^{(m)}_{{\mathfrak X},k,{\mathbb Q}}$-module. Then ${\mathscr E}$ is generated by its global sections as ${\widetilde{\sD}}^{(m)}_{{\mathfrak X},k,{\mathbb Q}}$-module. Furthermore, ${\mathscr E}$ has a resolution by finite free ${\widetilde{\sD}}^{(m)}_{{\mathfrak X},k,{\mathbb Q}}$-modules.
\vskip8pt
(ii) Let ${\mathscr E}$ be a coherent ${\widetilde{\sD}}^\dagger_{{\mathfrak X},k,{\mathbb Q}}$-module. Then ${\mathscr E}$ is generated by its global sections as ${\widetilde{\sD}}^\dagger_{{\mathfrak X},k,{\mathbb Q}}$-module. $H^0({\mathfrak X},{\mathscr E})$ is a $H^0({\mathfrak X},{\widetilde{\sD}}^\dagger_{{\mathfrak X},k,{\mathbb Q}})$-module of finite presentation.
Furthermore, ${\mathscr E}$ has a resolution by finite free ${\widetilde{\sD}}^{\dagger}_{{\mathfrak X},k,{\mathbb Q}}$-modules.
\end{prop}
{\it Proof. } (i) Using \ref{acycl_tsD} it remains to see that any ${\widetilde{\sD}}^{(m)}_{{\mathfrak X},k,{\mathbb Q}}$-module of type ${\widetilde{\sD}}^{(m)}_{{\mathfrak X},k,{\mathbb Q}}(-r)$ admits a linear surjection
$({\widetilde{\sD}}^{(m)}_{{\mathfrak X},k,{\mathbb Q}})^{\oplus s}\rightarrow{\widetilde{\sD}}^{(m)}_{{\mathfrak X},k,{\mathbb Q}}(-r)$ for suitable $s\geq 0$. We argue as in \cite[5.1]{Huyghe97}. Let $M:=H^0(X,{\widetilde{\cD}}^{(m)}_{X,k}(-r))$, a finitely generated $D^{(m)}({\mathbb G}(k))$-module by \ref{prop-auxiliaryII}. Consider the linear map of ${\widetilde{\cD}}^{(m)}_{X,k}$-modules equal to the composite
$$ {\widetilde{\cD}}^{(m)}_{X,k}\otimes_{D^{(m)}({\mathbb G}(k))} M \rightarrow {\widetilde{\cD}}^{(m)}_{X,k}\otimes_{H^0(X,{\widetilde{\cD}}^{(m)}_{X,k})} M\rightarrow {\widetilde{\cD}}^{(m)}_{X,k}(-r)$$
where the first map is the surjection induced by the map $Q^{(m)}_{X,k}$ appearing in \ref{global_sections_tcD}.
Let ${\mathcal E}$ be the cokernel of the composite map. Since $D^{(m)}({\mathbb G}(k))$ is noetherian, the source of the map is coherent and hence ${\mathcal E}$ is coherent. Moreover, ${\mathcal E}\otimes{\mathbb Q}=0$ since ${\widetilde{\cD}}^{(m)}_{X,k}(-r)\otimes{\mathbb Q}$ is generated by global sections \cite{BB81}. All in all, there is $i$ with $p^{i}{\mathcal E}=0$. Now choose a linear surjection $(D^{(m)}({\mathbb G}(k)))^{\oplus s}\rightarrow M$. We obtain the exact sequence of coherent modules
$$ ({\widetilde{\cD}}^{(m)}_{X,k})^{\oplus s}\rightarrow {\widetilde{\cD}}^{(m)}_{X,k}(-r)\rightarrow {\mathcal E}\rightarrow 0.$$ Passing to $p$-adic completions (which is exact in our situation \cite[3.2]{BerthelotDI}) and inverting $p$ yields the linear surjection $$({\widetilde{\sD}}^{(m)}_{{\mathfrak X},k,{\mathbb Q}})^{\oplus s}\rightarrow {\widetilde{\sD}}^{(m)}_{{\mathfrak X},k,{\mathbb Q}}(-r).$$ This shows (i).
\vskip8pt
(ii) This follows from (i) exactly as in \cite{Huyghe97}. \qed
\begin{para} {\it The functors ${\mathscr Loc}^{(m)}_{{\mathfrak X},k}$ and ${\mathscr Loc}^\dagger_{{\mathfrak X},k}$.} Let $E$ be a finitely generated $H^0({\mathfrak X},{\widetilde{\sD}}^{(m)}_{{\mathfrak X},k,{\mathbb Q}})$-module (resp. a finitely presented $H^0({\mathfrak X},{\widetilde{\sD}}^\dagger_{{\mathfrak X},k,{\mathbb Q}})$-module). Then we let ${\mathscr Loc}^{(m)}_{{\mathfrak X},k}(E)$ (resp. ${\mathscr Loc}^\dagger_{{\mathfrak X},k}(E)$) be the sheaf on ${\mathfrak X}$ associated to the presheaf
$$U \mapsto {\widetilde{\sD}}^{(m)}_{{\mathfrak X},k,{\mathbb Q}}(U) \otimes_{H^0({\mathfrak X},{\widetilde{\sD}}^{(m)}_{n,k,{\mathbb Q}})} E \hskip16pt ({\rm resp.} \;\; U \mapsto {\widetilde{\sD}}^\dagger_{{\mathfrak X},k,{\mathbb Q}}(U) \otimes_{H^0({\mathfrak X},{\widetilde{\sD}}^\dagger_{{\mathfrak X},k,{\mathbb Q}})} E \;) \;.$$
\vskip8pt
It is obvious that ${\mathscr Loc}^{(m)}_{{\mathfrak X},k}$ (resp. ${\mathscr Loc}^\dagger_{{\mathfrak X},k}$) is a functor from the category of finitely generated $H^0({\mathfrak X},{\widetilde{\sD}}^{(m)}_{{\mathfrak X},k,{\mathbb Q}})$-modules (resp. finitely presented $H^0({\mathfrak X},{\widetilde{\sD}}^\dagger_{{\mathfrak X},k,{\mathbb Q}})$-modules) to the category of sheaves of modules over ${\widetilde{\sD}}^{(m)}_{{\mathfrak X},k,{\mathbb Q}}$ (resp. ${\widetilde{\sD}}^\dagger_{{\mathfrak X},k,{\mathbb Q}}$).
\end{para}
\vskip8pt
\begin{thm}\label{thm-equivalence} (i) The functors ${\mathscr Loc}^{(m)}_{{\mathfrak X},k}$ and $H^0$ (resp. ${\mathscr Loc}^\dagger_{{\mathfrak X},k}$ and $H^0$) are quasi-inverse equivalences between the categories of finitely generated $H^0({\mathfrak X},{\widetilde{\sD}}^{(m)}_{{\mathfrak X},k,{\mathbb Q}})$-modules and coherent ${\widetilde{\sD}}^{(m)}_{{\mathfrak X},k,{\mathbb Q}}$-modules (resp. finitely presented $H^0({\mathfrak X},{\widetilde{\sD}}^\dagger_{{\mathfrak X},k,{\mathbb Q}})$-modules and coherent ${\widetilde{\sD}}^\dagger_{{\mathfrak X},k,{\mathbb Q}}$-modules).
\vskip8pt
(ii) The functor ${\mathscr Loc}^{(m)}_{{\mathfrak X},k}$ (resp. ${\mathscr Loc}^\dagger_{{\mathfrak X},k}$) is an exact functor.
\end{thm}
{\it Proof. } The proofs of \cite[5.2.1, 5.2.3]{Huyghe97} for the first and the second assertion, respectively, carry over word for word. \qed
\vskip12pt
\section{Localization of representations of ${\mathbb G}(L)$}\label{loc}
\subsection{Modules over ${\mathcal D}^{\rm an}({\mathbb G}(k)^\circ)_{\theta_0}$}
Let as before ${\mathfrak X}$ be the $p$-adic completion of an admissible blow up $X$ of $X_0$.
We recall that the algebra $H^0({\mathfrak X},{\widetilde{\sD}}^\dagger_{{\mathfrak X},k,{\mathbb Q}})$ is canonically isomorphic to the coherent $L$-algebra ${\mathcal D}^{\rm an}({\mathbb G}(k)^\circ)_{\theta_0}$, cf. \ref{global_sec_tsD}. According to \cite[Lem. 5.2.1]{PSS4} the
algebras ${\mathcal D}^{\rm an}({\mathbb G}(k)^\circ)$ and ${\mathcal D}^{\rm an}({\mathbb G}(k)^\circ)_{\theta_0}$ are compact type algebras with noetherian defining Banach algebras in the sense of loc.cit.
\vskip8pt
In the following we extend some notions appearing in \cite[5.2]{PSS4} to the more general situation considered here. We consider the locally $L$-analytic compact group $G_0={\rm {\mathbb G}}_0({\mathfrak o})$ with its series of congruence subgroups $G_{k+1}={\mathbb G}(k)^\circ(L)$. The group $G_0$ acts by translations on the space $C^{\rm cts}(G_0,K)$ of continuous $K$-valued functions. Following \cite[(5.3)]{EmertonA} let $D(\bbG(k)^\circ,\GN)$ be the strong dual of the space of ${\mathbb G}(n)^\circ$-analytic vectors
$$D(\bbG(k)^\circ,\GN):= (C^{\rm cts}(G_0,K)_{{\mathbb G}(k)^\circ-\rm an})'_b \;.$$
\vskip8pt
It is a locally convex topological $L$-algebra naturally isomorphic to the crossed product of the ring ${\mathcal D}^{\rm an}({\mathbb G}(k)^\circ)$ with the finite group $G_0/G_{k+1}$. In particular,
\begin{numequation}\label{equ-finitefree}D(\bbG(k)^\circ,\GN)=\oplus_{g\in G_0/G_{k+1}} {\mathcal D}^{\rm an}({\mathbb G}(k)^\circ)*\delta_g\end{numequation}
is a finitely generated free topological module over ${\mathcal D}^{\rm an}({\mathbb G}(k)^\circ)$. Denoting by $C^{\rm la}(G_0,K)$ the space of $K$-valued locally analytic functions and dualizing the isomorphism
$$ \varinjlim_k C^{\rm cts}(G_0,K)_{{\mathbb G}(k)^\circ-\rm an}\stackrel{\simeq}{\longrightarrow} C^{\rm la}(G_0,K)$$
\vskip8pt
yields an isomorphism of topological algebras
$$ D(G_0)\stackrel{\simeq}{\longrightarrow} \varprojlim_k D(\bbG(k)^\circ,\GN) \;.$$
\vskip8pt
This is the weak Fr\'echet-Stein structure on the locally analytic distribution algebra $D(G_0)$ as introduced by Emerton in \cite[Prop. 5.3.1]{EmertonA}. In an obviously similar manner, we may construct the ring $D(\bbG(k)^\circ,\GN)_{\theta_0}$ and obtain an isomorphism $D(\GN)_{\theta_0}\stackrel{\simeq}{\longrightarrow} \varprojlim_k D(\bbG(k)^\circ,\GN)_{\theta_0}.$
\vskip8pt
\vskip8pt
We consider an admissible locally analytic $G_0$-representation $V$, its coadmissible module $M:=V'_b$ and its subspace of $\bbG(k)^\circ$-analytic vectors $V_{\bbG(k)^\circ-\rm an}\subseteq V$.
The latter subspace is naturally a nuclear Fr\'echet space \cite[Lem. 6.1.6]{EmertonA} and we let $(V_{\bbG(k)^\circ-\rm an})'_b$ be its strong dual. It is a space of compact type and a topological $D(\bbG(k)^\circ,\GN)$-module which is finitely generated \cite[Lem. 6.1.13]{EmertonA}. According to \cite[Thm. 6.1.20]{EmertonA} the modules $M_k:= (V_{\bbG(k)^\circ-\rm an})'$ form a $(D(\bbG(k)^\circ,\GN))_{k\in{\mathbb N}}$-sequence, in the sense of \cite[Def. 1.3.8]{EmertonA}, for the coadmissible module $M$ relative to the weak Fr\'echet-Stein structure on $D(G_0).$ This implies that one has
\begin{numequation}\label{equ-weakfamily}
M_k=D(\bbG(k)^\circ,\GN)\hat{\otimes}_{D(G_0)} M
\end{numequation}
as $D(\bbG(k)^\circ,\GN)$-modules for any $k$. Here, the completed tensor product is understood in the sense of \cite[Lem. 1.2.3]{EmertonA}, as in \cite{PSS4}.
\begin{lemma}\label{lem-refine} (i) The $D(\bbG(k)^\circ,\GN)$-module $M_k$ is finitely presented.
\vskip8pt
(ii) There are natural isomorphisms
$$D({\mathbb G}(k-1)^\circ,G_0)\otimes_{D(\bbG(k)^\circ,\GN)} M_k\stackrel{\simeq}{\longrightarrow} M_{k-1}.$$
\vskip8pt
\end{lemma}
{\it Proof. } This can be proved exactly as \cite[Lem. 5.2.4]{PSS4}. \qed
\vskip8pt
Remark: These results have obvious analogues when the character $\theta_0$ is involved.
\subsection{$G_0$-equivariance and the functor ${\mathscr Loc}_{\infty}^\dagger$}\label{subsec_G0}
A $p$-adic completion ${\mathfrak X}$ of an admissible blow-up $X$ of $X_0$ will be called an {\it admissible formal blow-up} of ${\mathfrak X}_0$. We note here that any formal scheme ${\mathfrak X}$ which is obtained from ${\mathfrak X}_0$ by blowing-up a coherent open ideal on ${\mathfrak X}_0$ is an admissible blow-up in this sense. Indeed, if ${\mathcal I}\subset{\mathfrak X}_0$ is the ideal which is blown-up, then ${\mathcal I}\cap{\mathcal O}_{X_0}$ is a coherent open ideal on $X_0$. Blowing-up this ideal on $X_0$ and completing $p$-adically gives back ${\mathfrak X}$.
\vskip8pt
If the coherent open ideal ${\mathcal I}\subset X_0$ that is blown-up is $G_0$-stable, then there is an induced $G_0$-action on $X$ and ${\mathfrak X}$.
In this case, we will say that $X$ and ${\mathfrak X}$ are $G_0$-stable. In the following we suppose that $k$ is large enough for ${\mathfrak X}$, so that the sheaf
${\widetilde{\sD}}^\dagger_{{\mathfrak X},k,{\mathbb Q}}$ is defined on ${\mathfrak X}$.
\begin{prop}\label{prop-exactdirectimage}
Let $\pi:{\mathfrak X}'\rightarrow{\mathfrak X}$ be a morphism between admissible formal blow-ups of ${\mathfrak X}_0$ which is an isomorphism on corresponding rigid analytic spaces. If ${\mathscr M}'$ is a coherent ${\widetilde{\sD}}^\dagger_{{\mathfrak X}',k',{\mathbb Q}}$-module, then $R^j\pi_*{\mathscr M}'=0$ for $j>0$. Moreover, $\pi_*{\widetilde{\sD}}^\dagger_{{\mathfrak X}',k',{\mathbb Q}}={\widetilde{\sD}}^\dagger_{{\mathfrak X}',k',{\mathbb Q}}$, so that $\pi_*$ induces an exact functor between coherent modules over
${\widetilde{\sD}}^\dagger_{{\mathfrak X}',k',{\mathbb Q}}$ and ${\widetilde{\sD}}^\dagger_{{\mathfrak X},k',{\mathbb Q}}$ respectively.
\end{prop}
{\it Proof. } We denote the associated rigid analytic space of ${\mathfrak X}$ by ${\mathfrak X}_{\mathbb Q}$. Coherent modules over ${\mathfrak X}_{\mathbb Q}$ are equivalent to coherent modules over ${\mathcal O}_{{\mathfrak X},{\mathbb Q}}$ \cite[4.1.3]{BerthelotDI} and similarly for ${\mathfrak X}'$. This implies $R^j\pi_*{\mathcal O}_{{\mathfrak X}',{\mathbb Q}}=0$ for $j>0$ and $\pi_*{\mathcal O}_{{\mathfrak X}',{\mathbb Q}}={\mathcal O}_{{\mathfrak X},{\mathbb Q}}$. In particular, there is $N\geq 0$ such that
$p^NR^j\pi_*{\mathcal O}_{{\mathfrak X}'}=0$ for $j>0$ and such that the kernel and cokernel of the natural map ${\mathcal O}_{{\mathfrak X}}\rightarrow\pi_*{\mathcal O}_{{\mathfrak X}'}$ are killed by $p^N$. For any $i\geq 0$, let $X_i$ be the reduction of ${\mathfrak X}$ mod $p^{i+1}$ and similarly for ${\mathfrak X}'$ and denote by $\pi_i: X'_i\rightarrow X_i$ the morphism induced by $\pi$. For any $i$ we have then $p^NR^j\pi_{i*}{\mathcal O}_{X_i}=0$ for $j>0$ and such that the kernel and cokernel of the natural map ${\mathcal O}_{X_i}\rightarrow\pi_{i*}{\mathcal O}_{X_i'}$ are killed by $p^N$. We now prove the assertions. According to \ref{prop-genglobal} the module ${\mathscr M}'$ is generated by global sections. Induction on $j$ reduces us therefore to the case ${\mathscr M}={\widetilde{\sD}}^\dagger_{{\mathfrak X}',k',{\mathbb Q}}$. Since $R^j\pi_*$ commutes with inductive limits, it suffices to prove the claim for ${\widetilde{\sD}}^{(m)}_{{\mathfrak X}',k',{\mathbb Q}}$. Abbreviate ${\widetilde{\cD}}^{(m)}_{X_i,k'}$ for ${\widetilde{\sD}}^{(m)}_{{\mathfrak X},k'}/p^{i+1}{\widetilde{\sD}}^{(m)}_{{\mathfrak X},k'}$ and similarly for ${\mathfrak X}'$. Then ${\widetilde{\cD}}^{(m)}_{X'_i,k'}=\pi_i^*{\widetilde{\cD}}^{(m)}_{X_i,k'}$ by \ref{graded_tcD} and the projection formula implies $$R^j\pi_{i*}{\widetilde{\cD}}^{(m)}_{X'_i,k'}={\widetilde{\cD}}^{(m)}_{X_i,k'}\otimes R^j\pi_{i*}{\mathcal O}_{X'_i}.$$
We see for any $i\geq0$ that $$p^N R^j\pi_{i*}{\widetilde{\cD}}^{(m)}_{X'_i,k'}=0$$ for all $j>0$ and that the natural map $${\widetilde{\cD}}^{(m)}_{X_i,k'}\rightarrow \pi_{i*}{\widetilde{\cD}}^{(m)}_{X'_i,k'}$$ has kernel and cokernel killed by $p^N$. Taking inverse limits over $i$, arguing as in \ref{completion} and finally inverting $p$ yields the first claim and $\pi_*{\widetilde{\sD}}^\dagger_{{\mathfrak X}',k',{\mathbb Q}}={\widetilde{\sD}}^\dagger_{{\mathfrak X},k',{\mathbb Q}}$.\qed
\begin{para}
If ${\mathfrak X}$ is $G_0$-stable, then there is an induced (left) action of $G_0$ on the sheaf ${\widetilde{\sD}}^\dagger_{{\mathfrak X},k,{\mathbb Q}}$. Given $g \in G_0$ and a local section $s$ of ${\widetilde{\sD}}^\dagger_{{\mathfrak X},k,{\mathbb Q}}$, there is thus a local section $g.s$ of ${\widetilde{\sD}}^\dagger_{{\mathfrak X},k,{\mathbb Q}}$. We can then consider the abelian category ${\rm Coh}({\widetilde{\sD}}^\dagger_{{\mathfrak X},k,{\mathbb Q}},G_0)$ of (left) $G_0$-equivariant coherent ${\widetilde{\sD}}^\dagger_{{\mathfrak X},k,{\mathbb Q}}$-modules. Furthermore, the group $G_{k+1}$ is contained in ${\mathcal D}^{\rm an}({\mathbb G}(k)^\circ)$ as a set of delta distributions, and for $h \in G_{k+1}$ we write $\delta_h$ for its image in $H^0({\mathfrak X}, {\widetilde{\sD}}^\dagger_{{\mathfrak X},k,{\mathbb Q}}) = {\mathcal D}^{\rm an}({\mathbb G}(k)^\circ)_{\theta_0}$.
For $g \in G_0$, $h \in G_{k+1}$, we have $g.\delta_h = \delta_{ghg^{-1}}$, and for a local section $s$ of ${\widetilde{\sD}}^\dagger_{{\mathfrak X},k,{\mathbb Q}}$ we have then the relation
\begin{numequation}\label{non_commuting} g.(s\delta_h) = (g.s)(g.\delta_h) = (g.s) \delta_{ghg^{-1}} \;.
\end{numequation}
Suppose now that $\pi:{\mathfrak X}'\rightarrow{\mathfrak X}$ is a $G_0$-equivariant morphism between admissible formal blow-ups of ${\mathfrak X}_0$ which is an isomorphism on corresponding rigid analytic spaces and that $k'\geq k$ are sufficiently large. According to \ref{prop-exactdirectimage} there is then a natural morphism of sheaves of rings
\begin{numequation}\label{equ-transit_sheaf}
\pi_*{\widetilde{\sD}}^\dagger_{{\mathfrak X}',k',{\mathbb Q}}={\widetilde{\sD}}^\dagger_{{\mathfrak X},k',{\mathbb Q}}\stackrel{\subseteq}{\longrightarrow} {\widetilde{\sD}}^\dagger_{{\mathfrak X},k,{\mathbb Q}}
\end{numequation}
which is $G_0$-equivariant. Given a coherent ${\widetilde{\sD}}^\dagger_{{\mathfrak X}',k',{\mathbb Q}}$-module ${\mathscr M}_{{\mathfrak X}'}$, the ${\widetilde{\sD}}^\dagger_{{\mathfrak X},k,{\mathbb Q}}$-module
$${\widetilde{\sD}}^\dagger_{{\mathfrak X},k,{\mathbb Q}} \otimes_{\pi_* {\widetilde{\sD}}^\dagger_{{\mathfrak X}',k',{\mathbb Q}}} \pi_*{\mathscr M}_{{\mathfrak X}'}$$
\vskip8pt
is $G_0$-equivariant via $g.(s\otimes m)=(g.s) \otimes (g.m)$ for local sections $s,m$ and $g \in G_0$.
Consider its submodule ${\mathcal R}_{{\mathfrak X}}$ locally generated by all elements $s \delta_h \otimes m - s\otimes (h.m)$ for $h \in G_{k'}$. Because of \ref{non_commuting} the submodule ${\mathcal R}_{{\mathfrak X}}$ is $G_0$-stable. We put
$${\widetilde{\sD}}^\dagger_{{\mathfrak X},k,{\mathbb Q}} \otimes_{{\widetilde{\sD}}^\dagger_{{\mathfrak X}',k',{\mathbb Q}},G_{k'}} {\mathscr M}_{{\mathfrak X}'}:=
\left({\widetilde{\sD}}^\dagger_{{\mathfrak X},k,{\mathbb Q}} \otimes_{\pi_* {\widetilde{\sD}}^\dagger_{{\mathfrak X}',k',{\mathbb Q}}} \pi_*{\mathscr M}_{{\mathfrak X}'}\right)/\;{\mathcal R}_{{\mathfrak X}} \;.$$
\end{para}
\begin{para} We let $X={\mathbb G}/{\mathbb B}$ be the flag variety of ${\mathbb G}$ and denote by $${\mathcal X}:=X^{\rm ad}$$ the associated adic space. For simplicity, an admissible formal blow-up ${\mathfrak X}$ of ${\mathfrak X}_0$ will be called a {\it formal model for ${\mathcal X}$ over ${\mathfrak X}_0$}. This set of formal models is a projective system if it is indexed by the directed family of coherent open ideals on ${\mathfrak X}_0$, cf. \cite[9.3]{BoschLectures}. Any morphism in the projective system is an isomorphism on corresponding rigid analytic spaces. Given a subsystem ${\mathcal F}$ of this projective system, we will denote the corresponding projective limit by ${\mathfrak X}_\infty({\mathcal F})=\varprojlim_{\mathcal F} {\mathfrak X}$ or simply ${\mathfrak X}_\infty$. In the following we will work relative to such a fixed system ${\mathcal F}$. A {\it family of congruence levels for ${\mathcal F}$} is an increasing family $\underline{k}$ of natural numbers $k=k_{\mathfrak X}$ for each ${\mathfrak X}\in{\mathcal F}$ (increasing means that $k'\geq k $ whenever
there is a morphism ${\mathfrak X}'\rightarrow{\mathfrak X}$ in ${\mathcal F}$). In the following, we fix such a family $\underline{k}$ with the additional property that the sheaf
${\widetilde{\sD}}^\dagger_{{\mathfrak X},k_{\mathfrak X},{\mathbb Q}}$ is defined for each ${\mathfrak X}\in {\mathcal F}$. We finally assume that all models in ${\mathcal F}$ are $G_0$-stable and that all morphisms in ${\mathcal F}$ are $G_0$-equivariant.
\vskip8pt
\begin{prop}\label{prop-cofinal}
If ${\mathcal F}$ equals the set of {\it all} $G_0$-equivariant formal models of ${\mathcal X}$ over ${\mathfrak X}_0$, then ${\mathfrak X}_\infty={\mathcal X}$.
\end{prop}
{\it Proof. } According to \cite{SchnVPut} it suffices to see that any admissible formal blow-up ${\mathfrak X}$ of ${\mathfrak X}_0$ is dominated by one which is $G_0$-stable. If ${\mathcal I}$ is the ideal which is blown-up and if $p^k{\mathcal O}_{{\mathfrak X}_0}\subset{\mathcal I}$ for some $k$, then $G_k:={\mathbb G}(k)({\mathfrak o})$ stabilizes ${\mathcal I}$ and ${\mathcal I}':={\mathcal I}\cdot {\mathcal O}_{{\mathfrak X}}$. Let $g_1,...,g_N$ be a system of representatives for $G_0/G_k$ and let ${\mathcal J}$ be the product of the finitely many ideals $g_i{\mathcal I}'$. Then ${\mathcal J}$ is $G_0$-stable and blowing-up ${\mathcal J}$ on ${\mathfrak X}$ yields a $G_0$-stable model over ${\mathfrak X}$. \qed
\end{para}
\begin{dfn}\label{dfn-coadmod} A {\it $G_0$-equivariant coadmissible module} on ${\mathfrak X}_\infty$ consists of a family ${\mathscr M}:=({\mathscr M}_{\mathfrak X})_{{\mathfrak X}\in{\mathcal F}}$ of objects ${\mathscr M}_{\mathfrak X}\in {\rm Coh}({\widetilde{\sD}}^\dagger_{{\mathfrak X},k,{\mathbb Q}},G_0)$ together with isomorphisms
\begin{numequation}\label{equ-transit_mod}
{\widetilde{\sD}}^\dagger_{{\mathfrak X},k,{\mathbb Q}} \otimes_{{\widetilde{\sD}}^\dagger_{{\mathfrak X}',k',{\mathbb Q}},G_{k'}} {\mathscr M}_{{\mathfrak X}'}\stackrel{\simeq}{\longrightarrow} {\mathscr M}_{{\mathfrak X}}
\end{numequation}
of $G_0$-equivariant ${\widetilde{\sD}}^\dagger_{{\mathfrak X},k,{\mathbb Q}}$-modules whenever there is a morphism ${\mathfrak X}'\rightarrow{\mathfrak X}$ in ${\mathcal F}$.
The isomorphisms are required to satisfy the obvious transitivity condition whenever there are morphisms ${\mathfrak X}''\rightarrow{\mathfrak X}'\rightarrow{\mathfrak X}$ in ${\mathcal F}$.
\end{dfn}
A morphism ${\mathscr M}\rightarrow{\mathscr N}$ between two such modules consists of morphisms ${\mathscr M}_{\mathfrak X}\rightarrow{\mathscr N}_{\mathfrak X}$ in ${\rm Coh}({\widetilde{\sD}}^\dagger_{{\mathfrak X},k,{\mathbb Q}},G_0)$ compatible with the isomorphisms above.
\vskip8pt
Let ${\mathscr M}$ be a $G_0$-equivariant coadmissible module on ${\mathfrak X}_\infty$. The isomorphisms \ref{equ-transit_mod} induce morphisms $\pi_*{\mathscr M}_{{\mathfrak X}'}\rightarrow {\mathscr M}_{{\mathfrak X}}$ having global sections $H^0({\mathfrak X}',{\mathscr M}_{{\mathfrak X}'})\rightarrow H^0({\mathfrak X},{\mathscr M}_{{\mathfrak X}})$. We let
$$ H^0({\mathfrak X}_\infty,{\mathscr M}):=\varprojlim_{\mathfrak X} H^0({\mathfrak X},{\mathscr M}_{{\mathfrak X}}) \;.$$
\vskip8pt
\begin{para}
On the other hand, we consider the category of coadmissible $D(\GN)_{\theta_0}$-modules. Given such a module $M$ we have its associated admissible locally analytic $G_0$-representation $V=M'_b$ together with its subspace of $\bbG(k)^\circ$-analytic vectors $V_{\bbG(k)^\circ-\rm an}$. The latter is stable under the $G_0$-action and its dual $M_k:= (V_{\bbG(k)^\circ-\rm an})'$ is a finitely presented $D(\bbG(k)^\circ,\GN)_{\theta_0}$-module, cf. \ref{lem-refine}. Now consider a model ${\mathfrak X}$ in ${\mathcal F}$ and let $k=k_{\mathfrak X}$. According to Thm. \ref{thm-equivalence} we have the coherent ${\widetilde{\sD}}^\dagger_{{\mathfrak X},k,{\mathbb Q}}$-module
$${\mathscr Loc}^\dagger_{{\mathfrak X},k}(M_k)= {\widetilde{\sD}}^\dagger_{{\mathfrak X},k,{\mathbb Q}}\otimes_{\cD^{\rm an}(\bbG(k)^\circ)_{\theta_0}} M_k$$
\vskip8pt
on ${\mathfrak X}$. Using the contragredient $G_0$-action on the dual space $M_k$, we put
$$g.(s \otimes m) := (g.s) \otimes (g.m)$$
\vskip8pt
for $g\in G_0, m\in M_k$ and a local section $s$. In this way, ${\mathscr Loc}^\dagger_{{\mathfrak X},k}(M_k)$ becomes an object of ${\rm Coh}({\widetilde{\sD}}^\dagger_{{\mathfrak X},k,{\mathbb Q}},G_0)$.
\end{para}
\begin{prop}\label{prop-equivalenceII} (i) The family ${\mathscr Loc}^\dagger_{{\mathfrak X},k_{\mathfrak X}} (M_{k_{{\mathfrak X}}})$ forms a $G_0$-equivariant coadmissible module on ${\mathfrak X}_\infty$. Call it ${\mathscr Loc}^\dagger(M)$. The formation of ${\mathscr Loc}^\dagger(M)$ is functorial in $M$.
\vskip8pt
(ii) The functors ${\mathscr Loc}^\dagger$ and $H^0({\mathfrak X}_\infty,\cdot)$ are quasi-inverse equivalences between the categories of coadmissible $D(\GN)_{\theta_0}$-modules and $G_0$-equivariant coadmissible modules on ${\mathfrak X}_\infty$.
\end{prop}
\vskip8pt
{\it Proof. } Assume $k'\geq k$. We let $H:=G_{k+1}/G_{k'+1}$ and we denote a system of representatives in $G_{k+1}$ for the cosets in $H$ by the same symbol. For simplicity, we abbreviate in this proof
$$D(k):=\cD^{\rm an}(\bbG(k)^\circ)_{\theta_0} \hskip10pt {\rm and}\hskip10pt D(k,G_0):=D(\bbG(k)^\circ,\GN)_{\theta_0}$$
\vskip8pt
and similarly for $k'$. We have the natural inclusion $D(k)\hookrightarrow D(k,G_0)$ from \ref{equ-finitefree} which is compatible with variation in $k$. Now suppose $M$ is a $D(k',G_0)$-module. We then have the free $D(k)$-module $D(k)^{\oplus M\times H}$
on a basis $e_{m,h}$ indexed by the elements $(m,h)$ of the set $M\times H$. Its formation is functorial in $M$: if $M'$ is another module and $f: M\rightarrow M'$ a linear map, then $e_{m,h}\rightarrow e_{f(m),h}$ induces a linear map between the corresponding free modules. The free module comes with a linear map
$$f_M: D(k)^{\oplus M\times H}\rightarrow D(k)\otimes_{D(k')} M$$
\vskip8pt
given by
$$\oplus_{(m,h)}\lambda_{m,h}e_{m,h}\mapsto (\lambda_{m,h} \delta_h) \otimes m - \lambda_{m,h} \otimes (h \cdot m)$$
\vskip8pt
for $\lambda_{m,h}\in D(k)$ where we consider $M$ a $D(k')$-module via the inclusion $D(k')\hookrightarrow D(k',G_0)$.
The map is visibly functorial in $M$ and gives rise to the sequence of linear maps
$$ D(k)^{\oplus M\times H}\stackrel{f_M}{\longrightarrow} D(k)\otimes_{D(k')} M \stackrel{can_M}{\longrightarrow}
D(k,G_0)\otimes_{D(k',G_0)} M\longrightarrow 0$$
\vskip8pt
where the second map is induced from the inclusion $D(k')\hookrightarrow D(k',G_0)$.
The sequence is functorial in $M$, since so are both occuring maps.
\vskip8pt
{\it Claim 1: If $M$ is a finitely presented $D(k',G_0)$-module, then the above sequence is exact.}
\vskip8pt
{\it Proof. } This can be proved as in the proof of \cite[Prop. 5.3.5]{PSS4}. \qed
\vskip8pt
Let $\pi:{\mathfrak X}'\rightarrow{\mathfrak X}$ be a morphism in ${\mathcal F}$ and let $k=k_{{\mathfrak X}}, k'=k_{{\mathfrak X}'}$.
\vskip8pt
{\it Claim 2: Suppose $M$ is a finitely presented $D(k')$-module and let ${\mathscr M}:= {\mathscr Loc}^\dagger_{{\mathfrak X}',k'}(M)$. The natural morphism
$${\mathscr Loc}^\dagger_{{\mathfrak X},k}(D(k)\otimes_{D(k')} M)\stackrel{\simeq}{\longrightarrow} {\widetilde{\sD}}^\dagger_{{\mathfrak X},k,{\mathbb Q}} \otimes_{\pi_*{\widetilde{\sD}}^\dagger_{{\mathfrak X}',k',{\mathbb Q}}} \pi_*{\mathscr M} $$
\vskip8pt
is bijective.}
\vskip8pt
{\it Proof. } The functor $\pi_*$ is exact on coherent ${\widetilde{\sD}}^\dagger_{{\mathfrak X}',k',{\mathbb Q}}$-modules according to \ref{prop-exactdirectimage}.
Choosing a finite presentation of $M$ reduces to the case $M=D(k')$ which is obvious.\qed
\vskip8pt
Now let $M$ be a finitely presented $D(k',G_0)$-module. Let $m_1, \ldots ,m_r$ be generators for $M$ as a $D(k')$-module. We have a sequence of $D(k)$-modules
$$\bigoplus_{i,h} D(k)e_{m_i,h}\stackrel{f'_M}{\longrightarrow} D(k)\otimes_{D(k')} M \stackrel{can_M}{\longrightarrow}
D(k,G_0)\otimes_{D(k',G_0)} M\longrightarrow 0$$
\vskip8pt
where $f'_M$ denotes the restriction of the map $f_M$ to the free submodule of $D(k)^{\oplus M\times H}$
generated by the finitely many vectors $e_{m_i,h}, i=1,\ldots,r$, $h \in H_n$. Since ${\rm im}(f'_M)={\rm im}(f_M)$ the sequence is exact by the first claim. Since it consists of finitely presented $D(k)$-modules, we may apply the exact functor ${\mathscr Loc}^\dagger_{{\mathfrak X},k,{\mathbb Q}}$ to it. By the second claim, we get an exact sequence
$$ ({\widetilde{\sD}}^{\dagger}_{{\mathfrak X},k,{\mathbb Q}})^{\oplus r|H|} \rightarrow {\widetilde{\sD}}^\dagger_{{\mathfrak X},k,{\mathbb Q}} \otimes_{\pi_*{\widetilde{\sD}}^\dagger_{{\mathfrak X}',k',{\mathbb Q}}} \pi_*{\mathscr M} \rightarrow
{\mathscr Loc}^\dagger_{{\mathfrak X},k}(D(k,G_0)\otimes_{D(k',G_0)} M) \rightarrow 0$$
\vskip8pt
where ${\mathscr M} = {\mathscr Loc}^\dagger_{{\mathfrak X}',k'}(M)$. The cokernel of the first map in this sequence equals by definition
$${\widetilde{\sD}}^\dagger_{{\mathfrak X},k,{\mathbb Q}} \otimes_{\pi_* {\widetilde{\sD}}^\dagger_{{\mathfrak X}',k',{\mathbb Q}},G_{k'}} \pi_*{\mathscr M} \;,$$
\vskip8pt
whence an isomorphism
$$ {\widetilde{\sD}}^\dagger_{{\mathfrak X},k,{\mathbb Q}} \otimes_{\pi_* {\widetilde{\sD}}^\dagger_{{\mathfrak X}',k',{\mathbb Q}},G_k'} \pi_*{\mathscr M} \stackrel{\simeq}{\longrightarrow} {\mathscr Loc}^\dagger_{{\mathfrak X},k}(D(k,G_0)\otimes_{D(k',G_0)} M) \;.$$
\vskip8pt
This implies both parts of the proposition. \qed
\vskip8pt
\begin{para}\label{para-sheaf}
Denote the canonical projection map ${\mathfrak X}_\infty\rightarrow{\mathfrak X}$ by ${\rm sp}_{{\mathfrak X}}$ for each ${\mathfrak X}$. We define the following sheaf of rings on ${\mathfrak X}_\infty$. Assume $V\subseteq{\mathfrak X}_\infty$ is an open subset of the form ${\rm sp}_{{\mathfrak X}}^{-1}(U)$ with an open subset $U\subseteq{\mathfrak X}$ for a model ${\mathfrak X}$. We have that
$${\rm sp}_{{\mathfrak X}'}(V)=\pi^{-1}(U)$$
\vskip8pt
for any blow-up morphism $\pi:{\mathfrak X}'\rightarrow{\mathfrak X}$ in ${\mathcal F}$ and so, in particular, ${\rm sp}_{{\mathfrak X}'}(V)\subseteq{\mathfrak X}'$ is an open subset for such ${\mathfrak X}'$.
Moreover,
$$\pi^{-1}({\rm sp}_{{\mathfrak X}'}(V))={\rm sp}_{{\mathfrak X}''}(V)$$
\vskip8pt
whenever $\pi:{\mathfrak X}''\rightarrow{\mathfrak X}'$ is a blow-up morphism over ${\mathfrak X}$ in ${\mathcal F}$. In this situation, the morphism \ref{equ-transit_sheaf} induces the ring homomorphism
\begin{numequation}\label{equ-transit_hom} {\widetilde{\sD}}^{\dagger}_{{\mathfrak X}'',k'',{\mathbb Q}}({\rm sp}_{{\mathfrak X}''}(V))=\pi_*{\widetilde{\sD}}^{\dagger}_{{\mathfrak X}'',k'',{\mathbb Q}}(({\rm sp}_{{\mathfrak X}'}(V))\rightarrow
{\widetilde{\sD}}^{\dagger}_{{\mathfrak X}',k',{\mathbb Q}}({\rm sp}_{{\mathfrak X}'}(V))
\end{numequation}
and we form the projective limit
$${\widetilde{\sD}}^\dagger_{\infty,{\mathbb Q}}(V):=\varprojlim_{{\mathfrak X}'\rightarrow {\mathfrak X}} {\widetilde{\sD}}^{\dagger}_{{\mathfrak X}',k',{\mathbb Q}}({\rm sp}_{{\mathfrak X}'}(V))$$
\vskip8pt
over all these maps. The open subsets of the form $V$ form a
basis for the topology on ${\mathfrak X}_\infty$ and ${\widetilde{\sD}}^\dagger_{\infty,{\mathbb Q}}$ is a presheaf on this basis. We denote the associated sheaf
on ${\mathfrak X}_\infty$ by the symbol ${\widetilde{\sD}}^\dagger_{\infty,{\mathbb Q}}$ as well. It is a $G_0$-equivariant sheaf of rings on ${\mathfrak X}_\infty$.
\vskip8pt
Remark: In the case where ${\mathcal F}$ consists of {\it all} $G_0$-stable formal models of ${\mathcal X}$ over ${\mathfrak X}_0$, we have ${\mathfrak X}_\infty={\mathcal X}$ by \ref{prop-cofinal} and we denote the sheaf ${\widetilde{\sD}}^\dagger_{\infty,{\mathbb Q}}$ by ${\widetilde{\sD}}^\dagger_{{\mathcal X},{\mathbb Q}}$ (or simply ${\widetilde{\sD}}^{\dagger}_{{\mathcal X}}$).
\end{para}
\begin{para}
Suppose ${\mathscr M}:=({\mathscr M}_{\mathfrak X})_{{\mathfrak X}}$ is a $G_0$-equivariant coadmissible module on ${\mathfrak X}_\infty$ as defined in \ref{dfn-coadmod}.
The isomorphisms \ref{equ-transit_mod} induce $G_0$-equivariant maps $\pi_*{\mathscr M}_{{\mathfrak X}'}\rightarrow{\mathscr M}_{{\mathfrak X}}$ which are linear relative to the morphism \ref{equ-transit_sheaf}. In a completely analogous manner as above, we obtain a sheaf ${\mathscr M}_\infty$ on ${\mathfrak X}_\infty$.
It is a $G_0$-equivariant (left) ${\widetilde{\sD}}^\dagger_{\infty,{\mathbb Q}}$-module on ${\mathfrak X}_\infty$ whose formation is functorial in ${\mathscr M}$.
\end{para}
\begin{prop}
The functor ${\mathscr M}\rightarrow{\mathscr M}_\infty$ from $G_0$-equivariant coadmissible modules on ${\mathfrak X}_\infty$ to $G_0$-equivariant ${\widetilde{\sD}}^\dagger_{\infty,{\mathbb Q}}$-modules is a
fully faithful embedding.
\end{prop}
{\it Proof. } We have ${\rm sp}_{{\mathfrak X}}({\mathfrak X}_\infty)={\mathfrak X}$ for all ${\mathfrak X}$.
The global sections of $M_\infty$ are therefore equal to
$$\Gamma({\mathfrak X}_\infty,{\mathscr M}_\infty)=\varprojlim_{{\mathfrak X}}\Gamma({\mathfrak X},{\mathscr M}_{{\mathfrak X}})=H^0({\mathfrak X}_\infty,{\mathscr M})$$
\vskip8pt
in the notation of the previous section. Thus, the functor ${\mathscr Loc}^\dagger \circ \Gamma({\mathfrak X}_\infty,-)$ is a left quasi-inverse according to Prop. \ref{prop-equivalenceII}.\qed
\vskip8pt
We denote by ${\mathscr Loc}^\dagger_\infty$ the composite of the functor ${\mathscr Loc}^\dagger$ with $(\cdot)_\infty$, i.e.
$$ \{ ~coadmissible~ D(G_0)_{\theta_0}-modules~\}\stackrel{{\mathscr Loc}^\dagger_\infty}{\longrightarrow} \{ ~G_0-equivariant~ {\widetilde{\sD}}^\dagger_{\infty,{\mathbb Q}}-modules~ \}.$$
\vskip8pt
It is fully faithful. We tentatively call its essential image the {\it coadmissible} $G_0$-equivariant ${\widetilde{\sD}}^\dagger_{\infty,{\mathbb Q}}$-modules. It is an abelian category.
In the case where ${\mathfrak X}_\infty$ equals the whole adic flag variety ${\mathcal X}$ we write ${\mathscr Loc}^\dagger_{\mathcal X}$ for the functor ${\mathscr Loc}^\dagger_\infty$.
\vskip8pt
\subsection{$G$-equivariance and the main theorem}
Let $G:={\mathbb G}(L)$. Denote by ${\mathcal B}$ the (semi-simple) Bruhat-Tits building of the $p$-adic group $G$ together with its natural $G$-action.
\begin{para}
To each special vertex $v\in{\mathcal B}$ we have the associated smooth affine Bruhat-Tits group scheme ${\mathbb G}_v$ over ${\mathfrak o}$ and a smooth model $X_0(v)$ of the flag variety of ${\mathbb G}$. All constructions in sections \ref{models}-\ref{loc_n} are associated with the group scheme ${\mathbb G}_0$ with vertex, say $v_0$, but can be done canonically for any other of the reductive group schemes ${\mathbb G}_v$. We distinguish the various constructions from each other by adding the corresponding vertex $v$ to them, i.e. we write $X(v)$ for an admissible blow-up of the smooth model $X_0(v)$ and so on. We then choose a subset ${\mathcal F}(v)$ of models over ${\mathfrak X}_0(v)$ for ${\mathcal X}$ as in the preceding subsection, but relative to the vertex $v$. We assume that one of these subsets, say the one belonging to $v_0$, is stable under admissible blowing-up. According to \cite{BoschLectures}, any ${\mathfrak X}(v)\in{\mathcal F}(v), v\in{\mathcal B}$ is then dominated by an element from ${\mathcal F}(v_0)$. As before, we denote the projective limit over ${\mathcal F}(v_0)$ by ${\mathfrak X}_\infty$. According to the previous subsection, we have the $G_0$-equivariant sheaf ${\widetilde{\sD}}^\dagger_{\infty,{\mathbb Q}}$ on ${\mathfrak X}_\infty$.
\vskip8pt
An element $g\in G$ induces a morphism $X_0(v)\stackrel{g.}{\longrightarrow}X_0(gv)$
which satisfies $(gh).=(g.)\circ(h.)$ and $1.={\rm id}$ for $g,h\in G$. If $X(v)\rightarrow X_0(v)$ is an admissible blowing up of an ideal ${\mathcal I}\subset X_0(v)$, then the universal property of blowing-up induces an isomorphism $X(v)\stackrel{g.}{\longrightarrow}X(gv)$ onto the
blowing-up of $X_0(gv)$ at the ideal $g.{\mathcal I}$. We make the assumption that our union of models is {\it $G$-stable} in the sense that
$${\mathfrak X}(v) \in {\mathcal F}(v) \Longrightarrow {\mathfrak X}(gv)\in{\mathcal F}(gv)$$
for any ${\mathfrak X}(v)\in{\mathcal F}(v)$ and any $g\in G$. We also assume that $k_{{\mathfrak X}(v)}=k_{{\mathfrak X}(gv)}$ in this situation. We obtain thus a $G$-action on ${\mathfrak X}_\infty$. By definition of this action, there is an equality
\begin{numequation}\label{equivariant}{\rm sp}_{{\mathfrak X}(gv)}(g.V)=g.{\rm sp}_{{\mathfrak X}(v)}(V)
\end{numequation}
in ${\mathfrak X}(gv)$ for $g\in G$ and $V\subseteq{\mathfrak X}_\infty$.
\end{para}
\vskip8pt
\begin{prop}
The $G_0$-equivariant structure on the sheaf ${\widetilde{\sD}}^\dagger_{\infty,{\mathbb Q}}$ extends to a $G$-equivariant structure.
\end{prop}
{\it Proof. } Let $g\in G$. The isomorphism ${\mathfrak X}(v)\stackrel{g.}{\longrightarrow}{\mathfrak X}(gv)$ induces a ring isomorphism
\begin{numequation}\label{equ-ringiso}{\widetilde{\sD}}^\dagger_{{\mathfrak X}(v),k,{\mathbb Q}}(U)\stackrel{g.}{\longrightarrow}{\widetilde{\sD}}^\dagger_{{\mathfrak X}(gv),k,{\mathbb Q}}(g.U)\end{numequation}
for any open subset $U\subseteq{\mathfrak X}(v)$ where $k=k_{{\mathfrak X}(v)}$. In particular, for an open subset $V\subseteq{\mathfrak X}_\infty$ of the form $V={\rm sp}_{{\mathfrak X}(v)}^{-1}(U)$ with $U\subseteq{\mathfrak X}(v)$ open and a blow-up morphism ${\mathfrak X}'(v)\rightarrow{\mathfrak X}(v)$, this gives a ring homomorphism
\begin{numequation}\label{equ-action} {\widetilde{\sD}}^\dagger_{{\mathfrak X}'(v),k',{\mathbb Q}}({\rm sp}_{{\mathfrak X}'(v)}(V))\stackrel{g.}{\longrightarrow}{\widetilde{\sD}}^\dagger_{{\mathfrak X}'(gv),k',{\mathbb Q}}(g.{\rm sp}_{{\mathfrak X}'(v)}(V))
={\widetilde{\sD}}^\dagger_{{\mathfrak X}'(gv),k',{\mathbb Q}}({\rm sp}_{{\mathfrak X}'(gv)}(gV))
\end{numequation}
where we have used \ref{equivariant} and where $k'=k_{{\mathfrak X}'(v)}$. A given morphism $\pi: {\mathfrak X}(v')\rightarrow{\mathfrak X}(v)$ with ${\mathfrak X}(v')\in{\mathcal F}(v')$ and ${\mathfrak X}(v)\in{\mathcal F}(v)$ which is an isomorphism on corresponding rigid analytic spaces induces a morphism of sheaves of rings
\begin{numequation}\label{equ-transit_sheafII}
\pi_* {\widetilde{\sD}}^\dagger_{{\mathfrak X}(v'),k',{\mathbb Q}}={\widetilde{\sD}}^\dagger_{{\mathfrak X}(v),k',{\mathbb Q}}\stackrel{\subseteq}{\longrightarrow} {\widetilde{\sD}}^\dagger_{{\mathfrak X}(v),k,{\mathbb Q}}\end{numequation}
by \ref{prop-exactdirectimage}. Here, $k=k_{{\mathfrak X}(v)}$ and $k'=k_{{\mathfrak X}(v')}$. Given $V \subset {\mathfrak X}_\infty$ of the form $V={\rm sp}_{{\mathfrak X}(\tilde{v})}^{-1}(U)$ with an open set $U\subseteq{\mathfrak X}(\tilde{v})$, the morphism \ref{equ-transit_sheafII} induces a ring homomorphism
\begin{numequation}\label{equ-transit_homII} {\widetilde{\sD}}^{\dagger}_{{\mathfrak X}(v'),k',{\mathbb Q}}({\rm sp}_{{\mathfrak X}(v')}(V))
=\pi_*{\widetilde{\sD}}^{\dagger}_{{\mathfrak X}(v'),k',{\mathbb Q}}({\rm sp}_{{\mathfrak X}(v)}(V))\rightarrow {\widetilde{\sD}}^{\dagger}_{{\mathfrak X}(v),k,{\mathbb Q}}({\rm sp}_{{\mathfrak X}(v)}(V))
\end{numequation}
whenever the morphism $\pi:{\mathfrak X}(v')\rightarrow{\mathfrak X}(v)$ lies over ${\mathfrak X}(\tilde{v}).$ If we write ${\mathfrak X}(v')\geq {\mathfrak X}(v)$ in this situation, then
the family of all models ${\mathfrak X}(v)$ over ${\mathfrak X}(\tilde{v})$ becomes directed, the ${\widetilde{\sD}}^{\dagger}_{{\mathfrak X}(v),k,{\mathbb Q}}({\rm sp}_{{\mathfrak X}(v)}(V))$ become a projective system and we may form the projective limit
$$ \varprojlim_{{\mathfrak X}(v)\rightarrow{\mathfrak X}(\tilde{v})}{\widetilde{\sD}}^{\dagger}_{{\mathfrak X}(v),k,{\mathbb Q}}({\rm sp}_{{\mathfrak X}(v)}(V)) \;.$$
\vskip8pt
By cofinality, this projective limit equals ${\widetilde{\sD}}^\dagger_{\infty,{\mathbb Q}}(V)$.
Since the homomorphism \ref{equ-action} is compatible with varying ${\mathfrak X}'(v')$ in the directed family, we deduce for a given $g\in G$ a ring homomorphism
\vskip8pt
$${\widetilde{\sD}}^\dagger_{\infty,{\mathbb Q}}(V)=\varprojlim_{{\mathfrak X}(v)}{\widetilde{\sD}}^{\dagger}_{{\mathfrak X}(v),k,{\mathbb Q}}({\rm sp}_{{\mathfrak X}(v)}(V))\stackrel{g.}{\rightarrow}
\varprojlim_{{\mathfrak X}(gv)}{\widetilde{\sD}}^{\dagger}_{{\mathfrak X}(gv),k,{\mathbb Q}}({\rm sp}_{{\mathfrak X}(gv)}(gV))={\widetilde{\sD}}^\dagger_{\infty,{\mathbb Q}}(gV) \;.$$
\vskip8pt
It implies that the sheaf ${\widetilde{\sD}}^{\dagger}_{\infty,{\mathbb Q}}$ is $G$-equivariant. It is clear from the construction that the $G$-equivariant structure extends the $G_0$-structure.\qed
\vskip8pt
A coadmissible $G_0$-equivariant ${\widetilde{\sD}}^\dagger_{\infty,{\mathbb Q}}$-module whose equivariant structure extends to the full group $G$, will simply be called a coadmissible $G$-equivariant ${\widetilde{\sD}}^\dagger_{\infty,{\mathbb Q}}$-module.
\begin{thm}
The functors ${\mathscr Loc}^\dagger_\infty$ and $\Gamma({\mathfrak X}_\infty,\cdot)$ are quasi-inverse equivalences between the categories of coadmissible $D(G_0)_{\theta_0}$-modules and coadmissible $G_0$-equivariant ${\widetilde{\sD}}^\dagger_{\infty,{\mathbb Q}}$-modules. The subcategories
of coadmissible $D(G)_{\theta_0}$-modules and coadmissible $G$-equivariant ${\widetilde{\sD}}^\dagger_{\infty,{\mathbb Q}}$-modules correspond to each other.
\end{thm}
{\it Proof. } We only need to show the second statement. It is clear that a coadmissible $D(G_0)_{\theta_0}$-module which comes from a
coadmissible $G$-equivariant ${\widetilde{\sD}}^\dagger_{\infty,{\mathbb Q}}$-module is a $D(G)_{\theta_0}$-module. For the converse, we consider a special vertex $v\in{\mathcal B}$ and a model ${\mathfrak X}(v)$ and the corresponding localisation functor ${\mathscr Loc}^\dagger_{{\mathfrak X}(v),k,{\mathbb Q}}$ (where $k=k_{{\mathfrak X}(v)}$) which is an equivalence between finitely presented $D^{an}({\mathbb G}_v(k)^\circ)_{\theta_0}$-modules and coherent ${\widetilde{\sD}}^\dagger_{{\mathfrak X}(v),k,{\mathbb Q}}$-modules on ${\mathfrak X}(v)$. Here, ${\mathbb G}_v$ denotes as before the reductive Bruhat-Tits group scheme over ${\mathfrak o}$ associated with the special vertex $v$. The adjoint action of $G$ on its Lie algebra induces a ring isomorphism
\begin{numequation}\label{equ-ringisoII}
D^{an}({\mathbb G}_{v}(k)^\circ)\stackrel{g.}{\rightarrow}D^{an}({\mathbb G}_{gv}(k)^\circ)\end{numequation}
for any $g\in G$. Now consider a coadmissible $D(G)_{\theta_0}$-module $M$ with dual space $V=M'$. We have the family $({\mathscr M}_{{\mathfrak X}(v)})_{{\mathfrak X}(v)}$ where
$${\mathscr M}_{{\mathfrak X}(v)} = {\mathscr Loc}^\dagger_{{\mathfrak X}(v),k,{\mathbb Q}}(M_{k,v}) = {\widetilde{\sD}}^\dagger_{{\mathfrak X}(v),k,{\mathbb Q}}\otimes_{D^{an}({\mathbb G}_v(k)^\circ)_{\theta_0}} M_{k,v}$$
\vskip8pt
and $M_{k,v}=(V_{{\mathbb G}_v(k)^\circ-\rm an})'$ with $k=k_{{\mathfrak X}(v)}$.
Let $g\in G$. The map $m \mapsto gm$ on $M$ induces a map $M_{k,v}\rightarrow M_{k,gv}$ which is linear relative to \ref{equ-ringisoII}. We therefore have for any open subset $U\subseteq{\mathfrak X}(v)$ a homomorphism
$${\mathscr M}_{{\mathfrak X}(v)}(U)\stackrel{g.}{\longrightarrow}{\mathscr M}_{{\mathfrak X}(gv)}(g.U)$$
\vskip8pt
which is induced by the map
$$s \otimes m \mapsto (g.s) \otimes gm \;.$$
\vskip8pt
for $s\in {\widetilde{\sD}}^\dagger_{{\mathfrak X}(v),k,{\mathbb Q}}(U), m\in M_{k,v}$ and where
$g.$ is the ring isomorphism \ref{equ-ringiso}. In particular, for an open subset $V\subseteq{\mathfrak X}_\infty$ of the form $V={\rm sp}_{{\mathfrak X}(v)}^{-1}(U)$ with $U\subseteq{\mathfrak X}(v)$ open, this gives a homomorphism for
\begin{numequation}\label{equ-actionII}
{\mathscr M}_{{\mathfrak X}(v)}({\rm sp}_{{\mathfrak X}(v)}(V)) \stackrel{g.}{\longrightarrow} {\mathscr M}_{{\mathfrak X}(gv)}(g.{\rm sp}_{{\mathfrak X}(v)}(V)) = {\mathscr M}_{{\mathfrak X}(gv)}({\rm sp}_{{\mathfrak X}(gv)}(gV))
\end{numequation}
which is linear relative to the ring homomorphism \ref{equ-action}.
\vskip8pt
A given morphism $\pi: {\mathfrak X}(v')\rightarrow{\mathfrak X}(v)$ with ${\mathfrak X}(v')\in{\mathcal F}(v')$ and ${\mathfrak X}(v)\in{\mathcal F}(v)$ which is an isomorphism on corresponding rigid analytic spaces induces a morphism
\begin{numequation}\label{equ-mor}\pi_*{\mathscr M}_{{\mathfrak X}(v')}\longrightarrow {\mathscr M}_{{\mathfrak X}(v)}\end{numequation}
compatible with the morphism of rings \ref{equ-transit_sheafII} as follows. First of all, one has an isomorphism
$$\pi_*\left({\mathscr Loc}^\dagger_{{\mathfrak X}(v'),k',{\mathbb Q}}(M_{k',v'})\right)\stackrel{\simeq}{\longrightarrow}
\left(\pi_*{\widetilde{\sD}}^\dagger_{{\mathfrak X}(v'),k',{\mathbb Q}}\right)\otimes_{D^{an}({\mathbb G}_{v'}(k')^\circ)_{\theta_0}} M_{k',v'}.$$
Indeed, $\pi_*$ is exact by \ref{prop-exactdirectimage} and we may argue with finite presentations as usually.
Moreover, we have inclusions ${\mathbb G}_{v'}(k')\subseteq {\mathbb G}_{v}(k)$ and thus
$$V_{{\mathbb G}_{v}(k)^\circ-\rm an}\subseteq V_{{\mathbb G}_{v'}(k')^\circ-\rm an} \;.$$
\vskip8pt
The dual map $M_{k',v'}\rightarrow M_{k,v}$ is linear relative to the natural inclusion
$$D^{an}({\mathbb G}_{v'}(k')^\circ)\rightarrow D^{an}({\mathbb G}_{v}(k)^\circ) \;.$$
\vskip8pt
The latter inclusion is compatible with the morphism of rings \ref{equ-transit_sheafII} via taking global sections. Hence, we have a morphism \ref{equ-mor} as claimed. We now have everything at hand to follow the arguments in the proof of the preceding proposition word for word and to conclude
that the projective limit ${\mathscr M}_\infty$ has a $G$-action which extends its $G_0$-action and which makes it a $G$-equivariant ${\widetilde{\sD}}^\dagger_{\infty,{\mathbb Q}}$-module. This completes the proof of the theorem. \qed
\vskip8pt
We finally look at the special case of the whole adic flag variety ${\mathcal X}$ with its sheaf of infinite order differential operators ${\widetilde{\sD}}^\dagger_{{\mathcal X},{\mathbb Q}}$, cf. \ref{para-sheaf}. Recall the functor ${\mathscr Loc}^\dagger_{\mathcal X}$ from the end of \ref{subsec_G0}.
\begin{thm}\label{thm-main}
Let ${\mathcal X}$ be the adic analytic flag variety of ${\mathbb G}$ with its sheaf ${\widetilde{\sD}}^\dagger_{{\mathcal X},{\mathbb Q}}$ of infinite order differential operators.
The functors ${\mathscr Loc}^\dagger_{\mathcal X}$ and $\Gamma({\mathcal X},\cdot)$ are quasi-inverse equivalences between the categories of coadmissible $D(G_0)_{\theta_0}$-modules and coadmissible $G_0$-equivariant ${\widetilde{\sD}}^\dagger_{{\mathcal X},{\mathbb Q}}$-modules. The subcategories
of coadmissible $D(G)_{\theta_0}$-modules and coadmissible $G$-equivariant ${\widetilde{\sD}}^\dagger_{{\mathcal X},{\mathbb Q}}$-modules correspond to each other.
\end{thm}
{\it Proof. }
Taking each ${\mathcal F}(v)$ to be the set of {\it all} equivariant
formal models, we obtain ${\mathfrak X}_\infty={\mathcal X}$ according to \ref{prop-cofinal} and the result follows from the preceding theorem. \qed
\bibliographystyle{plain}
| -121,998.885099 |
[
-2.546875,
2.189453125
] | 52.12766 |
[
-3.0546875,
-0.1199951171875,
-2.39453125,
-5.61328125,
-0.76953125,
8.546875
] |
[
3.3203125,
9.9765625,
0.197509765625,
5.01171875
] | 824 | 11,758 |
[
-3.3984375,
3.904296875
] | 36.01351 |
[
-5.34375,
-3.763671875,
-5.65625,
-2.6875,
1.5810546875,
13.421875
] | 0.327251 | 40.992735 | 16.21024 | 3.350087 |
[
0.7058361768722534
] | -87,046.989946 | 6.003402 | -122,627.364535 | 1.396838 | 6.070471 |
[
-1.5673828125,
-3.3359375,
-4.2265625,
-5.7265625,
1.7060546875,
13.015625
] |
[
-5.734375,
-1.7099609375,
-1.77734375,
-0.71728515625,
3.34765625,
3.517578125
] | |
BkiUdhHxaKgQIQzdyQmI
|
\section{Introduction}
Microquasars are short-period stellar binaries in which one of the component is a compact object, i.e. a neutron star or a black-hole. The presence of such object is causing (sometimes persistent) strong radio jets and X-ray flares. In some cases, the relativistic jets appear to be superluminal, i.e. having an apparent speed larger than the speed of light. Microquasars are galactic laboratories of high-energy phenomena, and they must be seen as part of a large paradigm where AGNs, microquasars and possibly gamma-ray bursts (GRBs) share similar physics \citep{Mirabel-2004}. The main advantages of microquasars over their large-scale parents, is their short timescales, allowing much more detailed dynamical studies.
GRO~J1655--40 (a.k.a. Nova Sco 94) has been discovered as a Soft X-ray Transient (SXT) on July 27, 1994 with \emph{BATSE} on board the \emph{Compton Gamma Ray Observatory} \citep{Zhang-etal-1994}. Strong evidence that the compact object in GRO~J1655--40 ~is a black hole was presented by \citet{Bailyn-etal-1995b} \citep[see also][]{Balucinska-Church-2001}. They obtained through spectroscopy a mass function of the secondary (i.e. the absolute minimum mass of the compact object) of $f$ = 3.16$\pm$0.15 $M_{\odot}$; that has been later updated by \citet{Orosz-Bailyn-1997} to $f$ = 3.24$\pm$0.09 $M_{\odot}$. This value is above the theoretical maximum mass of a neutron star: $\sim$3 $M_{\odot}$ \citep{Arnett-Bowers-1977} \citep[but see also][ for a discussion on the dependance of this maximum mass with the equation of state]{Nauenberg-Chapline-1973,Prakash-etal-1988,vanKerkwijk-etal-1995}. Later, \citet{Shahbaz-etal-1999} revised down the mass function below this limit ($f$ = 2.73$\pm$0.09 $M_{\odot}$), but published the masses of the compact object and the secondary star: 5.5--7.9 $M_{\odot}$ and 1.7--3.3 $M_{\odot}$ respectively thanks to the inclination angle, clearly confirming the presence of a black-hole. The jets of GRO~J1655--40 were observed in radio, and given a distance of about 3 kpc, they appeared to be superluminal \citep[see e.g.][]{Rees-1966}: 1.5$\pm$0.4~$c$ \citep{Tingay-etal-1995}, 1.05~$c$ \citep{Hjellming-Rupen-1995}, where $c$ is the speed of light.
All studies until the most recent ones \citep[e.g.][]{Barret-etal-1996,Regos-etal-1998,vanderHooft-etal-1998,Phillips-etal-1999,Shahbaz-etal-1999,Kuulkers-etal-2000,Soria-etal-2000,Greene-etal-2001,Combi-etal-2001,Buxton-Vennes-2001,Yamaoka-etal-2001,Gierlinski-etal-2001,Kubota-etal-2001,Remillard-etal-2002,Kong-etal-2002,Kobayashi-etal-2003,Stevens-etal-2003,Willems-etal-2005,Brocksopp-etal-2006} usually quote the distance of 3.2$\pm$0.2~kpc determined by \citet{Hjellming-Rupen-1995}, although \citet{Mirabel-etal-2002} pointed out that a distance of about 1~kpc cannot be ruled out with the current data. At the time of writing, the latest discovery about GRO~J1655--40 is the fact that magnetic fields are the only explanation for the launch of jets in this object by \citet{Miller-etal-2006}, who also use the distance of 3.2 kpc. The purpose of this paper is to complement the work of \citet{Foellmi-etal-2006b-astroph}, who have shown that the distance of GRO~J1655--40 must be smaller than 1.7~kpc, but focusing on how the distance of 3.2~kpc has been determined, and to show that this value is far from being firm. Other distance methods will be discussed in a forthcoming paper.
\section{The Radio-Jet Kinematic Distance of GRO~J1655--40}
\label{radio-distance}
\citet{Hjellming-Rupen-1995} first mention three references to say that GRO~J1655--40 lies at a distance of roughly 3~kpc: \citet{Harmon-etal-1995}, \citet{Tingay-etal-1995} and \citet{McKay-Kesteven-1994}, which are discussed below. Then, the authors present new radio data obtained with the \emph{Very Large Array} (VLA) and the \emph{Very Long Baseline Array} (VLBA).
\subsection{The True Constraints from the Radio Data}
The 22 epochs of \emph{VLA} observations (see their Fig.~1) did not resolve the source at a level of 100 mas, but only a multi-cores object elongating with time. It is therefore necessary to use the more precise \emph{VLBA} observations. However, the authors emphasize clearly the lack of very-long-baseline interferometry calibrator, which implies that the data are self-calibrated, "eliminating all absolute positional information, and leaving the alignment of the different images a free parameter." Consequently, they must assume that the brightest point in each image is the stationary center of ejection. As mentioned in the paper, "these [VLA] and other [unspecified] data are consistent with constant intrinsic proper motion of 54 mas~d$^{-1}$" and later "the underlying proper motions appear constant". This is a very likely hypothesis, although it is only an hypothesis.
The authors mention that this hypothesis is strengthen by the fact that the (supposed) constant proper motions observed with the VLA and the VLBA agree. But what if this brightest point is not showing the true motion of the matter of the jets? Although very likely too, this hypothesis does not take into account the flux decays, and that the flux decay rate varies along the jet, as mentioned by the authors themselves. Moreover, the authors also mention the daily Southern Hemisphere VLBI Experiment (SHEVE) array observations of \citet{Tingay-etal-1995} which are consistent only with the major structures. The proper motion inferred from SHEVE data of 65$\pm$5 mas~d$^{-1}$ agrees with the 62 mas~d$^{-1}$ motion of the {\it outer edge} of the early NE ejecta. The relevance of choosing the brightest point of each VLBA image could then be questioned.
Finally, they build a kinematic model of the radio jets of GRO~J1655--40 ~based on these \emph{VLBA} observations. This is the method "C" of \citet{Jonker-Nelemans-2004}. They use the kinematic equation described in \cite{Mirabel-Rodriguez-1994} that link the apparent velocity of the receding and approaching radio jets ($\mu_+$ and $\mu_-$ respectively), and the distance $D$, the true jets velocity $\beta=v/c$ and the jet projection angle relative to the line of sight $\theta$:
\begin{equation}
\mu_{\pm} = \frac{\beta \sin(\theta)}{1 \pm \beta \cos(\theta)} \frac{c}{D}
\end{equation}
where $c$ is the speed of light. The problem with these relations is the number of known variables ($\mu_{\pm}$) and the number of unknowns ($\theta, D, v$). To find a distance, \citet{Hjellming-Rupen-1995} had to assume a value for the projection angle: $\theta = 85^{\circ}$. But a careful reading of this paper reveals that it is the maximum value allowed by the data, i.e. an upper limit only.
More precisely, the \emph{VLA} gives an estimation of the apparent true proper motion of the jets (assuming the two jets are moving apart with the same velocity) $v/D \sim 50$ mas~d$^{-1}$. It is then possible to rewrite the above equation as follows:
\begin{equation}
\mu_{\pm} = \frac{\sin(\theta)}{1 \pm \frac{v}{c} \cos(\theta)} \frac{v}{D}
\label{muplusmumoins2}
\end{equation}
and solving for $\theta$ by eliminating $v$. We obtain:
\begin{equation}
\sin \theta = \frac{1}{v/D} \, \left( \frac{2 \mu_+ \mu_-}{\mu_+ + \mu_-} \right) \leq 1
\label{eq4}
\end{equation}
From the measured values of $\mu_-$ and $\mu_+$ of 54 and 45 mas~d$^{-1}$ respectively with the \emph{VLBA} data, it means that $v/D \geq 49.1$ mas~d$^{-1}$, consistent with \emph{VLA} data.\footnote{The constraint on the inclination angle obtained by Hjellming \& Rupen is computed by eliminating $v/D$ in Eq.~\ref{muplusmumoins2}: $v/c = (\mu_- - \mu_+) (\mu_- + \mu_+)^{-1} \cos(\theta)^{-1} < 1$, which gives: $\theta \leq 84.8^{\circ}$.} Rewriting Eq.~\ref{eq4}, we obtain:
\begin{equation}
\frac{D}{c} \left( \frac{2 \mu_+ \mu_-}{\mu_+ + \mu_-} \right) \leq \frac{v}{c} < 1.
\end{equation}
where $c$ is the speed of light. Consequently, as noted by \citet{Mirabel-etal-2002}, only an upper limit to the distance can be obtained from the data: $D < c \, (\mu_+ + \mu_-)/(2 \mu_+ \mu_-) = 3.53 \; \textrm{kpc}$.
In summary, the data allow to derive an lower limit for the proper motion, and an upper limit for the inclination angle, and the distance. Quoting literally the paper of Hjellming \& Rupen: "For a distance of 3.2~kpc, this corresponds to $v \geq 0.91 c$, implying $84.3^{\circ} \leq \theta \leq 84.8^{\circ}$." Although not clearly stated, the reason why the authors chose 3.2 kpc is that this value is right in between their upper limit, and the lower limit given by {\it other} references.
\subsection{The True Constraints from the Quoted References}
As noted above, \citet{Hjellming-Rupen-1995} quote three other papers for a first estimation of the distance: \citet{Harmon-etal-1995}, \citet{Tingay-etal-1995} and \citet{McKay-Kesteven-1994}. Unfortunately, the first reference is quoting the two others for a distance value, and is therefore irrelevant for this issue. On the other hand, \citet{McKay-Kesteven-1994} is an IAU Circular in which it is simply stated that "HI observations of GRO~J1655--40 made with the \emph{AT Compact Array} show solid absorption in the velocity range $+10$ to $-30$ km~s$^{-1}$, with a further isolated weak feature at $-50$ km~s$^{-1}$. {\it The balance of probabilities is that the distance is around 3.5 kpc, unless the $-50$ km~s$^{-1}$ feature is due to an atypical cloud.}" This Circular is obviously not a measurement of the distance, and is based on the hypothesis that this feature at $-50$ km~s$^{-1}$ can be correctly interpreted as a HI cloud that is moving with the mean Galactic rotation.
This is exactly what \citet{Tingay-etal-1995} also do, independently. Interestingly, the authors seem to {\it expect} a rather large distance in order to agree with a supposedly "significant reddening due to absorption", which is quoted from another IAU Circular: \citet{dellaValle-1994}. What this latter Circular states is that: "[...] The [optical] spectrum exhibits prominent, broad Balmer lines [...] superimposed on a relatively red continuum. [...]" However, the spectrum has been taken during an active state of the object, whose characteristics were, at that time, unknown, and the reddening is not at all measured.
\citet{Tingay-etal-1995} present new \emph{VLBI} and \emph{ATCA} data of GRO~J1655--40. Their HI spectrum obtained with \emph{ATCA} (see their Fig.~2) shows a multi-component profile, with weak features at $-30$ and $-50$~km~s$^{-1}$ too. Attributing the absorption feature at $-30$~km~s$^{-1}$ to HI clouds participating in general Galactic rotation places a lower limit of 3.0 kpc on the distance. As noted by the authors, if the feature at $-50$~km~s$^{-1}$ was due to Galactic rotation, the implied minimum distance of GRO~J1655--40 would be of 4.2 kpc. But it couldn't be ruled out that this feature is due instead to HI driven by a $50$~km~s$^{-1}$ expanding shell or ionized material surrounding the adjacent Scorpius OB1 association, at a distance of 1.9 kpc \citep{Crawford-etal-1989}. They concluded that the minimum distance of GRO~J1655--40 is roughly around 3.0 kpc (obviously not providing uncertainties). Although it looks reasonable, this assumption might simply not be true. These measurements are very dependent on the distribution and velocities of various HI clouds along the line of sight; a difficulty that has been claimed to be important by \citet{Mirabel-etal-2002} who note that, in this direction, there are clouds with anomalous velocities up to $-50$ km~s$^{-1}$ \citep{Crawford-etal-1989}, i.e. with an amplitude similar to that of the weak feature observed in the \emph{ATCA} spectrum.
The distance of 3.2$\pm$0.2 kpc has certainly been chosen by \citet{Hjellming-Rupen-1995} as a reasonable value between the lower limit of 3.0 kpc given by \citet{Tingay-etal-1995} based on a very weak assumption about moving HI clouds (although their distance range seems to be strengthened by other quick observational reports), and the upper limit provided by their radio data.
\section{Conclusions and Prospects}
It is important to stress that the exact and accurate distance of GRO~J1655--40 is still an open question. Before the direct measurement by the european satellite \emph{GAIA}, it should be possible to test the distance value using methods that have never been used so far. In particular, we want to mention a new (and exploratory) possibility using bisectors.
First, \citet{Gray-2005} has shown that the bluemost point of single-line bisector is an indicator of the luminosity for late-type stars. This very interesting relation is weakened by the fact that high-quality single-line bisectors require a high spectroscopic resolution, and a very high Signal-to-Noise ratio (above, say, 300). However, the recent study by \citet{Dall-etal-2006} has shown that the bisector of the cross-correlation function (CCF) can be used as much the same way as single-line bisectors. The combination of these two results could lead to a first-order "direct" method of estimating the luminosity of the secondary star in GRO~J1655--40. Given the much larger brightness of GRO~J1655--40 in the near infrared (NIR, $m_K \sim 13$) than in the optical ($m_V \sim 17$), a CCF bisector could be obtained in a reasonable amount of time with the new near-IR echelle spectrograph on the VLT: CRIRES, which is being commissioned at the time of writing. Such investigation is already underway. This new method might also proved to be useful for many other faint and embedded stars in star-forming regions.
\acknowledgements
This work has been partly made in collaboration with T.H. Dall, E. Depagne and I.F. Mirabel. This research has made extensive use of NASA's Astrophysics Data System, of the ArXiv astro-ph, and the Central Bureau for Astronomical Telegrams of the IAU.
| -10,760.009423 |
[
-3.42578125,
3.14453125
] | 67.567568 |
[
-6.34765625,
-3.373046875,
-3.783203125,
-9.4140625,
2.34765625,
14.5390625
] |
[
2.6015625,
7.52734375,
5.23046875,
4.15625
] | 137 | 1,945 |
[
-2.587890625,
2.443359375
] | 27.77256 |
[
-5.94140625,
-4.5703125,
-2.73828125,
-0.84912109375,
2.78125,
9.390625
] | 1.04742 | 46.126153 | 36.195373 | 2.691512 |
[
3.230714797973633
] | -8,515.441563 | 5.399486 | -10,547.798584 | 0.52371 | 5.645962 |
[
-3.83984375,
-3.8984375,
-3.15625,
-3.46875,
2.71484375,
9.5078125
] |
[
-7.12109375,
-4.65234375,
-3.513671875,
-2.72265625,
5.28515625,
8.625
] | |
BkiUc-vxK7IDPXIhnLPg
|
\section{Introduction}
Recent progress in understanding specific risks faced by an entity is mainly rising from the emergence of models reflecting more precisely the entity and measures that are used to quantify and represent a company's global and granular structures. Risk measures are essential for insurance companies and financial institutions for several reasons such as quantifying capital requirements to protect against unexpected future losses and to set insurance premiums for all lines of business and risk categories. Different univariate risk measures have been proposed in the literature. The most common risk measures are Value-at-Risk (VaR) and Tail Value-at-Risk (TVaR). VaR, which represents an $\alpha$-level quantile, found its way through the G-30 report, see \citet{global1993derivatives} for details. \citet{heath1999coherent} show that VaR is not a coherent risk measure in addition to not providing any information about the tail of the distribution and thus suggests other specific risk measures such as TVaR, which evaluates the average value over all VaR values at confidence levels greater than $\alpha$, which is a significant measure for heavy-tailed distributions.
Dependencies between risks needs to be taken into account to obtain accurate capital allocation and systemic risk evaluation. For example, systemic risk refers to the risks imposed by interdependencies in a system. Univariate risk measures are not suitable to be employed for heterogeneous classes of homogeneous risks. Therefore, multivariate risk measures have been developed and gained popularity in the last decade.
The notion of quantile curves is employed in \citet{embrechts2006bounds}, \citet{nappo2009kendall} and \citet{prekopa2012multivariate} to define a multivariate risk measure called upper and lower orthant VaR. Based on the same idea, \citet{cossette2013bivariate} redefine the lower and upper orthant VaR and \citet{cossette2015vector} propose the lower and upper orthant TVaR. \citet{cousin2013multivariate} develop a finite vector version of the lower and upper orthant VaR. A drawback of multivariate VaR is that it represents the boundary of the $\alpha$-level set and no additional tail information is provided, similar to the univariate VaR. Furthermore, relationships holding for univariate risk measures can be different in a multivariate setting.
Most risk measures are defined as functions of the loss distribution which should be estimated from the data. In \citet{cont2010robustness}, risk measurement procedures are defined and analysis of robustness of different risk measures is performed. They point out the conflict between the subadditivity and robustness and propose a robust risk measure called weighted VaR (WVaR). The use of a truncated version of TVaR, defined as Range-Value-at-Risk (RVaR), is suggested by \citet{bignozzi2016parameter}. The lower and upper orthant RVaR in the multivariate setting are developed in this paper, in order to provide a new robust multivariate risk measure. We aim to study in details their properties and derive their estimators. We will also focus on extreme value distributions which can be used to model the heavy tail of the data.
The paper is organized as follows. In Section \ref{Section:Preliminaries}, definitions and properties of the univariate $\RVaR$ are given, with examples in the Extreme Value Theory (EVT) framework. Sections \ref{Section:LowerRVaR} and \ref{Section:UpperRVaR} define the multivariate lower and upper orthant $\RVaR$, respectively. Section \ref{Section:PropertiesRVaR} presents their interesting and desirable properties, such as their behavior under transformations or translation of the multivariate variables and monotonicity. We also develop asymptotic results, the behavior with aggregate risks and we prove their robustness. In Section \ref{Section:RVaREmpirical}, we define the empirical estimator of the lower and upper orthant $\RVaR$ and we illustrate the accuracy of this estimator graphically. Concluding remarks are given in Section \ref{Section:Conclusion}
\section{Preliminaries}\label{Section:Preliminaries}
In this section, we present the univariate definition of RVaR and provide resulting measures, based on univariate RVaR, in the EVT framework and in asymptotic scenarios.
\subsection{Univariate RVaR}
Consider a random loss variable $X$ on a probability space $(\Omega,\mathcal{F},\mathbb{P})$ with its cumulative distribution function (cdf) $F_X$. A risk measure $\rho(X)$ for a random variable (r.v.) $X$ corresponds to the required amount that has to be maintained such that the financial position $\rho(X)-X$ is acceptable. Since there are several definitions of risk measures, an appropriate choice becomes crucial for stakeholders.
\begin{Definition}
For a continuous random variable $X$ with cumulative distribution function (cdf) $F_X$, the univariate Range Value-at-Risk at significance level range $[\alpha_1, \alpha_2]\subseteq[0,1]$ is defined by
$$\RVaR_{\alpha_1,\alpha_2}(X)=\mathbb{E}\left[X|\VaR_{\alpha_1}(X)\leq X\leq \VaR_{\alpha_2}(X)\right]=\frac{1}{\alpha_2 -\alpha_1}\int^{\alpha_2}_{\alpha_1}\VaR_u(X)du,$$
where
$$\VaR_\alpha(X)=\inf{\{x\in\mathbb{R}:F_X(x)\geq\alpha\}}$$
is the univariate Value-at-Risk at significance level $\alpha\in[0,1]$.
\end{Definition}
For a continuous random variable $X$ with strictly increasing cdf, $\VaR_\alpha(X)=F^{-1}_X(\alpha)$, is also called the $\alpha$-quantile, where $F^{-1}_X$ is the inverse function of cdf. VaR fails to give any information beyond the level $\alpha$, However, RVaR quantifies the magnitude of the loss of the worst $100(1-\alpha_1)$ to $100(1-\alpha_2)$ cases. When $\alpha_2=1$, we obtain a special case of RVaR, which is referred to as TVaR in this article.
Robust statistics can be defined as statistics that are not unduly affected by outliers. In order to establish the robustness of RVaR, we need to define a measure of affectation. Consider a continuous random variable $X$ with cdf $F\in\mathbb{D}$ where $\mathbb{D}$ is the convex set of cdfs. Notice that a risk measure is distribution-based if $\rho(X_1)=\rho(X_2)$ when $F_{X_1}=F_{X_2}$. Hence, we use $\rho(F)\triangleq\rho(X)$ to represent the distribution-based risk measures. To quantify the sensitivity of a risk measure to the change in the distribution, we use the sensitivity function. This method is used by \citet{cont2010robustness} and can be explained as the one-sided directional derivative of the effective risk measure at $F$ in the direction $\delta_z$.
\begin{Definition}
Consider $\rho$, a distribution-based risk measure of a continuous random variable $X$ with distribution function $F\in\mathbb{D}$ where $\mathbb{D}$ is the convex set of cdfs. For $\varepsilon\in[0,1)$, set $F_\varepsilon=\varepsilon\delta_z+(1-\varepsilon)F$ such that $F_\varepsilon\in\mathbb{D}$. $\delta_z\in\mathbb{D}$ is the probability measure which gives a mass of 1 to $\{z\}$. The distribution $F_\varepsilon$ is differentiable at any $x\neq z$ and has a jump point at the point $x=z$. The sensitivity function is defined by
$$S(z)=S(z;F)\triangleq\lim_{\varepsilon\rightarrow 0^+}\frac{\rho(F_\varepsilon)-\rho(F)}{\varepsilon},$$
for any $z\in\mathbb{R}$ such that the limit exists.
\end{Definition}
The value of sensitivity function for a robust statistic will not go to infinity when $z$ becomes arbitrarily large. In other word, the bounded sensitivity function makes sure that the risk measure will not blow up when a small change happens. Accordingly, \citet{cont2010robustness} show that VaR and RVaR are robust statistics by showing that their respective sensitivity functions are bounded.
\subsection{Examples of univariate RVaR in Extreme Value Theory}\label{Section: EVT}
In this section, we will provide some examples for the discussed risk measures in the EVT framework. Most of the statistical techniques are focused on the behavior of the center of the distribution, usually the mean. However, EVT is a branch in statistics that is focused on the behavior of the tail of the distribution. There are two principle models for extreme values; the block maxima model and the peaks-over-threshold model. The \textit{block maxima} approach is used to model the largest observations from samples of identically distributed observations in successive periods.
The \textit{peaks-over-threshold} is used to model all large observations that exceed a given high threshold value, denoted $u$.
The limiting distribution of block maxima, from \citet{fisher1928limiting}, is given in the theorem below;
\begin{Theorem}\label{Thm: FisherTippet}
Let $X_1, \ldots, X_n$ be a sequence of independent random variables having a common distribution function $F$ and consider $M_n = \max\lbrace X_1,\ldots,X_n\rbrace$. If there exists norming constants $(a_n)$ and $(b_n)$ where $a_n\in\mathbb{R}$ and $b_n>0$ for all $n\in\mathbb{N}$ and some non-degenerate distribution function $H$ such that $$\frac{M_n-a_n}{b_n}\xrightarrow{\text{d}} H,$$
then $H$ is defined as the Generalized Extreme Value Distribution (GEV) given by
\begin{align*}
H_{\xi}(x)&=
\begin{dcases}
\exp \left\lbrace -\left(1+\xi x \right)^{-\frac{1}{\xi}}\right\rbrace, &\ \ \xi\neq0,\\
\exp\left\lbrace-\exp(-x)\right\rbrace, &\ \ \xi=0,\\
\end{dcases}
\end{align*}
where $1+\xi x>0$. A three-parameter family is obtained by defining $H_{\xi,\mu,\sigma} :=H_{\xi}\left( \frac{x-\mu }{\sigma} \right)$ for a location parameter $\mu \in \mathbb{R}$, a scale parameter $\sigma >0$, and a shape parameter $\xi \in \mathbb{R}$.
\begin{comment}
then $H$ belongs to one of the following three classes of distributions (up to location and scaling):
\begin{align*}
\text{Fr\'echet:} \ \ \ &\Phi_\alpha(x) = \begin{dcases}
0, &\ \ x\leq0,\\
\exp\left\lbrace-x^{-\alpha}\right\rbrace, &\ \ x>0,\\
\end{dcases}\ \ \ \alpha>0,\\
\text{Gumbel:} \ \ \ &\Lambda(x) = \exp\left\lbrace-\exp\lbrace-x\rbrace\right\rbrace, \ \ \ x\in\mathbb{R},\\
\text{Weibull:} \ \ \ &\Psi_\alpha(x) = \begin{dcases}
\exp\left\lbrace-(-x)^{\alpha}\right\rbrace, &\ x\leq0,\\
1, &\ x>0,\\
\end{dcases}\ \ \ \alpha>0.
\end{align*}
\end{comment}
\end{Theorem}
\begin{comment}
\noindent Given that in reality we do not know the distribution $F$ and its maxima limiting case, the below one-parameter distribution provides a generalization of Theorem \ref{Thm: FisherTippet}. Set $\xi=\alpha^{-1}$ for the Fr\'echet distribution, $\xi=-\alpha^{-1}$ for the Weibull distribution and interpret the Gumbel distribution as a limiting case as $\xi\to0$ \citet{jenkinson1955frequency,VonMises1954}, then we obtain the following definition.
\end{comment}
\begin{comment}
\begin{Definition}
The distribution function of a Generalized Extreme Value Distribution (GEV) is given by
\begin{align*}
H_{\xi}(x)&=
\begin{dcases}
\exp \left\lbrace -\left(1+\xi x \right)^{-\frac{1}{\xi}}\right\rbrace, &\ \ \xi\neq0,\\
\exp\left\lbrace-\exp(-x)\right\rbrace, &\ \ \xi=0,\\
\end{dcases}
\end{align*}
where $1+\xi x>0$. A three-parameter family is obtained by defining $H_{\xi,\mu,\sigma} :=H_{\xi}\left( \frac{x-\mu }{\sigma} \right)$ for a location parameter $\mu \in \mathbb{R}$, a scale parameter $\sigma >0$, and a shape parameter $\xi \in \mathbb{R}$.
\end{Definition}
\end{comment}
\noindent The one-parameter GEV is the limiting distribution of the normalized maxima, but in reality, we do not know the norming constants $(a_n)$ and $(b_n)$, therefore, the three-parameter GEV provides a more general and flexible approach as it is the limiting distribution of the unnomarlized maxima.
\citet{pickands1975statistical} and \citet{balkema1974residual} show that the theorem below provides a very powerful result regarding the excess distribution function;
\begin{Theorem}\label{Thm:GPD}
Let $X$ be a random variable with distribution function $F$ and an upper end-point $x_F \leq \infty$. If $F$ is a distribution function that belongs to the maximum domain attraction of a GEV distribution $H_{\xi,\mu,\sigma}$, then
$$\lim_{u\to x_F}\sup_{0\leq x<x_F-u}\left| F_u(x) -G_{\xi,\sigma}(x) \right|=0.$$
where \begin{align*}
F_u(x) = \Pr(X-u\leq x|X>u)=\frac{F(x+u)-F(u)}{1-F(u)}, \ \ \ 0\leq x< x_F-u.
\end{align*}
is the excess distribution over the threshold $u$ and \begin{align*}
G_{\xi,\sigma}(x)&=
\begin{dcases}
1-\left(1+\xi\frac{x}{\sigma} \right)^{-\frac{1}{\xi}}, &\ \ \xi\neq0,\\
1-\exp\left(-\frac{x}{\sigma} \right), &\ \ \xi=0.\\
\end{dcases}
\end{align*}
for $\sigma>0$, and $x\geq0$ when $\xi\geq0$, while $0\leq x\leq-\frac{\sigma}{\xi}$ when $\xi<0$. The parameters $\xi$ and $\sigma$ are referred to, respectively, as the shape and scale parameters.
\end{Theorem}
\noindent This essentially implies that $F_u \approx G_{\xi,\sigma}$ if $u$ is high enough, where $G_{\xi,\sigma}$ is called the Generalized Pareto Distribution (GPD).
\begin{Proposition}\label{Proposition:UnivariateRiskMeasuresGEV}
Assume $X \sim \text{GEV}(\mu,\sigma,\xi)$, then for $0\leq\alpha\leq1$
\begin{align*}
\VaR_{\alpha}(X)&=
\begin{dcases}
\mu -\frac{\sigma}{\xi}\left[1-\left(-\ln \alpha\right)^{-\xi}\right] &\ \ \xi\neq0,\\
\mathbb{E}[X] -\frac{\sigma}{\xi}\left[ \Gamma(1-\xi) -(-\ln\alpha)^{-\xi}\right]&\ \ \xi\neq0, \xi<1,\\
\mathbb{E}[X] - \sigma\gamma - \sigma\ln(-\ln\alpha)&\ \ \xi=0,
\end{dcases}
\end{align*}
where
\begin{align*}
\mathbb{E}[X] &= \begin{dcases}
\mu + \frac{\sigma}{\xi}(\Gamma(1-\xi)-1)&\ \ \xi\neq0, \xi<1,\\
\mu + \sigma \gamma &\ \ \xi=0,\\
\infty &\ \ \xi\geq1,\\
\end{dcases}
\end{align*}
where $\Gamma(x,a)$ is the incomplete Gamma function $\Gamma(x,a)=\int_a^\infty t^{x-1}e^{-t}dt$ such that $\Gamma(x)=\Gamma(x,0)$ and $\gamma$ is Euler's constant defined by $\gamma = \int_1^\infty \left(-\frac{1}{x}+\frac{1}{\lfloor x \rfloor}\right)dx$. VaR diverges for $\xi\geq1$.\\
Let $0\leq\alpha_1\leq\alpha_2\leq1$, then
\begin{align*}
\RVaR_{\alpha_1,\alpha_2}(X)=
\begin{cases}
\mu - \frac{\sigma}{\xi(\alpha_2-\alpha_1)}\left[(\alpha_2-\alpha_1)-\Gamma(1-\xi,-\log \alpha_2) +\Gamma(1-\xi,-\log \alpha_1) \right]&\ \ \xi\neq0,\\
\VaR_{\alpha_i}(X) - \frac{\sigma}{\alpha_2-\alpha_1} \left[\ln(-\ln \alpha_2))-\alpha_j(\ln(-\ln \alpha_1) - \text{li}(\alpha_2) + \text{li}(\alpha_1)\right] &\ \ \xi=0,\\
\end{cases}
\end{align*}
for $i,j\in \lbrace 1,2\rbrace$, $i\neq j$ and where $\text{li}(x)$ is the logarithmic integral $\text{li}(x) = \int_0^x \frac{1}{\ln(t)} dt$ for $0<x<1$ and has a singularity at $x=1$.\\
As a special case, let $\alpha_2=1$, then for $\xi \neq 0$ and $\xi<1$, we have that
\begin{eqnarray*}
\TVaR_{\alpha}(X)&=& \mathbb{E}[X] -\frac{\sigma}{\xi(1-\alpha)}\left[ \Gamma(1-\xi, -\ln\alpha) -\alpha\Gamma(1-\xi) \right] \\
&=& \VaR_{\alpha}(X) -\frac{\sigma}{\xi(1-\alpha)}\left[(1-\alpha)(-\ln\alpha)^{-\xi} +\Gamma(1-\xi, -\ln\alpha) - \Gamma(1-\xi) \right],
\end{eqnarray*}
and $\TVaR$ diverges for $\xi=0$ and $\xi\geq1$.\\
\end{Proposition}
\begin{proof}
See Appendix \ref{Appendix:Proof}.
\end{proof}
\noindent In addition to the risk measures, it is interesting to observe how the ratio of the risk measures behaves for large confidence levels $\alpha$.
\begin{Proposition}\label{Proposition:GEV_RVaR_limit_limit}
Assume $X\sim GEV(\mu,\sigma,\xi)$. Let $0\leq\alpha_1\leq\alpha_2\leq1$, then
\begin{align*}
\lim_{\alpha_1 \to 1}\left[\lim_{\alpha_2 \to 1} \frac{\RVaR_{\alpha_1,\alpha_2}(X)}{\VaR_{\alpha_1}(X)}\right]=\lim_{\alpha_1 \to 1}\left[ \frac{\TVaR_{\alpha_1}(X)}{\VaR_{\alpha_1}(X)}\right] &=\begin{dcases}
(1 -\xi)^{-1} &\ \ \xi>0,\\
1 &\ \ \xi<0.\\
\end{dcases}
\end{align*}
\end{Proposition}
\begin{proof}
See Appendix \ref{Appendix:Proof}.
\end{proof}
Hence, the shape parameter $\xi$ is a strong factor that affects this ratio for large values of $\alpha$. For $\xi<0$, TVaR approaches the value of VaR for high values of $\alpha$, while for $\xi>0$, TVaR becomes significantly larger than VaR.\\
\begin{Proposition}\label{Proposition:UnivariateRiskMeasuresGPD}
Consider the random variable $X$ with cdf $F$ and survival $\bar{F}$. Assume $F \in\text{MDA}(H_{\xi})$, then for $0\leq\alpha\leq1$ and $x\geq u$, where $u$ is a high threshold, we have the following
\begin{align*}
\VaR_{\alpha}(X)
&=\begin{dcases}
u+\frac{\sigma}{\xi}\left(\left(\frac{1-\alpha}{\zeta_u}\right)^{-\xi} -1\right)&\ \ \xi\neq0,\\
u-\sigma\log\left(\frac{1-\alpha}{\zeta_u}\right) &\ \ \xi=0.\\
\end{dcases}
\end{align*}
Let $0\leq\alpha_1\leq\alpha_2\leq1$, then for any value of $\xi$, we have that $$\RVaR_{\alpha_1,\alpha_2}(X) = \dfrac{(1-\alpha_1)\VaR_{\alpha_1}(X) - (1-\alpha_2)\VaR_{\alpha_2}(X) }{(\alpha_2-\alpha_1)(1-\xi)}+\frac{ (\sigma-\xi u)}{(1-\xi)}.$$
As a special case, let $\alpha_2=1$, then for $\xi<1$
$$ \TVaR_{\alpha}(X)=\frac{ \VaR_\alpha(X)}{1-\xi} + \frac{ \sigma-\xi u}{1-\xi},$$
where $\zeta_u=\bar F(u)=\mathbb{P}(X>u)$, and TVaR is infinite for $\xi\geq1$.
\end{Proposition}
\begin{proof}
See Appendix \ref{Appendix:Proof}.
\end{proof}
\noindent $\RVaR$ has not been explored in the literature of Extreme Value Theory. Therefore, we have derived a closed form expression for $\RVaR$. Even though TVaR is infinite for values of $\xi>1$, RVaR exists. Thus, RVaR is useful for the cases where $\xi\geq1$, due to its ability to capture an expected value over a range of high extremes. In theory, this might not be representative of the heavy tail, however, in practice, this can be used to eliminate the issue of having an infinite mean for real data, i.e. insurance companies and financial institutions would still be interested in calculating their reserves and economic capital, and it is not possible to hold an infinite amount of reserves. In this scenario, RVaR can be used with high values of $\alpha_1$ and $\alpha_2$.\\
\noindent In addition to the results of VaR and TVaR, it is interesting to observe how the ratio of the two risk measures behaves for large confidence levels $\alpha$.
\begin{comment}
\begin{Proposition}\label{Proposition:GPD_TVaR_Limit}
Consider the random variable $X$ with cdf $F$. Assume $F \in\text{MDA}(H_{\xi})$, then for $0\leq\alpha\leq1$, $\xi<1$ and $x\geq u$, where $u$ is a high threshold, we have the following
\begin{align*}
\lim_{\alpha \to 1} \frac{\TVaR_\alpha(X)}{\VaR_\alpha(X)}
&=\begin{dcases}
(1-\xi)^{-1}&\ \ \xi\geq0,\\
1 &\ \ \xi<0.\\
\end{dcases}
\end{align*}
\end{Proposition}
\begin{proof}
See Appendix \ref{Appendix:Proof}.
\end{proof}
\begin{Proposition}\label{Proposition:GPD_RVaR_limit}
Consider the random variable $X$ with cdf $F$. Assume $F \in\text{MDA}(H_{\xi})$, then for $0\leq\alpha\leq1$ and $x\geq u$, where $u$ is a high threshold, we have the following
\begin{align*}
\lim_{\alpha_2 \to 1} \frac{\RVaR_{\alpha_1,\alpha_2}(X)}{\VaR_{\alpha_2}(X)}
&=\begin{dcases}
0&\ \ \xi\geq0,\\
1 + \frac{\frac{\sigma}{\xi}\left(\frac{1-\alpha_1}{\zeta_u}\right)^{-\xi}}{(1-\xi)\left(u-\frac{\sigma}{\xi} \right)} &\ \ \xi<0,\\
\end{dcases}
\end{align*}
\end{Proposition}
\begin{proof}
See Appendix \ref{Appendix:Proof}.
\end{proof}
\end{comment}
\begin{Proposition}\label{Proposition:GPD_RVaR_limit_limit}
Consider the random variable $X$ with cdf $F$. Assume $F \in\text{MDA}(H_{\xi})$, then for $0\leq\alpha\leq1$, $\xi<1$ and $x\geq u$, where $u$ is a high threshold, we have the following
\begin{align*}
\lim_{\alpha_1 \to 1}\left[\lim_{\alpha_2 \to 1} \frac{\RVaR_{\alpha_2,\alpha_1}(X)}{\VaR_{\alpha_1}(X)}\right]=\lim_{\alpha_1 \to 1}\left[ \frac{\TVaR_{\alpha_1}(X)}{\VaR_{\alpha_1}(X)}\right]
&=\begin{dcases}
(1 -\xi)^{-1} &\ \ \xi\geq0,\\
1 &\ \ \xi<0.\\
\end{dcases}
\end{align*}
\end{Proposition}
\begin{proof}
See Appendix \ref{Appendix:Proof}.
\end{proof}
Hence, similar to the GEV distribution, the shape parameter $\xi$ is a strong factor that affects this ratio for large values of $\alpha$.\\
\section{Multivariate Lower and Upper Orthant RVaR}\label{Section:MultiRVaR}
In this section, we define the multivariate lower and upper orthant RVaR and study their properties. Examples and illustrations of the findings are provided. Finally, empirical estimators are presented.
\subsection{Lower Orthant RVaR}\label{Section:LowerRVaR}
Consider the continuous random vector $\boldsymbol{X}=(X_1,X_2,\ldots,X_d)\in \mathbb{R}_{+}^{d}$ with joint CDF $F$ and joint survival function $\bar{F}$. Define the random vector $\boldsymbol{X}_{\setminus i}=(X_1,\ldots,X_{i-1},X_{i+1},\ldots,X_d)$ with joint cdf $F_{\setminus i}$ and joint survival function $\bar{F}_{\setminus i}$, for $i=1,\ldots,d$. Let $\boldsymbol x=(x_1,\ldots,x_d)$ be a realization of $\boldsymbol X$ and consider the vector $\boldsymbol{x}_{\setminus i}=(x_1,\ldots,x_{i-1},x_{i+1},\ldots,x_d)$.
\begin{Definition}\label{Definition: LowerRVar}
Consider a continuous random vector $\boldsymbol{X}=(X_1,X_2)$ on
the probability space $(\Omega,\mathcal{F},\mathbb{P})$ with a joint cdf $F$. The lower orthant RVaR at significance level range $[\alpha_1, \alpha_2]\subseteq[0,1]$ is given by
$$\underline{\RVaR}_{\alpha_1,\alpha_2} (\boldsymbol{X})= \bigcup_{i=1}^d \left\lbrace \left(x_1,\ldots,x_{i-1},\underline{\RVaR}_{\alpha_1,\alpha_2,\boldsymbol{x}_{\setminus i}} (\boldsymbol{X})),x_{i+1},\ldots,x_d\right)\right\rbrace,$$
where
$$\underline{\RVaR}_{\alpha_1,\alpha_2,\boldsymbol{x}_{\setminus i}}(\boldsymbol{X})=\mathbb{E}[X_i|\underline{\VaR}_{\alpha_1,\boldsymbol{x}_{\setminus i}}(\boldsymbol{X})\leq X_i\leq \VaR_{\alpha_2}(X_i), \boldsymbol{X}_{\setminus i}\leq \boldsymbol{x}_{\setminus i}],$$
for
$$\underline{\VaR}_{\alpha_1,\VaR_{\alpha_2}(\boldsymbol{X}_{\setminus i})}(\boldsymbol{X})\leq x_j\leq \VaR_{\alpha_2}(X_j),\quad \text{for all } j=1,\ldots,d, i\neq j,$$
in which the lower orthant VaR at significance level $\alpha$ is defined by
$$\underline{\VaR}_{\alpha}(\boldsymbol{X}) = \bigcup_{i=1}^d \left\lbrace \left(x_1,\ldots,x_{i-1},\underline{\VaR}_{\alpha,\boldsymbol{x}_{\setminus i}} (\boldsymbol{X})),x_{i+1},\ldots,x_d\right):x_j \geq \VaR_{\alpha}(X_j),\forall j\neq i\right\rbrace,$$
where
$$\underline{\VaR}_{\alpha,\boldsymbol{x}_{\setminus i}} (\boldsymbol{X})=\inf\left\lbrace x_i \in \mathbb{R}: F_{\boldsymbol{x}_{\setminus i}}(X_i)\geq \alpha \right\rbrace.$$
\end{Definition}
\begin{Proposition}\label{Proposition:RVaRLowerIntegration}
For a continuous random vector $\boldsymbol{X}=(X_1,\ldots,X_d)$ with joint cdf $F$ and for the subvector $\boldsymbol{X}_{\setminus i}=(X_1,\ldots,X_{i-1},X_{i+1},\ldots,X_d)$ with joint cdf $F_{\setminus i}$, $\underline{\RVaR}_{\alpha_1,\alpha_2,\boldsymbol{x}_{\setminus i}}(\boldsymbol{X})$ can be restated as
$$\underline{\RVaR}_{\alpha_1,\alpha_2,\boldsymbol{x}_{\setminus i}} (\boldsymbol{X})=\frac{1}{F(\boldsymbol{x}_{\setminus i},\VaR_{\alpha_2}(X_i))-\alpha_1}\int_{\alpha_1}^{F(\boldsymbol{x}_{\setminus i},\VaR_{\alpha_2}(X_i))}\underline{\VaR}_{u,\boldsymbol{x}_{\setminus i}}(\boldsymbol{X})du,$$
for
$$\underline{\VaR}_{\alpha_1,\VaR_{\alpha_2}(\boldsymbol{X}_{\setminus i})}(\boldsymbol{X})\leq x_j\leq \VaR_{\alpha_2}(X_j),\quad \text{for all } j=1,\ldots,d, \quad i\neq j.$$
\end{Proposition}
\begin{proof}
See Appendix \ref{Appendix:Proof}.
\end{proof}
\begin{Example}\label{Example:RVaR_Lower}
Consider the random vector $(X_1,X_2)$ with joint cdf defined with a Gumbel copula with dependence parameter $\theta=1.5$ and marginals $X_1\sim$ Weibull (2, 50) and $X_2\sim$ Weibull (2, 150). Let the confidence level range be $\alpha_1=0.95$ and $\alpha_2=0.99$. Then, we get bivariate lower orthant $\RVaR$ in Figure \ref{Figure5}. For comparison, we plot $\underline{\VaR}_{0.95,x_i}(\boldsymbol{X})$ on the same graph.
\end{Example}
\begin{figure}%
\hfill
\subfigure[]{\includegraphics[width=8cm]{5}}%
\hfill
\subfigure[]{\includegraphics[width=8cm]{7}}%
\hfill
\caption{(a) Lower orthant VaR at level 0.95 and RVaR at level range $[0.95,0.99]$ for fixed values of $X_1$ and (b) Lower orthant VaR at level 0.95 and RVaR at level range $[0.95,0.99]$ for fixed values of $X_2$}%
\label{Figure5}%
\end{figure}
One can observe from Figure \ref{Figure5} that $\underline{\RVaR}_{\alpha_1,\alpha_2,x_i}(\boldsymbol{X})$ converges to the univariate RVaR when $x_i$ ($i=1,2$) approaches infinity. Also, when $x_i$ gets close to $\VaR_{\alpha_1}(X_i)$, $\underline{\RVaR}_{\alpha_1,\alpha_2,x_i}(\boldsymbol{X})$ approaches $\VaR_{\alpha_2}(X_j)$.\\
By letting $\alpha_2=1$, a special case of the lower orthant RVaR is obtained, namely the lower orthant TVaR, as defined, studied and illustrated by \citet{cossette2015vector}.
\subsection{Upper Orthant RVaR}\label{Section:UpperRVaR}
\begin{Definition}
Consider a continuous random vector $\boldsymbol{X}=(X_1,X_2)$ on
the probability space $(\Omega,\mathcal{F},\mathbb{P})$ with a joint cdf $F$. The upper orthant $\RVaR$ at significance level range $[\alpha_1, \alpha_2]\subseteq[0,1]$ is given by
$$\overline{\RVaR}_{\alpha_1,\alpha_2} (\boldsymbol{X})= \bigcup_{i=1}^d \left\lbrace \left(x_1,\ldots,x_{i-1},\overline{\RVaR}_{\alpha_1,\alpha_2,\boldsymbol{x}_{\setminus i}} (\boldsymbol{X})),x_{i+1},\ldots,x_d\right)\right\rbrace,$$
where
$$\overline{\RVaR}_{\alpha_1,\alpha_2,\boldsymbol{x}_{\setminus i}}(\boldsymbol{X})=\mathbb{E}[X_i|\VaR_{\alpha_1}(X_i)\leq X_i\leq \overline{\VaR}_{\alpha_2,\boldsymbol{x}_{\setminus i}}(\boldsymbol{X}), \boldsymbol{X}_{\setminus i}\geq \boldsymbol{x}_{\setminus i}],$$
for
$$\VaR_{\alpha_1}(X_j)\leq x_j\leq \overline{\VaR}_{\alpha_2,\VaR_{\alpha_1}(\boldsymbol{X}_{\setminus i})}(\boldsymbol{X}),\quad \text{for all } j=1,\ldots,d, i\neq j,$$
in which the upper orthant $\VaR$ at significance level $\alpha$ is defined by
$$\overline{\VaR}_{\alpha}(\boldsymbol{X}) = \bigcup_{i=1}^d \left\lbrace \left(x_1,\ldots,x_{i-1},\overline{\VaR}_{\alpha,\boldsymbol{x}_{\setminus i}} (\boldsymbol{X})),x_{i+1},\ldots,x_d\right):x_j \leq \VaR_{\alpha}(X_j),\forall j\neq i\right\rbrace,$$
where
$$\overline{\VaR}_{\alpha,\boldsymbol{x}_{\setminus i}} (\boldsymbol{X}))=\inf\left\lbrace x_i \in \mathbb{R}: \overline{F}_{\boldsymbol{x}_{\setminus i}}(X_i)\leq 1- \alpha \right\rbrace.$$
\end{Definition}
Similar to the lower orthant RVaR we can define the upper orthant RVaR
in the form of the integration of $\overline{\VaR}_{\alpha,x_i}(\boldsymbol{X})$.\\
\begin{Proposition}\label{Proposition:RVaRUpperIntegration}
For a continuous random vector $\boldsymbol{X}=(X_1,\ldots,X_d)$ with joint cdf $F$ and for the subvector $\boldsymbol{X}_{\setminus i}=(X_1,\ldots,X_{i-1},X_{i+1},\ldots,X_d)$ with joint cdf $F_{\setminus i}$, $\overline{\RVaR}_{\alpha_1,\alpha_2,\boldsymbol{x}_{\setminus i}}(\boldsymbol{X})$ can be restated as
$$\overline{\RVaR}_{\alpha_1,\alpha_2,\boldsymbol{x}_{\setminus i}} (\boldsymbol{X})=\frac{1}{\alpha_2 - (1-\overline{F}(\boldsymbol{x}_{\setminus i},\VaR_{\alpha_1}(X_i)))}\int^{\alpha_2}_{1-\overline{F}(\boldsymbol{x}_{\setminus i},\VaR_{\alpha_1}(X_i))}\overline{\VaR}_{u,\boldsymbol{x}_{\setminus i}}(\boldsymbol{X})\text{d}u,$$
for
$$\VaR_{\alpha_1}(X_j)\leq x_j\leq \overline{\VaR}_{\alpha_2,\VaR_{\alpha_1}(\boldsymbol{X}_{\setminus i})}(\boldsymbol{X}),\quad \text{for all } j=1,\ldots,d, i\neq j,$$
\end{Proposition}
\begin{proof}
See Appendix \ref{Appendix:Proof}.
\end{proof}
\begin{Example}
Consider the same random vector defined in Example~\ref{Example:RVaR_Lower}. Let the confidence level range be $\alpha_1=0.95$ and $\alpha_2=0.99$. Then, we get bivariate upper orthant $\RVaR$ in Figure \ref{Figure6}. For comparison, we plot $\overline{\VaR}_{0.99,x_i}(\boldsymbol{X})$ on the same graph.
\end{Example}
\begin{figure}%
\hfill
\subfigure[]{\includegraphics[width=8cm]{6}}%
\hfill
\subfigure[]{\includegraphics[width=8cm]{8}}%
\hfill
\caption{(a) Upper orthant VaR at level 0.99 and RVaR at level range $[0.95,0.99]$ for fixed values of $X_1$ and (b) Upper orthant VaR at level 0.99 and RVaR at level range $[0.95,0.99]$ for fixed values of $X_2$}%
\label{Figure6}%
\end{figure}
One observes from Figure \ref{Figure5} that $\overline{\RVaR}_{\alpha_1,\alpha_2,x_i}(\boldsymbol{X})$ converges to the univariate $\RVaR_{0.95,0.99}(X_j)$ when $x_i$ ($i=1,2$) gets close to the lower support of $X_i$. Also, when $x_i$ approaches $\VaR_{\alpha_2}(X_i)$, $\overline{\RVaR}_{\alpha_1,\alpha_2,x_i}(\boldsymbol{X})$ approaches $\VaR_{\alpha_1}(X_j)$. As a result, the curves of bivariate RVaR are bounded by the curves of univariate $\VaR$, which is similar to the univariate RVaR.\\
Analogously as for the lower orthant case, letting $\alpha_2=1$, is a special case of the upper orthant which leads to the upper orthant TVaR.
\subsection{Properties of Multivariate Lower and Upper Orthant RVaR}\label{Section:PropertiesRVaR}
For simplicity of notation and proofs, we will consider the bivariate case.
\begin{Proposition}\label{Proposition:Properties}
Let $\boldsymbol{X}=(X_1,X_2)$ be a continuous random vector.
\begin{itemize}
\item [1.](Translation invariance) For all $\textbf{c}=(c_1,c_2)\in\mathbb{R}^2$ and $i,j=1,2$, $i\neq j$, then
$$\underline{\RVaR}_{\alpha_1,\alpha_2,x_j+c_j}(\boldsymbol{X}+\textbf{c})=\underline{\RVaR}_{\alpha_1,\alpha_2,x_j}(\boldsymbol{X})+c_i,$$ $$\overline{\RVaR}_{\alpha_1,\alpha_2,x_j+c_j}(\boldsymbol{X}+\textbf{c})=\overline{\RVaR}_{\alpha_1,\alpha_2,x_j}(\boldsymbol{X})+c_i.$$
\item [2.](Positive homogeneity) For all $\textbf{c}=(c_1,c_2)\in\mathbb{R}^2_{+}$ and $i,j=1,2$, $i\neq j$, then
$$\underline{\RVaR}_{\alpha_1,\alpha_2,c_jx_j}(\textbf{c}\boldsymbol{X})=c_i\underline{\RVaR}_{\alpha_1,\alpha_2,x_j}(\boldsymbol{X}),$$ $$\overline{\RVaR}_{\alpha_1,\alpha_2,c_jx_j}(\textbf{c}\boldsymbol{X})=c_i\overline{\RVaR}_{\alpha_1,\alpha_2,x_j}(\boldsymbol{X}).$$
\item [3.](Monotonicity) Let $\boldsymbol{X}=(X_1,X_2)$ and $\boldsymbol{X}'=(X'_1,X'_2)$ be two pairs of risks with joint cdf's $F_{\boldsymbol{X}}$ and $F_{\boldsymbol{X}'}$, respectively. If $\boldsymbol{X}\prec_{co}\boldsymbol{X}'$, then
$$\underline{\RVaR}_{\alpha_1,\alpha_2}(\boldsymbol{X}')\prec\underline{\RVaR}_{\alpha_1,\alpha_2}(\boldsymbol{X}),$$ $$\overline{\RVaR}_{\alpha_1,\alpha_2}(\boldsymbol{X})\prec\overline{\RVaR}_{\alpha_1,\alpha_2}(\boldsymbol{X}').$$
\end{itemize}
\end{Proposition}
\begin{proof}
See Appendix \ref{Appendix:Proof}.
\end{proof}
Consider the random vectors $\boldsymbol{X}_M$, $\boldsymbol{X}_W$ and $\boldsymbol{X}_\Pi$ which denote the monotonic, countermonotonic and independent vector, respectively. They have the following relationship
$$\boldsymbol{X}_W\prec_{co}\boldsymbol{X}_\Pi\prec_{co}\boldsymbol{X}_M,$$
which means according to the Proposition \ref{Proposition:Properties}, we have
$$\underline{\RVaR}_{\alpha_1,\alpha_2}(\boldsymbol{X}_M)\prec\underline{\RVaR}_{\alpha_1,\alpha_2}(\boldsymbol{X}_\Pi)\prec\underline{\RVaR}_{\alpha_1,\alpha_2}(\boldsymbol{X}_W),$$
and
$$\overline{\RVaR}_{\alpha_1,\alpha_2}(\boldsymbol{X}_W)\prec\overline{\RVaR}_{\alpha_1,\alpha_2}(\boldsymbol{X}_\Pi)\prec\overline{\RVaR}_{\alpha_1,\alpha_2}(\boldsymbol{X}_M).$$
\begin{Example}
Consider a bivariate random vector $(X_1,X_2)$ which is either comonotonic, countermotonic or independent. We obtain the lower orthant RVaR based on Proposition \ref{Proposition:RVaRLowerIntegration}. For $i,j=1,2$ ($i\neq j$),\\
\begin{align*}
&\underline{\RVaR}_{\alpha_1,\alpha_2,x_i} (\boldsymbol{X}_\Pi)=\frac{1}{\alpha_2F_{X_i}(x_i)-\alpha_1}\int_{\alpha_1}^{\alpha_2F_{X_i}(x_i)}\VaR_{\frac{u}{F_{X_i}(x_i)}}(X_j)du,\\
&\underline{\RVaR}_{\alpha_1,\alpha_2,x_i} (\boldsymbol{X}_M)=\frac{1}{F_{X_i}(x_i)-\alpha_1}\int^{F_{X_i}(x_i)}_{\alpha_1}\VaR_{u}(X_j)du,\\
&and\\
&\underline{\RVaR}_{\alpha_1,\alpha_2,x_i} (\boldsymbol{X}_W)=\frac{1}{F_{X_i}(x_i)+\alpha_2-1-\alpha_1}\int_{\alpha_1}^{F_{X_i}(x_i)+\alpha_2-1}\VaR_{u-F_{X_i}(x_i)+1}(X_j)du .\end{align*}
Let the random vector above be defined with exponential marginal cdfs, i.e. $X_i\sim Exp(\lambda_i)$, then we get the following results.\\
\begin{align*}
\underline{\RVaR}_{\alpha_1,\alpha_2,x_i} (\boldsymbol{X}_\Pi)=&\frac{1}{\alpha_2F_{X_i}(x_i)-\alpha_1}\left(\frac{1}{\lambda_i}\Bigg[(F_{X_i}(x_i)-\alpha_2F_{X_i}(x_i))\ln (1-\alpha_2)\right.\\
&\left.\left.-(F_{X_i}(x_i)-\alpha_1)\ln \left(\frac{F_{X_i}(x_i)-\alpha_1}{F_{X_i}(x_i)}\right)+(\alpha_2F_{X_i}(x_i)-\alpha_1)\right]\right),
\\
\underline{\RVaR}_{\alpha_1,\alpha_2,x_i} (\boldsymbol{X}_M)=&\frac{1}{F_{X_i}(x_i)-\alpha_1}\left(\frac{1}{\lambda_i}\Bigg[(1-F_{X_i}(x_i))\ln (1-F_{X_i}(x_i))\right.\\
&-(1-\alpha_1)\ln \left(1-\alpha_1\right)+(F_{X_i}(x_i)-\alpha_1)\Bigg]\Bigg),
\\
\underline{\RVaR}_{\alpha_1,\alpha_2,x_i} (\boldsymbol{X}_W)=&\frac{1}{(F_{X_i}(x_i)+\alpha_2-1)-\alpha_1}\left(\frac{1}{\lambda_i}\Bigg[(1-\alpha_2)\ln (1-\alpha_2)\right.\\
&-(F_{X_i}(x_i)-\alpha_1)\ln \left(F_{X_i}(x_i)-\alpha_1\right)+(F_{X_i}(x_i)+\alpha_2-1-\alpha_1)\Bigg]\Bigg)
.\end{align*}
\end{Example}
Now, we illustrate some examples of multivariate RVaR in the context of EVT. We present closed form expressions obtained in the independence case. Dependence between random variables is considered in section \ref{Section:RVaREmpirical}.
\begin{Example}\label{Example:GEV_Independent}
Assume $F_{X_i}\sim \text{GEV}(\mu_i,\sigma_i,\xi_i)$ and $F_{X_j}\sim \text{GEV}(\mu_j,\sigma_j,\xi_j)$. Let $0\leq\alpha\leq1$ and consider the independent copula where $C(u,v)=uv$.
Let $A=F_{X_i}(x_i), B=F(x_i,\VaR_{\alpha_2}(X_j))$ and $C=1-\bar{F}(x_i,VaR_{\alpha_1}(X_j))$, then, \begin{align*}
\underline{\VaR}_{\alpha,x_i} (\boldsymbol{X})=&
\begin{dcases}
\mu_j-\frac{\sigma_j}{\xi_j}\left[1-\left(\ln\left(\frac{A}{\alpha}\right)\right)^{-\xi_j}\right], & \xi_i,\xi_j\neq0 \\
\mu_j-\sigma_j\ln\left[ \ln \left(\frac{A}{\alpha}\right) \right], & \xi_i=\xi_j=0,
\end{dcases}\\
\end{align*}
and
\begin{align*}
\overline{\VaR}_{\alpha,x_i} (\boldsymbol{X})=&
\begin{dcases}
\mu_j- \frac{\sigma_j}{\xi_j}\left(1-\left[\ln\left(\frac{1-A}{\alpha-A}\right)\right]^{-\xi_j} \right) , & \xi_i,\xi_j\neq0 \\
\mu_j-\sigma_j\ln\left[\ln\left(\frac{1-A)}{\alpha-A}\right)\right] , & \xi_i=\xi_j=0.
\end{dcases}\\
\end{align*}
Then by using the above results, and for $\xi_i,\xi_j \neq 0$, we obtain the multivariate lower and upper orthant RVaR, respectively represented by
\begin{align*}
\underline{\RVaR}_{\alpha_1,\alpha_2,x_i} (\boldsymbol{X})=& \mu_j-\frac{\sigma_j}{\xi_j}\left[1-\frac{A}{B-\alpha_1}\left[ \Gamma\left(1-\xi_j, \ln \left( \frac{A}{B}\right)\right)-\Gamma\left(1-\xi_j, \ln \left( \frac{A}{\alpha_1}\right)\right)\right]\right],\\
\overline{\RVaR}_{\alpha_1,\alpha_2,x_i} (\boldsymbol{X})=&\ \mu_j - \frac{\sigma_j}{\xi_j}\left[1-\frac{1-A}{\alpha_2-C} \left[\Gamma \left(1-\xi_j, \ln\left(\frac{1-A}{
\alpha_2-A} \right) \right) - \Gamma \left(1-\xi_j, \ln\left(\frac{1-A}{C -A} \right)\right)\right] \right],
\end{align*}
while for $\xi_i=\xi_j=0$,
\begin{align*}
\underline{\RVaR}_{\alpha_1,\alpha_2,x_i} (\boldsymbol{X})=&\ \mu_j - \frac{\sigma_j}{B-\alpha_1}\left[B\ln\left(\ln\left(\frac{A}{B}\right)\right) -\alpha_1 \ln\left( \ln\left(\frac{A}{\alpha_1}\right)\right)\right] \\
&-\frac{\sigma_jA}{B-\alpha_1}\left[ \operatorname{Ei}\left(\ln\left(\frac{A}{\alpha_1}\right) \right)-\operatorname{Ei}\left(\ln\left(\frac{A}{B}\right)\right) \right],\\
\overline{\RVaR}_{\alpha_1,\alpha_2,x_i} (\boldsymbol{X})=&\ \mu_i - \frac{\sigma_j}{\alpha_2-C}\left[\left(\alpha_2-A\right)\ln\left(\ln\left(\frac{1-A}{\alpha_2-A}\right)\right)- \left(C-A\right)\ln\left(\ln\left(\frac{1-A}{C-A}\right)\right)\right]\\
&- \frac{\sigma_j\left(1-A\right)}{\alpha_2-C}\left[\operatorname{E_1}\left(\ln\left(\frac{1-A}{\alpha_2-A}\right)\right)-\operatorname{E_1}\left(\ln\left(\frac{1-A}{C-A}\right)\right)\right],
\end{align*}
\begin{comment}
and
$\overline{\RVaR}_{\alpha_1,\alpha_2,x_i} (\boldsymbol{X})$
\begin{eqnarray*}
=&
\begin{dcases}
&&\mu_j - \frac{\sigma_j}{\xi_j}\left[1-\frac{1-A}{\alpha_2-C} \left[\Gamma \left(1-\xi_j, \ln\left(\frac{1-A}{
\alpha_2-A} \right) \right) - \Gamma \left(1-\xi_j, \ln\left(\frac{1-A}{C -A} \right)\right)\right] \right], & \xi_i,\xi_j\neq0,\xi_i,\xi_j\neq1 \\
&&\mu_j - \frac{\sigma_j}{\alpha_2-C}\left[\left(\alpha_2-A\right)\ln\left(\ln\left(\frac{1-A}{\alpha_2-A}\right)\right)- \left(C-A\right)\ln\left(\ln\left(\frac{1-A}{C-A}\right)\right)- \left(1-A\right)\left[ \operatorname{E_1}\left(\ln\left(\frac{1-A}{C-A}\right)\right)\right]-\operatorname{E_1}\left(\ln\left(\frac{1-A}{\alpha_2-A}\right)\right)\right], & \xi_i=\xi_j=0.
\end{dcases}\\
\end{eqnarray*}
\end{comment}
where $\operatorname{Ei}(x) = -\int_{-x}^\infty\frac{e^{-t}}{t}dt$.\\
As a special case of $\RVaR$, we have that when $\alpha_2=1$,
\begin{align*}
\underline{\TVaR}_{\alpha,x_i} (\boldsymbol{X})=&
\begin{dcases}
\mu_j - \frac{\sigma_j}{\xi_j}\left[1 - \frac{ A}{A-\alpha}\left[ \Gamma\left(1-\xi_j\right)- \Gamma\left(1-\xi_j,\ln \left(\frac{A}{ \alpha}\right)\right) \right]\right], & \xi_i,\xi_j\neq0, \\
\infty, & \xi_i=\xi_j=0,
\end{dcases}\\
\end{align*}
and
\begin{align*}
\overline{\TVaR}_{\alpha,x_i} (\boldsymbol{X})=&
\begin{dcases}
\mu_j - \frac{\sigma_j}{\xi_j}\left[1 - \dfrac{1-A}{1-\alpha}\left[\Gamma\left(1-\xi_j\right) - \Gamma\left(1-\xi_j,\ln\left(\frac{1-A}{\alpha-A}\right)\right)\right] \right] , & \xi_i,\xi_j\neq0, \\
\infty, & \xi_i=\xi_j=0.
\end{dcases}\\
\end{align*}
\begin{comment}
\robacomment{Do we need those three ratios below? I feel we do not have a "conclusion" from them.}
\melinacomment{I agree to remove them (do not forget to remove in appendix as well please.}
\begin{align*}
\lim_{\alpha\to 1}\frac{\underline{\TVaR}_{\alpha,x_i} (\boldsymbol{X})}{\underline{\VaR}_{\alpha,x_i} (\boldsymbol{X})} &=
\begin{dcases}
\frac{\mu_j - \frac{\sigma_j}{\xi_j}\left[1 - \frac{ A}{A-1}\left[ \Gamma\left(1-\xi_j\right)- \Gamma\left(1-\xi_j,\ln A\right) \right]\right]}{\mu_j-\frac{\sigma_j}{\xi_j}\left[1-\left(\ln{A}\right)^{-\xi_j}\right]}, & \xi_i,\xi_j\neq0,\xi_i,\xi_j\neq1 \\
\infty , & \xi_i,\xi_j=0 \\
\end{dcases}
\end{align*}
\begin{align*}
\lim_{\alpha_2\to 1}\frac{\underline{\RVaR}_{\alpha_1,\alpha_2,x_i} (\boldsymbol{X})}{\underline{\VaR}_{\alpha_2,x_i} (\boldsymbol{X})} &=\begin{dcases}
\frac{\mu_j - \frac{\sigma_j}{\xi_j}\left[1 - \frac{A}{A-\alpha_1}\left[ \Gamma\left(1-\xi_j \right)- \Gamma\left(1-\xi_j,\ln \left(\frac{A}{\alpha_1}\right)\right) \right]\right]}{\mu_j-\frac{\sigma_j}{\xi_j}\left[1-\left(\ln{A}\right)^{-\xi_j}\right]}, & \xi_i,\xi_j\neq0,\xi_i,\xi_j\neq1 \\
\infty , & \xi_i,\xi_j=0 \\
\end{dcases}
\end{align*}
\begin{align*}
\lim_{\alpha_1\to1}\left[\lim_{\alpha_2\to 1}\frac{\underline{\RVaR}_{\alpha_1,\alpha_2,x_i} (\boldsymbol{X})}{\underline{\VaR}_{\alpha_1,x_i} (\boldsymbol{X})}\right] &=
\begin{dcases}
\frac{\mu_j - \frac{\sigma_j}{\xi_j}\left[1 - \frac{A}{A-1}\left[ \Gamma\left(1-\xi_j\right)- \Gamma\left(1-\xi_j,\ln A\right) \right]\right]}{\mu_j-\frac{\sigma_j}{\xi_j}\left[1-\left(\ln{A}\right)^{-\xi_j}\right]}, & \xi_i,\xi_j\neq0,\xi_i,\xi_j\neq1 \\
\infty , & \xi_i,\xi_j=0 \\
\end{dcases}
\end{align*}
\end{comment}
\end{Example}
\begin{proof}
See Appendix \ref{Appendix:Proof}.
\end{proof}
\begin{Proposition}\label{Proposition:RVaRLimit}
Let $\boldsymbol{X}=(X_1,X_2)$ be a pair of random variables with cdf $F_{\boldsymbol{X}}$ and marginal distributions $F_{X_1}$ and $F_{X_2}$. Assume that $F_{\boldsymbol{X}}$ is continuous and strictly increasing. Then, for $i,j=1,2$ and $i\neq j$,
$$\lim_{x_i\rightarrow\underline{\VaR}_{\alpha_1,\VaR_{\alpha_2}(X_j)}(\boldsymbol{X}) }\underline{\RVaR}_{\alpha_1,\alpha_2,x_i}(\boldsymbol{X})=\VaR_{\alpha_2}(X_j),$$
$$\lim_{x_i\rightarrow \overline{\VaR}_{\alpha_2,\VaR_{\alpha_1}(X_j)}(\boldsymbol{X})}\overline{\RVaR}_{\alpha_1,\alpha_2,x_i}(\boldsymbol{X})=\VaR_{\alpha_1}(X_j).$$
Moreover,
$$\lim_{x_i\rightarrow u_{x_i} }\underline{\RVaR}_{\alpha_1,\alpha_2,x_i}(\boldsymbol{X})=\RVaR_{\alpha_1,\alpha_2}(X_j),$$
$$\lim_{x_i\rightarrow l_{x_i}}\overline{\RVaR}_{\alpha_1,\alpha_2,x_i}(\boldsymbol{X})=\RVaR_{\alpha_1,\alpha_2}(X_j),$$
where $u_{x_i}$ (or $l_{x_i}$) represents the upper (or lower) support of the rv $X_i$.
\end{Proposition}
\begin{proof}
See Appendix \ref{Appendix:Proof}.
\end{proof}
Now, we consider the behavior of aggregate risks defined as follows:
$$\textbf{S}=\binom{S_1}{S_2}=\sum_{i=1}^{n}\binom{X_i}{Y_i},$$
where $S_1$ and $S_2$ denote the aggregate amount of claims for two different business class respectively. $X_i$ and $Y_i$ represent the risks within each class, where $i=1,\ldots,n$, such that $S_1=\sum_{i=1}^{n}X_i$ and $S_2=\sum_{i=1}^{n}Y_i$.\\
Unlike univariate TVaR, the univariate RVaR does not satisfy the subadditivity. Hence, it seems impossible to prove that the bivariate RVaR is subadditive. However, if we suppose that $(X_1,\ldots,X_n)$ (respectively $(Y_1,\ldots,Y_n)$) is comonotonic, the following results can be obtained.\\
\begin{Proposition}\label{Proposition:Aggregate}
Let $(X_1,\ldots,X_n)$ (respectively $(Y_1,\ldots,Y_n)$) be comonotonic with cdf's $F_{X_1},\ldots,F_{X_n}$ (respectively $G_{Y_1},\ldots,G_{Y_n}$). The dependence structure between $(X_1,\ldots,X_n)$ and $(Y_1,\ldots,Y_n)$ is unknown. Then,
$$\underline{\RVaR}_{\alpha_1,\alpha_2,S_1}(\textbf{S})=\sum_{i=1}^{n}\underline{\RVaR}_{\alpha_1,\alpha_2,x_i}(X_i,Y_i),$$
$$\underline{\RVaR}_{\alpha_1,\alpha_2,S_2}(\textbf{S})=\sum_{i=1}^{n}\underline{\RVaR}_{\alpha_1,\alpha_2,y_i}(X_i,Y_i),$$
and
$$\overline{\RVaR}_{\alpha_1,\alpha_2,S_1}(\textbf{S})=\sum_{i=1}^{n}\overline{\RVaR}_{\alpha_1,\alpha_2,x_i}(X_i,Y_i),$$
$$\overline{\RVaR}_{\alpha_1,\alpha_2,S_2}(\textbf{S})=\sum_{i=1}^{n}\overline{\RVaR}_{\alpha_1,\alpha_2,y_i}(X_i,Y_i).$$
\end{Proposition}
\begin{proof}
See Appendix \ref{Appendix:Proof}.
\end{proof}
In conclusion, the bivariate RVaR has similar properties to the bivariate VaR and TVaR, such as translation invariance, positive homogeneity and monotonicity. Furthermore, it has an advantage over bivariate VaR and TVaR. Compared to bivariate VaR, bivariate TVaR and RVaR provide essential information about the tail of the distribution. Moreover, $\underline{\TVaR}_{\alpha,x_i}(\boldsymbol{X})$ and $\overline{\TVaR}_{\alpha,x_i}(\boldsymbol{X})$ will go to infinity when $X_i$ approaches $\VaR_{\alpha}(X_i)$ whereas the bivariate RVaR is bounded in the area $[\VaR_{\alpha_1}(X_i),\VaR_{\alpha_2}(X_i)]\times[\VaR_{\alpha_1}(X_j),\VaR_{\alpha_2}(X_j)]$. This measure could be useful for insurance companies that must set aside capital for risks that are sent to a reinsurer after having reached a certain level. Assume that the insurance company transfers the risks to the reinsurer when the total losses exceed VaR at level $\alpha_2$. Then, to comply to solvency capital requirements, the insurance company needs to measure the risks with truncated data. In this case, multivariate RVaR could be helpful.
\\
We will check the robustness of the estimator of bivariate RVaR. Since RVaR is distribution-based, the sensitivity function can be used to quantify the robustness.
\begin{Proposition}\label{Proposition:VaRRobust}
For a pair of continuous random variables $\boldsymbol{X}$ with joint cdf $F(x_1,x_2)$ and marginals $F_{X_1}(x_1)$ and $F_{X_2}(x_2)$, the sensitivity function of $\underline{\VaR}_{\alpha,x_i}(\boldsymbol{X})$ is given by
\begin{align*}
S(z) &=\begin{dcases}
-\frac{F_{X_i}(x_i)-\alpha}{f_{x_i}\left[\underline{\VaR}_{\alpha,x_i}(\boldsymbol{X})\right]F_{X_i}(x_i)},&\ \ z<\underline{\VaR}_{\alpha,x_i}(\boldsymbol{X}),\\
\frac{\alpha}{f_{x_i}\left[\underline{\VaR}_{\alpha,x_i}(\boldsymbol{X})\right]F_{X_i}(x_i)}, &\ \ z>\underline{\VaR}_{\alpha,x_i}(\boldsymbol{X}),\\
0, &\ \ \text{otherwise},\\
\end{dcases}
\end{align*}
which is bounded. Thus, $\underline{\VaR}_{\alpha,x_i}(\boldsymbol{X})$ is a robust risk measure.
\end{Proposition}
\begin{proof}
See Appendix \ref{Appendix:Proof}.
\end{proof}
\begin{Proposition}\label{Proposition:RVaRRobust}
For a pair of continuous random variables $\boldsymbol{X}$ with joint cdf $F(x_1,x_2)$ and marginals $F_{X_1}(x_1)$ and $F_{X_2}(x_2)$, let $A=F_{X_i}(x_i)$ and $B=F(x_i,\VaR_{\alpha_2}(X_j))$. Then the sensitivity function of $\underline{\RVaR}_{\alpha_1,\alpha_2,x_i}(\boldsymbol{X})$ is given by\\\\
$S(z)=S'(z) -\underline{\RVaR}_{\alpha_1,\alpha_2,x_i}(\boldsymbol{X}),$\\
where
\begin{align*}
S'(z)=
\begin{dcases}
\frac{(A-\alpha_1)\underline{\VaR}_{\alpha_1,x_i}(\boldsymbol{X})-(A-B)\VaR_{\alpha_2}(X_j)}{B-\alpha_1},&\ \ z<\underline{\VaR}_{\alpha_1,x_i}(\boldsymbol{X}),\\
\frac{zA-\alpha_1\underline{\VaR}_{\alpha_1,x_i}(\boldsymbol{X})-(A-B)\VaR_{\alpha_2}(X_j)}{B-\alpha_1}, &\ \ \underline{\VaR}_{\alpha_1,x_i}(\boldsymbol{X})\leq z\leq \VaR_{\alpha_2}(X_j),\\
\frac{B \VaR_{\alpha_2}(X_j)-\alpha_1\underline{\VaR}_{\alpha_1,x_i}(\boldsymbol{X})}{B-\alpha_1}, &\ \ z>\VaR_{\alpha_2}(X_j),\\
\end{dcases}
\end{align*}
is a bounded function. Thus, $\underline{\RVaR}_{\alpha_1,\alpha_2,x_i}(\boldsymbol{X})$ is robust.
\end{Proposition}
\begin{proof}
See Appendix \ref{Appendix:Proof}.
\end{proof}
\begin{Proposition}\label{Proposition:UpperVaRRobust}
For a pair of continuous random variables $\boldsymbol{X}$ with joint survival function $\bar{F}(x_1,x_2)$ and marginals $F_{X_1}(x_1)$ and $F_{X_2}(x_2)$, the sensitivity function of $\overline{\VaR}_{\alpha,x_i}(\boldsymbol{X})$ is given by
\begin{align*}
S(z) &=\begin{dcases}
-\frac{1-\alpha}{f_{\bar{x}_i}\left[\overline{\VaR}_{\alpha,x_i}(\boldsymbol{X})\right](1-F_{X_i}(x_i))},&\ \ z<\overline{\VaR}_{\alpha,x_i}(\boldsymbol{X}),\\
\frac{\alpha-F_{X_i}(x_i)}{f_{\bar{x}_i}\left[\overline{\VaR}_{\alpha,x_i}(\boldsymbol{X})\right](1-F_{X_i}(x_i))}, &\ \ z>\overline{\VaR}_{\alpha,x_i}(\boldsymbol{X}),\\
0, &\ \ z=\overline{\VaR}_{\alpha,x_i}(\boldsymbol{X}).\\
\end{dcases}
\end{align*}
The bounded sensitivity function implies $\overline{\VaR}_{\alpha,x_i}(\boldsymbol{X})$ is a robust risk measure.
\end{Proposition}
\begin{proof}
See Appendix \ref{Appendix:Proof}.
\end{proof}
\begin{Proposition}\label{Proposition:RVaRUpperRobust}
For a pair of continuous random variables $\boldsymbol{X}$ with joint survival function $\bar{F}(x_1,x_2)$ and marginals $F_{X_1}(x_1)$ and $F_{X_2}(x_2)$, let $A=F_{X_i}(x_i)$ and $C=1-\bar{F}(x_i,VaR_{\alpha_1}(X_j))$. Then the sensitivity function of $\overline{\RVaR}_{\alpha_1,\alpha_2,x_i}(\boldsymbol{X})$ is given by\\
$S(z)=S'(z) -\overline{\RVaR}_{\alpha_1,\alpha_2,x_i}(\boldsymbol{X}),$\\
where
\begin{align*}
S'(z)=
\begin{dcases}
\frac{(1-C)\VaR_{\alpha_1}(X_j)-(1-\alpha_2)\overline{\VaR}_{\alpha_2,x_i}(\boldsymbol{X})}{\alpha_2-C},&\ \ z<\VaR_{\alpha_1}(X_j),\\
\frac{z(1-A)-(C-A) \VaR_{\alpha_1}(X_j)-(1-\alpha_2)\overline{\VaR}_{\alpha_2,x_i}(\boldsymbol{X})}{\alpha_2-C}, &\ \ \VaR_{\alpha_1}(X_j)\leq z\leq \overline{\VaR}_{\alpha_2,x_i}(\boldsymbol{X}),\\
\frac{(\alpha_2-A) \overline{\VaR}_{\alpha_2,x_i}(\boldsymbol{X})-(C-A) \VaR_{\alpha_1}(X_j)}{\alpha_2-C}, &\ \ z>\overline{\VaR}_{\alpha_2,x_i}(\boldsymbol{X}).\\
\end{dcases}
\end{align*}
The bounded function proves that $\overline{\RVaR}_{\alpha_1,\alpha_2,x_i}(\boldsymbol{X})$ is robust.
\end{Proposition}
\begin{proof}
See Appendix \ref{Appendix:Proof}.
\end{proof}
\subsection{Empirical Estimator for Multivariate Lower and Upper Orthant RVaR}\label{Section:RVaREmpirical}
Next, we will propose empirical estimators for the lower and upper orthant RVaR, based on the estimators developed by \citet{beck2015multivariate}, and provide numerical examples.
\begin{Definition}
Consider a series of observations $\boldsymbol{X}=(\boldsymbol{X}_1,\ldots,\boldsymbol{X}_d)$ with $\boldsymbol{X}_i=(x_{1i},\ldots,x_{ni})$ and $\boldsymbol{X}_{\setminus i}=(\boldsymbol{X}_{1},\ldots,\boldsymbol{X}_{i-1},\boldsymbol{X}_{i+1},\ldots,\boldsymbol{X}_{d})$, $i=1,\ldots,d$. Denote $F_n$ and $F_{n,\setminus i}$, the empirical cdf's (ecdf) for $\boldsymbol{X}$ and $\boldsymbol{X}_{\setminus i}$, $i=1,\ldots,d$, respectively. We define the estimator for the lower orthant $RVaR$ for fixed $\boldsymbol{x}_{\setminus i}$, $i=1,\ldots,d$, by
$$\underline{\RVaR}^n_{\alpha_1,\alpha_2,\boldsymbol{x}_{\setminus i}} (\boldsymbol{X})=\frac{1}{F_n(\boldsymbol{x}_{\setminus i},\VaR_{\alpha_2}(\boldsymbol{X}_i))-\alpha_1}\int_{\alpha_1}^{F_n(\boldsymbol{x}_{\setminus i},\VaR_{\alpha_2}(\boldsymbol{X}_i))}\underline{\VaR}^n_{u,\boldsymbol{x}_{\setminus i}}(\boldsymbol{X})du,$$
For $m\in\mathbb{N}$ large enough, let $s=\frac{F_n(\boldsymbol{x}_{ \setminus i},\VaR_{\alpha_2}(\boldsymbol{X}_{ i}))-\alpha_1}{m}$ and $u_k=\alpha_1+ks$, then the above expression can be simplified into
\begin{align*}
\underline{\RVaR}^n_{\alpha_1,\alpha_2,\boldsymbol{x}_{\setminus i}} (\boldsymbol{X})&=\sum_{k=1}^{m}\frac{\underline{\VaR}^n_{u_k,\boldsymbol{x}_{\setminus i}}(\boldsymbol{X})\cdot s}{F_n(\boldsymbol{x}_{\setminus i},\VaR_{\alpha_2}(\boldsymbol{X}_i))-\alpha_1}\\
&=\sum_{k=1}^{m}\frac{\underline{\VaR}^n_{u_k,\boldsymbol{x}_{\setminus i}}(\boldsymbol{X})}{m},
\end{align*}
where $\underline{\VaR}^n_{u,\boldsymbol{x}_{\setminus i}}(\boldsymbol{X})=\inf{\{x_i\in\mathbb{R}_{+}:F_{n,\boldsymbol{x}_{\setminus i}}(x_i)\geq u\}}$ is the empirical lower orthant $\VaR$ for a given $\boldsymbol{x}_{\setminus i}$ and $F_{n,\boldsymbol{x}_{\setminus i}}$ is the ecdf of $\boldsymbol{X}$ given the same $\boldsymbol{x}_{\setminus i}$.
\end{Definition}
\begin{comment}
\begin{Definition}
Consider a series of observations $\boldsymbol{X}=(X_1,X_2)$ with $X_i=(x_{1i},\ldots,x_{ni})$, $i=1,2$. Additionally, we have $\boldsymbol{x}_l=(x_{l1},x_{l2})\in\mathbb{R}^2_{+}$, $l=1,\ldots,n$. Denote $F_n$ and $F_{n,i}$, the empirical cdf's (ecdf) for $\boldsymbol{X}$ and $X_{i}$, respectively. We define the estimator for the lower orthant $RVaR$, for a fixed $X_i$, by
$$\underline{\RVaR}^n_{\alpha_1,\alpha_2,x_{i}} (\boldsymbol{X})
=\frac{\int_{\alpha_1}^{F_n(x_{i},\VaR_{\alpha_2}(X_j))}\underline{\VaR}^n_{u,x_{i}}(\boldsymbol{X})du}{F_n(x_{i},\VaR_{\alpha_2}(X_j))-\alpha_1}.$$
For $m\in\mathbb{N}$ large enough, let $s=\frac{F_n(x_{i},\VaR_{\alpha_2}(X_j))-\alpha_1}{m}$ and $u_k=\alpha_1+ks$, then the above expression can be simplified into
\begin{align*}
\underline{\RVaR}^n_{\alpha_1,\alpha_2,x_{i}} (\boldsymbol{X})&=\sum_{k=1}^{m}\frac{\underline{\VaR}^n_{u_k,x_{i}}(\boldsymbol{X})\cdot s}{F_n(x_{i},\VaR_{\alpha_2}(X_j))-\alpha_1}\\
&=\sum_{k=1}^{m}\frac{\underline{\VaR}^n_{u_k,x_{i}}(\boldsymbol{X})}{m},
\end{align*}
where $\underline{\VaR}^n_{u,x_{i}}(\boldsymbol{X})=\inf{\{x_j\in\mathbb{R}_{+}:F_{n,x_{i}}(x_j)\geq u\}}$ is the empirical lower orthant $VaR$ given $X_{i}$ and $F_{n,x_{i}}$ the ecdf of $\boldsymbol{X}$ given the same $X_{i}$.
\end{Definition}
\end{comment}
Note, $\underline{\VaR}^n_{u,\boldsymbol{x}_{\setminus i}}(\boldsymbol{X})$ is the smallest value of $\boldsymbol{X}_i$ given $\boldsymbol{x}_{\setminus i}$ such that $F_n$ is larger than $u$. Similarly, we define the empirical estimator of upper orthant RVaR as follows.\\
\begin{Definition}
Consider a series of observations $\boldsymbol{X}=(\boldsymbol{X}_1,\ldots,\boldsymbol{X}_d)$ with $\boldsymbol{X}_i=(x_{1i},\ldots,x_{ni})$ and $\boldsymbol{X}_{\setminus i}=(\boldsymbol{X}_{1},\ldots,\boldsymbol{X}_{i-1},\boldsymbol{X}_{i+1},\ldots,\boldsymbol{X}_{d})$, $i=1,\ldots,d$. Denote $\bar{F}_n$ and $\bar{F}_{n,\setminus i}$, the empirical survival functions for $\boldsymbol{X}$ and $\boldsymbol{X}_{\setminus i}$, $i=1,\ldots,d$, respectively. We define the estimator for the upper orthant $\RVaR$ for fixed $\boldsymbol{x}_{\setminus i}$, $i=1,\ldots,d$, by
$$\overline{\RVaR}^n_{\alpha_1,\alpha_2,\boldsymbol{x}_{\setminus i}} (\boldsymbol{X})=\frac{1}{\alpha_2 - (1-\overline{F}_n(\boldsymbol{x}_{\setminus i},\VaR_{\alpha_1}(\boldsymbol{X}_i)))}\int^{\alpha_2}_{1-\overline{F}_n(\boldsymbol{x}_{\setminus i},\VaR_{\alpha_1}(\boldsymbol{X}_i))}\overline{\VaR}^n_{u,\boldsymbol{x}_{\setminus i}}(\boldsymbol{X})\text{d}u,$$
For $m\in\mathbb{N}$ large enough. Let $s=\frac{\alpha_2-(1-\bar{F}_n(\boldsymbol{x}_{\setminus i},\VaR_{\alpha_1}(\boldsymbol{X}_i)))}{m}$ and $v_k=1-\bar{F}_n(\boldsymbol{x}_{\setminus i},\VaR_{\alpha_1}(\boldsymbol{X}_i))+ks$, then the above expression can be simplified into
\begin{align*}
\overline{\RVaR}^n_{\alpha_1,\alpha_2,\boldsymbol{x}_{\setminus i}} (\boldsymbol{X}) & =\sum_{k=1}^{m}\frac{\overline{\VaR}^n_{v_k,\boldsymbol{x}_{\setminus i}}(\boldsymbol{X})\cdot s}{\alpha_2-(1-\bar{F}_n(\boldsymbol{x}_{\setminus i},\VaR_{\alpha_1}(\boldsymbol{X}_i)))} \\
&=\sum_{k=1}^{m}\frac{\overline{\VaR}^n_{v_k,\boldsymbol{x}_{\setminus i}}(\boldsymbol{X})}{m},
\end{align*}
where $\overline{\VaR}^n_{v,\boldsymbol{x}_{\setminus i}}(\boldsymbol{X})=\inf{\{x_i\in\mathbb{R}_{+}:\bar{F}_{n,\boldsymbol{x}_{\setminus i}}(\boldsymbol{x}_i)\leq 1-v\}}$ is the empirical upper orthant $\VaR$ given $\boldsymbol{x}_{\setminus i}$ and $\bar{F}_{n,\boldsymbol{x}_{\setminus i}}$ is the empirical survival function of $\boldsymbol{X}$ given the same $\boldsymbol{x}_{\setminus i}$.\\
\end{Definition}
\begin{comment}
\begin{Definition}
Consider a series of observations $\boldsymbol{X}=(X_1,X_2)$ with $X_i=(x_{1i},\ldots,x_{ni})$, $i=1,2$. Additionally, we have $\boldsymbol{x}_l=(x_{l1},x_{l2})\in\mathbb{R}^2_{+}$, $l=1,\ldots,n$. Denote by $F$ the distribution function of $\boldsymbol{X}$, and by $\bar{F}_n$ and $\bar{F}_{n,i}$, empirical survival functions for $\boldsymbol{X}$ and $X_{i}$, respectively. For a fixed $X_i$, the estimator for the lower orthant $RVaR$ is defined by
$$\overline{\RVaR}^n_{\alpha_1,\alpha_2,x_{i}} (\boldsymbol{X})
=\frac{\int^{\alpha_2}_{1-\bar{F}_n(x_{i},\VaR_{\alpha_1}(X_j))}\overline{\VaR}^n_{v,x_{i}}(\boldsymbol{X})du}{\alpha_2-(1-\bar{F}_n(x_{i},\VaR_{\alpha_1}(X_j)))}.$$
For $m\in\mathbb{N}$ large enough. Let $s=\frac{\alpha_2-(1-\bar{F}_n(x_{i},\VaR_{\alpha_1}(X_j)))}{m}$ and $v_k=1-\bar{F}_n(x_{i},\VaR_{\alpha_1}(X_j))+ks$, then the above expression can be simplified into
\begin{align*}
\overline{\RVaR}^n_{\alpha_1,\alpha_2,x_{i}} (\boldsymbol{X}) & =\sum_{k=1}^{m}\frac{\overline{\VaR}^n_{v_k,x_{i}}(\boldsymbol{X})\cdot s}{\alpha_2-(1-\bar{F}_n(x_{i},\VaR_{\alpha_1}(X_j)))} \\
&=\sum_{k=1}^{m}\frac{\overline{\VaR}^n_{v_k,x_{i}}(\boldsymbol{X})}{m},
\end{align*}
where $\overline{\VaR}^n_{v,x_{i}}(\boldsymbol{X})=\inf{\{x_j\in\mathbb{R}_{+}:\bar{F}_{n,x_{i}}(x_j)\leq 1-v\}}$ is the empirical upper orthant $VaR$ given $X_{i}$ and $\bar{F}_{n,x_{i}}$ the empirical survival function of $\boldsymbol{X}$ given the same $X_{i}$.\\
\end{Definition}
\end{comment}
The following proposition, based on the proof of the consistency of bivariate VaR by \citet{cousin2013multivariate}, shows the consistency of the bivariate RVaR in Hausdorff distance. For $\alpha\in(0,1)$ and $r,\zeta>0$, consider the ball
$$E=B(\{\boldsymbol{x}\in\mathbb{R}^2_+:|F(\boldsymbol{x})-\alpha|\leq r\},\zeta).$$
Denote $m^\nabla=\inf_{\boldsymbol{x}\in E}\parallel(\nabla F)_{\boldsymbol{x}} \parallel$ as the infimum of the Euclidean norm of the gradient vector and $M_H=\sup_{\boldsymbol{x}\in E}\parallel(HF)_{\boldsymbol{x}}\parallel$ as the matrix norm of the Hessian matrix evaluated at $\boldsymbol{x}$ for a twice differentiable $F(x_1,x_2)$.\\
\begin{Proposition}\label{Proposition:Empirical}
Let $[\alpha_1,\alpha_2]\subset(0,1)$ and $F(x_1,x_2)$ be twice differentiable on $\mathbb{R}^2$. Assume there exists $r,\zeta>0$ such that $m^\nabla>0$ and $M_H<\infty$. Assume for each $n$, $F_n$ is continuous with probability one (wp1) and
$$\parallel F-F_n\parallel\underset{n\rightarrow\infty}{\overset{wp1}{\longrightarrow}}0.$$
Also, let $F_{n,i}$ be the consistent estimator of $F_i$. Then, we have
$$\underline{\RVaR}^n_{\alpha_1,\alpha_2,x_i}(\boldsymbol{X})\underset{n\longrightarrow\infty}{\overset{wp1}{\longrightarrow}}\underline{\RVaR}_{\alpha_1,\alpha_2,x_{i}}(\boldsymbol{X}).$$
\end{Proposition}
\begin{proof}
See Appendix \ref{Appendix:Proof}.
\end{proof}
A simulation study is performed to compare the empirical estimators to the theoretical lower and upper orthant $\RVaR$. Marginally, the random variables are distributed from GEV distributions. The dependence is represented by an independent copula. 50 simulations are performed for samples of 4000 observations from each marginal distribution and for $m=250$. The results of the simulation are presented in Figure \ref{Figure:GEV_simulation_Lower}. As shown, the differeneces between the theoretical values and their empirical estimates are negligible. This could be attributed to the robustness and consistency of the empirical estimators of $\VaR$ and $\RVaR$. The accuracy of the estimates improves with the sample size and value of $m$.
\begin{comment}
\begin{figure}%
\hfill
\subfigure[]{\includegraphics[width=8cm]{RVaR_Lower_GEV.png}}%
\hfill
\subfigure[]{\includegraphics[width=8cm]{RVaR_Lower_GEV_xi0.png}}%
\hfill
\caption{(a) Lower orthant VaR at level 0.95 and RVaR at level range $[0.95,0.99]$ for fixed values of $X_1$ for GEV samples where $\xi_1,\xi_2\neq0$ and (b) Lower orthant VaR at level 0.95 and RVaR at level range $[0.95,0.99]$ for fixed values of $X_1$ for GEV samples where $\xi_1=\xi_2=0$}%
\label{Figure:GEV_simulation_Lower}%
\end{figure}
\end{comment}
\begin{figure}%
\hfill
\subfigure[]{\includegraphics[width=8cm]{RVaR_GEV_Lower_New.png}}%
\hfill
\subfigure[]{\includegraphics[width=8cm]{RVaR_GEV_Lower_xi0_New.png}}%
\hfill
\caption{(a) Lower orthant VaR at level 0.95 and RVaR at level range $[0.95,0.99]$ for fixed values of $X_1$ for GEV samples where $\xi_1,\xi_2\neq0$ and (b) Lower orthant VaR at level 0.95 and RVaR at level range $[0.95,0.99]$ for fixed values of $X_1$ for GEV samples where $\xi_1=\xi_2=0$}
\label{Figure:GEV_simulation_Lower}%
\end{figure}
\begin{comment}
\begin{figure}%
\hfill
\subfigure[]{\includegraphics[width=8cm]{RVaR_upper_GEV.png}}%
\hfill
\subfigure[]{\includegraphics[width=8cm]{RVaR_Lower_GEV_xi0.png}}%
\hfill
\caption{(a) Upper orthant VaR at level 0.95 and RVaR at level range $[0.95,0.99]$ for fixed values of $X_1$ for GEV samples where $\xi_1,\xi_2\neq0$ and (b) Lower orthant VaR at level 0.95 and RVaR at level range $[0.95,0.99]$ for fixed values of $X_1$ for GEV samples where $\xi_1=\xi_2=0$}%
\label{Figure:GEV_simulation_Upper}%
\end{figure}
\end{comment}
\section{Conclusion}\label{Section:Conclusion}
In this paper, we introduce the multivariate extension of the RVaR risk measure, which results in a tool employed to assess dependent risks. This tool is particularly useful for heavy-tail distributions. Similar to its univariate counterpart, multivariate lower and upper orthant $\RVaR$ are defined as the conditional expectation of the lower and upper orthant $\VaR$ for large confidence levels. Their properties are discussed, such as translation invariance, positive homogeneity and monotonicity. Subadditivity can be satisfied for aggregated risks if each risk class is monotonic. Moreover, we develop resulting measures with specific extreme value distributions. The method of sensitivity functions to study robustness is extended for distribution-based multivariate RVaR. Finally, the empirical estimators of multivariate RVaR are proposed. The robustness and consistency of such estimators are confirmed. Furthermore, the simulations illustrate the accuracy of the empirical estimators without the need to assume any statistical distributions. RVaR may be extremely relevant in some instances where the loss distribution is characterized with an infinite mean, which results in an infinite TVaR.
\section*{Acknowledgements}
M\'elina Mailhot acknowledges financial support from the Natural Sciences and Engineering Research Council. Roba Bairakdar acknowledges financial support from the Society of Actuaries Hickman Scholar Program.
\section{Introduction}
Recent progress in understanding specific risks faced by an entity is mainly rising from the emergence of models reflecting more precisely the entity and measures that are used to quantify and represent a company's global and granular structures. Risk measures are essential for insurance companies and financial institutions for several reasons such as quantifying capital requirements to protect against unexpected future losses and to set insurance premiums for all lines of business and risk categories. Different univariate risk measures have been proposed in the literature. The most common risk measures are Value-at-Risk (VaR) and Tail Value-at-Risk (TVaR). VaR, which represents an $\alpha$-level quantile, found its way through the G-30 report, see \citet{global1993derivatives} for details. \citet{heath1999coherent} show that VaR is not a coherent risk measure in addition to not providing any information about the tail of the distribution and thus suggests other specific risk measures such as TVaR, which evaluates the average value over all VaR values at confidence levels greater than $\alpha$, which is a significant measure for heavy-tailed distributions.
Dependencies between risks needs to be taken into account to obtain accurate capital allocation and systemic risk evaluation. For example, systemic risk refers to the risks imposed by interdependencies in a system. Univariate risk measures are not suitable to be employed for heterogeneous classes of homogeneous risks. Therefore, multivariate risk measures have been developed and gained popularity in the last decade.
The notion of quantile curves is employed in \citet{embrechts2006bounds}, \citet{nappo2009kendall} and \citet{prekopa2012multivariate} to define a multivariate risk measure called upper and lower orthant VaR. Based on the same idea, \citet{cossette2013bivariate} redefine the lower and upper orthant VaR and \citet{cossette2015vector} propose the lower and upper orthant TVaR. \citet{cousin2013multivariate} develop a finite vector version of the lower and upper orthant VaR. A drawback of multivariate VaR is that it represents the boundary of the $\alpha$-level set and no additional tail information is provided, similar to the univariate VaR. Furthermore, relationships holding for univariate risk measures can be different in a multivariate setting.
Most risk measures are defined as functions of the loss distribution which should be estimated from the data. In \citet{cont2010robustness}, risk measurement procedures are defined and analysis of robustness of different risk measures is performed. They point out the conflict between the subadditivity and robustness and propose a robust risk measure called weighted VaR (WVaR). The use of a truncated version of TVaR, defined as Range-Value-at-Risk (RVaR), is suggested by \citet{bignozzi2016parameter}. The lower and upper orthant RVaR in the multivariate setting are developed in this paper, in order to provide a new robust multivariate risk measure. We aim to study in details their properties and derive their estimators. We will also focus on extreme value distributions which can be used to model the heavy tail of the data.
The paper is organized as follows. In Section \ref{Section:Preliminaries}, definitions and properties of the univariate $\RVaR$ are given, with examples in the Extreme Value Theory (EVT) framework. Sections \ref{Section:LowerRVaR} and \ref{Section:UpperRVaR} define the multivariate lower and upper orthant $\RVaR$, respectively. Section \ref{Section:PropertiesRVaR} presents their interesting and desirable properties, such as their behavior under transformations or translation of the multivariate variables and monotonicity. We also develop asymptotic results, the behavior with aggregate risks and we prove their robustness. In Section \ref{Section:RVaREmpirical}, we define the empirical estimator of the lower and upper orthant $\RVaR$ and we illustrate the accuracy of this estimator graphically. Concluding remarks are given in Section \ref{Section:Conclusion}
\section{Preliminaries}\label{Section:Preliminaries}
In this section, we present the univariate definition of RVaR and provide resulting measures, based on univariate RVaR, in the EVT framework and in asymptotic scenarios.
\subsection{Univariate RVaR}
Consider a random loss variable $X$ on a probability space $(\Omega,\mathcal{F},\mathbb{P})$ with its cumulative distribution function (cdf) $F_X$. A risk measure $\rho(X)$ for a random variable (r.v.) $X$ corresponds to the required amount that has to be maintained such that the financial position $\rho(X)-X$ is acceptable. Since there are several definitions of risk measures, an appropriate choice becomes crucial for stakeholders.
\begin{Definition}
For a continuous random variable $X$ with cumulative distribution function (cdf) $F_X$, the univariate Range Value-at-Risk at significance level range $[\alpha_1, \alpha_2]\subseteq[0,1]$ is defined by
$$\RVaR_{\alpha_1,\alpha_2}(X)=\mathbb{E}\left[X|\VaR_{\alpha_1}(X)\leq X\leq \VaR_{\alpha_2}(X)\right]=\frac{1}{\alpha_2 -\alpha_1}\int^{\alpha_2}_{\alpha_1}\VaR_u(X)du,$$
where
$$\VaR_\alpha(X)=\inf{\{x\in\mathbb{R}:F_X(x)\geq\alpha\}}$$
is the univariate Value-at-Risk at significance level $\alpha\in[0,1]$.
\end{Definition}
For a continuous random variable $X$ with strictly increasing cdf, $\VaR_\alpha(X)=F^{-1}_X(\alpha)$, is also called the $\alpha$-quantile, where $F^{-1}_X$ is the inverse function of cdf. VaR fails to give any information beyond the level $\alpha$, However, RVaR quantifies the magnitude of the loss of the worst $100(1-\alpha_1)$ to $100(1-\alpha_2)$ cases. When $\alpha_2=1$, we obtain a special case of RVaR, which is referred to as TVaR in this article.
Robust statistics can be defined as statistics that are not unduly affected by outliers. In order to establish the robustness of RVaR, we need to define a measure of affectation. Consider a continuous random variable $X$ with cdf $F\in\mathbb{D}$ where $\mathbb{D}$ is the convex set of cdfs. Notice that a risk measure is distribution-based if $\rho(X_1)=\rho(X_2)$ when $F_{X_1}=F_{X_2}$. Hence, we use $\rho(F)\triangleq\rho(X)$ to represent the distribution-based risk measures. To quantify the sensitivity of a risk measure to the change in the distribution, we use the sensitivity function. This method is used by \citet{cont2010robustness} and can be explained as the one-sided directional derivative of the effective risk measure at $F$ in the direction $\delta_z$.
\begin{Definition}
Consider $\rho$, a distribution-based risk measure of a continuous random variable $X$ with distribution function $F\in\mathbb{D}$ where $\mathbb{D}$ is the convex set of cdfs. For $\varepsilon\in[0,1)$, set $F_\varepsilon=\varepsilon\delta_z+(1-\varepsilon)F$ such that $F_\varepsilon\in\mathbb{D}$. $\delta_z\in\mathbb{D}$ is the probability measure which gives a mass of 1 to $\{z\}$. The distribution $F_\varepsilon$ is differentiable at any $x\neq z$ and has a jump point at the point $x=z$. The sensitivity function is defined by
$$S(z)=S(z;F)\triangleq\lim_{\varepsilon\rightarrow 0^+}\frac{\rho(F_\varepsilon)-\rho(F)}{\varepsilon},$$
for any $z\in\mathbb{R}$ such that the limit exists.
\end{Definition}
The value of sensitivity function for a robust statistic will not go to infinity when $z$ becomes arbitrarily large. In other word, the bounded sensitivity function makes sure that the risk measure will not blow up when a small change happens. Accordingly, \citet{cont2010robustness} show that VaR and RVaR are robust statistics by showing that their respective sensitivity functions are bounded.
\subsection{Examples of univariate RVaR in Extreme Value Theory}\label{Section: EVT}
In this section, we will provide some examples for the discussed risk measures in the EVT framework. Most of the statistical techniques are focused on the behavior of the center of the distribution, usually the mean. However, EVT is a branch in statistics that is focused on the behavior of the tail of the distribution. There are two principle models for extreme values; the block maxima model and the peaks-over-threshold model. The \textit{block maxima} approach is used to model the largest observations from samples of identically distributed observations in successive periods.
The \textit{peaks-over-threshold} is used to model all large observations that exceed a given high threshold value, denoted $u$.
The limiting distribution of block maxima, from \citet{fisher1928limiting}, is given in the theorem below;
\begin{Theorem}\label{Thm: FisherTippet}
Let $X_1, \ldots, X_n$ be a sequence of independent random variables having a common distribution function $F$ and consider $M_n = \max\lbrace X_1,\ldots,X_n\rbrace$. If there exists norming constants $(a_n)$ and $(b_n)$ where $a_n\in\mathbb{R}$ and $b_n>0$ for all $n\in\mathbb{N}$ and some non-degenerate distribution function $H$ such that $$\frac{M_n-a_n}{b_n}\xrightarrow{\text{d}} H,$$
then $H$ is defined as the Generalized Extreme Value Distribution (GEV) given by
\begin{align*}
H_{\xi}(x)&=
\begin{dcases}
\exp \left\lbrace -\left(1+\xi x \right)^{-\frac{1}{\xi}}\right\rbrace, &\ \ \xi\neq0,\\
\exp\left\lbrace-\exp(-x)\right\rbrace, &\ \ \xi=0,\\
\end{dcases}
\end{align*}
where $1+\xi x>0$. A three-parameter family is obtained by defining $H_{\xi,\mu,\sigma} :=H_{\xi}\left( \frac{x-\mu }{\sigma} \right)$ for a location parameter $\mu \in \mathbb{R}$, a scale parameter $\sigma >0$, and a shape parameter $\xi \in \mathbb{R}$.
\begin{comment}
then $H$ belongs to one of the following three classes of distributions (up to location and scaling):
\begin{align*}
\text{Fr\'echet:} \ \ \ &\Phi_\alpha(x) = \begin{dcases}
0, &\ \ x\leq0,\\
\exp\left\lbrace-x^{-\alpha}\right\rbrace, &\ \ x>0,\\
\end{dcases}\ \ \ \alpha>0,\\
\text{Gumbel:} \ \ \ &\Lambda(x) = \exp\left\lbrace-\exp\lbrace-x\rbrace\right\rbrace, \ \ \ x\in\mathbb{R},\\
\text{Weibull:} \ \ \ &\Psi_\alpha(x) = \begin{dcases}
\exp\left\lbrace-(-x)^{\alpha}\right\rbrace, &\ x\leq0,\\
1, &\ x>0,\\
\end{dcases}\ \ \ \alpha>0.
\end{align*}
\end{comment}
\end{Theorem}
\begin{comment}
\noindent Given that in reality we do not know the distribution $F$ and its maxima limiting case, the below one-parameter distribution provides a generalization of Theorem \ref{Thm: FisherTippet}. Set $\xi=\alpha^{-1}$ for the Fr\'echet distribution, $\xi=-\alpha^{-1}$ for the Weibull distribution and interpret the Gumbel distribution as a limiting case as $\xi\to0$ \citet{jenkinson1955frequency,VonMises1954}, then we obtain the following definition.
\end{comment}
\begin{comment}
\begin{Definition}
The distribution function of a Generalized Extreme Value Distribution (GEV) is given by
\begin{align*}
H_{\xi}(x)&=
\begin{dcases}
\exp \left\lbrace -\left(1+\xi x \right)^{-\frac{1}{\xi}}\right\rbrace, &\ \ \xi\neq0,\\
\exp\left\lbrace-\exp(-x)\right\rbrace, &\ \ \xi=0,\\
\end{dcases}
\end{align*}
where $1+\xi x>0$. A three-parameter family is obtained by defining $H_{\xi,\mu,\sigma} :=H_{\xi}\left( \frac{x-\mu }{\sigma} \right)$ for a location parameter $\mu \in \mathbb{R}$, a scale parameter $\sigma >0$, and a shape parameter $\xi \in \mathbb{R}$.
\end{Definition}
\end{comment}
\noindent The one-parameter GEV is the limiting distribution of the normalized maxima, but in reality, we do not know the norming constants $(a_n)$ and $(b_n)$, therefore, the three-parameter GEV provides a more general and flexible approach as it is the limiting distribution of the unnomarlized maxima.
\citet{pickands1975statistical} and \citet{balkema1974residual} show that the theorem below provides a very powerful result regarding the excess distribution function;
\begin{Theorem}\label{Thm:GPD}
Let $X$ be a random variable with distribution function $F$ and an upper end-point $x_F \leq \infty$. If $F$ is a distribution function that belongs to the maximum domain attraction of a GEV distribution $H_{\xi,\mu,\sigma}$, then
$$\lim_{u\to x_F}\sup_{0\leq x<x_F-u}\left| F_u(x) -G_{\xi,\sigma}(x) \right|=0.$$
where \begin{align*}
F_u(x) = \Pr(X-u\leq x|X>u)=\frac{F(x+u)-F(u)}{1-F(u)}, \ \ \ 0\leq x< x_F-u.
\end{align*}
is the excess distribution over the threshold $u$ and \begin{align*}
G_{\xi,\sigma}(x)&=
\begin{dcases}
1-\left(1+\xi\frac{x}{\sigma} \right)^{-\frac{1}{\xi}}, &\ \ \xi\neq0,\\
1-\exp\left(-\frac{x}{\sigma} \right), &\ \ \xi=0.\\
\end{dcases}
\end{align*}
for $\sigma>0$, and $x\geq0$ when $\xi\geq0$, while $0\leq x\leq-\frac{\sigma}{\xi}$ when $\xi<0$. The parameters $\xi$ and $\sigma$ are referred to, respectively, as the shape and scale parameters.
\end{Theorem}
\noindent This essentially implies that $F_u \approx G_{\xi,\sigma}$ if $u$ is high enough, where $G_{\xi,\sigma}$ is called the Generalized Pareto Distribution (GPD).
\begin{Proposition}\label{Proposition:UnivariateRiskMeasuresGEV}
Assume $X \sim \text{GEV}(\mu,\sigma,\xi)$, then for $0\leq\alpha\leq1$
\begin{align*}
\VaR_{\alpha}(X)&=
\begin{dcases}
\mu -\frac{\sigma}{\xi}\left[1-\left(-\ln \alpha\right)^{-\xi}\right] &\ \ \xi\neq0,\\
\mathbb{E}[X] -\frac{\sigma}{\xi}\left[ \Gamma(1-\xi) -(-\ln\alpha)^{-\xi}\right]&\ \ \xi\neq0, \xi<1,\\
\mathbb{E}[X] - \sigma\gamma - \sigma\ln(-\ln\alpha)&\ \ \xi=0,
\end{dcases}
\end{align*}
where
\begin{align*}
\mathbb{E}[X] &= \begin{dcases}
\mu + \frac{\sigma}{\xi}(\Gamma(1-\xi)-1)&\ \ \xi\neq0, \xi<1,\\
\mu + \sigma \gamma &\ \ \xi=0,\\
\infty &\ \ \xi\geq1,\\
\end{dcases}
\end{align*}
where $\Gamma(x,a)$ is the incomplete Gamma function $\Gamma(x,a)=\int_a^\infty t^{x-1}e^{-t}dt$ such that $\Gamma(x)=\Gamma(x,0)$ and $\gamma$ is Euler's constant defined by $\gamma = \int_1^\infty \left(-\frac{1}{x}+\frac{1}{\lfloor x \rfloor}\right)dx$. VaR diverges for $\xi\geq1$.\\
Let $0\leq\alpha_1\leq\alpha_2\leq1$, then
\begin{align*}
\RVaR_{\alpha_1,\alpha_2}(X)=
\begin{cases}
\mu - \frac{\sigma}{\xi(\alpha_2-\alpha_1)}\left[(\alpha_2-\alpha_1)-\Gamma(1-\xi,-\log \alpha_2) +\Gamma(1-\xi,-\log \alpha_1) \right]&\ \ \xi\neq0,\\
\VaR_{\alpha_i}(X) - \frac{\sigma}{\alpha_2-\alpha_1} \left[\ln(-\ln \alpha_2))-\alpha_j(\ln(-\ln \alpha_1) - \text{li}(\alpha_2) + \text{li}(\alpha_1)\right] &\ \ \xi=0,\\
\end{cases}
\end{align*}
for $i,j\in \lbrace 1,2\rbrace$, $i\neq j$ and where $\text{li}(x)$ is the logarithmic integral $\text{li}(x) = \int_0^x \frac{1}{\ln(t)} dt$ for $0<x<1$ and has a singularity at $x=1$.\\
As a special case, let $\alpha_2=1$, then for $\xi \neq 0$ and $\xi<1$, we have that
\begin{eqnarray*}
\TVaR_{\alpha}(X)&=& \mathbb{E}[X] -\frac{\sigma}{\xi(1-\alpha)}\left[ \Gamma(1-\xi, -\ln\alpha) -\alpha\Gamma(1-\xi) \right] \\
&=& \VaR_{\alpha}(X) -\frac{\sigma}{\xi(1-\alpha)}\left[(1-\alpha)(-\ln\alpha)^{-\xi} +\Gamma(1-\xi, -\ln\alpha) - \Gamma(1-\xi) \right],
\end{eqnarray*}
and $\TVaR$ diverges for $\xi=0$ and $\xi\geq1$.\\
\end{Proposition}
\begin{proof}
See Appendix \ref{Appendix:Proof}.
\end{proof}
\noindent In addition to the risk measures, it is interesting to observe how the ratio of the risk measures behaves for large confidence levels $\alpha$.
\begin{Proposition}\label{Proposition:GEV_RVaR_limit_limit}
Assume $X\sim GEV(\mu,\sigma,\xi)$. Let $0\leq\alpha_1\leq\alpha_2\leq1$, then
\begin{align*}
\lim_{\alpha_1 \to 1}\left[\lim_{\alpha_2 \to 1} \frac{\RVaR_{\alpha_1,\alpha_2}(X)}{\VaR_{\alpha_1}(X)}\right]=\lim_{\alpha_1 \to 1}\left[ \frac{\TVaR_{\alpha_1}(X)}{\VaR_{\alpha_1}(X)}\right] &=\begin{dcases}
(1 -\xi)^{-1} &\ \ \xi>0,\\
1 &\ \ \xi<0.\\
\end{dcases}
\end{align*}
\end{Proposition}
\begin{proof}
See Appendix \ref{Appendix:Proof}.
\end{proof}
Hence, the shape parameter $\xi$ is a strong factor that affects this ratio for large values of $\alpha$. For $\xi<0$, TVaR approaches the value of VaR for high values of $\alpha$, while for $\xi>0$, TVaR becomes significantly larger than VaR.\\
\begin{Proposition}\label{Proposition:UnivariateRiskMeasuresGPD}
Consider the random variable $X$ with cdf $F$ and survival $\bar{F}$. Assume $F \in\text{MDA}(H_{\xi})$, then for $0\leq\alpha\leq1$ and $x\geq u$, where $u$ is a high threshold, we have the following
\begin{align*}
\VaR_{\alpha}(X)
&=\begin{dcases}
u+\frac{\sigma}{\xi}\left(\left(\frac{1-\alpha}{\zeta_u}\right)^{-\xi} -1\right)&\ \ \xi\neq0,\\
u-\sigma\log\left(\frac{1-\alpha}{\zeta_u}\right) &\ \ \xi=0.\\
\end{dcases}
\end{align*}
Let $0\leq\alpha_1\leq\alpha_2\leq1$, then for any value of $\xi$, we have that $$\RVaR_{\alpha_1,\alpha_2}(X) = \dfrac{(1-\alpha_1)\VaR_{\alpha_1}(X) - (1-\alpha_2)\VaR_{\alpha_2}(X) }{(\alpha_2-\alpha_1)(1-\xi)}+\frac{ (\sigma-\xi u)}{(1-\xi)}.$$
As a special case, let $\alpha_2=1$, then for $\xi<1$
$$ \TVaR_{\alpha}(X)=\frac{ \VaR_\alpha(X)}{1-\xi} + \frac{ \sigma-\xi u}{1-\xi},$$
where $\zeta_u=\bar F(u)=\mathbb{P}(X>u)$, and TVaR is infinite for $\xi\geq1$.
\end{Proposition}
\begin{proof}
See Appendix \ref{Appendix:Proof}.
\end{proof}
\noindent $\RVaR$ has not been explored in the literature of Extreme Value Theory. Therefore, we have derived a closed form expression for $\RVaR$. Even though TVaR is infinite for values of $\xi>1$, RVaR exists. Thus, RVaR is useful for the cases where $\xi\geq1$, due to its ability to capture an expected value over a range of high extremes. In theory, this might not be representative of the heavy tail, however, in practice, this can be used to eliminate the issue of having an infinite mean for real data, i.e. insurance companies and financial institutions would still be interested in calculating their reserves and economic capital, and it is not possible to hold an infinite amount of reserves. In this scenario, RVaR can be used with high values of $\alpha_1$ and $\alpha_2$.\\
\noindent In addition to the results of VaR and TVaR, it is interesting to observe how the ratio of the two risk measures behaves for large confidence levels $\alpha$.
\begin{comment}
\begin{Proposition}\label{Proposition:GPD_TVaR_Limit}
Consider the random variable $X$ with cdf $F$. Assume $F \in\text{MDA}(H_{\xi})$, then for $0\leq\alpha\leq1$, $\xi<1$ and $x\geq u$, where $u$ is a high threshold, we have the following
\begin{align*}
\lim_{\alpha \to 1} \frac{\TVaR_\alpha(X)}{\VaR_\alpha(X)}
&=\begin{dcases}
(1-\xi)^{-1}&\ \ \xi\geq0,\\
1 &\ \ \xi<0.\\
\end{dcases}
\end{align*}
\end{Proposition}
\begin{proof}
See Appendix \ref{Appendix:Proof}.
\end{proof}
\begin{Proposition}\label{Proposition:GPD_RVaR_limit}
Consider the random variable $X$ with cdf $F$. Assume $F \in\text{MDA}(H_{\xi})$, then for $0\leq\alpha\leq1$ and $x\geq u$, where $u$ is a high threshold, we have the following
\begin{align*}
\lim_{\alpha_2 \to 1} \frac{\RVaR_{\alpha_1,\alpha_2}(X)}{\VaR_{\alpha_2}(X)}
&=\begin{dcases}
0&\ \ \xi\geq0,\\
1 + \frac{\frac{\sigma}{\xi}\left(\frac{1-\alpha_1}{\zeta_u}\right)^{-\xi}}{(1-\xi)\left(u-\frac{\sigma}{\xi} \right)} &\ \ \xi<0,\\
\end{dcases}
\end{align*}
\end{Proposition}
\begin{proof}
See Appendix \ref{Appendix:Proof}.
\end{proof}
\end{comment}
\begin{Proposition}\label{Proposition:GPD_RVaR_limit_limit}
Consider the random variable $X$ with cdf $F$. Assume $F \in\text{MDA}(H_{\xi})$, then for $0\leq\alpha\leq1$, $\xi<1$ and $x\geq u$, where $u$ is a high threshold, we have the following
\begin{align*}
\lim_{\alpha_1 \to 1}\left[\lim_{\alpha_2 \to 1} \frac{\RVaR_{\alpha_2,\alpha_1}(X)}{\VaR_{\alpha_1}(X)}\right]=\lim_{\alpha_1 \to 1}\left[ \frac{\TVaR_{\alpha_1}(X)}{\VaR_{\alpha_1}(X)}\right]
&=\begin{dcases}
(1 -\xi)^{-1} &\ \ \xi\geq0,\\
1 &\ \ \xi<0.\\
\end{dcases}
\end{align*}
\end{Proposition}
\begin{proof}
See Appendix \ref{Appendix:Proof}.
\end{proof}
Hence, similar to the GEV distribution, the shape parameter $\xi$ is a strong factor that affects this ratio for large values of $\alpha$.\\
\section{Multivariate Lower and Upper Orthant RVaR}\label{Section:MultiRVaR}
In this section, we define the multivariate lower and upper orthant RVaR and study their properties. Examples and illustrations of the findings are provided. Finally, empirical estimators are presented.
\subsection{Lower Orthant RVaR}\label{Section:LowerRVaR}
Consider the continuous random vector $\boldsymbol{X}=(X_1,X_2,\ldots,X_d)\in \mathbb{R}_{+}^{d}$ with joint CDF $F$ and joint survival function $\bar{F}$. Define the random vector $\boldsymbol{X}_{\setminus i}=(X_1,\ldots,X_{i-1},X_{i+1},\ldots,X_d)$ with joint cdf $F_{\setminus i}$ and joint survival function $\bar{F}_{\setminus i}$, for $i=1,\ldots,d$. Let $\boldsymbol x=(x_1,\ldots,x_d)$ be a realization of $\boldsymbol X$ and consider the vector $\boldsymbol{x}_{\setminus i}=(x_1,\ldots,x_{i-1},x_{i+1},\ldots,x_d)$.
\begin{Definition}\label{Definition: LowerRVar}
Consider a continuous random vector $\boldsymbol{X}=(X_1,X_2)$ on
the probability space $(\Omega,\mathcal{F},\mathbb{P})$ with a joint cdf $F$. The lower orthant RVaR at significance level range $[\alpha_1, \alpha_2]\subseteq[0,1]$ is given by
$$\underline{\RVaR}_{\alpha_1,\alpha_2} (\boldsymbol{X})= \bigcup_{i=1}^d \left\lbrace \left(x_1,\ldots,x_{i-1},\underline{\RVaR}_{\alpha_1,\alpha_2,\boldsymbol{x}_{\setminus i}} (\boldsymbol{X})),x_{i+1},\ldots,x_d\right)\right\rbrace,$$
where
$$\underline{\RVaR}_{\alpha_1,\alpha_2,\boldsymbol{x}_{\setminus i}}(\boldsymbol{X})=\mathbb{E}[X_i|\underline{\VaR}_{\alpha_1,\boldsymbol{x}_{\setminus i}}(\boldsymbol{X})\leq X_i\leq \VaR_{\alpha_2}(X_i), \boldsymbol{X}_{\setminus i}\leq \boldsymbol{x}_{\setminus i}],$$
for
$$\underline{\VaR}_{\alpha_1,\VaR_{\alpha_2}(\boldsymbol{X}_{\setminus i})}(\boldsymbol{X})\leq x_j\leq \VaR_{\alpha_2}(X_j),\quad \text{for all } j=1,\ldots,d, i\neq j,$$
in which the lower orthant VaR at significance level $\alpha$ is defined by
$$\underline{\VaR}_{\alpha}(\boldsymbol{X}) = \bigcup_{i=1}^d \left\lbrace \left(x_1,\ldots,x_{i-1},\underline{\VaR}_{\alpha,\boldsymbol{x}_{\setminus i}} (\boldsymbol{X})),x_{i+1},\ldots,x_d\right):x_j \geq \VaR_{\alpha}(X_j),\forall j\neq i\right\rbrace,$$
where
$$\underline{\VaR}_{\alpha,\boldsymbol{x}_{\setminus i}} (\boldsymbol{X})=\inf\left\lbrace x_i \in \mathbb{R}: F_{\boldsymbol{x}_{\setminus i}}(X_i)\geq \alpha \right\rbrace.$$
\end{Definition}
\begin{Proposition}\label{Proposition:RVaRLowerIntegration}
For a continuous random vector $\boldsymbol{X}=(X_1,\ldots,X_d)$ with joint cdf $F$ and for the subvector $\boldsymbol{X}_{\setminus i}=(X_1,\ldots,X_{i-1},X_{i+1},\ldots,X_d)$ with joint cdf $F_{\setminus i}$, $\underline{\RVaR}_{\alpha_1,\alpha_2,\boldsymbol{x}_{\setminus i}}(\boldsymbol{X})$ can be restated as
$$\underline{\RVaR}_{\alpha_1,\alpha_2,\boldsymbol{x}_{\setminus i}} (\boldsymbol{X})=\frac{1}{F(\boldsymbol{x}_{\setminus i},\VaR_{\alpha_2}(X_i))-\alpha_1}\int_{\alpha_1}^{F(\boldsymbol{x}_{\setminus i},\VaR_{\alpha_2}(X_i))}\underline{\VaR}_{u,\boldsymbol{x}_{\setminus i}}(\boldsymbol{X})du,$$
for
$$\underline{\VaR}_{\alpha_1,\VaR_{\alpha_2}(\boldsymbol{X}_{\setminus i})}(\boldsymbol{X})\leq x_j\leq \VaR_{\alpha_2}(X_j),\quad \text{for all } j=1,\ldots,d, \quad i\neq j.$$
\end{Proposition}
\begin{proof}
See Appendix \ref{Appendix:Proof}.
\end{proof}
\begin{Example}\label{Example:RVaR_Lower}
Consider the random vector $(X_1,X_2)$ with joint cdf defined with a Gumbel copula with dependence parameter $\theta=1.5$ and marginals $X_1\sim$ Weibull (2, 50) and $X_2\sim$ Weibull (2, 150). Let the confidence level range be $\alpha_1=0.95$ and $\alpha_2=0.99$. Then, we get bivariate lower orthant $\RVaR$ in Figure \ref{Figure5}. For comparison, we plot $\underline{\VaR}_{0.95,x_i}(\boldsymbol{X})$ on the same graph.
\end{Example}
\begin{figure}%
\hfill
\subfigure[]{\includegraphics[width=8cm]{5}}%
\hfill
\subfigure[]{\includegraphics[width=8cm]{7}}%
\hfill
\caption{(a) Lower orthant VaR at level 0.95 and RVaR at level range $[0.95,0.99]$ for fixed values of $X_1$ and (b) Lower orthant VaR at level 0.95 and RVaR at level range $[0.95,0.99]$ for fixed values of $X_2$}%
\label{Figure5}%
\end{figure}
One can observe from Figure \ref{Figure5} that $\underline{\RVaR}_{\alpha_1,\alpha_2,x_i}(\boldsymbol{X})$ converges to the univariate RVaR when $x_i$ ($i=1,2$) approaches infinity. Also, when $x_i$ gets close to $\VaR_{\alpha_1}(X_i)$, $\underline{\RVaR}_{\alpha_1,\alpha_2,x_i}(\boldsymbol{X})$ approaches $\VaR_{\alpha_2}(X_j)$.\\
By letting $\alpha_2=1$, a special case of the lower orthant RVaR is obtained, namely the lower orthant TVaR, as defined, studied and illustrated by \citet{cossette2015vector}.
\subsection{Upper Orthant RVaR}\label{Section:UpperRVaR}
\begin{Definition}
Consider a continuous random vector $\boldsymbol{X}=(X_1,X_2)$ on
the probability space $(\Omega,\mathcal{F},\mathbb{P})$ with a joint cdf $F$. The upper orthant $\RVaR$ at significance level range $[\alpha_1, \alpha_2]\subseteq[0,1]$ is given by
$$\overline{\RVaR}_{\alpha_1,\alpha_2} (\boldsymbol{X})= \bigcup_{i=1}^d \left\lbrace \left(x_1,\ldots,x_{i-1},\overline{\RVaR}_{\alpha_1,\alpha_2,\boldsymbol{x}_{\setminus i}} (\boldsymbol{X})),x_{i+1},\ldots,x_d\right)\right\rbrace,$$
where
$$\overline{\RVaR}_{\alpha_1,\alpha_2,\boldsymbol{x}_{\setminus i}}(\boldsymbol{X})=\mathbb{E}[X_i|\VaR_{\alpha_1}(X_i)\leq X_i\leq \overline{\VaR}_{\alpha_2,\boldsymbol{x}_{\setminus i}}(\boldsymbol{X}), \boldsymbol{X}_{\setminus i}\geq \boldsymbol{x}_{\setminus i}],$$
for
$$\VaR_{\alpha_1}(X_j)\leq x_j\leq \overline{\VaR}_{\alpha_2,\VaR_{\alpha_1}(\boldsymbol{X}_{\setminus i})}(\boldsymbol{X}),\quad \text{for all } j=1,\ldots,d, i\neq j,$$
in which the upper orthant $\VaR$ at significance level $\alpha$ is defined by
$$\overline{\VaR}_{\alpha}(\boldsymbol{X}) = \bigcup_{i=1}^d \left\lbrace \left(x_1,\ldots,x_{i-1},\overline{\VaR}_{\alpha,\boldsymbol{x}_{\setminus i}} (\boldsymbol{X})),x_{i+1},\ldots,x_d\right):x_j \leq \VaR_{\alpha}(X_j),\forall j\neq i\right\rbrace,$$
where
$$\overline{\VaR}_{\alpha,\boldsymbol{x}_{\setminus i}} (\boldsymbol{X}))=\inf\left\lbrace x_i \in \mathbb{R}: \overline{F}_{\boldsymbol{x}_{\setminus i}}(X_i)\leq 1- \alpha \right\rbrace.$$
\end{Definition}
Similar to the lower orthant RVaR we can define the upper orthant RVaR
in the form of the integration of $\overline{\VaR}_{\alpha,x_i}(\boldsymbol{X})$.\\
\begin{Proposition}\label{Proposition:RVaRUpperIntegration}
For a continuous random vector $\boldsymbol{X}=(X_1,\ldots,X_d)$ with joint cdf $F$ and for the subvector $\boldsymbol{X}_{\setminus i}=(X_1,\ldots,X_{i-1},X_{i+1},\ldots,X_d)$ with joint cdf $F_{\setminus i}$, $\overline{\RVaR}_{\alpha_1,\alpha_2,\boldsymbol{x}_{\setminus i}}(\boldsymbol{X})$ can be restated as
$$\overline{\RVaR}_{\alpha_1,\alpha_2,\boldsymbol{x}_{\setminus i}} (\boldsymbol{X})=\frac{1}{\alpha_2 - (1-\overline{F}(\boldsymbol{x}_{\setminus i},\VaR_{\alpha_1}(X_i)))}\int^{\alpha_2}_{1-\overline{F}(\boldsymbol{x}_{\setminus i},\VaR_{\alpha_1}(X_i))}\overline{\VaR}_{u,\boldsymbol{x}_{\setminus i}}(\boldsymbol{X})\text{d}u,$$
for
$$\VaR_{\alpha_1}(X_j)\leq x_j\leq \overline{\VaR}_{\alpha_2,\VaR_{\alpha_1}(\boldsymbol{X}_{\setminus i})}(\boldsymbol{X}),\quad \text{for all } j=1,\ldots,d, i\neq j,$$
\end{Proposition}
\begin{proof}
See Appendix \ref{Appendix:Proof}.
\end{proof}
\begin{Example}
Consider the same random vector defined in Example~\ref{Example:RVaR_Lower}. Let the confidence level range be $\alpha_1=0.95$ and $\alpha_2=0.99$. Then, we get bivariate upper orthant $\RVaR$ in Figure \ref{Figure6}. For comparison, we plot $\overline{\VaR}_{0.99,x_i}(\boldsymbol{X})$ on the same graph.
\end{Example}
\begin{figure}%
\hfill
\subfigure[]{\includegraphics[width=8cm]{6}}%
\hfill
\subfigure[]{\includegraphics[width=8cm]{8}}%
\hfill
\caption{(a) Upper orthant VaR at level 0.99 and RVaR at level range $[0.95,0.99]$ for fixed values of $X_1$ and (b) Upper orthant VaR at level 0.99 and RVaR at level range $[0.95,0.99]$ for fixed values of $X_2$}%
\label{Figure6}%
\end{figure}
One observes from Figure \ref{Figure5} that $\overline{\RVaR}_{\alpha_1,\alpha_2,x_i}(\boldsymbol{X})$ converges to the univariate $\RVaR_{0.95,0.99}(X_j)$ when $x_i$ ($i=1,2$) gets close to the lower support of $X_i$. Also, when $x_i$ approaches $\VaR_{\alpha_2}(X_i)$, $\overline{\RVaR}_{\alpha_1,\alpha_2,x_i}(\boldsymbol{X})$ approaches $\VaR_{\alpha_1}(X_j)$. As a result, the curves of bivariate RVaR are bounded by the curves of univariate $\VaR$, which is similar to the univariate RVaR.\\
Analogously as for the lower orthant case, letting $\alpha_2=1$, is a special case of the upper orthant which leads to the upper orthant TVaR.
\subsection{Properties of Multivariate Lower and Upper Orthant RVaR}\label{Section:PropertiesRVaR}
For simplicity of notation and proofs, we will consider the bivariate case.
\begin{Proposition}\label{Proposition:Properties}
Let $\boldsymbol{X}=(X_1,X_2)$ be a continuous random vector.
\begin{itemize}
\item [1.](Translation invariance) For all $\textbf{c}=(c_1,c_2)\in\mathbb{R}^2$ and $i,j=1,2$, $i\neq j$, then
$$\underline{\RVaR}_{\alpha_1,\alpha_2,x_j+c_j}(\boldsymbol{X}+\textbf{c})=\underline{\RVaR}_{\alpha_1,\alpha_2,x_j}(\boldsymbol{X})+c_i,$$ $$\overline{\RVaR}_{\alpha_1,\alpha_2,x_j+c_j}(\boldsymbol{X}+\textbf{c})=\overline{\RVaR}_{\alpha_1,\alpha_2,x_j}(\boldsymbol{X})+c_i.$$
\item [2.](Positive homogeneity) For all $\textbf{c}=(c_1,c_2)\in\mathbb{R}^2_{+}$ and $i,j=1,2$, $i\neq j$, then
$$\underline{\RVaR}_{\alpha_1,\alpha_2,c_jx_j}(\textbf{c}\boldsymbol{X})=c_i\underline{\RVaR}_{\alpha_1,\alpha_2,x_j}(\boldsymbol{X}),$$ $$\overline{\RVaR}_{\alpha_1,\alpha_2,c_jx_j}(\textbf{c}\boldsymbol{X})=c_i\overline{\RVaR}_{\alpha_1,\alpha_2,x_j}(\boldsymbol{X}).$$
\item [3.](Monotonicity) Let $\boldsymbol{X}=(X_1,X_2)$ and $\boldsymbol{X}'=(X'_1,X'_2)$ be two pairs of risks with joint cdf's $F_{\boldsymbol{X}}$ and $F_{\boldsymbol{X}'}$, respectively. If $\boldsymbol{X}\prec_{co}\boldsymbol{X}'$, then
$$\underline{\RVaR}_{\alpha_1,\alpha_2}(\boldsymbol{X}')\prec\underline{\RVaR}_{\alpha_1,\alpha_2}(\boldsymbol{X}),$$ $$\overline{\RVaR}_{\alpha_1,\alpha_2}(\boldsymbol{X})\prec\overline{\RVaR}_{\alpha_1,\alpha_2}(\boldsymbol{X}').$$
\end{itemize}
\end{Proposition}
\begin{proof}
See Appendix \ref{Appendix:Proof}.
\end{proof}
Consider the random vectors $\boldsymbol{X}_M$, $\boldsymbol{X}_W$ and $\boldsymbol{X}_\Pi$ which denote the monotonic, countermonotonic and independent vector, respectively. They have the following relationship
$$\boldsymbol{X}_W\prec_{co}\boldsymbol{X}_\Pi\prec_{co}\boldsymbol{X}_M,$$
which means according to the Proposition \ref{Proposition:Properties}, we have
$$\underline{\RVaR}_{\alpha_1,\alpha_2}(\boldsymbol{X}_M)\prec\underline{\RVaR}_{\alpha_1,\alpha_2}(\boldsymbol{X}_\Pi)\prec\underline{\RVaR}_{\alpha_1,\alpha_2}(\boldsymbol{X}_W),$$
and
$$\overline{\RVaR}_{\alpha_1,\alpha_2}(\boldsymbol{X}_W)\prec\overline{\RVaR}_{\alpha_1,\alpha_2}(\boldsymbol{X}_\Pi)\prec\overline{\RVaR}_{\alpha_1,\alpha_2}(\boldsymbol{X}_M).$$
\begin{Example}
Consider a bivariate random vector $(X_1,X_2)$ which is either comonotonic, countermotonic or independent. We obtain the lower orthant RVaR based on Proposition \ref{Proposition:RVaRLowerIntegration}. For $i,j=1,2$ ($i\neq j$),\\
\begin{align*}
&\underline{\RVaR}_{\alpha_1,\alpha_2,x_i} (\boldsymbol{X}_\Pi)=\frac{1}{\alpha_2F_{X_i}(x_i)-\alpha_1}\int_{\alpha_1}^{\alpha_2F_{X_i}(x_i)}\VaR_{\frac{u}{F_{X_i}(x_i)}}(X_j)du,\\
&\underline{\RVaR}_{\alpha_1,\alpha_2,x_i} (\boldsymbol{X}_M)=\frac{1}{F_{X_i}(x_i)-\alpha_1}\int^{F_{X_i}(x_i)}_{\alpha_1}\VaR_{u}(X_j)du,\\
&and\\
&\underline{\RVaR}_{\alpha_1,\alpha_2,x_i} (\boldsymbol{X}_W)=\frac{1}{F_{X_i}(x_i)+\alpha_2-1-\alpha_1}\int_{\alpha_1}^{F_{X_i}(x_i)+\alpha_2-1}\VaR_{u-F_{X_i}(x_i)+1}(X_j)du .\end{align*}
Let the random vector above be defined with exponential marginal cdfs, i.e. $X_i\sim Exp(\lambda_i)$, then we get the following results.\\
\begin{align*}
\underline{\RVaR}_{\alpha_1,\alpha_2,x_i} (\boldsymbol{X}_\Pi)=&\frac{1}{\alpha_2F_{X_i}(x_i)-\alpha_1}\left(\frac{1}{\lambda_i}\Bigg[(F_{X_i}(x_i)-\alpha_2F_{X_i}(x_i))\ln (1-\alpha_2)\right.\\
&\left.\left.-(F_{X_i}(x_i)-\alpha_1)\ln \left(\frac{F_{X_i}(x_i)-\alpha_1}{F_{X_i}(x_i)}\right)+(\alpha_2F_{X_i}(x_i)-\alpha_1)\right]\right),
\\
\underline{\RVaR}_{\alpha_1,\alpha_2,x_i} (\boldsymbol{X}_M)=&\frac{1}{F_{X_i}(x_i)-\alpha_1}\left(\frac{1}{\lambda_i}\Bigg[(1-F_{X_i}(x_i))\ln (1-F_{X_i}(x_i))\right.\\
&-(1-\alpha_1)\ln \left(1-\alpha_1\right)+(F_{X_i}(x_i)-\alpha_1)\Bigg]\Bigg),
\\
\underline{\RVaR}_{\alpha_1,\alpha_2,x_i} (\boldsymbol{X}_W)=&\frac{1}{(F_{X_i}(x_i)+\alpha_2-1)-\alpha_1}\left(\frac{1}{\lambda_i}\Bigg[(1-\alpha_2)\ln (1-\alpha_2)\right.\\
&-(F_{X_i}(x_i)-\alpha_1)\ln \left(F_{X_i}(x_i)-\alpha_1\right)+(F_{X_i}(x_i)+\alpha_2-1-\alpha_1)\Bigg]\Bigg)
.\end{align*}
\end{Example}
Now, we illustrate some examples of multivariate RVaR in the context of EVT. We present closed form expressions obtained in the independence case. Dependence between random variables is considered in section \ref{Section:RVaREmpirical}.
\begin{Example}\label{Example:GEV_Independent}
Assume $F_{X_i}\sim \text{GEV}(\mu_i,\sigma_i,\xi_i)$ and $F_{X_j}\sim \text{GEV}(\mu_j,\sigma_j,\xi_j)$. Let $0\leq\alpha\leq1$ and consider the independent copula where $C(u,v)=uv$.
Let $A=F_{X_i}(x_i), B=F(x_i,\VaR_{\alpha_2}(X_j))$ and $C=1-\bar{F}(x_i,VaR_{\alpha_1}(X_j))$, then, \begin{align*}
\underline{\VaR}_{\alpha,x_i} (\boldsymbol{X})=&
\begin{dcases}
\mu_j-\frac{\sigma_j}{\xi_j}\left[1-\left(\ln\left(\frac{A}{\alpha}\right)\right)^{-\xi_j}\right], & \xi_i,\xi_j\neq0 \\
\mu_j-\sigma_j\ln\left[ \ln \left(\frac{A}{\alpha}\right) \right], & \xi_i=\xi_j=0,
\end{dcases}\\
\end{align*}
and
\begin{align*}
\overline{\VaR}_{\alpha,x_i} (\boldsymbol{X})=&
\begin{dcases}
\mu_j- \frac{\sigma_j}{\xi_j}\left(1-\left[\ln\left(\frac{1-A}{\alpha-A}\right)\right]^{-\xi_j} \right) , & \xi_i,\xi_j\neq0 \\
\mu_j-\sigma_j\ln\left[\ln\left(\frac{1-A)}{\alpha-A}\right)\right] , & \xi_i=\xi_j=0.
\end{dcases}\\
\end{align*}
Then by using the above results, and for $\xi_i,\xi_j \neq 0$, we obtain the multivariate lower and upper orthant RVaR, respectively represented by
\begin{align*}
\underline{\RVaR}_{\alpha_1,\alpha_2,x_i} (\boldsymbol{X})=& \mu_j-\frac{\sigma_j}{\xi_j}\left[1-\frac{A}{B-\alpha_1}\left[ \Gamma\left(1-\xi_j, \ln \left( \frac{A}{B}\right)\right)-\Gamma\left(1-\xi_j, \ln \left( \frac{A}{\alpha_1}\right)\right)\right]\right],\\
\overline{\RVaR}_{\alpha_1,\alpha_2,x_i} (\boldsymbol{X})=&\ \mu_j - \frac{\sigma_j}{\xi_j}\left[1-\frac{1-A}{\alpha_2-C} \left[\Gamma \left(1-\xi_j, \ln\left(\frac{1-A}{
\alpha_2-A} \right) \right) - \Gamma \left(1-\xi_j, \ln\left(\frac{1-A}{C -A} \right)\right)\right] \right],
\end{align*}
while for $\xi_i=\xi_j=0$,
\begin{align*}
\underline{\RVaR}_{\alpha_1,\alpha_2,x_i} (\boldsymbol{X})=&\ \mu_j - \frac{\sigma_j}{B-\alpha_1}\left[B\ln\left(\ln\left(\frac{A}{B}\right)\right) -\alpha_1 \ln\left( \ln\left(\frac{A}{\alpha_1}\right)\right)\right] \\
&-\frac{\sigma_jA}{B-\alpha_1}\left[ \operatorname{Ei}\left(\ln\left(\frac{A}{\alpha_1}\right) \right)-\operatorname{Ei}\left(\ln\left(\frac{A}{B}\right)\right) \right],\\
\overline{\RVaR}_{\alpha_1,\alpha_2,x_i} (\boldsymbol{X})=&\ \mu_i - \frac{\sigma_j}{\alpha_2-C}\left[\left(\alpha_2-A\right)\ln\left(\ln\left(\frac{1-A}{\alpha_2-A}\right)\right)- \left(C-A\right)\ln\left(\ln\left(\frac{1-A}{C-A}\right)\right)\right]\\
&- \frac{\sigma_j\left(1-A\right)}{\alpha_2-C}\left[\operatorname{E_1}\left(\ln\left(\frac{1-A}{\alpha_2-A}\right)\right)-\operatorname{E_1}\left(\ln\left(\frac{1-A}{C-A}\right)\right)\right],
\end{align*}
\begin{comment}
and
$\overline{\RVaR}_{\alpha_1,\alpha_2,x_i} (\boldsymbol{X})$
\begin{eqnarray*}
=&
\begin{dcases}
&&\mu_j - \frac{\sigma_j}{\xi_j}\left[1-\frac{1-A}{\alpha_2-C} \left[\Gamma \left(1-\xi_j, \ln\left(\frac{1-A}{
\alpha_2-A} \right) \right) - \Gamma \left(1-\xi_j, \ln\left(\frac{1-A}{C -A} \right)\right)\right] \right], & \xi_i,\xi_j\neq0,\xi_i,\xi_j\neq1 \\
&&\mu_j - \frac{\sigma_j}{\alpha_2-C}\left[\left(\alpha_2-A\right)\ln\left(\ln\left(\frac{1-A}{\alpha_2-A}\right)\right)- \left(C-A\right)\ln\left(\ln\left(\frac{1-A}{C-A}\right)\right)- \left(1-A\right)\left[ \operatorname{E_1}\left(\ln\left(\frac{1-A}{C-A}\right)\right)\right]-\operatorname{E_1}\left(\ln\left(\frac{1-A}{\alpha_2-A}\right)\right)\right], & \xi_i=\xi_j=0.
\end{dcases}\\
\end{eqnarray*}
\end{comment}
where $\operatorname{Ei}(x) = -\int_{-x}^\infty\frac{e^{-t}}{t}dt$.\\
As a special case of $\RVaR$, we have that when $\alpha_2=1$,
\begin{align*}
\underline{\TVaR}_{\alpha,x_i} (\boldsymbol{X})=&
\begin{dcases}
\mu_j - \frac{\sigma_j}{\xi_j}\left[1 - \frac{ A}{A-\alpha}\left[ \Gamma\left(1-\xi_j\right)- \Gamma\left(1-\xi_j,\ln \left(\frac{A}{ \alpha}\right)\right) \right]\right], & \xi_i,\xi_j\neq0, \\
\infty, & \xi_i=\xi_j=0,
\end{dcases}\\
\end{align*}
and
\begin{align*}
\overline{\TVaR}_{\alpha,x_i} (\boldsymbol{X})=&
\begin{dcases}
\mu_j - \frac{\sigma_j}{\xi_j}\left[1 - \dfrac{1-A}{1-\alpha}\left[\Gamma\left(1-\xi_j\right) - \Gamma\left(1-\xi_j,\ln\left(\frac{1-A}{\alpha-A}\right)\right)\right] \right] , & \xi_i,\xi_j\neq0, \\
\infty, & \xi_i=\xi_j=0.
\end{dcases}\\
\end{align*}
\begin{comment}
\robacomment{Do we need those three ratios below? I feel we do not have a "conclusion" from them.}
\melinacomment{I agree to remove them (do not forget to remove in appendix as well please.}
\begin{align*}
\lim_{\alpha\to 1}\frac{\underline{\TVaR}_{\alpha,x_i} (\boldsymbol{X})}{\underline{\VaR}_{\alpha,x_i} (\boldsymbol{X})} &=
\begin{dcases}
\frac{\mu_j - \frac{\sigma_j}{\xi_j}\left[1 - \frac{ A}{A-1}\left[ \Gamma\left(1-\xi_j\right)- \Gamma\left(1-\xi_j,\ln A\right) \right]\right]}{\mu_j-\frac{\sigma_j}{\xi_j}\left[1-\left(\ln{A}\right)^{-\xi_j}\right]}, & \xi_i,\xi_j\neq0,\xi_i,\xi_j\neq1 \\
\infty , & \xi_i,\xi_j=0 \\
\end{dcases}
\end{align*}
\begin{align*}
\lim_{\alpha_2\to 1}\frac{\underline{\RVaR}_{\alpha_1,\alpha_2,x_i} (\boldsymbol{X})}{\underline{\VaR}_{\alpha_2,x_i} (\boldsymbol{X})} &=\begin{dcases}
\frac{\mu_j - \frac{\sigma_j}{\xi_j}\left[1 - \frac{A}{A-\alpha_1}\left[ \Gamma\left(1-\xi_j \right)- \Gamma\left(1-\xi_j,\ln \left(\frac{A}{\alpha_1}\right)\right) \right]\right]}{\mu_j-\frac{\sigma_j}{\xi_j}\left[1-\left(\ln{A}\right)^{-\xi_j}\right]}, & \xi_i,\xi_j\neq0,\xi_i,\xi_j\neq1 \\
\infty , & \xi_i,\xi_j=0 \\
\end{dcases}
\end{align*}
\begin{align*}
\lim_{\alpha_1\to1}\left[\lim_{\alpha_2\to 1}\frac{\underline{\RVaR}_{\alpha_1,\alpha_2,x_i} (\boldsymbol{X})}{\underline{\VaR}_{\alpha_1,x_i} (\boldsymbol{X})}\right] &=
\begin{dcases}
\frac{\mu_j - \frac{\sigma_j}{\xi_j}\left[1 - \frac{A}{A-1}\left[ \Gamma\left(1-\xi_j\right)- \Gamma\left(1-\xi_j,\ln A\right) \right]\right]}{\mu_j-\frac{\sigma_j}{\xi_j}\left[1-\left(\ln{A}\right)^{-\xi_j}\right]}, & \xi_i,\xi_j\neq0,\xi_i,\xi_j\neq1 \\
\infty , & \xi_i,\xi_j=0 \\
\end{dcases}
\end{align*}
\end{comment}
\end{Example}
\begin{proof}
See Appendix \ref{Appendix:Proof}.
\end{proof}
\begin{Proposition}\label{Proposition:RVaRLimit}
Let $\boldsymbol{X}=(X_1,X_2)$ be a pair of random variables with cdf $F_{\boldsymbol{X}}$ and marginal distributions $F_{X_1}$ and $F_{X_2}$. Assume that $F_{\boldsymbol{X}}$ is continuous and strictly increasing. Then, for $i,j=1,2$ and $i\neq j$,
$$\lim_{x_i\rightarrow\underline{\VaR}_{\alpha_1,\VaR_{\alpha_2}(X_j)}(\boldsymbol{X}) }\underline{\RVaR}_{\alpha_1,\alpha_2,x_i}(\boldsymbol{X})=\VaR_{\alpha_2}(X_j),$$
$$\lim_{x_i\rightarrow \overline{\VaR}_{\alpha_2,\VaR_{\alpha_1}(X_j)}(\boldsymbol{X})}\overline{\RVaR}_{\alpha_1,\alpha_2,x_i}(\boldsymbol{X})=\VaR_{\alpha_1}(X_j).$$
Moreover,
$$\lim_{x_i\rightarrow u_{x_i} }\underline{\RVaR}_{\alpha_1,\alpha_2,x_i}(\boldsymbol{X})=\RVaR_{\alpha_1,\alpha_2}(X_j),$$
$$\lim_{x_i\rightarrow l_{x_i}}\overline{\RVaR}_{\alpha_1,\alpha_2,x_i}(\boldsymbol{X})=\RVaR_{\alpha_1,\alpha_2}(X_j),$$
where $u_{x_i}$ (or $l_{x_i}$) represents the upper (or lower) support of the rv $X_i$.
\end{Proposition}
\begin{proof}
See Appendix \ref{Appendix:Proof}.
\end{proof}
Now, we consider the behavior of aggregate risks defined as follows:
$$\textbf{S}=\binom{S_1}{S_2}=\sum_{i=1}^{n}\binom{X_i}{Y_i},$$
where $S_1$ and $S_2$ denote the aggregate amount of claims for two different business class respectively. $X_i$ and $Y_i$ represent the risks within each class, where $i=1,\ldots,n$, such that $S_1=\sum_{i=1}^{n}X_i$ and $S_2=\sum_{i=1}^{n}Y_i$.\\
Unlike univariate TVaR, the univariate RVaR does not satisfy the subadditivity. Hence, it seems impossible to prove that the bivariate RVaR is subadditive. However, if we suppose that $(X_1,\ldots,X_n)$ (respectively $(Y_1,\ldots,Y_n)$) is comonotonic, the following results can be obtained.\\
\begin{Proposition}\label{Proposition:Aggregate}
Let $(X_1,\ldots,X_n)$ (respectively $(Y_1,\ldots,Y_n)$) be comonotonic with cdf's $F_{X_1},\ldots,F_{X_n}$ (respectively $G_{Y_1},\ldots,G_{Y_n}$). The dependence structure between $(X_1,\ldots,X_n)$ and $(Y_1,\ldots,Y_n)$ is unknown. Then,
$$\underline{\RVaR}_{\alpha_1,\alpha_2,S_1}(\textbf{S})=\sum_{i=1}^{n}\underline{\RVaR}_{\alpha_1,\alpha_2,x_i}(X_i,Y_i),$$
$$\underline{\RVaR}_{\alpha_1,\alpha_2,S_2}(\textbf{S})=\sum_{i=1}^{n}\underline{\RVaR}_{\alpha_1,\alpha_2,y_i}(X_i,Y_i),$$
and
$$\overline{\RVaR}_{\alpha_1,\alpha_2,S_1}(\textbf{S})=\sum_{i=1}^{n}\overline{\RVaR}_{\alpha_1,\alpha_2,x_i}(X_i,Y_i),$$
$$\overline{\RVaR}_{\alpha_1,\alpha_2,S_2}(\textbf{S})=\sum_{i=1}^{n}\overline{\RVaR}_{\alpha_1,\alpha_2,y_i}(X_i,Y_i).$$
\end{Proposition}
\begin{proof}
See Appendix \ref{Appendix:Proof}.
\end{proof}
In conclusion, the bivariate RVaR has similar properties to the bivariate VaR and TVaR, such as translation invariance, positive homogeneity and monotonicity. Furthermore, it has an advantage over bivariate VaR and TVaR. Compared to bivariate VaR, bivariate TVaR and RVaR provide essential information about the tail of the distribution. Moreover, $\underline{\TVaR}_{\alpha,x_i}(\boldsymbol{X})$ and $\overline{\TVaR}_{\alpha,x_i}(\boldsymbol{X})$ will go to infinity when $X_i$ approaches $\VaR_{\alpha}(X_i)$ whereas the bivariate RVaR is bounded in the area $[\VaR_{\alpha_1}(X_i),\VaR_{\alpha_2}(X_i)]\times[\VaR_{\alpha_1}(X_j),\VaR_{\alpha_2}(X_j)]$. This measure could be useful for insurance companies that must set aside capital for risks that are sent to a reinsurer after having reached a certain level. Assume that the insurance company transfers the risks to the reinsurer when the total losses exceed VaR at level $\alpha_2$. Then, to comply to solvency capital requirements, the insurance company needs to measure the risks with truncated data. In this case, multivariate RVaR could be helpful.
\\
We will check the robustness of the estimator of bivariate RVaR. Since RVaR is distribution-based, the sensitivity function can be used to quantify the robustness.
\begin{Proposition}\label{Proposition:VaRRobust}
For a pair of continuous random variables $\boldsymbol{X}$ with joint cdf $F(x_1,x_2)$ and marginals $F_{X_1}(x_1)$ and $F_{X_2}(x_2)$, the sensitivity function of $\underline{\VaR}_{\alpha,x_i}(\boldsymbol{X})$ is given by
\begin{align*}
S(z) &=\begin{dcases}
-\frac{F_{X_i}(x_i)-\alpha}{f_{x_i}\left[\underline{\VaR}_{\alpha,x_i}(\boldsymbol{X})\right]F_{X_i}(x_i)},&\ \ z<\underline{\VaR}_{\alpha,x_i}(\boldsymbol{X}),\\
\frac{\alpha}{f_{x_i}\left[\underline{\VaR}_{\alpha,x_i}(\boldsymbol{X})\right]F_{X_i}(x_i)}, &\ \ z>\underline{\VaR}_{\alpha,x_i}(\boldsymbol{X}),\\
0, &\ \ \text{otherwise},\\
\end{dcases}
\end{align*}
which is bounded. Thus, $\underline{\VaR}_{\alpha,x_i}(\boldsymbol{X})$ is a robust risk measure.
\end{Proposition}
\begin{proof}
See Appendix \ref{Appendix:Proof}.
\end{proof}
\begin{Proposition}\label{Proposition:RVaRRobust}
For a pair of continuous random variables $\boldsymbol{X}$ with joint cdf $F(x_1,x_2)$ and marginals $F_{X_1}(x_1)$ and $F_{X_2}(x_2)$, let $A=F_{X_i}(x_i)$ and $B=F(x_i,\VaR_{\alpha_2}(X_j))$. Then the sensitivity function of $\underline{\RVaR}_{\alpha_1,\alpha_2,x_i}(\boldsymbol{X})$ is given by\\\\
$S(z)=S'(z) -\underline{\RVaR}_{\alpha_1,\alpha_2,x_i}(\boldsymbol{X}),$\\
where
\begin{align*}
S'(z)=
\begin{dcases}
\frac{(A-\alpha_1)\underline{\VaR}_{\alpha_1,x_i}(\boldsymbol{X})-(A-B)\VaR_{\alpha_2}(X_j)}{B-\alpha_1},&\ \ z<\underline{\VaR}_{\alpha_1,x_i}(\boldsymbol{X}),\\
\frac{zA-\alpha_1\underline{\VaR}_{\alpha_1,x_i}(\boldsymbol{X})-(A-B)\VaR_{\alpha_2}(X_j)}{B-\alpha_1}, &\ \ \underline{\VaR}_{\alpha_1,x_i}(\boldsymbol{X})\leq z\leq \VaR_{\alpha_2}(X_j),\\
\frac{B \VaR_{\alpha_2}(X_j)-\alpha_1\underline{\VaR}_{\alpha_1,x_i}(\boldsymbol{X})}{B-\alpha_1}, &\ \ z>\VaR_{\alpha_2}(X_j),\\
\end{dcases}
\end{align*}
is a bounded function. Thus, $\underline{\RVaR}_{\alpha_1,\alpha_2,x_i}(\boldsymbol{X})$ is robust.
\end{Proposition}
\begin{proof}
See Appendix \ref{Appendix:Proof}.
\end{proof}
\begin{Proposition}\label{Proposition:UpperVaRRobust}
For a pair of continuous random variables $\boldsymbol{X}$ with joint survival function $\bar{F}(x_1,x_2)$ and marginals $F_{X_1}(x_1)$ and $F_{X_2}(x_2)$, the sensitivity function of $\overline{\VaR}_{\alpha,x_i}(\boldsymbol{X})$ is given by
\begin{align*}
S(z) &=\begin{dcases}
-\frac{1-\alpha}{f_{\bar{x}_i}\left[\overline{\VaR}_{\alpha,x_i}(\boldsymbol{X})\right](1-F_{X_i}(x_i))},&\ \ z<\overline{\VaR}_{\alpha,x_i}(\boldsymbol{X}),\\
\frac{\alpha-F_{X_i}(x_i)}{f_{\bar{x}_i}\left[\overline{\VaR}_{\alpha,x_i}(\boldsymbol{X})\right](1-F_{X_i}(x_i))}, &\ \ z>\overline{\VaR}_{\alpha,x_i}(\boldsymbol{X}),\\
0, &\ \ z=\overline{\VaR}_{\alpha,x_i}(\boldsymbol{X}).\\
\end{dcases}
\end{align*}
The bounded sensitivity function implies $\overline{\VaR}_{\alpha,x_i}(\boldsymbol{X})$ is a robust risk measure.
\end{Proposition}
\begin{proof}
See Appendix \ref{Appendix:Proof}.
\end{proof}
\begin{Proposition}\label{Proposition:RVaRUpperRobust}
For a pair of continuous random variables $\boldsymbol{X}$ with joint survival function $\bar{F}(x_1,x_2)$ and marginals $F_{X_1}(x_1)$ and $F_{X_2}(x_2)$, let $A=F_{X_i}(x_i)$ and $C=1-\bar{F}(x_i,VaR_{\alpha_1}(X_j))$. Then the sensitivity function of $\overline{\RVaR}_{\alpha_1,\alpha_2,x_i}(\boldsymbol{X})$ is given by\\
$S(z)=S'(z) -\overline{\RVaR}_{\alpha_1,\alpha_2,x_i}(\boldsymbol{X}),$\\
where
\begin{align*}
S'(z)=
\begin{dcases}
\frac{(1-C)\VaR_{\alpha_1}(X_j)-(1-\alpha_2)\overline{\VaR}_{\alpha_2,x_i}(\boldsymbol{X})}{\alpha_2-C},&\ \ z<\VaR_{\alpha_1}(X_j),\\
\frac{z(1-A)-(C-A) \VaR_{\alpha_1}(X_j)-(1-\alpha_2)\overline{\VaR}_{\alpha_2,x_i}(\boldsymbol{X})}{\alpha_2-C}, &\ \ \VaR_{\alpha_1}(X_j)\leq z\leq \overline{\VaR}_{\alpha_2,x_i}(\boldsymbol{X}),\\
\frac{(\alpha_2-A) \overline{\VaR}_{\alpha_2,x_i}(\boldsymbol{X})-(C-A) \VaR_{\alpha_1}(X_j)}{\alpha_2-C}, &\ \ z>\overline{\VaR}_{\alpha_2,x_i}(\boldsymbol{X}).\\
\end{dcases}
\end{align*}
The bounded function proves that $\overline{\RVaR}_{\alpha_1,\alpha_2,x_i}(\boldsymbol{X})$ is robust.
\end{Proposition}
\begin{proof}
See Appendix \ref{Appendix:Proof}.
\end{proof}
\subsection{Empirical Estimator for Multivariate Lower and Upper Orthant RVaR}\label{Section:RVaREmpirical}
Next, we will propose empirical estimators for the lower and upper orthant RVaR, based on the estimators developed by \citet{beck2015multivariate}, and provide numerical examples.
\begin{Definition}
Consider a series of observations $\boldsymbol{X}=(\boldsymbol{X}_1,\ldots,\boldsymbol{X}_d)$ with $\boldsymbol{X}_i=(x_{1i},\ldots,x_{ni})$ and $\boldsymbol{X}_{\setminus i}=(\boldsymbol{X}_{1},\ldots,\boldsymbol{X}_{i-1},\boldsymbol{X}_{i+1},\ldots,\boldsymbol{X}_{d})$, $i=1,\ldots,d$. Denote $F_n$ and $F_{n,\setminus i}$, the empirical cdf's (ecdf) for $\boldsymbol{X}$ and $\boldsymbol{X}_{\setminus i}$, $i=1,\ldots,d$, respectively. We define the estimator for the lower orthant $RVaR$ for fixed $\boldsymbol{x}_{\setminus i}$, $i=1,\ldots,d$, by
$$\underline{\RVaR}^n_{\alpha_1,\alpha_2,\boldsymbol{x}_{\setminus i}} (\boldsymbol{X})=\frac{1}{F_n(\boldsymbol{x}_{\setminus i},\VaR_{\alpha_2}(\boldsymbol{X}_i))-\alpha_1}\int_{\alpha_1}^{F_n(\boldsymbol{x}_{\setminus i},\VaR_{\alpha_2}(\boldsymbol{X}_i))}\underline{\VaR}^n_{u,\boldsymbol{x}_{\setminus i}}(\boldsymbol{X})du,$$
For $m\in\mathbb{N}$ large enough, let $s=\frac{F_n(\boldsymbol{x}_{ \setminus i},\VaR_{\alpha_2}(\boldsymbol{X}_{ i}))-\alpha_1}{m}$ and $u_k=\alpha_1+ks$, then the above expression can be simplified into
\begin{align*}
\underline{\RVaR}^n_{\alpha_1,\alpha_2,\boldsymbol{x}_{\setminus i}} (\boldsymbol{X})&=\sum_{k=1}^{m}\frac{\underline{\VaR}^n_{u_k,\boldsymbol{x}_{\setminus i}}(\boldsymbol{X})\cdot s}{F_n(\boldsymbol{x}_{\setminus i},\VaR_{\alpha_2}(\boldsymbol{X}_i))-\alpha_1}\\
&=\sum_{k=1}^{m}\frac{\underline{\VaR}^n_{u_k,\boldsymbol{x}_{\setminus i}}(\boldsymbol{X})}{m},
\end{align*}
where $\underline{\VaR}^n_{u,\boldsymbol{x}_{\setminus i}}(\boldsymbol{X})=\inf{\{x_i\in\mathbb{R}_{+}:F_{n,\boldsymbol{x}_{\setminus i}}(x_i)\geq u\}}$ is the empirical lower orthant $\VaR$ for a given $\boldsymbol{x}_{\setminus i}$ and $F_{n,\boldsymbol{x}_{\setminus i}}$ is the ecdf of $\boldsymbol{X}$ given the same $\boldsymbol{x}_{\setminus i}$.
\end{Definition}
\begin{comment}
\begin{Definition}
Consider a series of observations $\boldsymbol{X}=(X_1,X_2)$ with $X_i=(x_{1i},\ldots,x_{ni})$, $i=1,2$. Additionally, we have $\boldsymbol{x}_l=(x_{l1},x_{l2})\in\mathbb{R}^2_{+}$, $l=1,\ldots,n$. Denote $F_n$ and $F_{n,i}$, the empirical cdf's (ecdf) for $\boldsymbol{X}$ and $X_{i}$, respectively. We define the estimator for the lower orthant $RVaR$, for a fixed $X_i$, by
$$\underline{\RVaR}^n_{\alpha_1,\alpha_2,x_{i}} (\boldsymbol{X})
=\frac{\int_{\alpha_1}^{F_n(x_{i},\VaR_{\alpha_2}(X_j))}\underline{\VaR}^n_{u,x_{i}}(\boldsymbol{X})du}{F_n(x_{i},\VaR_{\alpha_2}(X_j))-\alpha_1}.$$
For $m\in\mathbb{N}$ large enough, let $s=\frac{F_n(x_{i},\VaR_{\alpha_2}(X_j))-\alpha_1}{m}$ and $u_k=\alpha_1+ks$, then the above expression can be simplified into
\begin{align*}
\underline{\RVaR}^n_{\alpha_1,\alpha_2,x_{i}} (\boldsymbol{X})&=\sum_{k=1}^{m}\frac{\underline{\VaR}^n_{u_k,x_{i}}(\boldsymbol{X})\cdot s}{F_n(x_{i},\VaR_{\alpha_2}(X_j))-\alpha_1}\\
&=\sum_{k=1}^{m}\frac{\underline{\VaR}^n_{u_k,x_{i}}(\boldsymbol{X})}{m},
\end{align*}
where $\underline{\VaR}^n_{u,x_{i}}(\boldsymbol{X})=\inf{\{x_j\in\mathbb{R}_{+}:F_{n,x_{i}}(x_j)\geq u\}}$ is the empirical lower orthant $VaR$ given $X_{i}$ and $F_{n,x_{i}}$ the ecdf of $\boldsymbol{X}$ given the same $X_{i}$.
\end{Definition}
\end{comment}
Note, $\underline{\VaR}^n_{u,\boldsymbol{x}_{\setminus i}}(\boldsymbol{X})$ is the smallest value of $\boldsymbol{X}_i$ given $\boldsymbol{x}_{\setminus i}$ such that $F_n$ is larger than $u$. Similarly, we define the empirical estimator of upper orthant RVaR as follows.\\
\begin{Definition}
Consider a series of observations $\boldsymbol{X}=(\boldsymbol{X}_1,\ldots,\boldsymbol{X}_d)$ with $\boldsymbol{X}_i=(x_{1i},\ldots,x_{ni})$ and $\boldsymbol{X}_{\setminus i}=(\boldsymbol{X}_{1},\ldots,\boldsymbol{X}_{i-1},\boldsymbol{X}_{i+1},\ldots,\boldsymbol{X}_{d})$, $i=1,\ldots,d$. Denote $\bar{F}_n$ and $\bar{F}_{n,\setminus i}$, the empirical survival functions for $\boldsymbol{X}$ and $\boldsymbol{X}_{\setminus i}$, $i=1,\ldots,d$, respectively. We define the estimator for the upper orthant $\RVaR$ for fixed $\boldsymbol{x}_{\setminus i}$, $i=1,\ldots,d$, by
$$\overline{\RVaR}^n_{\alpha_1,\alpha_2,\boldsymbol{x}_{\setminus i}} (\boldsymbol{X})=\frac{1}{\alpha_2 - (1-\overline{F}_n(\boldsymbol{x}_{\setminus i},\VaR_{\alpha_1}(\boldsymbol{X}_i)))}\int^{\alpha_2}_{1-\overline{F}_n(\boldsymbol{x}_{\setminus i},\VaR_{\alpha_1}(\boldsymbol{X}_i))}\overline{\VaR}^n_{u,\boldsymbol{x}_{\setminus i}}(\boldsymbol{X})\text{d}u,$$
For $m\in\mathbb{N}$ large enough. Let $s=\frac{\alpha_2-(1-\bar{F}_n(\boldsymbol{x}_{\setminus i},\VaR_{\alpha_1}(\boldsymbol{X}_i)))}{m}$ and $v_k=1-\bar{F}_n(\boldsymbol{x}_{\setminus i},\VaR_{\alpha_1}(\boldsymbol{X}_i))+ks$, then the above expression can be simplified into
\begin{align*}
\overline{\RVaR}^n_{\alpha_1,\alpha_2,\boldsymbol{x}_{\setminus i}} (\boldsymbol{X}) & =\sum_{k=1}^{m}\frac{\overline{\VaR}^n_{v_k,\boldsymbol{x}_{\setminus i}}(\boldsymbol{X})\cdot s}{\alpha_2-(1-\bar{F}_n(\boldsymbol{x}_{\setminus i},\VaR_{\alpha_1}(\boldsymbol{X}_i)))} \\
&=\sum_{k=1}^{m}\frac{\overline{\VaR}^n_{v_k,\boldsymbol{x}_{\setminus i}}(\boldsymbol{X})}{m},
\end{align*}
where $\overline{\VaR}^n_{v,\boldsymbol{x}_{\setminus i}}(\boldsymbol{X})=\inf{\{x_i\in\mathbb{R}_{+}:\bar{F}_{n,\boldsymbol{x}_{\setminus i}}(\boldsymbol{x}_i)\leq 1-v\}}$ is the empirical upper orthant $\VaR$ given $\boldsymbol{x}_{\setminus i}$ and $\bar{F}_{n,\boldsymbol{x}_{\setminus i}}$ is the empirical survival function of $\boldsymbol{X}$ given the same $\boldsymbol{x}_{\setminus i}$.\\
\end{Definition}
\begin{comment}
\begin{Definition}
Consider a series of observations $\boldsymbol{X}=(X_1,X_2)$ with $X_i=(x_{1i},\ldots,x_{ni})$, $i=1,2$. Additionally, we have $\boldsymbol{x}_l=(x_{l1},x_{l2})\in\mathbb{R}^2_{+}$, $l=1,\ldots,n$. Denote by $F$ the distribution function of $\boldsymbol{X}$, and by $\bar{F}_n$ and $\bar{F}_{n,i}$, empirical survival functions for $\boldsymbol{X}$ and $X_{i}$, respectively. For a fixed $X_i$, the estimator for the lower orthant $RVaR$ is defined by
$$\overline{\RVaR}^n_{\alpha_1,\alpha_2,x_{i}} (\boldsymbol{X})
=\frac{\int^{\alpha_2}_{1-\bar{F}_n(x_{i},\VaR_{\alpha_1}(X_j))}\overline{\VaR}^n_{v,x_{i}}(\boldsymbol{X})du}{\alpha_2-(1-\bar{F}_n(x_{i},\VaR_{\alpha_1}(X_j)))}.$$
For $m\in\mathbb{N}$ large enough. Let $s=\frac{\alpha_2-(1-\bar{F}_n(x_{i},\VaR_{\alpha_1}(X_j)))}{m}$ and $v_k=1-\bar{F}_n(x_{i},\VaR_{\alpha_1}(X_j))+ks$, then the above expression can be simplified into
\begin{align*}
\overline{\RVaR}^n_{\alpha_1,\alpha_2,x_{i}} (\boldsymbol{X}) & =\sum_{k=1}^{m}\frac{\overline{\VaR}^n_{v_k,x_{i}}(\boldsymbol{X})\cdot s}{\alpha_2-(1-\bar{F}_n(x_{i},\VaR_{\alpha_1}(X_j)))} \\
&=\sum_{k=1}^{m}\frac{\overline{\VaR}^n_{v_k,x_{i}}(\boldsymbol{X})}{m},
\end{align*}
where $\overline{\VaR}^n_{v,x_{i}}(\boldsymbol{X})=\inf{\{x_j\in\mathbb{R}_{+}:\bar{F}_{n,x_{i}}(x_j)\leq 1-v\}}$ is the empirical upper orthant $VaR$ given $X_{i}$ and $\bar{F}_{n,x_{i}}$ the empirical survival function of $\boldsymbol{X}$ given the same $X_{i}$.\\
\end{Definition}
\end{comment}
The following proposition, based on the proof of the consistency of bivariate VaR by \citet{cousin2013multivariate}, shows the consistency of the bivariate RVaR in Hausdorff distance. For $\alpha\in(0,1)$ and $r,\zeta>0$, consider the ball
$$E=B(\{\boldsymbol{x}\in\mathbb{R}^2_+:|F(\boldsymbol{x})-\alpha|\leq r\},\zeta).$$
Denote $m^\nabla=\inf_{\boldsymbol{x}\in E}\parallel(\nabla F)_{\boldsymbol{x}} \parallel$ as the infimum of the Euclidean norm of the gradient vector and $M_H=\sup_{\boldsymbol{x}\in E}\parallel(HF)_{\boldsymbol{x}}\parallel$ as the matrix norm of the Hessian matrix evaluated at $\boldsymbol{x}$ for a twice differentiable $F(x_1,x_2)$.\\
\begin{Proposition}\label{Proposition:Empirical}
Let $[\alpha_1,\alpha_2]\subset(0,1)$ and $F(x_1,x_2)$ be twice differentiable on $\mathbb{R}^2$. Assume there exists $r,\zeta>0$ such that $m^\nabla>0$ and $M_H<\infty$. Assume for each $n$, $F_n$ is continuous with probability one (wp1) and
$$\parallel F-F_n\parallel\underset{n\rightarrow\infty}{\overset{wp1}{\longrightarrow}}0.$$
Also, let $F_{n,i}$ be the consistent estimator of $F_i$. Then, we have
$$\underline{\RVaR}^n_{\alpha_1,\alpha_2,x_i}(\boldsymbol{X})\underset{n\longrightarrow\infty}{\overset{wp1}{\longrightarrow}}\underline{\RVaR}_{\alpha_1,\alpha_2,x_{i}}(\boldsymbol{X}).$$
\end{Proposition}
\begin{proof}
See Appendix \ref{Appendix:Proof}.
\end{proof}
A simulation study is performed to compare the empirical estimators to the theoretical lower and upper orthant $\RVaR$. Marginally, the random variables are distributed from GEV distributions. The dependence is represented by an independent copula. 50 simulations are performed for samples of 4000 observations from each marginal distribution and for $m=250$. The results of the simulation are presented in Figure \ref{Figure:GEV_simulation_Lower}. As shown, the differeneces between the theoretical values and their empirical estimates are negligible. This could be attributed to the robustness and consistency of the empirical estimators of $\VaR$ and $\RVaR$. The accuracy of the estimates improves with the sample size and value of $m$.
\begin{comment}
\begin{figure}%
\hfill
\subfigure[]{\includegraphics[width=8cm]{RVaR_Lower_GEV.png}}%
\hfill
\subfigure[]{\includegraphics[width=8cm]{RVaR_Lower_GEV_xi0.png}}%
\hfill
\caption{(a) Lower orthant VaR at level 0.95 and RVaR at level range $[0.95,0.99]$ for fixed values of $X_1$ for GEV samples where $\xi_1,\xi_2\neq0$ and (b) Lower orthant VaR at level 0.95 and RVaR at level range $[0.95,0.99]$ for fixed values of $X_1$ for GEV samples where $\xi_1=\xi_2=0$}%
\label{Figure:GEV_simulation_Lower}%
\end{figure}
\end{comment}
\begin{figure}%
\hfill
\subfigure[]{\includegraphics[width=8cm]{RVaR_GEV_Lower_New.png}}%
\hfill
\subfigure[]{\includegraphics[width=8cm]{RVaR_GEV_Lower_xi0_New.png}}%
\hfill
\caption{(a) Lower orthant VaR at level 0.95 and RVaR at level range $[0.95,0.99]$ for fixed values of $X_1$ for GEV samples where $\xi_1,\xi_2\neq0$ and (b) Lower orthant VaR at level 0.95 and RVaR at level range $[0.95,0.99]$ for fixed values of $X_1$ for GEV samples where $\xi_1=\xi_2=0$}
\label{Figure:GEV_simulation_Lower}%
\end{figure}
\begin{comment}
\begin{figure}%
\hfill
\subfigure[]{\includegraphics[width=8cm]{RVaR_upper_GEV.png}}%
\hfill
\subfigure[]{\includegraphics[width=8cm]{RVaR_Lower_GEV_xi0.png}}%
\hfill
\caption{(a) Upper orthant VaR at level 0.95 and RVaR at level range $[0.95,0.99]$ for fixed values of $X_1$ for GEV samples where $\xi_1,\xi_2\neq0$ and (b) Lower orthant VaR at level 0.95 and RVaR at level range $[0.95,0.99]$ for fixed values of $X_1$ for GEV samples where $\xi_1=\xi_2=0$}%
\label{Figure:GEV_simulation_Upper}%
\end{figure}
\end{comment}
\section{Conclusion}\label{Section:Conclusion}
In this paper, we introduce the multivariate extension of the RVaR risk measure, which results in a tool employed to assess dependent risks. This tool is particularly useful for heavy-tail distributions. Similar to its univariate counterpart, multivariate lower and upper orthant $\RVaR$ are defined as the conditional expectation of the lower and upper orthant $\VaR$ for large confidence levels. Their properties are discussed, such as translation invariance, positive homogeneity and monotonicity. Subadditivity can be satisfied for aggregated risks if each risk class is monotonic. Moreover, we develop resulting measures with specific extreme value distributions. The method of sensitivity functions to study robustness is extended for distribution-based multivariate RVaR. Finally, the empirical estimators of multivariate RVaR are proposed. The robustness and consistency of such estimators are confirmed. Furthermore, the simulations illustrate the accuracy of the empirical estimators without the need to assume any statistical distributions. RVaR may be extremely relevant in some instances where the loss distribution is characterized with an infinite mean, which results in an infinite TVaR.
\section*{Acknowledgements}
M\'elina Mailhot acknowledges financial support from the Natural Sciences and Engineering Research Council. Roba Bairakdar acknowledges financial support from the Society of Actuaries Hickman Scholar Program.
| -148,471.688176 |
[
-3.05078125,
2.84375
] | 30.548303 |
[
-3.515625,
0.271484375,
-1.8818359375,
-6.37890625,
-0.625,
8.796875
] |
[
2.984375,
7.55859375,
1.69921875,
6.6015625
] | 618 | 10,358 |
[
-2.1015625,
2.2734375
] | 34.272784 |
[
-5.99609375,
-4.3046875,
-4.609375,
-2.16796875,
2.404296875,
12.5625
] | 0.490894 | 24.307259 | 12.454142 | 2.26533 |
[
2.5390982627868652
] | -97,875.515635 | 7.86677 | -148,099.750112 | 0.876246 | 6.008669 |
[
-2.501953125,
-3.5546875,
-3.84765625,
-5.08203125,
2.5234375,
12.2578125
] |
[
-5.625,
-2.427734375,
-2.5546875,
-2.197265625,
4.0625,
5.93359375
] | |
BkiUdfTxK6Ot9UjEBlvA
|
\section{Introduction}
Diffuse dwarf galaxies (DDGs) are low-luminosity galaxies
that seem to be characterized by
structural parameters (luminosity, stellar scale length)
fundamentally different from those found in
spiral and elliptical galaxies (\citealt{kormendy1985}) with the
dwarf spheroidal galaxies (dSphs) at the extreme end of this sequence.
In particular, dSphs are thought to be satellite galaxies in the Local Group
(see e.g., \citealt{mashchenko}), have approximately spheroidal shapes (sometimes typical of irregular
and late-type spiral galaxies), and are usually
at least two orders of magnitude less
luminous than the faintest known spiral galaxies. These systems have stellar contents in the range
$3\times10^3$ M$_{\odot}$ to $2\times10^7$ M$_{\odot}$ (\citealt{martin2008}) on
length scales of a few kpc or less. Additionally, they show evidence of
being dark-matter dominated at all radii (for a review see \citealt{mateo1997})
as shown by the measurements of the central velocity dispersion that are much larger than typical values
(see \citealt{mateo1998a}, \citealt{kleina2001},
\citealt{kleina2002} and references therein). The central velocity dispersion allows the
mass-to-light ratios to be estimated.
The differences between the normal galaxies and the dSphs families probably come from a different
formation history with the most favored theory being that dSphs have low mass density since past
supernova winds removed large amounts of gas (\citealt{silk1987}).
Despite being very different in their physical properties from spirals
and ellipticals, dSphs show kinematical properties that can be modeled using
dark matter (DM) halos with the same mass profiles as those which reproduce
the rotation curves of spirals (\citealt{salucci2012}). Thus, the derived central densities
and core radii for dSphs are consistent with the values
obtained by extrapolation of the relevant quantities from spiral galaxies.
The dSph group is also interesting since it provides an optimal laboratory to study the evolution of a particular
stellar population (of known metallicity and age) without suffering extreme crowding conditions as often
happens in globular clusters.
In this respect, the high-energy view of these galaxies, such as that offered by deep {\it XMM}-Newton observations,
allows the study of the faint end of the
X-ray luminosity function of an old stellar population. Furthermore, by studying
the low mass X-ray binary
(LMXB) population characteristics in dSphs and globular clusters, it is possible
to get information about the formation history (still challenging, see e.g.
\citealt{maccarone2005b}) of such systems. Since any persistently bright LMXB would entirely consume the mass of the
companion via accretion in a few hundred million years (see e.g. the discussion on the LMXB formation history
in \citealt{maccarone2005b}), the presence of
bright X-ray binaries in old stellar systems represents a problem that is yet to be solved. An example of
these challenging targets is the Sculptor dSph galaxy. When studying a deep Chandra survey of this dwarf galaxy, \citet{maccarone2005b}
found at least five X-ray sources with optical counterparts hence pushing towards alternative formation theories of the local LMXB population.
A push in this direction would be the observation of targets with no globular cluster contamination (like the Sculptor galaxy) and, possibly,
with a short epoch of star formation.
Based on the extrapolation to globular clusters of the fundamental $M_{BH}-M_{Bulge}$ relation derived from the study of super massive
black holes in galactic nuclei (see e.g. \citealt{magorrian1998}), one expects to find intermediate
mass black holes (hereafter IMBHs) in globular clusters, i.e., spherical systems of stars
which survived the interactions with the surroundings objects and now orbit the center of the hosting galaxy. Since it is commonly
accepted that the galaxies and associated globular clusters
formed at the same time (see e.g. \citealt{globular1} and \citealt{globular2}), it is
natural to expect that at least some of these spherical systems may host an IMBH.
Apart from the DM and stellar population issues, dSphs are intriguing places to search for
IMBHs, i.e., collapsed objects in the mass range $10^2$-$10^5$ M$_{\odot}$
which are considered to be the missing link between the observed stellar mass black holes (of a few tens of solar masses) and the super massive ones
($10^6-10^8$ M$_{\odot}$) residing at the center of most galaxies. One of the reasons why the existence of such objects is expected is
they might play a crucial role in the
formation of the super massive objects which are thought
to grow from a population of seed objects with masses in the IMBH range (see e.g. \citealt{ebisuzaki2001}).
Obviously, those seeds that did not accrete a
substantial amount of matter and/or did not merge to form a central super massive black hole remain as IMBHs.
We expect to find IMBHs in dSphs as well (\citealt{maccarone2005}). For example, \citet{reines2011} reported
that the nearby dwarf starburst galaxy Henize2-10 harbors a compact radio source at its dynamical
center spatially coincident with a hard X-ray source (possibly an
$\simeq 10^6$ M$_{\odot}$ accreting black hole). \citet{farrell2009} found the brightest
known ultra-luminous X-ray source HLX-1
(see e.g. \citealt{vandermarel2004} for a review) in the halo of the edge-on S0a galaxy ESO 243-49,
possibly associated with an IMBH whose mass was initially evaluated to be
$\ut> 9\times 10^3$ M$_{\odot}$ (\citealt{servillat2011}) and then better constrained to the range $9\times 10 ^3$-$9\times 10 ^4$ M$_{\odot}$
(\citealt{webb2012}). A further step towards the IMBH hypothesis was given by \citet{farrell2012} who detected evidence for a
young ($<$ 200 Myr) stellar cluster of total mass $\sim 10^6$ M$_{\odot}$ around the putative black hole and concluded that HLX-1
is likely to be the stripped remnant of a nucleated dwarf galaxy.
In the case of the Fornax dSph, \citet{volonteri} assumed that an IMBH of mass $M_{BH} \simeq 10^5$ M$_{\odot}$ is hosted in the galactic core
and suggested that measuring the dispersion velocity of the stars within $30$ pc from the center would
allow that hypothesis to be tested. \citet{jardel2012} recently constructed axisymmetric Schwarzschild
models in order to estimate the mass profile of the Fornax dSph and, once these models were tested versus the available kinematic data,
it was possible to put
a 1-$\sigma$ upper limit of $M_{BH} = 3.2\times 10^4$ M$_{\odot}$ on the IMBH mass.
In this work, we concentrate on the Fornax dSph re-analyzing a set of {\it XMM}-Newton data previously studied by \citet{orioproc} who searched
for the X-ray population in the Leo I and Fornax dSphs. In our work, we used the most recent analysis software and calibration files.
Apart from the characterization of the high-energy population
detected towards the galaxy and the cross correlation with the available databases, we discuss the
possible identification of a few genuine X-ray sources belonging to the Fornax dSph.
We also considered the possible existence of an IMBH in the galaxy core as suggested by
\citet{volonteri} and \citet{jardel2012} and show that one of the detected X-ray sources coincides with one of the possible Fornax dSph
centers of gravity. We then constrained the black hole accretion parameters and noted that additional important information may be obtained
by moderately deep radio observations and high-angular resolution X-ray data.
The paper is structured as follows: in Sect. 2 we describe the X-ray data analysis and in Sect. 3 we report our findings on the
X-ray population observed towards the galaxy, its correlation with known catalogs,
and the identification of a few objects as genuine X-ray sources in Fornax dSph.
We further discuss the IMBH hypothesis and address our conclusions in Sect. 4.
\section{X-ray observations and data processing}
The Fornax dwarf galaxy (at J2000 coordinates RA = 02$^{\rm h}$ 39$^{\rm m}$ 59$\fs$3 and Dec = $-$34$^\circ$ 26$'$
$57.1''$) was observed on August 8, 2005 for $\simeq 100$ ks (Observation ID 0302500101)
with the three European Photon Imaging Cameras (EPIC MOS 1, MOS 2, and pn) (\citealt{struder2001}, \citealt{turner2001})
on board the {\it XMM}-Newton satellite. The target was observed in imaging mode with the full-frame
window and medium filter.
\subsection{Data reduction and screening}
The observation data files (ODFs) were processed using the {\it XMM}-Science
Analysis System (SAS version $11.0.0$\footnote{{\tt http://xmm.esa.int/sas/}}) with the
latest available calibration constituent files. The event lists for the three cameras were obtained
by processing the raw data via the standard {\it emchain} and {\it epchain} tools.
We followed standard procedures in screening the data. In particular, for the spectral analysis
we rejected time intervals affected by high levels of background activity. These time intervals
(particularly evident in the energy range 10--12 keV) were flagged, strictly
following the instructions described in the XRPS User's Manual\footnote{{\tt http://xmm.esac.esa.int/external/xmm\_user\_support/ \\
/documentation/rpsman/index.html}} by selecting a threshold of
0.4 counts s$^{-1}$ and 0.35 counts s$^{-1}$ for the pn and MOS cameras, respectively.
This allowed us to compile lists of good time intervals (GTIs)
which were used to discard high background activity periods. The resulting exposure
times for the two MOS and pn cameras were $\simeq 82$
ks and $\simeq 62$ ks, respectively. We screened the data by using the filter expressions
${\rm \#XMMEA\_EM}$ (for MOS) and
$\#XMMEA\_EP$ (for pn). We also added the ${\rm FLAG==0}$ selection expression in order to
reject events close to CCD gaps or bad pixels, taking into account
all the valid patterns (PATTERN in [0:12]) for the two MOS cameras and only single and
double events (PATTERN in [0:4]) for pn.
\subsection{Source detection}
For each camera, the list of events was divided into 5 energy bands chosen according to
those used in the 2XMM catalog of serendipitous X-ray sources \citep{watson2009}, i.e., $B_1: 0.2-0.5$ keV, $B_2: 0.5-1.0$ keV,
$B_3: 1.0-2.0$ keV, $B_4: 2.0-4.5$ keV, and $B_5: 4.5-12.0$ keV. For the three EPIC cameras, we produced one image
for each energy band and a mosaic image in the $0.3-10$ keV energy band for inspection purposes only.
We then performed the source detection using the SAS task {\it edetect$\_$chain}. For each camera and input image, the tool first evaluates
the corresponding exposure map (via the task {\it eexpmap}) taking into account the calibration information
on the spatial quantum efficiency, filter transmission, and vignetting. In the next step, we produced image masks that delimited
the regions where the source searching was performed.
The sources identified with the local and map
searching algorithms\footnote{For more details, the reader is addressed to the on-line thread:\\ {\tt
http://xmm.esac.esa.int/sas/current/documentation}} were then used by the task {\it emldetect}
which performs a point spread function (PSF) fitting in each of the EPIC cameras for the five energy bands simultaneously. In particular, for any of the detected sources, the location and extent were fixed to the same values in all
bands while leaving the count rates free to vary among the different energy bands.
After the task completion, three lists of sources remained (one per camera), each containing
the refined coordinates, count rates, hardness ratios, and maximum detection likelihood of the source candidates.
For each of the detected sources, we derived the X-ray flux (in units of erg s$^{-1}$ cm$^{-2}$) in a given band as
\begin{equation}
F_i = \frac{B_{i}}{ECF_i}~,
\end{equation}
where $B_i$ is the count rate in the $i$ band and $ECF_i$ is an energy conversion factor
which has been calculated using the most recent calibration
matrices for the MOS 1, MOS 2, and pn. ECFs
for each camera, energy band, and filter are in units of $10^{11}$ counts cm$^{2}$ erg$^{-1}$.
In particular, we used the ECFs\footnote{For details on the adopted ECFs, see \citet{note2003}. Please note that,
since 2008, improvements in the calibration of EPIC cameras lead to changes in the ECFs. Thus,
the quoted ECFs must be multiplied by a correction factor to get a better agreement among the fluxes
evaluated in each camera \citep{note2006}.
The ECFs and associated correction factors are reported in the User's Guide of the 2XMM catalog of serendipitous sources available at\\
{\tt http://xmmssc-www.star.le.ac.uk/Catalogue/2XMMi-DR3}.}
obtained assuming a power-law model with photon index $\Gamma=1.7$ and
a Galactic foreground absorption of $N_H\simeq 3.0\times 10^{20}$ cm$^{-2}$ \citep{watson2009}. Note also that the adopted
hydrogen column density is of the same order of magnitude as the average value estimated towards the target
by using the ``N$_{\rm H}$" online
calculator\footnote{Available in the tool section of \tt http://heasarc.nasa.gov}, i.e.,
2.7$\times$10$^{20}$ cm$^{-2}$ \citep{kalberla2005}.
We refined the absolute astrometry by matching the candidate source lists to the USNO-B1 catalog \citep{usnob1}. From this catalog,
we extracted a sub-sample of stars with coordinates within 20$'$ from the Fornax center and used the SAS task {\it eposcorr} (with parameter of
maximum distance of 1$''$) to obtain the coordinate correction which was (on average) $-0.56'' \pm 0.50''$ in RA and $-1.77'' \pm 0.34''$ in Dec.
We purged the candidate source lists (one per EPIC camera) by accepting only sources with a maximum likelihood detection (as provided by the source detection
algorithm) larger than 10 (equivalent to 4$\sigma$) and then cross correlated the source lists
via the IDL routine {\it srcor.pro}\footnote{\tt{http://idlastro.gsfc.nasa.gov}} by requiring a critical radius of $1''$ outside which the correlations were rejected and
obtained the source parameters (i.e., coordinates, count rates, fluxes, and hardness ratios) by weighting (with the errors)
the values of interest associated with the sources identified in the three cameras.
After removing a few spurious sources (i.e., those positioned at the borders of the cameras and those not appearing as such in the 0.2-12 kev band),
our catalog resulted in
107 sources detected towards the Fornax dSph galaxy with, in particular, 32 sources identified contemporarily in MOS 1, MOS 2, and pn,
4 sources only in MOS 1 and MOS 2, 7 sources only in MOS 1 and pn, 18 sources only in MOS 2 and pn, 30 sources only in pn,
2 sources only in MOS 1, and 14 sources only in MOS 2.
Note that the number of detected
sources agrees very closely with the 104 sources found by \citet{orioproc} when analyzing the same data set, the discrepancy probably
arising from slightly different choices
in the data screening procedure and the detection threshold used.
\begin{figure*}[!t]
\begin{center}
\psfig{figure=fig1_ms_20152.eps,width=14.0cm,angle=0}
\end{center}
\caption{A mosaic image of the MOS 1, MOS 2, and pn exposures in the $0.3-10$ keV energy band (see text for details).}
\label{fig1}%
\end{figure*}
A mosaic image (logarithmically scaled and smoothed with a 3-pixel Gaussian kernel) of the MOS 1, MOS 2, and pn exposures
in the 0.3-10 keV energy band is given in Figure \ref{fig1}. Here, each of the
identified sources is indicated by a $35''$ radius circle (containing $\cong$ 90\% of energy at 1.5 keV in the pn camera, see e.g. the
\citealt{xrps}), labeled with a sequential number and with a color
depending on which camera (or set of cameras) the source was detected by: yellow (pn), magenta (pn and MOS 1), red (pn and MOS 2), black (pn, MOS 1, and MOS 2),
blue (MOS 1 and MOS 2), cyan (MOS 1), and green (MOS 2). The red-dashed ellipse indicates the extension of the galaxy
which is characterized by semi-major and semi-minor axes of $\simeq 17'$ and $\simeq 13'$
(see the NASA/IPAC extragalactic database \footnote{$\tt http://ned.ipac.caltech.edu/$} -NED-) and a major-axis position angle of $\simeq 48^0$ (\citealt{mateo1998b}).
In Table \ref{fornaxsources} we summarize our source analysis, ordering the list by increasing X-ray flux in the 0.2-12 keV energy band.
Here, we give a sequential number (Scr) for each source,
a label ($\#$) indicating which EPIC camera (or set of cameras) identified the source,
i.e., A (pn only), B (pn and MOS 1), C (pn and MOS 2), D (pn, MOS 1, and MOS 2), E (MOS 1 and MOS 2), F (MOS 1), and G (MOS 2).
We also give the J2000 coordinates with the associated errors (Column 3-5), the high-energy hardness ratios $HR_1$ and $HR_2$ (see next section)
(Columns 6-7),
the $0.2-12$ keV absorbed flux (Column 8) and the cross correlations (Column 9-10).
We used the Two Micron All-Sky Survey (2MASS), the Two Micron All-Sky Survey Extended objects (2MASX) \citep{2mass},
the United States Naval Observatory all-sky survey (USNO-B1) \citep{usnob1}, and the variable star catalog in the Fornax galaxy \citep{varstar}
to correlate the X-ray source catalog with optical counterparts. In doing this, we associated with the coordinates of each of the
identified X-ray source an error as resulting from the quadrature sum of the
{\it XMM}-Newton positional accuracy ($\simeq 2''$ at $2\sigma$ confidence level, see \citealt{kirsch2004}, and \citealt{guainazzi2010})
and the statistical error as determined by the {\it edetect$\_$chain} tool\footnote{Since the resulting
positional uncertainty is of a few arcseconds, we do not over-plot the source error circles in any of the figures appearing in the paper.}.
Similarly, the error associated with the optical counterpart was derived from the relevant catalog.
When an X-ray source is found to be within $1\sigma$ from an optical counterpart in a given catalog, we report in Table \ref{fornaxsources}
the corresponding distance in arcseconds in the relevant column. We remind the reader that the Fornax dSph galaxy was already the target
of a ROSAT X-ray observation (see \citealt{gizis1993} for details) with the purpose of characterizing the X-ray population (if any) in
the galaxy and of constraining the extended gas component.
The analysis of the ROSAT data resulted in the compilation of a catalog (hereinafter GMD093) listing 19
discrete sources congruous with the expected number of sources in the
extragalactic background. In Table \ref{tab} we give the GMD093 labels and coordinates (columns 1-3) of the sources
observed by ROSAT and the labels of the corresponding sources
in our catalog (column 4). The distance between the sources in the GMD093 catalog and the corresponding sources found in the
{\it XMM}-Newton list
is given in the fifth column with the error evaluated as the sum in quadrature between the {\it XMM}-Newton and ROSAT positional accuracies
\footnote{Because of the lack of any positional uncertainty on the sources
reported in \citet{gizis1993}, we associated each of the ROSAT detected
sources with an error of $\simeq 15''$ on both the celestial coordinates: we are aware
that this error may be underestimated since the positional accuracy of ROSAT PSPC
decreases with increasing offset from the detector axis.}. Finally, a possible classification
(based on the SIMBAD\footnote{$\tt http://simbad.u-strasbg.fr/simbad/$} and NED web sites)
is attempted in the sixth column (QSO -quasar-, EmG -emission line galaxy-, H2G -HII galaxy-, SyG -Seyfert galaxy-):
when a source is not cross correlated with SIMBAD and/or NED, this is simply labeled as {\it Xrs} (X-ray source).
Note that, among the 19 sources belonging to the GMD093 catalog, 6 are out of the {\it XMM}-Newton field of view.
Furthermore, from the $L_X$-$L_B$ relation ($L_B$ being the Fornax dSph B band luminosity), \citet{gizis1993}
estimated that at least one accreting binary should be present in the field of view.
For completeness, we mention that as far as the extended component is concerned, the ROSAT data did not show any diffuse
emission different from the normal X-ray background and no trend with radius was evident. Here, we concentrate on the point-like sources identified in the {\it XMM}-Newton observation of the Fornax dSph.
\addtocounter{table}{1}
\footnotesize{
\begin{center}
\begin{table*}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
GMD093 & RA & Dec & Src & Distance & Type & Ref. \\
ID & (J2000) & (J2000) & & (arcsec) & & \\
\hline
1 & 2 40 15.7 & -34 13 23.1 & 100 & 27 $\pm $ 15 & Xrs & -- \\
2 & 2 39 14.7 & -34 15 04.8 & -- &-- & Xrs & -- \\
3 & 2 39 12.2 & -34 18 18.2 & 101 & 3 $\pm $ 15 & Xrs & -- \\
4 & 2 39 08.5 & -34 18 38.1 & 95 & 19 $\pm $ 15 & Xrs & -- \\
5 & 2 39 50.0 & -34 20 10.9 & 104 & 18 $\pm $ 15 & QSO/EmG & 1,2 \\
6 & 2 39 34.8 & -34 22 01.9 & 82 & 15 $\pm $ 15 & Xrs & -- \\
7 & 2 39 33.5 & -34 25 39.0 & 105 & 16 $\pm $ 15 & QSO & 1,2 \\
8 & 2 40 34.2 & -34 27 11.0 & 84 & 8 $\pm $ 15 & Xrs & -- \\
9 & 2 38 20.8 & -34 30 17.6 & -- &-- & Xrs & -- \\
10 & 2 39 26.2 & -34 32 46.7 & 93 & 23 $\pm $ 16 & Xrs & -- \\
11 & 2 40 08.1 & -34 34 25.3 & 97 & 6 $\pm $ 15 & QSO & 2,3 \\
12 & 2 39 15.9 & -34 35 32.3 & 102 & 8 $\pm $ 15 & Xrs & -- \\
13 & 2 39 03.7 & -34 36 46.4 & 90 & 8 $\pm $ 15 & Xrs & -- \\
14 & 2 38 48.8 & -34 36 59.9 & -- &-- & Xrs & -- \\
15 & 2 40 19.0 & -34 37 27.1 & 107 & 7 $\pm $ 15 & QSO/H2G & 1,2,4 \\
16 & 2 40 38.9 & -34 39 06.2 & 106 & 8 $\pm $ 15 & QSO/SyG & 2,3 \\
17 & 2 42 08.1 & -34 40 06.8 & -- &-- & Xrs & -- \\
18 & 2 38 55.5 & -34 40 52.4 & -- &-- & QSO & 1,2 \\
19 & 2 40 39.1 & -34 48 10.7 & -- &-- & Xrs & -- \\
\hline
\end{tabular}
\caption {List of the detected X-ray sources cross-correlated with the catalog GMD093 (\citealt{gizis1993}). The GMD093 sources labeled as 2,9,14,17,18, and 19
are out of the {\it XMM}-Newton FOV.}
\tablebib{
(1)~\citet{tinney1999}; (2) \citet{vcv2006}; (3) \citet{tdz1997}; (4) \citet{jones2006}.
\label{tab}}
\end{table*}
\end{center}
}
\section{The high-energy view of the Fornax dSph}
\subsection{X-ray colors and X-ray-to-NIR flux ratios}
Using the results presented in the previous section and with the purpose of a tentative classification of all the sources identified towards the Fornax dSph, we constructed the
color-color diagram in Figure \ref{fig2}. Here, we considered the colors
\begin{equation}
HR_1 = \frac{H-M}{S+M+H}~~{\rm and}~~HR_2 = \frac{M-S}{S+M+H}~,
\end{equation}
following the convention used in \citet{ramsay2006} in studying the Sagittarius and Carina dwarf galaxies and first
introduced by \citet{prestwich2003} and \citet{soria2003} S, M, and H correspond to the count rates in 0.3-1 keV,
1-4 keV, and 4-10 keV energy bands.
Since we expect that the detected X-ray sources are mainly accreting compact objects such as background AGN (Active Galactic Nuclei), X-ray binaries, and the brighter end of cataclysmic variables (CVs), we
compared the source measured hardness ratios (red squares) with two spectral models
(power-law, for simulating AGN and X-ray binaries, and bremsstrahlung, for CVs) which may be used to have an overall, although
simplified, description of the source spectra (see e.g. \citealt{ramsay2006} for a similar analysis made on the
Sgr and Car dSphs).
The model tracks, appearing in the color-color diagram, were obtained by simulating synthetic spectra within XSPEC
(An X-Ray Spectral Fitting Package, \citealt{arnaud}), version 12.0.0. In particular, we give the
expected set of color-color contours for bremsstrahlung (grey
region) and power-law (black region) components. In both cases,
the equivalent hydrogen column density $N_H$ varies in the range
$10^{19}$cm$^{-2}$ to $10^{22}$ cm$^{-2}$: each of the almost horizontal
lines corresponds to models with equal $N_H$ which increases from bottom to top. The temperature kT of
the bremsstrahlung models (taken in the range 0.1 - 3 keV) and
the power-law index $\Gamma$ (in the range 0.1 - 3) is associated with
primarily vertical lines: the values of kT and $\Gamma$ increase from left to right and from
right to left, respectively.
A representative error bar, obtained by averaging all the
data point error bars, is also given. Most of the detected sources
have colors consistent with those of the absorbed power-law or
absorbed bremsstrahlung models, although a few of them may require combined spectra.
\begin{figure}[!t]
\begin{center}
\hspace{-1.08cm}
\psfig{figure=fig2_ms_20152.eps,width=10.0cm,angle=0}
\end{center}
\vspace{-0.7cm}
\caption[]{The color-color diagram of the sources detected by the EPIC cameras towards the Fornax dSph galaxy. The solid lines represent the theoretical tracks
expected for different emitting models (see text for details). A representative error bar (obtained averaging all data point error bars) is also shown.}
\label{fig2}%
\end{figure}
Most of the sources appear to have spectra consistent with that of a typical AGN (which in the hardness diagram of Figure \ref{fig2} would
correspond to a dot lying close to the harder color-color tracks, see e.g. \citealt{ramsay2006}).
However, because the large error bars affect the hardness ratio data, a classification based only on
the hardness ratio cannot constrain the nature of the objects in our sample.
A better diagnostic tool was found by \citet{haakonsen2009} when studying the cross associations of the ROSAT All-Sky Survey Bright Source Catalog
(RASS/BSC) with the infrared counterparts found in 2MASS. In particular, these authors found that a color-color diagram
(based on the ratio between the
$0.2-2.4$ keV flux ($F_X$) and the NIR flux in J band ($F_J$) versus the J-K color) is useful in studying the characteristics of the sources.
In this plane, the galaxies
(Quasars and Seyfert 1 objects) and the coronally active stars (including pre-main sequence and main-sequence stars, high proper-motion objects,
and binary stars)
occupy distinct regions in such a way that galaxies have $(J-K)>0.6$ and $F_X/F_J>3\times 10^{-2}$ (dashed red lines in Figure \ref{figclass}), while almost all the objects in the second class have
$(J-K)<1.1$ (dotted black line in Figure \ref{figclass}) and $F_X/F_J<3\times 10^{-2}$, respectively\footnote{Note that
a better diagnostic would use the unabsorbed X-ray flux in the 0.2-2.4 keV band and the extinction-corrected NIR magnitudes: in the
proposed color-color diagram of \citet{haakonsen2009},
the galaxies are positioned in the region described by $(J-K)>(J-K)_0>0.6$ and $(F_X/F_J)_0>F_X/F_J>3\times 10^{-2}$. On the contrary, the coronally active stars satisfy the relations
$(J-K)_0<(J-K)<1.1$ and $(F_X/F_J)<(F_X/F_J)_0<3\times 10^{-2}$, with the subscript {\it 0} indicating the result of the de-reddening procedure (see \citealt{haakonsen2009} for details).}.
For the sources of our sample which correlate with 2MASS
counterparts, we give in Table \ref{lasttable} the 0.2-2.4 keV band flux, the NIR J band flux and the J and K magnitudes. Note that the 0.2-2.4 keV band flux has been
obtained from that in the 0.2-12 keV band (given in Table 1) assuming in webPIMMS a power-law model with a spectral index $\Gamma=1.7$
and an absorption column density $N_H=3\,\times \,10^{20}\, {\rm cm^{-2}}$. In Figure \ref{figclass}, we give the color-color diagram
(X-ray to J band flux ratio
against J-K) for the sources listed in Table \ref{lasttable}. The sources lying above the horizontal dashed line are consistent with background AGN,
while the others seem to have characteristics similar to X-ray active stars and binary sources: three of them (19, 41, and 82) have measured
proper motions
in the PPMX catalog (see also Table 1), confirming the stellar nature of these objects. Note that the X-ray source labeled as 107 correlates
(within 1.0$''$) with a source in the
2MASX and, according to our color-color diagram, is possibly associated with a background AGN (see also \citealt{mendez2011} where it is used as a reference background AGN
for proper motion estimates), while it was probably reported in the PPMX catalog by mistake
(source labeled as 024019.0-343719): this confirms the predictive power of this color-color diagram.
\begin{table}
\footnotesize{
\begin{tabular}{|c|c|c|c|c|}
\hline
Src & F$_X$ & F$_{J}$ & J & K \\
& ($\times 10^{-14}$ cgs) & ($\times 10^{-13}$ cgs) & & \\
\hline
6 & 0.14 $\pm$ 0.12 & 1.82 $\pm$ 0.20 & 16.12 $\pm$ 0.11 & 15.18 $\pm$ 0.20 \\
10 & 0.18 $\pm$ 0.16 & 10.90 $\pm$ 0.32 & 14.17 $\pm$ 0.03 & 13.35 $\pm$ 0.04\\
19 & 0.34 $\pm$ 0.16 & 17.11 $\pm$ 0.43 & 13.68 $\pm$ 0.03 & 13.14 $\pm$ 0.04\\
40 & 0.57 $\pm$ 0.28 & 1.77 $\pm$ 0.16 & 16.15 $\pm$ 0.10 & 15.32 $\pm$ 0.19\\
41 & 0.59 $\pm$ 0.45 & 102.71 $\pm$ 2.30 & 11.74 $\pm$ 0.02 & 11.30 $\pm$ 0.02 \\
46 & 0.69 $\pm$ 0.28 & 8.86 $\pm$ 0.43 & 14.40 $\pm$ 0.05 & 13.42 $\pm$ 0.10 \\
48 & 0.74 $\pm$ 0.28 & 1.53 $\pm$ 0.20 & 16.30 $\pm$ 0.15 & 14.10 $\pm$ 0.15 \\
75 & 1.42 $\pm$ 0.47 & 1.50 $\pm$ 0.21 & 16.33 $\pm$ 0.15 & 15.14 $\pm$ 0.18\\
82 & 1.75 $\pm$ 0.26 & 76.42 $\pm$ 1.70 & 12.06 $\pm$ 0.02 & 11.47 $\pm$ 0.02\\
84 & 1.87 $\pm$ 0.25 & 54.25 $\pm$ 1.10 & 12.43 $\pm$ 0.02 & 11.58 $\pm$ 0.02 \\
86 & 2.03 $\pm$ 0.38 & 1.32 $\pm$ 0.22 & 16.46 $\pm$ 0.17 & 14.95 $\pm$ 0.20 \\
102& 4.25 $\pm$ 0.93 & 2.10 $\pm$ 0.19 & 15.97 $\pm$ 0.10 & 15.32 $\pm$ 0.20 \\
107& 9.84 $\pm$ 1.22 & 6.97 $\pm$ 0.48 & 14.66 $\pm$ 0.10 & 13.43 $\pm$ 0.10 \\
\hline
\end{tabular}
\caption{A few sources of the X-ray sample detected towards the Fornax dSph correlate with the 2MASS catalog. Following
\citet{haakonsen2009}, we can try to constrain the nature of the sources by using the X-ray (in the 0.2-2.4 keV band) and NIR (J and K bands) fluxes
(see text and Figure \ref{figclass} for details).
}
\label{lasttable}
}
\end{table}
\begin{figure}[!t]
\begin{center}
\hspace{-1.08cm}
\psfig{figure=fig3_ms_20152.eps,width=10.0cm,angle=0}
\end{center}
\vspace{-0.7cm}
\caption[]{The color-color diagram for the X-ray sources of our sample with a counterpart in the 2MASS catalog (see text for details).}
\label{figclass}%
\end{figure}
\subsection{Background sources from the log N - log S plot}
We estimated the number of background sources expected towards the Fornax dwarf galaxy through the log N - log S diagram \citep{Hasinger}
and by using
the minimum absorbed flux in the 0.2-12 keV band among the sources of our analysis, i.e.,
$F^{Abs}_{0.2-12}=2.24\,\times\,10^{-15} {\rm erg\, s^{-1}\, cm^{-2}}$.
Using webPIMMS v3.9\footnote{{\tt http://heasarc.gsfc.nasa.gov/Tools/w3pimms.html}} and assuming a power-law model with a spectral index $\Gamma=1.7$
and an absorption column density $N_H=3\,\times \,10^{20}\, {\rm cm^{-2}}$, we obtained an unabsorbed flux in
the same band $F^{Unabs}_{0.2-12}=2.47\,\times\,10^{-15} {\rm erg\, s^{-1}\, cm^{-2}}$,
hence $F^{Unabs}_{0.5-2}=6.95\,\times\,10^{-16} {\rm erg\, s^{-1}\, cm^{-2}}$ in the 0.5-2 keV band.
We used this value as an input parameter of the \citet{Hasinger} method, obtaining a theoretical number of AGN as a function of the angular distance from the galactic center. Then we
divided the Fornax field of view (FOV) into five rings and compared the number of sources observed in each annulus to the expected number using the
log N - log S
relation \citep{Hasinger}, see Table \ref{tab2}. The number of sources detected in each annulus is consistent with the expected
number except in the external annulus, as the expected value does not account for border effects although all the detected sources in the Fornax FOV may be background objects, because of the intrinsic
statistical meaning of the
log N - log S diagram,
it cannot be ruled out that some of them actually belong to the galaxy itself. This conclusion is somehow supported by Figure \ref{figclass} where the X-ray sources
in the bottom-left region are most likely of local origin.
\subsection{Hint of a local variable source associated with an X-ray source}
When searching for optical counterparts to the detected X-ray sources (see Table \ref{fornaxsources}) in
the available catalogs, we found that source number 61 is possibly associated, within 2$"$,
with one source (J023941.4-343340) belonging to a catalog of variable stars \citep{varstar}, making it a good candidate for a genuine
X-ray source in the Fornax galaxy. The angular sensitivity of {\it XMM}-Newton, however, makes it difficult to separate this source from the one labeled 80 (see Figure \ref{fig1}) which is only $\simeq 22"$ apart.
One solution is to reduce the contamination from the nearby source by reducing the source extraction radius and by selecting suitable
background regions. In the particular case of source 61, we used a source extraction circle
with a radius of $\simeq 11"$ (corresponding to $\simeq 50\%$ of the source encircled energy, see \citealt{xrps}) and extracted
the background from a circular region having the same radius as the source extraction area
and localized at a distance of $\simeq 22"$ from source 80. The latter choice allowed us to properly
remove the background noise as well as any pattern hidden in the data caused by the proximity of source 80.
We then extracted the pn spectrum of source 61 (which consisted of only $\simeq 200$ counts)
and fitted it with an absorbed power-law model with hydrogen column density fixed to the value 2.7$\times$10$^{20}$ cm$^{-2}$
(see Section 2.2). The best fit, which converged to a power-law index $\Gamma \simeq 2.0$, corresponded to a flux in the $0.2-12$ keV energy
band of $\simeq 2.0 \times 10^{-14}$ erg s$^{-1}$ cm$^{-2}$ consistent with that obtained
from the analysis discussed in Section 2 (see also Table 1).
Using the extraction regions previously described, we generated source and background pn light curves (with a bin-size of $500$ seconds)
from the cleaned event files and used the SAS task {\it epiclccorr} to account for absolute
and relative corrections and for the background subtraction. The resulting light curve was then
analyzed with the Lomb-Scargle method (\citealt{scargle1982}) to search for periodicities in the range $1000-10000$ seconds and
the significance of any possible features appearing in the Lomb-Scargle periodogram was evaluated by
comparing the peak height with the power threshold corresponding to a given false alarm probability in white noise simulations
\citep{scargle1982}. When applied to the presently available data, this method did not detect any clear periodicity.
\begin{table}
\footnotesize{
\begin{tabular}{|c|c|c|c|c|}
\hline
Annulus & R$_{in}$ & R$_{ex}$ & \# Exp & \# Obs \\
& (arcmin) & (arcmin) & & \\
\hline
1 & 0.00 & 0.76 & 0.4 $\pm$ 0.1 & 0 \\
2 & 0.76 & 3.60 & 10.2 $\pm$ 1.7 & 6 \\
3 & 3.60 & 6.50 & 24.2 $\pm$ 4.0 & 21 \\
4 & 6.50 & 9.00 & 32.1 $\pm$ 5.0 & 29 \\
5 & 9.00 & 16.00 & 144.5$\pm$ 20.0 & 51 \\
\hline
\end{tabular}
\caption {List of sources expected through the log N - log S diagram and observed in annuli around Fornax center. Here,
$R_{in}$ and $R_{ex}$ represent the interior and exterior annulus radii, respectively.}
\label{tab2}}
\end{table}
\subsection{The Fornax globular clusters}
Among all the other dSphs which are satellites of the Milky Way, the Fornax dSph has the peculiar characteristic of
hosting five globular clusters (GCs) whose
(J2000) coordinates and main structural parameters
(mass $M$, core radius $r_c$, projected distance $R_p$ from the galaxy center, tidal radius $r_t$,
and maximum radius $r_M$) are reported in Table \ref{tableGC}.
Apart from the puzzling question of why dynamical friction has not yet
dragged any of the Fornax GCs towards the center of the galaxy (see e.g. \citet{cole2012} for an introduction to this intriguing problem and
to \citealt{goerdt2006} for a possible solution\footnote{\citet{goerdt2006} have shown that dynamical friction cannot drag globular clusters
to the galaxy center in the case of cored dark-matter halo distributions. In this case the drag is stopped at the point where the dark-matter
density remains constant, i.e., $\simeq 200$ pc.}), a search for X-ray sources associated with the GCs is also interesting.
We performed this search towards the five Fornax dSph GSs and found that a few X-ray sources
are located close to GC 3 and GC 4 (see the zoomed views of the {\it XMM}-Newton field around
the two clusters in Figures \ref{figGC4} and \ref{figGC3}.
For GC 4, the three green dashed concentric circles are centered on the GC coordinates and have radii of $2.6''$, $43''$, and $64''$, corresponding to
the cluster core radius, tidal radius, and maximum radius (see also Table \ref{tableGC}) as given in \citet{cole2012} and \citet{mackey}. In the case of
GC 3, for clarity, we have plotted only the circles having a radius equal to the core radius ($2.4''$) and tidal radius ($77''$).
\begin{table}
\footnotesize{
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
GC & RA & Dec & M & $r_c$& $R_p$ & $r_T$ & $r_M$ \\
& (J2000) & (J2000)& & (pc) & (kpc) & (pc) & (pc) \\
\hline
1& 2 37 02 & -34 11 01 & 0.4& 10.0& 1.7 & {37.1} & {49.7} \\
2& 2 38 44 & -34 48 30 & 1.8& 5.8& 1.0 & {49.0} & {50.4} \\
3& 2 39 48 & -34 15 30 & 3.6& 1.6& 0.4 & {51.0} & {50.4} \\
4& 2 40 7.7 & -34 32 11 & 1.3& 1.7& 0.2 & {28.5} & {42.4} \\
5& 2 42 21 & -34 06 06 & 1.8& 1.4& 1.4 & {49.0} & {50.4} \\
\hline
\end{tabular}
\caption {The coordinates of the Fornax globular clusters are taken from the NASA/IPAC extragalactic database (NED). For each cluster we give
the mass $M$ (in units of $10^5$ M$_{\odot}$), core radius $r_c$, and projected distance $R_p$ from the galaxy center as found in \citet{cole2012},
while the values of the tidal radius $r_t$ and maximum radius $r_M$ (expressed in parsec for a distance of $0.138$ Mpc) were derived from \citet{mackey} and references therein.}
\label{tableGC}
}
\end{table}
All the other circles (with associated radii of $35''$) appearing in the figures indicate the X-ray sources detected in our analysis (see Sect. 2.2).
In the case of GC 4, we identified one X-ray source (labeled as 28 in the source list appearing in Table \ref{fornaxsources}) which
is at a distance of $\simeq 37''$ from the globular cluster center\footnote{ We checked that none of the X-ray sources of our sample
has a counterpart in the catalog of variable stars (indicated as blue crosses in Figure \ref{figGC4}) as reported by \citet{greco2007}.}. Consequently, considering the GC 4 structural parameters reported in
Table \ref{tableGC}, one observes that source 28 is well within the tidal radius ($\simeq 43''$) of the globular cluster and possibly associated with it, as already noted
by \citet{orioproc} when analyzing the same {\it XMM}-Newton data set.
In the case of GC 3, two variable Sx Phe sources\footnote{A Sx Phe object is a variable pulsating star whose magnitude
can vary from a few $0.001$ mag to several $0.1$ mag with typical periods of $P\ut < 0.10$ days and is often used as a distance estimator. For a description on the main properties
of this class of variable stars we refer the reader to \citet{pych} and references therein.} (6$\_$V40765 and 6$\_$V38403)
were identified by \citet{poretti} as belonging to the globular cluster. Both sources appear to be at a distance of $\simeq 30''$ and $\simeq 78''$ from the globular cluster center and, therefore,
are possibly associated with GC 3. As one can see from a close inspection of Figure \ref{figGC3},
there is no clear association of any of the variable stars (blue crosses)
observed in GC 3 with
the detected X-ray sources. Moreover, one X-ray source (labeled as 103 at $\simeq 72''$
from the globular cluster center) and identified in the {\it XMM}-Newton field of view is within
the globular cluster tidal radius ($\simeq 77''$) and, therefore, may be associated with it (as already claimed by \citealt{orioproc}).
Finally, we note that for an estimated distance of
0.138 Mpc to the Fornax dSph, the observed 0.2-12 keV band luminosities associated with the sources 28 and 103 are
$\simeq 2.3 \times 10^{34}$ erg s$^{-1}$ and $\simeq 2.6 \times 10^{35}$ erg s$^{-1}$, respectively. While the source possibly associated with GC 4 has a luminosity
comparable to that of a typical CV, the source possibly residing in GC3 has a luminosity well within the typical ranges
(although towards the lower limit)
for LMXBs and high-mass X-ray binaries (HMXBs) (see
e.g. \citealt{fabbiano}, \citealt{kuulkers}, and \citealt{tauris}).
\begin{figure}[ht]
\centering
\includegraphics[width=100mm]{fig4_ms_20152.eps}
\caption{A zoomed view of the region around the Fornax globular cluster GC 4 (see text for details).}
\label{figGC4}
\end{figure}
\begin{figure}[ht]
\centering
\includegraphics[width=100mm]{fig5_ms_20152.eps}
\caption{As in Figure \ref{figGC4}, but the circles are centered on the coordinates of the globular cluster GC 3.}
\label{figGC3}
\end{figure}
\subsection{An intermediate-mass black hole in the Fornax galaxy: high-energy constraints}
When the galactic globular clusters were identified in X-rays,
\citet{bahcall1975} and \citet{silk1976} suggested that the observed emission was due to the
presence of IMBHs in the mass range $\sim 10^2$ M$_{\odot}$ -- $10^5$M$_{\odot}$ accreting material from the
intracluster medium.
The discovery of ultra-luminous compact X-ray sources (ULXs, with
luminosity greater than $\sim 10^{39}$ erg s$^{-1}$) initially pushed the community to interpret such objects
as IMBHs. Note however that the current accepted sceario is that all the ULXs (with the exception of the highest luminosity objects still
contain an IMBH) are stellar-sized black holes which accrete at super-Eddington rates (\citealt{miller2003}).
More evidence comes from the study of the central velocity dispersion of stars in specific globular clusters
(as G1 in the Andromeda galaxy, see e.g. Gebhardt et al. \citeyear{gebhardt2002}, but also Pooley et al. \citeyear{pooley2006},
or M15 and $\omega$ Centauri in the Milky Way, see e.g. Gerssen et al. \citeyear{gerssen2002}, \citeyear{gerssen2003}, and
Miocchi \citeyear{miocchiomegacen}, respectively ) which may contain
central IMBHs. As far as M15 is concerned, milli-second pulsar timing studies have allowed us
to put an upper limit of $\simeq 3\times 10 ^{3}$ M$_{\odot}$ on the IMBH mass
(\citealt{depaolis1996}). Finally, compact objects of IMBH size are also predicted by N-body simulations
(see e.g. \citealt{potegiezwart2004}) as
a consequence of merging of massive stars.
As noted by several authors (see e.g. \citealt{baumgardt2005}, and \citealt{miocchi}), photometric studies may provide a further
hint for the existence of IMBHs in globular clusters.
It is expected
that the mass density profile of a stellar system with a central IMBH follows a cuspy $\rho \propto
r^{-7/4}$ law, so that the projected density profile, as well as the
surface brightness, should also have a cusp profile with slope $-3/4$.
As shown by Miocchi (\citeyear{miocchi}), the globular
clusters that most likely host a central IMBH are those characterized by a projected photometry well fitted by a King profile, except in the
central part where a power-law deviation ($\alpha\simeq -0.2$) from a flat behavior is expected. However, the errors with which the slopes
of central densities can be determined are approximately $0.1-0.2$ (\citealt{noyola}), so that optical surface density profiles do not give
clear evidence of the existence of IMBH in globular clusters.
It is then natural to expect that observations in different bands of the electromagnetic spectrum, such as radio and X-ray bands, would
permit further constraints on the IMBH parameters. This issue was considered by several authors. For example, \citet{grindlay2001} provided the census of the compact objects
and binary populations in the globular cluster 47 Tuc and obtained an upper limit
to the central IMBH of a few hundred solar masses; \citet{nucita2008} showed that
the core of the globular cluster NGC 6388 harbors several {\it X}-ray sources;
and \citet{cseh2010} refined the analysis putting an upper limit to the central IMBH mass of a few thousand solar masses
(see also \citealt{bozzo2011}, and \citealt{nucita2012}).
One expects to find IMBHs in dSph as well (\citealt{maccarone2005}).
In the specific case of the Fornax dSph, \citet{volonteri} assumed that an IMBH of mass $M_{BH} \simeq 10^5$ M$_{\odot}$ exists in the galactic core
and suggested that measuring the dispersion velocity of the stars within $30$ pc from the center would allow us to test that hypothesis.
Furthermore, \citet{jardel2012} recently constructed axisymmetric Schwarzschild
models in order to estimate the mass profile of the Fornax dSph. These models were tested versus the available kinematic data allowing the authors to put
a 1-$\sigma$ upper limit of $M_{BH} = 3.2\times 10^4$ M$_{\odot}$ on the IMBH mass. We will use the latter
value in the following discussions.
Since any Brownian motion of IMBH at the center of the galaxy is negligible\footnote{
From simulations, one expects an IMBH within a globular cluster (or a spheroidal galaxy) to move randomly when interacting with the surrounding stars.
In the assumption that all the stars have the same mass $m$, the IMBH
moves with an amplitude $\sim r_c(m/M_{BH})$ (see e.g. \citealt{bw}, \citealt{gurzadyan}, and \citealt{merritt}) where $r_c$ is the core radius
and $M_{BH}$ the black hole mass.}, we searched for X-ray sources close to the Fornax center. This search was made difficult owing to the
uncertainty with which the Fornax dSph center position is known. This is caused mainly by the large asymmetry in the galaxy surface brightness as first observed by \citet{hodge} (see also \citealt{hodge1974}). The asymmetry
was also confirmed by \citet{stetson} who
evaluated the global galaxy centroid using a circle moving on the sky plane until the median (x, y) position of all the stars contained within
the aperture coincided with the center of the circle.
Depending on the circle radius (500 or 2000 pixels), \citet{stetson} obtained two center positions (separated by $\simeq 3'$) indicated by the green
diamonds (with labels {\it SHS98 500 pxl} and {\it SHS98 2000 pxl}) in Figure \ref{figcenters}. In the same figure,
we also give the coordinates of the centroid as obtained by \citet{hodge1974} (red diamond) and \citet{demers} (black diamond). For comparison,
the yellow diamond (on a CCD gap of the pn camera) represents the centroid as recently determined by \citet{battaglia} and the blue diamond is centered on the NED coordinates of Fornax dSph.
\begin{figure}[ht]
\centering
\includegraphics[width=100mm]{fig6_ms_20152.eps}
\caption{A zoomed view around the Fornax dSph center. The diamonds indicate the global centroids of the galaxy obtained by several authors. In particular,
the green diamonds (with labels {\it SHS98 500 pxl} and {\it SHS98 2000 pxl}) correspond to the galaxy centers obtained by \citet{stetson}, the red and black diamonds
indicate the centroids as derived by \citet{hodge1974} and \citet{demers}, respectively (see text for details).}
\label{figcenters}
\end{figure}
Note that the centroid labeled {\it SHS98 2000 pxl}, as estimated by \citet{stetson} when considering the large structure of the galaxy,
is at a distance of $\simeq 28''$ from the centroid position according to \citet{battaglia} and is apparently very close to one
of the X-ray sources (labeled 2)
detected in our analysis, thus allowing us to estimate the X-ray luminosity from a putative IMBH. In all the other cases, the centroids fall in a region apparently
free of sources, so that only an upper limit on the IMBH flux can be obtained.
As a reference for this case we use {\it SHS98 500 pxl} as the position of the centroid.
In the following paragraphs we speculate about the possible existence of an IMBH in the Fornax
dSph and whether it is in the position labeled {\it SHS98 2000 pxl} or coincident with the centroid {\it SHS98 500 pxl}.
As in the case of the G1 \citep{pooley2006} and M15 \citep{ho2003} globular clusters,
the X-ray emission from the putative IMBH at the center of the Fornax dSph may be due to Bondi accretion (\citealt{bondi1994})
onto the black hole, either from the cluster gas or from stellar winds. Thus, assuming that low-angular momentum gas
close to the compact object accretes spherically, for a black hole of mass $M_{BH}$ moving with velocity $v$ through
a gaseous medium with hydrogen number density
$n$, the accretion rate is
\begin{equation}
\dot{M}\simeq 4\pi(GM_{BH})^2(v^2+c_s^2)^{-3/2}m_p n~,
\end{equation}
where $m_p$ and $c_s$ are the proton mass and sound speed in the medium, respectively.
The expected X-ray luminosity is then
\begin{equation}
L_X\simeq \epsilon \eta \dot{M} c^2~,
\end{equation}
which can be parametrized as
\begin{equation}
L_X\simeq \epsilon \eta 8.8\times10^{36}\left(\frac{M_{BH}}{10^3 ~{\rm M}_{\odot}}\right)^2 \left(\frac{V}{15~{\rm km~s^{-1}}}\right)^{-3}
\left(\frac{n}{0.1 ~{\rm cm^{-3}}}\right)~{\rm erg~s^{-1}~cm^{-2}}~,
\label{explum}
\end{equation}
where $V=(v^2+c_s^2)^{1/2}$, $\epsilon$ is the efficiency in converting mass to radiant energy and $\eta$ is
the fraction of the Bondi-Hoyle accretion rate onto the black hole.
The hydrogen number density of the mass feeding the black hole can be estimated by using the structural parameters of the Fornax dSph as given
in \citet{McConnachie2012}. In particular, it was found that the galaxy hosts at least $M_{HI}\simeq0.17\times 10^6$ M$_{\odot}$ of gas within
the observed half-light radius of $r_{h}\simeq710$ pc. Thus, a lower limit to the gas density can be evaluated as
\begin{equation}
n\simeq \frac{3 M_{HI}}{4\pi r_h^3 m_p}\simeq 5\times 10^{-3}~{\rm cm^{-3}}~.
\end{equation}
Assuming that $v \simeq c_s\simeq 10$ km s$^{-1}$, one has $V\simeq 14-15$ km s$^{-1}$. Thus, using the
IMBH upper limit of $M_{BH} = 3.2\times 10^4$ M$_{\odot}$ quoted above,
Eq. \ref{explum} finally gives the expected X-ray luminosity, i.e.,
\begin{equation}
L_X\simeq \epsilon \eta 4.50\times10^{38}~{\rm erg~s^{-1}}~.
\end{equation}
For an estimated distance of 0.138 Mpc to the Fornax dSph, the expected IMBH luminosity $L_X$ corresponds to an observable flux of
$F_X \simeq \epsilon \eta 1.98 \times 10^{-10}$ erg s$^{-1}$ cm$^{-2}$.
As one can see from a close inspection of Figure \ref{figcenters}, in the case when the putative IMBH position coincides
with the centroid {\it SHS98 500 pxl}
\footnote{In the case in which the Fornax centroid position coincides with one of those evaluated
by \citet{hodge1974}, \citet{demers}, or \citet{battaglia} (red, black, and yellow diamonds
in Figure \ref{figcenters}, respectively), we can set only an upper limit to the IMBH unabsorbed flux
of $\simeq 2.5\times 10^{-15}$ erg s$^{-1}$ cm$^{-2}$ which, at the distance of the Fornax dSph, corresponds to
a luminosity limit of $\simeq 5.7\times 10^{33}$ erg s$^{-1}$.},
there was no clear detection of X-ray counterparts in the $0.2-12$
keV band. To be conservative, the expected flux can be compared with the minimum (unabsorbed)
flux ($\simeq 2.5\times 10^{-15}$ erg s$^{-1}$ cm$^{-2}$,
see Section 2.4 for further details) detectable
in the {\it XMM}-Newton observation. Hence, one easily constrains the accretion efficiency of the IMBH to be $\epsilon \eta \leq 1.3\times 10^{-5}$.
On the contrary, a detection appears (source labeled 2) on the centroid {\it SHS98 2000 pxl} position.
In this case, using the flux values obtained by the automatic procedure described in Section 2,
we found that source 2 has an unabsorbed flux (obtained via webPIMMS) in the $0.2-12$ keV of
$\simeq 3.0\times 10^{-15}$ erg s$^{-1}$ cm$^{-2}$ which, at the Fornax dSph distance, corresponds
to an intrinsic luminosity of $L_X=7.0\times 10^{33}$ erg s$^{-1}$. Thus, in the IMBH hypothesis,
the accretion efficiency turns out to be $\epsilon \eta \simeq 1.6\times 10^{-5}$.
With a more detailed analysis, we further extracted the MOS 1, MOS 2, and pn spectra of source 2
using a circular region centered on the target coordinates and with radius
of $35''$. We then accumulated the background on a region located on the same chip and in
a position apparently free of sources. This resulted in MOS 1, MOS 2, and pn spectra with
$\simeq230$, $\simeq250$, and $\simeq480$ counts, respectively.
After grouping the data by 25 counts/bin, we imported the resulting
background-reduced spectrum within XSPEC. A model consisting of an absorbed power-law, with hydrogen column density
fixed to the average value observed towards the target (2.7$\times$10$^{20}$ cm$^{-2}$, see Section 2 for further details), resulted
in the best fit parameters $\Gamma=1.3^{+1.0}_{-0.8}$ and $N= (5.0 \pm 4.0)\times 10^{-7}$ kev$^{-1}$ cm$^{-2}$ s$^{-1}$ with
$\chi ^2_\nu =1.4$ (for 34 d.o.f.). The 0.2-12 keV band absorbed flux of source 2 is then $(0.6^{+1.7}_{-0.4})\times10^{-14}$ erg s$^{-1}$ cm$^{-2}$ (90$\%$ confidence level)
which corresponds to an unabsorbed flux of $\simeq 6.2 \times 10^{-15}$ erg s$^{-1}$ cm$^{-2}$ and an unabsorbed luminosity of
$L_X \simeq 1.4 \times 10^{34}$ erg s$^{-1}$. Assuming a spherical accretion, one finally has an
accretion efficiency of $\epsilon \eta \simeq 3\times 10^{-5}$.
Note also that for $\eta = 1$, the radiative efficiency $\epsilon$ of the putative Fornax dSph IMBH is similar to the values
found for other systems in the radiatively inefficient regime ($10^{-5}-1$, see \citealt{baganoff}, but also \citealt{fender2003}, and
\citealt{koerding2006a,koerding2006b}). The same conclusion can be reached when comparing the observed X-ray luminosity in the $0.2-12$
keV band of source 2 ($L_X\simeq 1.4\times 10^{34}$ erg s$^{-1}$) with the expected Eddington luminosity
(i.e., $L_{Edd}\simeq 1.3 \times 10^{38} \left(M_{BH}/M_{\odot}\right)$ erg s$^{-1}$)
in the IMBH hypothesis. In this case, $L_X/L_{Edd}\simeq 3\times 10^{-9}$ is obtained, thus
implying that the IMBH must be extremely radiatively inefficient (for a comparison see the M15 case discussed in
\citealt{ho2003}).
Accretion onto a black hole may be different from a simple spherical model, since the accreting gas has angular momentum. In such a case
a Keplerian disk forms and the accretion occurs due to the presence of viscous torques,
which transport the angular momentum outward from the inner to the outer regions of the disk
(see e.g. \citealt{shapiro}). In the simplified case of a thin disk structure (see \citealt{shakura}),
the total luminosity of the disk (mainly originating in its innermost parts) is
\begin{equation}
L\simeq\frac{1}{2}\frac{GM_{BH}\dot{M}}{r_I}~,
\label{diskluminosity}
\end{equation}
where $\dot{M}$ is the accretion rate and $r_I$ is the radius of the inner edge of the disk.
For a disk that extends inward to the last stable orbit ($r_I= 6GM_{BH}/c^2$ for a non-rotating black hole),
it is easy to verify that the efficiency in converting the accreting
mass to radiant energy is $\simeq 8.3\%$. When comparing the observed X-ray luminosity
in the $0.2-12$ keV band of source 2 with the value expected in the thin disk scenario
(Eq. \ref{diskluminosity}), we can estimate a mass accretion rate of $\dot{M}\simeq1.87\times 10^{14}$ g s$^{-1}$.
Assuming again the radiation efficiency to be $8.3\%$, a black hole of mass $3.2\times 10^4$ M$_{\odot}$ has an
Eddington mass accretion rate of $\dot{M}_{Edd}\simeq 5.5\times 10^{22}$ g s$^{-1}$. Thus,
the ratio $\dot{M}/\dot{M}_{Edd}\simeq4\times 10^{-9}$ implies that the IMBH in Fornax dSph (if any) accretes
very inefficiently even in the context of a Keplerian thin disk model.
The above considerations, the observed low luminosity, and the estimated power-law index
$\Gamma=1.3^{+1.0}_{-0.8}$ (marginally consistent with an index in the range 1.4-2.1, see e.g. \citealt{remilard} for a classification of black hole binaries)
allow us to depict a scenario in which the Fornax dSph IMBH may be in a quiescent state.
Recently, it has also been proposed that a relationship between black hole mass, X-ray luminosity, and radio
luminosity does exist (see e.g. \citealt{merloni} and \citealt{koerding2006a}). This {\it fundamental plane} can be
used to test the IMBH hypothesis in globular clusters and dwarf spheroidal galaxies (\citealt{maccarone2005}).
In particular, \citet{maccarone2004} scaled the fundamental-plane relation to values appropriate for an
IMBH host in a Galactic globular cluster; by rescaling their equation to estimate the expected IMBH radio
flux at $5$ GHz, i.e.,
\begin{equation}
F_{5~{\rm GHz}}=10\left(\frac{L_X}{3\times10^{31}~{\rm cgs}}\right)^{0.6}\left(\frac{M_{BH}}{100~{\rm M_{\odot}}}\right)^{0.78}\left(\frac{10~{\rm kpc}}{d}\right)^{2}~{\rm \mu Jy}~,
\label{radio}
\end{equation}
which, for the above X-ray flux estimates, corresponds to
\begin{equation}
F_{5~{\rm GHz}}\simeq0.1\left(\frac{\epsilon}{10^{-5}}\right)^{0.6}~{\rm mJy}~.
\label{radio2}
\end{equation}
For an accretion efficiency as low as a few $10^{-5}$, the expected radio flux is well within the
detection possibilities of the Australia Telescope Compact Array (ATCA) which, for an integration time of $\simeq 12$ hr, may reach
an RMS sensitivity of $\simeq 10$ $\mu$Jy/beam at 5 GHz.
\section{Conclusions}
In this paper we re-analyzed a deep archive {\it XMM}-Newton data of the Fornax dSph galaxy with the aim of characterizing the
X-ray point-like source population. By using a restrictive analysis, we detected 107 X-ray sources. Most of them are likely
to be background objects since the number of detected sources is statistically consistent with that expected from the logN-logS relation. However,
we cannot exclude that a few of the detected objects belong to the Fornax dSph. The color-color diagram (based on the ratio between the
$0.2-2.4$ keV flux ($F_X$) and the NIR flux in J band ($F_J$) versus the J-K color) for the X-ray sources with a counterpart in the 2MASS catalog shows the presence
of a few objects (see bottom-left part of Figure \ref{figclass}) possibly of stellar nature and local origin. Furthermore, source number 61 appears to be spatially
coincident (within $\simeq 2''$) with a long-period variable star found in the catalog compiled by \citet{varstar}.
As discussed previously, among all the other dwarf spheroidal satellites of the Milky Way, a peculiar characteristic of the Fornax dSph is that
of hosting five globular clusters. We noted that two X-ray sources, detected towards the galaxy with our analysis, are coincident with the two
Fornax globular clusters GC 3 and GC 4. In particular, for GC 4 we identified one X-ray source (labeled 28 in the source list appearing in
Table \ref{fornaxsources})
which is at a distance of $\simeq 37''$ from the globular cluster center. Consequently, considering the GC 4 structural parameters reported in
Table \ref{tableGC}, one observes that source 28 is well within the tidal radius ($\simeq 43''$) of the globular cluster and, hence, possibly associated with it (as already claimed by
\citealt{orioproc}). In the case of GC 3, we identified one X-ray source (labeled 103 at a distance of $\simeq 72''$ from the GC center)
that might be at the outskirts of the globular cluster.
Finally, we discussed the IMBH hypothesis and found that one of the X-ray sources (labeled 2) might be associated with one
of the possible galaxy centroids identified by \citet{stetson}. In this framework, we estimated the IMBH accretion parameter
to be $\epsilon \eta \simeq 10^{-5}$. However, since
there is a large uncertainty in the identification of the galaxy's center of gravity, the latter value can be considered as
an upper limit to the IMBH accretion parameters.
\begin{acknowledgements}
We are grateful to F. Strafella, V. Orofino, and B.M.T. Maiolo for the interesting discussions while preparing the manuscript. We also acknowledge the anonymous Referee
for his suggestions.
\end{acknowledgements}
| -33,433.100059 |
[
-3.609375,
3.255859375
] | 22.831633 |
[
-3.232421875,
0.63232421875,
-1.603515625,
-5.78125,
-0.47802734375,
7.9453125
] |
[
3.767578125,
7.53125,
4.30859375,
5.8515625
] | 690 | 8,902 |
[
-2.7578125,
2.9609375
] | 31.108376 |
[
-5.875,
-3.037109375,
-3.4296875,
-2.12109375,
1.1279296875,
10.8984375
] | 1.255662 | 11.701421 | 22.006291 | 6.876407 |
[
3.073646068572998
] | -22,307.662125 | 5.009885 | -31,986.07692 | 0.81618 | 6.172179 |
[
-3.34375,
-3.681640625,
-3.009765625,
-3.62890625,
2.45703125,
10.3984375
] |
[
-5.8671875,
-2.783203125,
-2.318359375,
-1.90234375,
4.13671875,
5.90234375
] | |
BkiUdG45qX_Bl-uWGa7i
|
\section{The Hubble Space Telescope}
The {\it Hubble Space Telescope\/} ({{\it HST}\/}) has recently celebrated its
16th anniversary in orbit, having been launched aboard the NASA space shuttle in
April 1990. Since then there have been four astronaut servicing missions between
1993 and 2002 (SM 1, 2, 3a, and 3b). We are currently hoping that there will be
an SM4 in 2008, during which two new instruments would be installed: the Cosmic
Origins Spectrograph (COS) and the Wide Field Camera 3 (WFC3). WFC3 will be a
remarkable instrument that provides direct imaging at UV, optical, and near-IR
wavelengths.
The {\it HST}\/ is uniquely suited for high-resolution imaging. In this paper I
would like to present some results from {\it HST\/} imaging of the remarkable
light echoes around V838 Monocerotis.
\section{HST Observations of V838 Mon}
Shortly after V838~Mon reached maximum light in February 2002, an expanding
light echo was discovered by Henden, Munari, \& Schwartz (2002). These light
echoes have evolved to become the most spectacular display of the phenomenon in
astronomical history. They have been the subject of extensive imaging by
ground-based observers (e.g., Crause et al.\ 2005 and references therein), and
are of course the inspiration for the conference here in La~Palma.
Based on the highly structured appearance of the initial ground-based images,
our team proposed for Director's Discretionary (DD) time on {\it HST}\/ for a program
of direct imaging and imaging polarimetry. The team members are as follows:
S.~Starrfield (Arizona State University); Z.~Levay, N.~Panagia, W.~Sparks,
B.~Sugerman, R.~White, and myself (STScI); A.~Henden (AAVSO); M.~Wagner
(University of Arizona); R.~Corradi (Issac Newton Group); U.~Munari (Padova
University); L.~Crause (SAAO); and M.~Dopita (ANU)\null.
We received {\it HST}\/ observing time at five epochs in 2002 through DD allocations:
April, May, September, October, and December. All of the observations were made
with the Advanced Camera for Surveys (ACS), which had been installed in {\it HST}\/
during SM3b in March 2002. I need not emphasize to this audience how
extraordinarily unfortunate it is that no {\it HST}\/ observations were obtained
during 2003---the loss of this opportunity is truly incalculable. However, the
echoes were imaged twice in 2004 through the Hubble Heritage program, in
February and October. More happily, the {\it HST}\/ Cycle~14 allocation committee did
award our team observing time for an intensive {\it HST}\/ imaging campaign from
October 2005 to January 2006, and we also have two more epochs of observations
scheduled in Cycle~15 for late 2006 and early 2007.
A summary of results from the 2002 imaging is given by Bond et~al.\ (2003). A
discussion of the imaging polarimetry is presented elsewhere in these conference
proceedings (Sparks et al.\ 2006).
It is nearly impossible to represent the extraordinary {\it HST}\/ images on paper,
especially in black and white. As a partial guide to what is available, Figure~1
presents a montage of the {\it HST}\/ images from 2002 and 2004, all at the same
angular scale. However, I urge readers to go to the STScI website
(www.stsci.edu), and of course to the Hubble Heritage site (heritage.stsci.edu),
where many full-color images of the light echoes can be viewed.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=\textwidth]{bond_b_new_fig1.eps}
\end{center}
\caption{{\it HST}\/ images of V838 Mon, 2002-2004. The images are all at the same
angular scale, and dramatically illustrate the apparent expansion of the echoes.
Images taken in late 2005 and early 2006, however, have nearly the same size as
the October 2004 image, indicating that we are about to enter the phase of
apparent contraction.}
\end{figure}
An animation showing all of the {\it HST}\/ images has been prepared by Zolt Levay
and myself. There is a link to this animation at the conference website,
www.ing.iac.es/conferences/v838mon, or you can view it at
www.stsci.edu\slash$^{\scriptscriptstyle\sim}$bond/v838mon\_2002-5\_anim.gif.
\section{Applications of the HST Images}
In addition to their aesthetic beauty, the {\it HST}\/ images of the V838~Mon light
echoes provide several scientific applications. These include direct geometrical
distance determinations, and information on the detailed morphology and
three-dimensional structure of the circumstellar dust. (The images will also
ultimately provide important information on the physics of interstellar dust
particles, since we are in an unique situation where both the spectrum and time
dependence of the illumination, and the scattering angle, are unambiguously
known.)
\subsection{Geometrical Distance Determinations}
Light echoes provide two independent means for geometrical distance
determinations. One method is based on apparent angular expansion rates, and is
described in detail by Bond et~al.\ (2003). This method requires an assumption
about the geometry of the dust. For the 2003 analysis, we assumed thin,
spherical sheets centered on the star, and were able to obtain a lower limit to
the distance of $\sim$2~kpc.
We will carry out a similar detailed analysis of the {\it HST}\/ images from the
2005-6 campaign, but an approximate calculation goes as follows: In late 2005,
the radius at which zero apparent expansion would be seen (for spherical shells
centered on the star) would be $\sim$1400~light-days. The corresponding angular
radius of zero expansion would be $40''\times(6\,{\rm kpc}/d)$, where $d$ is the
distance. The apparent expansion in the late 2005 {\it HST}\/ images is indeed close
to zero at a radius of about $40''$, consistent with an approximate distance of
$\sim$6~kpc (see next paragraph). At larger radii, the echoes are clearly still
expanding, indicating that the distance cannot be much less than $\sim$6~kpc.
The other geometrical method relies on polarimetry. This method was worked out
in detail more than a decade ago by our team member Bill Sparks (Sparks 2004).
His intention was to use the technique to find distances to extragalactic
supernovae, but the method now finds its application to a peculiar object in our
own Milky Way! The Sparks technique is based on the fact that maximum linear
polarization occurs for scattering off of dust particles at an angle of
$90^\circ$. Given the geometry of light echoes (e.g., Bond et~al.\ 2003;
Sugerman 2003), this location corresponds to a linear radius of $c\Delta t$,
where $\Delta t$ is the time since the outburst. The corresponding angular
radius of maximum linear polarization then yields the distance. As described
elsewhere in these proceedings, the {\it HST}\/ polarimetric images yield a distance
of 5.9~kpc (Sparks et al.\ 2006). Reassuringly, this agrees very well
with the distance of 6.2~kpc recently obtained from spectral classification and
photometry of three B-type stars in the sparse, young cluster surrounding
V838~Mon (Bond \& Af\c{s}ar\ 2006; Af\c{s}ar\ \& Bond 2006).
\subsection{Morphology of the Light Echoes: Is the Dust Interstellar or Ejected
from the Star?}
In this subsection, I want to discuss the issue whether the dust illuminated in
the light echoes arises from material ejected from V838~Mon in previous
outbursts, or is merely ambient dust with no special relation to V838~Mon (as
advocated, for example, by Tylenda 2004 and Crause et al.\ 2005). My approach is
not based on physics, but upon the morphological approach described so
brilliantly by Zwicky (1957). I conclude that the dust was indeed ejected from
V838~Mon. Apart from the fact that the star is copiously producing dust during
its current outburst, and that that dust will expand slowly away from the star
and would be illuminated by any future outburst, there are the following
morphological arguments:
\begin{enumerate}
\item {\it There are features in the dust that show a connection with the star.}
The most dramatic one is the ``double-helix'' structure seen in the {\it HST}\/
images of September, October, and December 2002 (which was discussed by
Carlqvist 2005). The axis of the helix points directly at the star in all three
images, one of which is shown in Figure~2. It would be remarkable to find such a
precise alignment if the dust were merely ambient interstellar material that
happens to lie in the vicinity of V838~Mon.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=2.25in]{bond_b_fig2.eps}
\end{center}
\caption{Detail from {\it HST}\/ image of the V838 Mon light echo on 2002
September~2. A dotted line drawn from V838~Mon through the ``double-helix''
feature at the upper right passes precisely through the axis of the helix.}
\end{figure}
Note that the three-dimensional location of any illuminated dust particle is
unambiguously known in a light echo. We plan to develop three-dimensional
visualizations of the dust distribution, which will doubtless provide further
information on the relationships between the dust and the star. At a given
epoch, the illuminated dust lies along a paraboloid with the star at its focus.
\item {\it The dust has a well-defined outer edge, and the distribution of outer
edges is centered near the star.} As a simple example of three-dimensional
mapping, I marked the outer edges of the dust distribution in each of the
available {\it HST}\/ images. As noted in the previous paragraph, the $(x,y)$
position of any feature in the echoes at a time $\Delta t$ corresponds exactly
to a $z$ location in the line of sight ($z$ does, of course, depend also on the
distance to the star, but as noted above $d$ is now well known).
The images in 2002 have very sharp outer edges (see Figure~1), indicating a
sharp drop in the dust density at the corresponding locations in
three-dimensional space (since the edges are much sharper than would be produced
by purely geometrical effects). The color images obtained by {\it HST}\/ are
particularly useful, because they contain obvious sharp blue rims, corresponding
to the sharp blue peak at maximum in the light curve. The edges are not quite as
sharp at later epochs, but are still easily located. Until we can present a
fully three-dimensional visualization of the dust, I have simply measured the
locations of the outermost edges along a north-south axis passing through the
star, and along an east-west axis. I then converted these locations in the plane
of the sky to the true three-dimensional locations. The resulting maps of the
outer edges are shown in Figure~3a (north-south), and Figure~3b (east-west).
\begin{figure}[htb]
\begin{center}
\includegraphics[width=2.45in]{bond_b_new_fig3a.eps} \hskip0.24in
\includegraphics[width=2.45in]{bond_b_new_fig3b.eps}
\end{center}
\caption{Maps showing the locations of the outermost edges of the light echoes
seen in the available {\it HST}\/ images from 2002 (inner parabolas), 2004 (two
intermediate parabolas), and 2005-6 (densely packed outer parabolas). The edges
are shown in a north-south plane centered on V838~Mon (left), and in an
east-west plane (right). Each parabola corresponds to the location of the
illuminated dust at that epoch, and the large black dots show the $(x,z)$
location of the outer edge along the corresponding parabola. Note that the $x$
and $z$ scales are in parsecs, and V838~Mon itself is located at the origin of
coordinates. Both maps suggest that the dust lies in an ellipsoidal
distribution, centered near the star.}
\end{figure}
Several authors concluded from the 2002 data that the dust edges defined a
plane, and suggested that this implied an interstellar origin for the dust.
However, with the inclusion of more recent observations, our Figures~3a and 3b
show that there is now clear curvature in the dust fronts. In fact, it appears
that the distribution is consistent with a roughly ellipsoidal structure, whose
center coincides approximately with the location of V838~Mon. Such a morphology
strongly suggests an origin from the star itself, rather than a feature that
arose by chance.
\item {\it Recent images appear to show an ``equatorial plane'' passing through
the star.} Figure~4a shows the {\it HST}\/ image obtained on 2005 November 17. A
dashed line has been added to guide the eye to regions of enhanced surface
brightness that lie on both sides of the star.
This image bears a remarkable similarity to that of a typical planetary nebula,
M27 or the ``Dumbbell Nebula.'' An amateur photograph of this nebula is shown
in Figure~4b, which clearly demonstrates an equatorial feature on either side
of the central star.
\begin{figure}[htb]
\begin{center}
\includegraphics[height=2.4in]{bond_b_new_fig4a.eps} \hskip0.25in
\includegraphics[height=2.4in]{bond_b_new_fig4b.eps}
\end{center}
\caption{Left: {\it HST}\/ image of V838~Mon on 2005 November 17. The surface
brightness is highest along an ``equatorial plane'' passing through the star,
marked with a dashed line to guide the eye. Right: amateur photograph of the
planetary nebula M27. Note the striking resemblance of the light echo's
structure to that of the planetary nebula.}
\end{figure}
There is no doubt that M27 was ejected from its central star. The striking
resemblance of the light echo in late 2005 to a genuine planetary nebula
provides another strong morphological argument that the illuminated dust was
indeed ejected from the star.
\end{enumerate}
\section{Conclusion}
V838 Monocerotis is illuminating the most spectacular light echoes seen in the
history of astronomy. The high spatial resolution provided by the {\it Hubble
Space Telescope\/} has yielded images of extraordinary beauty, as well as
providing unique scientific information.
The {\it HST}\/ images have provided a direct geometrical distance to V838~Mon, based
on polarimetric imaging. Angular expansion rates have also given limits to the
distance.
Several morphological features seen in the {\it HST}\/ images strongly suggest that
the illuminated dust was ejected from the star in a previous outburst, similar
to the current one. The {\it Hubble\/} images show an extremely rich array of
filaments and other features on small physical scales, suggestive of
magnetohydrodynamic forces.
Future work on the {\it HST}\/ images are expected to yield dust physics (i.e., the
angular and color dependence of the scattering function) and a fully
three-dimensional map of the dust distribution at high spatial resolution.
\acknowledgements
I thank numerous colleagues for discussions and encouragement, including
especially the members of the {\it HST\/} V838~Mon observing team, and the
participants in this conference. Based on observations made with the NASA/ESA
{\it Hubble Space Telescope}, obtained at the Space Telescope Science Institute.
STScI is operated by the Association of Universities for Research in Astronomy,
Inc., under NASA contract NAS5-26555. Support from STScI grants including
GO-10913 is gratefully acknowledged.
| -8,679.748858 |
[
-3.0546875,
2.87109375
] | 26.59176 |
[
-5.71484375,
-2.884765625,
-3.271484375,
-8.234375,
2.0390625,
12.6328125
] |
[
4.515625,
4.6171875,
4.63671875,
6.6640625
] | 149 | 2,244 |
[
-3.21484375,
3.45703125
] | 23.794837 |
[
-5.64453125,
-3.595703125,
-2.0234375,
-0.447265625,
1.6328125,
7.484375
] | 1.663113 | 18.494052 | 32.709447 | 2.561917 |
[
1.981815218925476
] | -7,872.13979 | 5.225045 | -8,518.117872 | 0.886994 | 5.62962 |
[
-4.5703125,
-4.26171875,
-2.251953125,
-2.25390625,
3.01953125,
7.62109375
] |
[
-8.15625,
-6.27734375,
-4.74609375,
-3.046875,
6.3046875,
11.2109375
] | |
BkiUfSvxK6nrxjHzDfaq
|
\section{Introduction}
Deep inelastic scattering (DIS) has been one of the primary experimental and
theoretical tools used to study the strong force and establish quantum
chromodynamics (QCD) as the theory of quarks and hadrons. The strong
dynamics of DIS processes are captured by the hadronic tensor, which
can be factored into infrared-safe perturbative coefficients and parton
distribution functions (PDFs). The PDFs, which are the leading contribution to
the nucleon structure functions in the twist expansion, characterise low-energy
physics and must be determined nonperturbatively.
PDFs cannot be directly calculated on a Euclidean lattice, because they are
defined in terms of light-cone matrix elements. So PDFs--which depend on the
target nucleon, but are independent of the scattering process--are usually
determined using global analyses
\cite{Jimenez-Delgado:2014xza}.
A direct nonperturbative method to compute PDFs would be highly desirable,
both as an important test of lattice QCD and as a means to
constrain global fits of PDFs in regions that are experimentally
inaccessible.
Traditionally, lattice calculations have focussed on the Mellin
moments of PDFs, determined via matrix elements of local
twist-2 operators, where twist is the dimension minus the spin of the
operator, that \emph{can} be directly computed in Euclidean space.
However, the reduced symmetry of the cubic group induces radiative mixing
between operators of different spin. Moreover, operators of different mass
dimension suffer
from power-divergent mixing, which means that the continuum limit cannot be
taken. Moments up to the fourth moment can be extracted using
carefully chosen external momenta, but beyond this power-divergent mixing is
inevitable.
A method was recently proposed to directly compute PDFs on the lattice
using
a large-momentum effective theory \cite{Ma:2014jla}.
The first lattice calculations were presented in
\cite{Lin:2014zya}, but there are some unsolved challenges
\cite{Monahan:inprep}:
the renormalisation of the nonperturbative matrix
elements has yet to be fully understood and the practical difficulty of
the resolution of sufficiently large momenta on the lattices has not been
addressed.
Here we propose a new formalism--the ``smeared Operator Product
Expansion'' (sOPE)-- that removes mixing in the continuum limit and, in
principle, enables the determination of higher moments of PDFs on the lattice.
This formalism applies to any lattice calculation that suffers from
power-divergent mixing. We expand non-local continuum operators in a basis of
smeared operators. Matrix elements of these
smeared operators can be determined directly on the lattice and these matrix
elements require no further renormalisation, up to wavefunction
renormalisation,
provided the physical smearing scale
is kept fixed as the continuum limit is taken.
Smearing has been widely applied in lattice computations to reduce ultraviolet
fluctuations, partially restore rotational symmetry and systematically
improve the precision of lattice calculations. In the sOPE, we implement
smearing
via the gradient flow, a classical evolution of the original degrees
of freedom towards the stationary points of the action in a new dimension, the
flow time
\cite{Narayanan:2006rf,Luscher:2010iy,Luscher:2013cpa}. The
gradient flow corresponds to a continuous stout-smearing procedure
\cite{Morningstar:2003gk} and generally enables the use of smearing lengths
of only one or two lattice spacings, which is much smaller than typical
hadronic
length scales and does not distort the low energy
physics \cite{Davoudi:2012ya}. Moreover, the gradient flow is computationally
very cheap to implement.
We demonstrate the feasibility of our approach by studying the sOPE applied to
scalar field theory. We introduce the gradient flow for scalar fields, the
simplicity of which facilitates instructive comparison between
the sOPE and the local OPE. We start with a brief discussion of Wilson's
formulation of the OPE applied to scalar field theory and
then outline the sOPE. In Section \ref{sec:phi4} we calculate smeared Wilson
coefficients and derive
renormalisation group equations in the small flow time limit. We conclude by
discussing the application
of the sOPE to realistic lattice computations of twist-2 matrix
elements.
\section{The operator product expansion}
The OPE for a non-local operator is well-known, so here we simply introduce the
notation
necessary for the following discussions. We write the OPE for a non-local
operator, ${\cal Q}(x)$, as
\begin{equation}
{\cal Q}(x) \stackrel{x\rightarrow 0}{\sim} \; \sum_k
c_k(x,\mu) {\cal O}_R^{(k)}(0,\mu).
\end{equation}
The perturbative Wilson coefficients, $c_k(x,\mu)$, are complex functions that
capture the short-distance physics associated with the renormalised
local operator ${\cal O}_R^{(k)}(0,\mu)$, which is a polynomial in the scalar
field and its derivatives. The renormalisation scale is $\mu$ and the
free-field mass dimension of the local operator governs the
leading spacetime dependence of the Wilson coefficients. This equation is
understood in the weak sense of holding between matrix elements.
The most straightforward example is the time-ordered two-point function, ${\cal
T}\{ \phi(x)\phi(0)\}$, with spacetime separation $x$. For free scalar field
theory, the OPE is a Laurent expansion around zero spacetime separation,
\begin{equation}
{\cal T}\{ \phi(x)\phi(0)\} = \frac{1}{4\pi^2x^2}\mathbb{I}+
\phi^2(0) + {\cal O}(x).
\end{equation}
Incorporating interactions, the OPE becomes
\begin{equation}\label{eq:opeint}
{\cal T}\{
\phi(x) \phi(0)\} = \frac{c_{\mathbb{I}}(\mu x,mx)}{4\pi^2x^2}\mathbb{I}+
c_{\phi^2}(\mu x,mx)\left[\phi^2(0,\mu) \right]_R +
{\cal O}(x),
\end{equation}
where we denote renormalised operators by $[\ldots]_R$ and we have factored out
the leading spacetime dependence from the Wilson
coefficients. Radiative corrections generate sub-leading dependence
on the spacetime separation and, written in this form, the Wilson coefficients
are dimensionless functions of the spacetime separation $x$, the (renormalised)
mass $m$, and the renormalisation scale, $\mu$. The ${\cal O}(x)$
represents terms of order $x$, up to logarithmic corrections.
\subsection{Our proposal: the \emph{smeared} OPE}
We propose a new expansion in terms of smeared operators, the sOPE:
\begin{equation}
{\cal Q}(x) \stackrel{x\rightarrow 0}{\sim} \; \sum_k
d_k(\tau,x,\mu) {\cal S}_R^{(k)}(\tau,0,\mu).
\end{equation}
The smeared Wilson coefficients $d_k(\tau,x,\mu)$ are now functions of three
scales: the smearing scale,
$\tau$; the spacetime separation, $x$; and the
renormalisation scale, $\mu$. The leading spacetime
dependence of the smeared Wilson coefficients is dictated by the mass
dimension of the corresponding smeared operator, ${\cal S}_R(\tau,0)$; hence
the leading spacetime dependence is unchanged.
Returning to the time-ordered two-point
function, the sOPE is
\begin{equation}
{\cal T}\{ \phi(x)\phi(0)\} =
\frac{d_{\mathbb{I}}(\mu x,\mu^2\tau,mx)}{4\pi^2x^2}\mathbb{I}+
d_{\rho^2}(\mu x,\mu^2\tau,mx)\rho^2(\tau,0,\mu)
+
{\cal O}(x),
\end{equation}
where we denote smeared fields by $\rho(\tau,x)$. The smeared
fields are often referred to as ``bulk'' fields and
the unsmeared fields as ``boundary'' fields and we note that
the flow time has mass dimension $[\tau] = [m]^{-2}$. The sOPE is only valid
for small flow time values, just as the OPE only converges for small spacetime
separations.
\section{\label{sec:phi4}Scalar field theory}
Scalar field theory is a particularly
straightforward arena in which to apply the sOPE, because we can
solve the flow time equations exactly and there are no gauge-fixing
\cite{Luscher:2010iy} or fermion renormalisation
complications \cite{Luscher:2013cpa}. Moreover, we can view the sOPE for scalar
fields as a resummation of the local OPE and, although it is not
necessary for our work, it is interesting to note that
the OPE converges for Euclidean $\phi^4$-theory in four dimensions
\cite{Hollands:2011gf}.
We work with $\phi^4$-theory in four-dimensional Euclidean spacetime, defined
by the action
\begin{equation}
S_\phi[\phi] = \frac{1}{2}\int \mathrm{d}^4x\, \left[(\partial_\nu
\phi)^2 + m^2\phi^2 + \frac{\lambda}{12}\phi^4\right],
\end{equation}
and introduce the scalar
gradient flow via the flow evolution equation
\begin{equation}
\frac{\partial \rho(\tau,x)}{\partial \tau}= \partial^2 \rho(\tau,x).
\end{equation}
Here $\partial^2$ is the Euclidean, four-dimensional Laplacian operator.
We impose the Dirichlet
boundary condition
$\rho(0,x) = \phi(x)$, so that the full solution is
\begin{equation}
\rho(\tau,x) = \int\mathrm{d}^4y\,\int \frac{\mathrm{d}^4p}{(2\pi)^4}
\,e^{ip\cdot(x-y)} e^{-\tau p^2} \phi(y) = \frac{1}{16\pi^2\tau^2}
\int\mathrm{d}^4y\,e^{-(x-y)^2/4\tau}\phi(y).
\label{eq:smearedfield}
\end{equation}
This demonstrates explicitly the ``smearing'' effect of the gradient flow: the
flow time exponentially damps ultraviolet fluctuations. We can
parameterise the smearing radius by the root-mean-square smearing length,
$s_{\mathrm{rms}}$, which is given by $s_{\mathrm{rms}}^2 = 8\tau$.
Formally we can write the solution to the flow time equation as $
\rho(\tau,x) = e^{\tau\partial^2}\phi(x)$,
which we can expand for sufficiently small flow times as
\begin{equation}\label{eq:smalltrho}
\rho_R(\tau,x,\mu) = \phi_R(x,\mu)+\tau \left[\partial^2 \phi (x,\mu)\right]_R
+ \tau^2\left[(\partial^2)^2\phi (x,\mu)\right]_R+{\cal O}(\tau^4).
\end{equation}
Therefore, the (renormalised) smeared fields that
appear in the smeared operators of the sOPE can be represented by an
infinite tower of Laplacian operators acting on an unsmeared field.
Clearly, in this case, the sOPE corresponds to nothing more than a
reorganisation of the local OPE.
At this stage it is worth commenting on the small flow-time expansion, which
has proved an important tool for analysing the gradient flow in several
different contexts, both perturbatively and nonperturbatively
\cite{Shindler:2013bia,Luscher:2014kea}. These
studies incorporate a small flow-time expansion of smeared fields in terms of
local fields. We can view such an expansion
as a local OPE in the flow time and thereby relate
smeared field correlation functions to the
corresponding renormalised
correlation functions in the original theory, which would otherwise
be difficult to compute.
In this work, we take a related approach with an interpretation that is
tailored to the study of power-divergent mixing in lattice calculations. We do
not expand the bulk fields in terms of boundary fields, but rather take
as the fundamental objects of study the (matrix elements of)
fields at positive flow time. We formally
expand matrix elements of non-local operators at vanishing flow time in terms
of a basis of ``smeared'' operators at positive, but small, flow time.
\subsection{Computing smeared Wilson coefficients}
We show Feynman diagrams that contribute
to the Wilson coefficient $d_{\rho^2}(\mu x,\mu^2\tau,mx)$ at one loop and
$d_{\mathbb{I}}(\mu x,\mu^2\tau,mx)$ at tree level in Figure \ref{fig:sope2pt}.
\begin{figure}
\centering
\begin{minipage}{0.25\textwidth}
\includegraphics[width=0.9\textwidth,keepaspectratio=true]{dphi1loop}
\end{minipage
\hspace*{1cm}
\begin{minipage}{0.25\textwidth}
\includegraphics[width=0.9\textwidth,keepaspectratio=true]{d1tree}
\end{minipage
\caption{\label{fig:sope2pt}
Diagrams representing the contributions to the
smeared Wilson coefficients: $d_{\rho^2}$ to one loop (left-hand
diagrams) and $d_{\mathbb{I}}$ at tree level (right-hand diagrams). Black and
blue
lines are propagators at vanishing and non-vanishing flow times, respectively.
The black squares are unsmeared fields $\phi(0)$, black dots are
interaction vertices at vanishing flow time and the blue blob represents
the smeared operator $\rho^2(\tau,0)$.}
\end{figure}
The smeared Wilson coefficient for the leading connected and
disconnected operators are
\begin{align}\label{eq:drhores}
d_{\rho^2} = {} & 1-\frac{\lambda}{2}\left[\gamma_{\;\mathrm{E}} -1 +
\log\left(\frac{x^2}{8\tau}\right)\right] +{\cal O}(\lambda^2) \\
d_{\mathbb{I}} = {} & 1 -
\frac{x^2}{8\tau} +\frac{m^2x^2}{4}\left[\gamma_{\;\mathrm{E}} -1 +
\log\left(\frac{x^2}{8\tau}\right)\right]+{\cal O}(\lambda)
\end{align}
Here $\lambda =\lambda_0/(4\pi)^2$, and $\gamma_{\;\mathrm{E}}\simeq 0.577216$
is the
Euler-Mascheroni constant. The corresponding local Wilson coefficients in the
$\ms$ scheme are
\begin{align}
\!\!\overline{c}_{\mathbb{I}} = {} &
1 +\frac{
m^2x^2}{4} \left[1 + 2\gamma_{\;\mathrm{E}} +
\log\left(\frac{\mu^2x^2}{16}\right)\right]+{\cal
O}(\lambda),\\
\!\!\overline{c}_{\phi^2} = {} & 1+\frac{\lambda}{2} \left[1
+2\gamma_{\;\mathrm{E}}+
\log\left(\frac{\mu^2x^2}{16}\right)\right]+{\cal
O}(\lambda^2). \label{eq:cphires}
\end{align}
Although we have expressed the smeared and local Wilson coefficients in
different renormalisation schemes, so that we cannot directly compare the
finite contributions, we note four features:
\vspace*{-1.5\baselineskip}\\
\begin{enumerate}[leftmargin=*]
\setlength{\itemsep}{1pt}
\setlength{\parskip}{0pt}
\setlength{\parsep}{-5pt}
\item The logarithmic dependence on the spacetime separation is the same.
\item The flow time serves as the renormalisation scale for
the leading order contributions.
\item Wilson coefficients must be independent of the
external states. We ensure this
by choosing $\tau< x^2$, or equivalently by taking the flow time
sufficiently small that terms of ${\cal O}(\tau)$ can be neglected. This
is easily seen
by considering the derivative with respect to the external momenta of the
integrand for $d_{\rho^2}$ \cite{Monahan:inprep2}. Choosing the smearing radius
smaller than the spacetime extent of the non-local operator ensures that
the sOPE remains an (exponentially-)local expansion. In other words, if the
gradient flow probes length scales on the order of the non-local operator, then
the sOPE becomes a poor expansion for the original operator. This physical
intuition underlies both the small flow-time expansion
\cite{Shindler:2013bia,Luscher:2014kea} and the sOPE.
\item The flow time cannot regularise the Wilson coefficients beyond the
leading order contributions, because the flow evolution is classical and
interactions necessarily appear at vanishing flow time
(a higher-order calculation is given in \cite{Monahan:inprep2}).
However, the
renormalisation scale dependence of the smeared operators is absorbed by the
renormalisation parameters of the original theory.
\end{enumerate}
\section{Renormalisation group equations}
The standard renormalisation group (RG) operator is
\begin{equation}
\mu\frac{\mathrm{d}}{\mathrm{d} \mu} = \mu\frac{\partial}{\partial
\mu}\bigg|_{\lambda,m} + \beta
\frac{\partial}{\partial \lambda}\bigg|_{\mu,m} - \gamma_m
m^2\frac{\partial}{\partial m^2}\bigg|_{\mu,\lambda},
\end{equation}
where
\begin{equation}
\beta = \mu \frac{\mathrm{d}\lambda}{\mathrm{d}\mu}, \qquad
\gamma_m = -\frac{\mu}{2}\frac{\mathrm{d}\log(m^2)}{\mathrm{d}\mu},
\;\; \mathrm{and}\;\;
\gamma = \frac{\mu}{2} \frac{\mathrm{d}\log(Z_\phi)}{\mathrm{d}\mu}.
\end{equation}
Here $Z_\phi$ is the wavefunction renormalisation. In general the beta
function, $\beta$, and anomalous dimensions, $\gamma$ and $\gamma_m$, are
functions of the renormalised mass, $m$, the
renormalisation scale, $\mu$, and the coupling constant, $\lambda$, and are
known to five loops for an $O(N)$-symmetric theory
\cite{Kleinert:2001ax}.
For the sOPE, we must account for the extra flow time scale. If we choose
$\tau = \kappa^2 x^2$, with $\kappa$ a real number that ensures
$s_{\mathrm{rms}}<x$, then the smeared RG operator
is $\mu\mathrm{d}/\mathrm{d}
\mu- 2\kappa \mathrm{d}/\mathrm{d} \kappa$.
For small flow times, the smeared Wilson
coefficient then obeys the smeared RG equation:
\begin{equation}\label{eq:srge}
\left[\mu\frac{\mathrm{d}}{\mathrm{d} \mu}-2\gamma\right]
d_{\rho^2} = 2\left[\kappa\frac{\mathrm{d}}{\mathrm{d}
\kappa}+\zeta_{\rho^2}\right]
d_{\rho^2},
\end{equation}
where $\zeta_{\rho^2}$ is an anomalous dimension that
parameterises the flow time dependence of the smeared operator $\rho^2(\tau,0)
$ \cite{Luscher:2010iy,Luscher:2014kea}. An analogous equation
holds for the renormalised matrix elements of $\rho^2(\tau,0)
$, which can be studied nonperturbatively.
\paragraph{Nonperturbative calculations:} We have so far considered the sOPE
for
$\lambda \phi^4$ in four spacetime dimensions, which is not asymptotically
free. Therefore we briefly consider how the sOPE connects
nonperturbative
calculations to perturbative results for DIS calculations in QCD. The sOPE
incorporates two scales, so a ``line of constant physics'' is a path for which
the scales are tied together, for example, by fixing their product,
$\mu^2\tau=\kappa^2$.
Our aim is to connect hadronic matrix elements with smeared Wilson
coefficients, which are determined in perturbation theory, at a suitably high
scale using a finite-size scaling analysis
\cite{Luscher:1991oxn}. We calculate
matrix elements of smeared operators at some low energy scale,
determined by the inverse box size, and take the continuum limit at fixed
physical flow time. The smearing radius should be less than any
hadronic scales present in the nonperturbative determination of the twist
operators.
The scaling of the nonperturbative matrix elements is now entirely determined
by the renormalisation scale dependence and we can
apply a standard finite volume step-scaling procedure (see, for example, the
step-scaling procedure for smeared operators in \cite{Monahan:2013lwa}) to some
high scale. We choose the end point so that the new flow time is small relative
to the experimentally determined
inverse momentum transfer of the DIS process. At this new scale,
the new
renormalisation scale is sufficiently high that the nonperturbative matrix
elements
can be reliably combined with smeared Wilson coefficients.
\section{Summary}
We have proposed a new formalism, which we call the smeared operator product
expansion (sOPE), to extract lattice matrix elements without power divergent
mixing. The most obvious application is to deep inelastic scattering, but the
sOPE can be applied to nonperturbative calculations that suffer
from power-divergent mixing, such as the computations of $K\rightarrow \pi\pi$
decays and $B$-meson mixing \cite{Dawson:1997ic}.
We expand non-local operators in a basis of smeared operators, generated via
the gradient flow. The continuum limit of these matrix elements is free from
power-divergent mixing, provided the smearing scale, or flow time, is kept
fixed in the continuum limit. The matrix elements are then functions of
both the renormalisation scale and the smearing length. The sOPE
systematically relates these matrix elements to smeared Wilson coefficients,
which are calculated perturbatively, and thus provides a complete framework in
which to extract phenomenologically-relevant physics.
\acknowledgments
We would like to thank Martin L\"uscher and Andrea Shindler for
enlightening and helpful discussions during
the course of this work. This project was supported by the
U.S. Department of Energy, Grant Number DE-FG02-04ER41302. K.O.~was also
supported by the U.S. Department of Energy through Grant Number
DE-AC05-06OR23177, under which JSA operates the Thomas Jefferson National
Accelerator Facility. \vspace*{-0.5\baselineskip}
| -15,574.008786 |
[
-3.6171875,
3.150390625
] | 19.806763 |
[
-2.880859375,
-0.228759765625,
-2.732421875,
-6.87109375,
-1.044921875,
10.0625
] |
[
3.37109375,
9.4609375,
3.376953125,
7.00390625
] | 114 | 2,437 |
[
-3.412109375,
3.875
] | 25.47648 |
[
-5.85546875,
-4.6015625,
-4.8203125,
-2.435546875,
1.8359375,
12.859375
] | 1.001736 | 10.929149 | 31.883463 | 2.094896 |
[
1.7098581790924072
] | -10,764.744857 | 6.14444 | -15,415.754118 | 0.734607 | 5.645513 |
[
-2.140625,
-3.8359375,
-3.86328125,
-5.10546875,
2.1640625,
12.828125
] |
[
-5.16796875,
-2.625,
-2.654296875,
-1.4951171875,
3.6640625,
5.6484375
] | |
BkiUdPc5qoYA0D8lKjOo
|
\section{Introduction}
Quantum information processing tasks always involve the preparation and manipulation of quantum systems. To be able to perform such tasks it is essential to be able to characterize both quantum states and quantum operations. The protocols for characterizing a quantum state are usually refered to as quantum state tomography (QST)\cite{NielsenChuang, QST, CompressedSensing, EQST, pazNature, paperConTerra}. In general, QST is a hard task since it involves an exponentially large number of measurements to be preformed (exponentially large on the number of subsystems). Not only that, but the type of measurements that one needs to perform on the systems are usually not easy to perform.
On the other hand, the characterization of quantum processes, known as quantum process tomography (QPT)\cite{NielsenChuang, AAPT, MohsRezLid, DCQD}, is also an exponentially hard task. However, there are some quantum algorithms that allow to efficiently extract important information about a given quantum process \cite{SCNQP, CeciLopez1, CeciLopez2, paper1conFer, paper2conFer, prlChris, paperConCeciLopez}. These algorithms do not require performing QST on the final states but measuring quantities such as survival probabilities (or transition probabilities). Other algorithms for QPT, however, do depend upon QST. This makes algorithms for QST an essential tool not just for state tomography, but for process tomography also.
In this paper we will present new methods for quantum state tomography. To be specific, it is useful to describe the quantum states we are to perform tomography in the following way:. Let $\mathcal H$ be the Hilbert space for the system in question, and let $\mathcal B=\left\lbrace \ket{\psi_a}, a=1,...,D \right\rbrace$ be a basis of $\mathcal H$, where $D=\text{dim} \mathcal H$. Then the density matrix $\rho$ of a state can be written in the basis $\mathcal B$ as
\begin{equation}\label{eqEstadoH}
\rho=\sum_{a,b=1}^D \alpha_{ab}\ket{\psi_a}\bra{\psi_b}
\end{equation}
where $\alpha_{ab}=\bra{\psi_a}\rho\ket{\psi_b}$.
In what follows, we will present a method for selective efficient quantum state tomography (SEQST) that allows to estimate any coefficient $\alpha_{ab}$ with resources scalling polynomially with the number of subsystems. For this to be possible, we need that any state from the basis $\mathcal B$ can be efficently prepared in a controlled manner. This method, when applied for implementing QPT results in a protocol for efficient and selective QPT that is equivalent to the one presented in \cite{paper1conFer, paper2conFer}, illustrating one of the virtues of such a selective and efficient QST scheme.
This paper if organized as follows. First we briefly review existing methods for QPT that rely on performing QST in the final states of a certain process. Then we present the selective and efficient algorithm for QST and show how it provides the right tool for efficient and selective QPT, as opposed to previous methods for QST. Finally, we compare that QPT algorithm to the one presented in \cite{paper1conFer, paper2conFer}, showing how both can be understood in a common theoretical frame.
\section{Quantum Process Tomography Based on State Tomography}
The goal of QPT is to identify the temporal evolution enforced by a certain quantum process. Any such process is mathematically represented by a linear map $\mathcal E$ transforming initial states into final states. In fact, the operation $\mathcal E$ is not only linear but also completely positive, and acts as
\begin{equation}
\mathcal E(\rho_{in})=\rho_{out}
\end{equation}
This operation represents the discrete (input--output) evolution of quantum states. We will focus on maps that are trace preserving and whose output dimension is the same as the input one. To describe the quantum map it is convenient to parametrize it in some way. It is simple to notice that any linear map can be written in terms of a certain matrix, known as the $\chi$--matrix. This is defined with respect to a certain basis of the space of operators. In fact, if we choose a basis $\left\lbrace E_m, m=0,...,D^2-1\right\rbrace$, the $\chi$--matrix representation for $\mathcal E$ is determined by the equation:
\begin{equation}\label{eq:ChiRepresentation}
\mathcal E(\rho) = \sum_{m n} \chi_{mn}E_m \rho E_{n}^\dagger.
\end{equation}
This description is completely general since the above expressions can be written for any linear channel. Properties of the channel are in one to one correspondence with properties of the $\chi$--matrix. In fact, the channel preserves the hermiticity if and only if the $\chi$--matrix is hermitian. Also, the channel preserves trace, if and only if the condition $\sum_{m n} \chi_{m n} E_{n}^\dagger E_m = I$ is satisfied. Fianlly, the channel is completely positive if and only if the $\chi$--matrix is positive. Thus, the $\chi$--matrix (which, as we mentioned above, depends on the choice of basis for the space of operators) fully describes the channel. Therefore, quantum process tomography is the task of estimating the matrix elements of $\chi$. To achieve this goal there are several methods, some of which involve performing quantum state tomography on final states. Let us review them now.
\subsection{Ancilla Assisted Quantum Process Tomography}\label{secAAPT}
The Ancilla Assisted Quantum Process Tomography (AAPT) \cite{AAPT, MohsRezLid} uses $n$ ancilliary qubits and allows to extract all the information about the channel. However, as presented in \cite{AAPT, MohsRezLid}, it only allows to obtain full information about the process, being unable to only useful partial information about the channel. Thus it is inherently inefficient as full QPT always is.
This is the first of two methods that we will mention here that exploit the state--channel duality given by Choi--Jamio\l{}kowski's isomorphism. Such isomorphism establishes a one to one relationship between linear operators from $\mathcal H \otimes \mathcal H$ to $\mathcal H \otimes \mathcal H$ and completely positive superoperators acting on the space of operators from $\mathcal H$ to $\mathcal H$. The Choi--Jamio\l{}kowski's isomorphisms establishes a correspondence between states and channels in the following way:
\begin{equation}\label{eqCJ}
\rho_{\mathcal E} = \mathcal{E}\otimes\mathbb I \left( \ket{I}\bra{I}\right).
\end{equation}
where $\ket{I}= \sum_i\ket{ii}/\sqrt{D}$ is the maximally entangled state. After the application of the channel to one of the parts, one can perform state tomography to the state $\rho_{\mathcal E}$, obtaining full information about the process ${\mathcal E}$. Figure \ref{figAAPT} ilustrates this algorithm.
\begin{figure}[ht]
\begin{center}
\epsfig{file=fig1-crop.pdf, angle=0, width=0.45\textwidth}
\end{center}
\caption{Scheme for the Ancilla Assisted Process Tomography quantum algorithm.\label{figAAPT}}
\end{figure}
One of the strenghts of AAPT is that the initial state can be another state and not necessarily the maximally entangled state. As the number of independent parameters defining the initial state (the Shmidt number) is $D^2$, such state can always encode the necessary information to define the quantum channel.
This method, besides requiring $n$ ancilliary qubits, has the following two troublesome properties: First, it is not clear if the information from the $\chi$ matrix can be directly accessed via measurements on the resulting states. Second but related to the previous point, it is not clear how to use QST on the final state to efficiently extract partial and relevant information on the channel. These two issues will be solved in what follows.
\subsection{Direct Characterization of Quantum Dynamics}\label{secDCQD}
The Direct Characterization of Quantum Dynamics (DQCD)\cite{MohsRezLid, DCQD} is a quantum algorithm similar to that of AAPT in many aspects. It also resorts to $n$ ancilliary qubits, and relies on the Choi--Jamio\l{}kowski isomorphism. Contrary to the AAPT, on the DCQD the authors explicitely showed a method to efficiently and selectively measure the diagonal coefficients of the $\chi$--matrix; however, off-diagonal elements still require to invert en exponentially large matrix. This makes the method inefficient.
To describe the method let us consider the operator basis consisting of the $n$--fold tensor product Pauli operators acting on each qubit. We denote these operators as $\left\lbrace E_m, m=0,...,D^2-1 \right\rbrace$. In that basis, the channel description is given by
\begin{equation}
{\mathcal E}\left(\rho\right)=\sum_{mn}\chi_{mn}E_m \rho E_n^\dagger
\end{equation}
In order to perform diagonal tomography, the algorithm proceeds as follows. First, as the AAPT, we have to generate the state that is isomorphic to the channel
\begin{equation}
\rho_{\mathcal E} = \frac{1}{D}\sum_{ijmn}\chi_{mn}E_m\ket{i}\bra{j}E_n^\dagger\otimes\ket{i}\bra{j}.
\end{equation}
Now, the probability of measuring the state $\frac{1}{\sqrt{D}}\sum_i\ket{ii}$ on the output is given by
\begin{equation}
\frac{1}{D}\sum_{kl}\bra{kk}\rho_{\mathcal E}\ket{ll}=\frac{1}{D^2}\sum_{mn}\chi_{mn}\mathrm{Tr}\left(E_m\right)\mathrm{Tr}\left(E_n^\dagger\right)=\chi_{00}.
\end{equation}
Thus, we see that the survival probability of the input state directly gives the coefficient $\chi_{00}$. It is easy to show that the very same method can be used to evaluate any diagonal coefficient of the $\chi$ matrix. In fact, the probability of obtaining the final state $\frac{1}{\sqrt{D}}\sum_iE_k\otimes\mathbb I\ket{ii}$ is nothing but $\chi_{kk}$. As the set
\begin{equation}\label{eqBaseCJ}
\mathcal R=\left\lbrace E_k\otimes\mathbb I\ket{I}, k=0,\ldots,D^2-1 \right\rbrace
\end{equation}
forms an orthonormal basis of $\mathcal H\otimes\mathcal H$, a measurement in that basis will suffice for diagonal tomography of the $\chi$--matrix.
The main problem arises when off-diagonal tomography is taken into account. The solution presented in \cite{MohsRezLid, DCQD} is to use a state other than a maximally entangled one. However, it can be shown that in the most general case, this procedure requires inverting an exponentally large matrix. Again, this makes the method inefficient.
Below, we will introduce a method for QST that provides not only a convenient tool for QST, but also would provide the necessary ingredient missing in AAPT and DCQD to obtain an efficient and selective QPT.
\section{Selective and Efficient Quantum State Tomography}\label{secSEQST}
The standard method for QST was clearly described in \cite{NielsenChuang}.
This method resorts to the description of the state in the Pauli operator basis as
\begin{equation}
\rho=\frac{1}{D} \sum_i \text{Tr}\left(\rho E_i\right) E_i
\end{equation}
where $E_i$ are the $n$--fold Pauli operator basis for an $n$ qubit system. It is straightforward to perform tomography in this basis by just measuring the expectation value of every $E_i$. Although this method is indeed selective and efficient, it is not well suited for selective and efficient QPT.
The method we are about to introduce, the Selective and Efficient Quantum State Tomography (SEQST), is also efficient and selective but, as opposed to the standard method, it is selective in any basis of the corresponding Hilbert space. That is, given the state written in the form
\begin{equation}
\rho=\sum_{a,b=1}^D \alpha_{ab}\ket{\psi_a}\bra{\psi_b}
\end{equation}
and provided we know how to prepare the states from the corresponding basis in a controlled manner, we will be able to selectively measure any given coefficient $\alpha_{ab}$. Such measurement will be efficient, meaning that given a precision, the number of required single click measurements doesn't scale with the size of the system.
Consider the circuit shown in Fig. \ref{figTomEstNoDiagRed} where the operators $V_a$ are the ones that prepare the states from the basis $\mathcal B=\left\lbrace\ket{\psi_a}, a=1,...,D\right\rbrace$ from the state $\ket{\psi_0}$. That is, $V_a\ket{\psi_0}=\ket{\psi_a}$.
\begin{figure}[ht]
\begin{center}
\epsfig{file=fig2-crop.pdf, angle=0, width=0.45\textwidth}
\end{center}
\caption{Quantum circuit for Selective and Efficient Quantum State Tomography.\label{figTomEstNoDiagRed}}
\end{figure}
It is straightforward to verify that by measuring the average value of the operator $\ket{\psi_0}\bra{\psi_0}\otimes\sigma_x$ (that is the average value of the operator $\sigma_x$ conditioned on the detection of
the state $\ket{\psi_0}$ on the main system) one obtains the real part of $\chi_{ab}$. Moreover, replacing $\sigma_x$ by $\sigma_y$, the same method provides the imaginary part of the same matrix element. Thus,
\begin{eqnarray}
\mathrm{Tr}\left(\rho_F \ket{\psi_0}\bra{\psi_0}\otimes\sigma_x \right)&=&\Re{\chi_{ab}}\\
\mathrm{Tr}\left(\rho_F \ket{\psi_0}\bra{\psi_0}\otimes\sigma_y \right)&=&\Im{\chi_{ab}}
\end{eqnarray}
where $\rho_F$ is the state prior to the measurement.
To discuss the efficiency of the method we should analyze the resources needed by this algorithm. First of all, the efficiency of the method is limited by that of the implementation of the controlled--$V_a^\dagger$ and controlled--$V_b^\dagger$ operators. In fact, if the implementation of such operations require $O\left(f\left(n\right)\right)$ operations, then the full circuit will also require $O\left(f\left(n\right)\right)$.
The other point to determine the efficiency of the method is to determine the number of experimental runs required to obtain the desired result to a given precision $\epsilon$ with a probability of success $p$. To answer that question we just need to consider that each experimental run gives one of three results ($+1$ corresponding to $\ket{\psi_0}$ on the system and $\ket{0}$ on the ancilla, $-1$ corresponding to $\ket{\psi_0}$ on the system and $\ket{1}$ on the ancilla, and $0$ corresponding to another result on the system). The $\chi$ matrix element is estimated by computing the average value these results after performing a certain number of repetitions $M$. Each of the results are detected at random with their corresponding probabilities. Therefore, one can use a Chernoff bound to show that to obtain the correct result with uncertainty $\epsilon$ and a probability $p$ of success, the number of experimental runs $M$ must be such that
\begin{equation}
M\geq \frac{2 \ln\left(\frac{2}{p}\right) }{\epsilon^2}
\end{equation}
which does not depend on $n$ or $D$. This implies that the method is efficient.
\subsection{Application to Quantum Process Tomography}
In this section we will show how the SEQST algorithm is the right tool to perform selective and efficient QPT when combined with the AAPT method reviewed in Sec. \ref{secAAPT}.
In order to proceed, we need to find out the way in which the quantum state isomorphic to the channel ${\mathcal E}$ depends on the $\chi$--matrix of such channel. Thus, we will show that the $\chi$--matrix of the channel is nothing but the matrix element of the quantum state in a particular basis. Therefore, by performing quantum state tomography in that basis we directly provide the information about the $\chi$--matrix of the channel. To show this, we use equation \eqref{eqCJ} and replace the expansion of ${\mathcal E}$ in the Pauli basis:
\begin{equation}
\rho_{\mathcal E} = \sum_{mn} \chi_{mn} E_m\otimes\mathbb I \ket{I}\bra{I} E_n^\dagger\otimes\mathbb I.
\end{equation}
Indeed, this shows that $\chi$ is the matrix representation of $\rho_{\mathcal E}$ in the basis $\mathcal R$ shown in equation \eqref{eqBaseCJ}. Therefore, to selectively measure a single $\chi$ coefficient one only needs to perform selective tomography in the basis $\mathcal R$.
Using the results already presented in the previous section, we see that to do this we should implement the circuit described in Figure \ref{figSEQSTQPT}. Here, the application of the channel ${\mathcal E}$ to one of the pieces of the maximally entangled state can be regarded as the preparation of the state isomoprhic to the channel. In turn, the rest of the circuit is nothing but the SEQST algorithm described above.
\begin{figure}[ht]
\epsfig{file=fig3-crop.pdf, angle=0, width=0.45\textwidth}
\caption{Application of SEQST to QPT.\label{figSEQSTQPT}}
\end{figure}
It is important to point out that, since the measurement is direct, the analysis of the resources required to implement the method (presented in the previous section) directly applies to this case. The only extra resources needed in this case are involved in the preparation and measurement of the maximally entangled state which require $O\left(n\right)$ aditional single and two qubit gates.
\section{Comparison with Other Selective and Efficient QPT Schemes}
Another quantum algorithm for selective and efficient quantum process tomography is the one known as SEQPT, precisely for Selective and Efficient Quantum Process Tomography\cite{paper1conFer, paper2conFer}. The main idea there is to follow the proceedure described by the circuit shown in Figure \ref{circ:offdiag}, and to estimate the average answer averaging over the entire Hilbert space of the system using the Haar measure for that purpose. As it is shown in \cite{paper1conFer, paper2conFer}, that average cn be directly related to the matrix element $\chi_{ab}$ as
\begin{eqnarray}\label{eqReImChi}
\int \left<\sigma_x\otimes \ket{\psi}\bra{\psi} \right> d\psi &=& \frac{D\text{Re}\left(\chi_{ab} \right) +\delta_{ab}}{D+1}\\
\int \left<\sigma_y\otimes \ket{\psi}\bra{\psi} \right> d\psi &=& \frac{D\text{Im}\left(\chi_{ab} \right)}{D+1}.
\end{eqnarray}
Moreover, it can be shown that the average over the entire Hilbert space can be efficiently estimated by randomly sampling over a special (and finite) set of states which is known as a $2$--designs. For these reasons, the method SEQPT is not only selective but also efficient.
\begin{figure}[ht]
\epsfig{file=fig4-crop.pdf, angle=0, width=0.45\textwidth}
\caption{Circuit for the Selective and Efficient Quantum Process Tomography algorithm. Depending on the measurement of $\sigma_x$ or $\sigma_y$, the real or imaginary part of $\chi_{ab}$ will be obtained.}
\label{circ:offdiag}
\end{figure}
As we mentioned above, in the SEQPT scheme, the average is estimated by randomly sampling states. In the scheme we proposed above, the average is obtained \emph{automatically} by the quantum correlation between both parts of the maximally entangled state, without the need to resort to randomly preparing and detecting the special states of the $2$--design.
\section{Summary}
In this paper we presented a novel quantum algorithm to perform selective and efficient quantum state tomography in any Hilbert space basis, given that the states from that basis can be efficiently prepared in a controlled manner.
We then showed that, when properly combined with the Ancilla Assisted Process Tomography, it yields a protocol for QPT that is both selective and efficient. Finally, we showed that this protocol shares some properties with SEQPT, a method presented in \cite{paper1conFer, paper2conFer}. The main difference is that the use of ancillas is a way to avoid the preparation of the special states of the $2$--design and sampling on them.
This work was partially supported by funds from CONICET, ANPCyT, UBACyT, EU Q-Essence and Spanish FIS2010-14830 projects. A.B. is funded by an ERC Starting Grant PERCENT.
| -14,547.317021 |
[
-2.703125,
2.53125
] | 53.672316 |
[
-3.103515625,
0.2213134765625,
-2.294921875,
-5.59375,
-0.55615234375,
8.0625
] |
[
3.140625,
8.4609375,
2.21484375,
8.3671875
] | 137 | 2,689 |
[
-2.982421875,
3.234375
] | 23.28007 |
[
-6.2109375,
-4.87109375,
-4.6796875,
-1.9130859375,
2.650390625,
12.5625
] | 1.331203 | 35.534574 | 25.399777 | 0.836404 |
[
2.559865951538086
] | -10,431.879112 | 5.587207 | -14,462.113975 | 1.817093 | 5.441307 |
[
-2.490234375,
-3.73046875,
-3.71875,
-4.80078125,
2.400390625,
12.0625
] |
[
-5.4765625,
-2.060546875,
-2.15625,
-1.33984375,
3.7734375,
4.74609375
] | |
BkiUbWY25YjgKLaE8IjZ
|
\section{Introduction}
The spin foam approach is a way to quantize gravity in a spacetime covariant setting \cite{Oriti,perez}.
Whereas in 3 dimensions, where gravity is a particular case of the so called BF theory,
the rules of spin foam quantization and the geometric interpretation of results
are more or less clear \cite{PonzanoRegge,Noui:2004iy},
the physically relevant case of 4 dimensions is still an arena for debates.
One of the main non-settled issues is how to implement the constraints which appear
in the formulations of 4-dimensional general relativity used in the spin foam quantization.
This is the problem which we are going to address in this paper.
\subsection{Simplicity constraints}
The classical formulation of general relativity, which is the most suitable to build spin foam models,
was suggested by Plebanski \cite{Plebanski:1977zz}. It represents gravity as a simple topological
BF theory supplemented by the so called {\it simplicity constraints}.
These constraints are conditions on the $B$ field of BF theory
that ensure that it is constructed from tetrad one-forms
\begin{equation}
B^{IJ}=*(e^I\wedge e^J),
\label{Bee}
\end{equation}
where $*$ is the Hodge operator acting on tangent space indices.
The constraints can be written in the following form
\begin{equation}
\varepsilon_{IJKL} B^{IJ}_{\mu\nu} B^{KL}_{\rho\sigma}=\sigma\,
{\cal V} \, \varepsilon_{\mu\nu\rho\sigma} ,
\label{simplicityconditions1}
\end{equation}
where ${\cal V}= \frac{1}{4!}\,\varepsilon_{IJKL}\varepsilon^{\mu\nu\rho\sigma} B_{\mu\nu}^{IJ} B_{\rho\sigma}^{KL}$ is the 4-volume
and $\sigma=\pm 1$ in the Riemannian/Lorentzian case.
The standard strategy of the spin foam quantization is first to quantize the BF theory,
and then to impose the simplicity constraints already at the level of the discretized path integral
\cite{De Pietri:1998mb,Freidel:1998pt}. In particular, this means that the constraints
are imposed not directly on the $B$ field, since it is integrated out, but on representations and
intertwiners of the gauge group which are assigned to elements of a simplicial decomposition of spacetime.
These group theoretic data are thought as quantum degrees of freedom corresponding to the $B$ field.
As a result, it is a hard task to find a consistent and correct way implementing the constraints.
For a long time the most popular model in 4 dimensions was the one proposed by Barrett and Crane (BC)
\cite{BCE,BC}. However, it has become clear that it is insufficient to describe the genuine general relativity.
Therefore, recently various modifications of the Barrett--Crane quantization procedure
have been suggested to improve the BC model
\cite{Livine:2007vk,Engle:2007uq,Alexandrov:2007pq,Engle:2007qf,Freidel:2007py,Livine:2007ya,Oriti:2007vf,Engle:2007mu,Pereira:2007nh,Engle:2007wy}.
All these proposals differ from the original model and between each other in the way
they treat the simplicity constraints.
Note that some of the new models suggest a way to overcome the above mentioned problem of disappearance
of the $B$ field in the quantum partition function.
In particular, the approach based on coherent states \cite{Livine:2007vk,Freidel:2007py,Livine:2007ya}
allows to give a faithful representation for the $B$ field at the quantum level in terms of
additional geometric data appearing as labels of these states.
Therefore the imposition of the simplicity constraints in this approach
becomes much more transparent and reliable comparing to previous studies.
Another interesting development was done in \cite{Oriti:2007vf} where a new class of
group field theories was proposed. These models keep the $B$ field in
the partition function explicitly which might be crucial for the correct implementation of
the constraints.
There is however one additional complication which is usually neglected in most of
the old and new spin foam models.
As it is clear from canonical analysis of Plebanski formulation \cite{Buffenoir:2004vx,ABR},
the simplicity constraints are supplemented by secondary constraints involving the gauge connection.
Altogether they form a system of second class constraints. The question now is: should these
secondary constraints be inserted explicitly into the path integral?
One usually assumes that one can start from the Lagrangian path integral with a trivial measure,
which does not involve contributions from any constraints. But it is generally believed that
the phase space path integral is more fundamental. The latter certainly contains delta-functions
of all constraints and a non-trivial measure. Therefore, whether the simple Lagrangian path integral
can be used depends on our possibility to derive it from the phase space path integral. Whereas
there is a general method for this purpose \cite{Henneaux:1994jf}, it turns out that it has some
restrictions and should be used with care. In particular, we will argue
that the secondary constraints of Plebanski formulation cannot be removed and must be taken into
account in the spin foam quantization.
The latter requires however to realize the secondary constraints at the discretized level
in terms of holonomies and bi-vectors. Unfortunately, we were not able to find such a realization
for the constraints relevant for the gravitational sector of Plebanski formulation.
Nevertheless, it is possible to show that it is these constraints that are responsible
for the restrictions on representations and intertwiners of a simplicial decomposition.
We demonstrate also how this leads to constructions appearing in
the framework of covariant loop quantum gravity (CLQG) \cite{SA,AV,SAcon,AlLiv,Livine:2006ix}.
\subsection{Closure constraint}
Besides the simplicity constraints, the spin foam quantization involves another type of constraints,
{\it the closure constraint}. The latter appears in all spin foam models and represents the Gauss constraint
of canonical formulation realized in the discrete setting. Thus, it ensures the fundamental gauge invariance
of quantized theory.
The closure constraint can be formulated as follows.
Let us consider a simplicial decomposition of spacetime which the spin foam quantization is based on.
In 4 dimensions such decomposition is formed by
4-simplices, tetrahedra, triangles, segments and points. The triangles are colored with irreducible representations
of the relevant gauge group $G$, which will be taken either SO(4) or SL(2,${\rm \bf C}$),
and the tetrahedra are equipped with intertwiners between the representations
assigned to its sides.
We label the triangles and the tetrahedra by $f$ and $t$, respectively.
If we denote the representation assigned to the $f$th triangle by $\lambda_f$,
then the intertwiner $N_t$ is a vector in the tensor product of four representation spaces,
$N_t\in {\bigotimes_{f\subset t}}\, H_G^{(\lambda_f)}$.
The closure constraint requires that the intertwiners are gauge invariant, {\it i.e.}
\begin{equation}
\vphantom{A\over B}
\smash{\sum_{f\subset\, t}}\, T_f\, N_t=0,
\label{clcon}
\end{equation}
where $T_f$ are generators of the gauge algebra acting on $H_G^{(\lambda_f)}$.
In this form the constraint was widely used in various constructions including the original
BC model and the recent proposals \cite{Engle:2007uq,Engle:2007qf,Engle:2007mu,Pereira:2007nh,Engle:2007wy}.
However, in the work \cite{Alexandrov:2007pq} a relaxed version of this constraint has been proposed.
It can be written as
\begin{equation}
\vphantom{A\over B_B}
\smash{\sum\limits_{{f\subset\, t}}}\, T_f\, N_t(x_t)=\hat T\cdot N_t\(x_t\),
\label{clconrel}
\end{equation}
where $x_t\in G/H$ ($H$ is the maximal compact subgroup of $G$) is interpreted as normal to the tetrahedron
and $\hat T$ acts on functions of $x_t$ as a generator of $G$
\begin{equation}
\hat T_{IJ}\cdot f(x)=\eta_{IK} x^K \partial_J f-\eta_{JK} x^K \partial_I f.
\label{generator}
\end{equation}
The intertwiners satisfying \eqref{clconrel} are not invariant anymore, but rather covariant.
The relaxed constraint was motivated by comparison with results of canonical quantization
obtained in the framework of CLQG.
The use of the constraint \eqref{clconrel} together with a modified identification
of bi-vectors with generators of the gauge algebra ensured the coincidence of boundary states
of the resulting spin foam model with kinematical states of CLQG \cite{Alexandrov:2007pq}.
In this paper we justify the new constraint \eqref{clconrel} from the pure
spin foam point of view. The key point for this is the consistent implementation of second class constraints
and, in particular, the secondary constraints mentioned above.
In the discrete setting these constraints involve the normals to tetrahedra.
It turns out that this fact requires the normals $x_t$ to be considered as
additional arguments of the boundary states and leads precisely to the intertwiners satisfying \eqref{clconrel}.
The usual invariant intertwiners \eqref{clcon} can be obtained by integrating over the normals $x_t$,
what however would make impossible implementation of the simplicity and secondary constraints.
\
The organization of the paper is as follows.
In the next section we discuss the general method to pass from
the phase space path integral to the Lagrangian one in the presence of second class constraints
and argue that it does not work for generic correlation functions.
In section \ref{sec_simconstr_new} we revise the derivation of the simplex
boundary state and express it in terms of projected spin networks of CLQG.
Then in section \ref{sec_conseq} we discuss
implications of our approach on the closure constraint and demonstrate that only its relaxed version
is consistent with the second class constraints.
Finally we consider an example which allows to make various aspects of the construction explicit.
\section{Path integral in presence of second class constraints}
\label{sec_constraint}
Let us consider a dynamical system with the phase space parameterized by $(q^a,p_a)$ and
subject to second class constraints. For the sake of simplicity, we suppose that there are only
two constraints one of which, say $\phi$, is primary and the second, $\psi$, is secondary.
This means that
\begin{equation}
\psi=\{ H, \phi\}, \qquad \{ \phi,\psi\}\ne 0,
\end{equation}
where $H$ is the Hamiltonian.
Correlation functions for this system are defined by the phase space path integral
\begin{equation}
\langle{\cal O}\rangle = Z^{-1}\int {\cal D} q\,{\cal D} p \, |\,{\rm det}\, \{\phi,\psi\}|\,\delta(\phi)\delta(\psi)\, e^{i\int dt\(p_a \dot q^a-H\)}{\cal O}(q,p),
\label{phspint}
\end{equation}
where ${\cal O}$ is an observable and $Z$ is the partition function.
The problem we would like to address is whether these correlation functions are equal to their analogues
defined by the configuration space path integral. Since we want to include into consideration also systems
with first order Lagrangians (like Plebanski or Palatini formulations of gravity), by latter we mean
\begin{equation}
\langle{\cal O}\rangle_{\rm c.s.} =
Z_{\rm c.s.}^{-1}\int {\cal D} q\,{\cal D} p \, {\cal D}\lambda\, \rho(q,p)\, e^{i\int dt\(p_a \dot q^a-H-\lambda\phi\)}{\cal O}(q,p),
\label{confspint}
\end{equation}
where $\lambda$ is the Lagrange multiplier for the primary constraint, $\rho(q,p)$ is a regular local measure,
and the expression in the exponential is the Lagrangian in its first order formulation.
Comparison of \eqref{phspint} and \eqref{confspint} shows that the difference between the two expressions
arises due to the secondary constraint $\psi$. To bring the correlation functions together,
one should somehow remove its contribution.
In \cite{Henneaux:1994jf} an elegant method has been suggested to achieve this goal for the partition function.
It makes use of the canonical transformation generated by $\mu\phi$ where $\mu$ is the Lagrange multiplier
which can be introduced for the secondary constraint.
Implemented in the action
$\int dt\(p_a \dot q^a-H-\lambda\phi-\mu\psi\)$,
it cancels the term linear in $\mu$ and therefore the integral over this Lagrange multiplier produces a regular
local measure instead of the $\delta$-function of the constraint.
However, this idea does not work so well for the correlation functions because the phase space function
${\cal O}(q,p)$ gets transformed together with the integral. As a result, one obtains a correlation function
of the following form
\begin{equation}
\langle{\cal O}\rangle = Z^{-1}\int {\cal D} q\,{\cal D} p \,{\cal D} \lambda \, {\cal D}\mu \, |\,{\rm det}\, \{\phi,\psi\}|\,
e^{i\int dt\(p_a \dot q^a-H-\lambda\phi-{1\over 2}\,\mu^2\{\phi,\psi\}+O(\mu^3)\)}
{\cal O}'(q,p,\mu),
\label{phspint_tr}
\end{equation}
where ${\cal O}'$ is the result of the canonical transformation of the initial function ${\cal O}(q,p)$.
Even if one succeeds to perform the integral over $\mu$, the result will differ from \eqref{confspint}.
Roughly speaking, one will obtain the correlation function of a different observable.
Thus, the two definitions, \eqref{phspint} and \eqref{confspint}, do not coincide in general.
This situation can be illustrated on very simple examples.
The simplest example is provided by the trivial system with the Lagrangian $L={1\over 2}\, q^2$.
It is clear that it corresponds to the situation considered above with $\phi=p$, $\psi=q$.
For this system $\langle q^2\rangle_{\rm c.s.}>\langle q^2\rangle=0$. The reason for the inequality can easily be traced back
to the non-commutativity of the observable ${\cal O}=q^2$ with the primary constraint $\phi$
used in the canonical transformation so that in this case ${\cal O}'=(q-\mu)^2$.
A similar example is given by the Lagrangian $L=p\dot q -{1\over 2}\, p^2-\lambda q$, which leads to
the constraints $\phi=q$ and $\psi=-p$. Now the two definitions do not coincide
for the correlation functions of observables dependent on the momentum $p$.
It is clear that in both examples the canonical quantization based on the introduction of a Dirac bracket
would lead to results consistent with the phase space path integral.
Thus, we conclude that if one wishes to consider correlation functions of observables dependent on all
phase space variables, one has to work with the path integral where all constraints, primary and secondary,
appear in $\delta$-functions.\footnote{We remark also that the modern approach to the Lagrangian path integral
originated from the works of Batalin and Vilkovisky \cite{Batalin:1981jr,Batalin:1984jr}
leads to similar conclusions \cite{Batalin:1997dy}.}
\section{Simplicity constraints revisited}
\label{sec_simconstr_new}
\subsection{Constraints on connection}
\label{subsec_connect}
The main lesson of the previous section is that one should not ignore the secondary second class constraints.
Four dimensional gravity in the first order formulation is an example of the system possessing such type
of constraints.
Therefore, quantizing gravity by the spin foam approach, one must include
them into the measure together with the usual simplicity constraints.
Let us briefly recall what the canonical analysis tells us about the secondary constraints of Plebanski formulation.
In fact, in \cite{ABR} it has been shown that the canonical formulation of Plebanski action \cite{Buffenoir:2004vx}
is essentially equivalent to the Lorentz covariant canonical formulation of the Hilbert--Palatini action \cite{SA}.
Therefore, we can use the results from the latter where the constraints have been extensively studied.
The secondary constraints, which appear by commuting the Hamiltonian with the simplicity constraints,
depend linearly on the spin connection. This connection however is not suitable for quantization
and it is more convenient to consider another connection, which we call ${\cal A}_i^{IJ}$,
differing from the original one by a shift by the Gauss constraint \cite{AV}.
Its holonomies have simple commutation relations with the smeared triad what has important consequences
for quantum theory. Due to this, it is advantageous to formulate the secondary second class constraints also
in terms of this new variable. They have the following form
\begin{equation}
I_{(Q)KL}^{IJ}{\cal A}_i^{KL}=\Gamma_i^{IJ}(B),
\label{contA}
\end{equation}
where $\Gamma_i^{IJ}$ is the Levi--Civita connection determined by the space components of the $B$ field
and we introduced two projectors
\begin{equation}
I_{(Q)}^{IJ,KL}(x)=
\eta^{I[K}\eta^{L]J}-2\sigma\, x^{[J}\eta^{I][K} x^{L]},
\qquad
I_{(P)}^{IJ,KL}(x)=2\sigma\,x^{[J}\eta^{I][K} x^{L]}.
\label{projxxx}
\end{equation}
They depend on a 4-dimensional unit vector $x^I$ ($x^I x_I=\sigma$), which defines the normal to three-dimensional
hypersurfaces foliating spacetime and it can be extracted from the $B$ field by solving the simplicity
constraints. This vector defines a subgroup $H_x={\rm SU}_x(2)$ of the gauge group $G$
which leaves it invariant. Then the geometric meaning of $I_{(Q)}^{IJ,KL}$ and $I_{(P)}^{IJ,KL}$ is
that they project on the algebra of $H_x$ and its orthogonal completion, respectively.
These projectors play an important role and appear also in the symplectic structure
determining the commutator of the shifted connection with the $B$ field
\begin{equation}
[\, \varepsilon^{jkl}B_{kl}^{IJ}(y),{\cal A}_i^{KL}(x)]
=-i\hbar\,I_{(P)}^{IJ,KL}\delta_i^j\delta(x,y).
\label{commPleb}
\end{equation}
Given the geometric interpretation of the projectors, it is clear that the meaning of the constraints
\eqref{contA} is to restrict the ``rotational" part of the shifted connection as a function of
the $B$ field.\footnote{A reader
familiar with the self-dual Ashtekar formulation of general relativity \cite{Ashtekar:1986yd,Ashtekar:1991hf}
may find similarities between the constraints \eqref{contA} and the so called reality conditions imposed on complex
Ashtekar connection \cite{Ashtekar:1991hf,Immirzi:1992ar}. And indeed the reality conditions can be shown to be
equivalent to our second class constraints \cite{Alexandrov:2005ng}.}
The ``rotational" part is defined by the vector $x^I$ as explained above.
Therefore, all independent physical degrees of freedom of the connection are contained only in its ``boost" part.
It is worth to notice that one can quantize the theory using another object, ${\bf A}_i^{IJ}$, instead of the shifted connection
${\cal A}_i^{IJ}$. This object has all properties of a gauge
connection except that it does not transform appropriately under time diffeomorphisms.
It has an advantage that it is commutative and in the time gauge $x^I=\delta^{I,0}$ the loop quantization based on
holonomies defined by ${\bf A}_i^{IJ}$ reproduces the results of LQG \cite{SAcon,AlLiv}. This can be traced back to the secondary constraints
which in terms of ${\bf A}_i^{IJ}$ become
\begin{equation}
I_{(P)KL}^{IJ} {\bf A}_i^{KL}=2\,x^{[J}\partial_i x^{I]},
\label{su2con}
\end{equation}
They imply that in the time gauge ${\bf A}_i^{IJ}$ becomes an SU(2) connection coinciding with the one used
in the Ashtekar--Barbero approach \cite{Ashtekar:1986yd,Barbero:1994an}. However, the failure of this quantity to
have correct spacetime transformations points in favor of quantization based on the true spacetime connection ${\cal A}_i^{IJ}$.
It is the latter quantization that gives rise to the CLQG approach.
\subsection{Discretization}
\label{subsec_discr}
Before quantizing gravity in the spin foam approach, one needs to realize its degrees of freedom at the discrete level.
Given a simplicial decomposition of spacetime, one considers the following data:
i) elements of the gauge group $g_{\sigma t}$ assigned to every couple of simplex $\sigma$ and tetrahedron $t$, which
represents holonomy of the gauge connection from the center of the former to the center of the latter;
ii) bi-vectors $B_f^{IJ}$, which can be thought as elements of the gauge algebra $\hat B_f=B_f^{IJ} T_{IJ}$,
assigned to every triangle $f$.
\noindent
These data are enough to encode all degrees of freedom and to represent the gravity action at the discretized
level (see, for example, \cite{Engle:2007qf}). However, one has to also rewrite
the second class constraints in terms of the discrete variables $B_f^{IJ}$ and $g_{\sigma t}$.
The discretization of the quadratic simplicity constraints \eqref{simplicityconditions1} has been extensively
studied in the literature. There is however a nice way to effectively linearize them. It makes use
of the projectors \eqref{projxxx}. It turns out that the simplicity constraints can be rewritten as \cite{Alexandrov:2007pq}
\begin{equation}
I_{(Q)KL}^{IJ}(x)B^{KL}=0.
\label{newsim}
\end{equation}
Although the constraint \eqref{newsim} seems to be linear, one should remember that $x$ is a part
of the $B$ field. What is really going on here is a decoupling of degrees of freedom: once $x$ has been extracted,
the rest of the $B$ field is constrained to satisfy \eqref{newsim} and the only remaining freedom
corresponds to the triad of canonical theory.
At the discrete level the $B$ field gives rise to bi-vectors normal to triangles.
Taking into account that $I_{(Q)}(x)$ annihilates bi-vectors coaligned with $x^I$,
it is easy to understand the geometric meaning of the condition \eqref{newsim}. It requires that
all triangles, for which the constraint is written with the same $x^I$, must lie in the hypersurface
normal to this 4-dimensional vector. Therefore, $x^I$ is naturally identified with the normals to tetrahedra
of the simplicial decomposition.
Regarding this, we introduce an additional set of geometric data:
iii) unit vectors $x_t^I$ playing the role of normals to tetrahedra;
iv) elements of the gauge group ${\rm g}_{ft}$ assigned to every couple of tetrahedron $t$ and triangle $f$
and constrained to satisfy ${\rm g}_{ft}\cdot x_t={\rm g}_{ft'}\cdot x_{t'}$ for two tetrahedra sharing $f$.
\noindent
The introduction of ${\rm g}_{ft}$ is related to the subtlety arising after discretization that
the normals $x_t$ and the bi-vectors $B_f$ all live in different frames. Therefore,
we need to parallel transport them to be able to compare. The elements ${\rm g}_{ft}$ just serve
to this purpose. They are not really important and will disappear from the action and the path integral.
Given these additional data, it is now easy to formulate the simplicity constraints at the discrete level.
Let us define the operator
\begin{equation}
\hat I_{(Q)t}^{IJ} = I_{(Q)}^{IJ,KL}(x_t) T_{KL}.
\end{equation}
Then the simplicity constraints \eqref{newsim} become
\begin{equation}
\phi_{f}^{IJ}=\,{\rm Tr}\, \( \hat I_{(Q)t}^{IJ}\, {\rm g}_{ft}^{-1} \, \hat B_f \, {\rm g}_{ft}^{\hphantom{1}} \).
\label{opsim}
\end{equation}
Due to the condition on $g_{ft}$, the r.h.s. does not depend on the chosen tetrahedron
and therefore we omitted the label $t$.
The secondary constraints \eqref{contA} are much more difficult to discretize.
The reason is that, whereas the simplicity constraints are formulated at the algebra level,
the constraints on connection should be realized as a condition on group elements.
In this paper we leave this problem unsolved. Nevertheless, it is possible to draw some important conclusions
just assuming that there is a measure on the space of holonomies incorporating the secondary constraints.
For this we will need only one property of this measure.
As we saw in the previous subsection, all versions of the secondary constraints, similarly to the simplicity
constraints \eqref{newsim}, depend on $x^I$.
This implies that the discrete measure should contain a dependence of the normals to tetrahedra and, what is important,
it should appropriately transform under the gauge group.
Thus, we assume that there exists a measure ${\cal D}^{(x_t)} [g_{\sigma t}]$ on the space of holonomies
dependent of the normal $x_t$. It may also have a dependence of $B_f$ which we do not display explicitly.
The measure is required to transform covariantly with respect to the group
transformations\footnote{Here the measure is defined for holonomies between a simplex and one of its tetrahedra.
This is the reason why only the transformation property under the right group action is imposed. To define the left group action,
one would need to introduce a 4-dimensional vector associated with a 4-simplex. But there are no such natural vectors.
In any case, the holonomies always appear in combinations $g_{\sigma t}^{-1} g_{\sigma t'}$ and therefore the group transformations acting
from the left are not important.}
\begin{equation}
{\cal D}^{(x)} \[g\, {\rm g}\]={\cal D}^{(x^{\rm g})} [g], \qquad x^{\rm g}={\rm g}\cdot x.
\label{trmeas}
\end{equation}
An important consequence of \eqref{trmeas} is that the measure defined by
the normal fixed to $x_0=(1,0,0,0)$ is invariant with respect to the diagonal SU(2) subgroup
\begin{equation}
{\cal D}^{(x_0)} \[g \,{\rm g}^h\]={\cal D}^{(x_0)} [g], \qquad {\rm g}^h=(h,h).
\label{invmeas}
\end{equation}
Finally, the unconstrained action at the discrete level reads
\begin{equation}
S_{BF}=\sum_f \,{\rm Tr}\, \( {\rm g}_{ft_1}^{-1} \, \hat B_f \, {\rm g}_{ft_1}^{\hphantom{1}} \,
g_{\sigma_{12} t_1}^{-1}g_{\sigma_{12} t_2}^{\hphantom{1}}
\cdots g_{\sigma_{n 1} t_n}^{-1}g_{\sigma_{n1} t_1}^{\hphantom{1}} \),
\label{dact}
\end{equation}
where the trace includes the product over all tetrahedra containing a given triangle and this product
does not depend on the ``reference" tetrahedron $t_1$.
Comparing with \eqref{opsim}, we see that choosing such a ``reference" tetrahedron $t_{f}$ for every triangle, it is natural to redefine
${\rm g}_{ft_{f}}^{-1} \, \hat B_f \, {\rm g}_{ft_{f}}^{\hphantom{1}}\to \hat B_f$ so that the dependence of auxiliary
group elements $g_{ft}$ disappears.
\subsection{Simplex boundary state}
\label{subsec_sbs}
Now we would like to reconsider the derivation of the simplex boundary state taking into account
the previous results. Thus, we include the contribution from both primary and secondary second class constraints
into the measure from the very beginning. This means in particular that we give up the usual strategy used
in the spin foam quantization: first quantize and then impose constraints.
The natural object to start with is the discretized path integral for a single 4-simplex
with fixed $B_f$ on the boundary. Inclusion of all constraints amounts to inserting the $\delta$-function
of $\phi_{f}^{IJ}$ \eqref{opsim} and taking the measure ${\cal D}^{(x_t)} [g_t]$ for holonomies. This gives
\begin{eqnarray}
A[B_f]&=&\int \prod_t {\cal D}^{(x_t)} [g_t]
\prod_f\[\delta\(\phi_{f} \)
e^{i \,{\rm Tr}\, \(B_f g_{u(f)}^{-1}g_{d(f)}^{\hphantom{1}}\)}\]
\nonumber \\
&=&\int \prod_f \[ d g_f\, e^{i\,{\rm Tr}\, (B_f g_f)}\delta\(\phi_{f}\)\]
\int\prod_t {\cal D}^{(x_t)} [g_t] \prod_f\delta\( g_{u(f)}^{\hphantom{1}}g_fg_{d(f)}^{-1}\),
\label{amplitgen}
\end{eqnarray}
where $u(f)$ and $d(f)$ denote two tetrahedra which share the $f$th triangle. Since the triangle is oriented,
one of them is considered as ``up" and the other as ``down".
From \eqref{amplitgen} one gets the simplex boundary state in the ``connection" representation
\begin{equation}
A[g_f;x_t]=\int\prod_t {\cal D}^{(x_t)} [g_t]
\prod_f\delta\( g_{u(f)}^{\hphantom{1}}g_fg_{d(f)}^{-1}\).
\label{simbs_init}
\end{equation}
Notice that the amplitude depends on the group elements $g_f$, playing the role of external holonomies,
as well as on the normals $x_t$.\footnote{If there is a dependence of $B_f$
in the measure ${\cal D}^{(x_t)} [g_t]$, one can formally replace all $B_f$
by the functional derivative $\frac{1}{i}\,\frac{\delta}{\delta g_f}$.}
The latter dependence could be removed by an explicit integral over $x_t$. However,
it would be inconsistent with gluing of different simplices and the proposed measure.
We refer to section \ref{sec_conseq} for more detailed discussion of this important issue
which has direct consequences for the closure constraint and the boundary state space.
The normals $x_t$ can be considered as elements of the homogeneous factor space $G/H$.
It will be convenient to denote by ${\rm g}_x$ a representative of $x\in X$ in $G$ so that ${\rm g}_x \cdot x_0=x$.
Changing the variable $g_t\to g_t{\rm g}_{x_t}^{-1}$, one can use the covariance property \eqref{trmeas}
to extract the dependence of the normals from the measure
\begin{eqnarray}
A[g_f;x_t]&=&
\int\prod_t {\cal D}^{(x_0)} [g_t]
\prod_f\delta\( g_{u(f)}^{\hphantom{1}}G_fg_{d(f)}^{-1}\)
\nonumber
\\
&=& \sum_{\lambda_f}\int\prod_t {\cal D}^{(x_0)} [g_t]
\prod_f d_{\lambda_f}\,{\rm tr}\, _{\lambda_f}\( g_{u(f)}^{\hphantom{1}}G_fg_{d(f)}^{-1}\),
\label{vertgen}
\end{eqnarray}
where we denoted $G_f={\rm g}_{x_{u(t)}}^{-1}\, g_f\, {\rm g}_{x_{d(t)}}^{\hphantom{1}}$,
$d_{\lambda}$ is the dimension of the representation $\lambda$
and we used the Plancherel decomposition of the $\delta$-function on the group into the sum over
irreducible representations $\lambda_f$.\footnote{In the Lorentzian case $\sum_{\lambda}d_{\lambda}$
must be replaced by an integral over the spectrum of irreducible representations with the Plancherel measure.}
To proceed further, one can insert an additional integral over the diagonal subgroup SU(2) with group elements inserted
between $g_t$ and $G_f$. This insertion does not change the amplitude
because the group elements are absorbed into $g_t$ and disappear due to the invariance property
of the measure \eqref{invmeas}.
This gives
\begin{equation}
A[g_f;x_t]=\sum_{\lambda_f}\int\prod_t {\cal D}^{(x_0)} [g_t]\int \prod_t dh_t
\prod_f d_{\lambda_f}\,{\rm tr}\, _{\lambda_f}
\( g_{u(f)}^{\hphantom{1}} {\rm g}^{h_{u(f)}} G_f ({\rm g}^{h_{d(f)}})^{-1} g_{d(f)}^{-1}\).
\label{simbs_inter}
\end{equation}
The following computations will be done assuming that we are working in the Riemannian case.
But the final conclusions will be true for the Lorentzian model as well.
For the Riemannian signature the relevant gauge group $G={\rm SO}(4)$
is the product of two SU(2) groups (factorized by ${\rm \bf Z}_2$), and
its irreducible representations are labeled by two SU(2) spins $\lambda=(j^+,j^-)$.
We will represent an element $g\in{\rm SO}(4)$ as $(g^+,g^-)$, $g^\pm\in{\rm SU}(2)$.
Thus, in \eqref{simbs_inter} one encounters group integrals of eight matrix coefficients
that can be represented as follows
\begin{equation}
\int dh_t \prod_{f_i\in \,t} D^{(j^+_{f_i})}_{m_i n_i^{\vphantom{-}}}(h_t)D^{(j^-_{f_i})}_{m'_i n'_i}(h_t)
=\prod_{f_i\in\, t}\(\sum_{j_{tf_i}} \ClG{j^+_{f_i}}{j^-_{f_i}}{j_{tf_i}^{\vphantom{+}}}{m_i^{\vphantom{+}}}{m'_i}{\ell_{f_i}}
\overline{\ClG{j^+_{f_i}}{j^-_{f_i}}{j_{tf_i}^{\vphantom{+}}}{n_i^{\vphantom{+}}}{n'_i}{l_{f_i}}} \)
\sum_{k_t}\iota^{\{ j_{tf_i}\} }_{\{ \ell_{f_i} \} }(k_t)\overline{\iota^{\{ j_{tf_i}\} }_{\{ l_{f_i} \} }(k_t)},
\end{equation}
where $C\lefteqn{\vphantom{\overline{\hat A}_-}}^{{j_1}\ {j_2}\ {j_3}}_{{m_1}{m_2}{m_3}}$ are SU(2) Clebsch-Gordan coefficients
and $\iota^{\{ j_i\} }_{\{ m_i \} }(k)$ are matrix elements of an invariant SU(2) intertwiner between 4 representations
with spins $j_i$, $i=1,\dots,4$, and with intermediate spin $k$.
The sum over repeated representation indices is implied.
Thus, the integral over $h_t$ yields the amplitude in the following factorized form
\begin{equation}
A[g_f;x_t]=\sum_{\lambda_f,j_{tf},k_t}\(\prod_f d_{\lambda_f}\)A(\lambda_f,j_{tf},k_t)\Psi^{(\lambda_f,j_{tf},k_t)}[g_f;x_t] ,
\label{reprampl}
\end{equation}
where
\begin{equation}
\Psi^{(\lambda_f,j_{tf},k_t)}[g_f;x_t]=\prod_f\(
\overline{\ClG{j^+_f}{j^-_f}{j_{u(f)f}^{\vphantom{+}}}{m^{\vphantom{+}}}{m'}{\ell_{u(f)f}}}
D^{(j^+_f)}_{mn^{\vphantom{+}}}(G^+_f)D^{(j^-_f)}_{m'n'}(G^-_f) \ClG{j^+_f }{j^-_f }{j_{d(f)f}}{n^{\vphantom{+}}}{n'}{\ell_{d(f)f}} \)
\prod_t \iota^{\{ j_{tf_i}\} }_{\{ \ell_{tf_i} \} }(k_t),
\label{projspin}
\end{equation}
\begin{equation}
A(\lambda_f,j_{tf},k_t)=\int \prod_t {\cal D}^{(x_0)} [g_t]\,
\Psi^{(\lambda_f,j_{tf},k_t)}\[g_{d(f)}^{-1}g_{u(f)}^{\hphantom{1}};x_0\].
\label{amplit_simplex}
\end{equation}
It is easy to realize that the function
$\Psi^{(\lambda_f,j_{tf},k_t)}[g_f;x_t]$ is the so called projected spin network. Such states have been introduced in the
context of CLQG \cite{psn,AlLiv} and form the (enlarged) kinematical Hilbert space of this approach.
They are functions on the full gauge group $G$ defined on the graph dual to the triangulated boundary.
At every vertex the holonomies taken in a representation of $G$ ($\lambda_f=(j^+_f,j^-_f)$) are projected
to irreducible SU(2) representations ($j_{tf}$) and then coupled by SU(2) invariant intertwiners
($\iota^{\{ j_{tf_i}\} } (k_t)$). The dependence of the normals $x_t$ can be pushed from the matrix coefficients
into the intertwiners. Its only effect is that the projection is done on the ``rotated" subgroup
$H_{x_t}$ introduced after eq. \eqref{projxxx}.
We see that here the projected spin networks also appear as a basis of spin foam boundary states in
the perfect agreement with the canonical approach.
The coefficient $A(\lambda_f,j_{tf},k_t)$ in \eqref{reprampl} is nothing else but the vertex
amplitude of spin foam quantization in the spin network basis.
It contains the crucial information about dynamics of the theory.
It is in this place where the concrete form of the measure ${\cal D}^{(x_0)} [g_t]$ becomes important.
Up to now the derivation worked well for any measure consistent with \eqref{trmeas}.
But without its knowledge we cannot evaluate the vertex amplitude.
There is actually another place where the explicit form of the measure is implicated.
We expect that the measure contains a $\delta$-function because the second class constraints
fix a half of components of the connection. From \eqref{simbs_init} one therefore
expects that some restrictions will arise on the group elements $g_f$. These restrictions are the same
discretized secondary second class constraints that have been imposed on $g_t$.
Their implementation corresponds in the canonical approach to the passage from the enlarged Hilbert
space consisting from {\it all} projected spin networks to the kinematical Hilbert space
with the second class constraints implemented at the quantum level \cite{AK,Alexandrov:2007pq}.
One expects that its effect is to fix some labels of the projected spin networks as,
for example, to restrict oneself to the sector with only {\it simple} representations.
\section{Closure constraint revisited}
\label{sec_conseq}
Usually the closure constraint \eqref{clcon} is considered as a quantization of a classical relation
between bi-vectors $B_f$ associated to triangles of a tetrahedron.
Since the triangles are not arbitrary, but form the tetrahedron, their bi-vectors must satisfy \cite{BCE}
\begin{equation}
\vphantom{A\over B}
\smash{\sum\limits_{f\subset\, t}}B_{f}=0.
\label{classclos}
\end{equation}
After quantization, $B_f$ are represented by generators $T_f$ acting on the representation spaces
assigned to the triangles. This gives the constraint \eqref{clcon} on the intertwiners which enter
boundary states and vertex amplitudes of spin foam models.
From our point of view this procedure is too naive to be able to capture possible corrections
which might appear at the quantum level. A more fair strategy would be to {\it derive} the closure constraint
from a path integral representation of a spin foam model.
Indeed, as was noticed in \cite{Freidel:2007py}, the closure constraint is imposed automatically
by integration over holonomy group elements in the partition function and thus it does not need to be
imposed by hand.\footnote{Although in \cite{Engle:2007qf} the closure constraint is imposed strongly
on the intertwiners of boundary states, it was also noticed that its classical counterpart \eqref{classclos}
can be obtained by the variation of the connection in the discrete action. This can be considered as another
indication in favor of treating the closure constraint as an equation of motion which can fluctuate at the quantum level.}
In other words, it is not an ingredient, but rather a consequence of the spin foam quantization.
This is consistent with the fact that its classical analogue, the Gauss constraint,
is first class and therefore it is generated simply by integration over the corresponding Lagrange
multiplier. In contrast, the second class constraints, supplied with an appropriate
determinant, must be inserted into the path integral measure from the very beginning
(see discussion in section \ref{sec_constraint}).
At the quantum level the closure constraint is nothing else but the invariance property of the intertwiners
entering the spin foam boundary states. If the basis in this state space is realized by the projected spin networks,
as in \eqref{reprampl}, the intertwiners can be read off from \eqref{projspin}.
Let ${\bf D}_{pq}^{(\lambda_f)}(g_f)$ denote matrix coefficients of an element of the group $G$ in the representation $\lambda_f$.
Then the intertwiner coupling them at the vertex $t$
depends on the normal $x_t$ and can be presented as \cite{Alexandrov:2007pq}
\begin{equation}
N^{(\{\lambda_{i}\},\{j_{i}\},k_t)}_{p_1 \cdots p_4}(x_t)=\mathop{\sum}\limits_{\ell_{j_1}\cdots\ell_{j_4}}
\iota^{\{ j_{i}\} }_{\{ \ell_{j_{i}} \} }(k_t) \prod_{i=1}^4 {\bf D}^{(\lambda_i)}_{p_i \ell_{j_{i}}}({\rm g}_{x_t}).
\label{genBC}
\end{equation}
where the indices $\ell_j$ label the basis of the subspace $H_{\rm SU(2)}^{(j)}$ appearing in the decomposition
of the representation $\lambda$ on the subgroup
\begin{equation}
H_G^{(\lambda)}=\bigoplus\limits_{j} H_{\rm SU(2)}^{(j)}.
\label{decompH}
\end{equation}
It is easy to see that the intertwiner \eqref{genBC} satisfies
\begin{equation}
\mathop{\sum}\limits_{q_{1}\cdots q_{4}}
\(\prod_{i=1}^4 {\bf D}^{(\lambda_i)}_{p_i q_i}({\rm g})\) N^{(\{\lambda_{i}\},\{j_{i}\},k_t)}_{q_1 \cdots q_4}(x_t)
=N^{(\{\lambda_{i}\},\{j_{i}\},k_t)}_{p_1 \cdots p_4}({\rm g}\cdot x_t).
\label{invN}
\end{equation}
The infinitesimal version of this transformation law gives precisely the relaxed closure constraint
\eqref{clconrel}.
It is clear that the usual invariance is spoiled due to the dependence of $x_t$.
An easy way to restore it is to integrate over these normals. This is what usually
done in spin foam models by inserting integrals $\int dx_t$ in the boundary states
\cite{Engle:2007qf,Engle:2007mu,Engle:2007wy}. Of course, this insertion produces $G$-invariant
intertwiners.
Thus, to decide which version of the closure constraint is the relevant one, we have to understand whether or not
one should integrate over the normals to tetrahedra.
For this it is necessary to consider gluing of several simplices.
In terms of the boundary states, the gluing is achieved by integrating over holonomies associated to common triangles.
Let us assume that we do insert the integration over $x_t$ in the simplex boundary states.
Then one immediately runs into two problems.
First, for a tetrahedron shared by two simplices, there will be two integrals over its normal.
This means that one actually introduced two normals to the same tetrahedron, one for each simplex.
This is inconsistent because the normals form a part of the $B$ field which associates unique data
to the elements of the simplicial decomposition.\footnote{The two normals cannot be interpreted as
the same normal seen from different reference frames related to the two simplices. We defined it
without any relation to the reference frame of the simplex which did not appear at all in our discretization.}
Second and may be more important, the integration over $x_t$ is inconsistent with the measure
on $g_f$ induced by the second class constraints. Indeed, as we discussed in the end of section \ref{subsec_sbs},
the presence of $\delta$-function in the measure ${\cal D}^{(x_t)} [g_t]$ leads to some conditions
on $g_f$. As a result, the measure for these group elements will also be non-trivial and in particular
it should depend on the normals to the tetrahedra sharing the triangle $f$.
However, once the integral over $x_t$ has been inserted, one does not have them at our disposal,
the non-trivial measure cannot be written and the gluing becomes impossible.
Thus, we conclude that the integration over $x_t$ in the boundary states is inconsistent with the second class
constraints and the closure constraint must be imposed in its relaxed form \eqref{clconrel}.
Several remarks are in order:
\begin{itemize}
\item
The reasoning we gave above can be applied only in the presence of the second class constraints.
If they are absent, the measure ${\cal D}^{(x_t)} [g_t]$ coincides with the usual one
and the dependence of the normals disappears from the very beginning.
Of course, the normals can be artificially introduced, but there is no way to measure them.
Thus, we are free to insert integrals over $x_t$ and find the standard closure constraint \eqref{clcon}.
This is precisely what happens in the models \cite{Livine:2007vk,Freidel:2007py,Livine:2007ya}
based on coherent states where the secondary constraints are ignored and the measure over
holonomies is taken as the standard Haar measure.
In this case it is independent of the normals so that the integration over $x_t$ and the gluing
are mutually consistent.
\item
There is another way to see why the dependence of the normals should be preserved in the presence of
the second class constraints and is auxiliary in the opposite case.
In canonical theory the vectors $x^I$ are conjugated to a part of the spin connection hidden in the Gauss constraint.
However, as we mentioned in section \ref{subsec_connect}, to quantize the theory one passes to another connection,
either ${\cal A}_i^{IJ}$ or ${\bf A}_i^{IJ}$. Both modified connections commute with $x^I$ \cite{SAcon}.
The reason for this is that they have 3 independent components less comparing to
the spin connection. (They satisfy 9 second class constraints, whereas the spin connection fulfills only 6.)
The missing components are precisely the ones which are conjugated to $x^I$.
Therefore, these unit vectors should be added to the list of configuration variables.
In contrast, if there are no second class constraints, the connection used in holonomy operators
coincides with the usual spin connection and does not commute with $x^I$.
\item
One can notice that even when two simplices are glued together, to obtain the common boundary state,
it is not needed to integrate over the normal to the tetrahedron shared by the simplices.
Indeed, the gluing gives rise to a contribution to the boundary state which can be schematically written
as follows
\begin{equation}
\Psi_{12}\[g_f;x_t\]
=\int \prod_{t_{12}}{\cal D}^{(x^{\hphantom{1}}_{t_{12}})} [\rho^{\hphantom{1}}_{f_{12}}]\,
\Psi_1\[g^{\hphantom{1}}_{f_1},g^{\hphantom{1}}_{f_{12}}\rho^{\hphantom{1}}_{f_{12}};x^{\hphantom{1}}_{t_1},x^{\hphantom{1}}_{t_{12}}\]
\Psi_2\[g^{\hphantom{1}}_{f_2},\rho_{f_{12}}^{-1};x^{\hphantom{1}}_{t_2},x^{\hphantom{1}}_{t_{12}}\],
\label{glue_simplex}
\end{equation}
where $f=(f_1,f_2,f_{12}),t=(t_1,t_2,t_{12})$, the labels 1,2 refer to non-shared faces and tetrahedra of
the corresponding glued simplices and the label 12 marks the shared faces and tetrahedron.
It is easy to see that, similarly to what happens in the vertex amplitude \eqref{amplit_simplex},
the dependence on $x_{t_{12}}$ drops out.
Moreover, an integral over $x_{t_{12}}$ would produce an infinite factor in the Lorentzian case.
Therefore, even in the complete partition function all normals should be kept fixed.
This corresponds to a gauge fixing of boosts in the discretized path integral.
Note that in the models proposed in \cite{Engle:2007qf,Freidel:2007py,Livine:2007ya} this issue does not arise
because all the ingredients, the boundary amplitudes to be glued and the measure to be used
for the gluing, are independent of the normals.
\item
The necessity to use the relaxed closure constraint resolves
a problem with the model proposed in \cite{Freidel:2007py}, which has been found in the work \cite{Engle:2007mu}.
It was noticed that in that model the space of intertwiners remains unrestricted despite the simplicity constraints
have been imposed. This is indeed true, but only when one considers the SO(4) invariant subspace
so that the correct statement is: the gauge-invariant projection of
the span of all intertwiners of \cite{Freidel:2007py} is equal to ${\cal I}^{(\lambda_i)}$, the space
of all SO(4) invariant intertwiners. In our language this originates from the fact that the intertwiners
\eqref{genBC} integrated with respect to $x_t$ form an over-complete basis in ${\cal I}^{(\lambda_i)}$ and hence choosing a subset of
these intertwiners does not necessarily reduce the projection of their span.
On the other hand, if one allows for dependence on the normals to tetrahedra,
the intertwiners satisfy the relaxed closure constraint and span a larger space ${\cal I}^{(\lambda_i)}_{\rm rel}$.
In this case any restriction on the labels reduces to a certain subspace of ${\cal I}^{(\lambda_i)}_{\rm rel}$ and one finds that
the model of \cite{Freidel:2007py} does put some constraints on these {\it covariant} intertwiners.
\item
Finally, we mention that the need for a modification of the relation between bi-vectors and generators of the gauge algebra,
which is closely related to the relaxation of the closure constraint \cite{Alexandrov:2007pq},
was observed in \cite{Oriti:2007vf}.
Besides, the intertwiners satisfying the relaxed closure constraint appeared also in
\cite{Livine:2005tp,Oriti_conf} in the group field theory context as well as in the case of coupling of particles.
Thus, they seem to be quite natural for the spin foam approach.
In this section we argued that in the presence of second class constraints they are actually needed
to describe the correct boundary state space.
\end{itemize}
\section{Example: SU(2) BF theory}
Since we do not know the correct discrete measure incorporating the secondary second class constraints
for gravity \eqref{contA}, our construction is somewhat inexplicit.
To highlight its various aspects, we illustrate it on a simple example, which is however closely related
to the model of \cite{Engle:2007qf} and LQG.
This example serves as a model for quantization based on the connection ${\bf A}_i^{IJ}$ relevant
to the Ashtekar--Barbero approach. In this case the secondary constraints are given by \eqref{su2con}.
They are much easier to discretize comparing to \eqref{contA} because they simply mean that,
for a constant $x^I$, ${\bf A}_i^{IJ}$
has vanishing boost part and its holonomies belong to an SU(2) subgroup.
Working in the Riemannian case, it is easy to write the corresponding measure as
\begin{equation}
{\cal D}^{(x_t)}[g_t ]=\delta\((g^+_{t})^{-1} g^-_{t}\, u_{x_t}\)\, dg_t^+ \, dg_t^-,
\label{measure_su2}
\end{equation}
where $u_{x_t}={\rm g}_{x_t}^-({\rm g}_{x_t}^+)^{-1}\in {\rm SU(2)}$.
It is important to notice that in this case the new version of the simplicity constraints \eqref{newsim}
is inconsistent with the constraints on the connection because it leads to the vanishing
discretized action \eqref{dact}. There is however a similar, but inequivalent way to write the
initial quadratic constraints
\begin{equation}
I_{(P)KL}^{IJ}(x)B^{KL}=0.
\label{newsim2}
\end{equation}
It implies that only the ${\rm SU}_x(2)$ part of the $B$ field is non-trivial and corresponds
to the topological sector of Plebanski formulation where $B^{IJ}=e^I\wedge e^J$.
After discretization, the simplicity constraints can be written as
\begin{equation}
\hat B_f^- = u_{x_{u(f)}}\,\hat B_f^+\, u_{x_{u(f)}}^{-1},
\label{opsim2}
\end{equation}
where we have chosen the ``up" tetrahedron as the reference one (see the end of section \ref{subsec_discr}).
Inserting both second class constraints into the action \eqref{dact}, one finds that it reduces
to the discretized action of SU(2) BF theory. Thus, this example provides a quantization of this simple topological
theory embedded into an SO(4) covariant framework. The fact that we used constraints leading to Ashtekar--Barbero
formulation does not imply of course that its quantization is equivalent to the BF theory.
The reason is twofold. First, the connection ${\bf A}_i^{IJ}$, in contrast with ${\cal A}_i^{IJ}$,
is not equal to the spin connection on the constraint surface. Hence, the
initial action expressed in terms of ${\bf A}_i^{IJ}$ will have a more complicated form than the action of BF theory.
And second, as we mentioned, ${\bf A}_i^{IJ}$ is not really a spacetime connection and therefore
it cannot be used in the spacetime covariant quantization: its path ordered exponentials cannot be interpreted as holonomies.
Despite the triviality of the example, it is instructive to see
how the construction described in the previous sections works.
First, the knowledge of the explicit form of the measure for
holonomies allows to evaluate the vertex amplitude \eqref{amplit_simplex}.
Substituting \eqref{measure_su2}, one obtains
\begin{equation}
A(\lambda_f,j_{tf},k_t)=\int \prod_t dh_t\,\Psi^{(\lambda_f,j_{tf},k_t)}\[({\rm g}^{h_{d(f)}})^{-1}{\rm g}^{h_{u(f)}};x_0\].
\label{amplit_SU2}
\end{equation}
Using the property
\begin{equation}
\mathop{\sum}\limits_{m,m',n,n'}\overline{\ClG{j^+}{j^-}{j_1^{\vphantom{+}}}{m^{\vphantom{+}}}{m'}{\ell_1}}
D^{(j^+)}_{mn^{\vphantom{+}}}(h)D^{(j^-)}_{m'n'}(h) \ClG{j^+}{j^-}{j_2}{n^{\vphantom{+}}}{n'}{\ell_2}
=\delta_{j_1j_2} d_{j_1} D^{(j_1)}_{\ell_1 \ell_2}(h),
\label{property}
\end{equation}
it is easy to realize that the vertex amplitude is given by
\begin{equation}
A(\lambda_f,j_{tf},k_t)=\(\prod_f\delta_{j_{u(f)f}^{\vphantom{+}}j_{d(f)f}^{\vphantom{+}}}\)15J(j_{tf};k_t).
\label{amplit_SU2res}
\end{equation}
Thus, the result indeed coincides with the vertex amplitude of SU(2) BF theory.
Notice that the dependence of the vertex amplitude on
the SO(4) representations $\lambda_f$ completely disappears and the SU(2) representations
$j_{u(f)f}$ and $j_{d(f)f}$ are required to coincide.
The same of course should be true for the boundary states.
The conditions on the labels of the projected spin networks
can be obtained taking into account the constraints on $g_f$ following from
\eqref{simbs_init} and \eqref{measure_su2}.
It follows that these group elements must satisfy
\begin{equation}
g_f^-=u_{x_{u(f)}}\, g^+_{f}\, u_{x_{d(f)}}^{-1}.
\label{congf}
\end{equation}
As a result, the element $G_f$ appearing in \eqref{projspin} belongs to the diagonal SU(2) subgroup
and the same property \eqref{property} can be used to get the restrictions.
The resulting boundary state coincides with the usual SU(2) spin network as it should be for SU(2)
BF theory.
This procedure gives an example of the explicit realization of imposing second class constraints
at the level of the Hilbert space. Moreover, precisely this example was employed in \cite{AlLiv}
to demonstrate that the Lorentz covariant
loop quantization based on ${\bf A}_i^{IJ}$ reproduces the kinematical Hilbert space of LQG.
This is not surprising because the kinematical Hilbert spaces of LQG and SU(2) BF theory are isomorphic.
Notice that if one integrates the simplex boundary state over $x_t$, the reduction to the SU(2) spin networks
would become impossible.
The reason is that the normals appearing in the boundary state and in the constraint \eqref{congf} would be
decoupled: the first one is integrated over and the second is fixed. As a result, they would not cancel and
the argument of the projected spin networks would not reduce to
the diagonal subgroup.\footnote{In \cite{Engle:2007qf} the restrictions on representations were
achieved by weakly imposing the simplicity constraints
on the $B$ field. In our approach the constraints on $B_f$
are already taken into account by the explicit $\delta$-function in the measure. Besides, although
the method of \cite{Engle:2007qf} leads to spin networks with the same set of labels as that of LQG, the states
of the two models are {\it physically} different
because the former depend on SO(4) holonomies and the latter are SU(2) spin networks.}
This shows the importance of keeping the normals $x_t$ as free arguments of the boundary states
and, as a consequence, of the use of the relaxed closure constraint.
\section*{Acknowledgements}
The author is grateful to Kirill Krasnov and Philippe Roche for interesting discussions.
This research is supported by CNRS and by the contract ANR-06-BLAN-0050.
| -30,187.869372 |
[
-3.224609375,
2.939453125
] | 40.249433 |
[
-2.169921875,
0.732421875,
-2.197265625,
-5.6875,
-1.1552734375,
8.140625
] |
[
3.1015625,
9.5234375,
1.8291015625,
6.13671875
] | 322 | 7,078 |
[
-3.59765625,
4.18359375
] | 25.994614 |
[
-6.13671875,
-5.09375,
-5.8984375,
-2.453125,
2.30859375,
14.1953125
] | 1.337477 | 17.905266 | 20.599039 | 1.148025 |
[
1.7373749017715454
] | -20,868.61649 | 5.545776 | -30,269.413493 | 1.401167 | 5.868168 |
[
-2.2109375,
-3.705078125,
-3.6171875,
-4.84765625,
2.05078125,
12.2109375
] |
[
-5.578125,
-2.447265625,
-2.587890625,
-1.357421875,
4.125,
5.265625
] | |
BkiUc-XxK3xgpbERC94a
|
\section{Introduction and summary of results}
Noncommutative geometry permits the semiclassical modelling of quantum physics by classical physics on `quantum' (noncommutative) spaces, whose underlying geometry serves as both generator of and recepticle for quantum corrections; indeed, in much of theoretical physics, this might as well be the operational definition of noncommutative geometry itself. From this perspective, it may be surprising to note that a faithful and more-or-less complete formulation of classical Yang--Mills gauge theory on `quantum' principal bundles does not yet exist. Nonetheless, partial foundations have already been laid by Brzezi\'{n}ski--Majid~\cite{BrM}, who make sense of principal (Ehresmann) connections on quantum principal bundles in terms of suitable first-order differential calculi (\textsc{fodc}) on principal comodule algebras in a manner compatible with the theory of quantum groups and quantum homogeneous spaces.
There are gaps, however, in these partial foundations. Principal connections correspond to gauge potentials---their curvature, which would then encode field strength, is typically only accessed in a piecemeal and somewhat indirect fashion through the curvature of induced module connections on quantum associated vector bundles. More significantly, gauge transformations are generally only defined in the theoretically convenient but geometrically and physically unnatural special case of universal \textsc{fodc}{}. Furthermore, to the author's best knowledge, almost all calculations of affine spaces of principal connections on quantum principal bundles are restricted to this special case; Brzezi\'{n}ski--Gaunt--Schenkel's computations on the \(\theta\)-deformed complex Hopf fibration~\cite{BGS} provide a valuable and instructive exception.
We aim to fill these gaps in a conceptually economical and computationally tractable fashion that readily interfaces with noncommutative Riemannian geometry, whether that mean Connes's functional-analytic theory of spectral triples~\citelist{\cite{Connes95}\cite{Connes96}} or, e.g., noncommutative K\"{a}hler geometry as championed by \'{O} Buachalla~\cite{OB}. We take our cue from \'{C}a\'{c}i\'{c}--Mesland~\cite{CaMe}, who construct a comprehensive theory of principal connections and global gauge transformations on noncommutative Riemannian principal bundles with compact connected Lie structure group in terms of spectral triples. For our purposes, their main formal innovation is to encode principal connections in terms of horizontal covariant derivatives (cf. \DJ{}ur\dj{}evi\'{c}~\cite{Dj97}), which permits a workable notion of gauge transformation acting affinely on principal connections in a manner that faithfully generalises the classical case. There is an apparent drawback to this change of perspective: neither variation of the principal connection nor application of a gauge transformation generally preserves the total \textsc{fodc}{}. However, we shall have reason to embrace this as a feature, not a bug.
We begin in Section~\ref{sec2} by effecting the aforementioned change of perspective at the level of \textsc{fodc}{}. Let \(H\) be a Hopf \(\ast\)-algebra and let \(P\) be a principal left \(H\)-comodule \(\ast\)-subalgebra. Fix a horizontal calculus on \(P\), which therefore consists of a \textsc{fodc}{} \((\Omega^1_B,\dt{B})\) on \(B \coloneqq \coinv{H}{P}\) and a suitable \emph{projectable horizontal lift} \(\Omega^1_{P,\mathrm{hor}} = P \cdot \Omega^1_B \cdot P = P \cdot \Omega^1_B\) of \(\Omega^1_B = \coinv{H}{\Omega^1_{P,\mathrm{hor}}}\). For us, a \emph{gauge transformation} is an \(H\)-covariant \(\ast\)-automorphism \(f\) of \(P\) satisfying \(\rest{f}{B} = \id_B\) (i.e., a \emph{vertical} \(\ast\)-automorphism) that induces via \(\id_{\Omega^1_B}\) an \(H\)-comodule automorphism \(f_{\ast,\mathrm{hor}}\) of \(\Omega^1_{P,\mathrm{hor}}\), while a \emph{gauge potential} is a lift of \(\dt{B} : B \to \Omega^1_B\) to an \(H\)-covariant \(\ast\)-derivation \(P \to \Omega^1_{P,\mathrm{hor}}\). The group \(\fr{G}\) of gauge transformations now acts affine-linearly on the real affine space \(\fr{At}\) of gauge potentials, whose space of translations \(\fr{at}\) consists of \emph{relative gauge potentials}. In turn, we can define the subgroup \(\Inn(\fr{G})\) of inner gauge transformations and the subspace \(\Inn(\fr{at})\) of inner relative gauge potentials and check that \(\Out(\fr{G}) \coloneqq \fr{G}/\Inn(\fr{G})\) still acts affine-linearly on \(\Out(\fr{At}) \coloneqq \fr{At}/\Inn(\fr{at})\).
We now relate these constructions to the standard theory. Fix a bicovariant \textsc{fodc}{} \((\Omega^1_H,\dt{H})\) on \(H\) with respect to which \(P\) is \emph{locally free}. We simultaneously consider all \(H\)-covariant \textsc{fodc}{} \((\Omega^1_P,\dt{P})\) on \(P\) that make \((P;\Omega^1_P,\dt{P})\) into a quantum principal \((H;\Omega^1_H,\dt{H})\)-bundle inducing the horizontal calculus \((\Omega^1_B,\dt{B};\Omega^1_{P,\mathrm{hor}})\) and admitting a \(\ast\)-preserving bimodule connection---for convenience, call such an \textsc{fodc}{} \emph{admissible}. Indeed, let \(\mathcal{G}[\Omega^1_H]\) be the groupoid of \emph{abstract gauge transformations}, whose objects are admissible \textsc{fodc}{} on \(P\) and whose arrows are vertical \(\ast\)-automorphisms of \(P\) that are differentiable with respect to the source \textsc{fodc}{} on the domain and the target \textsc{fodc}{} on the codomain; let \(\mu[\Omega^1_H] : \mathcal{G}[\Omega^1_H] \to \Aut(P)\) be the forgetful homomorphism. Likewise, let \(\mathcal{A}[\Omega^1_H]\) be the set of all triples \((\Omega^1_P,\dt{P};\Pi)\), where \((\Omega^1_P,\dt{P})\) is admissible and where \(\Pi\) is a (necessarily strong) \(\ast\)-preserving bimodule connection on \((P;\Omega^1_P,\dt{P})\). Thus, the classical affine action of gauge transformations on principal Ehresmann connections generalises to the action of \(\mathcal{G}[\Omega^1_H]\) on \(\mathcal{A}[\Omega^1_H]\) by conjugation of bimodule connections. We can now summarise the non-trivial results of Section~\ref{sec2} as follows:
\begin{theorem}
Reconstruction of \textsc{fodc}{} induces an explicit equivalence of groupoids
\[
\Sigma[\Omega^1_H] : \fr{G} \ltimes \fr{At} \to \mathcal{G}[\Omega^1_H] \ltimes \mathcal{A}[\Omega^1_H]
\]
with explicit homotopy inverse manifesting the action groupoid \(\fr{G} \ltimes \fr{At}\) as a deformation retraction of the action groupoid \(\mathcal{G}[\Omega^1_H] \ltimes \mathcal{A}[\Omega^1_H]\). Furthermore:
\begin{enumerate}
\item the forgetful homomorphism \(\mu[\Omega^1_H] : \mathcal{G}[\Omega^1_H] \to \Aut(P)\) maps surjectively onto \(\fr{G}\);
\item the equivalence of groupoids \(\Sigma[\Omega^1_H]\) descends to a groupoid isomorphism \[\fr{G} \ltimes \fr{At}/\fr{at}[\Omega^1_H] \xrightarrow{\sim} \mathcal{G}[\Omega^1_H]/\ker\mu[\Omega^1_H],\] where \(\fr{at}[\Omega^1_H]\) is the \(\fr{G}\)-invariant subspace of \emph{\((\Omega^1_H,\dt{H})\)-adapted} relative gauge potentials.
\end{enumerate}
Hence, in particular, the quotient affine space \(\fr{At}/\fr{at}[\Omega^1_H]\) defines a \(\fr{G}\)-equivariant moduli space of admissible \textsc{fodc}{}.
\end{theorem}
\noindent This justifies our definition of gauge transformation and proves that we can work directly with the group \(\fr{G}\) and real affine space \(\fr{At}\) without any loss of geometrical or physical information, at least to first order.
Next, in Section~\ref{sec3}, we turn to making sense of curvature---field strength---in a conceptually minimalistic fashion. This requires the careful refinement of the constructions and results from Section~\ref{sec2} to the context of second-order differential calculi (\textsc{sodc}{}), i.e., \(\ast\)-differential calculi truncated at degree \(2\). Suppose that we have prolonged our horizontal calculus to a \emph{second-order horizontal calculus} \((\Omega_B,\dt{B};\Omega_{P,\mathrm{hor}})\) on \(P\). A gauge transformation \(f\) is \emph{prolongable} if it extends via \(f_\ast : \Omega^1_{P,\mathrm{hor}} \to \Omega^1_{P,\mathrm{hor}}\) to an \(H\)-covariant \(\ast\)-automorphism of the graded left \(H\)-comodule \(\ast\)-algebra \(\Omega_{P,\mathrm{hor}}\); a gauge potential \(\nabla\) is \emph{prolongable} if it extends via \(\dt{B} : \Omega^1_B \to \Omega^2_B\) and \(0 : \Omega^2_{P,\mathrm{hor}} \to 0\) to an \(H\)-covariant degree \(1\) \(\ast\)-derivation of \(\Omega_{P,\mathrm{hor}}\), in which case its \emph{field strength} is \(\mathbf{F}[\nabla] \coloneqq \rest{\nabla^2}{P}\). The subgroup \(\pr{\fr{G}}\) of prolongable gauge transformations continues to act affine-linearly on the affine subspace \(\pr{\fr{At}}\) of prolongable gauge transformations, whose space of translations \(\pr{\fr{at}}\) consists of \emph{prolongable} relative gauge potentials.
Now, suppose that we have prolonged the bicovariant \textsc{fodc}{} \((\Omega^1_H,\dt{H})\) to a bicovariant \(\ast\)-differential calculus \((\Omega_H,\dt{H})\). We distill a proposal of Beggs--Majid~\cite{BeM}*{\S 5.5} into a notion of \emph{strong second-order quantum principal \((H;\Omega_H,\dt{H})\)-bundle} and propose a compatible notion of \emph{prolongable} \(\ast\)-preserving bimodule connection, which turns out to be related to \DJ{}ur\dj{}evi\'{c}'s notion of multiplicative connection~\cite{Dj97}. We can still define a groupoid \(\mathcal{G}[\Omega^{\leq 2}_H]\) of \emph{prolongable} abstract gauge transformations on \(P\), a set \(\mathcal{A}[\Omega^{\leq 2}_H]\) of prolongable \(\ast\)-preserving bimodule connections, and an action of \(\mathcal{G}[\Omega^{\leq 2}_H]\) on \(\mathcal{A}[\Omega^{\leq 2}_H]\) by conjugation. However, constructing an analogue \(\Sigma[\Omega^{\leq 2}_H]\) of the groupoid equivalence \(\Sigma[\Omega^1_H]\) turns out to be a subtle matter.
As it turns out, we can still reconstruct from \(\Omega_{P,\mathrm{hor}}\) and \((\Omega_H)^{\mathrm{co}H}\) a canonical \(H\)-covariant graded \(\ast\)-algebra \(\Omega_{P,\oplus}\) of total differential forms on \(P\) through degree \(2\) (cf.\ \DJ{}ur\dj{}evi\'{c}~\cite{Dj10}). However, a prolongable gauge potential \(\nabla\) will yield an object \((\Omega_{P,\oplus},\dt{P,\nabla})\) of \(\mathcal{G}[\Omega^{\leq 2}_H]\) if and only if its field strength factors as \(\mathbf{F}[\nabla] = F[\nabla] \circ \dv{P}\), where \((\Omega^1_{P,\mathrm{ver}},\dv{P})\) is the first-order \emph{vertical calculus} of \(P\) induced by \((\Omega^1_H,\dt{H})\) and \(F[\nabla] : \Omega^1_{P,\mathrm{ver}} \to \Omega^2_{P,\mathrm{hor}}\) is a (necessarily unique) \(H\)-covariant morphism of \(P\)-\(\ast\)-bimodules. We say that such a gauge potential \(\nabla\) is \emph{\((\Omega^1_H,\dt{H})\)-adapted}, in which case, we call \(F[\nabla]\) its \emph{curvature \(2\)-form}; we denote the \(\pr{\fr{G}}\)-invariant quadric subset of all \((\Omega^1_H,\dt{H})\)-adapted prolongable gauge potentials by \(\pr{\fr{At}}[\Omega^1_H]\). Reconstruction of \textsc{sodc}{} yields an explicit equivalence of groupoids
\[
\Sigma[\Omega^{\leq 2}_H] : \pr{\fr{G}} \ltimes \pr{\fr{At}}[\Omega^1_H] \to \mathcal{G}[\Omega^{\leq 2}_H] \ltimes \mathcal{A}[\Omega^{\leq 2}_H]
\]
with explicit homotopy inverse manifesting the action groupoid \(\pr{\fr{G}} \ltimes \pr{\fr{At}}[\Omega^1_H]\) as a deformation retraction of the action groupoid \(\mathcal{G}[\Omega^{\leq 2}_H] \ltimes \mathcal{A}[\Omega^{\leq 2}_H]\), so that one can work directly with \(\pr{\fr{G}}\) and \(\pr{\fr{At}}[\Omega^1_H]\) without any loss of geometric or physical information, at least to second order. Once more, it follows that the obvious forgetful homomorphism \(\mu[\Omega^{\leq 2}_H] : \mathcal{G}[\Omega^{\leq 2}_H] \to \Aut(P)\) maps surjectively onto \(\pr{\fr{G}}\), justifying our notion of prolongable gauge potential. However, the construction of a \(\pr{\fr{G}}\)-equivariant moduli space of admissible \textsc{sodc}{} on \(P\) becomes considerably more involved:
\begin{theorem}
Suppose that \((\Omega_H,\dt{H})\) is given through degree \(2\) by the canonical prolongation \`{a} la Woronowicz of \((\Omega^1_H,\dt{H})\). Then \(\Sigma[\Omega^{\leq 2}_H]\) descends to a groupoid isomorphism
\[
\pr{\fr{G}} \ltimes \pr{\fr{At}}[\Omega^1_H]/\pr{\fr{at}}_{\can}[\Omega^{1}_H] \xrightarrow{\sim} \mathcal{G}[\Omega^{\leq 2}_H]/\ker\mu[\Omega^{\leq 2}_H],
\]
where \(\pr{\fr{at}}_{\can}[\Omega^1_H]\) is the \(\pr{\fr{G}}\)-invariant subspace of \emph{canonically \((\Omega^1_H,\dt{H})\)-adapted} prolongable relative gauge potentials. Thus, the \(\pr{\fr{G}}\)-invariant quadric subset \(\pr{\fr{At}}[\Omega^1_H]/\pr{\fr{at}}_{\can}[\Omega^{1}_H]\) of the affine space \(\pr{\fr{At}}/\pr{\fr{at}}_{\can}[\Omega^{1}_H]\) defines a \(\pr{\fr{G}}\)-equivariant moduli space of admissible \textsc{sodc}{}.
\end{theorem}
Next, in Section~\ref{sec4}, we illustrate our definitions and results in the case of crossed product algebras, which one views as trivial quantum principal bundles. As \'{C}a\'{c}i\'{c}--Mesland showed in the context of spectral triples~\cite{CaMe}, the group \(\fr{G}\) of gauge transformations and the affine space \(\fr{At}\) of gauge potentials of a crossed product by \(\mathbf{Z}^n\) \emph{qua} trivial quantum principal \(\mathbf{T}^n\)-bundle can be computed in terms of the degree \(1\) group cohomology of \(\mathbf{Z}^n\) with coefficients in a certain group of unitaries and a certain \(\mathbf{R}[\mathbf{Z}^m]\)-module of noncommutative \(1\)-forms, respectively. Generalising their calculations from the commutative and cocommutative Hopf \(\ast\)-algebra \(\mathbf{C}[\mathbf{Z}^m] \cong \mathcal{O}{}(\mathbf{T}^m)\) to arbitary Hopf \(\ast\)-algebras requires the construction of novel \emph{ad hoc} degree \(1\) cohomology groups, which we term `lazy' on account of their formal resemblance to a construction of Bichon--Carnovale~\cite{BC}. These generalisations reduce appropriately to conventional group cohomology in the case of a group algebra and to Lie algebra cohomology in the case of the universal enveloping algebra of a real Lie algebra.
Let \(H\) be a Hopf \(\ast\)-algebra, let \(B\) be a right \(H\)-module \(\ast\)-algebra, and let \(M\) be an \(H\)-equivariant \(B\)-\(\ast\)-bimodule. On the one hand, we can define the degree \(1\) \emph{lazy \((B,M)\)-valued Sweedler cohomology}
\(
\HS^1_\ell(H;B,M) \coloneqq \ZS^1_\ell(H;B,M)/\BS^1_\ell(H;B,M)
\)
of \(H\), where \(\ZS^1_\ell(H;B,M)\) is the group of \emph{lazy Sweedler \(1\)-cocycles} and \(\BS^1_\ell(H;B,M)\) is the central subgroup of \emph{lazy Sweedler \(1\)-coboundaries}; this generalises Sweedler cohomology~\cite{Sweedler} through degree \(1\) to the case of not-necessarily-cocommutative Hopf \(\ast\)-algebras and non-trivial coefficients. On the other hand, we can refine the degree \(1\) Hoschchild cohomology of \(H\) with coefficients in \(M\) with the given right \(H\)-action and the trivial left \(H\)-action to degree \(1\) \emph{lazy \(M\)-valued Hoschchild cohomology}
\(
\HH^1_\ell(H;M) \coloneqq \ZH^1_\ell(H;M)/\BH^1_\ell(H;M),
\)
where \(\ZH^1_\ell(H;M)\) is the real vector space of \emph{lazy Hochschild \(1\)-cocycles} and \(\BH^1_\ell(H;M)\) is the subspace of \emph{lazy Hochschild \(1\)-coboundaries}. Conjugation with respect to convolution on \(H\) now yields a representation of \(\ZS^1_\ell(H;B,M)\) on \(\ZH^1_\ell(H;M)\) that descends to a representation of \(\HS^1_\ell(H;B,M)\) on \(\HH^1_\ell(H;M)\). Furthermore, any \(H\)-equivariant \(\ast\)-derivation \(\partial : B \to M\) induces a canonical group \(1\)-cocycle \(\MC[\partial] : \ZS^1_\ell(H;B,M) \to \ZH^1_\ell(H;M)\) that descends, in turn, to a group \(1\)-cocycle \(\widetilde{MC}[\partial] : \HS^1_\ell(H;B,M) \to \HH^1_\ell(H;M)\).
We can now exemplify the use of lazy Sweedler and Hochschild cohomology to compute the group \(\fr{G}\), the real affine space \(\fr{At}\), and the affine-linear action of \(\fr{G}\) on \(\fr{At}\) for a trivial quantum principal \(H\)-bundle \(B \rtimes H\); note that none of this structure would be visible if we did not consider all possible admissible \textsc{fodc}{} simultaneously.
\begin{proposition}
Let \(P \coloneqq B \rtimes H\), so that \(B = \coinv{H}{P}\). Suppose that \((\Omega^1_B,\dt{B})\) is a an \(H\)-equivariant \textsc{fodc}{} on \(B\), and endow \(P\) with the horizontal calculus \((\Omega^1_B,\dt{B};\Omega^1_B \rtimes H)\). We have a group isomorphism \(\Op : \ZS^1_\ell(H;B,\Omega^1_B) \to \fr{G}\) and an isomorphism \(\Op : \ZH^1_\ell(H;\Omega^1_B) \xrightarrow{\sim} \fr{At}\) of real affine spaces given, respectively, by
\begin{gather*}
\forall \sigma \in \ZS^1_\ell(H;B,\Omega^1_B), \, \forall h \in H, \, \forall b \in B, \quad \Op(\sigma)(hb) \coloneqq \cm{h}{1}\sigma(\cm{h}{2})b,\\
\forall \mu \in \ZH^1_\ell(H;\Omega^1_B), \, \forall h \in H, \, \forall b \in B, \quad \Op(\mu)(hb) \coloneqq h \cdot \dt{B}(b) + \cm{h}{1} \cdot \mu(\cm{h}{2}) \cdot b;
\end{gather*}
these descend, respectively, to a group isomorphism \(\widetilde{\Op} : \HS^1_\ell(H;B,\Omega^1_B) \xrightarrow{\sim} \Out(\fr{G})\) and an affine-linear isomorphism \(\widetilde{\Op} : \HH^1_\ell(H;\Omega^1_B) \xrightarrow{\sim} \Out(\fr{At})\). Moreover for every lazy Sweedler \(1\)-cocycle \(\sigma \in \ZS^1_\ell(H;B,\Omega^1_B)\) and lazy Hochschild \(1\)-cocycle \(\mu \in \ZH^1_\ell(H;\Omega^1_B)\),
\begin{gather*}
\Op(\sigma) \triangleright \Op(\mu) = \Op(\sigma \triangleright \mu + \MC[\dt{B}](\sigma)),\\
\widetilde{\Op}([\sigma]) \triangleright \widetilde{\Op}([\mu]) = \widetilde{\Op}\mleft([\sigma]\triangleright[\mu]+\widetilde{\MC}[\dt{B}]([\sigma])\mright).
\end{gather*}
\end{proposition}
\noindent We can similarly compute the second-order gauge theory of \(B \rtimes H\) in terms of suitable further refinements of lazy Sweedler and lazy Hochschild cohomology. As we shall show in forthcoming work~\cite{CaTi}, one can similarly analyse general quantum principal \(\Unit(1)\)-bundles with commutative base space \(C^\infty(M)\) in terms of the degree \(1\) group cohomology of \(\mathbf{Z}\) with coefficients in \(C^\infty(M,\Unit(1))\) and \(\Omega^1_{\mathrm{dR}}(M,\ensuremath{\mathrm{i}}{}\mathbf{R})\); although \(\fr{G} \ltimes \fr{At}\) will only depend on \(M\), all other groupoids of interest will turn out to be sensitive to the underlying dynamics.
Finally, in Section~\ref{sec5}, we apply our formalism to the example of the non-trivial quantum principal \(\mathcal{O}{}(\Unit(1))\)-bundle implicit to Manin's `Alterstraum'~\cite{Manin}. Let \(\theta \in \mathbf{R}\) be a quadratic irrationality, and let \(\mathcal{A}_\theta\) be the corresponding smooth noncommutative \(2\)-torus endowed with the canonical \textsc{sodc}{} \((\Omega_{\mathcal{A}_\theta},\dt{\mathcal{A}_\theta})\). Then, by combining results of Schwarz~\cite{Schwarz98}, Dieng--Schwarz~\cite{DiengSchwarz}, Polishchuk--Schwarz~\cite{PolishchukSchwarz}, Po\-li\-shchuk \cite{Polishchuk}, and Vlasenko~\cite{Vlasenko}, we canonically assemble the self-Morita equivalence bimodules among the basic Heisenberg modules over \(\mathcal{A}_\theta\) into a non-cleft principal \(\mathcal{O}{}(\Unit(1))\)-comodule \(\ast\)-algebra \(P\) with base \(\coinv{\mathcal{O}{}(\Unit(1))}{P} = \mathcal{A}_\theta\). Moreover, using results of Poli\-shchuk--Schwarz~\cite{PolishchukSchwarz}, we canonically assemble Connes's constant curvature connections~\cite{Connes80} on the isotypical components into a prolongable gauge potential \(\nabla_0\) on \(P\) with respect to a certain second-order horizontal calculus constructed from \((\Omega_{\mathcal{A}_\theta},\dt{\mathcal{A}_\theta})\). From there, we can show that \(\fr{G} = \pr{\fr{G}} \cong \Unit(1)\), where \(\Unit(1)\) acts as the structure group on \(P\), that \(\Inn(\fr{G}) = \set{1}\), that every relative gauge potential is inner prolongable and given by supercommutation by a \(1\)-form in \(\mathbf{R}^2 \subset \mathcal{A}_{\theta}^{\oplus 2} = \Omega^1_{\mathcal{A}_\theta}\), and that \(\fr{G} = \pr{\fr{G}}\) acts trivially on \(\fr{At} \cong \mathbf{R}^2\). In particular, we can prove the following:
\begin{theorem}
Let \(\epsilon = c_1\theta + d_1\) be the norm-positive fundamental unit of \(\mathbf{Q}[\theta]\), where \(c_1 \in \mathbf{N}\) and \(d_1 \in \mathbf{Z}\) are uniquely determined. Given \(q \in \mathbf{R}^\times\), let \((\Omega^1_q,\dt{q})\) denote the corresponding \(q\)-deformed bicovariant \textsc{fodc}{} on \(\mathcal{O}{}(\Unit(1))\), so that \((\Omega^1_1,\dt{1})\) is the de Rham calculus. Then
\[
\forall q \in \mathbf{R}^\times, \quad \pr{\fr{At}}[\Omega^1_q] = \begin{cases} \fr{At} &\text{if \(q = \epsilon^2\),}\\ \emptyset &\text{else,}\end{cases}
\]
and for every \(\nabla \in \fr{At} = \pr{\fr{At}}[\Omega^1_{\epsilon^2}]\), the curvature \(2\)-form \(F[\nabla]\) is non-zero and given by
\[
F[\nabla](\dt{\epsilon^2}t) = - \ensuremath{\mathrm{i}}{}\epsilon c_1\vol_{\mathcal{A}_\theta},
\]
where \(\dt{\epsilon^2}t \coloneqq \tfrac{1}{2\pi\ensuremath{\mathrm{i}}{}}\dt{\epsilon^2}(z) \cdot z^{-1}\) and where \(\vol_{\mathcal{A}_{\theta}} \coloneqq 1_{\mathcal{A}_\theta} \in \mathcal{A}_{\theta} = \Omega^2_{\mathcal{A}_\theta}\). Furthermore,
\[
\forall q \in \mathbf{R}^\times, \quad \fr{at}[\Omega^1_q] = \pr{\fr{at}}_{\can}[\Omega^{1}_q] = \begin{cases} \fr{at} &\text{if \(q = \epsilon\),}\\ 0 &\text{else,}\end{cases}
\]
so that \(\fr{At}/\fr{at}[\Omega^1_{\epsilon^2}] = \pr{\fr{At}}[\Omega^1_{\epsilon^2}]/\pr{\fr{at}}_{\can}[\Omega^{1}_{\epsilon^2}] = \fr{At} \cong \mathbf{R}^2\).
\end{theorem}
\noindent Thus, when \(q = \epsilon\), we obtain a \(q\)-monopole over \(\mathcal{A}_\theta\) exactly analogous to the \(q\)-monopole constructed by Brzezi\'{n}ski--Majid~\cite{BrM} on the \(q\)-deformed complex Hopf fibration; furthermore, distinct gauge potentials are gauge-inequivalent and yield non-isomorphic admissible \textsc{fodc}{}. Note that the elementary algebraic number theory of the real quadratic irrational \(\theta\) appears again and again in all constructions and calculations.
\addtocontents{toc}{\protect\setcounter{tocdepth}{0}}
\subsection*{Notation and conventions} We shall mostly follow the notation, terminology, and conventions of Beggs--Majid~\cite{BeM} with certain exceptions. In this work, unless otherwise stated, all algebras are unital \(\mathbf{C}\)-algebras and all modules are unital modules.
We shall use Sweedler notation as follows. If \(h\) is an element of a bialgebra \(H\), its coproduct will be denoted by \(\Delta(h) \eqqcolon \cm{h}{1} \otimes \cm{h}{2} \in H \otimes H\). If \(p\) is an element of a left \(H\)-comodule \(P\), we denote its left \(H\)-coaction by \(\delta(p) \eqqcolon \ca{p}{-1} \otimes \ca{p}{0} \in H \otimes P\), and we denote the subspace of left \(H\)-coinvariants of \(P\) by \(\coinv{H}{P}\); similarly, if \(q\) is an element of a right \(H\)-comodule \(Q\), we denote its right \(H\)-coaction by \(\delta(q) \eqqcolon \ca{q}{0} \otimes \ca{q}{1} \in P \otimes H\), and we denote the subspace of right \(H\)-coinvariants of \(Q\) by \(Q^{\mathrm{co}H}\). We shall also use the corresponding higher-order Sweedler notation, e.g.,
\begin{align*}
\forall h \in H, \quad (\id_H \otimes \Delta) \circ \Delta(h) &= (\Delta \otimes \id) \circ \Delta(h) \eqqcolon \cm{h}{1} \otimes \cm{h}{2} \otimes \cm{h}{3},\\
\forall p \in P, \quad (\id \otimes \delta) \circ \delta(p) &= (\Delta \otimes \id) \circ \delta(p) \eqqcolon \ca{p}{-2} \otimes \ca{p}{-1} \otimes \ca{p}{0},\\
\forall q \in Q, \quad (\delta \otimes \id) \circ \delta(q) &= (\id \otimes \Delta) \circ \delta(q) \eqqcolon \ca{q}{0} \otimes \ca{q}{1} \otimes \ca{q}{2}.
\end{align*}
We use the following conventions related to bimodules. Given a \(\ast\)-algebra \(B\), a \emph{\(B\)-\(\ast\)-bimodule} is a \(B\)-bimodule \(M\) with a \(\mathbf{C}\)-antilinear involution \(\ast : M \to M\) satisfying
\[
\forall b_1,b_2 \in B,\,\forall m \in M, \quad (b_1 \cdot m \cdot b_2)^\ast = b_2^\ast \cdot m^\ast \cdot b_1^\ast.
\]
In this case, we set \(M_{\mathrm{sa}} \coloneqq \set{m \in M \given m^\ast = m}\) and define the \emph{centre} of \(M\) with respect to \(B\) by
\(
\Zent_B(M) \coloneqq \set{m \in M \given \forall b \in B,\, b \cdot m = m \cdot b};
\)
moreover, given a subset \(S \subset M\), we define the \emph{centraliser} of \(S\) in \(B\) by
\(
\Cent_B(S) \coloneqq \set{b \in B \given \forall m \in S,\, b \cdot m = m \cdot b}
\). Given a \(\ast\)-algebra \(B\) and a \(B\)-\(\ast\)-module \(M\), we say that a derivation \(\partial : B \to M\) is a \emph{\(\ast\)-derivation} whenever
\[
\forall b \in B, \quad \partial(b^\ast) = -\partial(b)^\ast,
\]
and we denote by \(\Der_B(M)\) the \(\mathbf{R}\)-vector space of all \(\ast\)-derivations \(B \to M\); given \(m \in M\), the resulting \emph{inner} \(\ast\)-derivation \(\ad_m \in \Der_B(M)\) is defined by setting
\[
\forall b \in B, \quad \ad_m(b) \coloneqq [m,b].
\]
Note that Beggs--Majid use the opposite convention, where a \(\ast\)-derivation is \(\ast\)-preserving (i.e., intertwines \(\mathbf{C}\)-antilinear involutions). A \(\ast\)-algebra \(B\) admits an obvious category of \(B\)-\(\ast\)-bimodules, where a morphism is a left and right \(B\)-linear map that is \(\ast\)-preserving.
Finally, let us set the following terminology and conventions related to differential calculi. A \emph{graded \(\ast\)-algebra} is a \(\mathbf{Z}\)-graded algebra \(\Omega = \bigoplus_{k \in \mathbf{Z}} \Omega^k\) with \(\Omega^k = 0\) for \(k < 0\) together with a \(\mathbf{C}\)-linear involution \(\ast : \Omega \to \Omega\) such that \(\ast(\Omega^k) = \Omega^k\) for all \(k \in \mathbf{Z}\) and
\[
\forall m,n \in \mathbf{Z}_{\geq 0}, \, \forall \alpha \in \Omega^m, \, \forall \beta \in \Omega^n, \quad (\alpha \wedge \beta)^\ast = (-1)^{mn} \beta^\ast \wedge \alpha^\ast.
\]
Given a graded \(\ast\)-algebra \(\Omega\) and \(k \in \mathbf{Z}\), a \emph{degree \(k\) \(\ast\)-derivation} is a \(\mathbf{C}\)-linear map \(\partial : \Omega \to \Omega\) satisfying \(\partial(\Omega^m) \subseteq \Omega^{m+k}\) for \(m \in \mathbf{Z}_{\geq 0}\) and
\begin{gather*}
\forall m,n \in \mathbf{Z}_{\geq 0}, \, \forall \alpha \in \Omega^m, \, \forall \beta \in \Omega^n, \quad \partial(\alpha \wedge \beta) = \partial(\alpha) \wedge \beta + (-1)^{km} \alpha \wedge \partial(\beta),\\
\forall \alpha \in \Omega, \quad \partial(\alpha^\ast) = -\partial(\alpha)^\ast.
\end{gather*}
Hence, given a \(\ast\)-algebra \(B\), a \emph{\(\ast\)-differential calculus} on \(B\) is a pair \((\Omega_B,\dt{B})\), where \(\Omega_B\) is a graded \(\ast\)-algebra with \(\Omega^0_B = B\) and \(\dt{B} : \Omega_B \to \Omega_B\) is a degree \(1\) \(\ast\)-derivation with \(\dt{B}^2 = 0\), such that \(\Omega_B\) is generated over \(B\) by \(\dt{B}(B) \subset \Omega^1_B\); in this case, we say that \((\Omega_B,\dt{B})\) is a \emph{prolongation} of the first-order differential calculus (\textsc{fodc}{}) \((\Omega^1_B,\dt{B})\) on \(B\). In particular, a \emph{second-order differential calculus} (\emph{\textsc{sodc}{}}) on a \(\ast\)-algebra \(B\) is a \(\ast\)-differential calculus \((\Omega_B,\dt{B})\) on \(B\), such that \(\Omega_B\) is truncated at degree \(2\), i.e., \(\Omega^k_B = 0\) for \(k > 2\).
\subsection*{Acknowledgements} The author is grateful to Edwin Beggs, Viqar Husain, Will Jagy, Andrey Krutov, R\'{e}amonn \'{O} Buachalla, Matilde Marcolli, Bram Mesland, Pavle Pand\v{z}i\'{c}, Karen Strung, and V.\ Karthik Timmavajjula for useful comments and conversations. This research was supported by NSERC Discovery Grant RGPIN-2017-04249 and by a Harrison McCain Foundation Young Scholar Award.
\addtocontents{toc}{\protect\setcounter{tocdepth}{2}}
\section{Gauge theory to first order}\label{sec2}
In this section, we use explicit groupoid equivalences to [re]formulate the notions of gauge transformation and principal connection on quantum principal bundles in terms of horizontal covariant derivatives, thereby yielding a computationally tractable generalisation of the gauge action on principal connections to quantum principal bundles with general first-order differential calculi (\textsc{fodc}); in the process, we obtain a gauge-equivariant moduli space of all relevant \textsc{fodc}{} inducing the same vertical and horizontal calculi.
\subsection{Deconstruction of quantum principal bundles to first order}\label{firstorderdecon}
Let \(H\) be a Hopf \(\ast\)-algebra over \(\mathbf{C}\), and let \((\Omega^1_H,\dt{H})\) be a bicovariant \textsc{fodc}{} on \(H\). We give a novel review of the theory of quantum principal \(H\)-bundles \`{a} la Brzezi\'{n}ski--Majid~\cite{BrM} incorporating recent insights of Beggs--Majid~\cite{BeM} while recovering earlier insights of \DJ{}ur\dj{}evi\'{c}~\cite{Dj97}. In particular, we shall see how a strong bimodule connection decomposes the total \textsc{fodc} of a quantum principal \(H\)-bundle into the direct sum of independent vertical and horizontal calculi, where the latter can be viewed as a horizontal lift of the induced \textsc{fodc} on the base.
Let us first recall the relevant notion of topological quantum principal \(H\)-bundle.
\begin{definition}[Brzezi\'{n}ski--Hajac~\cite{BH}]
A left \(H\)-comodule \(\ast\)-algebra \(P\) is \emph{principal} if and only if both of the following conditions hold:
\begin{enumerate}
\item the canonical map \(P \otimes P \to H \otimes P\) defined by \(p \otimes \p{p} \mapsto \ca{p}{-1} \otimes \ca{p}{0}\p{p}\) descends to a bijection \(P \otimes_B P \to H \otimes P\);
\item there exists a unital bicovariant map \(\omega : H \to P \otimes P\), such that \(m_P \circ \omega = \epsilon(\cdot) 1_P\), where \(m_P : P \otimes P \to P\) is multiplication in \(P\) and \(\epsilon\) is the counit of \(H\).
\end{enumerate}
\end{definition}
\begin{example}[Schneider~\cite{Schneider}]
If the Hopf \(\ast\)-algebra \(H\) is cosemisimple, e.g., if \(H = \mathcal{O}{}(G)\) for \(G\) a compact Lie group or \(H = \mathbf{C}[\Gamma]\) for \(\Gamma\) a discrete group, then \(P\) is principal if and only if its canonical map is surjective.
\end{example}
\begin{remark}[Baum--De Commer--Hajac~\cite{BDH}]
One can interpret principality of a left \(H\)-co\-module \(\ast\)-algebra \(P\) as topological freeness of the \(H\)-coaction on \(P\) together with vestigial local triviality of the topological quantum principal \(H\)-bundle \(P\) to the extent that quantum associated vector bundles are finitely generated and projective as \(\coinv{H}{P}\)-modules \cite{DGH}*{Cor.\ 2.6}.
\end{remark}
We now consider differentiable quantum principal \(H\)-bundles compatible with the given bicovariant \textsc{fodc} \((\Omega^1_H,\dt{H})\) on \(H\). To this end, we will find it convenient to encode \((\Omega^1_H,\dt{H})\) in terms of the \(\ast\)-closed left \(H\)-subcomodule
\begin{equation}
\Lambda^1_H \coloneqq (\Omega^1_H)^{\operatorname{co}H}
\end{equation}
of right \(H\)-covariant \(1\)-forms and \emph{quantum Maurer--Cartan form} \(\varpi_H : H \to \Lambda^1_H\) given by
\begin{equation}
\forall h \in H, \quad \varpi_H(h) \coloneqq \dt{H}(\cm{h}{1})\cdot S(\cm{h}{2});
\end{equation}
the relevant properties of \(\Lambda^1_H\) and \(\varpi_H\) are given by the following definition.
\begin{definition}
A \emph{left crossed \(H\)-\(\ast\)-module} is a left \(H\)-module and comodule \(V\) over \(\mathbf{C}\) together with a conjugate-linear involution \(\ast : V \to V\), such that
\begin{gather*}
\forall h \in H, \, \forall v \in V, \quad \delta(h \triangleright v) = \cm{h}{1} \ca{v}{-1} S(\cm{h}{3}) \otimes \cm{h}{2} \triangleright \ca{v}{0},\\
\forall h \in H, \, \forall v \in V, \quad (h \triangleright v)^\ast = S(h)^\ast \triangleright v^\ast,\\
\forall v \in V, \quad \delta(v^\ast) = (\ca{v}{-1})^\ast \otimes (\ca{v}{0})^\ast;
\end{gather*}
in this case, a \(V\)-valued \emph{\(1\)-cocycle} is a \(\mathbf{C}\)-linear map \(\varpi : H \to V\) satisfying
\begin{gather*}
\forall h,k \in H, \quad \varpi(hk) = h \triangleright \varpi(k) + \varpi(h)\epsilon(k),\\
\forall h \in H, \quad \varpi(h)^\ast = \varpi(S(h)^\ast),
\end{gather*}
which we call \emph{\(\Ad\)-covariant} whenever it also satisfies
\[
\forall h \in H, \quad \delta(\varpi(h)) = \cm{h}{1}S(\cm{h}{3}) \otimes \varpi(\cm{h}{2}),
\]
\end{definition}
One can now show that \(\Lambda^1_H\) defines a left crossed \(H\)-\(\ast\)-module with respect to the left \emph{adjoint} action of \(H\) given by
\begin{equation}
\forall h \in H, \, \forall \omega \in \Lambda^1_H, \quad h \triangleright \omega \coloneqq \cm{h}{1} \cdot \omega \cdot S(\cm{h}{2})
\end{equation}
and that \(\varpi_H\) defines a surjective \(\Lambda^1_H\)-valued \(\Ad\)-covariant \(1\)-cocycle. Furthermore, one can show that the bicovariant \textsc{fodc} \((\Omega^1_H,\dt{H})\) can be recovered from the data \((\Lambda^1_H,\varpi_H)\) up to isomorphism~\cite{BeM}*{Proof of Thm.\ 2.26}. The proof of this fact, \emph{mutatis mutandis}, permits the following noncommutative generalisation---essentially due to \DJ{}ur\dj{}evi\'{c}---of a locally free action of a connected Lie group and its orbitwise differential calculus.
\begin{definition}[cf.\ \DJ{}ur\dj{}evi\'{c}~\cite{Dj97}*{Lemma 3.1}]
The \emph{vertical calculus} of a left \(H\)-comodule \(\ast\)-algebra \(P\) with respect to the bicovariant \textsc{fodc} \((\Omega^1_H,\dt{H})\) is the pair \((\Omega^1_{P,\mathrm{ver}},\dv{P})\), where:
\begin{enumerate}
\item \(\Omega^1_{P,\mathrm{ver}} \coloneqq \Lambda^1_H \otimes P\) is a left \(H\)-comodule \(P\)-\(\ast\)-bimodule with respect to the left \(H\)-coaction, \(P\)-bimodule structure, and \(\ast\)-structure given respectively by
\begin{gather}
\forall p \in P, \, \forall \omega \in \Lambda^1_H, \quad \delta(\omega \otimes p) \coloneqq \ca{\omega}{-1}\ca{p}{-1} \otimes \ca{\omega}{0} \otimes \ca{p}{0},\\
\forall p,q,\p{q} \in P, \, \forall \omega \in \Lambda^1_H, \quad q \cdot (\omega \otimes p) \cdot \p{q} \coloneqq \ca{q}{-1} \triangleright \omega \otimes \ca{q}{0}p\p{q},\\
\forall \omega \in \Lambda^1_H, \, \forall p \in P, \quad (\omega \otimes p)^\ast \coloneqq \ca{p}{-1}^\ast \triangleright \omega^\ast \otimes \ca{p}{0}^\ast;
\end{gather}
\item \(\dv{P} : P \to \Omega^1_{P,\mathrm{ver}}\) is the left \(H\)-covariant \(\ast\)-derivation defined by
\begin{equation}
\forall p \in P, \quad \dv{P}(p) \coloneqq (\varpi_H \otimes \id) \circ \delta(p) = \varpi_H(\ca{p}{-1}) \otimes \ca{p}{0}.
\end{equation}
\end{enumerate}
We say that the bicovariant \textsc{fodc} \((\Omega^1_H,\dt{H})\) is \emph{locally freeing} for \(P\) whenever
\begin{equation}
\Omega^1_{P,\mathrm{ver}} = P \cdot \dv{P}(P),
\end{equation}
so that \((\Omega^1_{P,\mathrm{ver}},\dv{P})\) defines a left \(H\)-covariant \textsc{fodc} on \(P\).
\end{definition}
\begin{example}
If \(P\) is a principal \(H\)-comodule \(\ast\)-algebra, then the universal \textsc{fodc}{} on \(H\) is locally freeing for \(P\).
\end{example}
\begin{remark}
If \((\Omega^1_H,\dt{H})\) is locally freeing for \(P\), we can interpret \((\Omega^1_{P,\mathrm{ver}},\dv{P})\) as the orbitwise differential calculus of the locally free \(H\)-coaction on \(P\); in particular, we can interpret \(\dv{P} : P \to \Omega^1_{P,\mathrm{ver}}\) as the dualised ``infinitesimal coaction'' of \(H\) on \(P\) with respect to \((\Omega^1_H,\dt{H})\).
\end{remark}
We can now give a suitable formulation of the standard notion of differentiable quantum principal \(H\)-bundle. From now on, let \(P\) be a principal left \(H\)-comodule \(\ast\)-algebra with \(\ast\)-subalgebra of coinvariants \(B \coloneqq \coinv{H}{P}\).
\begin{definition}[Brzezi\'{n}ski--Majid~\cite{BrM}*{Def.\ 4.9}, cf.\ Beggs--Majid~\cite{BeM}*{\S 5.4}]
Let \((\Omega^1_P,\dt{P})\) be a left \(H\)-covariant \textsc{fodc} on \(P\). Then \((P;\Omega^1_P,\dt{P})\) defines a \emph{quantum principal \((H;\Omega^1_H,\dt{H})\)-bundle} if and only if the \emph{vertical map} \(\mathrm{ver}[\dt{P}] : \Omega^1_P \to \Omega^1_{P,\mathrm{ver}}\) given by
\begin{equation}\label{verdef}
\forall p, \p{p} \in P, \quad \mathrm{ver}[\dt{P}]\mleft(p \cdot \dt{P}(\p{p})\mright) \coloneqq p \cdot \dv{P}(\p{p})
\end{equation}
is well-defined and surjective with kernel \(P \cdot \dt{P}(B) \cdot P\), where \(B \coloneqq \coinv{H}{P}\); in this case, we say that \((\Omega^1_P,\dt{P})\) is \emph{\((H;\Omega^1_H,\dt{H})\)-principal}.
\end{definition}
\begin{example}[Brzezi\'{n}ski--Majid~\cite{BrM}*{\S 4}]
Let \((\Omega^1_{H,u},\dt{H,u})\) be the universal \textsc{fodc}{} on \(H\). Then the universal \textsc{fodc}{} on \(P\) is \((H;\Omega^1_{H,u},\dt{H,u})\)-principal.
\end{example}
\begin{remark}[Brzezi\'{n}ski--Majid~\cite{BrM}*{\S 4.1}]
Suppose that \((\Omega^1_P,\dt{P})\) on \(P\) is an \((H;\Omega^1_H,\dt{H})\)-principal \textsc{fodc}{} on \(P\). We view its vertical map \(\mathrm{ver}[\dt{P}] : \Omega^1_{P} \to \Omega^1_{P,\mathrm{ver}}\) as encoding contraction of \(1\)-forms with fundamental vector fields, so that \(\ker\mathrm{ver}[\dt{P}] = P \cdot \dt{P}(B) \cdot P\) is the \(P\)-\(\ast\)-bimodule of horizontal \(1\)-forms; note that \(\ker[\dt{P}]\) is correctly generated as a \(P\)-bimodule by the basic \(1\)-forms \(B \cdot \dt{P}(B)\).
\end{remark}
\begin{remark}\label{locallyfreeremark}
If \(P\) admits an \((H;\Omega^1_H,\dt{H})\)-principal \textsc{fodc}{} \((\Omega^1_P,\dt{P})\), then \((\Omega^1_H,\dt{H})\) is locally freeing for \(P\). Indeed, since \(\dv{P} = \mathrm{ver}[\dt{P}] \circ \dt{P}\), it follows that
\[
\Omega^1_{P,\mathrm{ver}} = \mathrm{ver}[\dt{P}](\Omega^1_P) = \mathrm{ver}[\dt{P}](P \cdot \dt{P}) = P \cdot \dv{P}(P).
\]
Thus, following \DJ{}ur\dj{}evi\'{c}~\cite{Dj97}*{\S 3}, we can also view \(\mathrm{ver}[\dt{P}]\) as encoding restriction of \(1\)-forms to orbitwise \(1\)-forms.
\end{remark}
The following now gives the most commonly used notion of principal connection on a differentiable quantum principal \(H\)-bundle.
\begin{definition}[Brzezi\'{n}ski--Majid~\cite{BrM}*{\S 4.2}, Hajac~\cite{Hajac}*{Def.\ 2.1}, Beggs--Majid~\cite{BeM}*{\S 5.4}]
Suppose that \((\Omega^1_P,\dt{P})\) is an \((H;\Omega^1_H,\dt{H})\)-principal \textsc{fodc} on the principal left \(H\)-comodule \(\ast\)-algebra \(P\). A \emph{connection} on the quantum principal \((H;\Omega^1_H,\dt{H})\)-bundle \((P;\Omega^1_P,\dt{P})\) is a left \(H\)-covariant left \(P\)-linear map \(\Pi : \Omega^1_P \to \Omega^1_P\) satisfying \[\Pi^2 = \Pi, \quad \ker\Pi = \ker\mathrm{ver}[\dt{P}];\] in this case, \(\Pi\) is a \emph{bimodule connection} if and only if it is right \(P\)-linear and \(\ast\)-preserving, and it is \emph{strong} if and only if
\begin{equation}\label{strong}
(\id-\Pi) \circ \dt{P}(P) \subseteq P \cdot \dt{P}(B).
\end{equation}
\end{definition}
\begin{remark}[cf.\ Atiyah~\cite{Atiyah57}]\label{sesremark}
Suppose that \((\Omega^1_P,\dt{P})\) is an \((H;\Omega^1_H,\dt{H})\)-principal \textsc{fodc}{} on \(P\), so that we have a short exact sequence
\begin{equation}\label{ses}
0 \to P \cdot \dt{P}(B) \cdot P \to \Omega^1_P \xrightarrow{\mathrm{ver}[\dt{P}]} \Omega^1_{P,\mathrm{ver}}\to 0
\end{equation}
of left \(H\)-comodule \(P\)-\(\ast\)-bimodules generalising the Atiyah sequence of a smooth principal bundle~\cite{Atiyah57}. Then a connection \(\Pi\) corresponds to a splitting of~\eqref{ses} in the category of left \(H\)-comodule left \(P\)-modules, which is a splitting in the category of left \(H\)-comodule \(P\)-\(\ast\)-bimodules if and only if \(\Pi\) is a bimodule connection. In particular, a connection \(\Pi\) induces the right splitting \((\mathrm{ver}[\dt{P}] \circ \Pi)^{-1}\) and the left splitting \(\id-\Pi\).
\end{remark}
\begin{remark}
Suppose that \((\Omega^1_P,\dt{P})\) is an \((H;\Omega^1_H,\dt{H})\)-principal \textsc{fodc}{} on \(P\). If the quantum principal \((H;\Omega^1_H,\dt{H})\)-bundle \((P;\Omega^1_P,\dt{P})\) admits a strong connection, then every connection on \((P;\Omega^1_P,\dt{P})\) is strong.
\end{remark}
\begin{remark}
Suppose that \((\Omega^1_P,\dt{P})\) is an \((H;\Omega^1_H,\dt{H})\)-principal \textsc{fodc}{} on \(P\). If \(\Pi\) is a connection on \((P;\Omega^1_P,\dt{P})\), then the restriction of \[(\rest{\mathrm{ver}[\dt{P}] \circ \Pi}{\ran\Pi})^{-1} : \Omega^1_{P,\mathrm{ver}} \to \Omega^1_P\] to \(\Lambda^1_H\) is its (absolute) connection \(1\)-form in the sense of Brzezi\'{n}ski--Majid~\cite{BrM}*{Prop.\ 4.10} and \DJ{}ur\dj{}evi\'{c}~\cite{Dj97}*{Def.\ 4.1}. Thus, by a result of Beggs--Majid~\cite{BeM}*{Prop. 5.54}, bimodule connections correspond to regular connections in the sense of \DJ{}ur\dj{}evi\'{c}~\cite{Dj97}*{Def.\ 4.3}.
\end{remark}
The algebraic significance of the strong connection condition \eqref{strong} was already noted by Hajac~\cite{Hajac}, while its functional-analytic significance has recently been demonstrated by \'{C}a\'{c}i\'{c}--Mesland~\cite{CaMe}*{Appx.\ A}. However, a conceptual understanding of this condition has only recently been provided by Beggs--Majid~\cite{BeM}*{\S 5.4.2}: under standard hypotheses, \eqref{strong} is equivalent to requiring that \(\coinv{H}{\ker(\mathrm{ver}[\dt{P}])} = B \cdot \dt{P}(B)\), i.e., that a \(1\)-form is basic if and only if it is horizontal and \(H\)-coinvariant. We shall repeatedly use the following abstract restatement of this reformulation.
\begin{definition}
Recall that \(B \coloneqq \coinv{H}{P}\) for \(P\) a principal left \(H\)-comodule \(\ast\)-subalgebra. Let \(E\) be a \(B\)-\(\ast\)-bimodule. A \emph{horizontal lift} of \(E\) is a pair \((\tilde{E},\iota)\), where \(\tilde{E}\) is a left \(H\)-covariant \(P\)-\(\ast\)-bimodule and \(\iota : E \hookrightarrow \coinv{H}{\tilde{E}}\) is an injective morphism of \(B\)-\(\ast\)-bimodules, such that
\(
\tilde{E} = P \cdot \iota(E) \cdot P
\);
we say that \((\tilde{E},\iota)\) is \emph{projectable} whenever \(\coinv{H}{\tilde{E}} = \iota(E)\).
\end{definition}
\begin{proposition}[Beggs--Majid~\cite{BeM}*{Cor.\ 5.53}]\label{strongprop}
Let \(E\) be a \(B\)-\(\ast\)-bimodule and let \((\tilde{E},\iota)\) be a horizontal lift of \(E\). Then \((\tilde{E},\iota)\) is projectable if and only if \(\tilde{E} = P \cdot \iota(E)\).
\end{proposition}
We now expand upon Beggs--Majid's characterisation of strong connections to reinterpret a strong bimodule connection on a regular quantum principal \((H;\Omega^1_H,\dt{H})\)-bundle as a splitting of the total differential calculus into the direct sum of the vertical calculus and a projectible horizontal lift of the basic calculus.
\begin{proposition}[cf. \DJ{}ur\dj{}evi\'{c}~\cite{Dj97}*{Prop.\ 4.6, Thm.\ 4.12}, Beggs--Majid~\cite{BeM}*{Prop.\ 5.54}]\label{analysisthm}
Suppose that \((\Omega^1_P,\dt{P})\) is an \((H;\Omega^1_H,\dt{H})\)-principal \textsc{fodc} on the principal left \(H\)-comodule \(\ast\)-algebra \(P\), such that quantum principal \((H;\Omega^1_H,\dt{H})\)-bundle \((P;\Omega^1_P,\dt{P})\) admits a bimodule connection. Define the restriction of \((\Omega^1_P,\dt{P})\) to an \textsc{fodc} on \(B \coloneqq \coinv{H}{P}\) by
\begin{equation}
(\Omega^1_B,\dt{B}) \coloneqq (B \cdot \dt{P}(B),\rest{\dt{P}}{B}),
\end{equation}
Then the left \(H\)-covariant \(P\)-\(\ast\)-submodule
\begin{equation}
\Omega^1_{P,\mathrm{hor}} \coloneqq \ker \mathrm{ver}[\dt{P}] = P \cdot \dt{P}(B) \cdot P
\end{equation}
of \(\Omega^1_P\) together with the inclusion \(\Omega^1_B \hookrightarrow \coinv{H}{\Omega^1_{P,\mathrm{hor}}}\) defines a horizontal lift of \(\Omega^1_B\), which is projectable if and only if every bimodule connection on \((P;\Omega^1_P,\dt{P})\) is strong. In this case, for every strong bimodule connection \(\Pi\) on \((P;\Omega^1_P,\dt{P})\):
\begin{enumerate}
\item \(\nabla_\Pi \coloneqq (\id-\Pi) \circ \dt{P} : P \to \Omega^1_{P,\mathrm{hor}}\)
is a left \(H\)-covariant \(\ast\)-derivation satisfying
\[
\Omega^1_{P,\mathrm{hor}} = P \cdot \nabla_\Pi(P), \quad \rest{\nabla_\Pi}{B} = \dt{B};
\]
\item the map \(\psi_\Pi : \Omega^1_P \to \Omega^1_{P,\mathrm{ver}} \oplus \Omega^1_{P,\mathrm{hor}}\) given by
\begin{equation*}
\forall \omega \in \Omega^1_P, \quad \psi_\Pi(\omega) \coloneqq \left(\mathrm{ver}[\dt{P}](\omega),(\id-\Pi)(\omega)\right)
\end{equation*}
defines a left \(H\)-covariant isomorphism of \(P\)-\(\ast\)-bimodules, such that
\begin{equation*}
\forall p \in P, \quad \psi_\Pi \circ \dt{P}(p) = \left(\dv{P}(p),\nabla_\Pi(p)\right).
\end{equation*}
\end{enumerate}
\end{proposition}
\begin{proof}
By Proposition~\ref{strongprop}, \(\Omega^1_{P,\mathrm{hor}}\) is a projectable horizontal lift of \(\Omega^1_B\) if and only if \(\Omega^1_{P,\mathrm{hor}} = P \cdot \Omega^1_B\). On the one hand, if \((P;\Omega^1_P,\dt{P})\) admits a strong connection \(\Pi\), then
\[
\Omega^1_{P,\mathrm{hor}} = (\id-\Pi)(\Omega^1_P) = (\id-\Pi)(P \cdot \dt{P}(P)) = P \cdot (\id-\Pi) \circ \dt{P}(P) = P \cdot P \cdot \dt{P}(B) = P \cdot \Omega^1_B;
\]
on the other hand, if \(\Omega^1_{P,\mathrm{hor}} = P \cdot \Omega^1_B\), then, for every connection \(\Pi\) on \((P;\Omega^1_P,\dt{P})\),
\[
(\id-\Pi) \circ \dt{P}(B) \subset \Omega^1_{P,\mathrm{hor}} = P \cdot \Omega^1_B = P \cdot \dt{P}(B).
\]
Now, suppose that \(\Pi\) is a strong bimodule connection on \((P;\Omega^1_P,\dt{P})\). First, since \(\Pi\) is a left \(H\)-covariant morphism of \(P\)-\(\ast\)-bimodules, \(\nabla_\Pi \coloneqq (\id-\Pi) \circ \dt{P}\) is a left \(H\)-covariant \(\ast\)-derivation; since \(\ker \Pi = \Omega^1_{P,\mathrm{hor}}\) and \(\Pi^2 = \Pi\), it follows that \(\rest{\nabla_\Pi}{B} = \dt{B}\) and
\[
\Omega^1_{P,\mathrm{hor}} = (\id-\Pi)(P \cdot \dt{P}(P)) = P \cdot (\id-\Pi)\circ\dt{P}(P) = P \cdot \nabla_\Pi(P).
\]
Next, in light of Remark~\ref{sesremark}, the map \(\psi_\Pi : \Omega^1_P \to \Omega^1_{P,\mathrm{ver}} \oplus \Omega^1_{P,\mathrm{hor}}\) is simply the isomorphism of left \(H\)-comodule \(P\)-\(\ast\)-bimodules induced by splitting of the short exact sequence~\eqref{ses} of left \(H\)-comodule \(P\)-\(\ast\)-bimodules defined by \(\Pi\) as a bimodule connection. Finally, for all \(p \in P\),
\[
\psi_\Pi \circ \dt{P}(p) = (\mathrm{ver}[\dt{P}] \circ \dt{P}(p),(\id-\Pi) \circ \dt{P}(p)) = (\dv{P}(p),\nabla_\Pi(p)). \qedhere
\]
\end{proof}
Thus, if a quantum principal \((H;\Omega^1_H,\dt{H})\)-bundle \((P;\Omega^1_P,\dt{P})\) admits strong bimodule connection, then a choice of strong bimodule connection \(\Pi\) is equivalent to a decomposition (up to isomorphism) of \((\Omega^1_P,\dt{P})\) as a direct sum of left \(H\)-covariant \textsc{fodc}
\begin{equation}\label{decomposition}
(\Omega^1_P,\dt{P}) \cong (\Omega^1_{P,\mathrm{ver}},\dv{P}) \oplus (\Omega^1_{P,\mathrm{hor}},\nabla_\Pi).
\end{equation}
Here, the left \(H\)-covariant \textsc{fodc} \((\Omega^1_{P,\mathrm{ver}},\dv{P})\) on \(P\) is completely determined by the left \(H\)-coaction on \(P\) and by the bicovariant \textsc{fodc} \((\Omega^1_H,\dt{H})\) on \(H\), while the left \(H\)-covariant \textsc{fodc} \((\Omega^1_{P,\mathrm{hor}},\nabla_\Pi)\) on \(P\) is a projectable horizontal lift of \((\Omega^1_B,\dt{B})\) in the sense that
\[
\left(\coinv{H}{\Omega^1_{P,\mathrm{hor}}},\rest{\nabla_\Pi}{B}\right) = (\Omega^1_B,\dt{B}).
\]
This strongly suggests that variation of the (principal) connection \(\Pi\) can be viewed as tantamount to variation of the lift \(\nabla_\Pi : P \to \Omega^1_{P,\mathrm{hor}}\) of \(\dt{B} : B \to \Omega^1_B\), a change of perspective that we shall explore in the next section and justify in \S\ref{reconsec}.
\subsection{Gauge transformations and gauge potentials}
Let \(H\) be a Hopf \(\ast\)-algebra, and let \(P\) be a principal left \(H\)-comodule \(\ast\)-algebra with algebra of coinvariants \(B \coloneqq \coinv{H}{P}\). We shall now will study the problem of lifting a \textsc{fodc} \((\Omega^1_B,\dt{B})\) on \(B\) to a left \(H\)-covariant \textsc{fodc} \((\Omega^1_{P,\mathrm{hor}},\nabla)\) on \(P\), such that \((\coinv{H}{\Omega^1_{P,\mathrm{hor}}},\rest{\nabla}{B})\) recovers \((\Omega^1_B,\dt{B})\) up to isomorphism. As \DJ{}ur\dj{}evi\'{c} observed~\cite{Dj98}, this suggests encoding principal connections through their induced horizontal covariant derivatives; we shall see that this also yields a tentative definition of gauge transformation for quantum principal bundles with general \textsc{fodc}{} in the spirit of \'{C}a\'{c}i\'{c}--Mesland~\cite{CaMe}*{\S 3}. Our definitions will be rigorously justified in \S\ref{reconsec}.
We begin with the following definition, which encodes a choice of \textsc{fodc}{} on the base \(B\) together with compatible left \(H\)-covariant \(P\)-\(\ast\)-bimodule of horizontal \(1\)-forms.
\begin{definition}[cf.\ \DJ{}ur\dj{}evi\'{c}~\cite{Dj10}*{\S\ 3.1}]
A \emph{(first-order) horizontal calculus} on the principal left \(H\)-co\-module \(\ast\)-algebra \(P\) is a quadruple \((\Omega^1_B,\dt{B};\Omega^1_{P,\mathrm{hor}},\iota)\), where:
\begin{enumerate}
\item \((\Omega^1_B,\dt{B})\) is a (trivially left \(H\)-covariant) \textsc{fodc}{} on \(B \coloneqq \coinv{H}{P}\);
\item \((\Omega^1_{P,\mathrm{hor}},\iota)\) is a projectable horizontal lift of the \(B\)-\(\ast\)-bimodule \(\Omega^1_B\).
\end{enumerate}
\end{definition}
\begin{example}
Let \((\Omega^1_H,\dt{H})\) be a bicovariant \textsc{fodc}{} for \(H\), and suppose that \((\Omega^1_P,\dt{P})\) is an \((H;\Omega^1_H,\dt{H})\)-principal \textsc{fodc}{} on \(P\), such that the quantum principal \((H;\Omega^1_H,\dt{H})\)-bundle \((P;\Omega^1_P,\dt{P})\) admits a strong bimodule connection. Then, by Proposition~\ref{analysisthm}, the triple
\begin{equation}
(\Omega^1_B,\dt{B};\Omega^1_{P,\mathrm{hor}},\iota) \coloneqq \left(B \cdot \dt{P}(B),\rest{\dt{P}}{B};\ker\mathrm{ver}[\dt{P}],\rest{\id_{\Omega^1_P}}{B \cdot \dt{P}(B)}\right)
\end{equation}
defines a first-order horizontal calculus on \(P\), which we view as the \emph{canonical} first-order horizontal calculus induced by the \((H;\Omega^1_H,\dt{H})\)-principal \textsc{fodc}{} \((\Omega^1_P,\dt{P})\).
\end{example}
Assume, therefore, that \(P\) admits a first-order horizontal calculus \((\Omega^1_B,\dt{B};\Omega^1_{P,\mathrm{hor}},\iota)\), which we now fix. Let us also assume that \(\dt{B} : B \to \Omega^1_B\) admits a lift to a left \(H\)-covariant \(\ast\)-derivation \(P \to \Omega^1_{P,\mathrm{hor}}\). To simplify notation, we suppress the inclusion map \(\iota\) and identify \(\Omega^1_B\) with its image in \(\Omega^1_{P,\mathrm{hor}}\); hence, where convenient, we denote the first-order horizontal calculus \((\Omega^1_B,\dt{B};\Omega^1_{P,\mathrm{hor}},\iota)\) by the triple \((\Omega^1_B,\dt{B};\Omega^1_{P,\mathrm{hor}})\).
We begin by considering automorphisms of horizontal lifts; this will permit us to define gauge transformations as automorphisms of \(\Omega^1_{P,\mathrm{hor}}\) \emph{qua} horizontal lift of \(\Omega^1_B\).
\begin{definition}
Let \(E\) be a \(B\)-\(\ast\)-bimodule, where \(B \coloneqq \coinv{H}{P}\), and let \((\tilde{E},\iota)\) be a horizontal lift of \(E\). An \emph{automorphism} of \((\tilde{E},\iota)\) is a pair \((\phi,\phi_\ast)\), where \(\phi : P \to P\) is a left \(H\)-covariant \(\ast\)-automorphism satisfying \(\rest{\phi}{B} = \id_B\) and \(\phi_\ast : \tilde{E} \to \tilde{E}\) is a left \(H\)-covariant \(\ast\)-preserving \(\mathbf{C}\)-linear bijection satisfying \(\phi_\ast \circ \iota = \iota\) and
\begin{equation}
\forall p, \p{p} \in P, \, \forall \eta \in \tilde{E}, \quad \phi_\ast(p \cdot \eta \cdot \p{p}) = \phi(p) \cdot \phi_\ast(\eta) \cdot \phi(\p{p}).
\end{equation}
We denote the group of all automorphisms of \((\tilde{E},\iota)\) by \(\Aut(\tilde{E},\iota)\).
\end{definition}
\begin{remark}\label{asthor}
Suppose that \(E\) is a \(B\)-\(\ast\)-bimodule with horizontal lift \((\tilde{E},\iota)\). Since \(\iota(E)\) generates \(E\) as a \(P\)-bimodule, an automorphism \((\phi,\phi_\ast) \in \Aut(\tilde{E},\iota)\) is uniquely determined by \(\phi\). By mild abuse of notation, we can therefore identify \(\Aut(\tilde{E},\iota)\) with the subgroup of \(\Aut(P)\) consisting of left \(H\)-covariant \(\phi \in \Aut(P)\), such that \(\rest{P}{B} = \id_B\) and the map
\[
\phi_\ast : \tilde{E} \to \tilde{E}, \quad p \cdot \iota(e) \cdot \p{p} \mapsto \phi(p) \cdot \iota(e) \cdot \phi(\p{p})
\]
is well-defined and bijective; with this convention, it follows that
\[
(\phi \mapsto \phi_\ast) : \Aut(\tilde{E},\iota) \to \GL(\tilde{E})
\]
is a group homomorphism.
\end{remark}
If \(B \coloneqq \coinv{H}{P}\) is not central in \(P\) (or even if \(B\) is central in \(P\) but does not centralise the lifted \(B\)-\(\ast\)-bimodule), then the inner automorphisms will form a non-trivial central subgroup of the group of automorphisms of a horizontal lift. More broadly, the significance of inner automorphisms in noncommutative gauge theory has been pointed out by Connes~\cite{Connes96}*{p.\ 165} in the context of the spectral action on spectral triples.
\begin{definition}
Let \(E\) be a \(B\)-\(\ast\)-bimodule, and let \((\tilde{E},\iota)\) be a horizontal lift of \(E\). We say that an automorphism \(\phi \in \Aut(\tilde{E},\iota)\) is \emph{inner} if and only if
\[
\phi = \Ad_v \coloneqq (p \mapsto v p v^\ast), \quad \phi_\ast = \Ad_v \coloneqq (\eta \mapsto v \cdot \eta \cdot v^\ast)
\]
for some \(v \in \Unit(B)\). We denote the subset of all inner automorphisms of \((\tilde{E},\iota)\) by \(\Inn(\tilde{E},\iota)\).
\end{definition}
\begin{proposition}\label{innergaugeprop1}
Let \(E\) be a \(B\)-\(\ast\)-bimodule, and let \((\tilde{E},\iota)\) be a horizontal lift of \(E\). For every \(v \in \Unit(B)\), the automorphism \(\Ad_v\) of \(P\) defines an inner automorphism of \((\tilde{E},\iota)\) if and only if \(v \in \Cent_B(B \oplus E)\). Hence, \(\Inn(\tilde{E},\iota)\) defines a central subgroup of \(\Aut(\tilde{E},\iota)\).
\end{proposition}
\begin{proof}
First, \(\Ad_{v}\) restricts to the identity on \(B\) if and only if \(v \in \Zent(B)\). Now, given \(v \in \Unit(\Zent(B))\), for every \(p,\,\p{p}\in P\) and \(e \in E\),
\begin{align*}
\Ad_v(p \cdot e \cdot \p{p}) - \Ad_v(p) \cdot e \cdot \Ad_v(\p{p}) &= v \cdot (p \cdot e \cdot \p{p}) \cdot v^\ast - vpv^\ast \cdot e \cdot v\p{p}v^\ast\\
&= vp \cdot (e - v^\ast \cdot e \cdot v) \cdot \p{p} v^\ast,
\end{align*}
so that \(\Ad_v \in \Inn(\tilde{E},\iota)\) if and only if \(v \in \Cent_B(\Omega^1_B)\). Thus, in particular, the group homomorphism
\(
(v \mapsto \Ad_v) : \Unit(P) \to \Aut(P)
\)
restricts to a surjection \(\Unit(\Cent_B(B\oplus E)) \twoheadrightarrow \Inn(\tilde{E},\iota)\), so that \(\Inn(\tilde{E},\iota) \leq \Aut(\tilde{E},\iota)\) since \(\Unit(\Cent_B(B\oplus E)) \leq \Unit(P)\).
Let us now show that \(\Inn(\tilde{E},\iota)\) is central. Let \(\phi \in \Aut(\tilde{E},\iota)\) and let \(v \in \Unit(\Cent_B(B\oplus E))\). Then, for all \(p \in P\), since \(\rest{\phi}{B} = \id_B\),
\[
\phi \circ \Ad_v \circ \inv{\phi}(p) = \phi(v \inv{\phi}(p) v^\ast) = v \phi(\inv{\phi}(p)) v^\ast = \Ad_v(p). \qedhere
\]
\end{proof}
At last, we can tentatively define a gauge transformation to be an automorphism of \(\Omega^1_{P,\mathrm{hor}}\) as a horizontal lift of \(\Omega^1_B\) and check its differentiability with respect to any vertical calculus on \(P\). This definition, which is a non-trivial refinement of Brzezi\'{n}ski's notion of vertical automorphism~\cite{Br96}*{\S 5}, will be justified by Proposition~\ref{equiv1} and Corollary~\ref{equiv1cor}.
\begin{definition}
A \emph{gauge transformation} of the principal left \(H\)-comodule \(\ast\)-algebra \(P\) with respect to the horizontal calculus \((\Omega^1_B,\dt{B};\Omega^1_{P,\mathrm{hor}},\iota)\) is an automorphism \(\phi\) of the projectable horizontal lift \((\Omega^1_{P,\mathrm{hor}},\iota)\) of \(\Omega^1_B\); hence, the \emph{gauge group}, \emph{inner gauge group}, and \emph{outer gauge group} of \(P\) with respect to \((\Omega^1_B,\dt{B};\Omega^1_{P,\mathrm{hor}},\iota)\) are given by
\[
\fr{G} \coloneqq \Aut(\Omega^1_{P,\mathrm{hor}},\iota), \quad \Inn(\fr{G}) \coloneqq \Inn(\Omega^1_{P,\mathrm{hor}},\iota), \quad \Out(\fr{G}) \coloneqq \fr{G}/\Inn(\fr{G}),
\]
respectively. When clarity requires, for \(\phi \in \fr{G}\), we will denote \(\phi_\ast\) by \(\phi_{\ast,\mathrm{hor}}\).
\end{definition}
\begin{remark}
Thus, by Proposition~\ref{innergaugeprop1}, the map \((v \mapsto \Ad_v) : \Unit(\Cent_B(B\oplus\Omega^1_B)) \twoheadrightarrow \Inn(\fr{G})\) yields the short exact sequence of Abelian groups
\[
1 \to \Unit(\Cent_B(\Omega^1_B) \cap \Zent(P)) \to \Unit(\Cent_B(B \oplus \Omega^1_B)) \to \Inn(\fr{G}) \to 1,
\]
where \(\Inn(\fr{G})\) is central in \(\fr{G}\).
\end{remark}
We will find it useful to record the following straightforward observation, which guarantees that gauge transformations are differentiable with respect the vertical calculus induced by any locally freeing bicovariant \textsc{fodc}{} on \(H\).
\begin{proposition}\label{astver}
Suppose that \((\Omega^1_H,\dt{H})\) is a bicovariant \textsc{fodc} on \(H\) that is locally freeing for\(P\); let \((\Omega^1_{P,\mathrm{ver}},\dv{P})\) be the resulting vertical calculus. For every \(f \in \fr{G}\), the map
\begin{equation}
f_{\ast,\mathrm{ver}} \coloneqq \id_{\Lambda^1_H} \otimes f
\end{equation}
defines a left \(H\)-covariant \(\ast\)-preserving \(\mathbf{C}\)-linear endomorphism of \(\Omega^1_{P,\mathrm{ver}}\) satisfying
\begin{gather}
\forall p, \p{p} \in P, \, \forall \omega \in \Omega^1_{P,\mathrm{ver}}, \quad f_{\ast,\mathrm{ver}}(p \cdot \omega \cdot \p{p}) = f(p) \cdot f_{\ast,\mathrm{ver}}(\omega) \cdot f(\p{p}),\\
\dv{P} \circ f = f_{\ast,\mathrm{ver}} \circ \dv{P}.
\end{gather}
Furthermore, the map \((f \mapsto f_{\ast,\mathrm{ver}}) : \fr{G} \to \GL(\Omega^1_{P,\mathrm{ver}})\) is a group homomorphism.
\end{proposition}
Having tentatively defined gauge transformations to be symmetries of the projectable horizontal lift \(\Omega^1_{P,\mathrm{hor}}\) of \(\Omega^1_B\), we now tentatively define a gauge potential to be a lift of \(\dt{B} : B \to \Omega^1_B\) to a left \(H\)-covariant \(\ast\)-derivation \(P \to \Omega^1_{P,\mathrm{hor}}\), which, like \DJ{}ur\dj{}evi\'{c}~\cite{Dj98}, we view as the horizontal covariant derivative of a principal connection. The justification for this definition, which builds crucially on the role of (strong bimodule) connections in splitting the noncommutative Atiyah sequence \eqref{ses}, will be provided by Proposition~\ref{equiv1}.
\begin{definition}[cf.\ \DJ{}ur\dj{e}vi\'{c}~\cite{Dj98}*{Def.\ 1}]
A \emph{gauge potential} on the principal left \(H\)-comodule \(\ast\)-algebra \(P\) with respect to the first-order horizontal calculus \((\Omega^1_B,\dt{B};\Omega^1_{P,\mathrm{hor}})\) is a left \(H\)-covariant \(\ast\)-derivation \(\nabla : P \to \Omega^1_{P,\mathrm{hor}}\), such that
\(
\rest{\nabla}{B} = \dt{B}
\);
we define the \emph{Atiyah space} of \(P\) with respect to \((\Omega^1_B,\dt{B};\Omega^1_{P,\mathrm{hor}})\) to be the set \(\fr{At}\) of all gauge potentials.
\end{definition}
\begin{example}
Suppose that the \textsc{fodc}{} \((\Omega^1_B,\dt{B})\) is inner, so that \(\dt{B} = \ad_\alpha\) for some \(1\)-form \(\alpha \in (\Omega^1_B)_{\mathrm{sa}}\). Then \(\nabla \coloneqq \ad_{\iota(\alpha)}\) is a gauge potential on \(P\) with respect to \((\Omega^1_B,\dt{B};\Omega^1_{P,\mathrm{hor}})\).
\end{example}
Our assumption on the horizontal calculus \((\Omega^1_B,\dt{B};\Omega^1_{P,\mathrm{hor}})\) is that \(\fr{At} \neq \emptyset\). Since \(\Omega^1_{P,\mathrm{hor}}\) is projectable as a horizontal lift of \(\Omega^1_B\), it follows that \(\Omega^1_{P,\mathrm{hor}} = P \cdot \Omega^1_B = P \cdot \dt{B}(B)\). Thus for every \(\nabla \in \fr{At}\), the pair \((\Omega^1_{P,\mathrm{hor}},\nabla)\) defines a left \(H\)-covariant \textsc{fodc} on \(P\) satisfying
\[
(\Omega^1_B,\dt{B}) = (\coinv{H}{\Omega^1_{P,\mathrm{hor}}},\rest{\nabla}{\coinv{H}{\Omega^1_{P,\mathrm{hor}}}});
\]
in other words, it defines a projectable horizontal lift of \((\Omega^1_B,\dt{B})\).
It is now straightforward to check that the Atiyah space \(\fr{At}\) is an \(\fr{G}\)-invariant \(\mathbf{R}\)-affine subspace of the \(\mathbf{R}\)-vector space \(\Der_P(\Omega^1_{P,\mathrm{hor}})\) of \(\ast\)-derivations on the \(P\)-\(\ast\)-bimodule \(\Omega^1_{P,\mathrm{hor}}\).
\begin{definition}[cf. \DJ{}ur\dj{}evi\'{c}~\cite{Dj98}*{p.\ 98}]
A \emph{relative gauge potential} on \(P\) with respect to the horizontal calculus \((\Omega^1_B,\dt{B};\Omega^1_{P,\mathrm{hor}})\) is a left \(H\)-covariant \(\ast\)-derivation \(\mathbf{A}: P \to \Omega^1_{P,\mathrm{hor}}\), such that \(\rest{\mathbf{A}}{B} = 0\); we denote by \(\fr{at}\) the \(\mathbf{R}\)-vector space of all relative gauge potentials on \(P\) with respect to \((\Omega^1_B,\dt{B};\Omega^1_{P,\mathrm{hor}})\).
\end{definition}
\begin{proposition}
The Atiyah space \(\fr{At}\) of \(P\) with respect to \((\Omega^1_B,\dt{B};\Omega^1_{P,\mathrm{hor}})\) defines a \(\mathbf{R}\)-affine space modelled on the \(\mathbf{R}\)-vector space \(\fr{at}\) with respect to the subtraction map \(\fr{At} \times \fr{At} \to \fr{at}\) defined by \((\nabla,\nabla^\prime) \mapsto \nabla - \nabla^\prime\). Moreover, \(\fr{At}\) admits an affine action of the gauge group \(\fr{G}\) defined by
\begin{equation}
\forall \phi \in \fr{G}, \, \forall\, \nabla \in \fr{At}, \quad \phi \triangleright \nabla \coloneqq \phi_{\ast} \circ \nabla \circ \phi^{-1},
\end{equation}
whose linear part is the linear action of \(\fr{G}\) on \(\fr{at}\) defined by
\begin{equation}
\forall \phi \in \fr{G}, \, \forall \mathbf{A} \in \fr{at}, \quad \phi \triangleright \mathbf{A} \coloneqq \phi_{\ast} \circ \mathbf{A} \circ \phi^{-1}.
\end{equation}
\end{proposition}
If \(B \coloneqq \coinv{H}{P}\) is not central in \(P\) (or if \(B\) is central in \(P\) but does not centralise \(\Omega^1_{P,\mathrm{hor}}\)), one obtains a non-trivial \(\fr{G}\)-invariant \(\mathbf{R}\)-subspace of inner relative gauge potentials. More broadly, the significance of inner derivations in noncommutative gauge theory has been pointed out by Connes~\cite{Connes96}*{p.\ 166} in the context of the spectral action on spectral triples.
\begin{definition}
A relative gauge potential \(\mathbf{A} \in \fr{at}\) on \(P\) with respect to the horizontal calculus \((\Omega^1_B,\dt{B};\Omega^1_{P,\mathrm{hor}})\) is \emph{inner} whenever \(\mathbf{A} = \ad_\alpha\) for some \(\alpha \in (\Omega^1_B)_{\mathrm{sa}}\); we denote the subspace of all inner relative gauge potentials by \(\Inn(\fr{at})\). Thus, we define the \emph{outer Atiyah space} of \(P\) with respect to \((\Omega^1_B,\dt{B};\Omega^1_{P,\mathrm{hor}})\) to be the affine space \(\Out(\fr{At}) \coloneqq \fr{At}/\Inn(\fr{at})\) with space of translations \(\Out(\fr{at}) \coloneqq \fr{at}/\Inn(\fr{at})\).
\end{definition}
\begin{proposition}\label{innerprop}
A map \(\mathbf{A} : P \to \Omega^1_{P,\mathrm{hor}}\) is an inner relative gauge potential if and only if \(\mathbf{A} = \ad_\alpha\) for some \(\alpha \in \Zent_B(\Omega^1_B)_{\mathrm{sa}}\). Thus, the map \((\alpha \mapsto \ad_\alpha) : Z_B(\Omega^1_B)_{\mathrm{sa}} \to \Inn(\fr{at})\) yields the short exact sequence
\begin{equation}\label{innerpropeq}
0 \to \Zent_B(\Omega^1_B)_{\mathrm{sa}} \cap \Zent_P(\Omega^1_{P,\mathrm{hor}}) \to \Zent_B(\Omega^1_B)_{\mathrm{sa}} \to \Inn(\fr{at}) \to 0.
\end{equation}
Moreover, the subspace \(\Inn(\fr{at})\) of inner relative gauge potentials consists of \(\fr{G}\)-invariant vectors, so that the affine action of \(\fr{G}\) on \(\fr{At}\) descends to an affine action of \(\fr{G}\) on \(\Out(\fr{At})\).
\end{proposition}
\begin{proof}
First, given \(\alpha \in (\Omega^1_B)_{\mathrm{sa}}\), the left \(H\)-covariant \(\ast\)-derivation \(\mathbf{A} \coloneqq \ad_{\alpha}\) defines a relative gauge potential if and only if \(\ker \mathbf{A} \supseteq B\), if and only if \(\alpha \in \Zent_B(\Omega^1_B)\); this immediately yields the short exact sequence \eqref{innerpropeq}.
\end{proof}
In fact, it turns out that the affine action of the gauge group \(\fr{G}\) on the Atiyah space \(\fr{At}\) descends further to an affine action of the outer gauge group \(\Out(\fr{G})\) on the outer Atiyah space \(\Out(\fr{At})\), which, as we shall see, will yield a non-trivial invariant of the principal left \(H\)-comodule \(\ast\)-algebra \(P\) endowed with the horiziontal calculus \((\Omega^1_{B},\dt{B};\Omega^1_{P,\mathrm{hor}})\).
\begin{proposition}\label{inducedprop}
The inner gauge group \(\Inn(\fr{G})\) acts trivially on \(\Out(\fr{at})\), so that the induced action of \(\fr{G}\) on \(\Out(\fr{At})\) descends further to an affine action of \(\Out(\fr{G})\) on \(\Out(\fr{At})\).
\end{proposition}
\begin{proof}
Let \(\phi \in \Inn(\fr{G})\), so that by Proposition~\ref{innergaugeprop1}, \(\phi = \Ad_v\) and \(\phi_\ast = \Ad_v\) for some unitary \(v \in \Unit(Z(B) \cap \Cent_B(\Omega^1_B))\); let \(\nabla \in \fr{At}\). Then, for all \(p \in P\),
\begin{align*}
(\phi \triangleright \nabla - \nabla)(p) &= v \cdot \nabla(v^\ast p v) \cdot v^\ast - \nabla(p)\\
&= v \cdot \left(\dt{B}(v^\ast) \cdot pv + v^\ast \cdot \nabla(p) \cdot v + v^\ast p \cdot \dt{B}(v)\right) \cdot v^\ast - \nabla(p)\\
&= [-v^\ast \cdot \dt{B}(v),p]
\end{align*}
so that \(\phi \triangleright \nabla - \nabla = \ad_{-v^\ast \dt{B}(v)} \in \Inn(\fr{at})\). Hence, \(\Inn(\fr{G})\) acts trivially on \(\Out(\fr{At})\).
\end{proof}
\subsection{Reconstruction of quantum principal bundles to first order}\label{reconsec}
Let \(H\) be a Hopf \(\ast\)-algebra and let \(P\) be a principal left \(H\)-comodule \(\ast\)-algebra with \(\ast\)-subalgebra of coinvariants \(B \coloneqq \coinv{H}{P}\). Given a horizontal calculus \((\Omega^1_B,\dt{B};\Omega^1_{P,\mathrm{hor}})\) for \(P\) and a bicovariant \textsc{fodc}{} \((\Omega^1_H,\dt{H})\) on \(H\) that is locally freeing for \(P\), we consider \emph{all} \((H;\Omega^1_H,\dt{H})\)-principal \textsc{fodc}{} on \(P\) inducing the first-order horizontal calculus \((\Omega^1_B,\dt{B};\Omega^1_{P,\mathrm{hor}})\). This will allow us to justify our notions of gauge transformation, gauge potential, and gauge action relative to the theory of Brzezi\'{n}ski--Majid~\cite{BrM} by means of a conceptually transparent equivalence of relevant groupoids. Furthermore, this equivalence will yield a gauge-equivariant affine moduli space of \((H;\Omega^1_H,\dt{H})\)-principal \textsc{fodc}{} on \(P\) inducing \((\Omega^1_B,\dt{B};\Omega^1_{P,\mathrm{hor}})\). For relevant definitions from the basic theory of groupoids, see Appendix~\ref{Groupoids}.
From now on, fix a horizontal calculus \((\Omega^1_B,\dt{B};\Omega^1_{P,\mathrm{hor}},\iota)\) on the principal left \(H\)-co\-mod\-ule \(\ast\)-algebra \(P\). Given a bicovariant \textsc{fodc}{} \((\Omega^1_H,\dt{H})\) on \(H\) that is locally freeing for \(P\), we construct a groupoid whose objects are \((H;\Omega^1_H,\dt{H})\)-principal \textsc{fodc}{} on \(P\) inducing the horizontal calculus \((\Omega^1_B,\dt{B};\Omega^1_{P,\mathrm{hor}},\iota)\) on \(P\) and whose arrows will give an abstract notion of gauge transformation adapted to general \textsc{fodc}{}; note that we do not require an abstract gauge transformation to be differentiable with respect to the same \textsc{fodc}{} on \(P\) as both domain and codomain. In what follows, let \((\Omega^1_{P,\mathrm{ver}},\dv{P})\) denote the vertical calculus on \(P\) induced by a locally freeing bicovariant \textsc{fodc}{} \((\Omega^1_H,\dt{H})\).
\begin{definition}\label{qpbdef}
Let \((\Omega^1_H,\dt{H})\) be a bicovariant \textsc{fodc}{} on \(H\) that is locally freeing for \(P\). We define the \emph{abstract gauge groupoid} with respect to \((\Omega^1_H,\dt{H})\) to be the groupoid \(\mathcal{G}[\Omega^1_H]\) defined as follows:
\begin{enumerate}
\item an object is an \((H;\Omega^1_H,\dt{H})\)-principal \textsc{fodc}{} \((\Omega^1_P,\dt{P})\) on \(P\), such that the quantum principal \((H;\Omega^1_H,\dt{H})\)-bundle \((P;\Omega^1_P,\dt{P})\) admits bimodule connections, and
\[
(\ker\mathrm{ver}[\dt{P}], \dt{B}(b) \mapsto \dt{P}(b))
\]
defines a horizontal lift of \(\Omega^1_B\) admitting a (necessarily unique) left \(H\)-covariant isomorphism
\(
C[\Omega^1_P] : P \cdot \dt{B}(P) \to \Omega^1_{P,\mathrm{hor}}
\)
of \(P\)-\(\ast\)-bimodules satisfying \begin{equation}C[\Omega^1_P] \circ \rest{\dt{P}}{B} = \iota \circ \dt{B};\end{equation}
\item given objects \((\Omega_1,\dt{1})\) and \((\Omega_2,\dt{2})\), an arrow \(f : (\Omega_1,\dt{1}) \to (\Omega_2,\dt{2})\) consists of a left \(H\)-covariant \(\ast\)-automorphism \(f : P \to P\), such that \(\rest{f}{B} = \id\), and such that
\begin{equation}
f_\ast : \Omega_1 \to \Omega_2, \quad p \cdot \dt{1}(\p{p}) \cdot \pp{p} \mapsto f(p) \cdot \dt{2}(f(\p{p})) \cdot f(\pp{p})
\end{equation}
is a well-defined bijection;
\item composition of arrows is induced by composition of \(\ast\)-automorphisms of \(P\), and the identity of an object \((\Omega,\dt{})\) is given by \(\id_{(\Omega,\dt{})} \coloneqq \left(\id_P : (\Omega,\dt{}) \to (\Omega,\dt{})\right)\).
\end{enumerate}
Moreover, we define the star-injective homomorphism \(\mu[\Omega^1_H] : \mathcal{G}[\Omega^1_H] \to \Aut(P)\) by
\begin{equation}
\forall \left(f : (\Omega_1,\dt{1}) \to (\Omega_2,\dt{2})\right) \in \mathcal{G}[\Omega^1_H], \quad \mu[\Omega^1_H]\mleft(f : (\Omega_1,\dt{1}) \to (\Omega_2,\dt{2})\mright) \coloneqq f.
\end{equation}
\end{definition}
\begin{remark}
Thus, the canonical horizontal calculus of an object \((\Omega_P,\dt{P})\) of \(\mathcal{G}[\Omega^{1}_H]\) induces the given horizontal calculus \((\Omega^1_B,\dt{B};\Omega^1_{P,\mathrm{hor}})\) up to the canonical isomorphism \(C[\Omega_P]\) of left \(H\)-covariant \(P\)-\(\ast\)-bimodules.
\end{remark}
\begin{remark}
Suppose that \(f : (\Omega_1,\dt{1}) \to (\Omega_2,\dt{2})\) is an arrow in \(\mathcal{G}[\Omega^1_H]\). Then Proposition~\ref{astver} applies unchanged to the vertical automorphism \(f\), i.e., there exists a unique left \(H\)-covariant \(\ast\)-preserving bijection \(f_{\ast,\mathrm{ver}} : \Omega^1_{P,\mathrm{ver}} \to \Omega^1_{P,\mathrm{ver}}\), such that
\[
\forall p, q \in P, \, \forall \omega \in \Omega^1_{P,\mathrm{ver}}, \quad f_{\ast,\mathrm{ver}}(p \cdot \omega \cdot q) = f(p) \cdot f_{\ast,\mathrm{ver}}(\omega) \cdot f(q)
\]
and \(f_{\ast,\mathrm{ver}} \circ \dv{P} = \dv{P} \circ f_{\ast,\mathrm{ver}}\). A straightforward calculation now shows that
\[
\mathrm{ver}[\dt{2}] \circ f_\ast = f_{\ast,\mathrm{ver}} \circ \mathrm{ver}[\dt{1}].
\]
\end{remark}
In keeping with our stated philosophy, given a bicovariant \textsc{fodc}{} \((\Omega^1_H,\dt{H})\) on \(H\), we consider bimodule connections on all quantum principal \((H;\Omega^1_H,\dt{H})\)-bundles induced from \(P\) by \textsc{fodc}{} in \(\Ob(\mathcal{G}[\Omega^1_H])\). It is straightforward to check that the abstract gauge groupoid \(\mathcal{G}[\Omega^1_H]\) admits a canonical action on this set of bimodule connections.
\begin{propositiondefinition}
Let \((\Omega^1_H,\dt{H})\) be a bicovariant \textsc{fodc}{} on \(H\) that is locally freeing for \(P\). Let \(\mathcal{A}[\Omega^1_H]\) to be the set of all triples \((\Omega^1_P,\dt{P};\Pi)\), where \((\Omega^1_P,\dt{P}) \in \Ob(\mathcal{G}[\Omega^1_H])\) and \(\Pi\) is a bimodule connection on the quantum principal \((\Omega^1_H,\dt{H})\)-bundle \((P;\Omega^1_P,\dt{P})\); hence, let \(p[\Omega^1_H]:\mathcal{A}[\Omega^1_H] \to \Ob(\mathcal{G}[\Omega^1_H])\) be the canonical surjection given by
\[
\forall (\Omega^1_P,\dt{P};\Pi) \in \mathcal{A}[\Omega^1_H], \quad p[\Omega^1_H](\Omega^1_P,\dt{P};\Pi) \coloneqq (\Omega^1_P,\dt{P}).
\]
Then the \emph{abstract gauge action} is the action of \(\mathcal{G}[\Omega^1_H]\) on \(\mathcal{A}[\Omega^1_H]\) via \(p[\Omega^1_H]\) defined by
\begin{multline}
\forall \left(f : (\Omega_1,\dt{1}) \to (\Omega_2,\dt{2})\right) \in \mathcal{G}[\Omega^1_H], \, \forall (\Omega_1,\dt{1};\Pi) \in p[\Omega^1_H]^{-1}(\Omega_1,\dt{1}),\\ \left(f : (\Omega_1,\dt{1}) \to (\Omega_2,\dt{2})\right) \triangleright (\Omega_1,\dt{1};\Pi) \coloneqq (\Omega_2,\dt{2};f_\ast \circ \Pi \circ f_\ast^{-1}).
\end{multline}
Hence, the canonical covering \(\pi[\Omega^1_H] : \mathcal{G}[\Omega^1_H] \ltimes \mathcal{A}[\Omega^1_H] \to \mathcal{G}[\Omega^1_H]\) is given by
\begin{multline}
\forall \left(\left(f : (\Omega_1,\dt{1}) \to (\Omega_2,\dt{2})\right),(\Omega_1,\dt{1};\Pi)\right) \in \mathcal{G}[\Omega^1_H] \ltimes \mathcal{A}[\Omega^1_H], \\ \pi[\Omega^1_H]\mleft(\left(f : (\Omega_1,\dt{1}) \to (\Omega_2,\dt{2})\right),(\Omega_1,\dt{1};\Pi)\mright) \coloneqq \left(f : (\Omega_1,\dt{1}) \to (\Omega_2,\dt{2})\right).
\end{multline}
\end{propositiondefinition}
As a convenient abuse of notation, we will denote an arrow \[\left(\left(f : (\Omega_1,\dt{1}) \to (\Omega_2,\dt{2})\right),(\Omega_1,\dt{1};\Pi)\right)\] of the action groupoid \(\mathcal{G}[\Omega^1_H] \ltimes \mathcal{A}[\Omega^1_H]\) by
\[
f: (\Omega_1,\dt{1};\Pi_1) \to (\Omega_2,\dt{2};\Pi_2),
\]
where \(\Pi_2 \coloneqq f_\ast \circ \Pi_1 \circ f_\ast^{-1}\), so that, in particular
\[
\pi[\Omega^1_H]\mleft(f: (\Omega_1,\dt{1};\Pi_1) \to (\Omega_2,\dt{2};\Pi_2)\mright) \coloneqq \left(f : (\Omega_1,\dt{1}) \to (\Omega_2,\dt{2})\right).
\]
\begin{example}
Let \(\Omega^1_{P,\oplus} \coloneqq \Omega^1_{P,\mathrm{ver}} \oplus \Omega^1_{P,\mathrm{hor}}\), and let \(\Pi_\oplus : \Omega^1_{P,\oplus} \to \Omega^1_{P,\oplus}\) denote the projection onto \(\Omega^1_{P,\mathrm{ver}}\) along \(\Omega^1_{P,\mathrm{hor}}\). For every gauge potential \(\nabla \in \fr{At}\) on \(P\) with respect to \((\Omega^1_B,\dt{B};\Omega^1_{P,\mathrm{hor}})\), the left \(H\)-covariant \(\ast\)-derivation \(\dt{P,\nabla} : P \to \Omega^1_{P,\oplus}\) given by
\begin{equation}
\forall p \in P, \quad \dt{P,\nabla}(p) \coloneqq \left(\dv{P}(p),\nabla(p)\right)
\end{equation}
makes \((\Omega^1_{P,\oplus},\dt{P,\nabla};\Pi_{\oplus})\) into an element of \(\mathcal{A}[\Omega^1_H]\), such that \(\nabla_{\Pi_\oplus} = \nabla\); in particular, the resulting vertical map \(\mathrm{ver}[\dt{P,\nabla}]\) is simply the projection onto \(\Omega^1_{P,\mathrm{ver}}\) along \(\Omega^1_{P,\mathrm{hor}}\).
\end{example}
Let \(\fr{G}\) and \(\fr{At}\) respectively denote the gauge group and Atiyah space of the principal left \(H\)-comodule \(\ast\)-algebra \(P\) with respect to the horizontal calculus \((\Omega^1_B,\dt{B};\Omega^1_{P,\mathrm{hor}})\), where, by the usual abuse of notation, we suppress the inclusion \(\iota : \Omega^1_B \hookrightarrow \Omega^1_{P,\mathrm{hor}}\). We now promote the decomposition \eqref{decomposition} given by Proposition~\ref{analysisthm} to an explicit equivalence of categories that realises the action groupoid \(\fr{G} \ltimes \fr{At}\), which is independent of the choice of bicovariant \textsc{fodc}{} \((\Omega^1_H,\dt{H})\) on \(H\), as a deformation retraction of the action groupoid \(\mathcal{G}[\Omega^1_H] \ltimes \mathcal{A}[\Omega^1_H]\) of the abstract gauge action. This rigorously justifies identifying the action of the gauge group \(\fr{G}\) on the Atiyah space \(\fr{At}\) as the affine action of global gauge transformations on principal connections for the quantum principal \(H\)-bundle \(P\) with respect to the bicovariant \textsc{fodc}{} \((\Omega^1_H,\dt{H})\) on \(H\) and the horizontal calculus \((\Omega^1_B,\dt{B};\Omega^1_{P,\mathrm{hor}})\).
\begin{proposition}\label{equiv1}
Let \((\Omega^1_H,\dt{H})\) be a bicovariant \textsc{fodc}{} on \(H\) that is locally freeing for \(P\). The groupoid homomorphism \(\Sigma[\Omega^1_H] : \fr{G} \ltimes \fr{At} \to \mathcal{G}[\Omega^1_H] \ltimes \mathcal{A}[\Omega^1_H]\)
given by
\begin{equation}
\forall (\phi,\nabla) \in \fr{G} \ltimes \fr{At}, \quad \Sigma[\Omega^1_H](\phi, \nabla) \coloneqq \left(\phi : (\Omega^1_{P,\oplus},\dt{P,\nabla};\Pi_{\oplus}) \to (\Omega^1_{P,\oplus},\dt{P,\phi\triangleright\nabla};\Pi_{\oplus})\right)
\end{equation}
is an equivalence of groupoids with left inverse and homotopy inverse \(A[\Omega^1_H]\) given by
\begin{multline}
\forall \left(f: (\Omega_1,\dt{1};\Pi_1) \to (\Omega_2,\dt{2};\Pi_2)\right) \in \mathcal{G}[\Omega^1_H] \ltimes \mathcal{A}[\Omega^1_H], \\
A[\Omega^1_H]\mleft(f: (\Omega_1,\dt{1};\Pi_1) \to (\Omega_2,\dt{2};\Pi_2)\mright) \coloneqq \left(f, C[\Omega_1] \circ (\id-\Pi_1) \circ \dt{1}\right).
\end{multline}
In particular, there exists a homotopy \(\eta[\Omega^1_H] : \id_{\mathcal{G}[\Omega^1_H] \ltimes \mathcal{A}[\Omega^1_H]} \Rightarrow \Sigma[\Omega^1_H] \circ A[\Omega^1_H]\), which is necessarily unique, such that
\begin{equation}\label{natiso}
\forall (\Omega^1_P,\dt{P};\Pi) \in \mathcal{A}[\Omega^1_H], \quad \mu[\Omega^1_H] \circ \pi[\Omega^1_H]\mleft(\eta[\Omega^1_H]_{(\Omega^1_P,\dt{P};\Pi)}\mright) = \id_P.
\end{equation}
\end{proposition}
\begin{proof}
First, let us check that \(\Sigma[\Omega^1_H]\) is well-defined. Let \((\phi,\nabla) \in \fr{G} \ltimes \fr{At}\). Then, by Remark~\ref{asthor} and Proposition~\ref{astver}, for all \(p, \p{p}, \pp{p} \in P\),
\begin{align*}
\phi(p) \cdot \dt{P,\phi \triangleright \nabla}(\phi(\p{p})) \cdot \phi(\pp{p}) &= \phi(p) \cdot \left(\dv{P}(\phi(\p{p})),(\phi \triangleright \nabla)(\phi(\p{p}))\right) \cdot \phi(\pp{p})\\
&=\phi(p) \cdot \left(\phi_{\ast,\mathrm{ver}}(\dv{P}(\p{p})),\phi_{\ast,\mathrm{hor}}(\nabla(\p{p}))\right) \cdot \phi(\pp{p})\\
&= \left(\phi_{\ast,\mathrm{ver}}(p \cdot \dv{P}(\p{p}) \cdot \pp{p}),\phi_{\ast,\mathrm{hor}}(p \cdot \nabla(\p{p}) \cdot \pp{p})\right)\\
&= (\phi_{\ast,\mathrm{ver}}\oplus \phi_{\ast,\mathrm{hor}})\mleft(p \cdot \dt{P,\nabla}(\p{p}) \cdot \pp{p}\mright)
\end{align*}
so that \(\phi_\ast = \phi_{\ast,\mathrm{ver}} \oplus \phi_{\ast,\mathrm{hor}}\) is a well-defined bijection that satisfies
\begin{gather*}
\phi_{\ast,\mathrm{ver}} \circ \mathrm{ver}[\dt{P,\nabla}] = \phi_{\ast,\mathrm{ver}} \circ \Proj_1 = \Proj_1 \circ (\phi_{\ast,\mathrm{ver}}\oplus\phi_{\ast,\mathrm{hor}}) = \mathrm{ver}[\dt{P,\phi \triangleright \nabla}] \circ \phi_\ast, \\ \phi_\ast \circ \Pi_\oplus = (\phi_{\ast,\mathrm{ver}}\oplus\phi_{\ast,\mathrm{hor}}) \circ \Pi_\oplus = \Pi_\oplus \circ (\phi_{\ast,\mathrm{ver}}\oplus\phi_{\ast,\mathrm{hor}}) = \Pi_\oplus \circ \phi_\ast.
\end{gather*}
Hence, \(\phi : (\Omega^1_{P,\oplus},\dt{P,\nabla};\Pi_{\oplus}) \to (\Omega^1_{P,\oplus},\dt{P,\phi\triangleright\nabla};\Pi_{\oplus})\) is an arrow in \(\mathcal{G}[\Omega^1_H] \ltimes \mathcal{A}[\Omega^1_H]\). Thus, \(\Sigma[\Omega^1_H]\) is well-defined as a function between sets of arrows. That \(\Sigma[\Omega^1_H]\) is a groupoid homomorphism now follows from the fact that both of
\begin{gather*}
(\phi \mapsto \phi_{\ast,\mathrm{ver}}) : \fr{G} \to \GL(\Omega^1_{P,\mathrm{ver}}), \quad (\phi \mapsto \phi_{\ast,\mathrm{hor}}) : \fr{G} \to \GL(\Omega^1_{P,\mathrm{hor}})
\end{gather*}
are group homomorphisms.
Next, let us check that \(A[\Omega^1_H]\) is well-defined. Let \(f: (\Omega_1,\dt{1};\Pi_1) \to (\Omega_2,\dt{2};\Pi_2)\) be an arrow in \(\mathcal{G}[\Omega^1_H] \ltimes \mathcal{A}[\Omega^1_H]\), so that \(f_\ast : \Omega_1 \to \Omega_2\) is a well-defined bijection satisfying
\[
\mathrm{ver}[\dt{2}] \circ f_\ast = f_{\ast,\mathrm{ver}} \circ \mathrm{ver}[\dt{1}], \quad \Pi_2 \circ f_\ast = f_\ast \circ \Pi_2.
\]
For \(i=1,2\), let \(\nabla_i \coloneqq C[\Omega_i] \circ (\id-\Pi_i) \circ \dt{P} \in \fr{At}\); we need to show that \(f \in \fr{G}\) and that \(\nabla_2 = f \triangleright \nabla_1\). Now, for every \(p,\p{p} \in P\) and \(b \in B\),
\begin{align*}
f(p) \cdot \dt{B}(b) \cdot f(\p{p}) &= C[\Omega_2]\mleft(f(p) \cdot \dt{2}(b) \cdot f(\p{p})\mright)\\
&= C[\Omega_2] \circ f_\ast \mleft(p \cdot \dt{1}(b) \cdot \p{p}\mright)\\
&= C[\Omega_2] \circ f_\ast \circ C[\Omega_1]^{-1}\mleft(p \cdot \dt{B}(b) \cdot \p{p}\mright),
\end{align*}
so that \(f_{\ast,\mathrm{hor}} = C[\Omega_2] \circ f_\ast \circ C[\Omega_1]^{-1}\) is well-defined and bijective, hence \(f \in \fr{G}\), with
\begin{multline*}
f \triangleright \nabla_1 = f_{\ast,\mathrm{hor}} \circ \nabla_1 \circ f^{-1} = C[\Omega_2] \circ f_\ast \circ C[\Omega_1]^{-1} \circ C[\Omega_1] \circ (\id-\Pi_1) \circ \dt{1} \circ f^{-1}\\
= C[\Omega_2] \circ f_\ast \circ (\id-\Pi_1) \circ \dt{1} \circ f^{-1} = C[\Omega_2] \circ (\id-\Pi_2) \circ \dt{2} = \nabla_2.
\end{multline*}
As a result, \((f,\nabla_1) : \nabla_1 \to \nabla_2\) is a well-defined arrow in \(\fr{G} \ltimes \fr{At}\). Thus, \(A[\Omega^1_H]\) is well-defined as a function between sets of arrows. That \(A[\Omega^1_H]\) is a groupoid homomorphism now follows since \((\phi \mapsto \phi_{\ast,\mathrm{hor}}) : \fr{G} \to \GL(\Omega^1_{P,\mathrm{hor}})\) is a group homomorphism.
On the one hand, \(C[\Omega^1_{P,\oplus}] = \id_{\Omega^1_{P,\mathrm{hor}}}\) by uniqueness of \(C[\Omega^1_{P,\oplus}]\); hence, it follows that \(A[\Omega^1_H] \circ \Sigma[\Omega^1_H] = \id_{\fr{G}\ltimes\fr{At}}\). On the other hand, by the proof of Proposition~\ref{analysisthm}, \emph{mutatis mutandis}, we can define a homotopy \(\eta[\Omega^1_H] : \id_{\mathcal{G}[\Omega^1_H] \ltimes \mathcal{A}[\Omega^1_H]} \Rightarrow \Sigma[\Omega^1_H] \circ A[\Omega^1_H]\) by
\begin{multline*}
\forall (\Omega^1_P,\dt{P};\Pi) \in \mathcal{A}[\Omega^1_H], \\ \eta[\Omega^1_H]_{(\Omega^1_P,\dt{P};\Pi)} \coloneqq \left(\id_P : (\Omega^1_P,\dt{P};\Pi) \to A[\Omega^1_H] \circ \Sigma[\Omega^1_H](\Omega^1_P,\dt{P};\Pi)\right),
\end{multline*}
which, by construction, satisfies~\eqref{natiso}.
\end{proof}
In particular, this justifies the identification of \(\fr{G}\) as the group of global gauge transformations on the quantum principal \(H\)-bundle \(P\) with respect to the bicovariant \textsc{fodc}{} \((\Omega^1_H,\dt{H})\) on \(H\) and horizontal calculus \((\Omega^1_B,\dt{B};\Omega^1_{P,\mathrm{hor}})\).
\begin{corollary}\label{equiv1cor}
Let \((\Omega^1_H,\dt{H})\) be a bicovariant \textsc{fodc}{} on \(H\) that is locally freeing for \(P\). The range of the star-injective groupoid homomorphism \(\mu[\Omega^1_H] : \mathcal{G}[\Omega^1_H] \to \Aut(P)\) is \(\fr{G}\), so that, after restriction of codomain,
\[
\mu[\Omega^1_H] : \mathcal{G}[\Omega^1_H] \to \fr{G}, \quad \mu[\Omega^1_H] \circ \pi[\Omega^1_H] : \mathcal{G}[\Omega^1_H] \ltimes \mathcal{A}[\Omega^1_H] \to \fr{G}
\]
both define coverings of groupoids.
\end{corollary}
Given a bicovariant \textsc{fodc}{} \((\Omega^1_H,\dt{H})\) on \(H\), we now use the groupoid equivalence \(\Sigma[\Omega^1_H]\) of Proposition~\ref{equiv1} to construct a \(\fr{G}\)-equivariant moduli space of \((H;\Omega^1_H,\dt{H})\)-principal \textsc{fodc}{} on \(P\) inducing the horizontal calculus \((\Omega^1_B,\dt{B};\Omega^1_{P,\mathrm{hor}})\). Indeed, this moduli space will turn out to be a quotient of the Atiyah space \(\fr{At}\) by the the space of all relative gauge potentials of the following form.
\begin{definition}
Let \((\Omega^1_H,\dt{H})\) be a bicovariant \textsc{fodc} on \(H\) that is locally freeing for \(P\). We say that a relative gauge potential \(\mathbf{A} \in \fr{at}\) for \(P\) with respect to \((\Omega^1_B,\dt{B};\Omega^1_{P,\mathrm{hor}})\) is \emph{\((\Omega^1_H,\dt{H})\)-adapted} whenever
\begin{equation}
\mathbf{A} = \omega[\mathbf{A}] \circ \dv{P}
\end{equation}
for some (necessarily unique) left \(H\)-covariant morphism \(\omega[\mathbf{A}] : \Omega^1_{P,\mathrm{ver}} \to \Omega^1_{P,\mathrm{hor}}\) of \(P\)-\(\ast\)-bimodules; in this case, we call \(\omega[\mathbf{A}]\) the \emph{relative connection \(1\)-form} of \(\mathbf{A}\). We denote by \(\fr{at}[\Omega^1_H]\) the subspace of all \((\Omega^1_H,\dt{H})\)-adapted relative gauge potentials on \(P\) with respect to the horizontal calculus \((\Omega^1_B,\dt{B};\Omega^1_{P,\mathrm{hor}})\).
\end{definition}
\begin{remark}[cf.\ Brzezi\'{n}ski--Majid~\cite{BrM}*{\S 4.2}, \DJ{}ur\dj{}evi\'{c}~\cite{Dj97}*{\S 4}, Beggs--Majid~\cite{BeM}*{Prop.\ 5.54}]
Using uniqueness, one can now check that the map
\[
\omega : \fr{at}[\Omega^1_H] \to \Hom_{P}(\Omega^1_{P,\mathrm{ver}},\Omega^1_{P,\mathrm{hor}}), \quad \mathbf{A} \mapsto \omega[\mathbf{A}]
\]
is \(\mathbf{R}\)-linear; moreover, given \(\mathbf{A} \in \fr{at}[\Omega^1_H]\), the relative connection \(1\)-form \(\omega[\mathbf{A}]\) is completely determined by its restriction to a map \(\Lambda^1_H \cong \Lambda^1_H \otimes 1_P \to \Omega^1_{P,\mathrm{hor}}\), which can be viewed as a noncommutative Lie-valued \(1\)-form.
\end{remark}
Now, let \((\Omega_1,\dt{1}),(\Omega_2,\dt{2})\in\Ob(\mathcal{G}[\Omega^1_H])\), where \((\Omega^1_H,\dt{H})\) is a locally freeing bicovariant \textsc{fodc}{} on \(H\). Observe that \((\Omega_1,\dt{1})\) and \((\Omega_2,\dt{2})\) admit an isomorphism of left \(H\)-covariant \textsc{fodc} for \(P\) if and only if \(\id_P : (\Omega_1,\dt{1}) \to (\Omega_2,\dt{2})\) is an arrow in \(\mathcal{G}[\Omega^1_H]\). Since the subgroupoid of all such arrows is precisely \(\ker\mu[\Omega^1_H]\), it follows that \((\Omega_1,\dt{1})\) and \((\Omega_2,\dt{2})\) are isomorphic if and only if they define the same object in the quotient groupoid \(\mathcal{G}[\Omega^1_H]/\ker\mu[\Omega^1_H]\), which will turn out to be well-defined and canonically isomorphic to \(\fr{G} \ltimes \fr{At}/\fr{at}[\Omega^1_H]\). Thus, the quotient affine space \(\fr{At}/\fr{at}[\Omega^1_H]\) yields the desired \(\fr{G}\)-equivariant affine moduli space of \((\Omega^1_H,\dt{H})\)-principal \textsc{fodc}{} on \(P\) inducing \((\Omega^1_B,\dt{B};\Omega^1_{P,\mathrm{hor}})\).
\begin{theorem}\label{firstorderclassification}
Let \((\Omega^1_H,\dt{H})\) be a bicovariant \textsc{fodc}{} on \(H\). Suppose that the Atiyah space \(\fr{At}\) is non-empty. The subspace \(\fr{at}[\Omega^1_H]\) is \(\fr{G}\)-invariant; the subgroupoid \(\ker\mu[\Omega^1_H]\) of \(\mathcal{G}[\Omega^1_H]\) is wide and has trivial isotropy groups, so that the quotient groupoid \(\mathcal{G}[\Omega^1_H]/\ker\mu[\Omega^1_H]\) is well-defined; and there exists a unique groupoid isomorphism \[\tilde{\Sigma}[\Omega^1_H] : \fr{G} \ltimes \left(\fr{At}/\fr{at}[\Omega^1_H]\right) \xrightarrow{\sim} \mathcal{G}[\Omega^1_H]/\ker\mu[\Omega^1_H],\] such that
\begin{equation}
\forall (\phi,\nabla) \in \fr{G} \ltimes \fr{At}, \quad \tilde{\Sigma}[\Omega^1_H](\phi,\nabla+\fr{at}[\Omega^1_H]) = \left[\pi[\Omega^1_H] \circ \Sigma[\Omega^1_H](\phi,\nabla)\right]_{\ker\mu[\Omega^1_H]}.
\end{equation}
\end{theorem}
\begin{proof}
Let us first show that \(\fr{at}[\Omega^1_H]\) is \(\fr{G}\)-invariant. Let \(\mathbf{A} \in \fr{at}[\Omega^1_H]\), so that \(\mathbf{A} = \omega[\mathbf{A}] \circ \dt{P,\mathrm{ver}}\). Then, for any \(\phi \in \fr{G}\),
\[
\phi \triangleright \mathbf{A} = \phi_{\ast,\mathrm{hor}} \circ \omega[\mathbf{A}] \circ \dt{P,\mathrm{ver}} \circ \phi^{-1} = \left(\phi_{\ast} \circ \omega[\mathbf{A}] \circ \phi^{-1}_{\ast,\mathrm{ver}}\right) \circ \dt{P,\mathrm{ver}},
\]
where, by properties of \(\phi_{\ast}\) and \(\phi^{-1}\), the map \(\phi_{\ast,\mathrm{hor}} \circ \omega[\mathbf{A}] \circ \phi_{\ast,\mathrm{ver}}^{-1} : \Omega^1_{P,\mathrm{ver}} \to \Omega^1_{P,\mathrm{hor}}\) remains a left \(H\)-covariant morphism of \(P\)-\(\ast\)-bimodules, so that \(\phi \triangleright \mathbf{A} \in \fr{at}[\Omega^1_H]\), with \[\omega[\phi\triangleright\mathbf{A}] = \phi_{\ast,\mathrm{hor}} \circ \omega[\mathbf{A}] \circ \phi_{\ast,\mathrm{ver}}^{-1}.\]
Now, by Proposition~\ref{equiv1}, the surjective covering \(\pi[\Omega^1_H] : \mathcal{G}[\Omega^1_H] \ltimes \mathcal{A}[\Omega^1_H] \to \mathcal{G}[\Omega^1_H]\), the star-injective groupoid homomorphism \(\mu[\Omega^1_H] : \mathcal{G}[\Omega^1_H] \to \Aut(P)\), the injective group\-oid homomorphism \(\Sigma[\Omega^1_H] : \fr{G} \ltimes \fr{At} \to \mathcal{G}[\Omega^1_H] \ltimes \mathcal{A}[\Omega^1_H]\), and the left inverse \(A[\Omega^1_H]\) of \(\Sigma[\Omega^1_H]\) combine to satisfy the hypotheses of Lemma~\ref{groupoidlemma}. Thus, the subgroupoid \(\ker\mu[\Omega^1_H]\) is wide and has trivial isotropy groups, the quotient groupoid \(\mathcal{G}[\Omega^1_H]/\ker\mu[\Omega^1_H]\) is well-defined, the equivalence kernel \(\sim\) of the map
\[
\left(\nabla \mapsto \left[\pi[\Omega^1_H] \circ \Sigma[\Omega^1_H](\id_P,\nabla)\right]_{\ker\mu[\Omega^1_H]}\right) : \fr{At} \to \Ob(\mathcal{G}[\Omega^1_H]/\ker\mu[\Omega^1_H])
\]
is a \(\fr{G}\)-invariant equivalence relation on \(\fr{At}\), and there exists a unique groupoid isomorphism \(\tilde{\Sigma}[\Omega^1_H] : \fr{G} \ltimes \fr{At}/\sim \xrightarrow{\sim} \mathcal{G}[\Omega^1_H]/\ker\mu[\Omega^1_H]\), such that
\[
\forall (\phi,\nabla) \in \fr{G} \ltimes \fr{At}, \quad \tilde{\Sigma}[\Omega^1_H](\phi,[\nabla]_\sim) = \left[\Sigma[\Omega^1_H](\phi,\nabla)\right]_{\ker\mu[\Omega^1_H]}.
\]
Thus, it remains to show that \(\sim\) is the orbit equivalence with respect to the translation action of \(\fr{at}[\Omega^1_H] \leq \fr{at}\) on \(\fr{At}\).
On the one hand, suppose that \(\nabla_1,\nabla_2 \in \fr{At}\) satisfy \(\nabla_1 \sim \nabla_2\), so that \(\id_P\) defines an arrow
\[
\left(\id_P : (\Omega^1_{P,\oplus},\dt{P,\nabla_1}) \to (\Omega^1_{P,\oplus},\dt{P,\nabla_2})\right) \in \ker\mu[\Omega^1_H] \subset \mathcal{G}[\Omega^1_H].
\]
Thus, \((\id_P)_\ast : \Omega^1_{P,\oplus} \to \Omega^1_{P,\oplus}\) is a left \(H\)-covariant automorphism of the \(P\)-\(\ast\)-bimodule \(\Omega^1_{P,\oplus}\), such that \(\dt{P,\nabla_2} = (\id_P)_{\ast} \circ \dt{P,\nabla_1}\) and
\begin{gather*}
\Proj_1 \circ (\id_P)_\ast = \mathrm{ver}[\dt{P,\nabla_2}] \circ (\id_P)_{\ast} = (\id_P)_{\ast,\mathrm{ver}} \circ \mathrm{ver}[\dt{P,\nabla_1}] = \Proj_1.
\end{gather*}
Let \(\mathcal{N} \coloneqq (\id_P)_\ast - \id\). Now, for all \(p \in P\),
\[
0 = \dv{P}(p) - \dv{P}(p) = \Proj_1 \circ \dt{P,\nabla_2} - \Proj_1 \circ \dt{P,\nabla_1} = \Proj_1 \circ \mathcal{N} \circ \dt{P,\nabla_1},
\]
so that \(\ran \mathcal{N} \subset \Omega^1_{P,\mathrm{hor}}\), while for all \(p \in P\) and \(b \in B\),
\[
(\id_P)_\ast(p \cdot \dt{B}(b)) = p \cdot \dt{B}(b) = \id(p \cdot \dt{B}(b)),
\]
so that \(\Omega^1_{P,\mathrm{hor}} \subset \ker \mathcal{N}\), and hence \(\mathcal{N} \circ \Pi_\oplus = \mathcal{N}\). Thus, for all \(p \in P\),
\[
(\nabla_2-\nabla_1)(p) = \Proj_2 \circ (\dt{P,\nabla_2}-\dt{P,\nabla,1}) (p) = \Proj_2 \circ \mathcal{N} \circ \Pi_{\oplus} \circ \dt{P,\nabla_1} = \Proj_2 \circ \mathcal{N} \circ \dv{P}(p),
\]
so that \(\nabla_2 - \nabla_1 \in \fr{at}[\Omega^1_H]\) with \(\omega[\nabla_2-\nabla_1] = \rest{\Proj_2 \circ \mathcal{N}\,}{\Omega^1_{P,\mathrm{ver}}}\).
On the other hand, suppose that \(\nabla_1,\nabla_2 \in \fr{At}\) satisfy \(\nabla_2 - \nabla_1 \in \fr{at}[\Omega^1_H]\). Let \[N \coloneqq \omega[\nabla_2-\nabla_1],\] and observe that \((N \circ \Pi_{\oplus})^2 = 0\). Then, for all \(p, \p{p},\pp{p} \in P\),
\begin{align*}
p \cdot \dt{P,\nabla_2}(\p{p}) \cdot \pp{p} &= p \cdot \left(\dv{P}(\p{p}),\nabla_2(\p{p})\right) \cdot \pp{p} \\
&= p \cdot \left(\dv{P}(\p{p}),\nabla_1(\p{p}) + N \circ \dv{P}(\p{p}) \right) \cdot \pp{p}\\
&= p \cdot \left(\dt{P,\nabla_1}(\p{p}) + N \circ \Pi_{\oplus} \circ \dt{P,\nabla_1}(\p{p})\right) \cdot \pp{p}\\
&= \left(\id + (N \circ \Pi_{\oplus})\right)\mleft(p \cdot \dt{P,\nabla_1}(\p{p}) \cdot \pp{p}\mright),
\end{align*}
so that \((\id_P)_\ast = \id + (N \circ \Pi_{\oplus})\) is a well-defined bijection satisfying
\[
\mathrm{ver}[\dt{P,\nabla_2}] \circ (\id_P)_\ast = \Proj_1 \circ \left(\id + (N \circ \Pi_{\oplus})\right) = \Proj_1 = (\id_P)_{\ast,\mathrm{ver}} \circ \mathrm{ver}[\dt{P,\nabla_1}].
\]
Hence, \(\id_P : (\Omega^1_{P,\oplus},\dt{P,\nabla_1}) \to (\Omega^1_{P,\oplus},\dt{P,\nabla_2})\) defines an arrow in \(\ker\mu[\Omega^1_H] \subset \mathcal{G}[\Omega^1_H]\).
\end{proof}
\begin{remark}[cf.\ Zucca~\cite{Zucca}*{\S 8.4}]
Let \((\Omega^1_P,\dt{P}) \in \Ob(\mathcal{G}[\Omega^1_H])\). We can view the stabilizer subgroup of
\(
\tilde{\Sigma}[\Omega^1_H]^{-1}([(\Omega^1_P,\dt{P})]_{\ker\mu[\Omega^1_H]})
\)
in \(\fr{G}\) as the gauge group of the quantum principal \((H;\Omega^1_H,\dt{H})\)-bundle \((P;\Omega^1_P,\dt{P})\). Moreover, \(\Sigma[\Omega^1_H]\) restricts to a gauge-equivariant bijection from \(\fr{at}[\Omega^1_H]\) to the set of all strong bimodule connections on \((P;\Omega^1_P,\dt{P})\).
\end{remark}
\section{Gauge theory to second order}\label{sec3}
Our goal for this section is to develop and justify a conceptually economical notion of curvature of a principal connection on a quantum principal bundle that only involves differential calculus to second order and remains simultaneously compatible with both the theory of quantum principal \(H\)-bundles and strong bimodule connections \`{a} la Brzezi\'{n}ski--Majid and our notions of gauge transformations and (relative) gauge potentials. Our guiding principle will be prolongability of principal \textsc{fodc}{} to second order in a manner compatible with decomposition into vertical and horizontal calculi.
\subsection{Deconstruction of quantum principal bundles to second order}
We begin by prolonging the standard theory of quantum principal \(H\)-bundles and strong bimodule connections \`{a} la Brzezi\'{n}ski--Majid to second order in the spirit of Beggs--Brzezi\'{n}ski~\cite{BB} and Beggs--Majid~\cite{BeM}*{\S 5.5}; in the process, we recover certain insights of \DJ{}ur\dj{}evi\'{c}~\cite{Dj97}*{\S 4}.
Let \((\Omega^1_H,\dt{H})\) be a bicovariant \textsc{fodc} on \(H\) with corresponding left crossed \(H\)-\(\ast\)-module \(\Lambda^1_H\) of right coinvariant \(1\)-forms and quantum Maurer--Cartan form \(\varpi_H\); let \((\Omega_H,\dt{H})\) be a prolongation of \((\Omega^1_H,\dt{H})\) to a bicovariant \(\ast\)-differential calculus on \(H\). We wish to consider differentiable quantum principal \(H\)-bundles compatible with \((\Omega_H,\dt{H})\) through degree \(2\). To this end, we will find it convenient to encode \((\Omega_H,\dt{H})\) in terms of the graded left crossed \(H\)-module \(\ast\)-algebra of right \(H\)-coinvariant forms
\begin{equation}
\Lambda_H \coloneqq (\Omega_H)^{\operatorname{co}H}
\end{equation}
and the restricted differential \(\rest{\dt{H}}{\Lambda_H} : \Lambda_H \to \Lambda_H\). By results of Majid--Tao~\cite{MT}*{Prop.\ 3.3, 3.4}, it follows that \(\rest{\dt{H}}{
\Lambda^1_H} : \Lambda^1_H \to \Lambda^2_H\) is given by the Maurer--Cartan equation
\begin{equation}
\forall h \in H, \quad \dt{H}(\varpi_H(h)) \coloneqq \varpi_H(\cm{h}{1}) \wedge \varpi_H(\cm{h}{2}).
\end{equation}
and satisfies
\begin{multline}\label{deg2eq}
\forall h,k \in H, \\ \dt{H}\mleft(h \triangleright \varpi_H(k)\mright) - h \triangleright \dt{H}\varpi(k) = \cm{h}{1} \triangleright \varpi_H(k) \wedge \varpi_H(\cm{h}{2}) + \varpi_H(\cm{h}{1}) \wedge \cm{h}{2}\triangleright\varpi_H(k).
\end{multline}
The proof that \(\Omega_H\) can be recovered from \(\Lambda_H\) up to isomorphism~\cite{MT}*{Prop.\ 3.3}, \emph{mutatis mutandis}, lets us make the following definition, which we shall use to reconstruct graded \(\ast\)-algebras of (total) differential forms on a quantum principal \(H\)-bundle from relevant graded \(\ast\)-algebras of vertical and horizontal forms, respectively. Note that we shall only be concerned with \(\ast\)-differential calculi and graded \(\ast\)-algebras through degree \(2\).
\begin{definition}[cf.\ \DJ{}ur\dj{}evi\'{c}~\cite{Dj97}*{Eqq.\ 4.49, 4.50}]
Let \(\Omega\) be a graded left \(H\)-comodule \(\ast\)-algebra. We define the graded left \(H\)-comodule \(\ast\)-algebra \(\Lambda_H \widehat{\otimes}^{\leq 2} \Omega\) as follows:
\begin{enumerate}
\item \((\Lambda_H \widehat{\otimes}^{\leq 2} \Omega)^0 \coloneqq \Omega^0\) as a left \(H\)-comodule \(\ast\)-algebra;
\item \((\Lambda_H \widehat{\otimes}^{\leq 2} \Omega)^1 \coloneqq (\Lambda^1_H \otimes \Omega^0) \oplus \Omega^1\) as a left \(H\)-comodule right \(\Omega^0\)-module together with the left \(\Omega^0\)-module structure defined by
\begin{equation}
\forall p,q \in \Omega^0, \, \forall \omega \in \Lambda^1_H, \, \forall \alpha \in \Omega^1, \quad q \cdot (\omega \otimes p,\alpha) \coloneqq (\ca{q}{-1}\triangleright\omega \otimes \ca{q}{0}p,q\cdot\alpha)
\end{equation}
and the \(\ast\)-structure defined by
\begin{equation}
\forall p \in \Omega^0, \, \forall \omega \in \Lambda^1_H, \, \forall \alpha \in \Omega^1, \quad (\omega \otimes p,\alpha)^\ast \coloneqq (\ca{p}{-1}^\ast \triangleright \omega^\ast \otimes \ca{p}{0}^\ast,\alpha^\ast);
\end{equation}
\item \((\Lambda_H \widehat{\otimes}^{\leq 2} \Omega)^1 \coloneqq (\Lambda^2_H \otimes \Omega^0) \oplus (\Lambda^1_H \otimes \Omega^1) \oplus \Omega^2\) as a left \(H\)-comodule right \(\Omega^0\)-module together with the left \(\Omega^0\)-structure defined by
\begin{multline}
\forall \omega \in \Lambda^1_H, \, \forall \mu \in \Lambda^2_H, \,\forall p,q \in \Omega^0,\,\forall \alpha \in \Omega^1,\,\forall \beta \in \Omega^2,\\
q \cdot (\mu \otimes p,\omega \otimes \alpha,\beta) \coloneqq (\ca{q}{-1} \triangleright \mu \otimes \ca{q}{0}p,\ca{q}{-1}\triangleright\omega\otimes \ca{q}{0} \cdot \alpha,q \cdot \beta)
\end{multline}
and the \(\ast\)-structure defined by
\begin{multline}
\forall \omega \in \Lambda^1_H, \, \forall \mu \in \Lambda^2_H, \,\forall p \in \Omega^0,\,\forall \alpha \in \Omega^1,\,\forall \beta \in \Omega^2,\\
(\mu \otimes p,\omega \otimes \alpha,\beta)^\ast \coloneqq (\ca{p}{-1}^\ast \triangleright \mu^\ast \otimes \ca{p}{0}^\ast,\ca{p}{-1}^\ast \triangleright\omega^\ast \otimes \ca{p}{0}^\ast,\beta^\ast);
\end{multline}
\item the wedge product \(\wedge : (\Lambda_H \widehat{\otimes}^{\leq 2} \Omega)^1 \otimes_{\Omega^0} (\Lambda_H \widehat{\otimes}^{\leq 2} \Omega)^1 \to (\Lambda_H \widehat{\otimes}^{\leq 2} \Omega)^2\) is defined by
\begin{multline}
\forall \omega,\p{\omega}\in\Lambda^1_H,\,\forall p,\p{p}\in \Omega^0, \, \forall \alpha,\p{\alpha} \in \Omega^1,\\
(\omega \otimes p,\alpha) \wedge (\p{\omega}\otimes \p{p},\p{\alpha}) \coloneqq (\omega \wedge \ca{p}{-1}\triangleright\p{\omega} \otimes \ca{p}{0}\p{p},\omega \otimes p\cdot\p{\alpha}+\ca{\alpha}{-1}\triangleright\p{\omega}\otimes\ca{\omega}{0}\cdot\p{p},\alpha \wedge \p{\alpha});
\end{multline}
\item \((\Lambda_H \widehat{\otimes}^{\leq 2} \Omega)^k \coloneqq 0\) for \(k > 2\).
\end{enumerate}
\end{definition}
For greater notational simplicity, given a graded left \(H\)-comodule \(\ast\)-algebra \(\Omega\), we shall view \(\Lambda_H \widehat{\otimes}^{\leq 2} \Omega\) as the graded left \(H\)-comodule \(\ast\)-algebra, truncated at degree \(2\), generated by the graded left \(H\)-subcomodule \(\ast\)-subalgebras \(\Lambda_H\) and \(\Omega\) subject the relation \(1_{\Lambda_H} = 1_\Omega\) and the braided graded commutation relations
\begin{equation}
\forall \omega \in \Lambda_H, \, \forall \alpha \in \Omega, \quad \alpha \wedge \omega = (-1)^{\abs{\alpha}\abs{\omega}} \ca{\alpha}{-1} \triangleright \omega \wedge \ca{\alpha}{0}.
\end{equation}
Now, let \(P\) once more be a principal left \(H\)-comodule \(\ast\)-algebra over \(\mathbf{C}\) with \(B \coloneqq \coinv{H}{P}\). If the bicovariant \textsc{fodc}{} \((\Omega^1_H,\dt{H})\) on \(H\) is locally freeing for \(P\), then the proof~\cite{MT}*{\S 3.1} that \((\Omega_H,\dt{H})\) can be recovered from the data \((\Lambda_H,\rest{\dt{H}}{\Lambda_H})\), \emph{mutatis mutandis}, lets us extend the induced (first-order) vertical calculus on \(P\) to a left \(H\)-covariant \textsc{sodc} as follows.
\begin{definition}[\DJ{}ur\dj{}evi\'{c}~\cite{Dj97}*{Lemma 3.1}]
Suppose that the bicovariant \textsc{fodc}{} \((\Omega^1_H,\dt{H})\) is locally freeing for \(P\). The \emph{second-order vertical calculus} of \(P\) is the extension of the vertical calculus \((\Omega^1_{P,\mathrm{ver}},\dv{P})\) to a left \(H\)-covariant \textsc{sodc} \((\Omega_{P,\mathrm{ver}},\dv{P})\) on \(P\) defined as follows:
\begin{enumerate}
\item \(\Omega_{P,\mathrm{ver}} \coloneqq \Lambda_H \widehat{\otimes}^{\leq 2} P\) as a graded left \(H\)-comodule \(\ast\)-algebra, where \(P\) is trivially extended to a graded left \(H\)-comodule \(\ast\)-algebra;
\item the derivative \(\dv{P} : \Omega_{P,\mathrm{ver}} \to \Omega_{P,\mathrm{ver}}\) is given by
\begin{equation}
\forall \omega \in \Lambda_H, \, \forall p \in P, \quad \dv{P}(\omega \cdot p) \coloneqq \dt{H}(\omega) \cdot p+(-1)^{\abs{\omega}}\omega \cdot \dv{p}(p).
\end{equation}
\end{enumerate}
\end{definition}
We can now refine the standard definition of differentiable quantum principal \(H\)-bundle to account for differential calculus through degree \(2\). This can be viewed as a distillation of recent results of Beggs--Majid~\cite{BeM}*{\S 5.5} that specialise Beggs--Brzezi\'{n}ski's theory of noncommutative fibrations~\cite{BB} to the case of quantum principal bundles; it also echoes \DJ{}ur\dj{}evi\'{c}'s notion of differentiable quantum principal bundle~\cite{Dj97}*{\S 3}.
\begin{definition}[cf.\ Beggs--Majid~\cite{BeM}*{\S 5.5}]\label{sodc}
Suppose that the bicovariant \textsc{fodc}{} \((\Omega^1_H,\dt{H})\) on \(H\) is locally freeing for the principal left \(H\)-comodule \(\ast\)-algebra \(P\); hence, let \((\Omega_{P,\mathrm{ver}},\dv{P})\) be the second-order vertical calculus of \(P\) with respect to the bicovariant prolongation \((\Omega_H,\dt{H})\) of \((\Omega^1_H,\dt{H})\). Let \((\Omega_P,\dt{P})\) be a left \(H\)-covariant \textsc{sodc} on \(P\) . Define the restriction \((\Omega_B,\dt{B})\) of \((\Omega_P,\dt{P})\) to a \textsc{sodc} on \(B \coloneqq \coinv{H}{P}\) by
\[
\Omega^1_B \coloneqq B \cdot \dt{P}(B), \quad \Omega^2_B \coloneqq \Omega^1_B \wedge \Omega^1_B, \quad \dt{B} \coloneqq \rest{\dt{P}}{\Omega_B}.
\]
Then \((P;\Omega,\dt{P})\) defines a \emph{strong (second-order) quantum principal \((H;\Omega_H,\dt{H})\)-bundle} if and only if the following all hold:
\begin{enumerate}
\item\label{socd1} the pair \((\Omega^1_P,\dt{P})\) defines an \((H;\Omega^1_H,\dt{H})\)-principal \textsc{fodc}{} on \(P\) whose vertical map \(\mathrm{ver}[\dt{P}] : \Omega^1_P \to \Omega^1_{P,\mathrm{ver}}\) satsfies
\begin{equation}
\ker \mathrm{ver}[\dt{P}] = P \cdot \Omega^1_B;
\end{equation}
\item\label{socd2} the map \(\mathrm{ver}^{2,2}[\dt{P}] : \Omega^2_P \to \Omega^2_{P,\mathrm{ver}}\) given by
\begin{equation}
\forall \alpha,\beta \in \Omega^1_P, \quad \mathrm{ver}^{2,2}[\dt{P}](\alpha \wedge \beta) \coloneqq \mathrm{ver}[\dt{P}](\alpha) \wedge \mathrm{ver}[\dt{P}](\beta)
\end{equation}
is well-defined, is surjective, and satisfies
\begin{equation}
\ker(\mathrm{ver}^{2,2}[\dt{P}]) = \Omega^1_P \wedge \Omega^1_B;
\end{equation}
\item\label{socd3} the map \(\mathrm{ver}^{2,1}[\dt{P}] : \Omega^2_P \to \Lambda^1_H \otimes \Omega^1_{P} \subset (\Lambda_H \widehat{\otimes}^{\leq 2} \Omega_P)^2\) given by
\begin{equation}
\forall \alpha,\beta \in \Omega^1_P, \quad \mathrm{ver}^{2,1}[\dt{P}](\alpha\wedge\beta) \coloneqq \mathrm{ver}[\dt{P}](\alpha) \wedge \beta + \alpha \wedge \mathrm{ver}[\dt{P}](\beta)
\end{equation}
is well-defined and satisfies
\begin{align}
\ker(\mathrm{ver}^{2,1}[\dt{P}]) \cap \ker(\mathrm{ver}^{2,2}[\dt{P}])&= P \cdot \Omega^2_B, \\
\mathrm{ver}^{2,1}[\dt{P}](\ker(\mathrm{ver}^{2,2}[\dt{P}])) & = \Lambda^1_H \otimes P \cdot \Omega^1_B.
\end{align}
\end{enumerate}
In this case, we call \((\Omega_P,\dt{P})\) a \emph{strongly \((H;\Omega_H,\dt{H})\)-principal} \textsc{sodc}{} on \(P\), and we define the left \(H\)-covariant graded \(\ast\)-subalgebra \(\Omega_{P,\mathrm{hor}}\) of \emph{horizontal forms} in \(\Omega_P\) by
\[
\Omega^0_{P,\mathrm{hor}} \coloneqq P, \quad \Omega^1_{P,\mathrm{hor}} \coloneqq P \cdot \Omega^1_B, \quad \Omega^2_{P,\mathrm{hor}} \coloneqq P \cdot \Omega^2_B.
\]
\end{definition}
\begin{remark}\label{verremark}
We can provide the following conceptual interpretation of Definition~\ref{sodc}. On the one hand, by condition~\ref{socd2}, the map \(\mathrm{ver}[\dt{P}] : \Omega^1_P \to \Omega^1_{P,\mathrm{ver}}\) extends via \(\mathrm{ver}^{2,2}[\dt{P}]\) to a left \(H\)-covariant graded \(\ast\)-epimorphism \(\mathrm{ver}[\dt{P}] : \Omega_P \to \Omega_{P,\mathrm{ver}}\), such that
\[
\rest{\mathrm{ver}[\dt{P}]}{P} = \id_P, \quad \ker\mathrm{ver}[\dt{P}] = \Omega_P \wedge \Omega^1_{P,\mathrm{hor}};
\]
following \DJ{}ur\dj{}evi\'{c}~\cite{Dj97}*{\S 3}, we interpret this extension of \(\mathrm{ver}[\dt{P}]\) as encoding restriction of differential forms to orbitwise differential forms. On the other hand, by condition~\ref{socd3}, the rescaled vertical map \(-\ensuremath{\mathrm{i}}{}\,\mathrm{ver}[\dt{P}] : \Omega^1_P \to \Omega^1_{P,\mathrm{ver}}\) extends via \(-\ensuremath{\mathrm{i}}{}\,\mathrm{ver}^{2,1}[\dt{P}]\) to a degree \(0\) left \(H\)-covariant \(\ast\)-derivation \(\Int[\dt{P}] : \Omega_P \to \Lambda_H \widehat{\otimes}^{\leq 2} \Omega_P\), such that
\[
\rest{\Int[\dt{P}]}{P} = 0, \quad \ker \Int[\dt{P}] \cap \ker\mathrm{ver}[\dt{P}] = \Omega^2_{P,\mathrm{hor}}, \quad \Int[\dt{P}](\ker\mathrm{ver}[\dt{P}]) = \Lambda^1_H \otimes \Omega^1_{P,\mathrm{hor}};
\]
we interpret the map \(\Int[\dt{P}] : \Omega_P \to \Lambda_H \widehat{\otimes}^{\leq 2} \Omega_P\) as encoding contraction of differential forms with fundamental vector fields. Proposition~\ref{strongprop} now implies that
\[
\ker \Int[\dt{P}] \cap \ker\mathrm{ver}[\dt{P}] = \Omega^1_{P,\mathrm{hor}} \oplus \Omega^2_{P,\mathrm{hor}}
\]
recovers the graded \(\ast\)-ideal of horizontal forms of positive degree, so that the equality \[\coinv{H}{(\ker \Int[\dt{P}] \cap \ker\mathrm{ver}[\dt{P}])} = \Omega^1_B \oplus \Omega^2_B\] recovers the basic differential forms as the \(H\)-coinvariant horizontal differential forms.
\end{remark}
\begin{remark}
Suppose that \((\Omega_H,\dt{H})\) is Woronowicz's canonical prolongation~\cite{Woronowicz}*{\S\S 3--4} of the bicovariant \textsc{fodc}{} \((\Omega^1_H,\dt{H})\) on \(H\), so that \(\Omega_H\) defines a graded super-Hopf \(\ast\)-algebra~\cite{Br93}. Suppose that \((\Omega_P,\dt{P})\) is a \(\ast\)-differential calculus on \(P\), such that the left \(H\)-coaction of \(H\) on \(P\) extends to a differentiable left \(\Omega_H\)-coaction \(\hat{\delta}_{\Omega_P}\) of the graded super-Hopf \(\ast\)-algebra \(\Omega_H\) on \(\Omega_P\) (cf.\ Beggs--Majid~\cite{BeM}*{\S 5.5}), so that \((P;\Omega_P,\dt{P})\) is a quantum principal \((H;\Omega_H,\dt{H})\)-bundle \`{a} la \DJ{}ur\dj{}evi\'{c}~\cite{Dj97}*{\S 3}, and hence the map \(\mathrm{ver}[\dt{P}] : \Omega_P \to \Omega_{P,\mathrm{ver}}\) of Remark~\ref{verremark} is a well-defined surjective morphism of left \(H\)- and \(\Omega_H\)-covariant \(\ast\)-\textsc{dga}~\cite{Dj97}*{Prop.\ 3.9}. Let \(\Omega_B\) be the graded \(\ast\)-subalgebra of left \(\Omega_H\)-coinvariants generated by \(B\) and \(\dt{P}(B)\). Then the degree \(2\) truncation \((\Omega_P^{\leq 2},\dt{P})\) of \((\Omega_P,\dt{P})\) is a strongly \((H;\Omega_H,\dt{H})\)-principal \textsc{sodc}{} on \(P\) whenever
\[
\ker\mathrm{ver}[\dt{P}] = \Omega_P \wedge \Omega_B, \quad
\set{\omega \in \Omega_P \given \hat{\delta}_{\Omega_P}(\omega) = \delta_{\Omega_P}(\omega)} = P \cdot \Omega_B,
\]
in which case, \(\coinv{\Omega_H}{\Omega_P} = \Omega_B\) by Proposition \ref{strongprop}.
\end{remark}
\begin{remark}[Beggs--Majid~\cite{BeM}*{Lemma 5.60, Theorem 5.61}] In the context of Definition~\ref{sodc}, suppose that the \(B\)-\(\ast\)-bimodule of basic \(1\)-forms \(\Omega^1_B \coloneqq B \cdot \dt{P}(B)\) is flat as a left \(B\)-module (e.g., finitely generated and projective), that conditions~\ref{socd1} and~\ref{socd2} are satisfied, and that the map \(\mathrm{ver}^{2,1}[\dt{P}] : \Omega^2_P \to \Lambda^1_H \otimes \Omega^1_{P}\) of condition~\ref{socd3} is well-defined. Then \((\Omega_P,\dt{P})\) is a strongly \((H;\Omega_H,\dt{H})\)-principal \textsc{sodc}{} on the principal left \(H\)-comodule \(\ast\)-algebra \(P\), and the inclusion \(\Omega_B \hookrightarrow \Omega_P\) defines (to second order) a noncommutative fibration in the sense of Beggs--Brzezi\'{n}ski~\cite{BB}.
\end{remark}
We now refine the definition of strong bimodule connection to the context of strong second-order quantum principal bundles, thereby providing a straightforward notion of noncommutative principal connection compatible with non-universal \textsc{sodc}{}. We shall soon see that this refinement is related to \DJ{}ur\dj{}evi\'{c}'s notion of multiplicative connection~\cite{Dj97}*{Def.\ 4.2}, and as such resolves (through degree \(2\)) the following question implicitly flagged by Hajac~\cite{Hajac}*{\S 4}: when does a connection extend from \(\Omega^1_P\) to all of \(\Omega_P\)?
\begin{definition}\label{prolongcon}
Suppose that the bicovariant \textsc{fodc}{} \((\Omega^1_H,\dt{H})\) on \(H\) is locally freeing for \(P\) and that \((\Omega_P,\dt{P})\) is a strongly \((H;\Omega_H,\dt{H})\)-principal \textsc{sodc}{} on the principal left \(H\)-comodule \(\ast\)-algebra \(P\). A connection \(\Pi\) on the quantum principal \((H;\Omega^1_H,\dt{H})\)-bundle \((P;\Omega^1_P,\dt{P})\) is called \emph{prolongable} if and only if it is a bimodule connection that satisfies both of the following:
\begin{enumerate}
\item\label{prolongcon1} the map \(\Pi \wedge \Pi : \Omega^2_P \to \Omega^2_P\) given by
\begin{equation}
\forall \alpha, \beta \in \Omega^1_P, \quad \Pi \wedge \Pi(\alpha \wedge \beta) \coloneqq \Pi(\alpha) \wedge \Pi(\beta)
\end{equation}
is well-defined and satisfies \(\ker(\Pi \wedge \Pi) = \Omega^1_P \wedge \Omega^1_{P,\mathrm{hor}}\).
\item\label{prolongcon2} the map \(\Pi \wedge \id + \id{} \wedge \Pi : \Omega^2_P \to \Omega^2_P\) given by
\begin{equation}
\forall \alpha,\beta \in \Omega^1_P, \quad (\Pi \wedge \id + \id \wedge \Pi)(\alpha \wedge \beta) \coloneqq \Pi(\alpha) \wedge \beta + \alpha \wedge \Pi(\beta)
\end{equation}
is well-defined and satisfies \(\ker((\Pi \wedge \id + \id \wedge \Pi) - \Pi \wedge \Pi) = \Omega^2_{P,\mathrm{hor}}\).
\end{enumerate}
\end{definition}
\begin{remark}
Suppose that the bimodule connection \(\Pi\) satisfies Condition~\ref{prolongcon1}. Then \(\Pi\) satisfies condition~\ref{prolongcon2} if and only if the map \((\id-\Pi) \wedge (\id-\Pi) : \Omega^2_P \to \Omega^2_P\) given by
\[
\forall \alpha, \beta \in \Omega^1_P, \quad (\id-\Pi) \wedge (\id-\Pi)(\alpha \wedge \beta) \coloneqq (\id-\Pi)(\alpha) \wedge (\id-\Pi)(\beta)
\]
is well-defined and satisfies \(\ker(\id - (\id-\Pi)\wedge(\id-\Pi)) = \Omega^2_{P,\mathrm{hor}}\).
\end{remark}
Suppose that \((\Omega_P,\dt{P})\) is a strongly \((H;\Omega_H,\dt{H})\)-principal \textsc{sodc}{} on the principal left \(H\)-comodule \(\ast\)-algebra \(P\). Recall that a bimodule connection \(\Pi\) on the quantum princnipal \((H;\Omega^1_H,\dt{H})\)-bundle \((P;\Omega^1_P,\dt{P})\) yields a resolution of the identity \(\set{\Pi,\id-\Pi}\) on \(\Omega^1_P\) realising \(\ker\mathrm{ver}[\dt{P}] = \Omega^1_{P,\mathrm{hor}}\) as a complementable left \(H\)-subcomodule \(P\)-\(\ast\)-subbimodule of \(\Omega^1_P\). Straightforward calculations now show that a prolongable bimodule connection \(\Pi\) yields a compatible resolution of the identity \(\set{\Pi_{2,2},\Pi_{2,1},\Pi_{2,0}}\) on \(\Omega^2_P\) that realises
\[
\ker(\mathrm{ver}^{2,2}[\dt{P}]) = \Omega^1_P \wedge \Omega^1_{P,\mathrm{hor}}, \quad \ker(\mathrm{ver}^{2,1}[\dt{P}]) \cap \ker(\mathrm{ver}^{2,2}[\dt{P}]) = \Omega^2_{P,\mathrm{hor}}
\]
as complementable left \(H\)-subcomodule \(P\)-\(\ast\)-subbimodules of \(\Omega^2_P\).
\begin{proposition}
Suppose that the bicovariant \textsc{fodc}{} \((\Omega^1_H,\dt{H})\) on \(H\) is locally freeing for \(P\), that \((\Omega_P,\dt{P})\) is a strongly \((H;\Omega_H,\dt{H})\)-principal \textsc{sodc}{} on \(P\), and that \(\Pi\) is a prolongable bimodule connection on the strong quantum principal \((H;\Omega_H,\dt{H})\)-bundle \((P;\Omega_P,\dt{P})\).
\begin{enumerate}
\item The maps \(\Pi_{1,1},\Pi_{1,0} : \Omega^1_P \to \Omega^1_P\) given by
\[
\Pi_{1,1} \coloneqq \Pi, \quad \Pi_{1,0} \coloneqq \id-\Pi,
\]
respectively, define an orthogonal pair of idempotent left \(H\)-covariant morphisms of \(P\)-\(\ast\)-bimodules, such that \(\Pi_{1,1}+\Pi_{1,0}=\id\) and
\[
\ran \Pi_{1,0} = \ker \Pi_{1,1} = P \cdot \Omega^1_B.
\]
\item The maps \(\Pi_{2,2},\Pi_{2,1},\Pi_{2,0}: \Omega^2_P \to \Omega^2_P\) given by
\begin{gather*}
\Pi_{2,2} \coloneqq \Pi \wedge \Pi, \quad \Pi_{2,1} \coloneqq (\Pi\wedge\id{}+\id{}\wedge\Pi) -2(\Pi\wedge\Pi), \\ \Pi_{2,0} \coloneqq \id - (\Pi\wedge\id{}+\id{}\wedge\Pi) + \Pi \wedge \Pi,
\end{gather*}
respectively, define a pairwise orthogonal triple of idempotent left \(H\)-covariant morphisms of \(P\)-\(\ast\)-bimodules, such that \(\Pi_{2,2}+\Pi_{2,1}+\Pi_{2,0}=\id\) and
\[
\ran \Pi_{2,0} = \ker(\Pi_{2,2}+\Pi_{2,1}) = \Omega^2_{P,\mathrm{hor}}, \quad \ran(\Pi_{2,1}+\Pi_{2,0}) = \ker(\Pi_{2,2}) = \Omega^1_P \wedge \Omega^1_{P,\mathrm{hor}};
\]
\item For all \(\alpha,\beta \in \Omega^1_P\),
\begin{align*}
\Pi_{2,2}(\alpha\wedge\beta) &= \Pi_{1,1}(\alpha)\wedge\Pi_{1,1}(\beta), \\ \Pi_{2,1}(\alpha\wedge\beta) &= \Pi_{1,1}(\alpha)\wedge \Pi_{1,0}(\beta) + \Pi_{1,0}(\alpha) \wedge \Pi_{1,1}(\beta), \\ \Pi_{2,0}(\alpha \wedge \beta) &= \Pi_{1,0}(\alpha)\wedge\Pi_{1,0}(\beta).
\end{align*}
\end{enumerate}
\end{proposition}
\begin{remark}\label{horremark}
Recall the maps \(\mathrm{ver}[\dt{P}] : \Omega_P \to \Omega_{P,\mathrm{ver}}\) and \(\Int[\dt{P}] : \Omega_P \to \Lambda_H \widehat{\otimes}^{\leq 2} \Omega_P\) from Remark~\ref{verremark} induced by the vertical map \(\mathrm{ver}[\dt{P}] : \Omega^1_P \to \Omega^1_{P,\mathrm{ver}}\). On the one hand, the strong bimodule connection \(\Pi = \Pi_{1,1}\) extends via \(\Pi_{2,2}\) to a left \(H\)-covariant graded \(\ast\)-homomorphism \(\mathrm{ver}_\Pi : \Omega_P \to \Omega_P\), such that
\[
\rest{\mathrm{ver}_\Pi}{P} = \id, \quad (\mathrm{ver}_\Pi)^2 = \mathrm{ver}_\Pi, \quad \ker \mathrm{ver}_\Pi = \Omega^1_P \wedge \Omega^1_{P,\mathrm{hor}} = \ker \mathrm{ver}[\dt{P}];
\]
in particular, it follows that \(\mathrm{ver}[\dt{P}] \circ \mathrm{ver}_\Pi = \mathrm{ver}[\dt{P}]\). On the other, following an observation of \DJ{}ur\dj{}evi\'{c}~\cite{Dj97}*{Eq.\ 4.58}, we see that \(\id-\Pi = \Pi_{1,0}\) extends via \(\Pi_{2,0}\) to a left \(H\)-covariant graded \(\ast\)-homomorphism \(\mathrm{hor}_\Pi : \Omega_P \to \Omega_P\), such that
\[
\rest{\mathrm{hor}_\Pi}{P} = \id, \quad (\mathrm{hor}_\Pi)^2 = \mathrm{hor}_\Pi, \quad \ran \mathrm{hor}_\Pi = \Omega_{P,\mathrm{hor}} = P \oplus \ker \Int[\dt{P}] \cap \ker \mathrm{ver}[\dt{P}];
\]
in particular, it follows that \(\rest{\mathrm{hor}_\Pi}{\Omega_B} = \id\). Thus, in particular, the map \(\mathrm{ver}_\Pi\) yields the extension from \(\Omega^1_P\) to \(\Omega_P\) of the strong bimodule connection \(\Pi\) required by Hajac for the discussion of curvature~\cite{Hajac}*{\S 4}.
\end{remark}
We now see that if \(\Pi\) is a prolongable bimodule projection on a strong quantum principal \((H;\Omega_H,\dt{H})\)-bundle \((P;\Omega_P,\dt{P})\), then \(\nabla_\Pi \coloneqq \Pi \circ \dt{P}\) extends to a lift of \(\dt{B} : \Omega_B \to \Omega_B\) to a degree \(1\) left \(H\)-covariant \(\ast\)-derivation on \(\Omega_{P,\mathrm{hor}}\), thereby addressing---if only through degree \(2\)---an open issue for connections \`{a} la Brzezi\'{n}ski--Majid that was first flagged by Hajac~\cite{Hajac}*{\S 4}. Furthermore, we shall also see that \(\nabla_\Pi^2\) directly yields a well-defined curvature \(2\)-form for \(\Pi\) \emph{qua} principal connection without invoking (absolute) connection \(1\)-forms in the sense of Brzezi\'{n}ski--Majid~\cite{BrM}*{Prop.\ 4.10} or \DJ{}ur\dj{}evi\'{c}~\cite{Dj97}*{Def.\ 4.1}.
\begin{propositiondefinition}[cf.\ Brzezi\'{n}ski--Majid~\cite{BrM}*{Appx.\ A and \S 3}, Hajac~\cite{Hajac}*{\S 4}, \DJ{}ur\dj{}evi\'{c}~\cite{Dj97}*{Def.\ 4.5 and Prop.\ 4.6}]
Suppose that the bicovariant \textsc{fodc}{} \((\Omega^1_H,\dt{H})\) on \(H\) is locally freeing for the principal left \(H\)-comodule \(\ast\)-algebra \(P\), that \((\Omega_P,\dt{P})\) is a strongly \((H;\Omega_H,\dt{H})\)-principal \textsc{sodc}{} on \(P\), and that \(\Pi\) is a prolongable bimodule connection on the strong quantum principal \((H;\Omega_H,\dt{H})\)-bundle \((P;\Omega_P,\dt{P})\).
\begin{enumerate}
\item The left \(H\)-covariant map \(\nabla_\Pi : \Omega_{P} \to \Omega_{P,\mathrm{hor}}\) given by
\[
\rest{\nabla_\Pi}{P} \coloneqq \Pi_{1,0} \circ \dt{P}, \quad \rest{\nabla_\Pi}{\Omega^1_P} \coloneqq \Pi_{2,0} \circ \dt{P}
\]
restricts to a degree \(1\) left \(H\)-covariant \(\ast\)-derivation \(\Omega_{P,\mathrm{hor}} \to \Omega_{P,\mathrm{hor}}\), such that
\[
\rest{\nabla_\Pi}{\Omega_B} = \dt{B};
\]
we call \(\nabla_\Pi\) the \emph{exterior covariant derivative} induced by \(\Pi\).
\item The left \(H\)-covariant right \(P\)-linear map \(F_\Pi : \Omega^1_{P,\mathrm{ver}} \to \Omega^2_{P,\mathrm{hor}}\) given by
\begin{equation}
\rest{F_\Pi}{\Lambda^1_H} \coloneqq \ensuremath{\mathrm{i}}{}\,\Pi_{2,0} \circ \dt{P} \circ \rest{(\rest{\mathrm{ver}[\dt{P}]}{\ran \Pi_{1,1}})^{-1}}{\Lambda^1_H}
\end{equation}
defines the unique left \(H\)-covariant morphism of \(P\)-\(\ast\)-bimodules, such that
\begin{equation}
\rest{\nabla^2_\Pi}{P} = \ensuremath{\mathrm{i}}{} \,F_\Pi \circ \dv{P};
\end{equation}
we call \(F_\Pi\) the \emph{curvature} of \(\Pi\).
\end{enumerate}
\end{propositiondefinition}
\begin{proof}
Let us first check that the map \(\nabla_\Pi\), which is left \(H\)-covariant and satisfies \[\forall \alpha \in \Omega_P, \quad \nabla_\Pi(\alpha^\ast) = -\nabla(\alpha)^\ast,\] restricts to a degree \(1\) \(\ast\)-derivation on \(\Omega_{P,\mathrm{hor}}\). Since \(\rest{\nabla_\Pi}{P}\) is the horizontal derivative of Proposition~\ref{analysisthm}, it remains to show that \(\rest{\nabla_\Pi}{\Omega^1_P}\) satisfies the correct graded Leibniz rules with respect to the products \(P \otimes \Omega^1_{P,\mathrm{hor}} \to \Omega^1_{P,\mathrm{hor}}\) and \(\Omega^1_{P,\mathrm{hor}} \otimes P \to \Omega^1_{P,\mathrm{hor}}\). Indeed, for all \(p \in P\) and \(\alpha \in \Omega^1_{P,\mathrm{hor}}\),
\begin{gather*}
\begin{multlined}\nabla_\Pi(p \cdot \alpha) = \Pi_{2,0} \circ \dt{P}(p \cdot \alpha) = \Pi_{2,0}(\dt{P}(p) \wedge \alpha + p \cdot \dt{P}(\alpha))\\ = \Pi_{1,0}(\dt{P}(p)) \wedge \Pi_{1,0}(\alpha) + p \cdot \Pi_{2,0}(\dt{P}(\alpha)) = \nabla_\Pi(p) \wedge \alpha + p \cdot \nabla_\Pi(\alpha),\end{multlined}\\
\begin{multlined}
\nabla_\Pi(\alpha \cdot p) = \Pi_{2,0} \circ \dt{P}(\alpha \cdot p) = \Pi_{2,0}(\dt{P}(\alpha) \cdot p - \alpha \wedge \dt{P}(p))\\ = \Pi_{2,0}(\dt{P}(\alpha)) \cdot p - \Pi_{1,0}(\alpha) \wedge \Pi_{1,0}(\dt{P}(p)) = \nabla_\Pi(\alpha) \cdot p - \alpha \wedge \nabla_\Pi(p)
\end{multlined}
\end{gather*}
as required. Note that \(\rest{\nabla_\Pi}{\Omega_B} = \dt{B}\) because \(\rest{\Pi_{1,0}}{\Omega^1_B} = \id\) and \(\rest{\Pi_{2,0}}{\Omega^2_B} = \id\).
Let us now check that \(F_\Pi\) is left \(P\)-linear and \(\ast\)-preserving. Set \(\theta \coloneqq (\rest{\mathrm{ver}[\dt{P}]}{\ran \Pi_{1,1}})^{-1}\). Then, for all \(p \in P\) and \(\omega \in \Lambda^1_H\),
\begin{align*}
p \cdot F_\Pi(\omega) &= p \cdot \left(\ensuremath{\mathrm{i}}{}\, \Pi_{2,0} \circ \dt{P} \circ \theta(\omega)\right)\\
&= \ensuremath{\mathrm{i}}{}\, \Pi_{2,0}\mleft(\dt{P}(\theta(p \cdot \omega) - \dt{P}(p) \wedge \theta(\omega) \mright)\\
&= \ensuremath{\mathrm{i}}{}\, \Pi_{2,0} \mleft(\dt{P}(\theta(\ca{p}{-1} \triangleright \omega) \cdot \ca{p}{0}) \mright)\\
&= \ensuremath{\mathrm{i}}{}\, \Pi_{2,0} \mleft(\dt{P}(\theta(\ca{p}{-1}) \triangleright \omega) \cdot \ca{p}{0} - \theta(\ca{p}{-1}\triangleright\omega) \wedge \dt{P}(\ca{p}{0})\mright)\\
&= \ensuremath{\mathrm{i}}{}\, \Pi_{2,0} \circ\dt{P}\mleft(\theta(\ca{p}{-1}\triangleright\omega) \cdot \ca{p}{0}\mright)\\
&= F_\Pi\mleft((\ca{p}{-1}\triangleright\omega)\cdot\ca{p}{0}\mright),
\end{align*}
so that \(F_\Pi\) is left \(P\)-linear; that \(F_\Pi\) is \(\ast\)-preserving now follows from the observation that \(\rest{F_\Pi}{\Lambda^1_H} = -\ensuremath{\mathrm{i}}{}\,\Pi_{2,0} \circ \dt{P} \circ \rest{\theta}{\Lambda^1_H}\) is \(\ast\)-preserving.
Finally, let us check the relation between \(F_\Pi\) and \(\nabla^2\). For every \(p \in P\),
\begin{align*}
\nabla_\Pi^2(p) &= \nabla_\Pi(\dt{P}(p)-\theta \circ \dv{P}(p))\\
&= \Pi_{2,0} \circ \dt{P}\mleft(\dt{P}(p) - \theta(\varpi_H(\ca{p}{-1})) \cdot \ca{p}{0} \mright)\\
&= \Pi_{2,0}\mleft(-\dt{P} \circ \theta(\varpi_H(\ca{p}{-1})) \cdot \ca{p}{0} + \theta(\varpi_H(\ca{p}{-1})) \wedge \dt{P}(\ca{p}{0}) \mright)\\
&= -\Pi_{2,0} \circ \dt{P} \circ \theta(\varpi_H(\ca{p}{-1})) \cdot \ca{p}{0}\\
&= \ensuremath{\mathrm{i}}{}\, F_\Pi \circ \dv{P}(p),
\end{align*}
so that \(\rest{\nabla_\Pi^2}{P} = \ensuremath{\mathrm{i}}{} \,F_\Pi \circ \dv{P}\), as was claimed; since \(\Omega^1_{P,\mathrm{ver}} = P \cdot \dv{P}(P)\), it now follows that \(F_\Pi : \Omega^1_{P,\mathrm{ver}} \to \Omega^2_{P,\mathrm{hor}}\) is the unique such left \(H\)-covariant morphism of \(P\)-\(\ast\)-bimodules.
\end{proof}
\begin{remark}[cf.\ Brzezi\'{n}ski--Majid~\cite{BrM}*{Appx.\ A}, \DJ{}ur\dj{}evi\'{c}~\cite{Dj97}*{Prop.\ 4.16}]
In terms of the map \(\mathrm{hor}_\Pi : \Omega_P \to \Omega_P\) defined in Remark~\ref{horremark},
\[
\nabla_\Pi = \mathrm{hor}_\Pi \circ \dt{P}, \quad \rest{F_\Pi}{\Lambda^1_H} = \mathrm{hor}_\Pi \circ \dt{P} \circ \rest{(\rest{\mathrm{ver}[\dt{P}]}{\ran \Pi})^{-1}}{\Lambda^1_H}.
\]
\end{remark}
Given a prolongable connection \(\Pi\) on a strong quantum principal \((H;\Omega_H,\dt{H})\)-bundle \((P;\Omega_P,\dt{P})\), we can extend the decomposition of the \((H;\Omega^1_H,\dt{H})\)-principal \textsc{fodc}{} \((\Omega^1_P,\dt{P})\) on \(P\) given by Proposition~\ref{analysisthm} to a left \(H\)-covariant isomorphism of graded \(\ast\)-alge\-bras \(\psi_\Pi : \Omega_P \to \Lambda_H \mathbin{\widehat{\otimes}} \Omega_{P,\mathrm{hor}}\), such that \(\psi_\Pi \circ \dt{P} \circ \psi_\Pi^{-1}\) can be explicitly expressed in terms of the quantum Maurer--Cartan form \(\varpi_H\) of the bicovariant \textsc{fodc}{} \((\Omega^1_H,\dt{H})\) on \(H\), the gauge potential \(\nabla_\Pi\) induced by \(\Pi\), and the curvature \(F_\Pi\) of \(\Pi\). This will provide a roadmap for replacing prolongable connections with sufficiently well-behaved gauge potentials by the appropriate refinement of Proposition~\ref{equiv1}.
\begin{proposition}[cf.\ \DJ{}ur\dj{}evi\'{c}~\cite{Dj97}*{Thm.\ 4.12}]\label{isothm}
Suppose that the bicovariant first-order differential calculus \((\Omega^1_H,\dt{H})\) on \(H\) is locally freeing for the principal left \(H\)-comodule \(\ast\)-algebra \(P\), that \((\Omega_P,\dt{P})\) is a strongly \((H;\Omega_H,\dt{H})\)-principal \textsc{sodc}{} on \(P\), and that \(\Pi\) is a prolongable bimodule connection on the strong quantum principal \((H;\Omega_H,\dt{H})\)-bundle \((P;\Omega_P,\dt{P})\). The map \(\psi_\Pi : \Omega_P \to \Lambda_H \mathbin{\widehat{\otimes}} \Omega_{P,\mathrm{hor}}\) given by
\[
\rest{\psi_\Pi}{P} \coloneqq \id, \quad \rest{\psi_\Pi}{\Omega^1_P} \coloneqq \mathrm{ver}[\dt{P}] + \Pi_{1,0}, \quad \rest{\psi_\Pi}{\Omega^2_P} \coloneqq \mathrm{ver}^{2,2}[\dt{P}] + \mathrm{ver}^{2,1}[\dt{P}] \circ \Pi_{2,1} + \Pi_{2,0}
\]
defines a left \(H\)-covariant isomorphism of graded \(\ast\)-algebras, such that
\begin{gather}
\forall p \in P, \quad \psi_\Pi \circ \dt{P} \circ \psi_\Pi^{-1}(p) = \dv{P}(p) + \nabla_\Pi(p),\\
\forall \omega \in \Lambda^1_H, \quad \psi_\Pi \circ \dt{P} \circ \psi_\Pi^{-1}(\omega) = \dt{H}(\omega) - \ensuremath{\mathrm{i}}{}\,F_\Pi(\omega),\\
\forall \alpha \in \Omega^1_{P,\mathrm{hor}}, \quad \psi_\Pi \circ \dt{P} \circ \psi_\Pi^{-1}(\alpha) = \varpi_H(\ca{\alpha}{-1}) \wedge \ca{\alpha}{0} +\nabla_\Pi(\alpha),
\end{gather}
and hence, in particular, \(\rest{\psi_\Pi \circ \dt{P} \circ \psi_\Pi^{-1}}{\Omega_B} = \dt{B}\).
\end{proposition}
\begin{proof}
Before continuing, recall that the restriction \(\rest{\psi_\Pi}{\Omega^1_P}\) is a left \(H\)-covariant isomorphism of \(P\)-\(\ast\)-bimodules by Proposition~\ref{analysisthm}. First, observe the the left \(H\)-covariant graded \(\mathbf{C}\)-linear map \(\psi_\Pi : \Omega_P \to \Lambda_H \widehat{\otimes}^{\leq 2} \Omega_{P,\mathrm{hor}}\) is bijective: since \((P;\Omega_P,\dt{P})\) is a strong second-order quantum principal \((H;\Omega_H,\dt{H})\)-bundle and since \(\Pi\) is prolongable, it follows that
\[
\rest{\mathrm{ver}^{2,2}[\dt{P}]}{\ran \Pi_{2,2}} : \ran \Pi_{2,2} \to \Omega^2_{P,\mathrm{ver}}, \quad \rest{\mathrm{ver}^{2,1}}{\ran \Pi_{2,1}} : \ran \Pi_{2,1} \to \Lambda^1_H \otimes \Omega^1_{P,\mathrm{hor}},
\]
are both bijective, so that, in turn, the restriction
\[
\rest{\psi_\Pi}{\Omega^2_P} \coloneqq \mathrm{ver}^{2,2}[\dt{P}] + \mathrm{ver}^{2,1}[\dt{P}] \circ \Pi_{2,1} + \Pi_{2,0} = \mathrm{ver}^{2,2}[\dt{P}] \circ \Pi_{2,2} + \mathrm{ver}^{2,1}[\dt{P}] \circ \Pi_{2,1} + \Pi_{2,0}
\]
is also bijective. Next, to show that the map \(\psi_\Pi\) is a homomorphism, it suffices to show that it is multiplicative with respect to the product \(\Omega^1_P \otimes \Omega^1_P \to \Omega^2_P\); indeed, for all \(\alpha,\beta \in \Omega^1_P\),
\begin{align*}
\psi_\Pi(\alpha \wedge \beta) &= \mathrm{ver}^{2,2}[\dt{P}](\alpha\wedge\beta) + \mathrm{ver}^{2,1}[\dt{P}]\circ\Pi_{2,1}(\alpha\wedge\beta) + \Pi_{2,0}(\alpha\wedge\beta)\\
&= \mathrm{ver}^{2,2}[\dt{P}](\alpha\wedge\beta) + \mathrm{ver}^{2,1}[\dt{P}]\mleft(\Pi_{1,1}(\alpha)\wedge\Pi_{1,0}(\beta)+\Pi_{1,0}(\alpha)\wedge\Pi_{1,1}(\beta)\mright)\\ &\quad\quad + \Pi_{1,0}(\alpha)\wedge\Pi_{1,0}(\beta)\\
&= \mathrm{ver}[\dt{P}](\alpha)\wedge\mathrm{ver}[\dt{P}](\beta) + \mathrm{ver}[\dt{P}](\alpha)\wedge\Pi_{1,0}(\beta) + \Pi_{1,0}(\alpha)\wedge\mathrm{ver}[\dt{P}](\beta)\\ &\quad\quad+ \Pi_{1,0}(\alpha)\wedge\Pi_{1,0}(\beta)\\
&=\left(\mathrm{ver}[\dt{P}](\alpha)+\Pi_{1,0}(\alpha)\right)\wedge\left(\mathrm{ver}[\dt{P}](\beta)+\Pi_{1,0}(\beta)\right)\\
&=\psi_\Pi(\alpha)\wedge\psi_\Pi(\beta).
\end{align*}
Since \(\rest{\psi_\Pi}{\Omega^1_P}\) is \(\ast\)-preserving and \(\Omega^2_P = \Omega^1_P \wedge \Omega^1_P\), it now follows that \(\rest{\psi_\Pi}{\Omega^2_P}\) is \(\ast\)-preserving. Hence, the map \(\psi_\Pi\) is indeed a left \(H\)-covariant isomorphism of graded \(\ast\)-algebras.
Let us now compute \(\psi_\Pi \circ \dt{P} \circ \psi_\Pi^{-1}\). By Proposition~\ref{analysisthm}, we know that \[\rest{\psi_\Pi \circ \dt{P} \circ \psi_\Pi^{-1}}{P} = \psi_\Pi \circ \dt{P} = \dv{P} + \nabla,\] so it remains to compute \(\rest{\psi_\Pi \circ \dt{P} \circ \psi_\Pi^{-1}}{\Omega^1_P}\). Observe that
\begin{align*}
\psi_\Pi \circ \dt{P} &= \left(\mathrm{ver}^{2,2}[\dt{P}] + \mathrm{ver}^{2,1}[\dt{P}] \circ \Pi_{2,1} + \Pi_{2,0}\right) \circ \dt{P}\\
&= \mathrm{ver}^{2,2}[\dt{P}] \circ \dt{P} + \mathrm{ver}^{2,1}[\dt{P}] \circ \Pi_{2,1} \circ \dt{P} + \nabla_\Pi,
\end{align*}
On the one hand, for all \(p,q \in P\),
\[
\mathrm{ver}^{2,2}[\dt{P}] \circ \dt{P}(p \cdot \dt{P}q) = \mathrm{ver}^{2,2}[\dt{P}](\dt{P}(p) \wedge \dt{P}(q)) = \dv{P}(p) \wedge \dv{P}(q) = \dv{P}(p \cdot \dv{P}(q)),
\]
so that \(\mathrm{ver}^{2,2}[\dt{P}] \circ \dt{P} = \dv{P} \circ \mathrm{ver}[\dt{P}]\). On the other, for all \(p,q \in P\),
\begin{align*}
&\mathrm{ver}^{2,1}[\dt{P}] \circ \Pi_{2,1} \circ \dt{P}(p \cdot \dt{P}(q))\\ &= \mathrm{ver}^{2,1}[\dt{P}] \circ \Pi_{2,1}(\dt{P}(p) \wedge \dt{P}(q))\\
&= \mathrm{ver}^{2,1}[\dt{P}]\mleft(\Pi_{1,1}(\dt{P}(p)) \wedge \Pi_{1,0}(\dt{P}(q)) + \Pi_{1,0}(\dt{P}(p)) \wedge \Pi_{1,1}(\dt{P}(q))\mright)\\
&= (\mathrm{ver}[\dt{P}] \circ \Pi_{1,1})(\dt{P}(p)) \wedge \nabla_\Pi(q) + \nabla_\Pi(p) \wedge (\mathrm{ver}[\dt{P}] \circ \Pi_{1,1})(\dt{P}(q))\\
&= \dv{P}(p) \wedge \nabla_\Pi(q) + \nabla_\Pi(p) \wedge \dv{P}(q)\\
&= \varpi_H(\ca{p}{-1})\epsilon(\ca{q}{-1}) \otimes \ca{p}{0}\cdot\nabla_\Pi(\ca{q}{0}) - \ca{p}{-1} \triangleright \varpi_H(\ca{q}{-1}) \otimes \nabla_\Pi(\ca{p}{0}) \cdot \ca{q}{0}\\
&= \varpi_H(\ca{p}{-1}\ca{q}{-1}) \otimes \ca{p}{0}\cdot\nabla_\Pi(\ca{p}{0}) - \ca{p}{-1}\triangleright\varpi_H(\ca{q}{-1}) \otimes \nabla_\Pi(\ca{p}{0}\ca{q}{0})\\
&= \varpi_H(\ca{\Pi_{1,0}(p \cdot \dt{P}(q))}{-1}) \otimes \ca{\Pi_{1,0}(p \cdot \dt{P}(q))}{0} + (\id \mathbin{\widehat{\otimes}} \nabla_\Pi) \circ \mathrm{ver}[\dt{P}](p \cdot \dt{P}(q)),
\end{align*}
where \(\id \mathbin{\widehat{\otimes}} \nabla_\Pi : \Lambda^1_H \otimes P \to \Lambda^1_H \otimes \Omega^1_{P,\mathrm{hor}}\), yielding the projected Cartan's magic formula
\[
\forall \alpha \in \Omega^1_P, \quad \mathrm{ver}^{2,1}[\dt{P}] \circ \Pi_{2,1} \circ \dt{P}(\alpha) = \varpi_H(\ca{\Pi_{1,0}(\alpha)}{-1}) \otimes \ca{\Pi_{1,0}(\alpha)}{0} + (\id \mathbin{\widehat{\otimes}} \nabla_\Pi) \circ \mathrm{ver}[\dt{P}](\alpha).
\]
It therefore follows that
\begin{align*}
\forall \alpha \in \Omega^1_P, \quad \psi_\Pi \circ \dt{P}(\alpha) &= \dv{P} \circ \mathrm{ver}[\dt{P}](\alpha) + \varpi_H(\ca{\Pi_{1,0}(\alpha)}{-1}) \otimes \ca{\Pi_{1,0}(\alpha)}{0}\\&\quad\quad + (\id \mathbin{\widehat{\otimes}} \nabla_\Pi) \circ \mathrm{ver}[\dt{P}](\alpha) + \nabla_\Pi(\alpha).
\end{align*}
On the one hand, for all \(\omega \in \Lambda^1_H\), since \(\mathrm{ver}[\dt{P}] \circ \theta(\omega) = \omega\) and \(\Pi_{1,0} \circ \theta(\omega) = 0\), it follows that
\[
\psi_\Pi \circ \dt{P} \circ \psi_\Pi^{-1}(\omega) = \dt{H}(\omega) + \nabla_\Pi \circ \theta(\omega) = \dt{H}(\omega)-\ensuremath{\mathrm{i}}{}\,F_\Pi(\omega),
\]
while on the other, for all \(\alpha \in \Omega^1_{P,\mathrm{hor}}\), since \(\mathrm{ver}[\dt{P}](\alpha) = 0\) and \(\Pi_{1,0}(\alpha)=\alpha\), it follows that
\[
\psi_\Pi \circ \dt{P} \circ \psi_\Pi^{-1}(\alpha) = \varpi_H(\ca{\alpha}{-1}) \otimes \ca{\alpha}{0} + \nabla_\Pi(\alpha) = \varpi_H(\ca{\alpha}{-1}) \wedge \ca{\alpha}{0} + \nabla_\Pi(\alpha).
\]
Finally, since \(\rest{\nabla_\Pi}{\Omega_B} = \dt{B}\) and \(\Omega_B = \coinv{H}{\Omega_{P,\mathrm{hor}}}\), it follows that \(\rest{\psi_\Pi \circ \dt{P} \circ \psi_\Pi^{-1}}{\Omega_B} = \dt{B}\).
\end{proof}
\begin{remark}
The isomorphism \(\psi_\Pi\) was first constructed \DJ{}ur\dj{}evi\'{c}~\cite{Dj97}*{Thm.\ 4.12} for multiplicative regular connections in his sense~\cite{Dj97}*{Def.\ 4.2 and 4.3}. Since regular connections \`{a} la \DJ{}ur\dj{}evi\'{c} yield bimodule connections \`{a} la Beggs--Majid, this suggests that our notion of prolongable connection can be viewed as a variant of \DJ{}ur\dj{}evi\'{c}'s notion of multiplicative connection. Our computation of \(\psi_\Pi^{-1} \circ \dt{P} \circ \psi_\Pi\), however, seems to be novel.
\end{remark}
\begin{remark}
\emph{Mutatis mutandis}, one can also prove the following (unprojected) Cartan's magic formula for a strong quantum principal \((H;\Omega_H,\dt{H})\)-bundle \((P;\Omega_P,\dt{P})\):
\begin{equation}
\forall \alpha \in \Omega^1_P, \quad \mathrm{ver}^{2,1}[\dt{P}] \circ \dt{P}(\alpha) = \varpi(\ca{\alpha}{-1}) \wedge \ca{\alpha}{0} + (\id \mathbin{\widehat{\otimes}} \dt{P}) \circ \mathrm{ver}[\dt{P}](\alpha).
\end{equation}
\end{remark}
\subsection{Prolongability and field strength}
Having adapted the notions of quantum principal bundle and strong bimodule connection to the setting of second-order differential calculi, we now turn to the notions of gauge transformation and gauge potential with respect to a fixed horizontal calculus. In particular, we see that the standard noncommutative-geometric notion of curvature of a module connection can be adapted to gauge potentials \emph{qua} horizontal covariant derivatives. In contrast to \DJ{}ur\dj{}evi\'{c}~\cites{Dj98,Dj10}, we are explicitly concerned with the problem of extending constructions from degree \(1\) to degree \(2\).
From now on, let \(P\) be a principal \(H\)-comodule \(\ast\)-algebra, and let \(B \coloneqq \coinv{H}{P}\). We begin with the following refinement of the notion of horizontal calculus on \(P\), which encodes a choice of basic differential calculus (through degree \(2\)) together with compatible left \(H\)-covariant graded \(\ast\)-algebra over \(P\) of horizontal forms (through degree \(2\)).
\begin{definition}[cf.\ \DJ{}ur\dj{e}vi\'{c}~\cite{Dj10}*{\S 3.1}]
A \emph{second-order horizontal calculus} on the principal left \(H\)-comodule \(\ast\)-algebra \(P\) is a quadruple \((\Omega_B,\dt{B};\Omega_{P,\mathrm{hor}},\iota)\), where:
\begin{enumerate}
\item \((\Omega_B,\dt{B})\) is a \textsc{sodc}{} on \(B\);
\item \(\Omega_{P,\mathrm{hor}}\) is a left \(H\)-covariant graded \(\ast\)-algebra generated by \(\Omega^1_{P,\mathrm{hor}}\) over \(\Omega^0_P = P\) and truncated at degree \(2\);
\item \(\iota : \Omega_B \hookrightarrow \coinv{H}{\Omega_{P,\mathrm{hor}}}\) is an injective morphism of graded \(\ast\)-algebras, such that the pair \((\Omega^1_{P,\mathrm{hor}},\rest{\iota}{\Omega^1_{P,\mathrm{hor}}})\) defines a projectable horizontal lift of the \(B\)-\(\ast\)-bimodule \(\Omega^1_B\).
\end{enumerate}
\end{definition}
\begin{example}
Let \((\Omega_H,\dt{H})\) be a bicovariant \textsc{sodc}{} for \(H\), and suppose that the bicovariant \textsc{fodc}{} \((\Omega^1_H,\dt{H})\) on \(H\) is locally freeing for \(P\). Suppose that \((\Omega_P,\dt{P})\) is a strongly \((H;\Omega_H,\dt{H})\)-principal \textsc{sodc}{} on \(P\) admitting a prolongable bimodule connection; recall that \((\Omega_P,\dt{P})\) therefore restricts to the \textsc{sodc}{} \((\Omega_B,\dt{B})\) on \(B \coloneqq \coinv{H}{P}\) given by
\[
\Omega^1_B \coloneqq B \cdot \dt{P}(B), \quad \Omega^2_B \coloneqq \Omega^1_B \wedge \Omega^1_B, \quad \dt{B} \coloneqq \rest{\dt{P}}{\Omega_B}.
\]
Finally, let \(\Omega_{P,\mathrm{hor}}\) be the left \(H\)-subcomodule graded \(\ast\)-subalgebra of \(\Omega_P\) generated by \(\Omega^1_B\) over \(P\), and let \(\iota : \Omega_B \hookrightarrow \Omega_{P,\mathrm{hor}}\) be the inclusion map. Then, by Proposition~\ref{analysisthm} applied to the quantum principal \((H;\Omega^1_H,\dt{H})\)-bundle \((P;\Omega^1_P,\dt{P})\), the data
\(
(\Omega_B,\dt{B};\Omega_{P,\mathrm{hor}},\iota)
\)
define a second-order horizontal calculus on \(P\), which we can view as the \emph{canonical} second-order horizontal calculus on \(P\) induced by the strongly \((H;\Omega_H,\dt{H})\)-principal \textsc{sodc}{} \((\Omega_P,\dt{P})\) on \(P\).
\end{example}
\begin{remark}
If \((\Omega_B,\dt{B};\Omega_{P,\mathrm{hor}},\iota)\) is a second-order horizontal calculus on \(P\), then the data \((\Omega^1_B,\dt{B};\Omega^1_{P,\mathrm{hor}},\rest{\iota}{\Omega^1_B})\) define a first-order horizontal calculus on \(P\).
\end{remark}
Although it is not obvious, in a second-order horizontal calculus \((\Omega_B,\dt{B};\Omega_{P,\mathrm{hor}},\iota)\) on \(P\), the left \(H\)-covariant graded \(\ast\)-algebra \(\Omega_{P,\mathrm{hor}}\) of horizontal forms defines a projectable horizontal lift of the entire graded \(\ast\)-algebra \(\Omega_B\) of basic forms (through degree \(2\)) on \(P\).
\begin{proposition}\label{projectprop}
Suppose that \((\Omega_B,\dt{B};\Omega_{P,\mathrm{hor}},\iota)\) is a second-order horizontal calculus on \(P\); recall that \(B \coloneqq \coinv{H}{P}\). Then \((\Omega^2_{P,\mathrm{hor}},\rest{\iota}{\Omega^2_{P,\mathrm{hor}}})\) defines a projectable horizontal lift of the \(B\)-\(\ast\)-bimodule \(\Omega^2_B\), and hence \(\iota : \Omega_B \hookrightarrow \coinv{H}{\Omega_{P,\mathrm{hor}}}\) is an isomorphism of graded \(\ast\)-algebras.
\end{proposition}
\begin{proof}
First, by applying Proposition~\ref{strongprop} to the projectable horizontal lift \((\Omega^1_{P,\mathrm{hor}},\rest{\iota}{\Omega^1_{P,\mathrm{hor}}})\) of the \(B\)-\(\ast\)-bimodule \(\Omega^1_B\), we find that
\[
\Omega^2_{P,\mathrm{hor}} = \Omega^1_{P,\mathrm{hor}} \wedge \Omega^1_{P,\mathrm{hor}} = P \cdot \iota(\Omega^1_B) \wedge \iota(\Omega^1_B) \cdot P = P \cdot \iota(\Omega^2_B) \cdot P,
\]
so that \((\Omega^2_{P,\mathrm{hor}},\rest{\iota}{\Omega^2_{P,\mathrm{hor}}})\) is a horizontal lift of \(\Omega^2_B\).
Now, let \(p,\p{p} \in P\) and \(\omega,\p{\omega} \in \Omega^1_B\). Then \(\iota(\p{\omega})\cdot \p{p} = \sum_i q_i \cdot \iota(\p{\omega}_i)\) for some \(q_i \in P\) and \(\p{\omega}_i \in \Omega^1_B\), and for each \(i\), \(\iota(\omega) \cdot q_i = \sum_j q_{ij} \cdot \iota(\omega_{ij})\) for some \(q_{ij} \in P\) and \(\omega_{ij} \in \Omega^1_B\), so that
\[
p \cdot \iota(\omega) \wedge \iota(\p{\omega}) \cdot \p{p} = \sum_i p \cdot \iota(\omega) \wedge q_i \cdot \iota(\p{\omega}_i) = \sum_{i,j} p q_{ij} \cdot \iota(\omega_{ij} \wedge \p{\omega}_i) \in P \cdot \iota(\Omega^2_B).
\]
Hence, by Proposition~\ref{strongprop}, the horizontal lift \((\Omega^2_{P,\mathrm{hor}},\rest{\iota}{\Omega^2_{P,\mathrm{hor}}})\) of \(\Omega^2_B\) is projectable. In particular, it now follows that \(\iota(\Omega_B) = \coinv{H}{\Omega_{P,\mathrm{hor}}}\).
\end{proof}
Assume, therefore, that \(P\) admits a second-order horizontal calculus \((\Omega_B,\dt{B};\Omega_{P,\mathrm{hor}},\iota)\), which we now fix. To simplify notation, we suppress the inclusion map \(\iota\) and identify \(\Omega_B\) with its image in \(\Omega_{P,\mathrm{hor}}\); hence, where convenient, we denote the second-order horizontal calculus \((\Omega_B,\dt{B};\Omega_{P,\mathrm{hor}},\iota)\) by the triple \((\Omega_B,\dt{B};\Omega_{P,\mathrm{hor}})\). Since \((\Omega^1_B,\dt{B};\Omega^1_{P,\mathrm{hor}})\) is a first-order horizontal calculus on \(P\), we can define its gauge group \(\fr{G}\), its inner gauge group \(\Inn(\fr{G})\), and its Atiyah space \(\fr{At}\) with corresponding space of translations \(\fr{at}\).
We begin by characterizing those gauge transformations \(f \in \fr{G}\) that extend via the induced map \(f_{\ast} : \Omega^1_{P,\mathrm{hor}} \to \Omega^1_{P,\mathrm{hor}}\) to automorphisms of \(\Omega_{P,\mathrm{hor}}\).
\begin{definition}
We say that a gauge transformation \(\phi \in \fr{G}\) is \emph{prolongable} with respect to the second-order horizontal calculus \((\Omega_B,\dt{B};\Omega_{P,\mathrm{hor}},\iota)\) on \(P\) whenever \(\phi\) is also an automorphism of the projectable horizontal lift \((\Omega^2_{P,\mathrm{hor}},\rest{\iota}{\Omega^2_B})\) of \(\Omega^2_B\). Hence, we define the \emph{prolongable gauge group} of \(P\) with respect to \((\Omega_B,\dt{B};\Omega_{P,\mathrm{hor}},\iota)\) by
\[
\pr{\fr{G}} \coloneqq \fr{G} \cap \Aut\mleft(\Omega^2_{P,\mathrm{hor}},\rest{\iota}{\Omega^2_B}\mright),
\]
where \(\fr{G}\) is the gauge group of \(P\) with respect to \((\Omega^1_B,\dt{B};\Omega^1_{P,\mathrm{hor}},\rest{\iota}{\Omega^1_B})\) and \(\Aut(\Omega^2_{P,\mathrm{hor}},\rest{\iota}{\Omega^2_B})\) is the automorphism group of \((\Omega^2_{P,\mathrm{hor}},\rest{\iota}{\Omega^2_B})\). By abuse of notation, given \(\phi \in \dva{\fr{G}}\), we denote by \(\phi_{\ast}\) or by \(\phi_{\ast,\mathrm{hor}}\) the induced automorphism of the left \(H\)-comodule graded \(\ast\)-algebra \(\Omega_{P,\mathrm{hor}}\).
\end{definition}
\begin{remark}
That a prolongable gauge transformation \(\phi \in \dva{\fr{G}}\) induces an automorphism of the entire left \(H\)-covariant graded \(\ast\)-algebra \(\Omega_{P,\mathrm{hor}}\), viz, that
\[
\forall \omega, \p{\omega} \in \Omega^1_{P,\mathrm{hor}}, \quad \phi_\ast(\omega \wedge \p{\omega}) = \phi_\ast(\omega) \wedge \phi_\ast(\p{\omega}),
\]
is a consequence of the proof of Proposition~\ref{projectprop}.
\end{remark}
As an example, all inner gauge transformations are automatically prolongable.
\begin{propositiondefinition}
The prolongable gauge group \(\pr{\fr{G}}\) of \(P\) with respect to the second-order horizontal calculus \((\Omega_B,\dt{B};\Omega_{P,\mathrm{hor}})\) contains the inner gauge group \(\Inn(\fr{G})\) of \(P\) with respect to the first-order horizontal calculus \((\Omega^1_B,\dt{B};\Omega^1_{P,\mathrm{hor}})\) as a central subgroup. Hence, the \emph{outer prolongable gauge group} of \(P\) with respect to \((\Omega_B,\dt{B};\Omega_{P,\mathrm{hor}})\) is
\[
\Out(\pr{\fr{G}}) \coloneqq \pr{\fr{G}}/\Inn(\fr{G}).
\]
\end{propositiondefinition}
\begin{proof}
Since \(\Omega^2_{P,\mathrm{hor}}\) is a horizontal lift of \(\Omega^2_B = \Omega^1_B \wedge \Omega^1_B\), it follows that
\(
\Cent_B(\Omega^2_B) \supseteq \Cent_B(\Omega^1_B)
\), so that, by Proposition~\ref{innergaugeprop1},
\[
\Inn(\fr{G}) = \set{\Ad_v \given v \in \Unit(Z(B) \cap \Cent_B(\Omega^1_B)} \leq \set{\Ad_v \given v \in \Unit(Z(B) \cap \Cent_B(\Omega^2_B))} = \Inn(\Omega^2_{P,\mathrm{hor}},\iota),
\]
and hence, \(\Inn(\fr{G}) \leq \fr{G} \cap \Aut(\Omega^2_{P,\mathrm{hor}},\iota) \eqqcolon \pr{\fr{G}}\).
\end{proof}
We can now characterise those gauge potentials on \(P\) with respect to the first-order horizontal calculus \((\Omega^1_B,\dt{B};\Omega^1_{P,\mathrm{hor}})\) that extend to lifts of \(\dt{B} : \Omega_B \to \Omega_B\) to degree \(1\) left \(H\)-covariant \(\ast\)-derivations of \(\Omega_{P,\mathrm{hor}}\). Because we only work through degree \(2\), such extensions are completely determined by maps \(\Omega^1_{P,\mathrm{hor}} \to \Omega^2_{P,\mathrm{hor}}\) of the following form.
\begin{definition}
Let \(\Omega\) be a left \(H\)-comodule graded \(\ast\)-algebra truncated at degree \(2\). Let \(\partial : \Omega^0 \to \Omega^1\) be a left \(H\)-comodule \(\ast\)-derivation on the \(\Omega^0\)-\(\ast\)-bimodule \(\Omega^1\). A \emph{second-order prolongation} of \(\partial\) is a left \(H\)-covariant \(\mathbf{C}\)-linear map \(\partial^\prime : \Omega^1 \to \Omega^2\) satisfying
\begin{gather*}
\forall a,b \in \Omega^0, \, \forall \omega \in \Omega^1, \quad \p{\partial}(a \cdot \omega \cdot b) = \nabla(a) \wedge \omega \cdot b + a \cdot \p{\partial}(\omega) \cdot b - a \cdot \omega \wedge \partial(b),\\
\forall \alpha \in \Omega^1, \quad \p{\partial}(a)^\ast = -\p{\partial}(\alpha^\ast),
\end{gather*}
so that \(\partial : \Omega^0 \to \Omega^1\) extends via \(\partial^\prime : \Omega^1 \to \Omega^2\) and \(0 : \Omega^2 \to 0\) to a degree \(1\) left \(H\)-covariant \(\ast\)-derivation on the left \(H\)-comodule graded \(\ast\)-algebra \(\Omega\).
\end{definition}
We can characterise those gauge potentials on \(P\) with respect to \((\Omega^1_B,\dt{B};\Omega^1_{P,\mathrm{hor}})\) that suitably extend to all of \(\Omega_{P,\mathrm{hor}}\); this, in turn, yields a conceptually minimalistic notion of curvature compatible with the standard notion of curvature for module connections.
\begin{propositiondefinition}\label{canprol}
We say that a gauge potential \(\nabla \in \fr{At}\) is \emph{prolongable} with respect to the second-order horizontal calculus \((\Omega_B,\dt{B};\Omega_{P,\mathrm{hor}})\) whenever its \emph{canonical prolongation}
\[
\dva{\nabla} : \Omega^1_{P,\mathrm{hor}} \to \Omega^2_{P,\mathrm{hor}}, \quad p \cdot \dt{B}(b) \cdot \p{p} \mapsto \nabla(p) \wedge \dt{B}(b) \cdot \p{p} - p \cdot \dt{B}(b) \wedge \nabla(\p{p})
\]
is well-defined, in which case:
\begin{enumerate}
\item \(\dva{\nabla}\) is the unique second-order prolongation of \(\nabla\), such that \(\rest{\pr{\nabla}}{\Omega^1_B} = \dt{B}\);
\item the \emph{field strength} \(\mathbf{F}[\nabla] : P \to \Omega^2_{P,\mathrm{hor}}\) of \(\nabla\) defined by
\[
\mathbf{F}[\nabla] \coloneqq -\ensuremath{\mathrm{i}}{}\,\dva{\nabla} \circ \nabla.
\]
is a left \(H\)-covariant \(\ast\)-derivation, such that \(\rest{\mathbf{F}[\nabla]}{B} = 0\).
\end{enumerate}
Hence, we define the \emph{prolongable Atiyah space} \(\dva{\fr{At}}\) of \(P\) with respect to \((\Omega_B,\dt{B};\Omega_{P,\mathrm{hor}})\) to be the subset of all prolongable gauge potentials on \(P\) with respect to \((\Omega_B,\dt{B};\Omega_{P,\mathrm{hor}})\).
\end{propositiondefinition}
\begin{proof}
Let us first show that \(\dva{\nabla}\) is a secord-order prolongation of \(\nabla\). On the one hand, \(\dva{\nabla}\) is left \(H\)-covariant since \(\Omega^1_B = \coinv{H}{\Omega^1_{P,\mathrm{hor}}}\) and \(\nabla\) is left \(H\)-covariant. On the other, for all \(p,q_1,q_2 \in P\) and \(b \in \dt{B}(b)\), we have
\begin{align*}
\dva{\nabla}\mleft(q_1 \cdot (p\cdot \dt{B}(b)) \cdot q_2\mright)
&= \nabla(q_1 \cdot p) \wedge \dt{B}(b) \cdot q_2 - q_1 \cdot p \cdot \dt{B}(b) \wedge \nabla(q_2)\\
&= \left(\nabla(q_1) \cdot p + q_1 \cdot \nabla(p)\right) \wedge \dt{B}(b) \cdot q_2 - q_1 \cdot p \cdot \dt{B}(b) \wedge \nabla(q_2)\\
&= \nabla(q_1) \wedge (p \cdot \dt{B}(b)) \cdot q_2 + q_1 \cdot \dva{\nabla}(p \cdot \dt{B}(b)) \cdot q_2\\&\quad\quad - q_1 \cdot (p\cdot\dt{B}(b)) \wedge \nabla(q_2),
\end{align*}
while for all \(p \in P\) and \(b \in \dt{B}(b)\),
\[
\dva{\nabla}(p \cdot \dt{B}(b))^\ast = \left(\nabla(p) \wedge \dt{B}(b)\right)^\ast = -\dt{B}(b^\ast) \wedge \nabla(p^\ast) = \dva{\nabla}(\dt{B}(b^\ast) \cdot p^\ast) = -\dva{\nabla}\mleft((p \cdot \dt{B}(b))^\ast\mright).
\]
Next, observe that \(\rest{\dva{\nabla}}{\Omega^1_B} = \dt{B}\), since for all \(b_1,b_2 \in B\),
\[
\dva{\nabla}(b_1 \cdot \dt{B}(b_2)) = \nabla(b_1) \wedge \dt{B}(b_2) = \dt{B}(b_1) \wedge \dt{B}(b_2) = \dt{B}(b_1 \cdot \dt{B}(b_2));
\]
in particular, it now follows that \(\mathbf{F}[\nabla]\) vanishes on \(B\); thus, if \(\nabla^\prime\) is any second-order prolongation of \(\nabla\), then for all \(p,q \in P\) and \(\beta \in \Omega^1_B\),
\begin{align*}
\p{\nabla}(p \cdot \beta \cdot q) = \nabla(p) \wedge \dt{B}(b) \cdot q + p \cdot \dt{B}(\beta) \cdot q - p \cdot \dt{B}(b) \wedge \nabla(q) = \pr{\nabla}(p \cdot \beta \cdot q),
\end{align*}
so that \(\p{\nabla} = \pr{\nabla}\).
Finally, observe that the left \(H\)-covariant map \(\mathbf{F}[\nabla] : P \to \Omega^2_{P,\mathrm{hor}}\) is a \(\ast\)-derivation, since for all \(p,q \in P\),
\begin{align*}
\mathbf{F}[\nabla](pq) &= -\ensuremath{\mathrm{i}}{}\dva{\nabla}\mleft(\nabla(p) \cdot q + p \cdot \nabla(q)\mright)\\ &= \mathbf{F}[\nabla](p) \cdot q + \ensuremath{\mathrm{i}}{} \nabla(p) \wedge \nabla(q) - \ensuremath{\mathrm{i}}{} \nabla(p) \wedge \nabla(q) + \mathbf{F}[\nabla](q)\\ &= \mathbf{F}[\nabla](p) \cdot q + p \cdot \mathbf{F}[\nabla](q),
\end{align*}
while for all \(p \in P\),
\[
\mathbf{F}[\nabla](p)^\ast = \left(-\ensuremath{\mathrm{i}}{}\dva{\nabla}(\nabla(p))\right)^\ast = -\ensuremath{\mathrm{i}}{}\dva{\nabla}(\nabla(p)^\ast) = \ensuremath{\mathrm{i}}{}\dva{\nabla}(\nabla(p^\ast)) = -\mathbf{F}[\nabla](p^\ast). \qedhere
\]
\end{proof}
\begin{remark}
That the field strength of a prolongable gauge potential defines a left \(H\)-covariant \(\ast\)-derivation \(P \to \Omega^2_{P,\mathrm{hor}}\) was essentially first observed by \DJ{}ur\dj{}evi\'{c}~\cite{Dj98}*{p.\ 101}.
\end{remark}
We can now similarly characterise those relative gauge potentials on \(P\) with respect to \((\Omega^1_B,\dt{B};\Omega^1_{P,\mathrm{hor}})\) that suitably extend to all of \(\Omega_{P,\mathrm{hor}}\); this follows, \emph{mutatis mutandis}, from the proof of Proposition-Definition~\ref{canprol} .
\begin{propositiondefinition}
We say that \(\mathbf{A} \in \fr{at}\) is \emph{prolongable} with respect to the second-order horizontal calculus \((\Omega_B,\dt{B};\Omega_{P,\mathrm{hor}})\) whenever its \emph{canonical prolongation}
\[
\dva{\mathbf{A}} : \Omega^1_{P,\mathrm{hor}} \to \Omega^2_{P,\mathrm{hor}}, \quad p \cdot \dt{B}(b) \cdot \p{p} \mapsto \mathbf{A}(p) \wedge \dt{B}(b) \cdot \p{p} - p\cdot \dt{B}(b) \wedge \mathbf{A}(\p{p})
\]
is well-defined, in which case, the canonical prolongation \(\pr{\mathbf{A}}\) is the unique second-order prolongation of \(\mathbf{A}\), such that \(\rest{\pr{\mathbf{A}}}{\Omega^1_B} = 0\). We denote by \(\pr{\fr{at}}\) the subspace of all prolongable relative gauge potentials on \(P\) with respect to \((\Omega_B,\dt{B};\Omega_{P,\mathrm{hor}})\).
\end{propositiondefinition}
It now follows that the affine action of the gauge group \(\fr{G}\) on the Atiyah space \(\fr{At}\) restricts to an affine action of the subgroup \(\pr{\fr{G}}\) on the affine subspace \(\pr{\fr{At}}\) that is compatible with canonical prolongation.
\begin{proposition}\label{fieldstrengthprop}
Suppose that the prolongable Atiyah space \(\pr{\fr{At}}\) of \(P\) with respect to the second-order horizontal calculus \((\Omega_B,\dt{B};\Omega^1_{P,\mathrm{hor}})\) is non-empty. Then \(\dva{\fr{At}}\) is a \(\dva{\fr{G}}\)-invariant affine subspace of the Atiyah space \(\fr{At}\) with space of translations \(\dva{\fr{at}}\). Moreover:
\begin{enumerate}
\item \((\nabla \mapsto \dva{\nabla}) : \dva{\fr{At}} \to \Hom_{\mathbf{C}}(\Omega^1_{P,\mathrm{hor}},\Omega^2_{P,\mathrm{hor}})\) is \(\dva{\fr{G}}\)-equivariant and affine linear with \(\dva{\fr{G}}\)-equivariant linear part \((\mathbf{A} \mapsto \dva{\mathbf{A}}) : \dva{\fr{at}} \to \Hom_{\mathbf{C}}(\Omega^1_{P,\mathrm{hor}},\Omega^2_{P,\mathrm{hor}})\);
\item \(\mathbf{F} \coloneqq (\nabla \mapsto \mathbf{F}[\nabla]) : \dva{\fr{At}} \to \Der_P(\Omega^2_{P,\mathrm{hor}})\) is \(\dva{\fr{G}}\)-equivariant and affine quadratic, satisfying
\[
\forall \nabla \in \dva{\fr{At}}, \,\forall \mathbf{A} \in \dva{\fr{at}}, \quad \mathbf{F}[\nabla + \mathbf{A}] - \mathbf{F}[\nabla] = -\ensuremath{\mathrm{i}}{}\left(\dva{\nabla} \circ \mathbf{A} + \dva{\mathbf{A}} \circ \nabla + \dva{\mathbf{A}} \circ \mathbf{A}\right).
\]
\end{enumerate}
\end{proposition}
\begin{proof}
First, that \(\pr{\fr{At}}\) is an affine subspace of \(\fr{At}\) with space of translations \(\fr{at}\) follows from the observation that \(\pr{\fr{At}}\) is defined by an affine-linear condition whose linear part is defines \(\pr{\fr{at}}\). Next, let \(\phi \in \pr{\fr{G}}\), \(\nabla \in \pr{\fr{At}}\). Then, for all \(p, \p{p} \in P\) and \(b \in B\),
\begin{align*}
&(\phi \triangleright \nabla)(p) \wedge \dt{B}(b) \cdot \p{p} - p \cdot \dt{B}(b) \wedge (\phi \triangleright \nabla)(\p{p})\\
&= (\phi_{\ast} \circ \nabla \circ \inv{\phi})(p) \wedge \dt{B}(b) \cdot \p{p} - p \cdot \dt{B}(b) \wedge (\phi_{\ast} \circ \nabla \circ \inv{\phi})(\p{p})\\
&= \phi_\ast\mleft( \nabla(\phi^{-1}(p)) \wedge \phi^{-1}_{\ast}(\dt{B}(b) \cdot \p{p}) - \phi^{-1}_{\ast}(p \cdot \dt{B}(b)) \wedge \nabla(\phi^{-1}(\p{p}))\mright)\\
&= \phi_\ast \circ \pr{\nabla} \circ \phi^{-1}_\ast \mleft(p \cdot \dt{B}(b) \cdot \p{p}\mright),
\end{align*}
so that \(\phi \triangleright \nabla \in \pr{\fr{At}}\) with \(\pr{(\phi \triangleright \nabla)} = \phi_\ast \circ \pr{\nabla} \circ \phi_\ast^{-1}\). Thus, \(\pr{\fr{At}}\) is \(\pr{\fr{G}}\)-invariant and the map \((\nabla \mapsto \pr{\nabla}) : \pr{\fr{At}} \to \Hom_{\mathbf{C}}(\Omega^1_{P,\mathrm{hor}},\Omega^2_{P,\mathrm{hor}})\) is \(\pr{\fr{G}}\)-equivariant. A similary calculation shows that \(\pr{\fr{at}}\) is \(\pr{\fr{G}}\)-invariant and that \((\mathbf{A} \mapsto \pr{\mathbf{A}}) : \pr{\fr{at}} \to \Hom_{\mathbf{C}}(\Omega^1_{P,\mathrm{hor}},\Omega^2_{P,\mathrm{hor}})\) is \(\pr{\fr{G}}\)-equivariant. Finally, our claims about \(\mathbf{F} : \dva{\fr{At}} \to \Der_P^H(P,\Omega^2_{P,\mathrm{hor}})\) follow by straightforward calculation.
\end{proof}
While inner gauge transformations are automatically prolongable and yield inner automorphisms of the left \(H\)-comodule graded \(\ast\)-algebra \(\Omega_{P,\mathrm{hor}}\), the analogous statement is no longer generally true for inner relative gauge potentials. Instead, we shall consider those prolongable inner relative gauge potentials that yield inner derivations of \(\Omega_{P,\mathrm{hor}}\).
\begin{definition}
Let \(\mathbf{A} \in \pr{\fr{at}}\) be a prolongable gauge potential on \(P\) with respect to the second-order horizontal calculus \((\Omega_B,\dt{B};\Omega_{P,\mathrm{hor}})\). We say that \(\mathbf{A}\) is \emph{inner} if and only if there exists a \(1\)-form \(\alpha \in (\Omega^1_B)_{\mathrm{sa}}\), such that
\[
\mathbf{A} = \ad_\alpha \coloneqq (p \mapsto [\alpha,p]), \quad \pr{\mathbf{A}} = \ad_\alpha \coloneqq (\omega \mapsto [\alpha,\omega]).
\]
We denote by \(\Inn(\pr{\fr{at}})\) the subspace of all inner prolongable relative gauge potentials on \(P\) with respect to \((\Omega_B,\dt{B};\Omega_{P,\mathrm{hor}})\). Thus, we define the \emph{outer prolongable Atiyah space} of \(P\) with respect to \((\Omega_B,\dt{B};\Omega_{P,\mathrm{hor}})\) to be the quotient affine space \(\Out(\pr{\fr{At}}) \coloneqq \pr{\fr{At}}/\Inn(\pr{\fr{at}})\) with space of translations \(\Out(\pr{\fr{at}}) \coloneqq \pr{\fr{at}}/\Inn(\pr{\fr{at}})\).
\end{definition}
We can now characterize those elements of \(\Omega^1_B\) that yield inner prolongable relative gauge potentials on \(P\) with respect to \((\Omega_B,\dt{B};\Omega_{P,\mathrm{hor}})\); as an upshot, we find that the field strength of a prolongable gauge potential varies \emph{linearly}---not quadratically---under translation by inner prolongable relative gauge potentials.
\begin{proposition}\label{innpot}
Let \(\alpha \in (\Omega^1_B)_{\mathrm{sa}}\). Then the inner derivation \(\ad_\alpha\) defines an inner prolongable relative gauge potential on \(P\) with respect to \((\Omega_B,\dt{B};\Omega_{P,\mathrm{hor}})\) if and only if the \(1\)-form \(\alpha\) is central in \(\Omega_B\), in which case
\begin{equation}\label{innpoteq}
\forall \nabla \in \dva{\fr{At}}, \, \forall p \in P, \quad \quad \ensuremath{\mathrm{i}}{}\left(\mathbf{F}[\nabla+\ad_\alpha]-\mathbf{F}[\nabla]\right)(p) = \ad_{\dt{B}(\alpha)}(p) \coloneqq [\dt{B}(\alpha),p].
\end{equation}
Thus, the map \((\alpha \mapsto \ad_\alpha) : (\Omega^1_B)_{\mathrm{sa}} \cap \Zent(\Omega_B) \twoheadrightarrow \Inn(\pr{\fr{at}})\) yields a short exact sequence
\[
0 \to (\Omega^1_B)_{\mathrm{sa}} \cap \Zent(\Omega_{P,\mathrm{hor}}) \to (\Omega^1_B)_{\mathrm{sa}} \cap \Zent(\Omega_B) \to \Inn(\pr{\fr{at}}) \to 0.
\]
\end{proposition}
\begin{proof}
The first part of the claim follows from observing that for all \(p,\p{p} \in P\) and \(\beta \in \Omega^1_B\),
\[
[\alpha,p] \wedge \beta \cdot \p{p} - p\cdot \beta \wedge [\alpha,\p{p}] = [\alpha,p\cdot\beta\cdot\p{p}] - p \cdot [\alpha,\beta] \cdot \p{p}.
\]
Now, if \([\alpha,\Omega^1_B] = \set{0}\), then, by the above calculation, \(\ad_\alpha\) is prolongable with canonical prolongation \(\dva{(\ad_\alpha)} = [\alpha,\cdot]\); moreover, if \(\nabla \in \dva{\fr{At}}\), then for any \(p \in P\),
\begin{align*}
\ensuremath{\mathrm{i}}{}\left(\mathbf{F}[\nabla+\mathbf{A}]-\mathbf{F}[\nabla]\right)(p) &=(\dva{\nabla} \circ \ad_\alpha + \dva{(\ad_\alpha)} \circ \nabla)(p) + \dva{(\ad_\alpha)} \circ \ad_\alpha(p)\\
&= \dva{\nabla}([\alpha,p]) + [\alpha,\nabla(p)] + [\alpha,[\alpha,p]]\\
&= [\dt{B}(\alpha) + \alpha \wedge \alpha,p]\\
&= [\dt{B}(\alpha),p]. \qedhere
\end{align*}
\end{proof}
Just as in the first-order case, the affine action of the prolongable gauge group \(\fr{G}\) on the prolongable Atiyah space \(\pr{\fr{At}}\) descends further to an affine action of the outer prolongable gauge group \(\Out(\pr{\fr{G}})\) on the outer prolongable Atiyah space \(\Out(\pr{\fr{At}})\), which will yield a non-trivial invariant of the principal left \(H\)-comodule \(\ast\)-algebra \(P\) endowed with the second-order horizontal calculus \((\Omega_B,\dt{B};\Omega_{P,\mathrm{hor}})\).
\begin{proposition}\label{innprop}
The subspace \(\Inn(\pr{\fr{at}})\) of inner prolongable gauge potentials on \(P\) consists of \(\pr{\fr{G}}\)-invariant vectors, so that the affine action of \(\pr{\fr{G}}\) on \(\pr{\fr{At}}\) descends to an affine action of \(\pr{\fr{G}}\) on \(\Out(\pr{\fr{At}}) \coloneqq \pr{\fr{At}}/\Inn(\pr{\fr{at}})\). Moreover, the inner gauge group \(\Inn(\fr{G})\) acts trivially on \(\Out(\pr{\fr{at}})\), so that the affine action of \(\pr{\fr{G}}\) on \(\pr{\fr{At}}\) descends further to an affine action of \(\Out(\pr{\fr{G}})\) on \(\Out(\pr{\fr{At}})\).
\end{proposition}
\begin{proof}
Since \(\Inn(\fr{at})\) consists of \(\fr{G}\)-invariant vectors, it follows \emph{a fortiori} that \(\Inn(\pr{\fr{at}})\) consists of \(\pr{\fr{G}}\)-invariant vectors. Now, let \(\phi \in \Inn(\fr{G})\), so that by Proposition~\ref{innergaugeprop1}, \(\phi = \Ad_v\) and \(\phi_\ast = \Ad_v\) for some \(v \in \Unit(Z(B) \cap \Cent_B(\Omega^1_B))\); let \(\nabla \in \pr{\fr{At}}\). By the proof of Proposition~\ref{inducedprop}, we find that \(\phi \triangleright \nabla - \nabla = \ad_\alpha\) for \(\alpha \coloneqq - v^\ast \cdot \dt{B}(v)\), so that \(\phi \triangleright \nabla - \nabla \in \Inn(\pr{\fr{at}})\) if and only if \(-v^\ast \cdot \dt{B}(v) \in Z(\Omega_B)\), if and only if \([\dt{B}(v),\dt{B}(B)] = \set{0}\). However, for all \(b \in B\),
\[
[\dt{B}(v),\dt{B}(b)] = \dt{B}(v) \wedge \dt{B}(b) + \dt{B}(b) \wedge \dt{B}(v) = \dt{B}(v \cdot \dt{B}(b)) - \dt{B}(\dt{B}(b) \cdot v) = 0,
\]
so that, indeed, \(\phi \triangleright \nabla - \nabla \in \Inn(\pr{\fr{at}})\). Hence, \(\Inn(\fr{G})\) acts trivially on \(\Out(\pr{\fr{At}})\).
\end{proof}
Finally, by Proposition~\ref{innprop}, we can characterize the variation of field strength under translation by inner prolongable gauge potentials as follows---note that field strength, \emph{a priori}, is affine quadratic, not linear.
\begin{definition}
Let \(\mathbf{A}\) be an inner prolongable gauge potential on \(P\) with respect to the second-order horizontal calculus \((\Omega_B,\dt{B};\Omega_{P,\mathrm{hor}})\). The \emph{relative field strength} of \(\mathbf{A}\) is the unique \(H\)-covariant \(\ast\)-derivation \(\mathbf{F}_{\mathrm{rel}}[\mathbf{A}] : P \to \Omega^2_{P,\mathrm{hor}}\) vanishing on \(B\), such that
\[
\forall \nabla \in \pr{\fr{At}}, \quad \mathbf{F}[\nabla+\mathbf{A}]-\mathbf{F}[\nabla] = \mathbf{F}_{\mathrm{rel}}[\mathbf{A}].
\]
\end{definition}
\begin{corollary}\label{relcor}
Let \(\alpha \in (\Omega^1_B)_{\mathrm{sa}} \cap \Zent(\Omega_B)\), so that \(\ad_\alpha\) defines an inner prolongable gauge potential on \(P\) with respect to \((\Omega_B,\dt{B};\Omega_{P,\mathrm{hor}})\). Then
\begin{equation}
\mathbf{F}_{\mathrm{rel}}[\ad_\alpha] = \ad_{-\ensuremath{\mathrm{i}}{}\dt{B}(\alpha)}.
\end{equation}
Hence, the map \(\mathbf{F}_{\mathrm{rel}} \coloneqq (\mathbf{A} \mapsto \mathbf{F}_{\mathrm{rel}}[\mathbf{A}]) : \Inn(\pr{\fr{at}}) \to \Der^H(P,\Omega^2_{P,\mathrm{hor}})\) is \(\pr{\fr{G}}\)-equivariant, \(\Inn(\fr{G})\)-invariant, and \(\mathbf{R}\)-linear.
\end{corollary}
\begin{example}\label{relex}
Let \(\phi \in \Inn(\fr{G})\); we claim that
\[
\forall \nabla \in \pr{\fr{At}}, \quad \mathbf{F}_{\mathrm{rel}}[\phi \triangleright \nabla - \nabla] = 0.
\]
Indeed, by Proposition~\ref{innergaugeprop1}, choose \(v \in \Unit(\Cent_B(B\oplus\Omega^1_B))\), such that \(\phi = \Ad_v\) and \(\phi_\ast = \Ad_v\), and set \(\alpha \coloneqq -v^\ast \cdot \dt{B}(v)\); by Proposition~\ref{inducedprop}, it follows that
\[
\forall \nabla \in \pr{\fr{At}}, \quad \phi \triangleright \nabla - \nabla = \ad_{\alpha}.
\]
Hence, by Corollary~\ref{relcor}, it suffices to show that \(\dt{B}(\alpha) = 0\). Since \(v \in \Unit(\Cent_B(B\oplus\Omega^1_B))\),
\[
\dt{B}(\alpha) = \dt{B}(v^\ast \cdot \dt{B}(v)) = \dt{B}(v^\ast) \wedge \dt{B}(v) = - v^\ast \cdot \dt{B}(v) \cdot v^\ast \wedge \dt{B}(v) = 0.
\]
\end{example}
\subsection{Reconstruction of quantum principal bundles to second order}
Recall that \(H\) is a Hopf \(\ast\)-algebra; let \(P\) be a principal left \(H\)-comodule \(\ast\)-algebra with \(\ast\)-subalgebra of coinvariants \(B \coloneqq \coinv{H}{P}\). Given a second-order horizontal calculus \((\Omega_B,\dt{B};\Omega_{P,\mathrm{hor}})\) on \(P\) and a bicovariant \(\ast\)-differential calculus \((\Omega_H,\dt{H})\) on \(H\) whose \textsc{fodc}{} is locally freeing for \(P\), we consider \emph{all} possible strongly \((H;\Omega_H,\dt{H})\)-principal \textsc{sodc}{} on \(P\) inducing the second-order horizontal calculus \((\Omega_B,\dt{B};\Omega_{P,\mathrm{hor}})\). This will permit us to justify our notions of prolongable gauge transformation, prolongable gauge potential, and field strength by refining the equivalence of groupoids of Proposition~\ref{equiv1}. Furthermore, when \((\Omega_H,\dt{H})\) is Woronowicz's canonical prolongation~\cite{Woronowicz}, this will yield a gauge-equivariant moduli space of strongly \((H;\Omega_H,\dt{H})\)-principal \textsc{sodc}{} on \(P\) inducing \((\Omega_B,\dt{B};\Omega_{P,\mathrm{hor}})\). Again, for relevant definitions from the basic theory of groupoids, see Appendix~\ref{Groupoids}.
Let us now fix a second-order horizontal calculus \((\Omega_B,\dt{B};\Omega_{P,\mathrm{hor}},\iota)\) on the principal left \(H\)-comodule \(\ast\)-algebra \(P\). Given a bicovariant \(\ast\)-differential calculus \((\Omega_H,\dt{H})\) on \(H\) whose \textsc{fodc}{} is locally freeing for \(P\), we construct the groupoid analogous to the groupoid \(\mathcal{G}[\Omega^1_H]\) of Definition~\ref{qpbdef} whose objects are strongly \((H;\Omega_H,\dt{H})\)-principal \textsc{sodc}{} on \(P\) and whose arrows give a abstract notion of gauge transformation adapted to general \textsc{sodc}{}. In what follows, let \((\Omega^{\leq 2}_H,\dt{H})\) denote the truncation of \((\Omega_H,\dt{H})\) to a bicovariant \textsc{sodc}{} on \(H\).
\begin{definition}
Let \((\Omega_H,\dt{H})\) be a bicovariant \(\ast\)-differential calculus on \(H\) whose \textsc{fodc}{} is locally freeing for \(P\). We define the \emph{prolongable abstract gauge groupoid} \(\mathcal{G}[\Omega^{\leq 2}_H]\) with respect to \((\Omega_H,\dt{H})\) as follows:
\begin{enumerate}
\item an object is a strongly \((H;\Omega_H,\dt{H})\)-principal \textsc{sodc} \((\Omega_P,\dt{P})\) on \(P\), such that the resulting strong quantum principal \(H;\Omega_H,\dt{H})\)-principal bundle \((P;\Omega_P,\dt{P})\) admits prolongable bimodule connections, and
\[
(\ker\mathrm{ver}[\dt{P}], \dt{B}(b) \mapsto \dt{P}(b))
\]
defines a horizontal lift of \(\Omega^1_B\) admitting a (necessarily unique) left \(H\)-covariant isomorphism
\[
C[\Omega_P] : P \oplus \ker\mathrm{ver}[\dt{P}] \oplus \left(\ker(\mathrm{ver}^{2,2}[\dt{P}]) \cap \ker(\mathrm{ver}^{2,1}[\dt{P}])\right) \to \Omega_{P,\mathrm{hor}}
\]
of graded \(\ast\)-algebras satisfying \(\rest{C[\Omega_P]}{P} = \id\) and \(C[\Omega_P] \circ \rest{\dt{P}}{B} = \iota \circ \dt{B}\).
\item given objects \((\Omega_1,\dt{1})\) and \((\Omega_2,\dt{2})\), an arrow \(f : (\Omega_1,\dt{1}) \to (\Omega_2,\dt{2})\) consists of a left \(H\)-covariant \(\ast\)-automorphism \(f : P \to P\), such that \(\rest{f}{B}=\id\) and
\[
f_\ast : \Omega^1_1 \to \Omega^1_2, \quad p \cdot \dt{1}(\p{p}) \cdot \pp{p} \mapsto f(p) \cdot \dt{2}(f(\p{p})) \cdot f(\pp{p})
\]
is well-defined and extends multiplicatively to a bijection \(f_\ast : \Omega_1 \to \Omega_2\);
\item composition of arrows is induced by composition of \(\ast\)-automorphisms of \(P\), and the identity of an object \((\Omega,\dt{})\) is given by \(\id_{(\Omega,\dt{})} \coloneqq \left(\id_P : (\Omega,\dt{}) \to (\Omega,\dt{})\right)\).
\end{enumerate}
Moreover, we define the star-injective homomorphism \(\mu[\Omega^{\leq 2}_H] : \mathcal{G}[\Omega^{\leq 2}_H] \to \Aut(P)\) by
\[
\forall \left(f : (\Omega_1,\dt{1}) \to (\Omega_2,\dt{2}) \right) \in \mathcal{G}[\Omega^{\leq 2}_H], \quad \mu[\Omega^{\leq 2}_H]\mleft(f : (\Omega_1,\dt{1}) \to (\Omega_2,\dt{2}) \mright) \coloneqq f.
\]
\end{definition}
\begin{remark}
Thus, the canonical second-order horizontal calculus \((\Omega_P,\dt{P})\) of \(\mathcal{G}[\Omega^{\leq 2}_H]\) induces the second-order horizontal calculus \((\Omega_B,\dt{B};\Omega_{P,\mathrm{hor}})\) up to the canonical isomorphism \(C[\Omega_P]\) of graded left \(H\)-comodule \(\ast\)-algebras.
\end{remark}
Just as in the first-order case, given a bicovariant \(\ast\)-differential calculus \((\Omega_H,\dt{H})\) on \(H\), we simultaneously consider prolongable bimodule connections on all strong quantum principal \((H;\Omega^1_H,\dt{H})\)-bundles induced from \(P\) by \textsc{sodc}{} in \(\Ob(\mathcal{G}[\Omega^{\leq 2}_H])\). Once more, it is straightforward to check that the prolongable abstract gauge groupoid admits a canonical action on this set of bimodule connections.
\begin{propositiondefinition}
Let \((\Omega_H,\dt{H})\) be a bicovariant \(\ast\)-differential calculus on \(H\) whose \textsc{fodc}{} is locally freeing for \(P\). Let \(\mathcal{A}[\Omega^{\leq 2}_H])\) be the set of all triples \((\Omega_P,\dt{P};\Pi)\), where \((\Omega_P,\dt{P}) \in \Ob(\mathcal{G}[\Omega^{\leq 2}_H]\) and \(\Pi\) is a prolongable bimodule connection on the strong quantum principal \((H;\Omega_H,\dt{H})\)-bundle \((P;\Omega_P,\dt{P})\); hence, let \(p[\Omega^{\leq 2}_H] : \mathcal{A}[\Omega^{\leq 2}_H] \to \Ob(\mathcal{G}[\Omega^{\leq 2}_H])\) be the canonical surjection given by
\[
\forall (\Omega_P,\dt{P};\Pi) \in \mathcal{A}[\Omega^{\leq 2}_H], \quad p[\Omega^{\leq 2}_H](\Omega_P,\dt{P};\Pi) \coloneqq (\Omega_P,\dt{P}).
\]
Then the \emph{abstract gauge action} is the action of \(\mathcal{G}[\Omega^{\leq 2}_H]\) on \(\mathcal{A}[\Omega^{\leq 2}_H]\) via \(p[\Omega^{\leq 2}_H]\) defined by
\begin{multline}
\forall \left(f : (\Omega_1,\dt{1}) \to (\Omega_2,\dt{2})\right) \in \mathcal{G}[\Omega^{\leq 2}_H], \, \forall (\Omega_1,\dt{1};\Pi) \in p[\Omega^1_H]^{-1}(\Omega_1,\dt{1}),\\ \left(f : (\Omega_1,\dt{1}) \to (\Omega_2,\dt{2})\right) \triangleright (\Omega_1,\dt{1};\Pi) \coloneqq (\Omega_2,\dt{2};f_\ast \circ \Pi \circ f_\ast^{-1}).
\end{multline}
Hence, the canonical covering \(\pi[\Omega^{\leq 2}_H] : \mathcal{G}[\Omega^{\leq 2}_H] \ltimes \mathcal{A}[\Omega^{\leq 2}_H] \to \mathcal{G}[\Omega^{\leq 2}_H]\) is given by
\begin{multline}
\forall \left(\left(f : (\Omega_1,\dt{1}) \to (\Omega_2,\dt{2})\right),(\Omega_1,\dt{1};\Pi)\right) \in \mathcal{G}[\Omega^{\leq 2}_H] \ltimes \mathcal{A}[\Omega^{\leq 2}_H], \\ \pi[\Omega^{\leq 2}_H]\mleft(\left(f : (\Omega_1,\dt{1}) \to (\Omega_2,\dt{2})\right),(\Omega_1,\dt{1};\Pi)\mright) \coloneqq \left(f : (\Omega_1,\dt{1}) \to (\Omega_2,\dt{2})\right).
\end{multline}
\end{propositiondefinition}
Once more, as a convenient abuse of notation, we will denote an arrow
\[
\left(\left(f : (\Omega_1,\dt{1}) \to (\Omega_2,\dt{2})\right),(\Omega_1,\dt{1};\Pi)\right)
\]
of the action groupoid \(\mathcal{G}[\Omega^{\leq 2}_H] \ltimes \mathcal{A}[\Omega^{\leq 2}_H]\) by
\[
f: (\Omega_1,\dt{1};\Pi_1) \to (\Omega_2,\dt{2};\Pi_2),
\]
where \(\Pi_2 \coloneqq \rest{f_\ast \circ \Pi \circ f_\ast^{-1}}{\Omega^1_2}\), so that, in particular,
\[
\pi[\Omega^{\leq 2}_H]\mleft(f: (\Omega_1,\dt{1};\Pi_1) \to (\Omega_2,\dt{2};\Pi_2)\mright) \coloneqq \left(f : (\Omega_1,\dt{1}) \to (\Omega_2,\dt{2})\right).
\]
\begin{remark}
There are an obvious star-injective homomorphism \(\mathcal{G}[\Omega^{\leq 2}_H] \to \mathcal{G}[\Omega^{1}_H]\) and an obvious surjection \(\mathcal{A}[\Omega^{\leq 2}_H] \twoheadrightarrow \mathcal{A}[\Omega^1_H]\) defined by restricting \textsc{sodc} to \textsc{fodc}, which, in turn, yield the following commutative diagram of groupoid homomorphisms:
\[
\begin{tikzcd}
\mathcal{G}[\Omega^{\leq 2}_H] \ltimes \mathcal{A}[\Omega^{\leq 2}_H]\arrow[r] \arrow[d, "\pi\lbrack\Omega^{\leq 2}_H\rbrack"'] &\mathcal{G}[\Omega^1_H] \ltimes \mathcal{A}[\Omega^1_H] \arrow[d,"\pi\lbrack\Omega^1_H\rbrack"] \\
\mathcal{G}[\Omega^{\leq 2}_H] \arrow[r] \arrow[d, "\mu\lbrack\Omega^{\leq 2}_H\rbrack"'] &\mathcal{G}[\Omega^1_H] \arrow[d, "\mu\lbrack\Omega^1_H\rbrack"] \\
\Aut(P) \arrow[r, hookleftarrow] &\fr{G}
\end{tikzcd}
\]
here, \(\fr{G}\) is the gauge group of the principal left \(H\)-comodule \(\ast\)-algebra \(P\) with respect to the first-order horizontal calculus \((\Omega^1_B,\dt{B};\Omega^1_{P,\mathrm{hor}})\).
Hence, it follows that \(\ran \mu[\Omega^{\leq 2}_H] \subseteq \fr{G}\).
\end{remark}
Now, let \(\pr{\fr{G}}\) and \(\pr{\fr{At}}\) respectively denote the prolongable gauge group and prolongable Atiyah space of the principal left \(H\)-comodule \(\ast\)-algebra \(P\) with respect to the second-order horizontal calculus \((\Omega_B,\dt{B};\Omega_{P,\mathrm{hor}})\). Our goal is to promote the isomorphism of Proposition~\ref{isothm} to an explicit equivalence of categories that suitably refines the equivalence of Proposition~\ref{equiv1}. This will first require a characterisation of those prolongable gauge potentials on \(P\) with respect to \((\Omega_B,\dt{B};\Omega_{P,\mathrm{hor}})\) that correctly induce elements of \(\mathcal{A}[\Omega^{\leq 2}_H]\). In what follows, let \((\Omega_{P,\mathrm{ver}},\dv{P})\) denote the vertical calculus on \(P\) induced by a bicovariant \(\ast\)-differential calculus on \(H\) whose \textsc{fodc}{} is locally freeing for \(P\).
\begin{definition}
Let \((\Omega^1_H,\dt{H})\) be a bicovariant \textsc{fodc}{} on \(H\) that is locally freeing for \(P\). We say that a prolongable gauge potential \(\nabla \in \dva{\fr{At}}\) on \(P\) with respect to \((\Omega_B,\dt{B};\Omega_{P,\mathrm{hor}})\) is \emph{\((\Omega^1_H,\dt{H})\)-adapted} whenever its field strength \(\mathbf{F}[\nabla]\) is given by
\[\mathbf{F}[\nabla] = F[\nabla] \circ \dv{P}\]
for a (necessarily unique) left \(H\)-covariant morphism \(F[\nabla] : \Omega^1_{P,\mathrm{ver}} \to \Omega^2_{P,\mathrm{hor}}\) of \(P\)-\(\ast\)-bimod\-ules; in this case, we call \(F[\nabla]\) the \emph{curvature \(2\)-form} of \(\nabla\). We define the \emph{\((\Omega^1_H,\dt{H})\)-adapted prolongable Atiyah space} of \(P\) with respect to \((\Omega_B,\dt{B};\Omega_{P,\mathrm{hor}})\) to be the subset \(\pr{\fr{At}}[\Omega^1_H]\)of all \((\Omega^1_H,\dt{H})\)-adapted prolongable gauge potentials on \(P\).
\end{definition}
\begin{proposition}[cf.\ \DJ{}ur\dj{}evi\'{c}~\cite{Dj10}*{Prop. 26 and 27}]\label{sodciso}
Let \((\Omega_H,\dt{H})\) be a bicovariant \(\ast\)-differen\-tial calculus on \(H\) whose \textsc{fodc}{} is locally freeing for \(P\); let \(\Lambda_H\) be the resulting graded left crossed \(H\)-module \(\ast\)-algebra of right \(H\)-coinvariant forms, and set \(\Omega_{P,\oplus} \coloneqq \Lambda_H \widehat{\otimes}^{\leq 2} \Omega_{P,\mathrm{hor}}\). Let \(\nabla\) be a gauge potential on \(P\) with respect to \((\Omega^1_B,\dt{B};\Omega^1_{P,\mathrm{hor}})\). The following are equivalent:
\begin{enumerate}
\item\label{sodciso1} the left \(H\)-covariant \(\ast\)-derivation \(\dt{P,\nabla} : P \to \Omega^1_{P,\oplus}\) given by
\[
\forall p \in P, \quad \dt{P,\nabla}(p) \coloneqq \dv{P}(p) + \nabla(p)
\]
admits a (necessarily unique) extension to a left \(H\)-covariant degree \(1\) \(\ast\)-derivation \[\dt{P,\nabla} : \Omega_{P,\oplus} \to \Omega_{P,\oplus},\] such that \((\Omega_{P,\oplus},\dt{P,\nabla},\Pi_\oplus) \in \Ob(\mathcal{G}[\Omega^{\leq 2}_H] \ltimes \mathcal{A}[\Omega^{\leq 2}_H])\), where \(\Pi_\oplus : \Omega^1_{P,\oplus} \to \Omega^1_{P,\oplus}\) is the projection onto \(\Omega^1_{P,\mathrm{ver}}\) along \(\Omega^1_{P,\mathrm{hor}}\);
\item\label{sodciso2} the gauge potential \(\nabla\) is prolongable with respect to \((\Omega_B,\dt{B};\Omega_{P,\mathrm{hor}})\) and \((\Omega^1_H,\dt{H})\)-adapted.
\end{enumerate}
If either (and hence both) of the above conditions are satisfied, the prolongable bimodule connection \(\Pi_{\oplus}\) on the strong quantum principal \((H;\Omega_H,\dt{H})\)-bundle \((P;\Omega_{P,\oplus},\dt{P,\oplus})\) satisfies
\[
\nabla_{\Pi_\oplus} = \nabla, \quad F_{\Pi_\oplus} = F[\nabla].
\]
\end{proposition}
\begin{proof}
First, suppose that \(\dt{P,\nabla}\) extends to \(\Omega_{P,\oplus}\) and that \((\Omega_{P,\oplus},\dt{P,\nabla};\Pi_\oplus)\) defines an object of \(\mathcal{G}[\Omega^1_H] \ltimes \mathcal{A}[\Omega^1_H]\), so that \(\nabla_{\Pi} = \nabla\) and
\[
\mathrm{ver}[\dt{P,\nabla}] = \Proj_1 : \Omega^1_{P,\oplus}\ = \Omega^1_{P,\mathrm{ver}} \oplus \Omega^1_{P,\mathrm{hor}} \to \Omega^1_{P,\mathrm{ver}}.
\]
For every \(p,q \in P\) and \(b \in B\),
\begin{align*}
\nabla_{\Pi_\oplus}(p \cdot \dt{B}(b) \cdot q) &= \nabla_{\Pi_\oplus}(p) \wedge \dt{B}(b) \cdot q + p \cdot \nabla_{\Pi_\oplus}(\dt{B}(b)) \cdot q - p\cdot\dt{B}(b) \wedge \nabla_{\Proj_1}(q)\\
&= \nabla(p) \wedge \dt{B}(b) \cdot q - p \cdot \dt{B}(b) \wedge \nabla(q)
\end{align*}
so that \(\nabla\) is prolongable with \(\dva{\nabla} = \rest{\nabla_{\Pi_\oplus}}{\Omega^1_{P,\mathrm{hor}}}\) and \((\Omega^1_H,\dt{H})\)-adapted with \(F[\nabla] = F_{\Pi_\oplus}\).
Now, suppose that \(\nabla\) is prolongable and \((\Omega^1_H,\dt{H})\)-adapted. We know that \((P;\Omega^1_{P,\oplus};\Pi_\oplus)\) defines an element of \(\mathcal{A}[\Omega^1_H]\) with
\[
\mathrm{ver}[\dt{P,\nabla}] = \Proj_1 : \Omega^1_{P,\oplus} = \Omega^1_{P,\mathrm{ver}} \oplus \Omega^1_{P,\mathrm{hor}} \to \Omega^1_{P,\mathrm{ver}}.
\]
By mild abuse of notation, let \(\dt{P,\nabla}: \Omega^1_{P,\oplus} \to \Omega^2_{P,\oplus}\) be the left \(H\)-covariant degree \(1\) map defined by
\begin{multline*}
\forall \omega \in \Lambda^1_H, \, \forall p \in P, \, \forall \alpha \in \Omega^1_{P,\mathrm{hor}}, \\ \dt{P,\nabla}(\omega \cdot p + \alpha) \coloneqq \dv{P}(\omega \cdot p) - \ensuremath{\mathrm{i}}{}\,F[\nabla](\omega \cdot p)- \omega \wedge \nabla(p) + \varpi(\ca{\alpha}{-1}) \wedge \ca{\alpha}{0} + \pr{\nabla}(\alpha).
\end{multline*}
While a routine calculation using the properties of \(\dv{P}\), \(\nabla\) and \(\varpi\) shows that \(\dt{P,\nabla}\) as defined on \(P\) and \(\Omega^1_{P,\oplus}\), respectively, yields a \(\ast\)-derivation \(\dt{P,\nabla} : \Omega_{P,\oplus} \to \Omega_{P,\oplus}\) of degree \(1\), but it is less obvious that \(\dt{P,\nabla}^2 = 0\). However, for all \(p \in P\),
\begin{align*}
\dt{P,\nabla}^2(p) &= \dt{P,\nabla}(\varpi(\ca{p}{-1}) \cdot \ca{p}{0} + \nabla(p))\\
&= \dv{P}(\varpi(\ca{p}{-1}) \cdot \ca{p}{0}) - \ensuremath{\mathrm{i}}{}\,F[\nabla](\varpi(\ca{p}{-1}) \cdot p) - \varpi(\ca{p}{-1}) \wedge \nabla(\ca{p}{0})\\&\quad\quad + \varpi(\ca{\nabla(p)}{-1}) \wedge \ca{\nabla(p)}{0} + \pr{\nabla} \circ \nabla(p)\\
&= \dv{P}(\dv{P}(p)) -\ensuremath{\mathrm{i}}{}\,F[\nabla](\dv{P}(p))+\ensuremath{\mathrm{i}}{}\,F[\nabla](\dv{P}(p)) = 0,
\end{align*}
as required. Given the explicit form of \(\mathrm{ver}[\dt{P,\nabla}]\) and \(\rest{\dt{P,\nabla}}{\Omega^1_{P,\oplus}}\), one can now check that
\begin{gather*}
\mathrm{ver}^{2,2}[\dt{P,\nabla}] = \Proj_1 : \Omega^2_{P,\oplus} = \Omega^2_{P,\mathrm{ver}} \oplus \Lambda^1_H \otimes \Omega^1_{P,\mathrm{hor}} \oplus \Omega^2_{P,\mathrm{hor}} \to \Omega^2_{P,\mathrm{ver}},\\
\mathrm{ver}^{2,1}[\dt{P,\nabla}] = \Proj_2 : \Omega^2_{P,\oplus} = \Omega^2_{P,\mathrm{ver}} \oplus \Lambda^1_H \otimes \Omega^1_{P,\mathrm{hor}} \oplus \Omega^2_{P,\mathrm{hor}} \to \Lambda^1_H \otimes \Omega^1_{P,\mathrm{hor}},
\end{gather*}
so that \((\Omega_{P,\oplus},\dt{P,\nabla})\) defines an object of \(\mathcal{G}[\Omega^{\leq 2}_H]\); given the explicit form of \(\Pi_\oplus\), it now follows that the connection \(\Pi\) is totally prolongable with
\[
\Pi_\oplus \wedge \Pi_\oplus = \Proj_1 \oplus{} 0 \oplus{} 0, \quad \Pi_\oplus \wedge \id + \id \wedge \Pi_\oplus = \Proj_1 \oplus \Proj_2 \oplus{} 0
\]
with respect to the decomposition \(\Omega^2_{P,\oplus} = \Omega^2_{P,\mathrm{ver}} \oplus \Lambda^1_H \otimes \Omega^1_{P,\mathrm{hor}} \oplus \Omega^2_{P,\mathrm{hor}}\).
\end{proof}
Given a bicovariant \textsc{fodc}{} \((\Omega^1_H,\dt{H})\) on \(H\) that is locally freeing for \(P\), Propositions~\ref{fieldstrengthprop} and~\ref{astver} now guarantee that the \((\Omega^1_H,\dt{H})\)-adapted prolongable Atiyah space \(\pr{\fr{At}}[\Omega^{1}_H]\) is a \(\pr{\fr{G}}\)-invariant subset of the prolongable Atiyah space \(\pr{\fr{At}}\) on which the assignment of curvature \(2\)-forms defines a \(\pr{\fr{G}}\)-equivariant map.
\begin{proposition}
Let \((\Omega^1_H,\dt{H})\) be a bicovariant \textsc{fodc}{} on \(H\) that is locally freeing for \(P\). The \((\Omega^1_H,\dt{H})\)-adapted prolongable Atiyah space \(\pr{\fr{At}}[\Omega^{1}_H]\) is a \(\dva{\fr{G}}\)-invariant subset of the prolongable Atiyah space \(\pr{\fr{At}}\), and the assignment
\[
F \coloneqq (\nabla \mapsto F[\nabla]) : \pr{\fr{at}}_{\can}[\Omega^1_H] \to \Hom_P^H(\Omega^1_{P,\mathrm{ver}},\Omega^2_{P,\mathrm{hor}})
\]
defines a \(\dva{\fr{G}}\)-equivariant map.
\end{proposition}
\begin{remark}
The \(\pr{\fr{G}}\)-invariant subset \(\pr{\fr{At}}[\Omega^{1}_H]\) of the affine space \(\pr{\fr{At}}\) is defined by an affine-quadratic constraint, and as such can be viewed as a \(\pr{\fr{G}}\)-invariant affine quadric subset of \(\pr{\fr{At}}\). In general, one should not expect \(\pr{\fr{At}}[\Omega^1_H]\) to be an affine-linear subspace of \(\pr{\fr{At}}\).
\end{remark}
\begin{remark}
It follows that the resulting action groupoid \(\pr{\fr{G}} \ltimes \pr{\fr{At}}[\Omega^1_H]\) defines a subgroupoid of the action groupoid \(\fr{G} \ltimes \fr{At}\), where \(\fr{G}\) and \(\fr{At}\) are, respectively, the gauge group and Atiyah space of \(P\) with respect to the first-order horizontal calculus \((\Omega^1_B,\dt{B};\Omega^1_{P,\mathrm{hor}})\).
\end{remark}
Suppose that \((\Omega^1_H,\dt{H})\) is a bicovariant \textsc{fodc}{} on \(H\) that is locally freeing for \(P\), and let \((\Omega_H,\dt{H})\) be any bicovariant prolongation of \((\Omega^1_H,\dt{H})\). We now promote Proposition~\ref{isothm} to an explicit equivalence of categories---refining the equivalence of Proposition~\ref{equiv1}---that realises the concrete action groupoid \(\pr{\fr{G}} \ltimes \pr{\fr{At}}[\Omega^1_H]\), which is independent of the choice of bicovariant prolongation \((\Omega_H,\dt{H})\), as a deformation retraction of the action groupoid \(\mathcal{G}[\Omega^{\leq 2}_H] \ltimes \mathcal{A}[\Omega^{\leq 2}_H]\) of the abstract gauge action on prolongable bimodule connections. This rigorously justifies identifying the action of \(\pr{\fr{G}}\) on \(\pr{\fr{At}}[\Omega^1_H]\) as the affine action of global gauge transformations on principal connections for the quantum principal \(H\)-bundle \(P\) with respect to the bicovariant \(\ast\)-differential calculus \((\Omega_H,\dt{H})\) and the second-order horizontal calculus \((\Omega_B,\dt{B};\Omega_{P,\mathrm{hor}})\). This follows from the proof of Proposition~\ref{equiv1}, \emph{mutatis mutandis}, together with Proposition~\ref{sodciso} and the proof of Proposition~\ref{isothm}.
\begin{proposition}\label{sodcequiv}
Let \((\Omega_H,\dt{H})\) be a bicovariant \(\ast\)-differential calculus on \(H\) whole \textsc{fodc}{} is locally freeing for \(P\). The groupoid homomorphism
\[
\Sigma[\Omega^{\leq 2}_H] : \pr{\fr{G}} \ltimes \pr{\fr{At}}[\Omega^{1}_H] \to \mathcal{G}[\Omega^{\leq 2}_H] \ltimes \mathcal{A}[\Omega^{\leq 2}_H]
\]
given by
\begin{multline}
\forall (\phi,\nabla) \in \pr{\fr{G}} \ltimes \pr{\fr{At}}[\Omega^{1}_H], \\ \Sigma[\Omega^{\leq 2}_H](\phi, \nabla) \coloneqq \left(\phi : (\Omega_{P,\oplus},\dt{P,\nabla},\Pi_{\oplus}) \to (\Omega_{P,\oplus},\dt{P,\phi\triangleright\nabla},\Pi_{\oplus})\right)
\end{multline}
is an equivalence of groupoids with left inverse and homotopy inverse \(A[\Omega^{\leq 2}_H]\) given by
\begin{multline}
\forall \left(f : (\Omega_1,\dt{1};\Pi_1) \to (\Omega_2,\dt{2};\Pi_2)\right) \in \mathcal{G}[\Omega^{\leq 2}_H] \ltimes \mathcal{A}[\Omega^{\leq 2}_H],\\
A[\Omega^{\leq 2}_H]\mleft(f : (\Omega_1,\dt{1};\Pi_1) \to (\Omega_2,\dt{2};\Pi_2)\mright) \coloneqq \left(f, C[\Omega_1] \circ \nabla_{\Pi_1}\right).
\end{multline}
In particular, there exists a homotopy \(\eta[\Omega^{\leq 2}_H] : \id_{\mathcal{G}[\Omega^{\leq 2}_H] \ltimes \mathcal{A}[\Omega^{\leq 2}_H]} \Rightarrow \Sigma[\Omega^{\leq 2}_H] \circ A[\Omega^{\leq 2}_H]\), which is necessarily unique, such that
\begin{equation}
\forall (\Omega_P,\dt{P};\Pi) \in \Ob(\mathcal{G}[\Omega^{\leq 2}_H] \ltimes \mathcal{A}[\Omega^{\leq 2}_H]), \quad \mu[\Omega^{\leq 2}_H] \circ \pi[\Omega^{\leq 2}_H]\mleft(\eta[\Omega^{\leq 2}_H]_{(\Omega_P,\dt{P};\Pi)}\mright) = \id_P.
\end{equation}
\end{proposition}
This justifies the identification of the prolongable gauge group \(\pr{\fr{G}}\) as \emph{the} gauge group of the quantum principal \(H\)-bundle \(P\) with respect to the bicovariant \(\ast\)-differen\-tial calculus \((\Omega_H,\dt{H})\) on \(H\) and the second-order horizontal calculus \((\Omega_B,\dt{B};\Omega_{P,\mathrm{hor}})\).
\begin{corollary}
Let \((\Omega_H,\dt{H})\) be a bicovariant \(\ast\)-differential calculus on \(H\) whole \textsc{fodc}{} is locally freeing for \(P\). The star-injective groupoid homomorphism \(\mu[\Omega^{\leq 2}_H] : \mathcal{G}[\Omega^{\leq 2}_H] \to \Aut(P)\) has range \(\pr{\fr{G}}\), so that, after restriction of codomain,
\[
\mu[\Omega^{\leq 2}_H] : \mathcal{G}[\Omega^{\leq 2}_H] \to \pr{\fr{G}}, \quad \mu[\Omega^{\leq 2}_H] \circ \pi[\Omega^{\leq 2}_H] : \mathcal{G}[\Omega^{\leq 2}_H] \ltimes \mathcal{A}[\Omega^{\leq 2}_H] \to \pr{\fr{G}}
\]
both define coverings of groupoids.
\end{corollary}
\begin{remark}
The groupoid equivalences \(\Sigma[\Omega^1_H]\), \(\Sigma[\Omega^{\leq 2}_H]\), \(A[\Omega^1_H]\), and \(A[\Omega^{\leq 2}_H]\), the subgroupoid inclusion \(\pr{\fr{G}} \ltimes \pr{\fr{At}}[\Omega^1_H] \hookrightarrow \fr{G} \ltimes \fr{At}\), and the canonical star-injective groupoid homomorphisms \(\mathcal{G}[\Omega^{\leq 2}_H] \ltimes \mathcal{A}[\Omega^{\leq 2}_H] \to \mathcal{G}[\Omega^1_H] \ltimes \mathcal{A}[\Omega^1_H]\) and \(\mathcal{G}[\Omega^{\leq 2}_H] \to \mathcal{G}[\Omega^1_H]\) fit into the following commutative diagrams in the category of groupoids:
\[
\begin{tikzcd}[column sep=small]
\pr{\fr{G}} \ltimes \pr{\fr{At}}[\Omega^{1}_H] \arrow[r, hookrightarrow] \arrow[d,"\Sigma\lbrack\Omega^{\leq 2}_H\rbrack"'] & \fr{G} \ltimes \fr{At} \arrow[d,"\Sigma\lbrack\Omega^1_H\rbrack"] \\ \mathcal{G}[\Omega^{\leq 2}_H] \ltimes \mathcal{A}[\Omega^{\leq 2}_H] \arrow[r] & \mathcal{G}[\Omega^1_H] \ltimes \mathcal{A}[\Omega^1_H]
\end{tikzcd}\hfill
\begin{tikzcd}[column sep=small]
\pr{\fr{G}} \ltimes \pr{\fr{At}}[\Omega^1_H] \arrow[r, hookrightarrow] \arrow[d,leftarrow,"A\lbrack\Omega^{\leq 2}_H\rbrack"'] & \fr{G} \ltimes \fr{At} \arrow[d,leftarrow,"A\lbrack\Omega^1_H\rbrack"]\\ \mathcal{G}[\Omega^{\leq 2}_H] \ltimes \mathcal{A}[\Omega^{\leq 2}_H] \arrow[r] & \mathcal{G}[\Omega^1_H] \ltimes \mathcal{A}[\Omega^1_H]
\end{tikzcd}
\]
Moreover, the homotopies \[\eta[\Omega^1_H] : \id_{\mathcal{G}[\Omega^1_H] \ltimes \mathcal{A}[\Omega^1_H]} \Rightarrow \Sigma[\Omega^1_H] \circ A[\Omega^1_H], \quad \eta[\Omega^{\leq 2}_H] : \id_{\mathcal{G}[\Omega^{\leq 2}_H] \ltimes \mathcal{A}[\Omega^{\leq 2}_H]} \Rightarrow \Sigma[\Omega^{\leq 2}_H] \circ A[\Omega^{\leq 2}_H]\]
and the natural transformations
\[
\id_{\mathcal{G}[\Omega^{\leq 2}_H] \ltimes \mathcal{A}[\Omega^{\leq 2}_H]} \Rightarrow \id_{\mathcal{G}[\Omega^1_H] \ltimes \mathcal{A}[\Omega^1_H]}, \quad \Sigma[\Omega^{\leq 2}_H] \circ A[\Omega^{\leq 2}_H] \Rightarrow \Sigma[\Omega^1_H] \circ A[\Omega^1_H]
\]
induced by the subgroupoid inclusion \(\pr{\fr{G}} \ltimes \pr{\fr{At}}[\Omega^1_H] \hookrightarrow \fr{G} \ltimes \fr{At}\) and the canonical star-injective groupoid homomorphisms \(\mathcal{G}[\Omega^{\leq 2}_H] \ltimes \mathcal{A}[\Omega^{\leq 2}_H] \to \mathcal{G}[\Omega^1_H] \ltimes \mathcal{A}[\Omega^1_H]\) fit into the following commutative diagram in the category of functors:
\[
\begin{tikzcd}
\id_{\mathcal{G}[\Omega^{\leq 2}_H] \ltimes \mathcal{A}[\Omega^{\leq 2}_H]} \arrow[r, Rightarrow] \arrow[d,Rightarrow,"\eta\lbrack\Omega^{\leq 2}_H\rbrack"'] & \id_{\mathcal{G}[\Omega^1_H] \ltimes \mathcal{A}[\Omega^1_H]} \arrow[d,Rightarrow,"\eta\lbrack \Omega^1_H \rbrack"]\\
\Sigma[\Omega^{\leq 2}_H] \circ A[\Omega^{\leq 2}_H] \arrow[r,Rightarrow] & \Sigma[\Omega^1_H] \circ A[\Omega^1_H]
\end{tikzcd}
\]
\end{remark}
Given a bicovariant \(\ast\)-differential calculus \((\Omega_H,\dt{H})\) on \(H\) whose \textsc{fodc}{} is locally freeing for \(P\), we now use the weak equivalence \(\Sigma[\Omega^{\leq 2}_H]\) of Proposition~\ref{sodcequiv} to construct a \(\pr{\fr{G}}\)-equivariant moduli space of strongly \((H;\Omega_H,\dt{H})\)-principal \textsc{sodc}{} on \(P\) inducing the second-order horizontal calculus \((\Omega_B,\dt{B};\Omega_{P,\mathrm{hor}})\) on \(P\). To do so, we shall need \((\Omega_H,\dt{H})\) to be Woronowicz's \emph{canonical} prolongation~\cite{Woronowicz}*{\S\S 3--4} of its \textsc{fodc}{} \((\Omega^1_H,\dt{H})\), so that
\begin{equation}\label{wor}
\Lambda^2_H \coloneqq (\Omega^2_H)^{\operatorname{co}H} = \frac{\Lambda^1_H \otimes_{\mathbb{C}} \Lambda^1_H}{\Span\set{ \mu \otimes \nu + \ca{\mu}{-1} \triangleright \nu \otimes \ca{\mu}{0} \given \mu,\nu \in \Lambda^1_H}}
\end{equation}
by~\cite{Majid17}*{\S 2}. For us, the distinguishing feature of this particular prolongation (as opposed to, e.g., the maximal prolongation) will be the \(H\)-equivariance of \(\dt{H}\) when restricted to \(\Lambda \coloneqq \Omega^{\operatorname{co}H}\), which one can view as \(\Ad\)-equivariance of the dualised Lie bracket.
\begin{proposition}\label{worprop}
Let \((\Omega^1_H,\dt{H})\) be a bicovariant \textsc{fodc}{} on \(H\), let \((\Omega,\dt{})\) be a bicovariant prolongation of \((\Omega^1_H,\dt{H})\), and let \(\Lambda \coloneqq \Omega^{\operatorname{co}H}\) denote the corresponding graded left crossed \(H\)-module \(\ast\)-algebra of right coinvariant forms, where \(\Lambda^0 = \mathbf{C}\). The following are equivalent:
\begin{enumerate}
\item\label{wor1} the restriction of \(\dt{}\) to \(\Lambda^1 = \Lambda^1_H\) satisfies
\[
\forall h \in H, \, \forall \mu \in \Lambda^1_H, \quad \dt{H}(h \triangleright \mu) = h \triangleright \dt{H}(\mu);
\]
\item\label{wor2} the product \(\Lambda^1 \otimes_{\mathbf{C}} \Lambda^1 \to \Lambda^2\) satisfies the braided commutation relation
\[
\forall \mu,\nu \in \Lambda^1_H, \quad \mu \wedge \nu + \ca{\mu}{-1} \triangleright \nu \wedge \ca{\mu}{0} = 0.
\]
\end{enumerate}
\end{proposition}
\begin{proof}
On the one hand, suppose that Condition~\ref{wor1} holds. Then for all \(h,k \in H\), by \eqref{deg2eq},
\begin{align*}
\ca{\varpi(h)}{-1}\triangleright\varpi(k) \wedge \ca{\varpi(h)}{0}
&= \cm{h}{1}S(\cm{h}{3}) \triangleright \varpi(k) \wedge \varpi(\cm{h}{2})\\
&= - \cm{h}{1}S(\cm{h}{4}) \triangleright \varpi(k) \wedge \cm{h}{2} \triangleright \varpi(S(\cm{h}{3}))\\
&= - \cm{h}{1} \triangleright \left(S(\cm{h}{3}) \triangleright \varpi(k) \wedge \varpi(S(\cm{h}{2}))\right)\\
&= \cm{h}{1}\triangleright \left(\varpi(S(\cm{h}{3})) \wedge S(\cm{h}{2}) \triangleright \varpi(k)\right)\\
&= \cm{h}{1} \triangleright \varpi(S(\cm{h}{4})) \wedge \cm{h}{2} S(\cm{h}{3}) \triangleright \varpi(k)\\
&= - \varpi(h) \wedge \varpi(k),
\end{align*}
so that Condition~\ref{wor2} is satisfied. On the other hand, suppose that Condition~\ref{wor2} is satisfied. Then for all \(h, k \in H\), by \eqref{deg2eq},
\begin{align*}
\dt{H}(h \triangleright \varpi(k))
- h \triangleright \dt{H}\varpi(k) &= \cm{h}{1}\triangleright\varpi(k) \wedge\varpi(\cm{h}{2}) + \varpi(\cm{h}{1}) \wedge \cm{h}{2}\triangleright\varpi(k)\\
&= \cm{h}{1}\triangleright\varpi(k) \wedge\varpi(\cm{h}{2}) - \cm{h}{1}S(\cm{h}{3})\cm{h}{4}\triangleright\varpi(k)\wedge \varpi(\cm{h}{2})\\
&= 0
\end{align*}
so that Condition~\ref{wor1} is satisfied.
\end{proof}
In the case that \((\Omega_H,\dt{H})\) is the canonical prolongation of \((\Omega^1_H,\dt{H})\), we shall construct the aforementioned \(\pr{\fr{G}}\)-equivariant moduli space using the following distinguished set of relative gauge potentials, which will indeed turn out to be a \(\pr{\fr{G}}\)-invariant subspace of prolongable relative gauge potentials.
\begin{definition}
Let \((\Omega^1_H,\dt{H})\) be a bicovariant \textsc{fodc}{} on \(H\) that is locally freeing for \(P\). Let \(\mathbf{A}\) be a prolongable relative gauge potential on \(P\) with respect to the second-order horizontal calculus \((\Omega_B,\dt{B};\Omega_{P,\mathrm{hor}})\). We say that \(\mathbf{A}\) is \emph{canonically \((\Omega_H^{1},\dt{H})\)-adapted} if and only if it is \((\Omega^1_H,\dt{H})\)-adapted and satisfies both of the following:
\begin{gather}
\forall \mu \in \Lambda^1_H, \, \forall \alpha \in \Omega^1_B, \quad \left[\omega[\mathbf{A}](\mu),\alpha\right] = 0,\label{toteq1}\\
\forall \mu,\nu \in \Lambda^1_H, \quad \omega[\mathbf{A}](\mu) \wedge \omega[\mathbf{A}](\nu) + \omega[\mathbf{A}](\ca{\mu}{-1}\triangleright\nu) \wedge \omega[\mathbf{A}](\ca{\mu}{0}) = 0.\label{toteq2}
\end{gather}
We denote by \(\pr{\fr{at}}_{\can}[\Omega^{1}_H]\) the subset of all canonically \((\Omega^{1}_H,\dt{H})\)-adapted relative gauge potentials on \(P\) with respect to \((\Omega_B,\dt{B};\Omega_{P,\mathrm{hor}})\).
\end{definition}
Now, let \((\Omega_1,\dt{1})\), \((\Omega_2,\dt{2}) \in \Ob(\mathcal{G}[\Omega^{\leq 2}_H])\), where \((\Omega_H,\dt{H})\) is the canonical prolongation of a bicovariant \textsc{fodc}{} \((\Omega^1_H,\dt{H})\) on \(H\) that is locally freeing for \(P\). Observe that the left \(H\)-covariant \textsc{sodc} \((\Omega_1,\dt{1})\) and \((\Omega_2,\dt{2})\) on the principal left \(H\)-comodule \(\ast\)-algebra \(P\) are isomorphic if and only if \(\id_P : (\Omega_1,\dt{1}) \to (\Omega_2,\dt{2})\) is an arrow in \(\mathcal{G}[\Omega^{\leq 2}_H]\). Since the subgroupoid of such arrows is precisely \(\ker\mu[\Omega^{\leq 2}_H]\), it follows that \((\Omega_1,\dt{1})\) and \((\Omega_2,\dt{2})\) are isomorphic if and only if they define the same object in the quotient \(\mathcal{G}[\Omega^{\leq 2}_H]/\ker\mu[\Omega^{\leq 2}_H]\), which will turn out to be well-defined and canonically isomorphic to the action groupoid \(\pr{\fr{G}} \ltimes \left(\pr{\fr{At}}[\Omega^1_H]/\pr{\fr{at}}_{\can}[\Omega^{1}_H]\right)\). Thus, the quadric subset \(\pr{\fr{At}}[\Omega^{\leq 1}_H]/\pr{\fr{at}}_{\can}[\Omega^{1}_H]\) of the quotient affine space \(\pr{\fr{At}}/\pr{\fr{at}}_{\can}[\Omega^{1}_H]\) yields the desired \(\pr{\fr{G}}\)-equivariant affine moduli space of strongly \((\Omega_H,\dt{H})\)-principal \textsc{sodc}{} on \(P\) inducing \((\Omega_B,\dt{B};\Omega_{P,\mathrm{hor}})\).
\begin{theorem}\label{qpbthm}
Let \((\Omega^1_H,\dt{H})\) be a bicovariant \textsc{fodc}{} on \(H\) that is locally freeing for \(P\), and let \((\Omega_H,\dt{H})\) be its canonical prolongation. Suppose that \(\pr{\fr{At}}[\Omega^1_H]\) is non-empty. Let \(\pr{\fr{at}}\) be the space of prolongable gauge potentials on \(P\) with respect to \((\Omega_B,\dt{B};\Omega_{P,\mathrm{hor}})\), and let \(\fr{at}[\Omega^1_H]\) be subspace of \((\Omega^1_H,\dt{H})\)-adapted elements. The set \(\pr{\fr{at}}_{\can}[\Omega^{1}_H]\) defines a \(\pr{\fr{G}}\)-invariant subspace of \(\fr{at}[\Omega^1_H] \cap \pr{\fr{at}}\), such that \(\pr{\fr{At}}[\Omega_H]\) is invariant under translation by \(\pr{\fr{at}}_{\can}[\Omega^{1}_H]\); the subgroupoid \(\ker\mu[\Omega^{\leq 2}_H]\) of \(\mathcal{G}[\Omega^{\leq 2}_H]\) is wide and has trivial isotropy groups, so that the quotient groupoid \(\mathcal{G}[\Omega^{\leq 2}_H]/\ker\mu[\Omega^{\leq 2}_H]\) is well-defined; and there exists a unique isomorphism \[\pr{\tilde{\Sigma}}[\Omega^{\leq 2}_H] : \pr{\fr{G}} \ltimes \left(\pr{\fr{At}}[\Omega^1_H]/\pr{\fr{at}}_{\can}[\Omega^{1}_H]\right) \to \mathcal{G}[\Omega^{\leq 2}_H]/\ker\mu[\Omega^{\leq 2}_H],\] such that
\begin{multline}
\forall (\phi,\nabla) \in \pr{\fr{G}} \ltimes \pr{\fr{At}}[\Omega^1_H], \\ \pr{\tilde{\Sigma}}[\Omega^{\leq 2}_H]\mleft(\phi,\nabla+\pr{\fr{at}}_{\can}[\Omega^{1}_H]\mright) = \left[\pi[\Omega^{\leq 2}_H] \circ \Sigma[\Omega^{\leq 2}_H](\phi,\nabla)\right]_{\ker\mu[\Omega^{\leq 2}_H]}.
\end{multline}
\end{theorem}
As a preliminary to the proof of this theorem, we shall prove the following sequence of lemmata. In what follows, we shall assume the hypotheses of Theorem~\ref{qpbthm}; in particular, \((\Omega_{P,\mathrm{ver}},\dv{P})\) will denote the second-order vertical calculus induced by \((\Omega_H,\dt{H})\).
\begin{lemma}\label{qpblem1}
Let \(N : \Omega^1_{P,\mathrm{ver}} \to \Omega^1_{P,\mathrm{hor}}\) be a left \(H\)-covariant morphism of \(P\)-\(\ast\)-bimodules satisfying
\[
\forall \mu \in \Lambda^1_H, \, \forall \alpha \in \Omega^1_B, \quad [N(\mu),\alpha] = 0.
\]
Then \(N \circ \dv{P} \in \pr{\fr{at}}\). Moreover, for all \(\nabla \in \pr{\fr{At}}\), the map \(\nabla N : \Omega^1_{P,\mathrm{ver}} \to \Omega^2_{P,\mathrm{hor}}\) given by
\[
\forall \mu \in \Lambda^1_H, \, \forall p \in P, \quad (\nabla N)(\mu \cdot p) \coloneqq \nabla(N(\mu)) \cdot p
\]
defines a left \(H\)-covariant morphism of \(P\)-bimodules, such that \(-\ensuremath{\mathrm{i}}{}\nabla N\) is \(\ast\)-preserving.
\end{lemma}
\begin{proof}
Let us first show that \(\mathbf{A} \coloneqq N \circ \dv{P} \in \fr{at}[\Omega^1_H]\) is prolongable. Indeed, for all \(p, q \in P\) and \(b \in B\), we have
\begin{align*}
\mathbf{A}(p) \cdot \dt{B}(b) \cdot q - p \cdot \dt{B} \cdot \mathbf{A}(q) &= N(\varpi(\ca{p}{-1})) \wedge \ca{p}{0} \dt{B}(b) \cdot q - p \cdot \dt{B}(b) \cdot \wedge \varpi(\ca{q}{-1}) \cdot \ca{q}{0}\\
&= N\mleft(\varpi(\ca{p}{-1})\epsilon(\ca{q}{-1})+\ca{p}{-1}\triangleright\varpi(\ca{q}{-1})\mright)\wedge \ca{p}{0}\dt{B}(b)\ca{q}{0}\\
&= N\mleft(\varpi(\ca{p}{-1}\ca{q}{-1})\mright) \wedge \ca{p}{0} \cdot \dt{B}(b) \cdot \ca{q}{0}\\
&= N\mleft(\varpi(\ca{(p \cdot \dt{B}(b) \cdot q)}{-1})\mright) \wedge \ca{(p \cdot \dt{B}(b) \cdot q)}{0},
\end{align*}
so that \(\mathbf{A}\) is prolongable with \(\pr{\mathbf{A}}\) given by
\[
\forall \alpha \in \Omega^1_{P,\mathrm{hor}}, \quad \pr{\mathbf{A}}(\alpha) = N(\varpi(\ca{\alpha}{-1})) \wedge \ca{\alpha}{0}.
\]
Now, given \(\nabla \in \pr{\fr{At}}\), let us show that the left \(H\)-covariant right \(P\)-module map \(\nabla N\) is left \(P\)-linear and \(\ast\)-preserving. First, since \([N(\Lambda^1_H),\Omega^1_B]=\set{0}\), it follows that for all \(\mu \in \Lambda^1_H\), \(p \in P\), and \(\beta \in \Omega^1_B\),
\begin{multline*}
p \cdot \dt{B}(b) \wedge N(\mu) = - p \cdot N(\mu) \wedge \dt{B}(b) = -N(\ca{p}{-1}\triangleright\mu) \cdot \ca{p}{0} \wedge \dt{B}(b)\\ = -N(\ca{(p \cdot \dt{B}(b))}{-1} \triangleright \mu) \wedge \ca{(p \cdot\dt{B}(b))}{0},
\end{multline*}
so that
\[
\forall \mu \in \Lambda^1_H, \, \forall \alpha \in \Omega^1_{P,\mathrm{hor}}, \quad \alpha \wedge N(\mu) + N(\ca{\alpha}{-1}\triangleright\mu) \wedge \ca{\alpha}{0} = 0;
\]
Hence, for all \(p \in P\) and \(\mu \in \Lambda^1_H\),
\begin{align*}
p \cdot (\nabla N)(\mu)
&= \pr{\nabla}\mleft(p \cdot N(\mu)\mright) - \nabla(p) \wedge N(\mu)\\
&= \pr{\nabla}\mleft(N(\ca{p}{-1}\triangleright\mu)\mright) \cdot \ca{p}{0} - N(\ca{p}{-1}\triangleright\mu) \wedge \nabla(\ca{p}{0}) - \nabla(p) \wedge N(\mu)\\
&= (\nabla N)(p \cdot \mu) - N(\ca{p}{-1}\triangleright\mu) \wedge \nabla(\ca{p}{0}) + N(\ca{\nabla(p)}{-1} \triangleright \mu) \wedge \ca{\nabla(p)}{0}\\
&= (\nabla N)(p \cdot \mu),
\end{align*}
which shows that \(\nabla N\) is left \(P\)-linear; a similar calculation then shows that the map \(-\ensuremath{\mathrm{i}}{}\nabla N\) is also \(\ast\)-preserving.
\end{proof}
\begin{lemma}\label{qpblem2}
Recall that \((\Omega_H,\dt{H})\) is the canonical prolongation of \((\Omega^1_H,\dt{H})\). Suppose that \(N : \Omega^1_{P,\mathrm{ver}} \to \Omega^1_{P,\mathrm{hor}}\) be a left \(H\)-covariant morphism of \(P\)-\(\ast\)-bimodules satisfying
\begin{equation}\label{qpblem2eq}
\forall \mu,\nu \in \Lambda^1_H, \quad N(\mu) \wedge N(\nu) + N(\ca{\mu}{-1}\triangleright\nu) \wedge N(\ca{\mu}{0}) = 0,
\end{equation}
Then, the map \([N,N] : \Omega^1_{P,\mathrm{ver}} \to \Omega^2_{P,\mathrm{hor}}\) defined by
\[
\forall h \in H, \, \forall p \in P, \quad [N,N](\varpi(h) \cdot p) \coloneqq N(\varpi(\cm{h}{1})) \wedge N(\varpi(\cm{h}{2})) \cdot p
\]
is a left \(H\)-covariant morphism of \(P\)-bimodules, such that \(-\ensuremath{\mathrm{i}}{}[N,N]\) is \(\ast\)-preserving.
\end{lemma}
\begin{proof}
First, by \eqref{qpblem2eq} together with \eqref{wor}, the map
\[
(\mu \otimes \nu \mapsto N(\mu) \wedge N(\nu)) : \Lambda^1_H \otimes_{\mathbf{C}} \Lambda^1_H \to \Omega^2_{P,\mathrm{hor}}
\]
descends to a map \(\tilde{N} : \Lambda^2_H \to \Omega^2_{P,\mathrm{hor}}\), so that \([N,N] : \Omega^1_{P,\mathrm{ver}} \to \Omega^2_{P,\mathrm{hor}}\) is well-defined as left \(H\)-covariant right \(P\)-linear map and given by
\[
\forall \mu \in \Lambda^1_H, \, \forall p \in P, \quad [N,N](\mu \cdot p) \coloneqq \tilde{N}(\dt{H}\mu) \cdot p.
\]
Now, for all \(p \in P\) and \(h \in H\),
\begin{align*}
p \cdot [N,N](\varpi(h)) &= p \cdot N(\varpi(\cm{h}{1})) \wedge N(\varpi(\cm{h}{2}))\\
&= N(\ca{p}{-2} \triangleright \varpi(\cm{h}{1})) \wedge N(\ca{p}{-1} \triangleright \varpi(\cm{h}{2})) \cdot \ca{p}{0}\\
&= \tilde{N}\mleft(\ca{p}{-2}\triangleright\varpi(\cm{h}{1}) \wedge \ca{p}{-1} \triangleright\varpi(\cm{h}{2})\mright) \cdot \ca{p}{0}\\
&= \tilde{N}\mleft(\ca{p}{-1}\triangleright\dt{H}\varpi(h)\mright) \cdot \ca{p}{0}\\
&= \tilde{N}\mleft(\dt{H}(\ca{p}{-1}\triangleright\varpi(h))\mright) \cdot \ca{p}{0}\\
&= [N,N](p \cdot \varpi(h))
\end{align*}
by Proposition~\ref{wor}, so that \([N,N]\) is left \(P\)-linear. A qualitatively identical calculation now shows that \(-\ensuremath{\mathrm{i}}{}[N,N]\) is \(\ast\)-preserving.
\end{proof}
\begin{lemma}\label{qpblem3}
Recall that \((\Omega_H,\dt{H})\) is the canonical prolongation of \((\Omega^1_H,\dt{H})\) and that the subset \(\pr{\fr{At}}[\Omega^1_H]\) of \(\pr{\fr{At}}\) is assumed to be non-empty. Let \(\mathbf{A}\) be a gauge potential on \(P\) with respect to the first-order horizontal calculus \((\Omega^1_B,\dt{B};\Omega^1_{P,\mathrm{hor}})\). The following are equivalent:
\begin{enumerate}
\item\label{qpblem3c} the relative gauge potential \(\mathbf{A}\) is prolongable and canonically \((\Omega^{1}_H,\dt{H})\)-adapted;
\item\label{qpblem3a} for every \(\nabla \in \pr{\fr{At}}[\Omega^{1}_H]\), the gauge potential \(\nabla + \mathbf{A}\) is prolongable and \((\Omega^1_H,\dt{H})\)-adapted, and \(\id_P\) induces an arrow
\[
\left(\id_P : \pi[\Omega^{\leq 2}_H]\circ\Sigma[\Omega^{\leq 2}_H](\nabla) \to \pi[\Omega^{\leq 2}_H]\circ\Sigma[\Omega^{\leq 2}_H](\nabla+\mathbf{A})\right) \in \mathcal{G}[\Omega^{\leq 2}_H];
\]
\item\label{qpblem3b} there exists \(\nabla \in \pr{\fr{At}}[\Omega^{1}_H]\), such that \(\nabla+\mathbf{A} \in \pr{\fr{At}}[\Omega^{1}_H]\) and \(\id_P\) induces an arrow
\[
\left(\id_P : \pi[\Omega^{\leq 2}_H]\circ\Sigma[\Omega^{\leq 2}_H](\nabla) \to \pi[\Omega^{\leq 2}_H]\circ\Sigma[\Omega^{\leq 2}_H](\nabla+\mathbf{A})\right) \in \mathcal{G}[\Omega^{\leq 2}_H].
\]
\end{enumerate}
\end{lemma}
\begin{proof}
First, suppose that condition~\ref{qpblem3c} holds; let \(N \coloneqq \omega[\mathbf{A}]\), so that \(\mathbf{A} = N \circ \dv{P}\). Let \(\nabla \in \pr{\fr{At}}[\Omega^1_H]\) be given. First, by Lemma~\ref{qpblem1}, \(\mathbf{A} \in \dva{\fr{at}}\), so that \(\nabla + \mathbf{A} \in \pr{\fr{At}}\). Next, by Lemma~\ref{qpblem1}, \(\pr{\mathbf{A}}\) is given by
\[
\forall \alpha \in \Omega^1_{P,\mathrm{hor}}, \quad \pr{\mathbf{A}}(\alpha) = N(\varpi(\ca{\alpha}{-1})) \wedge \ca{\alpha}{0},
\]
while by Lemmata~\ref{qpblem1} and~\ref{qpblem2}, respectively, \(-\ensuremath{\mathrm{i}}{}\nabla N\) and \(-\ensuremath{\mathrm{i}}{}[N,N]\) are well-defined left \(H\)-covariant morphisms of \(P\)-\(\ast\)-algebras; hence, for all \(p \in P\),
\begin{align*}
\ensuremath{\mathrm{i}}{}\left(\mathbf{F}[\nabla+\mathbf{A}]-\mathbf{F}[\nabla]\right)(p) &= \left(\pr{\nabla} \circ \mathbf{A} + \pr{\mathbf{A}} \circ \nabla + \pr{\mathbf{A}} \circ \mathbf{A}\right)(p)\\
&= \pr{\nabla}\mleft(N(\varpi(\ca{p}{-1})) \cdot \ca{p}{0}\mright) + \pr{\mathbf{A}}\mleft(\nabla(p)\mright) + \pr{\mathbf{A}}\mleft(\mathbf{A}(p)\mright)\\
&\begin{multlined}= \pr{\nabla} \circ N\mleft(\varpi(\ca{p}{-1})\mright) \cdot \ca{p}{0} - N\mleft(\varpi(\ca{p}{-1})\mright) \wedge \nabla(\ca{p}{0}) \\+ N\mleft(\varpi(\ca{\nabla(p)}{-1})\mright) \wedge \ca{\nabla(p)}{0} + N\mleft(\varpi(\ca{\mathbf{A}(p)}{-1})\mright) \wedge \ca{\mathbf{A}(p)}{0}\end{multlined}\\
&= \nabla N\mleft(\varpi(\ca{p}{0})\mright) \cdot \ca{p}{0} + N(\varpi(\ca{p}{-2})) \wedge N(\varpi(\ca{p}{-1})) \cdot \ca{p}{0}\\
&= \left(\nabla N + [N,N]\right) \circ \dv{P}(p),
\end{align*}
so that \(\nabla + \mathbf{A} \in \pr{\fr{At}}[\Omega^{1}_H]\) with curvature \(2\)-form
\[
F[\nabla+\mathbf{A}] = F[\nabla]-\ensuremath{\mathrm{i}}{}\left(\nabla N + [N,N]\right).
\]
Let us now show that \(\id_P\) induces a morphism \[\left(\id_P : \pi[\Omega^{\leq 2}_H]\circ\Sigma[\Omega^{\leq 2}_H](\nabla) \to \pi[\Omega^{\leq 2}_H]\circ\Sigma[\Omega^{\leq 2}_H](\nabla+\mathbf{A})\right) \in \mathcal{G}[\Omega^{\leq 2}_H].\] Since \(\Omega_{P,\oplus} \coloneqq \Lambda_H \widehat{\otimes}^{\leq 2} \Omega_{P,\mathrm{hor}}\) is generated as a left \(H\)-covariant graded \(\ast\)-algebra over \(P\) by \(\Lambda^1_H\) and \(\Omega^1_{P,\mathrm{hor}}\) subject to the relations
\begin{gather*}
\forall \mu \in \Lambda^1_H, \, \forall \alpha \in \Omega^1_{P,\mathrm{hor}}, \quad \alpha \wedge \mu + \ca{\alpha}{-1} \triangleright \mu \wedge \ca{\alpha}{0} = 0,\\
\forall \mu,\nu \in \Lambda^1_H, \quad \forall \mu \wedge \nu + \ca{\mu}{-1}\triangleright\nu \wedge \ca{\mu}{0},
\end{gather*}
and since, by the proof of Lemma~\ref{qpblem2}, the map \(N : \Omega^1_{P,\mathrm{ver}} \to \Omega^1_{P,\mathrm{hor}}\) satisfies
\begin{gather*}
\forall \mu \in \Lambda^1_H, \, \forall \alpha \in \Omega^1_{P,\mathrm{hor}}, \quad \alpha \wedge N(\mu) + N(\ca{\alpha}{-1} \triangleright \mu) \wedge \ca{\alpha}{0} = 0,\\
\forall \mu,\nu \in \Lambda^1_H, \quad N(\mu) \wedge N(\nu) + N(\ca{\mu}{-1}\triangleright\nu) \wedge N(\ca{\mu}{0}),
\end{gather*}
the left \(H\)-covariant morphisms \(\phi : \Omega^1_{P,\oplus} \to \Omega^1_{P,\oplus}\) and \(\psi : \Omega^1_{P,\oplus} \to \Omega^1_{P,\oplus}\) given by
\begin{gather*}
\forall \omega \in \Omega^1_{P,\mathrm{ver}}, \, \forall \alpha \in \Omega^1_{P,\mathrm{hor}}, \quad \phi(\omega+\alpha) \coloneqq \omega + N(\omega) + \alpha,\\
\forall \omega \in \Omega^1_{P,\mathrm{ver}}, \, \forall \alpha \in \Omega^1_{P,\mathrm{hor}}, \quad \psi(\omega+\alpha) \coloneqq \omega - N(\omega) + \alpha
\end{gather*}
respectively, extend uniquely to left \(H\)-covariant graded \(\ast\)-endomorphisms of \(\Omega_{P,\oplus}\), such that \(\rest{\phi}{P} = \rest{\psi}{P} = \id_P\). Moreover, for all \(\omega \in \Omega^1_{P,\mathrm{ver}}\) and \(\alpha \in \Omega^1_{P,\mathrm{hor}}\),
\begin{gather*}
\psi \circ \phi(\omega+\alpha) = \psi(\omega+N(\omega)+\alpha) = \omega - N(\omega) + N(\omega) + \alpha = \omega + \alpha,\\
\phi \circ \psi(\omega+\alpha) = \phi(\omega-N(\omega)+\alpha) = \omega+N(\omega)-N(\omega)+\alpha=\omega+\alpha,
\end{gather*}
so that \(\phi\) and \(\psi\) are automorphisms with \(\phi^{-1}=\psi\). Finally, on the one hand,
\[
\mathrm{ver}[\dt{P,\nabla+\mathbf{A}}] \circ \phi = \Proj_1 \circ \phi = \Proj_1 = \mathrm{ver}[\dt{P,\nabla}],
\]
where \(\Proj_1 : \Omega^1_{P,\oplus} \to \Omega^1_{P,\mathrm{ver}}\) is the projection onto \(\Omega^1_{P,\mathrm{ver}}\) along \(\Omega^1_{P,\mathrm{hor}}\), while on the other, for all \(p \in P\),
\begin{multline*}
\phi \circ \dt{P,\nabla}(p) = \phi(\dv{P}+\nabla(p)) = \dv{P}(p) + N(\dv{P}(p)) + \nabla(p)\\ = \dv{P}(p) + \mathbf{A}(p) + \nabla(p) = \dt{P,\nabla+\mathbf{A}} \circ \phi(p).
\end{multline*}
Hence, \(\id_P\) induces an arrow \(\id_P : \pi[\Omega^{\leq 2}_H]\circ \Sigma[\Omega^{\leq 2}_H](\nabla) \to \pi[\Omega^{\leq 2}_H]\circ \Sigma[\Omega^{\leq 2}_H](\nabla+\mathbf{A})\) in \(\mathcal{G}[\Omega^{\leq 2}_H]\) with \((\id_P)_\ast = \phi\), as required. Since \(\nabla \in \pr{\fr{At}}[\Omega^1_H]\) was arbitrary, condition~\ref{qpblem3a} is satsified.
Now, condition~\ref{qpblem3a} trivially implies condition~\ref{qpblem3b}, so suppose that condition~\ref{qpblem3b} is satisfied. Fix \(\nabla \in \pr{\fr{At}}[\Omega^{1}_H]\), such that \(\id_P\) induces an arrow
\[
\left(\id_P : \pi[\Omega^{\leq 2}_H]\circ\Sigma[\Omega^{\leq 2}_H](\nabla) \to \pi[\Omega^{\leq 2}_H]\circ\Sigma[\Omega^{\leq 2}_H](\nabla+\mathbf{A})\right) \in \mathcal{G}[\Omega^{\leq 2}_H].
\]
Hence, let \(\Phi \coloneqq (\id_P)_\ast : \Omega_{P,\oplus} \to \Omega_{P,\oplus}\), so that \(\Phi\) is an automorphism of the graded \(H\)-comodule \(\ast\)-algebra \(\Omega_{P,\oplus}\), such that \(\rest{\Phi}{P} = \id_P\) and
\begin{gather*}
\Proj_1 \circ \rest{\Phi}{\Omega^1_{P,\oplus}} = \mathrm{ver}[\dt{P,\nabla+\mathbf{A}}] \circ (\id_P)_\ast = (\id_P)_{\ast,\mathrm{ver}} \circ \mathrm{ver}[\dt{P,\nabla}] = \Proj_1,\\
\dt{P,\nabla+\mathbf{A}} = \dt{P,\nabla+\mathbf{A}} \circ \id_P = (\id_P)_\ast \circ \dt{P,\nabla} = \Phi \circ \dt{P,\nabla},
\end{gather*}
where \(\Proj_1 : \Omega^1_{P,\oplus} \to \Omega^1_{P,\mathrm{ver}}\) is the projection onto \(\Omega^1_{P,\mathrm{ver}}\) along \(\Omega^1_{P,\mathrm{hor}}\). On the one hand, since \(\Proj_1 \circ \rest{\Phi}{\Omega^1_{P,\oplus}} = \Proj_1\), it follows that \((\id-\Phi)(\Omega^1_{P,\mathrm{ver}}) \subseteq \ker \Proj_1 = \Omega^1_{P,\mathrm{hor}}\). On the other hand, for all \(b \in B\), we see that
\[
\Phi(\dt{B}(b)) = \Phi \circ \dt{P,\nabla}(b) = \dt{P,\nabla+\mathbf{A}}(b) = \dt{B}(b),
\]
so that, in turn,
\(
\rest{\Phi}{\Omega^1_{P,\mathrm{hor}}} = \id_{\Omega^1_{P,\mathrm{hor}}}
\).
Thus we can define a left \(H\)-covariant morphism \(N : \Omega^1_{P,\mathrm{ver}} \to \Omega^1_{P,\mathrm{hor}}\) of \(P\)-\(\ast\)-bimodules by
\[
\forall \omega \in \Omega^1_{P,\mathrm{ver}}, \quad N(\omega) \coloneqq \omega - \Phi(\omega);
\]
we claim that \(\mathbf{A} = N \circ \dv{P} \in \pr{\fr{at}}[\Omega_H]\). First, for every \(p \in P\),
\begin{multline*}
N \circ \dv{P}(p) = \dv{P}(p) - \Phi(\dv{P}(p)) = \dv{P}(p) - \Phi(\dt{P,\nabla}(p)-\nabla(p))\\ = \dv{P}(p) - \dt{P,\nabla+\mathbf{A}}(p) - \nabla(p) = \mathbf{A}(p),
\end{multline*}
so that \(\mathbf{A} = N \circ \dv{P} \in \fr{at}[\Omega^1_H]\). Next, for all \(\mu \in \Lambda^1_H\) and \(\alpha \in \Omega^1_B\),
\[
N(\mu) \wedge \alpha = (\mu - \Phi(\mu)) \wedge \alpha = \mu \wedge \alpha - \Phi(\mu \wedge \alpha) = -\alpha \wedge \mu + \Phi(\alpha \wedge \mu) = -\alpha \wedge (\id-\Phi)(\mu) = -\alpha \wedge N(\mu).
\]
Finally, for all \(\mu,\nu \in \Lambda^1_H\),
\begin{align*}
N(\mu) \wedge N(\nu) &= \mu \wedge \nu - \Phi(\mu) \wedge \nu - \mu \wedge \Phi(\nu) + \Phi(\mu) \wedge \Phi(\nu)\\
&= \mu \wedge \nu - \Phi(\mu) \wedge \nu - \Phi(\Phi^{-1}(\mu) \wedge \nu) + \Phi(\mu \wedge \nu)\\
&= -\ca{\mu}{-1}\triangleright\nu \wedge \ca{\mu}{0} + \ca{\Phi(\mu)}{-1}\triangleright\nu \wedge \ca{\Phi(\mu)}{0} + \Phi\mleft(\ca{\Phi^{-1}(\mu)}{-1}\triangleright\nu \wedge \ca{\Phi^{-1}(\mu)}{0}\mright)\\&\quad\quad + \Phi\mleft(\ca{\mu}{-1}\triangleright\nu \wedge \ca{\mu}{0}\mright)\\
&= -\ca{\mu}{-1}\triangleright\nu \wedge \ca{\mu}{0} + \ca{\mu}{-1}\triangleright \nu \wedge \Phi(\ca{\mu}{0}) + \Phi\mleft(\ca{\mu}{-1} \triangleright \nu \wedge \inv{\Phi}(\ca{\mu}{0})\mright)\\&\quad\quad + \Phi\mleft(\ca{\mu}{-1}\triangleright\nu \wedge \ca{\mu}{0}\mright)\\
&= -N(\ca{\mu}{-1}\triangleright\nu) \wedge N(\ca{\mu}{0}).
\end{align*}
Hence, \(\mathbf{A} = N \circ \dv{P} \in \pr{\fr{at}}_{\can}[\Omega^{1}_H]\), as required. Thus, condition~\ref{qpblem3c} holds.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{qpbthm}]
Let us first show that the subset \(\pr{\fr{at}}_{\can}[\Omega^{1}_H]\) is \(\pr{\fr{G}}\)-invariant. Let \(\mathbf{A} \in \pr{\fr{at}}_{\can}[\Omega^{1}_H]\) and \(N \coloneqq \omega[\mathbf{A}]\); let \(\phi \in \pr{\fr{G}}\), so that \(\phi \triangleright \mathbf{A} = (\phi \triangleright N) \circ \dv{P} \in \fr{at}[\Omega^1_H]\) for
\[
\phi \triangleright N \coloneqq \phi_\ast \circ N \circ (\id \otimes \inv{\phi}).
\]
Then, for all \(\mu \in \Lambda^1_H\) and \(\alpha \in \Omega^1_B\),
\[
(\phi \triangleright N)(\mu) \wedge \alpha + \alpha \wedge (\phi \triangleright N)(\nu) = \phi_\ast \mleft(N(\mu)\wedge \alpha + \alpha \wedge N(\nu)\mright) = 0,
\]
while for all \(\mu,\nu \in \Lambda^1_H\),
\begin{multline*}
(\phi \triangleright N)(\mu) \wedge (\phi \triangleright N)(\nu) + (\phi \triangleright N)(\ca{\mu}{-1}\triangleright\nu) \wedge (\phi \triangleright N)(\ca{\mu}{0})\\ = \phi_\ast\mleft(N(\mu) \wedge N(\nu) + N(\ca{\mu}{-1}\triangleright\nu) \wedge N(\ca{\mu}{0}) \mright) = 0,
\end{multline*}
so that \(\phi \triangleright \mathbf{A} \in \pr{\fr{at}}_{\can}[\Omega^{1}_H]\).
Next, let us show that \(\pr{\fr{at}}_{\can}[\Omega^1_H]\) is a subspace of \(\fr{at}[\Omega^1_H] \cap \pr{\fr{at}}\). First, by Lemma~\ref{qpblem1}, \(\pr{\fr{at}}_{\can}[\Omega^1_H] \subset \fr{at}[\Omega^1_H] \cap \pr{\fr{at}}\). Next, since \(0 \in \pr{\fr{at}}_{\can}[\Omega^1_H]\) and since equations \eqref{toteq1} and \eqref{toteq2} are homogeneous of degree \(1\) and \(2\) respectively, it follows that \(\pr{\fr{at}}_{\can}[\Omega^1_H]\) is non-empty and closed under scalar multiplication by \(\mathbf{R}\). Finally, let \(\mathbf{A}_1\), \(\mathbf{A}_2 \in \pr{\fr{at}}_{\can}[\Omega^1_H]\). Let \(\nabla \in \pr{\fr{At}}[\Omega^1_H]\) be given. By Lemma~\ref{qpblem3}, \(\nabla+\mathbf{A}_1 \in \pr{\fr{At}}[\Omega^1_H]\), so that \[\left(\id_P : \pi[\Omega^{\leq 2}_H] \circ \Sigma[\Omega^{\leq 2}_H](\nabla) \to \pi[\Omega^{\leq 2}_H] \circ \Sigma[\Omega^{\leq 2}_H](\nabla+\mathbf{A}_1)\right) \in \mathcal{G}[\Omega^{\leq 2}_H];\] let \(\Phi \coloneqq (\id_P)_\ast\) be the unique left \(H\)-covariant automorphism of the \(P\)-\(\ast\)-bimodule \(\Omega^1_{P,\oplus}\), such that \(\Phi \circ \dt{P,\nabla} = \dt{P,\nabla+\mathbf{A}_1}\). Hence, by Lemma~\ref{qpblem3}, \(\nabla+\mathbf{A}_1 + \mathbf{A}_2 \in \pr{\fr{At}}[\Omega^1_H]\), so that \[\left(\id_P : \pi[\Omega^{\leq 2}_H]\circ\Sigma[\Omega^{\leq 2}_H](\nabla+\mathbf{A}_1) \to \pi[\Omega^{\leq 2}_H]\circ\Sigma[\Omega^{\leq 2}_H](\nabla+\mathbf{A}_1+\mathbf{A}_2)\right) \in \mathcal{G}[\Omega^{\leq 2}_H];\] let \(\Psi \coloneqq (\id_P)_\ast\) be the unique left \(H\)-covariant automorphism of the \(P\)-\(\ast\)-bimodule \(\Omega^1_{P,\oplus}\), such that \(\Psi \circ \dt{P,\nabla+\mathbf{A}_1} = \dt{P,\nabla+\mathbf{A}_1+\mathbf{A}_2}\). Then \(\Psi \circ \Phi\) is a left \(H\)-covariant automorphism of the \(P\)-\(\ast\)-bimodule \(\Omega^1_{P,\oplus}\) satisfying
\begin{gather*}
(\Psi \circ \Phi) \circ \dt{P,\nabla} = \Phi \circ \dt{P,\nabla+\mathbf{A}_1} = \dt{P,\nabla+\mathbf{A}_1+\mathbf{A}_2},\\
\mathrm{ver}[\dt{P,\nabla+\mathbf{A}_1+\mathbf{A}_2}] \circ (\Psi \circ \Phi) = \mathrm{ver}[\dt{P,\nabla+\mathbf{A}_1}] \circ \Phi = \mathrm{ver}[\dt{P,\nabla}],
\end{gather*}
so that \(\id_P : \pi[\Omega^{\leq 2}_H] \circ \Sigma[\Omega^{\leq 2}_H](\nabla) \to \pi[\Omega^{\leq 2}_H]\circ\Sigma[\Omega^{\leq 2}_H](\nabla+\mathbf{A}_1+\mathbf{A}_2)\) defines an arrow in \(\mathcal{G}[\Omega^{\leq 2}_H]\). Hence, by Lemma~\ref{qpblem3}, \(\mathbf{A}_1+\mathbf{A}_2 \in \pr{\fr{at}}_{\can}[\Omega^1_H]\). Thus, \(\pr{\fr{at}}_{\can}[\Omega^1_H]\) is a subspace of \(\pr{\fr{at}} \cap \fr{at}[\Omega^1_H]\). Note that \(\pr{\fr{At}}[\Omega^1_H] \subset \pr{\fr{At}}\) is invariant under translation by the subspace \(\pr{\fr{at}}_{\can}[\Omega^1_H] \subset \pr{\fr{at}}\) by Lemma~\ref{qpblem3}.
Finally, by Proposition~\ref{sodcequiv}, the covering \(\pi[\Omega^{\leq 2}_H] : \mathcal{G}[\Omega^{\leq 2}_H] \ltimes \mathcal{A}[\Omega^{\leq 2}_H] \to \mathcal{G}[\Omega^{\leq 2}_H]\), the star-injective groupoid homomorphism \(\mu[\Omega^{\leq 2}_H] : \mathcal{G}[\Omega^{\leq 2}_H] \to \Aut(P)\), the injective groupoid homomorphism \(\Sigma[\Omega^{\leq 2}_H] : \pr{\fr{G}} \ltimes \pr{\fr{At}}[\Omega^1_H] \to \mathcal{G}[\Omega^{\leq 2}_H] \ltimes \mathcal{A}[\Omega^{\leq 2}_H]\), and the left inverse \(A[\Omega^{\leq 2}_H]\) of \(\Sigma[\Omega^{\leq 2}_H]\) satisfy the hypotheses of Lemma~\ref{groupoidlemma}. Thus, the equivalence kernel \(\sim\) of the map
\[
\left(\nabla \mapsto \left[\pi[\Omega^{\leq 2}_H] \circ \Sigma[\Omega^{\leq 2}_H](\id_P,\nabla)\right]_{\ker\mu[\Omega^{\leq 2}_H]}\right) : \pr{\fr{At}}[\Omega^1_H] \to \Ob(\mathcal{G}[\Omega^{\leq 2}_H]/\ker\mu[\Omega^{\leq 2}_H])
\]
is a \(\pr{\fr{G}}\)-invariant equivalence relation, the subgroupoid \(\ker\mu[\Omega^{\leq 2}_H]\) is wide and has trivial isotropy groups, the quotient groupoid \(\mathcal{G}[\Omega^{\leq 2}_H]/\ker\mu[\Omega^{\leq 2}_H]\) is well-defined, and there exists a unique isomorphism \(\pr{\tilde{\Sigma}}[\Omega^{\leq 2}_H] : \pr{\fr{G}} \ltimes \pr{\fr{At}}[\Omega^1_H]/\sim \, \xrightarrow{\sim} \mathcal{G}[\Omega^{\leq 2}_H]/\ker\mu[\Omega^{\leq 2}_H]\), such that
\[
\forall (\phi,\nabla) \in \pr{\fr{G}} \ltimes \pr{\fr{At}}[\Omega^1_H], \quad \pr{\tilde{\Sigma}}[\Omega^{\leq 2}_H](\phi,[\nabla]_\sim) = \left[\pi[\Omega^{\leq 2}_H] \circ \Sigma[\Omega^{\leq 2}_H](\phi,\nabla)\right]_{\ker\mu[\Omega^{\leq 2}_H]}.
\]
Lemma~\ref{qpblem3} now implies that \(\sim\) is the orbit equivalence relation with respect to the translation action of \(\pr{\fr{at}}_{\can}[\Omega^1_H] \leq \pr{\fr{at}}\) on \(\pr{\fr{At}}[\Omega^1_H] \subset \pr{\fr{At}}\).
\end{proof}
Finally, observe that the subspace \(\Inn(\pr{\fr{at}})\) of inner prolongable gauge potentials acts by translation on the affine space \(\pr{\fr{At}}\). Given a bicovariant \textsc{fodc}{} \((\Omega^1_H,\dt{H})\) on \(H\) that is locally freeing for \(P\), we can now characterise the stabiliser subgroup in \(\Inn(\pr{\fr{at}})\) of the affine quadric subset \(\pr{\fr{At}}[\Omega^1_H]\) of \(\pr{\fr{At}}\). This will turn out to be the \(\pr{\fr{G}}\)-invariant \(\mathbf{R}\)-linear subspace of all inner prolongable relative gauge potentials of the following form.
\begin{definition}
Let \((\Omega^1_H,\dt{H})\) be a bicovariant \textsc{fodc}{} on \(H\) that is locally freeing for \(P\). Let \(\mathbf{A}\) be an inner prolongable relative gauge potential on \(P\) with respect to the second-order horizontal calculus \((\Omega_B,\dt{B};\Omega_{P,\mathrm{hor}})\). We say that \(\mathbf{A}\) is \emph{\((\Omega^1_H,\dt{H})\)-semi-adapted} whenever
\[
\mathbf{F}_{\mathrm{rel}}[\mathbf{A}] = F_{\mathrm{rel}}[\mathbf{A}] \circ \dv{P},
\]
for some (necessarily unique) left \(H\)-covariant morphism \(F_{\mathrm{rel}}[\mathbf{A}] : \Omega^1_{P,\mathrm{ver}} \to \Omega^2_{P,\mathrm{hor}}\) of \(P\)-\(\ast\)-bimodules, in which case, we call \(F_{\mathrm{rel}}[\mathbf{A}]\) the \emph{relative curvature \(2\)-form} of \(\mathbf{A}\). We denote by \(\Inn(\pr{\fr{at}};\Omega^1_H)\) the subspace of all \((\Omega^1_H,\dt{H})\)-semi-adapted inner prolongable gauge potentials on \(P\) with respect to \((\Omega_B,\dt{B};\Omega_{P,\mathrm{hor}})\).
\end{definition}
\begin{example}
By Example~\ref{relex}, for every \(\phi \in \Inn(\fr{G})\) and \(\nabla \in \pr{\fr{At}}\), we have
\[
\phi \triangleright \nabla -\nabla \in \Inn(\pr{\fr{at}};\Omega^1_H), \quad F_{\mathrm{rel}}[\phi \triangleright \nabla-\nabla] = 0.
\]
\end{example}
\begin{proposition}\label{semiadapt}
Let \((\Omega^1_H,\dt{H})\) be a bicovariant \textsc{fodc}{} on \(H\) that is locally freeing for \(P\). Suppose that the \((\Omega^1_H,\dt{H})\)-adapted Atiyah space \(\pr{\fr{At}}[\Omega^1_H]\) is non-empty. Let \(\mathbf{A}\) be an inner prolongable relative gauge potential on \(P\) with respect to \((\Omega_B,\dt{B};\Omega_{P,\mathrm{hor}})\). The following are equivalent:
\begin{enumerate}
\item the inner prolongable relative gauge potential \(\mathbf{A}\) is \((\Omega^1_H,\dt{H})\)-semi-adapted;
\item for all \(\nabla \in \pr{\fr{At}}[\Omega^1_H]\), the prolongable gauge potential \(\nabla+\mathbf{A}\) is also \((\Omega^1_H,\dt{H})\)-adapted;
\item there exists \(\nabla \in \pr{\fr{At}}[\Omega^1_H]\), such that \(\nabla+\mathbf{A} \in \pr{\fr{At}}\) is also \((\Omega^1_H,\dt{H})\)-adapted.
\end{enumerate}
Thus, in particular, the affine quadric subset \(\pr{\fr{At}}[\Omega^1_H]\) of the affine space \(\pr{\fr{At}}\) is invariant under translation by \(\Inn(\pr{\fr{at}};\Omega^1_H)\).
\end{proposition}
\begin{proof}
First, suppose that \(\mathbf{A} \in \Inn(\pr{\fr{at}};\Omega^1_H)\). Let \(\nabla \in \pr{\fr{At}}[\Omega^1_H]\). Then \(\nabla + \mathbf{A} \in \pr{\fr{At}}\) with
\[
\mathbf{F}[\nabla+\mathbf{A}] = \mathbf{F}[\nabla] + \mathbf{F}_{\mathrm{rel}}[\mathbf{A}] = F[\nabla] \circ \dv{P} + F_{\mathrm{rel}}[\mathbf{A}] \circ \dv{P} = \left(F[\nabla] + F_{\mathrm{rel}}[\mathbf{A}]\right) \circ \dv{P},
\]
so that \(\nabla + \mathbf{A} \in \pr{\fr{At}}[\Omega^1_H]\) with \(F[\nabla+\mathbf{A}] = F[\nabla]+F_{\mathrm{rel}}[\mathbf{A}]\). Now, suppose that there exists \(\nabla \in \pr{\fr{At}}[\Omega^1_H]\), such that \(\nabla+\mathbf{A} \in \pr{\fr{At}}[\Omega^1_H]\). Then
\[
\mathbf{F}_{\mathrm{rel}}[\mathbf{A}] = \mathbf{F}[\nabla+\mathbf{A}] - \mathbf{F}[\nabla] = F[\nabla+\mathbf{A}] \circ \dv{P} - F[\nabla] \circ \dv{P} = \left(F[\nabla+\mathbf{A}]-F[\nabla]\right)\circ\dv{P},
\]
so that \(\mathbf{A} \in \Inn(\pr{\fr{at}};\Omega^1_H)\) with \(F_{\mathrm{rel}}[\mathbf{A}] = F[\nabla+\mathbf{A}]-F[\nabla]\).
\end{proof}
We now summarise the basic properties of the subspace \(\Inn(\pr{\fr{at}};\Omega^1_H)\) and observe that the action of \(\pr{\fr{G}}\) on \(\pr{\fr{At}}[\Omega^1_H]\) descends to a second-order analogue of the action of \(\Out(\fr{G})\) on \(\Out(\fr{At})\); this all follows, \emph{mutatis mutandis}, from the proof of Proposition~\ref{innprop}.
\begin{proposition}
Let \((\Omega^1_H,\dt{H})\) be a bicovariant \textsc{fodc}{} on \(H\) that is locally freeing for \(P\). Suppose that \(\pr{\fr{At}}[\Omega^1_H]\) is non-empty. The subspace \(\Inn(\pr{\fr{at}};\Omega^1_H)\) of \(\Inn(\pr{\fr{at}})\) consists of \(\pr{\fr{G}}\)-invariant vectors, so that the affine action of \(\pr{\fr{G}}\) on the affine space \(\pr{\fr{At}}\) descends to an affine action of \(\pr{\fr{G}}\) on the quotient affine space
\(
\pr{\fr{At}}/\Inn(\pr{\fr{at}};\Omega^1_H)
\).
Furthermore, the inner gauge group \(\Inn(\fr{G})\) acts trivially on \(\pr{\fr{At}}/\Inn(\pr{\fr{at}};\Omega^1_H)\), so that the action of \(\pr{\fr{G}}\) on \(\pr{\fr{At}}/\Inn(\pr{\fr{at}};\Omega^1_H)\) further descends to an affine action of the outer prolongable gauge group \(\Out(\pr{\fr{G}})\) on \(\pr{\fr{At}}/\Inn(\pr{\fr{at}};\Omega^1_H)\) that restricts, in turn, to an action on the quadric subset\[
\Out(\pr{\fr{At}}[\Omega^1_H]) \coloneqq \pr{\fr{At}}[\Omega^1_H]/\Inn(\pr{\fr{at}};\Omega^1_H).
\]
\end{proposition}
Finally, we record the basic properties of the relative curvature \(2\)-form.
\begin{corollary}
Let \((\Omega^1_H,\dt{H})\) be a bicovariant \textsc{fodc}{} on \(H\) that is locally freeing for \(P\). The map \(F_{\mathrm{rel}} \coloneqq (\mathbf{A} \mapsto F_{\mathrm{rel}}[\mathbf{A}]) : \Inn(\pr{\fr{at}};\Omega^1_H) \to \Hom_P^H(\Omega^1_{P,\mathrm{ver}},\Omega^2_{P,\mathrm{hor}})\) is linear with range contained in \(\Hom_P^H(\Omega^1_{P,\mathrm{ver}},\Omega^2_{P,\mathrm{hor}})^{\pr{\fr{G}}}\). In particular, it satisfies
\[
\forall \nabla \in \pr{\fr{At}}[\Omega^1_H], \, \forall \mathbf{A} \in \Inn(\pr{\fr{at}};\Omega^1_H), \quad F[\nabla+\mathbf{A}] = F[\nabla] + F_{\mathrm{rel}}[\mathbf{A}].
\]
\end{corollary}
\begin{proof}
The map \(F_{\mathrm{rel}}\) is \(\mathbf{R}\)-linear and \(\pr{\fr{G}}\)-equivariant by Corollary~\ref{relcor} and \(\pr{\fr{G}}\)-equivar\-iance of \(\dv{P}\). Since \(\pr{\fr{G}}\) acts trivially on \(\Inn(\pr{\fr{at}};\Omega^1_H)\), it follows that the range of \(F_{\mathrm{rel}}\) is contained in \(\Hom_P^H(\Omega^1_{P,\mathrm{ver}},\Omega^2_{P,\mathrm{hor}})^{\pr{\fr{G}}}\). The relation between the maps \(F\) on \(\pr{\fr{At}}[\Omega^1_H]\) and \(F_\mathrm{rel}\) on \(\Inn(\pr{\fr{at}};\Omega^1_H)\) now follows by the proof of Proposition~\ref{semiadapt}.
\end{proof}
\section{Gauge theory on crossed products as lazy cohomology}\label{sec4}
Unlike in the commutative case, trivial quantum principal bundles can encode non-trivial dynamical information. As \'{C}a\'{c}i\'{c}--Mesland have already observed in the context of spectral triples~\cite{CaMe}*{\S 3.4}, the noncommutative \(\mathbf{T}^m\)-gauge theory of crossed products by \(\mathbf{Z}^m\) can be related to the degree \(1\) group cohomology of \(\mathbf{Z}^m\) with certain geometrically meaningful coefficients. In light of the groupoid equivalences of Proposition~\ref{equiv1} and~\ref{sodcequiv}, we shall now similarly relate the gauge theory of (non-twisted) crossed product algebras to certain generalisations of group cohomology in degree \(1\).
\subsection{Cohomological preliminaries}
In effect, \'{C}a\'{c}i\'{c}--Mesland computed the gauge group \(\fr{G}\) and Atiyah space \(\fr{At}\) of a crossed product by \(\mathbf{Z}^m\) in terms of the degree \(1\) group cohomology of \(\mathbf{Z}^m\) with coefficients in a certain group of unitaries and a certain \(\mathbf{R}[\mathbf{Z}^m]\)-module of noncommutative \(1\)-forms, respectively~\cite{CaMe}*{Thm.\ 3.36}. In this section, we construct suitable generalisations of these two distinct but interrelated cases of degree \(1\) group cohomology to arbitrary Hopf \(\ast\)-algebras; in the process, we provide an \emph{ad hoc} generalisation of degree \(1\) Sweedler cohomology to not-necessarily-cocommutative Hopf \(\ast\)-algebras and non-trivial coefficients. From now on, let \(H\) be a Hopf \(\ast\)-algebra.
We begin by recalling more-or-less standard constructions of convolution algebras and bimodules on \(H\) with suitable coefficients.
\begin{definition}
Let \(B\) be a [graded] \(\ast\)-algebra. The \emph{\(B\)-valued convolution algebra on \(H\)} is the [graded] unital \(\ast\)-algebra \(\mathcal{C}(H;B)\) defined by endowing \(\Hom_{\mathbf{C}}(H;B)\) with the product and \(\ast\)-structure defined by
\begin{gather}
\forall f,g \in \mathcal{C}(H;B), \, \forall h \in H, \quad (f \star g)(h) \coloneqq f(\cm{h}{1})g(\cm{h}{2}),\\
\forall f \in \mathcal{C}(H;B),\, \forall h \in H, \quad f^\ast(h) \coloneqq f(S(h)^\ast)^\ast,
\end{gather}
respectively, and the unit \(1_{\mathcal{C}(H,B)} \coloneqq \epsilon(\cdot)1_B\).
\end{definition}
\begin{definition}
Let \(B\) be a \(\ast\)-algebra, and let \(M\) be a \(B\)-\(\ast\)-bimodule. The \emph{\(M\)-valued convolution bimodule on \(H\)} is the \(\mathcal{C}(H,B)\)-\(\ast\)-bimodule \(\mathcal{C}(H;M)\) defined by endowing the \(\mathbf{C}\)-vector space \(\Hom_{\mathbf{C}}(H,M)\) with the left \(\mathcal{C}(H,B)\)-module structure, right \(\mathcal{C}(H,B)\)-module structure, and \(\ast\)-structure defined, respectively, by
\begin{gather}
\forall f \in \mathcal{C}(H,B), \, \forall \mu \in \mathcal{C}(H,M), \, \forall h \in H, \quad (f \star \mu)(h) \coloneqq f(\cm{h}{1}) \cdot \mu(\cm{h}{2}),\\
\forall f \in \mathcal{C}(H,B), \, \forall \mu \in \mathcal{C}(H,M), \, \forall h \in H, \quad (\mu \star f)(h) \coloneqq \mu(\cm{h}{1}) \cdot f(\cm{h}{2}),\\
\forall \mu \in \mathcal{C}(H,M), \, \forall h \in H, \quad \mu^\ast(h) \coloneqq \mu(S(h)^\ast)^\ast.
\end{gather}
\end{definition}
In the case of \(H\)-module \(\ast\)-algebras and \(H\)-equivariant \(\ast\)-bimodules, one can canonically embed coefficient algebras and bimodules into the resulting convolution algebras and bimodules, respectively, in a manner respecting \(\ast\)-bimodule structures. When \(H\) is no longer a group algebra, this will yield subtler notions of commutation than pointwise commutation.
\begin{proposition}
Let \(B\) be a [graded] \(H\)-module \(\ast\)-algebra. Then \(\rho_B : B \to \mathcal{C}(H;B)\) defined by
\[
\forall b \in B, \, \forall h \in H, \quad \rho_B(b)(h) \coloneqq b \triangleleft h
\]
is an injective [graded] \(\ast\)-homomorphism.
\end{proposition}
\begin{proposition}
Let \(B\) be a right \(H\)-module \(\ast\)-algebra, and let \(M\) be a right \(H\)-equivariant \(B\)-\(\ast\)-bimodule. Then the map \(\rho_M : M \to \mathcal{C}(H;M)\) defined by
\[
\forall m \in M, \, \forall h \in H, \quad \rho_M(m)(h) \coloneqq m \triangleleft h
\]
is an injective \(\ast\)-preserving \(\mathbf{C}\)-linear map satisfying
\[
\forall a,b \in B, \, \forall m \in M, \quad \rho_B(a) \star \rho_M(m) \star \rho_B(b) = \rho_M(a \cdot m \cdot b).
\]
In particular, \(\mathcal{C}(H;M)\) defines a \(B\)-\(\ast\)-bimodule with respect to \(\rho_B\).
\end{proposition}
We can now provide a suitable generalisation of the degree \(1\) group cohomology of a group \(\Gamma\) with coefficients in the unitary group of a commutative \(\Gamma\)-\(\ast\)-algebra. It can be viewed as an \emph{ad hoc} generalisation of degree \(1\) Sweedler cohomology~\cite{Sweedler} in the spirit of Bichon--Carnovale's `lazy' cohomology~\cite{BC} to the case of non-cocommutative Hopf algebras and non-commutative coefficient algebras. We shall use it to compute the gauge group of a crossed product by \(H\).
\begin{propositiondefinition}[cf.\ Sweedler~\cite{Sweedler}]\label{lazysweedler}
Let \(B\) be a right \(H\)-module \(\ast\)-algebra, and let \(M\) be a right \(H\)-equivariant \(B\)-\(\ast\)-bimodule.
\begin{enumerate}
\item A \emph{lazy \((B,M)\)-valued Sweedler \(0\)-cochain on \(H\)} is an element of \[\CS_\ell^0(H;B,M) \coloneqq \Unit(\Cent_B(B\oplus M)) \leq \Unit(B).\]
\item A \emph{lazy \((B,M)\)-valued Sweedler \(1\)-cocycle on \(H\)} is \(\sigma \in \Unit\mleft(\Cent_{\mathcal{C}(H,B)}(\rho_B(B) \oplus \rho_M(M))\mright)\), such that \(\sigma(1) = 1\) and
\begin{equation}\label{scocycle}
\forall h,k \in H, \quad \sigma(hk) = \left(\sigma(h) \triangleleft \cm{k}{1}\right)\sigma(\cm{k}{2});
\end{equation}
we denote by \(\ZS_\ell^1(H;B,M)\) the set of all lazy \((B,M)\)-valued Sweedler \(1\)-cocycles on \(H\), which defines a subgroup of \(\Unit(\mathcal{C}(H,B))\).
\item The \emph{coboundary map} is the homomorphism \(D: \ZS_\ell^0(H;B,M) \to \ZS_\ell^1(H;B,M)\) defined by
\begin{equation}
\forall \upsilon \in \CS_\ell^0(H;B,M), \, \forall h \in H, \quad D \upsilon(h) \coloneqq (\upsilon \triangleleft h) \cdot \upsilon^\ast;
\end{equation}
thus, a \emph{lazy \((B,M)\)-valued Sweedler \(1\)-coboundary on \(H\)} is an element of
\[
\BS_\ell^1(H;B,M) \coloneqq D\mleft(\CS_\ell^0(H;B,M)\mright) \leq \Zent\mleft(\ZS^1_\ell(H;B,M)\mright).
\]
\item The \emph{lazy degree \(1\) Sweedler cohomology of \(H\) with coefficients in \((B,M)\)} is the group
\[
\HS_\ell^1(H;B,M) \coloneqq \ZS_\ell^1(H;B,M)/\BS_\ell^1(H;B,M).
\]
\end{enumerate}
\end{propositiondefinition}
The proof of this result---and several others besides---will require the following straightforward technical lemma.
\begin{lemma}\label{centrallem}
Let \(B\) be a right \(H\)-module \(\ast\)-algebra, and let \(M\) be a right \(H\)-equivariant \(B\)-\(\ast\)-bimodule.
\begin{enumerate}
\item Let \(b \in B\) and let \(\mu \in \mathcal{C}(H,M)\). Then \([\rho_B(b),\mu] = 0\) if and only if
\[
\forall h \in H, \quad (b \triangleleft \cm{h}{1}) \cdot \mu(\cm{h}{2}) = \mu(\cm{h}{1}) \cdot (b \triangleleft \cm{h}{2}).
\]
\item Let \(\beta \in \mathcal{C}(H,B)\) and let \(m \in M\). Then \([\beta,\rho_M(m)] = 0\) if and only if
\[
\forall h \in H, \quad \beta(\cm{h}{1}) \cdot (m \triangleleft \cm{h}{2}) = (m \triangleleft \cm{h}{1}) \cdot \beta(\cm{h}{2}).
\]
\end{enumerate}
\end{lemma}
\begin{proof}
On the one hand, for all \(b \in B\), \(\mu \in \mathcal{C}(H,M)\), and \(h \in H\),
\begin{align*}
[\rho_B(b),\mu](h) &= \rho_B(b)(\cm{h}{1}) \cdot \mu(\cm{h}{2}) - \mu(\cm{h}{1}) \cdot \rho_B(b)(\cm{h}{2})\\ &= (b \triangleleft \cm{h}{1}) \cdot \mu(\cm{h}{2}) - \mu(\cm{h}{1}) \cdot (b \triangleleft \cm{h}{2}).
\end{align*}
On the other hand, for all \(\beta \in \mathcal{C}(H,B)\), \(m \in M\), and \(h \in H\),
\begin{align*}
[\beta,\rho_M(m)](h) &= \beta(\cm{h}{1}) \cdot \rho_M(m)(\cm{h}{2}) - \rho_M(m)(\cm{h}{1}) \cdot \beta(\cm{h}{2})\\
&= \beta(\cm{h}{1}) \cdot (m \triangleleft \cm{h}{2}) - (m \triangleleft \cm{h}{1}) \cdot \beta(\cm{h}{2}). \qedhere
\end{align*}
\end{proof}
\begin{proof}[Proof of Proposition~\ref{lazysweedler}]
Before continuing, note that by Lemma~\ref{centrallem}, \(f \in \mathcal{C}(H,B)\) centralises the subset \(\rho_B(B) \oplus \rho_M(B)\) of \(\mathcal{C}(H;B\oplus M)\) if and only if
\begin{gather*}
\forall b \in B,\,\forall h \in H, \quad f(\cm{h}{1}) \cdot (b \triangleleft \cm{h}{2}) = (b \triangleleft \cm{h}{1}) \cdot f(\cm{h}{2}),\\
\forall m \in M,\,\forall h \in H, \quad f(\cm{h}{1}) \cdot (m \triangleleft \cm{h}{2}) = (m \triangleleft \cm{h}{1}) \cdot f(\cm{h}{2}).
\end{gather*}
We first show that \(\ZS_\ell^1(H;B,M)\) is a subgroup of \(\Unit\mleft(\Cent_{\mathcal{C}(H,B)}(\rho_B(B) \oplus \rho_M(M))\mright)\). First, observe that \(\ZS_\ell^1(H;B,M) \ni 1_{\mathcal{C}(H,B)}\). Next, let \(\sigma_1,\sigma_2 \in \ZS_\ell^1(H;B,M)\). Then for all \(h, k \in H\),
\begin{align*}
\sigma_1 \star \sigma_2(hk) &= \sigma_1(\cm{h}{1}\cm{k}{1}) \cdot \sigma_2(\cm{h}{2}\cm{k}{2})\\
&= (\sigma_1(\cm{h}{1}) \triangleleft \cm{k}{1}) \cdot \sigma_1(\cm{k}{2}) \cdot (\sigma_2(\cm{h}{2}) \triangleleft \cm{k}{3}) \cdot \sigma_2(\cm{k}{4})\\
&= (\sigma_1(\cm{h}{1}) \triangleleft \cm{k}{1}) \cdot (\sigma_2(\cm{h}{2}) \triangleleft \cm{k}{2}) \cdot \sigma_1(\cm{k}{3}) \cdot \sigma_2(\cm{k}{4})\\
&= (\sigma_1 \star \sigma_2(h) \triangleleft \cm{k}{1}) \cdot \sigma_1\star\sigma_2(\cm{k}{2}),
\end{align*}
while \(\sigma_1 \star \sigma_2(1) = \sigma_1(1) \cdot \sigma_2(1)=1\), so that, indeed, \(\sigma_1 \star \sigma_2 \in \ZS^1_\ell(H;B,M)\). Finally, let \(\sigma \in \ZS^1_\ell(H;B,M)\); note that \(\sigma^{-1} \in \Unit\mleft(\Cent_{\mathcal{C}(H,B)}(\rho_B(B) \oplus \rho_M(M))\mright)\). Then, for all \(h,k \in H\),
\begin{align*}
\sigma^{-1}(hk) &= \sigma(S(h)^\ast S(k)^\ast)^\ast\\
&= \left((\sigma(S(h)^\ast) \triangleleft \cm{(S(k)^\ast)}{1}) \cdot \sigma(\cm{(S(k)^\ast)}{2}) \right)^\ast\\
&=\left(\sigma(\cm{(S(k)^\ast)}{1}) \cdot (\sigma(S(h)^\ast) \triangleleft \cm{(S(k)^\ast)}{2})\right)^\ast\\
&= (\sigma^\ast(h) \triangleleft \cm{k}{1})\cdot \sigma^\ast(\cm{k}{2}),
\end{align*}
while \(\sigma^\ast(1) = \sigma(1)^\ast = 1\), so that \(\sigma^{-1} \in \ZS^1_\ell(H;B,M)\).
Next, we show that \(D: \ZS_\ell^0(H;B,M) \to \ZS_\ell^1(H;B,M)\) is well-defined as a function. Let \(\upsilon \in \CS^0_\ell(H;B,M)\), and set \(\sigma \coloneqq D\upsilon\). First, \(\sigma\) is unitary, since for all \(h \in H\),
\begin{gather*}
\sigma \star \sigma^\ast (h) = (\upsilon \triangleleft \cm{h}{1}) \cdot \upsilon^\ast \cdot \left((\upsilon \triangleleft S(\cm{h}{2})^\ast) \cdot \upsilon^\ast \right)^\ast = (\upsilon \triangleleft \cm{h}{1}) \cdot \upsilon^\ast \cdot \upsilon \cdot (\upsilon^\ast \triangleleft \cm{h}{2}) = \epsilon(h) 1_B,\\
\sigma^\ast \star \sigma(h) = \left((\upsilon \triangleleft S(\cm{h}{1})^\ast) \cdot \upsilon^\ast \right)^\ast \cdot (\upsilon \triangleleft \cm{h}{2}) \cdot \upsilon^\ast = \upsilon \cdot (\upsilon^\ast \triangleleft \cm{h}{1}) \cdot (\upsilon \triangleleft \cm{h}{2}) \cdot \upsilon^\ast = \epsilon(h) 1_B;
\end{gather*}
note, moreover, that \(\sigma(1) = (\upsilon \triangleleft 1) \cdot \upsilon^\ast = \upsilon \cdot \upsilon^\ast = 1\). Next, \(\sigma \in \Cent_{\mathcal{C}(H,B)}(\rho_B(B) \oplus \rho_M(M))\) since for all \(b \in B\), \(m \in M\), and \(h \in H\),
\begin{align*}
\sigma(\cm{h}{1}) \cdot (b \triangleleft \cm{h}{2}) - (b \triangleleft \cm{h}{1}) \cdot \sigma(\cm{h}{2}) &= (\upsilon \triangleleft \cm{h}{1}) \cdot \upsilon^\ast \cdot (b \triangleleft \cm{h}{2}) - (b \triangleleft \cm{h}{1}) \cdot (\upsilon \triangleleft \cm{h}{2}) \cdot \upsilon^\ast\\
&= \left((\upsilon \cdot b - b \cdot \upsilon) \triangleleft h \right)\cdot \upsilon^\ast\\
&= 0,
\end{align*}
and hence, \emph{mutatis mutandis}, \(\sigma(\cm{h}{1})\cdot(m\triangleleft\cm{h}{2}) - (m \triangleleft \cm{h}{1}) \cdot \sigma(\cm{h}{2})\). Finally, for all \(h,k\in H\),
\[
\sigma(hk) = (\upsilon \triangleleft hk) \cdot \upsilon^\ast = \left((\upsilon \triangleleft h) \triangleleft \cm{k}{1}\right) \cdot (\upsilon^\ast \triangleleft \cm{k}{2}) \cdot (\upsilon \triangleleft \cm{k}{3}) \cdot \upsilon^\ast = (\sigma(h) \triangleleft \cm{k}{1}) \cdot \sigma(\cm{k}{2}).
\]
Hence, \(D\upsilon \eqqcolon \sigma \in \ZS^1_{\ell}(H;B,M)\).
Next, we show that \(D\) is a group homomorphism. First, note that
\[
D1_B = \left(h \mapsto (1_B \triangleleft h) \cdot 1_B\right) = \epsilon(\cdot)1_B = 1_{\mathcal{C}(H,B)}.
\]
Now, let \(\upsilon_1,\upsilon_2 \in \CS^0_\ell(H,B;M)\). Then for all \(h \in H\),
\begin{align*}
D(\upsilon_1 \cdot \upsilon_2)(h) &= \left((\upsilon_1 \cdot \upsilon_2) \triangleleft h\right) \cdot (\upsilon_1 \cdot \upsilon_2)^\ast \\
&= (\upsilon_1 \triangleleft \cm{h}{1}) \cdot (\upsilon_2 \triangleleft \cm{h}{2}) \cdot \upsilon_2^\ast \upsilon_1^\ast \\
&= D\upsilon_1(\cm{h}{1}) \cdot D\upsilon_2(\cm{h}{2}),
\end{align*}
so that, indeed, \(D(\upsilon_1 \cdot \upsilon_2) = D\upsilon_1 \star D\upsilon_2\).
Finally, we show that \(\BS_\ell^1(H;B,M)\) is central in \(\ZS_\ell^1(H;B,M)\). Let \(\upsilon \in \CS^0_\ell(H;B,M)\) and \(\sigma \in \ZS^1_\ell(H;B,M)\). Then, for all \(h \in H\),
\begin{multline*}
\sigma \star D\upsilon \star \sigma^{-1} (h) = \sigma(\cm{h}{1}) \cdot (\upsilon \triangleleft \cm{h}{2}) \cdot \upsilon^\ast \cdot \sigma^{-1}(\cm{h}{3}) = (\upsilon \triangleleft \cm{h}{1}) \cdot \upsilon^\ast \cdot \sigma(\cm{h}{2}) \cdot \sigma^{-1}(\cm{h}{3}) = D\upsilon(h),
\end{multline*}
so that, indeed, \(\sigma \star D\upsilon \star \sigma^{-1} = D\upsilon\).
\end{proof}
We now provide a suitable generalisation of the degree \(1\) group cohomology of a group \(\Gamma\) with coefficients in an \(\mathbf{R}\)-linear representation of \(\Gamma\). That degree \(1\) group cohomology can be generalised using the appropriate Hochschild cohomology was first observed by Sch\"{u}rmann~\cite{Schurmann}*{\S 3}; this observation has been used, for instance, to characterise the Haagerup property on locally compact quantum groups~\cite{DFSW}. We shall use the following variation on this theme to compute the Atiyah space of a crossed product by \(H\).
\begin{propositiondefinition}
Let \(B\) be a right \(H\)-module \(\ast\)-algebra, and let \(M\) be a right \(H\)-equivariant \(B\)-\(\ast\)-bimodule.
\begin{enumerate}
\item A \emph{lazy \(M\)-valued Hochschild \(0\)-cochain on \(H\)} is an element of
\[
\CH^0_\ell(H;M) \coloneqq \Zent_B(M)_{\mathrm{sa}}.
\]
\item A \emph{lazy \(M\)-valued Hochschild \(1\)-cocycle on \(H\)} is
\(
\mu \in \Zent_B\mleft(\mathcal{C}(H,M)\mright)_{\mathrm{sa}}
\),
such that
\begin{equation}\label{hcocycle}
\forall h,k \in H, \quad \mu(hk) = \mu(h) \triangleleft k + \epsilon(h)\mu(k)
\end{equation}
and \(\mu(1) = 0\); we denote by \(\ZH^1_\ell(H;M)\) the set of all lazy \(M\)-valued Hochschild \(1\)-cocycles on \(H\), which defines a \(\mathbf{R}\)-subspace of \(\mathcal{C}(H,M)\).
\item The \emph{coboundary map} is the homomorphism \(D : \CH^0_\ell(H;M) \to \ZH^1_\ell(H;M)\) with
\begin{equation}
\forall m \in \CH^0_\ell(H;M), \, \forall h \in H, \quad D(m)(h) \coloneqq m \triangleleft h - m \epsilon(h);
\end{equation}
thus, a \emph{lazy \(M\)-valued Hochschild \(1\)-coboundary on \(H\)} is an element of
\[
\BH^1_\ell(H;M) \coloneqq D(\CH^0_\ell(H;M)) \leq \ZH^1_\ell(H;M).
\]
\item The \emph{lazy degree \(1\) Hochschild cohomology of \(H\) with coefficients in \(M\)} is the group
\[
\HH^1_\ell(H;M) \coloneqq \ZH^1_\ell(H;M)/\BH^1_\ell(H;M).
\]
\end{enumerate}
\end{propositiondefinition}
\begin{proof}
Note that the right \(H\)-module \(M\) becomes an \(H\)-bimodule with respect to the trivial left \(H\)-action. From this perspective, the set \(\ZH^1_\ell(H;B,M)\) is an \(\mathbf{R}\)-subspace of the \(\mathbf{C}\)-vector space of \(M\)-valued Hochschild \(1\)-cocycles on \(H\), the set \(\BH^1_\ell(H;B,M)\) is an \(\mathbf{R}\)-subspace of the \(\mathbf{C}\)-vector space of \(M\)-valued Hochschild \(1\)-coboundaries on \(H\), and the map \(D\) is the restriction of the usual Hochschild coboundary operator. Thus, it suffices to check the inclusion
\(
\BH^1_\ell(H;B,M) \subseteq \Zent_B(\mathcal{C}(H,M))_{\mathrm{sa}}
\).
Let \(m \in \CH^0_\ell(H;M)\), and set \(\mu \coloneqq Dm\). Then, for all \(b \in B\) and \(h \in H\),
\begin{align*}
[\mu,\rho_B(b)](h) &= (m \triangleleft \cm{h}{1} - m \epsilon(\cm{h}{1})) \cdot (b \triangleleft \cm{h}{2}) - (b \triangleleft \cm{h}{1}) \cdot (m \triangleleft \cm{h}{2} - m \epsilon(\cm{h}{2})) \\
&= [m,b] \triangleleft h - [m,b \triangleleft h] =0.
\end{align*}
Hence, indeed, \([\mu,\rho_B(b)] = 0\).
\end{proof}
We now show that the lazy Sweedler cohomology of \(H\) admits a canonical \(\mathbf{R}\)-linear action on the lazy Hochschild cohomology of \(H\) with relevant coefficients, with respect to which \(H\)-equvariant \(\ast\)-derivations canonically yield \(1\)-cocycles in the sense of ordinary group cohomology. We shall compute the gauge action of gauge transformations on gauge potentials for a crossed product by \(H\) in terms of the resulting \(\mathbf{R}\)-affine actions of lazy Sweedler cohomology on lazy Hochschild cohomology.
\begin{theorem}\label{lazymcthm}
Let \(B\) be a right \(H\)-module \(\ast\)-algebra, and let \(M\) be a right \(H\)-equivariant \(B\)-\(\ast\)-bimodule.
\begin{enumerate}
\item The group \(\ZS^1_\ell(H;B,M)\) acts \(\mathbf{R}\)-linearly on \(\ZH^1_\ell(H;M)\) via conjugation, i.e.,
\begin{equation}
\forall \sigma \in \ZS^1_\ell(H;B,M), \, \forall \mu \in \ZH^1_\ell(H;M), \quad \sigma \triangleright \mu \coloneqq \sigma \star \mu \star \sigma^{-1}.
\end{equation}
\item The subgroup \(\BS^1_\ell(H;B,M)\) acts trivially on \(\ZH^1_\ell(H;M)\) and the subspace \(\BH^1_\ell(H;M)\) consists of \(\ZS^1_\ell(H;B,M)\)-invariant vectors, so that the linear action of \(\ZS^1_\ell(H;B,M)\) on \(\ZH^1_\ell(H;M)\) descends to a linear action of \(\HS^1_\ell(H;B,M)\) on \(\HH^1_\ell(H;M)\).
\item Let \(\partial : B \to M\) be an \(H\)-equivariant \(\ast\)-derivation. The action of \(\ZS^1_{\ell}(H;B,M)\) on \(\ZH^1_\ell(H;M)\) admits the \(1\)-cocycle \(\MC[\partial] : \ZS^1_\ell(H;B,M) \to \ZH^1_\ell(H;M)\) given by
\begin{equation}
\forall \sigma \in \ZS^1_\ell(H;B,M), \, \forall h \in H, \quad \MC[\partial](\sigma)(h) \coloneqq -\partial\sigma(\cm{h}{1}) \cdot \sigma^\ast(\cm{h}{2}).
\end{equation}
Furthermore, the \(1\)-cocycle \(\MC[\partial]\) satisfies
\begin{equation}\label{coboundaryeq}
\forall \upsilon \in \CS^0_\ell(H;B,M), \quad \MC[\partial](D\upsilon) = D(-\partial(\upsilon)\upsilon^\ast),
\end{equation}
and hence descends to a \(1\)-cocycle \(\tilde{\MC}[\partial] : \HS^1_\ell(H;B,M) \to \HH^1_\ell(H;M)\) for the induced action of \(\HS^1_{\ell}(H;B,M)\) on \(\HH^1_\ell(H;M)\).
\end{enumerate}
\end{theorem}
\begin{proof}
Let us first show that \(\ZH^1_\ell(H;M)\) is invariant under conjugation by \(\ZS^1_\ell(H;B,M)\); note that
\(
\set{\sigma \triangleright \mu \given \sigma \in \ZS^1_\ell(H;B,M), \, \mu \in \ZH^1_\ell(H;M)} \subset Z_B(M)
\) since \(\ZS^1_\ell(H,M)\) centralises \(\rho_B(B)\). Let \(\sigma \in \ZS^1_\ell(H;B,M)\) and let \(\mu \in \ZH^1_\ell(H;M)\). First,
\[
(\sigma \triangleright \mu)(1) = \sigma(1) \cdot \mu(1) \cdot \inv{\sigma}(1) = 0.
\] Next, since \(\sigma\) is unitary in the \(\ast\)-algebra \(\mathcal{C}(H;B)\) and \(\mu\) is self-adjoint in the \(\mathcal{C}(H;B)\)-\(\ast\)-bimodule \(\mathcal{C}(H;M)\), it follows that \(\sigma \triangleright \mu = \sigma \star \mu \star \sigma^\ast\) is self-adjoint. Finally, for all \(h,k \in H\),
\begin{align*}
&(\sigma \triangleright \mu)(hk)\\ &\quad= \sigma(\cm{h}{1}\cm{k}{1}) \cdot \mu(\cm{h}{2}\cm{k}{2}) \cdot \inv{\sigma}(\cm{h}{3}\cm{k}{3})\\
&\quad= (\sigma(\cm{h}{1}) \triangleleft \cm{k}{1}) \cdot \sigma(\cm{k}{2}) \cdot \left(\mu(\cm{h}{2}) \triangleleft \cm{k}{3} - \epsilon(\cm{h}{2})\mu(\cm{k}{3}) \right) \cdot (\inv{\sigma}(\cm{h}{3}) \triangleleft \cm{k}{4}) \cdot \inv{\sigma}(\cm{k}{5})\\
&\quad= (\sigma(\cm{h}{1}) \triangleleft \cm{k}{1}) \cdot (\mu(\cm{h}{2}) \triangleleft \cm{k}{2}) \cdot \sigma(\cm{k}{3}) \cdot (\inv{\sigma}(\cm{h}{3})\triangleleft\cm{k}{4}) \cdot \inv{\sigma}(\cm{k}{5})\\
&\quad\quad\quad\quad -(\sigma(\cm{h}{1}) \triangleleft \cm{k}{1}) \cdot \sigma(\cm{k}{2}) \cdot (\inv{\sigma}(\cm{h}{2} \triangleleft \cm{k}{3}) \cdot \mu(\cm{k}{4}) \cdot \inv{\sigma}(\cm{k}{5})\\
&\quad= (\sigma(\cm{h}{1}) \triangleleft \cm{k}{1}) \cdot (\mu(\cm{h}{2}) \triangleleft \cm{k}{2}) \cdot (\inv{\sigma}(\cm{h}{3})\triangleleft\cm{k}{3}) \cdot \sigma(\cm{k}{4}) \cdot \inv{\sigma}(\cm{k}{5})\\
&\quad\quad\quad\quad - (\sigma(\cm{h}{1}) \triangleleft \cm{k}{1}) \cdot (\inv{\sigma}(\cm{h}{2} \triangleleft \cm{k}{2}) \cdot \sigma(\cm{k}{3}) \cdot \mu(\cm{k}{4}) \cdot \inv{\sigma}(\cm{k}{5})\\
&\quad= (\sigma\triangleright\mu)(h)\triangleleft k - \epsilon(h)(\sigma\triangleright\mu)(k)
\end{align*}
by repeated application of Lemma~\ref{centrallem}. Hence, \(\sigma \triangleright \mu \in \ZH^1_\ell(H;M)\).
Next, let us show that \(\BS^1_\ell(H;B,M)\) acts trivially by conjugation on \(\ZH^1_\ell(H;M)\) and that \(\BH^1_\ell(H;M)\) consists of \(\ZS^1_\ell(H;B,M)\)-invariant vectors. First, let \(\upsilon \in \CS^0_\ell(H;B,M)\) and let \(\mu \in \ZH^1_\ell(H;M)\). For all \(h \in H\), by Lemma~\ref{centrallem},
\begin{multline*}
D\upsilon \triangleright \mu(h) = (\upsilon \triangleleft \cm{h}{1})\upsilon^\ast \cdot \mu(\cm{h}{2}) \cdot (\upsilon^\ast \triangleleft \cm{h}{3})\upsilon
= (\upsilon \triangleleft \cm{h}{1})\upsilon^\ast (\upsilon^\ast \triangleleft \cm{h}{2}) \cdot \mu(\cm{h}{3})\cdot\upsilon \\
= (\upsilon \triangleleft \cm{h}{1})(\upsilon^\ast \triangleleft \cm{h}{2}) \cdot \mu(\cm{h}{3}) \cdot \upsilon^\ast\upsilon = \epsilon(\cm{h}{1})\mu(\cm{h}{2}) = \mu(h).
\end{multline*}
so that, indeed, \(D\upsilon \triangleright \mu = \mu\). Now, let \(m \in \CH^0_\ell(H;M)\) and \(\sigma \in \ZS^1_\ell(H;B,M)\). Then for all \(h \in H\), by Lemma~\ref{centrallem},
\begin{multline*}
(\sigma \triangleright Dm)(h) = \sigma(\cm{h}{1}) \cdot (m \triangleleft \cm{h}{2} - \epsilon(\cm{h}{2})m) \cdot \inv{\sigma}(\cm{h}{3})\\
= (m \triangleleft \cm{h}{1}) \cdot \sigma(\cm{h}{2})\inv{\sigma}(\cm{h}{3}) - \sigma(\cm{h}{1})\inv{\sigma}(\cm{h}{2}) \cdot m = m \triangleleft h - \epsilon(h)m = Dm(h).
\end{multline*}
From now on, let \(\partial : B \to M\) be an \(H\)-equivariant \(\ast\)-derivation. Let us now show that \(\MC[\partial]\) is a well-defined function. Let \(\sigma \in \ZS^1_\ell(H;B,M)\), and let \(\mu \coloneqq \MC[\partial](\sigma) \in \mathcal{C}(H,M)\). First, \(\mu(1) = -\partial\sigma(1) \cdot \sigma^\ast(1) = 0\). Next, for all \(b \in B\) and \(h \in H\), by Lemma~\ref{centrallem},
\begin{align*}
&(b \triangleleft \cm{h}{1}) \cdot \mu(\cm{h}{2})\\ &\quad= - (b \triangleleft \cm{h}{1}) \cdot \partial\sigma(\cm{h}{2}) \cdot \sigma^\ast(\cm{h}{3})\\
&\quad= -\partial\mleft((b\triangleleft\cm{h}{1}) \cdot \sigma(\cm{h}{2})\mright) \cdot \sigma^\ast(\cm{h}{3}) + \partial(b \triangleleft \cm{h}{1}) \cdot \sigma(\cm{h}{2})\sigma^\ast(\cm{h}{3})\\
&\quad= -\partial\mleft(\sigma(\cm{h}{1}) \cdot (b\triangleleft\cm{h}{2})\mright) \cdot \sigma^\ast(\cm{h}{3}) + \partial(b) \triangleleft h\\
&\quad= -\partial\sigma(\cm{h}{1}) \cdot (b \triangleleft \cm{h}{2})\sigma^\ast(\cm{h}{3}) -\sigma(\cm{h}{1}) \cdot \left(\partial(b) \triangleleft \cm{h}{2}\right) \cdot \sigma^\ast(\cm{h}{3}) + \partial(b) \triangleleft h\\
&\quad= - \partial\sigma(\cm{h}{1}) \cdot \sigma^\ast(\cm{h}{2}) (b \triangleleft \cm{h}{3}) - \sigma(\cm{h}{1})\sigma^\ast(\cm{h}{2}) \cdot (\partial(b) \triangleleft h) +\partial(b)\triangleleft h\\
&\quad= \mu(\cm{h}{1}) \cdot (b \triangleleft \cm{h}{2}),
\end{align*}
so that \(\mu \in \Zent_B(\mathcal{C}(H,M))\). Next, for all \(h \in H\), since \(\sigma(\cm{h}{1})\sigma^\ast(\cm{h}{2}) = \epsilon(h)1_B\),
\[
\mu^\ast(h) = \left(-\partial\sigma(\cm{S(h)}{1}^\ast) \cdot \sigma(S(\cm{S(h)}{2}^\ast)^\ast)^\ast \right)^\ast
= \sigma(\cm{h}{1}) \cdot \partial\sigma^\ast(\cm{h}{2}) = -\partial \sigma(\cm{h}{1}) \cdot \sigma^\ast(\cm{h}{2}) = \mu(h),
\]
so that \(\mu^\ast = \mu\). Finally, for all \(h, k \in H\), by Lemma~\ref{centrallem} and the fact that \(\sigma \in \Unit(\mathcal{C}(H,B))\),
\begin{align*}
&\mu(hk)\\ &\quad= -\partial\sigma(\cm{h}{1}\cm{k}{1}) \cdot \sigma^\ast(\cm{h}{2}\cm{k}{2})\\
&\quad= -\partial\mleft((\sigma(\cm{h}{1}) \triangleleft \cm{k}{1}) \cdot \sigma(\cm{k}{2})\mright) \cdot (\sigma^\ast(\cm{h}{2}) \triangleleft \cm{k}{3}) \sigma^\ast(\cm{k}{4})\\
&\quad= -\partial\mleft((\sigma(\cm{h}{1}) \triangleleft \cm{k}{1}) \cdot \sigma(\cm{k}{2})\mright) \cdot \sigma^\ast(\cm{k}{3}) (\sigma^\ast(\cm{h}{2}) \triangleleft \cm{k}{4})\\
&\quad= -(\partial\sigma(\cm{h}{1}) \triangleleft \cm{k}{1}) \cdot \sigma(\cm{k}{2}) \sigma^\ast(\cm{k}{3}) (\sigma^\ast(\cm{h}{2}) \triangleleft \cm{k}{4}) \\
&\quad\quad\quad\quad - (\sigma(\cm{h}{1})\triangleleft \cm{k}{1}) \cdot \partial\sigma(\cm{k}{2}) \cdot \sigma^\ast(\cm{k}{3}) (\sigma^\ast(\cm{h}{2}) \triangleleft \cm{k}{4})\\
&\quad= -(\partial\sigma(\cm{h}{1}) \triangleleft \cm{k}{1}) \cdot \epsilon(\cm{k}{2})(\sigma^\ast(\cm{h}{2}) \triangleleft \cm{k}{3}) - (\sigma(\cm{h}{1}) \triangleleft \cm{k}{1}) \cdot\mu(\cm{k}{2}) \cdot (\sigma^\ast(\cm{h}{2}) \triangleleft \cm{k}{3})\\
&\quad= -(\partial\sigma(\cm{h}{1}) \triangleleft \cm{k}{1}) (\sigma^\ast(\cm{h}{2}) \triangleleft \cm{k}{2}) - (\sigma(\cm{h}{1}) \triangleleft \cm{k}{1})(\sigma^\ast(\cm{h}{2}) \triangleleft \cm{k}{2})\cdot \mu(\cm{k}{3})\\
&\quad = \mu(h)\triangleleft k + \epsilon(h)\mu(k).
\end{align*}
Hence, \(\MC[\partial](\sigma) \eqqcolon \mu \in \ZH^1_\ell(H;M)\).
Next, let us show that \(\MC[\partial]\) is a \(1\)-cocycle for the conjugation action of \(\ZS^1_\ell(H;B,M)\) on \(\ZH^1_\ell(H;M)\). Let \(\sigma,\tau \in \ZS^1_\ell(H;B,M)\). Then, for all \(h \in H\),
\begin{align*}
\MC[\partial](\sigma \star \tau) (h) &= -\partial\mleft(\sigma(\cm{h}{1})\tau(\cm{h}{2})\mright) \cdot \tau^\ast(\cm{h}{3})\sigma^\ast(\cm{h}{4})\\
&= -\partial\sigma(\cm{h}{1}) \cdot \tau(\cm{h}{2})\tau^\ast(\cm{h}{3})\sigma^\ast(\cm{h}{4}) - \sigma(\cm{h}{1}) \cdot \partial\tau(\cm{h}{2}) \cdot \tau^\ast(\cm{h}{3})\sigma^\ast(\cm{h}{4})\\
&= \MC[\partial](\sigma)(h) + \sigma \triangleright \MC[\partial](\tau)(h),
\end{align*}
so that, indeed, \(\MC[\partial](\sigma \star \tau) = \MC[\partial](\sigma) + \sigma \triangleright \MC[\partial](\tau)\).
Finally, let us show that \(\MC[\partial]\) satisfies \eqref{coboundaryeq}. First, observe that \(\partial(\Cent_B(M)) \subset \Zent_B(M)\), since for all \(b \in \Cent_B(M)\) and \(c \in B\),
\[
\partial(b) \cdot c = \partial(bc) - b \cdot \partial(c) = \partial(cb) - b \cdot \partial(c) = \partial(c)\cdot b + c \cdot \partial(b) - b \cdot \partial(c) = c \cdot \partial(b).
\]
Now, let \(\upsilon \in \CS^0_\ell(H;B,M)\). Then, for all \(h \in H\), by the above observation,
\begin{align*}
\MC[D\upsilon](h) &= - \partial\mleft((\upsilon \triangleleft \cm{h}{1}) \cdot \upsilon^\ast\mright) \cdot \left((\upsilon^\ast \triangleleft \cm{h}{2}) \cdot \upsilon\right)\\
&= -(\partial(\upsilon) \triangleleft \cm{h}{1}) \cdot \upsilon^\ast (\upsilon^\ast \triangleleft \cm{h}{2}) \cdot \upsilon - (\upsilon \triangleleft \cm{h}{1}) \cdot \partial(\upsilon^\ast) \cdot (\upsilon^\ast \triangleleft \cm{h}{2}) \cdot \upsilon\\
&= -(\partial(\upsilon) \cdot \upsilon^\ast) \triangleleft h - \epsilon(h)\upsilon\cdot\partial(\upsilon^\ast)\\
&= -(\partial(\upsilon) \cdot \upsilon^\ast) \triangleleft h + \epsilon(h) \partial(\upsilon) \cdot \upsilon^\ast\\
&= D(-\partial(\upsilon) \cdot \upsilon^\ast)(h),
\end{align*}
as was claimed.
\end{proof}
\begin{definition}
Let \(B\) be a right \(H\)-module \(\ast\)-algebra, let \(M\) be a right \(H\)-equivariant \(B\)-\(\ast\)-bimodule, and let \(\partial : B \to M\) be a right \(H\)-equivariant \(\ast\)-derivation. We define the \emph{Maurer--Cartan \(1\)-cocycle} of \(\partial\) to be the \(1\)-cocycle \(\MC[\partial] : \ZS^1(H;B,M) \to \ZH^1(H;M)\) of Theorem~\ref{lazymcthm} induced by \(\partial\).
\end{definition}
When \(H\) is a group algebra, our constructions reduce to degree \(1\) group cohomology in a manner compatible, e.g., with the results of \'{C}a\'{c}i\'{c}--Mesland~\cite{CaMe}*{Thm.\ 3.11}.
\begin{proposition}[cf.\ Sweedler~\cite{Sweedler}*{Thm.\ 3.1}]\label{lazygrp}
Suppose that \(H = \mathbf{C}[\Gamma]\) for \(\Gamma\) a group. Let \(B\) be a right \(\Gamma\)-\(\ast\)-algebra, and let \(M\) be a right \(\Gamma\)-equivariant \(B\)-\(\ast\)-bimodule, so that \(\Unit(\Cent_B(B \oplus M))\) defines a multiplicative \(\Gamma\)-module and \(Z_B(M)_{\mathrm{sa}}\) defines a \(\mathbf{R}\)-linear representation of \(\Gamma\).
\begin{enumerate}
\item The map
\(
r_B \coloneqq (f \mapsto \rest{f}{\Gamma}) : \ZS^1_\ell(H;B,M) \to \operatorname{Z}^1\mleft(\Gamma,\Unit(\Cent_B(B \oplus M))\mright)
\)
is a group isomorphism, such that \(r_B \circ D = \dt{\Gamma}\) and
\(
r_B\mleft(\BS^1_\ell(H;B,M)\mright) = \operatorname{B}^1\mleft(\Gamma,\Unit(\Cent_B(B \oplus M))\mright);
\)
hence, in particular, it descends to an isomorphism of Abelian groups
\[
\HS^1_\ell(H;M) \xrightarrow{\sim} \operatorname{H}^1\mleft(\Gamma,\Unit(\Cent_B(B\oplus M))\mright).
\]
\item The map
\(
r_M \coloneqq (\mu \mapsto \rest{\mu}{\Gamma}) : \ZH^1_\ell(H;M) \to \operatorname{Z}^1(\Gamma,\Zent_B(M)_\mathrm{sa})
\)
defines an isomorphism of \(\mathbf{R}\)-vector spaces, such that \(r_M \circ D = \dt{\Gamma}\) and
\(
r_M\mleft(\BH^1_\ell(H;M)\mright) = \operatorname{B}^1(\Gamma,\Zent_B(M)_\mathrm{sa});
\)
hence, in particular, it descends to an isomorphism of Abelian groups
\[
\HH^1_\ell(H;M) \xrightarrow{\sim} \operatorname{H}^1(\Gamma,\Zent_B(M)_\mathrm{sa}).
\]
\item The conjugation action of \(\ZS^1_\ell(H;B,M)\) on \(\ZH^1_\ell(H;M)\) is trivial. Thus, for every \(\Gamma\)-equivariant \(\ast\)-derivation \(\partial : B \to M\), the induced map
\[
\MC_\Gamma \coloneqq r_M \circ \MC[\partial] \circ r_B^{-1} : \operatorname{Z}^1\mleft(\Gamma,\Unit(\Cent_B(B \oplus M))\mright)
\to \operatorname{Z}^1(\Gamma,\Zent_B(M)_\mathrm{sa})
\]
is a group homomorphism satisfying
\begin{gather*}
\forall \sigma \in \operatorname{Z}^1\mleft(\Gamma,\Unit(\Cent_B(B \oplus M))\mright), \,\forall \gamma \in \Gamma, \quad \MC_\Gamma(\sigma)(\gamma) = -\partial \sigma(\gamma) \cdot \sigma(\gamma)^\ast,\\
\forall \upsilon \in \Unit(\Cent_B(B \oplus M)), \quad
\MC_\Gamma(\dt{\Gamma}\upsilon) = \dt{\Gamma}(-\partial(\upsilon) \cdot \upsilon^\ast),
\end{gather*}
so that, in particular, it descends to a group homomorphism \[\operatorname{H}^1\mleft(\Gamma,\Unit(\Cent_B(B \oplus M))\mright)
\to \operatorname{H}^1(\Gamma,\Zent_B(M)_\mathrm{sa}).\]
\end{enumerate}
\end{proposition}
\begin{proof}
Given sets \(X\) and \(Y\), let \(\mathcal{F}(X,Y)\) be the set of all functions from \(X\) to \(Y\). It follows that \(\mathcal{F}(\Gamma,B)\) is a unital \(\ast\)-algebra with respect to the pointwise operations, that any unital \(\ast\)-subalgebra \(A\) of \(B\) yields a unital \(\ast\)-subalgebra \(\mathcal{F}(\Gamma,A)\) of \(\mathcal{F}(\Gamma,B)\), that \(\mathcal{F}(\Gamma,M)\) defines a \(\mathcal{F}(\Gamma,B)\)-\(\ast\)-bimodule with respect to pointwise operations, and that any \(B\)-\(\ast\)-sub-bimodule \(L\) of \(M\) yields a \(\mathcal{F}(\Gamma,B)\)-\(\ast\)-sub-bimodule of \(\mathcal{F}(\Gamma,M)\). By abuse of notation, let
\[
r_B \coloneqq (f \mapsto \rest{f}{\Gamma}) : \mathcal{C}(H,B) \to \mathcal{F}(\Gamma,B), \quad r_M \coloneqq (\mu \mapsto \rest{\mu}{\Gamma}) : \mathcal{C}(H,M) \to \mathcal{F}(\Gamma,M).
\]
Since \(\Gamma\) is a basis for \(H\), it follows that \(r_B\) and \(r_M\) are bijections, and since \(\Gamma\) consists of group-like elements and satisfies \(\rest{S \circ \ast}{\Gamma} = \id_\Gamma\), it follows that \(r_B\) is a \(\ast\)-isomorphism and that \(r_M\) is a \(\ast\)-preserving \(\mathbf{C}\)-linear map satisfying
\[
\forall \sigma,\tau \in \mathcal{C}(H,B), \, \forall \mu \in \mathcal{C}(H,M), \quad r_M(\sigma \star \mu \star \tau) = r_B(\sigma) \cdot r_M(\mu) \cdot r_B(\tau).
\]
The fact that \(\Gamma\) consists of group-like elements now implies that
\begin{gather*}
r_B\mleft(\Unit(\Cent_{\mathcal{C}(H,B)}(\rho_B(B) \oplus \rho_M(M)))\mright) = \mathcal{F}(\Gamma,\Unit(\Cent_B(B\oplus M))),\\
r_M\mleft(\Zent_B(\mathcal{C}(H,M))_\mathrm{sa}\mright) = \mathcal{F}(\Gamma,\Zent_B(M)_\mathrm{sa}),
\end{gather*}
from which our claims now follow by routine calculations.
\end{proof}
Similarly, when \(H\) is the universal enveloping algebra of a real Lie algebra, our constructions reduce to standard degree \(1\) Lie cohomology (with a small caveat related to lazy Sweedler cohomology) when the coefficient algebra satisfies the following condition.
\begin{definition}
Let \(\mathfrak{g}\) be a real Lie algebra, and let \(A\) be a commutative right \(\mathfrak{g}\)-module \(\ast\)-algebra. We say that \(A\) \emph{has \(\mathfrak{g}\)-equivariant exponentials} if for every \(a \in A_\mathrm{sa}\) there exists a unitary \(\upsilon_a \in \Unit(A)\), such that
\[
\forall X \in \mathfrak{g}, \quad (\upsilon_a) \triangleleft X = \ensuremath{\mathrm{i}}{} (a \triangleleft X) \cdot \upsilon_a,
\]
in which case we call \(\upsilon_a\) a \emph{\(\mathfrak{g}\)-equivariant exponential} of \(a\).
\end{definition}
\begin{proposition}[cf.\ Sweedler~\cite{Sweedler}*{Prop.\ 4.2}]
Suppose that \(H = \mathcal{U}(\mathfrak{g})\) for \(\mathfrak{g}\) a real Lie algebra. Let \(B\) be a right \(\mathfrak{g}\)-module \(\ast\)-algebra, and let \(M\) be a right \(\mathfrak{g}\)-equivariant \(B\)-\(\ast\)-bimodule, so that \(\Cent_B(B \oplus M)_\mathrm{sa}\) and \(Z_B(M)_\mathrm{sa}\) both define right \(\mathfrak{g}\)-modules over \(\mathbf{R}\).
\begin{enumerate}
\item The map
\(
r_B \coloneqq (f \mapsto \rest{-\ensuremath{\mathrm{i}}{}f}{\mathfrak{g}}) : \ZS^1_\ell(H;B,M) \to \operatorname{Z}^1\mleft(\mathfrak{g},\Cent_B(B \oplus M)_{\mathrm{sa}}\mright)
\)
defines an isomorphism of Abelian groups, such that
\[
r_B\mleft(\BS^1_\ell(H;B,M)\mright) = \set{(X \mapsto -\ensuremath{\mathrm{i}}{}(\upsilon \triangleleft X)\upsilon^\ast) \given \upsilon \in \Unit(\Cent_B(B \oplus M))}.
\]
Moreover, if \(\Cent_B(B \oplus M)\) has \(\mathfrak{g}\)-equivariant exponentials, then for every \(b \in \Cent_B(B \oplus M)_\mathrm{sa}\) and every \(\mathfrak{g}\)-equivariant exponential \(\upsilon_b \in \Unit(\Cent_B(B \oplus M))\) of \(b\),
\[
r_B \circ D(\upsilon_b) = \dt{\mathfrak{g}}(b),
\]
so that \(B^1\mleft(\mathfrak{g},\Cent_B(B \oplus M)_{\mathrm{sa}}\mright) \leq r_B\mleft(\BS^1_\ell(H;B,M)\mright)\), and
hence, \(r_B\) induces a short exact sequence of Abelian groups
\[
0 \to \frac{r_B\mleft(\BS^1_\ell(H;B,M)\mright)}{B^1\mleft(\mathfrak{g},\Cent_B(B \oplus M)_{\mathrm{sa}}\mright)} \to \operatorname{H}^1\mleft(\mathfrak{g},\Cent_B(B \oplus M)_{\mathrm{sa}}\mright) \to \HS^1_\ell(H;B,M) \to 0.
\]
\item The map
\(
r_M \coloneqq (\mu \mapsto \rest{\mu}{\mathfrak{g}}) : \ZH^1_\ell(H;M) \to \operatorname{Z}^1(\mathfrak{g},\Zent_B(M)_\mathrm{sa})
\)
is an isomorphism of \(\mathbf{R}\)-vector spaces, such that \(r_M \circ D = \dt{\mathfrak{g}}\) and
\(
r_M\mleft(\BH^1_\ell(H;M)\mright) = \operatorname{B}^1(\mathfrak{g},\Zent_B(M)_\mathrm{sa})
\);
hence, in particular, \(r_M\) descends to an isomorphism of Abelian groups
\[
\HH^1_\ell(H;M) \xrightarrow{\sim} \operatorname{H}^1(\mathfrak{g},\Zent_B(M)_\mathrm{sa}).
\]
\item The conjugation action of \(\ZS^1_\ell(H;B,M)\) on \(\ZH^1_\ell(H;M)\) is trivial. Thus, for every \(\mathfrak{g}\)-equivariant \(\ast\)-derivation \(\partial : B \to M\), the induced map
\[
\MC_\mathfrak{g} \coloneqq r_M \circ \MC[\partial] \circ r_B^{-1} : \operatorname{Z}^1\mleft(\mathfrak{g},\Cent_B(B \oplus M)_{\mathrm{sa}}\mright) \to \operatorname{Z}^1(\mathfrak{g},\Zent_B(M)_\mathrm{sa})
\]
is a group homomorphism satisfying
\begin{gather*}
\forall c \in \operatorname{Z}^1\mleft(\mathfrak{g},\Cent_B(B \oplus M)_{\mathrm{sa}}\mright), \,\forall X \in \mathfrak{g}, \quad \MC_\mathfrak{g}(\sigma)(X) = -\ensuremath{\mathrm{i}}{}\partial c(X),\\
\forall b \in \Cent_B(B \oplus M)_\mathrm{sa}, \, \quad \MC_\mathfrak{g}(\dt{\mathfrak{g}}b) = \dt{\mathfrak{g}}(-\ensuremath{\mathrm{i}}{}\partial b),
\end{gather*}
so that, in particular, it descends to a group homomorphism \[\operatorname{H}^1\mleft(\mathfrak{g},\Cent_B(B \oplus M)_{\mathrm{sa}}\mright) \to \operatorname{H}^1(\mathfrak{g},\Zent_B(M)_\mathrm{sa}).\]
\end{enumerate}
\end{proposition}
\begin{proof}
Recall that \(H = \mathcal{U}(\mathfrak{g})\) inherits a filtration from the complexified tensor algebra of \(\mathfrak{g}\).
Let \(r_B \coloneqq (\sigma \mapsto \rest{-\ensuremath{\mathrm{i}}{}\sigma}{\mathfrak{g}}) : \ZS^1_\ell(H;B,M) \to \Hom(\mathfrak{g},B)\). First, given \(\sigma \in \ZS^1_\ell(H;B,M)\), for all \(k \in \mathbf{N} \cup \set{0}\), \(h \in H_k\), and \(X \in \mathfrak{g}\),
\[
\sigma(h \cdot X) = \left(\sigma(h) \triangleleft \cm{X}{1}\right)\sigma(\cm{X}{2}) = \left(\sigma(h) \triangleleft X\right)\sigma(1) + \left(\sigma(h) \triangleleft 1\right)\sigma(X) = \sigma(h) \triangleleft X + \sigma(h)\sigma(X),
\]
so that by induction on \(k \in \mathbf{N} \cup \set{0}\), the map \(\sigma\) is uniquely determined on \(H = \bigcup_{k \geq 0} H_k\) by \(r_B(\sigma)\); hence, \(r_B\) is injective. Next, note that for all \(\sigma, \tau \in \ZS^1_\ell(H;B,M)\) and \(X \in \mathfrak{g}\),
\[
(\sigma \star \tau)(X) = \sigma(\cm{X}{1})\tau(\cm{X}{2}) = \sigma(X)\tau(1) + \sigma(1)\tau(X) = \sigma(X) + \tau(X),
\]
and that \(\rest{1_{\mathcal{C}(H,B)}}{\mathfrak{g}} = \rest{-\ensuremath{\mathrm{i}}{}\epsilon(\cdot)1_B}{\mathfrak{g}} = 0\); hence, \(r_B\) is an injective group homomorphism from \(\ZS^1_\ell(H;B,M)\) to the additive group \(\Hom(\mathfrak{g},B)\). Next, for \(\sigma \in \ZS^1_\ell(H;B,M)\) and \(X \in \mathfrak{g}\),
\[
0 = \epsilon(X)1_B = \sigma(\cm{X}{1})\sigma^\ast(\cm{X}{2}) = \sigma(X)\sigma(S(1)^\ast)^\ast + \sigma(1)\sigma(S(X)^\ast)^\ast = \sigma(X) + \sigma(X)^\ast;
\]
hence, \(r_B\) maps into \(\Hom(\mathfrak{g},B_\mathrm{sa})\). Next, for \(\sigma \in \ZS^1_\ell(H;B,M)\), \(b \in B\), and \(X \in \mathfrak{g}\),
\begin{align*}
0 &= [\sigma,\rho_B(b)](X)\\
&= \sigma(\cm{X}{1})(b \triangleleft \cm{X}{2}) - (b \triangleleft \cm{X}{1})\sigma(\cm{X}{2})\\
&= \sigma(X)(b \triangleleft 1) - (b \triangleleft X)\sigma(1) +\sigma(1)(b \triangleleft X) - (b \triangleleft 1)\sigma(X)\\
&= [\sigma(X),b],
\end{align*}
so that \(r_B\) maps into \(\Hom(\mathfrak{g},\Zent(B)_\mathrm{sa})\); in fact, \emph{mutatis mutandis}, this shows that \(r_B\) actually maps into \(\Hom(\mathfrak{g},\Cent_B(B \oplus M)_\mathrm{sa})\). Finally, for all \(\sigma \in \ZS^1_\ell(H;B,M)\) and \(X,Y \in \mathfrak{g}\),
\begin{align*}
0 &= \sigma(XY-YX) - \sigma([X,Y])\\
&= \left(\sigma(X) \triangleleft \cm{Y}{1}\right)\sigma(\cm{Y}{2}) -\left(\sigma(Y) \triangleleft \cm{X}{1}\right)\sigma(\cm{X}{2}) - \sigma([X,Y])\\
&= \left(\sigma(X)\triangleleft Y\right)\sigma(1) + \left(\sigma(X)\triangleleft 1\right)\sigma(Y) - \left(\sigma(Y) \triangleleft X\right)\sigma(1) - \left(\sigma(Y) \triangleleft 1\right)\sigma(X) - \sigma([X,Y])\\
&= \sigma(X) \triangleleft Y - \sigma(Y) \triangleleft X - \sigma([X,Y])\\
&= \ensuremath{\mathrm{i}}{}\dt{\mathfrak{g}}(r_B(\sigma))
\end{align*}
so that \(r_B\) maps into \(\operatorname{Z}^1(\mathfrak{g},\Cent_B(B \oplus M)_\mathrm{sa}\). Conversely, given \(c \in \operatorname{Z}^1(\mathfrak{g},\Cent_B(B \oplus M)_\mathrm{sa}\), one can construct \(r_B^{-1}(c) \in \ZS^1_\ell(H;B,M)\) inductively by setting \(r_B^{-1}(c)(1) \coloneqq 1\) and setting
\[
\forall k \in \mathbf{N} \cup \set{0}, \, \forall h \in H_k, \, \forall X \in \mathfrak{g}, \quad r_B^{-1}(c)(h \cdot X) \coloneqq r_B^{-1}(c)(h) \triangleleft X+ \ensuremath{\mathrm{i}}{} r_B^{-1}(c)(h) \cdot c(X);
\]
in particular, that \(r_B^{-1}(c)\) is well-defined on \(H = \mathcal{U}(\mathfrak{g})\) follows from the \(1\)-cocycle identity
\[
\forall X,Y \in \mathfrak{g}, \quad c(X) \triangleleft Y - c(Y) \triangleleft X - c([X,Y]) = 0
\]
together with the universal property of \(\mathcal{U}(\mathfrak{g})\). Thus, \(r_B : \ZS^1_\ell(H;B,M) \to \operatorname{Z}^1(\mathfrak{g},\Cent_B(B\oplus M)_\mathrm{sa})\) is a group isomorphism.
Next, by definition of \(D : \CS^0_\ell(H;B,M) \to \ZS^1_\ell(H;B,M)\) and of \(r_B\), it follows that
\[
r_B(\BS^1_\ell(H;B,M)) = \set{(X \mapsto -\ensuremath{\mathrm{i}}{}(\upsilon \triangleleft X)\upsilon^\ast \given \upsilon \in \Unit(\Cent_B(B\oplus M)) } \leq \operatorname{Z}(\mathfrak{g},\Cent_B(B\oplus M)_\mathrm{sa}).
\]
Suppose, now, that \(B\) has \(\mathfrak{g}\)-equivariant exponentials. Given \(b \in \Cent_B(B \oplus M)_\mathrm{sa}\), for every \(\mathfrak{g}\)-equivariant exponential \(\upsilon_b \in \Unit(\Cent_B(B\oplus M))\) of \(b\) and every \(X \in \mathfrak{g}\),
\begin{align*}
D\upsilon_b(X) = (\upsilon_b \triangleleft X)\upsilon_b^\ast = \ensuremath{\mathrm{i}}{}(b \triangleleft X)\upsilon_b\upsilon_b^\ast = \ensuremath{\mathrm{i}}{}(b\triangleleft X) = \ensuremath{\mathrm{i}}{}\dt{\mathfrak{g}}(b)(X),
\end{align*}
so that \(\dt{\mathfrak{g}}(b) = r_B \circ D(\upsilon_b) \in r_B(\ZS^1_\ell(H;B,M))\); hence, as additive groups,
\[
\operatorname{B}^1(\mathfrak{g};\Cent_B(B \oplus M)_\mathrm{sa}) \leq r_B\mleft(\BS^1_\ell(H;B,M)\mright).
\]
Now, the above proof that \(r_B : \ZS^1_\ell(H;B,M) \to \operatorname{Z}^1(\mathfrak{g},\Cent_B(B\oplus M)_\mathrm{sa})\) is a group isomorphism implies, \emph{mutatis mutandis}, that
\[
r_M \coloneqq (\mu \mapsto \rest{-\ensuremath{\mathrm{i}}{}\mu}{\mathfrak{g}}) : \ZH^1_\ell(H;M) \to \operatorname{Z}^1(\mathfrak{g},\Zent_B(M)_\mathrm{sa})
\]
is an isomorphism of \(\mathbf{R}\)-vector spaces; indeed, for all \(m \in \Zent_B(M)_\mathrm{sa}\) and \(X \in \mathfrak{g}\),
\[
Dm(X) = m \triangleleft X - \epsilon(X)m = m \triangleleft X = \dt{\mathfrak{g}}(m)(X),
\]
so that \(r_M \circ D = \dt{\mathfrak{g}}\), and hence \(\operatorname{B}^1(\mathfrak{g},\Zent_B(M)_\mathrm{sa}) = r_M\mleft(\BH^1_\ell(H;M)\mright)\). Moreover, \(\ZS^1_\ell(H;B,M)\) acts trivially on \(\ZH^1_\ell(H;M)\), since for all \(\sigma \in \ZS^1_\ell(H;B,M)\), \(\mu \in \ZH^1_\ell(H;M)\), and \(X \in \mathfrak{g}\),
\begin{align*}
\sigma \star \mu \star \inv{\sigma}(X) &= \sigma(\cm{X}{1})\mu(\cm{X}{2})\inv{\sigma}(\cm{X}{3})\\
&= \sigma(X)\mu(1)\inv{\sigma}(1) + \sigma(1)\mu(X)\inv{\sigma}(1) + \sigma(1)\mu(1)\inv{\sigma}(X)\\
&= \mu(X).
\end{align*}
Finally, let \(\partial : B \to M\) be a \(\mathfrak{g}\)-equivariant \(\ast\)-derivation. For all \(c \in \operatorname{Z}^1(\mathfrak{g},\Zent_B(M)_\mathrm{sa})\) and \(X \in \mathfrak{g}\), we find that
\begin{align*}
\MC_\mathfrak{g}[\partial](c)(X) &= -\partial\mleft(r_B^{-1}(c)(\cm{X}{1})\mright) \cdot r_B^{-1}(c)^\ast(\cm{X}{2})\\
&= -\partial\mleft(r_B^{-1}(c)(X)\mright) \cdot r_B^{-1}(c)^\ast(1) -\partial\mleft(r_B^{-1}(c)(1)\mright) \cdot r_B^{-1}(c)^\ast(X)\\
&= -\ensuremath{\mathrm{i}}{}\partial c(X).
\end{align*}
Hence, in particular, for all \(b \in B\) and \(X \in \mathfrak{g}\),
\[
\MC_\mathfrak{g}[\partial](\dt{\mathfrak{g}}(b))(X) = -\ensuremath{\mathrm{i}}{}\partial(b \triangleleft X) = \partial(-\ensuremath{\mathrm{i}}{}b) \triangleleft X = \dt{\mathfrak{g}}(-\ensuremath{\mathrm{i}}{}\partial b). \qedhere
\]
\end{proof}
We now refine our construction of lazy Hochschild cohomology in the manner that, as we shall see, will encode prolongability of (relative) gauge potentials.
\begin{propositiondefinition}
Let \(B\) be a right \(H\)-module \(\ast\)-algebra, let \(M\) be a right \(H\)-equivariant \(B\)-\(\ast\)-module, and let \(\Omega\) be a right \(H\)-module graded \(\ast\)-algebra \(\Omega\) with \(\Omega^0 = B\) and \(\Omega^1 = M\) that is generated over \(B\) by \(M\). Then
\begin{gather*}
\ZH^1_\ell(H;M,\Omega) \coloneqq \ZH^1_\ell(H;M) \cap \Cent_{\mathcal{C}(H;\Omega)}(\rho_\Omega(\Omega)),\\
\BH^1_\ell(H;M,\Omega) \coloneqq D(\Zent_{B}(M)_\mathrm{sa} \cap \Zent(\Omega)) \subset \BH^1_\ell(H;M) \cap \Cent_{\mathcal{C}(H;\Omega)}(\rho_\Omega(\Omega)),\\
\HH^1_\ell(H;M,\Omega) \coloneqq \ZH^1_\ell(H;M,\Omega)/\BH^1_\ell(H;M,\Omega).
\end{gather*}
Moreover, \(\ZH^1_\ell(H;M,\Omega)\) is a \(\ZS^1_\ell(H;B,M)\)-invariant subspace of \(\ZH^1_\ell(H;M)\), and hence the conjugation action of \(\ZS^1_\ell(H;B,M)\) on \(\ZH^1_\ell(H;M,\Omega)\) descends to a \(\mathbf{R}\)-linear action on the quotient space \(\HH^1_\ell(H;M,\Omega)\).
\end{propositiondefinition}
\begin{proof}
The first non-trivial claim to check is the inclusion \(\BH^1_\ell(H;M,\Omega) \subset \Cent_{\mathcal{C}(H;\Omega)}(\rho_\Omega(\Omega))\). Let \(m \in \Cent_\Omega(M)_\mathrm{sa}\). Then for all \(\p{m} \in M\) and \(h \in H\),
\begin{align*}
[Dm,\rho_M(\p{m})](h) &= (m \triangleleft \cm{h}{1} - \epsilon(\cm{h}{1})m) \wedge (\p{m} \triangleleft \cm{h}{2}) + (\p{m} \triangleleft \cm{h}{1}) \wedge (m \triangleleft \cm{h}{2} - \epsilon(\cm{h}{2})m)\\
&= [m,\p{m}] \triangleleft h - [m, \p{m}\triangleleft h]= 0,
\end{align*}
so that, indeed, \(Dm \in \Cent_{\mathcal{C}(H;\Omega)}(\rho_\Omega(\Omega))\).
The other non-trivial claim to check is \(\ZS^1_\ell(H;B,M)\)-invariance of \(\ZH^1_\ell(H;M,\Omega)\). Let \(\sigma \in \ZS^1_\ell(H;B,M)\) and let \(\mu \in \ZH^1_\ell(H;M,\Omega)\). Then, for all \(m \in M\) and \(h \in H\),
\begin{align*}
[\sigma \triangleright \mu,\rho_M(m)](h) &= \sigma(\cm{h}{1}) \cdot \mu(\cm{h}{2}) \cdot \inv{\sigma}(\cm{h}{3}) \wedge m \triangleleft \cm{h}{4}\\
&\quad\quad\quad - m \triangleleft \cm{h}{1} \wedge \sigma(\cm{h}{2}) \cdot \mu(\cm{h}{3}) \cdot \inv{\sigma}(\cm{h}{4})\\
&= \sigma(\cm{h}{1}) \cdot \mu(\cm{h}{2}) \wedge m \triangleleft \cm{h}{3} \cdot \inv{\sigma}(\cm{h}{4})\\
&\quad\quad\quad - \sigma(\cm{h}{1}) \cdot m \triangleleft \cm{h}{2} \wedge \mu(\cm{h}{3}) \cdot \inv{\sigma}(\cm{h}{4})\\
&= \sigma \star [\mu,\rho_M(m)] \star \inv{\sigma}(h)=0,
\end{align*}
so that, indeed, \(\sigma \triangleright \mu \in \Cent_{\mathcal{C}(H;\Omega)}(\rho_\Omega(\Omega))\).
\end{proof}
We conclude by observing that Maurer--Cartan \(1\)-cocycles are automatically compatible with the above refinement.
\begin{corollary}
Let \(B\) be a right \(H\)-module \(\ast\)-algebra, and let \((\Omega_B,\dt{B})\) be an \(H\)-equivariant \textsc{sodc} over \(B\). Then
\[
\MC[\dt{B}]\mleft(\ZS^1_\ell(H;B,\Omega^1_B)\mright) \subset \ZH^1_\ell(H;\Omega^1_B,\Omega_B), \quad \MC[\dt{B}]\mleft(\BS^1_\ell(H;B,\Omega^1_B)\mright) \subset \BH^1_\ell(H;\Omega^1_B,\Omega_B),
\]
so that \(\MC[\dt{B}]\) is a \(1\)-cocycle for the restricted action of \(\ZS^1_\ell(H;B,\Omega^1_B)\) on \(\ZH^1_\ell(H;\Omega^1_B,\Omega_B)\), and hence descends to a \(1\)-cocycle \(\widetilde{\MC}[\dt{B},\Omega_B] : \HS^1_\ell(H;B,\Omega^1_B) \to \HH^1_\ell(H;\Omega^1_B,\Omega_B)\) for the action of \(\HS^1_\ell(H;B,\Omega^1_B)\) on \(\HH^1_\ell(H;\Omega^1_B,\Omega_B)\).
\end{corollary}
\begin{proof}
First, let \(\sigma \in \ZS^1_\ell(H;B,\Omega^1_B)\). Then, for all \(b \in B\) and \(h \in H\),
\begin{align*}
&[\MC[\dt{B}](\sigma),\rho_M(\dt{B} b)](h)\\ &\quad= -\dt{B} \sigma(\cm{h}{1}) \cdot \inv{\sigma}(\cm{h}{2}) \wedge (\dt{B}(b) \triangleleft \cm{h}{3}) - (\dt{B}(b) \triangleleft \cm{h}{1}) \wedge \dt{B} \sigma(\cm{h}{2}) \cdot \inv{\sigma}(\cm{h}{3})\\
&\quad = -\dt{B}\sigma(\cm{h}{1}) \wedge \dt{B}(b \triangleleft \cm{h}{2}) \cdot \inv{\sigma}(\cm{h}{3})- \dt{B}(b \triangleleft \cm{h}{1}) \wedge \dt{B} \sigma(\cm{h}{2}) \cdot \inv{\sigma}(\cm{h}{3})\\
&\quad= - \dt{B}\mleft(\sigma(\cm{h}{1}) \cdot (\dt{B}(b) \triangleleft \cm{h}{2})-(\dt{B}(b)\triangleleft\cm{h}{1}) \cdot \sigma(\cm{h}{2})\mright) \cdot \inv{\sigma}(\cm{h}{3})\\
&\quad=0,
\end{align*}
so that, indeed, \(\MC[\dt{B}](\sigma) \in \Cent_{\mathcal{C}(H;\Omega_B)}(\rho_{\Omega_B}(\Omega_B))\).
Now, let \(\upsilon \in \CS^0_\ell(H,B,\Omega^1_B) = \Unit(\Cent_B(B\oplus \Omega^1_B))\), so that \(\MC[\dt{B}](D\upsilon) = D(-\dt{B}(\upsilon)\upsilon^\ast)\) by Theorem~\ref{lazymcthm}. Then, for all \(b \in B\),
\begin{align*}
[-\dt{B}(\upsilon)\upsilon^\ast,\dt{B}(b)] &= -\dt{B}(\upsilon)\upsilon^\ast \wedge \dt{B}(b) - \dt{B}(b) \wedge \dt{B}(\upsilon)\upsilon^\ast \\
&= -\left(\dt{B}(\upsilon) \wedge \dt{B}(b) + \dt{B}(b) \wedge \dt{B}(\upsilon) \right)\upsilon^\ast\\
&= -\dt{B}\mleft(\upsilon \cdot \dt{B}(b) - \dt{B}(b)\cdot\upsilon \mright)\upsilon^\ast = 0,
\end{align*}
so that, indeed, \(-\dt{B}(\upsilon)\upsilon^\ast \in \Zent(\Omega)\).
\end{proof}
\subsection{Gauge transformations and (relative) gauge potentials}
In this section, we use lazy Sweedler cohomology, lazy Hochschild cohomology, and Maurer--Cartan \(1\)-cocycles to compute the gauge group \(\fr{G}\), the Atiyah space \(\fr{At}\), and the affine action of \(\fr{G}\) on \(\fr{At}\) in the case of a crossed product by the Hopf \(\ast\)-algebra \(H\). This yields a far-reaching generalisation of the computation by \'{C}a\'{c}i\'{c}--Mesland~\cite{CaMe}*{\S 3.4} of the noncommutative \(\mathbf{T}^m\)-gauge theory in this sense of a crossed product by \(\mathbf{Z}^m\).
Let \(B\) be a right \(H\)-module \(\ast\)-algebra with fixed right \(H\)-invariant \textsc{sodc} \((\Omega_B,\dt{B})\). Let \[P \coloneqq B \rtimes H,\] so that \(P = H \otimes_\mathbf{C} B\) together with the multiplication, \(\ast\)-structure, and left \(H\)-coaction defined, respectively, by
\begin{gather*}
\forall h,\p{h} \in H, \, \forall b,\p{b} \in B, \quad (h \otimes b)(\p{h} \otimes \p{b}) \coloneqq h\cm{h}{1}^\prime \otimes (b \triangleleft \cm{h}{2}^\prime)\p{b},\\
\forall h \in H, \, \forall b \in B, \quad (h \otimes b)^\ast \coloneqq \cm{h}{1}^\ast \otimes b^\ast \triangleleft \cm{h}{2}^\ast,\\
\forall h \in H, \, \forall b \in B, \quad \delta_P(h \otimes b) \coloneqq \cm{h}{1} \otimes \left(\cm{h}{2} \otimes b\right);
\end{gather*}
it follows that \(P\) defines a principal left \(H\)-comodule \(\ast\)-algebra, such that \(\coinv{H}{P} = B\); in fact, \(P\) can be viewed as a globally trivial (\emph{cleft}) principal \(H\)-comodule algebra with trivialisation given by the injective left \(H\)-covariant \(\ast\)-homomorphism
\(
(h \mapsto h \otimes 1_B) : H \hookrightarrow P
\).
There is a canonical second-order horizontal calculus on \(P\), which we shall use exclusively from now on; it can be straightforwardly constructed as follows.
\begin{proposition}
Let \(\Omega_{P,\mathrm{hor}} \coloneqq \Omega_B \rtimes H\), so that \(\Omega_{P,\mathrm{hor}} = H \otimes_{\mathbf{C}} \Omega_B\) with the grading
\[
\forall k \in \set{0,1,2}, \quad \Omega^k_{P,\mathrm{hor}} = H \otimes_{\mathbf{C}} \Omega_B^k
\]
and the multiplication, \(\ast\)-structure, and left \(H\)-coaction defined, respectively, by
\begin{gather*}
\forall h,\p{h} \in H,\, \forall \beta,\p{\beta} \in \Omega_B, \quad (h \otimes \beta) \wedge (\p{h} \otimes \p{\beta}) \coloneqq h\cm{h}{1}^\prime \otimes (\beta \triangleleft \cm{h}{2}^\prime)\p{\beta},\\
\forall h \in H, \, \forall \beta \in \Omega_B, \quad (h \otimes \beta)^\ast \coloneqq \cm{h}{1}^\ast \otimes \beta^\ast \triangleleft \cm{h}{2}^\ast,\\
\forall h \in H, \, \forall \beta \in B, \quad \delta_{\Omega_{P,\mathrm{hor}}}(h \otimes \beta) \coloneqq \cm{h}{1} \otimes \left(\cm{h}{2} \otimes \beta\right).
\end{gather*}
Then \((\Omega_B,\dt{B};\Omega_{P,\mathrm{hor}},\beta\mapsto 1\otimes\beta)\) defines a second-order horizontal calculus on \(P \coloneqq B \rtimes H\).
\end{proposition}
In what follows, we will find it notationally convenient to view \(\Omega_{P,\mathrm{hor}}\) as the graded left \(H\)-comodule \(\ast\)-algebra generated by the graded \(\ast\)-subal\-gebra \(\Omega_B\) of \(H\)-coinvariants together with the left \(H\)-subcomodule \(\ast\)-subalgebra \(H\) in degree \(0\), subject to the relation \(1_H = 1_{\Omega_B}\) and the braided supercommutation relation
\[
\forall h \in H, \, \forall \beta \in \Omega_B, \quad \beta h = \cm{h}{1} (\beta \triangleleft \cm{h}{2}).
\]
We begin by expressing the gauge group \(\fr{G}\) of \(P\) with respect to the canonical first-order horizontal calculus \((\Omega^1_B,\dt{B};\Omega^1_{P,\mathrm{hor}})\) in terms of lazy Sweedler cohomology; note that inner gauge transformations will correspond precisely to lazy Sweedler \(1\)-coboundaries.
\begin{proposition}[cf.\ Brzezi\'{n}ski~\cite{Br96}*{Thm.\ 5.4}, \'{C}a\'{c}i\'{c}--Mesland~\cite{CaMe}*{Thm.\ 3.36.(2)}]\label{trivialgaugetrans}
The function \(\Op : \ZS^1_\ell(H;B,\Omega^1_B) \to \fr{G}\) given by
\begin{equation}
\forall \sigma \in \ZS^1_\ell(H;B,\Omega^1_B), \, \forall h \in H, \, \forall b \in B, \quad \Op(\sigma)(hb) \coloneqq \cm{h}{1}\sigma(\cm{h}{2})b
\end{equation}
is a group isomorphism. Moreover,
\begin{equation}
\forall \upsilon \in \CS^0_\ell(H;B,\Omega^1_B), \quad \Op(D\upsilon) = \Ad_\upsilon,
\end{equation}
so that \(\Op(\BS^1_\ell(H;B,\Omega^1_B)) = \Inn(\fr{G})\) is the central subgroup of inner gauge transformations on \(P\) with respect to \((\Omega^1_B,\dt{B};\Omega^1_{P,\mathrm{hor}})\), and hence \(\Op\) descends to a group isomorphism
\[
\widetilde{\Op}: \HS^1_\ell(H;B,\Omega^1_B) \xrightarrow{\sim} \fr{G}/\Inn(\fr{G}) \eqqcolon \Out(\fr{G}).
\]
\end{proposition}
\begin{proof}
Let \(\Aut_B(P)\) denote the group of all automorphisms \(f : P \to P\) of \(P\) as a left \(H\)-comodule right \(B\)-module, such that \(\rest{f}{B} =\id_B\), and let \(\mathcal{A}(P)\) denote the group of all invertible elements \(\sigma \in \mathcal{C}(H,B)\), such that \(\sigma(1)=1\); observe that
\[
\Inn(\fr{G}) \leq \fr{G} \leq \Aut_B(P), \quad \BS^1_\ell(H;B,\Omega^1_B) \leq \ZS^1_\ell(H;B,\Omega^1_B) \leq \mathcal{A}(P).
\]
By~\cite{Br96}*{Thm.\ 5.4}, \emph{mutatis mutandis}, the map \(\Op : \mathcal{A}(P) \to \Aut_B(P)\) defined by
\[
\forall \sigma \in \mathcal{A}(P), \, \forall h \in H, \, \forall b \in B, \quad \Op(\sigma)(hb) \coloneqq \cm{h}{1}\sigma(\cm{h}{2})b
\]
is a well-defined group isomorphism with inverse given by
\[
\forall f \in \Aut_B(P), \, \forall h \in H, \quad \inv{\Op}(f)(h) \coloneqq S(\cm{h}{1})f(\cm{h}{2}).
\]
Note, moreover, that for all \(\sigma \in \mathcal{A}(P)\), the inverse \(\inv{\sigma}\) is given by
\[
\forall h \in H, \quad \inv{\sigma}(h) \coloneqq \sigma(S(\cm{h}{1})) \triangleleft \cm{h}{2},
\]
since, for all \(h \in H\),
\[
\epsilon(h)1_B = \sigma(S(\cm{h}{1})\cm{h}{2}) = \left(\sigma(S(\cm{h}{1}) \triangleleft \cm{h}{2}\right)\cdot\sigma(\cm{h}{3}).
\]
Let us first show that \(\Op^{-1}(\fr{G}) = \ZS^1_\ell(H;B,\Omega^1_B)\). Let \(\sigma \in \mathcal{A}(P)\) and set \(f \coloneqq \Op(\sigma)\). First, for all \(h,k \in H\),
\begin{align*}
f(hk)-f(h)f(k) &= \cm{h}{1}\cm{k}{1}\sigma(\cm{h}{2}\cm{k}{2}) - \cm{h}{1}\sigma(\cm{h}{2})\cm{k}{1}\sigma(\cm{k}{2})\\
&= \cm{h}{1}\cm{k}{1}\left(\sigma(\cm{h}{2}\cm{k}{2})\epsilon(\cm{k}{3}) - (\sigma(\cm{h}{2}) \triangleleft \cm{k}{2})\sigma(\cm{k}{3})\right),
\end{align*}
while for all \(h \in H\) and \(b \in B\), since \(bh = \cm{h}{1}(b \triangleleft \cm{h}{2})\),
\begin{align*}
f(bh) - f(b)f(h) &= \cm{h}{1}\sigma(\cm{h}{2})(b \triangleleft \cm{h}{3}) - b \cm{h}{1}\sigma(\cm{h}{2})\\ &= \cm{h}{1}\left(\sigma(\cm{h}{2})(b \triangleleft \cm{h}{3}) - (b \triangleleft \cm{h}{2})\sigma(\cm{h}{3})\right),
\end{align*}
so that \(f \in \Aut_B(P)\) is an algebra automorphism if and only if \(\sigma \in \Cent_{\mathcal{C}(H,B)}(\rho_B(B))\) and
\[
\forall h,k \in H, \quad \sigma(hk) = (\sigma(h) \triangleleft \cm{k}{1}) \sigma(\cm{k}{2}).
\]
In particular, if \(f\) is an algebra automorphism, then \(\inv{\sigma}\) is given by
\[
\forall h \in H, \quad \inv{\sigma}(h) \coloneqq \sigma(S(\cm{h}{1})) \triangleleft \cm{h}{2},
\]
since, for all \(h \in H\),
\[
\epsilon(h)1_B = \sigma(S(\cm{h}{1})\cm{h}{2}) = \left(\sigma(S(\cm{h}{1}) \triangleleft \cm{h}{2}\right)\cdot\sigma(\cm{h}{3}).
\]
Now, suppose that \(f\) is an algebra automorphism. Then, for all \(h \in H\),
\begin{align*}
f(S(h)^\ast)^\ast - f(S(h))
&= \left(S(\cm{h}{2})^\ast \sigma(S(\cm{h}{1})^\ast)\right)^\ast - S(\cm{h}{2}) \sigma(S(\cm{h}{1}))\\
&= \sigma(S(\cm{h}{1})^\ast)^\ast S(\cm{h}{2})^\ast - \left(\sigma(S(\cm{h}{1}))^\ast S(\cm{h}{2})^\ast\right)^\ast\\
&= \sigma(S(\cm{h}{1})^\ast)^\ast S(\cm{h}{2})^\ast - \left(S(\cm{h}{3})^\ast (\sigma(S(\cm{h}{1}))^\ast \triangleleft S(\cm{h}{2})^\ast)\right)^\ast\\
&= \left(\sigma(S(\cm{h}{1})^\ast)^\ast \epsilon(\cm{h}{2}) - \sigma(S(\cm{h}{1})) \triangleleft \cm{h}{2}\right)S(\cm{h}{2})^\ast\\
&= \left(\sigma^\ast(\cm{h}{1}) - \inv{\sigma}(\cm{h}{1})\right) S(\cm{h}{2})^\ast,
\end{align*}
so that \(f\) is a \(\ast\)-automorphism if and only if \(\sigma\) is unitary. Finally, suppose that \(f\) is a \(\ast\)-automorphism. Then, for all \(h \in H\) and \(\beta \in B\), since \(\beta h = \cm{h}{1} \cdot (\beta \triangleleft \cm{h}{2})\),
\begin{align*}
f(\cm{h}{1}) \cdot (\beta \triangleleft \cm{h}{2}) - \beta \cdot f(h) &= \cm{h}{1}\sigma(\cm{h}{2}) \cdot (\beta \triangleleft \cm{h}{2}) - \beta \cdot \cm{h}{1}\sigma(\cm{h}{2})\\
&= \cm{h}{1} \cdot \left(\sigma(\cm{h}{2}) \cdot (\beta \triangleleft \cm{h}{2}) - (\beta \triangleleft \cm{h}{2}) \cdot \sigma(\cm{h}{3})\right),
\end{align*}
so that \(f \in \fr{G}\) if and only if \(\sigma \in \Cent_{\mathcal{C}(H,B)}(\rho_{\Omega^1_B}(\Omega^1_B))\).
Let us now show that \(\Op \circ D = (\upsilon \mapsto \Ad_\upsilon)\) on \(\CS^0_\ell(H;B,\Omega^1_B) = \Unit(\Cent_B(B\oplus\Omega^1_B))\), which will imply the rest of the claim. Let \(\upsilon \in \Unit(\Cent_B(B\oplus\Omega^1_B))\). Then, for all \(h \in H\) and \(b \in B\),
\[
\Op(D\upsilon)(hb) = \cm{h}{1}(\upsilon \triangleleft \cm{h}{2})\upsilon^\ast b = \upsilon hb \upsilon^\ast = \Ad_\upsilon(hb). \qedhere
\]
\end{proof}
Next, we express the Atiyah space \(\fr{At}\) of \(P\) with respect to the canonical first-order horizontal calculus \((\Omega^1_B,\dt{B};\Omega^1_{P,\mathrm{hor}})\) and its space of translations \(\fr{at}\) in terms of lazy Hochschild cohomology; note that inner relative gauge potentials will correspond precisely to lazy Hoch\-schild \(1\)-coboundaries.
\begin{proposition}[cf.\ \'{C}a\'{c}i\'{c}--Mesland~\cite{CaMe}*{Thm.\ 3.36.(1)}]\label{trivialgaugepot}
We have an isomorphism of \(\mathbf{R}\)-affine spaces \(\Op : \ZH^1_\ell(H;\Omega^1_B) \to \fr{At}\) given by
\begin{equation}
\forall \mu \in \ZH^1_\ell(H;\Omega^1_B)\, \forall h \in H, \, \forall b \in B, \quad \Op(\mu)(hb) \coloneqq h \cdot \dt{B}(b) + \cm{h}{1} \cdot \mu(\cm{h}{2}) \cdot b,
\end{equation}
whose linear part \(\Op_0 : \ZH^1_\ell(H;\Omega^1_B) \xrightarrow{\sim} \fr{at}\) is given by
\begin{equation}
\forall \mu \in \ZH^1_\ell(H;\Omega^1_B),\, \forall h \in H, \, \forall b \in B, \quad \Op_0(\mu)(hb) \coloneqq \cm{h}{1} \cdot \mu(\cm{h}{2}) \cdot b.
\end{equation}
Moreover,
\begin{equation}
\forall \alpha \in \CH^0_\ell(H;\Omega^1_B), \quad \Op_0(D\alpha) = \ad_\alpha,
\end{equation}
so that \(\Op_0(\BH^1_\ell(H;\Omega^1_B) = \Inn(\fr{at})\) is the subspace of inner relative gauge potentials on \(P\) with respect to \((\Omega^1_B,\dt{B};\Omega^1_{P,\mathrm{hor}})\), and hence \(\Op\) descends to an isomorphism of \(\mathbf{R}\)-affine spaces
\[
\widetilde{\Op} : \HH^1_\ell(H;\Omega^1_B) \xrightarrow{\sim} \fr{At}/\Inn(\fr{at}) \eqqcolon \Out(\fr{At})
\]
with linear part \(\widetilde{\Op_0} : \HH^1_\ell(H;\Omega^1_B) \xrightarrow{\sim} \fr{at}/\Inn(\fr{at}) \eqqcolon \Out(\fr{at})\) descending from \(\Op_0\).
\end{proposition}
\begin{proof}
Let \(\Hom_B(P,\Omega^1_{P,\mathrm{hor}})_0\) denote the \(\mathbf{R}\)-vector space of all morphisms \(f : P \to \Omega^1_{P,\mathrm{hor}}\) of left \(H\)-comodule right \(B\)-modules, such that \(\rest{f}{B} = 0\), and let \(\mathcal{C}(H,\Omega^1_B)_0\) denote the \(\mathbf{R}\)-vector space of all \(\mu \in \mathcal{C}(H,\Omega^1_B)\), such that \(\mu(1) = 0\); observe that
\[
\Inn(\fr{at}) \leq \fr{At} \leq \Hom_B(P,\Omega^1_{P,\mathrm{hor}})_0, \quad \BH^1_\ell(H;\Omega^1_B) \leq \ZH^1_\ell(H;\Omega^1_B) \leq \mathcal{C}(H,\Omega^1_B)_0.
\]
By~\cite{Br96}*{Thm.\ 5.4}, \emph{mutatis mutandis}, the map \(\Op_0 : \mathcal{C}(H,\Omega^1_B)_0 \to \Hom_B(P,\Omega^1_{P,\mathrm{hor}})_0\) defined by
\[
\forall \mu \in \mathcal{C}(H,\Omega^1_B)_0, \, \forall h \in H, \,\forall b \in B, \quad \Op_0(\mu)(hb) \coloneqq \cm{h}{1}\mu(\cm{h}{2}) \cdot b
\]
is a well-defined \(\mathbf{R}\)-linear isomorphism with inverse given by
\[
\forall \mathbf{A} \in \Hom_B(P,\Omega^1_{P,\mathrm{hor}})_0,\, \forall h \in H, \quad \Op_0^{-1}(\mathbf{A})(h) \coloneqq S(\cm{h}{1})\mathbf{A}(\cm{h}{2}).
\]
Let us first show that \(\Op_0^{-1}(\fr{at}) = \ZH^1_\ell(H;\Omega^1_B)\) and that \(\Op_0 \circ D = (\alpha \mapsto \ad_\alpha)\) on the domain \(\CH^0_\ell(H;\Omega^1_B) = \Zent_B(\Omega^1_B)_\mathrm{sa}\). Let \(\mu \in \mathcal{C}(H,\Omega^1_B)_0\) and set \(\mathbf{A} \coloneqq \Op(\mu)\). By the proof of Proposition~\ref{trivialgaugetrans}, \emph{mutatis mutandis}, it follows that \(\mathbf{A} \in \Hom_B(P,\Omega^1_{P,\mathrm{hor}})_0\) is a \(P\)-bimodule derivation if and only if \(\mu \in \Zent_B(\mathcal{C}(H,\Omega^1_B))\) and
\[
\forall h, k \in H, \quad \mu(hk) = \mu(h) \triangleleft k + \epsilon(h)\mu(k),
\]
in which case, for all \(h \in H\),
\[
0 = \mu(S(\cm{h}{1})\cm{h}{2}) = \mu(S(\cm{h}{1})) \triangleleft \cm{h}{2} + \epsilon(S(\cm{h}{1}))\mu(\cm{h}{2}) = \mu(S(\cm{h}{1})) \triangleleft \cm{h}{2} + \mu(h).
\]
Thus, if \(\mathbf{A}\) is a \(P\)-bimodule derivation, then, by the proof of Proposition~\ref{trivialgaugetrans},
\[
\forall h \in H, \quad \mathbf{A}(S(h)^\ast)^\ast + \mathbf{A}(S(h)) = \left(\mu^\ast(\cm{h}{1})-\mu(\cm{h}{1})\right)S(\cm{h}{2})^\ast,
\]
so that the \(\mathbf{A}\) is a \(\ast\)-derivation if and only if \(\mu = \mu^\ast\). The rest now follows from the proof of Proposition~\ref{trivialgaugetrans}, \emph{mutatis mutandis}.
Let us now show that \(\Op : \ZH^1_\ell(H;\Omega^1_B) \to \fr{At}\) is a well-defined isomorphism of \(\mathbf{R}\)-affine spaces with linear part \(\Op_0\); it suffices to show that the map \(\nabla_0 : P \to \Omega^1_{P,\mathrm{hor}}\) defined by
\[
\forall h \in H, \, \forall b \in B, \quad \nabla_0(hb) \coloneqq h \cdot \dt{B}(b)
\]
is a gauge potential, so that \(\Op(0) = \nabla_0\) is well-defined. First, for all \(h,k \in H\) and \(b,c \in B\), since \(hbkc = h\cm{k}{1}(b\triangleleft\cm{h}{2})c\),
\begin{align*}
\nabla_0(hbkc) - \nabla_0(hb)kc - hb\nabla_0(kc) &= h\cm{k}{1}\dt{B}((b \triangleleft \cm{k}{2})c) - h\dt{B}(b)kc - hbk\dt{B}(c)\\
&= h\cm{k}{1}\cdot\left(\dt{B}((b \triangleleft \cm{k}{2})c) - \dt{B}(b \triangleleft \cm{k}{1}) \cdot (b \triangleleft \cm{k}{1})\cdot\dt{B}(c)\right)\\
&=0.
\end{align*}
Next, for all \(h \in H\) and \(b \in B\), since \((hb)^\ast = \cm{h}{1}^\ast (b^\ast \triangleleft \cm{h}{2}^\ast)\) and \((h\dt{B}(b))^\ast = \cm{h}{1}^\ast \cdot \dt{B}(b)^\ast \triangleleft \cm{h}{2}^\ast\),
\[
\nabla_0((hb)^\ast) + \nabla_0(hb)^\ast = \cm{h}{1}^\ast \dt{B}(b^\ast \triangleleft \cm{h}{2}^\ast) +\left(h\dt{B}(b)\right)^\ast
= \cm{h}{1}^\ast \cdot \left(\dt{B}(b^\ast) +\dt{B}(b)^\ast\right) \triangleleft \cm{h}{2}^\ast = 0.
\]
Finally, by construction, \(\rest{\nabla_0}{B} = \dt{B}\).
\end{proof}
We now relate the affine action of the gauge group \(\fr{G}\) on the Atiyah space \(\fr{At}\) to the affine action of lazy Sweedler cohomology with coefficients in \((B,\Omega^1_B)\) on lazy Hochschild cohomology with coefficients in \(\Omega^1_B\) induced by the Maurer--Cartan \(1\)-cocycle \(\MC[\dt{B}]\) of the derivation \(\dt{B} : B \to \Omega^1_B\).
\begin{proposition}[cf.\ Brzezi\'{n}ski--Majid~\cite{BrM}*{Prop.\ 4.8}, \'{C}a\'{c}i\'{c}--Mesland~\cite{CaMe}*{Thm.\ 3.36.(3)}]\label{trivialgaugepotprop}
For every \(\sigma \in \ZS^1_\ell(H;B,\Omega^1_B)\) and every \(\mu \in \ZH^1_\ell(H;\Omega^1_B)\), we have
\begin{gather}
\Op(\sigma) \triangleright \Op(\mu) = \Op\mleft(\sigma \triangleright \mu + \MC[\dt{B}](\sigma)\mright),\\
\Op(\sigma) \triangleright \Op_0(\mu) = \Op_0(\sigma \triangleright \mu).
\end{gather}
and hence, at the level of cohomology,
\begin{gather}
\widetilde{\Op}([\sigma]) \triangleright \widetilde{\Op}([\mu]) = \widetilde{\Op}\mleft([\sigma] \triangleright [\mu] + \widetilde{\MC}[\dt{B}]([\sigma])\mright),\\
\widetilde{\Op}([\sigma]) \triangleright \widetilde{\Op_0}([\mu]) = \widetilde{\Op_0}([\sigma] \triangleright [\mu]).
\end{gather}
Thus, the maps \(\Op \times \Op\) and \(\widetilde{\Op} \times \widetilde{\Op}\) define groupoid isomorphisms
\begin{gather*}
\ZS^1_\ell(H;B,\Omega^1_B) \ltimes \ZH^1_\ell(H;\Omega^1_B) \to \fr{G} \ltimes \fr{At}, \\ \HS^1_\ell(H;B,\Omega^1_B) \ltimes \HH^1_\ell(H;\Omega^1_B) \to \Out(\fr{G}) \ltimes \Out(\fr{At}),
\end{gather*}
respectively, where \(\ZS^1_\ell(H;B,\Omega^1_B)\) acts affine-linearly on \(\ZH^1_\ell(H;\Omega^1_B)\) with \(1\)-cocycle \(\MC[\dt{B}]\) and \(\HS^1_\ell(H;B,\Omega^1_B)\) acts affine-linearly on \(\HH^1_\ell(H;\Omega^1_B)\) with \(1\)-cocycle \(\widetilde{\MC}[\dt{B}]\).
\end{proposition}
\begin{proof}
Let \(\sigma \in \ZS^1_\ell(H;B,\Omega^1_B)\) and \(\mu \in \ZH^1_\ell(H;\Omega^1_B)\); note that
\[
\Op(\sigma) \triangleright \Op(\mu) = \Op(\sigma) \triangleright \left(\Op(0) + \Op_0(\mu)\right) = \Op(\sigma) \triangleright \Op(0) + \Op(\sigma) \triangleright \Op_0(\mu).
\]
On the one hand, for all \(h \in H\) and \(b \in B\), since \(\sigma \star \sigma^\ast = \epsilon(\cdot)1_B\),
\begin{align*}
\Op(\sigma) \triangleright \Op(0)(hb) &= \Op(\sigma)_\ast \circ \Op(0) \circ \Op(\sigma^\ast)(hb)\\
&= \cm{h}{1}\sigma(\cm{h}{2}) \cdot \left(\dt{B}\sigma^\ast(\cm{h}{3}) \cdot b + \sigma^\ast(\cm{h}{3})\dt{B}(b)\right)\\
&= -\cm{h}{1} \cdot \dt{B}\sigma(\cm{h}{2}) \cdot \sigma^\ast(\cm{h}{3})b + \cm{h}{1}\sigma(\cm{h}{2})\sigma^\ast(\cm{h}{3}) \cdot \dt{B}(b)\\
&= \cm{h}{1} \cdot \MC[\dt{B}](\sigma)(\cm{h}{2}) \cdot b + h \cdot \dt{B}(b)\\
&= \Op(\MC[\dt{B}](\sigma))(hb),
\end{align*}
so that \(\Op(\sigma) \triangleright \Op(0) = \Op(\MC[\dt{B}](\sigma))\). On the other hand, for all \(h \in H\) and \(b \in B\),
\begin{align*}
\Op(\sigma) \triangleright \Op_0(\mu)(hb) &= \Op(\sigma)_\ast \circ \Op_0(\mu) \circ \Op(\sigma^\ast)(hb)\\
&= \cm{h}{1}\sigma(\cm{h}{2}) \cdot \mu(\cm{h}{3}) \cdot \sigma^\ast(\cm{h}{4})b\\
&= \cm{h}{1} \cdot (\sigma \triangleright \mu)(\cm{h}{2}) \cdot b\\
&= \Op_0(\sigma \triangleright \mu)(hb),
\end{align*}
so that \(\Op(\sigma) \triangleright \Op_0(\mu) = \Op_0(\sigma \triangleright \mu)\).
\end{proof}
All of the above results yield straightforward characterisations of prolongable gauge transformations and prolongable (relative) gauge potentials in terms of lazy Sweedler cohomology nd lazy Hochschild cohomology, respectively; in particular, we shall find that every gauge transformation is automatically prolongable.
\begin{corollary}\label{trivialprcor}
Let \(\pr{\fr{G}}\) be the prolongable gauge group of \(P\) with respect to the canonical second-order horizontal calculus \((\Omega_B,\dt{B};\Omega_{P,\mathrm{hor}})\). Let \(\pr{\fr{At}}\) be the prolongable Atiyah space of \(P\) with respect to \((\Omega_B,\dt{B};\Omega_{P,\mathrm{hor}})\), let \(\pr{\fr{at}}\) be its translation space, and let \(\Inn(\pr{\fr{at}}) \subset \pr{\fr{at}}\) be the subspace of all inner prolongable gauge potentials. Then \(\pr{\fr{G}} = \fr{G}\) and
\begin{gather*}
\inv{\Op}\mleft(\pr{\fr{At}}\mright) = \Op_0^{-1}\mleft(\pr{\fr{at}}\mright) = \ZH^1_\ell(H;\Omega^1_B,\Omega_B),\\
\Op_0^{-1}\mleft(\Inn(\pr{\fr{at}})\mright) = \BH^1_\ell(H;\Omega^1_B,\Omega_B),
\end{gather*}
hence \(\Op\) induces an isomorphism \(\pr{\widetilde{\Op}} : \HH^1_\ell(H;\Omega^1_B,\Omega_B) \xrightarrow{\sim} \pr{\fr{At}}/\Inn(\pr{\fr{at}}) \eqqcolon \Out(\pr{\fr{At}})\) of \(\mathbf{R}\)-affine spaces with linear part \(\pr{\widetilde{\Op_0}} : \HH^1_\ell(H;\Omega^1_B,\Omega_B) \xrightarrow{\sim} \pr{\fr{at}}/\Inn(\pr{\fr{at}}) \eqqcolon \Out(\pr{\fr{at}})\) induced by \(\Op_0\). Moreover, for all \(\sigma \in \ZS^1_\ell(H;B,\Omega^1_B,\Omega_B)\) and \(\mu \in \ZH^1_\ell(H;\Omega^1_B,\Omega_B)\),
\begin{gather}
\widetilde{\Op}([\sigma]) \triangleright \pr{\widetilde{\Op}}([\mu]) = \pr{\widetilde{\Op}}\mleft([\sigma] \triangleright [\mu] + \widetilde{\MC}[\dt{B},\Omega_B]([\sigma])\mright),\\
\widetilde{\Op}([\sigma]) \triangleright \pr{\widetilde{\Op_0}}([\mu]) = \pr{\widetilde{\Op_0}}([\sigma] \triangleright [\mu]).
\end{gather}
Thus, the maps \(\Op \times \rest{\Op}{\ZH^1_\ell(H;\Omega^1_B,\Omega_B)}\) and \(\widetilde{\Op} \times \pr{\widetilde{\Op}}\) define groupoid isomorphisms
\begin{gather*}
\ZS^1_\ell(H;B,\Omega^1_B) \ltimes \ZH^1_\ell(H;\Omega^1_B,\Omega_B) \to \pr{\fr{G}} \ltimes \pr{\fr{At}}, \\ \HS^1_\ell(H;B,\Omega^1_B) \ltimes \HH^1_\ell(H;\Omega^1_B,\Omega_B) \to \Out(\pr{\fr{G}}) \ltimes \Out(\pr{\fr{At}}),
\end{gather*}
respectively, where \(\ZS^1_\ell(H;B,\Omega^1_B)\) acts affinely on \(\ZH^1_\ell(H;\Omega^1_B,\Omega_B)\) with \(1\)-cocycle \(\MC[\dt{B}]\) and \(\HS^1_\ell(H;B,\Omega^1_B)\) acts affinely on \(\HH^1_\ell(H;\Omega^1_B,\Omega_B)\) with \(1\)-cocycle \(\widetilde{\MC}[\dt{B}]\).
\end{corollary}
\begin{proof}
Before continuing, note that for all \(h,k \in H\) and \(b,\p{b},c \in B\),
\[
hb \cdot \dt{B}(\p{b}) \cdot kc = h\cm{k}{1}(b \triangleleft \cm{k}{2})\cdot \dt{B}(\p{b} \triangleleft \cm{k}{3}) \cdot c.
\]
Let us first show that \(\fr{G} = \pr{\fr{G}}\). Let \(\sigma \in \ZS^1_\ell(H;B,\Omega^1_B)\) be given. Then, for all \(h,k \in H\), \(b,c \in B\), and \(\alpha,\beta \in \Omega^1_B\),
\begin{align*}
&\Op(\sigma)(hb) \cdot \alpha \wedge \beta \cdot \Op(\sigma)(kc) \\
&\quad= \cm{h}{1} \sigma(\cm{h}{2}) b \cdot\alpha \wedge \beta \cdot \cm{k}{1}\sigma(\cm{k}{2})c\\
&\quad= \cm{h}{1}\cm{k}{1} (\sigma(\cm{h}{2}) \triangleleft \cm{k}{2})(b \triangleleft \cm{k}{3}) \cdot (\alpha \triangleleft \cm{k}{4}) \wedge (\beta \triangleleft \cm{k}{5}) \cdot \sigma(\cm{k}{6})c\\
&\quad= \cm{h}{1}\cm{k}{1} (\sigma(\cm{h}{2}) \triangleleft \cm{k}{2})\sigma(\cm{k}{3})(b \triangleleft \cm{k}{4}) \cdot (\alpha \triangleleft \cm{k}{5}) \wedge (\beta \triangleleft \cm{k}{6}) \cdot c\\
&\quad= \cm{h}{1}\cm{k}{1} \sigma(\cm{h}{2}\cm{k}{2})(b \triangleleft \cm{k}{3}) \cdot (\alpha \triangleleft \cm{k}{4}) \wedge (\beta \triangleleft \cm{k}{5}) \cdot c\\
&\quad= \Op(\sigma)(h\cm{k}{1}) \cdot \left((b \triangleleft \cm{k}{2})\cdot (\alpha \triangleleft \cm{k}{3}) \wedge (\beta \triangleleft \cm{k}{4}) \cdot c\right),
\end{align*}
where \(hb \cdot \alpha \wedge \beta \cdot kc = h\cm{k}{1} \cdot \left((b \triangleleft \cm{k}{2})\cdot (\alpha \triangleleft \cm{k}{3}) \wedge (\beta \triangleleft \cm{k}{4}) \cdot c\right)\), so that \(\Op(\sigma) \in \pr{\fr{G}}\) with
\[
\forall h \in H, \, \forall \eta \in \Omega^2_B, \quad \Op(\sigma)_\ast(h\eta) = \cm{h}{1}\sigma(\cm{h}{2}) \cdot \eta.
\]
Next, observe that \(\Op(0) \in \pr{\fr{At}}\). Indeed, for all \(h,k \in H\) and \(b,\p{b},c \in B\),
\begin{align*}
&\Op(0)(hb) \wedge \dt{B}(\p{b}) \cdot kc - hb \cdot \dt{B}(\p{b}) \wedge \Op(0)(kc)\\
&= h \cdot \dt{B}(b) \wedge \dt{B}(\p{b}) \cdot kc - hb \cdot \dt{B}(\p{b}) \wedge k \cdot \dt{B}(c)\\
&= h \cm{k}{1} \cdot \dt{B}\mleft((b \triangleleft \cm{k}{2}) \cdot \dt{B}(\p{b} \triangleleft \cm{k}{3}) \cdot c \mright),
\end{align*}
where \(hb \cdot \dt{B}(\p{b}) \cdot kc = h\cm{k}{1}(b \triangleleft \cm{k}{2})\cdot \dt{B}(\p{b} \triangleleft \cm{k}{3}) \cdot c\), so that \(\Op(0) \in \pr{\fr{At}}\) with
\[
\forall h \in H, \, \forall \beta \in \Omega^1_B, \quad \pr{\Op(0)}(h \cdot \beta) = h \cdot \dt{B}(\beta).
\]
Next, let us show that \(\Op_0^{-1}(\pr{\fr{at}}) = \ZH^1_\ell(H;\Omega^1_B,\Omega_B)\); since \(\Op(0) \in \pr{\fr{at}}\), this will also imply that \(\Op^{-1}(\pr{\fr{At}}) = \ZH^1_\ell(H;\Omega^1_B,\Omega_B)\). Let \(\mu \in \ZH^1_\ell(H;\Omega^1_B)\) be given. Then, for all \(h \in H\) and \(\beta \in \Omega^1_B\), so that \(\beta \cdot h = \cm{h}{1} \cdot \beta \triangleleft \cm{h}{2}\),
\begin{align*}
\Op_0(\mu)(\cm{h}{1}) \wedge \beta \triangleleft \cm{h}{2} + \beta \wedge \Op_0(\mu)(h) &= \cm{h}{1}\cdot \mu(\cm{h}{2}) \wedge \beta \triangleleft \cm{h}{3} + \beta \wedge \cm{h}{1} \cdot \mu(\cm{h}{2})\\
&= \cm{h}{1} \cdot \left(\mu(\cm{h}{2}) \wedge \beta \triangleleft \cm{h}{3} + \beta \triangleleft \cm{h}{2} \wedge \mu(\cm{h}{3})\right)\\
&= \cm{h}{1} \cdot [\mu,\rho_{\Omega^1_B}(\beta)](\cm{h}{2}).
\end{align*}
Thus, \(\Op_0(\mu) \in \pr{\fr{at}}\) if and only if \(\mu \in \ZH^1_\ell(H;\Omega^1_B,\Omega_B)\), in which case
\[
\forall h \in H, \, \forall \beta \in \Omega^1_B, \quad \pr{\Op_0(\mu)}(h \cdot \beta) \coloneqq \cm{h}{1} \cdot \mu(\cm{h}{2}) \wedge \beta.
\]
Finally, let us show that \(\Op_0^{-1}(\Inn(\pr{\fr{at}})) = \BH^1_\ell(H;\Omega^1_B,\Omega_B)\). Let \(\alpha \in \Zent_B(\Omega^1_B)_\mathrm{sa}\). Then, for all \(h \in H\) and \(\beta \in \Omega^1_B\),
\begin{align*}
\pr{\Op_0(D\alpha)}(h \cdot \beta) - [\alpha,h \cdot \beta] &= \cm{h}{1} \cdot (\alpha \triangleleft \cm{h}{2} - \epsilon(\cm{h}{2})\alpha) \wedge \beta - \alpha \wedge h \cdot \beta - h \cdot \beta \wedge \alpha\\
&= - h \cdot [\alpha,\beta]_{\mathcal{C}(H,\Omega_B)},
\end{align*}
so that \(\Op_0(D\alpha) \in \Inn(\pr{\fr{at}})\) if and only if \(\alpha \in (\Omega^1_B)_{\mathrm{sa}} \cap \Zent(\Omega_B) = \CH^0_\ell(H;\Omega^1_B,\Omega_B)\).
\end{proof}
We conclude by observing that field strength as a map on the prolongable Atiyah space \(\pr{\fr{At}}\) admits a straightforward reinterpretation as an \(\mathbf{R}\)-affine quadratic map between spaces of lazy Hochschild \(1\)-cocycles that, in turn, descends to a map between lazy Hochschild cohomology groups. Given a choice of bicovariant \textsc{fodc}{} \((\Omega^1_H,\dt{H})\) on \(H\), this reinterpretation will prove crucial to characterising \((\Omega^1_H,\dt{H})\)-adapted prolongable gauge potentials in terms of lazy Hochschild cohomology.
\begin{proposition}\label{curveprop}
The map \(\mathcal{F} : \ZH^1_\ell(H;\Omega^1_B,\Omega_B) \to \ZH^1_\ell(H;\Omega^2_B)\) defined by
\begin{equation*}
\forall h \in H, \, \forall \mu \in \ZH^1_\ell(H;\Omega^1_B,\Omega_B), \quad \mathcal{F}[\mu](h) \coloneqq S(\cm{h}{1}) \cdot \mathbf{F}[\Op(\mu)](h)
\end{equation*}
is an \(\mathbf{R}\)-affine quadratic map, satisfying
\begin{gather*}
\forall \mu \in \ZH^1_\ell(H;\Omega^1_B,\Omega_B), \quad \mathcal{F}[\mu] = -\ensuremath{\mathrm{i}}{}\left(\dt{B}\mu(\cdot) + \tfrac{1}{2}[\mu,\mu]_{\mathcal{C}(H,\Omega_B)}\right),\\
\forall \mu,\nu \in \ZH^1_\ell(H;\Omega^1_B,\Omega_B), \quad \mathcal{F}[\mu+\nu] - \mathcal{F}[\mu] -\mathcal{F}[\nu] = -\ensuremath{\mathrm{i}}{}[\mu,\nu]_{\mathcal{C}(H,\Omega_B)}, \\
\forall \alpha \in (\Omega^1_B)_\mathrm{sa} \cap \Zent(\Omega_B), \quad \mathcal{F}[D\alpha] = D(-\ensuremath{\mathrm{i}}{}\dt{B}\alpha), \label{coboundcurve}\\
\forall \sigma \in \ZS^1_\ell(H;B,\Omega^1_B), \, \forall \mu \in \ZH^1_\ell(H;\Omega^1_B,\Omega_B), \quad \mathcal{F}[\sigma \triangleright \mu + \MC[\dt{B}](\sigma)] = \sigma \triangleright \mathcal{F}[\mu].
\end{gather*}
Hence, \(\mathcal{F}\) descends to an \(\mathbf{R}\)-affine quadratic map \(\widetilde{\mathcal{F}} : \HH^1_\ell(H;\Omega^1_B,\Omega_B) \to \HH^1_\ell(H;
\Omega^2_B)\), satisfying
\begin{multline*}
\forall \sigma \in \ZS^1_\ell(H;B,\Omega^1_B), \, \forall \mu \in \ZH^1_\ell(H;\Omega^1_B,\Omega_B), \\
\widetilde{\mathcal{F}}\mleft[[\sigma] \triangleright [\mu] + \widetilde{\MC}[\dt{B},\Omega_B]([\sigma])\mright] = [\sigma] \triangleright \widetilde{\mathcal{F}}\mleft[[\mu]\mright].
\end{multline*}
\end{proposition}
\begin{proof}
First, note that \(\mathcal{F} : \ZH^1_\ell(H;\Omega^1_B,\Omega_B) \to \ZH^1_\ell(H;\Omega^2_B)\) is well-defined by the proof of Proposition~\ref{trivialgaugepot}, \emph{mutatis mutandis}, hence \(\mathbf{R}\)-affine quadratic by Proposition~\ref{fieldstrengthprop}.
Next, let \(\mu \in \ZH^1_\ell(H;\Omega^1_B,\Omega_B)\). Then, for all \(h \in H\),
\begin{align*}
\ensuremath{\mathrm{i}}{}\mathcal{F}[\mu](h) &= S(\cm{h}{1}) \cdot \pr{\Op(\mu)} \circ \Op(\mu)(\cm{h}{2})\\
&= S(\cm{h}{1}) \cdot \pr{\Op(\mu)}(\cm{h}{2}\cdot\mu(\cm{h}{3})\\
&= S(\cm{h}{1}) \cdot \left(\cm{h}{2} \cdot \mu(\cm{h}{3}) \wedge \mu(\cm{h}{4})+\cm{h}{2} \cdot \dt{B}\mu(\cm{h}{3})\right)\\
&= \dt{B}\mu(h) +\mu(\cm{h}{1})\wedge\mu(\cm{h}{2}),
\end{align*}
so that \(\mathcal{F}[\mu] = -\ensuremath{\mathrm{i}}{}\left(\dt{B}\mu(\cdot) + \tfrac{1}{2}[\mu,\mu]_{\mathcal{C}(H,\Omega_B)}\right)\). Thus, for all \(\mu,\nu \in \ZH^1_\ell(H;\Omega^1_B,\Omega_B)\),
\[
\mathcal{F}[\mu+\nu] = \dt{B}(\mu+\nu)(\cdot)+\tfrac{1}{2}[\mu+\nu,\mu+\nu]_{\mathcal{C}(H,\Omega_B)} = \mathcal{F}[\mu] + \mathcal{F}[\nu] +[\mu,\nu]_{\mathcal{C}(H,\Omega_B)}.
\]
Now, let \(\alpha \in (\Omega^1_B)_\mathrm{sa} \cap \Zent(\Omega_B)\); note that \(-\ensuremath{\mathrm{i}}{}\dt{B}\alpha \in \Zent_B(\Omega^2_B)_\mathrm{sa}\), since for all \(b \in B\),
\[
b \cdot \dt{B}\alpha - \dt{B}\alpha \cdot b = \dt{B}(b \cdot \alpha)-\dt{B}b\wedge\alpha-\dt{B}(\alpha \cdot b)-\alpha \wedge \dt{B}b = 0.
\]
Then, for all \(h \in H\),
\begin{align*}
\ensuremath{\mathrm{i}}{}\mathcal{F}[D\alpha](h) &= \dt{B}(\alpha \triangleleft h+\epsilon(h)\alpha) + (\alpha \triangleleft \cm{h}{1}+\epsilon(\cm{h}{1})\alpha) \wedge (\alpha \triangleleft \cm{h}{2}+\epsilon(\cm{h}{2})\alpha) \\
&= \left(\dt{B}\alpha + \tfrac{1}{2}[\alpha,\alpha]_{\Omega_B}\right) \triangleleft h + \epsilon(h)\left(\dt{B}\alpha + \tfrac{1}{2}[\alpha,\alpha]_{\Omega_B}\right) + [\alpha,\alpha \triangleleft h]_{\Omega_B}\\
&= \dt{B}(\alpha) \triangleleft h + \epsilon(h)\dt{B}(\alpha),
\end{align*}
so that \(\mathcal{F}[D\alpha] = D(-\ensuremath{\mathrm{i}}{}\dt{B}\alpha) \in \BH^1_\ell(H;\Omega^2_B)\).
Finally, let \(\sigma \in \ZS^1_\ell(H;B,\Omega^1_B)\) and \(\mu \in \ZH^1_\ell(H;\Omega^1_B,\Omega_B)\). Then, for all \(h \in H\),
\begin{align*}
\mathcal{F}[\sigma \triangleright \mu + \MC[\dt{B}](\sigma)](h) &= S(\cm{h}{1}) \cdot \mathbf{F}[\Op(\sigma) \triangleright \Op(\mu)](\cm{h}{2})\\
&= S(\cm{h}{1}) \cdot \Op(\sigma)_\ast \circ \mathbf{F}[\Op(\mu)] \circ \Op(\sigma^\ast)(\cm{h}{2})\\
&= S(\cm{h}{1}) \cdot \Op(\sigma)_\ast \circ \mathbf{F}[\Op(\mu)]\mleft(\cm{h}{2}\sigma^\ast(\cm{h}{3})\mright)\\
&= S(\cm{h}{1}) \cdot \Op(\sigma)_\ast\mleft(\cm{h}{2} \cdot \mathcal{F}[\mu](\cm{h}{3}) \cdot \sigma^\ast(\cm{h}{4})\mright)\\
&= S(\cm{h}{1}) \cm{h}{2}\sigma(\cm{h}{3}) \cdot \mathcal{F}[\mu](\cm{h}{4}) \cdot \sigma^\ast(\cm{h}{5})\\
&= \sigma(\cm{h}{1}) \cdot \mathcal{F}[\mu](\cm{h}{2}) \cdot \sigma^\ast(\cm{h}{3})\\
&= \sigma \triangleright \mathcal{F}[\mu](h),
\end{align*}
so that, indeed, \(\mathcal{F}[\sigma \triangleright \mu + \MC[\dt{B}](\sigma)](h) = \sigma \triangleright \mathcal{F}[\mu]\).
\end{proof}
\subsection{Reconstruction of total calculi}
As we have just seen, all purely horizontal aspects of noncommutative gauge theory on the trivial quantum principal \(H\)-bundle \(B \rtimes H\) can be computed solely in terms of lazy Sweedler cohomology and lazy Hochschild cohomology on \(H\) with coefficients arising from the basic calculus \((\Omega_B,\dt{B})\). We now turn to those aspects that depend on a choice of bicovariant \textsc{fodc}{} \((\Omega^1_H,\dt{H})\) on \(H\), e.g., the \(\fr{G}\)-equivariant moduli space \(\fr{At}/\fr{at}[\Omega^1_H] \cong \Ob\mleft(\mathcal{G}[\Omega^1_H]/\ker\mu[\Omega^1_H]\mright)\) of total \textsc{fodc}{} on \(P\) and the \(\pr{\fr{G}}\)-invariant quadratic subset \(\pr{\fr{At}}[\Omega^1_H]\) of \(\Omega^1_H\)-adapted prolongable gauge potentials.
Let us fix a bicovariant \textsc{fodc} \((\Omega^1_H,\dt{H})\) on \(H\) with left crossed \(H\)-\(\ast\)-module \(\Lambda^1_H\) of right \(H\)-covariant \(1\)-forms and quantum Maurer--Cartan form \(\varpi : H \to \Lambda^1_H\). Let \((\Omega_H,\dt{H})\) denote its canonical prolongation to a bicovariant \textsc{sodc} on \(H\), let \((\Omega_{P,\mathrm{ver}},\dt{P,\mathrm{ver}})\) denote the resulting second-order vertical calculus of \(P\) , and let \(\Omega_{P,\oplus} \coloneqq \Lambda_H \mathbin{\widehat{\otimes}}^{\leq 2} \Omega_{P,\mathrm{hor}}\), which therefore contains \(\Omega_{P,\mathrm{ver}}\) as a left \(H\)-subcomodule graded \(\ast\)-algebra.
Using the multiplication map \(
(\omega \otimes h \mapsto \omega \cdot h) : \Lambda_H \otimes H \to \Omega_H
\), we can identify \(\Omega_H\) with the graded \(\ast\)-subalgebra of \(\Omega_{P,\oplus}\) generated by \(H \subset P\) and \(\Lambda^1_H \mathbin{\widehat{\otimes}} 1_P \subset \Omega^1_{P,\mathrm{ver}}\). Hence, we can view \(\Omega_{P,\oplus}\) as the graded left \(H\)-comodule \(\ast\)-algebra, truncated at degree \(2\), generated by the graded left \(H\)-subcomodule \(\ast\)-subalgebras \(\Omega_H\) and \(\Omega_B\) subject to the relation \(1_{\Omega_H} = 1_{\Omega_B}\) and the braided graded commutation relations
\begin{equation}
\forall \omega \in \Omega_H, \, \forall \alpha \in \Omega_B, \quad \alpha \wedge \omega \coloneqq (-1)^{\abs{\alpha}\abs{\omega}} \ca{\omega}{0} \wedge \alpha \triangleleft \ca{\omega}{1};
\end{equation}
indeed, \(\Omega_{P,\oplus} = \Omega_H \cdot \Omega_B\) is freely generated as a graded right \(\Omega_B\)-module by \(\Omega_H\). In particular, the graded left \(H\)-subcomodule \(\ast\)-subalgebra \(\Omega_{P,\mathrm{ver}} = \Omega_H \cdot B\) is freely generated as a right \(B\)-module by \(\Omega_H\), and
\begin{equation}
\forall \omega \in \Omega_H, \, \forall b \in B, \quad \dv{P}(\omega \cdot b) = \dt{H}(\omega) \cdot b,
\end{equation}
so that \((\Omega^1_H,\dt{H})\) is necessarily locally freeing for \(P\).
We can now characterise the space \(\fr{at}[\Omega^1_H]\) of \((\Omega^1_H,\dt{H})\)-adapted relative gauge potentials on \(P\) with respect to the canonical second-order horizontal calculus \((\Omega_B,\dt{B};\Omega_{P,\mathrm{hor}})\), and hence relate the \(\fr{G}\)-equivariant moduli space \(\fr{At}/\fr{at}[\Omega^1_H] \cong \Ob\mleft(\mathcal{G}[\Omega^1_H]/\ker\mu[\Omega^1_H]\mright)\) of strongly \((H;\Omega^1_H,\dt{H})\)-principal \textsc{sodc}{} on \(P\) inducing \((\Omega_B,\dt{B};\Omega_{P,\mathrm{hor}})\) to lazy Hochschild cohomology on \(H\).
\begin{proposition}\label{trivialadaptprop}
The groupoid isomorphism \(\ZS^1_\ell(H;B,\Omega^1_B) \ltimes \ZH^1_\ell(H;\Omega^1_B) \xrightarrow{\sim} \fr{G} \ltimes \fr{At}\) of Proposition~\ref{trivialgaugepotprop} descends to a groupoid isomorphism
\[
\ZS^1_\ell(H;B,\Omega^1_B) \ltimes \left(\frac{\ZH^1_\ell(H;\Omega^1_B)}{\Op_0^{-1}\mleft(\fr{at}[\Omega^1_H]\mright)} \right) \xrightarrow{\sim} \fr{G} \ltimes \left(\frac{\fr{At}}{\fr{at}[\Omega^1_H]}\right),
\]
where
\[
\Op_0^{-1}\mleft(\fr{at}[\Omega^1_H]\mright) = \set{\mu \in \ZH^1_\ell(H;\Omega^1_B) \given \ker(\mu\circ\inv{S}) \supseteq \ker\varpi}.
\]
\end{proposition}
\begin{proof}
Before continuing, recall that \(\varpi : H \to \Lambda^1_H\) satisfies
\[
\forall h \in H, \quad \ca{\varpi(h)}{-1} \otimes \ca{\varpi(h)}{0} = \cm{h}{1}S(\cm{h}{3}) \otimes \varpi(\cm{h}{2}).
\]
so that that
\(
\Delta(\ker\varpi) \subseteq \ker(\id_H \otimes \varpi) = H \otimes \ker\varpi
\). Let \(\mu \in \ZH^1_\ell(H;\Omega^1_B)\) and set \(\mathbf{A} \coloneqq \Op_0(\mu)\).
First, suppose that \(\mathbf{A} \in \fr{at}[\Omega^1_H]\). Then for all \(h \in H\),
\begin{align*}
\mu \circ \inv{S}(h) &= S(\cm{\inv{S}(h)}{1}) \cdot \mathbf{A}(\cm{\inv{S}(h)}{2})\\
&= \cm{h}{2} \cdot \mathbf{A}(\inv{S}(\cm{h}{1}))\\
&= \mathbf{A}(\inv{S}(\cm{h}{1}S(\cm{h}{2})) - \mathbf{A}(\cm{h}{2})\cdot\inv{S}(\cm{h}{1})\\
&= -\omega[\mathbf{A}]\mleft(\varpi(\cm{h}{2})\cdot\cm{h}{3}\mright) \cdot \inv{S}(\cm{h}{1})\\
&= -\omega[\mathbf{A}]\mleft(\ca{\varpi(h)}{0} \cdot \inv{S}(\ca{\varpi(h)}{-1}) \mright),
\end{align*}
so that, indeed, \(\ker (\mu \circ \inv{S}) \supseteq \ker\varpi\).
Now, suppose that \(\ker (\mu \circ \inv{S}) \supseteq \ker\varpi\). Then, for all \(h\in H\),
\begin{multline*}
\mathbf{A}(\cm{h}{1}) \cdot S(\cm{h}{2}) = -\mathbf{A}(\cm{h}{1}^\ast)^\ast \cdot S(\cm{h}{2})
= -\left(\cm{h}{1}^\ast \cdot \mu (\cm{h}{2})^\ast\right)^\ast \cdot S(\cm{h}{3})
= -\mu(\cm{h}{2}^\ast)^\ast \cdot \cm{h}{1}S(\cm{h}{3})\\
= -\mu(S(\cm{h}{2}^\ast)^\ast) \cdot \cm{h}{1}S(\cm{h}{3})
= -\mu(\inv{S}(\cm{h}{2})) \cdot \cm{h}{1}S(\cm{h}{3});
\end{multline*}
hence, if \(h \in \ker\varpi\), then \(\cm{h}{1}S(\cm{h}{3}) \otimes \cm{h}{2} \in H \otimes \ker\varpi\), so that \(\mathbf{A}(\cm{h}{1}) \cdot S(\cm{h}{2}) = 0\). Since \(\varpi : H \to \Lambda^1_H\) is surjective, \(\mathbf{A} = N \circ \dv{P}\) for the morphism of left \(H\)-comodule right \(P\)-modules \(N : \Omega^1_{P,\mathrm{ver}} \to \Omega^1_{P,\mathrm{hor}}\) defined by
\[
\forall h,k \in H, \, \forall b \in B, \quad N(\varpi(k) \cdot hb) \coloneqq \mathbf{A}(\cm{k}{1}) \cdot S(\cm{k}{2})hb;
\]
it therefore suffices to show that \(N\) is left \(P\)-linear and \(\ast\)-preserving, for then \(\mathbf{A} \in \fr{at}[\Omega^1_H]\) with relative connection \(1\)-form \(\omega[\mathbf{A}] = N\). Let \(h,k \in H\) and \(b \in B\), so that
\begin{align*}
hb \cdot \varpi(k) &= h \cdot \varpi(k) \cdot \left(b \triangleleft 1\right)\\
&= \left(\varpi(\cm{h}{1}k) - \epsilon(k)\varpi(\cm{h}{1})\right) \cdot \cm{h}{2}b\\
&= \varpi(\cm{h}{1}k) \cdot \cm{h}{2} b - \varpi(\cm{h}{1}) \cdot \cm{h}{2}\epsilon(k)b.
\end{align*}
On the one hand,
\begin{align*}
&N\mleft(\varpi(\cm{h}{1}k) \cdot \cm{h}{2} b - \varpi(\cm{h}{1}) \cdot \cm{h}{2}\epsilon(k)b\mright)\\
&= \mathbf{A}(\cm{h}{1}\cm{k}{1}) \cdot S(\cm{h}{2}\cm{k}{2}) \cdot \cm{h}{3}b -\mathbf{A}(\cm{h}{1}) \cdot S(\cm{h}{2})\cm{h}{3}\epsilon(k)b \\
&= \mathbf{A}(\cm{h}{1}\cm{k}{1}) \cdot S(\cm{k}{2})S(\cm{h}{2})\cm{h}{3}b - \epsilon(k)\mathbf{A}(h)b\\
&= \mathbf{A}(h)\cm{k}{1} \cdot S(\cm{k}{2})b +h\mathbf{A}(\cm{k}{1}) \cdot S(\cm{k}{2})b - \epsilon(k)\mathbf{A}(h)b\\
&= h \mathbf{A}(\cm{k}{1}) \cdot S(\cm{k}{2}) b,
\end{align*}
while on the other,
\begin{align*}
hb N(\varpi(k)) &= hb \cdot \mathbf{A}(\cm{k}{1})S(\cm{k}{2})\\
&= -hb\cm{k}{1})\mathbf{A}(S(\cm{k}{2}))\\
&= -h\cm{k}{1}S(\cm{k}{5})\left((b\triangleleft\cm{k}{2})\triangleleft S(\cm{k}{4})\right) \cdot \mu(S(\cm{k}{3}))\\
&= -h\cm{k}{1}S(\cm{k}{5}) \cdot \mu(S(\cm{k}{4})) \cdot \left((b\triangleleft\cm{k}{2})\triangleleft S(\cm{k}{3})\right)\\
&= -h\cm{k}{1}S(\cm{k}{3})\mu(S(\cm{k}{2}))b\\
&= -h\cm{k}{1} \cdot \mathbf{A}(S(\cm{k}{2})) \cdot b\\
&= h \cdot \mathbf{A}(\cm{k}{1}) \cdot S(\cm{k}{2})b,
\end{align*}
so that
\(
hb N(\varpi(k)) = N\mleft(\varpi(\cm{h}{1}k) \cdot \cm{h}{2} b - \varpi(\cm{h}{1}) \cdot \cm{h}{2}\epsilon(k)b\mright)
\).
Hence, \(N\) is left \(P\)-linear. Similar calculations now show that \(N\) is also \(\ast\)-preserving.
\end{proof}
The techniques of this proof can also be used to characterise the quadric set \(\pr{\fr{At}}[\Omega^1_H]\) of \((\Omega^1_H,\dt{H})\)-adapted prolongable gauge potentials on \(P\) with respect to \((\Omega_B,\dt{B};\Omega_{P,\mathrm{hor}})\)---hence, also, the \(\pr{\fr{G}}\)-equivariant moduli space \[\pr{\fr{At}}[\Omega^1_H]/\pr{\fr{at}}_{\can}[\Omega^1_H] \cong \Ob\mleft(\mathcal{G}[\Omega^{\leq 2}_H]/\ker\mu[\Omega^{\leq 2}_H]\mright)\] of strongly \((H;\Omega_H,\dt{H})\)-principal \textsc{sodc}{} on \(P\) with respect to \((\Omega_B,\dt{B};\Omega_{P,\mathrm{hor}})\)---in terms of lazy Hochschild cohomology. Recall that \(\pr{\fr{at}}_{\can}[\Omega^1_H]\) denotes the space of all \((\Omega_H,\dt{H})\)-adapted prolongable relative gauge potentials on \(P\) with respect to \((\Omega_B,\dt{B};\Omega_{P,\mathrm{hor}})\) and that \(\Inn(\pr{\fr{at}};\Omega^1_H)\) denotes the space of all \((\Omega^1_H,\dt{H})\)-semi-adapted inner prolongable relative gauge potentials on \(P\) with respect to \((\Omega_B,\dt{B};\Omega_{P,\mathrm{hor}})\), so that \[\Out(\pr{\fr{at}}[\Omega^1_H]) \coloneqq \pr{\fr{At}}[\Omega^1_H]/\Inn(\pr{\fr{at}};\Omega^1_H).\]
\begin{theorem}\label{trivialthm}
The groupoid isomorphism \(\ZS^1_\ell(H;B,\Omega^1_B) \ltimes \ZH^1_\ell(H;\Omega^1_B,\Omega_B) \xrightarrow{\sim} \pr{\fr{G}} \ltimes \pr{\fr{At}}\) of Corollary~\ref{trivialprcor} restricts to a groupoid isomorphism
\[
\ZS^1_\ell(H;B,\Omega^1_B) \ltimes \Op^{-1}\mleft(\pr{\fr{At}}[\Omega^1_H]\mright) \xrightarrow{\sim} \pr{\fr{G}} \ltimes \pr{\fr{At}}[\Omega^1_H],
\]
where
\begin{equation}\label{triveq1}
\Op^{-1}\mleft(\pr{\fr{At}}[\Omega^1_H]\mright) = \set{\mu \in \ZH^1_\ell(H;\Omega^1_B,\Omega_B) \given \ker(\mathcal{F}[\mu] \circ \inv{S}) \supseteq \ker\varpi}.
\end{equation}
In turn, this restricted groupoid isomorphism descends to respective groupoid isomorphisms
\begin{gather*}
\ZS^1_\ell(H;B,\Omega^1_B) \ltimes \left(\frac{\Op^{-1}\mleft(\pr{\fr{At}}[\Omega_H]\mright)}{\Op_0^{-1}\mleft(\pr{\fr{at}}_{\can}[\Omega^1_H]\mright)}\right) \xrightarrow{\sim} \pr{\fr{G}} \ltimes \left(\frac{\pr{\fr{At}}[\Omega^1_H]}{\pr{\fr{at}}_{\can}[\Omega^1_H]}\right),\label{triveq2}\\
\HS^1_\ell(H;B,\Omega^1_B) \ltimes \left(\frac{\Op^{-1}\mleft(\pr{\fr{At}}[\Omega^1_H]\mright)}{\Op_0^{-1}\mleft(\Inn(\pr{\fr{at}};\Omega^1_H)\mright)}\right) \xrightarrow{\sim} \Out(\pr{\fr{G}}) \ltimes \Out(\pr{\fr{At}}[\Omega^1_H]),\label{triveq3}
\end{gather*}
where
\begin{gather}
\Op_0^{-1}\mleft(\pr{\fr{at}}_{\can}[\Omega^1_H]\mright) = \set{\mu \in \ZH^1_\ell(H;\Omega^1_B,\Omega_B) \given \ker(\mu \circ \inv{S}) \supseteq \ker\varpi},\\
\Op_0^{-1}\mleft(\Inn(\pr{\fr{at}};\Omega^1_H)\mright) = \set{D\beta \given \beta \in Z_B(\Omega^1_B)_\mathrm{sa} \cap Z(\Omega_B), \,\ker(D(-\ensuremath{\mathrm{i}}{}\dt{B}\beta) \circ \inv{S}) \supseteq \ker\varpi}.
\end{gather}
In particular, \(\pr{\fr{at}}_{\can}[\Omega^1_H] = \pr{\fr{at}} \cap \fr{at}[\Omega^1_H]\).
\end{theorem}
\begin{proof}
Let us first prove \eqref{triveq1}. Let \(\mu \in \ZH^1_\ell(H;\Omega^1_B,\Omega_B)\), so that
\[
\forall h \in H, \, \forall b \in B, \quad \mathbf{F}[\Op(\mu)](hb) = \cm{h}{1}\cdot\mathcal{F}[\mu](\cm{h}{2})\cdot b.
\]
By applying the proof of Proposition~\ref{trivialadaptprop}, \emph{mutatis mutandis}, to the left \(H\)-covariant \(\ast\)-deriva\-tion \(\mathbf{F}[\Op(\mu)] : P \to \Omega^2_{P,\mathrm{hor}}\), it follows that \(\Op(\mu) \in \tot{\fr{At}}\) if and only if
\[
\ker(\mathcal{F}[\mu] \circ \inv{S}) \supseteq \ker\varpi.
\]
This argument now also proves \eqref{triveq3}, since for all \(\beta \in Z_B(\Omega^1_B)_\mathrm{sa} \cap Z(\Omega_B)\) and \(h \in H\)
\[
D(-\ensuremath{\mathrm{i}}{}\dt{B}\beta)(h) = \mathcal{F}[D\alpha](h) = S(\cm{h}{1}) \cdot \mathbf{F}[\Op(\alpha)](\cm{h}{2}) = S(\cm{h}{1}) \cdot \mathbf{F}_\mathrm{rel}[\Op_0(\alpha)](\cm{h}{2})
\]
by \eqref{coboundcurve} and the fact that \(\mathbf{F}[\Op(0)] = 0\). Hence, it remains to prove \eqref{triveq3}.
On the one hand, \(\pr{\fr{At}}[\Omega_H] \subseteq \pr{\fr{at}} \cap \fr{at}[\Omega^1_H]\) by Theorem~\ref{qpbthm}; on the other hand, by Corollary~\ref{trivialprcor} and Proposition~\ref{trivialadaptprop},
\[
\Op_0^{-1}(\pr{\fr{at}} \cap \fr{at}[\Omega^1_H]) = \set{ \mu \in \ZH^1_\ell(H;\Omega^1_B,\Omega_B) \given \ker (\mu \circ \inv{S}) \supseteq \ker\varpi} \eqqcolon \ZH^1_\ell(H;\Omega^1_B,\Omega_B)[\Omega^1_H] .
\]
Hence, it suffices to show that \(\pr{\fr{At}}[\Omega_H] \supseteq \pr{\fr{at}} \cap \fr{at}[\Omega^1_H]\).
Let \(\mathbf{A} \in \pr{\fr{at}} \cap \fr{at}[\Omega^1_H]\) be given; set \(\mu \coloneqq \Op_0^{-1}(\mathbf{A})\) and \(N \coloneqq \omega[\mathbf{A}]\), so that for all \(h \in H\),
\[
N(\varpi(h)) = \mathbf{A}(\cm{h}{1}) \cdot S(\cm{h}{2}) = -\cm{h}{1} \cdot \mathbf{A}(S(\cm{h}{2})) = -\cm{h}{1}S(\cm{h}{3}) \cdot \mu(S(\cm{h}{2}))
\]
by the proof of Proposition~\ref{trivialadaptprop}. We will show that \(\mathbf{A} \in \pr{\fr{At}}[\Omega_H]\) by showing that \(N\) satisfies both \eqref{toteq1} and \eqref{toteq2}.
First, let \(h \in H\) and \(\beta \in \Omega^1_B\). Since \(\mu \in \ZH^1_\ell(H;\Omega^1_B,\Omega_B)\), it follows that
\begin{align*}
\beta \wedge N(\varpi(h)) &= -\beta \wedge \cm{h}{1}S(\cm{h}{3}) \cdot \mu(S(\cm{h}{2}))\\
&= -\cm{h}{1}S(\cm{h}{5}))\left((\beta \triangleleft \cm{h}{2}) \triangleleft S(\cm{h}{4})\right) \wedge \mu(S(\cm{h}{3}))\\
&= \cm{h}{1}S(\cm{h}{5})) \cdot \mu(S(\cm{h}{4})) \wedge \left((\beta \triangleleft \cm{h}{2}) \triangleleft S(\cm{h}{3})\right) \\
&= \cm{h}{1}S(\cm{h}{3}) \cdot \mu(S(\cm{h}{3})) \cdot \beta\\
&= -N(\varpi(h)) \wedge \beta
\end{align*}
by Lemma~\ref{centrallem}. Hence, \(N\) satisfies \eqref{toteq1}.
Let us now check that \(N\) satisfies \eqref{toteq2}. Let \(h, k \in H\); note that
\begin{align*}
\ca{\varpi(k)}{-1} \triangleright\varpi(h) \otimes \ca{\varpi(k)}{0} &= \cm{k}{1}S(\cm{h}{k}) \triangleright\varpi(h) \otimes \varpi(\cm{k}{2})\\
&= \varpi(\cm{h}{1}S(\cm{h}{3})k) \otimes \varpi(\cm{h}{2}) - \epsilon(k)\varpi(\cm{h}{1}S(\cm{h}{3})) \otimes \varpi(\cm{h}{2}),
\end{align*}
so that it suffices to show that
\[
N(\varpi(h)) \wedge N(\varpi(k)) + N(\varpi(\cm{h}{1}S(\cm{h}{3})k)) \wedge N(\varpi(\cm{h}{2})) - \epsilon(k)N(\varpi(\cm{h}{1}S(\cm{h}{3})))N(\varpi(\cm{h}{2})) = 0.
\]
First, by repeated applications of Lemma~\ref{centrallem} together with the fact that \(\varpi\) is a \(1\)-cocycle valued in the left crossed \(H\)-\(\ast\)-module \(\Lambda^1_H\), we find that
\begin{align*}
N(\varpi(h)) \wedge N(\varpi(k)) &= \mathbf{A}(\cm{h}{1}) \cdot S(\cm{h}{2}) \wedge \mathbf{A}(\cm{k}{1}) \cdot S(\cm{k}{2})\\
&= \cm{h}{1} \cdot \mathbf{A}(S(\cm{h}{2})) \wedge \cm{k}{1} \cdot \mathbf{A}(S(\cm{k}{2}))\\
&= \cm{h}{1}\cdot \mathbf{A}(S(\cm{h}{2})\cm{k}{1}) \wedge \mathbf{A}(\cm{k}{2}) - \cm{h}{1}S(\cm{h}{2}) \cdot \mathbf{A}(\cm{k}{1}) \wedge \mathbf{A}(S(\cm{k}{2}))\\
&= \cm{h}{1}S(\cm{h}{3})\cm{k}{1} \cdot \mu(S(\cm{h}{2})\cm{k}{2}) \wedge S(\cm{k}{4}) \cdot \mu(S(\cm{k}{3}))\\
&\quad\quad\quad-\epsilon(h)\cm{k}{1}\cdot\mu(\cm{k}{2})\wedge S(\cm{k}{4})\cdot\mu(S(\cm{k}{3}))\\
&= \cm{h}{1}S(\cm{h}{3})\cm{k}{1}S(\cm{k}{5}) \cdot \mu(S(\cm{h}{2})\cm{k}{2}) \triangleleft S(\cm{k}{4}) \wedge \mu(S(\cm{k}{3}))\\
&\quad\quad\quad-\epsilon(h)\cm{k}{1}S(\cm{k}{5})\cdot\mu(\cm{k}{2})\triangleleft S(\cm{k}{4})\wedge\mu(S(\cm{k}{3}))\\
&= - \cm{h}{1}S(\cm{h}{3})\cm{k}{1}S(\cm{k}{5}) \cdot \mu(S(\cm{k}{4})) \wedge \mu(S(\cm{h}{2})\cm{k}{2}) \triangleleft S(\cm{k}{3})\\
&\quad\quad\quad+\epsilon(h)\cm{k}{1}S(\cm{k}{5})\cdot\mu(S(\cm{k}{4})) \wedge \mu(\cm{k}{2})\triangleleft S(\cm{k}{3})\\
&=- \cm{h}{1}S(\cm{h}{3})\cm{k}{1}S(\cm{k}{3}) \cdot \mu(S(\cm{k}{2})) \wedge \mu(S(\cm{h}{2}))\\
&\quad\quad\quad+\epsilon(h)\cm{k}{1}S(\cm{k}{4}) \cdot \mu(S(\cm{k}{3}))\wedge\mu(S(\cm{k}{2}))\\
&\quad\quad\quad-\epsilon(h)\cm{k}{1}S(\cm{k}{4}) \cdot \mu(S(\cm{k}{3}))\wedge\mu(S(\cm{k}{2}))\\
&= -\cm{h}{1}S(\cm{h}{3})\cm{k}{1}S(\cm{k}{3}) \cdot \mu(S(\cm{k}{2}) \wedge \mu(S(\cm{h}{2})).
\end{align*}
Next, by applying the last calculation, \emph{mutatis mutandis}, we find that
\begin{align*}
&N(\varpi(\cm{h}{1}S(\cm{h}{3})k)) \wedge N(\varpi(\cm{h}{2}))\\
&=- \cm{h}{1}S(\cm{h}{9})\cm{k}{1}S(\cm{k}{3})S^2(\cm{h}{7})S(\cm{h}{3})\cm{h}{4}S(\cm{h}{6}) \cdot\mu(S(\cm{h}{5})\wedge \mu(S(\cm{k}{2})S^2(\cm{h}{8})S(\cm{h}{2}))\\
&= -\cm{h}{1}S(\cm{h}{5})\cm{k}{1}S(\cm{k}{3})\cdot\mu(S(\cm{h}{3}))\wedge\mu(S(\cm{k}{2})S^2(\cm{h}{4})S(\cm{h}{2}))\\
&= -\cm{h}{1}S(\cm{h}{5})\cm{k}{1}S(\cm{k}{3})\cdot\mu(S(\cm{h}{3}))\wedge\left(\mu(S(\cm{k}{2})S^2(\cm{h}{4})) \triangleleft S(\cm{h}{2}) + \epsilon(\cm{k}{2})\epsilon(\cm{h}{4})\mu(S(\cm{h}{2}) \right) \\
&= \cm{h}{1}S(\cm{h}{5})\cm{k}{1}S(\cm{k}{3}) \cdot \mu(S(\cm{k}{2})S^2(\cm{h}{4})) \triangleleft S(\cm{h}{3}) \wedge \mu(S(\cm{h}{2})) \\
&\quad\quad\quad-\epsilon(k)\cm{h}{1}S(\cm{h}{4}) \cdot \mu(S(\cm{h}{3})) \wedge \mu(S(\cm{h}{2}))\\
&= \cm{h}{1}S(\cm{h}{3})\cm{k}{1}S(\cm{k}{3}) \cdot \mu(S(\cm{k}{2}) \wedge \mu(S(\cm{h}{2})) - \epsilon(k)\cm{h}{1}S(\cm{h}{4}) \cdot \mu(S(\cm{h}{3})) \wedge \mu(S(\cm{h}{2}))\\
&\quad\quad\quad-\epsilon(k)\cm{h}{1}S(\cm{h}{4}) \cdot \mu(S(\cm{h}{3})) \wedge \mu(S(\cm{h}{2}))\\
&= -N(\varpi(h)) \wedge N(\varpi(k)) - 2\epsilon(k)\cm{h}{1}S(\cm{h}{4}) \cdot \mu(S(\cm{h}{3})) \wedge \mu(S(\cm{h}{2})).
\end{align*}
Finally, by applying the last calculation, \emph{mutatis mutandis}, we find that
\[
N(\varpi(\cm{h}{1}S(\cm{h}{3}))) \wedge N(\varpi(\cm{h}{2})) = -2\cm{h}{1}S(\cm{h}{4}) \cdot \mu(S(\cm{h}{3})) \wedge \mu(S(\cm{h}{2})),
\]
which we can substitute into the calculation of \(N(\varpi(\cm{h}{1}S(\cm{h}{3})k)) \wedge N(\varpi(\cm{h}{2}))\) to obtain
\[
N(\varpi(\cm{h}{1}S(\cm{h}{3})k)) \wedge N(\varpi(\cm{h}{2})) = -N(\varpi(h)) \wedge N(\varpi(k)) + 2\epsilon(k)N(\varpi(\cm{h}{1}S(\cm{h}{3}))) \wedge N(\varpi(\cm{h}{2})),
\]
which, in turn, yields our claim.
\end{proof}
\begin{remark}
Note, in particular, that \(\pr{\fr{At}}[\Omega^1_H]\) necessarily contains \(\Op(0)\), which corresponds to the trivial flat connection.
\end{remark}
The proof of the above result almost immediately yields the following expression for the curvature \(2\)-form of an \((\Omega^1_H,\dt{H})\)-adapted prolongable gauge potential.
\begin{corollary}\label{trivcurvcor}
Let \(\mu \in \Op^{-1}\mleft(\pr{\fr{At}}[\Omega^1_H]\mright)\). Then
\[
\forall h \in H, \quad F[\Op(\mu)](\varpi(h)) = \cm{h}{1}S(\cm{h}{4}) \cdot \ensuremath{\mathrm{i}}{}\left(\dt{B}\mu(S(\cm{h}{3}))\epsilon(\cm{h}{2}) + \mu(S(\cm{h}{3})) \wedge \mu(S(\cm{h}{2}))\right)
\]
\end{corollary}
\section{\texorpdfstring{\(q\)}{q}-Monopoles over real multiplication noncommutative \texorpdfstring{\(2\)}{2}-tori}\label{sec5}
Let \(\theta \in \mathbf{R} \setminus \mathbf{Q}\) be a quadratic irrationality, and let \(\mathcal{A}_\theta\) be the corresponding smooth noncommutative \(2\)-torus. In this section, we show how to subsume Connes's constant curvature connections~\cite{Connes80} and Polishchuk--Schwarz's holomorphic structures~\cite{PolishchukSchwarz} on the self-Morita equivalence bimodules amongst the basic Heisenberg modules over \(\mathcal{A}_\theta\) into gauge theory on a certain canonical non-trivial principal \(\mathcal{O}{}(\Unit(1))\)-module algebra \(P\) over \(\mathcal{A}_\theta\) implicit in Manin's `Alterstraum'~\cite{Manin}. In the process, we will encounter striking formal similarities with the \(q\)-deformed complex Hopf fibration; in this case, however, there is a canonical value for the parameter \(q\) arising from the algebraic number theory of the real quadratic irrationality \(\theta\).
\subsection{Number-theoretic preliminaries} We begin by recalling relevant folklore about real quadratic number fields; our primary reference is the monograph of Halter--Koch~\cite{HK}. Let \(\theta \in \mathbf{R} \setminus \mathbf{Q}\) be a quadratic irrationality. Recall that \(\SL(2,\mathbf{Z})\) acts on \(\mathbf{R} \setminus \mathbf{Q}\) by fractional linear transformations, i.e., by
\[
\forall g \in \SL(2,\mathbf{Z}), \, \forall \xi \in \mathbf{R} \setminus \mathbf{Q}, \quad g \triangleright \xi \coloneqq \frac{g_{11}\xi+g_{12}}{g_{21}\xi+g_{22}},
\]
and observe that this action descends to an action of \(\PSL(2,\mathbf{Z})\) on \(\mathbf{R} \setminus \mathbf{Q}\); denote the stabilizers of \(\theta\) in \(\SL(2,\mathbf{Z})\) and \(\PSL(2,\mathbf{Z})\) by \(\SL(2,\mathbf{Z})_\theta\) and \(\PSL(2,\mathbf{Z})_\theta\). Our goal is to compute \(\SL(2,\mathbf{Z})_\theta\) (and hence \(\PSL(2,\mathbf{Z})_\theta\)) in terms of the solutions of the positive Pell's equation associated with the real quadratic irrationality \(\theta\).
First, the \emph{type} of \(\theta\) is the unique coprime triple \((a,b,c) \in (\mathbf{Z} \setminus \set{0}) \times \mathbf{Z}^2\), such that
\[
\theta = \frac{b+\sqrt{b^2-4ac}}{2a}
\]
and \(b^2-4ac\) is not a square, so that \(\Delta \coloneqq b^2-4ac\) is the \emph{discriminant} of \(\theta\). On the one hand, then, the \emph{norm} on the real quadratic number field \(\mathbf{Q}[\theta] = \mathbf{Q}[\sqrt{\Delta}]\) is the multiplicative unit-preserving map \(\mathcal{N} : \mathbf{Q}[\theta] \to \mathbf{Q}\) defined by
\[
\forall r,s \in \mathbf{Q}, \quad \mathcal{N}(r+s\sqrt{\Delta}) \coloneqq r^2-\Delta s^2;
\]
on the other hand, the \emph{quadratic order} of discriminant \(\Delta\) in \(\mathbf{Q}[\theta] = \mathbf{Q}[\!\sqrt{\Delta}]\) is the subring
\[
\mathcal{O}_\Delta = \set{\tfrac{u+v\sqrt{\Delta}}{2} \given u,v \in \mathbf{Z}, \, u \equiv v \Delta \bmod 2}.
\]
Thus, we can define the multiplicative group of \emph{norm-positive} units in \(\mathcal{O}{}_\Delta\) by
\[
\mathcal{O}_\Delta^{\times,+} \coloneqq \set{\epsilon \in \mathcal{O}_\Delta^\times \given \mathcal{N}(\epsilon) > 0} = \set{\tfrac{u+v\sqrt{\Delta}}{2} \given u,v \in \mathbf{Z}, \, u^2 - \Delta v^2 = 4},
\]
which can therefore be viewed as the group of positive solutions of the Pell's equation
\[x^2 - \Delta y^2 = 4.\]
By applying Dirichlet's unit theorem to the quadratic order \(\mathcal{O}{}_\Delta\) of the real quadratic field \(\mathbf{Q}[\theta]\), one finds that \(\mathcal{O}{}_\Delta^\times \cap \mathbf{R}_{>0}\) is infinite cyclic and generated by the \emph{fundamental unit}
\[
\epsilon_\Delta \coloneqq \min\set{\epsilon \in \mathcal{O}{}_\Delta^\times \cap \mathbf{R}_{>0} \given \epsilon > 1} = \min\set{\epsilon \in \mathcal{O}{}_\Delta^\times \given \epsilon > 1},
\]
so that, in turn, the subgroup \(\mathcal{O}{}_\Delta^{\times,+} \cap \mathbf{R}_{>0}\) is also infinite cyclic and generated by the \emph{norm-positive fundamental unit} (or \emph{Pell's unit})
\[
\epsilon_\Delta^+ \coloneqq \begin{cases} \epsilon_\Delta &\text{if \(\mathcal{N}(\epsilon_\Delta)=1\),}\\ \epsilon_\Delta^2 &\text{if \(\mathcal{N}(\epsilon_\Delta) = -1\).} \end{cases}
\]
The following folkloric result now gives the desired number-theoretic characterization of the stabilizer groups \(\SL(2,\mathbf{Z})_\theta\) and \(\PSL(2,\mathbf{Z})_\theta\).
\begin{proposition}[Folklore~\cite{HK}*{Thm.\ 5.2.10}]\label{stabthm}
The map \(\Phi : \mathcal{O}{}_\Delta^{\times,+} \to \SL(2,\mathbf{Z})_\theta\) given by
\begin{equation}
\forall \tfrac{u+v\sqrt{\Delta}}{2} \in \mathcal{O}{}_\Delta^{\times,+}, \quad \Phi\left(\tfrac{u+v\sqrt{\Delta}}{2}\right) \coloneqq \begin{pmatrix} \tfrac{u+bv}{2} & -cv \\ av & \tfrac{u-bv}{2} \end{pmatrix}
\end{equation}
defines a group isomorphism satisfying \(\Phi(-1) = -I\) and
\begin{equation}\label{stabthmeq}
\forall g \in \SL(2,\mathbf{Z})_\theta, \quad \inv{\Phi}(g) \coloneqq g_{21}\theta+g_{22}.
\end{equation}
Thus, in particular, \(\SL(2,\mathbf{Z})_\theta\) decomposes as the internal direct product
\[
\SL(2,\mathbf{Z})_\theta = \Phi\mleft(\mathcal{O}{}^{\times,+}_\Delta \cap \mathbf{R}_{>0}\mright) \times \set{\pm I},
\]
where \(\Phi\mleft(\mathcal{O}{}^{\times,+}_\Delta \cap \mathbf{R}_{>0}\mright)\) is infinite cyclic with canonical generator \(\Phi(\epsilon_\Delta^+)\), so that \(\PSL(2,\mathbf{Z})_\theta\) is infinite cyclic with canonical generator \([\Phi(\epsilon_\Delta^+)]\).
\end{proposition}
\subsection{Basic Heisenberg modules over irrational noncommutative \texorpdfstring{\(2\)}{2}-tori} In this section, we recall the construction of basic Heisenberg modules over an irrational noncommutative \(2\)-torus \(C^\infty(\mathbf{T}^2_\theta)\); when \(\theta \in \mathbf{R} \setminus \mathbf{Q}\) is a quadratic irrationality, we assemble the self-Morita equivalence bimodules of positive rank among the basic Heisenberg modules into a canonical non-trivial principal \(\mathcal{O}{}(\Unit(1))\)-comodule algebra \(P\) over \(C^\infty(\mathbf{T}^2_\theta)\).
Let us first recall the relevant algebraic notions of (Morita) equivalence bimodule and self-Morita equivalence bimodule.
\begin{definition}
Let \(A\) and \(B\) be unital \(\mathbf{C}\)-algebras. An \emph{\((A,B)\)-equivalence bimodule} is an \((A,B)\)-bimodule \({}_A E_B\) satisfying the following:
\begin{enumerate}
\item the right \(B\)-module \(E_B\) is finitely generated, projective, and full in the sense that
\[
B = \Span_{\mathbf{C}}\set{f(e) \given e \in E, f \in \Hom_B(E_B,B_B)};
\]
\item the left \(A\)-module structure on \({}_A E_B\) is an algebra isomorphism \(A \xrightarrow{\sim} \End_B(E_B)\).
\end{enumerate}
In particular, a \emph{self-Morita equivalence bimodule} over \(B\) is an \((B,B)\)-equiva\-lence bimodule.
\end{definition}
\begin{example}
Let \(B\) be a unital \(\mathbf{C}\)-algebra. The \emph{trivial} self-Morita equivalence bimodule over \(B\) is \(B\) itself equipped with the left and right \(B\)-module structures induced by multiplication in the algebra \(B\).
\end{example}
Next, recall that the \emph{smooth noncommutative \(2\)-torus} with deformation parameter \(\theta \in \mathbf{R}\) is the unital \(\ast\)-algebra \(\mathcal{A}_\theta\) of rapidly decaying Laurent series in two unitary generators \(U_\theta\) and \(V_\theta\) satisfying the commutation relation
\begin{equation}
V_{\theta} U_\theta = \ensuremath{\mathrm{e}}^{2\pi\ensuremath{\mathrm{i}}{}\theta} U_\theta V_\theta;
\end{equation}
note that \(\mathcal{A}_\theta\) can be canonically topologised as a Fr\'{e}chet pre-\(C^\ast\)-algebra, whose \(C^\ast\)-algebraic completion can be identified with the rotation algebra \(A_\theta \coloneqq C(\Unit(1)) \rtimes_\theta \mathbf{Z}\). As Rieffel famously observed~\cite{Rieffel83}*{Thm.\ 1.1}, the action of \(\SL(2,\mathbf{Z})\) on \(\mathbf{R} \setminus \mathbf{Q}\) by fractional linear transformations manifests itself as distinguished family of equivalence bimodules, the \emph{basic Heisenberg modules}, for smooth noncommutative \(2\)-tori with irrational deformation parameter. These bimodules had already been constructed \emph{qua} right modules by Connes~\cite{Connes80} as the very first examples of noncommutative smooth vector bundles in noncommutative differential geometry.
\begin{theoremdefinition}[Connes~\cite{Connes80}, Rieffel~\citelist{\cite{Rieffel81}\cite{Rieffel83}*{Thm.\ 1.1}\cite{Rieffel88}*{\S\S 2--3, 5}}]
Let \(\theta \in \mathbf{R} \setminus \mathbf{Q}\). For every \(g \in \SL(2,\mathbf{Z})\), we can construct a \(\left(\mathcal{A}_{g \triangleright \theta},\mathcal{A}_\theta\right)\)-equivalence bimodule \(\mathcal{E}(g,\theta)\) as follows:
\begin{enumerate}
\item if \(g_{21} = 0\), so that \(\mathcal{A}_{g \triangleright \theta} = \mathcal{A}_{\theta}\) and \(g_{22} \neq 0\), let \(\mathcal{E}(g,\theta) \coloneqq \mathcal{A}_{\theta}\) be the trivial self-Morita equivalence bimodule over \(\mathcal{A}_\theta\);
\item if \(g_{22} \neq 0\), let \(\mathcal{E}(g,\theta) \coloneqq \mathcal{S}(\mathbf{R}) \otimes \mathbf{C}[\mathbf{Z}_{g_{21}}]\) with the unique right \(\mathcal{A}_{\theta}\)-module structure satisfying
\begin{align*}
\forall f \in \mathcal{E}(g,\theta), \, \forall (x,k) \in \mathbf{R} \times \mathbf{Z}_{g_{21}}, \quad (f \cdot U_\theta)(x,k) &= \exp\mleft(2\pi\ensuremath{\mathrm{i}}{}\left(x-\tfrac{k g_{22}}{g_{21}}\right)\mright) f(x,k),\\
\forall f \in \mathcal{E}(g,\theta), \, \forall (x,k) \in \mathbf{R} \times \mathbf{Z}_{g_{21}}, \quad (f \cdot V_\theta)(x,k) &= f\mleft(x-\tfrac{g_{21}\theta+g_{22}}{g_{21}},k-1\mright),
\end{align*}
and the unique left \(\mathcal{A}_{g\triangleright\theta}\)-module structure satisfying
\begin{align*}
\forall f \in \mathcal{E}(g,\theta), \, \forall (x,k) \in \mathbf{R} \times \mathbf{Z}_{g_{21}}, \quad (U_{g\triangleright\theta} \cdot f)(x,k) &= \exp\mleft(2\pi\ensuremath{\mathrm{i}}{}\left(\tfrac{x}{g_{21}\theta+g_{22}} - \tfrac{k}{g_{21}}\right)\mright) f(x,k),\\
\forall f \in \mathcal{E}(g,\theta), \, \forall (x,k) \in \mathbf{R} \times \mathbf{Z}_{g_{21}}, \quad (V_{g\triangleright\theta} \cdot f)(x,k) &= f\mleft(x-\tfrac{1}{g_{21}},k-g_{11}\mright).
\end{align*}
\end{enumerate}
We call \(\mathcal{E}(g,\theta)\) the \emph{basic Heisenberg module} over \(\mathcal{A}_\theta\) with \emph{rank} \(g_{21}\theta+g_{22}\) and \emph{degree} \(g_{21}\).
\end{theoremdefinition}
Now, given \(\theta \in \mathbf{R} \setminus \mathbf{Q}\) and \(g \in \SL(2,\mathbf{Z})\), we see that \(g \triangleright \theta = \theta\) if and only if \(\theta\) is a quadratic irrationality and \(g \in \SL(2,\mathbf{Z})_\theta\). Thus, in light of Proposition~\ref{stabthm}, if \(\theta\) is a real quadratic irrationality, then the family of basic Heisenberg bimodules over \(\mathcal{A}_\theta\) contains a canonical family of self-Morita equivalence bimodules, which we can now assemble into a principal \(\mathcal{O}{}(\Unit(1))\)-module \(\ast\)-algebra \(P\) with \(\coinv{\mathcal{O}{}(\Unit(1))}{P} = \mathcal{A}_\theta\).
\begin{theorem}[Schwarz~\cite{Schwarz98}*{\S 3}, Dieng--Schwarz~\cite{DiengSchwarz}, Polishchuk--Schwarz~\cite{PolishchukSchwarz}*{\S 1.3}, Po\-li\-shchuk \cite{Polishchuk}*{\S 2.2}, Vlasenko~\cite{Vlasenko}*{Thm.\ 6.1}]\label{quadprincipalthm} Let \(\theta \in \mathbf{R} \setminus \mathbf{Q}\) be a quadratic irrationality with norm-positive fundamental unit \(\epsilon\), and let \(\Phi : \langle \epsilon \rangle \times \set{\pm 1} \to \SL(2,\mathbf{Z})_\theta\) be the isomorphism of Proposition~\ref{stabthm}; hence, given \(m \in \mathbf{Z}\), let
\[
a_m \coloneqq \Phi(\epsilon^m)_{11}, \quad b_m \coloneqq \Phi(\epsilon^m)_{12}, \quad c_m \coloneqq \Phi(\epsilon^m)_{21}, \quad d_m \coloneqq \Phi(\epsilon^m)_{22}.
\]
For each \(n \in \mathbf{Z}\), let \(P_n \coloneqq \mathcal{E}(\Phi(\epsilon^n),\theta)\), and define a \(\mathcal{O}{}(\Unit(1))\)-comodule \(\mathcal{A}_\theta\)-bimodule
\[
P \coloneqq \bigoplus_{n \in \mathbf{Z}} P_n,
\]
where the corepresentation of \(\mathcal{O}{}(\Unit(1))\) is induced by the obvious \(\mathbf{Z}\)-grading. Then \(P\) defines a principal \(\mathcal{O}{}(\Unit(1))\)-comodule \(\ast\)-algebra over \(\coinv{\mathcal{O}{}(\Unit(1))}{P} = P_0 = \mathcal{A}_\theta\) when endowed with the \(\ast\)-operation defined by
\[
\forall m \in \mathbf{Z}, \, \forall f \in P_m, \, \forall (x,k) \in \mathbf{R} \times \mathbf{Z}_{c_m}, \quad f^\ast(x,k) \coloneqq \overline{f(\epsilon^m x,-a_mk)}
\]
and the multiplication \(P \otimes_\mathbf{C} P \to P\) defined as follows:
\begin{enumerate}
\item the restrictions of the multiplication to \(P_0 \otimes_{\mathbf{C}} P = \mathcal{A}_\theta \otimes_{\mathbf{C}} P\) and \(P \otimes_{\mathbf{C}} P_0 = P \otimes_{\mathbf{C}} \mathcal{A}_\theta\) are given by the left and right \(\mathcal{A}_\theta\)-module structures on \(P\), respectively;
\item for all non-zero \(m \in \mathbf{Z}\), for all \(f \in P_{-m}\), and for all \(g \in P_m\),
\[
f \cdot g \coloneqq \sum_{(n_1,n_2) \in \mathbf{Z}^2} U^{n_1} V^{n_2} \sum_{k \in \mathbf{Z}_{c_m}} \int_{\mathbf{R}} (V_\theta^{-n_2}U_\theta^{-n_1} \cdot f)\mleft(\tfrac{x}{\epsilon^m},k\mright) g\mleft(x,-a_m k\mright) \,\mathrm{d}x;
\]
\item for all non-zero \(m,n \in \mathbf{Z}\) with \(m+n \neq 0\), for all \(f \in P_m\), and for all \(g \in P_n\),
\begin{multline*}
\forall (x,k) \in \mathbf{R} \times \mathbf{Z}_{c_{m+n}},\\
(f\cdot g)(x,k) \coloneqq \sum_{j\in\mathbf{Z}} f\mleft(\tfrac{x}{\epsilon^{n}} + \epsilon^m\left(\tfrac{d_{m+n}k}{c_{m+n}}-\tfrac{j}{c_m}\right),a_{m}d_{m+n}k-j\mright) \cdot g\mleft(x-\left(\tfrac{d_{m+n}k}{c_{m+n}}-\tfrac{j}{c_n}\right),a_n j \mright).
\end{multline*}
\end{enumerate}
\end{theorem}
\begin{proof}
First, observe that the results of Schwarz~\cite{Schwarz98}*{\S 3}, Dieng--Schwarz~\cite{DiengSchwarz}, and Poli\-shchuk--Schwarz~\cite{PolishchukSchwarz}*{\S 1.3} specialise to yield the multiplication \(P \otimes P \to P\) that makes \(P\) into a unital \(\mathbf{C}\)-algebra. Next, the results of Po\-li\-shchuk~\cite{Polishchuk}*{\S 2.2} specialise to show that \(\ast : P \to P\) makes \(P\) into a \(B\)-\(\ast\)-bimodule, while results of Vlasenko~\cite{Vlasenko}*{Thm.\ 6.1}, with our notation and conventions, specialise to show that
\[
\forall m,n \in \mathbf{Z}, \, \forall p_1,q_1 \in P_m, \, \forall p_2,q_2 \in P_n, \quad (p_1 \cdot p_2) \cdot (q_1 \cdot q_2)^\ast = (p_1 \cdot (p_2 \cdot q_2^\ast)) \cdot q_1^\ast,
\]
so that, by associativity of the multiplication on \(P\),
\begin{equation}\label{vlasenkoeq}
\forall m,n \in \mathbf{Z}, \, \forall p_1,q_1 \in P_m, \, \forall p_2,q_2 \in P_n, \quad (p_1 \cdot p_2) \cdot (q_2^\ast \cdot q_1).
\end{equation}
Finally, by observations of Dieng--Schwarz~\cite{DiengSchwarz}*{\S 2} and Polishchuk--Schwarz~\cite{PolishchukSchwarz}*{Proof of Prop.\ 1.2}, for all \(m,n \in \mathbf{Z}\), multiplication in \(P\) yields an isomorphism \(P_m \otimes_B P_n \xrightarrow{\sim} P_{m+n}\). On the one hand, for all \(m,n \in \mathbf{Z}\), \(q_1 \in P_m\), and \(q_2 \in P_n\), since \(P_{m+n} = P_m \cdot P_n\) and since the multiplication on \(P\) restricts to an isomorphism of \(B\)-bimodules \(P_{m+n} \otimes_B P_{-m-n} \xrightarrow{\sim} B\), Equation~\ref{vlasenkoeq} implies that \((q_1 \cdot q_2)^\ast = q_2^\ast \cdot q_1\), so that \(\ast : P \to P\) makes \(P\) into a unital \(\ast\)-algebra. On the other hand, since \(P_m \cdot P_n = P_{m+n}\) for all \(m,n \in \mathbf{Z}\), it follows that the \(\ast\)-algebra \(P\) defines a principal \(\mathcal{O}{}(\Unit(1))\)-comodule \(\ast\)-algebra~\cite{ADL}*{Thm.\ 4.4}.
\end{proof}
Now, if \(B\) is a \(\mathcal{O}{}(\Unit(1))\)-module \(\ast\)-algebra, then the trivial principal \(\mathcal{O}{}(\Unit(1))\)-comodule \(\ast\)-algebra \(B \rtimes \mathcal{O}{}(\Unit(1))\) admits the left \(\mathcal{O}{}(\Unit(1))\)-covariant \(\ast\)-homomorphism
\[
(h \mapsto h \otimes 1_P) : \mathcal{O}{}(\Unit(1)) \to B \rtimes \mathcal{O}{}(\Unit(1)),
\]
which one views as a global trivialisation of \(B \rtimes \mathcal{O}{}(\Unit(1))\). Indeed, more generally, a principal \(\mathcal{O}{}(\Unit(1))\)-comodule \(\ast\)-algebra \(P\) is left \(\mathcal{O}{}(\Unit(1))\)-covariant \(\ast\)-isomorphic to such a trivial principal \(\mathcal{O}{}(\Unit(1))\)-comodule \(\ast\)-algebra if and only if it admits a left \(\mathcal{O}{}(\Unit(1))\)-covariant \(\ast\)-homomorphism \(\mathcal{O}{}(\Unit(1)) : \to P\). We now show that Theorem~\ref{quadprincipalthm} yields non-trivial principal \(\mathcal{O}{}(\Unit(1))\)-comodule \(\ast\)-algebra in this technically precise sense.
\begin{proposition}
Let \(\theta \in \mathbf{R} \setminus \mathbf{Q}\) be a quadratic irrationality, and let \(P\) be the resulting principal \(\mathcal{O}{}(\Unit(1))\)-comodule \(\ast\)-algebra of Theorem~\ref{quadprincipalthm}. There does not exist a left \(H\)-covariant unital \(\ast\)-homomorphism \(\mathcal{O}{}(\Unit(1)) \to P\).
\end{proposition}
\begin{proof}
Let \(\epsilon\) be the norm-positive fundamental unit of \(\mathbf{Q}[\theta]\) induced by the real quadratic irrationality \(\theta\), and let \(\Phi : \langle \epsilon \rangle \times \set{\pm 1} \to \SL(2,\mathbf{Z})_\theta\) is the isomorphism of Proposition~\ref{stabthm}; note, in particular, that \(\Phi_{21}(\epsilon)\epsilon + \Phi_{22}(\epsilon) = \epsilon \in \mathbf{R} \setminus \mathbf{Q}\), where \(\Phi_{21}(\epsilon),\Phi_{22}(\epsilon)\in\mathbf{Z}\), so that, necessarily, \(\Phi_{21}(\epsilon) \neq 0\). Thus, by a result of Connes~\cite{Connes80}*{Thm.\ 7}, it follows \(P_1 \coloneqq \mathcal{E}(\Phi(\epsilon),\theta)\) is not free as a right \(\mathcal{A}_\theta\)-module. But now, if \(\psi : \mathcal{O}{}(\Unit(1)) \to P\) were a left \(H\)-covariant \(\ast\)-homomor\-phism, then \(P_1\) would be freely generated as a right \(\mathcal{A}_\theta\)-module by the unitary element \(\psi((z \mapsto z)) \in P_1 \cap \Unit(P)\).
\end{proof}
\begin{remark}
In fact, this shows that \(P\) is not cleft, i.e., that there does not exist a unital convolution-invertible left \(\mathcal{O}{}(\Unit(1))\)-covariant map \(\mathcal{O}{}(\Unit(1)) \to P\). Indeed, \(P_1\) is the quantum vector bundle (in the sense of Brzezi\'{n}ski--Majid~\cite{BrM}*{Def.\ A.3}) associated to \(P\) induced by the standard irreducible corepresentation of \(\mathcal{O}{}(\Unit(1))\), so that \(P_1\) would be free as a right \(\mathcal{A}_\theta\)-module if \(P\) were cleft~\cite{BrM}*{Appx.\ A}.
\end{remark}
From now on, we will be concerned with the \(\mathcal{O}{}(\Unit(1))\)-gauge theory of the canonical non-trivial principal left \(\mathcal{O}{}(\Unit(1))\)-comodule \(\ast\)-algebra of Theorem~\ref{quadprincipalthm} induced by a real quadratic irrationality.
\subsection{Constant curvature connections as \texorpdfstring{\(q\)}{q}-monopoles}
From now on, let \(\theta \in \mathbf{R} \setminus \mathbf{Q}\) be a real quadratic irrationality, let \(\epsilon \in \mathbf{Q}[\theta]^\times \cap (1,\infty)\) be its norm-positive fundamental unit, let \(B \coloneqq \mathcal{A}_\theta\) be the smooth noncommutative \(2\)-torus with deformation parameter \(\theta\), and let \(P\) be the non-trivial principal \(\mathcal{O}{}(\Unit(1))\)-comodule \(\ast\)-algebra with \(\coinv{\mathcal{O}{}(\Unit(1))}{P} = \mathcal{A}_\theta\) of Theorem~\ref{quadprincipalthm}. We shall now show that Connes's constant curvature connections~\cite{Connes80} on the isotypical subspaces of \(P\) \emph{qua} basic Heisenberg modules on \(\mathcal{A}_\theta\) combine to yield a \(\epsilon^{-1}\)-monopole closely analogous to Brzezi\'{n}ski--Majid's \(q\)-monopole on the \(q\)-deformed complex Hopf fibration~\cite{BrM}*{\S 5.2}. In fact, we shall see that Polishchuk--Schwarz's holomorphic structures~\cite{PolishchukSchwarz} exhaust the \(\mathcal{O}{}(\Unit(1))\)-gauge theory of \(P\) in a manner that demonstrates the necessity of considering \emph{all} principal \textsc{fodc}{} on \(P\) compatible with given vertical and horizontal calculi.
Before continuing, let us fix notation. On the one hand, define a left \(\mathcal{O}{}(\Unit(1))\)-covariant unital \(\mathbf{C}\)-algebra automorphism \(\sigma : P \to P\) by
\begin{equation}
\forall m \in \mathbf{Z}, \, \forall p \in P_m, \quad \sigma(p) \coloneqq \epsilon^{-m}p.
\end{equation}
Although neither \(\sigma\) nor \(\sigma^2\) is a \(\ast\)-automorphism, they do respectively satisfy
\[
(\sigma \circ \ast)^2 = \id_P, \quad (\sigma^2 \circ \ast)^2 = \id_P.
\]
On the other hand, given \(m \in \mathbf{Z}\), define \(a_m,b_m,c_m,d_m \in \mathbf{Z}\) by
\begin{equation}
\begin{pmatrix}
a_m & b_m \\ c_m & d_m
\end{pmatrix}
\coloneqq \Phi(\epsilon^m),
\end{equation}
where \(\Phi : \langle \epsilon \rangle \times \set{\pm 1} \xrightarrow{\sim} \SL(2,\mathbf{Z})_\theta\) is the isomorphism of Proposition~\ref{stabthm}. Thus, by \eqref{stabthmeq},
\[
\forall m \in \mathbf{Z}, \quad c_m \epsilon + d_m = \epsilon^m,
\]
so that \(\set{m \in \mathbf{Z} \given c_m = 0} = \set{0}\). Moreover, by an observation of Polishchuk--Schwarz~\cite{PolishchukSchwarz}*{Eq.\ 1.2}, the map \((m \mapsto c_m) : \mathbf{Z} \to \mathbf{Z}\) satisfies
\begin{equation}
\forall m, n \in \mathbf{Z},\quad c_{m+n} = c_m\epsilon^{-n} + \epsilon^m c_n
\end{equation}
Let us now recall the construction of the standard \(\ast\)-differential calculus on the smooth noncommutative \(2\)-torus \(B \coloneqq \mathcal{A}_\theta\), which consists of rapidly decaying Laurent series in unitary generators \(U_\theta\) and \(V_\theta\) satisfying \(V_\theta U_\theta = \epsilon^{2\pi\ensuremath{\mathrm{i}}{}\theta}U_\theta V_\theta\). Recall that \(B\) admits a commuting pair of \(\ast\)-derivations \(\delta_1,\delta_2 : B \to B\) uniquely determined by
\[
\delta_1(U) \coloneqq 2\pi U, \quad \delta_1(V) \coloneqq 0, \quad \delta_2(U) \coloneqq 0, \quad \delta_2(V) \coloneqq 2\pi V.
\]
Hence, the canonical \(\ast\)-differential calculus \((\Omega_B,\dt{B})\) on \(B\) is given by
\[
\forall k \in \mathbf{Z}_{\geq 0}, \quad \Omega_B^k \coloneqq \begin{cases} B &\text{if \(k=0,2\)},\\ B^{\oplus 2} &\text{if \(k=1\),}\\ 0 &\text{else,} \end{cases}
\]
where \(B\) is viewed as the trivial \(B\)-\(\ast\)-bimodule, the non-trivial product \(\wedge : \Omega^1_B \otimes_B \Omega^1_B \to \Omega^2_B\) is given by
\[
\forall (b_1,b_2), (c_1,c_2) \in \Omega^1_B, \quad (b_1,b_2) \wedge (c_1,c_2) \coloneqq b_2 c_1 - b_1 c_2,
\]
and the exterior derivative \(\dt{B}\) given by
\begin{gather*}
\forall b \in \Omega^0_B, \quad \dt{B}(b) \coloneqq (\delta_1(b),\delta_2(b)),\\
\forall (b_1,b_2) \in \Omega^1_B, \quad \dt{B}(b_1,b_2) \coloneqq \delta_2(b_1)-\delta_1(b_2).
\end{gather*}
For notational convenience, let
\[
\ensuremath{\mathrm{d}}{\tau}^1 \coloneqq \frac{1}{2\pi\ensuremath{\mathrm{i}}{}} U_\theta^\ast \dt{B}(U_\theta) = (-\ensuremath{\mathrm{i}},0), \quad \ensuremath{\mathrm{d}}{\tau}^2 \coloneqq \frac{1}{2\pi\ensuremath{\mathrm{i}}{}} V_\theta^\ast \dt{B}(V_\theta) = (0,-\ensuremath{\mathrm{i}}), \quad \vol_B \coloneqq \ensuremath{\mathrm{d}}{\tau}^1 \wedge \ensuremath{\mathrm{d}}{\tau}^2 = 1,
\]
so that \(\ensuremath{\mathrm{d}}{\tau}^1\) and \(\ensuremath{\mathrm{d}}{\tau^2}\) are central in the graded \(\ast\)-algebra \(\Omega_B\) and skew-adjoint, \(\vol_B\) is central in \(\Omega_B\) and self-adjoint, \(\set{\ensuremath{\mathrm{d}}{\tau}^1,\ensuremath{\mathrm{d}}{\tau}^2}\) is a basis for \(\Omega^1_B\) as both a left and right \(B\)-module, and \(\set{\vol_B}\) is a basis for \(\Omega^2_B\) as both a left and right \(B\)-module. In particular, can write
\begin{gather*}
\forall b \in B, \quad \dt{B}(b) = \ensuremath{\mathrm{i}}{}\partial_1(b)\ensuremath{\mathrm{d}}{\tau}^1 +\ensuremath{\mathrm{i}}{}\partial_2(b)\ensuremath{\mathrm{d}}{\tau}^2,\\
\forall (b_1,b_2) \in B^{\oplus 2}, \quad \dt{B}(b_1 \ensuremath{\mathrm{d}}{\tau}^1 + b_2 \ensuremath{\mathrm{d}}{\tau}^2) = -\ensuremath{\mathrm{i}}{}(\partial_2(b_1)-\partial_1(b_2))\vol_B
\end{gather*}
Now, suppose that we have completed \((\Omega_B,\dt{B})\) to a second-order horizontal calculus \((\Omega_B,\dt{B};\Omega_{P,\mathrm{hor}})\) on the principal left \(H\)-comodule \(\ast\)-algebra \(P\). Since \(P\) is principal and since \(\Omega_{P,\mathrm{hor}}^1\) and \(\Omega_{P,\mathrm{hor}}^2\) are projectable horizontal lifts for \(\Omega^1_B\) and \(\Omega^2_B\), respectively, it follows by the generalised Hopf module lemma~\cite{BeM}*{Lemma 5.29} that \(\Omega^1_{P,\mathrm{hor}}\) and \(\Omega^2_{P,\mathrm{hor}}\) are free as left \(P\)-modules with respective bases \(\set{\ensuremath{\mathrm{d}}{\tau}^1,\ensuremath{\mathrm{d}}{\tau}^2}\) and \(\set{\vol_B}\). Thus, for every prolongable gauge potential \(\nabla\) on \(P\) with respect to \((\Omega_B,\dt{B};\Omega_{P,\mathrm{hor}})\), there necessarily exist unique left \(H\)-covariant maps \(\partial_1,\partial_2 : P \to P\) extending \(\delta_1,\delta_2 : B \to B\), respectively, such that
\begin{equation}\label{connectioneq1}
\forall p \in P, \quad \nabla(p) = \ensuremath{\mathrm{i}}{}\partial_1(p)\ensuremath{\mathrm{d}}{\tau}^1+\ensuremath{\mathrm{i}}{}\partial_2(p)\ensuremath{\mathrm{d}}{\tau}^2;
\end{equation}
in this case, it follows that the canonical prolongation \(\pr{\nabla}\) of \(\nabla\) is given by
\begin{equation}\label{connectioneq2}
\forall (p_1,p_2) \in P^{\oplus 2}, \quad \nabla(p_1\ensuremath{\mathrm{d}}{\tau}^1+p_2\ensuremath{\mathrm{d}}{\tau}^2) = -\ensuremath{\mathrm{i}}{}(\partial_2(p_1)-\partial_1(p_2)) \vol_B,
\end{equation}
and hence that the field strength \(\mathbf{F}[\nabla]\) of \(\nabla\) is given by
\begin{equation}\label{connectioneq3}
\forall p \in P, \quad \mathbf{F}[\nabla](p) = \ensuremath{\mathrm{i}}{}[\partial_1,\partial_2](p)\vol_B.
\end{equation}
As it turns out, we have the following canonical candidate for \(\set{\partial_1,\partial_2}\) due originally to Connes~\cite{Connes80}*{Thm.\ 7}; our immediate goal, then, will be to reverse-engineer \(\Omega_{P,\mathrm{hor}}\) so that \eqref{connectioneq1} defines a prolongable gauge potential \(\nabla\) on \(P\) with respect to \((\Omega_B,\dt{B};\Omega_{P,\mathrm{hor}})\).
\begin{theorem}[Connes~\cite{Connes80}*{Thm.\ 7}, Polishchuk--Schwarz~\cite{PolishchukSchwarz}*{Propp.\ 2.1, 2.2}]\label{cpsthm}
Define \(\mathcal{O}{}(\Unit(1))\)-covariant \(\mathbf{C}\)-linear extensions \(\partial_1,\partial_2 : P \to P\) of \(\delta_1\) and \(\delta_2\), respectively, by
\begin{gather}
\forall m \in \mathbf{Z} \setminus \set{0}, \, \forall f \in P_m, \, \forall (x,k) \in \mathbf{R} \times \mathbf{Z}_{c_m}, \quad \partial_1f(x,k) \coloneqq -\ensuremath{\mathrm{i}}{}\tfrac{\partial}{\partial t}{f}(x,k), \quad \\
\forall m \in \mathbf{Z} \setminus \set{0}, \, \forall f \in P_m, \, \forall (x,k) \in \mathbf{R} \times \mathbf{Z}_{c_m}, \quad \partial_2f(x,k) \coloneqq 2\pi \epsilon^{-n}c_n x f(x,\alpha).
\end{gather}
where \(\tfrac{\partial}{\partial t}{}\) denotes differentiation on \(\mathcal{S}(\mathbf{R}) \otimes \mathbf{C}[\mathbf{Z}_{c_m}]\) with respect to the continuous variable.
Then the maps \(\partial_1\) and \(\partial_2\) satisfy the following relations:
\begin{gather}
\forall j \in \set{1,2}, \forall p, \p{p} \in P, \quad \partial_j(p\p{p}) = \partial_j(p)\cdot \sigma(\p{p}) + p \cdot \partial_j(\p{p}),\label{twist1}\\
\forall j \in \set{1,2}, \, \forall p \in P, \quad \partial_j(p^\ast) = -\sigma(\partial_j(p)^\ast). \label{twist2}
\end{gather}
Moreover, the commutator \([\partial_1,\partial_2] \coloneqq \partial_1 \circ \partial_2 - \partial_2 \circ \partial_1\) satisfies
\begin{equation}\label{twist3}
\forall m \in \mathbf{Z}, \, \forall f \in P_m, \quad [\partial_1,\partial_2](f) = -2\pi\ensuremath{\mathrm{i}}{}\epsilon^{-m}c_m f.
\end{equation}
\end{theorem}
\begin{remark}[Polishchuk~\cite{PolishchukJGP}]
Let \(\tau \in \set{z \in \mathbf{C} \given \Im z < 0}\); let \(g \coloneqq \Phi(\epsilon)\). Then \[B_g(\theta,\tau) \coloneqq\ker(\partial_1 + \tau \partial_2)\] is a unital (non-\(\ast\)-closed) subalgebra of \(P\), which can be interpreted as the homogeneous coordinate ring of \(\mathcal{A}_\theta\) with the complex structure \(\tau\) \emph{qua} noncommutative projective variety.
\end{remark}
We can now use \eqref{twist1} and \eqref{twist2} to reverse-engineer a canonical second-order horizontal calculus on \(P\) of the form \((\Omega_B,\dt{B};\Omega_{P,\mathrm{hor}})\) encoding the canonical twisted derivations \(\partial_1\) and \(\partial_2\) on \(P\) as a prolongable gauge potential. This will require the following definition.
\begin{definition}
Let \(H\) be a Hopf \(\ast\)-algebra, let \(A\) be a left \(H\)-comodule \(\ast\)-algebra, and let \(E\) be a left \(H\)-covariant \(A\)-\(\ast\)-bimodule. Let \(\phi : A \to A\) be a left \(H\)-covariant unital \(\mathbf{C}\)-algebra automorphism, such that \((\phi \circ \ast)^2 = \id_A\). We define the left \(H\)-covariant \(A\)-\(\ast\)-bimodule \(E_\phi\) to be the left \(H\)-covariant left \(A\)-module \(E\) together with right \(A\)-module structure \(\cdot_\phi : E \otimes_\mathbf{C} A \to E\) and \(\ast\)-structure \(\ast_\phi : E \to E\) defined, respectively, by
\begin{gather*}
\forall e \in E, \, \forall a \in A, \quad e \cdot_\phi a \coloneqq e \cdot \phi(a),\\
\forall e \in E, \quad e^{\ast_\phi} \coloneqq \phi(e^\ast).
\end{gather*}
\end{definition}
\begin{proposition}\label{constantcurvatureprop}
Define a \(\mathbf{Z}_{\geq 0}\)-graded left \(\mathcal{O}{}(\Unit(1))\)-comodule \(P\)-\(\ast\)-bimodule \(\Omega_{P,\mathrm{hor}}\) by
\[
\forall k \in \mathbf{Z}_{\geq 0}, \quad \Omega^k_{P,\mathrm{hor}} \coloneqq \begin{cases} P &\text{if \(k = 0\),} \\ (P_\sigma)^{\oplus 2} &\text{if \(k = 1\),} \\ P_{\sigma^2} & \text{if \(k=2\),}\\0 &\text{else;}\end{cases}
\]
let \(\iota : \Omega_B \hookrightarrow \Omega_{P,\mathrm{hor}}\) be the map induced by the inclusion \(B \hookrightarrow P\). Then \((\Omega_B,\dt{B};\Omega_{P,\mathrm{hor}},\iota)\) defines a second-order horizontal calculus on \(P\) when \(\Omega_{P,\mathrm{hor}}\) is endowed with the extension of the \(P\)-bimodule structure to a multiplication \(\Omega_{P,\mathrm{hor}} \otimes \Omega_{P,\mathrm{hor}} \to \Omega_{P,\mathrm{hor}}\) given by
\begin{multline*}
\forall (p_1,p_2), (\p{p}_1,\p{p}_2) \in \Omega^1_{P,\mathrm{hor}}, \\ (p_1 \cdot \ensuremath{\mathrm{d}}{\tau}^1+p_2 \cdot \ensuremath{\mathrm{d}}{\tau}^2) \wedge (\p{p}_1 \cdot \ensuremath{\mathrm{d}}{\tau}^1 + \p{p}_2 \cdot \ensuremath{\mathrm{d}}{\tau}^2) \coloneqq \left(p_1 \sigma(\p{p}_2) - p_2 \sigma(\p{p}_1)\right) \cdot \vol_B.
\end{multline*}
Moreover, the left \(\mathcal{O}{}(\Unit(1))\)-covariant map \(\nabla_0 : P \to \Omega^1_{P,\mathrm{hor}}\) given by
\begin{equation}
\forall p \in P, \quad \nabla_0(p) = \ensuremath{\mathrm{i}}{}\partial_1(p)\cdot\ensuremath{\mathrm{d}}{\tau}^1+\ensuremath{\mathrm{i}}{}\partial_2(p)\cdot\ensuremath{\mathrm{d}}{\tau}^2,
\end{equation}
where \(\partial_1,\partial_2 : P \to P\) are the maps constructed in Theorem~\ref{cpsthm}, defines a prolongable gauge potential \(\nabla_0\) on \(P\) with respect to \((\Omega_B,\dt{B};\Omega_{P,\mathrm{hor}})\) with canonical prolongation \(\pr{\nabla}_0\) given by
\begin{equation}
\forall (p_1,p_2) \in P^{\oplus 2}, \quad \pr{\nabla}_0(p_1\cdot\ensuremath{\mathrm{d}}{\tau}^1+p_2\cdot\ensuremath{\mathrm{d}}{\tau}^2) = -\ensuremath{\mathrm{i}}{}(\partial_2(p_1)-\partial_1(p_2)) \cdot\vol_B,
\end{equation}
and field strength \(\mathbf{F}[\nabla_0]\) given by
\begin{equation}
\forall p \in P, \quad \mathbf{F}[\nabla_0](p) = \left[2\pi\frac{\epsilon c_1}{\epsilon^2-1}\vol_B,p\right].
\end{equation}
\end{proposition}
\begin{proof}
Let us first show that \((\Omega_B,\dt{B};\Omega_{P,\mathrm{hor}},\iota)\) correctly defines a second-order horizontal calculus on \(P\); since \(\rest{\sigma}{B} = \id_B\), the only non-trivial point is that the \(\ast\)-structure on \(\Omega_{P,\mathrm{hor}}\) is graded-antimultiplicative with respect to the multiplication \(\Omega^1_{P,\mathrm{hor}} \times \Omega^1_{P,\mathrm{hor}} \to \Omega^2_{P,\mathrm{hor}}\). Indeed, for all \((p_1,p_2),(\p{p}_1,\p{p}_2) \in P^{\oplus 2}\),
\begin{align*}
&\left((p_1 \cdot \ensuremath{\mathrm{d}}{\tau}^1 + p_2 \cdot \ensuremath{\mathrm{d}}{\tau}^2) \wedge (\p{p}_1 \cdot \ensuremath{\mathrm{d}}{\tau}^1 + \p{p}_2 \cdot \ensuremath{\mathrm{d}}{\tau}^2) \right)^\ast \\
&\quad\quad = \left((p_2\sigma(\p{p}_1)-p_1\sigma(\p{p}_2))\cdot\vol_B\right)^\ast\\
&\quad\quad = \vol_B \cdot (\sigma(\p{p}_1)^\ast p_2^\ast - \sigma(\p{p}_2)^\ast p_1^\ast)\\
&\quad\quad = \left(\sigma^2(\sigma(\p{p}_1)^\ast) \sigma^2(p_2^\ast)-\sigma^2(\sigma(\p{p}_2)^\ast)\sigma^2(p_1^\ast)\right) \cdot \vol_B\\
&\quad\quad = -\left(\sigma((\p{p}_1)^\ast) \cdot \ensuremath{\mathrm{d}}{\tau}^1 + \sigma((\p{p}_2)^\ast) \cdot \ensuremath{\mathrm{d}}{\tau}^2\right) \wedge \left(\sigma(p_1^\ast) \cdot\ensuremath{\mathrm{d}}{\tau}^1 + \sigma(p_2^\ast) \cdot \ensuremath{\mathrm{d}}{\tau}^2\right)\\
&\quad\quad = -\left(\ensuremath{\mathrm{d}}{\tau}^1 \cdot (\p{p}_1)^\ast + \ensuremath{\mathrm{d}}{\tau}^2 \cdot (\p{p}_2)^\ast\right) \wedge \left( \ensuremath{\mathrm{d}}{\tau}^1 \cdot p_1^\ast + \ensuremath{\mathrm{d}}{\tau}^2 \cdot p_2^\ast\right)\\
&\quad\quad = -(\p{p}_1 \cdot \ensuremath{\mathrm{d}}{\tau}^1 + \p{p}_2 \cdot \ensuremath{\mathrm{d}}{\tau}^2)^\ast \wedge (p_1 \cdot \ensuremath{\mathrm{d}}{\tau}^1 + p_2 \cdot \ensuremath{\mathrm{d}}{\tau}^2)^\ast.
\end{align*}
Let us now show that the map \(\nabla_0\) defines a prolongable gauge potential on \(P\) with respect to \((\Omega_B,\dt{B};\Omega_{P,\mathrm{hor}})\) with the correct canonical prolongation and field strength. Before continuing, note that \(\delta_1\), \(\delta_2\), and \(\sigma\) are all block-diagonal with respect to the decomposition \(P = \bigoplus_{m\in\mathbf{Z}}P_m\) and that \(\sigma\) acts as multiplication by a scalar on each isotypical subspace \(P_m\), so that \(\sigma \circ \delta_1 = \delta_1 \circ \sigma\) and \(\sigma \circ \delta_2 = \delta_2 \circ \sigma\).
First, equations \eqref{twist1} and \eqref{twist2} together with the construction of \(\Omega_{P,\mathrm{hor}}\) imply that the left \(\mathcal{O}{}(\Unit(1))\)-covariant map \(\nabla_0 : P \to \Omega^1_{P,\mathrm{hor}}\) is a \(\ast\)-derivation, while the fact that \(\rest{\partial_j}{B} = \delta_j\) for \(j=1,2\) implies that \(\rest{\nabla_0}{B} = \dt{B}\). Hence, \(\nabla_0\) defines a gauge potential with respect to the first-order horizontal calculus \((\Omega^1_B,\dt{B};\Omega^1_{P,\mathrm{hor}})\) induced by \((\Omega_B,\dt{B};\Omega_{P,\mathrm{hor}})\).
Next, for all \(p,q \in P\) and \(b \in B\),
\begin{align*}
&\nabla_0(p) \wedge \dt{B}(b) \cdot q - p \cdot \dt{B}(b) \wedge \nabla_0(q) \\
&\quad\quad = \ensuremath{\mathrm{i}}{}\left(\partial_1(p)\cdot\ensuremath{\mathrm{d}}{\tau}^1 + \partial_2(p)\cdot\ensuremath{\mathrm{d}}{\tau}^2\right) \wedge \ensuremath{\mathrm{i}}{}\left(\delta_1(b) \cdot \ensuremath{\mathrm{d}}{\tau}^1+\delta_2(b)\cdot\ensuremath{\mathrm{d}}{\tau}^2\right) \cdot q\\
&\quad\quad\quad\quad - p \cdot \ensuremath{\mathrm{i}}{}\left(\delta_1(b) \cdot \ensuremath{\mathrm{d}}{\tau}^1+\delta_2(b)\cdot\ensuremath{\mathrm{d}}{\tau}^2\right) \wedge \ensuremath{\mathrm{i}}{}\left(\partial_1(q)\cdot\ensuremath{\mathrm{d}}{\tau}^1 + \partial_2(q)\cdot\ensuremath{\mathrm{d}}{\tau}^2\right)\\
&\quad\quad = -\left(\partial_1(p) \delta_2(b) - \delta_2(p)\delta_1(b)\right)\sigma^2(q)\cdot \vol_B + p \left(\delta_1(b)\sigma(\partial_2(q))-\delta_2(b)\sigma(\partial_1(q))\right) \cdot \vol_B\\
&\quad\quad = \left(\partial_2(p\delta_1(b)\sigma(q))-\partial_1(p\delta_2(b)\sigma(q))\right) \cdot \vol_B,
\end{align*}
where
\[
p \cdot \dt{B}(b) \cdot q = p \cdot \ensuremath{\mathrm{i}}{}\left(\delta_1(b) \cdot \ensuremath{\mathrm{d}}{\tau}^1+\delta_2(b)\cdot\ensuremath{\mathrm{d}}{\tau}^2\right) \cdot q = \ensuremath{\mathrm{i}}{}p \delta_1(b)\sigma(q) \cdot \ensuremath{\mathrm{d}}{\tau}^1 + \ensuremath{\mathrm{i}}{} p\delta_2(b)\sigma(q) \cdot \ensuremath{\mathrm{d}}{\tau}^2.
\]
Hence, the gauge potential \(\nabla_0\) is prolongable with canonical prolongation \(\pr{\nabla}_0\) given by
\[
\forall (p_1,p_2) \in P^{\oplus 2}, \quad \pr{\nabla}_0(p_1 \cdot \ensuremath{\mathrm{d}}{\tau}^1+ p_2 \cdot \ensuremath{\mathrm{d}}{\tau}^2) = -\ensuremath{\mathrm{i}}{}\left(\partial_2(p_1)-\partial_1(p_2)\right) \cdot \vol_B.
\]
Finally, by Equation~\ref{twist3}, we see that for all \(m \in \mathbf{Z}\) and \(p \in P_m\),
\begin{align*}
\mathbf{F}[\nabla_0](p) &= -\ensuremath{\mathrm{i}}{}\nabla_0(\ensuremath{\mathrm{i}}{}\partial_1(p)\cdot\ensuremath{\mathrm{d}}{\tau}^1 + \ensuremath{\mathrm{i}}{}\partial_2(p) \cdot \ensuremath{\mathrm{d}}{\tau}^2)\\
&= -\ensuremath{\mathrm{i}}{}\left(\partial_2(\partial_1(p))-\partial_1(\partial_2(p))\right) \cdot \vol_B
= 2\pi \epsilon^{-m} c_m p \cdot \vol_B.
\end{align*}
But now, since
\[
\forall m,n \in \mathbf{Z}, \quad \epsilon^{-(m+n)}c_{m+n} = \epsilon^{-m-n}(q^{n}c_m+q^{-m}c_n) = (\epsilon^{-m}c_m) + (\epsilon^{-2})^m (\epsilon^{-n}c_n),
\]
it follows that for all \(m \in \mathbf{Z}\),
\[
\epsilon^{-m} c_m = \frac{1-\epsilon^{-2m}}{1-\epsilon^{-2}}\epsilon^{-1}c_1 = -(1-\epsilon^{-2m})\frac{\epsilon c_1}{\epsilon^2-1},
\]
so that, in turn, for all \(m \in \mathbf{Z}\) and \(p \in P_m\),
\[
F[\nabla_0](p) = 2\pi\epsilon^{-m}c_m p \cdot \vol_B = 2\pi (1-\epsilon^{-2m}) \frac{\epsilon c_1}{\epsilon^2-1} p \cdot \vol_B = \left[2\pi\frac{\epsilon c_1}{\epsilon^2-1}\vol_B,p\right],
\]
as was claimed.
\end{proof}
We can now readily compute the Atiyah space \(\fr{At}\) and gauge group \(\fr{G}\) of \(P\) with respect to \((\Omega^1_B,\dt{B};\Omega^1_{P,\mathrm{hor}})\) as well as their prolongable analogues with respect to \((\Omega_B,\dt{B};\Omega_{P,\mathrm{hor}})\). In fact, we shall see that every gauge potential and gauge transformation is automatically prolongable and that the group \(\fr{G} \cong \Unit(1)\) acts trivially on the affine space \(\fr{At} \cong \mathbf{R}^2\).
\begin{proposition}\label{heisthm}
Let \((\Omega_B,\dt{B};\Omega_{P,\mathrm{hor}})\) be the second-order horizontal calculus on \(P\) and let \(\nabla_0\) be the prolongable gauge potential on \(P\) with respect to \((\Omega_B,\dt{B};\Omega_{P,\mathrm{hor}})\) of Proposition~\ref{constantcurvatureprop}.
\begin{enumerate}
\item\label{heis1} Let \(\fr{at}\) be the space of relative gauge potentials on \(P\) with respect to the first-order horizontal calculus \((\Omega^1_B,\dt{B};\Omega^1_{P,\mathrm{hor}})\). The map \(\psi_{\fr{at}} : \mathbf{R}^2 \to \fr{at}\) defined by
\[
\forall (s_1,s_2) \in \mathbf{R}^2, \, \forall p \in P, \quad \psi_{\fr{at}}(s_1,s_2)(p) \coloneqq \left[\ensuremath{\mathrm{i}}{}(s_1\ensuremath{\mathrm{d}}{\tau}^1+s_2\ensuremath{\mathrm{d}}{\tau}^2),p\right]
\]
is an isomorphism of \(\mathbf{R}\)-vector spaces; hence, in particular, the subspace \(\Inn(\fr{at})\) of inner relative gauge potentials satisfies \(\Inn(\fr{at}) = \fr{at}\).
\item\label{heis2} Let \(\pr{\fr{at}}\) be the space of prolongable relative gauge potentials on \(P\) with respect to the second-order horizontal calculus \((\Omega_B,\dt{B};\Omega_{P,\mathrm{hor}})\), and let \(\Inn(\pr{\fr{at}})\) be the subspace of inner prolongable gauge potentials. Then \(\pr{\fr{at}} = \Inn(\pr{\fr{at}}) = \fr{at}\), where
\[
\forall (s_1,s_2) \in \mathbf{R}^2, \quad \mathbf{F}[\nabla_0 + \sigma(s_1,s_2)] = \mathbf{F}[\nabla_0].
\]
\item\label{heis3} Let \(\fr{G}\) be the gauge group of \(P\) with respect to \((\Omega^1_B,\dt{B};\Omega^1_{P,\mathrm{hor}})\). Then the function \(\psi_{\fr{G}} : \Unit(1) \to \fr{G}\) defined by
\[
\forall \zeta \in \Unit(1), \, \forall m \in \mathbf{Z}, \, \forall p \in P_m, \quad \psi_{\fr{G}}(\zeta)(p) \coloneqq \zeta^m p
\]
is a group isomorphism; hence, in particular, the subgroup \(\Inn(\fr{G})\) of inner gauge transformations and the subgroup \(\pr{\fr{G}}\) of prolongable gauge transformations with respect to \((\Omega_B,\dt{B};\Omega_{P,\mathrm{hor}})\) satisfy \(\Inn(\fr{G}) = \set{\id_P}\) and \(\pr{\fr{G}} = \fr{G}\). Moreover, the group \(\fr{G}\) acts trivially on the Atiyah space \(\fr{At}\) of \(P\) with respect to \((\Omega^1_B,\dt{B};\Omega^1_{P,\mathrm{hor}})\).
\end{enumerate}
\end{proposition}
\begin{proof}
Let us begin by proving part~\ref{heis1}. Note that \(\psi_{\fr{at}}\) is a well-defined \(\mathbf{R}\)-linear map with range contained in \(\Inn(\fr{at})\). Note, moreover, that \(\psi_{\fr{at}}\) is injective: indeed, if \((s_1,s_2) \in \ker\psi_{\fr{at}}\), then for all \(p \in P_1\),
\[
0 = \psi_{\fr{at}}(s_1,s_2)(p) = \ensuremath{\mathrm{i}}{}(s_1\ensuremath{\mathrm{d}}{\tau}^1+s_2\ensuremath{\mathrm{d}}{\tau}^2) \cdot p - p \cdot \ensuremath{\mathrm{i}}{}(s_1\ensuremath{\mathrm{d}}{\tau}^1+s_2\ensuremath{\mathrm{d}}{\tau}^2) = \ensuremath{\mathrm{i}}{}(\epsilon^{-1}-1)(s_1 p \cdot \ensuremath{\mathrm{d}}{\tau}^1 + s_2 p \cdot \ensuremath{\mathrm{d}}{\tau}^2),
\]
so that \((s_1,s_2) = (0,0)\) as was claimed. Thus, it suffices to show that \(\psi_{\fr{at}}\) is surjective.
Let \(\mathbf{A} \in \fr{at}\) be given. Given \(m \in \mathbf{Z}\), since \(\mathbf{A}\) is left \(\mathcal{O}{}(\Unit(1))\)-covariant and right \(B\)-linear, since the left \(P\)-module \(\Omega^1_{P,\mathrm{hor}}\) is free with basis \(\set{\ensuremath{\mathrm{d}}{\tau}^1,\ensuremath{\mathrm{d}}{\tau}^2} \subset \Zent_B(\Omega^1_B)\), and since \(P_m\) is a self-Morita equivalence bimodule for \(B\), there exist unique \(\alpha_{m,1},\alpha_{m,2} \in B\), such that
\[
\forall p \in P_m, \quad \mathbf{A}(p) = \alpha_{m,1} p \cdot \ensuremath{\mathrm{d}}{\tau}^1 + \alpha_{m,2}p \cdot \ensuremath{\mathrm{d}}{\tau}^2;
\]
note that \(\alpha_{0,1} = \alpha_{0,2} = 0\) since \(\rest{\mathbf{A}}{P_0} = 0\). First, for all \(m \in \mathbf{Z}\), we have \(\alpha_{m,1},\alpha_{m,2} \in \mathbf{C}\); indeed, given \(b \in B\), for all \(p \in P_m\),
\begin{align*}
0 &= \mathbf{A}(b p) - b \cdot \mathbf{A}(p)\\
&= \alpha_{m,1}bp \cdot \ensuremath{\mathrm{d}}{\tau}^1 + \alpha_{m,2}bp \cdot \ensuremath{\mathrm{d}}{\tau}^2 -b\alpha_{m,1}p \cdot \ensuremath{\mathrm{d}}{\tau}^1 - b\alpha_{m,2}p \cdot \ensuremath{\mathrm{d}}{\tau}^2\\
&= [\alpha_{m,1},b]p \cdot \ensuremath{\mathrm{d}}{\tau}^1+ [\alpha_{m,2},b]p \cdot \ensuremath{\mathrm{d}}{\tau}^2,
\end{align*}
so that \([\alpha_{m,1},b] = [\alpha_{m,2},b] = 0\) since \(P_{m}\) is a self-Morita equivalence bimodule for \(B\), and hence \(\alpha_{m,1},\alpha_{m,2} \in \mathbf{C}\) since the algebra \(B \coloneqq \mathcal{A}_\theta\) is central. Next, given \(m,n \in \mathbf{Z}\), we see that for all \(p \in P_m\) and \(q \in P_n\),
\begin{align*}
0 &= \mathbf{A}(pq) - \mathbf{A}(p)\cdot q - p \cdot \mathbf{A}(q)\\
&=(\alpha_{m+n,1}pq \cdot \ensuremath{\mathrm{d}}{\tau}^1+\alpha_{m+n,2}pq \cdot \ensuremath{\mathrm{d}}{\tau}^2) - (\alpha_{m,1}p\cdot\ensuremath{\mathrm{d}}{\tau}^1 + \alpha_{m,2}p \cdot \ensuremath{\mathrm{d}}{\tau}^2) \cdot q\\
&\quad\quad-p \cdot (\alpha_{n,1}q \cdot \ensuremath{\mathrm{d}}{\tau}^1 + \alpha_{n,2}q \cdot \ensuremath{\mathrm{d}}{\tau}^2)\\
&= (\alpha_{m+n,1}-\epsilon^{-n}\alpha_{m,1}-\alpha_{n,1})p \cdot \ensuremath{\mathrm{d}}{\tau}^1 + (\alpha_{m+n,2}-\epsilon^{-n}\alpha_{m,2}-\alpha_{n,2})p \cdot \ensuremath{\mathrm{d}}{\tau}^2;
\end{align*}
again, since \(P_{m+n} = P_m \cdot P_n\) is a self-Morita equivalence bimodule, it follows that
\begin{equation}\label{cocycle}
\forall j \in \set{1,2}, \quad \alpha_{m+n,j} = \epsilon^{-n}\alpha_{m,j} + \alpha_{n,j}
\end{equation}
On the one hand, given \(m \in \mathbf{Z}\), for all \(p \in P_m\),
\begin{align*}
0 &= \mathbf{A}(p) + \mathbf{A}(p^\ast)^\ast\\
&= (\alpha_{m,1}p \cdot \ensuremath{\mathrm{d}}{\tau}^1+\alpha_{m,2}p\cdot\ensuremath{\mathrm{d}}{\tau}^2) + (\alpha_{-m,1}p^\ast \cdot \ensuremath{\mathrm{d}}{\tau}^1 + \alpha_{-m,2}p^\ast \cdot\ensuremath{\mathrm{d}}{\tau}^2)^\ast\\
&= (\alpha_{m,1}p \cdot \ensuremath{\mathrm{d}}{\tau}^1+\alpha_{m,2}p\cdot\ensuremath{\mathrm{d}}{\tau}^2) + (-\epsilon^{-m}\alpha_{m,1}p^\ast \cdot \ensuremath{\mathrm{d}}{\tau}^1 - \epsilon^{-m}\alpha_{m,2}p^\ast \cdot\ensuremath{\mathrm{d}}{\tau}^2)^\ast\\
&= (\alpha_{m,1}p \cdot \ensuremath{\mathrm{d}}{\tau}^1+\alpha_{m,2}p\cdot\ensuremath{\mathrm{d}}{\tau}^2) + (\alpha_{m,1} \ensuremath{\mathrm{d}}{\tau}^1 \cdot p^\ast + \alpha_{m,2}\ensuremath{\mathrm{d}}{\tau}^2 \cdot p^\ast)^\ast \\
&= (\alpha_{m,1}+\overline{\alpha_{m,1}})p \cdot \ensuremath{\mathrm{d}}{\tau}^1 + (\alpha_{m,2}+\overline{\alpha_{m,2}})p \cdot \ensuremath{\mathrm{d}}{\tau}^2,
\end{align*}
so that again, since \(P_m\) is a self-Morita equivalence bimodule, \(\alpha_{m,1},\alpha_{m,2} \in \ensuremath{\mathrm{i}}{}\mathbf{R}\). On the other hand, by induction together with the \(1\)-cocycle identity \eqref{cocycle}, for all \(m \in \mathbf{Z}\) and \(j \in \set{1,2}\),
\[
\alpha_{m,j} = \frac{1-\epsilon^{-m}}{1-\epsilon^{-1}}\alpha_{1,j} = (1-\epsilon^{-m}) \frac{\alpha_{1,j}}{1-\epsilon^{-1}}.
\]
Hence, for all \(m \in \mathbf{Z}\) and \(p \in P_m\),
\[
\mathbf{A}(p) = \sum_{j=1}^2 \alpha_{m,j}p \cdot \ensuremath{\mathrm{d}}{\tau}^j = \sum_{j=1}^2(1-\epsilon^{-m}) \frac{\alpha_{1,j}}{1-\epsilon^{-1}}p \cdot \ensuremath{\mathrm{d}}{\tau}^j = \left[\sum_{j=1}^2\frac{\alpha_{1,j}}{1-\epsilon^{-1}}\ensuremath{\mathrm{d}}{\tau}^j,p\right],
\]
so that \(\mathbf{A} = \sigma(s_1,s_2)\) for \(s = (-\ensuremath{\mathrm{i}}{}(1-\epsilon^{-1})^{-1}\alpha_{1,1},-\ensuremath{\mathrm{i}}{}(1-\epsilon^{-1})^{-1}\alpha_{1,2}) \in \mathbf{R}^2\).
Let us now turn to part~\ref{heis2}. First, since \(\ensuremath{\mathrm{d}}{\tau}^1,\ensuremath{\mathrm{d}}{\tau}^2 \in \Zent_B(\Omega^1_B)\), it follows that
\[
\fr{at} = \ran \psi_{\fr{at}} \subseteq \Inn(\pr{\fr{at}}),
\]
so that, indeed, \(\pr{\fr{at}} = \Inn(\pr{\fr{at}}) = \fr{at}\). Now, given \((s_1,s_2) \in \mathbf{R}^2\), it follows by Corollary~\ref{relcor} that for all \(p \in P\),
\[
\left(\mathbf{F}[\nabla_0+\sigma(s_1,s_2)]-\mathbf{F}[\nabla_0]\right)(p) = \left[-\ensuremath{\mathrm{i}}{}\ensuremath{\mathrm{d}}_B\mleft(\ensuremath{\mathrm{i}}{}(s_1\ensuremath{\mathrm{d}}{\tau}^2+s_2\ensuremath{\mathrm{d}}{\tau}^2)\mright),p\right] = 0.
\]
Finally, let us consider part~\ref{heis3}. Before continuing, note that \(\Inn(\fr{G}) = \set{\id_P}\) since \(B\) is central. By construction, the map \(\psi_{\fr{G}} : \Unit(1) \to \fr{G}\) is a well-defined injective group homomorphism. Furthermore, for every \(\zeta \in \Unit(1)\), the gauge transformation \(\psi_{\fr{G}}(\zeta)\) acts diagonally on \(P = \bigoplus_{m\in\mathbf{Z}}P_m\) by scalar multiplication; hence, the range of \(\psi_{\fr{G}}\) is contained in \(\pr{\fr{G}}\) and acts trivially on \(\fr{At}\). The proof that \(\psi_{\fr{at}} : \mathbf{R}^2 \to \fr{at}\) is an \(\mathbf{R}\)-vector space isomorphism, \emph{mutatis mutandis}, now shows that \(\psi_{\fr{G}}\) is surjective.
\end{proof}
Now, recall that the usual de Rham calculus on \(\mathcal{O}{}(\Unit(1))\) fits into a canonical \(1\)-parameter family of \(1\)-dimensional bicovariant \(\ast\)-differential calculi on \(\mathcal{O}{}(\Unit(1))\) defined as follows. Let \(q \in \mathbf{R}^\times\) be given, and recall that the corresponding \emph{\(q\)-numbers} are defined by
\[
\forall n \in \mathbf{Z}, \quad [n]_q \coloneqq \begin{cases} \frac{1-q^n}{1-q} &\text{if \(q \neq 1\),}\\ n &\text{if \(q = 1\).}\end{cases}
\]
We define \((\Omega_q,\dt{q})\) to be the unique \(\ast\)-differential calculus on \(\mathcal{O}{}(\Unit(1))\) with \(\ast\)-closed left \(\mathcal{O}{}(\Unit(1))\)-comodule \(\Lambda^1_q \eqqcolon (\Omega^1_q)^{\mathcal{O}{}(\Unit(1))}\) of right \(\mathcal{O}{}(\Unit(1))\)-covariant \(1\)-forms and quantum Maurer--Cartan form \(\varpi_q : \mathcal{O}{}(\Unit(1)) \to \Lambda^1_q\) defined as follows:
\begin{enumerate}
\item the left crossed \(\mathcal{O}{}(\Unit(1))\)-\(\ast\)-module \(\Lambda^1_q\) is defined to be \(\mathbf{C}\) with the left \(\mathcal{O}{}(\Unit(1))\)-module structure given by
\[
\forall m \in \mathbf{Z}, \, \forall \mu \in \Lambda^1_q, \quad (z \mapsto z^m) \triangleright \mu \coloneqq q^m\mu
\]
and the \(\ast\)-structure given by complex conjugation;
\item the \(\Lambda^1_q\)-valued \(\Ad\)-invariant \(1\)-cocycle \(\varpi_q : \mathcal{O}{}(\Unit(1)) \to \Lambda^1_q\) is given by
\[
\forall m \in \mathbf{Z}, \quad \varpi_q((z\mapsto z^m)) \coloneqq 2\pi[m]_{q}.
\]
\end{enumerate}
Since elements of \(\Lambda^1_q\) are also bicoinvariant and since \(\Lambda^1_q\) is spanned by the image under \(\varpi_q\) of the group-like unitary \((z \mapsto z) \in \mathcal{O}{}(\Unit(1))\), it follows \(\Omega^k_q = 0\) for \(k \geq 2\). Note that \(\Omega^1_q\) is freely generated as both a left and right \(\mathcal{O}{}(\Unit(1))\)-module by the skew-adjoint bicoinvariant element \(\ensuremath{\mathrm{d}}_q t \coloneqq (2\pi\ensuremath{\mathrm{i}}{})^{-1}\varpi_q((z \mapsto z)) = -\ensuremath{\mathrm{i}}{}\);
note also that setting \(q = 1\) recovers the de Rham calculus on \(\mathcal{O}{}(\Unit(1))\). We now show that there exists a unique value of \(q\), such that \(P\) admits \((\Omega^1_q,\dt{q})\)-adapted prolongable gauge potentials with respect to the second-order horizontal calculus \((\Omega_B,\dt{B};\Omega_{P,\mathrm{hor}})\).
\begin{theorem}
Let \(q \in \mathbf{R}^\times\).
\begin{enumerate}
\item\label{heiscor1} The quadric subset \(\pr{\fr{At}}[\Omega^1_q]\) of all \((\Omega^1_q,\dt{q})\)-adapted prolongable gauge potentials on \(P\) with respect to \((\Omega_B,\dt{B};\Omega_{P,\mathrm{hor}})\) satisfies
\[
\pr{\fr{At}}[\Omega^1_q] = \begin{cases}\fr{At},&\text{if \(q = \epsilon^{2}\),}\\ \emptyset, &\text{else.}\end{cases}
\]
In particular, for every \(\nabla \in \fr{At} = \pr{\fr{At}}[\Omega^1_{\epsilon^2}]\), the curvature \(2\)-form \(F[\nabla]\) is non-zero, independent of \(\nabla\), and uniquely determined by
\[
F[\nabla](\ensuremath{\mathrm{d}}_{\epsilon^2}t) = -\ensuremath{\mathrm{i}}{}\epsilon c_1 \vol_B.
\]
\item\label{heiscor2} The space \(\fr{at}[\Omega^1_q]\) of all \((\Omega^1_q,\ensuremath{\mathrm{d}}_q t)\)-adapted relative gauge potentials on \(P\) with respect to \((\Omega^1_B,\dt{B};\Omega^1_{P,\mathrm{hor}})\) and the subspace \(\pr{\fr{at}}[\Omega^{\leq 2}_q]\) of all \((\Omega_q,\dt{q})\)-adapted prolongable relative gauge potentials on \(P\) with respect to \((\Omega_B,\dt{B};\Omega_{P,\mathrm{hor}})\) satisfy
\[
\fr{at}[\Omega^1_q] = \pr{\fr{at}}[\Omega^{\leq 2}_q] = \begin{cases}\fr{at},&\text{if \(q = \epsilon\),}\\0 &\text{else;}\end{cases}
\]
hence, in particular, it follows that
\[
\fr{At}/\pr{\fr{at}}[\Omega^1_{\epsilon^{2}}] = \pr{\fr{At}}[\Omega^1_{\epsilon^{2}}]/\pr{\fr{at}}[\Omega^{\leq 2}_{\epsilon^{2}}] = \fr{At}.
\]
\item\label{heiscor3} The space \(\Inn(\pr{\fr{at}};\Omega^1_q)\) of all \((\Omega^1_q,\dt{q})\)-semi-adapted inner prolongable gauge potentials on \(P\) with respect to \((\Omega_B,\dt{B};\Omega_{P,\mathrm{hor}})\) satisfies \(\Inn(\pr{\fr{at}};\Omega^1_q) = \fr{at}\); hence, in particular,
\[
\Out(\pr{\fr{At}}[\Omega^1_{\epsilon^{2}}]) \coloneqq \pr{\fr{At}}[\Omega^1_{\epsilon^{2}}]/\Inn(\pr{\fr{at}};\Omega^1_{\epsilon^{2}}) = \fr{At}/\fr{at} =\set{\nabla_0 + \fr{at}}.
\]
\end{enumerate}
\end{theorem}
\begin{proof}
Let \((\Omega_{P,\mathrm{ver}},\dv{P})\) be the second-order vertical calculus on \(P\) induced by the unique bicovariant prolongation \((\Omega_q,\dt{q})\) of \((\Omega^1_q,\dt{q})\). Since \(\Omega^1_{P,\mathrm{ver}}\) and \(\Omega^2_{P,\mathrm{hor}}\) are free as left \(P\)-modules with respective bases \(\set{\ensuremath{\mathrm{d}}_q t} \subset \coinv{\mathcal{O}{}(\Unit(1))}{\Omega^1_{P,\mathrm{ver}}}\) and \(\set{\vol_B} \subset \coinv{\mathcal{O}{}(\Unit(1))}{\Omega^2_{P,\mathrm{hor}}} = \Omega^2_B\), it follows that a left \(\mathcal{O}{}(\Unit(1))\)-covariant left \(P\)-linear map \(\phi : \Omega^1_{P,\mathrm{ver}} \to \Omega^2_{P,\mathrm{hor}}\) is completely determined by the unique element \(c \in \coinv{\mathcal{O}{}(\Unit(1))}{P} = B\), such that \(\phi(\ensuremath{\mathrm{d}}_q t) = c\vol_B\), and \emph{vice versa}. Thus, given \(c \in B\) with corresponding map \(\phi\), for all \(b \in B\), \(m \in \mathbf{Z}\), and \(p \in P_m\),
\begin{gather*}
\phi(\ensuremath{\mathrm{d}}_q t \cdot b) - \phi(\ensuremath{\mathrm{d}}_q t) \cdot b = \phi(b\cdot \ensuremath{\mathrm{d}}_q t) - c\vol_B \cdot b = (b c - c b) \vol_B,\\
\phi(\ensuremath{\mathrm{d}}_q t \cdot p) - \phi(\ensuremath{\mathrm{d}}_q t) \cdot p = \phi(q^{-m} p \cdot \ensuremath{\mathrm{d}}_q t) - c\vol_B \cdot p = (q^{-m} p c - c \epsilon^{-2m} p) \vol_B,
\end{gather*}
so that \(\phi\) is a \(P\)-bimodule map if and only if \(c = 0\) or \(c \in \mathbf{C} \setminus \set{0}\) and \(q = \epsilon^{2}\). We can now proceed with the proof of this corollary.
Let us first check part \ref{heiscor1}. On the one hand, suppose that there exists \(\nabla \in \pr{\fr{At}}[\Omega^1_q]\); recall that \(F[\nabla]\) denotes its curvature \(2\)-form. Since \(\mathbf{F}[\nabla] = \mathbf{F}[\nabla_0] \neq 0\) by Propositions~\ref{constantcurvatureprop} and~\ref{heisthm}, it follows that \(F[\nabla] : \Omega^1_{P,\mathrm{ver}} \to \Omega^2_{P,\mathrm{hor}}\) is a non-zero left \(\mathcal{O}{}(\Unit(1))\)-covariant morphism of \(P\)-bimodules, so that \(q = \epsilon^{2}\) by the above discussion. On the other hand, let \(\nabla \in \fr{At}\) be given. By the proof of Proposition~\ref{constantcurvatureprop} together with Proposition~\ref{heisthm}, for all \(m \in \mathbf{Z}\) and \(p \in P_m\),
\begin{gather*}
\mathbf{F}[\nabla](p) = \mathbf{F}[\nabla_0](p) = 2\pi [m]_{\epsilon^{-2}} \epsilon^{-1}c_1 p \cdot \vol_B, \\ \dv{P}(p) = 2\pi\ensuremath{\mathrm{i}}{}[m]_q\,\ensuremath{\mathrm{d}}_q t \cdot p = 2\pi \ensuremath{\mathrm{i}}{}[m]_q q^{-m}p\cdot\ensuremath{\mathrm{d}}_q t = 2\pi \ensuremath{\mathrm{i}}{} q^{-1} [m]_{-q} p \cdot \ensuremath{\mathrm{d}}_q t
\end{gather*}
so that \(\nabla \in \pr{\fr{At}}[\Omega^1_{\epsilon^2}]\) with \(F[\nabla] : \Omega^1_{P,\mathrm{ver}} \to \Omega^2_{P,\mathrm{hor}}\) uniquely defined by
\[
F[\nabla](\ensuremath{\mathrm{d}}_{\epsilon^2} t) \coloneqq -\ensuremath{\mathrm{i}}{} \epsilon c_1\vol_B.
\]
Let us now turn to parts \ref{heiscor2} and \ref{heiscor3}. Note that \(\Omega^1_{P,\mathrm{hor}}\) is free as a left \(P\)-module with basis given by \(\set{\ensuremath{\mathrm{d}}\tau^1,\ensuremath{\mathrm{d}}\tau^2} \subset \coinv{\mathcal{O}{}(\Unit(1))}{\Omega^1_{P,\mathrm{hor}}} = \Omega^1_B\). Hence, the above argument, \emph{mutatis mutandis}, shows that \(\fr{at}[\Omega^1_q] = \fr{at}\) when \(q = \epsilon\) and \(\fr{at}[\Omega^1_q] = 0\) otherwise; indeed, for all \((s_1,s_2) \in \mathbf{R}^2\), the relative connection \(1\)-form \(\omega[\sigma(s_1,s_2)]\) of \(\sigma(s_1,s_2) \in \fr{at}[\Omega^1_\epsilon]\) is given by
\[
\omega[\sigma(s_1,s_2)](\ensuremath{\mathrm{d}}_\epsilon t) \coloneqq - \frac{\epsilon-1}{2\pi}(s_1\ensuremath{\mathrm{d}}\tau^1+s_2\ensuremath{\mathrm{d}}\tau^2) \in \Span_{\mathbf{R}}\set{\ensuremath{\mathrm{d}}\tau^1,\ensuremath{\mathrm{d}}\tau^2}.
\]
Since \(\pr{\fr{at}} = \fr{at}\), since \(\Omega^2_q = 0\), and since \(\ensuremath{\mathrm{d}}\tau^1\wedge\ensuremath{\mathrm{d}}\tau^1 = \ensuremath{\mathrm{d}}\tau^1 \wedge \ensuremath{\mathrm{d}}\tau^2+\ensuremath{\mathrm{d}}\tau^2 \wedge\ensuremath{\mathrm{d}}\tau^1 = \ensuremath{\mathrm{d}}\tau^2\wedge\ensuremath{\mathrm{d}}\tau^2 = 0\), it now follows that \(\pr{\fr{at}}[\Omega^{\leq 2}_q] = \fr{at}[\Omega^1_q]\) in each case. Since \(\mathbf{F}[\nabla] = \mathbf{F}[\nabla_0]\) for all \(\nabla \in \pr{\fr{At}} = \fr{At}\), it now follows by Proposition~\ref{heisthm} that \(\Inn(\pr{\fr{at}};\Omega^1_q) = \fr{at}\) in general.
\end{proof}
It therefore follows that the norm-positive fundamental unit \(\epsilon\) of the real quadratic field \(\mathbf{Q}[\theta]\) induced by \(\theta\) is the unique value of \(q \in \mathbf{R}^\times\) for which \(P\) admits \((\Omega^1_{q^2},\dt{q^2})\)-adapted prolongable gauge potentials. In this case, every gauge potential is prolongable \((\Omega^1_{q^2},\dt{q^2})\)-adapted with the same non-zero constant curvature \(2\)-form
\[
\ensuremath{\mathrm{d}}_{q^2} t \mapsto -\ensuremath{\mathrm{i}}{}\epsilon c_1 \vol_B
\]
thereby defining a \(q\)-monopole analogous to the \(q\)-monopole of Brzezi\'{n}ski--Majid~\cite{BrM}*{\S 5.2} on the \(q\)-deformed complex Hopf fibration; mdistinct gauge potentials are gauge-inequivalent and yield non-isomorphic \((\mathcal{O}{}(\Unit(1));\Omega^1_{q^2},\dt{q^2})\)-principal \textsc{fodc}{} on \(P\).
| -515,535.229629 |
[
-3.203125,
2.921875
] | 25.832187 |
[
-2.392578125,
0.8076171875,
-2.19140625,
-6.74609375,
-1.5869140625,
9.671875
] |
[
5.36328125,
10.3046875,
2.4453125,
6.68359375
] | 1,267 | 35,940 |
[
-3.021484375,
3.505859375
] | 39.245745 |
[
-5.3359375,
-4.6640625,
-5.9765625,
-2.50390625,
1.9365234375,
14.4453125
] | 0.380196 | 12.456582 | 11.825264 | 2.048739 |
[
1.2953181266784668
] | -344,707.390541 | 6.542627 | -512,842.420438 | 0.624729 | 6.374321 |
[
-1.2607421875,
-3.265625,
-4.59375,
-6.11328125,
1.8291015625,
13.5625
] |
[
-5.33203125,
-2.134765625,
-2.3359375,
-1.220703125,
3.8515625,
4.79296875
] | |
BkiUftw5qhDCYT-7h2Ps
|
\section{Introduction}
The preprint \cite{1} reviews some recent experiments on measuring
the Casimir force. The author makes a reservation that he will
consider only ``credible'' experiments. As incredible, the
experiments which have claimed ``1\% or better agreement'' are
meant with a generic reference to review \cite{2}.
In different places of the preprint these unspecified
incredible experiments are characterized as ``1\% level work''
(pp.\ 1, 3), experiments ``that claim 1\% accuracy''
(p.\ 4), and ``experiments claiming 1\% precision'' (p.\ 27).
In the first part of Ref.~\cite{1} the author provides
arguments why it is unclear to him ``what these experiments
really mean''. The second part of Ref.~\cite{1} is largely
devoted to different aspects of author's own work employing
spherical lenses of more than 10\,cm curvature radius.
Below we demonstrate that the author's arguments against
what he
calls a ``1\% level work'' are based on incorrect or
incomplete information. With respect to measurements of
the Casimir force using lenses of centimeter-size
curvature radii,
we show that they are fundamentally flawed. According to
our calculations, experiments of this type may lead to
unpredictable results for the Casimir force, due to
unavoidable deviations from a spherical shape of
mechanically polished and ground surfaces.
The paper is organized as follows. In Sec.~2 we
explain what is incorrect in the argumentation of
Ref.~\cite{1} against precise experiments on measuring
the Casimir force. Here we consider a relationship among
the concepts of accuracy, precision and measure of
agreement with theory, discuss total experimental error,
its constituents, and rules of their combination,
explain the misjudgement of constraints on long-range forces
made in Ref.~\cite{1}. In Sec.~3 it is shown that all
measurements of the Casimir force employing
centimeter-size spherical lenses are fundamentally
flawed. We discuss both the electrostatic calibrations
and the measurement of the Casimir force.
We demonstrate that commonly used simplified form of the
proximity force approximation (PFA) is inapplicable
in the presence of standard imperfections
on the optical surfaces (bubbles and pits) and derive
new expressions for the Casimir force
valid in the presence
of these imperfections. In Sec.~4 some further
objectionable features of Ref.~\cite{1} are discussed.
Section~5 contains our conclusions and discussion.
\section{What is incorrect in the arguments against
precise experiments}
\subsection{Confusion between accuracy, precision and measure of agreement
with theory}
The present state of the art in experiments on measuring the
Casimir force is reflected in the review \cite{2}. Some results
in this review do not necessarily coincide with respective
formulations in original publications because several experiments
were later reanalyzed using more reliable methods of data
processing. At the moment only these updated results of
Ref.~\cite{2} should be used in all discussions.
According to Ref.~\cite{2}, each experiment on measuring the
Casimir force is characterized by a total experimental error,
total theoretical error, and measure of agreement between
experiment and theory determined at some high (usually 95\%)
confidence level. In this respect the above cited
characterizations
of precise experiments in Ref.~\cite{1}, which confuse 1\%
agreement, 1\% accuracy and 1\% precision are completely
misleading. According to Ref.~\cite{2}, the best measure of
agreement between the Casimir pressure measured in the most
precise experiment \cite{3} using a micromachined oscillator
and theory is equal to 1.8\% at the separation $a=400\,$nm
between the test bodies.
As to the best measure of agreement between the measured
Casimir force in sphere-plate geometry and theory in the
most precise experiment \cite{4} using an atomic force
microscope, it is equal to 5.4\% at separations around
$a=80\,$nm \cite{5}.
Hence the ``1\% or better agreement'' is an incorrect
information.
Of even greater concern is the mention of experiments
``that claim 1\% accuracy'' \cite{1}. According to the
review \cite{2}, there are no such experiments. What's
more, measuring of accuracy in percents, as is done in
Ref.~\cite{1} for many times, is in contradiction with
the rigorous understanding of this concept.
According to the International vocabulary of
metrology \cite{6} produced by the Joint Committee for
Guides in Metrology, measurement accuracy is the
closeness of agreement between a measured quantity value
and a true quantity value. As underlined in Ref.~\cite{6},
the concept ``measurement accuracy'' is not a quantity
and is not given a numerical value. This interpretation
is well founded because a true quantity value is in
principle unknown. A measurement is said to be more
accurate when it offers a smaller measurement error.
Thus, it is meaningless to speak about 1\% or any other
numerical degree of accuracy, as is done and erroneously
attributed to some unspecified experiments in Ref.~\cite{1}.
Another concept used in Ref.~\cite{1} in the same context
is the concept of precision. Measurement precision is the
closeness of agreement between measured quantity values
obtained by replicate measurements on the same or similar
objects under specified conditions \cite{6}. Precision
is expressed numerically by measures of imprecision
\cite{6}. Specifically, the experimental errors
can be used as such measures.
In the experiment \cite{3} the highest measurement
precision is achieved at the separation $a=162\,$nm
where the total relative experimental error is equal to
\cite{2}
\begin{equation}\delta_t\Pi^{\rm expt}=
\frac{\Delta_t\Pi^{\rm expt}}{|\Pi^{\rm expt}|}
=0.19\mbox{\%},
\label{eq0}
\end{equation}
\noindent
where $\Pi^{\rm expt}$ is the measured value of the
physical quantity $\Pi$ (the Casimir pressure between two
parallel plates), and $\Delta_t\Pi^{\rm expt}$
is the total absolute experimental error.
In the experiment \cite{4} the highest precision is
achieved at 63\,nm and corresponds to the total relative
experimental error $\delta_t\Pi^{\rm expt}=1.5$\% \cite{2},
where $\Pi^{\rm expt}$ is the measured Casimir force
between a sphere and a plate.
Although the author of Ref.~\cite{1} writes that
``There is a tendency among workers in this field to
confuse precision with accuracy, of which I am guilty
myself'',
the definitions presented in Ref.~\cite{1} continue to be
unrelated to the rigorous formulations \cite{6}.
Specifically, precision relates not a number of significant
figures provided by a measurement device, as erroneously
stated in Ref.~\cite{1}, but to a closeness between measured
quantity values in replicate measurements (the same
voltmeter, for instance, can be used in different experiments
leading to different precisions).
\subsection{Total experimental error and its constituents}
Reference \cite{1} argues that ``to obtain a given experimental
accuracy, say 1\%, requires that the calibrations and force
measurements must be done to much better than 1\% accuracy...''
However, the suggested arguments that the latter is yet not
possible contain several incorrect statements.
Before we indicate each of the specific mistakes made, let us
emphasize that the word ``accuracy'' used in Ref.~\cite{1}
must be replaced with the word ``precision'' because, as explained
above, accuracy is not given a numerical value and there are no
experiments claiming a 1\% accuracy.
The author of Ref.~\cite{1} is right that if, for instance,
the total experimental error is equal to 1\% all calibration
errors must be smaller accordingly depending on their size and
number. In the list of these errors, however, Ref.~\cite{1}
again confuses by mixing experimental errors,
theoretical errors and agreement between experiment and theory.
Here, we illustrate what are the constituents of the lowest
total experimental error $\delta_t\Pi^{\rm expt}=0.19$\%
at a separation $a=162\,$nm in the most precise experiment
\cite{3}. The relative random error in the Casimir pressure at
$a=162\,$nm is $\delta_r\Pi^{\rm expt}=0.04$\%
\cite{2}. It is determined from the standard statistical procedure
using Student distribution \cite{7}. The systematic error is
caused by the errors in the measurement of the sphere radius,
of the frequency shift, and of the proximity force approximation
(which is a part of experimental procedure in the indirect
measurement of the Casimir pressure). The resulting relative
systematic error at a separation of 162\,nm is
$\delta_s\Pi^{\rm expt}=0.19$\%.
According to Ref.~\cite{1}, for achieving a 1\% experimental
accuracy (read precision) the sphere radius needs to be measured
to 0.5\% accuracy (precision). Reference~\cite{1} claims that
``the radius measurement is not discussed in sufficient details
in any of papers...'' This is, however, not so. The value of the
sphere radius in the experiment \cite{3} was determined to be
$R=151.2\pm 0.2\,\mu$m leading to the relative error of only
0.13\% \cite{2,3}, i.e., smaller error than is demanded in
Ref.~\cite{1}.
All the details for determination of sphere radius by
means of electrostatic calibrations are provided in
Refs.~\cite{3,8} (specifically, the calibration details are
presented in full in Ref.~\cite{9}).
The other sources of errors considered in Ref.~\cite{1} are
unrelated to the experimental precision. Thus, the knowledge
of the optical properties of the surfaces is not needed for
the determination of precision. The discussion of errors in
the Casimir force induced by the errors in absolute separations
bears no relation to force and pressure measurements as well.
The separation distance is an independent quantity and is
measured with its own measurement error (in Ref.~\cite{3}
the latter is equal to 0.6\,nm).
Both these errors are important for the comparison
between experiment and theory, but have nothing to do with
the achieved experimental precision of force and pressure
measurements reviewed in Ref.~\cite{2}.
\subsection{Is it really uncertain how to combine different errors
and uncertainties?}
As discussed above, the total experimental error results from
the combination of random and systematic errors. In its turn,
the systematic error has several constituents.
According to Ref.~\cite{1}, precision measurement experts still
debate whether these errors and uncertainties can be added
in quadrature or be simply added.
Regarding this statement we suggest that the author of Ref.~\cite{1}
was guided by outdated information.
It is common knowledge that errors and uncertainties are random
quantities and are characterized by some distributions \cite{10}.
The composition law of several random quantities depends on
the specific form of these distributions.
In the measurement of the Casimir force it is usually supposed
that all systematic errors in the form of systematic deviations
(i.e., biases in a measurement which always make the measured
value higher or lower than the true value) are already removed
using some known process, i.e., through a calibration.
The remaining systematic errors are the errors of a calibration
device and have the meaning of the smallest fractional devision
of the scale of the device. Such systematic errors are random
quantities characterized by a uniform distribution (equal
probability). The errors in an approximate theoretical formula
used to convert a directly measured quantity into an indirectly
measured one are also distributed uniformly.
Then the resulting systematic error at a chosen confidence
level $\beta$ is obtained from its constituents
$\Delta_s^{\!(i)}\Pi^{\rm expt}$ ($i=1,2,\,\ldots ,J$) using the
following statistical rule \cite{10}
\begin{equation}
\Delta_s\Pi^{\rm expt}=\min\left[
\sum_{i=1}^{J}\Delta_s^{\!(i)}\Pi^{\rm expt},
k_{\beta}^{(J)}\sqrt{
\sum_{i=1}^{J}(\Delta_s^{\!(i)}\Pi^{\rm expt})^2\,\,\,}\right].
\label{eq1}
\end{equation}
\noindent
Here, $k_{\beta}^{(J)}$ is a tabulated coefficient. The above
value of $\delta_s\Pi^{\rm expt}=0.19$\% at $a=162\,$nm in the
experiment \cite{3} (see Sec.~2.2) was obtained using Eq.~(\ref{eq1})
with $J=3$, $\beta=0.95$, and $k_{0.95}^{(3)}=1.1$ \cite{2,5}.
Contrary to Ref.~\cite{1}, statistical rules for the combination
of random and systematic errors have also been much studied.
The random error is described by the normal or Student
distribution. The resulting systematic error is described by
a combination of uniform distributions. It can be shown that
if the resulting systematic error is also assumed to be
distributed uniformly, the total experimental error will be
overestimated. Thus, this assumption is conservative and can be
used safely. There are several methods in statistcs how to
combine errors described by normal and uniform distributions
\cite{10}. A widely used method puts
\begin{equation}
\Delta_t\Pi^{\rm expt}=\Delta_r\Pi^{\rm expt}, \qquad
\Delta_t\Pi^{\rm expt}=\Delta_s\Pi^{\rm expt},
\label{eq2}
\end{equation}
\noindent
or
\begin{equation}
\Delta_t\Pi^{\rm expt}=q_{\beta}(r)\left[\Delta_r\Pi^{\rm expt}
+\Delta_s\Pi^{\rm expt}\right]
\label{eq3}
\end{equation}
\noindent
depending on what respective inequality is fulfiled for all $a$
over the entire measurement range
\begin{equation}
r(a)<0.8, \qquad r(a)>8, \qquad
\mbox{or}\qquad
0.8\leq r(a)\leq 8.
\label{eq4}
\end{equation}
\noindent
Here, the quantity $r(a)$ is defined as
\begin{equation}
r(a)=\frac{\Delta_s\Pi^{\rm expt}(a)}{s_{\bar\Pi}(a)},
\label{eq4a}
\end{equation}
\noindent
where $s_{\bar\Pi}(a)$ is the variance of the mean
of a measured quantity $\Pi^{\rm expt}$.
The coefficient $q_{\beta}(r)$ at a confidence level
$\beta=0.95$ varies between 0.71 and 0.81 depending on the
value of $r(a)$. Note that the value of the relative total error
in the experiment \cite{3} at $a=162\,$nm was obtained using the
second equality in Eq.~(\ref{eq2}).
We emphasize that the dominance of the resulting systematic error
over the random error within the entire measurement range
achieved in the experiment of Ref.~\cite{3} is the distinguishing
feature of precise experiments of a metrological quality.
{}From the above facts one can conclude that precision measurement
experts have gone far beyond debates whether uncertainties can be added
in quadrature or simply added.
\subsection{Misjudgement of constraints on long-range forces following
from the most precise Casimir experiment}
The measure of agreement between the Casimir pressures measured in
the most precise experiment \cite{3} and calculated theoretically
was used to obtain the strongest constraints on the parameters of
long-range Yukawa-type forces in the interaction range of several
tens of nanometers \cite{3,5}. In doing so the Yukawa pressure was
calculated \cite{3} by the application of the PFA.
Reference \cite{1} informs the reader that the use of this
approximation has been criticized in Ref.~\cite{11}.
The author of Ref.~\cite{1} repeats the conclusion of
Ref.~\cite{11} that the PFA ``only applies to a force that depends
on the location of body surfaces'' and ``is not valid for
the volume integral required for calculating the anomalous
force''.
This conclusion is, however, incorrect as is demonstrated in
available literature overlooked by the author of Ref.~\cite{1}.
In Ref.~\cite{12} it is shown that the PFA is applicable
for the calculation of the Yukawa force under conditions
that the separation $a$ and interaction range $\lambda$ are much
smaller than the sphere radius $R$ and the plate thickness $D$.
All these conditions are satisfied with a large safety margin
in the experimental setup of Ref.~\cite{3}. In Ref.~\cite{13}
the respective Yukawa pressure in the setup of Ref.~\cite{3}
was calculated both exactly and using the PFA with coinciding
results. The purported ``corrections'' to the calculation of
Ref.~\cite{12} pointed out in Ref.~\cite{11} were shown to
be invalid and based on a simple misunderstanding \cite{13}.
What's more, one of the authors of Ref.~\cite{11} (R.O.)
recognized \cite{14} that the issue raised in their paper
``is not of practical concern for current experiments''.
Thus, the reliability of constraints on the parameters
of Yukawa interactions obtained in Ref.~\cite{3} is
beyond doubt. However, the author of Ref.~\cite{1}
included only an incorrect reference to the paper \cite{12}
(the title is taken from one paper and the publication
data from another; see Ref.~[24] in \cite{1}) in his list
of references. As to important Refs.~\cite{13,14},
where the validity of constraints of Ref.~\cite{3} is
reinforced in an unambiguous way, the author of
Ref.~\cite{1} didnot mention them.
\section{Why Casimir force measurements using centimeter-size
spherical lenses are fundamentally flawed}
\subsection{Anomalies in electrostatic calibrations}
Observations of anomalous electrostatic forces in the lens-plate
geometry for lenses of centimeter-size curvature radii is the
subject of wide speculation (see, e.g., Refs.~\cite{15,16,17}).
In Ref.~\cite{18} it was shown that anomalous behavior of the
electrostatic force can be explained due to deviations
of the mechanically polished and ground surfaces of
centimeter-size lenses from a perfect spherical form.
The point is that the typical surface of a centimeter-size lens
is characterized in terms of scratch and dig optical
surface specification data. In particular, depending on the
quality of lens used, bubbles and pits with a diameter
varying from $30\,\mu$m to 1.2\,mm are allowed on the
surface \cite{19}. There may be also scratches with a width
varying from 3 to $120\,\mu$m \cite{19}. The problem of
bubbles on the centimeter-size lens surface should not be
reduced to the fact that lens curvature radius $R$ is
determined with some error. The thickness of each bubble should
of course be less than the absolute error in the measurement of lens
curvature radius (for a lens with $R=15.10\,$cm in Ref.~\cite{16},
for instance, $\Delta R=0.05\,$cm). The crucial point is that
curvature radii of bubbles can be orders of magnitude different,
as compared to $R$. This allows one to suggest models leading to
quite different (``anomalous'') dependence of electrostatic force
on separation in comparison with the case of perfect spherical
surfaces \cite{18}.
Reference \cite{1} mentions the possibility that the anomalous
electrostatic forces are due to simple geometrical effects
without reference to the source of this idea (Ref.~\cite{18} is
missing in the list of references in \cite{1}).
According to Ref.~\cite{1}, this possibility ``is credibly
discarded''
in Ref.~\cite{20}. The author of Ref.~\cite{1} does not inform the
reader that computations of Ref.~\cite{20} were repeated in e-print
\cite{21} and shown to be not reproducible.
Thus, there is no scientific objection against the possibility
that anomalous electrostatic forces are due to deviations of
mechanically polished and ground surfaces from perfect sphericity.
Furthermore, some of the authors of Ref.~\cite{20}
(D.A.R.D.\ and R.O.) recently recognized \cite{22} that local
geometrical deformations of the surface can really lead to an
anomalous electrostatic force not only in sphere-plate
geometry, but for a cylindrical lens in close proximity to
the plate as well. According to Ref.~\cite{22},
``this is certainly a crucial point to be taken into account
in future experiments''. This reference, however, is missing
in the list of references in \cite{1}.
An extensive consideration of the electrostatic calibrations
in Ref.~\cite{1} always assumes perfect sphericity of the
lens surface.
Another misrepresented point directly relevant to electrostatic
calibrations is the dependence of the contact potential on the
separation distance. According to Ref.~\cite{1}, every paper
on the Casimir effect ``that has bothered measuring the contact
potential as a function of distance has shown an apparent
distance dependence of that potential''. This is, however, not
the case. In Ref.~\cite{3} the contact potential was carefully
measured as a function of separation and found to be constant.
The respective measurement data of the electrostatic
calibrations are published in Refs.~\cite{9,18}. Constant
contact potential was observed in all other experiments by
R.\ S.\ Decca as well (review of these experiments can be found
in Refs.~\cite{2,5}). Independent on separation contact
potential was also reported in Refs.~\cite{4,23,24,25}
and in all other experiments by U.\ Mohideen (see
Refs.~\cite{2,5} for a review). It is notable that
all these experiments were performed in high vacuum with small
spheres of order $100\,\mu$m curvature radii.
\subsection{Influence of surface imperfections on the Casimir force
for lenses of centimeter-size curvature radius}
The Casimir force is far more sensitive than the electrostatic
force to the bubbles and pits that are unavoidably present
on the mechanically polished and ground surface of any lens
of centimeter-size curvature radius. The physical reason is that
the Casimir force falls with the increase of separation distance
more rapidly than the electric force.
As a result, it is determined by smaller regions near the points
of closest approach of the surfaces. If the local curvature
radius on the lens surface near the point of closest approach to
the plate is significantly different from the mean lens
curvature radius $R$, the impact on the Casimir force can be
tremendous. Below we demonstrate that just this happens due to
the presence of bubbles and pits on a lens surface.
For the sake of simplicity, we consider ideal metal surfaces.
However, it is easily seen that all conclusions obtained are
preserved for real bodies as well.
The Casimir force in sphere-plate geometry under the experimental
conditions $a\ll R$ is usually calculated using the PFA \cite{2,5}.
According to the most general formulation of the PFA \cite{27},
the unknown force between the elements of curved surfaces is
approximately replaced with a known force per unit area of the plane
surfaces (i.e., a pressure) at the respective separation
multiplied by an area element.
Applied to a spherical lens of thickness $D$ above a plane
$z=0$, the PFA represents the force between them in the form
\begin{equation}
F_{sp}(a,T)=\int_{\Sigma}d\sigma P(z,T).
\label{eq5}
\end{equation}
\noindent
Here, $d\sigma$ is the element of plate area, $\Sigma$ is the
projection of the lens onto the plate, $a$ is the shortest
separation between them, $z=z(x,y)$ is the equation of a lens
surface, and $P(z,T)$ is the pressure for two
plane parallel plates at a separation $z$ at temperature $T$.
We choose the origin of a cylindrical coordinate system on
the plane $z=0$ under the lens center. Then for a perfectly
shaped spherical lens the coordinate $z$ of any point
of its surface is given by
\begin{equation}
z=R+a-(R^2-\rho^2)^{1/2}, \quad
\rho^2=x^2+y^2.
\label{eq6}
\end{equation}
\noindent
In this case Eq.~(\ref{eq5}) leads to
\begin{equation}
\!\!\!\!\!\!\!\!\!\!\!\!\!\!
F_{sp}^{\rm perf}(a,T)=2\pi\int_{0}^{\sqrt{2RD-D^2}}\!\!\!\rho d\rho P(z,T)=
2\pi\int_{a}^{D+a}(R+a-z)P(z,T)dz.
\label{eq7}
\end{equation}
\noindent
Keeping in mind that the Casimir pressure is expressed as
\begin{equation}
P(z,T)=-\frac{\partial{\cal F}_{pp}(z,T)}{\partial z},
\label{eq8}
\end{equation}
\noindent
where ${\cal F}_{pp}(z,T)$ is the free energy per unit area of parallel
plates, and integrating by parts in Eq.~(\ref{eq7}), one arrives
at
\begin{equation}
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
F_{sp}^{\rm perf}(a,T)=2\pi R{\cal F}_{pp}(a,T)-2\pi(R-D){\cal F}_{pp}(D+a,T)-
2\pi\int_{a}^{D+a}\!\!{\cal F}_{pp}(z,T)dz.
\label{eq9}
\end{equation}
\noindent
We consider centimeter-size spherical lenses satisfying
a condition $a\ll D$. For such lenses
${\cal F}_{pp}(D+a,T)\ll{\cal F}_{pp}(a,T)$.
Because of this, one can neglect the second term on the right-hand
side of Eq.~(\ref{eq9}) in comparison with the first \cite{28}.
It can be also shown \cite{28,29} that the first term on the right-hand
side of Eq.~(\ref{eq9}) is in excess of the third by a factor of $R/a$.
This allows one to neglect the third term and arrive to what is
called the simplified formulation of the PFA \cite{28,29}
\begin{equation}
F_{sp}^{\rm perf}(a,T)\approx 2\pi R{\cal F}_{pp}(a,T)
\label{eq10}
\end{equation}
\noindent
widely used for both spherical lenses and for spheres [note that for
a semisphere the second term on the right-hand
side of Eq.~(\ref{eq9}) is identically equal to zero].
For two parallel ideal metal plates spaced $z$ apart the Casimir
free energy per unit area is given by \cite{5,26}
\begin{equation}
{\cal F}_{pp}(z,T)=\frac{k_BT}{\pi}\sum_{l=0}^{\infty}
{\vphantom{\sum}}^{\!\prime}\int_{0}^{\infty}k_{\bot}dk_{\bot}
\ln(1-e^{-2zq_l}).
\label{eq11}
\end{equation}
\noindent
Here, $k_B$ is the Boltzmann constant, $k_{\bot}$ is the magnitude
of the projection of the wave vector on the plates,
$q_l^2=k_{\bot}^2+\xi_l^2/c^2$, $\xi_l=2\pi k_BTl/\hbar$ with
$l=0,\,1,\,2,\,\ldots$ are the Matsubara frequencies, and the
primed summation sign means that the term with $l=0$ is
multiplied by 1/2. For the sake of convenience in computations,
we rewrite Eq.~(\ref{eq11}) in terms of a dimensionless
integration variable $y=2aq_l$ and expand the logarithm in power
series
\begin{equation}
{\cal F}_{pp}(z,T)=-\frac{k_BT}{4\pi z^2}\sum_{l=0}^{\infty}
{\vphantom{\sum}}^{\!\prime}\int_{\tau_z l}^{\infty}ydy
\sum_{n=1}^{\infty}\frac{e^{-ny}}{n}.
\label{eq12}
\end{equation}
\noindent
Here, the dimensionless parameter $\tau_z$ is defined as
$\tau_z=4\pi zk_BT/(\hbar c)$. After performing integration
and then the summation with respect to $l$, the following
result is obtained:
\begin{equation}
{\cal F}_{pp}(z,T)=-\frac{k_BT}{4\pi z^2}\left[
\frac{\zeta(3)}{2}+
\sum_{n=1}^{\infty}\frac{e^{-\tau_zn}}{n^2(1-e^{-\tau_zn})}
\left(\frac{1}{n}+\frac{\tau_z}{1-e^{-\tau_zn}}\right)\right],
\label{eq13}
\end{equation}
\noindent
where $\zeta(x)$ is the Riemann zeta function. Note that the
first contribution on the right-hand side of Eq.~(\ref{eq13})
coincides with the high temperature limit of the free energy.
This is quite reasonable if to take into account that
$\tau_z=2\pi T/T_{\rm eff}$, where the effective temperature
is defined from $k_BT_{\rm eff}=\hbar c/(2z)$.
Now we are in a position to compute the Casimir force between real
spherical lens of large curvature radius with bubbles and pits
of different types and a plane plate. It is common to use the
simplified formulation of the PFA (\ref{eq10}) in sphere-plate
geometry for both small spheres of about $100\,\mu$m radii
and large spherical lenses (see, for instance,
Refs.~\cite{3,4,16,30}). In doing so the role of bubbles and
pits on the surface of lenses of centimeter-size curvature
radii is simply disregarded. Equation (\ref{eq10}), however,
is not applicable for real lenses with large curvature radii
because it assumes perfect spherical surface.
For such lenses one should use a general formulation of the
PFA in Eq.~(\ref{eq5}). To illustrate this fact, we perform
calculations for three typical model imperfections on the
spherical surface near the point of closest approach to the
plate allowed by the optical surface specification data
\cite{19}.
\begin{figure*}[t]
\vspace*{-18.5cm}
\hspace*{-2.5cm}\includegraphics{figLS-1.ps}
\vspace*{-8.cm}
\caption{
The configuration of a spherical lens with curvature radius $R$
possessing a surface imperfection at the point of closest
approach to a plate. The bubble curvature radius is $R_1>R$.
The relative sizes of the lens and imperfection are shown
not to scale.
}
\end{figure*}
As the first example, we consider a bubble of the curvature radius
$R_1=25\,$cm which is larger than the curvature radius $R=15\,$cm
of the lens used (see Fig.~1). The thickness of the spherical
lens formed by the bubble is chosen to be $D_1=0.5\,\mu$m
(this is much less than typical absolute error
$\Delta R=0.05\,$cm in the measurement of centimeter-size
lenses curvature radius). The radius of the bubble is
determined from $r^2=2R_1D_1-D_1^2\approx 0.25\,\mbox{mm}^2$,
leading to $2r=1\,\mbox{mm}<1.2\,$mm, i.e., less than
a maximum value allowed by the optical surface specification data
\cite{19}. Respective quantity $d$ defined in Fig.~1 is equal to
$d\approx r^2/(2R)\approx 0.83\,\mu$m. Then the flattening
of a lens surface at the point of closest approach to the
plate is $d-D_1\approx 0.33\,\mu$m which is much less than
$\Delta R$.
The general formulation of the PFA (\ref{eq5}) should be applied
taking into account that the surface of the bubble is described
by the equation
\begin{equation}
z=R_1+a-(R_1^2-\rho^2)^{1/2},
\label{eq14}
\end{equation}
\noindent
where $a$ is the distance between the bottom point of the bubble
and the plate (see Fig.~1). In this notation the surface of the
lens is described by the equation
\begin{equation}
z=R+D_1-d+a-(R^2-\rho^2)^{1/2}.
\label{eq14a}
\end{equation}
\noindent
Using Eqs.~(\ref{eq14}) and (\ref{eq14a}) one arrives, instead
of Eq.~(\ref{eq7}), at
\begin{eqnarray}
F_{sp}(a,T)&=&2\pi\int_{a+D_1}^{a+D_1-d+D}\!\!\!
(R-z+D_1-d+a)P(z,T)dz
\nonumber \\
&&+
2\pi\int_{a}^{a+D_1}(R_1-z+a)P(z,T)dz.
\label{eq15}
\end{eqnarray}
\noindent
Now we take into consideration that the quantities
$a$, $d$, and $D_1$ are smaller than the error in the
determination of large radii $R$ and $R_1$.
Then one can rearrange Eq.~(\ref{eq15}) to the form
\begin{equation}
\!\!\!\!\!\!\!\!\!\!
F_{sp}(a,T)\approx 2\pi\int_{a+D_1}^{a+D_1-d+D}\!\!\!
(R-z)P(z,T)dz
+2\pi R_1\int_{a}^{a+D_1}P(z,T)dz.
\label{eq16}
\end{equation}
\noindent
Here, the first integral on the right-hand side is calculated
similar to Eqs.~(\ref{eq7}) and (\ref{eq9}) leading to
$2\pi R{\cal F}_{pp}(a+D_1,T)$. Calculating the second integral
with the help of Eq.~(\ref{eq8}), one finally obtains
\begin{equation}
F_{sp}(a,T)\approx 2\pi (R-R_1){\cal F}_{pp}(a+D_1,T)+
2\pi R_1{\cal F}_{pp}(a,T).
\label{eq17}
\end{equation}
\begin{figure*}[t]
\vspace*{-15.6cm}
\hspace*{-1.cm}\includegraphics{figLS-2.ps}
\vspace*{-7.cm}
\caption{
The normalized Casimir force acting between a sphere with
surface imperfections of different types and a plate as
a function of separation. Lines 1, 2, and 3 are for the
surface inperfections shown in Figs.~1 and 3(a,b),
respectively.
}
\end{figure*}
We present a few computational results demonstrating that the
Casimir force in Eq.~(\ref{eq17}) taking the flattening of a
lens surface into account deviates significantly from the Casimir
force $F_{sp}^{\rm perf}$ in Eq.~(\ref{eq10}) obtained for
perfect spherical surface. Computations of the quantity
$F_{sp}(a,T)/F_{sp}^{\rm perf}(a,T)$ were performed using
Eq.~(\ref{eq13}) at $T=300\,$K within the separation region
from 1 to $3\,\mu$m (see the line labeled 1 in Fig.~2).
As can be seen in Fig.~2, in the presence of a bubble leading to
a flattening of lens surface shown in Fig.~1, the use of
Eq.~(\ref{eq10}) for perfect spherical surface instead of
Eq.~(\ref{eq17}) considerably underestimates the magnitude of
the Casimir force. Thus, at separations $a=1.0$, 1.5, 2.0,
2.5, and $3.0\,\mu$m the quantity $F_{sp}/F_{sp}^{\rm perf}$
is equal to 1.458, 1.361, 1.287, 1.233, and 1.193,
respectively, i.e., the underestimation varies from 46\% at
$a=1\,\mu$m to 19\% at $a=3\,\mu$m.
\begin{figure*}[b]
\vspace*{-11.cm}
\hspace*{-5.cm}\includegraphics{figLS-3.ps}
\vspace*{-15.5cm}
\caption{
The configuration of a spherical lens with curvature radius $R$
possessing a surface imperfection at the point of closest
approach to a plate. (a) The bubble curvature radius is $R_1<R$.
(b) The pit curvature radius is $R_1<R$.
The relative sizes of the lens and imperfection are shown
not to scale.
}
\end{figure*}
Now we consider two more examples of surface imperfection,
specifically, a bubble with the curvature radius $R_1=5\,$cm
[see Fig.~3(a)] and a pit with the curvature radius $R_1=12\,$cm
[see Fig.~3(b)]. In both cases the curvature radius of the
lens remains the same $R=15\,$cm. For the bubble we choose
$D_1=1\,\mu$m which results in $r\approx 0.32\,$mm,
$d\approx 0.33\,\mu$m, and $D_1-d\approx 0.67\,\mu$m
in agreement with allowed values.
Equation (\ref{eq17}) is evidently preserved with new values
of parameters. The computed values of the quantity
$F_{sp}(a,T)/F_{sp}^{\rm perf}(a,T)$ as a function of separation
are shown by the line labeled 2 in Fig.~2. It can be seen that
in this case the assumption of perfect sphericity of a lens
surface considerably overestimates the magnitude of the
Casimir force. Thus, at separations $a=1.0$, 1.5, 2.0,
2.5, and $3.0\,\mu$m the values of the
quantity $F_{sp}/F_{sp}^{\rm perf}$
are equal to 0.429, 0.507, 0.580, 0.641, and 0.689,
respectively, i.e., overestimation varies from 57\% at
$a=1\,\mu$m to 36\% at $a=3\,\mu$m.
Now we deal with a pit shown in Fig.~3(b). Here, the lens surface
near the point of closest approach to the plate is concave up,
i.e., in the direction of lens center. The related parameters are
$D_1=1\,\mu$m, $r\approx 0.49\,$mm,
$d\approx 0.8\,\mu$m, and $d+D_1\approx 1.8\,\mu$m.
The pit surface is described by the equation
\begin{equation}
z=a+D_1-R_1+(R_1^2-\rho^2)^{1/2}.
\label{eq18}
\end{equation}
\noindent
Here, $a$ is the separation distance between the plate and points
of a circle on the lens surface closest to it.
The surface of the
lens is described as
\begin{equation}
z=R+a-d-(R^2-\rho^2)^{1/2}.
\label{eq19}
\end{equation}
\noindent
Repeating calculations that have led to Eq.~(\ref{eq17})
with the help of Eqs.~(\ref{eq18}) and (\ref{eq19}),
we obtain
\begin{equation}
F_{sp}(a,T)\approx 2\pi (R-R_1){\cal F}_{pp}(a,T)+
2\pi R_1{\cal F}_{pp}(a+D_1,T).
\label{eq20}
\end{equation}
The computational results using Eqs.~(\ref{eq10}), (\ref{eq13})
and (\ref{eq20}) are shown by the line labeled 3 in Fig.~2.
Once again, the assumption of perfect lens sphericity
significantly overestimates the magnitude of the
Casimir force. Thus, at separations $a=1.0$, 1.5, 2.0,
2.5, and $3.0\,\mu$m the
ratio $F_{sp}/F_{sp}^{\rm perf}$
is equal to 0.314, 0.409, 0.496, 0.570, and 0.627,
respectively, i.e., overestimation varies from 69\% at
$a=1\,\mu$m to 37\% at $a=3\,\mu$m.
To conclude this section, we have shown that depending on the
character of imperfections on the lens surface near the
points of closest approach to the plate, the use of the PFA
in the simplest form (\ref{eq10}) can lead to either
underestimated or overestimated Casimir force by a few tens of
percent. Keeping in mind that the exact position of the point
of closest approach on a spherical surface cannot be controlled
with sufficient precision, it seems impossible to determine
the character of surface imperfections near the
point of closest approach microscopically for subsequent
use of Eqs.~(\ref{eq17}) or (\ref{eq20}). This leads us
to the conclusion that measurements of the Casimir force
by using spherical lenses of centimeter-size
curvature radii are
fundamentally flawed in the sense that they can lead to
unpredictable measurement results which cannot be reliably
compared with theory.
\section{Further objectionable features}
\subsection{Whether or not the Casimir force is simply the
retarded van der Waals force?}
Reference \cite{1} gives negative answer to this question
because ``the Casimir force does not depend on the properties of
the individual atoms of the plates, but on their bulk
properties''.
The same is, however, correct for the van der Waals force.
It is common knowledge that the Lifshitz theory expresses
both the van der Waals and Casimir forces in terms of the
dielectric permittivity of plate material which is a bulk
property. The Lifshitz theory presents the unified
description of the van der Waals and Casimir forces for
both dielectric and metallic plates. According to this
theory, the van der Waals force occurs in the nonretarded
limit. With the increase of separation after some transition
regime, the van der Waals force transforms into the Casimir force
in the retarded limit. Reference \cite{1} argues that
``if the Casimir force was simply the retarded van der Waals
force it would make little sense consider modifying the
Casimir force, in a fundamental way, by altering the mode
structure imposed by specially tailored boundary
conditions''. This argument, however, does not work
because in the classical Ref.~\cite{31} the nonretarded
van der Waals force was just obtained from the mode structure
determined by boundary conditions. Because of this,
in the configuration of two material bodies
separated with a gap
it is beyond reason to make difference between the
Casimir force and retarded van der Waals force.
\subsection{Is there conflict between the Casimir effect and
electrical engineering?}
Reference \cite{1} applies classical Maxwell equations in
combination with the plasma model at low frequency and
arrives at an effect that is not experimentally observed.
Basing on this, it is concluded that ``we are faced with
discarding over a century of electrical engineering
knowledge in order to explain a few 1\% level Casimir
force of questionable accuracy''. This conclusion is,
however, unjustified. Electrical engineering deals with
real electromagnetic fields. It is a matter of common
knowledge that
classical Maxwell equations in the quasistatic limit lead to
the Drude model dielectric permittivity which is inverse
proportional to the frequency, and not to the plasma
model which is only applicable in the range of infrared
frequencies. This fact is underlined (see, for instance,
Refs.~\cite{2,5,32}) when the plasma model is used for
the theoretical description of the Casimir force.
The delicate point, overlooked in Ref.~\cite{1},
is that both classical electrodynamics and
electrical engineering deal with
real electromagnetic fields, whereas the Casimir effect
deals with fluctuating electromagnetic fields possessing
zero mean value. One of the postulates of quantum
statistical physics that physical system reacts in the
same way on real and fluctuating electromagnetic fields
is presently under question from both theoretical and
experimental sides. This does not touch any fact related
to real electromagnetic fields. In view of this, it
seems baseless to write about discarding over a centure
of electrical engineering knowledge.
\subsection{Technical mistakes}
Many references, equations, and formulations in Ref.~\cite{1}
are incorrect. Below we indicate only a few examples.
In Sec.~2.4 we already wrote that Ref.~[24] in Ref.~\cite{1}
is incorrect. Another example is Ref.~[7] in Ref.~\cite{1}.
According to it, there is a Comment by three authors
on Lamoreaux's paper \cite{30}. In fact there was a
Comment \cite{33} by the two authors and
Lamoreaux's reply \cite{34}.
Equation (5) in Ref.~\cite{1} for the Matsubara frequencies
is incorrect because of missing multiple 2 on the right-hand
side.
In Eq.~(6) of Ref.~\cite{1} the velocity of light $c$ in
the denominator on the right-hand side must be deleted
because the electrical conductivity $\sigma$ is measured
in $\mbox{s}^{-1}$ [the author uses CGSE units, where the
dielectric permittivity expressed by his Eq.~(6) must be
dimensionless]. For the same reason $c$ in
the denominator on the right-hand side of Eq.~(8)
in Ref.~\cite{1} must be
replaced with $c^2$.
According to the explanations below Eq.~(7) of Ref.~\cite{1},
which introduces the generalized plasma model,
``$\varepsilon$ is the usual Drude model permittivity,
for example...'' This is, however, incorrect.
Here, $\varepsilon$ is the dielectric permittivity
describing the interband transitions of core electrons
with nonzero oscillator frequencies. Thus, the
Drude-like term is excluded. Then, according to Ref.~\cite{1},
Eq.~(7) ``is assumed to be valid at very high frequencies,
much above the resonances in the system of atoms and charges
that comprise the plates''. This is also incorrect.
For real electromagnetic fields the generalized plasma
model is valid in the region of infrared optics, which is
below the resonances describing interband transitions,
and in the region of these resonances as well.
According to Ref.~\cite{1}, ``Until now, no significant
or non-trivial corrections to the Casimir force due to
boundary modifications have been observed
experimentally.'' Concerning the work \cite{35} on
the Casimir force between a Au coated sphere and a Si
plate structured with rectangular corrugations, it is
recognized that it presents ``a convincing measurement
of a non-trivial geometrical influence on the Casimir
force'' (the reference to this work in Ref.~\cite{1}
contains mistakes). As was noted in Ref.~\cite{1},
however, the calculations in Ref.~\cite{35}
``were not for real materials.''
Contrary to what is stated in Ref.~\cite{1}, there is
a work \cite{25}, where nontrivial geometrical effects
were observed and found to be in agreement with exact
scattering theory in the configuration of a
sinusoidally corrugated sphere above a sinusoidally
corrugated plate. In so doing real material properties
were taken into account and computations were done at
the laboratory temperature $T=300\,$K.
This work was not mentioned
in Ref.~\cite{1}.
\section{Conclusions and discussion}
In the foregoing we have drawn attention to a few facts
relevant to the progress in measurements of the Casimir
force. We have discussed what is incorrect in the
argumentation of Ref.~\cite{1} against precise experiments
performed up to date and clarified some terminology
of metrological character.
The main new result of this paper is that all measurements
of the Casimir force with centimeter-size spherical lenses are
fundamentally flawed due to unavoidable deviations from
a spherical shape arising from their manufacture.
We have demonstrated that bubbles and pits satisfying
constraints imposed by the optical surface specification
data make inapplicable the simplified formulation of
the PFA commonly used in the literature. We have also derived the
expressions for the Casimir force applicable in the presence
of bubbles or pits. It was shown that surface imperfections
may lead to both a decrease and increase in the magnitude
of the Casimir force up to a few tens of percent
in the range of separations
from 1 to $3\,\mu$m. Keeping in mind that experimentally
it is impossible to determine
the position of points of closest approach between
interacting surfaces with sufficient precision, this in fact
renders lenses of centimeter-size curvature radii
impractical for measurements of the Casimir force.
There might be additional problems in the application of
spherical lenses of centimeter-size curvature radii
for measuring the Casimir force which are not discussed
here. We assert, however, that until the problem of
deviations of lens surface from perfect spherical shape
is somehow resolved any further measurements of the
Casimir force using large spherical lenses are meaningless.
Thus, the single remaining candidate for the registration
of thermal effects in the Casimir force at micrometer
separations is the configuration of two parallel plates.
The fundamental flaw discussed in this paper is not peculiar for
spheres with much smaller radii used in numerous experiments
performed by different authors
by means of an atomic force microscope and micromachined
oscillator (see review \cite{2}).
For instance, surfaces of polystyrene spheres of about
$100\,\mu$m radii made from liquid phase preserve perfect
spherical shape due to surface tension. The surface
quality of such spheres was investigated using a scanning
electron microscope and did not reveal any bubbles and
scratches.
To conclude on a positive note, we agree with Ref.~\cite{1}
that it is the reader who will finally decide which experiments in
the Casimir force area are credible and which are not
``based on verifiable facts and independent scientific
analysis.''
\ack{G.L.K.\ and V.M.M.\ are grateful to the Federal University of
Para\'{\i}ba (Jo\~{a}o Pessoa, Brazil) for kind hospitality.
They were partially supported by CNPq (Brazil).
}
\medskip
\section*{References}
\section{Introduction}
Experiments on measuring the Casimir force attracted considerable
interest and raised heated debate.
The paper\cite{1} reviews some recent work on this subject
with a reservation that
only ``credible'' experiments are considered.
As incredible, the
experiments which have claimed ``1\% or better agreement'' are
meant with a generic reference to review.\cite{2}
In different places, these unspecified
incredible experiments are characterized\cite{1} as ``1\% level work'',
experiments ``that claim 1\% accuracy'',
and ``experiments claiming 1\% precision''.
In the first part of Ref.~\refcite{1} the author provides
arguments why it is unclear to him ``what these experiments
really mean''. The second part of Ref.~\refcite{1} is largely
devoted to different aspects of author's own work employing
spherical lenses of more than 10\,cm curvature radius.
Below we demonstrate that the author's arguments against
what he
calls a ``1\% level work'' are based on incorrect or
incomplete information. With respect to measurements of
the Casimir force using lenses of centimeter-size
curvature radii,
we show that they are fundamentally flawed. According to
our calculations, experiments of this type may lead to
unpredictable results, due to
unavoidable deviations from a spherical shape of
mechanically polished and ground surfaces.
The paper is organized as follows. In Sec.~2 we
explain what is incorrect in the argumentation\cite{1}
against precise experiments on measuring
the Casimir force. We consider a relationship among
the concepts of accuracy, precision and measure of
agreement with theory, discuss total experimental error,
its constituents, and rules of their combination,
explain the misjudgement of constraints on long-range forces
made.\cite{1} In Sec.~3 it is shown that all
measurements of the Casimir force employing
centimeter-size spherical lenses are fundamentally
flawed. We discuss both the electrostatic calibrations
and the measurement of the Casimir force.
We demonstrate that commonly used simplified form of the
proximity force approximation (PFA) is inapplicable
in the presence of standard imperfections
on the optical surfaces (bubbles and pits) and derive
new expressions for the Casimir force
valid in the presence
of these imperfections. In Sec.~4 some further
objectionable features of Ref.~\refcite{1} are discussed.
Sec.~5 contains our conclusions and discussion.
\section{What Is Incorrect in the Arguments Against
Precise Experiments}
\subsection{Confusion between accuracy, precision and measure of agreement
with theory}
The present state of the art in experiments on measuring the
Casimir force is reflected in the review.\cite{2} Some results
in this review do not necessarily coincide with respective
formulations in original publications because several experiments
were later reanalyzed using more reliable methods of data
processing. At the moment only these updated results\cite{2}
should be used in all discussions.
In so doing, each experiment on measuring the
Casimir force is characterized by a total experimental error,
total theoretical error, and measure of agreement between
experiment and theory determined at some high (usually 95\%)
confidence level.\cite{2} Therefore the above cited
characterizations
of precise experiments,\cite{1} which confuse 1\%
agreement, 1\% accuracy and 1\% precision are completely
misleading. According to Ref.~\refcite{2}, the best measure of
agreement between the Casimir pressure measured in the most
precise experiment\cite{3} using a micromachined oscillator
and theory is equal to 1.8\% at the separation $a=400\,$nm
between the test bodies.
As to the best measure of agreement between the measured
Casimir force in sphere-plate geometry and theory in the
most precise experiment\cite{4} using an atomic force
microscope, it is equal\cite{5} to 5.4\% at separations around
$a=80\,$nm.
Hence the ``1\% or better agreement'' is an incorrect
information.
Of even greater concern is the mention\cite{1} of experiments
``that claim 1\% accuracy''. Actually,
there are no such experiments.\cite{2} What's
more, measuring of accuracy in percents\cite{1}
is in contradiction with
the rigorous understanding of this concept.
According to the International vocabulary of
metrology\cite{6} produced by the Joint Committee for
Guides in Metrology, measurement accuracy is the
closeness of agreement between a measured quantity value
and a true quantity value. It is underlined\cite{6}
that the concept ``measurement accuracy'' is not a quantity
and is not given a numerical value. This interpretation
is well founded because a true quantity value is in
principle unknown. A measurement is said to be more
accurate when it offers a smaller measurement error.
Thus, it is meaningless to speak about 1\% or any other
numerical degree of accuracy, and
attribute it to some unspecified experiments.\cite{1}
Another concept used\cite{1} in the same context
is the concept of precision. Measurement precision is the
closeness of agreement between measured quantity values
obtained by replicate measurements on the same or similar
objects under specified conditions.\cite{6} Precision
is expressed numerically by measures of
imprecision.\cite{6} Specifically, the experimental errors
can be used as such measures.
In the experiment\cite{3} the highest measurement
precision is achieved at the separation $a=162\,$nm
where the total relative experimental error is equal
to\cite{2}
\begin{equation}\delta_t\Pi^{\rm expt}=
\frac{\Delta_t\Pi^{\rm expt}}{|\Pi^{\rm expt}|}
=0.19\mbox{\%},
\label{eq0}
\end{equation}
\noindent
where $\Pi^{\rm expt}$ is the measured value of the
physical quantity $\Pi$ (the Casimir pressure between two
parallel plates), and $\Delta_t\Pi^{\rm expt}$
is the total absolute experimental error.
In the experiment\cite{4} the highest precision is
achieved at 63\,nm and corresponds to the total relative
experimental error\cite{2} $\delta_t\Pi^{\rm expt}=1.5$\%,
where $\Pi^{\rm expt}$ is the measured Casimir force
between a sphere and a plate.
Although it is recognized\cite{1} that
``There is a tendency among workers in this field to
confuse precision with accuracy, of which I am guilty
myself,''
the definitions presented\cite{1} continue to be
unrelated to the rigorous formulations.\cite{6}
Specifically, precision relates not to
the number of significant
figures provided by a measurement device, as erroneously
stated,\cite{1} but to a closeness between measured
quantity values in replicate measurements (the same
voltmeter, for instance, can be used in different experiments
leading to different precisions).
\subsection{Total experimental error and its constituents}
It is argued\cite{1} that ``to obtain a given experimental
accuracy, say 1\%, requires that the calibrations and force
measurements must be done to much better than 1\% accuracy...''
However, the suggested arguments that the latter is yet not
possible contain several incorrect statements.
Before we indicate each of the specific mistakes made, let us
emphasize that the word ``accuracy'' used\cite{1}
must be replaced with the word ``precision'' because, as explained
above, accuracy is not given a numerical value and there are no
experiments claiming a 1\% accuracy.
It is right that if, for instance,
the total experimental error is equal to 1\% all calibration
errors must be smaller accordingly depending on their size and
number. The list of these errors\cite{1} is, however,
again confusing by mixing experimental errors,
theoretical errors and agreement between experiment and theory.
Here, we illustrate what are the constituents of the lowest
total experimental error $\delta_t\Pi^{\rm expt}=0.19$\%
at a separation $a=162\,$nm in the most precise experiment.\cite{3}
The relative random error in the Casimir pressure at
$a=162\,$nm is\cite{2} $\delta_r\Pi^{\rm expt}=0.04$\%.
It is determined from the standard statistical procedure
using Student distribution.\cite{7} The systematic error is
caused by the errors in the measurement of the sphere radius,
of the frequency shift, and of the proximity force approximation
(which is a part of experimental procedure in the indirect
measurement of the Casimir pressure). The resulting relative
systematic error at a separation of 162\,nm is
$\delta_s\Pi^{\rm expt}=0.19$\%.
It is claimed\cite{1} that for achieving a 1\% experimental
accuracy (read precision) the sphere radius needs to be measured
to 0.5\% accuracy (precision), and that
``the radius measurement is not discussed in sufficient details
in any of papers...'' This is, however, not so. The value of the
sphere radius in the experiment\cite{3} was determined to be
$R=151.2\pm 0.2\,\mu$m leading\cite{2,3} to the relative error of only
0.13\%, i.e., smaller error than is demanded.\cite{1}
In fact all the details for determination of sphere radius by
means of electrostatic calibrations are provided.\cite{3,8,9}
The other sources of errors considered\cite{1} are
unrelated to the experimental precision. Thus, the knowledge
of the optical properties of the surfaces is not needed for
the determination of precision. The discussion of errors in
the Casimir force induced by the errors in absolute separations
bears no relation to force and pressure measurements as well.
The separation distance is an independent quantity and is
measured with its own measurement error (in Ref.~\refcite{3}
the latter is equal to 0.6\,nm).
Both these errors are important for the comparison
between experiment and theory, but have nothing to do with
the achieved experimental precision of force and pressure
measurements.\cite{2}
\subsection{Is it really uncertain how to combine different errors
and uncertainties?}
As discussed above, the total experimental error results from
the combination of random and systematic errors. In its turn,
the systematic error has several constituents.
According to Ref.~\refcite{1}, precision measurement experts still
debate whether these errors and uncertainties can be added
in quadrature or be simply added.
Regarding this statement we suggest that the author\cite{1}
was guided by outdated information.
It is common knowledge that errors and uncertainties are random
quantities and are characterized by some distributions.\cite{10}
The composition law of several random quantities depends on
the specific form of these distributions.
In the measurements of the Casimir force it is usually supposed
that all systematic errors in the form of systematic deviations
(i.e., biases in a measurement which always make the measured
value higher or lower than the true value) are already removed
using some known process, i.e., through a calibration.
The remaining systematic errors are the errors of a calibration
device and have the meaning of the smallest fractional devision
of the scale of the device. Such systematic errors are random
quantities characterized by a uniform distribution (equal
probability). The errors in an approximate theoretical formula
used to convert a directly measured quantity into an indirectly
measured one are also distributed uniformly.
Then the resulting systematic error at a chosen confidence
level $\beta$ is obtained from its constituents
$\Delta_s^{\!(i)}\Pi^{\rm expt}$ ($i=1,2,\,\ldots ,J$) using the
following statistical rule\cite{10}
\begin{equation}
\Delta_s\Pi^{\rm expt}=\min\left[
\sum_{i=1}^{J}\Delta_s^{\!(i)}\Pi^{\rm expt},
k_{\beta}^{(J)}\sqrt{
\sum_{i=1}^{J}(\Delta_s^{\!(i)}\Pi^{\rm expt})^2\,\,\,}\right].
\label{eq1}
\end{equation}
\noindent
Here, $k_{\beta}^{(J)}$ is a tabulated coefficient. The above
value of $\delta_s\Pi^{\rm expt}=0.19$\% at $a=162\,$nm in the
experiment\cite{3} (see Sec.~2.2) was obtained\cite{2,5} using Eq.~(\ref{eq1})
with $J=3$, $\beta=0.95$, and $k_{0.95}^{(3)}=1.1$.
Contrary to Ref.~\refcite{1}, statistical rules for the combination
of random and systematic errors have also been much studied.
The random error is described by the normal or Student
distribution. The resulting systematic error is described by
a combination of uniform distributions. It can be shown that
if the resulting systematic error is also assumed to be
distributed uniformly, the total experimental error will be
overestimated. Thus, this assumption is conservative and can be
used safely. There are several methods in statistcs how to
combine errors described by normal and uniform
distributions.\cite{10} A widely used method puts
\begin{equation}
\Delta_t\Pi^{\rm expt}=\Delta_r\Pi^{\rm expt}, \quad
\Delta_t\Pi^{\rm expt}=\Delta_s\Pi^{\rm expt},\quad
\Delta_t\Pi^{\rm expt}=q_{\beta}(r)\left[\Delta_r\Pi^{\rm expt}
+\Delta_s\Pi^{\rm expt}\right]
\label{eq2}
\end{equation}
\noindent
depending on what respective inequality is fulfiled for all $a$
over the entire measurement range
$r(a)<0.8$, $r(a)>8$, or
$0.8\leq r(a)\leq 8$.
Here, the quantity $r(a)$ is defined as
$r(a)={\Delta_s\Pi^{\rm expt}(a)}/{s_{\bar\Pi}(a)}$,
where $s_{\bar\Pi}(a)$ is the variance of the mean
of a quantity $\Pi^{\rm expt}$.
The coefficient $q_{\beta}(r)$ at a confidence level
$\beta=0.95$ varies between 0.71 and 0.81 depending on the
value of $r(a)$. Note that the value of the relative total error
in the experiment\cite{3} at $a=162\,$nm was obtained using the
second equality in (\ref{eq2}).
We emphasize that the dominance of the resulting systematic error
over the random error within the entire measurement range
achieved in the experiment\cite{3} is the distinguishing
feature of precise experiments of a metrological quality.
{}From the above it is seen that precision measurement
experts have gone far beyond debates whether uncertainties can be added
in quadrature or simply added.
\subsection{Misjudgement of constraints on long-range forces following
from the most precise Casimir experiment}
The measure of agreement between the Casimir pressures measured in
the most precise experiment\cite{3} and calculated theoretically
was used to obtain the strongest constraints on the parameters of
long-range Yukawa-type forces in the interaction range of several
tens of nanometers.\cite{3,5} In doing so the Yukawa pressure was
calculated\cite{3} by the application of the PFA.
The paper\cite{1} informs the reader that the use of this
approximation has been criticized.\cite{11}
The conclusions\cite{11}
that the PFA ``only applies to a force that depends
on the location of body surfaces'' and ``is not valid for
the volume integral required for calculating the anomalous
force'' are repeated.\cite{1}
This conclusion is, however, incorrect as is demonstrated in
available literature overlooked by the author.\cite{1}
Thus, it is shown\cite{12} that the PFA is applicable
for the calculation of the Yukawa force under conditions
that the separation $a$ and interaction range $\lambda$ are much
smaller than the sphere radius $R$ and the plate thickness $D$.
All these conditions are satisfied with a large safety margin
in the experimental setup.\cite{3}
The respective Yukawa pressure in the setup\cite{3}
was calculated\cite{13} both exactly and using the PFA with coinciding
results. The purported ``corrections'' to the calculation\cite{12}
pointed out\cite{11} were shown to
be invalid and based on a simple misunderstanding.\cite{13}
What's more, one of the authors\cite{11} (R.O.)
recognized\cite{14} that the issue raised in their paper
``is not of practical concern for current experiments''.
Thus, the reliability of constraints on the parameters
of Yukawa interactions obtained\cite{3} is
beyond doubt.
\section{Why Casimir Force Measurements Using Centimeter-Size
Spherical Lenses Are Fundamentally Flawed}
\subsection{Anomalies in electrostatic calibrations}
Observations of anomalous electrostatic forces in the lens-plate
geometry for lenses of centimeter-size curvature radii is the
subject of wide speculation.\cite{15}\cdash\cite{17}
It was shown\cite{18} that anomalous behavior of the
electrostatic force can be explained due to deviations
of the mechanically polished and ground surfaces of
centimeter-size lenses from a perfect spherical form.
The point is that the typical surface of a centimeter-size lens
is characterized in terms of scratch and dig optical
surface specification data. In particular, depending on the
quality of lens used, bubbles and pits with a diameter
varying from $30\,\mu$m to 1.2\,mm are allowed on the
surface.\cite{19} There may be also scratches\cite{19} with a width
varying from 3 to $120\,\mu$m. The problem of
bubbles on the centimeter-size lens surface should not be
reduced to the fact that lens curvature radius $R$ is
determined with some error. The thickness of each bubble should
of course be less than the absolute error in the measurement of lens
curvature radius (for a lens\cite{16} with $R=15.10\,$cm,
for instance, $\Delta R=0.05\,$cm). The crucial point is that
curvature radii of bubbles can be orders of magnitude different,
as compared to $R$. This allows one to suggest models leading to
quite different (``anomalous'') dependence of electrostatic force
on separation in comparison with the case of perfect spherical
surfaces.\cite{18}
Paper\cite{1} mentions the possibility that the anomalous
electrostatic forces are due to simple geometrical effects
without citing the source of this idea,
but claims that this possibility ``is credibly
discarded''
in Ref.~\refcite{20}. The author\cite{1} does not inform the
reader that computations\cite{20} were repeated\cite{21}
and shown to be not reproducible.
Thus, there is no scientific objection against the possibility
that anomalous electrostatic forces are due to deviations of
mechanically polished and ground surfaces from perfect sphericity.
Furthermore, some of the authors\cite{20}
(D.A.R.D.\ and R.O.) recently recognized\cite{22} that local
geometrical deformations of the surface can really lead to an
anomalous electrostatic force not only in sphere-plate
geometry, but for a cylindrical lens in close proximity to
the plate as well. It was underlined\cite{22} that
``this is certainly a crucial point to be taken into account
in future experiments.'' This reference, however, is missing
in the list of references.\cite{1}
An extensive consideration\cite{1} of the electrostatic
calibrations always assumes perfect sphericity of the
lens surface.
Another misrepresented point directly relevant to electrostatic
calibrations is the dependence of the contact potential on the
separation distance. It is claimed\cite{1} that every paper
on the Casimir effect ``that has bothered measuring the contact
potential as a function of distance has shown an apparent
distance dependence of that potential''. This is, however, not
the case. Thus, the contact potential was carefully
measured as a function of separation\cite{3} and found to be constant.
The respective measurement data of the electrostatic
calibrations are published.\cite{9,18} Constant
contact potential was observed in all other experiments by
R.\ S.\ Decca as well.\cite{2,5} Independent on separation contact
potential was also reported in experiments
by U.~Mohideen.\cite{2,4,5,23}\cdash\cite{25}
It is notable that
all these experiments were performed in high vacuum with small
spheres of order $100\,\mu$m curvature radii.
\subsection{Influence of surface imperfections on the Casimir force
for lenses of centimeter-size curvature radius}
The Casimir force is far more sensitive than the electrostatic
force to the bubbles and pits that are unavoidably present
on the mechanically polished and ground surface of any lens
of centimeter-size curvature radius. The physical reason is that
the Casimir force falls with the increase of separation distance
more rapidly than the electric force.
As a result, it is determined by smaller regions near the points
of closest approach of the surfaces. If the local curvature
radius on the lens surface near the point of closest approach to
the plate is significantly different from the mean lens
curvature radius $R$, the impact on the Casimir force can be
tremendous. Below we demonstrate that just this happens due to
the presence of bubbles and pits on a lens surface.
For simplicity, we consider ideal metal surfaces.
However, it is shown\cite{25a} that all conclusions obtained are
preserved for real bodies as well.
The Casimir force in sphere-plate geometry under the experimental
conditions $a\ll R$ is usually calculated using the PFA.\cite{2,5}
According to the most general formulation of the PFA,\cite{27}
the unknown force between the elements of curved surfaces is
approximately replaced with a known force per unit area of the plane
surfaces (i.e., a pressure) at the respective separation
multiplied by an area element.
Applied to a spherical lens of thickness $D$ above a plane
$z=0$, the PFA represents the force between them in the form
\begin{equation}
F_{sp}(a,T)=\int_{\Sigma}d\sigma P(z,T).
\label{eq5}
\end{equation}
\noindent
Here, $d\sigma$ is the element of plate area, $\Sigma$ is the
projection of the lens onto the plate, $a$ is the shortest
separation between them, $z=z(x,y)$ is the equation of a lens
surface, and $P(z,T)$ is the pressure for two
plane parallel plates at a separation $z$ at temperature $T$.
We choose the origin of a cylindrical coordinate system on
the plane $z=0$ under the lens center. Then for a perfectly
shaped spherical lens the coordinate $z$ of any point
of its surface is given by
\begin{equation}
z=R+a-(R^2-\rho^2)^{1/2}, \quad
\rho^2=x^2+y^2.
\label{eq6}
\end{equation}
\noindent
In this case (\ref{eq5}) leads to
\begin{equation}
\!\!\!\!\!\!\!\!\!\!\!\!\!\!
F_{sp}^{\rm perf}(a,T)=2\pi\int_{0}^{\sqrt{2RD-D^2}}\!\!\!\rho d\rho P(z,T)=
2\pi\int_{a}^{D+a}(R+a-z)P(z,T)dz.
\label{eq7}
\end{equation}
\noindent
Keeping in mind that the Casimir pressure is expressed as
$P(z,T)=-{\partial{\cal F}_{pp}(z,T)}/{\partial z}$,
where ${\cal F}_{pp}(z,T)$ is the free energy per unit area of parallel
plates, and integrating by parts in (\ref{eq7}), one arrives
at
\begin{eqnarray}
F_{sp}^{\rm perf}(a,T)&=&2\pi R{\cal F}_{pp}(a,T)-2\pi(R-D){\cal F}_{pp}(D+a,T)
\nonumber\\
&&
-
2\pi\int_{a}^{D+a}\!\!{\cal F}_{pp}(z,T)dz.
\label{eq9}
\end{eqnarray}
\noindent
We consider centimeter-size spherical lenses satisfying
a condition $a\ll D$. For such lenses
${\cal F}_{pp}(D+a,T)\ll{\cal F}_{pp}(a,T)$.
Because of this, one can neglect the second term on the right-hand
side of (\ref{eq9}) in comparison with the first.\cite{28}
It can be also shown\cite{28,29} that the first term on the right-hand
side of (\ref{eq9}) is in excess of the third by a factor of $R/a$.
This allows one to neglect the third term and arrive to what is
called the simplified formulation of the PFA\cite{28,29}
\begin{equation}
F_{sp}^{\rm perf}(a,T)\approx 2\pi R{\cal F}_{pp}(a,T)
\label{eq10}
\end{equation}
\noindent
widely used for both spherical lenses and for spheres [note that for
a semisphere the second term on the right-hand
side of (\ref{eq9}) is identically equal to zero].
For two parallel ideal metal plates spaced $z$ apart the Casimir
free energy per unit area is given by\cite{5,26}
\begin{equation}
{\cal F}_{pp}(z,T)=\frac{k_BT}{\pi}\sum_{l=0}^{\infty}
{\vphantom{\sum}}^{\!\prime}\int_{0}^{\infty}k_{\bot}dk_{\bot}
\ln(1-e^{-2zq_l}).
\label{eq11}
\end{equation}
\noindent
Here, $k_B$ is the Boltzmann constant, $k_{\bot}$ is the magnitude
of the projection of the wave vector on the plates,
$q_l^2=k_{\bot}^2+\xi_l^2/c^2$, $\xi_l=2\pi k_BTl/\hbar$ with
$l=0,\,1,\,2,\,\ldots$ are the Matsubara frequencies, and the
primed summation sign means that the term with $l=0$ is
multiplied by 1/2. For the sake of convenience in computations,
we rewrite Eq.~(\ref{eq11}) in terms of a dimensionless
integration variable $y=2aq_l$ and expand the logarithm in power
series.
Then, after performing integration
and the summation with respect to $l$, the following
result is obtained:
\begin{equation}
{\cal F}_{pp}(z,T)=-\frac{k_BT}{4\pi z^2}\left[
\frac{\zeta(3)}{2}+
\sum_{n=1}^{\infty}\frac{e^{-\tau_zn}}{n^2(1-e^{-\tau_zn})}
\left(\frac{1}{n}+\frac{\tau_z}{1-e^{-\tau_zn}}\right)\right],
\label{eq13}
\end{equation}
\noindent
where $\zeta(x)$ is the Riemann zeta function, and
$\tau_z=4\pi zk_BT/(\hbar c)$. Note that the
first contribution on the right-hand side of (\ref{eq13})
coincides with the high temperature limit of the free energy.
This is quite reasonable if to take into account that
$\tau_z=2\pi T/T_{\rm eff}$, where the effective temperature
is defined from $k_BT_{\rm eff}=\hbar c/(2z)$.
Now we are in a position to compute the Casimir force between real
spherical lens of large curvature radius with bubbles and pits
of different types and a plane plate. It is common to use the
simplified formulation of the PFA (\ref{eq10}) in sphere-plate
geometry for both small spheres of about $100\,\mu$m radii
and large spherical lenses (see, for instance,
Refs.~\refcite{3,4,16,30,30a}). In doing so the role of bubbles and
pits on the surface of lenses of centimeter-size curvature
radii is simply disregarded. Equation (\ref{eq10}), however,
is not applicable for real lenses with large curvature radii
because it assumes perfect spherical surface.
For such lenses one should use a general formulation of the
PFA in Eq.~(\ref{eq5}). To illustrate this fact, we perform
calculations for three typical model imperfections on the
spherical surface near the point of closest approach to the
plate allowed by the optical surface specification
data.\cite{19}
\begin{figure*}[t]
\vspace*{-4.5cm}
\centerline{\hspace*{1cm}\psfig{file=figVM-1.ps,width=20cm}}
\vspace*{-19.6cm}
\caption{
(a) The configuration of a spherical lens with curvature radius $R$
possessing a surface imperfection at the point of closest
approach to a plate. The bubble curvature radius is $R_1>R$.
The relative sizes of the lens and imperfection are shown
not to scale.
(b) The normalized Casimir force acting between a sphere with
surface imperfections of different types and a plate as
a function of separation. Lines 1, 2, and 3 are for the
surface inperfections shown in Figs.~1(a) and 2(a,b),
respectively.
}
\end{figure*}
As the first example, we consider a bubble of the curvature radius
$R_1=25\,$cm which is larger than the curvature radius $R=15\,$cm
of the lens used [see Fig.~1(a)]. The thickness of the spherical
lens formed by the bubble is chosen to be $D_1=0.5\,\mu$m
(this is much less than typical absolute error
$\Delta R=0.05\,$cm in the measurement of centimeter-size
lenses curvature radius). The radius of the bubble is
determined from $r^2=2R_1D_1-D_1^2\approx 0.25\,\mbox{mm}^2$,
leading to $2r=1\,\mbox{mm}<1.2\,$mm, i.e., less than
a maximum value allowed by the optical surface specification
data.\cite{19} Respective quantity $d$ defined in Fig.~1(a) is equal to
$d\approx r^2/(2R)\approx 0.83\,\mu$m. Then the flattening
of a lens surface at the point of closest approach to the
plate is $d-D_1\approx 0.33\,\mu$m which is much less than
$\Delta R$.
The general formulation of the PFA (\ref{eq5}) should be applied
taking into account that the surfaces of the bubble and of the
lens are described
by the equations
\begin{equation}
z=R_1+a-(R_1^2-\rho^2)^{1/2},\quad
z=R+D_1-d+a-(R^2-\rho^2)^{1/2},
\label{eq14}
\end{equation}
\noindent
respetively.
Here $a$ is the distance between the bottom point of the bubble
and the plate [see Fig.~1(a)].
Using (\ref{eq14}) one arrives, instead
of (\ref{eq7}), at
\begin{equation}
F_{sp}(a,T)=2\pi\int_{a+D_1}^{a+D_1-d+D}\!\!\!\!\!\!\!\!\!\!\!
(R-z+D_1-d+a)P(z,T)dz
+
2\pi\int_{a}^{a+D_1}\!\!\!\!\!\!(R_1-z+a)P(z,T)dz.
\label{eq15}
\end{equation}
\noindent
Now we take into consideration that the quantities
$a$, $d$, and $D_1$ are smaller than the error in the
determination of large radii $R$ and $R_1$.
Then, calculating
similar to (\ref{eq7}) and (\ref{eq9}), one finally obtains
\begin{equation}
F_{sp}(a,T)\approx 2\pi (R-R_1){\cal F}_{pp}(a+D_1,T)+
2\pi R_1{\cal F}_{pp}(a,T).
\label{eq17}
\end{equation}
We present a few computational results demonstrating that the
Casimir force in Eq.~(\ref{eq17}) taking the flattening of a
lens surface into account deviates significantly from the Casimir
force $F_{sp}^{\rm perf}$ in Eq.~(\ref{eq10}) obtained for
perfect spherical surface. Computations of the quantity
$F_{sp}(a,T)/F_{sp}^{\rm perf}(a,T)$ were performed using
(\ref{eq13}) at $T=300\,$K within the separation region
from 1 to $3\,\mu$m [see the line labeled 1 in Fig.~1(b)].
As can be seen in Fig.~1(b), in the presence of a bubble leading to
a flattening of lens surface shown in Fig.~1(a), the use of
(\ref{eq10}) for a perfect spherical surface instead of
(\ref{eq17}) considerably underestimates the magnitude of
the Casimir force. Thus, at separations $a=1.0$, 2.0,
and $3.0\,\mu$m the quantity $F_{sp}/F_{sp}^{\rm perf}$
is equal to 1.458, 1.287, and 1.193,
respectively, i.e., the underestimation varies from 46\% at
$a=1\,\mu$m to 19\% at $a=3\,\mu$m.
\begin{figure*}[t]
\vspace*{-4.cm}
\centerline{\hspace*{-1.45cm}\psfig{file=figVM-2.ps,width=18cm}}
\vspace*{-17.5cm}
\caption{
The configuration of a spherical lens with curvature radius $R$
possessing a surface imperfection at the point of closest
approach to a plate. (a) The bubble curvature radius is $R_1<R$.
(b) The pit curvature radius is $R_1<R$.
The relative sizes of the lens and imperfection are shown
not to scale.
}
\end{figure*}
Now we consider two more examples of surface imperfections,
specifically, a bubble with the curvature radius $R_1=5\,$cm
[see Fig.~2(a)] and a pit with the curvature radius $R_1=12\,$cm
[see Fig.~2(b)]. In both cases the curvature radius of the
lens remains the same $R=15\,$cm. For the bubble we choose
$D_1=1\,\mu$m which results in $r\approx 0.32\,$mm,
$d\approx 0.33\,\mu$m, and $D_1-d\approx 0.67\,\mu$m
in agreement with allowed values.
Equation (\ref{eq17}) is evidently preserved with the new values
of parameters. The computed values of the quantity
$F_{sp}(a,T)/F_{sp}^{\rm perf}(a,T)$ as a function of separation
are shown by the line labeled 2 in Fig.~1(b). It can be seen that
in this case the assumption of perfect sphericity of a lens
surface considerably overestimates the magnitude of the
Casimir force. Thus, at separations $a=1.0$, 2.0,
and $3.0\,\mu$m the values of the
quantity $F_{sp}/F_{sp}^{\rm perf}$
are equal to 0.429, 0.580, and 0.689,
respectively, i.e., overestimation varies from 57\% at
$a=1\,\mu$m to 36\% at $a=3\,\mu$m.
Now we deal with a pit shown in Fig.~2(b). Here, the lens surface
near the point of closest approach to the plate is concave up,
i.e., in the direction of lens center. The related parameters are
$D_1=1\,\mu$m, $r\approx 0.49\,$mm,
$d\approx 0.8\,\mu$m, and $d+D_1\approx 1.8\,\mu$m.
The pit and lens surfaces are described by the equations
\begin{equation}
z=a+D_1-R_1+(R_1^2-\rho^2)^{1/2},\quad
z=R+a-d-(R^2-\rho^2)^{1/2},
\label{eq18}
\end{equation}
\noindent
respectively.
Here, $a$ is the separation distance between the plate and points
of a circle on the lens surface closest to it.
Repeating calculations that have led to (\ref{eq17})
with the help of (\ref{eq18}),
we obtain
\begin{equation}
F_{sp}(a,T)\approx 2\pi (R-R_1){\cal F}_{pp}(a,T)+
2\pi R_1{\cal F}_{pp}(a+D_1,T).
\label{eq20}
\end{equation}
The computational results using (\ref{eq10}), (\ref{eq13})
and (\ref{eq20}) are shown by the line labeled 3 in Fig.~1(b).
Once again, the assumption of perfect lens sphericity
significantly overestimates the magnitude of the
Casimir force. Thus, at separations $a=1.0$, 2.0,
and $3.0\,\mu$m the
ratio $F_{sp}/F_{sp}^{\rm perf}$
is equal to 0.314, 0.496, and 0.627,
respectively, i.e., overestimation varies from 69\% at
$a=1\,\mu$m to 37\% at $a=3\,\mu$m.
To conclude this section, we have shown that depending on the
character of imperfections on the lens surface near the
points of closest approach to the plate, the use of the PFA
in the simplest form (\ref{eq10}) can lead to either
underestimated or overestimated Casimir force by a few tens of
percent. Keeping in mind that the exact position of the point
of closest approach on a spherical surface cannot be controlled
with sufficient precision, it seems impossible to determine
the character of surface imperfections near the
point of closest approach microscopically for subsequent
use of (\ref{eq17}) or (\ref{eq20}). This leads us
to the conclusion that measurements of the Casimir force
by using spherical lenses of centimeter-size
curvature radii are
fundamentally flawed in the sense that they can lead to
unpredictable measurement results which cannot be reliably
compared with theory.
\section{Further Objectionable Features}
\subsection{Whether or not the Casimir force is simply the
retarded van der Waals force?}
The paper\cite{1} gives negative answer to this question
because ``the Casimir force does not depend on the properties of
the individual atoms of the plates, but on their bulk
properties''.
The same is, however, correct for the van der Waals force.
It is common knowledge that the Lifshitz theory expresses
both the van der Waals and Casimir forces between
macrobodies in terms of
dielectric permittivity of their materials which is a bulk
property. The Lifshitz theory presents the unified
description of both forces for
dielectric and metallic plates. According to this
theory, the van der Waals force occurs in the nonretarded
limit. With the increase of separation after some transition
regime, the van der Waals force transforms into the Casimir force
in the retarded limit. The paper\cite{1} argues that
``if the Casimir force was simply the retarded van der Waals
force it would make little sense consider modifying the
Casimir force, in a fundamental way, by altering the mode
structure imposed by specially tailored boundary
conditions''. This argument, however, does not work
because the nonretarded
van der Waals force was just obtained\cite{31} from the mode structure
determined by boundary conditions. Because of this,
in the configuration of two material bodies
separated with a gap
it is beyond reason to make difference between the
Casimir force and retarded van der Waals force.
\subsection{Is there conflict between the Casimir effect and
electrical engineering?}
The paper\cite{1} applies classical Maxwell equations in
combination with the plasma model at low frequencies and
arrives at an effect that is not experimentally observed.
Basing on this, it is concluded that ``we are faced with
discarding over a century of electrical engineering
knowledge in order to explain a few 1\% level Casimir
force of questionable accuracy''. This conclusion is,
however, unjustified. Electrical engineering deals with
real electromagnetic fields. It is a matter of common
knowledge that
classical Maxwell equations in the quasistatic limit lead to
the Drude model dielectric permittivity which is inverse
proportional to the frequency, and not to the plasma
model which is only applicable in the range of infrared
frequencies. This fact is underlined\cite{2,5,32}
when the plasma model is used for
the theoretical description of the Casimir force.
The delicate point overlooked\cite{1}
is that both classical electrodynamics and
electrical engineering deal with
real electromagnetic fields, whereas the Casimir effect
deals with fluctuating electromagnetic fields possessing
zero mean value. One of the postulates of quantum
statistical physics that physical system reacts in the
same way on real and fluctuating electromagnetic fields
is presently under question from both theoretical and
experimental sides. This does not touch any fact related
to real electromagnetic fields. In view of this, it
seems baseless to write about discarding over a centure
of electrical engineering knowledge.
\section{Conclusions and Discussion}
In the foregoing we have drawn attention to a few facts
relevant to the progress in measurements of the Casimir
force. We have discussed what is incorrect in the
argumentation\cite{1} against precise experiments
performed up to date and clarified some terminology
of metrological character.
The main new result of this paper is that all measurements
of the Casimir force with centimeter-size spherical lenses are
fundamentally flawed due to unavoidable deviations from
a spherical shape arising from their manufacture.
We have demonstrated that bubbles and pits satisfying
constraints imposed by the optical surface specification
data make inapplicable the simplified formulation of
the PFA commonly used in the literature. We have also derived the
expressions for the Casimir force applicable in the presence
of bubbles or pits. It was shown that surface imperfections
may lead to both a decrease and increase in the magnitude
of the Casimir force up to a few tens of percent
in the range of separations
from 1 to $3\,\mu$m. Keeping in mind that experimentally
it is impossible to determine
the position of points of closest approach between
interacting surfaces with sufficient precision, this in fact
renders lenses of centimeter-size curvature radii
impractical for measurements of the Casimir force.
There might be additional problems in the application of
spherical lenses of centimeter-size curvature radii
for measuring the Casimir force which are not discussed
here. We assert, however, that until the problem of
deviations of lens surface from perfect spherical shape
is somehow resolved any further measurements of the
Casimir force using large spherical lenses are meaningless.
Thus, the single remaining candidate for the registration
of thermal effects in the Casimir force at micrometer
separations is the configuration of two parallel plates.
The fundamental flaw discussed in this paper is not peculiar for
spheres with much smaller radii used in numerous experiments
performed by different authors
by means of an atomic force microscope and micromachined
oscillator (see review Ref.~\refcite{2}).
For instance, surfaces of polystyrene spheres of about
$100\,\mu$m radii made from liquid phase preserve perfect
spherical shape due to surface tension. The surface
quality of such spheres was investigated using a scanning
electron microscope and did not reveal any bubbles and
scratches.
To conclude on a positive note, we agree with Ref.~\refcite{1}
that it is the reader who will finally decide which experiments in
the Casimir force area are credible and which are not
``based on verifiable facts and independent scientific
analysis.''
\section*{Acknowledgments}
The authors are grateful to the Federal University of
Para\'{\i}ba (Jo\~{a}o Pessoa, Brazil) for kind hospitality.
They were partially supported by CNPq (Brazil).
| -48,883.277926 |
[
-2.44921875,
2.2890625
] | 19.497542 |
[
-3.087890625,
0.72216796875,
-2.46484375,
-4.77734375,
-0.352783203125,
7.19140625
] |
[
2.103515625,
6.84765625,
2.57421875,
5.75
] | 891 | 11,615 |
[
-1.1533203125,
1.119140625
] | 24.935027 |
[
-5.6640625,
-3.341796875,
-3.568359375,
-2.09375,
1.6982421875,
11.34375
] | 1.195777 | 7.821266 | 13.00043 | 2.930552 |
[
2.1497178077697754
] | -34,103.371205 | 5.471976 | -47,937.011664 | 1.888069 | 5.868797 |
[
-3.162109375,
-3.2578125,
-2.685546875,
-3.84375,
2.359375,
9.96875
] |
[
-5.2734375,
-2.01953125,
-1.8251953125,
-0.77587890625,
3.783203125,
3.98046875
] | |
BkiUduA4uzlhgyWvJ1dJ
|
\section{Introduction \& Formalism}
The recent popularity of machine learning calls for a deeper understanding of AI security. Amongst the numerous AI threats published so far, poisoning attacks currently attract considerable attention.\smallskip
An ML algorithm $\mathcal{A}$ is a state machine with a two-phase life-cycle: during the first phase, called \textsl{training}, $\mathcal{A}$ builds a \textsl{model} (captured by a state variable $\sigma_i$) based on sample data $D$, called ``training data'':
$$D=\{d_1,\ldots,d_k\} \mbox{~where~} d_i=\{\mbox{data}_i,\mbox{label}_i\}$$
Learning is hence defined by:
$$\sigma_i\leftarrow\mathcal{A}(\mbox{learn},\sigma_{i-1},d_i)$$
e.g. $\mbox{data}_i$ can be human face images and the $\mbox{label}_i\in\{\mars,\female\}$.
During the second phase, called \textsl{testing}\footnote{Or \textsl{inference}.}, $\mathcal{A}$ is given an unlabelled $\mbox{data}$. $\mathcal{A}$'s goal is to predict as accurately as possible the corresponding $\mbox{label}$ given the distribution $\mathfrak{D}$ inferred from $D$.
$$\underline{\mbox{label}}=\mathcal{A}(\mbox{test},\sigma_{k},\mbox{data})$$
We denote by $T$ the dataset $\{\mbox{data}_i,\mbox{label}_i\}$ used during testing where $\mbox{label}_i$ is the correct label (solution) corresponding to $\mbox{data}_i$ and $\underline{\mbox{label}}_i$ is the label predicted by $\mathcal{A}(\mbox{test},\sigma_{k},\mbox{data}_i)$\footnote{i.e. if $\mathcal{A}$ is perfect then \underline{\mbox{label}}=\mbox{label}.}.\smallskip
In a poisoning attack the opponent partially tampers $D$ to influence $\sigma_k$ and mislead $\mathcal{A}$ during testing. Formally, letting $\overline{d}=\{\overline{\mbox{data}},\overline{\mbox{label}}\}$, the attacker generates a \textsl{poison} dataset
$$\Tilde{D}=\{\Tilde{d}_1,\ldots,\Tilde{d}_k\}$$
resulting in a corrupted model $\Tilde{\sigma}_k$ such that
$$\overline{\mbox{label}}\neq\underline{\overline{\mbox{label}}}=\mathcal{A}(\mbox{test},\Tilde{\sigma}_k,\overline{\mbox{data}})$$
Poisoning attacks were successfully implemented by tampering both incremental and periodic training models. In the \textsl{incremental training model} \footnote{Also called the \textsl{incremental update model}.}, whenever a new $d_i$ is seen during testing, $\mathcal{A}$'s performance on $d_i$ is evaluated and $\sigma$ is updated. In the \textsl{periodic retraining model}, data is stored in a buffer. When $\mathcal{A}$ falls below a performance threshold (or after a fixed number of queries) the buffer's data is used to retrain $\mathcal{A}$ anew. Retraining is either done using the buffer alone (resulting in a totally new $\sigma$) or by merging the buffer with previous information (updating $\sigma$).\smallskip
Protections against poisoning attacks can be categorized into two types: \textsl{robustification} and \textsl{sanitizing}:
\subsubsection{Robustification} (built-in resistance) modifies $\mathcal{A}$ so that it takes into account the poison but tolerates its effect. Note that $\mathcal{A}$ does not need to identify the poisoned data as such but the effect of poisonous data must be diminished, dampened or nullified up to a point fit for purpose.\smallskip
The two main robustification techniques discussed in the literature are:\smallskip
\textsl{Feature squeezing} \cite{XEQ17,SGY+18} is a model hardening technique that reduces data complexity so that adversarial perturbations disappear because of low sensitivity. Usually the quality of the incoming data is degraded by encoding colors with fewer values or by using a smoothing filter over the images. This maps several inputs onto one ``characteristic'' or ``canonical input'' and reduces the perturbations introduced by the attacker. While useful in practice, those techniques inevitably degrade the $\mathcal{A}$'s accuracy.\smallskip
\textsl{Defense-GANs} \cite{SKC18} use Generative Adversarial Networks \cite{GPM+14} to reduce the poison's efficiency. Informally, the GAN builds a model of the learned data and projects the input onto it.\smallskip
\subsubsection{Sanitizing} detects (by various methods e.g. \cite{CSL+08,LP16}) and discards poisoned $d_i$s. Note that sanitizing necessarily decreases $\mathcal{A}$'s ability to learn. \smallskip
This work prevents poisoning by sanitizing.\smallskip
Figure \ref{learning} shows a generic abstraction of sanitizing. $\mathcal{A}$ takes $D$ (periodically or incrementally) and outputs a $\sigma$ for the testing phase. But $d_i$s go through the poisoning detection module $\mbox{\textsf{Det}}$ before entering $\mathcal{A}$. If $\mbox{\textsf{Det}}$ decides that the probability that some $d_i$ is poisoned is too high, the suspicious $d_i$ is trashed to avoid corrupting $\sigma$.\smallskip
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{crop.pdf}
\caption{Before entering $\mathcal{A}$, $D$ is given to the poison detection module $\mbox{\textsf{Det}}$. If $\mbox{\textsf{Det}}$ decides that $D$ is poisoned, $D$ is trashed. Otherwise $D$ is fed into $\mathcal{A}$ who updates $\sigma$.} \label{learning}
\end{figure}
Because under normal circumstances $D$ and $T$ are drawn from the same distribution $\mathfrak{D}$ it is natural to implement $\mbox{\textsf{Det}}$ using standard algorithms allowing to test the hypothesis $D\in\mathfrak{D}$.\smallskip
The most natural tool allowing one to do so is \textsl{nonparametric hypothesis tests} (NPHTs, hereafter denoted by $G$). Let $A,B$ be two datasets. $G(A,B)\in\{\mbox{\textsf{T}},\mbox{\textsf{F}}\}$ allows to judge how compatible is a difference observed between $A$ and $B$ with the hypothesis that $A,B$ were drawn from the same distribution $\mathfrak{D}$.\smallskip
It is important to underline that $G$ is nonparametric, i.e. $G$ makes no assumptions on $\mathfrak{D}$.\smallskip
The above makes NPHTs natural candidates for detecting poison. However, whilst NPHTs are very good for natural hypothesis testing, they succumb spectacularly in adversarial scenarios where the attacker has full knowledge of the target's specification \cite{Kerk}. Indeed, section \ref{whatever} illustrates such a collapse.\smallskip
To regain a head-up over the attacker, our strategy will consist in mapping $A$ and $B$ into a \textsl{secret} space unpredictable by the adversary where $G$ can work confidentially. This mapping is defined by a key $\kappa$ making it hard for the adversary to design $A\in\mathfrak{D}$ and $B\in\mathfrak{D}'$ such that
$$G(A,B)=\mbox{\textsf{T}}\mbox{~~and~~}\mathfrak{D}\neq\mathfrak{D}'$$
\section{A Brief Overview of Poisoning Attacks}
Barreno et al. \cite{BNJT10} were the first to coin the term ``poisoning attacks''. Follow-up works such as Kearns et al. \cite{KL93} sophisticated and theorized this approach.\smallskip
Classical references introducing poisoning are \cite{PDL+06,NKS06,NBC+08,RNH+09,BNL12,XXE12,NPXN14,XBB+15,MZ15,BL17,KL17}. At times (e.g. \cite{KL17}) the opponent does not create or modify $d_i$'s but rather adds legitimate but carefully chosen $d_i$'s to $D$ to bias learning. Those inputs are usually computed using gradient descent. This was later generalized by \cite{SKC18}.\smallskip
During a poisoning attack, data modifications can either concern data or labels. \cite{BNL11} showed that a random flip of 40\% of labels suffices to seriously affect SVMs. \cite{KL10} showed that inserting malicious points into $D$ could gradually shift the decision boundary of an anomaly detection classifier. Poisoning points were obtained by solving a linear programming problem maximizing the mean of the displacement of the mass center of $D$. For a more complete overview we recommend \cite{BR18}.\smallskip
\subsubsection{Adversarial Goals.} Poisoning may seek to influence the classifier's decision when presented with a later target query or to leak information about $D$ or $\sigma$.\smallskip
The attacker's goals always apply to the testing phase and may be:
\begin{itemize}
\item \textsl{Confidence Reduction}: Have $\mathcal{A}$ make more errors. In many cases, ``less confidence'' can clear suspicious instances at the benefit of doubt (\textsl{in dubio pro reo}).
\item \textsl{Mis-classification attacks}: are defined by replacing $\mbox{\textsf{adj}}_1,\mbox{\textsf{adj}}_2$ in the definition:\smallskip
\begin{center}
``Make $\mathcal{A}$ conclude that a $\mbox{\textsf{adj}}_1$ $\mbox{data}_i$ belongs to a $\mbox{\textsf{adj}}_2$ wrong label.''
\end{center}
\end{itemize}
\begin{center}
\begin{tabular}{|l|c|c|}\hline
\textbf{~~Attack} & $\mbox{\textsf{adj}}_1$ & $\mbox{\textsf{adj}}_2$ \\\hline
~~Mis-classification\footnote{This is useful if any mistake may serve the opponent's interests e.g. any navigation error would crash a drone with high probability.} &~~random~~&~~random~~\\\hline
~~Targeted Mis-classification &~~chosen~~&~~random~~\\\hline
~~Source-Target Mis-classification~~&~~chosen~~&~~chosen~~\\\hline
\end{tabular}
\end{center}
\subsubsection{Adversarial capabilities} designate the degree of knowledge that the attacker has on the target system. Bibliography distinguishes between \textsl{training phase capabilities} and \textsl{testing phase capabilities}. Poisoning assumes training phase capabilities.\smallskip
The attacker's capabilities may be:
\begin{itemize}
\item \textsl{Data Injection}: Add new data to $D$.
\item \textsl{Data Modification}: Modify $D$ before training.
\item \textsl{Logic Corruption}: Modify the code (behavior) of $\mathcal{A}$\footnote{This is the equivalent of fault attacks in cryptography.}.
\end{itemize}
\section{Keyed Anti-Poisoning}
To illustrate our strategy, we use Mann-Whitney's $U$-test and Stouffer's method that we recall in the appendix.\smallskip
We assume that when training starts, we are given a safe subset of $D$ denoted $D_s$ (where the subscript $s$ stands for ``safe''). Our goal is to assess the safety of the upcoming subset of $D$ denoted $D_u$ (where the subscript $u$ stands for ``unknown'').
We assume that $D_s$ and $T$ come from the same distribution $\mathfrak{D}$. As mentioned before, the idea is to map $D_s$ and $D_u$ to a space $f_{\kappa}(\mathfrak{D})$ hidden from the opponent. $f$ is keyed to prevent the attacker from predicting how to create adversarial input fooling $\mathcal{A}$.\smallskip
Figure \ref{keyed} shows the details of the $\mbox{\textsf{Det}}$ plugin added to $\mathcal{A}$ in Figure \ref{learning}. $\mbox{\textsf{Det}}$ takes a key $\kappa$, reads $D_s,D_u$, performs the keyed transformation, calls $G$ on $f_{\kappa}(D_u),f_{\kappa}(D_s)$ and outputs a decision.\smallskip
$G$ can be Mann-Whitney's test (illustrated in this paper) or any other NPHT e.g. the location test, the paired $T$ test, Siegel-Turkey's test, the variance test, or multidimensional tests such as deep gaussianization \cite{Tolpin:2019:PAD:3297280.3297414}.\smallskip
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{crop2.pdf}
\caption{Implementing \mbox{\textsf{Det}}~using keying. \textcolor{red}{Red}: unknown to the opponent. \textcolor{orange}{Orange}: known by the opponent.
\textcolor{blue}{Blue}: controlled by the opponent.} \label{keyed}
\end{figure}
\subsection{Trivial Mann-Whitney Poisoning}
\label{whatever}
Let $G$ be Mann-Whitney's $U$-Test returning a $p$-value:
\[
p = G(A,B) = G(\{a_0,\ldots,a_{n_1-1}\}, \{b_0,\ldots,b_{n_2-1}\})
\]
$G$ is, between others, susceptible to poisoning as follows: assume that $A$ is sampled from a Gaussian distribution $A\in_R\mathcal{N}(\mu_A,\sigma_A)$ and that $B$ is sampled from $\{-q,q\}$ where $\mu_A\ll q$ (Figure \ref{attackMW}). While $A$ and $B$ are totally different, $G$ will be misled.\smallskip
For instance, after picking $10^6$ samples $A \in_R \mathcal{N}(0,3)$ and $10^6$ samples $B \in_R \{-15,15\}$ (i.e. we took $q=15$), we get a $p$-value of $0.99$. From Mann-Whitney's perspective, $A,B$ come from the same parent distribution with a very high degree of confidence while, in all evidence, they do not.\smallskip
\begin{figure}
\centering
\includegraphics[width=0.7\textwidth]{attackmw-eps-converted-to.pdf}
\caption{Trivial Mann-Whitney poisoning. Samples drawn from the blue distribution are Mann-Whitney-indistinguishable from samples drawn from the orange one.} \label{attackMW}
\end{figure}
\subsection{Keyed Mann-Whitney}
We instantiate $f_{\kappa}$ by secret random polynomials i.e. polynomials $R(x)$ whose coefficients are randomly and secretly refreshed before each invocation of $G$. Instead of returning $G(A,B)$, $\mbox{\textsf{Det}}$ returns $G(R(A),R(B))$ where:
\[
R(A) = \Big\{R(a_0),\dots,R(a_{n_1-1})\Big\} \quad\mbox{and}\quad R(B) = \Big\{R(b_0),\dots,R(b_{n_2-1})\Big\}
\]
The rationale is that $R$ will map the attacker's input to an unpredictable location in which the Mann-Whitney is very likely to be safe.\smallskip
$\ell$ random polynomials $R_1(x),\dots,R_{\ell}(x)$ are selected as keys and $\mbox{\textsf{Det}}$ calls $G$ for each polynomial. To aggregate all resulting $p$-values, $\mbox{\textsf{Det}}$ computes:
\[
\Delta = \mbox{Stouffer}\Big(G\Big(R_1(A), R_1(B)\Big),\ldots,G\Big(R_{\ell}(A), R_{\ell}(B)\Big)\Big)
\]
If $\Delta \approx 0$, the sample is rejected as poisonous with very high probability.\smallskip
Note that any smooth function can be used as $R$, e.g. B-splines. The criterion on $R$ is that the random selection process must yield significantly different functions.
\subsection{Experiments}
We illustrate the above by protecting $\mathcal{N}(0,1)$. The good thing about $\mathcal{N}(0,1)$ is that random polynomials tend to diverge when $x=1$ but adapt well to the central interval in which the Gaussian is not negligible.\smallskip
We attack $\mathcal{N}(0,1)$ by poisoning with $\{-q, q\}$, where $q$ is set to 3, 2, 1, and 0.5, respectively. For each value of $q$, two sets of 50 samples are drawn from the two distributions. Those samples are then transformed into other sets by applying a random polynomial of degree 4 and then fed into $G$ to obtain a $p$-value (using the two-sided mode). This $p$-value predicts whether these two sets of transformed samples come from the same distribution: a $p$-value close to 0 is a strong evidence against the null hypothesis. In each of our experiments, we apply nine secret random polynomials of degree 4 and aggregate the resulting $p$-values using Stouffer's method. For each setting, we run 1000 simulations. Similarly, for the same polynomials and $q$, we run a ``honest'' test, where both samples come from the same distribution.\smallskip
We thus retrieve 1000 ``attack'' $p$-values, which we sort by ascending order. Similarly, we sort the ``honest'' $p$-values. It is a classic result that, under the null hypothesis, a $p$-value follows a random uniform distribution over $[0, 1]$, hence a plot of the sorted ``honest'' $p$-values is a linear curve.\smallskip
An attack is successful if, on average, the ``attack'' sample is accepted as least as often as the ``honest'' sample. This can be rewritten as $\mathrm E(p^{\mbox{\scriptsize{attack}}}) \geq \mathrm E(p^{\mbox{\scriptsize{honest}}})$, with $E$ the . Hence, a sufficient condition for the validity is that the curve of sorted attack $p$-values (solid lines in our figures) is above the curve of sorted honest $p$-values (dashed lines). \smallskip
Experimental results are summarized in Figures \ref{gamma3},~\ref{gamma2},~\ref{gamma1} and~\ref{gammahalf}. \smallskip
The first quadrant illustrates the polynomials used in the simulation and two bars for $\{-q, q\}$. The same random polynomials were used for each experiment. For simplicity, the coefficients of the polynomials were uniformly selected from $\{-1, 0, 1\}$, and (useless) polynomials of degree lower than 2 were excluded from the random selection. Then, we also added the identity polynomial (\texttt{poly0}), as a witness of what happens when there is no protection.\smallskip
The following nine quadrants give the distribution of $p$-values for each polynomial, over 1000 simulations, sorted in increasing order. The dotted distribution corresponds to what an honest user would obtain, whereas the plain line simulation is based on poisoned datasets.\smallskip
The last quadrant contains the sorted distribution of the aggregated $p$-values using Stouffer. Experimental results show that for poisoned datasets, the aggregated $p$-values remain close to zero, while a honest dataset does not appear to be significantly affected. In other words, with very high probability, keyed testing detects poisoning. \smallskip
\begin{figure}
\includegraphics[width=0.9\textwidth]{gamma3v4-eps-converted-to.pdf}%
\caption{Attack with $q=3$. Defense with polynomials of degree 4.} \label{gamma3}
\end{figure}
\begin{figure}
\includegraphics[width=0.9\textwidth]{gamma2v4-eps-converted-to.pdf}
\caption{Attack with $q=2$. Defense with polynomials of degree 4.} \label{gamma2}
\end{figure}
\begin{figure}
\includegraphics[width=0.9\textwidth]{gamma1v4-eps-converted-to.pdf}
\caption{Attack with $q=1$. Defense with polynomials of degree 4.} \label{gamma1}
\end{figure}
\begin{figure}
\includegraphics[width=0.9\textwidth]{gammahalfv4-eps-converted-to.pdf}
\caption{Attack with $q=0.5$. Defense with polynomials of degree 4.} \label{gammahalf}
\end{figure}
\subsection{Discussion}
We observe a saturation when $q$ is too far from $\mu_A$, this is due to the fact that even after passing through $R$ the attack samples remain at the extremes. Hence if $R$ is of odd degree, nothing changes. If the degree of $R$ is even then the two extremes are projected to the same side and Mann-Whitney detects 100\% of the poisonings. It follows that at saturation a keyed Mann-Whitney gives either very high or very low $p$-value. This means that polynomials or B-splines must be carefully chosen to make keying effective.\smallskip
The advantage of combining the $p$-values with Stouffer's method is that the weak $p$-values are very penalizing (by opposition to Pearson's method whose combined $p$-value degrades much slower). A more conservative aggregation would be using Fisher's method.\smallskip
All in all, experimental results reveal that keying managed to endow the test with a significant level of immunity. \smallskip
Interestingly, $\mbox{\textsf{Det}}$ can be implemented independently of $\mathcal{A}$.\smallskip
A cautionary note: Our scenario assumes that testing does not start before learning is complete. If the opponent can alternate learning and testing then he may infer that a poisonous sample was taken into account (if $\sigma$ was updated and $\mathcal{A}$'s behavior was modified). This may open the door to attacks on $\mathcal{A}$.
\section{Notes and Further Research}
This paper opens perspectives beyond the specific poisoning problem. e.g. cryptographers frequently use randomness tests $\mathcal{R}$ to assess the quality of random number generators. In a strong attack model where the attacker knows $\mathcal{R}$ and controls the random source it becomes possible to trick many $\mathcal{R}$s into declaring flagrantly non random data as random. Hence, the authors believe that developing \textsl{keyed randomness tests} $\mathcal{R}_{\kappa}$ is important and useful as such.\smallskip
For instance, in the original minimum distance test 8000 points (a set $S$) sampled from the tested randomness source $\mathcal{S}$ are placed in a $10000\times 10000$ square. Let $\delta$ be the minimum distance between the pairs. If $\mathcal{S}$ is random then $\delta^2$ is exponentially distributed with mean $0.995$. To key the test a secret permutation $R_{\kappa}$ of the plan can generated and the test can be applied to $R_{\kappa}(S)$.\smallskip
To the best of our knowledge such primitives were not proposed yet.\smallskip
We note, however, that keyed protections to different (non cryptographic!) decision problems in very diverse areas do emerge independently e.g. \cite{NaorY14,Kenny,Marius,TRV18}.
\bibliographystyle{splncs04}
| -15,862.96873 |
[
-1.09375,
1.462890625
] | 42.922374 |
[
-2.927734375,
0.96142578125,
-1.5029296875,
-5.515625,
-0.39794921875,
6.96875
] |
[
2.16796875,
7.85546875,
1.2509765625,
6.8203125
] | 191 | 2,482 |
[
-2.0859375,
2.03125
] | 26.067293 |
[
-5.328125,
-3.2734375,
-3.611328125,
-1.951171875,
1.921875,
10.46875
] | 0.397693 | 39.435672 | 39.000806 | 1.306329 |
[
2.6619248390197754
] | -11,163.801973 | 6.078566 | -15,620.821936 | 0.437463 | 6.015727 |
[
-2.826171875,
-2.994140625,
-3.001953125,
-4.234375,
2.37890625,
10.1953125
] |
[
-5.2734375,
-1.4697265625,
-1.8876953125,
-1.1259765625,
3.001953125,
3.69140625
] | |
BkiUfN_xK7IAE_K9ytSd
|
\section*{}
\vspace{-1cm}
\footnotetext{\textit{$^{a}$~G. W. Gray Centre for Advanced Materials, Department of Physics \& Mathematics, University of Hull, Hull HU6 7RX, United Kingdom. }}
\footnotetext{\textit{$^{b}$~The MacDiarmid Institute for Advanced Materials and Nanotechnology, School of Chemical and Physical Sciences, Victoria University of Wellington, Kelburn Parade, Wellington 6012, New Zealand. }}
\footnotetext{\textit{$^{c}$~medPhoton GmbH, Strubergasse 16, 5020 Salzburg, Austria. }}
\footnotetext{\textit{$^{d}$~Institute of Particle Technology, Friedrich-Alexander University Erlangen-Nuernberg, Cauerstrasse 4, 91058 Erlangen, Germany. }}
\footnotetext{\textit{$^{e}$~Interdisciplinary Center for Functional Particle Systems, Friedrich-Alexander University Erlangen-Nuernberg, Ha-berstrasse 9a, 91058 Erlangen, Germany. }}
\footnotetext{\textit{$^{f}$~Department of Mathematical Sciences, Loughborough University, Loughborough LE11 3TU, United Kingdom. }}
\footnotetext{\textit{$^{g}$~E-mail: [email protected] }}
\footnotetext{\ddag~These authors contributed equally to this work}
\section{Introduction}
One of the distinguishing features of soft matter systems is the presence of a wide variety of interactions. The subtle interplay between the different types of interactions leads to a very rich self-assembly behaviour. An outstanding example of this is the case of hard-core/soft-shell (HCSS) particles which interact through an isotropic pair potential consisting of two characteristic length-scales, a short range hard-core potential and a longer-range soft shoulder repulsion (see Figure 1(a)). Experimentally, such a potential can arise for example in microgel particles with a higher cross-link density in the core compared to the corona\cite{Rey2016,Geisel2014}, rigid colloidal particles decorated with a soft polymeric layer\cite{Vogel2012,Rey2017,Rey2018,ElTawargy2018}, or block copolymer micelles consisting of a core of hydrophobic blocks surrounded by a shell of hydrophilic blocks.\cite{Fischer2011}
Jagla\cite{Jagla1998,Jagla1999} was the first to show that when HCSS particles are compressed in two dimensions (2D), they can self-assemble into a surprisingly rich variety of \emph{anisotropic} structures beyond the simple hexagonal structures that one would expect for spherical particles with isotropic interactions. These structures include stripes, squares, rhomboids and other complex crystal structures containing more than one particle per unit cell. Since the pioneering work of Jagla, a great variety of other structures have been found theoretically for HCSS particles in 2D including labyrinths,\cite{Camp2003} cluster and inverse cluster phases\cite{Malescio2003,Glaser2007,Fornleitner2008,Fornleitner2010} and quasicrystals of various symmetries.\cite{Jagla1998,Dotera2014,Pattabhiraman2015,Schoberth2016,Pattabhiraman2017} Remarkably, this rich variety of patterns is generated from a simple competition between the hard-core and soft-shoulder length scales in the interaction.\cite{Rey2017,Glaser2007,Dotera2014,Archer2013} Specifically, when the core-shell particles are compressed such that their shells begin to touch, provided the profile of the soft repulsive shoulder is flat enough so that the energy difference between fully and partially overlapping shells is small, the system can minimise its energy by fully overlapping neighbouring shells in some directions (i.e. so that the cores touch) in order to prevent the overlap of shells in others (i.e., so that the shells touch).
Most theoretical studies of HCSS particles in the literature model the soft shell interaction as a square shoulder potential. Although appealing from a theoretical point of view, this potential is physically unrealistic in the sense that it does not generate a repulsive force in the soft shell regime because the potential is flat.\cite{Schoberth2016} This situation is in contrast to real HCSS systems, e.g., colloids decorated by grafted polymers, where the interaction potential is not flat\cite{Witten1986,Milner1988,Norizoe2005}, and a finite repulsive force therefore exists when the soft shells of neighbouring particles overlap. To overcome this, here we consider HCSS particles with a range soft shell potential profiles beyond the standard square shoulder potential, including the linear ramp potential which is experimentally more realistic.\cite{Rey2017} Note that a recent study by Schoberth et al\cite{Schoberth2016} also considers HCSS particles with non-square soft shoulder interactions, although the range of soft shoulder potentials considered in that study are different from the ones considered here.
To simplify our discussion, we focus on HCSS particles in the `thin' shell regime, which we define as the case where the soft shoulder to hard core length scale ratio $r_1/r_0 < \sqrt{3}$. The significance of $\sqrt{3}$ in this context is that shell overlaps beyond nearest neighbour particles are only possible when $r_1/r_0 > \sqrt{3}$. We study the self-assembly of two-dimensional HCSS particles in this regime by performing both minimum energy calculations and finite temperature Monte Carlo (MC) simulations.
For our minimum energy calculations, our goal is to find a minimal model that captures the essential physics for the self-assembly of the system into periodic structures. In ref.\cite{Rey2017,Rey2018}, we performed a preliminary analysis where we considered minimum energy configurations (MECs) containing only one particle per unit cell. The one-particle model is able to reproduce the hexagonal, square and chain structures that are found experimentally in ref.\cite{Rey2017,Rey2018}, but it is not capable of generating the more complex periodic structures found in more elaborate models.\cite{Jagla1998,Jagla1999,Fornleitner2008,Fornleitner2010} In this paper, we therefore extend our calculations to include structures that contain up to two particles per unit cell.
Surprisingly, we find that this simple extension is sufficient to reproduce the periodic structures found from more sophisticated Genetic Algorithm calculations (which include up to 15 particles per unit cell).\cite{Fornleitner2008,Fornleitner2010} The MECs we find are also corroborated by our MC simulations. This good agreement demonstrates that our two-particle model is sufficient to capture many of the periodic structures that are accessible to HCSS particles in the thin shell regime. Our results for the phase diagram reproduce the broad features found by Jagla in his seminal studies,\cite{Jagla1998,Jagla1999} but correct a number of important discrepancies in those studies arising from the incomplete set of MECs that were used.\cite{Jagla1999} The model also allows us to predict the conditions under which HCSS particles in the thin shell regime can form the honeycomb lattice.
In addition to using MC simulations to check the accuracy of our minimum energy calculations, the MC simulations also allow us to study non-periodic structures such as quasicrystals. Specifically, by tuning both the ratio $r_1/r_0$ and density, we show that HCSS particles with a linear ramp soft shoulder potential can form dodecagonal (i.e., 12-fold symmetric) and decagonal (i.e., 10-fold symmetric) quasicrystals. These results are in excellent agreement with previous studies of HCSS particles with a square shoulder repulsion.\cite{Dotera2014,Pattabhiraman2015} However, since linear ramp potentials are more experimentally realistic,\cite{Rey2017} our study suggests the exciting possibility of forming these 2D quasicrystalline structures experimentally through colloidal self-assembly.
\section{Theoretical Methods}
\subsection{Hard-Core/Soft-Shoulder Potential}
We assume that the particles in our system interact through the generic hard-core/soft-shoulder potential proposed by Jagla\cite{Jagla1999}
\begin{equation}\label{hcss}
U_g(r)= \begin{cases}
\infty, & r < r_0 \\
U_0 \frac{g+\left[ \left( \frac{r-r_0}{r_1-r_0} \right) (g-g^{-1})-g \right]^{-1}}{g-g^{-1}}, & r_0 \leq r \leq r_1 \\
0, & r>r_1
\end{cases}
\end{equation}
where $r_0$, $r_1$ are the range of the hard-core and soft-shoulder repulsion respectively, $U_0$ is the shoulder height and the parameter $g$ controls the profile of the soft shoulder, going from no shoulder ($g=0$), via a linear ramp ($g=1$) to a square shoulder ($g=\infty$) as we increase $g$ (see Figure 1(a)). As mentioned in the Introduction, we consider thin shell HCSS particles in this paper where $r_1/r_0 < \sqrt{3}$. Note that most theoretical studies of HCSS particles in the literature focus on hard-core/square shoulder potentials, i.e., $g=\infty$. In this paper, we consider variable $g$ in order to study the impact of $g$ on the phase behaviour of HCSS particles.
\subsection{Minimum Energy Calculations}
In order to determine the equilibrium structures formed by HCSS particles when they are compressed in two dimensions, we first calculate the minimum energy configurations (MECs) of the system, i.e., the equilibrium structure at zero temperature. The zero temperature regime is relevant so long as the energy scale of the soft-shell repulsion is much greater than the thermal energy, i.e., $U_0 \gg k_B T$, or equivalently the reduced temperature $T^* \equiv k_B T/U_0 \ll 1$, a condition that is easily satisfied in many experimental systems.\cite{Rey2017,Rey2018} As mentioned in the Introduction, we perform a comprehensive exploration of all two-dimensional structures containing (up to) two particles per unit cell. Specifically, we can define the unit cell as a parallelogram spanned by two lattice vectors $\mathbf{a}=a(1,0)$, $\mathbf{b}=a \gamma (\cos \phi, \sin \phi)$, where $\phi$ is the angle between the lattice vectors, $\gamma = b/a$ is the aspect ratio of the unit cell and $a,b$ are the lattice constants (see Figure 1(b)). Within this unit cell, the first particle is at $(0,0)$ (without loss of generality) while the second particle is at $\mathbf{r}=\alpha \mathbf{a} + \beta \mathbf{b}$, where $\alpha,\beta \in (0,1)$ are the coordinates of the second particle in the lattice basis set.
When calculating the zero temperature phase diagram, it is convenient to work in the NPT ensemble where the area per particle is variable. This is because the system exists as a single phase in the NPT ensemble except at the coexistence pressure between two or more phases. Specifically, parameterising the area per particle as $\frac{\sqrt{3}}{2} \ell^2$, where $\ell$ is the lattice constant of the system in the hexagonal phase, and noting that the area per unit cell is $a^2 \gamma \sin \phi$, for two particles per unit cell, the lattice constant $a$ is fixed by the condition $(a^2 \gamma \sin \phi)/2=\frac{\sqrt{3}}{2} \ell^2$. We can therefore express $a$ as a function of $\ell$, $\gamma$ and $\phi$ as $a=\ell \left(\frac{\sqrt{3}}{\gamma \sin \phi} \right)^{1/2}$. Note that the number density of HCSS particles (i.e., number of particles per unit area) is given in terms of the parameter $\ell$ by $\rho = 2/(\sqrt{3} \ell^2)$ while the core area fraction is given by $\eta=\pi r_0^2/(2 \sqrt{3} \ell^2)$.
\begin{figure*}
\centering
\includegraphics[width=1.8\columnwidth]{Figure1.png}
\caption{(a) Jagla potential with energy scale $U_0$, hard-core range $r_0$, soft shell range $r_1$ and different values for the soft shell profile parameter $g$. The dotted lines on the left and upper right correspond to $g=0$ (no shoulder) and $g=\infty$ (square shoulder), respectively. (b) Unit cell containing two particles used in our minimum energy calculations, where $\mathbf{a}, \mathbf{b}$ are the lattice vectors, $\phi$ is the unit cell angle and the thick and thin circles represent the particle core and corona, respectively. (c) Zero temperature phase diagram for HCSS particles with $r_1/r_0=1.5$ in the $g$ and reduced pressure $P^*$ plane. The dotted, dashed and dotted dashed vertical lines correspond to the state points for which we performed Monte Carlo simulations in Figure 2. (d) Representative minimum energy configurations (MECs) for $r_1/r_0=1.5$. Note that the hexagonal structure (HEX) shown is the low density hexagonal lattice (i.e., HEXL, no overlap of shells).}
\label{figure1}
\end{figure*}
In the NPT ensemble, the equilibrium state is found by minimising the Gibbs free energy, or more specifically by minimising the enthalpy when we are at zero temperature. For our HCSS particles, the enthalpy per particle for crystal structures containing one particle per unit cell is given in ref.\cite{Rey2017}, while for crystals containing two particles per unit cell it is
\begin{equation}\label{gibbs2}
H = \frac{1}{2} \left[ \sum_{h,k \neq 0} U_g (|h \mathbf{a} + k \mathbf{b}|) + \sum_{h,k} U_g (|h \mathbf{a} + k \mathbf{b}+\alpha \mathbf{a} + \beta \mathbf{b}|) \right] + \frac{\sqrt{3}}{2} P \ell^2
\end{equation}
where $U_g (r)$ is the HCSS potential given by Eq.~\eqref{hcss}. The first term on the right hand side of Eq.\eqref{gibbs2} represents the interaction energy per particle calculated via lattice sums, while the second term is the surface pressure $P$ (i.e., force per unit length) times the area per particle. Specifically, the first and second sum inside the square brackets represent the interactions between like particles (i.e., particle 1--particle 1 or particle 2--particle 2) and unlike particles (i.e., particle 1--particle 2) respectively and the factor $\frac{1}{2}$ outside the square bracket converts the interaction energy per unit cell to the interaction energy per particle. Both summations run over all integer values of $h,k$ satisfying $|h \mathbf{a}+k \mathbf{b}|\leq r_c$ (except for $h,k=0$ in the first sum), where $r_c$ is the cut-off radius for interactions. Since we are considering the thin shell regime where $r_1/r_0 < \sqrt{3}$, we are able to use a very short cut-off length of $r_c/r_0=2$.
In order to calculate the zero temperature phase diagram, we determined the MECs of the system as a function of $r_1/r_0$, $g$ and $P$ (the parameter $U_0$ is irrelevant at zero temperature) by minimising $H$ given in Eq.~\eqref{gibbs2} with respect to the lattice parameters $\phi$, $\gamma$, $\ell$, $\alpha$ and $\beta$. Since this is a relatively high dimensional minimisation, the minimisation proceeded via several stages. We first minimised $H$ over a relatively wide range of values for the lattice parameters to obtain an initial estimate for their equilibrium values. We then further minimised $H$ over a much smaller range around these initial estimates to obtain more refined estimates for the equilibrium lattice parameters. Finally, these refined estimates were used to guide us in finding exact values for the equilibrium lattice parameters using geometry (see Supplementary Information).
\subsection{Monte Carlo Simulations}\label{MC}
Monte Carlo (MC) Simulations were performed in the NVT ensemble in order to mimic the experiments in ref.\cite{Rey2017,Rey2018} where monolayer area rather than surface pressure was controlled. The simulations were carried out using a standard Metropolis scheme for $1024$ particles contained in a rectangular box with aspect ratio $2:\sqrt{3}$ and periodic boundary conditions. One of the challenges of studying the thin shell regime is the fact that the area fraction of the hard-cores is high, making it difficult to equilibrate the system.\cite{Jagla1998} In order to overcome this problem, long simulations and very gradual quenches were used. Specifically, for each particle area fraction, the particles were first disordered at $T^*=\infty$ (i.e., hard core repulsion only) and then brought to the final reduced temperature of $T^* = 0.01$ through successive stages of $T^*=0.3, 0.2, 0.1, 0.06, 0.03, 0.01$. At each stage, the system was equilibrated for $10^5$ attempted moves per particle with an acceptance probability of around $30\%$.
\section{Results \& Discussions}
\subsection{Phase Behaviour in the $g-P^*$ Plane}
Figure 1(c) shows the zero temperature phase diagram in the $g$ versus reduced pressure $P^*=r_0^2 P/U_0$ plane for $r_1/r_0 = 1.5$ while Figure 1(d) shows the corresponding MEC structures.
For $g \geq 1$, the soft shoulder profile is flat enough so that (under compression) it is energetically favourable for neighbouring shells to be either fully overlapped or not overlapped. This is shown for example by the fact that for $g \geq 1$, the hexagonal phase at low pressures corresponds to the low density hexagonal phase (HEXL, no overlap of corona) and at high pressures to the high density hexagonal phase (HEXH, full overlap of corona). This simple interplay between the hard-core and soft-shoulder length scales also leads to anisotropic MECs at intermediate pressures where all the lattice parameters of the MEC can be calculated exactly from geometry for $g \geq 1$ (see Supplementary Information).
For $g \geq 2$, the phase boundaries in Figure 1(c) become independent of $g$, with the same sequence of phases being observed as we increase $P^*$, going from the low density hexagonal phase (HEXL) to chains (CH), to what we shall call the zig-zag 1 phase (ZZ1) and finally to high density hexagons (HEXH), see Figure 1(d). On the other hand, for $1 \leq g \leq 2$, an additional phase, which we shall call the zig-zag 2 phase (ZZ2), emerges in between HEXH and ZZ1. Both ZZ1 and ZZ2 contain two particles per unit cell (see Figure 1 in Supplementary Information) and have their particles aligned along chains. However, the chains in both phases are not straight like in the CH phase, but have a zig-zag shape in order to accommodate the larger number of shell overlaps arising from the higher compression. Alternatively, ZZ1 and ZZ2 can be viewed as being made up of elongated six-particle rings, and these six-particle rings are a characteristic motif for both zig-zag phases. The main difference between ZZ1 and ZZ2 is the fact that the chains are slightly staggered with respect to each other in ZZ1 (so that there is no rectangular order), while they are in register with each other in ZZ2 (so that there is rectangular order), see Figure 1(d).
In contrast, for $g<1$, the repulsive shoulder is concave enough so that partial overlap of the corona becomes energetically favourable. This leads to more subtle MECs where only some lattice parameters can be fixed by geometry, whilst others have to be obtained via numerical minimization of the enthalpy. Specifically, the region $0.6 <g < 1$ represents a transitional zone where rhomboidal (RH) and square (SQ) phases with non-touching cores become more energetically favourable than the phases listed in the Supplementary Information. Finally, anisotropic structures disappear altogether for $g < 0.6$ so that the hexagonal phase goes continuously from HEXL to HEXH as we increase compression.
The broad features of the phase diagram shown in Figure 1(c) agree with the corresponding phase diagram calculated by Jagla in ref.\cite{Jagla1999} though there are also some important differences. Firstly, while the square phase (SQ) in the Jagla phase diagram extends considerably into the $g>1$ region (up to $g \approx 1.5$), it is confined to the $g<1$ region in Figure 1(c). Secondly, the ZZ1 phase is absent from Jagla's phase diagram. Finally, Jagla predicts the existence of an intricate phase containing five particles per unit cell (i.e., the S4 phase in ref.\cite{Jagla1999}) for $g \geq 1.5$, which is obviously absent from our phase diagram as we only consider up to two particles per unit cell. However, we note that the presence of the ZZ1 phase and absence of the S4 phase are both confirmed by very accurate Genetic Algorithm (GA) calculations containing up to 15 particles per unit cell by Fornleitner and Kahl for $r_1/r_0 = 1.5$ and $g=\infty$,\cite{Fornleitner2008,Fornleitner2010} providing independent validation for the accuracy of our two particle per unit cell calculations for the above state point. We believe that the discrepancy between our phase diagram and Jagla's is due to the incomplete set of MECs used by Jagla.
To verify that the zero temperature phase diagram in Figure 1(c) is correct, we performed Monte Carlo simulations along the dotted line ($g=0.75$), dashed line ($g=1$) and dotted-dashed line ($g=2$) in Figure 1(c) for $r_1/r_0=1.5$ and reduced temperature of $T^*=0.01$. Representative structures at different core area fractions $\eta=\rho \pi r_0^2/4$ along each of these lines are shown in Figure 2, where $\rho$ is the number density of the HCSS particles. In the insets of Figure 2, we also show a magnified view of selected regions of each snapshot to highlight local structure and help identify the underlying symmetry of the structure.
Note that the phase diagram in Figure 1(c) is calculated in the NPT ensemble, where the phase behaviour is controlled by changing pressure $P^*$, while the MC simulations in Figure 2 are performed in the NVT ensemble, where phase behaviour is controlled by changing density $\eta$. However, it is straightforward to find the pressure $P^*$ corresponding to different densities $\eta$ and vice-versa. Specifically, the coexistence pressure between two phases (i.e., the pressure at the phase boundary between the two phases) corresponds to $\eta$ lying between the $\eta$ values for the two phases. For example, for $g=1$, $r_1/r_0=1.5$ (i.e., dashed line in Figure 1(c)), the coexistence pressure between HEXL and CH is $P^*_{coex}=1.871$, the coexistence pressure between CH and ZZ1 is $P^*_{coex}=2.369$, while $\eta_{HEXL}=0.403$, $\eta_{CH}=0.555$ and $\eta_{ZZ1}=0.653$ (see Table 1 in Supplementary Information). Therefore, the pressure of the system is $P^*=1.871$ in the density range $0.403 < \eta <0.555$ while $P^*=2.369$ in the density range $0.555 < \eta <0.653$. On the other hand, for $\eta=0.403$, the pressure of the system can be any value in the range $P^*<1.871$, while for $\eta=0.555$, the pressure of the system can be any value in the range $1.871 < P^* < 2.369$ etc..
For $g=0.75$, as we increase compression, the system transitions from HEXL to RH (unit cell angle $\phi \neq 90^\circ$) to SQ to HEXH, see Figures 2(a)-(d) respectively, noting in particular the local structure shown in the insets. Apart from the higher density RH phase which we could not identify from the MC simulations, this sequence of structures is exactly that predicted by Figure 1(c) along the dotted line.
\begin{figure*}
\centering
\includegraphics[width=1.8\columnwidth]{Figure2.png}
\caption{(a-l) Snapshots from Monte Carlo simulations of HCSS particles for $r_1/r_0=1.5$, reduced temperature of $T^*=0.01$ and different values of $g$ and core area fractions $\eta$. Note that the $g=0.75$, $g=1$ and $g=2$ simulations correspond to the dotted, dashed and dotted-dashed vertical lines respectively in Figure 1(c). See main text for a discussion of how to find the reduced pressure $P^*$ corresponding the different values of $\eta$ in the snapshots. The solid disks in the snapshots represent the particle cores while particle shells are omitted for clarity. In the insets, we show a magnified view of selected regions of each snapshot to highlight local structure.}
\label{figure2}
\end{figure*}
For $g=1$, as we increase compression, the system transitions from HEXL to CH to ZZ1 to ZZ2, see Figures 2(e)-(h) respectively, and finally to a coexistence between ZZ2 and HEXH at the highest density we could access (not shown). We can identify that Figures 2(g) and (h) correspond to zig-zag phases (either ZZ1 or ZZ2) from the fact that the characteristic elongated six-particle ring motif is present in the local structure (see insets). We can further distinguish between ZZ1 and ZZ2 in these snapshots from the fact that there is no evident rectangular order in Figure 2(g) main panel (i.e., the particle chains are staggered, hence ZZ1), while the rectangular order is evident in Figure 2(h) main panel (i.e., the particle chains are in register, hence ZZ2). In Figure 2(f), the system is in the CH phase as can be seen from the local structure in the inset (compare this to the CH structure in Figure 1(d)). We note that the sequence of structures shown in Figure 2(e)-(h) is exactly that predicted by Figure 1(c) along the dotted line.
For $g=2$, as we increase compression, the system transitions from HEXL to CH to ZZ1 to a coexistence between ZZ1 and HEXH at the highest density we could access, see Figure 2(i)-(l) respectively. Note that the ZZ1 phase in Figures 2(k),(l) are much less ordered than in Figure 2(g) but the phase can still be discerned from isolated instances of the characteristic six-particle rings in the local structure (see for example the inset of Figure 2(k)). The sequence of structures shown in Figure 2(i)-(l) is exactly that predicted by Figure 1(c) along the dotted-dashed line.
Finally, we note that the core area fractions of all the snapshots in Figure 2 are in good agreement with the area fractions of the corresponding MECs given in Table 1 in the Supplementary Information. The MC results in Figure 2 therefore confirm the accuracy of the phase diagram in Figure 1(c).
\subsection{Phase Behaviour in the $r_1/r_0 - P^*$ Plane}
In Figures 3(a),(b), we show the zero temperature phase diagram in the $r_1/r_0$ vs reduced pressure $P^*$ plane for $g=1$ and $g=10$ respectively. These values of $g$ were chosen because $g=1$ is relevant to experimental two-dimensional core-shell particles\cite{Rey2017,Rey2018} while $g=10$ is essentially equivalent to the square shoulder repulsion $g=\infty$ (see Figure 1(a)) which has been extensively studied theoretically in the literature.
\begin{figure}[h]
\centering
\includegraphics[width=1.0\columnwidth]{Figure3.png}
\caption{Zero temperature phase diagram for core-shell particles in the $r_1/r_0$--$P^*$ plane for (a) $g=1$ (b) $g=10$. The minimum energy configurations (MECs) labelled in the phase diagrams are low density hexagons (HEXL, no overlap of corona), high density hexagons (HEXH, full overlap of corona), chains (CH), rhomboids (RH), zig-zag 1 (ZZ1) and zig-zag 2 (ZZ2). See main text for more details.}
\label{figure3}
\end{figure}
We see that there are some differences between the phase diagrams in Figures 3(a) and 3(b), e.g., for $g=\infty$ the ZZ1 phase occupies a larger area of the phase diagram (at the expense of the ZZ2 phase) compared to $g=1$. However, apart from these relatively minor differences, the two phase diagrams are broadly similar. In what follows, we therefore focus on the phase behaviour of HCSS particles with $g=1$ (linear ramp soft shoulder) since this potential is more realistic experimentally.\cite{Rey2017,Rey2018}
In Figures 3(a), several of the phase boundaries are vertical, occurring at certain significant values of $r_1/r_0$. These correspond to values of $r_1/r_0$ where changes in the geometry of the particle arrangements occur. For example $r_1/r_0=\sqrt{2}$ forms the phase boundary between RH and ZZ2. For this value of $r_1/r_0$, both the RH and ZZ2 phases approach the square structure so that there is a continuous phase transition from RH to ZZ2 as we increase $r_1/r_0$. The formation of the square lattices for HCSS particles with $g=1$, $r_1/r_0 \approx \sqrt{2}$ has been confirmed in our previous MC simulations.\cite{Rey2018}
More interestingly, $r_1/r_0 = \sqrt{2}$ corresponds to one of the special shell to core ratio where the HCSS particles can form quasicrystals.\cite{Dotera2014} This is because for $r_1/r_0=\sqrt{2}$, the unit cell of the RH phase is a square with side length $r_0$ while the unit cell of the HEXH phase consists of two compact equilateral triangles with side length $r_0$. At the phase boundary between the RH and HEXH phase (i.e., $P^*=7.464$ for $g=1$ and $r_1/r_2=\sqrt{2}$, see Figure 3(a)), the enthalpy per particle of both phases are equal. In the $NVT$ ensemble, the RH and HEXH phases are in coexistence for the core area fraction range $0.785 \leq \eta \leq 0.907$ (see Table 1 in Supplementary Information). Random tilings of squares and triangles can therefore occur in this range\cite{Henley1988} and the resultant random tiling phases are degenerate ground states of the system at $T^*=0$.\cite{Jagla1998} When the number ratio of squares to triangles is $\sqrt{3}/4$, the structure obtained is a random dodecagonal (12-fold symmetric) quasicrystal.\cite{Leung1989,Kawamura1991,Widom1993} Taking into account the fact that a square and an equilateral triangle contain $1$ particle and $\frac{1}{2}$ a particle respectively, we expect to find dodecagonal quasicrystals for core area fractions around $\eta \approx 0.848$ at $T^*=0$.
\begin{figure*}
\centering
\includegraphics[width=1.8\columnwidth]{Figure4.png}
\caption{(a) Diffraction pattern calculated from Monte Carlo (MC) simulations of HCSS particles with $r_1/r_0=1.41$, $\eta=0.778$, $T^*=0.01$ and $g=1$. The innermost circle corresponds to the wavenumber where the first set of 12-fold diffraction peaks occurs while the radius ratio for successive circles is 1.93. (b) Real space image of a MC snapshot of the system where the particle centres are denoted by small grey disks while the thin lines are the Delaunay triangulation of the particle centres. We have also drawn thick lines between particles whose centres are closer than $1.3 r_0$ to accentuate the polygonal tiles in the system. (c) The Delaunay triangles found in (b), including compact equilateral triangles (C) and short (S) and long (L) isosceles triangles. (d) Some of the tiles found in (b), compact equilateral triangles (Tr), squares (Sq) and pentagons whose internal angles are not $108^\circ$ (P).}
\label{figure4}
\end{figure*}
In Figure 4(a), we show the 2D diffraction pattern calculated from MC simulations of HCSS particles with $r_1/r_0=1.41$, $\eta=0.778$, $T^*=0.01$ and $g=1$. The diffraction pattern clearly shows 12-fold symmetry, indicating that the system is a dodecagonal quasicrystal. In addition, the wavenumber ratio for the successive circles of 12-fold peaks shown in Figure 4(a) is 1.93, further confirming the dodecagonal symmetry.\cite{Rucklidge2012,Barkan2014,Lifshitz1997} Our results agree with those of Dotera et al\cite{Dotera2014} who obtained 12-fold quasicrystals for HCSS particles with $r_1/r_0=1.4$, $\eta=0.77$, $T^*=0.278$ and $g=\infty$ (i.e., square shoulder repulsion). Interestingly, the core area fractions for dodecagonal quasicrystals in both studies are considerably lower than the $T^*=0$ value of $\eta \approx 0.848$. This fact is consistent with the study of Pattabhiraman et al \cite{Pattabhiraman2015} who found that for HCSS particles with $r_1/r_0=1.4$, configurational entropy causes the dodecagonal quasicrystal phase to shift to slightly lower core area fractions at finite temperatures.
In order to analyse the real-space structure of the quasicrystal, in Figure 4(b) we show the Delaunay triangulation of the particle centres from a MC snapshot of our system (thin lines between particle centres). We see that the Delaunay triangles primarily consist of compact equilateral triangles with side length $r_0$ (C), short isosceles triangles with apex angle $90^\circ$ and side length $r_0$ for the two equal sides (S) and long isosceles triangles with apex angle $41.4^\circ$ and side length $r_1$ for the two equal sides (L)(see Figure 4(c)). Note that two S triangles can be combined along their long edge to form a square (See Figure 4(d)).
Another instructive way to analyse the real space structure is to break the pattern up into tiles by joining together particle pairs whose cores are close to contact. Specifically, in Figure 4(b) we join together particle pairs whose centres are closer than $1.3 r_0$ (thick lines). We see that the resultant tiles consist mainly of compact equilateral triangles (Tr) and squares of side length $r_0$ (Sq), together with a small number of pentagons whose internal angles are not $108^\circ$ (P) and irregular polygons,\cite{Dotera2014} see Figure 4(d). The predominance of square and triangular tiles is not surprising since, as discussed earlier, squares and equilateral triangles can be used as the fundamental building blocks for constructing random dodecagonal quasicrystals.\cite{Dotera2014,Leung1989,Widom1993,Oxborrow1993}
Our results in Figure 4 demonstrate that HCSS particles with a linear ramp shoulder repulsion can form 12-fold quasicrystals. These results are significant since in our previous study,\cite{Rey2017} we showed that the phase behaviour of mixed monolayers of polystyrene microspheres (diameter 1.5 $\mu$m) and poly(N-isoprpylacrylamide) (PNiPAm) microgels (diameter 150 nm) at the air/water interface could be quantitatively modelled using HCSS particles with $g=1$ and $r_1/r_0 \approx 1.4$. Our study suggests that when compressed to a suitable density, this system could form 12-fold quasicrystals. One important caveat here is the fact that the experimental system in ref.\cite{Rey2017} is not ergodic because the energy scale of the soft shoulder repulsion is much greater than $k_BT$ and the density of the 12-fold quasicrystalline state is relatively high. It may therefore be necessary to provide external energy input into the system, e.g., through mechanical vibrations, in order to help the system to find its equilibrium state.
Another significant value for $r_1/r_0$ is the golden ratio $r_1/r_0=\tau \equiv (1+\sqrt{5})/2 \approx 1.618$ which forms the phase boundary between ZZ1 and ZZ2. For this core-shell ratio, both ZZ1 and ZZ2 approach the same structure (see Figure 5(b)) so that there is a continuous phase transition from ZZ1 to ZZ2 as we increase $r_1/r_0$. The reason why the ZZ1 phase comes abruptly to an end at this point is because for $r_1/r_0>1.618$, the lattice constant $b$ in Figure 1(e) in Supplementary Information becomes smaller than $r_1$, leading to an additional overlap of two shells which is energetically unfavourable.
\begin{figure}[h]
\centering
\includegraphics[width=1.0\columnwidth]{Figure5.png}
\caption{(a) CH phase for $r_1/r_0=1.618$, which can be built up from thin Penrose tiles (black rhombi), or more fundamentally from long Robinson triangles L (red triangle) shown in Figure 6(c). Note that the thin Penrose tile can be built up from two L triangles. (b) ZZ2 phase for $r_1/r_0=1.618$, which can be built up from fat Penrose tiles (black rhombi), or more fundamentally from both long (L) and short (S) Robinson triangles (red and blue triangles respectively) shown in Figure 6(c). Note the fat Penrose tile can be built up from two L and two S triangles.}
\label{figure5}
\end{figure}
More interestingly, $r_1/r_0=\tau=1.618$ is another shell to core ratios where the system can form quasicrystals.\cite{Jagla1998,Dotera2014} This is because for $r_1/r_0=1.618$, we can use the fat and thin Penrose tiles as the unit cells for the ZZ2 phase and the CH phase respectively,\cite{Jagla1998} see black rhombi in Figure 5. Furthermore, at the phase boundary between ZZ2 and CH (i.e., $P^*=1.701$ for $g=1$, $r_1/r_2=1.618$, see Figure 3(a)), the enthalpy per particle of both phases are equal. In the $NVT$ ensemble, the ZZ2 and CH phases are in coexistence for the core area fraction range $0.510 \leq \eta \leq 0.631$ (see Table 1 in Supplementary Information), we therefore expect a random tiling of the two Penrose tiles to occur in this range. When the number fractions of fat and thin tiles are $\tau^{-1}$ and $\tau^{-2}$ respectively, the structure obtained is a random quasicrystal with decagonal (i.e., 10-fold) symmetry.\cite{Henley1988} Taking into account the fact that the fat and thin tiles in Figure 5 contain 2 particles and 1 particle respectively, this means that we should expect to find decagonal quasicrystals for core area fractions around $\eta \approx 0.60$.
In Figure 6(a), we show the 2D diffraction pattern calculated from MC simulation snapshots of HCSS particles with $r_1/r_0=1.618$, $\eta=0.59$, $T^*=0.01$ and $g=1$ (linear ramp potential). Notwithstanding the small degree of noise due to some disorder in the MC snapshot, the diffraction pattern clearly shows non-crystallographic 10-fold symmetry, indicating that the system is a decagonal quasicrystal. In addition, the wavenumber ratio for the successive circles of 10-fold peaks shown in Figure 6(a) is 1.618, further confirming the decagonal symmetry.\cite{Rucklidge2012,Barkan2014,Lifshitz1997} We note that 10-fold quasicrystals were also obtained by Jagla\cite{Jagla1998} for $r_1/r_0=1.65$, $P^*1.7$, $T^*<0.05$, $g=1$ and by Dotera et al\cite{Dotera2014} for $r_1/r_0=1.6$, $\eta=0.55$, $T^*=0.133$, $g=\infty$.
\begin{figure*}
\centering
\includegraphics[width=1.8\columnwidth]{Figure6.png}
\caption{(a) Diffraction pattern calculated from Monte Carlo (MC) simulations of HCSS particles with $r_1/r_0=1.618$, $\eta=0.59$, $T^*=0.01$ and $g=1$. The innermost circle corresponds to the wavenumber where the first set of 10-fold diffraction peaks occurs while the radius ratio for successive circles is 1.618. (b) Real space image of a MC snapshot of the system where the particle centres are denoted by small grey disks while the thin black lines are the Delaunay triangulation of the particle centres. We have also drawn thick lines between particles whose centres are closer than $1.4 r_0$ to accentuate the polygonal tiles in the system. (c) The Delaunay triangles found in (b), including short (S) and long (L) Robinson triangles and compact (C) and expanded (E) equilateral triangles. (d) The standard decagonal tiles found in (b), including Pentagons (P), Hexagons (H) and Nonagons (N).}
\label{figure6}
\end{figure*}
In order to analyse the real-space structure of the quasicrystal, in Figure 6(b) we show the Delaunay triangulation of the particle centres from a MC snapshot of our system (thin lines between particle centres). It is striking that the Delaunay triangles consist overwhelmingly of short (S) and long (L) Robinson triangles with apex angles of $108^\circ$ and $36^\circ$ respectively (see Figure 6(c)), and that these triangles also appear in both the CH and ZZ2 phases in Figure 5 (red (L) and blue (S) triangles). This is not surprising since Robinson triangles can be seen as the fundamental building blocks for constructing decagonal quasicrystals.\cite{Dotera2014} Indeed the thin Penrose tile in Figure 5(a) can be built up from two L triangles while the fat Penrose tile in Figure 5(b) can be built up from two L and two S triangles. In addition to the S and L triangles, the Delaunay triangles also consist of a very small number of compact equilateral triangles with side length $r_0$ (C) and expanded equilateral triangles with side length $r_1$ (E) (see Figure 6(c)).
As before, we also analyse the real space structure by joining together particle pairs whose centres are closer than $1.4 r_0$ (thick lines). We see that the resultant tiles are not in fact the idealised fat and thin Penrose tiles shown in Figure 5, but consist of a larger palette of standard decagonal tiles (including Pentagons (P), Hexagons (H), Nonagons (N), see Figure 6(d)),\cite{Dotera2014,Engel2007} less regular decagonal tiles and a small number of compact equilateral triangles (C). Apart from C, all the other tiles can essentially be built up from obtuse and acute Robinson triangles.
It is useful at this point to compare our results for the quasicrystalline phases with those of Schoberth et al\cite{Schoberth2016} which were also obtained for HCSS particles with non-square soft shoulder interactions. Specifically, these authors introduce a parameter $\alpha$ which controls the soft shoulder profile, with $\alpha = 0$ resembling the square shoulder potential and $\alpha \geq 10$ resembling the linear ramp potential. Interestingly, for linear-ramp-like potentials ($\alpha \geq 10$), these authors did not observe any quasicrystalline phases for shell to core ratios of $\lambda \approx 1.4$, while they observed an interesting mixed quasicrystal state (with relatively weak 10-fold symmetry locally and much stronger 12-fold symmetry on larger length scales) for $\lambda \approx 1.6$. These results are significantly different to our linear ramp results where we observe 12-fold quasi-crystals for $r_1/r_0 = 1.41$ and pure 10-fold quasi-crystals for $r_1/r_0 = 1.618$. The difference between the two sets of results suggest that the formation of quasicrystalline states may be relatively sensitive to subtle differences in the soft shell profile.
\begin{figure}[h]
\centering
\includegraphics[width=0.8\columnwidth]{Figure7.png}
\caption{Snapshot from Monte Carlo simulations of HCSS particles with $r_1/r_0=1.73$, $g=1$, $\eta=0.605$ and $T^*=0.01$. Note the clear honeycomb structure in the snapshot. The inset shows the MEC for $r_1/r_0=1.73$.}
\label{figure7}
\end{figure}
Finally, as $r_1/r_0 \rightarrow \sqrt{3}$, the ZZ2 structure approaches the honeycomb structure, see inset of Figure 7. To check whether this structure is accessible via MC simulations, in Figure 7, we show a representative snapshot from MC simulations of HCSS particles for $g=1$, $r_1/r_0=1.73$, $T^*=0.01$ and $\eta=0.605$ (the core area fraction of the ZZ2 phase for $r_1/r_0=1.73$, see Table 1). We clearly see the formation of honeycomb structures over large domains in agreement with the minimum energy calculations. Interestingly, honeycomb structures were also observed by Pattabhiraman and Dijkstra\cite{Pattabhiraman2017} for HCSS particles with a slightly larger shell to core ratio of $r_1/r_0=1.95$. However, the physics behind honeycomb formation is slightly different here as we are in the thin shell regime ($r_1/r_0 < \sqrt{3}$) where there are no shell overlaps beyond nearest neighbour particles.
\section{Conclusions}
We have studied the self-assembly of hard-core/soft shell (HCSS) particles in two dimensions and in the thin shell regime using both minimum energy calculations and Monte Carlo simulations. In contrast to most studies in this area, we have considered a range soft shell potential profiles beyond the standard square shoulder potential.
Our results for the phase diagram of HCSS particles agree with the broad features found in the seminal studies by Jagla,\cite{Jagla1998,Jagla1999} but also correct a number of important discrepancies in those studies. In particular, we show that by tuning the soft shoulder profile, the shell to core ratio $r_1/r_0$ and density, it is possible to generate hexagonal close packed structures, chains, rhomboids, squares and two distinct zig-zag structures. We also show that HCSS particles in the thin shell regime can form honeycombs when $r_1/r_0 \rightarrow \sqrt{3}$.
In addition to periodic structures, we show that by tuning both shell to core ratios and densities, HCSS particles with a linear ramp soft shoulder potentials can be engineered to form a variety of quasicrystalline structures, including 10-fold quasicrystals for $r_1/r_0=1.618$ and 12-fold quasicrystals for $r_1/r_0=1.41$. These results agree with the previous results of Dotera et al for HCSS particles with a square shoulder repulsion.\cite{Dotera2014} However, since linear ramp potentials are more experimentally realistic,\cite{Rey2017} these results serve as an important road map for the experimental realization of complex assembly phases using isotropic building blocks.
\section*{Conflicts of interest}
There are no conflicts to declare.
\section*{Acknowledgements}
WRCS and DMAB acknowledge funding from the EPSRC (Grant number EP/L025078/1) and the European Union's Horizon 2020 research and innovation programme under grant agreement No 861950, project POSEIDON. They also acknowledge the Viper High Performance Computing facility of the University of Hull and its support team. MR and NV acknowledge funding from the Deutsche Forschungsgemeinschaft (DFG) (grant number VO 1824/6-1) and and the European Union's Horizon 2020 research and innovation programme under grant agreement No 861950, project POSEIDON. AJA acknowledges funding from the EPSRC (Grant number EP/P015689/1 Quasicrystals: how and why do they form?).
\balance
\providecommand*{\mcitethebibliography}{\thebibliography}
\csname @ifundefined\endcsname{endmcitethebibliography}
{\let\endmcitethebibliography\endthebibliography}{}
\begin{mcitethebibliography}{31}
\providecommand*{\natexlab}[1]{#1}
\providecommand*{\mciteSetBstSublistMode}[1]{}
\providecommand*{\mciteSetBstMaxWidthForm}[2]{}
\providecommand*{\mciteBstWouldAddEndPuncttrue}
{\def\unskip.}{\unskip.}}
\providecommand*{\mciteBstWouldAddEndPunctfalse}
{\let\unskip.}\relax}
\providecommand*{\mciteSetBstMidEndSepPunct}[3]{}
\providecommand*{\mciteSetBstSublistLabelBeginEnd}[3]{}
\providecommand*{\unskip.}}{}
\mciteSetBstSublistMode{f}
\mciteSetBstMaxWidthForm{subitem}
{(\emph{\alph{mcitesubitemcount}})}
\mciteSetBstSublistLabelBeginEnd{\mcitemaxwidthsubitemform\space}
{\relax}{\relax}
\bibitem[Rey \emph{et~al.}(2016)Rey, Fernandez-Rodriguez, Steinacher,
Scheidegger, Geisel, Richtering, Squires, and Isa]{Rey2016}
M.~Rey, M.~A. Fernandez-Rodriguez, M.~Steinacher, L.~Scheidegger, K.~Geisel,
W.~Richtering, T.~M. Squires and L.~Isa, \emph{Soft Matter}, 2016,
\textbf{12}, 3545\relax
\mciteBstWouldAddEndPuncttrue
\mciteSetBstMidEndSepPunct{\mcitedefaultmidpunct}
{\mcitedefaultendpunct}{\mcitedefaultseppunct}\relax
\unskip.}
\bibitem[Geisel \emph{et~al.}(2014)Geisel, Isa, and Richtering]{Geisel2014}
K.~Geisel, L.~Isa and W.~Richtering, \emph{Angew. Chemie - Int. Ed.}, 2014,
\textbf{53}, 4905\relax
\mciteBstWouldAddEndPuncttrue
\mciteSetBstMidEndSepPunct{\mcitedefaultmidpunct}
{\mcitedefaultendpunct}{\mcitedefaultseppunct}\relax
\unskip.}
\bibitem[Vogel \emph{et~al.}(2012)Vogel, Fernandez-Lopez, Perez-Juste,
Liz-Marzan, Landfester, and Weiss]{Vogel2012}
N.~Vogel, C.~Fernandez-Lopez, J.~Perez-Juste, L.~M. Liz-Marzan, K.~Landfester
and C.~K. Weiss, \emph{Langmuir}, 2012, \textbf{28}, 8985\relax
\mciteBstWouldAddEndPuncttrue
\mciteSetBstMidEndSepPunct{\mcitedefaultmidpunct}
{\mcitedefaultendpunct}{\mcitedefaultseppunct}\relax
\unskip.}
\bibitem[Rey \emph{et~al.}(2017)Rey, Law, Buzza, and Vogel]{Rey2017}
M.~Rey, A.~D. Law, D.~M.~A. Buzza and N.~Vogel, \emph{J. Am. Chem. Soc.}, 2017,
\textbf{139}, 17464\relax
\mciteBstWouldAddEndPuncttrue
\mciteSetBstMidEndSepPunct{\mcitedefaultmidpunct}
{\mcitedefaultendpunct}{\mcitedefaultseppunct}\relax
\unskip.}
\bibitem[Rey \emph{et~al.}(2018)Rey, Yu, Bley, Landfester, Buzza, and
Vogel]{Rey2018}
M.~Rey, T.~Yu, K.~Bley, K.~Landfester, D.~M.~A. Buzza and N.~Vogel,
\emph{Langmuir}, 2018, \textbf{34}, 9990\relax
\mciteBstWouldAddEndPuncttrue
\mciteSetBstMidEndSepPunct{\mcitedefaultmidpunct}
{\mcitedefaultendpunct}{\mcitedefaultseppunct}\relax
\unskip.}
\bibitem[El-Tawargy \emph{et~al.}(2018)El-Tawargy, Stock, Gallei, Ramadan,
Shams El-Din, Reiter, and Reiter]{ElTawargy2018}
A.~S. El-Tawargy, D.~Stock, M.~Gallei, W.~A. Ramadan, M.~A. Shams El-Din,
G.~Reiter and R.~Reiter, \emph{Langmuir}, 2018, \textbf{34}, 3909\relax
\mciteBstWouldAddEndPuncttrue
\mciteSetBstMidEndSepPunct{\mcitedefaultmidpunct}
{\mcitedefaultendpunct}{\mcitedefaultseppunct}\relax
\unskip.}
\bibitem[Fischer and Exner(2011)]{Fischer2011}
S.~Fischer and A.~Exner, \emph{Proc. Natl. Acad. Sci.}, 2011, \textbf{108},
1810\relax
\mciteBstWouldAddEndPuncttrue
\mciteSetBstMidEndSepPunct{\mcitedefaultmidpunct}
{\mcitedefaultendpunct}{\mcitedefaultseppunct}\relax
\unskip.}
\bibitem[Jagla(1998)]{Jagla1998}
E.~A. Jagla, \emph{Phys. Rev. E}, 1998, \textbf{58}, 1478\relax
\mciteBstWouldAddEndPuncttrue
\mciteSetBstMidEndSepPunct{\mcitedefaultmidpunct}
{\mcitedefaultendpunct}{\mcitedefaultseppunct}\relax
\unskip.}
\bibitem[Jagla(1999)]{Jagla1999}
E.~A. Jagla, \emph{J. Chem. Phys.}, 1999, \textbf{110}, 451\relax
\mciteBstWouldAddEndPuncttrue
\mciteSetBstMidEndSepPunct{\mcitedefaultmidpunct}
{\mcitedefaultendpunct}{\mcitedefaultseppunct}\relax
\unskip.}
\bibitem[Camp(2003)]{Camp2003}
P.~J. Camp, \emph{Phys. Rev. E}, 2003, \textbf{68}, 061506\relax
\mciteBstWouldAddEndPuncttrue
\mciteSetBstMidEndSepPunct{\mcitedefaultmidpunct}
{\mcitedefaultendpunct}{\mcitedefaultseppunct}\relax
\unskip.}
\bibitem[Malescio and Pellicane(2003)]{Malescio2003}
G.~Malescio and G.~Pellicane, \emph{Nat. Mater.}, 2003, \textbf{2}, 97\relax
\mciteBstWouldAddEndPuncttrue
\mciteSetBstMidEndSepPunct{\mcitedefaultmidpunct}
{\mcitedefaultendpunct}{\mcitedefaultseppunct}\relax
\unskip.}
\bibitem[Glaser \emph{et~al.}(2007)Glaser, Grason, Kamien, Kosmrlj, Santangelo,
and Ziherl]{Glaser2007}
M.~A. Glaser, G.~M. Grason, R.~D. Kamien, A.~Kosmrlj, C.~D. Santangelo and
P.~Ziherl, \emph{Europhys. Lett.}, 2007, \textbf{78}, 46004\relax
\mciteBstWouldAddEndPuncttrue
\mciteSetBstMidEndSepPunct{\mcitedefaultmidpunct}
{\mcitedefaultendpunct}{\mcitedefaultseppunct}\relax
\unskip.}
\bibitem[Fornleitner and Kahl(2008)]{Fornleitner2008}
J.~Fornleitner and G.~Kahl, \emph{EPL}, 2008, \textbf{82}, 18001\relax
\mciteBstWouldAddEndPuncttrue
\mciteSetBstMidEndSepPunct{\mcitedefaultmidpunct}
{\mcitedefaultendpunct}{\mcitedefaultseppunct}\relax
\unskip.}
\bibitem[Fornleitner and Kahl(2010)]{Fornleitner2010}
J.~Fornleitner and G.~Kahl, \emph{J. Phys.: Condens. Matter}, 2010,
\textbf{22}, 104118\relax
\mciteBstWouldAddEndPuncttrue
\mciteSetBstMidEndSepPunct{\mcitedefaultmidpunct}
{\mcitedefaultendpunct}{\mcitedefaultseppunct}\relax
\unskip.}
\bibitem[Dotera \emph{et~al.}(2014)Dotera, Oshiro, and Ziherl]{Dotera2014}
T.~Dotera, T.~Oshiro and P.~Ziherl, \emph{Nature}, 2014, \textbf{506},
208\relax
\mciteBstWouldAddEndPuncttrue
\mciteSetBstMidEndSepPunct{\mcitedefaultmidpunct}
{\mcitedefaultendpunct}{\mcitedefaultseppunct}\relax
\unskip.}
\bibitem[Pattabhiraman \emph{et~al.}(2015)Pattabhiraman, Gantapara, and
Dijkstra]{Pattabhiraman2015}
H.~Pattabhiraman, A.~P. Gantapara and M.~Dijkstra, \emph{J. Chem. Phys.}, 2015,
\textbf{143}, 164905\relax
\mciteBstWouldAddEndPuncttrue
\mciteSetBstMidEndSepPunct{\mcitedefaultmidpunct}
{\mcitedefaultendpunct}{\mcitedefaultseppunct}\relax
\unskip.}
\bibitem[Schoberth \emph{et~al.}(2016)Schoberth, Emmerich, Holzinger, Dulle,
F\"{o}rster, and Gruhn]{Schoberth2016}
H.~G. Schoberth, H.~Emmerich, M.~Holzinger, M.~Dulle, S.~F\"{o}rster and
T.~Gruhn, \emph{Soft Matter}, 2016, \textbf{12}, 7644\relax
\mciteBstWouldAddEndPuncttrue
\mciteSetBstMidEndSepPunct{\mcitedefaultmidpunct}
{\mcitedefaultendpunct}{\mcitedefaultseppunct}\relax
\unskip.}
\bibitem[Pattabhiraman and Dijkstra(2017)]{Pattabhiraman2017}
H.~Pattabhiraman and M.~Dijkstra, \emph{Soft Matter}, 2017, \textbf{13},
4418\relax
\mciteBstWouldAddEndPuncttrue
\mciteSetBstMidEndSepPunct{\mcitedefaultmidpunct}
{\mcitedefaultendpunct}{\mcitedefaultseppunct}\relax
\unskip.}
\bibitem[Archer \emph{et~al.}(2013)Archer, Rucklidge, and Knobloch]{Archer2013}
A.~J. Archer, A.~M. Rucklidge and E.~Knobloch, \emph{Phys. Rev. Lett.}, 2013,
\textbf{111}, 165501\relax
\mciteBstWouldAddEndPuncttrue
\mciteSetBstMidEndSepPunct{\mcitedefaultmidpunct}
{\mcitedefaultendpunct}{\mcitedefaultseppunct}\relax
\unskip.}
\bibitem[Witten and Pincus(1986)]{Witten1986}
T.~A. Witten and P.~A. Pincus, \emph{Macromolecules}, 1986, \textbf{19},
2509\relax
\mciteBstWouldAddEndPuncttrue
\mciteSetBstMidEndSepPunct{\mcitedefaultmidpunct}
{\mcitedefaultendpunct}{\mcitedefaultseppunct}\relax
\unskip.}
\bibitem[Milner \emph{et~al.}(1988)Milner, Witten, and Cates]{Milner1988}
S.~T. Milner, T.~A. Witten and M.~E. Cates, \emph{Macromolecules}, 1988,
\textbf{21}, 2610\relax
\mciteBstWouldAddEndPuncttrue
\mciteSetBstMidEndSepPunct{\mcitedefaultmidpunct}
{\mcitedefaultendpunct}{\mcitedefaultseppunct}\relax
\unskip.}
\bibitem[Norizoe and Kawakatsu(2005)]{Norizoe2005}
Y.~Norizoe and T.~Kawakatsu, \emph{Europhys. Lett.}, 2005, \textbf{72},
583\relax
\mciteBstWouldAddEndPuncttrue
\mciteSetBstMidEndSepPunct{\mcitedefaultmidpunct}
{\mcitedefaultendpunct}{\mcitedefaultseppunct}\relax
\unskip.}
\bibitem[Henley(1988)]{Henley1988}
C.~L. Henley, \emph{J. Phys. A}, 1988, \textbf{21}, 1649\relax
\mciteBstWouldAddEndPuncttrue
\mciteSetBstMidEndSepPunct{\mcitedefaultmidpunct}
{\mcitedefaultendpunct}{\mcitedefaultseppunct}\relax
\unskip.}
\bibitem[Leung \emph{et~al.}(1989)Leung, Henley, and Chester]{Leung1989}
P.~W. Leung, C.~L. Henley and G.~V. Chester, \emph{Phys. Rev. B}, 1989,
\textbf{39}, 446\relax
\mciteBstWouldAddEndPuncttrue
\mciteSetBstMidEndSepPunct{\mcitedefaultmidpunct}
{\mcitedefaultendpunct}{\mcitedefaultseppunct}\relax
\unskip.}
\bibitem[Kawamura(1991)]{Kawamura1991}
H.~Kawamura, \emph{Physica A}, 1991, \textbf{177}, 73\relax
\mciteBstWouldAddEndPuncttrue
\mciteSetBstMidEndSepPunct{\mcitedefaultmidpunct}
{\mcitedefaultendpunct}{\mcitedefaultseppunct}\relax
\unskip.}
\bibitem[Widom(1993)]{Widom1993}
M.~Widom, \emph{Phys. Rev. Lett.}, 1993, \textbf{70}, 2094\relax
\mciteBstWouldAddEndPuncttrue
\mciteSetBstMidEndSepPunct{\mcitedefaultmidpunct}
{\mcitedefaultendpunct}{\mcitedefaultseppunct}\relax
\unskip.}
\bibitem[Rucklidge \emph{et~al.}(2012)Rucklidge, Silber, and
Skeldon]{Rucklidge2012}
A.~M. Rucklidge, M.~Silber and A.~C. Skeldon, \emph{Phys. Rev. Lett.}, 2012,
\textbf{108}, 074504\relax
\mciteBstWouldAddEndPuncttrue
\mciteSetBstMidEndSepPunct{\mcitedefaultmidpunct}
{\mcitedefaultendpunct}{\mcitedefaultseppunct}\relax
\unskip.}
\bibitem[Barkan \emph{et~al.}(2014)Barkan, Engel, and Lifshitz]{Barkan2014}
K.~Barkan, M.~Engel and R.~Lifshitz, \emph{Phys. Rev. Lett.}, 2014,
\textbf{113}, 098304\relax
\mciteBstWouldAddEndPuncttrue
\mciteSetBstMidEndSepPunct{\mcitedefaultmidpunct}
{\mcitedefaultendpunct}{\mcitedefaultseppunct}\relax
\unskip.}
\bibitem[Lifshitz and Petrich(1997)]{Lifshitz1997}
R.~Lifshitz and D.~M. Petrich, \emph{Phys. Rev. Lett.}, 1997, \textbf{79},
1261\relax
\mciteBstWouldAddEndPuncttrue
\mciteSetBstMidEndSepPunct{\mcitedefaultmidpunct}
{\mcitedefaultendpunct}{\mcitedefaultseppunct}\relax
\unskip.}
\bibitem[Oxborrow and Henley(1993)]{Oxborrow1993}
M.~Oxborrow and C.~L. Henley, \emph{Phys. Rev. B}, 1993, \textbf{48},
6966\relax
\mciteBstWouldAddEndPuncttrue
\mciteSetBstMidEndSepPunct{\mcitedefaultmidpunct}
{\mcitedefaultendpunct}{\mcitedefaultseppunct}\relax
\unskip.}
\bibitem[Engel and Trebin(2007)]{Engel2007}
M.~Engel and H.-R. Trebin, \emph{Phys. Rev. Lett.}, 2007, \textbf{98},
225505\relax
\mciteBstWouldAddEndPuncttrue
\mciteSetBstMidEndSepPunct{\mcitedefaultmidpunct}
{\mcitedefaultendpunct}{\mcitedefaultseppunct}\relax
\unskip.}
\end{mcitethebibliography}
\end{document}
| -32,358.463615 |
[
0.332763671875,
0.1138916015625
] | 28.475336 |
[
-2.771484375,
0.6845703125,
-1.8798828125,
-4.984375,
-1.1318359375,
7.61328125
] |
[
0.70556640625,
3.201171875,
-0.7685546875,
1.39453125
] | 639 | 6,999 |
[
-2.884765625,
3.12890625
] | 25.79796 |
[
-6.0625,
-4.0234375,
-4.41015625,
-2.6796875,
1.67578125,
12.5078125
] | 0.671094 | 24.073596 | 22.560366 | 6.206444 |
[
1.4715725183486938
] | -22,933.887039 | 6.067724 | -31,476.501652 | 8.613544 | 6.131828 |
[
-2.939453125,
-3.69140625,
-3.80078125,
-4.69921875,
2.43359375,
12.0546875
] |
[
-5.68359375,
-2.291015625,
-2.54296875,
-1.8818359375,
3.77734375,
5.5390625
] | |
BkiUdb45qdmDD94KRNW6
|
\section{Introduction}
Gravitational lensing is an astrophysical phenomenon, that can be observed on galactic to cosmic spatial scales, through which distant images are distorted by the intervening mass density field. Due to its nature, lensing is sensitive to the total mass distribution (both visible and invisible) along a line of sight \citep{[1],[2],[23],[24]}. Therefore, as the majority of massive structures in the universe are predominantly dark matter, lensing provides a novel way to probe the nature of dark matter itself.
\par
Weak gravitational lensing (WL) is a regime in which one makes the approximation that lensed sources have (at no time) come radially closer than an Einstein radius to the intervening mass concentrations \nobreak--\nobreak\hskip3pt which ensures that sources are not multiply imaged. The effect of weak lensing on distant source galaxies is two-fold: the galaxy size is magnified by a convergence field $\kappa$; and the galaxy ellipticity (third-flattening) is perturbed from an underlying intrinsic value by a shearing field $\gamma$. Direct observation of the convergence field is ill-defined \nobreak--\nobreak\hskip3pt as a result of the mass-sheet degeneracy \nobreak--\nobreak\hskip3pt and so typically measurements of the shearing field are inverted to produce estimators for $\kappa$. These estimators are colloquially named \textit{dark matter mass-maps}, and constitute one of the principle observables for cosmology \citep{[25]}.
Standard cosmological protocol is to extract weak lensing information in the form of second order statistics \citep{[26],[49],kilbinger2015} which are then compared to theory. In this approach mass-maps are not required. However, as two-point global statistics are by definition sensitive only to Gaussian contributions, and weak lensing is inherently non-Gaussian, it is informative to consider higher-order statistics \citep{[27], [28]}. Many higher-order statistical techniques can be performed directly on mass-maps ($\kappa$-fields), which motivates investigation into alternate mass-map reconstruction methodologies.
\par
Reconstructing mass-maps from shear observations requires solving an ill-posed (often seriously) inverse problem. Many approaches to solving this lensing inverse problem have been developed \citep[\textit{e.g.}][]{[29],[5],[6],[3],[15],[22]}, with the industry standard being Kaiser-Squires \citep[KS,][]{[5]}. Although these approaches often produce reliable convergence estimators, they lack principled statistical approaches to uncertainty quantification and often assume or impose Gaussianty during the reconstruction process (Gaussian smoothing in the KS case \nobreak--\nobreak\hskip3pt which is sub-optimal when one wishes to analyze small-scale non-Gaussian structure).
\par
Most methods refrain from quantifying uncertainties in reconstructions, but those that do often do so by assuming Gaussian priors and adopting Markov-chain Monte-Carlo (MCMC) techniques \citep{[30],[26],[43]}.The computational cost of MCMC approaches is large. Recent developments in probability concentration theory have led to advancements in fast approximate uncertainty quantification techniques \citep{[10],[11],[12]}.
\par
In this article we present a new mass-mapping formalism. We formulate the lensing inverse problem as a sparse hierarchical Bayesian inference problem from which we derive an unconstrained convex optimization problem. We solve this optimization problem in the analysis setting, with a wavelet-based, sparsity-promoting, $l_1$-norm prior \nobreak--\nobreak\hskip3pt similar priors have been shown to be effective in the weak lensing setting \citep{[15],[6],[9],[46]}. Formulating the problem in this way allows us, for the first time, to recover \textit{maximum a posteriori} (MAP) estimators, from which we can exploit analytic methods \citep{[10],[12]} to recover approximate highest posterior density (HPD) credible regions, and perform hypothesis testing of structure in a variety of ways. We apply our algorithm to a range of catalogs drawn from N-body simulations \nobreak--\nobreak\hskip3pt Buzzard v1.6 \citep{[22]} and Bolshoi \citep{[20]} \nobreak--\nobreak\hskip3pt and the debated A520 cluster catalogs \citep{[36], [32]}. We then demonstrate the aforementioned uncertainty quantification techniques on our MAP reconstructions from these catalogs.
\par
The structure of this article is as follows. In section \ref{Background} we provide a brief overview of the weak lensing paradigm and motivate a sparsity-based approach. In section \ref{Sparsity} we provide the details of our algorithm, as well as some updates to super-resolution image recovery. In section \ref{Bayesian Uncertainties} we present the uncertainty quantification techniques, both mathematically and mechanistically. In sections \ref{Simulation Results} and \ref{A520} we apply both our reconstruction algorithm and the uncertainty quantification techniques to the aforementioned datasets and analyze the results. Finally, in section \ref{Conclusions} we draw conclusions from this work and propose future avenues of research.
\par
Section \ref{Sparsity} relies on a moderate level of understanding in the fields of proximal calculus and compressed sensing, and section \ref{Bayesian Uncertainties} relies on a general understanding of Bayesian inference. As such, for the reader solely interested in practical application of these techniques we recommend sections \ref{Simulation Results} onwards.
\section{Weak Gravitational Lensing} \label{Background}
The following section presents a brief review of the mathematical background relevant to the weak lensing formalism, though a deeper description can be found in popular review articles \citep{[1],[2]}.
\subsection{Weak lensing regime}
Gravitational lensing refers to the deflection of distant photons as they propagate from their origin to us, the observer. This deflection is caused by local Newtonian potentials which are, in turn, sourced by the total local matter over or under density. As such, weak lensing is sensitive to both the visible and invisible matter distribution \nobreak--\nobreak\hskip3pt making it an ideal probe of dark matter in the Universe.
\par
The WL regime is satisfied when propagating photons (from a distant source) have an angular position on the source plane $\beta$ (relative to the line-of-sight from observer through the lensing mass) greater than the Einstein radius $\theta_E$ of the intervening mass. This assertion ensures that the solution of the first order lens equation is singular:
\begin{equation}
\beta = \theta - \theta_E^2 \frac{\theta}{|\theta|^2}.
\end{equation}
Where the Einstein radius is defined to be:
\begin{equation}
\theta_E = \sqrt{\frac{4GM}{c^2}\frac{f_K(r - r^{\prime})}{f_K(r)f_K(r^{\prime})}},
\end{equation}
where $f_K$ is the angular diameter distance in a cosmology with curvature $K$, $c$ is the speed of light in a vacuum, $G$ is the gravitational constant and $M$ is the lensing mass. Perhaps more generally the weak lensing regime can be defined as convergence fields for which $\kappa \ll 1$ -- ensuring that the shear signal remains linear.
\par
Due to the sparse nature of the distribution of galaxies across the sky, most sources are (to a good approximation) within the WL regime. The WL effect is best expressed in terms of a lensing potential $\phi$, defined to be the integral of the Newtonian potential $\Phi$ along a given line of sight:
\begin{equation}
\phi(r,\omega) = \frac{2}{c^2} \int_0^r dr^{\prime} \frac{f_K(r - r^{\prime})}{f_K(r) f_K(r^{\prime})} \Phi(r^{\prime}, \omega),
\end{equation}
where $r$ and $r^{\prime}$ are comoving distances, and $\omega = (\theta,\psi)$ are angular spherical co-ordinates. The local Newtonian potential must satisfy the Poisson equation and as such is related to the matter over-density field:
\begin{equation}
\nabla^2 \Phi(r,\omega) = \frac{3 \Omega_M H_0^2}{2a(r)} \delta(r,\omega),
\end{equation}
where $\Omega_M$ is the matter density parameter, $H_0$ is the current Hubble constant, $a(r)$ is the scale factor, and $\delta$ is the fractional over-density.
\par
To first order, there are two primary ways in which light from distant sources is distorted by this lensing potential. Images are magnified by a spin-0 convergence field $\kappa$ and sheared by a spin-2 shear field $\gamma$. These quantities can be shown \citep{[1]} to be related to the lensing potential by:
\begin{align}
&\kappa(r,\omega) = \frac{1}{4}(\eth \bar{\eth} + \bar{\eth} \eth) \; \phi(r,\omega), \label{eq:kappatophi} \\
&\gamma(r,\omega) = \frac{1}{2} \eth \eth \; \phi(r,\omega),\label{eq:gammatophi}
\end{align}
where $\eth$ and $\bar{\eth}$ are the spin $s$ raising and lowering operators respectively and are in general defined to be,
\begin{align}
\eth \equiv -\sin^s\theta \Big ( \frac{\partial}{\partial \theta} + \frac{i \partial}{\sin\theta \partial \psi} \Big ) \sin^{-s}\theta, \\
\bar{\eth} \equiv -\sin^{-s}\theta \Big ( \frac{\partial}{\partial \theta} - \frac{i \partial}{\sin\theta \partial \psi} \Big ) \sin^{s}\theta.
\end{align}
Where we have omitted spin subscripts for clarity.
\subsection{Standard mass-mapping techniques}
Typically we wish to make inferences about the projected matter over-density $\delta(r,\omega)$ which is most directly accessible by inverting the integral equation \citep{[2]}:
\begin{equation}
\kappa(r, \omega) = \frac{3\Omega_M H_0^2}{2c^2} \int_0^{r} dr^{\prime}\frac{f_K(r^{\prime})f_K(r - r^{\prime})}{f_K(r)} \frac{\delta(f_K(r^{\prime})r^{\prime}, r^{\prime})}{a(r)}.
\end{equation}
\par
This poses a difficulty as there is no \textit{a priori} way to determine the intrinsic brightness of galaxies before they are lensed \nobreak--\nobreak\hskip3pt referred to as the \textit{mass-sheet degeneracy}. This makes $\kappa$ an unobservable quantity. However, as the intrinsic ellipticity distribution of galaxies has zero mean, if one averages many galaxy ellipticities within a given pixel the true shear $\gamma$ can be recovered -- which makes $\gamma$ the observable field. As such we measure $\gamma$, and subsequently map to $\kappa$.
\par
For small sky fractions we can approximate the field of view as a plane (though this approximation degrades quickly with sky fraction; \citeauthor{[3]} \citeyear{[3]}). In this planar approximation $\eth$ and $\bar{\eth}$ reduce to \citep{[4]}:
\begin{equation}
\eth \approx -(\partial_x + i \partial_y) \quad \mbox{and} \quad \bar{\eth} \approx -(\partial_x - i \partial_y).
\end{equation}
Combining equations (\ref{eq:kappatophi}) and (\ref{eq:gammatophi}) we find the planar forward model in Fourier space:
\begin{equation} \label{eq:forwardmodel}
\hat{\gamma}(k_x,k_y) = \bm{\mathsf{D}}_{k_x,k_y} \hat{\kappa}(k_x, k_y),
\end{equation}
with the mapping operator being,
\begin{equation} \label{eq:KSD}
\bm{\mathsf{D}}_{k_x,k_y} = \frac{k_x^2-k_y^2+2ik_xk_y}{k_x^2+k_y^2}.
\end{equation}
Hereafter we drop the $k_x,k_y$ subscripts for clarity. It is informative to note that this forward model is undefined at the origin ($k=\sqrt{k_x^2+k_y^2}=0$) \nobreak--\nobreak\hskip3pt which corresponds to the mass-sheet degeneracy \citep{[1]}
\par
The most naive inversion of this forward model is Kaiser-squires (KS) inversion,
\begin{equation} \label{eq:Kaiser-Squires}
\hat{\kappa}^{\text{KS}} = \bm{\mathsf{D}}^{-1} \hat{\gamma},
\end{equation}
which is direct inversion in Fourier space \citep{[5]}. KS inversion of the forward model, given by equation (\ref{eq:forwardmodel}), performs adequately, provided the space over which it is defined is complete, and the sky fraction is small. However, masking and survey boundaries are inherent in typical WL surveys, leading to significant contamination of the KS estimator. Often maps recovered with the KS estimator are convolved with a Gaussian kernel to reduce the impact of these contaminations but this is sub-optimal. This smooths away a large fraction of the small-scale non-Gaussian information, which cosmologists are increasingly interested in extracting from WL surveys.
\section{Sparse MAP Estimators} \label{Sparsity}
Several alternate approaches for solving the inverse problem between convergence $\kappa$ and shear $\gamma$ which do not assume or impose Gaussianity have been proposed, some of which are based on the concept of wavelets and sparsity \citep{[6], [7], [8], [9]}.
\par
We propose a mass-mapping algorithm that relies on sparsity in a given wavelet dictionary. Moreover, we formulate the problem such that we can exploit recent developments in the theory of probability concentration, which have been developed further to produce novel uncertainty quantification techniques \citep{[10]}. Crucially, this allows us to recover principled statistical uncertainties on our MAP reconstructions \citep[as in][]{[11], [12]} as will be discussed in detail in the following section.
\par
As mentioned previously, galaxies have an intrinsic ellipticity. To mitigate the effect of intrinsic ellipticity we choose to project the ellipticity measurements onto a grid and average. If we assume that galaxies have no preferential orientation in the absence of lensing effects, then the average intrinsic ellipticity tends to zero. This is a good approximation for the purposes of this paper, but weak correlation between the intrinsic alignments of galaxies has been observed \citep{[13],[45]}.
\subsection{Hierarchical Bayesian Framework}
Hierarchical Bayesian inference provides a rigorous mathematical framework through which theoretically optimal solutions can be recovered. Moreover it allows one to construct measures of the uncertainty on recovered point estimates.
\par
As is common for hierarchical Bayesian models, we begin from Bayes' theorem for the posterior distribution,
\begin{equation} \label{eq:bayes}
p(\kappa|\gamma) = \frac{p(\gamma|\kappa)p(\kappa)}{\int_{\mathbb{C}^N} p(\gamma|\kappa)p(\kappa)d\kappa},
\end{equation}
where $p(\gamma|\kappa)$ is the likelihood function representing data fidelity, $N$ is the dimensionality of $\kappa$ and $p(\kappa)$ is a prior on the statistical nature of $\kappa$. The denominator is called the \textit{Bayesian evidence} which is constant and so can be dropped for our purposes. Typically the Bayesian evidence is used for model comparison, which we will not be considering within the context of this paper. Given Bayes' theorem, and the monotonicity of the logarithm function, we can easily show that the maximum posterior solution is defined by,
\begin{equation} \label{eq:logposterior}
\argmaxT_{\kappa} \lbrace p(\kappa|\gamma) \rbrace = \argminT_{\kappa} \lbrace -\log ( \; p(\kappa|\gamma) \;) \rbrace.
\end{equation}
This step is crucial, as it allows us to solve the more straightforward problem of minimizing the log-posterior rather than maximizing the full posterior. Conveniently, in most physical situations the operators associated with the log-posterior are convex. Drawing from the field of convex optimization, the optimal solution for the posterior can be recovered extremely quickly \nobreak--\nobreak\hskip3pt even in high dimensional settings.
\subsection{Sparsity and Inverse problems}
Let $\gamma \in \mathbb{C}^M$ be the discretized complex shear field extracted from an underlying discretized convergence field $\kappa \in \mathbb{C}^N$ by a \textit{measurement operator} $\bm{\Phi} \in \mathbb{C}^{M \times N}: \kappa \mapsto \gamma$. In the planar setting $\bm{\Phi}$ can be modeled by,
\begin{equation} \label{eq:measurement_operator}
\bm{\Phi} = \bm{\mathsf{F}}^{-1} \bm{\mathsf{D}} \bm{\mathsf{F}}.
\end{equation}
Here $\bm{\mathsf{F}}$ is the discrete fast Fourier transform (FFT), $\bm{\mathsf{F}}^{-1}$ is the inverse discrete fast Fourier transform (IFFT), and $\bm{\mathsf{D}}$ is a diagonal matrix applying the scaling of the forward model in Fourier space as defined in equation (\ref{eq:KSD}). Suppose Gaussian noise with variance $\sigma_n^2$ is present, realistic measurement of $\gamma$ will be contaminated such that:
\begin{equation}
\gamma = \bm{\Phi} \kappa + \mathcal{N}(0,\sigma_n^2),
\end{equation}
where $\mathcal{N}(0,\sigma_n^2) \in \mathbb{C}^M$ is additive i.i.d. Gaussian noise of variance $\sigma_n^2$. Often in WL experiments the total number of binned measurements is less than the number of pixels to be recovered, $M < N$, and the inverse problem becomes ill-posed.
\par
The likelihood function $p(\gamma|\kappa)$ for this scenario is:
\begin{align} \label{eq:likelihood}
p(\gamma|\kappa) &\propto \exp \Bigg(\frac{(\bm{\Phi} \kappa - \gamma)^{\dagger} \Sigma^{-1}(\bm{\Phi} \kappa - \gamma)}{2} \Bigg), \\
p(\gamma|\kappa) &\propto \exp \Bigg(\frac{-\norm{\bm{\Phi} \kappa - \gamma}_2^2}{2\sigma_n^2} \Bigg),
\end{align}
where the second line follows trivially as the covariance $\Sigma$ is (by construction) proportional to the identity, with constant of proportionality $\sigma_n^2$ -- note that $\norm{\cdot}_x$ is the $l_x$-norm. To regularize this inverse problem, we then define a sparsity promoting Laplace-type prior:
\begin{equation}
p(\kappa) \propto \exp \bigg(-\mu \norm{\bm{\Psi}^{\dag}\kappa}_1 \bigg),
\end{equation}
where $\bm{\Psi}$ is an appropriately selected wavelet dictionary, and $\mu \in \mathbb{R}_{+}$ is a regularization parameter \nobreak--\nobreak\hskip3pt effectively a weighting between likelihood and prior. Note that one may choose any convex log-prior within this formalism \textit{e.g.} an $\ell_2$-norm prior from which one essentially recovers Weiner filtering \citep[see][for alternate iterative Weiner filtering approaches]{Seljak2003,Horowitz2018}. From equations (\ref{eq:bayes}) and (\ref{eq:logposterior}) the unconstrained optimization problem which minimizes the log-posterior is,
\begin{equation} \label{eq:optimization-problem}
\kappa^{\text{map}} = \argminT_{\kappa} \Bigg \lbrace \mu \norm{ \bm{\Psi}^{\dag}\kappa}_1 + \frac{\norm{\bm{\Phi} \kappa - \gamma}_2^2}{2\sigma_n^2} \Bigg \rbrace,
\end{equation}
where the bracketed term is called the \textit{objective function}. To solve this convex optimization problem we adopt a forward-backward splitting algorithm \citep[\textit{e.g.}][]{[40]}. A full description of this algorithm applied in the current context is outlined in \citet{[12]}.
\par
Let $f(\kappa) = \mu \norm{\bm{\Psi}^{\dag} \kappa}_1$ denote our prior term, and $g(\kappa) = \norm{\bm{\Phi} \kappa - \gamma}_2^2 / 2\sigma_n^2$ denote our data fidelity term. Then our optimization problem can be re-written compactly as,
\begin{equation}
\argminT_{\kappa} \big \lbrace f(\kappa) + g(\kappa) \big \rbrace.
\end{equation}
The forward-backward iteration step is then defined to be,
\begin{equation}
\kappa^{(i+1)} = \text{prox}_{\lambda^{(i)}f} \bigg (\kappa^{(i)} - \lambda^{(i)} \nabla g(\kappa^{(i)}) \bigg ),
\end{equation}
for iteration $i$, with gradient,
\begin{equation}
\nabla g(\kappa) = \frac{\bm{\Phi}^{\dag} (\bm{\Phi} \kappa - \gamma)}{\sigma_n^2}.
\end{equation}
If the wavelet dictionary $\bm{\Psi}$ is a tight frame (i.e. $\bm{\Psi}^{\dag}\bm{\Psi} = \mathbb{I}$) the proximity operator is given by,
\begin{equation}
\text{prox}_{\lambda f}(z) = z + \bm{\Psi} \bigg ( \text{soft}_{\lambda\mu} (\bm{\Psi}^{\dag}z) - \bm{\Psi}^{\dag}z \bigg ),
\end{equation}
where $\text{soft}_{\lambda}(z)$ is the point-wise soft-thresholding operator \citep{[40]} and $\lambda$ is a parameter related to the step-size (which is in turn related to the Lipschitz differentiability of the log-prior) which should be set according to \citet{[12]}. The iterative algorithm is given explicitly in the primary iterations of algorithm \ref{alg:forward_back}. Adaptations for frames which are not tight can be found in \citet{[12]} and are readily available within our framework.
\par
Our algorithm has distinct similarities to the GLIMPSE algorithm presented by \citet{[6]}, but crucially differs in several aspects. Most importantly we formulate the problem in a hierarchical Bayesian framework which allows us to recover principled statistical uncertainties. In addition to this we include Bayesian inference of the regularization parameter, a robust estimate of the noise-level (which can be folded into the hierarchical model), and we use super-resolution operators instead of non-discrete fast Fourier transforms.
\begin{algorithm}
\setstretch{1.2}
\caption{Forward-backward analysis algorithm}
\hspace*{\algorithmicindent} \textbf{Input:} $\gamma \in \mathbb{C}^M$, $\kappa^{(0)} \in \mathbb{C}^N$, $\sigma_n$, $\lambda$, $\mu^{(0)} = i = t = 0$, $T_1,T_2 \in \mathbb{R}_+$ \\
\hspace*{\algorithmicindent} \textbf{Output:} $\kappa^{\text{map}} \in \mathbb{C}^N$, $\mu \in \mathbb{R}_{+}$ \\
\textbf{Precomputation:}\\
\textbf{Do:}
\begin{algorithmic}[1]
\State Calculate $\kappa^{(t)} = \argminT_{\kappa} \big \lbrace f(\kappa) + g(\kappa) \big \rbrace$,
\State Update $\mu^{(t+1)} = \frac{(N/k)+\alpha-1}{f(\kappa^{(t)}) + \beta}$,
\State $t = t + 1$,
\State On convergence, $\mu$ becomes fixed.
\end{algorithmic}
\textbf{Until:} Iteration limit reached.\\
\textbf{Primary Iterations:}\\
\textbf{Do:}
\begin{algorithmic}[1]
\State update $\nu^{(i+1)} = \kappa^{(i)} - \lambda \frac{\bm{\Phi}^{\dagger}(\bm{\Phi} \kappa^{(i)} - \gamma)}{\sigma_n^2}$,
\State compute $\eta = \bm{\Psi}^{\dagger}\nu^{(i+1)}$,
\State update $\kappa^{(i+1)} = \nu^{(i+1)} + \bm{\Psi} ( \text{soft}_{\lambda\mu}(\eta)-\eta)$,
\State $i = i + 1$.
\end{algorithmic}
\textbf{Until:} Stopping criterion satisfied. \\
\textit{i.e.} $\frac{\norm{\kappa^{(i)} - \kappa^{(i+1)}}_2}{\norm{\kappa^{(i)}}_2} < T_1$ and $\frac{\text{obj}(\kappa^{(i)}) - \text{obj}(\kappa^{(i+1)})}{ \text{obj}(\kappa^{(i)})} < T_2$.
\label{alg:forward_back}
\end{algorithm}
\subsection{Reduced Shear}
Due to a degeneracy between $\gamma$ and $\kappa$ the true observable quantity is in fact the \textit{reduced shear} $g$ \citep{[1]},
\begin{equation}
g = \frac{\gamma}{1-\kappa}.
\end{equation}
Deep in the weak lensing regime one can safely approximate $\gamma \approx g \ll 1$ which ensures that the optimization problem remains linear. However, when reconstructing regions close to massive structures (galaxy clusters) this approximation is no longer strictly valid and we must unravel this additional factor. We adopt the procedure outlined in \citet{[3]}, which we also outline schematically in Figure \ref{fig:schematic-of-reduced-shear} \nobreak--\nobreak\hskip3pt this method can be found in detail in \citeauthor{[47]}, pg 153. We find that these corrections typically converge after $\sim$ 5-10 iterations.
\begin{figure}
\begin{center}
\begin{tikzpicture}[node distance = 2cm, auto]
\node [block, text width=8em] (1) {Let: $\gamma^{i=0} = g $};
\node [block, below of=1, node distance=1.6cm, text width=8em] (2) {Calculate: $\kappa^{(i)}$};
\node [block, below of=2, node distance=1.6cm] (3) {Update: $\gamma^{i+1} = \gamma^{i} ( 1 - \kappa^{(i)})$};
\node [decision, below of=3] (4) {Stopping Criterion Met?};
\node [block, right of=2, node distance=3.0cm, text width=6em] (5) {$i \rightarrow i+1$};
\node [block, below of=4] (4b) {Return: $\kappa^{(i)}$.};
\path [line] (1) -- (2);
\path [line] (2) -- (3);
\path [line] (3) -- (4);
\path [line,dashed] (4) -| node[right] {No}(5);
\path [line,dashed] (5) -- (2);
\path [line,dashed] (4) -- node[near start] {Yes}(4b);
\end{tikzpicture}
\caption{Schematic of reduced shear iterations. An initial guess of the MAP solution $\kappa^{\text{map}}_i$ is constructed, the current best shear estimates $\gamma_i$ are then used in tandem to construct a new estimate of the true shear field $\gamma_{i+1}$.}
\label{fig:schematic-of-reduced-shear}
\end{center}
\end{figure}
\subsection{Regularization Parameter Selection}
One key issue of sparsity-based reconstruction methods is the selection of the regularization parameter $\mu$. Several methodologies have arisen \citep{[6], [9], [14], [15]} for selecting $\mu$, though often the regularization parameter is chosen somewhat arbitrarily \nobreak--\nobreak\hskip3pt as the integrity of the MAP solution is assumed to be weakly dependent on the choice of $\mu$. However, to extract principled statistical uncertainties on the recovered images, one must select this parameter in a principled statistical manner.
\par
We apply the hierarchical Bayesian formalism developed by \citet{[16]} \nobreak--\nobreak\hskip3pt the details of which are elegantly presented by the authors. Though we will outline roughly the underlying argument here.
\par
First define a sufficient statistic $f$ to be $k$-homogeneous if $\exists \; k \in \mathbb{R}_{+}$ such that,
\begin{equation}
f(\eta x) = \eta^kf(x), \: \forall x \in \mathbb{R}^N, \: \forall \eta > 0.
\end{equation}
All norms, composite norms and composition of norms with linear operators are 1-homogeneous \nobreak--\nobreak\hskip3pt and so our $\ell_1$-norm has $k$ of 1. If a sufficient statistic $f$ is $k$-homogeneous, then the normalization factor $C(\mu)$ of $p(\kappa | \mu)$ is given by \citep{[16]},
\begin{equation} \label{eq:proposition}
C(\mu) = A \mu^{-N/k},
\end{equation}
where $A$ is a constant independent from $\mu$. The proposed Bayesian inference model then implements a gamma-type hyper-prior \nobreak--\nobreak\hskip3pt which is a typical hyper-prior for scale-parameters,
\begin{equation} \label{eq:hyper-prior}
p(\mu) = \frac{\beta^{\alpha}}{\Gamma(\alpha)}\mu^{\alpha - 1} e^{-\beta \mu} \mathbb{I}_{\mathbb{R}_+}(\mu),
\end{equation}
where without loss of generality $\alpha = \beta = 1$. The result is effectively insensitive to their value (in numerical experiments values of $\alpha, \beta \in [10^{-2}, 10^{5}]$ produced essentially no difference in $\mu$).
\par
Now, let us extend the inference problem of the log-posterior to the case where $\mu$ is an additionally unknown parameter. In this context we compute the joint MAP estimator $(\kappa^{\text{map}}, \mu^{\text{map}}) \in \mathbb{C}^{N} \times \mathbb{R}_+$ which maximizes $p(\kappa, \mu | \gamma)$ such that,
\begin{equation}
\mathbf{0}_{N+1} \in \partial_{\kappa, \mu} \log p(\kappa^{\text{map}}, \mu^{\text{map}} | \gamma),
\end{equation}
where $\mathbf{0}_{i}$ is the $i$-dimensional null vector and $\partial_{s}h(s^{\prime})$ is the set of sub-gradients of function $h(s)$ at $s^{\prime}$. This in turn implies both that,
\begin{equation} \label{eq:initial-opt}
\mathbf{0}_N \in \partial_{\kappa} \log p(\kappa^{\text{map}}, \mu^{\text{map}} | \gamma),
\end{equation}
and
\begin{equation} \label{eq:new-opt}
\mathbf{0} \in \partial_{\mu} \log p(\kappa^{\text{map}}, \mu^{\text{map}} | \gamma).
\end{equation}
\par
From equation (\ref{eq:initial-opt}) we recover the optimization problem with known regularization parameter $\mu$ given in equation (\ref{eq:optimization-problem}). However, from equations (\ref{eq:proposition}, \ref{eq:hyper-prior}, \ref{eq:new-opt}) it follows that the MAP regularization parameter $\mu$ is given by \citep{[16]},
\begin{equation}
\mu^{\text{map}} = \frac{\frac{N}{k} + \alpha - 1}{f(\kappa^{\text{map}}) + \beta},
\end{equation}
where we recall that $N$ is the total dimension of our convergence space.
\par
It is precisely this optimal $\mu$ value which we wish to use in our hierarchical Bayesian model. Hereafter we drop the map superscript from $\mu$ for clarity. To calculate $\mu$ we perform preliminary iterations defined by:
\begin{equation}
\kappa^{(t)} = \argminT_{\kappa} \bigg \lbrace f(\kappa; \mu^{(t)}) + g(\kappa) \bigg \rbrace,
\end{equation}
where $g(\kappa)$ is our likelihood term and,
\begin{equation}
\mu^{(t+1)} = \frac{\frac{N}{k}+\alpha-1}{f(\kappa^{(t)}) + \beta}.
\end{equation}
Typically we find that these preliminary iterations take $\sim$ 5-10 iterations to converge, and recover close to optimal parameter selection for a range of test cases -- note that here the optimal selection of $\mu$ is that which maximizes the SNR of a recovered image.
\par
Another factor which can influence the quality of reconstructions is the selection of wavelet dictionary. In this paper we consider Daubechies (8 levels) and SARA dictionaries \citep{[39],[50]}, though a wide variety of wavelet dictionaries exist \nobreak--\nobreak\hskip3pt such as the starlets in \citet{[17]} or discrete cosine transforms in \citet{[7]}. The 8-level SARA dictionary is a combination of the Dirac and Daubechies 1 to 8 wavelet dictionaries. It is important to note that we use the SARA dictionary, not the complete SARA scheme \citep{[39],[50]}, which involves an iterative re-weighting scheme that is not considered here.
\subsection{Super-Resolution Image Recovery}
Griding of weak lensing data is advantageous in that it can provide a good understanding of the noise properties \nobreak--\nobreak\hskip3pt a necessary feature for principled uncertainty quantification. However, an inherent drawback of projecting data into a grid is the possibility of creating an incomplete space due to low sampling density -- often referred to as masking. Decomposition of spin signals on bounded manifolds is inherently degenerate \citep{bunn2003}; specifically the orthogonality of eigenfunctions is locally lost at the manifold boundaries, leading to signal leakage between Fourier (or on the sphere, harmonic) modes.
\par
One approach to mitigate this problem is to avoid the necessity of griding by substituting a \textit{non-uniform discrete Fourier transform} (NFFT) into the RHS of equation (\ref{eq:measurement_operator}) as presented by \citet{[6]}. A downside of this NFFT approach is that the noise is more difficult to handle, leading to complications when considering uncertainty quantification. Another approach is to perform super-resolution image recovery, which we present in the context of our algorithm.
\par
Suppose the dimension of our gridded measurement space is $M$ \nobreak--\nobreak\hskip3pt as before \nobreak--\nobreak\hskip3pt and the desired dimension of our solution space is $N^{\prime}$ where $N^{\prime} \geq N$. In this setting our shear measurements $\gamma \in \mathbb{C}^{M}$ and recovered convergence $\kappa \in \mathbb{C}^{N^{\prime}}$. Let us now define a \textit{super-resolution} (subscript SR) measurement operator to be,
\begin{equation}
\bm{\Phi}_{\text{SR}} = \bm{\mathsf{F}}^{-1}_{\text{lr}} \; \bm{\mathsf{D}} \; \bm{\mathsf{Z}} \; \bm{\mathsf{F}}_{\text{hr}}
\end{equation}
where $\bm{\mathsf{F}}_{\text{hr}}$ is a high resolution (dimension $N^{\prime}$) fast Fourier transform, $\bm{\mathsf{Z}} \in \mathbb{C}^{N \times N^{\prime}}$ is a Fourier space down-sampling which maps $\tilde{\kappa}^{\prime} \in \mathbb{C}^{N^{\prime}}$ on to $\tilde{\kappa} \in \mathbb{C}^{N}$, where tilde represents Fourier coefficients, and $\bm{\mathsf{D}}$ is the planar forward model given by equation (\ref{eq:forwardmodel}). Finally, $\bm{\mathsf{F}}^{-1}_{\text{lr}}$ is a low resolution (dimension $M$) inverse fast Fourier transform. For completeness the super-resolution adjoint measurement operator is given by,
\begin{equation}
\bm{\Phi}^{\dagger}_{\text{SR}} = \bm{\mathsf{F}}^{-1}_{\text{hr}} \; \bm{\mathsf{Z}}^{\dagger} \; \bm{\mathsf{D}}^{\dagger} \; \bm{\mathsf{F}}_{\text{lr}},
\end{equation}
where $\bm{\mathsf{D}}^{\dagger}$ is the adjoint of $\bm{\mathsf{D}}$ (which is self-adjoint hence $\bm{\mathsf{D}}^{\dagger}=\bm{\mathsf{D}}$) and $\bm{\mathsf{Z}}^{\dagger} \in \mathbb{C}^{M^{\prime} \times M}$ is
zero padding in Fourier space which acts by mapping $\tilde{\gamma} \in \mathbb{C}^{M}$ to $\tilde{\gamma}^{\prime} \in \mathbb{C}^{M^{\prime}}$. Note that when considering the KS estimate in the super-resolution setting a rescaling function to account for the different Fourier normalization factors must be introduced.
\subsection{Noise Estimation} \label{sig_estimates}
Working with simulation data presents several advantages not present when analyzing real observational data. The most obvious difference being that, in simulated datasets the noise level is added artificially in a well defined way. In the case of independent and identically distributed (i.i.d.) Gaussian noise the experimenter knows the value of $\sigma_n$ precisely, whereas on observational data $\sigma_n$ is an unknown parameter.
\par
One way to address this issue is to absorb $\sigma_n$ into the regularization parameter $\mu$ \citep[or implicitly into a covariance matrix as in][]{[6]}. However, for uncertainties on reconstructed images to be principled, the noise level must be known to high precision independently from $\mu$.
\par
We propose a variety of estimators for $\sigma_n$ based on the \textit{median absolute deviation} (MAD) methodology, and subsequently extend these approaches to masked fields in a representative variety of cases. For non-normal data, MAD methods provide a more robust measure of the spread (variance) as it is based on the median which has linear response to outliers as opposed to the mean which has a quadratic response to outliers.
\par
For notational ease we introduce the MAD operator defined on complex fields $X \in \mathbb{C}^M$,
\begin{equation} \label{eq:general_MAD}
\text{MAD}(X) = \sqrt{ \Bigg [ \frac{\text{Med} ( \; | \;\Re X\; | \; )}{0.67449}\Bigg ]^2 + \Bigg [ \frac{\text{Med} ( \; | \;\Im X\; | \; )}{0.67449} \Bigg ]^2 }
\end{equation}
\subsubsection{Method 1: Standard MAD}
Our first method is a straightforward application of the MAD estimator \citep{[41]} given in equation (\ref{eq:general_MAD}). We construct a field $\gamma^{\prime}$ which is representative of the noise field by subtracting the median from the real and imaginary parts of a given field $\gamma$, before evaluating the corresponding MAD estimator $\sigma_{n,1}$. Mathematically this is,
\begin{align}
& \gamma^{\prime} = \big [\Re\gamma - \text{Med}(\Re\gamma) \big ] + i \big [ \Im\gamma - \text{Med}(\Im\gamma) \big ] \\
& \sigma_{n,1} = \text{MAD}(\gamma^{\prime}). \label{eq:MAD}
\end{align}
Med is simply shorthand for median, and $\Re$, $\Im$ are shorthand for the Real$(\cdot)$ and Imaginary$(\cdot)$ components respectively and the prime on $\gamma$ indicates that the field median has been subtracted. Typically this naive $\sigma_n$ estimator has a percent error of $> 100\%$, motivating the need for more robust estimates. This is displayed in Figure \ref{fig:no_mask_sigma_estimation_methods}.
\subsubsection{Method 2: Wavelet MAD}
Our second approach is an extension to standard MAD approach by first performing a wavelet transform of the data and extracting the high frequency detail coefficients - on which the MAD estimator is applied. The choice of wavelet dictionary, $\bm{\Psi}$, was fixed as DB8 (1 and 2 levels), though in theory most dictionaries are valid choices. Mathematically this approach simply updates equation (\eqref{eq:MAD}) to read,
\begin{equation}\label{eq:MAD_wavelet}
\sigma_{n,2} = \text{MAD}(\bm{\Psi}\gamma),
\end{equation}
where the wavelet detail coefficients are considered to be representative of the noise field and the MAD($\cdot$) is given in equation (\ref{eq:general_MAD}). In the absence of masking \nobreak--\nobreak\hskip3pt or for trivial masking geometries \nobreak--\nobreak\hskip3pt the error in this estimate of $\sigma_n$ is typically of $\mathcal{O}(3\%)$, as can be seen in Figure \ref{fig:no_mask_sigma_estimation_methods} in Appendix \ref{sec:appendix_a}.
\subsubsection{Method 3: MAD via B-mode Noise Estimation} \label{sec: B-Mode Noise Estimation}
A convenient symmetry of the weak lensing formalism is that when the shear field is noiseless and defined at all points (\textit{i.e.} the space is complete), the imaginary component of the convergence field is necessarily a null field (for the standard cosmological model). However, typical noise fields do \textbf{not} share this intrinsic symmetry. Thus, the residual imaginary component of a noisy convergence reconstruction reflects the statistics of the noise field present on the underlying shear measurements.
\par
We therefore propose a novel $\sigma_n$ estimator \nobreak--\nobreak\hskip3pt schematically presented in Figure \ref{fig:schematic-of-sigma_method3} \nobreak--\nobreak\hskip3pt which exploits this symmetry. First we transform the noisy $\gamma$ field to a noisy realization of the $\kappa$ field, with the adjoint measurement operator $\bm{\Phi}^{\dagger}$. The imaginary component is then extracted and copied into the real component to form a $\kappa_{n}$ field \nobreak--\nobreak\hskip3pt the precise configuration of the noise field is irrelevant, only the underlying statistical distribution. This $\kappa_{n}$ is then transformed to form $\gamma_{n}$ by application of $\bm{\Phi}$, and the estimator for $\sigma_{n,3}$ is constructed by equation (\ref{eq:general_MAD}) such that,
\begin{equation}\label{eq:MAD_B_mode}
\sigma_{n,3} = \text{MAD}(\gamma_n)
\end{equation}
\begin{figure}
\begin{center}
\begin{tikzpicture}[node distance = 2cm, auto]
\node [block, text width=14em] (1) {Transform: $\kappa = \bm{\Phi}^{\dagger}\gamma$};
\node [block, below of=1, node distance=1.4cm] (2) {$\kappa_n = \Im\kappa + \text{i} \Im\kappa$};
\node [decision, below of=2, node distance=2.0cm, text width=6em] (5) {Masking?};
\node [below of=5, node distance=1.4cm] (dummy) {};
\node [block, left of=dummy, node distance=2.8cm, text width=8em] (5a) {$\kappa_n \leftarrow \mathcal{M} \kappa_n$};
\node [block, below of=5, node distance=2.8cm, text width=14em] (3) {Transform $\gamma_n = \bm{\Phi} \kappa_n$};
\node [block, below of=3, node distance=1.4cm, text width=20em] (4) {Return MAD B-mode estimator \eqref{eq:MAD_B_mode} for $\sigma_n$};
\path [line] (1) -- (2);
\path [line] (2) -- (5);
\path [line, dashed] (5) -| node[above] {Yes} (5a);
\path [line, dashed] (5) -- node[right] {No} (3);
\path [line, dashed] (5a) |- (3);
\path [line] (3) -- (4);
\end{tikzpicture}
\caption{Schematic representation of method 3 \nobreak--\nobreak\hskip3pt MAD via B-mode noise Estimation (\ref{sec: B-Mode Noise Estimation}). The symbols $\Im$ is shorthand for the imaginary component, and $\mathcal{M}$ is a given mask.}
\label{fig:schematic-of-sigma_method3}
\end{center}
\end{figure}
In the absence of masking \nobreak--\nobreak\hskip3pt or for trivial masking geometries \nobreak--\nobreak\hskip3pt the error in this estimate of $\sigma_n$ is typically of $\mathcal{O}(0.3\%)$, an order of magnitude smaller than method 2 given by equation (\ref{eq:MAD_wavelet}). Graphically this is displayed in Figure \ref{fig:no_mask_sigma_estimation_methods} in Appendix \ref{sec:appendix_a}.
\section{Bayesian Uncertainty Quantification} \label{Bayesian Uncertainties}
Estimators recovered from algorithms of the form presented in the previous section are MAP solutions to, in general, ill-conditioned inverse problems, and as such have significant intrinsic uncertainty. Theoretically, MCMC techniques could be applied to recover the complete posterior in the context of Gaussian \citep{[26],[43]} and sparsity-promoting \citep{[11],[44]} priors but these approaches are computationally demanding for high dimensional problems where $N$ is large. As $N$ can easily be larger than $10^6$ (\textit{e.g.} when considering $1024 \times 1024$ resolution images), MCMC approaches are often not feasible.
\par
In \citet{[10]} a methodology based on probability concentration is presented, which uses MAP estimators to estimate theoretically conservative approximate Bayesian credible regions (specifically highest posterior density credible regions) of the posterior, $p(\kappa|\gamma)$. As this approach requires only knowledge of the MAP solution and the objective function, the Bayesian credible regions can be approximated efficiently in high dimensional settings.
\subsection{Highest Posterior Density Regions }
A posterior credible region at confidence level $100(1-\alpha)\%$ is a sub-set $C_{\alpha} \in \mathbb{C}^N$ which satisfies the integral,
\begin{equation} \label{CredibleIntegral}
p(\kappa \in C_{\alpha}|\gamma) = \int_{\kappa \in \mathbb{C}^N} p(\kappa|\gamma)\mathbb{I}_{C_{\alpha}}d\kappa = 1 - \alpha,
\end{equation}
where $\mathbb{I}_{C_{\alpha}}$ is the set indicator function for $C_{\alpha}$ defined by $\mathbb{I}_{C_{\alpha}}(\kappa) = 1 \; \forall \kappa \in C_{\alpha}$ and $0$ elsewhere. One possible region which satisfies this property is the \textit{Highest Posterior Density} (HPD) region defined by,
\begin{equation}
C_{\alpha} := \lbrace \kappa : f(\kappa) + g(\kappa) \leq \epsilon_{\alpha} \rbrace,
\end{equation}
where $\epsilon_{\alpha}$ defines an isocontour (\textit{i.e.} level-set) of the log-posterior set such that the integral in (\ref{CredibleIntegral}) is satisfied. This region can be shown \citep{[19]} to have minimum volume and is thus decision-theoretically optimal. However, due to the dimensionality of the integral in (\ref{CredibleIntegral}) calculation of the HPD credible region is difficult. A conservative approximation of $C_{\alpha}$ was recently proposed \citep{[10]} and shown to be effective in the inverse imaging setting of radio interferometric imaging \citep{[12]}. This approximate HPD is defined by
\begin{equation}
C^{\prime}_{\alpha} := \lbrace \kappa : f(\kappa) + g(\kappa) \leq \epsilon^{\prime}_{\alpha} \rbrace,
\end{equation}
where the approximate threshold $\epsilon^{\prime}_{\alpha}$ is given by
\begin{equation}
\epsilon^{\prime}_{\alpha} = f(\kappa^{\text{map}}) + g(\kappa^{\text{map}}) + \tau_{\alpha} \sqrt{N} + N,
\end{equation}
with constant $\tau_{\alpha} = \sqrt{16 \log(3 / \alpha)}$. For a detailed derivation of this approximation see \citet{[10]}. Provided $\alpha \in \big ( 4\exp(-N/3) \;, 1 \big )$ the deviation of this adapted threshold is bounded and grows at most linearly with respect to $N$. The error of this approximate threshold is bounded by
\begin{equation}
0 \leq \epsilon^{\prime}_{\alpha} - \epsilon_{\alpha} \leq \eta_{\alpha} \sqrt{N} + N,
\end{equation}
where $\eta_{\alpha} = \sqrt{16 \log (3/\alpha)} + \sqrt{1/\alpha}$. In high dimensional settings ($N$ large) this error may naively appear large, however in practice the error is relatively small.
\subsection{Hypothesis Testing} \label{HypothesisTesting}
Extending the concept of HPD credible regions, one can perform \textit{knock-out} hypothesis testing of the posterior to determine the physicality of recovered structure \citep{[12]}.
\par
To perform such tests one first creates a surrogate image $\kappa^{\text{sgt}}$ by masking a feature of interest $\Omega_D \subset \Omega$ in the MAP estimator $\kappa^{\text{map}}$. It is then sufficient to check if,
\begin{equation}
f(\kappa^{\text{sgt}}) + g(\kappa^{\text{sgt}}) \leq \epsilon_{\alpha}^{\prime}.
\end{equation}
If this inequality holds, we interpret that the physicality of $\Omega_D$ is undetermined and so no strong statistical statement can be made. Should the objective function evaluated at $\kappa^{\text{sgt}}$ be larger than $\epsilon_{\alpha}^{\prime}$ then it no longer belongs to the approximate credible set $C_{\alpha}^{\prime}$ and therefore (as $\epsilon_{\alpha}^{\prime}$ is conservative) it \textbf{cannot} belong to the HPD credible set $C_{\alpha}$. Therefore, for $\kappa^{\text{sgt}}$ which do not satisfy the above inequality we determine the structure $\Omega_D$ to be strictly physical at $100(1-\alpha)\%$ confidence level. A schematic of hypothesis testing is provided in Figure \ref{fig:HypthosisTesting}.
\begin{figure}
\begin{center}
\begin{tikzpicture}[node distance = 2cm, auto]
\node [block, text width=8em] (1) {Calculate MAP solution: $\kappa^{\text{map}}$};
\node [block, below of=1, node distance=1.6cm, text width=8em] (2) {Construct surrogate: $\kappa^{\text{sgt}}$};
\node [block, below of=2, node distance=1.6cm, text width=14em] (3a) {Calculate: $\mathcal{E} = f(\kappa^{\text{sgt}}) + g(\kappa^{\text{sgt}})$};
\node [decision, below of=3a, node distance=1.6cm] (3) {$\mathcal{E} \leq \epsilon_{\alpha}^{\prime}$ ? };
\node [block, below of=3, right of=3, text width=8em] (4) {$\mathcal{Z}$ is Physical};
\node [block, below of=3, left of=3, text width=8em] (5) {$\mathcal{Z}$ is Inconclusive};
\path [line] (1) -- node {Remove feature $\mathcal{Z}$} (2);
\path [line] (2) -- (3a);
\path [line] (3a) -- (3);
\path [line,dashed] (3) -| node [near end] {No}(4);
\path [line,dashed] (3) -| node [near end] {Yes}(5);
\end{tikzpicture}
\caption{Schematic of hypothesis testing. The feature $\mathcal{Z}$ is entirely general and can be constructed by any well defined operator on the MAP solution $\kappa^{\text{map}}$.}
\label{fig:HypthosisTesting}
\end{center}
\end{figure}
\par
In pixel-space we begin by masking out a feature of interest, creating a rough surrogate image \nobreak--\nobreak\hskip3pt setting the pixels associated with a selected structure to 0 \nobreak--\nobreak\hskip3pt this rough surrogate is then passed through an appropriate wavelet filter $\Lambda$ as part of \textit{segmentation-inpainting} to replace generic background structure into the masked region. Mathematically, this amounts to the iterations,
\begin{equation}
\kappa^{(i+1),\text{sgt}} = \kappa^{\text{map}}\mathbb{I}_{\Omega - \Omega_D} + \Lambda^{\dag}\text{soft}_{\lambda_t}(\Lambda \kappa^{(i),\text{sgt}})\mathbb{I}_{\Omega_D},
\end{equation}
where $\Omega_D \subset \Omega$ is the sub-set of masked pixels, $\mathbb{I}_{\Omega-\Omega_D}$ is the set indicator function and $\lambda_t$ is a thresholding parameter which should be chosen appropriately for the image.
\par
A second straightforward method for generating surrogate images is to blur local pixel substructure into one collective structure \nobreak--\nobreak\hskip3pt in a process called \textit{segmentation-smoothing}. This approach provides a simple way to determine if the substructure in a given region is physical or likely to be an artifact of the reconstruction process.
\par
For example, if several massive peaks are located near one another, one can blur these structures into a single cohesive peak. This would be useful when considering peak statistics on convergence maps \nobreak--\nobreak\hskip3pt which is often used to constrain the cosmological parameters associated with dark matter.
\par
One can conduct such blurring of structure by: specifying a a subset of the reconstructed pixels $\Omega_D \subset \Omega$; convolving $\kappa^{\text{map}}$ with a Gaussian smoothing kernel; and replacing pixels that belong to $\Omega_D$ with their smoothed counterparts. This can be displayed algorithmically as,
\begin{equation}
\kappa^{\text{sgt}} = \kappa^{\text{map}}\mathbb{I}_{\Omega-\Omega_D} + \big (\kappa^{\text{map}} \ast \mathcal{G}(0,\chi) \big ) \mathbb{I}_{\Omega_D},
\end{equation}
where $\mathcal{G}(0,\chi)$ is a chosen Gaussian smoothing kernel and $\ast$ is a trivially extended 2D version of the the usual 1D Fourier convolution operator,
\par
In the scope of this paper we focus primarily on pixel-space features, but it is important to stress that \textit{knock-out} approach is entirely general and can be applied to any well defined feature of a MAP estimator \nobreak--\nobreak\hskip3pt \textit{i.e.} masking certain Fourier space features, removal of global small scale structure \textit{etc.}
\section{Illustration on ideal simulations} \label{Simulation Results}
We now consider a few idealized simulations to illustrate our sparse reconstruction method on cluster and LSS scales. Further to this, we showcase the aforementioned uncertainty quantification methods in a range of cluster and LSS scale MAP reconstructions.
\subsection{Datasets}
In this paper we focus primarily on two simulation datasets: 3 large clusters extracted from the Bolshoi N-body simulation \citep{[20]}, and the Buzzard V-1.6 N-body simulation catalogs \citep{DeRose2018,wechsler2018}.
\par
On the cluster scale we showcase our formalism on a set of Bolshoi N-body simulation data sets \nobreak--\nobreak\hskip3pt see Figure \ref{fig:bolshoi_input_maps}. The Bolshoi N-body cluster simulation catalogs we work with in this paper are those used in \citet{[6]}, which were extracted using the CosmoSim web-tool\footnote{https://www.cosmosim.org}. Construction of these weak lensing realisations assumed a redshift of $0.3$, with a $10 \times 10$ arcmin$^2$ field of view, and have convergence normalized with respect to lensing sources at infinity. Due to the relatively low particle density, these images were subsequently denoised by a multi-scale Poisson denoising algorithm.
\par
To illustrate sparse MAP reconstructions on a larger scale the Buzzard v-1.6 shear catalogs which are constructed from high resolution N-body simulations, with a quarter-sky coverage which were provided by the LSST-DESC collaboration\footnote{http://lsst-desc.org}. From this large scale N-body simulation we extract a set of 60 smaller independent planar regions, upon which we apply our reconstruction algorithm \nobreak--\nobreak\hskip3pt see Figure \ref{fig:buzzard_input_maps} for 3 randomly selected extracted maps.
\begin{figure}
\centering
\textbf{\Large Bolshoi-1 $\kappa$} \par
\includegraphics[width=0.45\textwidth, trim={0 22.5cm 0 1.6cm},clip]{Bolshoi_clean_plots/Bolshoi_clean_plots.png}
\textbf{\Large Bolshoi-2 $\kappa$} \par
\includegraphics[width=0.45\textwidth, trim={0 11.5cm 0 12.8cm},clip]{Bolshoi_clean_plots/Bolshoi_clean_plots.png}
\textbf{\Large Bolshoi-3 $\kappa$} \par
\includegraphics[width=0.45\textwidth, trim={0 0 0 23.8cm},clip]{Bolshoi_clean_plots/Bolshoi_clean_plots.png}
\caption{Input convergence maps extracted from the Bolshoi N-body simulation, as presented in \protect \citet{[6]}.}
\label{fig:bolshoi_input_maps}
\end{figure}
\begin{figure}
\centering
\textbf{\Large Buzzard-4 $\kappa$} \par
\includegraphics[width=0.48\textwidth, trim={0 22.5cm 0 1.6cm},clip]{Bolshoi_clean_plots/Buzzard_clean_plots.png}
\textbf{\Large Buzzard-10 $\kappa$} \par
\includegraphics[width=0.48\textwidth, trim={0 11cm 0 13.0cm},clip]{Bolshoi_clean_plots/Buzzard_clean_plots.png}
\textbf{\Large Buzzard-52 $\kappa$} \par
\includegraphics[width=0.48\textwidth, trim={0 0 0 24.3cm},clip]{Bolshoi_clean_plots/Buzzard_clean_plots.png}
\caption{Input convergence maps extracted from the Buzzard N-body LSS simulation \textit{via} ray-tracing.}
\label{fig:buzzard_input_maps}
\end{figure}
\subsection{Methodology}
Typically, we begin by creating an artificial shear field $\hat{\gamma} \in \mathbb{C}^M$ from a known \textit{ground-truth} convergence field $\kappa$, that is extracted from a given dataset. This is a common approach in the imaging community and presents a closed scenario in which the true input is known. These $\hat{\gamma}$ fields are created by,
\begin{equation}
\hat{\gamma} = \bm{\Phi} \kappa + \mathcal{N}(0,\sigma_n^2),
\end{equation}
In turn $\sigma_n$ is defined relative to a \textit{signal-to-noise ratio} (SNR) in dB as,
\begin{equation} \label{eq:noise_dB}
\sigma_n = \sqrt{\frac{\norm{\bm{\Phi} \kappa}_2^2}{N}} \times 10^{-\frac{\text{SNR}}{20.0}}.
\end{equation}
First, we estimate $\sigma_n$ from the noisy data by Method 3 (section \ref{sec: B-Mode Noise Estimation}). Using this estimate of the noise-level we then utilize the SOPT\footnote{A highly optimized sparse optimization solver, https://github.com/astro-informatics/SOPT} framework to perform our reconstruction algorithm on $\hat{\gamma}$ such that we recover a MAP estimator of the convergence $\kappa^{\text{map}}$. From this reconstructed convergence field a recovered SNR is computed and a selection of hypothesis tests are conducted to showcase the power of this formalism.
\par
In the case where the underlying clean $\gamma$ are unavailable (\textit{i.e.} application to A520 data) we conduct the same analysis as before but instead of creating artificial noisy $\hat{\gamma}$ maps we used the real noisy observational data.
\par
Throughout our analysis the recovered SNR (dB) is defined to be,
\begin{equation}
\text{SNR} = 20 \times \log_{10}\Bigg (\frac{\norm{\kappa}_2}{\norm{\kappa - \kappa^{\text{map}}}_2} \Bigg ),
\end{equation}
when the ground-truth convergence is known. When working with observational data, clearly the ground truth is not available. See appendix \ref{sec:noise_selection} for details on how one may convert from decibels to typical observational limitations -- \textit{e.g.} galaxy number density \textit{e.t.c.}
\begin{table}
\centering
\caption{Contains reconstruction SNR metric for both KS and sparse reconstructions of the Bolshoi-3 cluster in the standard ($256\times256$) setting for a range of input SNR values. The difference column is calculated as the difference between the Sparse and smoothed KS recovered SNR.}
\label{tab:SNR_data}
\begin{tabular}{l|c|c|c|c|c|r}
\hline
\hline
\multicolumn{5}{|c|}{\textbf{SNR (dB)}} \\
\hline
\hline
\textbf{Input} & \multirow{2}{*}{\textbf{KS}} & \textbf{KS} & \multirow{2}{*}{\textbf{Sparse}} & \multirow{2}{*}{\textbf{Difference}}\\
\textbf{SNR} & & \textbf{Smooth} & & \\
\hline
\hline
20.0 & 3.986 & 3.988 & 9.298 & + 5.310\\
15.0 & 3.844 & 3.912 & 9.906 & + 5.993\\
10.0 & 3.480 & 3.831 & 9.230 & + 5.399\\
5.0 & 2.670 & 3.0305 & 7.296 & + 4.265\\
\hline
\end{tabular}
\end{table}
\begin{figure}
\centering
\textbf{\large SNR Gain of Sparsity over KS}
\includegraphics[width=0.52\textwidth,trim=1cm 0 0 1.0cm, clip=true]{SNR_Comparison/Buzzard_DB8_SARA.png}
\caption{SNR gain of sparse hierarchical Bayesian reconstructions with SARA and DB8 wavelet dictionaries over the standard KS estimator. The solid lines represent the mean SNR gain and shaded regions represent the 1$\sigma$ levels. Clearly, both sparse wavelet dictionaries on average outperform KS on SNR $\in [5,20]$. The SNR gain is high when the noise level is high, and gradually decreases as the noise-level decreases. Notably, for reconstructions of dimension $512 \times 512$ and lager the SNR gain for the SARA dictionary becomes significantly larger. It should be noted that in practice the noise level may extend lower than the range considered here, in such cases the trend shown here is quite likely to continue however this preliminary comparison should not be used as hard evidence through such extrapolation. However, a larger range of noise-levels is beyond this paper and so is not considered.}
\label{fig:Buzzard_SNR_spread}
\end{figure}
\begin{figure*}
\centering
\includegraphics[width=1.45\columnwidth]{SNR_Comparison/No_super_res_SNR_spread_256_no_title.png}
\put(-255,522){\textbf{\Large \rotatebox[origin=c]{90}{Ground Truth $\kappa$}}}
\put(-370,408){\textbf{\Large \rotatebox[origin=c]{90}{SNR: 20}}}
\put(-370,290){\textbf{\Large \rotatebox[origin=c]{90}{SNR: 15}}}
\put(-370,175){\textbf{\Large \rotatebox[origin=c]{90}{SNR: 10}}}
\put(-370,60){\textbf{\Large \rotatebox[origin=c]{90}{SNR: 5}}}
\put(-297,-2){\textbf{\Large KS}}
\put(-183,-2){\textbf{\Large KS}}
\put(-195,-14){\textbf{\Large Smooth}}
\put(-72,-2){\textbf{\Large Sparse}}
\newline
\centering
\caption{\textit{Top to bottom:} Input convergence, SNR 20, 15, 10 and 5 recoveries respectively. Series of plots displaying the effectiveness of sparse reconstruction over the standard KS method for a range of input SNR values. The numerical details can be found in Table \ref{tab:SNR_data}. The vertical labels indicate the input SNR for a given row, whereas horizontal labels indicate the reconstruction type. In each case a near optimal (manually selected to maximize the recovered SNR) Gaussian smoothing kernel was applied to the KS recovery to yield the KS (smooth) recovery in an attempt to remove noise from the KS estimator. Clearly, in all cases, the sparse approach performs better.}
\label{fig:bolshoi_3_SNR_figure_no_super_res}
\end{figure*}
\begin{table*}
\centering
\caption{Displays the MAP objective function, level-set threshold at 99\% confidence, Surrogate objective function and whether the removed region was successfully identified as being physical. This data-set corresponds to Figures \ref{fig:bolshoi_1_hypothesis_testing}, \ref{fig:bolshoi_2_hypothesis_testing}, \ref{fig:bolshoi_3_hypothesis_testing} and \ref{fig:Buzzard}.}
\label{tab:bolshoi_hyp_data}
\begin{tabular}{lccccr}
\hline
\hline
\textbf{Masked} & \textbf{Initial} & \textbf{Threshold} & \textbf{Surrogate} & \textbf{Statistically} \\
\textbf{Region} & $f(\kappa) + g(\kappa)$ & $\epsilon_{99\%}^{\prime}$ & $f(\kappa^\text{sgt}) + g(\kappa^\text{sgt})$ & \textbf{Significant?}\\
\hline
\hline
\multicolumn{5}{|c|}{\textbf{Bolshoi-1}}\\
\hline
1. & 95426 & 163408 & 805513 & \checkmark \\
2. & 95426 & 163408 & 134080 & $\times$ \\
3. & 95426 & 163408 & 100582 & $\times$ \\
\hline
\multicolumn{5}{|c|}{\textbf{Bolshoi-2}}\\
\hline
1. & 97121 & 165103 & 824260 & \checkmark \\
2. & 97121 & 165103 & 221492 & \checkmark \\
3. & 97121 & 165103 & 366981 & \checkmark \\
\hline
\multicolumn{5}{|c|}{\textbf{Bolshoi-3}}\\
\hline
1. & 83419 & 151401 & 369939 & \checkmark \\
2. & 83419 & 151401 & 234305 & \checkmark \\
3. & 83419 & 151401 & 314089 & \checkmark \\
\hline
\multicolumn{5}{|c|}{\textbf{Buzzard}}\\
\hline
1. & 117614 & 185595 & 386175 & \checkmark \\
2. & 117614 & 185595 & 148475 & $\times$ \\
3. & 117614 & 185595 & 187056 & \checkmark \\
\hline
\end{tabular}
\end{table*}
\subsection{Bolshoi Cluster Catalogs} \label{bolshoi}
The Bolshoi cluster data used consists of 3 large clusters extracted from the Bolshoi N-body simulation \citep{[20],[6]}. These images were then multi-scale Poisson denoised to create suitable ground truth simulations. We choose to analyze the same clusters considered in \citet{[6]}, as they showcase a wide variety of structure on all scales. Hereafter, we restrict ourselves to the SARA dictionary \citep{[39]} truncated at DB4 for simplicity \nobreak--\nobreak\hskip3pt \textit{i.e.} the combination of the Dirac, DB1,..., DB4 wavelet dictionaries only.
To investigate the SNR gain of our formalism over KS in the cluster scale setting, we created realisations of noisy pseudo-shear maps for input SNR $\in [5,10,15,20]$ from the third Bolshoi cluster map, upon which we applied our reconstruction algorithm pipeline. The results of which are presented in Table \ref{tab:SNR_data}. It should be noted that for comparisons sake the KS estimate without convolution with a Gaussian smoothing kernel is provided in addition to the standard smoothed KS estimator. This has been done to display reconstruction results when no assumption of Gaussianity is enforced \nobreak--\nobreak\hskip3pt an important caveat when performing weak lensing reconstructions, as for convergence maps cosmologists are primarily interested in the non-Gaussian information content which is severely degraded by the Gaussian convolution step.
\par
As can be seen in Figure \ref{fig:bolshoi_3_SNR_figure_no_super_res} and Table \ref{tab:SNR_data}, for very clean measurements both reconstructions are good representations of the ground-truth convergence map. However, when the noise is increased our sparse MAP estimate mitigates the propagation of noise into the $\kappa$ estimate, whereas any noise present during the KS reconstruction propagates almost entirely into the $\kappa$ estimate.
\par
Regardless of the input SNR the sparse algorithm displays an SNR gain of $\sim$ 4-6 dB over KS \nobreak--\nobreak\hskip3pt for input SNR's $\ll 5$ dB the SNR gain decreases as the information content of the data is lost entirely (one cannot make inferences without information). We also find that as the dimension of the maps is increased to $512 \times 512$ and above the SNR gain becomes significantly larger.
\subsubsection{Hypothesis Testing: Bolshoi Clusters}
Perhaps more interestingly, we now perform a series of hypothesis tests as discussed in Section \ref{HypothesisTesting}. For each Bolshoi cluster we construct three possible example hypothesis tests which one may wish to perform. In this case these hypothesis' were either: structure removal followed by segmentation-inpainting; or Gaussian smoothing of certain structures (\textit{i.e.} smoothing multiple peaks into a single larger peak which may be of interest when conducting peak-count analysis). Though these are both extremely useful tools, it is important to stress the generality of our approach such that any well defined operation on the reconstructed image, with a clear understandable hypothesis, is applicable.
\par
To ensure the method behind hypothesis testing is clear, we will walk through a typical application. Figure \ref{fig:bolshoi_1_hypothesis_testing} displays the hypothesis tests applied to the Bolshoi-1 cluster. Conceptually, the correct way to interpret Hypothesis 1 (red) is: \textit{`The central dark core is likely just an artifact of the reconstruction'}.
\par
This structure is then removed from the image by segmentation-inpainting (lower left image), and the objective function is then recalculated. It is found that the objective function is now larger than the approximate level-set threshold $\epsilon^{\prime}_{99\%}$ and so the hypothesis is \textbf{rejected}. This implies that the structure is not simply an artifact, but is necessary to the integrity of the reconstruction, \textit{i.e.} this structure is now determined to be physical at $99\%$ confidence. However, had removing this region \textbf{not} raised the objective function above $\epsilon^{\prime}_{99\%}$, then the conclusion is that their is insufficient evidence to reject the hypothesis (which is \textbf{not} equivalent to saying that the region is strictly not physical).
\par
An identical thought process can be applied to hypothesis tests 2 and 3 in Figure \ref{fig:bolshoi_1_hypothesis_testing}, hypothesis test 1 in Figure \ref{fig:bolshoi_2_hypothesis_testing}, and all three hypothesis tests presented in Figure \ref{fig:bolshoi_3_hypothesis_testing}. The results of these demonstration hypothesis tests are presented in their corresponding tables.
\par
Hypothesis tests 2 and 3 of the Bolshoi-2 cluster \nobreak--\nobreak\hskip3pt Figure \ref{fig:bolshoi_2_hypothesis_testing} \nobreak--\nobreak\hskip3pt must be interpreted differently to the previous example. In these cases the central region has been blurred by segmentation-smoothing (convolution with a Gaussian smoothing kernel) \nobreak--\nobreak\hskip3pt the difference between these two cases being simply the degree of smoothing. Here the hypothesis is: \textit{`The central region is likely to be just a single peak, rather than two'}.
\par
As in the previous example, the objective function is recalculated and is now greater than $\epsilon^{\prime}_{99\%}$ and so the hypothesis is rejected. The natural conclusion is thus that the data \textbf{is} sufficient to determine that at least two peaks are physically present at $99\%$ confidence. All numerical data related to hypothesis testing of the Bolshoi cluster reconstructions can be found in Table \ref{tab:bolshoi_hyp_data}.
\begin{figure*}
\centering
\includegraphics[width=\textwidth, trim={0 0 0 0.5cm},clip]{Hypothesis_testing/Bolshoi_1_hypothesis.png}
\put(-495,125){\textbf{\Large Bolshoi-1 Sparse $\kappa$}}
\put(-340,125){\textbf{\Large Test 1}}
\put(-220,125){\textbf{\Large Test 2}}
\put(-100,125){\textbf{\Large Test 3}}
\caption{Hypothesis testing of three selected structures in the Bolshoi-1 cluster convergence field. The SNR of added Gaussian noise was 20 dB. The SNR of the sparse recovery was $\sim$ 6 dB (an increase in SNR of $\sim 3.5$ dB over the KS reconstruction). We correctly determine that region 1 (\textit{red}) is physical with $99\%$ confidence. Regions 2 (\textit{blue}) and 3 (\textit{green}) remain within the HPD region and are therefore inconclusive, given the data and noise level. The numerical details can be found in Table \ref{tab:bolshoi_hyp_data}.}
\label{fig:bolshoi_1_hypothesis_testing}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=\textwidth, trim={0 0 0 0.5cm},clip]{Hypothesis_testing/Bolshoi_2_hypothesis.png}
\put(-495,125){\textbf{\Large Bolshoi-2 Sparse $\kappa$}}
\put(-340,125){\textbf{\Large Test 1}}
\put(-220,125){\textbf{\Large Test 2}}
\put(-100,125){\textbf{\Large Test 3}}
\caption{Hypothesis testing of three selected structures in the Bolshoi-2 cluster convergence field. The SNR of added Gaussian noise was 20 dB. The SNR of the sparse recovery was $\sim$ 12 dB (an increase in SNR of $\sim 7$ dB over the KS reconstruction). We correctly determine that all three null hypothesis' (\textit{red, blue} and \textit{green}) are rejected at $99\%$ confidence. In test 1 the conclusion is that the left hand peak was statistically significant. In tests 2 and 3 the conclusions is that an image with the two peaks merged it unacceptable, and therefore the peaks are distinct at $99\%$ confidence. The numerical details can be found in Table \ref{tab:bolshoi_hyp_data}.}
\label{fig:bolshoi_2_hypothesis_testing}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=\textwidth, trim={0 0 0 2.0cm},clip]{Hypothesis_testing/Bolshoi_3_hypothesis.png}
\put(-495,125){\textbf{\Large Bolshoi-3 Sparse $\kappa$}}
\put(-340,125){\textbf{\Large Test 1}}
\put(-220,125){\textbf{\Large Test 2}}
\put(-100,125){\textbf{\Large Test 3}}
\caption{Hypothesis testing of three selected structures in the Bolshoi-3 cluster convergence field. The SNR of added Gaussian noise was 20 dB. The SNR of the sparse recovery was $\sim$ 9 dB (an increase in SNR of $\sim 6$ dB over the KS reconstruction). We correctly determine that all three hypothesis regions (\textit{red, blue} and \textit{green}) $\Omega_D$ are physical with $99\%$ confidence. The numerical details can be found in Table \ref{tab:bolshoi_hyp_data}.}
\label{fig:bolshoi_3_hypothesis_testing}
\end{figure*}
\subsection{Buzzard Simulation Catalogs} \label{Buzzard}
The Buzzard V-1.6 simulation catalog \citep{DeRose2018,wechsler2018} is generated via a full end-to-end N-body simulation, extracted by ray-tracing from the simulation box corner leading to a sky fraction of $1/4$. For the purpose of this paper we wish to consider the planar setting. This relies on the flat-sky approximation which is not satisfied for large sky-fractions \citep{[3]}, and so we extract smaller planar regions.
\pagebreak
To extract planar regions, we project the shear catalog into a HEALPix\footnote{http://healpix.sourceforge.net/documentation.php} pixelisation ($N_{\text{side}} = 16$). We then tessellate the largest square area into each HEALPix pixel which then defines $\sim 1.2 \deg^{2}$ planar patches each with $\sim 4 \times 10^6$ galaxies. The full Buzzard v1.6 catalog could generate $\sim 3 \times 10^3$ planar regions, though for our purposes we choose to extract $60$ random, independent planar regions.
\par
As a preliminary exercise we run the full reconstruction pipeline on all $60$ planar convergence maps, for a range of added noise-levels corresponding to SNR's of $\lbrace 5,7,10,12,15,17,20 \rbrace$, for two example wavelet dictionaries. We compare the SNR gain (over the standard KS reconstruction) of sparse reconstruction with Daubechies 8 (8-levels) and SARA dictionary \citep{[39],[50]} \nobreak--\nobreak\hskip3pt the results of which can be seen in Figure \ref{fig:Buzzard_SNR_spread}. Note that this preliminary comparison does not constitute a detailed analysis, thus results presented in Figure \ref{fig:Buzzard_SNR_spread} should not be taken as rigorous numerical benchmarking.
\par
For SNR $\in$ [5, 20] both SARA and DB8 wavelets are shown to outperform the KS estimator. At somewhat lower SNR's (<10 dB) both wavelet dictionaries produce a mean SNR gain of 1.5 to 2 dB, however this decays with SNR and at higher SNR (>17 dB) the mean SNR gain drops to $\sim 0.5$ dB -- as the problem become less seriously ill-posed.
\subsubsection{Hypothesis Testing: Buzzard LSS Extractions}
As in section \ref{bolshoi} we then conducted a series of hypothesis tests, all with an input SNR of 20 dB for consistency. In hypothesis 1 and 3 we removed the structure of interest (in this case 2 large over-dense regions) and then segment-inpainted the surrogate image as described in section \ref{HypothesisTesting}. Both null hypotheses were rejected, indicating that the structures considered are statistically significant at $99\%$ confidence. Hypothesis 2 removes a massive void (under-dense region), followed by the usual segment-inpainting. The hypothesis test was inconclusive in this case, though it's important to stress that this is simply because the void was of relatively low absolute intensity and spread over only a few pixels. The results can be found in Figure \ref{fig:Buzzard} and enumerated in Table \ref{tab:bolshoi_hyp_data}.
\begin{figure*}
\centering
\includegraphics[width=\textwidth, trim={0 0 0 1.5cm},clip]{Hypothesis_testing/Buzzard_hypothesis.png}
\put(-495,132){\textbf{\Large Buzzard Sparse $\kappa$}}
\put(-340,132){\textbf{\Large Test 1}}
\put(-220,132){\textbf{\Large Test 2}}
\put(-100,132){\textbf{\Large Test 3}}
\caption{Hypothesis testing of structure in an $\sim 1.2 \deg^{2}$ planar Buzzard extract. Both over-densities 1 and 3 are deemed to be physical, whereas the void structure 2 is inconclusive.}
\label{fig:Buzzard}
\end{figure*}
\section{Application to Abel-520 Observational Catalogs} \label{A520}
\defcitealias{[32]}{J14}
\defcitealias{[36]}{C12}
We perform an application of our entire reconstruction pipeline to real observational datasets. We select two observational datasets of the A520 cluster \citep{[32],[36]} \nobreak--\nobreak\hskip3pt hereafter for clarity we refer to them as \citetalias{[36]} and \citetalias{[32]} \citep[as in ][]{[9]}\footnote{http://www.cosmostat.org/software/glimpse}. For a full description of the datasets, how they were constructed, and how they account for different systematics we recommend the reader look to the respective papers.
\par
The \citetalias{[32]} catalog contains approximately twice the number of galaxies than \citetalias{[36]}, though both are derived from the same ACS (four pointings) and Magellan images. In addition, \citetalias{[32]} combines these images with the CFHT catalog used in the authors previous work \citep{[37]}.
\par
The \citetalias{[36]} observing area extends over a larger angular surface than the \citetalias{[32]} so for this analysis we limit both datasets to the region spanned by both sets. Due to the number density of measurements being very low we are forced to project the measurements into a $32\times32$ grid \nobreak--\nobreak\hskip3pt to ensures that the average number of galaxies in each grid pixel is at least above 1, though ideally we want many galaxies in each pixel to minimize the noise contribution from intrinsic ellipticity. In fact, even in this resolution the space is incomplete in several pixels, but we draw a compromise between the completeness of the space and the resolution of the data. We define an overall mask $M$ which is simply the union of the \citetalias{[36]} and \citetalias{[32]} masks.
\par
We then propose an extension to the IMT technique (section \ref{sec:IMT}) to this extremely low resolution grid ($32\times32$). The extension is to first upscale the initial mask and the data to a higher resolution grid ($64\times64$), certain pixel step-size's for the morphological operators are then fine-tuned. Specifically, the pixel step size used for opening and closing of the mask set is set to 2 (twice that of the default step-size for a typical $64\times64$ mask). This extended IMT technique was applied to both the \citetalias{[36]} and \citetalias{[32]} datasets, in both cases using $M$ as the initial mask. Following the IMT correction, we calculate an estimate of the noise-level (sigma value) of each dataset.
\par
Using these sigma estimates, the associated gridded datasets, and the combined mask $M$, MAP reconstructions of the \citetalias{[36]} and \citetalias{[32]} convergence maps were recovered and are presented in Figure \ref{fig:J14_C12_reconstructions}. For completeness we also performed reconstructions using each datasets individual mask ($M^{\text{J14}}$ and $M^{\text{C12}}$).
\subsection{Hypothesis Testing of Local Structure: A520 Datasets}
We conducted hypothesis tests on both the \citetalias{[36]} and \citetalias{[32]} datasets. Due to the high estimated noise-level present in the data, and the limited data resolution, no local massive structure of interest within either image could increase the objective function sufficiently to reject the hypothesis with any meaningful confidence. This is to say that; given the limited, noisy data and using the measurement operator and prior ($\ell_1$-term) presented in this paper we can say that the data is insufficient to statistically determine the physicality of local small scale structure in both the \citetalias{[36]} and \citetalias{[32]} datasets.
\par
The initial conflict between \citetalias{[36]} and \citetalias{[32]} was over the existence and position of a small, central convergence peak \nobreak--\nobreak\hskip3pt with a notably large mass-to-light ratio, indicated the possibility of self-interacting dark matter. A subsequent inquiry was conducted \citep{[9]} using the GLIMPSE reconstruction algorithm \citep{[6]} and concluded that this peculiar peak existed in the \citetalias{[32]} dataset but not in the \citetalias{[36]} dataset \nobreak--\nobreak\hskip3pt however as the GLIMPSE algorithm is not posed in a complete statistical form, this roughly speaking was the extent of their statistical analysis.
\par
As such, our conclusions agree well with \citeauthor{[9]} (and generally with those drawn in both \citetalias{[36]} and \citetalias{[32]}). However, within our Bayesian hierarchical formalism (which constitutes a principled statistical framework) we push this conclusion further to say that the data are insufficient to determine the physicality of these peaks, let alone their position.
\subsection{Hypothesis Testing of Global Structure: A520 Datasets}
However, we can draw somewhat meaningful conclusions on global structure. To do so, we manually over-regularize the reconstruction (manually set $\mu$ to be a factor $\sim 5$ larger than the automatically set $\mu$) which has the effect of removing low intensity substructure. The remaining structure is then removed via segmentation-inpainting from the automatically regularized reconstruction to form a surrogate $\kappa^{\prime}$. In both the \citetalias{[36]} and \citetalias{[32]} reconstructions this over-regularized structure is determined to be physical, see Figure \ref{fig:J14_C12_overregularized}. We can therefore (weakly) constrain the MAP solution by concluding that the structure present in the over-regularized surrogate is \textbf{collectively} physical, at $99\%$ confidence \nobreak--\nobreak\hskip3pt this can be seen in Figure \ref{fig:J14_C12_overregularized}.
\begin{figure*}
\centering
\textbf{\Large J14 KS $\kappa$} $\hspace{3.0cm}$ \textbf{\Large J14 Sparse $\kappa$} \\
\includegraphics[width=0.7\textwidth, trim={0 9.0cm 0 2.6cm},clip]{Hypothesis_testing/C12_J14_reconstruction.png} \\
\textbf{\Large C12 KS $\kappa$} $\hspace{3.0cm}$ \textbf{\Large C12 Sparse $\kappa$} \\
\includegraphics[width=0.7\textwidth, trim={0 0 0 11.35cm},clip]{Hypothesis_testing/C12_J14_reconstruction.png} \\
\caption{\textbf{Top}: \textit{(left to right)}: Heavily smoothed KS convergence reconstruction of \citetalias{[32]}, sparse convergence reconstruction of \citetalias{[32]} using IMT for sigma estimation. \textbf{Bottom}: \textit{(left to right)}: Heavily smoothed KS convergence reconstruction of \citetalias{[36]}, sparse convergence reconstruction of \citetalias{[36]} using IMT for sigma estimation. Clearly the two reconstructions are visually similar, particularly the two separated large over-dense regions \nobreak--\nobreak\hskip3pt \textit{upper left and lower right}. In a Bayesian manner it is found that the two datasets do not globally disagree at $99\%$ confidence. However, given the data resolution and noise-levels, no local small scale structures (peak over-dense regions) can be determined to be statistically significant. This is \textbf{not} to say they do not exist, but implies that the data quantity and quality is insufficient to make a robust, principled statistical statement which could be used as evidence of their existence. Interestingly, note the highlighted region (\textit{green}), this dual structure coincides with a particularly obvious bright patch on the original \citetalias{[32]} science image \citeauthor[Figure 1 in][]{[9]} and can be seen in the \citetalias{[36]} reconstruction but \textbf{not} the \citetalias{[32]} reconstruction. Additionally, the highlighted region (\textit{red}) is a feature present in both sparse datasets which was \textbf{not} present in the KS reconstructions \nobreak--\nobreak\hskip3pt this feature is also seen in the bootstrapping analysis (Appendix) of \citeauthor{[9]}.}
\label{fig:J14_C12_reconstructions}
\end{figure*}
\begin{table*}
\centering
\caption{Displays the MAP objective function, level-set threshold at 99\% confidence, surrogate objective function and whether the null hypothesis is rejected. As can be seen, both MAP solutions fail to reject the null hypothesis in the others objective function. This leads us to conclude that the two datasets do not disagree at 99\% confidence.}
\label{tab:C12_J14_hypothesis_data}
\begin{tabular}{lccccr}
\hline
\hline
\textbf{Hypothesis} & \textbf{Initial} & \textbf{Threshold} & \textbf{Surrogate} & \textbf{Statistically} \\
\textbf{Test} & $f(\kappa) + g(\kappa)$ & $\epsilon_{99\%}^{\prime}$ & $f(\kappa^\text{sgt}) + g(\kappa^\text{sgt})$ & \textbf{Significant?}\\
\hline
\hline
\citetalias{[36]} $\Leftrightarrow$ \citetalias{[32]} & 99231 & 168044 & 125601 & $\times$ \\
\citetalias{[32]} $\Leftrightarrow$ \citetalias{[36]} & 98943 & 167243 & 134391 & $\times$ \\
\hline
\end{tabular}
\end{table*}
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth, trim={0 0 0 2.6cm},clip]{Hypothesis_testing/C12_J14_SARA_overregularized.png}
\caption{\textbf{Top}: Reconstruction of \citetalias{[32]} with regularization parameter $\mu$ manually increased by a factor of $\sim 5$. \textbf{Bottom}: Reconstruction of \citetalias{[36]} with regularization parameter $\mu$ manually increased by a factor of $\sim 4$. \textbf{Top and Bottom:} In both cases the structure which remains is collectively determined to be physical at $99\%$ confidence. Of interest, note the large set of two peaks (\textit{green}) present in \citetalias{[36]} but not in \citetalias{[32]}. These anomalous peaks seem to coincide with bright objects in the original \citetalias{[32]} science image \citep[Figure 1 in][]{[9]}.}
\label{fig:J14_C12_overregularized}
\end{figure}
\par
Interestingly we can perform a final novel hypothesis test of global structure. This hypothesis is as follows: \textit{`The two MAP estimates are consistent with both sets of data',} \textit{i.e.} the MAP convergence estimate recovered from the \citetalias{[32]} (\citetalias{[36]}) data is within the credible-set (at $99\%$ confidence) of the \citetalias{[36]} (\citetalias{[32]}) objective function. We find that the \citetalias{[32]} (\citetalias{[36]}) MAP reconstruction is an acceptable solution to the \citetalias{[36]} (\citetalias{[32]}) inverse problem and so the MAP solutions do not disagree \nobreak--\nobreak\hskip3pt numerically this is shown in Table \ref{tab:C12_J14_hypothesis_data}.
\par
Given the inherent limitations of the data we are forced to conclude: \textit{`The data are insufficient to determine the existence of individual massive regions at high confidence \nobreak--\nobreak\hskip3pt though collectively these massive regions are found to be globally physical at $99\%$ confidence. The two MAP estimates are also found to be consistent at $99\%$ confidence.'}
\par
There is one further caveat which may be of interest. Here we have considered an average noise level within the \citetalias{[32]} and \citetalias{[36]} datasets, which may be generalized further to include spatially varying noise-levels. This can be folded quite easily into our hierarchical model by adopting the likelihood defined in equation (\ref{eq:likelihood}) explicitly as a multivariate Gaussian -- rather than making the assumption that the covariance $\Sigma \propto \mathbb{I}$. With this extension the likelihood term (and therefore the objective function) will be more sensitive to pixels $i$ in which the covariance $\Sigma_{ii}$ takes smaller values. As for cluster data-sets, such as \citetalias{[32]} and \citetalias{[36]}, more galaxies are inherently observed closer to large dark matter halos this may well increase the sensitivity of hypothesis testing on cluster scales -- additionally this may well increase the reconstruction quality. However for this first application the full covariance is not considered.
\section{Conclusions} \label{Conclusions}
We have presented a sparse hierarchical Bayesian mass-mapping algorithm which provides a principled statistical framework through which, for the first time, we can conduct uncertainty quantification on recovered convergence maps without relying on any assumptions of Gaussianity. Moreover, the presented formalism draws on ideas from convex optimization (rather than MCMC techniques) which makes it notably fast and allows it to scale well to big data, \textit{i.e.} high resolution and wide-field convergence reconstructions (which will be essential for future stage \rom{4} surveys, such as LSST and Euclid).
\par
Additionally, we demonstrate a hierarchical Bayesian inference approach to automatically approximate the regularization parameter, and show that it produces near optimal results in a variety of cases.
\par
To support this formalism we construct several novel noise-level ($\sigma_n$) estimation techniques, and compare them for a range of plausible weak lensing scenarios. We find that method 3 (a bespoke noise estimation algorithm which exploits the intrinsic symmetry of weak-lensing mass-mapping) produces excellent $\sigma_n$ estimates. Further to this, we develop an iterative morphological mask correction algorithm to extend these noise estimate techniques to non-trivially masked spaces.
\par
We showcase our complete mass-mapping pipeline (Estimate the noise-level $\Rightarrow$ Automatically set the regularization parameter $\Rightarrow$ Recover the \textit{maximum a posteriori} convergence map $\Rightarrow$ Conduct hypothesis testing of structure on the recovered $\kappa$-map) on both simulation datasets and observational data. Our mass-mapping formalism is shown to produce more accurate convergence reconstruction than the KS estimator on all simulations considered \nobreak--\nobreak\hskip3pt most notably for cluster level datasets. Hypothesis tests of substructure are demonstrated.
\par
It is found that neither of the two A520 datasets considered could provide sufficient evidence to determine the physicality of local massive substructure. However, global hypothesis tests indicate a good agreement between the two sets of data. These conclusions are roughly in agreement with those drawn previously but go further to demonstrate just how uncertain these types of cluster-scale weak lensing reconstruction inherently are (typically as a limitation of the relative information content of low-resolution, noisy datasets).
\par
It is now natural to extend this formalism to the entire celestial sphere \nobreak--\nobreak\hskip3pt a necessity of large-scale reconstruction techniques which aim to fully utilize the forthcoming Euclid and LSST\footnote{https://www.lsst.org} survey data.
\section*{Acknowledgements}
Author contributions are summarised as follows.
MAP: methodology, data curation, investigation, software, visualisation, writing - original draft;
JDM: conceptualisation, methodology, project administration, supervision, writing - review \& editing;
XC: methodology, investigation, writing - review \& editing;
TDK: methodology, supervision, writing - review \& editing;
CGRW: methodology.
This paper has undergone internal review in the LSST Dark Energy Science Collaboration. The internal reviewers were Chihway Chang, Tim Eifler, and François Lanusse.
The authors would like to thank Luke Pratley for an introduction to the C++ SOPT framework and the internal reviewers for valuable discussions. MAP is supported by the Science and Technology Facilities Council (STFC). TDK is supported by a Royal Society University Research Fellowship (URF). This work was also supported by the Engineering and Physical Sciences Research Council (EPSRC) through grant EP/M0110891 and by the Leverhulme Trust.
The DESC acknowledges ongoing support from the Institut National de Physique Nucl\'eaire et de Physique des Particules in France; the Science \& Technology Facilities Council in the United Kingdom; and the Department of Energy, the National Science Foundation, and the LSST Corporation in the United States. DESC uses resources of the IN2P3 Computing Center (CC-IN2P3--Lyon/Villeurbanne - France) funded by the Centre National de la Recherche Scientifique; the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the U.S.\ Department of Energy under Contract No.\ DE-AC02-05CH11231; STFC DiRAC HPC Facilities, funded by UK BIS National E-infrastructure capital grants; and the UK particle physics grid, supported by the GridPP Collaboration. This work was performed in part under DOE Contract DE-AC02-76SF00515.
\input{Massmapping_hypothesis_testing.bbl}
| -51,339.500444 |
[
-3.390625,
3.10546875
] | 26.372155 |
[
-3.580078125,
0.1358642578125,
-2.09375,
-6.015625,
-0.55615234375,
8.296875
] |
[
4.1953125,
8.1796875,
3.65625,
6.671875
] | 557 | 10,534 |
[
-2.68359375,
2.625
] | 26.865585 |
[
-6.375,
-4.7578125,
-5.265625,
-2.849609375,
2.228515625,
14.0703125
] | 0.58632 | 12.612993 | 21.945895 | 4.154973 |
[
3.0975608825683594
] | -31,160.184512 | 6.152554 | -50,329.460861 | 0.302418 | 6.453612 |
[
-2.76171875,
-3.939453125,
-3.755859375,
-4.796875,
2.419921875,
12.4296875
] |
[
-5.67578125,
-2.89453125,
-2.72265625,
-2.134765625,
3.939453125,
5.859375
] | |
BkiUc_I5qYVBWXyireYI
|
\section{Conclusion}\label{sec:conclusion}
We have studied the problem of user localization using the RSS measurements collected by a UAV. To do so, we first proposed a hybrid channel model which aims to accurately learn the path loss parameters as well as the UAV antenna pattern. A PSO technique then was employed by exploiting the learned hybrid channel model and leveraging the 3D map of the environment to accurately localize the ground users. The performance of the developed algorithm was evaluated through simulations and also real-world experiments.
\section{Acknowledgments}
This work was funded via the HUAWEI
France supported Chair on Future Wireless Networks at EURECOM.
\bibliographystyle{IEEEtran}
\section{Introduction}\label{sec:Intro}
In a wireless localization system, nodes with perfectly-known positions known as anchor nodes (which can be stationary or mobile) collect various radio measurements from the emitted radio frequency (RF) signals from the users in the network, and use them for localization purposes. Various measurements such as received signal strength (RSS), time-of-arrival (TOA), angle of arrival (AOA), etc., can be obtained from the RF signals by the anchor nodes \cite{zekavat2011handbook,delRauLop}.
On the other hand, advancement in robotic technologies and miniaturization of wireless equipment have made it possible to have flying radio networks (FRANs), where wireless connectivity to ground users can be provided by aerial base stations (BSs) that are mounted on unmanned aerial vehicles (UAVs) \cite{GanEsaGes,MozSaadBennNamDebb}.
Advantage of FRANs include, fast and dynamic network deployment during an emergency or temporary crowded events, providing connectivity in areas lacking network infrastructure, etc. While in terrestrial radio access networks static BSs are used as anchor nodes, in FRANs UAV BSs can be used as mobile anchor nodes.
Localization of ground users using RSS measurements collected by
aerial UAV anchor nodes
has gained interest recently \cite{Ref5_sallouha2017aerial,Ref19_lima2019support,Ref9_artemenko2015evaluation,Ref10_ji2019fair,Ref6_koohifar2018autonomous,ShahidSoltan,EsrGanGesAsil,esrafilian20203d}.
The main advantage of using UAV BS anchors in localization compared to static BSs is that UAV BSs with their inherent 3D mobility can collect
radio measurements in difference geographic locations which improves the localization performance. Generally, RSS measurements are easy to obtain in many wireless networks and does not require stringent synchronization and calibration constraints associated with timing based measurements. The works in \cite{Ref10_ji2019fair,Ref6_koohifar2018autonomous,ShahidSoltan}
assumed that the UAV flies high enough so that the RSS of the
UAV-user or Air-to-ground (A2G) link is modeled as a simple {\text{}}{los} channel. This LoS assumption is generally not valid in urban scenarios as A2G links are often blocked by city buildings. To over come this, the authors of \cite{EsrGanGesAsil,esrafilian20203d} have used a segmented pathloss model that differentiates between LoS and {\text{}}{nlos} channel conditions and showed improvement in localization performance. Moreover, in \cite{esrafilian20203d} it is shown that by exploiting the 3D city map which contains the building locations and height information, one can significantly improve the localization performance.
One common assumption in all the works in \cite{Ref5_sallouha2017aerial,Ref19_lima2019support,Ref9_artemenko2015evaluation,Ref10_ji2019fair,Ref6_koohifar2018autonomous,ShahidSoltan,EsrGanGesAsil,esrafilian20203d}, is that the UAV is assumed to have a perfect isotropic radiating antennas. However, in reality this is not true and several complications arise with UAV BSs as opposed to static BSs:
a) The UAV altitude and heading changes depending on its mobility pattern, hence the antenna gain changes with the UAV location and orientation \cite{7228721}
b) The radiation pattern of the antenna mounted on UAV is affected by the chassis, and hence it is difficult to measure the actual antenna pattern while UAV is flying \cite{7510362,8402597}.
The work in \cite{CheDevSinGuv} has demonstrated that the 3D radiation pattern of an
antenna mounted on a drone can significantly influence the RSS of the A2G link. Therefore, it is important to consider this in a practical localization system using UAV BS anchors.
The impact of the antenna
radiation pattern in A2G channels in a 3D localization system using time-difference-of-arrival (TDOA) measurements has been studied in
\cite {SinYapISm}. However, the authors assume that the radiation pattern of the UAV antenna is known. To the best of our knowledge, localizing users with a UAV with an unknown radiation pattern and RSS measurements has not been studied before. Specifically, our contributions are as follows:
\begin{itemize}
\item The unknown antenna radiation pattern of the UAV during the flight is parameterized by a neural network by feeding RSS measurements in the training phase.
\item An optimization framework is proposed capable of using the trained model, which characterizes the path loss and the antenna gain pattern, along with the 3D map of the environment to improve the localization performance.
\end{itemize}
\section{System Model and Problem Formulation}\label{sec:SysModel}
We consider a scenario similar to the one illustrated in Fig. \ref{fig:SystemModel}, where a UAV BS that is
connected to $K$ ground level users in an urban area
consisting of a number of city buildings. The users are
spread over the city and ${\bf u}_{k}=[x_{k},y_{k}]^{{\Transpose}}\in\mathbb{R}^{2},\,k\in[1,K]$
denotes the $k$-th user's location. The users are considered static and their locations are unknown.
A 3D map of the environment where the UAV and users are located is assumed to be available.
The aim of the UAV is to estimate
the unknown user locations based on RSS measurements taken in $N$ different time steps from the users. In the $n$-th time step, the UAV/drone position is denoted by
${\bf{v}}_n=[x[n],y[n],z[n]]^{{\Transpose}}\in\mathbb{R}^{3}$.
We assume that the UAV is equipped with a GPS receiver, hence
${\bf{v}}_n,\forall n$ is known.
\subsection{Channel Model}
We now describe the radio channel model between the UAV and ground users. Note that the channel parameters and the UAV antenna pattern are unknown and need to be learned. In general, the channel between UAV position ${\bf{v}}$ and user location ${\bf{u}}$ in dB can be modeled as
\begin{equation}
g_{z}=\phi_{z}({\bf{v}}, {\bf{u}}) + \gamma({\bf{v}}, {\bf{u}}, \psi) + \eta_{z},
\label{eq:CH_Model_dB}
\end{equation}
where $\phi_{z}({\bf{v}}, {\bf{u}})$ is the path loss between the UAV and the user link, $\gamma({\bf{v}}, {\bf{u}}, \psi)$ stands for the antenna gain of the UAV with $\psi$ denoting the heading angle of the UAV with respect to the north pole, and $\eta_{z}$ is the shadowing component that is modeled as a Gaussian random variable with $\mathcal{N}(0,\sigma_{z}^{2})$. $z\in\left\{ \text{LoS},\text{NLoS}\right\}$ emphasizes the strong dependence of the propagation parameters on the {\text{}}{los} or {\text{}}{nlos} segments. The variance of the shadowing component $(\sigma_{z}^{2})$ is assumed to be known for both segments. Note that \eqref{eq:CH_Model_dB} represents the logarithm of the channel gain which is averaged over the small scale fading of unit variance.
Classically, the path loss $\phi_{z}({\bf{v}}, {\bf{u}})$ between two radio nodes is modeled as \cite{ChenYanGes}
\begin{equation}
\phi_{z}({\bf{v}}, {\bf{u}})\triangleq \phi_{z}(d) =\ss_{z}-10\,\alpha_{z}\log_{10}\left(d\right), \label{eq:classical_ch_gain}
\end{equation}
where $d = \|{\bf{v}} - {\bf{u}}\|_2$, $\alpha_{z}$ is the path loss exponent, and $\ss_{z}$ is the log of average path loss at the reference point $d=1\si{m}$. We assume that the users are equipped with an omnidirectional antenna. However, the antenna mounted on the UAV does not have a specific gain pattern and can have a very complex form depending on the design of the antenna and also the type of materials used in the UAV itself.
\section{Radio channel Learning and User Localization \label{sec:LearningLocalization}}
In this section, we propose a map-based algorithm to estimate the user locations from the channel gain measurements collected by the UAV. Let us denote an arbitrary set of measurements taken by the
UAV during the mission by a sequence
$\chi = \left\{{\bf{v}}_n, n\in [1,N]\right\}$.
From each of these
locations, the UAV collects radio measurements form all $K$ users.
We denote the channel gain or RSS measurement (in dB scale) obtained from
the $k$-th user by the UAV in the $n$-th time step with $g_{n,k}$. Using the channel model in \eqref{eq:CH_Model_dB} we can write
\begin{equation}\label{eq:MeasurementModel}
g_{n,k} {=}
\begin{cases}
\phi_{{\text{}}{los}}(d_{n, k}) + \gamma({\bf{v}}_n, {\bf{u}}_k, \psi_n) + \eta_{n,k,{\text{}}{los}} & \small{\text{if} \text{ LoS}}\\
\phi_{{\text{}}{nlos}}(d_{n, k}) + \gamma({\bf{v}}_n, {\bf{u}}_k, \psi_n) + \eta_{n,k,{\text{}}{nlos}} & \small{\text{if} \text{ NLoS}},
\end{cases}
\end{equation}
where $d_{n,k} = \| {\bf{v}}_n - {\bf{u}}_k \|_2$, and $\psi_n$ is the UAV heading angle at time step $n$. The function $\gamma (.)$ is the antenna gain which is unknown and it needs to be learned. The probability distribution of a single measurement in \eqref{eq:MeasurementModel} is modeled as
\begin{equation} \label{eq:mixdit}
p(g_{n,k}) = (f_{n,k,\text{LoS}})^{w_{n,k}} (f_{n,k,\text{NLoS}})^{(1-w_{n,k})},
\end{equation}
where $\omega_{n,k} \in \{0,1\}$ is the classifier binary variable (yet unknown) indicating whether a measurement falls into the LoS or NLoS category, and $f_{n,k,z}$ has a Gaussian distribution with $\mathcal{N}(\phi_{z}(d_{n, k}) + \gamma({\bf{v}}_n, {\bf{u}}_k, \psi_n),\sigma_{z}^{2})$.
Assuming that collected measurements conditioned on the channel and user positions are independent and identically distributed (i.i.d) \cite{ChenYanGes}, using \eqref{eq:mixdit}, the negative log-likelihood of measurements leads to
\noindent
\begin{equation}
\begin{aligned}\label{eq:localization_liklihood}
\mathcal{L} &= \log\left(\frac{\sigma_{{\text{}}{los}}^2}{\sigma_{{\text{}}{nlos}}^2}\right)\sum_{k=1}^{K}\sum_{n=1}^{N} \omega_{n,k}+ \\ &\sum_{k=1}^{K}\sum_{n=1}^{N}
\frac{\omega_{n,k}}{\sigma_{{\text{}}{los}}^2}\left |g_{n,k}{-}\phi_{{\text{}}{los}}(d_{n, k}) - \gamma({\bf{v}}_n, {\bf{u}}_k, \psi_n)\right|^2 + \\
& \sum_{k=1}^{K}\sum_{n=1}^{N} \frac{(1-\omega_{n,k})}{\sigma_{{\text{}}{nlos}}^2} \left | g_{n,k}{-}\phi_{{\text{}}{nlos}}(d_{n, k}) - \gamma({\bf{v}}_n, {\bf{u}}_k, \psi_n)\right|^2.
\end{aligned}
\end{equation}
The estimate of the unknown channel parameters $\{\alpha_z, \ss_{z} \}$, $\bf{u}_k$, and $\gamma (.)$ can then be obtained by solving
\begin{subequations}
\begin{align}
\begin{split}
\min_{\substack{\omega_{n,k},\,{{\bf{u}}}_k\\ {\alpha_z, \ss_{z}, \gamma (.)}}
} & \quad \mathcal{L}
\end{split}\\
\begin{split}
\text{s.t.}&\quad \omega_{n,k} \in \{0,1\}, \forall n, \forall k.
\end{split}
\end{align}\label{eq:Localization_Opt_Org}%
\end{subequations}
The binary variables $\omega_{n,k}$ in objective function \eqref{eq:localization_liklihood}, and the fact that $\gamma(.)$ is not explicitly known and is a function of user locations, make problem \eqref{eq:Localization_Opt_Org} challenging to solve since it is a joint classification, channel learning and user localization problem. To tackle this difficulty, we split \eqref{eq:Localization_Opt_Org} into two sub-problems of offline channel learning and online user localization . We also exploit the 3D map of the city for the measurements classification which will be elaborated next.
\subsection{Offline Radio Channel Learning} \label{sec:cahnnel_learning}
We aim to learn the radio channel using a set of offline radio measurements which are collected from users with known locations in advance. In this manner, we have a set of training data set to learn the radio channel. Since the characteristic of the radio channel is independent of the user location and only affected by the structure of the city (i.e. the blocking objects in the environment) and the UAV antenna pattern, therefore learning the radio channel from a set of offline training data set can provide a good approximation for the radio channel. We also exploit the 3D map of the city to perform the LoS/NLoS classification of the measurements, since for a user with a known location the classification variables $\omega_{n,k}$ can be directly inferred from a trivial geometry argument: for a given UAV position, the user is considered in LoS to the UAV if the straight line passing through the UAV's and the user position lies higher than any buildings in between. Moreover, we use a neural network with parameters ${{\boldsymbol{\theta}}}$ as an approximation of the UAV antenna gain $\gamma_{{{\boldsymbol{\theta}}}}(.)$. We call this channel model a \textit{hybrid} channel model, since it consists of a traditional path loss representation $\phi_{z}(.)$ along with a neural network approximating the UAV antenna gain. The main reason for choosing such a channel model lies in the fact that a rough estimation of the channel can be obtained using the classical path loss model \eqref{eq:classical_ch_gain}, while all the uncertainties which can not be captured by the path loss function are then modeled using a neural network.
Now having classified the measurement and using the hybrid channel model, problem \eqref{eq:Localization_Opt_Org} just by considering the offline training data set (with known user locations) can be rewritten as follows
\begin{equation}
\begin{aligned}
\min_{\substack{{\alpha_z, \ss_{z}, {\boldsymbol{\theta}}}}
} & \quad \mathcal{L},
\end{aligned} \label{eq:cahnnel_learning}
\end{equation}
where $\theta$ is the parameters of the neural network for estimating of the antenna gain. Solving this problem is still challenging since the UAV antenna gain is the same for both LoS and NLoS measurements. To alleviate this burden, we split up our problem into two phases. In the first phase, we only find the path loss parameters by solving the following optimization problem
\begin{equation}
\begin{aligned}
{\alpha_z^*, \ss_{z}^*} := \argmin_{\substack{{\alpha_z, \ss_{z}}}
} & \quad \Bar{\mathcal{L}},
\end{aligned} \label{eq:cahnnel_learning_p1}
\end{equation}
where
\begin{equation}
\begin{aligned}\label{eq:L_p1}
\Bar{\mathcal{L}} =& \sum_{k=1}^{K}\sum_{n=1}^{N}
\frac{\omega_{n,k}}{\sigma_{{\text{}}{los}}^2}\left |g_{n,k}{-}\phi_{{\text{}}{los}}(d_{n, k}) \right|^2 + \\
& \sum_{k=1}^{K}\sum_{n=1}^{N} \frac{(1-\omega_{n,k})}{\sigma_{{\text{}}{nlos}}^2} \left | g_{n,k}{-}\phi_{{\text{}}{nlos}}(d_{n, k})\right|^2.
\end{aligned}
\end{equation}
In \eqref{eq:L_p1} the effect of the UAV antenna gain is ignored which allows us to find the closest estimate to the measurements using the path loss model. The parameters obtained by solving \eqref{eq:cahnnel_learning_p1} are denoted as ${\alpha_z^*, \ss_{z}^*}$.
In the second phase, the path loss parameters are fixed to ${\alpha_z^*, \ss_{z}^*}$ and the UAV antenna gain parameters are obtained as follows
\begin{equation}
\begin{aligned}
{\boldsymbol{\theta}}^* := \argmin_{\substack{{{\boldsymbol{\theta}}}}
} & \quad \mathcal{L}|_{\alpha_z^*, \ss_{z}^*}.
\end{aligned} \label{eq:cahnnel_learning_p2}
\end{equation}
Note that, both problems \eqref{eq:cahnnel_learning_p1}, \eqref{eq:cahnnel_learning_p2} can be solved using standard optimization frameworks (i.e. any gradient-based optimizer).
\subsection{User Localization} \label{sec:localization}
Having learned the radio channel, we continue to localize the unknown users in the online data set. The optimization problem \eqref{eq:Localization_Opt_Org} by utilizing the learned radio channel can be reformulated as follows:
\begin{subequations}
\begin{align}
\begin{split}
\min_{\substack{\omega_{n,k},\,{{\bf{u}}}_k}
} & \quad \mathcal{L}^*
\end{split}\\
\begin{split}
\text{s.t.}&\quad \omega_{n,k} \in \{0,1\}, \forall k , \forall n,
\end{split}
\end{align}\label{eq:Localization_unkown_users}%
\end{subequations}
where $\mathcal{L}^* $ is obtained by substituting the channel model with learned parameters ${\alpha_z^*, \ss_{z}^*}, {\boldsymbol{\theta}}^*$ in \eqref{eq:localization_liklihood}. It is hard to find a closed form and analytical solution to problem \eqref{eq:Localization_unkown_users} due to the binary random variables $\omega_{n,k}$, and the non-linear and non-convex objective function $\mathcal{L}^*$. We employ the particle swarm optimization (PSO) technique to solve this problem since PSO is suitable for solving various non-convex and non-linear optimization problems. More specifically, PSO is a population-based optimization technique that tries to find a solution to an optimization problem by iteratively trying to improve a candidate solution with regard to a given measure of quality (or objective function). The algorithm is initialized with a population of random solutions, called particles, and a search for the optimal solution is performed by iteratively updating each particle's velocity and position based on a simple mathematical formula (for more details on PSO see \cite{KenEbe}). As will be clear later, the PSO algorithm is enhanced to exploit the side information stemming from the 3D map of the environment which improves the performance of user localization and reduce the complexity of solving $\eqref{eq:Localization_unkown_users}$, since the binary variables $\omega_{n,k}$ can be obtained directly from the 3D map \cite{esrafilian20203d}.
For ease of exposition, we first solve $\eqref{eq:Localization_unkown_users}$ by assuming only one unknown user. Then we will generalize our proposed solution to the multi-user case. To apply the PSO algorithm, we define each particle to have the following form
\begin{equation}
{\bf{c}}_j = [ x_j, y_j]^{\text{T}} \in \mathbb{R}^2, j \in [1, C],
\end{equation}
where $C$ is the number of particles and each particle is an instance of the possible user location in the city. Therefore, by treating each particle as a potential candidate for the user location, the negative log-likelihood $\eqref{eq:localization_liklihood}$ for a given particle and learned parameters ${\alpha_z^*, \ss_{z}^*}, {\boldsymbol{\theta}}^*$ can be rewritten as follows
\begin{equation}
\begin{small}
\begin{aligned}
\mathcal{L}^*(&{\bf{c}}_{j}^{(i)}) = \log\left(\frac{\sigma^2_{{\text{}}{los}}}{\sigma^2_{{\text{}}{nlos}}}\right) \left |\mathcal{M}_{{\text{}}{los},1,j}\right| + \\ & \small{\sum_{z\in \{{\text{}}{los},{\text{}}{nlos}\}} \, \sum_{n\in \mathcal{M}_{z,1,j}}} \frac{1}{\sigma^{2}_{z}}
\left |g_{n,1}{-}\phi_{z}(d_{n, j}) - \gamma_{\theta^*}({\bf{v}}_n, {\bf{c}}_{j}^{(i)}, \psi_n)\right|^2 ,\label{eq:SSE_PSO_single_Ue}
\end{aligned}
\end{small}
\end{equation}
\noindent where ${\bf{c}}_{j}^{(i)}$ is the $j$-th particle at the $i$-th iteration of the PSO algorithm, $d_{n, j} = \|{\bf{v}}_n - {\bf{c}}_{j}^{(i)}\|_2$, and $\mathcal{M}_{z,1,j}$ is a set of time indices of measurements collected from user 1 which are in segment $z$ by assuming that the location of user 1 is the same as particle $j$. To form $\mathcal{M}_{z,1,j}$, a 3D map of the city is utilized. For example, measurement $g_{n,1}$ is considered LoS, if the straight line passing through ${\bf{c}}_{j}^{(i)}$ and the drone location ${\bf{v}}_n$ lies higher than any buildings in between. Therefore, the best particle minimizing \eqref{eq:SSE_PSO_single_Ue} can be obtained from solving the following optimization
\noindent
\begin{equation}
j^* := \arg \min_{j\in [1,C]} {\mathcal{L}^{*}({\bf{c}}_{j}^{(i)})}, \label{eq:best_particle}
\end{equation}
where $j^* $ is the index of the best particle which minimizes the objective function in \eqref{eq:best_particle}. In the next iteration of the PSO algorithm, the position and the velocity of particles are updated and the algorithm repeats for $\tau$ iterations. The best particle position in the last iteration is considered as the estimate of the user location.
Note that for the multi-user case, without loss of optimality, the problem can be transformed into multiple single-user localization problems, and then each problem can be solved individually. This stems from the fact that the radio channel is learned beforehand and is assumed to have the same characteristics for all the UAV-user links (the radio channel parameters and the UAV antenna pattern are independent of user locations).
\section{Numerical Results}\label{sec:simulations}
We consider a dense urban city neighborhood comprising buildings and streets as shown in Fig. \ref{fig:hybrid_localziation}-a. The height of the buildings is Rayleigh distributed in the range of 5 to \SI{40}{m} \cite{Ref26_HourKandeepJamail}. The true propagation parameters are chosen as $\alpha_{{\text{}}{los}}=2.2,\,\alpha_{{\text{}}{nlos}}=3.2,\,\ss_{{\text{}}{los}}=-32\,\text{dB},\,\ss_{{\text{}}{nlos}}=-35\,\text{dB}$
according to an urban micro scenario \cite{3GPP}. The variances
of the shadowing components in {\text{}}{los} and {\text{}}{nlos} scenarios are
$\sigma_{{\text{}}{los}}^{2}=2\,\text{dB}$, and $\sigma_{{\text{}}{nlos}}^{2}=5\,\text{dB}$, respectively. The following UAV antenna gain is considered to conduct the simulation
\begin{equation}
\gamma({\bf{v}}_n, {\bf{u}}_k, \psi_n) = 15 \left(| cos(\rho_{n, k})| + 2 |sin(\varphi_{n, k} + \psi_n)|\right),
\end{equation}
where $\rho_{n, k}, \varphi_{n, k}$ are , respectively, the elevation and azimuth angles between the UAV and the user, and $\psi_n$ is the heading angle of the UAV.
The hybrid channel model is trained in the same city as shown in Fig. \ref{fig:hybrid_localziation}-a by collecting radio measurements from $K=10$ different random users and over $N=200$ individual random UAV locations. To train the hybrid model, we first need to learn the pathloss parameters. To do so, we use the training data set $\mathcal{D}^{tr}_{pl} = \{ (d_{n, k}, g_{n, k}), \forall n, k \}$. The path loss parameters can then be obtained by solving \eqref{eq:cahnnel_learning_p1}.
Now we continue to learn the UAV antenna gain. To estimate the UAV antenna gain, a neural network with four hidden layers is used where the first and the second layers have $60$ neurons with the $tanh$ activation function, and the third and the fourth layers with $40$ neurons and the $relu$ activation function. To train this network the training data set $\mathcal{D}^{tr}_{ag} = \{ (d_{n, k}, {\bf{x}}_{n, k}, g_{n, k}), \forall n, k \}$ is used where ${\bf{x}}_{n, k}$ is the input vector to the neural network and is defined as follows
\begin{equation}
{\bf{x}}_{n, k} = [\frac{x[n] - x_k}{\|{\bf{v}}_n - {\bf{u}}_k\|_2}, \frac{y[n] - y_k}{\|{\bf{v}}_n - {\bf{u}}_k\|_2}, \frac{z[n]}{\|{\bf{v}}_n - {\bf{u}}_k\|_2}, \psi_n]^{\text{T}}.
\end{equation}
The parameters of the neural network $(\theta)$ can be obtained by solving the optimization problem \eqref{eq:cahnnel_learning_p2} where $\mathcal{L}$ is defined as follows
\begin{equation}
\begin{aligned}
\mathcal{L} &= \log\left(\frac{\sigma_{{\text{}}{los}}^2}{\sigma_{{\text{}}{nlos}}^2}\right)\sum_{k=1}^{K}\sum_{n=1}^{N} \omega_{n,k}+ \\ &\sum_{k=1}^{K}\sum_{n=1}^{N}
\frac{\omega_{n,k}}{\sigma_{{\text{}}{los}}^2}\left |g_{n,k}{-}\phi_{{\text{}}{los}}(d_{n, k}) - \gamma_{\theta}({\bf{x}}_{n, k})\right|^2 + \\
& \sum_{k=1}^{K}\sum_{n=1}^{N} \frac{(1-\omega_{n,k})}{\sigma_{{\text{}}{nlos}}^2} \left | g_{n,k}{-}\phi_{{\text{}}{nlos}}(d_{n, k}) - \gamma_{\theta}({\bf{x}}_{n, k})\right|^2.
\end{aligned}
\end{equation}
Having trained the hybrid channel model, the estimate of each measurement in accordance with \eqref{eq:MeasurementModel} is given by:
\begin{equation}\label{eq:mixed_ch_model_with_input}
\hat{g}_{n,k} {=}
\begin{cases}
\phi_{{\text{}}{los}}(d_{n, k}) + \gamma_{\theta}({\bf{x}}_{n, k}) & \small{\text{if} \text{ LoS}}\\
\phi_{{\text{}}{nlos}}(d_{n, k}) + \gamma_{\theta}({\bf{x}}_{n, k}) & \small{\text{if} \text{ NLoS}}.
\end{cases}
\end{equation}
In Fig. \ref{fig:hybrid_localziation}-a, the result of the user localization after the training phase is shown. We also compared the performance of the proposed algorithm with \cite{esrafilian20203d} which uses the conventional channel model consisting of the path loss model without considering the effect of the UAV antenna gain. Moreover, in Fig. \ref{fig:hybrid_localziation}-b the results of the channel model estimation is shown for different algorithms. It is clear that by using the hybrid channel model we can obtain a better estimation of the channel which results in a more precise user localization. Moreover, the user location estimated using the proposed algorithm for the multi-user scenario is illustrated in Fig. \ref{fig:multiUser_localization} and confirmed to be very close to the true user positions.
\begin{figure}[t]
\begin{centering}
\includegraphics[width=0.8\columnwidth]{IMG/MultiUser.eps}
\par\end{centering}
\caption{The performance of proposed localization algorithm for multi-user case.
\label{fig:multiUser_localization}}
\vspace{-6pt}
\end{figure}
In Fig. \ref{fig:cdf_localization}, the cumulative distribution function (CDF) of user localization error of our proposed algorithm with comparison to \cite{esrafilian20203d} over Monte-Carlo simulations is shown. We can see that the localization accuracy is considerably improved by using the hybrid channel model.
\begin{figure}[t]
\begin{centering}
\includegraphics[width=0.8\columnwidth]{IMG/CDF.eps}
\par\end{centering}
\caption{The CDF of user localization error for different algorithms.
\label{fig:cdf_localization}}
\vspace{-6pt}
\end{figure}
\section{Experimental Results}\label{sec:experiment}
We have also validated the performance of the proposed algorithm through real-world experimentation. We established a Wi-Fi mesh network comprising sets of outdoor ground nodes and a UAV node. all the nodes are equipped with a MicroTick Wi-Fi card, which is configured on channel 48 in the 5 GHz band, with two omnidirectional vertically polarized dipole antennas. Prior to applying the localization algorithm, we learned the wireless channel by training the hybrid channel model over the
training measurements collected from different ground users in the environment where the experiment is conducted.
Having learned the channel model, different test user locations are chosen to be localized. In Fig. \ref{fig:experiment_localization}, the top view of the UAV trajectory taken to collect the test data, the performance of the localization, and the estimate of the channel using our proposed method as well as the method in \cite{esrafilian20203d} for different scenarios are shown. In both trajectories, the heading of the UAV is set to a fixed angle facing towards the south over the course of the trajectory. It is worth noting that when the relative angle between the UAV and the ground node changes drastically, Fig. \ref{fig:experiment_localization}-b, the algorithm in [12] which uses the convectional channel model fails to localize the users. This stems from the fact that the UAV antenna pattern is changed considerably and is no longer symmetric and omnidirectional due to the proximity to all the components on the UAV (i.e. the body frame, propellers, motors, etc.) which makes it difficult to be precisely modeled by conventional channel models.
Moreover, in \cite{VideoClip}, a video recording
of the experiment in EURECOM campus is captured, illustrating the localization of two ground nodes while flying the UAV in the environment. As the UAV collects more measurements from the ground nodes, the estimate of the user location becomes more accurate.
\begin{figure}[t]
\centering
\subfloat[]{
\includegraphics[clip,width=0.49\columnwidth]{IMG/Exp1_Trj.eps}
\includegraphics[clip,width=0.48\columnwidth]{IMG/Exp1_rss.eps}
}
\newline
\subfloat[]{
\includegraphics[clip,width=0.49\columnwidth]{IMG/Exp2_Trj.eps}
\includegraphics[clip,width=0.48\columnwidth]{IMG/Exp2_rss.eps}
}
\caption{Left: top view of the UAV trajectory, true user and estimated user location. Right: corresponding test measurements collected from the user and the channel estimate using different models. \label{fig:experiment_localization}}
\end{figure}
| -24,411.164606 |
[
-2.525390625,
2.265625
] | 35.191638 |
[
-3.515625,
0.39599609375,
-2.037109375,
-5.41796875,
-0.405029296875,
7.36328125
] |
[
1.2041015625,
4.859375,
0.293212890625,
4.7578125
] | 166 | 3,504 |
[
-1.7763671875,
1.583984375
] | 27.784368 |
[
-6.8203125,
-5.1875,
-4.53125,
-1.712890625,
3.013671875,
12.375
] | 1.310397 | 21.477828 | 25.998858 | 1.091 |
[
2.2711126804351807
] | -16,805.756795 | 5.989155 | -24,041.558175 | 0.867245 | 5.650304 |
[
-3.150390625,
-3.6484375,
-3.677734375,
-4.4609375,
2.736328125,
11.5078125
] |
[
-6.01171875,
-1.8916015625,
-2.412109375,
-1.5830078125,
3.6796875,
4.8359375
] | |
BkiUav3xK6nrxqmoxKc4
|
\section{Some computations and proofs}\label{app:computations}
\begin{proof}[Proof of Theorem~\ref{thm:splitmatrel}]
It is known that the whiskers are Lagrangian manifolds \cite{EliassonBiasymptotic,DelshamsPotential,LochakMemoirs}; they are graphs over the angles and, moreover, the actions are gradients of potentials. In other words, there exist generating funcions $S^{u,s}$ such that
\begin{equation*}
I^{u,s}\equiv \partial_\phi S^{u,s}(\phi,\psi)\quad\text{and}\quad A^{u,s}\equiv \partial_\psi S^{u,s}(\phi,\psi).
\end{equation*}
We consider the latter row vectors. Differentiating both sides of
\begin{equation*}
\mathcal{H}^{u,s}\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=}\mathcal{H}(\phi,\psi,\partial_\phi S^{u,s},\partial_\psi S^{u,s})\equiv E
\end{equation*}
with respect to the $i$th component of $(\phi,\psi)\mathrel{=\mkern-4.2mu\raise.095ex\hbox{$:$}}\varphi=(\varphi_0,\varphi_1,\dots,\varphi_d)$ yields
\begin{equation}\label{eq:differentiated}
0=\mathcal{H}^{u,s}_i+\sum_{j=0}^d \mathcal{H}^{u,s}_{d+1+j}\,\partial_{\varphi_i}\partial_{\varphi_j}S^{u,s}=\mathcal{H}^{u,s}_i+\sum_{j=0}^d (\partial_\varphi^2 S^{u,s})_{ij} \partial_{J_j}\mathcal{H}^{u,s}.
\end{equation}
Here $J\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=}(I,A)$ and $\partial_{J_j}\mathcal{H}^{u,s}$ means $\mathcal{H}^{u,s}_{d+1+j}=\partial_{J_j}\mathcal{H}(\phi,\psi,\partial_\phi S^{u,s},\partial_\psi S^{u,s})$.
From \eqref{eq:differentiated} it follows that, \emph{on the homoclinic trajectory},
\begin{equation}\label{eq:identity}
\partial_J \mathcal{H}^{u,s}\,\partial_\varphi^2(S^u-S^s)=0
\end{equation}
because one has $\partial_\varphi S^u=\partial_\varphi S^s$ and therefore also $\mathcal{H}^u_i=\mathcal{H}^s_i$ for $i=0,\dots,2d+1$. For our particular Hamiltonian, $\partial_J \mathcal{H}^{u,s}=(I,A)^{u,s}$, where we now drop the superscripts $u,s$ and evaluate everything at the homoclinic point
\begin{equation}\label{eq:hp}
\varphi=(\phi,\psi)=(0,\omega t)+X^u(e^{\gamma t},\omega t) =(0,\omega t)+X^s(e^{\gamma t},\omega t)
\end{equation}
below. The $\psi$ component of \eqref{eq:identity} reads
\begin{equation*}
I\,\partial_\psi\partial_\phi(S^u-S^s) + A\,\partial^2_\psi(S^u-S^s) = 0,
\end{equation*}
or, recalling $I=I^0+\order{\epsilon}=2g/\cosh{gt}+\order{\epsilon}\neq 0$ (taking $\tilde \epsilon\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=} g^{-1} e^{g\abs{t}}\epsilon$ small),
\begin{equation}\label{eq:consequence}
\partial_\psi\partial_\phi(S^u-S^s)=-I^{-1} A \,\partial^2_\psi(S^u-S^s).
\end{equation}
Here $A$ is the row vector $(A_1,\dots,A_d)$, such that both sides of the equality are row vectors.
Next, let us perform the coordinate transformations
$
(\phi,\psi)=F^{u,s}(z,\theta)\mapsto(z,\theta).
$
To this end, we observe that the splitting vector satisfies
\begin{equation*}
\Delta^T(\phi,\psi) =A^u-A^s = \partial_\psi (S^u-S^s)(\phi,\psi).
\end{equation*}
Then, the $\theta$ derivative of the column vector $\Delta$ is the square matrix
\begin{equation*}
\partial_\theta\Delta=\partial_\psi\Delta\,\partial_\theta\psi+\partial_\phi\Delta\,\partial_\theta\phi,
\end{equation*}
by the chain rule. In particular, by \eqref{eq:consequence},
\begin{align}\label{eq:derivative-relation}
\partial_\phi\Delta &= [\partial_\psi\partial_\phi(S^u-S^s)]^T = -\partial^2_\psi(S^u-S^s)\,I^{-1} A^T =-\partial_\psi\Delta\,\,I^{-1} A^T
\end{align}
holds at a homoclinic point (see \eqref{eq:hp}), such that
\begin{equation*}
\partial_\theta\Delta=\partial_\psi\Delta\,\bigl(\partial_\theta\psi - I^{-1} A^T\partial_\theta\phi\bigr).
\end{equation*}
The matrix
$
M \mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=} \partial_\theta\psi - I^{-1} A^T\partial_\theta\phi
$
has the asymptotic expression
\begin{equation*}
M =\bigl(\one+\order{\epsilon}\bigr)-\Bigl(\frac{2g}{\cosh{gt}}+\order\epsilon\Bigr)^{-1} \cdot \bigl(\omega+\order{\epsilon}\bigr)^T \cdot \order{\epsilon}
\end{equation*}
with $\epsilon= ge^{-g\abs{t}}\tilde\epsilon$, as before, and $\tilde\epsilon$ small.
\end{proof}
\begin{proof}[Proof of Lemma~\ref{lem:Lazutkin}. (Adapted from \cite{Sauzin}).]
The Fourier transform of \eqref{eq:Lazutkin-id} yields
\begin{equation}\label{eq:Lazutkin-Fourier}
\hat F(s,q)=\hat F(0,q)\, e^{-i(g^{-1} \omega\cdot q) s},
\end{equation}
which is entire in $s$. But
$
\abs{\hat F(0,q)}\,e^{(g^{-1} \omega\cdot q)\,{\operatorname{\Im\mathfrak{m}}\,} s}=\abs{\hat F(s,q)}\leq B(g) e^{-\eta\abs{q}}
$
for $s\in [-i\vartheta,i\vartheta]$, and
\begin{equation*}
\abs{\hat F(0,q)}\leq B(g) e^{-\vartheta g^{-1} \abs{\omega\cdot q}-\eta\abs{q}}.
\end{equation*}
Finally, plugging this into \eqref{eq:Lazutkin-Fourier}, we get
\begin{equation*}
\abs{\hat F(s,q)}\leq B(g) e^{-(\vartheta-\abs{{\operatorname{\Im\mathfrak{m}}\,}{s}})g^{-1} \abs{\omega\cdot q}-\eta\abs{q}}\qquad(s\in\mathbb{C}).
\end{equation*}
For $\abs{{\operatorname{\Im\mathfrak{m}}\,}{s}}\leq \vartheta$ and $\abs{{\operatorname{\Im\mathfrak{m}}\,}{\theta}}\leq\eta'<\eta$, the series $\sum_{q\in\mathbb{Z}^d}\hat F(s,q)e^{iq\cdot \theta}$ is uniformly convergent and, as such, provides the analytic extension.
Since $\alpha x^{-\nu}+\beta x\geq \alpha(\nu+1)\bigl(\frac{\alpha\nu}{\beta}\bigr)^{-\nu/(\nu+1)}$ for positive $\alpha$, $\beta$, $\nu$, and $x$, we get from the Diophantine condition \eqref{Dio} that
\begin{equation*}
\abs{\hat F(s,q)}\leq B(g) e^{-\vartheta g^{-1} a\abs{q}^{-\nu}-\eta\abs{q}}\leq B(g) e^{-\delta\abs{q}}e^{-w(\vartheta,\eta-\delta)g^{-1/(\nu+1)}}
\end{equation*}
holds if $s\in\mathbb{R},\,q\in\mathbb{Z}^d\setminus\{0\}$, $0<\delta<\eta$, and $w(\vartheta,\eta-\delta)\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=} (\vartheta a)^{1/(\nu+1)}(\eta-\delta)^{\nu/(\nu+1)}(\nu+1)\nu^{-\nu/(\nu+1)}$. Moreover,
$
\sum_{q\in\mathbb{Z}^d\setminus\{0\}} e^{-\delta\abs{q}}\leq C\delta^{-d},
$
where $C$ only depends on the dimension $d$.
By \eqref{eq:Lazutkin-Fourier}, $\widetilde F\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=}\average{F(s,\,\cdot\,)}=\hat F(s,0)=\hat F(0,0)$, such that $\widetilde F$ does not depend on $(s,\theta)$.
\end{proof}
\begin{proof}[Proof of Lemma~\ref{lem:shift-of-contour}]
By shifting the contour of integration from $\mathbb{R}$ to the complex plane by $i t_q\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=} i\sgn(\omega\cdot q)\vartheta g^{-1}$ units, we compute
\begin{align*}
\Xint-_{-\infty}^\infty t^p \hat h(e^{gt},q) e^{i q\cdot \omega t}\,dt &=\res_{R=0}\frac{1}{R}\int_{-\infty}^\infty e^{-R\abs{t}} t^p \hat h(e^{gt},q) e^{i q\cdot \omega t}\,dt \\
&=\res_{R=0}\frac{1}{R}\Bigl\{e^{-iR t_q}H_q(R)+e^{iR t_q}I_q(R)\Bigr\} e^{-\vartheta g^{-1}\abs{\omega\cdot q}},
\end{align*}
where $H_q$ is defined in \eqref{eq:H_q} and
$
I_q(R)\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=}\int_{-\infty}^0 e^{(iq\cdot\omega+R)t}(t+it_q)^p\hat h(e^{g(t+it_q)},q)\,dt.
$
There are, \textit{a priori}, two additional line integrals $\int_0^{it_q}$, but they cancel due to the residue at $R=0$, as is easily checked. Because $\hat h(\,\cdot\,,0)=0$, $I_q(R)$ does not have a pole at $R=0$. Hence,
\begin{equation*}
\res_{R=0}\frac{e^{iR t_q}I_q(R)}{R}=\Xint-_{-\infty}^0 e^{iq\cdot\omega t}(t+it_q)^p\hat h(e^{g(t+it_q)},q)\,dt.
\end{equation*}
If $H_q(R)$ has a pole of order $k$ at $R=0$, then
\begin{equation*}
\biggl|\res_{R=0}\frac{e^{-iR t_q}H_q(R)}{R}\biggr|=\biggl|\sum_{j=0}^k \frac{(-it_q)^j H_{q,-j}}{j!}\biggr|\leq \sum_{j=0}^k \frac{1}{j!}\Bigl(\frac{\rho\vartheta}{g}\Bigr)^j\sup_{\abs{R}=\rho}\abs{H_q(R)},
\end{equation*}
because the Laurent coefficients
$
H_{q,-j}\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=} \frac{1}{2\pi}\oint_{\abs{R}=\rho}\frac{H_q(R)}{R^{-j+1}}\,dR
$
can be bounded from above by
$
\abs{H_{q,-j}}\leq \rho^j\sup_{\abs{R}=\rho}\abs{H_q(R)},
$
whenever the circle $\abs{R}=\rho$ is inside the domain of $H_q$.
Under the assumptions of the lemma, for any $0<\delta<\sigma$ and $q\in\mathbb{Z}^d\setminus\{0\}$,
\begin{equation*}
\biggl|\,\Xint-_{-\infty}^\infty t^p \hat h(e^{gt},q) e^{i q\cdot \omega t}\,dt \biggr|\leq \biggl[A(g)+B(g)\sum_{j=0}^k \frac{1}{j!}\Bigl(\frac{\rho\vartheta}{g}\Bigr)^j\biggr]e^{-\delta\abs{q}}e^{-w(\vartheta,\sigma-\delta)g^{-1/(\nu+1)}},
\end{equation*}
by mimicking the proof of Lemma~\ref{lem:Lazutkin}. Here $w(\vartheta,\sigma-\delta)\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=} (\vartheta a)^{1/(\nu+1)}\bigl(\frac{\sigma-\delta}{\nu}\bigr)^{\nu/(\nu+1)}(\nu+1)$. Summation over $q$ produces a factor $C\delta^{-d}$.
\end{proof}
\begin{proof}[Proof of Lemma~\ref{lem:multiplier-bound}]
Here we usually omit the subindices $i$, $v$, and $v_0$. According to Lemma~\ref{lem:r_v}, $u_v$ is analytic on $\{ \abs{z} \geq \tau^{-1} \} \times \{ \abs{{\operatorname{\Im\mathfrak{m}}\,}{\theta}}\leq \sigma \}$ with a pole of order $r$ at $z=\infty$. A Cauchy estimate then reads
$
\abs{u_{v,k}(\theta)}\leq \tau^k \sup_{\abs{z}=\tau^{-1}}\abs{u_v(z,\theta)}\leq C \tau^{k-r}.
$
Now the first bound follows from $\abs{u_{v,k}(q)}\leq e^{-\sigma\abs{q}}\sup_{\abs{{\operatorname{\Im\mathfrak{m}}\,}{\theta}}\leq\sigma}\abs{u_{v,k}(\theta)}$.
The second inequality is a trivial consequence of the first one for $\abs{z}>2\tau^{-1}$. For the other values of $z$, one uses \eqref{eq:multiplier-split} and \eqref{eq:index-sum} as well as $\abs{z}^{-r}\abs{u_v(z,q)}\leq C e^{-\sigma\abs{q}}$ to bound
\begin{align*}
\abs{u_{v,< -\tilde k-s}(z,q)} & \leq \abs{z}^{-\tilde k-s-1}\Bigl\{\abs{z}^{\tilde k+s+1} \abs{u_v(z,q)}+ C e^{-\sigma\abs{q}} \tau^{-\tilde k-s-1} \sum_{l=0}^{r+\tilde k+s} \abs{z\tau}^{l+1} \Bigr\} \\
& \leq C e^{-\sigma\abs{q}} \abs{z}^{-\tilde k-s-1} \Bigl\{(2\tau^{-1})^{\tilde k+s+r+1} + \tau^{-\tilde k-s-r-1} 2^{r+\tilde k+s} \Bigr\}.
\end{align*}
The last inequality in the lemma is obvious and is stated for completeness.
\end{proof}%
\section{Analytic Continuation of the Solution}\label{sec:continuation}
Here we present the proof of Theorem~\ref{thm:extension}. We drop the superscript $u$ from the notation and single out the uncoupled part $X^0$ of the complete solution $X$;
\begin{equation*}
X=X^0+\widetilde X\quad\text{with}\quad{\widetilde X}{\rvert}_{\epsilon=0}\equiv 0.
\end{equation*}
Equation~\eqref{xeq} and $\mathcal{L}^2X^0=(\gamma^2\sin\Phi^0,0)$ imply
$
\mathcal{L}^2\widetilde X=-(\gamma^2\sin\Phi^0,0)+\Omega(X^0+\widetilde X).
$
Thus,
\begin{equation}\label{til}
\mathcal{K}\widetilde X=\widetilde W(\widetilde X),
\end{equation}
when we define the linear operator
\begin{equation}\label{K}
\mathcal{K}\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=}
\begin{pmatrix}
L & 0\\
0 & \mathcal{L}^2
\end{pmatrix}
\quad\text{with}\quad
L\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=}\mathcal{L}^2-\gamma^2\cos\Phi^0
\end{equation}
and the nonlinear operator
\begin{equation}\label{tilW}
\widetilde W(\widetilde X)\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=}(-\gamma^2\sin\Phi^0-\gamma^2(\cos\Phi^0)\widetilde\Phi,0)+\Omega(X^0+\widetilde X).
\end{equation}
Throughout the rest of the work, we shall refer to different parts of the Taylor expansion of a suitable function $h(z,\theta)$ around $z=0$ using the notation
\begin{equation*}
h_k(\theta)\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=} \frac{\partial_z^k h(0,\theta)}{k!},\quad h_{\leq k}(z,\theta)\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=} \sum_{j=0}^k z^kh_k(\theta)\quad\text{and}\quad \delta_k h\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=} h-h_{\leq k-1}.
\end{equation*}
By~\eqref{til}, $\delta_2\widetilde X$ satisfies
\begin{equation}\label{Zeq}
\mathcal{K} \delta_2\widetilde X=W(\delta_2\widetilde X),
\end{equation}
where
\begin{equation}\label{Wdef}
W(Z)\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=}\delta_2\left[\widetilde W(\widetilde X_{\leq 1}+Z)+
\begin{pmatrix}
\gamma^2(\cos\Phi^0)\widetilde\Phi_{\leq 1} \\
0
\end{pmatrix}
\right].
\end{equation}
Let us consider analytic functions $Z$ on a small, compact and complex, neighbourhood $\Pi$ of the set $[-1,1]\times{\mathbb{T}^d}$. It is a complex Banach space $\mathcal{A}$, once equipped with the supremum norm, and has the closed subspace
\begin{equation}\label{eq:A1}
\mathcal{A}_1\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=}\left\{Z\in\mathcal{A}\;|\;Z_{\leq 1}=0\right\}.
\end{equation}
We have shown in \cite{Stenlund-whiskers} that $\mathcal{K}$ maps $\mathcal{A}_1$ to itself and has a bounded, $\order{\gamma^{-2}}$, inverse.
By virtue of \eqref{Zeq} and $\widetilde W$'s analyticity, $\delta_2\widetilde X$ admits the representation
\begin{equation}\label{eq:Zrec}
\begin{split}
\delta_2\widetilde X=\mathcal{K}^{-1} \delta_2 \biggl[
\begin{pmatrix}
\gamma^2 \cos\Phi^0 & 0 \\
0 & 0
\end{pmatrix}
\widetilde X_{\leq 1}+\sum_{k=0}^\infty w^{(k)}\bigl(\widetilde X_{\leq 1}\bigr)^{\otimes k}
\biggr]+ \\
+\,\mathcal{K}^{-1}\sum_{k=1}^\infty \Bigl[w^{(k)}\bigl(\widetilde X_{\leq 1}+\delta_2\widetilde X\bigr)^{\otimes k}-w^{(k)}\bigl(\widetilde X_{\leq 1}\bigr)^{\otimes k}\Bigr]
\end{split}
\end{equation}
on the set $\Pi$, taking $\epsilon$ small enough, and denoting
\begin{equation}\label{eq:wdefinition}
w^{(k)}\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=} \frac{1}{k!}D^k\widetilde W(0)
\end{equation}
as well as a repeated argument of such a symmetric $k$-linear operator by
$
(x)^{\otimes k}\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=} (x,\dots,x),
$
for the sake of brevity. Observe that we have omitted a $\delta_2$ in front of the square brackets on the second line of \eqref{eq:Zrec} as redundant.
Equation~\eqref{eq:Zrec} may be viewed as a recursion relation for $\delta_2\widetilde X$. It is crucial that
\begin{equation}\label{eq:nodeorder}
w^{(0)},\,w^{(1)}=\order{\epsilon g^2},
\end{equation}
when $(\epsilon,g)\in D$; see \eqref{eq:D}. Namely, any given order $\delta_2\widetilde X^\ell$ in the convergent expansion
$
\delta_2\widetilde X=\sum_{\ell=1}^\infty \epsilon^\ell \,\delta_2\widetilde X^\ell
$
is then completely determined by $\widetilde X_{\leq 1}$ and the \emph{lower orders} $\delta_2\widetilde X^l$ ($1\leq l\leq\ell-1$) through the right-hand side of \eqref{eq:Zrec}. Moreover, since $\widetilde X_{\leq 1}=\order{\epsilon}$, only \emph{finitely many} terms in the sum over the index $k$ are involved. Together these facts imply that only \emph{finitely many} recursive steps using \eqref{eq:Zrec} are needed to completely describe any given order $\delta_2\widetilde X^\ell$ in terms of $\widetilde X_{\leq 1}$ alone and that, at each such step, only \emph{finitely many} terms from the $k$-sum contribute.
It is important to understand that $\widetilde X_{\leq 1}$ is a predetermined function. As we shall see, the recursion procedure will then provide the analytic continuation of each $X^{u,\ell}=\widetilde X^\ell_{\leq 1}+\delta_2\widetilde X^\ell$ ($\ell\geq 1$) to the large region $\mathbb{U}_{\tau,\vartheta}\times\{\abs{{\operatorname{\Im\mathfrak{m}}\,}{\theta}}\leq\sigma\}$ of Theorem~\ref{thm:extension}.
\subsection{Tree expansion}
We next give a pictorial representation of the above recursion. It involves tree diagrams similar to those of Gallavotti, \textit{et al.} (see, \textit{e.g.}, \cite{GallavottiTwistless,ChierchiaGallavotti}), with one difference: there will be no resummations nor cancellations, as the expansion in \eqref{eq:Zrec} contains no resonances and is instead well converging. This so-called tree expansion is needed for bookkeeping and pedagogical purposes; we simply choose to draw a tree instead of spelling out a formula.
Let us first define the auxiliary functions
\begin{equation*}
h^{(k)}\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=}
\begin{cases}
w^{(0)}+\bigl[\bigl(\begin{smallmatrix}
\gamma^2 \cos\Phi^0 & 0 \\
0 & 0
\end{smallmatrix}\bigr)
+ w^{(1)}
\bigr]\widetilde X_{\leq 1} & \text{if $k=1$,}
\vspace{1mm}\\
w^{(k)}\bigl(\widetilde X_{\leq 1}\bigr)^{\otimes k} & \text{if $k=2,3,\dots$},
\end{cases}
\end{equation*}
and make the identifications
\begin{equation}\label{eq:endnode}
\raisebox{3pt}{
\pstree{\Tp}{\Tcircle{\tiny $k$}}
}
\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=}\mathcal{K}^{-1}\delta_2 h^{(k)}
\quad\text{and}\quad
\raisebox{3pt}{
\pstree{\Tp}{\Tc[fillstyle=hlines,fillcolor=black,hatchsep=2.5pt]{7pt}}
}
\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=}\mathcal{K}^{-1}\delta_2 \,\sum_{k=0}^\infty h^{(k)}.
\end{equation}
Furthermore, let
\begin{equation*}
\raisebox{3pt}{
\pstree{\Tp}{\Tc{7pt}}
}
\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=}\delta_2\widetilde X,\quad
\raisebox{3pt}{
\pstree{\Tp}{\TC*}
}
\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=} \widetilde X_{\leq 1}, \quad\text{and}\quad
\raisebox{3pt}{
\pstree[treesep=1.2cm]{\Tp}{\pstree{\TC*~[tnpos=r,tnsep=7.5pt]{\raisebox{1pt}{\tiny $k$ lines}}}{\Tp[name=top] \Tp[name=bot]}}
\ncarc[linestyle=dotted,nodesep=5pt]{top}{bot}
}
\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=} \mathcal{K}^{-1} w^{(k)}.
\vspace{1mm}
\end{equation*}
In the diagram representing the $k$-linear $w^{(k)}$, the $k$ ``free'' lines to the right of the node stand for the arguments. We say that these lines \emph{enter} the \emph{internal} node, whereas the single line to the left of the node \emph{leaves} it. For instance,
\begin{equation*}
\raisebox{3pt}{
\pstree[treesep=0.5cm]{\Tp}{\pstree{\TC*}{\TC* \Tc[bbh=12pt]{7pt} \Tcircle[]{\tiny 4}}}
}
=\mathcal{K}^{-1} w^{(3)}\bigl(\widetilde X_{\leq 1},\,\delta_2\widetilde X,\,\mathcal{K}^{-1}\delta_2 h^{(4)}\bigr).
\end{equation*}
Notice that, as $w^{(k)}$ is symmetric, permuting the lines entering a node does not change the resulting function. We emphasize that all of the functions introduced above are analytic on $\Pi$ and $\abs{\epsilon}<\epsilon_0$.
In terms of such \emph{tree diagrams}, or simply \emph{trees}, equation~\eqref{eq:Zrec} reads
\begin{equation}\label{eq:tree-rec}
\begin{split}
\raisebox{3pt}{
\pstree{\Tp}{\Tc{7pt}}
}
= \, &
\raisebox{3pt}{
\pstree{\Tp}{\Tc[fillstyle=hlines,fillcolor=black,hatchsep=2.5pt]{7pt}}
}
+
\raisebox{3pt}{
\pstree{\Tp}{\pstree{\TC*}{\Tc{7pt}}}
}
+
\raisebox{3pt}{
\pstree{\Tp}{\pstree{\TC*}{\Tc{7pt} \TC*}}
}
+
\\
& +
\raisebox{3pt}{
\pstree{\Tp}{\pstree{\TC*}{\TC* \Tc{7pt}}}
}
+
\raisebox{3pt}{
\pstree{\Tp}{\pstree{\TC*}{\Tc{7pt} \Tc{7pt}}}
}
+\,\cdots,
\end{split}
\end{equation}
using multilinearity to split the sums $\widetilde X_{\leq 1}+\delta_2\widetilde X$ into pieces. Above, the sum after the first tree consists of \emph{all} trees having one internal node and an arbitrary number of \emph{end nodes, at least one of which, however, is a white circle}. This rule encodes the fact that on the second line of \eqref{eq:Zrec} the summation starts from $k=1$ and that the contributions with only $\widetilde X_{\leq 1}$ in the argument (\textit{i.e.}, trees with only black dots as end nodes) are cancelled.
Using \eqref{eq:Zrec} recursively now amounts to replacing each of the lines with a white-circled end node by the complete expansion of such a tree above. This is to be understood additively, so that replacing one end node, together with the line leaving it, by a sum of two trees results in a sum of two new trees. For example, such a replacement in the third tree on the right-hand side of \eqref{eq:tree-rec} by the first two trees gives the sum
\begin{equation*}
\raisebox{3pt}{
\pstree{\Tp}{\pstree{\TC*}{\Tc[fillstyle=hlines,fillcolor=black,hatchsep=2.5pt]{7pt} \TC*}}
}
+
\raisebox{3pt}{
\pstree{\Tp}{\pstree{\TC*}{\pstree{\TC*}{\Tc{7pt}}
\TC*}}
}.
\end{equation*}
Before proceeding, we introduce a little bit of terminology. The leftmost line in a tree is called the \emph{root line}, whereas the node it leaves (\textit{i.e.}, the uniquely defined leftmost node) is called the \emph{root}. A line leaving a node $v$ and entering a node $v'$ can always be interpreted as the root line of a \emph{subtree}, the maximal tree consisting of lines and nodes in the original tree with $v$ as its root. We call $v$ a (not necessarily unique) \emph{successor} of $v'$, whereas $v'$ is the unique \emph{predecessor} of $v$.
The recursion \eqref{eq:tree-rec} can be repeated on a given tree if it has at least one white circle left. Otherwise, the tree in question must satisfy
\begin{itemize}
\item[{\bf (R$\mathbf 1'$)}] The tree has only filled circles (\raisebox{3pt}{\Tc[fillstyle=hlines,fillcolor=black,hatchsep=2.5pt]{7pt}}) and black dots (\raisebox{3pt}{\TC*}) as its end nodes,
\end{itemize}
together with
\begin{itemize}
\item[{\bf (R$\mathbf 2'$)}] Any internal node has an entering (line that is the root line of a) subtree containing at least one filled circle as an end node.
\end{itemize}
After all, the recursion can only stop by replacing an existing white circle with a filled one. Continuing \textit{ad infinitum} yields the expansion
\begin{equation}\label{eq:tree-exp1}
\raisebox{3pt}{
\pstree{\Tp}{\Tc{7pt}}
}
=\,\sum{\bigl(\text{Trees satisfying (R$1'$) and (R$2'$)}\bigr)}=\, \sideset{}{'}\sum_{\text{trees $T$}}T,
\end{equation}
where the prime restricts the summation to trees $T$ satisfying (R$1'$) and (R$2'$). We point out that each admissible tree appears precisely once in this sum, considering different two trees that can be superposed by a (nontrivial) permutation of subtrees that enter the same node.
The earlier discussion concerning the description of $\delta_2\widetilde X^\ell$ in terms of a finite sum involving only $\widetilde X_{\leq 1}$ translates to the language of trees in a straightforward fashion. First, the second part of \eqref{eq:nodeorder} and $\widetilde X_{\leq 1}=\order{\epsilon}$ amount pictorially to
\vspace{1mm}
\begin{equation*}
\raisebox{3pt}{
\pstree{\Tp}{\pstree{\TC*}{\Tp}}
}
=\order{\epsilon}
\quad\text{and}\quad
\raisebox{3pt}{
\pstree{\Tp}{\TC*}
}
=\order{\epsilon},
\vspace{1mm}
\end{equation*}
since $\mathcal{K}^{-1}$ produces a factor of $g^{-2}$. Second, $w^{(k)}=\order{g^2}$ and the first part of \eqref{eq:nodeorder} yield
\begin{equation*}
\raisebox{3pt}{
\pstree{\Tp}{\Tcircle{\tiny $k$}}
}
=\order{\epsilon^k}\quad(k\geq 1)
\qquad \text{and} \qquad
\raisebox{3pt}{
\pstree[treesep=1.2cm]{\Tp}{\pstree{\TC*~[tnpos=r,tnsep=7.5pt]{\raisebox{1pt}{\tiny $k$ lines}}}{\Tp[name=top] \Tp[name=bot]}}
\ncarc[linestyle=dotted,nodesep=5pt]{top}{bot}
}
=\order{1}\quad(k\geq 2).
\vspace{2mm}
\end{equation*}
Expanding the filled end nodes
\begin{equation}\label{eq:node-exp}
\raisebox{3pt}{
\pstree{\Tp}{\Tc[fillstyle=hlines,fillcolor=black,hatchsep=2.5pt]{7pt}}
}
=\,\sum_{k=1}^\infty
\raisebox{3pt}{
\pstree{\Tp}{\Tcircle{\tiny $k$}}
},
\end{equation}
according to \eqref{eq:endnode}, on the right-hand side of \eqref{eq:tree-exp1}, we get a new version of the latter by replacing the rules (R$1'$) and (R$2'$), respectively, with
\begin{itemize}
\item[{\bf (R1)}] The tree has only numbered circles (\raisebox{3pt}{{\tiny \Tcircle{$k$}}} with arbitrary values of $k$) and black dots (\raisebox{3pt}{\TC*}) as its end nodes,
\end{itemize}
and
\begin{itemize}
\item[{\bf (R2)}] Any internal node has an entering (line that is the root line of a) subtree containing at least one numbered circle as an end node.
\end{itemize}
Let us define the \emph{degree} of a tree as the positive integer
\begin{equation}\label{eq:deg}
\deg{T}\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=}\#(\!
\raisebox{3pt}{
\pstree{\Tp}{\pstree{\TC*}{\Tp}}
}
\!)\,+\,\#(\!
\raisebox{3pt}{
\pstree{\Tp}{\TC*}
}
\!)\,+\,\sum_{k=1}^\infty \,k\,\#(\!
\raisebox{3pt}{
\pstree{\Tp}{\Tcircle{\tiny $k$}}
}
\!)
\end{equation}
for any tree $T$ satisfying (R1) and (R2). By $\#(G)$ we mean the number of occurrences of the graph $G$ in the tree $T$. That is, the degree of a tree is the number of its end nodes with suitable weights plus the number of nodes with precisely one entering line. Since a tree has finitely many nodes, its degree is well-defined. Then a rearrangement of the sum arising from \eqref{eq:node-exp} being inserted into \eqref{eq:tree-exp1} yields formally
\begin{equation}\label{eq:tree-exp2}
\raisebox{3pt}{
\pstree{\Tp}{\Tc{7pt}}
}
=\,\sum_{l=1}^\infty\, \sideset{}{^*}\sum_{\substack{\text{trees $T$} \vspace{1pt} \\ \deg{T}=l}}T,
\end{equation}
where the asterisk reminds us that the rules (R1) and (R2) are being respected.
According to the analysis above, the particular graphs appearing in the definition of $\deg{T}$ are the only possible single-node subgraphs of $T$ proportional to a positive power of $\epsilon$. Since each tree is analytic in $\epsilon$, writing $(\,\cdot\,)^k$ for the $k$th coefficient of the power series in $\epsilon$, we have
\begin{equation*}
T=\sum_{k=\deg T}^\infty \epsilon^k \,T^k=\epsilon^{\deg T}\,\sum_{k=0}^\infty \epsilon^k \,T^{k+\deg T}.
\end{equation*}
Hence, only trees with degree \emph{at most} equal to $\ell$ can contribute to $\delta_2\widetilde X^\ell$:
\begin{equation}\label{eq:tree-exp-order}
\delta_2\widetilde X^\ell=\,\sum_{l=1}^\ell\, \sideset{}{^*}\sum_{\deg{T}=l}T^\ell=\, {\Bigl(\;
\sideset{}{^*}\sum_{\deg{T}\leq \ell}T
\Bigr)}^{\ell}
\end{equation}
or, alternatively,
\begin{equation}\label{eq:tree-exp3}
\delta_2\widetilde X=\, \sideset{}{^*}\sum_{\deg{T}\leq \ell}T+\order{\epsilon^{\ell+1}}\qquad(\epsilon\to 0)
\end{equation}
for \emph{each and every} $\ell=1,2,\dots$. The expansion in \eqref{eq:tree-exp2} is just a compact way of writing \eqref{eq:tree-exp3}. We emphasize that the latter can be derived completely rigorously, for each value of $\ell$ separately, but resorting to the use of formal series allowed us to treat all orders of $\delta_2\widetilde X$ at once. We call the series \eqref{eq:tree-exp2} an \emph{asymptotic expansion} of $\delta_2\widetilde X$; the partial sums ${\sum_{\deg{T}\leq \ell}^*}\,T$ need not converge to $\delta_2\widetilde X$ for any fixed $\epsilon$ as $\ell\to\infty$, but for a fixed $\ell$ the error is bounded by an $\ell$-dependent constant times $\abs{\epsilon}^{\ell+1}$ on the mutual domain of analyticity, $\abs{\epsilon}<\epsilon_0$.
\begin{example}
The beginning of the asymptotic expansion \eqref{eq:tree-exp3} reads
\vspace{1mm}
\begin{equation*}
\psset{levelsep=1cm}
\begin{split}
\delta_2\widetilde X
\quad = \quad \, & \raisebox{3pt}{
\pstree{\Tp}{\Tcircle{\tiny 1}}
}
+\order{\epsilon^2} \quad = \quad
\raisebox{3pt}{
\pstree{\Tp}{\Tcircle{\tiny 1}}
}
+
\raisebox{3pt}{
\pstree{\Tp}{\Tcircle{\tiny 2}}
}
+
\raisebox{3pt}{
\pstree{\Tp}{\pstree{\TC*}{\Tcircle{\tiny 1}}}
}
+ \\[2mm]
& \, +
\raisebox{3pt}{
\pstree{\Tp}{\pstree{\TC*}{\Tcircle{\tiny 1} \TC*}}
}
+
\raisebox{3pt}{
\pstree{\Tp}{\pstree{\TC*}{\TC* \Tcircle{\tiny 1}}}
}
+
\raisebox{3pt}{
\pstree{\Tp}{\pstree{\TC*}{\Tcircle{\tiny 1} \Tcircle{\tiny 1}}}
}
+\order{\epsilon^3}.
\end{split}
\end{equation*}
\end{example}
\subsection{Analyticity domain of trees}
As already pointed out, all trees $T$ above are analytic functions of $(z,\theta,\epsilon)$ on $\Pi\times\{\abs{\epsilon}<\epsilon_0\}$. Due to the projections $\delta_2$ appearing in \eqref{eq:endnode}, they also satisfy $T|_{z=0}=\partial_zT|_{z=0}=0$, \textit{i.e.}, are elements of the space $\mathcal{A}_1$ defined in \eqref{eq:A1}. On this space, the operator $\mathcal{K}$ in \eqref{K} has a bounded inverse given by a simple integral kernel \cite{Stenlund-whiskers}. Consequently, the analyticity domain of a tree in the $z$-variable is in fact much larger than the neighbourhood of $[-1,1]$ included in $\Pi$; it contains the wedgelike region $\mathbb{U}_{\tau,\vartheta}$ in \eqref{eq:treedomain}.
\begin{lemma}[Analytic continuation of trees]\label{lem:an-cont}
Without affecting the analyticity domain with respect to $\epsilon$, there exist numbers $0<\tau<1$, $0<\vartheta<\pi/2$, and $0<\sigma<\eta$ such that each tree in the sums \eqref{eq:tree-exp2} and \eqref{eq:tree-exp3} extends to an analytic function of $(z,\theta)$ on $\mathbb{U}_{\tau,\vartheta}\times\{\abs{{\operatorname{\Im\mathfrak{m}}\,}{\theta}}\leq\sigma\}$.
\end{lemma}
Since the sum in \eqref{eq:tree-exp-order} is finite and the functions $\widetilde X_{\leq 1}$ and $X^0$ in $X=X^0+\widetilde X_{\leq 1}+\delta_2\widetilde X$ are analytic on $\mathbb{U}_{\tau,\vartheta}\times\{\abs{{\operatorname{\Im\mathfrak{m}}\,}{\theta}}\leq\sigma\}$, Theorem~\ref{thm:extension} follows from Lemma~\ref{lem:an-cont}.
\section{Discussion}
Each term in the asymptotic expansion \eqref{eq:as-expansion} of the splitting matrix $\Upsilon$ is proportional to an integral of the form $\Xint-_{-\infty}^\infty \partial_\theta F(X^u;\tau,z,\theta)\,d\tau$. These we showed to be exponentially small in the limit $g\to 0$ at all orders $\ell$, by extending the integrands analytically into a wedge $\{\abs{\arg{z}}\leq \vartheta\}$ on the complex plane and then shifting the contour of integration. A key point is that we had to do this \emph{order by order}, because the series $\sum_{\ell=0}^\infty \epsilon^\ell X^{u,\ell}(z,\theta)$ is not expected to converge for large values of $\abs{z}$ if $\epsilon$ is fixed.
The large powers of the factorial $\ell!$, associated with the regularized integrals, are produced by accumulation of poles at the origin in the $R$ plane. To some extent the factorials are artifact, as is shown by the following simple example. In order to study, say, the integral
\begin{equation*}
\Xint-_0^{\infty}u(\theta+\omega t)\int_{-\infty}^t v(\theta+\omega \tau) ze^{\gamma\tau}\,d\tau\,dt,
\end{equation*}
we have to show that the sums
\begin{equation*}
\sum_p \hat u(p)\hat v(q-p)\int_0^{\infty}e^{(ip\cdot\omega-R) t}\int_{-\infty}^t e^{(i(q-p)\cdot \omega+\gamma) \tau} \,d\tau\,dt
\end{equation*}
extend analytically to a (punctured) neighbourhood of the origin, $R=0$. We integrate by parts, just as in \eqref{eq:int-by-parts} when extending analytically the tree integral \eqref{eq:reg-tree-int}, and get
\begin{align*}
\sum_p \hat u(p)\hat v(q-p)\frac{1}{R-ip\cdot\omega}\Biggl\{\frac{1}{(i(q-p)\cdot \omega+\gamma)}+\frac{1}{R-(\gamma+iq\cdot\omega)}\Biggr\}.
\end{align*}
Here the pole at $ip\cdot \omega$ gets arbitrarily close to the origin, unless we restrict $p$ somehow---for instance, by considering trigonometric polynomials. Either by simplification, or by computing the same expression directly by starting from the inner integral, we obtain
\begin{equation*}
\sum_p\hat u(p) \hat v(q-p) \frac{1}{i(q-p)\cdot \omega+\gamma}\cdot \frac{1}{R-(\gamma+iq\cdot\omega)}.
\end{equation*}
In the latter form there is no problem; the pole has cancelled. Of course, this is a naive example and in general it is hard to see whether a given pole popping out of the integration-by-parts procedure should really be there.
Also the coefficients $c_{ij}^p$ appearing in Proposition~\ref{prop:asymptotic} produced large powers of $\ell!$. Even though integration by parts was not exploited, the source of the factorials was again the accumulation of poles at the origin in the $R$ plane. In both this case and the previous, the type of ``divergence'' is very similar to what is encountered in KAM theory. There repeated resonances, or arbitrarily many occurrences of the operator $\mathcal{D}^{-1}$ in convolutions, ruin absolute convergence of the Fourier--Taylor expansion of a solution by producing high powers of the factorial $\ell!$. On the other hand, the state of affairs can be cured by well-known resummations, as in \cite{GallavottiTwistless}. Such resummations still escape us in the context of homoclinic splitting.
\section{Main Concepts and Results}
\subsection{Background and history}
The study of ``separatrix splitting'' dates back to Poincar\'e's classic \textit{Les M\'ethodes Nouvelles de la M\'ecanique C\'eleste} \cite{Poincare3}.
Starting with Kolmogorov's 1954 note \cite{Kolmogorov}, it was proved in a series of papers over a period of twenty years that quasiperiodic motions (invariant tori) are typical for nearly integrable Hamiltonians \cite{Moser,Arnold2,MoserRapid1,MoserRapid2,Moser67}, and that motions which become quasiperiodic asymptotically in time (stable/unstable manifolds) are stable under small perturbations \cite{Moser67,Graff}.
Arnold \cite{ArnoldDiffusion} described a mechanism how a chain of such ``whiskered" tori could provide a way of escape for special trajectories, resulting in instability in the system. (A trajectory would typically lie on a torus and therefore stay eternally within a bounded region in phase space.) The latter is often called Arnold mechanism and the general idea of instability goes by the name Arnold diffusion. It is conjectured in \cite{ArnoldAvez} that Arnold diffusion due to Arnold mechanism is present quite generically, for instance in the three body problem.
Arnold mechanism is based on Poincar\'e's concept of biasymptotic solutions, discussed in the last chapter of \cite{Poincare3}, that are formed at intersections of whiskers of tori. Following such intersections, a trajectory can ``diffuse" in a finite time from a neighbourhood of one torus to a neighbourhood of another, and so on.
Chirikov's report \cite{Chirikov} is a very nice physical account on Arnold diffusion, while Lochak's compendium \cite{LochakCompendium} discusses more recent developments in a readable fashion and is a good point to start learning about diffusion. Gelfreich's introduction \cite{GelfreichLondon} to splitting of separatrices is excellent, and we recommend it to anyone intending to study the topic. From there one should advance to \cite{Gelfreich-review}, which covers more topics with more details. The extensive memoir \cite{LochakMemoirs} by Lochak, Marco, and Sauzin is written from the geometric point of view. It has a historical flavor, making it interesting and accessible to virtually anyone.
For the separatrix splitting of the \emph{periodically} forced pendulum, see \cite{Holmes,Scheurle,DelshamsPeriodic,EllisonKummerSaenz}. In the case of the standard map, an asymptotic expression of the splitting has been obtained in \cite{GelfreichStandard}, as the culmination of a series of works starting with Lazutkin's \cite{Lazutkin}.
For the \emph{quasiperiodically} forced pendulum, several studies exist. We mention \cite{DelshamsGelfreichJorbaSeara,DelshamsEtal97,DelshamsJorbaSearaGelfreich,DelshamsGutierrez}. Especially \cite{Treschev,LochakMemoirs,RudnevWiggins2,Sauzin,RudnevTen} emphasize geometrical aspects of separatrix splitting in the quasiperiodic setting. A lower bound on the splitting has been obtained in \cite{GallavottiSplit}, where the pendulum is coupled to two rotators---one speeding up and one slowing down. A generalization to the case of several slow rotators can be found in \cite{Procesi}. These results extend \cite{ChierchiaGallavotti,GallavottiTwistless,GentileQuasiflat}. See also \cite{GGMMelnikovDominance}.
\subsection{The model}
We consider the Hamiltonian
\begin{equation}
\mathcal{H}(\phi,\psi,I,A)=\tfrac{1}{2} I^2+g^2\cos\phi+\tfrac{1}{2} A^2- \lambda f(\phi,\psi)
\label{eq:H}
\end{equation}
of a pendulum coupled to $d$ rotators, with $\phi\in\mathbb{S}^1\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=}\mathbb{R}/2\pi\mathbb{Z}$ and $I\in\mathbb{R}$ the coordinate and momentum of the pendulum, and $\psi\in{\mathbb{T}^d}\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=}(\mathbb{S}^1)^d$ and $A\in\mathbb{R}^d$ the angles and actions of the rotators, respectively. The perturbation $f$ is assumed to be real-valued and real-analytic in its arguments, and $\lambda$ is a (small) real number, whereas $g>0$. This Hamiltonian is sometimes called \emph{the generalized Arnold model} or the \emph{Thirring model}. It is \emph{the} prototype of a nearly integrable Hamiltonian system close to a simple resonance, as is explained in the introduction of \cite{GentileExponent}. A review of applications can be found in \cite{Chirikov}.
The equations of motion are
\begin{equation}
\dot\phi=I, \quad \dot\psi=A, \quad
\dot I=g^2\sin\phi+\lambda\,\partial_\phi f,\quad
\dot A=\lambda\,\partial_\psi f.
\label{eqm}
\end{equation}
For the parameter value $\lambda=0$, which is addressed as the unperturbed case, the pendulum and the rotators decouple. The former then has the separatrix flow $\phi:\mathbb{R}\to \mathbb{S}^1$ given by
\begin{equation*}
\phi(t)=\Phi^0(e^{gt}) \quad\text{with}\quad
\Phi^0(z)=4\arctan z.
\end{equation*}
By elementary trigonometry, the odd function $\Phi^0$ possesses the symmetry property
\begin{equation}\label{eq:arctan-sym}
\Phi^0(z)=2\pi-\Phi^0(z^{-1}).
\end{equation}
In the phase space of the pendulum, the separatrix---given by $\Phi^0$---separates closed trajectories (libration) from open ones (rotation).
On the other hand, $\psi:\mathbb{R}\to{\mathbb{T}^d}$ is quasiperiodic: the vector $
\omega \mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=} A(0) \equiv A(t)
$
in
\begin{equation*}
\psi(t)=\psi(0)+\omega t\pmod{2\pi}
\end{equation*}
is assumed to satisfy for some positive numbers $a$ and $\nu$ the Diophantine condition
\begin{equation}
|\omega\cdot q| > a\,|q|^{-\nu}\quad{\rm for}\quad q\in \mathbb{Z}^d\setminus\{0\}.
\label{Dio}
\end{equation}
Thus, at the instability point of the pendulum, the flow possesses the invariant tori
\begin{equation*}
\mathcal{T}_0\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=}\bigl\{(\phi,\psi,I,A)=(0,\theta, 0, \omega)\;\big|\; \theta\in T^d\bigr\}
\end{equation*}
indexed by $\omega$, with stable and unstable manifolds ($\mathcal{W}^s_0$ and $\mathcal{W}^u_0$, respectively) coinciding:
\begin{equation}
\mathcal{W}^{s,u}_0=\bigl\{(\phi,\psi,I,A)=\left(\Phi^0(z), \theta, gz\partial_z\Phi^0(z),\omega\right)\;\big|\; z\in [-\infty,\infty],\;\theta\in T^d\bigr\}.
\label{unpert}
\end{equation}
\begin{remark}\label{rem:timescale}
The constant $g$ is the Lyapunov exponent for the unstable fixed point of the pendulum motion; in the limit $s\to-\infty$ two nearby initial angles $\phi(s)$ and $\phi(s+\delta s)$ separate at the exponential rate $e^{gs}$. As $\phi(t)=\Phi^0(e^{t/g^{-1}})$, this fixes a natural time scale of $g^{-1}$ units, characteristic of the pendulum motion in the unperturbed Hamiltonian system \eqref{eq:H}.
\end{remark}
When the perturbation is switched on ($\lambda\neq 0$), it is known that most of the invariant tori survive and have stable and unstable manifolds---or ``whiskers'' as Arnold has called them---that may not coincide anymore. We prove a bound on their splitting.
\subsection{Main theorem}
In \cite{Stenlund-whiskers} we construct the perturbed manifolds in a form similar to \eqref{unpert} as graphs of analytic functions over a piece of $[-\infty,\infty]\times{\mathbb{T}^d}$.
To that end, we look for solutions of the form
\begin{equation}
(\phi(t),\psi(t))=(\Phi(e^{\gamma t},\omega t),\omega t+\Psi(e^{\gamma t},\omega t))=(0,\omega t)+(\Phi,\Psi)(e^{\gamma t},\omega t)
\label{trial}
\end{equation}
with quasiperiodic behavior in \emph{one} of the two limits $t\to \pm\infty$. Note especially that the Lyapunov exponent $\gamma>0$ depends on $\lambda$, with $\gamma{\rvert}_{\lambda=0}=g$.
\begin{remark}
One should not expect asymptotic quasiperiodicity in both limits $t\to\pm\infty$, as the unstable and stable manifolds, $\mathcal{W}^u_\lambda$ and $\mathcal{W}^s_\lambda$, are generically expected to depart for nonzero values of the perturbation parameter $\lambda$. Therefore, either the past \emph{or} future asymptotic of a trajectory will evolve so as to ultimately reach the deformed invariant torus $\mathcal{T}_\lambda$.
\end{remark}
Let us denote the total derivative $d/dt$ by $\partial_t$ and the complete angular gradient $(\partial_\phi,\partial_\psi)$ by $\partial$ for short. Substituting \eqref{trial} into the equations of motion,
we get the equation
\begin{equation*}
(\omega\cdot\partial_\theta+\gamma e^{\gamma t}\partial_z)^2 X(e^{\gamma t},\omega t)=[(g^2\sin\Phi,0)+\lambda\,\partial f(X+(0,\theta))](e^{\gamma t},\omega t)
\end{equation*}
for $X\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=}(\Phi,\Psi)$, where $\theta$ stands for the canonical projection $[-\infty,\infty]\times{\mathbb{T}^d}\to{\mathbb{T}^d}$.
Notice that the partial differential operator
\begin{equation*}
\mathcal{L}\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=}\omega\cdot\partial_\theta+\gamma z\partial_z
\label{el}
\end{equation*}
satisfies the characteristic identity
\begin{equation}\label{Lid}
\mathcal{L} F(ze^{\gamma t},\theta+\omega t)
=\partial_t F(ze^{\gamma t},\theta+\omega t),
\end{equation}
which reflects the time derivative nature of $\mathcal{L}$. In fact, if $T$ is the ``time-reversal map''
\begin{equation}\label{eq:time-reversal-map}
T(z,\theta)\equiv (z^{-1},-\theta),
\end{equation}
then, by the chain rule,
\begin{equation}\label{eq:time-reversal}
\mathcal{L} (F\circ T)=- (\mathcal{L} F)\circ T.
\end{equation}
Let us abbreviate
\begin{equation}\label{eq:Omega}
\Omega(X) \mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=} (g^2\sin\Phi,0)+\lambda\,\widetilde\Omega(X)\quad\text{with}\quad
\widetilde\Omega(X)\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=}\partial f(X+(0,\theta)).
\end{equation}
We have then encoded the equations of motion into the PDE
\begin{equation}\label{xeq}
\mathcal{L}^2 X=\Omega(X).
\end{equation}
The action variables, or momenta, trivially follow from the knowledge of $X(z,\theta)$:
\begin{equation*}
(I(t),A(t))=(0,\omega)+Y(e^{\gamma t},\omega t),\quad Y\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=}\mathcal{L} X.
\end{equation*}
The solutions $X$ provide a parametrization of the deformed tori and their stable and unstable manifolds.
As hinted below \eqref{trial}, we find two kinds of solutions, $X^u(z,\theta)$ defined for $z\in [-z_0,z_0]$ and $X^s(z,\theta)$ defined for $z\in [-\infty,-z_0^{-1}]\cup [z_0^{-1},\infty]$. Here, $z_0>1$.
We will consider $\lambda$ small in a $g$-dependent fashion, taking
\begin{equation}
\epsilon\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=}\lambda g^{-2}
\label{eq:epsilon}
\end{equation}
small. Such a choice is needed for studying the limit $g\to \infty$, which corresponds to rapid forcing; see Remark~\ref{rem:timescale}. The domain we restrict ourselves to is
\begin{equation}\label{eq:D}
D\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=}\bigl\{(\epsilon,g)\in \mathbb{C}\times\mathbb{R}\;\big|\;\abs{\epsilon}<\epsilon_0,\; 0<g<g_0\bigr\},
\end{equation}
for some positive values of $\epsilon_0$ and $g_0$.
The following theorem is from \cite{Stenlund-whiskers}. It is a version of a classical result, and by no means new; earlier treatments include \cite{Melnikov63,Moser67, Graff,EliassonBiasymptotic,GallavottiTwistless,GentileQuasiflat,GentileExponent}.
\begin{result}[Tori and their whiskers]\label{thm:manifolds}
Let $f$ be real-analytic and even, \textit{i.e.},
\begin{equation*}
f(\phi,\psi)=f(-\phi,-\psi).
\end{equation*}
Also, suppose $\omega$ satisfies the Diophantine condition \eqref{Dio}, and fix $g_0>0$. Then there exist a positive number $\epsilon_0$ and a function $\gamma(\epsilon,g)$ on $D$, analytic in $\epsilon$ with $\abs{\gamma-g}<Cg\abs{\epsilon}$, such that equation~\eqref{xeq} has a solution $X^u$ which is analytic in $\epsilon$ as well as in $(z,\theta)$ in a neighbourhood of $[-1,1]\times{\mathbb{T}^d}$ and which satisfies ($X^0\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=}(\Phi^0,0)$)
\begin{equation}
X^u(1,0)=(\pi,0),\quad X^u(z,\theta)=X^0(z)+\order{\epsilon}.
\label{norm}
\end{equation}
Corresponding to the same $\gamma$, there exists a solution $X^s(z,\theta)=X^0(z)+\order{\epsilon}$ which is an analytic function of $(z^{-1},-\theta)$ in a neighbourhood of $[-1,1]\times{\mathbb{T}^d}$. The maps
\begin{equation}\label{eq:param}
W^{s,u}(z,\theta)=(X^{s,u},Y^{s,u})(z,\theta)+((0,\theta),(0,\omega)),\quad Y^{s,u}\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=}\mathcal{L} X^{s,u},
\end{equation}
provide analytic parametrizations of the stable and unstable manifolds $\mathcal{W}^{s,u}_\lambda$ of the torus $\mathcal{T}_\lambda$.
\end{result}
\begin{remark}\label{rem:manifolds}
The number $\epsilon_0$ above depends on the Diophantine exponent $\nu$ and on $f$. The perturbation $(\phi,\psi)\mapsto f(\phi,\psi)$ is analytic on the compact set $\mathbb{S}^1\times{\mathbb{T}^d}$. By Abel's Lemma (multivariate power series converge on polydisks), it extends to an analytic map on a ``strip'' $\abs{{\operatorname{\Im\mathfrak{m}}\,}\phi},\abs{{\operatorname{\Im\mathfrak{m}}\,}\psi}\leq \eta$ ($\eta>0$) around $\mathbb{S}^1\times{\mathbb{T}^d}$. By Theorem~\ref{thm:manifolds}, there exists some $0<\sigma<\eta$ such that each $\theta\mapsto X^{s,u}(\,\cdot\,,\theta)$ is analytic on $\abs{{\operatorname{\Im\mathfrak{m}}\,}\theta}\leq \sigma$.
\end{remark}
An important part of Theorem~\ref{thm:manifolds} is that the domains of $X^u$ and $X^s$ overlap. Namely, if $X$ solves equation~\eqref{xeq}, then so does $(2\pi,0)-X\circ T$. This is due to \eqref{eq:time-reversal} and the parity of $f$. Consequently, by time-reversal, the stable and unstable manifolds are related through
\begin{equation}\label{Xsym}
X^s=(2\pi,0)-X^u\circ T.
\end{equation}
In particular, as $T(1,0)=(1,0)$,
$
X^s(1,0)=X^u(1,0).
$
The actions $Y^{s,u}=\mathcal{L} X^{s,u}$ satisfy
$
Y^s=Y^u\circ T,
$
yielding
$
Y^s(1,0)=Y^u(1,0).
$
In other words, a \emph{homoclinic intersection} of the stable and the unstable manifolds $\mathcal{W}^{s,u}_\lambda$ occurs at $(z,\theta)=(1,0)$, as their parametrizations \eqref{eq:param} coincide at this \emph{homoclinic point}. Since the manifolds $\mathcal{W}^{s,u}_\lambda$ are invariant, there in fact exists a \emph{homoclinic trajectory} on which the parametrizations agree:
\begin{equation}\label{eq:homoclinic-trajectory}
W^s(e^{\gamma t},\omega t)\equiv W^u(e^{\gamma t},\omega t).
\end{equation}
Coming to the second result of \cite{Stenlund-whiskers}, let us expand
$
X^u=\sum_{\ell=0}^\infty \epsilon^\ell X^{u,\ell}.
$
It turns out that the common analyticity domain of each $X^{u,\ell}$ in the $z$-variable is in fact much larger than the (small) neighbourhood of $[-1,1]$---the corresponding domain of $X^u$ according to Theorem~\ref{thm:manifolds}. Namely, it includes the wedgelike region
\begin{equation}\label{eq:treedomain}
\mathbb{U}_{\tau,\vartheta}\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=} \bigl\{\abs{z}\leq \tau\bigr\}\,\bigcup\,\bigl\{\arg{z}\in [-\vartheta,\vartheta]\cup [\pi-\vartheta,\pi+\vartheta]\bigr\}\subset\mathbb{C}
\end{equation}
(with some positive $\tau$ and $\vartheta$). We repeat parts of the argument in Section~\ref{sec:continuation} for convenience.
\begin{result}[Analytic continuation]\label{thm:extension}
Each order $X^{u,\ell}$ of the solution extends analytically to a common region $\mathbb{U}_{\tau,\vartheta}\times\{\abs{{\operatorname{\Im\mathfrak{m}}\,}{\theta}}\leq\sigma\}$. Moreover, if $\psi\mapsto f(\,\cdot\,,\psi)$ is a trigonometric polynomial of degree $N$, \textit{i.e.}, $N$ is the smallest integer such that $\hat f(\,\cdot\,,q)=0$ whenever $\abs{q}>N$, then $\theta\mapsto X^{u,\ell}(\,\cdot\,,\theta)$ is a trigonometric polynomial of degree $\ell N$, at most.
\end{result}
\begin{remark}\label{rem:extension}
With $\eta$ and $\sigma$ as in Remark~\ref{rem:manifolds}, the numbers $\tau$ and $\vartheta$ are specified by the following observation: $\Phi^0(z)=4\arctan z$ implies that $\abs{{\operatorname{\Im\mathfrak{m}}\,}\Phi^0(z)}\leq\eta$ in $\mathbb{U}_{\tau,\vartheta}$ with $\tau$ and $\vartheta$ sufficiently small. By Remark~\ref{rem:manifolds}, $(z,\theta)\mapsto f(\Phi^0(z),\theta)$ is analytic on $\mathbb{U}_{\tau,\vartheta}\times\{\abs{{\operatorname{\Im\mathfrak{m}}\,}{\theta}}\leq\sigma\}$, which is the basis of the proof.
\end{remark}
In spite of Theorem~\ref{thm:extension}, (a straightforward upper bound on) $X^{u,\ell}$ grows without a limit as $\abs{{\operatorname{\Re\mathfrak{e}}\,}{z}}\to \infty$, such that there is no reason whatsoever to expect absolute convergence of the series $\sum_{\ell=0}^\infty \epsilon^\ell X^{u,\ell}$ in an unbounded $z$-domain with a fixed $\epsilon$. In fact, it is known that the behavior of the unstable manifold gets extremely complicated for large values of $z$ even with innocent looking Hamiltonian systems. Still, it seems to us that the possibility of a uniform analytic extension of the coefficients $X^{u,\ell}$ has not been appreciated in the literature.
Due to \eqref{Xsym}, an analog of Theorem~\ref{thm:extension} is seen to hold for $X^s$, with $z$ replaced by $z^{-1}$.
Theorem~\ref{thm:extension} allows one (at each order in $\epsilon$) to track trajectories $t\mapsto W^{s,u}(e^{\gamma t},\theta+\omega t)$ on the invariant manifolds $\mathcal{W}_\lambda^{s,u}$ for arbitrarily long times in a uniform complex neighbourhood $\abs{{\operatorname{\Im\mathfrak{m}}\,}{t}}\leq g^{-1}\vartheta$ of the real line, for arbitrary $\theta\in{\mathbb{T}^d}$. The motivation for doing this stems from studying the splitting of the manifolds $\mathcal{W}_\lambda^{s,u}$ in the vicinity of the homoclinic trajectory \eqref{eq:homoclinic-trajectory}. The general ideology that, being able to extend ``splitting related functions" to a large complex domain yields good estimates, is due to Lazutkin \cite{Lazutkin}, as is emphasized in \cite{LochakMemoirs}.
In order to study the intersection more closely, we express the actions as functions of the original angle variables $(\phi,\psi)=X^{s,u}(z,\theta)+(0,\theta)$ appearing in the Hamiltonian \eqref{eq:H}. To this end, let $F^{s,u}:(z,\theta)\mapsto(\phi,\psi)$ be the above coordinate transformations, and write $Y^{s,u}= \bar Y^{s,u}\circ F^{s,u}$. We set
$
\partial\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=}(\partial_\phi,\partial_\psi)$ and $D\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=}(\partial_z,\partial_\theta).
$
By the chain rule,
\begin{equation}\label{eq:chain}
D Y^{s,u}=\left(\partial\bar Y^{s,u}\circ F^{s,u}\right)D F^{s,u}.
\end{equation}
The invertibility of the matrix $D F^{s,u}$ is a consequence of Theorem~\ref{thm:manifolds}.
At the homoclinic point $(z,\theta)=(1,0)$, equation~\eqref{eq:homoclinic-trajectory} implies that $F^s(e^{\gamma t},\omega t) \equiv F^u(e^{\gamma t},\omega t)$ and $F^{s,u}(1,0)=(\pi,0)$.
Casting in the obvious manner $\bar Y^{u,s}=(\bar Y^{u,s}_\Phi,\bar Y^{u,s}_\Psi)$, we define the \emph{splitting vector}:
\begin{equation*}
\Delta(\phi,\psi)\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=} (A^u-A^s)^T=\bigl(\bar Y^{u}_\Psi-\bar Y^{s}_\Psi\bigr)^T(\phi,\psi).
\end{equation*}
Transposition indicates that we consider $\Delta$ a column vector. We define the \emph{splitting matrices}:
\begin{align}\label{eq:splitmatdef1}
\Upsilon(t) & \mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=}\partial_\theta\!\left(Y^u_\Psi-Y^s_\Psi\right)^T(e^{\gamma t},\omega t), \\\widetilde\Upsilon(t) & \mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=} \partial_\psi\!\left(\bar Y^{u}_\Psi-\bar Y^{s}_\Psi\right)^T(F^{u,s}(e^{\gamma t},\omega t)).
\end{align}
Notice that if $(\phi,\psi)(t)$ stands for the moment for the angles on the homoclinic trajectory at time $t$, with the particular initial condition $(\phi,\psi)(0)=(\pi,0)$, then
$
\widetilde{\Upsilon}=\partial_\psi\Delta(\phi,\psi).
$
It is nontrivial but straightforward to establish that $\widetilde\Upsilon$ and $\Upsilon$ differ by a close-to-identity \emph{factor}, if $\epsilon$ is taken to be small proportionally to $g$:
\begin{result}\label{thm:splitmatrel}
Fix $t\in \mathbb{R}$ and define $\tilde\epsilon\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=} g^{-1} e^{g{\abs{t}}}\epsilon$. Then, as $g\to 0$,
\begin{equation*}\label{eq:relation}
\Upsilon(t)=\widetilde\Upsilon(t)\,\bigl(\one+\order{\tilde\epsilon}\bigr),
\end{equation*}
if $\tilde\epsilon$ is sufficiently small (independently of $g$ and $t$).
\end{result}
Theorem~\ref{thm:splitmatrel} is our first result. Its proof in Appendix~\ref{app:computations} is based on energy conservation and the fact that the actions are given by scalar potentials. We can now state the main theorem:
\begin{result}[Homoclinic splitting]\label{thm:splitting}
Let us assume that $\psi\mapsto f(\,\cdot\,,\psi)$ is a trigonometric polynomial in addition to the assumptions of Theorem~\ref{thm:manifolds}. Then for each $t\in\mathbb{R}$ there exist positive constants $C$ and $c$, such that the exponentially small upper bound
\begin{equation*}
\abs{\Upsilon_{ij}(t)}\leq C\abs{\epsilon} e^{-c g^{-1/(\nu+1)}},
\end{equation*}
where $\nu$ is the Diophantine exponent, holds, provided $\epsilon g^{-4}$ is sufficiently small.
\end{result}
Theorem~\ref{thm:splitting} is derived from Proposition~\ref{prop:asymptotic}---the centerpiece of this work---stated in Subsection~\ref{subsec:asymptotic}. Technicalities aside, we believe that Subsection~\ref{subsec:asymptotic} can be understood without any further introduction, and we urge the reader to go through it before moving on to Section~\ref{sec:continuation}.
Why just study a $d\times d$ submatrix of the full $(d+1)\times(d+1)$ Jacobi matrix of $(I^u-I^s,A^u-A^s)$ when measuring \emph{transversality} of the homoclinic intersection $\mathcal{W}^s_\lambda\cap\mathcal{W}^u_\lambda$? Because the latter is singular. As a matter of fact, \eqref{eq:identity} simply means that the splitting vanishes in the direction of the homoclinic trajectory. The author is grateful to Dr Mischa Rudnev for pointing this out. For further motivation, see item R1 of Appendix R and Section~8 in \cite{GallavottiSplit}.
\subsection*{Acknowledgements}
I am indebted to Antti Kupiainen for all his advice. I express my gratitude to Guido Gentile, Kari Astala, and Jean Bricmont for their comments. I thank Giovanni Gallavotti and Emiliano De Simone for discussions at Rutgers University and University of Helsinki, respectively. Mischa Rudnev, Pierre Lochak, and Michela Procesi were kind enough to explain me some of their works on the subject. I am grateful to Joel Lebowitz and Rutgers University for hospitality during the final stages of writing this work. This work was supported by the Finnish Cultural Foundation and NSF Grant DMR-01-279-26.
\section{Size of the Homoclinic Splitting}
The Lyapunov exponent $\gamma=\gamma(\epsilon,g)$ is an analytic function of $\epsilon$ in a neighbourhood of the origin ($\abs{\epsilon}<\epsilon_0$). In fact,
$
\gamma=g(1+\order{\epsilon})
$
for small $\epsilon$. Further, since $\mathcal{L}$ depends on $\gamma$, we have
$
\mathcal{L}=\mathcal{L}_0+\order{\epsilon}
$
with
$
\mathcal{L}_0= \omega\cdot\partial_\theta+gz\partial_z
$
. In particular, as $X^0(z,\theta)\equiv(4\arctan z,0)^T$,
the expansion of the splitting matrix $\Upsilon$---see \eqref{eq:splitmatdef1}---begins linearly with respect to $\epsilon$:
\begin{equation*}
\Upsilon=\sum_{\ell=1}^\infty\epsilon^\ell\Upsilon^\ell.
\end{equation*}
It is customary to call the first order coefficient, $\Upsilon^1$, the \emph{Melnikov term} or the \emph{Melnikov matrix}.
\subsection{The Melnikov term}
Let us study $\Upsilon^1$. The symmetry \eqref{eq:arctan-sym} comes in very handy, since it implies, together with the even parity of $f$, that
$
f(\Phi^0(z^{-1}),-\theta)\equiv f(\Phi^0(z),\theta).
$
Setting
\begin{equation}\label{eq:melnikov}
F(s,\theta)\equiv\int_{-\infty}^\infty \partial_\psi^2 f(\Phi^0(e^{g t+s}),\theta+\omega t)-\partial_\psi^2 f(0,\theta+\omega t)\,dt
\end{equation}
then yields, after some work,
\begin{equation*}
\Upsilon^1(t)= g^2 F(0,0)
\end{equation*}
for all $t$. Here the second term in the integrand removes the quasiperiodic limit of the first one, making the integral absolutely convergent. In fact, the integrand tends to zero at an exponential rate as $\abs{t}\to\infty$.
We now give a result, due to Lazutkin in it's original form, \cite{Lazutkin}, and extended by \cite{DelshamsGelfreichJorbaSeara}, \cite{Sauzin}, and \cite{LochakMemoirs} to the quasiperiodic setting. See Appendix~\ref{app:computations} for the proof.
\begin{lemma}[Lazutkin]\label{lem:Lazutkin}
Suppose a function $F$ is analytic on $S\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=}[-i\vartheta,i\vartheta]\times\{\abs{{\operatorname{\Im\mathfrak{m}}\,} \theta}<\eta\}$, continuous on the closure $\bar S$, and satisfies the identity
\begin{equation}\label{eq:Lazutkin-id}
F(s,\theta)\equiv F(0,\theta-g^{-1}\omega s).
\end{equation}
Then $F$ extends analytically to $\{\abs{{\operatorname{\Im\mathfrak{m}}\,} s}\leq\vartheta\}\times \{\abs{{\operatorname{\Im\mathfrak{m}}\,} \theta}<\eta\}$, with \eqref{eq:Lazutkin-id} still holding. On $\mathbb{R}\times{\mathbb{T}^d}$ it obeys a bound of the form
\begin{equation*}
\abs{F(s,\theta)-\widetilde F}\leq C B(g) e^{-c\,g^{-1/(\nu+1)}},\quad B(g)\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=} \sup_{(s,\theta)\in\bar S}{\abs{F(s,\theta)}},
\end{equation*}
where $\widetilde F$ stands for the $\theta$-average $\average{F(s,\,\cdot\,)}$ and is independent of $s$.
\end{lemma}
\begin{remark}
Notice that, for small values of $g$, the analyticity domain of $F$ in Lemma~\ref{lem:Lazutkin} is much larger than what the right-hand side of \eqref{eq:Lazutkin-id} suggests.
\end{remark}
The lemma applies to the function defined in \eqref{eq:melnikov}, due to Remarks~\ref{rem:manifolds} and \ref{rem:extension} regarding the analyticity of $f$. Because the $\psi$-derivatives in the integrand are interchangeable with total $\theta$-derivatives, $\widetilde F=0$, and we get
\begin{proposition}\label{prop:melnikov}
There exist positive constants $c$ and $C$ such that the Melnikov term satisfies
\begin{equation*}
\abs{\Upsilon^1}\leq C g^2 e^{-c\,g^{-1/(\nu+1)}}.
\end{equation*}
\end{proposition}
\begin{remark}
Identity~\eqref{eq:Lazutkin-id} in Lazutkin's lemma is equivalent to
\begin{equation}\label{eq:LazutkinPDE}
(\partial_s+g^{-1}\omega\cdot\partial_\theta)F=0,
\end{equation}
or, calling $\bar F(z,\theta)\equiv F(g^{-1}\ln z,\theta)$, to $\mathcal{L}_0\bar F=0$. There exists a whole industry trying to push through the argument given above replacing $\Upsilon^1$ with a ``better measure'' of the splitting, such as $\Upsilon$, by searching for a coordinate system in which the ``measure'' satisfies \eqref{eq:LazutkinPDE}. The state of this quest is best described in \cite{LochakMemoirs}.
\end{remark}
Neither Lemma~\ref{lem:Lazutkin} nor Proposition~\ref{prop:melnikov} is optimal. There is a refinement in \cite{Sauzin} which, however, is not optimal. In many special cases one can derive a much stronger bound.
\subsection{Regularized integrals}
This subsection borrows heavily from Gallavotti,~\textit{et al.}; see for instance \cite{ChierchiaGallavotti} and \cite{GallavottiSplit}. We adhere to the notation
\begin{equation*}
X(t;z,\theta)\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=} X(ze^{\gamma t},\theta+\omega t),
\end{equation*}
which makes the dependence on the Lyapunov exponent $\gamma$ \emph{implicit} (Recall that $\gamma$ is a function of $\epsilon$), such that the perturbation expansion reads compactly
$
X(t;z,\theta)=\sum_{\ell=0}^\infty \epsilon^\ell X^\ell(t;z,\theta).
$
With little trickery, the equations of motion retain their appearance: \emph{defining}
$
\Omega(X)(t;z,\theta) \equiv [(g^2\sin\Phi,0)+\lambda\,\widetilde\Omega(X)](t;z,\theta)
$
together with
$
\widetilde\Omega(X)(t;z,\theta) \equiv\partial f(X(t;z,\theta)+(0,\theta+\omega t)),
$
and implementing \eqref{Lid}, we obtain the analogue of \eqref{xeq}:
\begin{equation}\label{eq:eqm}
\partial_t Y(t;z,\theta)\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=}\partial_t^2 X(t;z,\theta)=\Omega(X)(t;z,\theta).
\end{equation}
There is a need to consider functions $h(t;z,\theta)$ that can be expanded as finite sums
\begin{equation}\label{eq:t-expansion}
\sum_{p=0}^P t^p h^p(z e^{\gamma t},\theta+\omega t),
\end{equation}
where $h^p$ is analytic on $\{0<\abs{z}<\tau\}\times{\mathbb{T}^d}$---a punctured neighbourhood of the origin times the $d$-torus---with at worst a \emph{finite} pole at $z=0$.
Let us set $T_N h(t;z,\theta)\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=} \sum_{\abs{q}\leq N}\hat h(t;z,q)e^{iq\cdot\theta}$ and define a \emph{regularized integral}
\begin{equation*}
\Xint- h\,(t;z,\theta)\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=} \lim_{N\to\infty}\underset{R=0}{\textrm{res}}\,\frac{1}{R}\int_{-\infty}^t e^{-R\abs{\tau}} T_N h(\tau;z,\theta)\,d\tau,
\end{equation*}
the residue at $R=0$ meaning that of the analytic extension from the complex half-plane with ${\operatorname{\Re\mathfrak{e}}\,}{R}$ sufficiently large. This is well-known from the theory of Laplace transforms. The truncation, $T_N$, is needed to insure that the origin of the $R$-plane is not an accumulation point for poles, since factors $(R+i\omega\cdot q)^{-1}$ appear from the integral.
By the following proposition, it is natural to employ the compelling notation
\begin{equation*}
\Xint-_{-\infty}^t h(\tau;z,\theta)\,d\tau\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=} \Xint- h\,(t;z,\theta),
\end{equation*}
and to view $z$ and $\theta$ as mere parameters:
\begin{proposition}\label{prop:regint}
The regularized integral is linear. Moreover,
\begin{enumerate}
\item $\Xint- h\,(t;z,\theta)=\Xint- h\,(t_0;z,\theta)+\int_{t_0}^t h(\tau;z,\theta)\,d\tau$ for any real $t_0$,
\item if $h$ and $h'$ are trigonometric polynomials (in $\theta$) and $h(t;z,\theta)=h'(t;z,\theta)$ for all $t$, then $\Xint- h\,(t;z,\theta)=\Xint- h'\,(t;z,\theta)$ for all $t$,
\item $T_N\,\Xint- h=\Xint- T_Nh$,
\item $
\partial_t\,\Xint- h\,(t;z,\theta)=\frac{d}{dt}\,\Xint-_{-\infty}^t h(\tau;z,\theta)\,d\tau=h(t;z,\theta),
$
\item $
\Xint-\!\!\Xint- h\,(t;z,\theta) \mathrel{=\mkern-4.2mu\raise.095ex\hbox{$:$}} \Xint-_{-\infty}^t\Xint-_{-\infty}^\tau h(s;z,\theta)\,ds\,d\tau=\Xint-_{-\infty}^t (t-\tau)\,h(\tau;z,\theta)\,d\tau.
$
\end{enumerate}
\end{proposition}
\begin{proof}
For a finite $t_0$, $\textrm{res}_{R=0} \frac{1}{R}\int_{t_0}^t e^{-R\abs{\tau}} u(\tau)\,d\tau$ equals the evaluation $\int_{t_0}^t e^{-R\abs{\tau}} u(\tau)\,d\tau\big|_{R=0}=\int_{t_0}^t u(\tau)\,d\tau$, since the integral is entire in $R$, from which (1) follows. If $h$ is a trigonometric polynomial, then $\Xint- h\,(t;z,\theta)=\underset{R=0}{\textrm{res}}\,\frac{1}{R}\int_{-\infty}^t e^{-R\abs{\tau}} h(\tau;z,\theta)\,d\tau$, yielding (2). Claim (3) is trivial and (4) follows from (1). In order to prove (5), one first checks that it holds for trigonometric polynomials and then deals with the general case.
\end{proof}
We also define
\begin{equation*}
\Xint-_t^\infty h(\tau;z,\theta)\,d\tau \mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=} \lim_{N\to\infty}\underset{R=0}{\textrm{res}}\,\frac{1}{R}\int_t^{\infty} e^{-R\abs{\tau}} T_N h(\tau;z,\theta)\,d\tau
\end{equation*}
when it makes sense,
and write
\begin{equation*}
\Xint-_{\pm\infty}^t=-\,\Xint-_t^{\pm\infty}\quad\text{and}\quad \Xint-_{-\infty}^t+\,\,\Xint-_t^\infty=\Xint-_{-\infty}^\infty.
\end{equation*}
The operator $\mathcal{D}^{-1}$ is defined by giving the expression of its Fourier kernel:
\begin{equation*}
\mathcal{D}^{-1} (p,q)=
\begin{cases}
(i\omega\cdot q)^{-1} & \text{if $p=q\in\mathbb{Z}^d\setminus\{0\}$}, \\
0 & \text{otherwise}.
\end{cases}
\end{equation*}
It is the formal inverse of $\mathcal{D}$ on functions $h(z,\theta)$ with vanishing $\theta$-average. Finally,
\begin{equation*}
\mathcal{I} h(z,\theta)\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=}\int_{-\infty}^0 h(ze^{\gamma\tau},\theta+\omega\tau)\,d\tau.
\end{equation*}
Applying \eqref{Lid} in the case $F=\mathcal{I} h$, the formal identity
$
\mathcal{L}\mathcal{I}=\one
$
follows. Recalling $\delta_1 h(z,\theta)\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=} h(z,\theta)-h_0(\theta)=h(z,\theta)-h(0,\theta)$, one also gets
$
\mathcal{I}\mathcal{L}\delta_1 =\delta_1
$
by a direct computation.
\begin{proposition}\label{prop:regint&I}
Let $h$ be analytic on $\{\abs{z}<\tau\}\times{\mathbb{T}^d}$, and denote $h(t;z,\theta)\equiv h(ze^{\gamma t},\theta+\omega t)$.
\begin{equation*}
\Xint-_{-\infty}^t h(\tau;z,\theta)\,d\tau=\average{h_0}t+\mathcal{D}^{-1} h_0(\theta+\omega t)+\mathcal{I}\delta_1 h(ze^{\gamma t},\theta+\omega t)
\end{equation*}
and
\begin{equation*}
\Xint-_{-\infty}^t \partial_\tau h(\tau;z,\theta)\,d\tau=h(t;z,\theta)-\average{h_0}.
\end{equation*}
\end{proposition}
\begin{proof}
Decompose $h=\average{h_0}+[h_0-\average{h_0}]+\delta_1 h$ and use the linearity of $\Xint-$. For the second identity, notice $\partial_\tau h(\tau;z,\theta)=\mathcal{D} h_0(\theta+\omega \tau)+\mathcal{L}\delta_1 h(ze^{\gamma \tau},\theta+\omega \tau)$, and recall the identities $\mathcal{D}^{-1}\mathcal{D} h_0=h_0-\average{h_0}$ and $\mathcal{I}\mathcal{L}\delta_1 h=\delta_1 h$.
\end{proof}
\begin{remark}
Observe that, unless the integrals converge in the traditional sense, $\Xint-_{-\infty}^t$ does generically \emph{not} tend to $\Xint-_{-\infty}^\infty$ or 0 as $t$ tends to $\infty$ or $-\infty$, respectively.
\end{remark}
A detailed proof of the following elementary result can be found in \cite{Thesis}.
\begin{proposition}\label{prop:Kinverse}
Let $\mathcal{S}_0$ and $\mathcal{S}_1$ be the spaces of analytic functions $h$ on $\{\abs{z}<\tau\}\times{\mathbb{T}^d}$ satisfying $\average{h_0}=0$ and $\average{h_1}=0$, respectively. Denote $h(t;z,\theta)\equiv h(ze^{\gamma t},\theta+\omega t)$. The operators $L_t=\partial_t^2-\gamma^2\cos\Phi^0(z e^{\gamma t})$, $L=\mathcal{L}^2-\gamma^2\cos\Phi^0$, and $\mathcal{L}=\mathcal{D}+\gamma z\partial_z$ satisfy
\begin{equation*}
\partial_t h(t;z,\theta)\equiv \mathcal{L} h(ze^{\gamma t},\theta+\omega t) \quad\text{and}\quad L_th(t;z,\theta)\equiv Lh(ze^{\gamma t},\theta+\omega t).
\end{equation*}
The operator $\Xint-$ maps $\mathcal{S}_0$ into itself and, in fact,
$\Xint-=\partial_t^{-1}$ on $\mathcal{S}_0$.
Similarly, the operator defined by $\Xint-_{-\infty}^t K_\Phi(t,\tau;z)\,h(\tau;z,\theta)\,d\tau$ maps $\mathcal{S}_1$ into itself and is the inverse of $L_t$ on $\mathcal{S}_1$.
In particular, the operator $\mathcal{K}=\bigl(\begin{smallmatrix} L & 0 \\ 0 & \mathcal{L}^2 \end{smallmatrix}\bigr)=\bigl(\begin{smallmatrix} L_t & 0 \\ 0 & \partial_t^2 \end{smallmatrix}\bigr)$ is invertible on the space $\mathcal{S}_1\times\mathcal{S}_0$, where
\begin{equation*}
\mathcal{K}^{-1} h(t;z,\theta)\equiv \Xint-_{-\infty}^t K(t,\tau;z)h(\tau;z,\theta)\,d\tau.
\end{equation*}
\end{proposition}
Above, the kernel $K=\bigl(\begin{smallmatrix} K_\Phi & 0 \\ 0 & K_\Psi \end{smallmatrix}\bigr)$ splits into the useful sum
\begin{equation}\label{eq:Ksplit}
K(t,\tau;z)=\sum_{i=0}^1 \sum_{j=1}^2 (t-\tau)^i K_{ij}(t;z)\bar K_{ij}(\tau;z),
\end{equation}
setting $P(t;z) \mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=} (z^2 e^{2\gamma t}+1)^{-1} ze^{\gamma t}$, $Q(t;z) \mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=} z^{-1} e^{-\gamma t} (z^2 e^{2\gamma t}-1)$, and
\begin{xalignat*}{3}
K_{01} &=\frac{1}{2\gamma}
\begin{pmatrix}
Q & 0 \\
0 & 0
\end{pmatrix},
&
\bar K_{01} &= \bar K_{11} =
\begin{pmatrix}
P & 0 \\
0 & 0
\end{pmatrix},
&
K_{02} &=-\frac{1}{2\gamma}
\begin{pmatrix}
P & 0 \\
0 & 0
\end{pmatrix},
\\
\bar K_{02} &=
\begin{pmatrix}
Q & 0 \\
0 & 0
\end{pmatrix},
&
K_{11} &=2
\begin{pmatrix}
P & 0 \\
0 & 0
\end{pmatrix},
&
K_{12}&=\bar K_{12}=
\begin{pmatrix}
0 & 0 \\
0 & \one
\end{pmatrix}.
\end{xalignat*}
\subsection{Asymptotic expansion for the splitting matrix} \label{subsec:asymptotic}
Starting from \eqref{eq:eqm} and using Proposition~\ref{prop:Kinverse} with $\average{Y_0^u}=\average{\mathcal{D} X_0^u}=0$, we have
\begin{equation*}
Y^u(t;z,\theta)=\Xint-_{-\infty}^t\Omega(X^u)(\tau;z,\theta) \,d\tau
\end{equation*}
The identity $X^s(t;z,\theta)\equiv (2\pi,0)-X^u(-t;z^{-1},-\theta)$ yields
$
Y^s(t;z,\theta)\equiv Y^u(-t;z^{-1},-\theta),
$
such that defining the $d$ component column vector
$
\Delta(t;z,\theta)\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=} (Y_\Psi^u-Y_\Psi^s)(t;z,\theta)
$
and the functions
$
f^{s,u}(t;z,\theta)\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=} f(X^{s,u}(t;z,\theta)+(0,\theta+\omega t))
$
thus results in an expression for the splitting matrix $\Upsilon$ of \eqref{eq:splitmatdef1} in terms of regularized integrals: we get
\begin{equation}\label{eq:splitting-homoclinic}
\Upsilon(t)=\partial_\theta\Delta(t;1,0)=\partial_\theta\Delta(0;e^{\gamma t},\omega t)
\end{equation}
because
\begin{equation*}
\partial_\theta\Delta(t;z,\theta)=\lambda\,\Xint-_{-\infty}^t \partial_\theta f^u_\psi(\tau;z,\theta)\,d\tau +\lambda\,\Xint-_{t}^\infty \partial_\theta f^s_\psi(\tau;z,\theta)\,d\tau,
\end{equation*}
where the subindices attached to $f^{s,u}$ stand for partial derivatives (our convention is $\varphi=(\phi,\psi)$ and $\partial=\partial_{\varphi}=(\partial_\phi,\partial_\psi)$).
\begin{remark}
Extracting here the first order in $\epsilon=g^{-2}\lambda$ casts the Melnikov matrix in a compact form in terms of the regularized integrals:
\begin{equation*}
\Upsilon^1= g^2\,\Xint-_{-\infty}^\infty \partial_\psi^2 f(\Phi^0(e^{g\tau}),\omega \tau)\,d\tau.
\end{equation*}
The reader is invited to compare this with \eqref{eq:melnikov} and the equation following it. We point out that a similar procedure, using Lemma~\ref{lem:Lazutkin}, that resulted in the exponentially small bound on $\Upsilon^1$ can be applied here, despite the unusual integrals.
\end{remark}
Suppose that, even with the superscript $u$, the integral
\begin{equation}\label{eq:problem-integral}
\Xint-_{t}^\infty \partial_\theta f^u_\psi(\tau;z,\theta)\,d\tau
\end{equation}
exists. If the integrand is a trigonometric polynomial in $\theta$, then Proposition~\ref{prop:regint} implies that the value of the integral only depends on the integrand's restriction to $\{(t;z,\theta)\,|\,t\in\mathbb{R}\}$, as the notation suggests \footnote{This is the first of the two places where we need to restrict ourselves to trigonometric polynomials\ldots}. Now, \emph{fixing} $(z,\theta)=(1,0)$ and dropping it from the notation,
\begin{equation}\label{eq:splitting}
\Upsilon(t)=\lambda\,\Xint-_{-\infty}^\infty \partial_\theta f^u_\psi(\tau)\,d\tau +\lambda\,\Xint-_{t}^\infty \bigl[f^s_{\psi\varphi}\,\partial_\theta(X^s-X^u)\bigr](\tau)\,d\tau,
\end{equation}
since $X^u(\tau;1,0)\equiv X^s(\tau;1,0)$.
Let us make the assumption that $\psi\mapsto f(\,\cdot\,,\psi)$ is a trigonometric polynomial, such that, by Theorem~\ref{thm:extension}, each order of $X$ in $\epsilon$ is a trigonometric polynomial in $\theta$. Even so, equation~\eqref{eq:splitting} is \emph{formal} in the sense that the integrands are defined for $\tau\gg 0$ only up to an arbitrarily high order in $\epsilon$. However, it is \emph{asymptotic}, which is to say that at each order the identity is \emph{exact}. Put differently, \eqref{eq:splitting} is a collection of exact identities, one for each order in $\epsilon$, written in closed form. Moreover, at each order, the integrands are \emph{analytic functions} by virtue of the extension result in Theorem~\ref{thm:extension}. Of course, we need to check that \emph{each order of} \eqref{eq:problem-integral}, \textit{i.e.}, the integral $\Xint-_{t}^\infty \bigl(\partial_\theta f^u_\psi\bigr)^\ell(\tau;z,\theta)\,d\tau$ with arbitrary $\ell\in\mathbb{N}$, exists \footnote{\ldots and this is the second.}.
The point of the formula above is twofold. First, the integral $\Xint-_{-\infty}^\infty \partial_\theta f^u_\psi(\tau)\,d\tau$ (at each order in $\epsilon$) turns out to be exponentially small in the limit $g\to 0$. Second, we can actually construct a (formal) recursion relation for the function $\partial_\theta(X^s-X^u)(\tau)$, taking us to ever-increasing orders in $\epsilon$, as follows. Differentiating both sides of \eqref{eq:eqm} with respect to $\theta$ one obtains
\begin{equation*}
\begin{pmatrix}
L_t & 0 \\
0 & \partial_t^2
\end{pmatrix}
\partial_\theta X^{s,u}=
M^{s,u}
\partial_\theta X^{s,u}+\lambda f^{s,u}_{\varphi\psi}
\end{equation*}
with $L_t=\partial_t^2-\gamma^2\cos\Phi^0(z e^{\gamma t})$, as in Proposition~\ref{prop:Kinverse}, and
\begin{equation*}
M^{s,u}(t;z,\theta)\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=}
\begin{pmatrix}
g^2\cos\Phi^{s,u}(t;z,\theta)-\gamma^2\cos\Phi^0(z e^{\gamma t}) & 0 \\
0 & 0
\end{pmatrix}
+\lambda f^{s,u}_{\varphi \varphi}(t;z,\theta)
.
\end{equation*}
Owing to the $\theta$-derivative acting on $X^{s,u}$ and Proposition~\ref{prop:Kinverse}, we get
\begin{equation*}
\partial_\theta X^{s,u}(t;z,\theta)=
\Xint-_{\pm\infty}^t K(t,t';z)\,\bigl[M^{s,u}
\partial_\theta X^{s,u}+\lambda f^{s,u}_{\varphi\psi}\bigr](t';z,\theta)\,d t',
\end{equation*}
upon choosing $\infty$ in conjunction with $X^s$ and $-\infty$ with $X^u$. The most convenient way to see this is to infer the identity for $\partial_\theta X^u$ and employing $K(-t,-\tau;z^{-1})\equiv -K(t,\tau;z)$ and $X^s(t;z,\theta)\equiv (2\pi,0)-X^u(-t;z^{-1},-\theta)$.
At $(z,\theta)=(1,0)$, $M^s(t)\equiv M^u(t)$ and $f^{s}_{\varphi\psi}(t)\equiv f^{u}_{\varphi\psi}(t)$, such that (2) of Proposition~\ref{prop:regint} validates
\begin{equation}\label{eq:recursion}
\begin{split}
\partial_\theta(X^s-X^u)(\tau) = \,\, &\Xint-_{\infty}^{-\infty} K(\tau,\tau')\,\bigl[M^u\partial_\theta X^u+\lambda f^{u}_{\varphi\psi} \bigr](\tau')\,d\tau'\,+ \\
& +\Xint-_{\infty}^\tau K(\tau,\tau')\,\bigl[M^s\partial_\theta(X^s-X^u)\bigr](\tau')\,d\tau'.
\end{split}
\end{equation}
By construction, the quantities appearing in square brackets above are $\theta$-gradients:
\begin{equation*}
M^{s,u}\partial_\theta X^{s,u}+\lambda f^{s,u}_{\varphi\psi}=
\partial_\theta\mathfrak{A}^{s,u}
\end{equation*}
with, for instance \footnote{We could make $\mathfrak{A}^{s,u}$ of order $\epsilon$ by deleting its lowest order, because the latter does not depend on $\theta$.},
\begin{equation*}
\mathfrak{A}^{s,u} \mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=}
\begin{pmatrix}
L_t & 0 \\
0 & \partial_t^2
\end{pmatrix}\!
X^{s,u}
=
\Omega(X^{s,u})-
\begin{pmatrix}
\gamma^2\cos\Phi^0(z e^{\gamma t}) & 0 \\
0 & 0
\end{pmatrix}\!
X^{s,u},
\end{equation*}
regardless of the value of $(z,\theta)$. In addition, but only at $(z,\theta)=(1,0)$,
\begin{equation*}
\bigl[M^s\partial_\theta(X^s-X^u)\bigr](t)
=\bigl[\partial_\theta(\mathfrak{A}^s-\mathfrak{A}^u)\bigr](t).
\end{equation*}
Notice that \eqref{eq:recursion} is a fixed point equation of the form $\zeta=\mathcal{E}+BM^s\zeta$ solved formally by $\zeta(t)=\partial_\theta(X^s-X^u)(t)$. This is highly interesting from the point of view of asymptotic analysis, since $M^s=\order{\epsilon}$ multiplies $\zeta$ on the right-hand side. Indeed, truncating the Taylor series in $\epsilon$, \textit{e.g.},
$
\zeta^{\leq k}=\sum_{i=0}^k\epsilon^i\zeta^i,
$
the identities $\zeta^{k}=(\mathcal{E}+BM^s\zeta)^{k}=(\mathcal{E}+BM^s\zeta^{\leq k-1})^{k}$ become exact. Hence, we can recursively construct any order of $\partial_\theta(X^s-X^u)(t)$ from its lower orders, without the need of diverting from $\{(t;z,\theta)=(t;1,0)\,|\,t\in\mathbb{R}\}$ in order to compute the $\theta$-derivative. Cumulatively, we have $\zeta^{k}=\bigl[\sum_{j=0}^{k-1}(BM^s)^j \mathcal{E}\bigr]^{k}$, where $j$ is a power.
\begin{proposition}\label{prop:Neumann-asympt}
In brief, the (presumably) divergent Neumann series $\sum_{j=0}^{\infty}(BM^s)^j \mathcal{E}$ is an asymptotic expansion of $\partial_\theta(X^s-X^u)(t)$ in the sense that
\begin{enumerate}
\item At each order in $\epsilon$ it terminates after finitely many well-defined terms, and
\item $\Bigl[\partial_\theta(X^s-X^u)-\bigl[\sum_{j=0}^{\infty}(BM^s)^j \mathcal{E}\bigr]^{\leq k}\Bigr](t)=\order{\epsilon^{k+1}}$ for each $k\in\mathbb{N}$.
\end{enumerate}
\end{proposition}
Once inserted into \eqref{eq:splitting}, the latter series provides us with an \emph{asymptotic expansion for the splitting matrix $\Upsilon$ in \eqref{eq:splitting-homoclinic}}:
\begin{corollary}\label{cor:almost-there}
In the asymptotic sense of Proposition~\ref{prop:Neumann-asympt},
\begin{equation*}
\Upsilon(t)=\lambda\,\Xint-_{-\infty}^\infty \partial_\theta f^u_\psi(\tau)\,d\tau +\lambda\,\Xint-_{t}^\infty \Bigl[f^s_{\psi\varphi}\sum_{j=0}^{\infty}(BM^s)^j \mathcal{E}\Bigr](\tau)\,d\tau.
\end{equation*}
\end{corollary}
The operator $B$ above has the expression
\begin{equation}\label{eq:Bexpression}
Bh(t;z,\theta)=\Xint-_{\infty}^t K(t,\tau;z)h(\tau;z,\theta)\,d\tau,
\end{equation}
whereas $\mathcal{E}$ is the restriction of
$
\mathcal{E}(t;z,\theta)\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=}\Xint-_{\infty}^{-\infty}K(t,\tau;z)\,\partial_\theta\mathfrak{A}^u(\tau;z,\theta)\,d\tau
$
to $(z,\theta)= (1,0)$.
Using \eqref{eq:Ksplit}, also $\mathcal{E}$ splits into pieces:
$
\mathcal{E}(t;z,\theta)=\sum_{i=0}^1 \sum_{p=0}^i\sum_{j=1}^2 t^p K_{ij}(t;z)\,\mathcal{E}_{ij}^p(z,\theta),
$
where
\begin{equation*}
\mathcal{E}_{ij}^p(z,\theta)\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=} -\Xint-_{-\infty}^{\infty}(-\tau)^{\delta_{i1}-p}\bar K_{ij}(\tau;z)\,\partial_\theta\mathfrak{A}^u(\tau;z,\theta)\,d\tau.
\end{equation*}
We shall see that these \emph{$t$-independent factors are exponentially small} with respect to $g$, as the latter tends to zero. Furthermore, denoting
$
K_{ij}^p(t;z)\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=} t^p K_{ij}(t;z),
$
\begin{equation*}
(BM^s)^l\mathcal{E}=\sum_{i=0}^1 \sum_{p=0}^i\sum_{j=1}^2 \bigl[(BM^s)^l K_{ij}^p\bigr]\,\mathcal{E}_{ij}^p.
\end{equation*}
This is so because, by Proposition~\ref{prop:regint}, all integrals due to \eqref{eq:Bexpression} \emph{factorize} formally: if $h$ is a trigonometric polynomial at each order, then
\begin{equation*}
\Xint-_t^\infty (h\mathcal{E})(\tau;z,\theta)\,d\tau=\sum_{i=0}^1 \sum_{p=0}^i\sum_{j=1}^2 \biggl[\,\Xint-_t^\infty h(\tau;z,\theta)\tau^p K_{ij}(\tau;z)\,d\tau\biggr] \mathcal{E}_{ij}^p(z,\theta).
\end{equation*}
By virtue of Corollary~\ref{cor:almost-there}, we infer
\begin{proposition}\label{prop:asymptotic}
On the homoclinic trajectory, \textit{i.e.}, setting $(z,\theta)=(1,0)$, the following asymptotic expansion of the splitting matrix holds:
\begin{equation}\label{eq:as-expansion}
\Upsilon(t)=\Xint-_{-\infty}^\infty \lambda\,\partial_\theta f^u_\psi(\tau)\,d\tau +\lambda \, c_{ij}^p(t) \mathcal{E}_{ij}^p,
\end{equation}
where repeated indices are contracted ($i\in\{0,1\}$, $p\in\{0,i\}$, $j\in\{1,2\}$) and
\begin{equation*}
c_{ij}^p(t)\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=} \Xint-_{t}^\infty \Bigl[f^s_{\psi\varphi}\sum_{l=0}^{\infty}(BM^s)^l K_{ij}^p\Bigr](\tau)\,d\tau.
\end{equation*}
\end{proposition}
As already pointed out, the terms appearing in the asymptotic expansion of Proposition~\ref{prop:asymptotic} will turn out exponentially small with respect to $g$. The factors $c_{ij}^p$ shall pose no problems, the functions $M^s$ and $K_{ij}^p$ being explicit and simple. In brief, Theorem~\ref{thm:splitting} begins to emerge!
For the record, writing $B^uh(t;z,\theta)\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=} \Xint-_{-\infty}^t K(t,\tau;z)h(\tau;z,\theta)\,d\tau$, we get (at $(z,\theta)=(1,0)$)
\begin{equation}\label{eq:c_ij}
c_{ij}^p(t)=(-1)^{p+j\delta_{i0}}\,\Xint-_{-\infty}^{-t}\Bigl[f^u_{\psi\varphi}\sum_{l=0}^{\infty}(B^uM^u)^l K_{ij}^p\Bigr](\tau)\,d\tau.
\end{equation}
\subsection{Emergence of exponential smallness}\label{subsec:exp-small}
The ``asymptotic'' integrals in \eqref{eq:as-expansion}, including $\lambda\,\mathcal{E}_{ij}^p$, are of the form
\begin{equation*}
\Xint-_{-\infty}^\infty \partial_\theta F(X^u;t;z,\theta)\,dt \equiv \sum_{\ell=1}^\infty\epsilon^\ell\sum_{0<\abs{q}\leq \ell N} iq\, e^{iq\cdot \theta}\,\Xint-_{-\infty}^\infty \bigl[\hat F(\hat X^u;t;z,q)\bigr]^\ell\,dt,
\end{equation*}
evaluated at $(z,\theta)=(1,0)$, where the integrand on the right-hand side is a trigonometric polynomial of degree $\leq\ell N$ by Theorem~\ref{thm:extension}. We also point out that, by construction, the latter are $\theta$-gradients, which allows us to omit the harmful $q=0$ terms. Above, F depends on $X=X^u$ locally, \textit{i.e.}, only through $X(t;z,\theta)$, as well as analytically near $X^0$, \textit{i.e.}, the series
\begin{equation}\label{eq:Fexpansion}
F(X)=F(X^0+\widetilde X)=\sum_{k=0}^\infty F^{(k)} \bigl(\widetilde X\bigr)^{\otimes k}
\end{equation}
converges. In fact, since $F$ is one of the functions in
\begin{equation}\label{eq:Fcollection}
\bigl\{\lambda f_\psi^u\bigr\}\, \bigcup\, \bigl\{-\lambda(-t)^{\delta_{i1}-p}\bar K_{ij}\,\mathfrak{A}^u \;\big|\;\text{$0\leq p\leq i\leq 1$ and $1\leq j\leq 2$}\bigr\},
\end{equation}
we observe that
\begin{equation}\label{eq:F-identity}
F(X^u;t;z,\theta)\equiv t^p F(X^u;0;ze^{\gamma t},\theta+\omega t)\qquad(p\in\{0,1\}),
\end{equation}
with $(z,\theta)\mapsto [F(X^u;0;z,\theta)]^\ell$ analytic on $(\mathbb{U}_{\tau,\vartheta}\setminus\{0\})\times\{\abs{{\operatorname{\Im\mathfrak{m}}\,}\theta}\leq\sigma\}$ for all $\ell$ by virtue of Theorem~\ref{thm:extension}. At $z=0$ there is a simple pole, due to $\bar K_{02}$, at worst.
We use the following lemma for analyzing such integrals. Its proof is given in Appendix~\ref{app:computations}.
\begin{lemma}[Shift of contour]\label{lem:shift-of-contour}
Suppose that the function $h(t;z,\theta) \linebreak[0] \equiv t^p h(ze^{gt},\theta+\omega t)$ is analytic with respect to $(z,\theta)\in(\mathbb{U}_{\tau,\vartheta}\setminus\{0\})\times\{\abs{{\operatorname{\Im\mathfrak{m}}\,}\theta}\leq\sigma\}$ and $\hat h(\,\cdot\,,0)=0$. Moreover, set $t_q\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=} \sgn(\omega\cdot q)\vartheta g^{-1}$ and
\begin{equation}\label{eq:H_q}
H_q(R)\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=}\int_0^\infty e^{(iq\cdot\omega-R)t}(t+it_q)^p\hat h(e^{g(t+it_q)},q)\,dt
\end{equation}
for each $q\in\mathbb{Z}^d\setminus\{0\}$. If
\begin{equation*}
\biggl |\, \Xint-_{-\infty}^0 e^{iq\cdot\omega t}(t+it_q)^p\hat h(e^{g(t+it_q)},q)\,dt\biggr | \leq A(g)e^{-\sigma\abs{q}}
\end{equation*}
and if $H_q(R)$ admits an analytic continuation to $\{0<\abs{R}\leq \rho\}$ with a pole of order $k$ at $R=0$, respecting the bound
\begin{equation*}
\sup_{\abs{R}=\rho}\abs{H_q(R)}\leq B(g)e^{-\sigma\abs{q}},
\end{equation*}
then we obtain the exponentially small ($c>0$) bound
\begin{equation*}
\biggl| \,\Xint-_{-\infty}^\infty h(t;1,0)\,dt\biggr|\leq C\biggl[A(g)+B(g)\sum_{j=0}^k \frac{1}{j!}\Bigl(\frac{\rho\vartheta}{g}\Bigr)^j\biggr]e^{-cg^{-1/(\nu+1)}}.
\end{equation*}
\end{lemma}
With the aid of Lemma~\ref{lem:shift-of-contour}, we shall prove in Section~\ref{sec:splitting-proof} the following key result:
\begin{proposition}[Convergence vs. exponential smallness]\label{prop:dichotomy} Fix a $t\in\mathbb{R}$. There exist positive constants $c$, $\epsilon_1$ and $C$, such that the estimates
\begin{equation*}
|\Upsilon^\ell(t)|\leq
C\begin{cases}
\epsilon_1^{-\ell}\ell!^{4(\nu+1)} e^{-cg^{-1/(\nu+1)}}\\
\epsilon_1^{-\ell}
\end{cases}
\end{equation*}
both hold true for all $\ell=1,2,\ldots$.
\end{proposition}
This dichotomy, in which the exponential smallness competes with the usual bound due to convergence, is not new; see \cite{GallavottiTwistless,GallavottiSplit,Procesi}. Remarkably, we get the same exponent of the factorial, $4(\nu+1)$, as the latter articles---even though our method is quite different.
Theorem~\ref{thm:splitting} follows immediately from Proposition~\ref{prop:dichotomy} by an argument due to Gallavotti, \textit{et al.}: For each $g$, let $n(g)$ be a positive integer. If $\abs{\tilde\epsilon}<\tfrac{1}{2}$ and $\epsilon=\tilde\epsilon\epsilon_1 n(g)^{-4(\nu+1)}$,
\begin{align*}
|\Upsilon(t)| & \leq C e^{-cg^{-1/(\nu+1)}}\sum_{\ell=1}^{n(g)}\Bigl(\frac{\abs{\epsilon}}{\epsilon_1}\Bigr)^\ell\ell!^{4(\nu+1)}+ C\sum_{\ell=n(g)+1}^\infty \Bigl(\frac{\abs{\epsilon}}{\epsilon_1}\Bigr)^\ell \\
& \leq C e^{-cg^{-1/(\nu+1)}}\frac{\abs{\epsilon}}{\epsilon_1} n(g)^{4(\nu+1)}+C\Bigl(\frac{\abs{\epsilon}}{\epsilon_1}\Bigr)^{n(g)+1} \\
& \leq C \abs{\tilde\epsilon}e^{-cg^{-1/(\nu+1)}}+C \abs{\tilde \epsilon} e^{n(g)\ln\abs{\tilde\epsilon}},
\end{align*}
since $\ell!\leq \ell^\ell\leq n(g)^\ell$ in the first sum. Now, set $n(g)=cg^{-1/(\nu+1)}/\ln 2$, such that $n(g)\ln\abs{\tilde\epsilon}\leq -cg^{-1/(\nu+1)}$ and $
\tilde\epsilon=(\epsilon/\epsilon_1) (c/\ln 2)^{4(\nu+1)}g^{-4}$ (demanding in particular that $\epsilon g^{-4}$ be small).
\begin{remark}
We used the exponentially small but diverging estimate to bound the partial sum $\sum_{\ell=1}^{n(g)}\epsilon^\ell\Upsilon^\ell$, whereas the remainder of the series was easily controlled by convergence. As $n(g)\to\infty$ with $g\to 0$, the important thing here is to have the exponentially small bound on $\Upsilon^\ell$ for \emph{arbitrarily} large $\ell$, in addition to the $\epsilon$-analyticity of $\Upsilon$.
\end{remark}
\section{Proof of Theorem~\ref{thm:splitting}}\label{sec:splitting-proof}
We are left with proving Proposition~\ref{prop:dichotomy}, since Theorem~\ref{thm:splitting} was already shown to be its corollary. Here things are most conveniently explained using tree diagrams. However, each tree will be treated as an \emph{individual}, solely for bookkeeping benefits, and no cancellations nor regroupings are forced upon them.
By Proposition~\ref{prop:asymptotic}, we need to consider the simple factors $c_{ij}^p$, as well as the integrals
\begin{equation}\label{eq:Fintegral}
\Xint-_{-\infty}^\infty \hat F(\hat X^u;t,z,q)\,dt\equiv \sum_{\ell=1}^\infty\epsilon^\ell\,\Xint-_{-\infty}^\infty \bigl[\hat F(\hat X^u;t;z,q)\bigr]^\ell\,dt
\end{equation}
of Subsection~\ref{subsec:exp-small}, at each order $\ell\geq\abs{q}/N>0$. As $F$ comes from the collection in \eqref{eq:Fcollection}, all integrals of the latter type shall be controlled with the aid of Lemma~\ref{lem:shift-of-contour}.
Due to the superscript $u$---referring to the unstable manifold---in the integrand above, the required bounds on the integrals over $\mathbb{R}_-$ in $\Xint-_{-\infty}^\infty=\Xint-_{-\infty}^0+\Xint-_{0}^\infty$ are straightforward, and are discussed later.
In order to deal with $\Xint-_0^\infty$, we present the procedure below, which amounts to \emph{little more than integration by parts}. First, we expand $F$ according to \eqref{eq:Fexpansion} and, like in Section~\ref{sec:continuation}, split
\begin{equation*}
\widetilde X=\widetilde X_{\leq 1}+\delta_2 \widetilde X=\,\raisebox{3pt}{\pstree{\Tp}{\TC*}} \, + \, \raisebox{3pt}{\pstree{\Tp}{\Tc{7pt}}} \; ,
\end{equation*}
dropping the superscript $u$ from the notation. We can then express $F$ pictorially as
\vspace{-2mm}
\begin{equation*}
\sum_{m=0}^\infty\,\sum_{0\leq m'\leq m}\binom{m}{m'}\raisebox{3pt}{
\pstree{\Tp}{
\pstree[treesep=3mm]{\TC*[name=n0]}{
\Tc[name=n1]{7pt} \Tc[name=n2]{7pt} \TC*[name=n3] \TC*[name=n4]}}
\ncarc[linestyle=dotted,nodesep=1pt]{n1}{n2}\trput{\tiny $m'$}
\ncarc[linestyle=dotted,nodesep=1pt]{n3}{n4}\trput{\tiny $m-m'$}
\nput[labelsep=1mm]{115}{n0}{\tiny $F^{(m)}$}
}\qquad,
\end{equation*}
where the binomial coefficient comes from the combinatorics of shuffling the arguments of the symmetric $F^{(m)}$ \footnote{This convention merely facilitates drawing.}, which is attached to the root (node). Next, we replace the subtrees $\raisebox{3pt}{\pstree{\Tp}{\Tc{7pt}}}=\delta_2 \widetilde X$ with the expansion \eqref{eq:tree-exp2} derived from \eqref{eq:tree-rec}.
In a tree $T_{v_0}$ with root $v_0$, a generic subtree $T_v$ with root $v$ has then the expression
\begin{equation}\label{eq:subtree}
T_v=U_v\bigl(\mathcal{K}^{-1} T_{w_1},\dots,\mathcal{K}^{-1} T_{w_{m_v'}};\bigl(\widetilde X_{\leq 1}\bigr)^{\otimes (m_v-m'_v)}\bigr),
\end{equation}
where $T_{w_j}$ is a subtree---with root $w_j$---entering $v$ and the \emph{node function}
\begin{equation*}
U_v=
\begin{cases}
\text{$F^{(m_v)}$ coming from \eqref{eq:Fcollection}}, & \text{if $v=v_0$}, \\
\text{$w^{(m_v)}$ as in \eqref{eq:wdefinition}}, & \text{otherwise}.
\end{cases}
\end{equation*}
In a pictorial representation, there are $m_v$ lines entering the node $v$ and precisely $m_v-m_v'$ of the latter are leaving a black dot (\raisebox{3pt}{\TC*}) end node:
\vspace{-2mm}
\begin{equation*}
\raisebox{3pt}{
\pstree{\Tp}{\tiny \Tcircle{\text{$T_{v}$}}}
}
=
\raisebox{3pt}{
\pstree{\Tp}{
\pstree[treesep=3mm]{\TC*[name=n0]}{
\Tcircle[name=n1]{\tiny \text{$T_{w_1}$}} \Tcircle[name=n2]{\tiny \text{$T_{w_{m_v'}}$}} \TC*[name=n3] \TC*[name=n4]}}
\ncarc[linestyle=dotted,nodesep=1pt]{n1}{n2}\naput[nrot=0]{\tiny $m'_v$}
\ncarc[linestyle=dotted,nodesep=1pt]{n3}{n4}\trput{\tiny $m_v-m'_v$}
\nput[labelsep=1mm]{115}{n0}{$U_v$}
}\qquad.
\end{equation*}
The ``whole'' tree $T_{v_0}$ contributes at orders $\ell$ satisfying
\begin{equation}\label{eq:ell-bound}
\ell\geq 1+(m_{v_0}-m'_{v_0})+\sum_{j=1}^{m_{v_0}'}\deg{T_{w_j}}\mathrel{=\mkern-4.2mu\raise.095ex\hbox{$:$}} \operatorname{d}(T_{v_0}),
\end{equation}
where $\deg{T_w}$---defined in \eqref{eq:deg}---counts end nodes with suitable weights as well as nodes with exactly one entering line in the subtree $T_w$, and the constant term $1$ counts the root ($F^{(m_{v_0})}=\order{\epsilon}$). By this we mean that $\operatorname{d}(T_{v_0})$ is the largest integer such that
\begin{equation*}
T_{v_0}=\order{\epsilon^{\operatorname{d}(T_{v_0})}}\quad\text{as}\quad \epsilon\to 0.
\end{equation*}
\subsection{Some combinatorics}
\begin{lemma}\label{lem:number-of-nodes}
The number of nodes, $\operatorname{n}(T_{v_0})$, of a tree $T_{v_0}$ contributing at order $\ell\geq 2$ obeys
\begin{equation}\label{eq:number-of-nodes}
\operatorname{n}(T_{v_0})\leq 2\bigl(\operatorname{d}(T_{v_0})-1\bigr)\leq 2(\ell-1).
\end{equation}
The number of end nodes is at most $\operatorname{d}(T_{v_0})-1\leq \ell-1$.
\end{lemma}
\begin{proof}
$\operatorname{n}(T_{v_0})$ attains its maximum with respect to the ``degree'' $\operatorname{d}(T_{v_0})$ as follows:
\begin{enumerate}
\item If $\operatorname{n}(T_{v_0})$ is even, there is only one line entering the root $v_0$ and the rest of the tree is binary, \textit{i.e.}, contains only end nodes and nodes with exactly two entering lines.
\item If $\operatorname{n}(T_{v_0})$ is odd, the tree is binary.
\end{enumerate}
Moreover, each of the end nodes is either \raisebox{3pt}{\TC*} or \raisebox{3pt}{\Tcircle{\tiny $1$}}, which contribute the least to the degree; see \eqref{eq:deg}. These choices minimize the number of end nodes when the $\order{\epsilon}$ nodes having exactly one entering line are excluded (except at $v_0$ which is always $\order{\epsilon}$). Therefore, $\operatorname{d}(T_{v_0})$ gets minimized with respect to the number of all nodes, $\operatorname{n}(T_{v_0})$. Since in a binary tree of $j$ end nodes there are $2j-1$ nodes, we infer \eqref{eq:number-of-nodes}.
The root and each of the end nodes increases $\operatorname{d}(T_{v_0})$ by at least one, which implies the bound on the number of end nodes.
\end{proof}
\begin{corollary}\label{cor:number-of-trees}
At most $2^{6\ell}$ trees contribute at order $\ell$.
\end{corollary}
\begin{proof}
It is well-known that the number of (rooted) trees with $k$ indistinguishable nodes is
$
N(k)\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=}\frac{1}{k}\binom{2k-2}{k-1}\leq \frac{1}{k}4^{k-1},
$
which follows from generating functions (see \cite{Drmota}) and the bound $(2m)!\leq 4^m(m!)^2$. By Lemma~\ref{lem:number-of-nodes}, the number of end nodes is less than $\ell$. We label the latter arbitrarily by the labels in $\{\raisebox{3pt}{\TC*}\}\bigcup \bigl\{\raisebox{3pt}{\Tcircle{\text{\tiny $m$}}}\;|\; \text{$1\leq m\leq\ell-1$}\bigr\}$ in order to form an upper bound on the number of our trees; these are the only possible labels, as otherwise $\operatorname{d}(T_{v_0})$ certainly exceeds the order $\ell$---contradicting \eqref{eq:ell-bound}. The labeling of a tree with $j$ end nodes can be carried out in at most $\binom{j+\ell-1}{j-1}\leq 2^{j+\ell-1}$ ways. The desired bound is thus obtained from
$
1+\sum_{k=2}^{\ell-1}N(k)\binom{\ell+k-2}{k-2}+\binom{2\ell-2}{\ell-2}\sum_{k=\ell}^{2(\ell-1)}N(k)\leq 2^{6\ell},
$
because by Lemma~\ref{lem:number-of-nodes} the number of end nodes is at most $\ell-1$ even though the total number of nodes can be as large as $2(\ell-1)$. The term 1 on the left-hand side counts the single node tree $F^{(0)}$.
\end{proof}
\subsection{Simplification of integrals: scalar trees}
We take a preliminary step towards bounding the values of the trees.
Let us split the kernels $K$ of the operators $\mathcal{K}^{-1}$ appearing in \eqref{eq:subtree} into four pieces according to \eqref{eq:Ksplit}. These operators are attached to the lines between the nodes of a tree. Each of the $2^{6\ell}$ trees counted in Corollary~\ref{cor:number-of-trees} thus breaks into at most $4^{2\ell}$ new trees, as there are no more than $2\ell$ such lines by Lemma~\ref{lem:number-of-nodes}.
At the same time, we also expand the matrix products due to the coordinate representation of \eqref{eq:subtree} at each node:
$
(U_v)^i(\widetilde T_1,\dots,\widetilde T_m)=\sum_{j_1=1}^{d+1}\dots\sum_{j_m=1}^{d+1} (U_v)^i_{j_1\dots j_m}\widetilde T_1^{j_1}\dots \widetilde T_m^{j_m},
$
where the $\widetilde T_k$'s represent all arguments of $U_v$ (\textit{i.e.}, subtrees entering the node $v$)---including $\widetilde X_{\leq 1}$---and the superindices specify vector components. We next separate each scalar term into its own tree, thus getting up to $(d+1)^{2\ell}$ of these \emph{scalar trees} from each old tree.
There are lines carrying a factor $t-\tau$, coming from the $i=1$ terms of \eqref{eq:Ksplit}. They correspond to double integrals: $\iint_{-\infty}^t (d\tau)^2 =\int_{-\infty}^t d\tau\,(t-\tau)$. We remove these factors, insert a new node $\tilde v$ on the line with a node function $U_{\tilde v}\equiv 1$ and an integral sign both on the line leaving and on the line entering $\tilde v$. This operation can be depicted as
\vspace{1mm}
\begin{equation}\label{eq:iint}
\raisebox{3pt}{
\pstree[levelsep=1.3cm]{\Tp}{\Tp \tbput{$\iint_{-\infty}^t$}}
} \quad \longmapsto \quad
\raisebox{3pt}{
\pstree[levelsep=1.3cm]{\Tp}{
\pstree{\TC*[name=v] \tbput{$\int_{-\infty}^t$}}{\Tp \tbput{$\int_{-\infty}^\tau$}}}
\nput[labelsep=1mm]{100}{v}{$1$}
}.
\vspace{4mm}
\end{equation}
It does not affect the number of trees, but slightly simplifies the discussion below, even though the number of lines in a single tree can be as much as doubled.
Altogether, we arrive at the following conclusion:
\begin{proposition}\label{prop:tree-count}
There are less than $C_d^{\ell}$ scalar trees to be considered, with $C_d=2^{10}(d+1)^2$, at each order $\ell$. Each of these trees has at most $4(\ell-1)$ lines.
\end{proposition}
\begin{remark}[Some conventions]\label{rem:convention}
From now on, by a tree we will always refer to a scalar tree, where all the decompositions above have been carried out. In order to alleviate notation, we systematically omit all vector component indices.
\end{remark}
Recall that the node $w'$ is the unique predecessor of the node $w$. Let $T_{v_0}$ be a generic (scalar) tree with root $v_0$. Denote by $V$ the set of all nodes and by $V_\text{int}$ the set of \emph{integrated nodes}, \textit{i.e.}, the nodes whose leaving line carries an integral. We consider the root $v_0$ an integrated node, such that $V_{\text{int}}$ consists of all nodes of $T_{v_0}$ except black dot (\raisebox{3pt}{\TC*}) end nodes. We can describe $T_{v_0}$ by giving its structure recursively: if $T_v$ is the subtree of $T$ with root $v\in V_{\text{int}}$, then
\begin{equation}\label{eq:scalar-subtree}
T_v(t;z,\theta)=u_v(t;z,\theta)\prod_{\substack{w\in V_{\text{int}}\\w'=v}} \int_{-\infty}^{t} T_w(\tau;z,\theta)\,d\tau.
\end{equation}
We call $u_v$ the \emph{multiplier} of the node $v$. It is a scalar function comprising all factors in the expression of the tree carrying the same variable of integration (``time label''), constricted in-between integral signs. In particular, it contains all subtrees $\raisebox{3pt}{\pstree{\Tp}{\TC*}}$ entering $v$, as these are the functions $\widetilde X_{\leq 1}$ involving no integrals.
To be completely explicit, the multiplier $u_v$ of a node $v\in V_{\text{int}}$ is one of the functions below:
\begin{enumerate}
\item\label{list:multipliers} $\bar K_v\delta_2h^{(k_v)}$, if $v$ is the end node \raisebox{3pt}{\Tcircle{\tiny $k_v$}}; see \eqref{eq:endnode}.
\item $\bar K_v w^{(m_v)}\Bigl(\prod_{\substack{w\notin V_{\text{int}}\\w'=v}}\widetilde X_{\leq 1}\Bigr) \Bigl(\prod_{\substack{w\in V_{\text{int}}\\w'=v}}K_{w}\Bigr)$, if $v$ is neither an end node nor the root $v_0$, and didn't appear from splitting a double integral according to \eqref{eq:iint}.
\item\label{item:double-int-split} $1$, if $v$ appeared from splitting a double integral; see \eqref{eq:iint}.
\item\label{item:root-t} $F^{(m_v)}\Bigl(\prod_{\substack{w\notin V_{\text{int}}\\w'=v}}\widetilde X_{\leq 1}\Bigr)\Bigl(\prod_{\substack{w\in V_{\text{int}}\\w'=v}}K_{w}\Bigr)$, if $v=v_0$; see \eqref{eq:Fcollection}.
\end{enumerate}
The functions $K_v$ and $\bar K_v$ refer to diagonal elements of $K_{ij}$ and $\bar K_{ij}$ in \eqref{eq:Ksplit}, respectively. $m_v'$ is the cardinality of $\{w\in V_\text{int}\:|\:w'=v\}$ and $m_v-m_v'$ the cardinality of $\{w\in V\setminus V_\text{int}\:|\:w'=v\}$.
We draw the scalar (sub)tree in \eqref{eq:scalar-subtree}---originated from \eqref{eq:subtree}---as
\begin{equation*}
\raisebox{3pt}{
\pstree{\Tp}{\Tcircle{\tiny \text{$T_{v}$}}}
}
=
\raisebox{3pt}{
\pstree{\Tp}{
\pstree[treesep=3mm]{\TC*[name=n0]}{
\Tcircle[name=n1]{\tiny \text{$T_{w_1}$}} \Tcircle[name=n2]{\tiny \text{$T_{w_{m_v'}}$}} }}
\ncarc[linestyle=dotted,nodesep=1pt]{n1}{n2}\naput[nrot=0]{\tiny $m'_v$}
\nput[labelsep=1mm]{115}{n0}{$u_v$}
}\qquad.
\vspace{2mm}
\end{equation*}
This diagram is reminiscent of the one below \eqref{eq:subtree}, except that the operator $U_v$ has changed into the multiplier $u_v$ and the subtrees $\raisebox{3pt}{\pstree{\Tp}{\TC*}}\:$ have been absorbed into $u_v$. Moreover, recalling the decompositions above, the present diagram carries $\int_{-\infty}^t$ on its lines instead of $\mathcal{K}^{-1}$---compare \eqref{eq:scalar-subtree} with \eqref{eq:subtree}---and has possibly more nodes due to the diagram in \eqref{eq:iint}.
We do not consider the end nodes with a black dot (\raisebox{3pt}{\TC*}) nodes anymore. Subsequently, we will refer to ``integrated nodes'' (elements of $V_\text{int}$) as just ``nodes''.
\subsection{Integration by parts: one step}
Reinserting the implicit arguments $(z,\theta)$,
\begin{equation}\label{eq:multiplier-prop}
u_v(t;z,\theta)\equiv t^p u_v(ze^{\gamma t},\theta+\omega t)
\end{equation}
for the obviously defined $u_v(z,\theta)$. The power $p$ can be nonzero only at the root, $v=v_0$, where it possibly assumes the value 1 due to case (\ref{item:root-t}) in the list of all possible multipliers above.
We now turn our attention to the Fourier transformed integrals
\begin{equation}
\Xint-_{0}^\infty T_{v_0}(t;z,q)\,dt =\Xint-_0^{\infty}\,\sum_{q_{v_0}+\sum_{w'=v_0} \tilde q_w=q} u_{v_0}(t;z,q_{v_0})\prod_{w'={v_0}} \int_{-\infty}^{t} T_w(\tau;z,\tilde q_w)\,d\tau\,dt,
\end{equation}
which---recalling \eqref{eq:scalar-subtree}---arise from \eqref{eq:Fintegral}, at each order $\ell\geq \abs{q}/N>0$. We omit the usual $\widehat{\text{hat}}$ in Fourier transforms, since there is no danger of confusion. Due to Theorem~\ref{thm:extension}, we may impose the finiteness condition
\begin{equation}\label{eq:Fourier-condition}
\abs{q_{v_0}} + \sum_{w'=v_0}\abs{\tilde q_w} \leq \ell N,
\end{equation}
such that $\Xint-_{0}^\infty T_{v_0}(t;z,q)\,dt$ agrees at order $\ell$ with
\begin{equation}\label{eq:reg-tree-int}
\sideset{}{^\star}\sum_{q_{v_0}+\sum_{w'=v_0} \tilde q_w=q} \res_{R=0}\frac{1}{R}\int_0^{\infty}e^{-Rt}\,u_{v_0}(t;z,q_{v_0})\prod_{w'={v_0}} \int_{-\infty}^{t} T_w(\tau;z,\tilde q_w)\,d\tau\,dt.
\end{equation}
Here the asterisk reminds us that \eqref{eq:Fourier-condition} is being respected by the sum.
\begin{remark}\label{rem:int-task}
The task is to show that the $t$-integral in \eqref{eq:reg-tree-int} extends analytically from large positive values of ${\operatorname{\Re\mathfrak{e}}\,}{R}$ to a punctured neighbourhood of $R=0$, such that the residue can be computed. For this, we need the specific structure of the multipliers $u_v$.
\end{remark}
Let us define
$
\xi_{01}=\bar\xi_{02}=1,$
$\bar\xi_{01}=\xi_{02}=\xi_{11}=\bar\xi_{11}=-1,$ and
$\xi_{12}=\bar\xi_{12}=0.
$
There exist positive, continuous, functions $a_{ij}$ and $\bar a_{ij}$ such that
\begin{equation}\label{eq:K-bounds}
\abs{K_{ij}(t;z)}\leq a_{ij}(z) e^{\xi_{ij}\gamma\abs{t}}\quad\text{and}\quad\abs{\bar K_{ij}(t;z)}\leq \bar a_{ij}(z) e^{\bar\xi_{ij}\gamma\abs{t}}
\end{equation}
hold on $\{(t,z)\in\mathbb{R}\times \mathbb{C}\;|\;\text{$\abs{z}=1$ and $ze^{\gamma t}\neq\pm i$}\}$.
Moreover,
\begin{equation}\label{eq:a-prod}
a_{ij}\bar a_{ij} \leq \frac{A}{\gamma^{1-i}},
\end{equation}
where the constant $A$ is independent of $\gamma$. Notice that, in the Kronecker delta notation,
\begin{equation}\label{eq:xi-sum}
\xi_{ij}+\bar\xi_{ij}=-2\,\delta_{i1}\delta_{j1}\leq 0.
\end{equation}
Emphasizing the node in question, instead of specifying the sub\-indices $ij$ we write $\xi_v$ and $\bar\xi_v$.
For each node $v$, we define the numbers
\begin{equation*}
r_v\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=}\sup_{(z,\theta)\in B}\min{\bigl\{k\in\mathbb{Z}\;\big|\;\forall\,\delta>0:\text{$u_v(t;z,\theta)e^{-(k+\delta)\gamma t}\to 0$ as $t\to\infty$}\bigr\}}
\end{equation*}
with
\begin{equation}\label{eq:ok-domain}
B\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=} \{\abs{z}=1, \abs{\arg{z}}\leq\vartheta\}\times\{\abs{{\operatorname{\Im\mathfrak{m}}\,}\theta}\leq\sigma\}.
\end{equation}
Thus, recalling \eqref{eq:multiplier-prop}, we are inside the analyticity domain $\mathbb{U}_{\tau,\vartheta}\times\{\abs{{\operatorname{\Im\mathfrak{m}}\,}\theta}\leq\sigma\}$ of Lemma~\ref{lem:an-cont}. These numbers measure the divergence rate of the multipliers $u_v$ in the limit $t\to\infty$.
\begin{lemma}\label{lem:r_v}
Ordered according to the list of possible $u_v$'s on p.~\pageref{list:multipliers},
\begin{equation*}
r_v=
\begin{cases}
\bar\xi_v+k_v, & \text{case (1),} \\
\bar\xi_v +(m_v-m_v')+\sum_{w'=v}\xi_w, & \text{case (2),} \\
0, & \text{case (3),} \\
\bar\xi_F+(m_v-m_v')+\sum_{w'=v}\xi_w, & \text{case (4)},
\end{cases}
\end{equation*}
where $\bar\xi_F\in\{0,1\}$ depends on the choice of $F$ in \eqref{eq:Fcollection}. Moreover $u_v(z,\theta)$, see \eqref{eq:multiplier-prop}, is analytic on $\{\abs{{\operatorname{\Im\mathfrak{m}}\,}\theta}\leq\sigma\}$ with respect to $\theta$ and on $\mathbb{U}_{\tau,\vartheta}\setminus\{0\}$ with respect to $z$. It is also analytic in the punctured neighbourhood $\{\abs{z}\geq \tau^{-1}\}$ of $z=\infty$, at which point there is a (possible) pole of order $r_v$ (if $r_v>0$).
\end{lemma}
\begin{proof}
The maps $\Phi^0(z)$ and $f(\Phi^0(z),\theta)$ are analytic in these domains, without singularities at $z=0,\infty$; see \eqref{eq:arctan-sym}. The rest follows by staring at the expression of $u_v$ in each case.
\end{proof}
Starting from the end nodes of a tree---setting $s_v\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=} 0$ for them---we recursively define
\begin{equation}\label{eq:n_v-def}
s_v\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=} \sum_{w'=v}\max(0,n_w) \quad\text{and}\quad n_v\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=} r_v+s_v.
\end{equation}
These numbers measure the divergence rate of (sub)trees in the limit $t\to\infty$:
\begin{lemma}
For $T_v$ as in \eqref{eq:scalar-subtree}, and any $\delta>0$, the estimates
\begin{equation}\label{eq:n_v-bound}
T_v(t;z,\theta)e^{-(n_v+\delta)\gamma t}\to 0\quad\text{as}\quad t\to\infty,
\end{equation}
\begin{equation}\label{eq:s_v-bound}
\biggl(\,\prod_{w'=v} \int_{-\infty}^{t} T_w(\tau;z,\theta)\,d\tau\biggr) e^{-(s_v+\delta)\gamma t}\to 0\quad\text{as}\quad t\to\infty
\end{equation}
hold true in the region $B$ of \eqref{eq:ok-domain}.
\end{lemma}
\begin{proof}
Assume that \eqref{eq:n_v-bound} is true for each successor $w$ of $v$ ($w'=v$). By \eqref{eq:scalar-subtree}, we only need to observe that, given $\delta>0$,
$
\bigl |\int_{-\infty}^t T_w(\tau;z,\theta)\,d\tau\bigr |\leq C_1+C_2\int_0^t e^{(n_w+\delta)\gamma \tau}\,d\tau=\order{e^{[\max(0,n_w)+\delta]\gamma t}}
$
in the limit $t\to\infty$. For then \eqref{eq:n_v-def} implies \eqref{eq:s_v-bound} for the node $v$. Now, \eqref{eq:n_v-bound} follows from the definition of $r_v$. Since \eqref{eq:n_v-bound} holds for end nodes ($n_v=r_v$), induction proves the claim.
\end{proof}
Recalling Remark~\ref{rem:int-task}, let us now come back to the integral in \eqref{eq:reg-tree-int}, setting $v=v_0$, \textit{i.e.},
\begin{equation}\label{eq:integraali}
\int_0^{\infty}e^{-Rt}\,u_v(t;z,q_v)\prod_{w'=v} \int_{-\infty}^{t} T_w(\tau;z,\tilde q_w)\,d\tau\,dt.
\end{equation}
An obvious problem is the exponential divergence of the integrand; see \eqref{eq:n_v-bound} and \eqref{eq:s_v-bound}. Our cure is the following. Since $u_v$ is meromorphic at $z=\infty$, by Lemma~\ref{lem:r_v}, we may expand
\begin{equation}\label{eq:multiplier-split}
u_v(z,\theta)=\sum_{k=-s}^{r_v} z^k u_{v,k}(\theta)+u_{v,< -s}(z,\theta)
\end{equation}
for any integer $s\geq -r_v$, with
$
u_{v,< -s}(z,\theta)=\order{z^{-s-1}}$ as $z\to\infty$. Extending \eqref{eq:multiplier-prop}, we write
\begin{equation*}
u_{v,k}(t;\theta)\equiv t^p \, e^{k\gamma t}\, u_{v,k}(\theta+\omega t)
\quad\text{and}\quad
u_{v,< -s}(t;z,\theta)\equiv t^p \, u_{v,< -s}(ze^{\gamma t},\theta+\omega t),
\end{equation*}
where $p\in\{0,1\}$ depends on the choice of $F$ in case (\ref{item:root-t}) on p.~\pageref{item:root-t}. In particular, the integral
$
\int_0^{\infty}e^{-Rt}\,u_{v,< -s_v}(t;z,q_v)\prod_{w'=v} \int_{-\infty}^{t} T_w(\tau;z,\tilde q_w)\,d\tau\,dt
$
is convergent for ${\operatorname{\Re\mathfrak{e}}\,}{R}>-\gamma$, by virtue of \eqref{eq:s_v-bound}, and can be estimated on a circle $\abs{R}=\rho<\gamma$ for the purpose of Lemma~\ref{lem:shift-of-contour}.
The rest of the integral \eqref{eq:integraali} is \emph{integrated by parts}: for $-s_v\leq k\leq r_v$ and sufficiently large positive values of ${\operatorname{\Re\mathfrak{e}}\,}{R}$,
\begin{equation}
\begin{split}
& \int_0^{\infty} e^{-Rt}\, z^k u_{v,k}(t;q_v) \prod_{w'=v} \int_{-\infty}^{t} T_w(\tau;z,\tilde q_w)\,d\tau\,dt \\
& = \frac{z^k u_{v,k}(q_v)}{R_v}\Biggl\{\frac{E_v(0)}{R_v^p} +\sum_{p'=0}^p\frac{1}{R_v^{p-p'}} \int_0^\infty e^{-R_v t}\,t^{p'} \frac{dE_v}{dt}\,dt\Biggr\}, \label{eq:int-by-parts}
\end{split}
\end{equation}
where
$
R_v\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=} R-k\gamma-i\omega\cdot q_v
$
and
$
E_v(t)\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=} E_v(t;z,\{\tilde q_{w} \, |\, w'=v\})\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=} \prod_{w'=v} \int_{-\infty}^{t} T_w(\tau;z,\tilde q_w)\,d\tau
$.
But $\frac{dE_v}{dt}=\sum_{w'=v} T_{w}(t)\prod_{{\bar w'=v, \bar w\neq w}}\int_{-\infty}^{t} T_{\bar w}(\tau)\,d\tau $, so that the recursion relation \eqref{eq:scalar-subtree} yields
\begin{align*}
\frac{dE_v}{dt}
&=\sum_{w'=v} \sum_{q_{w}+\sum_{\bar w'=w} \tilde q_{\bar w} = \tilde q_{w}} u_w(t;z,q_w) \biggl(\,\prod_{\bar w'=w}\int_{-\infty}^{t} T_{\bar w}(\tau;z,\tilde q_{\bar w})\,d\tau\biggr) \biggl(\, \prod_{\substack{\bar w'=v\\ \bar w\neq w}}\int_{-\infty}^{t} T_{\bar w}(\tau;z,\tilde q_{\bar w})\,d\tau \biggr).
\end{align*}
We now fix $w$, and collect all the Fourier sums together so that they run over the set specified by
$
q_{v}+q_{w} + \sum_{\bar w'=w}\tilde q_{\bar w} + \sum_{\bar w'=v, \bar w\neq w}\tilde q_{\bar w} = q.
$
Again, without affecting the $\ell$th order, we also impose the finiteness condition
$
|q_{v}|+|q_{w}| + \sum_{\bar w'=w} | \tilde q_{\bar w}| + \sum_{\bar w'=v, \bar w\neq w} | \tilde q_{\bar w}| \leq \ell N
$
similar to \eqref{eq:Fourier-condition}, which allows us to bring the sum out of the $t$-integral.
We observe that the remaining integral in \eqref{eq:int-by-parts} produces integrals similar to the original \eqref{eq:integraali}, with the following changes:
(i) $u_v$ changes to $u_w$,
(ii) $R$ changes to $R_v$,
(iii) $p$ changes to $p'$, and
(iv) the integral on the line \emph{leaving} $w$, $\int_{-\infty}^{t} T_{w}(\tau)\,d\tau$, is replaced by the ones on the lines \emph{entering} $w$, the product $\prod_{\bar w'=w}\int_{-\infty}^{t} T_{\bar w}(\tau)\,d\tau$.
Thus, a single step in the integration-by-parts scheme can be described in terms of trees as follows: Given a tree, consider one of the successors, $w$, of the root, $v=w'$. The line from $w$ to $v$ is ``contracted'' by erasing the node $w$, reattaching to $v$ all subtrees originally entering $w$, and replacing the multiplier of the root, $v$, by $u_w$. Finally, we rename the root $w$. Pictorially,
\begin{equation*}
\raisebox{3pt}{
\pstree[treesep=0.7cm]
{\Tp}{\pstree{\TC*[name=v]}{\Tp[name=top] \pstree{\TC*[name=cen,bbh=2pt]
}
{\Tp[name=centop] \Tp[name=cenbot]} \Tp[name=bot]}}
\ncarc[linestyle=dotted,nodesep=3pt]{top}{cen}
\ncarc[linestyle=dotted,nodesep=3pt]{cen}{bot}
\ncarc[linestyle=dotted,nodesep=3pt]{centop}{cenbot}
\nput[labelsep=1mm]{100}{v}{$u_{v}$}
\nput[labelsep=1mm]{120}{cen}{$u_{w}$}
}
\quad \longmapsto \quad
\raisebox{3pt}{
\pstree[treesep=0.5cm]
{\Tp}{\pstree{\TC*[name=v]}{\Tp[name=top1] \Tp[name=centop1] \Tp[name=cenbot1] \Tp[name=bot1]}}
\ncarc[linestyle=dotted,nodesep=3pt]{top1}{centop1}
\ncarc[linestyle=dotted,nodesep=3pt]{centop1}{cenbot1}
\ncarc[linestyle=dotted,nodesep=3pt]{cenbot1}{bot1}
\nput[labelsep=1mm]{100}{v}{$u_w$}
}\quad .
\end{equation*}
\subsection{Integration by parts: exhausting the entire tree}
We can proceed recursively and repeat the procedure in the previous subsection until there are no nodes left in the tree. Start with a tree $T_v$ having root $v$. First, call $T_0\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=} T_v$ and set $v_0\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=} v$. Then choose a successor $w$ of $v$ and define $v_1\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=} w$, contracting the line from $w$ to $v$. Next, in the \emph{new tree} (the rightmost diagram above) called $T_1$, choose a successor of $w$ and call it $v_2$. Contract the line from $v_2$ to $v_1$. Repeat until the tree has been exhausted and all nodes have been numbered.
We can express the sequence of trees formed as
\begin{equation}\label{eq:T_i}
T_i(t)=u_{v_i}\prod_{w\in T_i:\,w'=v_i}\int_{-\infty}^t T_w(\tau)\,d\tau,
\end{equation}
where in the product we consider the tree $T_i$ with the root $v_i$ having the successors $w$.
Let us define the numbers, the divergence rates of the multipliers $u_{v_i}$,
\begin{equation*}
r_i\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=} r_{v_i}.
\end{equation*}
Analogously to the numbers $s_v$ in \eqref{eq:n_v-def}, we set
\begin{equation*}
s_i\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=} \sum_{w\in T_i:\,w'=v_i}\max{(0,n_w)}.
\end{equation*}
The numbers $n_w$ are the ones defined in $\eqref{eq:n_v-def}$ for the original tree, $T_0$. Notice that, although $s_0=s_{v_0}$, $s_i$ is not simply equal to $s_{v_i}$, but is the analogue in the tree $T_i$ of which $v_i$ is the \emph{root}. Similarly to \eqref{eq:s_v-bound}, $s_i$ bounds the divergence rate of the product in \eqref{eq:T_i}: if $\delta>0$,
\begin{equation}\label{eq:s_i-bound}
\biggl(\, \prod_{w\in T_i:\,w'=v_i}\int_{-\infty}^{t} T_w(\tau)\,d\tau\biggr) e^{-(s_i+\delta)\gamma t}\to 0\quad\text{as}\quad t\to\infty.
\end{equation}
The following, elementary, property is useful.
\begin{lemma}\label{lem:s+r}
For every $i$,
$
s_{i+1}+r_{i+1}\leq s_i.
$
\end{lemma}
\begin{proof}
Since $r_{i+1}\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=} r_{v_{i+1}}$, $n_{v_{i+1}}\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=} r_{v_{i+1}}+s_{v_{i+1}}$, and
$
\sum_{w\in T_{i}:w'=v_{i+1}}\max{(0,n_w)}=s_{v_{i+1}},
$
and
\begin{equation*}
s_{i+1}\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=}\!\sum_{w\in T_{i+1} : w'=v_{i+1}}\max{(0,n_w)}=\!\!\sum_{w\in T_{i}:\,w\neq v_{i+1}, w'=v_{i}}\max{(0,n_w)} +\!\!\sum_{w\in T_{i} : w'=v_{i+1}}\max{(0,n_w)},
\end{equation*}
we get
$
s_{i+1}+r_{i+1}=\sum_{w\in T_{i}:\,w\neq v_{i+1}, w'=v_{i}}\max{(0,n_w)}+n_{v_{i+1}}\leq s_i.
$
Here we used the fact that $v_{i+1}'=v_i$ in the tree $T_i$.
\end{proof}
Let $\tilde k_{0}\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=} 0$ and $Q_0\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=} 0$. Suppose that at the $i$th step we are considering the integral
\begin{equation*}
\int_0^\infty e^{(iQ_i\cdot \omega+\tilde k_i\gamma-R)t}\,t^{p_i}u_{v_i}(ze^{\gamma t},q_{v_i})e^{iq_{v_i}\cdot \omega t}\,\prod_{w\in T_i:\,w'=v_i}\int_{-\infty}^{t} T_w(\tau;z,\tilde q_w)\,d\tau \,dt;
\end{equation*}
\textit{cf}.\@ \eqref{eq:integraali}. Then the integration-by-parts procedure described above takes us at the $(i+1)$st step to a similar integral with $i$ replaced by $i+1$, defining
\begin{equation}\label{eq:k-tilde-rec}
\tilde k_{i+1}\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=} \tilde k_{i}+k_i\quad\text{and}\quad Q_{i+1}\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=} Q_i+q_{v_i},
\end{equation}
multiplied by the factor
\begin{equation}\label{eq:R-fraction}
\frac{z^{k_i}u_{v_i,k_i}(q_{v_i})}{{\bigl(R-\tilde k_{i+1}\gamma-i\omega\cdot Q_{i+1}\bigr)}^{1+p_i-p_{i+1}}}.
\end{equation}
The integer indices $p_{i+1}$ and $k_i$ can assume the values
\begin{equation}\label{eq:k-range}
0\leq p_{i+1}\leq p_i\leq 1\quad\text{and}\quad k_i=-\tilde k_i-s_i,\dots,r_i.
\end{equation}
With the aid of Lemma~\ref{lem:s+r}, it is straightforward to see that
\begin{equation}\label{eq:index-sum}
0\leq \tilde k_i+r_i+s_i\leq n_{v_0},
\end{equation}
where $n_{v_0}=s_{v_0}+r_{v_0}=s_0+r_0$ is the number describing the divergence rate of the original tree, $T_0$, in the sense of \eqref{eq:n_v-bound}. The number of possible values of $k_i$ above is thus at most $1+n_{v_0}$. This implies the two-sided (equivalent) bounds
\begin{equation*}
r_i-n_{v_0}\leq k_i\leq r_i \quad\text{and}\quad 0\leq r_i-k_i\leq n_{v_0}.
\end{equation*}
We continue recursively until there are no nodes---alternatively, integrals that require regularization\linebreak[1]---left. The result is a sum with terms of two different species. We set
\begin{equation*}
R_i\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=} R-\tilde k_{i}\gamma-i\omega\cdot Q_{i},
\end{equation*}
in order to make the presentation more compact, and also define (\textit{cf.} $E_v$ in \eqref{eq:int-by-parts})
\begin{equation*}
E_j(t)\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=} E_j(t;z,\{\tilde q_w\,|\, w\in T_j, w'= v_j\})\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=} \!\!\prod_{w\in T_j:\,w'=v_j}\int_{-\infty}^{t} T_w(\tau;z,\tilde q_w)\,d\tau,
\end{equation*}
with the understanding that $E_{\abs{V_\text{int}}-1}\equiv 1$. The first class of terms is
\begin{equation}\label{eq:second-case}
\Biggl(\,\prod_{i=0}^{j-1}\frac{z^{k_i}u_{v_i,k_i}(q_{v_i})}{R_{i+1}^{1+p_i-p_{i+1}}}\Biggr) \int_0^\infty e^{(iq_{v_j}\cdot \omega-R_j) t} \,t^{p_j} \,u_{{v_j},<-\tilde k_j-s_j}(z e^{\gamma t},q_{v_j}) \,E_j(t) \,dt,
\end{equation}
for $0<j< \abs{V_\text{int}}$. Notice that $s_{\abs{V_\text{int}}-1}=0$. Second, there are the terms ($0<j< \abs{V_\text{int}}$)
\begin{equation}\label{eq:third-case}
\Biggl(\,\prod_{i=0}^{j-1}\frac{z^{k_i}u_{v_i,k_i}(q_{v_i})}{R_{i+1}^{1+p_i-p_{i+1}}}\Biggr)\,\frac{z^{k_j}u_{v_j,k_j}(q_{v_j})}{R_{j+1}^{1+p_j}} \, E_j(0).
\end{equation}
\subsection{Estimates}\label{subsec:estimates}
By the definition of $s_j$, we can bound
\begin{equation*}
\abs{E_j(t)}\leq C_{E_j}e^{(s_j+\frac 14)\gamma t}e^{-\sigma\sum{\abs{\tilde q_w}}}\qquad(t\geq 0),
\end{equation*}
for $\abs{z}=1$ with $\abs{\arg{z}}\leq \vartheta$. Deferring the proof to Appendix~\ref{app:computations}, we formulate
\begin{lemma}\label{lem:multiplier-bound}
The coefficients $u_{v_i,k_i}(q)$ satisfy
\begin{equation*}
\abs{u_{v_i,k_i}(q_{v_i})}\leq C_{v_i}\tau^{k_i-r_i} e^{-\sigma\abs{q_{v_i}}},
\end{equation*}
where $\tau$ is as in Lemma~\ref{lem:r_v}. On $\{1\leq\abs{z}\leq \tau^{-1}\,\,\text{\rm with}\,\,\abs{\arg{z}}\leq\vartheta\}\bigcup \,\{\abs{z}\geq \tau^{-1}\}$,
\begin{equation*}
\abs{u_{v_i,< -\tilde k_i-s_i}(z,q_{v_i})}\leq C_{v_i}(\tau\abs{z})^{-\tilde k_i-s_i-1} 2^{n_{v_0}} \tau^{-r_i} e^{-\sigma\abs{q_{v_i}}}.
\end{equation*}
When $\abs{z}=1, \abs{\arg{z}}\leq \vartheta$, and $\abs{{\operatorname{\Im\mathfrak{m}}\,}\theta}\leq\sigma$ it holds true, for suitable $\bar r_i\in\mathbb{Z}$, that
\begin{equation*}
\abs{u_{v_i}(ze^{\gamma t},\theta+\omega t)}\leq C_{v_i}
\begin{cases}
e^{\bar r_i \gamma t}, & t<0, \\
e^{r_i \gamma t}, & t\geq 0.
\end{cases}
\end{equation*}
\end{lemma}
Recalling \eqref{eq:index-sum}, the integral in \eqref{eq:second-case} is bounded by
$
(2\tau^{-1})^{n_{v_0}+1}\frac{C_{v_j}C_{E_{j}}}{\gamma^{1+p_j}} e^{-\sigma\abs{q_{v_j}}-\sigma\sum\abs{\tilde q_w}},
$
when ${\operatorname{\Re\mathfrak{e}}\,}{R}\geq - \tfrac{1}{2}\gamma$, $\abs{z}=1$, and $\abs{\arg{z}}\leq \vartheta$. Thus, the bounds on \eqref{eq:second-case} and \eqref{eq:third-case} read
\begin{equation} \label{eq:second-case1}
\Biggl(\,\prod_{i=0}^{j-1}\frac{C_{v_i}}{\bigl|R_{i+1}^{1+p_i-p_{i+1}}\bigr|}\Biggr)\frac{C_{v_j}C_{E_{j}}}{\gamma^{1+p_j}}(2\tau^{-1})^{n_{v_0}+1}\tau^{\sum_{i=0}^{j-1}(k_i-r_i)}e^{-\sigma\sum_{i=0}^j\abs{q_{v_i}}-\sigma\sum\abs{\tilde q_w}}
\end{equation}
and
\begin{equation}\label{eq:third-case1}
\Biggl(\,\prod_{i=0}^{j-1}\frac{C_{v_i}}{\bigl|R_{i+1}^{1+p_i-p_{i+1}}\bigr|}\Biggr)\frac{C_{v_j}C_{E_{j}}}{\bigl|R_{j+1}^{1+p_j}\bigr|}\tau^{\sum_{i=0}^{j}(k_i-r_i)}e^{-\sigma\sum_{i=0}^j\abs{q_{v_i}}-\sigma\sum\abs{\tilde q_w}} ,
\end{equation}
respectively. Here $e^{-\sigma\sum_{i=0}^j\abs{q_{v_i}}-\sigma\sum\abs{\tilde q_w}}\leq e^{-\sigma\abs{q}}$, because the tree is a Fourier transform with index $q$, and the $q_{v_i},\tilde q_w$ come from convolutions. The factors of $\tau$ are controlled by
\begin{lemma}
For all $j$,
$
\sum_{i=0}^j(k_i-r_i)\geq -n_{v_0}$. Moreover, $n_{v_0}\leq \operatorname{d}(T_{v_0})$
(see \eqref{eq:ell-bound}).
\end{lemma}
\begin{proof}
From \eqref{eq:k-tilde-rec} and \eqref{eq:k-range}, we clearly have $\tilde k_{i+1}\geq -s_i$. Then,
\begin{align*}
\sum_{i=0}^j(k_i-r_i) & =\tilde k_{j+1}-\sum_{i=0}^j r_i\geq -s_j-r_j-\sum_{i=0}^{j-1} r_i
& \geq -s_{j-1}-\sum_{i=0}^{j-1} r_i\geq\dots\geq -s_0-r_0=-n_{v_0},
\end{align*}
with the aid of Lemma~\ref{lem:s+r}. The estimate $n_{v_0}\leq \operatorname{d}(T_{v_0})$ follows easily from \eqref{eq:ell-bound} and \eqref{eq:xi-sum}.
\end{proof}
By the last bound of Lemma~\ref{lem:multiplier-bound} it is clear that $C_{E_j}=\bigl(\,\prod_{i=j+1}^{\abs{V_\text{int}}-1} C_{v_i} \bigr)\widetilde C_{E_j}$. In fact, each integral in $E_j$ produces a $\gamma^{-1}$ in the upper bound, and we get
$
\widetilde C_{E_j}\leq (C \gamma^{-1} )^{\abs{V_\text{int}}-j-1}.
$
Starting from the explicit expressions of the multipliers, we estimate
$
\prod_{i=0}^{\abs{V_\text{int}}-1} C_{v_i} \leq (C g)^{\abs{V}} (C \epsilon)^{\operatorname{d}(T_{v_0})}
$
with the aid of \eqref{eq:K-bounds} and \eqref{eq:a-prod}. Here $V$, defined above \eqref{eq:scalar-subtree}, is the set of all nodes including the black dot (\raisebox{3pt}{\TC*}) end nodes. Since $\abs{V_\text{int}}\leq \abs{V}\leq 2\operatorname{n}(T_{v_0})\leq 4\bigl(\operatorname{d}(T_{v_0})-1\bigr)$,
\begin{equation*}
\Biggl(\,\prod_{i=0}^{j}C_{v_i}\Biggr) C_{E_j} \leq g^{\abs{V}+1+j-\abs{V_\text{int}}} (C \epsilon)^{\operatorname{d}(T_{v_0})}
\qquad (0<j<\abs{V_\text{int}}).
\end{equation*}
Notice that the fraction $1/R_{i}$ is analytic on the domain where
$
\abs{R}\neq \abs{\tilde k_{i}\gamma+i\omega\cdot Q_{i}}.
$
Thus, all the $1/R_i$ are analytic in the punctured neighbourhood of the origin
\begin{equation*}
0<\abs{R}\leq \rho\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=} \tfrac{1}{2} \min \bigl\{\{\gamma \} \cup\{ \abs{\omega\cdot Q_i}\;|\;Q_i\neq 0\} \bigr\}.
\end{equation*}
On the circle $\abs{R}=\rho$,
$
\abs{R_{i}^{1+p}}^{-1} \leq \rho^{-1-p},
$
provided $p\geq -1$.
According to \eqref{eq:k-range},
\begin{equation*}
\prod_{i=0}^{j-1}\biggl| \frac{1}{R_{i+1}^{1+p_i-p_{i+1}}} \biggr| \leq \rho^{-j-p_0+p_j} \qquad(0<j<\abs{V_\text{int}}).
\end{equation*}
Subsequently, summing over $j$ the bounds on \eqref{eq:second-case1} and \eqref{eq:third-case1} we get
\begin{equation*}
\sum_{j=1}^{\abs{V_\text{int}}-1} (C \epsilon)^{\operatorname{d}(T_{v_0})} g^{\abs{V}+1+j-\abs{V_\text{int}}} \rho^{-j-1-p_0} e^{-\sigma\abs{q}}
\leq (C \epsilon)^{\operatorname{d}(T_{v_0})} (g/\rho)^{\abs{V}+1} g^{-1} e^{-\sigma\abs{q}}
\mathrel{=\mkern-4.2mu\raise.095ex\hbox{$:$}} B(g)e^{-\sigma\abs{q}}.
\end{equation*}
At the $\ell$th order, $\abs{Q_i}\leq \ell N$ for each $i$, and thus $\rho\geq \tfrac{1}{2} \min(\gamma, a (\ell N)^{-\nu})$.
From the above it is also clear that the order of the pole at $R=0$ in our integrals does not exceed $\abs{V_\text{int}}+1$. For the purposes of Lemma~\ref{lem:shift-of-contour}, we compute (using $g,\gamma\leq C$)
\begin{equation*}
\begin{split}
& B(g)\sum_{J=0}^{\abs{V_\text{int}}+1}\frac{1}{J!} \Bigl(\frac{\rho \vartheta}{g}\Bigr)^J \leq C B(g)
\leq (C_N \epsilon)^{\operatorname{d}(T_{v_0})} g^{-1} (\ell!)^{4\nu}.
\end{split}
\end{equation*}
There are at most
$
(4\ell)!\leq 4^{4\ell}(\ell!)^4
$
orders in which we can exhaust all of the up to $4\ell$ lines of a tree contributing at order $\ell$.
\subsection{Remaining integrals}
The integral over $\mathbb{R}_-$ in \eqref{eq:Fintegral} is simple, because the integrand satisfies the identity \eqref{eq:F-identity} and has the analyticity properties stated below that equation. In other words, we can separate the (possible) pole and the constant term from the rest: writing $F(X^u;0,z,\theta)\equiv F(z,\theta)$,
\begin{equation*}
F(X^u;t,z,\theta)=t^p z^{-1} e^{-\gamma t} F_{-1}(\theta+\omega t)+t^p F_{0}(\theta+\omega t)+t^p \delta_1 F(ze^{\gamma t},\theta+\omega t).
\end{equation*}
Here $p=0,1$. Applying $\Xint-_{-\infty}^0$ on each term separately, we get that $\Xint-_{-\infty}^0 F(X^u;t,z,\theta)$ equals
\begin{align*}
(-1)^p\sum_{q'}\frac{z^{-1} e^{iq'\cdot\theta} \hat F_{-1}(q')}{(iq'\cdot\omega-\gamma)^{1+p}}
\,+\, (-1)^p\sum_{q'\neq 0}\frac{e^{iq'\cdot\theta} \hat F_{0}(q')}{(iq'\cdot\omega)^{1+p}} \,+\, \int_{-\infty}^0 t^p\delta_1F(ze^{\gamma t},\theta+\omega t)\,dt.
\end{align*}
These terms are small compared to to the large bounds obtained for the $\Xint-_0^\infty$ part above.
We also have to study the coefficients $c_{ij}^p$ appearing in \eqref{eq:as-expansion}. This is most conveniently done in terms of the representation \eqref{eq:c_ij}. Since $M^u=\order{\epsilon}$, only
\begin{equation*}
\Xint-_{-\infty}^{-t}\Bigl[f^u_{\psi\varphi}(B^uM^u)^l K_{ij}^p\Bigr](\tau)\,d\tau
\end{equation*}
with $l \leq \ell$ can contribute to $c_{ij}^p$ at order $\ell$. We only need to consider $t\geq 0$, as $\Upsilon(t)=\Upsilon(-t)$.
The above integral consists in obvious shorthand notation, through \eqref{eq:Ksplit}, of $4^l$ terms like
{\small
\begin{equation*}
\Xint-_{-\infty}^{-t} (-t-\tau_l)^{p_{l+1}}(fK_l)(\tau_l)\Xint-_{-\infty}^{\tau_l}(\tau_l-\tau_{l-1})^{p_l} (\bar K_l M)(\tau_{l-1})\cdots K_1(\tau_1)\Xint-_{-\infty}^{\tau_1}(\tau_1-\tau_0)^{p_1}(\bar K_1 M)(\tau_{0})\tau_0^{p_0}K_0(\tau_0),
\end{equation*}
}%
where $p_{l+1}=0$. If $p_0=1$ (instead of 0), we use $\tau_0=(\tau_0-\tau_1)+\dots+(\tau_{l-1}-\tau_l)-(-t-\tau_l)-t$, getting $l+2$ terms of the original form except that $p_0=0$ and either there is a factor $t$ or precisely one change $p_i\mapsto p_i+1$ occurs. Due to $M=\order{\epsilon g^2}$, \eqref{eq:K-bounds} and \eqref{eq:a-prod}, each $K_i\bar K_i M$ produces a factor $CA\abs{\epsilon} g^{p_i+1}$ to the upper bound, whereas analyticity yields $e^{-\sigma\abs{q_i}}$ (which we prefer although at each order we reduce to trigonometric polynomials);
$
\prod_{i=0}^l e^{-\sigma\abs{q_i}}\leq e^{-\sigma\abs{q}}.
$
Last, the integral involves a total of $\sum_{i=1}^{l+1}(1+p_i)\leq 2l+3$ denominators of the form $\tilde k_i \gamma + i\omega\cdot Q_i$, where $\tilde k_i\in\mathbb{Z}$ and $Q_i\mathrel{\raise.095ex\hbox{$:$}\mkern-4.2mu=} \sum_{j=0}^i q_j$ with $Q_l=q\neq 0$. We bound these by
\begin{equation*}
\left( \frac{1}{\min\bigl\{\{\gamma\}\cup \{|\omega\cdot Q_i| \;|\; Q_i\neq 0\}\bigr\} }\right)^{\sum_{i=1}^{l+1}(1+p_i)} \leq \left(\max(\gamma^{-1},a^{-1} (N\ell)^\nu) \right)^{\sum_{i=1}^{l+1}(1+p_i)}.
\end{equation*}
Altogether, the bounds above yield
$
C_N^\ell \abs{\epsilon}^l g^{-4} e^{-\sigma\abs{q}}(\ell!)^{2\nu}.
$
Skipping further details, this is the upper bound on the integral. It is smaller than what was derived for the integral $\Xint-_{-\infty}^\infty F$---and thus for $\mathcal{E}^p_{ij}$ in \eqref{eq:as-expansion}---above. In conclusion, among the contributions
$
\bigl(c_{ij}^p\bigr)^{\ell_1}\bigl(\mathcal{E}_{ij}^p\bigr)^{\ell_2}
$
($\ell_1+\ell_2=\ell-1$) to $\Upsilon^\ell$, $\bigl(\mathcal{E}_{ij}^p\bigr)^{\ell-1}$ is the most dangerous one.
\enlargethispage{5mm}
\begin{remark}
Above, the sums over $q_i$, with $\sum_i q_i=q$ and $\sum_i \abs{q_i}\leq \ell N$ were dealt with as follows. Since the analyticity domain with respect to $\theta$ is the compact $\{\abs{{\operatorname{\Im\mathfrak{m}}\,}{\theta}}\leq \sigma\}$, it can be substituted by some $\{\abs{{\operatorname{\Im\mathfrak{m}}\,}{\theta}}\leq \sigma'\}$ with $\sigma<\sigma'$. Then, for every $i$ we actually have the factor $e^{-\sigma'\abs{q_i}}$ in the estimates above. While $e^{-\sigma\sum_i\abs{q_i}}\leq e^{-\sigma\abs{q}}$, we get rid of the sums over $q_i$:
\begin{equation*}
\sum_{q_i}e^{-(\sigma'-\sigma)\abs{q_i}}\leq C_{\sigma'-\sigma}.
\end{equation*}
A similar remark applies to the Fourier indices in Subsection~\ref{subsec:estimates}. \qed
\end{remark}
| -127,364.894963 |
[
-2.4765625,
2.115234375
] | 25.626305 |
[
-3.474609375,
0.1920166015625,
-2.048828125,
-6.91796875,
-1.0185546875,
9.4765625
] |
[
1.302734375,
7.6171875,
-0.470947265625,
4.51953125
] | 825 | 12,028 |
[
-3.14453125,
3.349609375
] | 33.811534 |
[
-5.48828125,
-3.798828125,
-4.4140625,
-2.650390625,
1.693359375,
11.9375
] | 0.681167 | 13.297588 | 27.49418 | 3.235329 |
[
1.9204424619674683
] | -74,548.798452 | 7.079149 | -126,559.036755 | 0.19378 | 6.645085 |
[
-1.85546875,
-3.513671875,
-4.43359375,
-6.09765625,
2.255859375,
13.75
] |
[
-4.48046875,
-1.0234375,
-1.478515625,
-1.201171875,
2.58984375,
2.494140625
] | |
BkiUfHs5qhLBvG9b_Ccl
|
\section{
\section{Introduction}
The prototypical instability in resistive magnetohydrodynamics (MHD) is the {tearing instability}. As the name suggests, this instability describes the growth of the ``tearing'' of magnetic field or, to be more precise, the change in the field's magnetic topology. In two dimensions, the tearing instability creates a series of magnetic islands, similar to Kelvin's cats eyes \citep[\emph{e.g.}][]{schindler06}. In three dimensions, the change in magnetic topology can be more complicated \citep[\emph{e.g.}][]{priest14}.
The standard magnetic field configuration for the tearing instability is the {current sheet}. The name derives from a thin layer of intense (compared to the surrounding environment) current density located in a highly sheared magnetic field. Normally, the magnetic field points in opposite directions on either side of the current sheet, with the width of the current sheet (where the change takes place) being much smaller than the typical length scale of the large-scale system.
Since the seminal work of \cite{furth63}, there have been many studies of the tearing instability that consider effects such as different geometries or the inclusion of extra physics \citep[\emph{e.g.}][]{pritchett80,terasawa83,tassi07,tenerani15}. In the context of solar physics, magnetic reconnection (the change of magnetic topology) is a fundamental physical process, so the tearing instability is of great interest in this field. Solar eruptions, ranging from flares to jets to coronal mass ejections (CMEs) are often believed to be triggered by magnetic reconnection \citep[\emph{e.g.}][]{macneice04,dmac14,dmac15}. MHD simulations have demonstrated that fast eruptive behaviour is strongly linked to the tearing of current sheets.
Although large-scale MHD simulations, such as those cited above, can describe the nonlinear evolution of the tearing instability, they are not so effective when it comes to analysing the onset of instability. The complex geometries, sensitivity to boundary conditions and low (compared to the corona) Lundquist numbers make a detailed analysis of the onset of instability very challenging. Therefore, studies that focus only on the (linear) onset of the instability are still very important.
When studying the onset of the tearing instability, the vast majority of studies have focussed on {normal-mode analysis} \citep[\emph{e.g.}][]{chandra61}. That is, solutions are sought with a time dependence of the form
\begin{equation}
\phi \sim \exp(-{\rm i}\omega t),
\end{equation}
where $\phi$ represents a variable of the system, $\omega$ is the frequency and $t$ is time. If $\Im(\omega)>0$, then $\phi$ grows exponentially as $t\rightarrow\infty$. Otherwise, if $\Im(\omega)<0$, then $\phi$ decays exponentially as $t\rightarrow\infty$. The objective of normal-mode analysis is to find the largest value of $\Im(\omega)$, which corresponds to the fastest growing mode. The onset of the instablilty can, therefore, be recast as an eigenvalue problem for eigenvalues $\omega$.
For the tearing instability, there is one eigenvalue such that $\Im(\omega)>0$. Hence, there is only one mode that causes exponential growth in the linearized system and is referred to as the {tearing mode}. It can be shown analytically \citep{furth63,schindler06} that the growth rate of the tearing mode depends on the magnetic Lundquist number $S$ (which we shall define later) in the form $S^{-\alpha}$, where $0<\alpha<1$. For environments such as the solar corona, where the magnetic Lundquist number is ${\rm O}(10^{8})$ and above \citep[\emph{e.g.}][]{hood11}, the tearing mode growth rate is very slow compared to rapidly occuring phenomena like flares. This has led researchers to study the nonlinear tearing instability in order to find faster dynamics. However, it may be the case that a faster onset of the instability can be found in the analysis of the linearized system by including the energy growth ignored by normal-mode analysis.
As mentioned above, eigenvalues describe the behaviour of growth or decay as $t\rightarrow\infty$. In normal-mode analysis, all eigenvalues satisfying $\Im(\omega)<0$ (exponential decay) are ignored. However, modes associated with these rejected eigenvalues can produce {transient growth} which, although it decays exponentially as $t\rightarrow \infty$, can produce significant energy growth within a finite time. If such transient growth is large enough, the growth of the linear system could enter the nonlinear regime much faster than by the growth rate of the tearing mode alone. {Therefore, the transient growth due to the damped modes of the system could lead to current sheet disruption much faster than by the growth rate of the (unstable) tearing mode.}
There has been a lot of interest in the study of transient growth of the linearlized Navier-Stokes equations for shear flows \citep[\emph{e.g.}][]{reddy93a,reddy93b,schmid94,hanifi96}. Mathematically, transient growth corresponds to the {non-normality} of the system. What characterises a non-normal system is the non-orthogonality of eigenmodes. To analyse the non-normal behaviour of such systems, a generalization of the eigenvalue spectrum, known as the {pseudospectrum}, can be used \citep{trefethen05}. \cite{bobra94} use pseudospectra to relate ideal and resistive MHD spectra. They show that the resistive MHD eigenmodes, for sheared background fields, are strongly non-orthogonal and, hence, can exhibit transient growth. The effects of non-normal behaviour in MHD have also been studied in the context of magnetic field generation \citep[\emph{e.g.}][]{farrell99a,farrell99b,livermore06}. In solar physics, transient energy growth has attracted attention in solar wind applications \citep[\emph{e.g.}][]{camporeale09,camporeale12}.
The effects of non-normal behaviour have not, to our knowledge, been applied to eruptive behaviour in the corona, which is the focus of this paper. By considering a sheared background magnetic field (a current sheet) we will study the effects of transient behaviour in the cases when the system is (spectrally) stable and unstable to the tearing instability. We illustrate the non-normality of the associated operator using a particular form of the pseudospectrum that is simple to calculate once the eigenvalue spectrum has been obtained. The paper is outlined as follows: the initial model equations and boundary conditions are introduced, the background theory for calculating the optimal energy growth is discussed, the spectra and energy growth envelopes are displayed for several cases, and the paper concludes with a discussion of potential applications and further work.
\section{Model Description}
To study the tearing instability, we consider the 2D incompressible MHD equations
\begin{equation}\label{mhd1}
\rho\left(\frac{{\partial}\mathbf{u}}{{\partial} t}+(\mathbf{u}\cdot{\bf\nabla})\mathbf{u}\right) = - {\bf\nabla} p+\mu^{-1}({\bf\nabla}\times\mathbf{B})\times\mathbf{B},
\end{equation}
\begin{equation}\label{mhd2}
\frac{{\partial}\mathbf{B}}{{\partial} t} = {\bf\nabla}\times(\mathbf{u}\times\mathbf{B})+\eta{\bf\nabla}^2\mathbf{B},
\end{equation}
\begin{equation}\label{mhd3}
{\bf\nabla}\cdot\mathbf{B} = {\bf\nabla}\cdot\mathbf{u} = 0,
\end{equation}
where $\mathbf{B}$ is the magnetic field, $\mathbf{u}$ is the velocity, $\rho$ is the (constant) density, $p$ is the plasma pressure, $\eta$ is the constant magnetic diffusivity and $\mu$ is the magnetic permeability. Although compressible MHD would be a more suitable model for the solar atmosphere, we choose to use incompressible MHD for two reasons. The first reason is simplicity - to illustrate our procedure, incompressible MHD allows for an obvious measure of the disturbance energy. The theory that we shall develop, however, could be extended to compressible MHD and more complicated models. The second reason is that most of the literature on the tearing instability uses incompressible MHD. Therefore, comparison with previous work can be made more directly.
For our background (static) equilibrium,
\begin{equation}
\quad p_0 = p_0(x), \quad {\mathbf{B}}_0 = {B_{0z}}(x)\mathbf{e}_z, \quad \mathbf{u}_0={\bf 0},
\end{equation}
where the zero subscript corresponds to the equilibrium and
\begin{equation}\label{equil}
p_0(x) + \frac{1}{2\mu} B_{0z}^2(x) = {\rm const.}
\end{equation}
Before choosing a particular form for $B_{0z}(x)$, let us linearize the MHD equations. Setting $(\mathbf{u},\mathbf{B},p) = (\mathbf{u}_0,\mathbf{B}_0,p_0) + (\mathbf{u}_1,\mathbf{B}_1,p_1)$ results in the linearization
\begin{equation}\label{lmhd1}
\rho\frac{{\partial}\mathbf{u}_1}{{\partial} t} = -{\bf\nabla} p_1 + \mu^{-1}({\bf\nabla}\times\mathbf{B}_1)\times\mathbf{B}_0 + \mu^{-1}({\bf\nabla}\times\mathbf{B}_0)\times\mathbf{B}_1,
\end{equation}
\begin{equation}\label{lmhd2}
\frac{{\partial}\mathbf{B}_1}{{\partial} t} = {\bf\nabla}\times(\mathbf{u}_1\times\mathbf{B}_0) + \eta{\bf\nabla}^2\mathbf{B}_1
\end{equation}
\begin{equation}\label{lmhd3}
{\bf\nabla}\cdot\mathbf{B}_1 = {\bf\nabla}\cdot\mathbf{u}_1 = 0.
\end{equation}
Note that we are assuming $\eta\ll1$ which is typical in many solar and astrophysical applications. We therefore ignore the contribution of diffusion on the background equilibrium in Equation \ref{lmhd2}, expecting the dynamics of the instability to to occur on a much shorter time scale than the diffusion time.
We now look for solutions of the form
\begin{equation}
\mathbf{u}_1 = [u(x,t),0,u_z(x,t)]^{\rm T} {\rm e}^{ikz}, \quad \mathbf{B}_1 = [b(x,t),0,b_z(x,t)]^{\rm T} {\rm e}^{ikz},
\end{equation}
where $k$ is the wavenumber of disturbances in the $z$-direction. Taking the curl of Equation \ref{lmhd1}, we eliminate $p_1$. Using the solenoidal constraints in Equation \ref{lmhd3}, we can eliminate $u_z$ and $b_z$. This leaves:
\begin{equation}\label{lin1}
\frac{{\partial}}{{\partial} t}\left(\frac{{\partial}^2u}{{\partial} x^2}-k^2u\right) = \frac{ik{B_{0z}}}{\mu\rho}\left(\frac{{\partial}^2b}{{\partial} x^2}-k^2b\right) - \frac{ikB''_{0z}}{\mu\rho}b,
\end{equation}
\begin{equation}\label{lin2}
\frac{{\partial} b}{{\partial} t} = ik{B_{0z}} u + \eta\left(\frac{{\partial}^2b}{{\partial} x^2}-k^2b\right),
\end{equation}
where a prime denotes differentiation with respect to $x$.
\subsection{Equilibrium}
We choose a classic form for the background magnetic field known as the {Harris sheet}. The magnetic field of the Harris sheet is given by
\begin{equation}
{B_{0z}}(x) = B_0\tanh\left(\frac{x}{x_0}\right), \quad B''_{0z}(x) = -\frac{B_0}{x_0^2}\frac{2}{\cosh^2(x/x_0)}\tanh\left(\frac{x}{x_0}\right),
\end{equation}
where $B_0$ is the maximal field strength and $x_0$ measures the thickness of the current sheet. The equilibrium pressure then comes from Equation \ref{equil} but is not important for our calculations.
\subsection{Non-dimensionalization}
To non-dimensionalize the equations, consider
\begin{equation}\label{nd}
u = u_0u^*, \quad b = B_0b^*, \quad t = t_0t^*, \quad x=x_0 x^*,
\end{equation}
with
\begin{equation}
t_0 = \frac{x_0}{u_0}, \quad u_0 = \frac{B_0}{\sqrt{\mu\rho}},
\end{equation}
where $u_0$ is the Alfv\'en speed. The linearized MHD equations become (after dropping the asterisks)
\begin{equation}\label{nlmhd1}
\frac{{\partial}}{{\partial} t}\left(\frac{{\partial}^2u}{{\partial} x^2}-k^2u\right) = {ik{B_{0z}}}\left(\frac{{\partial}^2b}{{\partial} x^2}-k^2b\right) - {ikB''_{0z}}b,
\end{equation}
\begin{equation} \label{nlmhd2}
\frac{{\partial} b}{{\partial} t} = ik{B_{0z}} u + S^{-1}\left(\frac{{\partial}^2b}{{\partial} x^2}-k^2b\right),
\end{equation}
where
\begin{equation}
S = \frac{x_0u_0}{\eta}
\end{equation}
is the (non-dimensional) Lundquist number.
\subsection{Boundary Conditions}
We require that $b\rightarrow 0$ and $u\rightarrow 0$ as $x\rightarrow\pm\infty$. However, since numerical simulations typically range bewteen finite values, we shall approximate the boundary conditions as $b=u=0$ at $x=\pm d$ for some $d>0$. This approach will make possible comparisons to simulations of tearing instabilities more feasible. Also, since the tearing instability develops in a thin boundary layer near $x=0$, a value of $d$ much larger than the width of the boundary layer will result in a good approximation. In the Appendix, we perform one of our subsequent calculations in the half-plane $(x,z)\in[0,\infty)\times(-\infty,\infty)$. Comparing this to the corresponding result from the closed domain reveals that the exact form of the boundary conditions is not of vital importance for the results of this paper.
\section{Background Theory}
In this section we discuss the background theory for determining the optimal energy growth and how non-normal contributions are included. { Our aim is to solve the full initial value problem, rather than just the eigenvalue problem. However, in order to determine the effects of different modes on energy growth, we recast the initial value problem in terms of a selection of eigenvalues and (corresponding) eigenmodes. By considering the kinetic and magnetic energies, we define a (physically) suitable norm for the system and use this to determine the optimal energy growth.}
\subsection{Operator Equations}
In anticipation of the numerical approach that we shall describe later, we write the linearized MHD equations as a maxtrix-vector system. Equations \ref{nlmhd1} and \ref{nlmhd2} can be written in the form
\begin{equation}\label{ivp}
\frac{{\partial}}{{\partial} t}{\bf{\mathcal{M}}}\mathbf{v} = {\mathcal{L}}\mathbf{v},
\end{equation}
with
\begin{equation}\label{mm}
{\bf{\mathcal{M}}}=\left(\begin{array}{cc}
{\mathcal{D}}^2-k^2 & 0\\
0&\mathcal{I}
\end{array}\right), \quad {\mathcal{L}} = \left(\begin{array}{cc}
0 & {\mathcal{L}}_I\\
ikB_z&{\mathcal{L}}_R
\end{array}\right), \quad \mathbf{v} = \left(\begin{array}{c}
u\\
b
\end{array}\right),
\end{equation}
and
\begin{equation}
{\mathcal{L}}_I = ikB_z({\mathcal{D}}^2-k^2) - ikB_z'', \quad {\mathcal{L}}_R = S^{-1}({\mathcal{D}}^2-k^2), \quad {\mathcal{D}} = \frac{{\partial} }{{\partial} x},
\end{equation}
where $\mathcal{I}$ represents the identity operator. If we consider solutions of the form
\begin{equation}\label{et}
\mathbf{v} = \widetilde{\mathbf{v}}\exp(-i{\omega} t), \quad \omega \in \mathbb{C},
\end{equation}
we can transform the initial value problem of Equation \ref{ivp} into the generalized eigenvalue problem
\begin{equation}\label{evalue}
-i{\omega} {\bf{\mathcal{M}}}\widetilde{\mathbf{v}} = {\mathcal{L}}\widetilde{\mathbf{v}}.
\end{equation}
Making the assumption of Equation \ref{et} restricts us to examining growth or decay in the limit of $t\rightarrow\infty$ only. In normal-mode analysis, we would solve Equation \ref{evalue} for the eigenvalue with the largest value of $\Im(\omega)>0$. This approach, however, misses the possibilty of transient growth due to eigenmodes with corresponding eigenvalues satisfying $\Im(\omega)<0$, {\emph{i.e.} damped modes.}
In our calculations of transient growth, we shall make use of the eigenvalue spectrum calculated from Equation \ref{evalue} and the corresponding eigenmodes. In practice, however, we shall only need to consider a finite number of eigenmodes since not all eigenfuctions will contribute non-normal behaviour. Therefore, we restrict ourselves to the space $\mathbb{S}^N$ spanned by the first $N$ {least damped} eigenmodes of ${\bf{\mathcal{M}}}^{-1}{\mathcal{L}}$:
\begin{equation}
\mathbb{S}^{N} = {\rm span}\{\widetilde{\mathbf{v}}_1,\dots,\widetilde{\mathbf{v}}_N\}.
\end{equation}
We expand the vector functions $\mathbf{v}\in\mathbb{S}^N$ in terms of the basis $\{\widetilde{\mathbf{v}}_1,\dots,\widetilde{\mathbf{v}}_N\}$:
\begin{equation}\label{kap}
\mathbf{v} = \sum^N_{n=1}{\kappa}_n(t)\widetilde{\mathbf{v}}_n.
\end{equation}
Note that the expansion coefficients $\kappa_n$ are functions of $t$ since we are solving the full initial value problem of Equation \ref{ivp} and not the restricted problem of Equation \ref{evalue}. We can restate Equation \ref{ivp} in the simple form
\begin{equation}
\frac{{\rm d} {\boldsymbol\kappa}}{{\rm d} t} = -i\Lambda{\boldsymbol\kappa}, \quad \Lambda\in\mathbb{C}^{N\times N}, \quad {\boldsymbol\kappa} \in \mathbb{C}^{N},
\end{equation}
with
\begin{equation}
{\boldsymbol\kappa} = [\kappa_1\dots,\kappa_N], \quad {\Lambda} = {\rm diag}[{\omega}_1\dots,{\omega}_N].
\end{equation}
The operator $\Lambda$ represents the linear evolution operator, ${\bf{\mathcal{M}}}^{-1}{\mathcal{L}}$, projected onto the space $\mathbb{S}^N$.
\subsection{Energy Norm}
In order to complete the transformation of the vector functions $\mathbf{v}$ to coefficients $\boldsymbol{\kappa}$, we must consider the scalar product and its associated norm. To measure the disturbance energy, we consider the combination of the (nondimensional) disturbance kinetic and magnetic energies
\begin{equation}
E_V = \frac12\int_V(|\mathbf{u}|^2+|\mathbf{b}|^2)\,{\rm d} V.
\end{equation}
From Equation (\ref{lmhd3}) we have
\begin{equation}
{\mathcal{D}} u + iku_z = 0, \quad {\mathcal{D}} b + ikb_z = 0.
\end{equation}
Therefore, by virtue of Parseval's equivalence \citep[\emph{e.g.}][]{tichmarsh48}, we can write
\begin{equation}\
E_V = \int_k\frac{1}{2k^2}\int_{-d}^{d}(|{\mathcal{D}} u|^2 + k^2|u|^2 + |{\mathcal{D}} b|^2 + k^2|b|^2)\, {\rm d} x\,{\rm d} k .
\end{equation}
Following previous works \citep[\emph{e.g.}][]{reddy93a}, we take the energy density $E$ as
\begin{equation}\label{eden}
E= \frac{1}{2k^2}\int_{-d}^{d}(|{\mathcal{D}} u|^2 + k^2|u|^2 + |{\mathcal{D}} b|^2 + k^2|b|^2)\, {\rm d} x.
\end{equation}
Since Equation \ref{eden} provides a sensible measure of the energy for a given $k$, we define the energy norm as
\begin{equation}\label{norm}
\|\mathbf{v}\|_{\rm E}^2 = \frac{1}{2k^2}\int_{-d}^{d}(|{\mathcal{D}} u|^2 + k^2|u|^2 + |{\mathcal{D}} b|^2 + k^2|b|^2)\, {\rm d} x
\end{equation}
For any $\mathbf{v}_1$,$\mathbf{v}_2\in\mathbb{S}^N$, the inner product associated with the above energy norm can be written as
\begin{equation}\label{inner}
(\mathbf{v}_1,\mathbf{v}_2)_{\rm E} = \frac{1}{2k^2}\int_{-d}^{d}\mathbf{v}_2^{H}{\mathcal{Q}}\mathbf{v}_1\,{\rm d} x,
\end{equation}
where
\begin{equation}
{\mathcal{Q}} = \left(\begin{array}{cc}
k^2-{\mathcal{D}}^2&0\\
0&k^2-{\mathcal{D}}^2
\end{array}\right),
\end{equation}
and the superscript $H$ represents the complex-conjugate transpose. The integrands in Equations \ref{norm} and \ref{inner} can be related \emph{via} integration by parts. Equation \ref{inner} can be written as
\begin{equation}
(\mathbf{v}_1,\mathbf{v}_2)_{\rm E} = \frac{1}{2k^2}\int_{-d}^{d}\mathbf{v}_2^{H}{\mathcal{Q}}\mathbf{v}_1\,{\rm d} x = {\boldsymbol\kappa}^{H}{Q}{\boldsymbol\kappa},
\end{equation}
where the matrix ${Q}$ has components
\begin{equation}
Q_{ij} = (\widetilde{\mathbf{v}}_i,\widetilde{\mathbf{v}}_j)_{\rm E} = \frac{1}{2k^2}\int_{-d}^{d}\widetilde{\mathbf{v}}_j^H{\mathcal{Q}}\widetilde{\mathbf{v}}_i\, {\rm d} x.
\end{equation}
The matrix ${Q}$ is both Hermitian and positive definite. We can, therefore, factor ${Q}$ according to ${Q}={F}^H{F}$ \citep[\emph{e.g.}][]{trefethen97}, leading to
\begin{eqnarray}
(\mathbf{v}_1,\mathbf{v}_2)_{\rm E} &=& {\boldsymbol\kappa}_2^H{Q}{\boldsymbol\kappa}_1 \\
&=& {\boldsymbol\kappa}_2^H{F}^H{F}{\boldsymbol\kappa}_1 \\
&=& ({F}{\boldsymbol\kappa}_1,{F}{\boldsymbol\kappa}_2)_2.\label{inner_F}
\end{eqnarray}
The associated vector norm satisfies
\begin{equation}
\|\mathbf{v}\|_{\rm E} = \|{F}{\boldsymbol\kappa}\|_2, \quad \mathbf{v}\in\mathbb{S}^N.
\end{equation}
This relationship between the energy norm and the $L^2$ norm will be useful for the practical calculation of the optimal energy growth that we shall discuss shortly.
\subsection{Optimal Growth}
The formal solution of the initial value problem \ref{ivp} can be written as
\begin{equation}
\mathbf{v} = \exp({\bf{\mathcal{M}}}^{-1}{\mathcal{L}} t)\mathbf{v}_0, \quad \mathbf{v}_0 = \mathbf{v}(0).
\end{equation}
Using Equation \ref{kap} we can transform the above result to
\begin{equation}
{\boldsymbol\kappa} = \exp(-i{\Lambda}t){\boldsymbol\kappa}_0, \quad {\boldsymbol\kappa}_0 = {\boldsymbol\kappa}(0).
\end{equation}
The optimal transient growth of the disturbance energy is given by the norm of the matrix exponential
\begin{eqnarray}
G(t) \equiv G(t,S,k) &=& \max_{\mathbf{v}_0\ne\bf{0}}\frac{\|\mathbf{v}(t)\|_{\rm E}^2}{\|\mathbf{v}_0\|_{\rm E}^2} \\
&=& \max_{{\boldsymbol\kappa}_0\ne\bf{0}}\frac{\|F{\boldsymbol\kappa}(t)\|_2^2}{\|F{\boldsymbol\kappa}_0\|_2^2} \\
&=& \max_{{\boldsymbol\kappa}_0\ne\bf{0}}\frac{\|F\exp(-i{\Lambda}t){\boldsymbol\kappa}_0\|_2^2}{\|F{\boldsymbol\kappa}_0\|_2^2} \\
&=& \max_{{\boldsymbol\kappa}_0\ne\bf{0}}\frac{\|F\exp(-i{\Lambda}t)F^{-1}F{\boldsymbol\kappa}_0\|_2^2}{\|F{\boldsymbol\kappa}_0\|_2^2} \label{pogr}\\
&=&\|{F}\exp(-i{\Lambda}t){F}^{-1}\|_2^2. \label{ogr}
\end{eqnarray}
Equation \ref{ogr} follows from Equation \ref{pogr} {via the definition of an induced norm}.
The curve traced out by $G(t)$ {\it vs.} $t$ represents the maximum possible energy amplification, which for each instant of time is optimized over all possible initial conditions with unit energy norm \citep{schmid94}. The initial disturbance that optimizes the amplification factor can be different for different times. Therefore, $G(t)$ should be thought of as the {envelope} of the energy growth of individual initial conditions with unit energy norm. Henceforth, we shall refer to $G(t)$ as the {optimal energy envelope}.
\section{Numerical Procedure}
In this section we briefly outline the main numerical procedures for the required calculations. Until now, we have presented the theory in terms of the underlying {operators}. Since a practical solution requires a (finite) discretization of the problem, we shall henceforth refer to {matrices} rather than operators and {eigenvectors} rather than eigenmodes. When referring back to an equation containing operators, it will be implicit that we are now considering the discretized version of that equation and, hence, are strictly dealing with finite matrices rather than operators.
\subsection{Discretization for the Eigenvalue Problem}
We follow previous works on non-normal stability by expanding the variables in terms of Chebyshev polynomials. These functions are defined in the interval $[-1,1]$. It is trivial to convert from the problem domain $[-d,d]$ to the Chebyshev domain \emph{via} $y=x/d$, with $y\in[-1,1]$. A function can be approximated on the Chebyshev interval as
\begin{equation}\label{cheb}
f(y) = \sum_{n=0}^Na_nT_n(y),
\end{equation}
where
\begin{equation}
T_n(y) = \cos[n\cos^{-1}(y)]
\end{equation}
and the $a_n$ are constants. The unknown variables, $u$ and $b$ in Equation \ref{ivp} are expanded in the form of Equation \ref{cheb}. Derivatives are also expressed in terms of Chebyshev polynomials and make use of standard recurrence relations \citep[\emph{e.g.}][]{as64}. In order to use these recurrence relations, the expanded equations are then required to be satisfied at the Gauss-Lobatto collocation points
\begin{equation}
y_j=\cos\left(\frac{\pi j}{N}\right).
\end{equation}
If we consider the eigenvalue problem of Equation \ref{evalue}, the expansion in terms of Chebyshev polynomials produces a matrix-vector system where the matrices (for the generalized eigenvalue problem) contain spectral differentiation matrices and the vector contains the expansion coefficients $a_n$.
Boundary conditions are included in rows of one of the matrices of the discretized generalized eigenvalue problem. The corresponding rows in the other matrix are chosen to be a complex multiple of these rows. By choosing a large complex multiple, spurious modes associated with the boundary conditions can be mapped to a part on the complex plane far from the region of interest (far below the eigenvalues near $\Im(\omega)=0)$. { To illustrate this approach, consider the discrete form of Equation \ref{evalue}
\begin{equation}
-i\omega M\mathbf{x} = L\mathbf{x}
\end{equation}
where $M$ and $L$ are finite matrices and $\mathbf{x}$ represents an eigenvector. We can write
\begin{equation}
M = \left(\begin{array}{ccc}
T_0(1) & T_1(1) & \cdots \\
T_0''(y_1)-k^2T_0(y_1) & T_1''(y_1)-k^2T_1(y_1) & \cdots \\
\vdots & \vdots & \vdots \\
T_0''(y_{N-1})-k^2T_0(y_{N-1}) & T_1''(y_{N-1})-k^2T_1(y_{N-1}) & \cdots \\
T_0(-1) & T_1(-1) & \cdots \\
\vdots & \vdots & \ddots \end{array}\right),
\end{equation}
where we indicate the layout of the top-left section of the matrix (see the definition of ${\bf{\mathcal{M}}}$ in Equation \ref{mm}). Boundary conditions have been included in the 1st and $N$th rows. The same rows in $L$ are chosen as a complex multiple of the corresponding rows in $M$ \citep{reddy93b}. In this paper, we multiply the rows by $-8000i$. For brevity, we do not display the full matrix of the discretized problem.
}
Once the system is fully discretized, the generalized eigenvalue problem can be solved by standard methods. In this paper, we perform the calculations in MATLAB.
\subsection{Optimal Quantities}
\subsubsection{Energy Growth}
To calculate the optimal energy growth, we make use of singular value decomposition (SVD). Writing $A={F}\exp(-i{\Lambda}t){F}^{-1}$, we can decompose this matrix as
\begin{equation}
AV=\Sigma U,
\end{equation}
where $U$ and $V$ are unitary matrices and $\Sigma$ is a matrix containing the singular values, ordered by size. It can be shown that $\|A\|_2=\sigma_1$, where $\sigma_1$ is the largest singular value of $A$ \citep[\emph{e.g.}][]{trefethen97}. \emph{Via} Equation \ref{ogr}, we use this property to determine the optimal energy growth. Again, we use MATLAB to calculate the SVD.
\subsubsection{Optimal Disturbances}\label{opt}
In order to determine the initial disturbance that will create the maximum possible amplification at a given time $t_0$, we can make further use of the SVD. Let ${A}={F}\exp(-i{\Lambda}t_0){F}^{-1}$. If $\sigma_1$ is the largest singular value of ${A}$ then, as described above,
\begin{equation}
\sigma_1=\|{F}\exp(-i{\Lambda}t_0){F}^{-1}\|_2=\|\exp(-i{\Lambda}t_0)\|_{\rm E}.
\end{equation}
If we perform a decomposition, as before, and now focus only on the column vectors of ${U}$ and ${V}$ corresponding to $\sigma_1$, we obtain
\begin{equation}
{A}{\mathbf{v}_1}=\sigma_1\mathbf{u}_1.
\end{equation}
The effect of ${A}$ on an input vector $\mathbf{v}_1$ results in an output vector $\mathbf{u}_1$ stretched by a factor of $\sigma_1$. That is, $\mathbf{v}_1$ represents an initial condition that will be amplified by a factor $\sigma_1$ due to the mapping ${F}\exp(-i{\Lambda}t_0){F}^{-1}$, where $t_0$ is the time when the amplification is reached \citep[\emph{e.g.}][]{schmid94}. On the subspace $\mathbb{S}^N$, the optimal initial disturbance can be expressed as
\begin{equation}
{\boldsymbol\kappa}_1={F}^{-1}\mathbf{v}_1.
\end{equation}
\section{Spectra and Perturbed Matrices}
In this section we present some of the results from solving the generalized eigenvalue problem of Equation \ref{evalue}. To be more precise, we solve the discretized version of Equation \ref{evalue} subject to the numerical scheme outlined in the previous section. Throughout the rest of the paper, unless specified otherwise, we will set $d=10$. Let us consider $S=1000$ and examine the spectra for the cases $k=0.5$ and $k=1.2$. Figure \ref{spectra} displays these two spectra.
\begin{figure}[h!]
\centering
{\includegraphics[width=9.5cm]{spectrum_R1000_k05.pdf}}
\centerline{
\hspace{0.4525 \textwidth} {(a)}
\hfill}
{\includegraphics[width=9.5cm]{spectrum_R1000_k12_2.pdf}}
\centerline{
\hspace{0.4525 \textwidth} {(b)}
\hfill}
\caption{Spectra for the discrete generalized eigenvalue problem, Equation \ref{evalue}, for $S=1000$ and (a) $k=0.5$ and (b) $k=1.2$. In (a), the unique eigenvalue corresponding to the tearing mode is highlighted.}
\label{spectra}
\end{figure}
For the tearing problem set up in this paper, it can be shown analytically that the equilibrium can only become tearing-unstable for $0<k<1$. The spectrum in Figure \ref{spectra}a is for $k=0.5$ and the system is, therefore, linearly unstable to the tearing instability. As can be seen from this spectrum, there is only one unstable eigenvalue, labelled as corresponding to the tearing mode. This eigenvalue is $\approx$0.0131, which is equivalent to the value obtained from a finite difference solution of the same problem (Hood, private communication). The layout of the spectrum is qualitatively similar to other tearing-unstable spectra that have been calculated for similar boundary conditions and background equilibria \citep[\emph{e.g.}][]{goedbloed10}. There is a distinct branching structure that is found in the spectra of many non-normal matrices \citep{reddy93b}.
In the spectrum for $k=1.2$, in Figure \ref{spectra}b, there are no eigenvalues with $\Im(\omega)>0$. There is still, however, a branching structure similar to the previous spectrum. The branch points of the spectra indicate the non-normal behaviour of this resistive MHD problem. This means that eigenvectors with eigenvalues satisfying $\Im(\omega)<0$ can contribute transient growth to the amplification of energy. In order to reveal this non-normal behaviour, consider the following description. Let ${A}$ be a matrix from which the eigenvalues of the problem are found, and let ${E}$ be a matrix such that $\|{E}\|_2\le 1$. Consider, also, a small parameter $\epsilon\ll 1$. A complex number, $z$, is in the pseudospectrum of ${A}$, $\sigma_{\epsilon}({A})$, if $z$ is in the spectrum of ${A}+\epsilon{E}$ (a similar statement can be made for finite operators). For a normal matrix, points $z\in\sigma_{\epsilon}$ can differ from corresponding points in the spectrum of ${A}$ by ${\rm O}(\epsilon)$, i.e. by the size of the perturbation \citep[\emph{e.g.}][]{trefethen05}.
For a non-normal matrix, however, the difference can be much larger. Instead of the eigenvalues of $A+\epsilon E$ differing from those of $A$ by, at most, ${\rm O}(\epsilon)$, they can differ by ${\rm O}(1)$. Such behaviour is particularly present at the branch points of spectra.
If ${A}$ represents the unperturbed matrix of the spectra displayed in Figure \ref{spectra}, Figure \ref{per} displays the spectra of ${A}+\epsilon{E}$ (for $k=0.5,1.2$) where $\epsilon={\rm O}(10^{-6})$ and the entries of ${E}$ are random and taken from a normal distribution. The eigenvalues of ${A}+\epsilon{E}$, for six different random matrices ${E}$, are shown in red.
\begin{figure}[h!]
\centering
{\includegraphics[width=9.5cm]{per_spectrum_R1000_k05.pdf}}
\centerline{
\hspace{0.4525 \textwidth} {(a)}
\hfill}
{\includegraphics[width=9.5cm]{per_spectrum_R1000_k12.pdf}}
\centerline{
\hspace{0.4525 \textwidth} {(b)}
\hfill}
\caption{Same spectra as in Figure \ref{spectra} but now with eigenvalues of the perturbed matrices included and shown in red.}
\label{per}
\end{figure}
The pseudospectrum of ${A}$ would be the subset of the complex plane given by
\begin{equation}\label{pseudospectrum}
\sigma_{\epsilon} = \bigcup_{\|{E}\|_2\le 1}\sigma({A}+\epsilon{E}).
\end{equation}
As demonstrated in Figure \ref{per}, however, only a few matrices ${E}$ are required to reveal the non-normal character of the matrix ${A}$.
There are several equivalent definitions of pseudospectra \citep{trefethen05}. The definition we have presented here gives the simplest and most practical demonstration of non-normal behaviour. For our current purposes, this version of the pseudospectrum will suffice.
Looking at the eigenvalues of the perturbed matrix, there are two main features that emerge. The first is that for large parts of the spectra, the eigenvalues of $A+\epsilon E$ differ from the eigenvalues $A$ by ${\rm O}(\epsilon)$, indicating normal behaviour. The second feature is that near the branch points of the spectra, the difference is now much larger. For both spectra displayed, a perturbation of ${\rm O}(10^{-6})$ produces a difference of ${\rm O}(10^{-1})$ between the eigenvalues of the matrices $A$ and $A+\epsilon E$ at the branch locations. This jump of five orders of magnitude is a clear signal of non-normality and, hence, the possibility of significant transient growth. Estimating the pseudospectrum of Equation \ref{pseudospectrum} with just a few random matrices ${E}$ is the recommended approach for determining if the system in question is non-normal since it is easily determined from the spectrum which we use for determining the optimal transient growth. Plotting the pseudospectrum estimate, as done in Figure \ref{per}, also reveals what eigenvectors will produce non-normal effects and then should, therefore, be included in the subspace $\mathbb{S}^N$.
\section{Optimal Energy Growth}
\subsection{Spectrally-Stable $k$}
As stated previously, the onset of the tearing instability, for the present setup, occurs only for $0<k<1$ in normal-mode analysis. However, as demonstrated in the previous section, the system is non-normal and allows for the possibility of transient growth, even for $k>1$. To get an overview of the optimal energy growth for spectrally stable $k$, we calculate $\max_{t\ge 0}G(t)$ for different $k$. For the calculation of $G(t)$, we only consider contributions from eigenvalues with $-1.4<\Im(\omega)<0$. This will mean that for different values of $k$, different numbers of eigenvalues (and therefore eigenvectors) will be used in the calculations. However, this range captures most of the effects of the non-normality, as suggested by the pseudospectra, and does not disguise the main results. The values of $\max_{t\ge 0}G(t)$ for a range of $k>1$ and for the cases $S=100$ and $S=1000$ are displayed in Table \ref{table}.
\begin{table}[h]
\begin{center}
\def~{\hphantom{0}}
\caption{Maxima of $G(t)$ in time for $k>1$ and magnetic Lundquist numbers $S=100$, 1000.}
\label{table}
\begin{tabular}{lcc}
\hline
$k$ & $\max G(t,S=100)$ & $\max G(t, S=1000)$ \\[3pt]
\hline
1.1 & 1.6 & 8.48 \\
1.2 & 1.51 & 10.86\\
1.3 & 1.41 & 11.79 \\
1.4 & 1.35 & 11.42 \\
1.5 & 1.29 & 11.57 \\
\hline
\end{tabular}
\end{center}
\end{table}
For $S=100$, the optimal energy growth is small and does not even double in size for the values of $k$ displayed. This result is important as the magnetic Lundquist number for many simulations can be of ${\rm O}(100)$. Therefore, any transient growth would not be noticed. Moving up to $S=1000$, the optimal energy growth can increase by an order of magnitude. In the solar corona, where $S\approx {\rm O}(10^8)$ and higher, it is therefore possible that transient growth for spectrally stable $k$ could become large enough to excite the nonlinear phase of the tearing instability. An example of a $G(t)$ envelope for $k=1.1$, $S=1000$ is shown in Figure \ref{gt_1}.
\begin{figure}[h]
\centering
{\includegraphics[width=9.5cm]{G_R1000_k11.pdf}}
\caption{An example of an optimal energy envelope $G(t)$ for $S=1000$ and $k=1.1$.}
\label{gt_1}
\end{figure}
{In light of the results of Table \ref{table}, we may ask how the transient energy growth can increase with increasing $S$? One way to answer this question is to consider a simple upper bound for the energy growth. For an initial value problem, suppose that $\omega_{\rm I}$ is the imaginary part of the least damped eigenvalue of $\Lambda$. It then follows that
\begin{eqnarray}
\exp(\omega_{\rm I}t) &\le& \|\exp(-i\Lambda t)\|_{\rm E} \\
&\le& \|F\|_2\|F^{-1}\|_2\exp(\omega_{\rm I}t)\\
&\le&\kappa(F)\exp(\omega_{\rm I}t),\label{bound}
\end{eqnarray}
where $\kappa(F)= \|F\|_2\|F^{-1}\|_2$ is the standard notation for the condition number of the matrix $F$ (not to be confused with $\kappa$ from Equation \ref{kap}). If $\kappa(F)=1$ in Equation \ref{bound}, we have equality and the energy bound is determined by the least damped eigenvalue alone. If, however, $\kappa(F) \gg 1$, then there is the potential for substantially larger energy growth at early times, even though it may be that $\omega_{\rm I} <0$. For the tearing-stable case studied above, $\omega_I\approx 0$ and so the energy bound is given by $\kappa(F)$. Table \ref{table2} shows how the condition number varies for some values of $S$ when $k=1.1$.
\begin{table}[h]
\begin{center}
\def~{\hphantom{0}}
\caption{$S$ \emph{vs.} $\kappa(F)$ for $k=1.1$.}
\label{table2}
\begin{tabular}{ccccccc}
\hline
$S$ & 10 & 50 & 100 & 500 & 1000 & 5000 \\
\hline
$\kappa(F)$ & 20 & 690 & 1.6$\times 10^4$ & 2$\times10^8$ & 3$\times 10^8$ & 3.8$\times 10^8$\\
\hline
\end{tabular}
\end{center}
\end{table}
Clearly, using $\kappa(F)$ as an upper bound for the energy is too loose for practical considerations. However, the purpose of displaying these results is to convey the following: as $S$ increases and, hence, the diffusion term in the induction equation is multiplied by a smaller coefficient $S^{-1}$, it may reasonably be expected that the energy bound tends to an ideal MHD limit, where the onset of instability is governed entirely by eigenvalues. However, the opposite is true, allowing for (non-normal) transient effects to play a significant role. As $S$ increases, the eigenvectors (related to $F$ \emph{via} the inner product in Equation \ref{inner_F}) become more ill-conditioned, as discussed in \cite{bobra94}.
Stricter bounds (both upper and lower) for the energy growth can be determined using pseudospectral theory \citep{trefethen05}. However, such considerations go beyond the scope of the present paper and will be considered in future work.
}
\subsection{Spectrally-Unstable $k$}
For $0<k<1$, a normal-mode analysis would produce the eigenvalue with the highest positive value of $\Im(\omega)$, which would represent the growth rate of the linearly unstable system. For the tearing instability, the growth rate behaves as $S^{-\alpha}$ for $0<\alpha<1$, which, for coronal values, is very slow. {For a discussion the various values of $\alpha$, determined by eigenvalue analysis in different regimes, the interested reader is directed to \cite{tenerani16}.}
Since normal-mode analysis ignores any energy growth that decays as $t\rightarrow 0$, the possibility of faster energy growth due to transient effects is often neglected. To demonstrate the possible effect of transients on the growth rate, Figure \ref{gt_2} displays the optimal energy growth envelopes for two cases: the optimal energy growth due to the tearing mode alone and the optimal energy growth due to the combination of the tearing mode and spectrally stable eigenvectors. This example is calculated for $S=1000$ and $k=0.2$ and when transient effects are included, we consider eigenvectors with corresponding eigenvalues with imaginary parts bounded below by $\Im(\omega)=-0.6$.
\begin{figure}[h]
\centering
{\includegraphics[width=9.5cm]{G2_R1000_k02.pdf}}
\caption{Optimal energy growth curves $G(t)$ for $S=1000$ and $k=0.2$. Key: $\Im(\omega)>-0.6$ (solid), $\Im(\omega)>0$ (dash).}
\label{gt_2}
\end{figure}
The dashed curve represents the optimal energy growth using only the tearing mode. This envelope could be produced if we performed a normal-mode analysis. Comparing this curve to the case where other eigenvectors are included in the calculation reveals very interesting behaviour. By $t\approx20$, $G(t)$ from the solid curve increases to $\approx30$ (note that the Figure displays $\log G(t)$), compared to that of the dashed curve which only increases to $\approx1$. Including the effects of transient growth has resulted in an optimal energy growth that proceeds much more rapidly, at short times, compared the contribution from the linearly unstable mode alone. By $t\approx 40$, the solid curve begins to plateau and the growth rate is now less than the dashed curve. This is due to the initial transients decaying and having less effect on the energy growth. From $t\approx 60$ and beyond, both curves become parallel. This behaviour is to be expected as the contribution from the unstable mode dominates as $t\rightarrow\infty$. It is clear from Figure \ref{gt_2} that including the effects of the transients can increase the optimal energy growth substantially.
As mentioned before, the curves of $G(t)$ are envelopes of the optimal energy growth and so, in practice, they may not be reached if the initial perturbation is not optimal. However, what Figure \ref{gt_2} reveals is that even if the optimal energy growth is not attained, the gap between the envelopes for growth with and without transient effects can be large. Hence, even a non-optimal perturbation can produce fast energy growth that could amplify the energy to an order of magnitude (or more) greater than that predicted by normal-mode analysis, within a given time.
\subsection{Optimal Distubances}
The optimal energy envelopes described in the last section represent, at every point in time, the energy amplification optimized over all initial conditions with unit energy norm. As described in Section \ref{opt}, we can determine the optimal perturbation from the same analysis used to calculate $G(t)$. That is, for a given time, we can determine the initial perturbation that produces the optimal energy amplification at that time. To illustrate this, Figure \ref{ini} shows the optimal initial values for the $x$-component of the velocity at times $t=30$, 40 for the case $S=1000$, $k=0.2$.
\begin{figure}[h]
\centering
{\includegraphics[width=9.5cm]{ux_ini_t30.png}}
\centerline{
\hspace{0.4525 \textwidth} {(a)}
\hfill}
{\includegraphics[width=9.5cm]{ux_ini_t40.png}}
\centerline{
\hspace{0.4525 \textwidth} {(b)}
\hfill}
\caption{Optimal initial $u_x$ for (a) $t=30$, (b) $t=40$.}
\label{ini}
\end{figure}
The other components of $\mathbf{u}$ and $\mathbf{b}$ at $t=0$ can also be found. For brevity, we omit displaying them here. The purpose of calculating the optimal initial conditions is described in the following section.
\section{Discussion}
\subsection{Summary}
In this paper, we have demonstrated that the linear onset of the tearing instability can exhibit large transient energy growth due to the non-normality of the associated resistive MHD operator. This energy amplification is found by solving the full initial value problem rather than just the eigenvalue problem of normal-mode analysis. The latter theory is only concerned with the asymptotic growth of the linear system and ignores transient effects. From our illustrative examples we have shown that transient energy growth can be amplified much faster than than that determined purely from normal-mode analysis. This behaviour has been demonstrated for both tearing-stable and tearing-unstable values of the wavenumber.
To determine the optimal energy growth, we have made use of the eigenvalues and eigenvectors of the system. By plotting pseudospectra, we reveal that a subset of eigenvectors contributes to transient energy growth. The eigenvectors of this subset have eigenvalues $\omega$ with $\Im(\omega)<0$, which are ignored by normal-mode analysis.
The optimal energy envelopes that we calculate increase in amplitude with the magnetic Lundquist number $S$. These curves represent the possible energy amplification that can be achieved if the initial condition is optimal. However, even if the initial condition is not optimal, there is still the possibility for energy growth that is much faster than the growth rate determined from normal-mode analysis. This means that transient energy growth could, potentially, trigger the nonlinear phase of the tearing instability much sooner than previously expected. If this is the case, the implications for the tearing instability in solar physics would be substantial.
\subsection{Solar Applications}
\subsubsection{Coronal Phenomena}
In the solar corona, two important phenomena that are often linked to current sheets and their dissipation are {coronal heating} and {solar eruptions}. For the first of these, the ``nanoflare'' theory suggests that the corona is heated by many ``small'' heating events (or flares) spread throughout the coronal magnetic field \citep{parker88}. The tearing of current sheets, that develop from the complex deformation of magnetic fields, is one possible way that the magnetic field can release its energy as heat. Recent models of the nonlinear development of the MHD kink instability have revealed the development of many small-scale current features that could act as nanoflares \citep[\emph{e.g.}][]{hood16}. Our results support the idea of coronal heating \emph{via} tearing instabilities as perturbations could excite large transient growth which, in turn, could potentially readily generate nanoflares.
For the second phenomenon, current sheets are believed to play an important role at the onset, and subsequent nonlinear development, of solar eruptions. Such current sheets would be manifest in the flares associated with the initiation of CMEs, jets and surges. Simulations of CME-type eruptions often reveal a combination of reconnection above and below the CME, referred to as the {breakout theory} of CMEs. In particular, simulations, both 2D and 3D, demonstrate that tearing reconnection above and below the CME heralds the onset of an eruption \citep[\emph{e.g.}][]{macneice04,dmac14}. The onset of jets and surges has also been linked to the tearing of current sheets \citep[\emph{e.g.}][]{dmac15}. The onset of jets and eruptions is an important topic, not only for theoretical interest but for space weather applications. Therefore, understanding all aspects (normal and non-normal) of the onset of the tearing instability is vital.
\subsubsection{Quasi-Singular Current Sheets and the Plasmoid Instability}
Recent work by \cite{pucci14} has highlighted that the aspect ratio of current sheets has a threshold value, after which, equilibrium cannot be reached and the current sheet must reconnect. Various simulations have revealed that a fast tearing instability can develop for large $S$ and have growth rates proportional to $S^{1/4}$ \citep{lourerio07,lapenta08}. Hence, in the limit as $S\rightarrow\infty$, there would be, in the words of \cite{pucci14}, an ``infinitely unstable mode'' which is impossible in ideal MHD. By a simple and clever scaling argument, they show that once the current sheet aspect ratio is ${\rm O}(S^{1/3})$, a laminar current sheet cannot be supported and fast tearing must proceed.
Although we agree with main conclusion of \cite{pucci14}, we would suggest an alternative path to reaching their result. Their analysis is based entirely on eigenvalues and eigenvectors and so ignores the contribution of any transient growth. As the possible energy amplification of transient growth increases with $S$, a much faster onset of the tearing instability could be found that is due to transient growth. Such transient growth depends on the initial perturbation. Hence, the result of \cite{pucci14} can be thought of as a lower bound, when there are no effects of transient growth. As soon as there are perturbations that can induce transient growth, energy amplification will grow faster, thus exciting the tearing instability faster, as shown in the example in Figure \ref{gt_2}.
{Further recent work by \cite{comisso16} attempts to describe a general theory of the plasmoid instability, formulated by means of a principle of least time. In their analysis, they find that the scaling relationships for the final aspect ratio, the transition time to rapid onset, the growth rate and the number of plasmoids depend on the size of the initial disturbance amplitude, the rate of current sheet evolution, and the Lundquist number. We agree that the initial conditions are important for the onset of the instability, however, we would suggest that the theory of \cite{comisso16} could be extended to include transient effects like those described in this paper. Using scalings for the tearing mode alone will not give a complete description of the transient phase of the instability.
}
\subsection{Future Work}
This work can proceed in two main directions. The first is to include extra physics (\emph{e.g.} two fluid effects) to study how this would effect transient growth. The second, and perhaps most important, is to use optimal initial perturbations as initial conditions in nonlinear resistive MHD simulations. This task will determine if transient growth can lead to a fast nonlinear phase of the tearing instability or if nonlinear terms saturate the transient growth. It will be particularly interesting to determine if the nonlinear tearing instability can be excited by perturbations with $k>1$, \emph{i.e.} spectrally-stable perturbations.
{Although we have suggested that our results can extend those of previous studies (such as \cite{pucci14} and \cite{comisso16}) there remains much further work to understand how the damped part of the eigenvalue spectrum perturbs the current sheet and drives reconnection, particularly at very high values of $S$.}
\section*{Appendix}
Throughout this paper we have performed calculations with boundary conditions $u=b=0$ at $x=\pm d$. This has been done so that our results can be easily compared to other works and to nonlinear simulations which typically use such boundary conditions. Since the tearing instability develops in a boundary layer near $x=0$, the precise nature of the boundary conditions should not play a strong role on the onset of the instability. To illustrate this, we solve the discrete form of Equation \ref{evalue}, with $S=1000$ and $k=0.5$, in the {half-plane} and compare the resulting spectrum to that in Figure \ref{spectra}a.
Anticipating a symmetric solution in $\mathbf{b}$ and an antisymmetric solution in $\mathbf{u}$ about $x=0$, we set the boundary conditions at $x=0$ to be
\begin{equation}
u=\frac{{\rm d}b}{{\rm d}x}=0.
\end{equation}
As $x\rightarrow\infty$, we set
\begin{equation}
u=b=0.
\end{equation}
In order to represent this boundary numerically, we consider a large domain denoted by $0\le x\le x_{\rm max}$. In order to expand the variables using Chebyshev polynomials, we need to map our coordinates to the domain $-1\le y \le 1$. This is achieved through
\begin{equation}
x = a\frac{1+y}{b-y},
\end{equation}
where
\begin{equation}
a = \frac{x_{\rm max}x_i}{x_{\rm max}-2x_i} \quad \mbox{and} \quad b=1+\frac{2a}{x_{\rm max}}.
\end{equation}
This mapping clusters the grid points near the boundary layer at $x=0$ and places half of the grid points in the region $0\le x\le x_i$ \citep{hanifi96}. In this example, we take $x_{\rm max}$=100 and $x_i=15$. The resulting spectrum is displayed in Figure \ref{app}.
\begin{figure}[h]
\centering
{\includegraphics[width=9.5cm]{appendix.pdf}}
\caption{Spectrum of the discrete eigenvalue problem with half-plane boundary conditions for $S=1000$, $k=0.5$.}
\label{app}
\end{figure}
By inspection, the comparison with Figure \ref{spectra}a yields few differences. The eigenvalue corresponding to the tearing mode now has a value $\approx0.0111$, which is still similar to that calculated for the other boundary conditions. Two isolated eigenvalues near $\Im(\omega)=0$ in Figure \ref{spectra}a are now pushed nearer the main branches in Figure \ref{app}. Apart from these minor differences, the spectra calculated from different boundary conditions are very similar. This result suggests that the exact form of boundary conditions, assuming they do not interfere dynamically with the boundary layer at $x=0$, will not radically change the behaviour of the onset of the tearing instability.
| -36,974.307535 |
[
-3.3984375,
3.130859375
] | 34.030418 |
[
-3.439453125,
0.92041015625,
-2.095703125,
-5.60546875,
-0.84423828125,
7.34765625
] |
[
3.798828125,
8.1875,
3.490234375,
6.85546875
] | 418 | 6,612 |
[
-3.0859375,
3.40234375
] | 27.19758 |
[
-6.546875,
-4.62890625,
-4.65625,
-2.625,
2.28125,
12.9375
] | 1.42428 | 19.76943 | 22.489413 | 4.135527 |
[
3.2136998176574707
] | -22,506.237014 | 5.84029 | -36,686.33203 | 1.35954 | 5.935282 |
[
-2.9140625,
-3.869140625,
-3.7265625,
-4.73828125,
2.486328125,
12.0859375
] |
[
-5.359375,
-2.15234375,
-2.697265625,
-2.072265625,
3.724609375,
5.06640625
] | |
BkiUfBXxK3YB9i3RNEBb
|
\section{Detection Algorithms}
\label{sec:alg}
This section describes the technical details of each of the detection algorithms explored in this paper.
At each time step, each detector computes a test statistic, $a_{*}(R_n)$, based on current and previous residuals.
The subscript $*$ is a placeholder for the detector designation, and $n$ is the discrete time step.
If, for a given detector, the test statistic exceeds a predefined threshold, then the detector raises an alarm.
These thresholds are denoted by $\tau_*$.
\subsection{$\chi^2$ Detector}
The $\chi^2$ detector uses the normalized residual $\bar{r}_n$ to develop a statistical test.
Since, under the assumption of no attack, $\bar{r}_n\sim\mathcal{N}(0,I)$, the \emph{$\chi^2$ detector} is defined as:
\begin{align}
a_{\chi^2}(R_n)=\bar{r}_n^T\bar{r}_n<\tau_{\chi^2},
\end{align}
where the test statistic $a_{\chi^2}(R_n)\sim\chi^2(q)$ when the system is not under attack.
A change to the distribution of the residual, as a result of an attack, may change the resulting distribution of the $\chi^2$ test value, but this is not true for all attacks.
For instance an attack could replace the residual vectors with:
\begin{align}
r^\prime_n=\Sigma_r^{1/2}\bar{r}_n^\prime=\Sigma_r^{1/2}\begin{bmatrix}\sqrt{c_n}&0&\hdots&0\end{bmatrix}^T
\end{align}
where $c_n\sim\chi^2(q)$.
Then $\bar{r}_n^{\prime T}\bar{r}_n^\prime=c_n\sim\chi^2(q)$.
Similarly some attacks, such as the one described in Section \ref{sec:exp}, can generate a false set of residuals that have the same distribution as the residual when no attack is taking place.
Such attacks would be indistinguishable by the $\chi^2$ detector while increasing the error in the observed state.
Furthermore, the $\chi^2$ detector is memoryless, which can make detecting small increases in the norm of the residuals difficult.
\subsection{CUSUM Detector}
The CUSUM detector addresses the difficulty in detecting small but persistent increases in the norm of the normalized residual by introducing dynamics to its test statistic:
\begin{align}
a_{C}(R_n)=\max(a_{C}(R_{n-1})+\bar{r}_n^T\bar{r}_n-\gamma,0)<\tau_C,
\end{align}
where $a_{C}(R_{-1})=0$, and $\gamma$ is a parameter called the \emph{forgetting factor}.
To ensure that the test statistic is stable, $\gamma>q$ where $q$ is the dimension of the residual \cite[Theorem 1]{Murguia2016CUSUMSensors}.
Similar to the $\chi^2$ detector, the CUSUM detector bounds the norm of the normalized residual under the assumption of no alarms:
\begin{align}
\bar{r}_n^T\bar{r}_n-\gamma\leq a_{C}(R_{n-1})+\bar{r}_n^T\bar{r}_n-\gamma<\tau_C.
\end{align}
For the $CUSUM$ detector, a persistent increase in the norm of the normalized residual, increases the likelihood that $\bar{r}_n^T\bar{r}_n^T>\gamma$.
When this is true for several steps, the CUSUM test value increases cumulatively, triggering an alarm.
\subsection{MEWMA Detector}
The MEWMA detector test statistic also incorporates dynamics.
The MEWMA detector uses the exponentially weighted moving average of the normalized residual:
\begin{align}
G_{n}=&\beta\bar{r}_n+(1-\beta)G_{n-1}
\end{align}
where $G_{-1}=0$ and the parameter $\beta\in(0,1]$ is also called the \emph{forgetting factor}.
The MEWMA detector is then defined as:
\begin{align}
a_{M}(R_n)=&\frac{2-\beta}{\beta}G_{n}^TG_n<\tau_M.
\end{align}
When $\beta=1$ the test statistic is equal to the $\chi^2$ detector's test statistic.
For smaller $\beta$, one gets a similar effect to that of the CUSUM detector, because, for a forgetting factor $\beta\in(0,1)$, a persistent increase in the norm of the residual results in a larger value of $G$ which results in a higher test statistic value.
However, the MEWMA test statistic does not increase for all persistent changes.
For instance, if the covariance of the residuals under attack are $\alpha\Sigma_r$ for some $\alpha\in(0,1)$, we expect a lower test value.
\subsection{Dynamic Watermarking}
Dynamic Watermarking is designed to address the shortcomings of the previous detection algorithms by sounding an alarm not only for changes in the distribution of the norm of the normalized residual (as the $\chi^2$ and CUSUM detectors are able to do), but also for persistent changes in the distribution of the normalized residual as the MEWMA detector is able to do.
To illustrate this, we now briefly summarize the results of \cite{Hespanhol2017}.
In its statistical limit form, Dynamic Watermarking was developed to detect any persistent change made to the residual:
\begin{thm}\label{thm:asymp_attack_conv_to_0_hespanhol_thm1}\cite[Theorem 1] {Hespanhol2017}
Suppose (A,B) is stabilizable, (A,C) is detectable, $\Sigma_e$ is full rank, and:
\begin{align}
k^\prime=\min\{k\geq0~|~C(A+BK)^kB\neq0\}\label{eq:k}.
\end{align}
If:
\begin{align}
\underset{N\rightarrow\infty}{aslim}\frac{1}{N}\sum_{i=0}^{N-1}r_nr_n^T&=\Sigma_r\quad \text{and} \label{eq:cond1}\\
\underset{N\rightarrow\infty}{aslim}\frac{1}{N}\sum_{i=0}^{N-1}r_ne_{n-k}^T&=0,\label{eq:cond2}
\end{align}
then the asymptotic attack power, defined as:
\begin{align}
\underset{N\rightarrow\infty}{\lim}\frac{1}{N}\sum_{i=0}^{N-1}v_n^Tv_n
\end{align}
must converge to 0.
\end{thm}
For an attack to have a persistent effect on the system, the asymptotic attack power must be greater than 0.
Since the test described in Theorem \ref{thm:asymp_attack_conv_to_0_hespanhol_thm1} requires evaluating an infinite sum, it does not consider attacks that are not persistent for all time.
To detect attacks which are not persistent in time, the infinite time tests are transformed into finite time tests by taking a sliding window of the combined values of the sums in \eqref{eq:cond1} and \eqref{eq:cond2}:
\begin{align}
D_n=&\sum_{n-\ell+1}^n\psi_n\psi_n^T\\
\psi_n^T=\Sigma_\psi^{-1/2}&\begin{bmatrix}r_n^T & (e_{n-k})^T\end{bmatrix}\label{eq:normalize}
\end{align}
where $\ell$ is the size of the sliding window and:
\begin{align}
\Sigma_\psi=\begin{bmatrix}\Sigma_r & 0\\0 & \Sigma_e\end{bmatrix}.
\end{align}
In this paper, we have normalized the vector $\psi_n$ in \eqref{eq:normalize} for the convenience of later derivations.
The matrix quantity $D_n$ is then approximately distributed according to a Wishart Distribution, $\mathcal{W}(I,\ell)$ \cite{Hespanhol2017}, giving rise to the Dynamic Watermarking detector we use in this work:
\begin{align}
a_{\mathcal{D}}(R_n)=&\mathcal{L}_{m+q}^\ell(D_n)<\tau_\mathcal{D}
\end{align}
where $\mathcal{L}$ is the negative log likelihood function:
\begin{align}
\mathcal{L}_{i}^j(X)=&\frac{(i+1-j)}{2}\cdot\log(|X|)+\frac{1}{2}\text{trace}\left(X\right)+\nonumber\\
&+\log\left(2^{ij/2}\Gamma_{(i)}(\frac{j}{2})\right).\label{eq:lik}
\end{align}
Note, we have also modified the formula from \cite{Hespanhol2017}, by including a constant term, and removing the scale matrix inverse from the trace in \eqref{eq:lik} due to the normalization carried out in \eqref{eq:normalize}.
Note that for $n<\ell+k^\prime$, the test statistic value is defined as $0$.
\section{Analytical Comparisons}
\label{sec:ana}
This section describes the metric for the attack capability, and derives methods for approximating the attack capability and the $R_{FA}$ for the $\chi^2$, CUSUM, MEWMA, and Dynamic Watermarking detectors.\add{ Due to space considerations proofs for the Theorems and Lemmas in this section can be found in the technical report \cite{Porter2018}.}
\subsection{Attack Capability}
\label{sec:reach}
Assuming the $A$ matrix is Schur Stable, the capability of an attack can be measured by its ability to affect the observer error $\delta_n$.
A reachable set of the observer error can evaluate the attack capability, but, since the noise is supported over an infinitely large set, this reachable set would have infinite volume.
As a result, this work focuses on computing the volume of the reachable set of the portion of the observer error corresponding to the residual under the condition of no alarms being raised.
To provide a rigorous definition of this set, we introduce some additional notation and definitions.
Using superposition, one can split the observer error described in \eqref{eq:observe_error} into two pieces:
\begin{align}
\delta^{(a)}_{n+1}&=(A+LC)\delta_n^{(a)}-Lz_n-Lv_n\label{eq:observer_sep}\\
\delta^{(b)}_{n+1}&=(A+LC)\delta_n^{(b)}-w_n.
\end{align}
The observer error is then $\delta_n=\delta_n^{(a)}+\delta_n^{(b)}$.
Here $\delta_n^{(a)}$ is the portion related to the residual, which can be seen by applying \eqref{eq:res} and \eqref{eq:norm_res} to \eqref{eq:observer_sep}:
\begin{align}
\delta^{(a)}_{n+1}&=A\delta^{(a)}_n+L\Sigma_r^{1/2}\bar{r}_n.\label{eq:errdyn2}
\end{align}
Since an attack is only able to affect the $\delta_n^{(a)}$ portion of the observer error, the other portion is ignored while evaluating attack capability.
For each $n\in\mathbb{N}$, denote the \emph{reachable set of $\delta_n^{(a)}$ at a given time step $n$ under the condition of no alarms for a threshold $\tau_*$} as $\mathcal{R}_n^{\tau_*}$ and define it as:
\begin{align}
\mathcal{R}_n^{\tau_*}=\{\delta_n^{(a)}~|~\delta^{(a)}_n=\bar{A}_{n-1}R_{n-1},~R_{n-1}\in\Omega_n^{\tau_*}\}\label{eq:Rnset}
\end{align}
where:
\begin{align}
\Omega_{n-1}^{\tau_*}=\{R_{n-1}~|~ a_{*}(R_{n-1})<\tau_*~\forall i<n\}\label{eq:omega},
\end{align}
and:
\begin{align}
\bar{A}_n&=\begin{bmatrix}L\Sigma_r^{1/2}&AL\Sigma_r^{1/2}\hdots~ A^{n}L\Sigma_r^{1/2}\end{bmatrix}.\label{eq:abar}
\end{align}
Furthermore, we denote the \emph{steady state reachable set under the condition of no alarms for a threshold $\tau_*$} as $\mathcal{R}^{\tau_*}$ and define it as:
\begin{align}
\mathcal{R}^{\tau_*}=\{\delta^{(a)}~|~\forall n\in\mathbb{N},~\exists m\in\mathbb{N} \text{ s.t.} ~m>n, \delta^{(a)}\in\mathcal{R}_m^{\tau_*} \}.\label{eq:Rset}
\end{align}
Finally, we evaluate the attack capability by measuring the volume of the steady state reachable set under the condition of no alarms under a threshold $\tau_*$, which is defined as:
\begin{align}\label{eq:VRS}
V_{RS}(\tau_*)=\mu(\mathcal{R}^{\tau_*}),
\end{align}
where $\mu$ denotes the Lebesgue measure.
Calculating the set $\mathcal{R}^{\tau_*}$ can be difficult, so we first derive a method for calculating $\mathcal{R}^{\tau_*}_n$:
\begin{thm}
Suppose $\tau_*\in\mathbb{R}$ and $\bar{A}_{n-1},~R_{n-1}$, $\mathcal{R}_n^{\tau_*}$, and $\Omega_{n-1}^{\tau_*}$ are as in \eqref{eq:abar}, \eqref{eq:R}, \eqref{eq:Rnset}, and \eqref{eq:omega}, respectively.
Suppose $v:\mathbb{R}^q\rightarrow \mathbb{R}$ is the solution to:
\begin{flalign}
& & \underset{v\in \mathcal{C}}{\text{inf}}\hspace*{0.25cm} & \int v(\delta) ~d\delta &&\\
& & \text{s.t.}\hspace*{0.25cm} & v(\delta)\geq 0 && \delta\in\mathbb{R}^q\\
& & & v(\bar{A}_{n-1}R_{n-1})-1\geq 0 && R_{n-1}\in \Omega^{\tau_*}_{n-1}\label{eq:const}
\end{flalign}
where $\mathcal{C}$ is the space of continuous functions.
Then the 1 super-level set of $v$ is an outer approximation to $\mathcal{R}_n^{\tau_*}$
\end{thm}
\remove{\begin{proof}
Let $\delta\in\mathcal{R}_n^{\tau_*}$.
Then, from \eqref{eq:Rnset}, there exists an $R_{n-1}\in\Omega_{n-1}^{\tau_*}$ such that $\delta=\bar{A}_{n-1}R_{n-1}$.
The constraint in \eqref{eq:const} then gives $v(\bar{A}_{n-1}R_{n-1})=v(\delta)\geq 1$.
\end{proof}}
To make this problem computationally tractable, we optimize over polynomial functions of fixed degree instead of continuous functions, and describe the positivity constraint, \eqref{eq:const}, with a Sums-of-Squares constraint.
We then apply Sums-of-Squares Programming to generate an outer approximation to the reachable set.
To replace \eqref{eq:const} with a Sums of Squares constraint, $\Omega_n^{\tau_*}$ must first be replaced with a semi-algebraic set \cite[Theorem 2.14]{Lasserre2010MomentsApplications}.
To simplify our exposition, we denote by $\Theta_n^{\tau_*}$ a collection of semi-algebraic constraints such that $\Omega_{n}^{\tau_*}\subseteq\Theta_n^{\tau_*}$.
In fact, as we show next, for many detectors, $\Omega_{n}^{\tau_*} = \Theta_n^{\tau_*}$.
For the $\chi^2$ detector, the constraint of no alarms is a quadratic constraint on the residual, so:
\begin{align}
\Theta_n^{\tau_{\chi^2}} = \left\{R_n~|~R_n^TQ_{(i,n)}^{\tau_{\chi^2}}R_n<1~i=0,...,n\right\}\label{eq:theta1}
\end{align}
where:
\begin{align}\label{eq:chi_con_set_quad2}
Q_{(i,n)}^{\tau_{\chi^2}}&=\frac{1}{\tau_{\chi^2}}\begin{bmatrix}0_{q(n-i)}&0&0\\0&I_{q}&0\\0&0&0_{q(i)}\end{bmatrix}.
\end{align}
Note that $\Theta_n^{\tau_{\chi^2}}=\Omega_n^{\tau_{\chi^2}}$ since $\frac{a_{\chi^2}(R_i)}{\tau_{\chi^2}}=R_n^TQ_{(i,n)}^{\tau_{\chi^2}}R_n$ for all $i\leq n$.
For the CUSUM detector:
\begin{align}
\Theta_n^{\tau_{C}} = \left\{R_n~|~R_n^TQ_{(i,j,n)}^{\tau_{C}}R_n<1~i=0,...,n~j\leq i\right\}\label{eq:theta2}
\end{align}
where:
\begin{align}\label{eq:CUSUM_con_set_quad2}
Q_{(i,j,n)}^{\tau_C}&=\frac{1}{\tau_C+\gamma (j+1)}\begin{bmatrix}0_{q(n-i)}&0&0\\0&I_{q(j+1)}&0\\0&0&0_{q(i-j)}\end{bmatrix}.
\end{align}
Note that $\Theta_n^{\tau_{C}}=\Omega_n^{\tau_{C}}$, since:
\begin{align}
a_{C}(R_i)=\max\left(\left\{\sum_{h=i-j}^i(\bar{r}_h^T\bar{r}_h-\gamma)~|~j\leq i\right\},0\right)
\end{align}
and:
\begin{align}
R_n^TQ_{(i,j,n)}R_n=\frac{1}{\tau_C+\gamma (j+1)}\sum_{h=i-j}^{i}\bar{r_h}^T\bar{r}_h<1
\end{align}
can be rearranged to form:
\begin{align}
\sum_{h=i-j}^i(\bar{r}_h^T\bar{r}_h-\gamma)<\tau_C.
\end{align}
For the MEWMA detector note that:
\begin{align}
\Theta_n^{\tau_M} = \left\{R_n~|~R_n^TQ_{(i,n)}^{\tau_M}R_n<1~i=0,...n\right\}\label{eq:theta3}
\end{align}
where:
\begin{align}\label{eq:MEWMA_con_set_quad2}
Q_{(i,n)}^{\tau_M} &= \frac{2-\beta}{\beta\tau_M}\begin{bmatrix}0_{q(n-i)\times q}\\\beta I_{q}\\(1-\beta)\beta I_q\\\vdots\\(1-\beta)^{i}\beta I_q\end{bmatrix}\begin{bmatrix}0_{q(n-i)\times q}\\\beta I_q\\(1-\beta)\beta I_q\\\vdots\\(1-\beta)^{i}\beta I_q\end{bmatrix}^T.
\end{align}
Note that $\Theta_n^{\tau_{M}}=\Omega_n^{\tau_{M}}$ since $\frac{a_{M}(R_i)}{\tau_{M}}=R_n^TQ_{(i,n)}^{\tau_{M}}R_n$ for all $i\leq n$.
While $\Omega_{n}^{\tau_*}$ is already a semi-algebraic set for the $\chi^2$, CUSUM and the MEWMA detectors, this is not true for the Dynamic Watermarking detector due to the log function in \eqref{eq:lik}.
Therefore, we consider an outer approximation to $\Omega_n^{\tau_\mathcal{D}}$ described via a quadratic constraint:
\begin{thm}\label{thm:SHREYAS_LABELED_THIS}
Suppose $\Omega_n^{\tau_\mathcal{D}}$ is as in \eqref{eq:omega}, $\ell$ is the window size of the Dynamic Watermarking detector, $\tau_\mathcal{D}$ is the threshold of the detector, $k^\prime$ is as in \eqref{eq:k}, $q$ is the dimension of the residual, $m$ is the dimension of the input signal, and:
\begin{align}
\Theta_n^{\tau_\mathcal{D}} = \left\{R_n~|~R_n^TQ_{(i,n)}^{\tau_\mathcal{D}}R_n<1~i=\ell+k^\prime,...,n\right\},\label{eq:theta4}
\end{align}
where:
\begin{align}\label{eq:DW_con_set_quad2}
Q_{(i,n)}^{\tau_\mathcal{D}}=\frac{1}{(m+q)\epsilon}\begin{bmatrix}0_{q(n-i)} & 0 & 0\\0 & I_{q\ell} & 0\\0 & 0 & 0_{q(i-\ell)}\end{bmatrix}
\end{align}
and where $\epsilon>\ell-1-q-m$ is a solution to:
\begin{align}
\tau_\mathcal{D}=\frac{(q+m)\epsilon}{2} &+\frac{(q+m+1-\ell)}{2}\log(\epsilon^{q+m}) + \nonumber \\&+\log\left(2^{(q+m)\ell/2}\Gamma_{(q+m)}(\frac{\ell}{2})\right).\label{eq:epsilonD}
\end{align}
Then $\Omega_n^{\tau_\mathcal{D}} \subset \Theta_n^{\tau_\mathcal{D}}$.
\end{thm}
\remove{To prove this theorem, consider the following lemma:
\begin{lm}
\label{lm:convex}
Suppose $(g_i){i \in \mathbb{N}}$ is a sequence of vectors where $g_i\in\mathbb{R}^q$, $\tau\in\mathbb{R}$ such that $\tau>0$, and $\ell\in\mathbb{N}$ such that $\ell>q+1$. Furthermore suppose that:
\begin{align}
\mathcal{L}_q^\ell(\sum_{i=1}^\ell g_ig_i^T)<\tau
\end{align}
where the function $\mathcal{L}_q^\ell$ is as in \eqref{eq:lik}. Then:
\begin{align}
\sum_{i=1}^\ell g_i^Tg_i<\epsilon q,
\end{align}
where $\epsilon>\ell-1-q$ is a solution to:
\begin{align}
\tau=\frac{(q)\epsilon}{2} &+\frac{(q+1-\ell)}{2}\log(\epsilon^{q}) + \nonumber \\&+\log\left(2^{(q)\ell/2}\Gamma_{(q)}(\frac{\ell}{2})\right).\label{eq:epsilon}
\end{align}
\end{lm}
\begin{proof} (Lemma \ref{lm:convex})
Denote the eigenvalues of $\sum_{i-1}^\ell g_ig_i^T$ as $\lambda_1,...\lambda_q$.
The eigenvalues are all non-negative due to the construction of the matrix.
Note that we can rewrite \eqref{eq:lik} as a new function $\mathfrak{L}_i^j$ in terms of these eigenvalues:
\begin{align}
\mathcal{L}_q^\ell\left(\sum_{i-1}^\ell g_ig_i^T\right)&=\mathfrak{L}_q^\ell(\lambda_1,...,\lambda_q)\\
&=\sum_{i=1}^q\frac{(q+1-\ell)}{2}\cdot\log(\lambda_i)+\frac{\lambda_i}{2}+\nonumber\\
&+\log\left(2^{(q)\ell/2}\Gamma_{(q)}(\frac{\ell}{2})\right).
\end{align}
Furthermore we have that $\mathfrak{L}_q^\ell$ is convex since:
\begin{align}
\nabla^2\mathfrak{L}_q^\ell(\lambda_1,...,\lambda_q)=\begin{bmatrix}\frac{(\ell-1-q)}{2\lambda_1^2} & 0 & 0\\0 &\ddots&0\\0&0&\frac{\ell-1-q}{2\lambda_q^2}\end{bmatrix}
\end{align}
is positive definite for $\lambda_i>0$. Also note that the function achieves a global minimum at $\lambda_i=\ell-1-q~i=1,...,q$ since:
\begin{align}
\nabla\mathfrak{L}_q^\ell(\lambda_1,...,\lambda_q)=\begin{bmatrix}\frac{q+1-\ell}{2\lambda_1}+\frac{1}{2}\\\vdots\\
\frac{q+1-\ell}{2\lambda_q}+\frac{1}{2}\end{bmatrix}
\end{align}
is zero at this point.
If we consider the particular case where $\lambda_1=...=\lambda_q=\epsilon$. Then $(\epsilon,...,\epsilon)$ is a boundary point to the $\tau$ level set of $\mathfrak{L}_q^\ell$. Furthermore we have that the derivative at that point is:
\begin{align}
\nabla\mathfrak{L}_q^\ell(\epsilon,...,\epsilon)=\begin{bmatrix}\frac{q+1-\ell}{2\epsilon}+\frac{1}{2}\\\vdots\\
\frac{q+1-\ell}{2\epsilon}+\frac{1}{2}\end{bmatrix}
\end{align}
which is some positive scalar times the vector $\begin{bmatrix}1&\hdots&1\end{bmatrix}^T$. Since the tangent plane at this point is a supporting hyperplane to the $\tau$ sublevel set of $\mathfrak{L}_q^\ell$ we then have that:
\begin{align}
\sum_{i=1}^\ell g_i^Tg_i=\sum_{i=1}^q\lambda_i<\epsilon q
\end{align}
for all $g_i$ such that $\mathcal{L}_q^\ell(\sum_{i=1}^\ell g_ig_i^T)<\tau$.
\end{proof}
Now we return to the prove the theorem
\begin{proof} (Theorem \ref{thm:SHREYAS_LABELED_THIS})
For a given $R\in\Omega_n^{\tau_\mathcal{D}}$:
\begin{align}
a_{\mathcal{D}}(R_i)=\mathcal{L}_{(m+q)}^\ell\left(\sum_{j=i-\ell+1}^i\psi_j\psi_j^T\right)<\tau_\mathcal{D}
\end{align}
for $i=\ell+k^\prime,...,n$. Lemma \ref{lm:convex} then gives us that:
\begin{align}
\sum_{j=i-\ell+1}^i\psi_j^T\psi_j=\sum_{j=i-\ell+1}^i\bar{r}_j^T\bar{r}_j+\sum_{j=i-\ell+1}^ie_j^Te_j<(m+q)\epsilon
\end{align}
for $i=\ell+k^\prime,...,n$. Furthermore we have that:
\begin{align}
R_n^TQ_{i,n}^{\tau_\mathcal{D}}R_n=\sum_{j=i-\ell+1}^i\bar{r}_j^T\bar{r}_j<(m+q)\epsilon
\end{align}
for $i=\ell+k^\prime,...,n$. Therefore $R\in\Theta_n^{\tau_\mathcal{D}}$.
\end{proof}}
Now, we construct an outer approximation to $\mathcal{R}_n^{\tau_*}$ using the constraint sets $\Theta_n^{\tau_*}$:
\begin{thm}
\label{thm:sos}
Suppose $\bar{A}_{n-1}$ and $R_{n-1}$ are defined as in \eqref{eq:abar} and \eqref{eq:R} respectively, $\mathcal{R}_n^{\tau_*}$ is the set in \eqref{eq:Rnset}, $\Phi$ is a compact semi-algebraic set such that $\mathcal{R}_n^{\tau_*} \subset \Phi$, $\Theta_{n-1}^{\tau_*}$ is defined based on the choice of detector
and:
\begin{align}
H_n^{\tau_*}=\frac{1}{1-c}H
\end{align}
where $H$ and $c$ are the solution to:
\begin{flalign}
& & \underset{H\in S~c\in\mathbb{R}}{\inf}\hspace*{0.25cm} & \int_{\Phi} \left( \delta^TH\delta+c \right) ~d\delta &&\\
& & \text{s.t.}\hspace*{0.25cm} & \delta^TH\delta+c\geq 0 && \delta\in\Phi\\
& & & R_{n-1}^T\bar{A}_{n-1}^TH\bar{A}_{n-1}R_{n-1}+c-1\geq 0 && R_{n-1}\in \Theta_n^{\tau_*}\label{eq:const2}
\end{flalign}
where $S\subset\mathbb{R}^{p\times p}$ is the set of symmetric matrices.
Then $\mathcal{R}_n^{\tau_*}\subseteq\{\delta~|~\delta^TH_n^{\tau_*}\delta\leq1\}$.
\end{thm}
\remove{\begin{proof}
Let $\delta\in\mathcal{R}_n^{\tau_*}$.
Then, from \eqref{eq:Rnset}, we have that there exists an $R_{n-1}\in\Omega_{n-1}^{\tau_*}\subseteq\Theta_n^{\tau_*}$ such that $\delta=\bar{A}_{n-1}R_{n-1}$.
Constraint \eqref{eq:const2} then gives $R_{n-1}^T\bar{A}_{n-1}^TH\bar{A}_{n-1}R_{n-1}+c\geq1$.
Furthermore $c>1$ since $0\in\Omega_{n-1}^{\tau_*}$, so we can rearrange the inequality resulting in $\delta^T\frac{1}{1-c}H\delta=\delta^TH_n^{\tau_*}\delta\leq1$.
\end{proof}}
One can solve the program in Theorem \ref{thm:sos} using the Spotless optimization toolbox \cite{Tobenkin2018} which formulates the problem as a Semi-Definite Program that can be solved using commercial solvers such as MOSEK \cite{mosek}.
This program assumes that we can find a compact semi-algebraic set $\Phi$ that outer approximates $\mathcal{R}_n^{\tau_*}$, which can be done using the following lemma under the specific case that $N=n$:
\add{\setcounter{lm}{1}}\begin{lm}
\label{lm:bound2}
Suppose $N,n\in\mathbb{N}$, such that $N\geq n$ and if applicable, suppose $N>\ell+k^\prime$ if the detector is the Dynamic Watermarking detector. Furthermore suppose $\tau_*\in\mathbb{R}$ such that $\tau_*>0$, $\bar{A}_{n-1}$ and $R_{N-1}$
are as in \eqref{eq:abar},\eqref{eq:R}.
Then there exists a $\eta\in\mathbb{R}$ such that:
\begin{align}
\{\delta=[0_{q\times q(N-n)}~\bar{A}_{n-1}]R_{N-1}~|~R_{N-1}\in\Theta_{N-1}^{\tau_*}\}\subset \mathcal{B}_{\eta}.
\end{align}
\end{lm}
\remove{\begin{proof}(Lemma \ref{lm:bound2})
First we show that $\Theta_{N-1}^{\tau_*}$ is bounded. We denote the upper bounds for the norm of elements in $\Theta_{N-1}^{\tau_*}$ as $\sigma^{\tau_*}$, and we use the decomposition of $R_{N-1}=[\bar{r}_{N-1}^T~\hdots~\bar{r}_0^T]^T\in\Theta_{N-1}^{\tau_*}$. For the $\chi^2$ detector we have that $\sigma^{\tau_{\chi^2}}=\sqrt{N\tau_{\chi^2}}$ since:
\begin{align}
\|[\bar{r}_{N-1}^T~\hdots~\bar{r}_0^T]^T\|=\sqrt{\sum_{i=0}^{N-1}\bar{r}_{i}^T\bar{r}_i}\leq\sqrt{N\tau_{\chi^2}}.
\end{align}
Similarly for the CUSUM detector we have that $\sigma^{\tau_{C}}=\sqrt{N(\tau_C+\delta)}$ since:
\begin{align}
\|[\bar{r}_{N-1}^T~\hdots~\bar{r}_0^T]^T\|=\sqrt{\sum_{i=0}^{N-1}\bar{r}_{i}^T\bar{r}_i}\leq\sqrt{N(\tau_{C}+\delta)}.
\end{align}
In the case of the MEWMA detector we have that $\sigma^{\tau_{M}}=\sqrt{\frac{N\tau_M(2-\beta)}{\beta}}$ since:
\begin{align}\label{eq:MEWMA_Bound1}
\|G_i\|=\|\beta\bar{r}_i+(1-\beta)G_{i-1}\|\leq\sqrt{\frac{\tau_M\beta}{2-\beta}}
\end{align}
and:
\begin{align}\label{eq:MEWMA_Bound2}
\beta\|\bar{r}_i\|-(1-\beta)\sqrt{\frac{\tau_M\beta}{2-\beta}}\leq\|\beta\bar{r}_i+(1-\beta)G_{i-1}\|.
\end{align}
Combining \eqref{eq:MEWMA_Bound1} and \eqref{eq:MEWMA_Bound2} we get:
\begin{align}
\|\bar{r}_i\|\leq\sqrt{\frac{\tau_M(2-\beta)}{\beta}}.
\end{align}
Then:
\begin{align}
\|[\bar{r}_{N-1}^T~\hdots~\bar{r}_0^T]^T\|=\sqrt{\sum_{i=0}^{N-1}\bar{r}_{i}^T\bar{r}_i}\leq\sqrt{\frac{N\tau_M(2-\beta)}{\beta}}.
\end{align}
In the case of the Dynamic Watermarking detector, we have that $\sigma^{\tau_{\mathcal{D}}}=\sqrt{N(m+q)\epsilon}$, where $\epsilon>\ell-1-q-m$ is the solution to \eqref{eq:epsilonD}, since:
\begin{align}
\|[\bar{r}_{N-1}^T~\hdots~\bar{r}_0^T]^T\|=\sqrt{\sum_{i=0}^{N-1}\bar{r}_{i}^T\bar{r}_i}\leq\sqrt{N(m+q)\epsilon}.
\end{align}
Then, since:
\begin{align}
\|[0_{q\times q(N-n)}&~\bar{A}_{n-1}]R_{N-1}\|\leq\nonumber\\
&\|[0_{q\times q(N-n)}~\bar{A}_{n-1}]\|~\|R_{N-1}\|,
\end{align}
let $\eta=\|[0_{q\times q(N-n)}~\bar{A}_{n-1}]\|\sigma^{\tau_*}$.
Then:
\begin{align}
\{\delta=[0_{q\times q(N-n)}~\bar{A}_{n-1}]R_{N-1}~|~R_{N-1}\in\Theta_{N-1}^{\tau_*}\}\subset \mathcal{B}_{\eta}.
\end{align}
\end{proof}}
The program in Theorem \ref{thm:sos} gives an upper bound to $\mathcal{R}_n^{\tau_*}$, which we denote by:
\begin{align}
\mathcal{T}_n^{\tau_*}=\{\delta~|~\delta^TH_n^{\tau_*}\delta\leq1\}. \label{eq:T}
\end{align}
We dilate $\mathcal{T}_n^{\tau_*}$ to obtain an outer approximation to $\mathcal{R}^{\tau_*}$:
\begin{thm}
\label{thm:dil}
Suppose $\tau_*\in\mathbb{R}$ such that $\tau_*>0$, $\mathcal{R}^{\tau}$ is as in \eqref{eq:Rset}, $\mathcal{T}_n^{\tau_*}$ is as in \eqref{eq:T}, and:
\begin{align}
\mathcal{E}_n^{\tau_*}=\mathcal{T}_n^{\tau_*}\oplus \mathcal{B}_\epsilon,
\end{align}
where:
\begin{align}
\epsilon=\frac{\|A^n\|}{\sqrt{s_1(H_n^{\tau_*})}(1-\|A^n\|)}.
\end{align}
Then $\mathcal{R}^{\tau_*}\subset \mathcal{E}_n^{\tau_*}$.
\end{thm}
\remove{To prove this result we must first consider the lemma:
\begin{lm}
\label{lm:bound1}
Suppose $n,N,h\in\mathbb{N}$ such that $0<n
\leq h\leq N$, $R=[r_N^T ... r_0^T]^T\in\Theta_N^{\tau_*}$ where $\Theta_N^{\tau_*}$ is defined based on the choice of detector.
Then $R^\prime=[r_{h}^T ... r_{h-n}^T]^T\in\Theta_n^{\tau_*}$.
\end{lm}
\begin{proof}(of Lemma \ref{lm:bound1})
To prove that $R^\prime$ is in $\Theta_n^{\tau_*}$ we show that each of the constraints associated with $\Theta_n^{\tau_*}$ are included as a constraint associated with $\Theta_N^{\tau_*}$ or that there exists a more restrictive constraint in $\Theta_N^{\tau_*}$.
For the $\chi^2$ test we have the inclusion of all constraints since using \eqref{eq:theta1} and \eqref{eq:chi_con_set_quad2} we have:
\begin{align}
R^{\prime T}Q_{(i,n)}^{\tau_{\chi^2}}R^\prime=R^TQ_{(i+h-n,N)}^{\tau_{\chi^2}}R<1~\forall i=0,...,n.
\end{align}
Similarly for the CUSUM detector we have that using \eqref{eq:theta2} and \eqref{eq:CUSUM_con_set_quad2} we have:
\begin{align}
R^{\prime T}Q_{(i,j,n)}^{\tau_{C}}R^\prime&=R^TQ_{(i+h-n,j,N)}^{\tau_C}R<1\nonumber\\
&\hspace*{1cm}\forall~i=0,...,n \text{ and } j=0,...,i.
\end{align}
For the MEWMA we have that $\Theta_N^{\tau_*}$ has more restrictive constraints since using \eqref{eq:theta3} and \eqref{eq:MEWMA_con_set_quad2} we have:
\begin{align}
R^{\prime T}Q_{(i,n)}^{\tau_{M}}R^\prime\leq R^TQ_{(i+h-n,N)}^{\tau_{M}}R<1~\forall i=0,...,n.
\end{align}
For The Dynamic Watermarking Detector we have the inclusion of all constraints since for \eqref{eq:theta4} and \eqref{eq:DW_con_set_quad2} we have:
\begin{align}
R^{\prime T}Q_{(i,n)}^{\tau_\mathcal{D}}R^\prime=R^TQ_{(i+h-n,N)}^{\tau_\mathcal{D}}R<1~\forall i=\ell+k^\prime,...,n.
\end{align}
\end{proof}
Now we return to proving Theorem \ref{thm:dil}.
\begin{proof}(Theorem \ref{thm:dil})
Let $\delta^\prime\in \mathcal{R}^{\tau_*}$, and assume that $\delta^\prime\notin\mathcal{E}_n^{\tau_*}$. Furthermore let:
\begin{align}
\epsilon_1=\inf\{\|\delta-\delta^\prime\|~|~\delta\in\mathcal{E}_n^{\tau_*}\}.
\end{align}
Now consider that, for a given $N>n$:
\begin{align}
\mathcal{R}_N^{\tau_*}\subseteq\{\delta~|~\delta=A_{N-1}X,~X\in\Theta_{N-1}^{\tau_*}\}.
\end{align}
Using Minkowski sums we over-approximate this set further as:
\begin{align}
\mathcal{R}_N^{\tau_*}&\subseteq\{\delta=[\bar{A}_{n-1}~0_{p\times p(N-n)}]R_{N-1}~|~R_{N-1}\in\Theta_{N-1}^{\tau_*}\}\oplus\nonumber\\
&\oplus\Bigg(\bigoplus_{i=1}^{j}A^{ni}\{\delta=[0_{p\times pi}~\bar{A}_{n-1}~0_{p\times p(N-n-i)}]R_{N-1}~|\nonumber\\
&\hspace*{4.6cm}R_{N-1}\in\Theta_{N-1}^{\tau_*}\}\Bigg)\oplus\nonumber\\
&\oplus A^{n(j+1)}\{\delta=[0_{p\times p(N-nj)}~\bar{A}_{h}]R_{N-1}~|\nonumber\\
&\hspace*{4.6cm}R_{N-1}\in\Theta_{N-1}^{\tau_*}\}.
\end{align}
where $N$ is evenly divisible by $n$, $j+1$ times and $h$ is the remainder. Applying Lemma \ref{lm:bound2} and \ref{lm:bound1}, we have:
\begin{align}
\mathcal{R}_N^{\tau_*}&\subseteq\{\delta=\bar{A}_{n-1}R_{n-1}~|~R_{n-1}\in\Theta_{n-1}^{\tau_*}\}\oplus\nonumber\\
&\oplus\left(\bigoplus_{i=1}^{j}A^{ni}\{\delta=\bar{A}_{n-1}R_{n-1}~|~R_{n-1}\in\Theta_{n-1}^{\tau_*}\}\right)\oplus\nonumber\\
&\oplus \mathcal{B}_{\eta\|A^{n(j+1)}\|}
\end{align}
where $\eta$ is the maximum radius when applying Lemma \ref{lm:bound2} for $h=0,...,n$.
Let $\sigma=\frac{1}{\sqrt{s_1(H_n^{\tau_*})}}$ then:
\begin{align}
\{\delta=\bar{A}_{n-1}R_{n-1}~|~R_{n-1}\in\Theta_{n-1}^{\tau_*}\}&\subset\mathcal{T}_i^{\tau_*}\\
=\{\delta~|~\delta^TH_n^{\tau_*}\delta\leq 1\}&\subset \mathcal{B}_\sigma.
\end{align}
This means that:
\begin{align}
\mathcal{R}_N^{\tau_*}\subseteq\mathcal{T}_n^{\tau_*}\oplus\left(\bigoplus_{i=1}^{j}\mathcal{B}_{\sigma\|A^{ni}\|}\right)\oplus \mathcal{B}_{\eta\|A^{n(j+1)}\|}.
\end{align}
Since the Minkowski sum of balls is a ball with its radius as the sum of the radii, we can increase the outer approximation by allowing the summation to extend towards infinity:
\begin{align}
\mathcal{R}_N^{\tau_*}\subseteq\mathcal{T}_n^{\tau_*}\oplus \mathcal{B}_\epsilon\oplus \mathcal{B}_{\eta\|A^{n(j+1)}\|},
\end{align}
where:
\begin{align}
\epsilon=\frac{\sigma\|A^n\|}{(1-\|A^n\|)}\geq\sum_{i=1}^\infty\sigma\|A^{ni}\|.
\end{align}
Since $j+1>\frac{N}{n}$, there exists an $N_2$ such that for $N>N_2$ we have that $\eta\|A^{n(j+1)}\|<\epsilon_1$ which contradicts $\delta\in\mathcal{R}^{\tau_*}$.
\end{proof}}
\subsection{False Alarm Rate}
\label{sec:false}
While decreasing the threshold for a detector decreases the attack capability, it increases the $R_{FA}$.
As described in the introduction, to compute this false alarm rate, it is typically assumed that the residuals are independent \cite{Murguia2016CUSUMSensors,Umsonst2018}.
Unfortunately, simulated experiments have noted that this assumption results in a consistent error between the expected and simulated results \cite{Murguia2016CUSUMSensors}.
This is because the residuals are not independent, which can be confirmed by computing the auto correlation of the sequence of residuals when no attack is present:
\begin{align}
\mathds{E}[r_nr_{n-1}^T]&=\mathds{E}[(C\delta_n-z_n)(C\delta_{n-1}-z_{n-1})^T].
\end{align}
By expanding the product and removing uncorrelated terms one can show that:
\begin{align}
\mathds{E}[r_nr_{n-1}^T]=\mathds{E}[C\delta_n\delta_{n-1}^TC^T]-\mathds{E}[C\delta_nz_{n-1}^T].\label{eq:corr2}
\end{align}
By applying \eqref{eq:observe_error} to \eqref{eq:corr2} and once again canceling uncorrelated terms one can show that:
\begin{align}
\mathds{E}[r_nr_{n-1}^T]=C(A+LC)\mathds{E}[\delta_n\delta_n^T] C^T+L\Sigma_z.
\end{align}
In fact, correlation affects the rate of false alarms \cite{Harris1991StatisticalObservations}.
To compute the threshold with a specified false alarm rate, we first fix a specific threshold for each detector and simulate the behavior of the system.
By simulating the system for a long enough time, we can estimate the rate of false alarms associated with the fixed threshold.
By repeating this approach for a range of thresholds, we can then build a lookup table that associates different thresholds with different false alarm rates.
By linearly interpolating between these thresholds, we can select a threshold that achieves a user-specified false alarm rate.
\section{Simulation and Real-World Comparison}
\label{sec:compare}
This section describes a simulated comparison in which the attack capability for a range of false alarm rates is approximated for each detector, and a real-world comparison in which the ability of each detection algorithm to detect particular attacks is explored.
\subsection{Simulation-Based Comparison of Attack Capability}
\label{sec:simp}
To illustrate the trade-off between the rate of false alarms and attack capability, we provide a comparison of each of the detection algorithms using a 2 dimensional model from \cite{Murguia2018OnAttacks}:
\begin{align*}
A&=\begin{bmatrix}0.84&0.23\\-0.47&0.12 \end{bmatrix}&
~B&=\begin{bmatrix}0.07& -0.32\\0.23&0.58\end{bmatrix}\\
C&=\begin{bmatrix}1&0\\2&1\end{bmatrix}&
~K&=\begin{bmatrix}1.404&-1.042\\1.842&1.008\end{bmatrix}\\
L&=\begin{bmatrix}0.0276&0.0448\\-0.01998&-0.0290\end{bmatrix}&
~\Sigma_z&=\begin{bmatrix}2&0\\0&2\end{bmatrix}\\
\Sigma_w&=\begin{bmatrix}0.035&-0.011\\-0.011&0.02\end{bmatrix}&
~\Sigma_r&=\begin{bmatrix}2.086&0.134\\0.134&2.230\end{bmatrix}
\end{align*}
with the addition of a watermark with covariance $\Sigma_e=10^{-2}I$. Thresholds for the false alarm rates between $0.01$ and $0.3$ were found by running the simulation under no attack for $10^6$ time steps.
Using these values, the reachable sets at time step $n=12$ were outer approximated using the optimization program stated in Theorem \ref{thm:sos} and dilated as stated in Theorem \ref{thm:dil} to provide outer approximations of the steady state reachable sets.
The resulting approximations for the $V_{RS}$, defined in \eqref{eq:VRS}, are plotted against the false alarm rate in Figure \ref{fig:reachcomp} for each of the detectors using various detector specific parameter selections.
\begin{figure}[t]
\centering
\includegraphics[trim={2.0in 3.6in 2.2in 3.8in},clip,width=0.44\textwidth]{figures/curves.pdf}
\caption{Approximate reachable set volume for varying false alarm rates for the example system in section \ref{sec:simp}}
\label{fig:reachcomp}
\vspace*{-0.5cm}
\end{figure}
One may note that while these methods provide smooth curves for the $\chi^2$, MEWMA, and Dynamic Watermarking detectors, the curves for the CUSUM detector appears discontinuous, and do not span the entire range of $R_{FA}$ values.
The apparent discontinuity is attributed to the fact that, for the $\chi^2$, MEWMA, and Dynamic Watermarking detectors, increasing the threshold $\tau_*$ results in a proportional scaling of the outer approximation of $\mathcal{R}_n^{\tau_*}$.
However, For the CUSUM detector, changing the threshold does not have this affect. In fact, changing the threshold for the CUSUM detector alters the shape of $\Theta_n^{\tau_C}$, resulting in the outer approximation of $\mathcal{R}_n^{\tau_C}$ being less conservative for certain threshold values. Furthermore, the shortened span of the curves for the CUSUM detector are a result of certain $R_{FA}$ values being un-achievable for a given forgetting factor.
To determine whether the outer approximation is tight for the $\chi^2$, CUSUM, and MEWMA detectors, simulations were run for 60 $R_{FA}$ values uniformly spaced between 0.01 and 0.3.
In these simulations, the portion of the observer related to the residual was propagated forward for $10^5$ steps using the dynamics \eqref{eq:errdyn2}.
The residuals were sampled from a normal distribution with 0 mean and covariance $5I$ and scaled if necessary, to avoid alarms.
The area of the convex hull of the observer error for the entire simulation was calculated.
The difference between the over approximated area and the simulated area ranged from $0.0156-0.1408$ for the $\chi^2$, $0.0032-0.1143$ for the CUSUM, and $0.0075-0.1301$ for the MEWMA.
The results indicate that the attack capability under the Dynamic Watermarking detector is comparable to the classic anomaly detectors as a function of false alarm rate.
\subsection{Real World Implementation}
\label{sec:exp}
\remove{\begin{table*}[t]
\centering
\begin{tabular}{|c|c|c|c|c|c|}
\hline
\multirow{2}{*}{$R_{FA}$} & \multirow{2}{*}{$\chi^2$} & CUSUM & MEWMA & Dyn. Wat. \\
& & $\gamma=(15~/~17~/~19)$ & $\beta=(0.6~/~0.7~/~0.8)$ &$\ell=(20~/~25~/~30)$ \\\hline
\rowcolor{gray!30} 0.05 & 12.08 & ( 51.04 / 12.27 / 2.21 ) & ( 13.09 / 15.00 / 16.85 ) & ( 99.570 / 103.74 / 105.59 ) \\\hline
\rowcolor{gray!15}0.03 & 14.29 & ( 655.81 / 384.73 / 158.68 ) & ( 15.05 / 17.59 / 20.17 ) & ( 103.05 / 106.73 / 108.58 ) \\\hline
0.01 & 21.38 & ( 1535.94 / 1445.58 / 1355.22 ) & ( 20.68 / 24.85 / 28.88 ) & ( 108.58 / 110.79 / 113.94 ) \\\hline
\end{tabular}
\caption[caption]{Experimentally Found Thresholds for Various False Alarm Rates and\\ Detector Specific Parameters for the Real World Implementation in Section \ref{sec:exp}}
\label{tab:thresh_exp}
\vspace*{-2em}
\end{table*}}
In this section, we evaluate the ability of each of the anomaly detection schemes to detect attacks using a Segway Robotics Mobility Platform performing a path-following task.
In addition, we illustrate that adding a watermark to the system leads to an imperceptible reduction
in performance, while significantly improving the detectability of an attack that was missed by classic anomaly detectors.
Localization was provided by Google Cartographer \cite{Hess2016} using planar lidar and wheel odometry measurements.
For the purpose of control, a LTV model was fit to the observed data yielding:
\begin{align}
\label{eq:error}
\begin{bmatrix}
e_{\ell,n+1}\\
e_{s,n+1}\\
e_{\theta,n+1}\\
e_{v,n+1}\\
e_{\dot{\theta},n+1}
\end{bmatrix}
=\begin{bmatrix}
e_{\ell,n}+(0.0478\tilde{v}_n)e_{\theta,n}-(0.045\dot{\tilde{\theta}}_n)e_{s,n}\\
e_{s,n}+(0.0478)e_{v,n}+(0.045\dot{\tilde{\theta}}_n)e_{\ell,n}\\
e_{\theta,n}+0.045e_{\dot{\theta},n}\\
e_{v,n}-0.1e_{v,n-4}+0.1u_{v,n}\\
0.6e_{\dot{\theta},n}+0.15e_{\dot{\theta},n-4}+0.24u_{\dot{\theta},n}
\end{bmatrix}
\end{align}
where the state is represented in trajectory error coordinates for a given nominal trajectory where $e_{\ell,n},e_{s,n}, \text{ and } e_{\theta,n}$ are the lateral, longitudinal, and heading error, $e_{v,n},e_{\dot{\theta},n}$ are the error in the velocity and angular velocity, $\tilde{v}_n,\dot{\tilde{\theta}}_n$ are the nominal velocity and angular velocity and $u_{v,n},u_{\dot{\theta},n}$ are the deviation from the nominal inputs.
For a constant nominal velocity of $0.6$ m/s and angular velocity of $0$ rad/s, this model can be represented as an LTI model with state vector:
\begin{align}
x&=\left[\begin{matrix}e_{\ell,n} & e_{s,n} & e_{\theta,n} & e_{v,n} & e_{\dot{\theta},n} & e_{v,n-1} \end{matrix}\right.\nonumber\\
&\hspace{1cm}\left.\begin{matrix} e_{v,n-2} & e_{v,n-3} & e_{\dot{\theta},n-1} & e_{\dot{\theta},n-2} & e_{\dot{\theta},n-3}\end{matrix}\right]^T.
\end{align}
For the sake of brevity, the A and B matrices are not stated explicitly but can be found by expanding \eqref{eq:error}.
The feedback gain matrix $K$ was found to make the closed loop system Schur Stable\add{.}\remove{ and is approximately:
\begin{align}
K=\begin{bmatrix}
0 & -1.639 \\ 0 & -1.984 \\ -0.313 & 0\\ -0.212 & 0 \\ 0 & -0.384 \\ 0.019 & 0\\ 0.020 & 0 \\ 0.021 & 0 \\0.022 & 0 \\0 & -0.039 \\ 0 & -0.043 \\ 0 & -0.052 \\ 0 & -0.065
\end{bmatrix}
\end{align}}
Similarly the observer gain matrix $L$ was found to make the observer Schur Stable\add{. Both the $K$ and $L$ matrices can be found in the technical report \cite{Porter2018}.}\remove{ and is approximately:
\begin{align}
L=\begin{bmatrix}
-0.791 & -0.016 & 0 & 0 & 0 \\
-0.002 & -0.501 & 0 & 0 & -0.022 \\
0 & 0 & -0.272 & -0.025 & 0 \\
0 & 0 & -0.011 & -0.252 & 0 \\
0 & -0.003 & 0 & 0 & -0.240 \\
0 & 0 & -0.013 & -0.258 & 0 \\
0 & 0 & -0.015 & -0.187 & 0 \\
0 & 0 & -0.015 & -0.133 & 0 \\
0 & 0 & -0.014 & -0.091 & 0 \\
0 & -0.004 & 0 & 0 & -0.395 \\
0 & -0.010 & 0 & 0 & -0.145 \\
0 & -0.008 & 0 & 0 & -0.054 \\
0 & -0.005 & 0 & 0 & -0.024 \\
\end{bmatrix}
\end{align}}
The steady state covariance of the residuals, $\Sigma_r$, was approximated using the sample covariance from data generated by the Segway following a straight line down a 16 m hallway 40 times.
To avoid the effects of the transient behavior at the start of each run, the beginning of each run was ignored.
This experiment was repeated a second time after the introduction of a watermark into the control input with covariance:
\begin{align}
\Sigma_e=\begin{bmatrix}0.02 & 0\\0&0.03\end{bmatrix},
\end{align}
in order to approximate $\Sigma_\psi$.
The average location error was 0.0262 m for the non-watermarked runs and 0.0506 m for the watermarked runs.
While adding the watermark increased the location error, the average error did not hinder overall performance during the lane-following task.
Threshold values for the false alarm rates of 0.01, 0.03, and 0.05 were approximated for the $\chi^2$, CUSUM, and MEWMA detectors using the residuals from the un-watermarked runs.
Thresholds for the Dynamic Watermarking detector and the same false alarm rates were found using the residuals from the watermarked runs.
The resulting threshold values are displayed in \add{the technical report \cite{Porter2018}.}\remove{ Table \ref{tab:thresh_exp}.}
Two attacks, following differing models, were then applied.
Attack model 1 assumes that the attacker adds random noise to the system such that $v_n\sim\mathcal{N}(0,10^{-5} I)$.
Attack model 2 takes the form:
\begin{align}
v_n&=-(Cx_n+z_n) +C\xi_n+\zeta_n.\label{eq:attack}
\end{align}
For this model, the attack measurement noise $\zeta_n$ is added to the false state such that $\zeta_n\sim\mathcal{N}(0,\Sigma_\zeta)$ and the false state $\xi_n\in\mathbb{R}^p$ is updated according to the closed loop dynamics of the system:
\begin{align}
\xi_{n+1}&=(A+BK)\xi_n+\omega_n
\end{align}
with attack process noise $\omega_n\sim\mathcal{N}(0,\Sigma_\omega)$.
The attack process and measurement noise were chosen to leave the distribution of the residuals unchanged.
For each attack, 10 experimental runs were completed without a watermark and 10 with a watermark for a total of 40 experimental runs.
The runs with a watermark were used in evaluating the Dynamic Watermarking detector, while all other detectors used the un-watermarked data.
The resulting detection rates, defined as the number of alarms divided by the total number of time steps in the attacked runs, are displayed in Table \ref{tab:exp_exam}.
\begin{table}[ht]
\centering
\begin{tabular}{|c|c|c|c|}
\hline
\multirow{2}{*}{Method} & \multirow{2}{*}{$R_{FA}$} & Attack Model 1 & Attack Model 2\\
& & Detection Rates & Detection Rates\\\hline
\multirow{3}{*}{$\chi^2$} & 0.05 \cellcolor{gray!30}& 0.69 \cellcolor{gray!30}& 0.03 \cellcolor{gray!30}\\\cline{2-4}
& 0.03 \cellcolor{gray!15}& 0.62 \cellcolor{gray!15}& 0.01 \cellcolor{gray!15}\\\cline{2-4}
& 0.01 & 0.47 & 0.00 \\\hline
\multirow{3}{*}{$\underset{\gamma=(15/17/19)}{\text{CUSUM}}$} &\cellcolor{gray!30}0.05 &\cellcolor{gray!30}( 0.98 / 0.99 / 0.99 ) &\cellcolor{gray!30}( 0.00 / 0.00 / 0.00 )\\\cline{2-4}
&\cellcolor{gray!15}0.03 &\cellcolor{gray!15}( 0.81 / 0.86 / 0.92 ) &\cellcolor{gray!15}( 0.00 / 0.00 / 0.00 ) \\\cline{2-4}
& 0.01 & ( 0.58 / 0.54 / 0.51 ) & ( 0.00 / 0.00 / 0.00 )\\\hline
\multirow{3}{*}{$\underset{\beta=(0.6/0.7/0.8)}{\text{MEWMA}}$} &\cellcolor{gray!30}0.05 &\cellcolor{gray!30}( 0.51 / 0.57 / 0.62 ) &\cellcolor{gray!30}( 0.03 / 0.03 / 0.03 ) \\\cline{2-4}
&\cellcolor{gray!15}0.03 &\cellcolor{gray!15}( 0.45 / 0.50 / 0.54 ) &\cellcolor{gray!15}( 0.01 / 0.01 / 0.01 ) \\\cline{2-4}
& 0.01 & ( 0.32 / 0.35 / 0.39 ) & ( 0.00 / 0.00 / 0.00 ) \\\hline
\multirow{3}{*}{$\underset{\ell=(20/25/30)}{\text{Dyn. Wat.}}$} &\cellcolor{gray!30}0.05 &\cellcolor{gray!30}( \textbf{1.00} / \textbf{1.00} / \textbf{1.00} ) &\cellcolor{gray!30}( 0.98 / \textbf{1.00} / \textbf{1.00} )\\\cline{2-4}
&\cellcolor{gray!15}0.03 &\cellcolor{gray!15}( \textbf{1.00} / \textbf{1.00} / \textbf{1.00} ) &\cellcolor{gray!15}( 0.97 / 0.99 / \textbf{1.00} )\\\cline{2-4}
& 0.01 & ( \textbf{1.00} / \textbf{1.00} / \textbf{1.00} ) & ( 0.95 / 0.98 / \textbf{1.00} )\\\hline
\end{tabular}
\caption{Experimentally Found Alarm Rates For Various Detector Specific Parameters for the Attack Models from Section \ref{sec:exp}}
\label{tab:exp_exam}
\vspace*{-0.5cm}
\end{table}
For attack model 1,
all of the detectors are able to reliably detect the attack, confirming that the implementation of the detectors is correct.
For attack model 2 the detection rate decrease from the $R_{FA}$ for the $\chi^2$, CUSUM, and MEWMA detectors.
This may be due to the residuals for the un-attacked system not being distributed as a Gaussian distribution resulting in higher threshold values.
Since attack model 2 replaces the feedback completely, the resulting residuals, when under attack, do follow a Gaussian distribution which then results in lower detection rates.
The Dynamic Watermarking detector in the presence of the second attack provides a high detection rates for each set of parameters, and in some cases achieves a perfect detection rate.
\section{Conclusion}
\label{sec:con}
This paper derives a method to evaluate the capability of an attacker without raising an alarm for $\chi^2$, CUSUM, MEWMA, and Dynamic Watermarking detectors.
Using this notion of attack capability, this paper illustrates that all considered detectors have comparable performance.
However, on a real-world system, Dynamic Watermarking is the only detector that is capable of detecting the presence of a certain class of attacks.
\section{Real World Implementation}
\label{sec:exp}
In this section we draw further comparison between the older algorithms and the newer Dynamic Watermarking detector, using a segway robot performing path following. Because the real world system explored in this section is not open loop stable, we no longer make use of the notion of the attack capability. Instead we use two sample attacks to demonstrate each detectors capabilities. Furthermore we show that adding a watermark to a dynamic system leads to only marginal reduction in performance, while increasing the detectability of certain attacks. The experimental setup, includes a segway robot with path following control. Localization is provided by google cartographer \cite{Hess2016} using planar lidar, and wheel odometry measurements. For the purpose of control an LTI model was fitted to the observed data yielded the following model:
\begin{figure}[ht]
\centering
\includegraphics[trim={0 0 0 0},clip,width=\columnwidth]{figures/slam_spoofing_1.png}
\caption{Caption}
\label{fig:error_cord}
\end{figure}
\Ram{Please be kind to your reader and explain what this section will show before going into detail. Also please make sure to reference all figures in the text.}
\begin{equation}
\label{eq:error}
\begin{bmatrix}
e_{\ell,n+1}\\
e_{s,n+1}\\
e_{\theta,n+1}\\
e_{v,n+1}\\
e_{\dot{\theta},n+1}
\end{bmatrix}
=\begin{bmatrix}
e_{\ell,n}+(0.0478\tilde{v}_n)e_{\theta,n}-(0.045\dot{\tilde{\theta}}_n)e_{s,n}\\
e_{s,n}+(0.0478)e_{v,n}+(0.045\dot{\tilde{\theta}}_n)e_{\ell,n}\\
e_{\theta,n}+0.045e_{\dot{\theta},n}\\
e_{v,n}-0.1e_{v,n-4}+0.1u_{v,n}\\
0.6e_{\dot{\theta},n}+0.15e_{\dot{\theta},n-4}+0.24u_{\dot{\theta},n}
\end{bmatrix}
\end{equation}
where the state is represented in trajectory error coordinates for a given nominal trajectory as shown in figure \ref{fig:error_cord}, $e_{v,n},e_{\dot{\theta},n}$ are the error in the velocity and angular velocity, $\tilde{v}_n,\dot{\tilde{\theta}}_n$ are the nominal velocity and angular velocity and $u_{v,n},u_{\dot{\theta},n}$ are the deviation from the nominal inputs.
For a constant nominal velocity of $0.6m/s$ and angular velocity of $0 rad/s$ this model can be represented as an LTI model with state vector $x=[e_{\ell,n}~
e_{s,n}~
e_{\theta,n}~
e_{v,n}~
e_{\dot{\theta},n}~
e_{v,n-1}~
e_{v,n-2}~
e_{v,n-3}~
e_{\dot{\theta},n-1}~
e_{\dot{\theta},n-2}~
e_{\dot{\theta},n-3}]^T$. For the sake of brevity the A and B matrices will not be stated explicitly. The feedback gain matrix $K$ was found to make the closed loop system schurr stable and is approximately:
\begin{align}
K=\begin{bmatrix}
0 & -1.639 \\ 0 & -1.984 \\ -0.313 & 0\\ -0.212 & 0 \\ 0 & -0.384 \\ 0.019 & 0\\ 0.020 & 0 \\ 0.021 & 0 \\0.022 & 0 \\0 & -0.039 \\ 0 & -0.043 \\ 0 & -0.052 \\ 0 & -0.065
\end{bmatrix}
\end{align}
Similarly the observer gain matrix $L$ was found to make observer schurr stable and is approximately:
\begin{align}
L=\begin{bmatrix}
-0.791 & -0.016 & 0 & 0 & 0 \\
-0.002 & -0.501 & 0 & 0 & -0.022 \\
0 & 0 & -0.272 & -0.025 & 0 \\
0 & 0 & -0.011 & -0.252 & 0 \\
0 & -0.003 & 0 & 0 & -0.240 \\
0 & 0 & -0.013 & -0.258 & 0 \\
0 & 0 & -0.015 & -0.187 & 0 \\
0 & 0 & -0.015 & -0.133 & 0 \\
0 & 0 & -0.014 & -0.091 & 0 \\
0 & -0.004 & 0 & 0 & -0.395 \\
0 & -0.010 & 0 & 0 & -0.145 \\
0 & -0.008 & 0 & 0 & -0.054 \\
0 & -0.005 & 0 & 0 & -0.024 \\
\end{bmatrix}
\end{align}
The steady state covariance of the residuals, $\Sigma_r$ was approximated by running trajectory following for a straight line down a hallway of approximately 16 meters 40 times. To avoid the effects of the transient behavior at the start of each run, the beggining of each run was ignored. Runs where the localization from google cartographer jumped when updating, were also ignored. The steady state covariance was then approximated by the sample covariance of the remaining data. These experiments were repeated a second time, this time with a watermark being added to the control input. The resulting affect of adding the watermark is shown in figure \ref{fig:traj}. Furthermore each of the detectors was used in post processing on these experimental residuals to approximate the rate of false alarms under various parameters and threshold values. The experimental residuals without watermark were used for all detectors except for Dynamic Watermarking detector which used the second set of experimental residuals. Threshold values for each set of parameters were then found using linear interpolation. Threshold values for the CUSUM at $0.01$ were ignored. These values can be found in table \ref{tab:thresh_exp}.
\begin{table*}[t]
\centering
\begin{tabular}{|c|c|c|c|c|c|}
\hline
\multirow{2}{*}{$R_{FA}$} & \multirow{2}{*}{$\chi^2$} & CUSUM & MEWMA & Dyn. Wat. \\
& & $\gamma=(15~/~17~/~19)$ & $\beta=(0.6~/~0.7~/~0.8)$ &$\ell=(20~/~25~/~30)$ \\\hline
0.05 & 12.08 & ( 51.04 / 12.27 / 2.21 ) & ( 13.09 / 15.00 / 16.85 ) & ( 99.570 / 103.74 / 105.59 ) \\\hline
0.03 & 14.29 & ( 655.81 / 384.73 / 158.68 ) & ( 15.05 / 17.59 / 20.17 ) & ( 103.05 / 106.73 / 108.58 ) \\\hline
0.01 & 21.38 & ( - / - / - ) & ( 20.68 / 24.85 / 28.88 ) & ( 108.582 / 110.79 / 113.94 ) \\\hline
\end{tabular}
\caption{Experimental thresholds for various parameters}
\label{tab:thresh_exp}
\end{table*}
\begin{figure}[h]
\centering
\caption{Effect of watermark on performance}
\label{fig:traj}
\end{figure}
Two attacks following differing models were then applied. Attack model 1 takes the form:
\begin{align}
v_n&=-(Cx_n+z_n) +C\xi_n+\zeta_n.\label{eq:attack}
\end{align}
For this style of attack the attack measurement noise $\zeta$ is added to the false state such that $\zeta\sim\mathcal{N}(0,\Sigma_\zeta)$,and the false state $\xi\in\mathbb{R}^p$ is updated according to the closed loop dynamics of the system:
\begin{align}
\xi_{n+1}&=(A+BK)\xi_n+\omega_n
\end{align}
where the attack process noise $\omega$ is added such that $\omega\sim\mathcal{N}(0,\Sigma_\omega)$.
The attack process and measurement noise were chosen to leave the distribution of the residuals unchanged.
Attack model 2 assumes that the attacker adds random noise to the system such that $v_n\sim\mathcal{N}(0,1E-5 I)$.
For each of attack, 10 experimental runs were completed without a watermark and 10 with a watermark for a total of 40 experimental runs. The beginning of each run, and runs in which cartographer jumps were once again discarded. The runs with a watermark were used in evaluating the dynamic watermarking detector, while all other detectors used the un-watermarked data. The resulting rate of alarm for each of the detection algorithms in the presence of these attacks are displayed in table \ref{tab:exp_exam}.
\begin{table}[t]
\centering
\begin{tabular}{|c|c|c|c|}
\hline
Method & $R_{FA}$ & Attack 1 & Attack 2\\\hline
\multirow{3}{*}{$\chi^2$} & 0.05 & 0.029 & 0.686\\\cline{2-4}
& 0.03 & 0.012 & 0.622\\\cline{2-4}
& 0.01 & 0.000 & 0.467\\\hline
\multirow{3}{*}{$\underset{\gamma=(15/17/19)}{\text{CUSUM}}$} & 0.05 & ( 0 / 0 / 0 )& ( 0.979 / 0.986 / 0.989 )\\\cline{2-4}
& 0.03 & ( 0 / 0 / 0 ) & ( 0.812 / 0.856 / 0.921 )\\\cline{2-4}
& 0.01 & ( - / - / - ) & ( - / - / - )\\\hline
\multirow{3}{*}{$\underset{\beta=(0.6/0.7/0.8)}{\text{MEWMA}}$} & 0.05 &( 0.026 / 0.026 / 0.027 ) &( 0.510 / 0.568 / 0.621 )\\\cline{2-4}
& 0.03 & ( 0.013 / 0.011 / 0.011 ) & ( 0.450 / 0.500 / 0.545 )\\\cline{2-4}
& 0.01 & ( 0.001 / 0.001 / 0.000 ) & ( 0.319 / 0.354 / 0.394 )\\\hline
\multirow{3}{*}{$\underset{\ell=(20/25/30)}{\text{Dyn. Wat.}}$} & 0.05 & ( 0.984 / 0.998 / 1.00 )& ( 1.00 / 1.00 / 1.00 )\\\cline{2-4}
& 0.03 & ( 0.973 / 0.994 / 0.999 )& ( 1.00 / 1.00 / 1.00 )\\\cline{2-4}
& 0.01 & ( 0.949 / 0.983 / 0.996 )& ( 1.00 / 1.00 / 1.00 )\\\hline
\end{tabular}
\caption{Experimental Results}
\label{tab:exp_exam}
\end{table}
Note that for the first attack model the rate of alarms appear to drop for the $\chi^2$, CUSUM, and MEWMA detectors. This may be due to the residuals for the un-attacked system not being distributed as a Gaussian distribution resulting in higher threshold values. Since Attack model 1 replaces the feedback completely, the resulting residuals when under attack do follow a Gaussian distribution which then results in lower detection rates. The Dynamic Watermarking detector in the presence of the first attack provides a large increase in the rate of alarms for each set of parameters. For Attack 2 we see that all of the detectors are able to reliably detect the attack.
\section{INTRODUCTION}
This template provides authors with most of the formatting specifications needed for preparing electronic versions of their papers. All standard paper components have been specified for three reasons: (1) ease of use when formatting individual papers, (2) automatic compliance to electronic requirements that facilitate the concurrent or later production of electronic products, and (3) conformity of style throughout a conference proceedings. Margins, column widths, line spacing, and type styles are built-in; examples of the type styles are provided throughout this document and are identified in italic type, within parentheses, following the example. Some components, such as multi-leveled equations, graphics, and tables are not prescribed, although the various table text styles are provided. The formatter will need to create these components, incorporating the applicable criteria that follow.
\section{PROCEDURE FOR PAPER SUBMISSION}
\subsection{Selecting a Template (Heading 2)}
First, confirm that you have the correct template for your paper size. This template has been tailored for output on the US-letter paper size.
It may be used for A4 paper size if the paper size setting is suitably modified.
\subsection{Maintaining the Integrity of the Specifications}
The template is used to format your paper and style the text. All margins, column widths, line spaces, and text fonts are prescribed; please do not alter them. You may note peculiarities. For example, the head margin in this template measures proportionately more than is customary. This measurement and others are deliberate, using specifications that anticipate your paper as one part of the entire proceedings, and not as an independent document. Please do not revise any of the current designations
\section{MATH}
Before you begin to format your paper, first write and save the content as a separate text file. Keep your text and graphic files separate until after the text has been formatted and styled. Do not use hard tabs, and limit use of hard returns to only one return at the end of a paragraph. Do not add any kind of pagination anywhere in the paper. Do not number text heads-the template will do that for you.
Finally, complete content and organizational editing before formatting. Please take note of the following items when proofreading spelling and grammar:
\subsection{Abbreviations and Acronyms} Define abbreviations and acronyms the first time they are used in the text, even after they have been defined in the abstract. Abbreviations such as IEEE, SI, MKS, CGS, sc, dc, and rms do not have to be defined. Do not use abbreviations in the title or heads unless they are unavoidable.
\subsection{Units}
\begin{itemize}
\item Use either SI (MKS) or CGS as primary units. (SI units are encouraged.) English units may be used as secondary units (in parentheses). An exception would be the use of English units as identifiers in trade, such as �3.5-inch disk drive�.
\item Avoid combining SI and CGS units, such as current in amperes and magnetic field in oersteds. This often leads to confusion because equations do not balance dimensionally. If you must use mixed units, clearly state the units for each quantity that you use in an equation.
\item Do not mix complete spellings and abbreviations of units: �Wb/m2� or �webers per square meter�, not �webers/m2�. Spell out units when they appear in text: �. . . a few henries�, not �. . . a few H�.
\item Use a zero before decimal points: �0.25�, not �.25�. Use �cm3�, not �cc�. (bullet list)
\end{itemize}
\subsection{Equations}
The equations are an exception to the prescribed specifications of this template. You will need to determine whether or not your equation should be typed using either the Times New Roman or the Symbol font (please no other font). To create multileveled equations, it may be necessary to treat the equation as a graphic and insert it into the text after your paper is styled. Number equations consecutively. Equation numbers, within parentheses, are to position flush right, as in (1), using a right tab stop. To make your equations more compact, you may use the solidus ( / ), the exp function, or appropriate exponents. Italicize Roman symbols for quantities and variables, but not Greek symbols. Use a long dash rather than a hyphen for a minus sign. Punctuate equations with commas or periods when they are part of a sentence, as in
$$
\alpha + \beta = \chi \eqno{(1)}
$$
Note that the equation is centered using a center tab stop. Be sure that the symbols in your equation have been defined before or immediately following the equation. Use �(1)�, not �Eq. (1)� or �equation (1)�, except at the beginning of a sentence: �Equation (1) is . . .�
\subsection{Some Common Mistakes}
\begin{itemize}
\item The word �data� is plural, not singular.
\item The subscript for the permeability of vacuum ?0, and other common scientific constants, is zero with subscript formatting, not a lowercase letter �o�.
\item In American English, commas, semi-/colons, periods, question and exclamation marks are located within quotation marks only when a complete thought or name is cited, such as a title or full quotation. When quotation marks are used, instead of a bold or italic typeface, to highlight a word or phrase, punctuation should appear outside of the quotation marks. A parenthetical phrase or statement at the end of a sentence is punctuated outside of the closing parenthesis (like this). (A parenthetical sentence is punctuated within the parentheses.)
\item A graph within a graph is an �inset�, not an �insert�. The word alternatively is preferred to the word �alternately� (unless you really mean something that alternates).
\item Do not use the word �essentially� to mean �approximately� or �effectively�.
\item In your paper title, if the words �that uses� can accurately replace the word �using�, capitalize the �u�; if not, keep using lower-cased.
\item Be aware of the different meanings of the homophones �affect� and �effect�, �complement� and �compliment�, �discreet� and �discrete�, �principal� and �principle�.
\item Do not confuse �imply� and �infer�.
\item The prefix �non� is not a word; it should be joined to the word it modifies, usually without a hyphen.
\item There is no period after the �et� in the Latin abbreviation �et al.�.
\item The abbreviation �i.e.� means �that is�, and the abbreviation �e.g.� means �for example�.
\end{itemize}
\section{USING THE TEMPLATE}
Use this sample document as your LaTeX source file to create your document. Save this file as {\bf root.tex}. You have to make sure to use the cls file that came with this distribution. If you use a different style file, you cannot expect to get required margins. Note also that when you are creating your out PDF file, the source file is only part of the equation. {\it Your \TeX\ $\rightarrow$ PDF filter determines the output file size. Even if you make all the specifications to output a letter file in the source - if your filter is set to produce A4, you will only get A4 output. }
It is impossible to account for all possible situation, one would encounter using \TeX. If you are using multiple \TeX\ files you must make sure that the ``MAIN`` source file is called root.tex - this is particularly important if your conference is using PaperPlaza's built in \TeX\ to PDF conversion tool.
\subsection{Headings, etc}
Text heads organize the topics on a relational, hierarchical basis. For example, the paper title is the primary text head because all subsequent material relates and elaborates on this one topic. If there are two or more sub-topics, the next level head (uppercase Roman numerals) should be used and, conversely, if there are not at least two sub-topics, then no subheads should be introduced. Styles named �Heading 1�, �Heading 2�, �Heading 3�, and �Heading 4� are prescribed.
\subsection{Figures and Tables}
Positioning Figures and Tables: Place figures and tables at the top and bottom of columns. Avoid placing them in the middle of columns. Large figures and tables may span across both columns. Figure captions should be below the figures; table heads should appear above the tables. Insert figures and tables after they are cited in the text. Use the abbreviation �Fig. 1�, even at the beginning of a sentence.
\begin{table}[h]
\caption{An Example of a Table}
\label{table_example}
\begin{center}
\begin{tabular}{|c||c|}
\hline
One & Two\\
\hline
Three & Four\\
\hline
\end{tabular}
\end{center}
\end{table}
\begin{figure}[thpb]
\centering
\framebox{\parbox{3in}{We suggest that you use a text box to insert a graphic (which is ideally a 300 dpi TIFF or EPS file, with all fonts embedded) because, in an document, this method is somewhat more stable than directly inserting a picture.
}}
\caption{Inductance of oscillation winding on amorphous
magnetic core versus DC bias magnetic field}
\label{figurelabel}
\end{figure}
Figure Labels: Use 8 point Times New Roman for Figure labels. Use words rather than symbols or abbreviations when writing Figure axis labels to avoid confusing the reader. As an example, write the quantity �Magnetization�, or �Magnetization, M�, not just �M�. If including units in the label, present them within parentheses. Do not label axes only with units. In the example, write �Magnetization (A/m)� or �Magnetization {A[m(1)]}�, not just �A/m�. Do not label axes with a ratio of quantities and units. For example, write �Temperature (K)�, not �Temperature/K.�
\section{CONCLUSIONS}
A conclusion section is not required. Although a conclusion may review the main points of the paper, do not replicate the abstract as the conclusion. A conclusion might elaborate on the importance of the work or suggest applications and extensions.
\addtolength{\textheight}{-12cm}
\section*{APPENDIX}
Appendixes should appear before the acknowledgment.
\section*{ACKNOWLEDGMENT}
The preferred spelling of the word �acknowledgment� in America is without an �e� after the �g�. Avoid the stilted expression, �One of us (R. B. G.) thanks . . .� Instead, try �R. B. G. thanks�. Put sponsor acknowledgments in the unnumbered footnote on the first page.
References are important to the reader; therefore, each citation must be complete and correct. If at all possible, references should be commonly available publications.
\section{Introduction}
Cyber-Physical Systems (CPS) have proven difficult to secure due to the ever-present risk of malicious attacks.
Failure to detect such attacks can have catastrophic consequences \cite{Abrams2008,Langner2011, Lee2016AnalysisGrid}.
Though these real-world attacks, such as the Stuxnet Worm or the attack on the Ukranian Power Grid, highlight the threat to existing industrial facilities and public utilities, researchers speculate that attacks on next generation transportation systems could be even more dangerous due to the reliance on communication between infrastructure and privately owned vehicles \cite{Amoozadeh2015,Dominic2016,Chen2018ExposingControl}.
To counter the growing risks of attacks on CPS, researchers have attempted to develop techniques to detect attacks while they are being conducted.
Rather than address all possible attacks, researchers have focused on detecting additive attacks in which an attacker alters a measurement that is used while performing feedback control.
To identify these attacks at run-time, detection schemes compute the residual between a state observer and measurements and then analyze this residual signal to determine whether an attack is taking place.
Three detectors, which were originally proposed for anomaly detection in quality control applications \cite{Hotelling1947MultivariateControl}, analyze the residual signal:
the $\chi^2$, cumulative sum (CUSUM), and multivariate exponentially weighted moving average (MEWMA) detectors \cite{Mo2010FalseSystems,Murguia2016CUSUMSensors}.
\begin{figure}[t]
\centering
\includegraphics[trim={0.2in 0.5in 0.2in 0.05in},clip,width=0.4\textwidth]{figures/ACC_graphic6.jpg}
\caption{This paper describes a real-world implementation of malicious attack detection on a Segway Robotics Mobility Platform performing a path-following task (top).
An additive attack is applied to this system as shown in the block diagram (bottom).
Four different detectors ($\chi^2$, CUSUM, MEWMA, and the presented Dynamic Watermarking method) are applied in the ``Detector'' block.
The Dynamic Watermarking method is shown to detect attacks that the others cannot, without sacrificing tracking performance relative to the desired path.
Video of this example can be found at \cite{Porter_vid_2018}.}
\label{fig:overview}
\vspace*{-1.25em}
\end{figure}
To detect an attack, each of the detectors evaluates a test statistic that is a function of the residual.
If this statistics value rises above some user-specified threshold, then the detector triggers an alarm.
To evaluate and design the threshold for these detectors, researchers have proposed the following three metrics: first, the \emph{attack capability} or the amount of perturbation to the state of the system that an attack can induce without either inducing an alarm \cite{Murguia2016CUSUMSensors} or without increasing the rate of alarms \cite{Murguia2018OnAttacks,Mo2016}; second, the rate of false alarms ($R_{FA}$) given by the detector when no attack is occurring; and third, the ability of the detector to reliably detect specific attack models.
For an open-loop stable system, the attack capability can be evaluated by computing the reachable set of the error in the observed state.
Since computing this reachable set can be challenging, researchers have instead attempted to evaluate surrogates for the attack capability such as the expected value of the state vector \cite{Murguia2016CUSUMSensors} or the norm of the largest time invariant residual \cite{Umsonst2018}.
However, these surrogates are unable to accurately characterize the attack capability of attacks that have large residuals for short amounts of time.
Note that, by reducing the threshold in any detector, one can reduce the attack capability; however, this can increase the rate of false alarms.
To compute this false alarm rate for classic anomaly detectors, it is typically assumed that the residuals are independent \cite{Murguia2016CUSUMSensors,Umsonst2018}.
Unfortunately, simulated experiments have noted that this assumption can lead to a consistent error while computing the false alarm rate \cite{Murguia2016CUSUMSensors}.
Researchers have also shown that a $\chi^2$ detector is always capable of identifying a specific type of additive attack where a measured signal is replayed \cite{Mo2009}; however, the ability to detect specific attacks has been less studied for the other detectors.
More recently, researchers have begun exploring techniques to detect more sophisticated attacks, which exploit knowledge of the system dynamics.
These detectors rely on \emph{Dynamic Watermarking} wherein an excitation signal that is only known to the control system is introduced into the input.
The detector then evaluates the covariance of the residual signal with the watermark to determine whether the system is under attack.
These Dynamic Watermarking based detectors are theoretically proven to detect attacks that exploit knowledge of the system dynamics.
Initially, Dynamic Watermarking was developed for LTI systems with full rank input matrices and full state observations.
These methods were proven to detect arbitrary attack models \cite{Satchidanandan2017} including attacks that replayed a measured signal \cite{Weerakkody2014}.
These methods were later extended to generalized LTI systems \cite{Hespanhol2017},
and to networked control systems \cite{Hespanhol2018}.
Though Dynamic Watermarking is proven to be capable of detecting a larger class of attacks when compared to prior detection algorithms, to the best of our knowledge, no one has conducted a real-world evaluation of any of these attack detection algorithms.
Moreover, no one has evaluated the attack capability of a system employing a Dynamic Watermarking scheme as a function of false alarm rate or developed a technique to design a detector using Dynamic Watermarking that achieves a user-specified false alarm rate.
The contributions of this paper are three-fold.
First, in Section \ref{sec:reach}, we construct a metric for quantifying the attack capability based on the reachable set of the portion of the observer error related to the residual without triggering an alarm, and we develop a technique to compute an outer approximation of this reachable set for the $\chi^2$, CUSUM, MEWMA, and Dynamic Watermarking detectors.
Second, in Section \ref{sec:false}, we develop an empirical method to design a Dynamic Watermarking detector that achieves a user-specified false alarm rate.
Third, in Section \ref{sec:simp}, the attack capability for each detector is compared, and in Section \ref{sec:exp} a real world example, shown in Figure \ref{fig:overview}, is presented to evaluate the performance of the $\chi^2$, CUSUM, MEWMA, and Dynamic Watermarking detectors for specific attacks.
See \cite{Porter_vid_2018} for a video of this real-world test.
The rest of this document is outlined as follows.
In Section \ref{sec:prelim}, we define the notation used in the paper, along with the LTI system model and assumptions.
In Section \ref{sec:alg} we provide notation for each of the detectors.
We draw conclusions in Section \ref{sec:con}.
\section{Preliminaries}
\label{sec:prelim}
This section describes the notation and assumptions used throughout the paper for the LTI system model.
\subsection{Notation}
\label{subsec:notation}
This paper uses three different probability distributions:
the multivariate Gaussian distribution, denoted $\mathcal{N}(\mu,\Sigma)$, with mean $\mu$ and covariance $\Sigma$;
the $\chi^2$ distribution, denoted $\chi^2(i)$, with $i$ degrees of freedom; and the the Wishart distribution, denoted $\mathcal{W}(\Sigma,i)$, with scale matrix $\Sigma$ and $i$ degrees of freedom \cite[Section 7.2]{Anderson2003AnAnalysis}.
The Euclidean Norm of a vector $X \in \mathbb{R}^n$ is denoted $\|X\|$.
Similarly, the induced operator norm for a matrix $X \in \mathbb{R}^{n \times m}$ is denoted $\|X\|$.
The expectation of a variable $X$ is denoted $\mathds{E}[X]$.
Zero matrices of dimension $i\times j$ are denoted $0_{i\times j}$, and in the case that $i=j$ the notation is simplified to $0_i$.
The identity matrix of dimension $i$ is denoted $I_i$.
The closed unit ball of radius $\epsilon$ is denoted $\mathcal{B}_\epsilon$.
The Minkowsi sum is denoted $\oplus$.
The minimum singular value of a matrix $X \in \mathbb{R}^{n\times m}$ is denoted $s_1(X)$.
\subsection{LTI Model}
This paper considers a discrete linear time invariant systems where the discrete time step is indexed by $n \in \mathbb{N}$:
\begin{align}
x_{n+1} &= Ax_n+Bu_n+w_n\\
y_n &= Cx_n+z_n+v_n
\end{align}
with state $x_n\in\mathbb{R}^p$, measurement $y_n\in\mathbb{R}^q$, and input $u_n\in\mathbb{R}^m$.
The process noise $w_n\in\mathbb{R}^p$ and the measurement noise $z_n\in\mathbb{R}^q$ are assumed to be Gaussian with $w_n\sim\mathcal{N}(0,\Sigma_w)$ and $z_n\sim\mathcal{N}(0,\Sigma_z)$.
At each time step, the attacker adds $v_n\in\mathbb{R}^q$ to the measurement.
For each $n \in \mathbb{N}$ an observer recovers the full observed state $\hat{x}_n$ from the measurements:
\begin{align}
\hat{x}_{n+1}=&(A+LC)\hat{x}_n+Bu_n-Ly_n.
\end{align}
The observed state is then used in full state feedback:
\begin{align}
u_n=&K\hat{x}_n+e_n
\end{align}
where $e_n\sim\mathcal{N}(0,\Sigma_e)$ is a private watermark and is only added when running the Dynamic Watermarking detector.
The controller gain matrix $K$ and the observer gain matrix $L$ are chosen such that the closed loop matrices $(A+BK)$ and $(A+LC)$ are Schur Stable.
For each $n \in \mathbb{N}$, the observer error $\delta_{n}=\hat{x}_n-x_n$ then has the following update equation:
\begin{equation}
\delta_{n+1}=(A+LC)\delta_n-w_n-Lz_n-Lv_n.\label{eq:observe_error}
\end{equation}
Denote the residual as $r_n$ which is defined as:
\begin{equation}
r_n=C\hat{x}_n-y_n=C\delta_n-z_n-v_n.\label{eq:res}
\end{equation}
When the system is not being attacked, i.e. $v_n=0,~\forall n\in \mathbb{N}$, the steady state covariance for the observer error $\Sigma_\delta=\lim_{n\rightarrow \infty}\mathds{E}[\delta_n\delta_n^T]$ can be found by a Discrete Lyapunov Equation as described in \cite[Section 2.2.2]{Lewis2007}:
\begin{align}
\Sigma_\delta=&(A+LC)\Sigma_\delta(A+LC)^T+\Sigma_w+L\Sigma_zL^T.\label{eq:lyap}
\end{align}
We can then use the steady state covariance of the observer error to find the steady state covariance of the residual $\Sigma_r$ when the system is not being attacked:
\begin{align}
\Sigma_r&=\lim_{n\rightarrow \infty}\mathds{E}[r_nr_n^T]\\
&=\lim_{n\rightarrow \infty}\mathds{E}[C\delta_n\delta_n^TC^T-C\delta_nz_n^T-z_n\delta_n^TC^T+z_nz_n^T]\label{eq:cov_cross}\\
&=C\Sigma_\delta C^T+\Sigma_z,
\end{align}
where the expectation of the cross terms in \eqref{eq:cov_cross} are zero due to causality.
Using the steady state covariance of the residuals, define the normalized residual $\bar{r}_n$:
\begin{align}
\bar{r}_n=\Sigma_r^{-1/2}r_n.\label{eq:norm_res}
\end{align}
Furthermore, we define the vector of current and previous normalized residuals:
\begin{align}
R_n&=[\bar{r}_n^T\hdots \bar{r}_0^T]^T.\label{eq:R}
\end{align}
| -63,705.792087 |
[
-2.458984375,
2.23828125
] | 36.490251 |
[
-3.189453125,
0.666015625,
-1.818359375,
-6.0625,
-1.3154296875,
8.09375
] |
[
0.55615234375,
6.2734375,
1.009765625,
6.4453125
] | 846 | 8,561 |
[
-3.56640625,
4.1953125
] | 33.128036 |
[
-6.2578125,
-4.26171875,
-4.1484375,
-2.15234375,
2.267578125,
11.8046875
] | 0.898226 | 15.649707 | 24.214094 | 6.237332 |
[
2.3521645069122314
] | -36,373.921037 | 6.177082 | -62,145.819112 | 0.816913 | 6.24405 |
[
-2.341796875,
-3.505859375,
-3.853515625,
-5.125,
2.421875,
12.125
] |
[
-5.59765625,
-1.94921875,
-1.8955078125,
-1.6806640625,
3.34375,
4.4453125
] | |
BkiUdl84uBhjAAuFmfLG
|
\section{Introduction}
In many occasions, cosmology has been and still is an invaluable means to constrain particle physics models. These constraints can arise by using information from homogeneous cosmology, such as the constraints on the light degrees of freedom at nucleosynthesis. With the discovery of the Cosmic Microwave Background (CMB) fluctuations, new constraints arise from the observed power spectrum of linear perturbations. Even more recently, the upper bounds on primordial non-Gaussianities have started to be used to constrain early Universe scenarios.
Although the simplest early Universe models are based on inflationary models with a single scalar field, many models involve additional scalar fields, which can play a dynamical role during inflation or simply be spectactor fields (see e.g. \cite{Langlois:2010xc} for introductory lectures). The existence of several degrees of freedom opens up the possibility of isocurvature perturbations, i.e. perturbations in the particle density ratio between two fluids, for example cold dark matter (CDM) isocurvature perturbations (between CDM and radiation) or baryon isocurvature perturbations (between baryons and radiation). Since primordial isocurvature perturbations leave distinctive features in the CMB anisotropies, they can be in principle disentangled from the usual adiabatic mode. The present upper bound on the isocurvature contribution to the power spectrum provides a stringent constraint.
This is the case for the curvaton scenario~\cite{curvaton} where large residual isocurvature perturbations (for CDM or baryons) can be generated, depending on how and when CDM or baryons are produced~\cite{Lyth:2002my, Lyth:2003ip} (see also \cite{Lemoine:2006sc,Lemoine:2008qj} for more detailed scenarios). The same constraints apply to moduli that are light during inflation, and thus acquire super-Hubble fluctuations, as discussed recently in \cite{Lemoine:2009is}.
Another potentially useful information on primordial perturbations is the amplitude and shape of their non-Gaussianity. So far, the current CMB data seem to favour a non-zero amount of so-called local non-Gaussianity~\cite{Komatsu:2010fb}, but Planck data will be needed to confirm or infirm this trend. Several models can generate local non-Gaussianity (see e.g. \cite{Wands:2010af} for a recent review): multiple field inflation (during inflation or at the end of inflation: see e.g. \cite{Byrnes:2010em}), modulated reheating~\cite{Zaldarriaga:2003my,Vernizzi:2003vs}, curvaton, modulated trapping~\cite{Langlois:2009jp}, etc.
It is thus interesting to combine the constraints on isocurvature modes and non-Gaussianity to explore the early Universe physics, as has been done recently in various scenarios \cite{beltran,Kawasaki:2008sn,Kawasaki:2008jy,Langlois:2008vk,Kawasaki:2008pa,Hikage:2008sk, Kawakami:2009iu,Takahashi:2009cx}.
The purpose of the present work is to give a unified treatment of linear and nonlinear perturbations, which enables to compute their evolution through one or several cosmological transitions, such as the decay of some particle species.
Our treatment takes into account the various decay products and their branching ratio.
Our formalism can thus be applied to a large class of early Universe scenarios, in order to compute automatically their predictions for adiabatic and isocurvature perturbations, and their non-Gaussianities. As input, one simply needs
parameters that depend on the homogeneous evolution. This thus provides a simple way to confront
an early Universe scenario, and its underlying particle physics model, with the present and future cosmological data.
As applications to our general formalism, we consider two specific examples. The first example is a more refined treatment of the isocurvature perturbations and their non-Gaussianity in the mixed curvaton-inflation scenario~\cite{Langlois:2004nn,Ferrer:2004nv,Lazarides:2004we}.
The second example deals with a multiple-curvaton scenario \cite{Hamaguchi:2003dc,Choi:2007fya,Assadullahi:2007uw,Huang:2008rj}. In both examples, we generalize the results that have been obtained in previous works, allowing the curvaton to decay into several species.
This paper is organized as follows. In Section 2, we introduce the non-linear curvature and isocurvature perturbations.
Section 3 is devoted to the general treatment of a cosmological transition, such as the decay of some particle species.
In Section 4, we focus on the first application, namely the mixed curvaton-inflaton scenario with a single curvaton.
In Section 5, we consider scenarios with two curvatons. We conclude in the final Section.
\section{Non-linear curvature perturbations}
We first introduce the notion of non-linear curvature perturbation. Several definitions have been proposed, which turn out to be equivalent on large scales, and we will follow here the covariant approach introduced in
\cite{Langlois:2005ii,Langlois:2005qp}, and reviewed recently in \cite{Langlois:2010vx}.
For a perfect fluid characterized by the energy density $\rho$, the pressure $P$ and the four-velocity $u^a$, the conservation law $\nabla_a T^{a}_{\ b}=0$ for the energy-momentum tensor,
$T_{ab}=\left(\rho+P\right) u_a u_b
+P g_{ab}$,
implies that the covector
\begin{equation}
\zeta_a\equiv
\nabla_aN-\frac{\dotN}{\dot\rho}\nabla_a\rho
\label{zeta_a}
\end{equation}
satisfies the relation
\begin{equation}
\label{dot_zeta}
\dot\zeta_a\equiv {\cal L}_u\zeta_a=
-\frac{\Theta}{3(\rho+p)}\left( \nabla_a p -
\frac{\dot p}{\dot \rho} \nabla_a\rho\right) \;,
\end{equation}
with the definitions
\begin{equation}
\Theta\equiv\nabla_a u^a, \quad N\equiv\frac{1}{3}\int d\tau \,
\Theta \;,
\end{equation}
where $\tau$ is the proper time along the fluid worldlines and a dot denotes a Lie derivative along $u^a$, which is equivalent to an ordinary
derivative for {\it scalar} quantities (e.g. $\dot\rho\equiv u^a\nabla_a\rho$). $N$ can be interpreted as the number of e-folds of the local scale factor associated with an observer following the fluid.
The covector $\zeta_a$ can be defined for the global cosmological fluid or for any of the
individual cosmological fluids, as long as they are non-interacting (the case of interacting fluids is discussed in \cite{Langlois:2006iq}). Using
the non-linear conservation equation
\begin{equation}
\dot\rho=-3\dotN(\rho+P)\;,
\end{equation}
which follows from $u^b\nabla_a T^a_{\ b}=0$,
one can re-express $\zeta_a$ in the form
\begin{equation}
\label{zeta_a2}
\zeta_a=\nabla_aN+\frac{\nabla_a\rho}{3(\rho+P)} \;.
\end{equation}
If $w\equiv P/\rho$ is constant,
the above covector is a total gradient and can be written as
\begin{equation}
\zeta_a=\nabla_a\left[N+\frac{1}{3(1+w)}\ln \rho\right]\, .
\label{zeta_a_w}
\end{equation}
On scales larger than the Hubble radius, the above definitions are
equivalent to the non-linear curvature perturbation on uniform
density hypersurfaces as defined in
\cite{Lyth:2004gb},
\begin{equation} \zeta = \delta N -
\int_{\bar \rho}^{\rho} H \frac{d \tilde \rho}{\dot{ \tilde \rho}}
=
\delta N + \frac13\int_{\bar \rho}^{\rho} \frac{d
\tilde{\rho}}{(1+w)\tilde{ \rho}}\;, \label{zeta}
\end{equation}
where $H=\dot a /a$ is the Hubble parameter.
It will be useful to distinguish the non-linear curvature
perturbation $\zeta$ of the total fluid, from the individual non-linear
perturbation $\zeta_A$ that describes the cosmological fluid $A$ (with $w_A\equiv P_A/\rho_A=0$ for a pressureless fluid or
$w_A=1/3$ for a relativistic fluid),
defined by
\begin{equation}
\label{defzeta}
\zeta_{_A} = \delta N +
\frac{1}{3(1+w_{_A})} \ln\left( \frac{\rho_{_A}}{\bar
\rho_{_A}} \right)\; ,
\end{equation}
where a bar denotes a homogeneous quantity.
Inverting this relation yields the expression of the inhomogeneous energy density as a function of the background energy density and of the curvature perturbation $\zeta_{_A}$,
\begin{equation}
\label{zeta_deltaN}
\rho_{_A}={\bar
\rho_{_A}} e^{3(1+w_{_A}) ( \zeta_{_A} -\delta N)}\, ,
\end{equation}
which we will use many times in the following.
The non-linear isocurvature (or entropy) perturbation between two fluids $A$ and $B$ is defined by
\begin{equation}
S_{A,B}\equiv 3(\zeta_A-\zeta_B).
\end{equation}
In the following, we will always define the isocurvature perturbations with respect to the radiation fluid, so that our definition for the isocurvature perturbation of the fluid $A$ will be
\begin{equation}
\label{iso_pert}
S_A\equiv 3(\zeta_A-\zeta_r),
\end{equation}
where $\zeta_r$ is the uniform-density curvature perturbation of the radiation fluid.
\section{Decay}
Let us now consider a cosmological transition associated with the decay of some species of particles, which we will call $\sigma$.
In the sudden decay approximation, the decay takes place on the hypersurface characterized by the condition
\begin{equation}
H_{\rm d}=\Gamma_\sigma\, .
\end{equation}
Therefore, since $H$ depends only on the {\it total} energy density, the decay hypersurface is a hypersurface of uniform total energy density, with $\delta N_{\rm d}=\zeta$, where $\zeta$ is the global curvature perturbation. Using
(\ref{zeta_deltaN}), the equality between the total energy densities, respectively before and after the decay, thus reads
\begin{equation}
\label{bilan_decay}
\sum_A\bar{\rho}_{A-}e^{3(1+w_A)(\zeta_{A-}-\zeta)}=\bar{\rho}_{\rm decay}=\sum_B\bar{\rho}_{B+}e^{3(1+w_B)(\zeta_{B+}-\zeta)},
\end{equation}
where the subscripts
$-$ and $+$ label quantities defined, respectively, {\it before} and {\it after} the transition.
\subsection{Before the decay}
The first equality in (\ref{bilan_decay}) leads to
\begin{equation}
\label{zeta_non_linear}
\sum_A \Omega_{A}e^{3(1+w_A)(\zeta_{A-}-\zeta)}=1,
\end{equation}
where we have defined
$\Omega_A\equiv {\bar\rho_{A-}}/\bar\rho_{\rm decay}$
(to avoid confusion, the $\Omega_A$ are always defined just {\it before} the decay).
The above relation determines $\zeta$ as a function of the $\zeta_{A-}$.
At linear order, this gives
\begin{equation}
\label{zeta_decay}
\zeta=
\frac{1}{{\tilde\Omega}} \sum_A{\tilde\Omega}_A\, \zeta_{A-}\qquad ({\rm first}\ {\rm order})
\end{equation}
with the notation
\begin{equation}
{\tilde\Omega}_A\equiv (1+w_A) \Omega_A, \qquad {\tilde\Omega}\equiv\sum_A{\tilde\Omega}_A\, .
\end{equation}
Expanding (\ref{zeta_non_linear})
up to second order, one finds
\begin{equation}
\zeta=\frac{1}{{\tilde\Omega}} \sum_A{\tilde\Omega}_A\, \left[\zeta_{A-}+\frac32 (1+w_A)\left(\zeta_{A-}-\zeta\right)^2\right] \qquad ({\rm second}\ {\rm order})
\end{equation}
where, on the right hand side, $\zeta$ is to be replaced by its first order expression (\ref{zeta_decay}).
\subsection{After the decay}
We now consider the outcome of the decay. In general, the species $\sigma$ decays
into various species $A$, with respective decay widths $\Gamma_{A\sigma}$.
Defining the relative branching ratios
\begin{equation}
\gamma_{A\sigma}\equiv\frac{\Gamma_{A\sigma}}{\Gamma_\sigma}, \quad \Gamma_\sigma\equiv \sum_A \Gamma_{A\sigma}\, ,
\end{equation}
one can write the energy density of the fluid $A$ after the decay in terms of the energy densities of $A$ and of $\sigma$ as
\begin{equation}
\rho_{A+}=\rho_{A-}+\gamma_{A\sigma}\rho_{\sigma -}\, .
\end{equation}
Using (\ref{zeta_deltaN}), one can rewrite this nonlinear equation in terms of the curvature perturbations $\zeta_{A+}$, $\zeta_{A-}$ and $\zeta_{\sigma-}$, which yields
\begin{equation}
\label{master_eq}
e^{3(1+w_A)(\zeta_{A+}-\zeta)}=\frac{\bar{\rho}_{A-}e^{3(1+w_A)(\zeta_{A-}-\zeta)}+\gamma_{A\sigma}\bar{\rho}_{\sigma -}e^{3(1+w_\sigma)(\zeta_{\sigma -}-\zeta)}}{\bar{\rho}_{A-} +\gamma_{A\sigma}\bar{\rho}_{\sigma -}} \, .
\end{equation}
This expression thus gives $\zeta_{A+}$ as a function of $\zeta_{A-}$, $\zeta_{\sigma}$ and of the global $\zeta$. Substituting $\zeta$ in terms of $\zeta_{\sigma}$ and of all the $\zeta_{B-}$, one finally obtains $\zeta_{A+}$ as a function of all the $\zeta_{B-}$.
Following this procedure, one finds that the linear curvature perturbation for any given fluid $A$ is given by
\begin{equation}
\label{decay_linear}
\zeta_{A+}=\sum_B T_{A}^{\ B}\zeta_{B-}\qquad ({\rm first}\ {\rm order})
\end{equation}
with
\begin{eqnarray}
T_{A}^{\ A}&=& 1-f_A+ f_A\frac{(w_A-w_\sigma ){\tilde\Omega}_A}{(1+w_A){\tilde\Omega}}
\label{T1}
\\
T_{A}^{\ \sigma}&=& f_A\, \frac{1+w_\sigma}{1+w_A}+f_A\frac{(w_A-w_\sigma ){\tilde\Omega}_\sigma}{(1+w_A){\tilde\Omega}}
\label{T2}
\\
T_{A}^{\ C}&=& f_A\frac{(w_A-w_\sigma ){\tilde\Omega}_C}{(1+w_A){\tilde\Omega}}, \qquad C\neq A, \sigma\, .
\label{T3}
\end{eqnarray}
In the above expressions, we have introduced the parameter
\begin{equation}
\label{f_A}
f_A\equiv \frac{\gamma_{A\sigma}\Omega_{\sigma}}{\Omega_{A}+\gamma_{A\sigma}\Omega_{\sigma}}\, ,
\end{equation}
which represents the fraction of the fluid $A$ that has been created by the decay. If $A$ does not belong to the decay products of $\sigma$, then $f_A=0$. The opposite limit, $f_A=1$, occurs when all the fluid $A$ is produced by the decay. For the intermediate values of $f_A$, part of $A$ is produced by the decay while the other part is preexistent.
In the following, we will assume that
the decaying species behaves like non-relativistic matter (this is the case for a curvaton or modulus field that oscillates in a quadratic potential)
and we will thus always use
$ w_\sigma = 0$.
From the above expressions (\ref{T1}-\ref{T3}), it is straightforward to check that
\begin{equation}
\label{sum1}
\sum_B T_{A}^{\ B}=1.
\end{equation}
The post-decay perturbation $\zeta_{A+}$ can thus be seen as the barycenter of the pre-decay perturbations $\zeta_{B-}$ with the weights $T_{A}^{\ B}$ (all these coefficients satisfy $0\leq T_{A}^{\ B}\leq 1$
for $w_\sigma=0$). Note that
if the fluid $A$ is not produced in the decay (i.e. $f_A=0$), then the transfer coefficients are trivial: $T_{A}^{\ B}=\delta_{A}^{\ B}$.
Since it is convenient to use the same range of species indices {\it before and after} the transition, we also
introduce the coefficients
$T_{\sigma}^{\ B}=0$, which imply that $\zeta_{\sigma+}=0$. This convention will be especially useful when one needs to combine several transitions, as we will discuss soon.
At second order, expanding (\ref{master_eq}) and substituting the first order expression (\ref{zeta_decay}) for $\zeta$, one obtains
\begin{equation}
\label{zeta_2ndorder}
\zeta_{A+}=\sum_BT_A^{\ B}\zeta_{B-}+\sum_{B, C}U_A^{BC}\zeta_{B-}\zeta_{C-},
\qquad ({\rm second}\ {\rm order})
\end{equation}
with
\begin{eqnarray}
U_A^{BC}&\equiv&\frac32 \left[T_{AB}(1+w_B)\delta_{BC}+2\frac{{\tilde\Omega}_C}{{\tilde\Omega}}(w_A-w_B)T_{AB}
-(1+w_A) T_{AB} T_{AC}
\right.
\cr
&&\left. \quad
-\frac{{\tilde\Omega}_B {\tilde\Omega}_C}{{\tilde\Omega}^2}\left(1+w_A-\sum_D T_{AD}(1+w_D)\right)\right].
\label{U_ABC}
\end{eqnarray}
The change of the various isocurvature perturbations, defined in (\ref{iso_pert}), can also be determined by using the above expressions.
In particular, at linear order, one finds, using the property (\ref{sum1}), the simple expression
\begin{equation}
\label{S_barycenter}
S_{A+}=\sum_B \left(T_A^{\ B}-T_r^{\ B}\right) S_{B-}\,
\qquad ({\rm first}\ {\rm order}).
\end{equation}
\subsection{Several transitions}
If the early Universe scenario involves several cosmological transitions, for example several particle decays, one can use the above expressions successively to determine the final ``primordial'' perturbations, i.e. the initial conditions at the onset of the standard cosmological era.
For linear perturbations, the expression of the final perturbations as a function of the initial ones, is simply given by
\begin{equation}
\zeta^{(f)}_A=\sum_B T_A^{\ B} \zeta^{(i)}_B, \qquad T=\prod_k T_{[k]}
\end{equation}
where $T$ is the matricial product of all transfer matrices $T_{[k]}$, which describe the successive transitions.
The cosmological transitions can result from the decay of some particle species but they can be of other types.
For example, if CDM consists of WIMPs (Weakly Interacting Massive Particles), the freeze-out can be treated as a cosmological transition.
If radiation is the dominant species at freeze-out, then $\zeta_{c+}=\zeta_r$. But,
if other species are significant in the energy budget of the universe at the time of freeze-out, any entropy perturbation between the extra species and radiation will modify the above relation. The presence of a pressureless component, like a curvaton, leads to \cite{Lyth:2003ip}
\begin{equation}
\label{freeze_out}
\zeta_{c+}=\zeta_{r-}+\frac{(\alpha_f -3)\Omega_{\sigma}}{2(\alpha_f-2)+\Omega_{\sigma}}\left(\zeta_{\sigma -}-\zeta_{r-}\right),\qquad \alpha_f\equiv \frac{m_c}{T_f}+\frac32
\end{equation}
at linear order, while the other $\zeta_A$ remain unchanged.
The symbol ``$\sigma$" denotes here the
conglomerate of all pressureless matter at the time of freeze-out, except of course the CDM species that is freezing out.
\section{Scenario with a single curvaton}
\defx_r{x_r}
\defx_c{x_c}
\deff_c{f_c}
Let us now apply our formalism to a simple scenario with only three initial species: radiation ($r$), CDM ($c$) and a curvaton ($\sigma$), considered in e.g. \cite{Gupta:2003jc}. After the decay of the curvaton, the radiation and CDM perturbations remain unchanged and provide the initial conditions for the perturbations at the onset of the standard cosmological phase (let us say around $T\sim 1$ MeV).
\subsection{Perturbations after the decay}
\subsubsection{Linear order}
According to the expressions (\ref{T1}-\ref{T3}), the linear transfer matrix $T_{AB}$ is given in this case by
\begin{equation}
\label{T}
T=
\left( \begin{array}{ccccc}
1-x_r && x_c && x_r-x_c \\
0 && 1-f_c && f_c \\
0 && 0 && 0
\end{array} \right),
\quad x_r\equiv \frac{f_r}{{\tilde\Omega}}, \quad x_c\equiv \frac14\Omega_c\, x_r
\end{equation}
where the order of the species is $(r,c,\sigma)$. This means that
the linear curvature perturbations for radiation and for CDM, after the
curvaton decay, are given respectively by
\begin{equation}
\zeta_{r+}= (1-x_r)\, \zeta_{r-}+x_c\, \zeta_{c-}+(x_r-x_c)\, \zeta_{\sigma-}
\end{equation}
and
\begin{equation}
\zeta_{c+}=(1-f_c)\, \zeta_{c-}+f_c \, \zeta_{\sigma-}.
\end{equation}
The entropy perturbation after the decay is thus
\begin{equation}
\frac13 S_{c+}\equiv \zeta_{c+}-\zeta_{r+}=(1-f_c-x_c)\zeta_{c-}+(x_r-1)\zeta_{r-}+(f_c+x_c-x_r)\zeta_{\sigma-}\,,
\end{equation}
which can also be expressed directly in terms of the pre-decay entropy perturbations, following (\ref{S_barycenter}),
\begin{equation}
S_{c+}=(1-f_c-x_c) \, S_{c-}+(f_c+x_c-x_r)\, S_{\sigma-}\, .
\end{equation}
Note that, if many CDM particles are created by the decay of the curvaton, a significant fraction of them could annihilate, leading to an effective suppression of the final isocurvature perturbation. This effect has been studied in \cite{Lemoine:2006sc} and can easily be incorporated in our formalism.
In practice, we will need the above expressions only in the limit $x_c=0$ since $\Omega_c$ is usually negligible when the decay occurs.
The coefficient $x_r$, which we will shorten into $r$ from now on, can then be expressed as
\begin{equation}
r\equiv x_r= \frac{f_r}{\Omega_\sigma}\left(\frac{ 3\Omega_\sigma}{4-\Omega_\sigma}\right)\equiv \xi \, {\tilde r}.
\end{equation}
The first factor,
\begin{equation}
\xi \equiv
\frac{f_r}{\Omega_\sigma}=
\frac{\gamma_{r \, \sigma}}{1-(1- \gamma_{r \, \sigma}) \Omega_\sigma}
\end{equation}
can be interpreted as the transfer efficiency between the curvaton and radiation. Its maximal value, $\xi=1$, is reached when all the energy stored in the curvaton is converted into radiation, i.e. when $\gamma_{r \, \sigma}=1$, as usually assumed in most works on the curvaton. However, if a fraction of the curvaton energy goes into species other than radiation, then the transfer efficiency $\xi$ is reduced.
The second factor,
\begin{equation}
{\tilde r}\equiv\frac{
3\, \Omega_\sigma}{4-\Omega_\sigma},
\end{equation}
is the familiar coefficient that appears in the literature on the curvaton, which coincides with our $r$ only if $\xi=1$.
\subsubsection{Second order}
The expressions for the curvature perturbations up to second order are obtained from the general expression
(\ref{zeta_2ndorder}-\ref{U_ABC}), using the transfer matrix (\ref{T}). The expression for CDM is relatively simple:
\begin{equation}
\label{iso_mixed}
\zeta_{c+}=(1-f_c)\zeta_{c-}+f_c \zeta_{\sigma-}+\frac32 f_c (1-f_c)\left(\zeta_{c-}-\zeta_{\sigma-}\right)^2.
\end{equation}
The expression for radiation is much more complicated in general, but in the limit $x_c=0$, which is of interest to us, the radiation perturbation reduces to
\begin{equation}
\label{curv_mixed2}
\zeta_{r+}= \zeta_{r-}+\frac{r}{3} \, S_{\sigma-}+\frac{r}{18}
\left[3
-4r
+\frac{2r}{\xi} - \frac{r^2}{\xi^2}
\right] S_{\sigma-}^2 \,.
\end{equation}
In the limit $ \gamma_{r\sigma}=1$, i.e. $\xi=1$, one recovers the usual expression.
Note that, although $\Omega_c$ is assumed to be very small, it cannot be neglected in the expression for $f_c$ [see (\ref{f_A})] because $\gamma_{c\sigma}$ or $\Omega_\sigma$ can be very small, and $f_c$ can thus take any value between $0$ and $1$.
\subsection{Initial curvaton perturbation}
We now need to relate the perturbation of the curvaton fluid with the fluctuations of the curvaton scalar field during inflation. For simplicity, we assume here that the potential of the curvaton is quadratic.
Before its decay, the oscillating curvaton (with mass $m\gg H$) is described by a pressureless, non-interacting fluid with
energy density
\begin{equation}
\rho_\sigma = m^2 \sigma^2 \;,
\end{equation}
where $\sigma$ is the rms amplitude of the curvaton field. Making use of Eq.~(\ref{zeta_deltaN}), the inhomogeneous
energy density of the curvaton can be reexpressed as
\begin{equation}
\rho_\sigma = \bar\rho_\sigma e^{3(\zeta_\sigma-\delta N)} \label{rho_curvaton}\;.
\end{equation}
In the post-inflation era where the curvaton is still subdominant, the spatially flat
hypersurfaces are characterized by $\delta N=\zeta_r$ (since CDM is also subdominant). On such a hypersurface, the
curvaton energy density can be written as
\begin{equation}
\bar\rho_\sigma e^{3(\zeta_\sigma-\zeta_r)}=\bar\rho_\sigma e^{S_\sigma}
=
m^2 \left( \bar\sigma+\delta\sigma \right)^2 \,. \label{rhorhobarcurv}
\end{equation}
Expanding this expression up to second
order, and using the conservation of $\delta\sigma/\sigma$ in a quadratic potential, we obtain
\begin{equation}
S_\sigma= 2\frac{\delta\sigma_*}{\bar\sigma_*}
-\left(\frac{\delta\sigma_*}{\bar\sigma_*} \right)^2 \,,
\label{S_dchi}
\end{equation}
where the initial curvaton field perturbation, $\delta\sigma_*$, is assumed to be
Gaussian, as would be expected for a weakly coupled field.
The curvaton entropy perturbation (\ref{S_dchi}) thus contains a linear part $\hat S$
which is Gaussian and a second order part which is quadratic in
$\hat S$:
\begin{equation}
\label{S_G}
S_\sigma=\hat S-\frac14 \hat S^2\, ,
\quad {\rm with}\quad
\hat S \equiv 2\frac{\delta\sigma_*}{\bar\sigma_*}\,
\end{equation}
where the power spectrum of $\hat S$, generated during inflation, is given by
\begin{equation}
\langle \hat S(\vec{k}) \hat S (\vec{k}')\rangle=
(2\pi)^3 \, \frac{2\pi^2}{k^3}\,{\cal P}_{\hat S}(k) \, \delta(\vec k + \vec k') ,
\quad
{\cal P}_{\hat S}(k)=\frac{4}{\sigma_*^2}\left(\frac{H_*}{2\pi }\right)^2
\;.
\end{equation}
The subscript $*$ means that the quantity is evaluated at the time when the corresponding scale crossed out the Hubble radius during inflation.
\subsection{Primordial adiabatic and isocurvature perturbations}
For simplicity, we now assume that the post-inflation perturbations for dark matter and radiation, i.e. before the curvaton decay, are the same and depend only on the inflaton fluctuations,
\begin{equation}
\label{init_inf}
\zeta_{c-}=\zeta_{r-}=\zeta_{\rm inf}\,,
\end{equation}
so that there are only two independent degrees of freedom from the inflationary epoch, $\zeta_{\rm inf}$ and $\hat S$.
Substituting (\ref{S_G}) and (\ref{init_inf}) into (\ref{curv_mixed2}) and (\ref{iso_mixed}) yields
\begin{equation}
\label{zetarad}
\zeta_{\rm r} = \zeta_{\rm inf}+\frac{r}{3} \hat S+ \frac{r}{36}\left[3 -8r
+\frac{4r}{\xi} -2\frac{r^2}{\xi^2} \right]
\hat S^2\end{equation}
and
\begin{eqnarray}
\label{S_c}
S_c=(f_c-r) \hat S+\frac{1}{12}\left[3 f_c (1-2f_c)-r\left(3 -8r
+\frac{4r}{\xi} -2\frac{r^2}{\xi^2} \right)\right] \hat S^2,
\end{eqnarray}
In the limit $ \gamma_{r\sigma}=1$, i.e. $\xi=1$, one recovers the well-known expression for $\zeta_r$.
Considering only the linear part of (\ref{zetarad}), one finds that
the power spectrum for the primordial adiabatic perturbation
$\zeta_{\rm r}$ can be expressed as
\begin{equation}
\label{finalzetar}
{\cal P}_{\zeta_{\rm r}}={\cal P}_{\zeta_{\rm inf}}+\frac{r^2}{9}{\cal P}_{\hat S} \equiv (1+\lambda){\cal P}_{\zeta_{\rm inf}}\equiv
\, \Xi^{-1}\, \frac{r^2}{9}{\cal P}_{\hat S}
\end{equation}
where $\lambda$ is defined as the ratio between the curvaton and inflaton contributions
and
$\Xi = (1+\lambda^{-1})^{-1}$ as the ratio between the curvaton contribution and the total curvature power spectrum.
The limit $\lambda\gg 1$, or $\Xi\simeq 1$, corresponds to the standard curvaton scenario, where
the inflaton perturbation is ignored.
The cases where the inflaton contribution is not negligible correspond to the mixed inflaton-curvaton scenario~\cite{Langlois:2004nn}. The curvaton contribution is subdominant when $\lambda\ll 1$, i.e. $\Xi\ll 1$.
Let us now turn to the primordial isocurvature perturbation. As can be read from the linear part of (\ref{S_c}), its power spectrum is given by
\begin{equation}
{\cal P}_{S_c}=(f_c-r)^2 {\cal P}_{\hat S}\, .
\end{equation}
and the correlation between adiabatic and isocurvature fluctuations is
\begin{equation}
\label{corr_ad_is}
{\cal C}
\equiv \frac{{\cal P}_{S_c, \zeta_r}}{\sqrt{{\cal P}_{S_c} {\cal P}_{\zeta_r}}}= \varepsilon_f
\, \Xi^{1/2}\,, \qquad \varepsilon_f \equiv {\rm sgn}(f_c-r)\, .
\end{equation}
In the pure curvaton limit ($\Xi\simeq 1$), adiabatic and isocurvature perturbations are either fully correlated, if $\varepsilon_f>0$, or fully anti-correlated, if $\varepsilon_f<0$.
In the opposite limit ($\Xi\ll 1$), the correlation vanishes.
For intermediate values of $\Xi$, the correlation is only partial, as can be also obtained in multifield inflation
\cite{Langlois:1999dw}.
As combined adiabatic and isocurvature perturbations lead to a distortion of the acoustic peaks, which depends on their correlation~\cite{Langlois:2000ar}, it is in principle possible to distinguish, in the observed fluctuations, the adiabatic and isocurvature contributions. So far, there is no detection of any isocurvature component, but only an upper bound on the ratio between isocurvature and adiabatic power spectra, which, in our case, is given by
\begin{equation}
\label{alpha}
\alpha\equiv \frac{{\cal P}_{S_c}}{{\cal P}_{{\zeta}_{\rm r}}} = 9 \left(1-\frac{f_c}{r}\right)^2 \, \Xi \, .
\end{equation}
The observational constraints on $\alpha$ depend on the correlation. Writing $\alpha\equiv a/(1-a)$ (note that $\alpha\simeq a$ if $\alpha$ is small), the constraints (WMAP+BAO+SN) given in \cite{Komatsu:2010fb} are
\begin{equation}
a_0<0.064\quad (95 \% {\rm CL}), \qquad a_1< 0.0037 \quad (95 \% {\rm CL})
\end{equation}
respectively for the uncorrelated case and for the fully correlated case
\footnote{Our notations differ from those of \cite{Komatsu:2010fb}. Our $a$ corresponds to their $\alpha$ and our fully {\sl correlated} limit corresponds to their fully {\sl anti-correlated} limit, because their definition of the correlation has the opposite sign.}.
One sees that the observational constraint $\alpha \ll 1$ can be satisfied in only two cases:
\begin{itemize}
\item $|f_c-r|\ll r$, i.e. a fine-tuning between the two parameters $f_c$ and $r$. This includes the case $f_c=1$ with $r\simeq 1$, considered in \cite{Langlois:2008vk}.
\item $\Xi \ll 1$, i.e. the curvaton contribution to the observed power spectrum is very small.
\end{itemize}
\subsection{Non-Gaussianities}
\label{sec-non-gauss}
Let us now examine the amplitude of the non-Gaussianities that can be generated in our model.
We recall that our observable quantities $\zeta$ and $S$ are of the form
\begin{equation}
\zeta=\zeta_{\rm inf}+z_1\, \hat S+\frac12 z_2\, \hat S^2, \qquad S=s_1 \, \hat S+\frac12 s_2\, \hat S^2,
\end{equation}
where $\zeta_{\rm inf}$ and $\hat S$ are two independent Gaussian fields and where the coefficients can be read explicitly from (\ref{zetarad}) and (\ref{S_c}).
Applying the general results of the Appendix to the present situation, we can easily compute for our model the ``reduced" angular bispectrum, which is of direct interest for a comparison with CMB observations and which generalizes the analysis of \cite{Komatsu:2001rj} in the purely adiabatic case. Specializing
(\ref{reduced_bispectrum_general}) to our case, one finds
\begin{eqnarray}
\label{b_l}
b_{l_1l_2 l_3}= 3 \sum_{I,J,K}\, b_{NL}^{I, JK} \int_0^\infty r^2 dr \tilde\beta^I_{(l_1}(r)\beta^{J}_{l_2}(r)\beta^{K}_{l_3)}(r)
\end{eqnarray}
with
\begin{equation}
b_{NL}^{I, JK} \equiv N^I_{(2)} N^J_{(1)} N^K_{(1)},
\end{equation}
where $N^\zeta_{(2)}=z_2$, $N^S_{(2)}=s_2$, $N^\zeta_{(1)}=z_1$, $N^S_{(1)}=
s_1$, respectively, and
\begin{equation}
\label{beta}
\tilde\beta^I_{l}(r)\equiv \frac{2}{\pi} \int k^2 dk \, j_l(kr) g^I_{l}(k) , \quad
\beta^{I}_{l}(r)\equiv \frac{2}{\pi}\int k^2 dk\, j_l(kr) g^I_{l}(k) P_{\hat S}(k)\, ,\quad
\end{equation}
where the $ g^I_{l}(k)$ denote the adiabatic ($I=\zeta$) and isocurvature ($I=S$) transfer functions.
Because of the symmetry under exchange of the last two indices, the reduced bispectrum contains six different contributions, whose amplitude is parametrized by the six coefficients $b_{NL}^{I, JK} $.
In order to compare these coefficients with the usual parameter $f_{NL}$ defined in the purely adiabatic case, one must recall that $f_{NL}$ is proportional to the bispectrum of $\zeta$ divided by the square of its power spectrum. By noting that the $\beta_l^I(r)$ introduced in (\ref{beta}) involve $P_{\hat S}$, this implies that the analogs of $f_{NL}$ can be defined by dividing the coefficient $b_{NL}^{I, JK}$ by the square of the ratio
$P_\zeta/P_{\hat S}=z_1^2\Xi^{-1}$. We thus introduce
the parameters
\begin{equation}
\tilde f_{NL}^{I, JK} \equiv \frac65 f_{NL}^{I, JK} \equiv \frac{\Xi^2}{z_1^4} \, b_{NL}^{I, JK} ,
\end{equation}
explicitly given by the expressions
\begin{eqnarray}
&\tilde f_{NL}
^{\,\zeta, \zeta\zeta}=\frac{z_2}{z_1^2}\, \Xi^2, \quad \tilde f_{NL}^{\zeta, \zeta S}=\frac{s_1 z_2}{z_1^3}\, \Xi^2, \quad
\tilde f_{NL}^{\zeta, S S}=\frac{s_1^2 z_2}{z_1^4}\, \Xi^2, &
\\\cr
&\tilde f_{NL}^{S, \zeta\zeta} = \frac{s_2}{z_1^2}\, \Xi^2,
\quad \tilde f_{NL}^{S, \zeta S}= \frac{s_1 s_2}{z_1^3}\, \Xi^2, \quad
\tilde f_{NL}^{S, SS}= \frac{s_1^2 s_2}{z_1^4}\, \Xi^2\, . &
\end{eqnarray}
In the absence of isocurvature perturbations, the above non-linear parameters vanish
except the first one, yielding
\begin{equation}
f_{NL}^{\zeta, \zeta\zeta} =\frac56 \,
\left(\frac{3}{2r} +\frac{2}{\xi}-4 -\frac{r}{\xi^2} \right)\, \Xi^2
\label{f_ad}\;,
\end{equation}
which exactly coincides with the familiar parameter $f_{NL}$.
The amplitude of the non-gaussianities is determined by the three parameters $r $, $\xi$ and $\Xi$ (note that one recovers the usual prediction of the pure curvaton scenario for $\xi=1$ and $\Xi=1$), which take values between $0$ and $1$.
The present constraints on $f_{NL}$, calculated from WMAP data by assuming purely adiabatic perturbations, are \cite{Komatsu:2010fb}:
\begin{eqnarray}
-10 \leq f_{NL}^{(\mathrm{local})} \leq 74.
\end{eqnarray}
A sufficiently small $r$, or even $\xi$, leads to a significant non-Gaussianity from the adiabatic component, whereas a small $\Xi$ tends to suppress it.
If isocurvature modes are present, however, the five other terms in the reduced bispectrum (\ref{b_l}) will also contribute in general. Interestingly, the six functions
on the right hand side of (\ref{b_l}) have distinct dependences on the $l_i$, because they involve different combinations of the adiabatic and isocurvature transfer functions. Therefore, this allows in principle to measure, or constrain, independently the corresponding six non-linear parameters from the CMB data.
The precise determination of constraints on the $f_{NL}^{I,JK}$ is beyond the scope of the present work,
but since all the functions multiplying the $b_{NL}^{I,JK}$ in (\ref{b_l}) are of similar amplitude, one can a priori expect the constraints on the $f_{NL}^{I,JK}$ to be of the same order of magnitude as those on $f_{NL}$\footnote{Observational constraints on isocurvature non-Gaussianities are given in \cite{Hikage:2008sk} , for an isocurvature perturbation of the form $S=S_L+f_{NL}^{(iso)}S_L^2$,
where $S_L$ is Gaussian. Their non-linear parameter $f_{NL}^{(iso)}$ is related to ours according to $\tilde f_{NL}^{S, SS}= 2 f_{NL}^{(iso)}\alpha^2$, $\tilde f_{NL}^{S, \zeta S}=2f_{NL}^{(iso)} \alpha^{3/2} |{\cal C}|$ and
$\tilde f_{NL}^{S, \zeta\zeta} =2 f_{NL}^{(iso)}\alpha\, {\cal C}^2$, where ${\cal C}$ is the correlation defined in (\ref{corr_ad_is}). }.
Let us now explore the amplitude of the non-linear parameters in our model. First of all, let us stress that finding significant non-Gaussianities (typically $f_{NL}\sim 10-100$) requires, in all cases, a small denominator $z_1$, i.e. $r\ll 1$, which will thus be assumed below.
Second, it is worth noting that all the coefficients are related via the two rules
\begin{equation}
f_{NL}^{I, JS}= \frac{s_1}{z_1} f_{NL}^{I, J\zeta}\ (R1), \qquad f_{NL}^{S, IJ}= \frac{s_2}{z_2} f_{NL}^{\zeta, IJ}, \ (R2)\, .
\end{equation}
Therefore, the hierarchy between the parameters can be deduced from the value of the first order ratio
\begin{equation}
\label{ratios_sz}
\frac{s_1}{z_1}=3\left(\frac{f_c}{r}-1\right)=\varepsilon_f \sqrt{\frac{\alpha}{\Xi}}\,, \qquad \varepsilon_f\equiv {\rm sgn}(f_c-r),
\end{equation}
where we have used (\ref{alpha}), as well as the second order ratio $s_2/z_2$, which is a more complicated expression in general.
We now consider successively the two limits for which the isocurvature bound is satisfied.
\subsubsection{Limit $|f_c-r|\ll r$, with $\Xi\simeq1$ (pure curvaton scenario)}
In this case,
the isocurvature-adiabatic ratio $\alpha$ must satisfy the observational constraint $\alpha \simeq a_1 \leq 0.0037$, since we are in the fully correlated case.
The relevant ratios are given here by
\begin{equation}
\frac{s_1}{z_1}\simeq \varepsilon_f \sqrt{\alpha}, \qquad
\frac{s_2}{z_2}\simeq
\frac{\varepsilon_f\sqrt{\alpha} -2\tilde r (2-\tilde r)}{1+2\, \tilde r \left(2-\tilde r\right)/3}
\end{equation}
where we have taken the limit $r=\xi\,\tilde r\ll 1$ (although $r$ cannot be smaller than $10^{-2}$, to be compatible with observational constraint on $f_{NL}$). If $r$ is small because $\tilde r \ll 1$, then the denominator in the expression for $s_2/z_2$ reduces to $1$. However, if $\xi\ll 1$ while
$\tilde r$ is of order $1$, the full expression for $s_2/z_2$ is needed.
The value of the first ratio implies that, with respect to $f_{NL}^{\zeta, \zeta\zeta}$, the coefficients $f_{NL}^{\zeta, \zeta S}$ and $f_{NL}^{\zeta, S S}$ are suppressed with factors
$\sqrt{\alpha}$ and $\alpha$, respectively, according to (R1).
Analogously the coefficients $f_{NL}^{S,\zeta S}$ and $f_{NL}^{S,SS}$ are suppressed, respectively with factors
$\sqrt{\alpha}$ and $\alpha$, with respect to $f_{NL}^{S,\zeta \zeta}$.
By contrast, using (R2), one sees that $f_{NL}^{S,\zeta \zeta}$ could be of the same order of magnitude as $f_{NL}^{\zeta, \zeta\zeta}$, if $\tilde r\sim 1$, or suppressed if $\tilde r$ is small.
To conclude, in the pure curvaton scenario, it is possible to satisfy the isocurvature constraint and to get measurable non-gaussianities only by assuming a fine-tuning between $f_c$ and $r$ at the percent level. In this situation, only the purely adiabatic parameter is significant, while the other parameters are suppressed, with increasing powers of $\alpha$.
\subsubsection{Limit $\Xi \ll 1$}
In this limit $\alpha$ must satisfy the constraint $\alpha\simeq a_0<0.064$ (uncorrelated case).
In the regime $f_c\ll r\ll 1$, one finds that both ratios $s_1/z_1$ and $s_2/z_2$ reduce to (-3), independently of the value of $\tilde r$.
Therefore, the relation between the non-linear parameters is simply
\begin{equation}
\label{f_tilde_1c_2}
\tilde f_{NL}^{\zeta, \zeta\zeta} \simeq \frac{\alpha^2}{54r}, \quad \tilde f_{NL}^{I, JK}\simeq (-3)^{I_S}\tilde f_{NL}^{\zeta, \zeta\zeta} \qquad (f_c\ll r\ll 1)
\end{equation}
where $I_S$ is the number of $S$ among the indices $(I, J K)$.
This is the result obtained in \cite{Langlois:2008vk} for $f_c=0$.
For $\alpha$ close to its present upper bound, one sees that detectable
non-Gaussianity can be generated with $r \sim 10^{-5}$.
By contrast, in the regime $f_c\gg r$, the purely adiabatic coefficient is strongly suppressed since
\begin{equation}
\tilde f_{NL}^{\zeta, \zeta\zeta} \simeq \frac{\alpha^2r^3}{54 f_c^4}\, .
\end{equation}
However, the other coefficients are now enhanced with respect to the purely adiabatic coefficient, via the large factors
\begin{equation}
\frac{s_1}{z_1}\simeq 3 \frac{f_c}{r}, \qquad
\frac{s_2}{z_2}\simeq 3\frac{f_c}{r}(1-2f_c)
\, .
\end{equation}
where, for simplicity, we have assumed $\tilde r \ll 1$ (the other possibility $\xi\ll 1$ yields a more complicated expression for the second ratio, with a dependence on $\tilde r$).
The dominant term is therefore the purely isocurvature term
\begin{equation}
\tilde f_{NL}^{S, SS} \simeq \alpha^2\, \frac{1-2 f_c}{2f_c}
\end{equation}
If $f_c\sim 1$, this purely isocurvature non-Gaussianity, although enhanced with respect to all the other contributions, remains negligible since it is suppressed by the very small factor
$\alpha^2$. This was the conclusion reached in \cite{Langlois:2008vk} (for $f_c=1$).
However, we now see that this suppression can be compensated if $f_c$ is smaller than $\alpha^2$. The purely isocurvature parameter and the other ones are then given by
\begin{equation}
\tilde f_{NL}^{S, SS}\simeq \frac{\alpha^2}{2f_c},\quad \tilde f_{NL}^{I, JK}\simeq \left(\frac{r}{3f_c}\right)^{I_\zeta}\tilde f_{NL}^{S, SS} \qquad (r\ll f_c \ll 1)
\end{equation}
where $I_\zeta$ is the number of $\zeta$ among the three indices. One can notice that the amplitude of the purely isocurvature non-Gaussianity does not depend on the parameter $r$, but only on $\alpha$ and $f_c$. For instance, with $\alpha=0.05$ which satisfies the current observational bound, a value $f_c=10^{-5}$ yields
$\tilde f_{NL}^{S, SS}\sim 100$. In such a scenario, one gets observable non-Gaussianity that comes essentially from isocurvature modes, even if the latter are subdominant in the power spectrum.
\section{Scenario with two curvatons}
We now apply our formalism to the models where two curvatons are present in the early Universe (see e.g. \cite{Hamaguchi:2003dc,Choi:2007fya,Assadullahi:2007uw}).
The curvaton $\sigma$ will be assumed to decay first, followed later by the curvaton denoted $\chi$.
\subsection{First order}
At linear order, the decay of the first curvaton can be characterized by the transfer matrix
\begin{equation}
T_{[1]}=
\left( \begin{array}{ccccccc}
1-x_{r1} && x_{c1} && x_{\cs 1} && x_{r1}-x_{c1} -x_{\cs 1} \\
0 && 1-f_{c1} && 0 && f_{c1} \\
0 && 0 && 1-f_{\cs 1} && f_{\cs 1} \\
0 && 0 && 0 && 0
\end{array} \right),
\label{MT1}
\end{equation}
where the order of the species is $(r,c,\chi, \sigma)$, while the decay of the second curvaton is characterized by the transfer matrix
\begin{equation}
T_{[2]}=
\left( \begin{array}{ccccccc}
1-x_{r2} && x_{c2} && x_{r2}-x_{c2} && 0 \\
0 && 1-f_{c2} && f_{c2} && 0 \\
0 && 0 && 0 && 0 \\
0 && 0 && 0 && 0
\end{array} \right)\, .
\label{MT2}
\end{equation}
In the above matrices, the definitions of the parameters are analogous to the definitions introduced in (\ref{T}), i.e. $x_{r1} \equiv f_{r1}/{{\tilde\Omega}_1}$, $x_{c1}\equiv \Omega_{c1}\, x_{r1} /4$, $x_{\cs 1}\equiv \Omega_{\chi 1}\, x_{r1} /4$, etc, and the indices $1$ and $2$ refer respectively to the first and second decays.
We have also allowed the possibility that the first curvaton $\sigma$ decays into the second curvaton $\chi$,
hence the presence of the parameter $f_{\cs 1}$.
The expression of the perturbations for radiation and CDM, after the two transitions, are expressed in terms of the initial perturbations $\zeta_{B0}$ via the product of the two transfer matrices given above, i.e.
\begin{equation}
\zeta_A=\sum_B \left(T_{[2]}\cdot T_{[1]}\right)_A^{\ B} \zeta_{B0}.
\end{equation}
At first order, the radiation curvature perturbation, after the second curvaton decay, reads
\begin{equation}
\zeta_{\rm r}=\zeta_{{\rm r}0}+
z_\sigma \, S_{\sigma 0}+
z_\chi \, S_{\chi 0}+
z_c \, S_{c 0},
\end{equation}
with
\begin{eqnarray}
3z_\sigma&=& (1-x_{r2})(x_{r1}-x_{c1}-x_{\cs 1})+f_{c1} x_{c2}+f_{\cs 1} (x_{r2}-x_{c2}),
\label{A}
\\
3z_\chi&=&(1-f_{\cs 1})(x_{r2}-x_{c2})+(1-x_{r2}) x_{\cs 1},
\label{B}
\\
3z_c &=& (1-f_{c1}) x_{c2}+(1-x_{r2}) x_{c1}\, .
\end{eqnarray}
Combining this expression with that of the CDM curvature perturbation, according to (\ref{iso_pert}), we find that the CDM entropy perturbation is given by
\begin{equation}
S_c=s_\sigma \, S_{\sigma 0}+s_\chi \, S_{\chi 0}+s_c \, S_{c 0},
\end{equation}
with
\begin{eqnarray}
s_\sigma&=& -3z_\sigma +f_{c1} (1-f_{c2})+f_{c2} f_{\cs 1},
\label{F}
\\
s_\chi&=& -3z_\chi +f_{c2} (1-f_{\cs 1}),
\label{G}
\\
s_c&=& -3z_c + (1-f_{c1})(1-f_{c2}).
\end{eqnarray}
For simplicity, we will restrict ourselves, from now on, to the case where $S_{c0}=0$.
Defining $\Lambda$ as the ratio between the two curvaton power spectra, such that
\begin{equation}
P_{S_{\cs0}}\equiv\Lambda P_{S_{\cf0}},
\end{equation}
one easily finds that the ratio between the isocurvature and the adiabatic spectra is given by
\begin{eqnarray}
\label{ratio_spec_beta}
\alpha =
\frac{P_{S_c}}{P_{\zeta_{\rm r}}} =
\frac{s_\sigma^2 + \Lambda s_\chi^2}{z_\sigma^2 + \Lambda z_\chi^2}
\; \Xi\,,
\qquad
\Xi \equiv \frac{\lambda_\chi+ \lambda_\sigma}{1+\lambda_\chi+\lambda_\sigma}
\end{eqnarray}
where $\lambda_\chi$ and $\lambda_\sigma$ are defined as in (\ref{finalzetar}), i.e.
\begin{equation}
{\cal P}_{\zeta_{\rm r}}={\cal P}_{\zeta_{r0}}+
z_\sigma^2{\cal P}_{S_{\sigma 0}} +
z_\chi^2{\cal P}_{S_{\chi 0}} \equiv (1+\lambda_\sigma+\lambda_\chi){\cal P}_{\zeta_{r0}}.
\end{equation}
The correlation between $\zeta_{\rm r}$ and $S_c$ can be expressed as
\begin{equation}
{\cal C}
=\frac{z_\sigmas_\sigma+\Lambda z_\chis_\chi}{\sqrt{(s_\sigma^2 + \Lambda s_\chi^2)(z_\sigma^2 + \Lambda z_\chi^2)}}\, \sqrt{\Xi}\, .
\end{equation}
The observational constraints on $\alpha$ impose that at least one of the following conditions must be satisfied:
\begin{equation}
\Xi
\ll 1
\quad
{\rm or}
\quad
s_\sigma^2 + \Lambda s_\chi^2\ll z_\sigma^2 + \Lambda z_\chi^2
\, .
\end{equation}
The first possibility, $\Xi \ll 1$, corresponds to a power spectrum dominated by the
inflaton, whereas the second possibility requires special cancellations in (\ref{F}-\ref{G}) so that $s_\sigma$ and $s_\chi$ are suppressed.
\subsection{Second order}
We now consider the perturbations up to the second order, in order to compute the non-Gaussianities.
First, let us decompose the curvaton entropy perturbations as in (\ref{S_G}), so that
\begin{eqnarray}
S_{\sigma 0} = \hat S_{\sigma} - \frac{1}{4} \hat S_{\sigma}^2
\hspace{1.5cm}
S_{\chi 0} = \hat S_{\chi} - \frac{1}{4} \hat S_{\chi}^2,
\end{eqnarray}
where $\hat S_{\sigma}$ and $\hat S_{\chi}$ are two independent Gaussian quantities.
The radiation curvature perturbation and the dark matter entropy perturbation after the second decay, up to second order, are given in our notation by
\begin{eqnarray}
\zeta_{\rm r} &=& \zeta_{{\rm r}0} +
z_\sigma \hat S_{\,\sigma} +
z_\chi \hat S_{\,\chi} + z_{\sigma\chi} \hat S_{\,\sigma} \hat S_{\,\chi}
+ \frac{1}{2} \D \hat S_{\,\sigma}^2 + \frac{1}{2} z_{\chi\chi} \hat S_{\,\chi}^2 \label{zetar}
\\ \cr
S_c &=& s_\sigma \hat S_{\,\sigma} + s_\chi \hat S_{\,\chi} + s_{\sigma\chi} \hat S_{\,\sigma} \hat S_{\,\chi}
+ \frac{1}{2} s_{\sigma\sigma} \hat S_{\,\sigma}^2 + \frac{1}{2} s_{\chi\chi} \hat S_{\,\chi}^2 \label{Sc}
\end{eqnarray}
where the coefficients $z_\sigma$, $z_\chi$, $s_\sigma$ and $s_\chi$ have already been defined in (\ref{A}-\ref{B}) and (\ref{F}-\ref{G}), respectively.
We do not give explicitly the full expressions for the second order coefficients because they are very lengthy, but they are straightforward to compute by using the general expressions (\ref{zeta_2ndorder}-\ref{U_ABC}) with the transfer matrices (\ref{MT1}-\ref{MT2}).
Let us calculate the reduced bispectrum by using the general expression given in the Appendix. In our model, ignoring the inflaton which does not produce significant non-Gaussianities, the relevant power spectra are independent so that
\begin{eqnarray}
P^{ab}(k)=\left( \begin{array}{ccc}
1 && 0\\ 0 && \Lambda
\end{array} \right)\, P_{\hat S_{\,\sigma} }\,,
\end{eqnarray}
where we furthermore assume that $\Lambda$ is strictly independent of $k$ (this is indeed the case if the masses of both curvatons are negligible with respect to $H$ during inflation).
As a consequence, the reduced bispectrum can be reduced to the same expression as that already given in Eq. (\ref{b_l}) with
\begin{equation}
\label{beta_2}
\beta^{I}_{l}(r)\equiv \frac{2}{\pi}\int k^2 dk j_l(kr) g^I_{l}(k) P_{\hat S_{\,\sigma} }(k)\,
\end{equation}
and the six parameters
\begin{eqnarray}
\label{b_coeff_2c}
b_{NL}^{I,JK} \equiv N_{\sigma\cf}^I N_{\sigma}^J N_{\sigma}^K
+ \Lambda N_{\sigma\chi}^I \left(N_{\sigma}^J N_{\chi}^K+N_{\chi}^J N_{\sigma}^K\right)
+\Lambda^2 N_{\chi\cs}^I N_{\chi}^J N_{\chi}^K\,,
\end{eqnarray}
where the coefficients $N_{ab}^I$, which are defined as in (\ref{X_I}), can be read off directly from (\ref{zetar}) and (\ref{Sc}).
In complete analogy with the model with one curvaton, to be compared with the standard $f_{NL}$, these coefficients must be divided by the square of the ratio $P_{\zeta}/P_{S_{\cf0}} = (z_\sigma^2+\betaLz_\chi^2)/\Xi$, hence:
\begin{eqnarray}
\label{f_tilde_2c}
\tilde f_{NL}^{I,JK} \equiv \left(\frac{\Xi}{z_\sigma^2+\betaLz_\chi^2}\right)^2b_{NL}^{I,JK}.
\end{eqnarray}
The six non-linearity coefficients are thus given by
\begin{eqnarray}
\label{b_NL_2C_lambda}
\tilde f_{NL}^{\zeta, \zeta \zeta} &=&
\left(\frac{\Xi}{z_\sigma^2+\betaLz_\chi^2}\right)^2
\left[\Dz_\sigma^2 + 2 \Lambda \Cz_\sigmaz_\chi + \Lambda^2 \Ez_\chi^2 \right],
\cr\cr
\tilde f_{NL}^{\zeta,\zeta S} &=& \left(\frac{\Xi}{z_\sigma^2+\betaLz_\chi^2}\right)^2
\left[\Dz_\sigmas_\sigma+\Lambda\,z_{\sigma\chi}(z_\sigmas_\chi+z_\chis_\sigma)+\Lambda^2\Ez_\chis_\chi \right],
\cr\cr
\tilde f_{NL}^{\zeta, SS} &=& \left(\frac{\Xi}{z_\sigma^2+\betaLz_\chi^2}\right)^2
\left[\Ds_\sigma^2+2\Lambda\,z_{\sigma\chi}s_\sigmas_\chi+\Lambda^2z_{\chi\chi}s_\chi^2 \right],
\cr\cr
\tilde f_{NL}^{S,\zeta\zeta} &=& \left(\frac{\Xi}{z_\sigma^2+\betaLz_\chi^2}\right)^2
\left[\Iz_\sigma^2 + 2\Lambda\Hz_\sigmaz_\chi+\Lambda^2\Jz_\chi^2 \right],
\cr\cr
\tilde f_{NL}^{S,S\zeta} &=& \left(\frac{\Xi}{z_\sigma^2+\betaLz_\chi^2}\right)^2
\left[\Iz_\sigmas_\sigma + \betaLs_{\sigma\chi}(\Fz_\chi+\Gz_\sigma) + \Lambda^2s_{\chi\chi}\Gz_\chi \right],
\cr\cr
\tilde f_{NL}^{S, SS} &=&
\left(\frac{\Xi}{z_\sigma^2+\betaLz_\chi^2}\right)^2
\left[s_{\sigma\sigma} s_\sigma^2 + 2 \Lambda \Hs_\sigmas_\chi + \Lambda^2 \Js_\chi^2 \right]\,.
\end{eqnarray}
In the following we analyze explicitly some limiting cases.
\subsection{Various limits}
We now explore the parameter space, in order to see whether it is possible to obtain significant non-Gaussianities.
Let us first mention that we have checked that our results agree with those of \cite{Assadullahi:2007uw} in the limit where the curvatons decay only into radiation (i.e. $f_{c1}=f_{c2}=f_{\cs 1}=0$), the dark matter abundance is neglected (i.e. $x_{c1}=x_{c2}=0$) and the inflaton contribution is ignored (i.e. $\Xi=1$).
\subsubsection{Limit $\Lambda \ll 1$}
In this limit where the contributions from the second curvaton are negligible, one finds
\begin{equation}
\label{limit_beta_0}
\alpha\simeq \Xi \, \frac{s_\sigma^2}{z_\sigma^2}, \qquad \tilde f_{NL}^{\zeta \zeta \zeta}\simeq \Xi^2 \frac{\D}{z_\sigma^2},
\end{equation}
while the other five non-linear coefficients can be deduced from $\tilde f_{NL}^{\zeta \zeta \zeta}$ according to the relations
\begin{equation}
\label{rules_c2_1}
f_{NL}^{I, JS}\simeq \frac{s_\sigma}{z_\sigma} f_{NL}^{I, J\zeta}\, \qquad f_{NL}^{S, IJ}\simeq \frac{s_{\sigma\sigma}}{\D} f_{NL}^{\zeta, IJ} \, .
\end{equation}
The quantity $\alpha$ is constrained by observations to be small, which requires either $\Xi\ll 1$ or $|s_\sigma|\ll |z_\sigma|$.
\paragraph{First possibility: $\Xi \ll 1$,} while $|z_\sigma| \sim |s_\sigma|$.
This leads to a suppression of all the non-Gaussianity coefficients by a factor $\Xi^2$. However, the coefficients $\tilde f_{NL}^{\zeta, JK}$ can still be significant if the ratio $\D/z_\sigma^2$ can compensate the $\Xi^2$ suppression (similarly for the $f_{NL}^{S,JK}$ if $s_{\sigma\sigma}/z_\sigma^2 $ compensates the $\Xi^2$ suppression).
Let us consider a specific example, with the simplifying assumptions
\begin{eqnarray}
\label{par_ex_2c}
x_{c1}=x_{c2}=f_{\cs 1}=x_{\cs 1}=0\,,
\end{eqnarray}
that is, we neglect the energy fraction of dark matter and assume that the curvaton $\sigma$ does not decay into $\chi$ and that $\chi$ is subdominant when $\sigma$ decays.
Under these assumptions,
$z_\sigma= x_{r1}(1-x_{r2})/3$ and we further assume $f_{c1} \llz_\sigma$ so that $s_\sigma\simeq -3z_\sigma$.
In the two limits $x_{r1} = \tilde r_1\xi_1\ll1$ and $(1-x_{r2})\ll1$, $z_\sigma$ is small and
the adiabatic non-Gaussianity behaves as
\begin{eqnarray}
\label{fNL_ex_2c}
\tilde f_{NL}^{\zeta, \zeta \zeta}=
\frac{1}{1-x_{r2}} \left[\tilde f_{NL\, 1}^{\zeta, \zeta \zeta} + \frac{x_{r2}}{1-x_{r2}}
\left(\frac32+x_{r2} \,\tilde f_{NL\, 2}^{\zeta, \zeta \zeta} \right) \right]\,,
\end{eqnarray}
where $\tilde f_{NL\, 1,2}^{\zeta, \zeta \zeta}$ correspond to single-curvaton coefficient, equation (\ref{f_ad}), but calculated with the parameters $\xi_{1,2}$ and $x_{r1,r2}$ respectively.
If we assume $x_{r2}\ll1$, the above equation corresponds to the single-curvaton result (\ref{f_tilde_1c_2}).
The other coefficients also follow the relations given in (\ref{f_tilde_1c_2}), since $s_{\sigma\sigma}/\D=-3$ with the assumptions (\ref{par_ex_2c}) and $f_{c1} \ll 1$, and are thus of comparable magnitude.
\paragraph{Second possibility:} $|s_\sigma|\ll |z_\sigma|$
When a small $\alpha$ is the consequence of $|s_\sigma|\ll |z_\sigma|$, one sees from the first relation in (\ref{rules_c2_1}) that all the $f_{NL}^{I, JS}$ are strongly suppressed with respect to $f_{NL}^{I, J\zeta}$. However, the two coefficients $f_{NL}^{I,\zeta\zeta}$ can still be important if $ |z_\sigma|$ is sufficiently small.
By examining (\ref{A}) and (\ref{F}), one sees that getting $|s_\sigma|\ll |z_\sigma|\ll 1$ requires some fine-tuning between the coefficients, which we now discuss.
In order to get $|z_\sigma|\ll 1$, the first possibility is that the first curvaton is subdominant, i.e. $x_{r1}={\cal O}(\epsilon)$, where $\epsilon$ is some small number (we neglect $x_{c2}$ which must be small because we are deep in the radiation era), which then requires either $x_{r2}={\cal O}(\epsilon)$ or
$f_{\cs 1}={\cal O}(\epsilon)$. The second possibility is that the second curvaton dominates at decay, i.e. $x_{r2}=1-{\cal O}(\epsilon)$, which also requires that $f_{\cs 1}={\cal O}(\epsilon)$.
Then, to obtain $|s_\sigma|\ll |z_\sigma|$, the terms of the right hand side of (\ref{F}), which are of order $\epsilon$ must compensate each other so that their sum is at most of order ${\cal O}(\alpha\,\epsilon)$, which necessitates some special relation between the $f_A$ and the $x_A$.
If we consider again the assumptions (\ref{par_ex_2c}) and neglect $f_{c2}$, one finds that the fine-tuning condition to get $|s_\sigma|\ll |z_\sigma|$ is
\begin{eqnarray}
f_{c1}-x_{r1}(1-x_{r2})\leq {\cal O}(\alpha \epsilon)\,.
\end{eqnarray}
The adiabatic parameter $\tilde f_{NL}^{\zeta, \zeta \zeta}$
is given in equation (\ref{fNL_ex_2c}), with now $\Xi\sim 1$, and its value is of order $10$ when $\epsilon\sim x_{r1}(1-x_{r2}) \sim 0.1$. Since $s_{\sigma\sigma}/\D\simeq - 3 +{\cal O}(f_{c1}/x_{r1}(1-x_{r2}))$, we also have $\tilde f_{NL}^{S,\zeta\zeta}\sim\tilde f_{NL}^{\zeta, \zeta \zeta}$.
Note that a significant non-Gaussianity generated by a dominant curvaton ($x_{r2}=1-{\cal O}(\epsilon)$) has already been pointed out in \cite{Assadullahi:2007uw}, but we see here that satisfying the isocurvature bound requires additional constraints on the branching ratios of the curvatons.
\subsubsection{Limit $\Lambda \gg 1$}
In this limit, one obtains
\begin{equation}
\alpha\sim \Xi \, \frac{s_\chi^2}{z_\chi^2}, \qquad \tilde f_{NL}^{\zeta \zeta \zeta}\sim \Xi^2\, \frac{z_{\chi\chi}}{z_\chi^2},
\qquad f_{NL}^{I, JS}\simeq \frac{s_\chi}{z_\chi} f_{NL}^{I, J\zeta}\, ,
\qquad f_{NL}^{S, IJ}\simeq \frac{s_{\chi\chi}}{z_{\chi\chi}} f_{NL}^{\zeta, IJ}.
\end{equation}
By comparing with (\ref{limit_beta_0}) and (\ref{rules_c2_1}), one sees that the analysis is analogous to the previous case, by replacing $z_\sigma$, $\D$, $s_\sigma$ and $s_{\sigma\sigma}$ by $z_\chi$, $z_{\chi\chi}$, $s_\chi$ and $s_{\chi\chi}$, respectively.
When the curvaton contribution to the power spectrum is not negligible, significant non-Gaussianity, while satisfying the isocurvature bound, is obtained when $|s_\chi|\ll |z_\chi|\ll 1$. This constraint is satisfied if one assumes $f_{\cs 1}=1-{\cal O}(\epsilon)$, which means that the second curvaton is created mainly by the decay of the first, while $x_{r2}=1-{\cal O}(\epsilon)$, $x_{\cs 1}\lesssim{\cal O}(\epsilon)$ and $f_{c2}=1-{\cal O}(\epsilon)$.
Other possibilities exist but require some fine-tuning between the parameters, in analogy with the previous analysis in the case $\Lambda \ll 1$.
\subsubsection{Intermediate values of $\Lambda$}
In this case, one must satisfy simultaneously the constraints $|s_\sigma|\ll |z_\sigma|$ and $|s_\chi|\ll |z_\chi|$, due to the isocurvature bound.
The relative strength of the different $\tilde f_{NL}$ coefficients cannot be expressed in such a simple form as in (\ref{rules_c2_1}), but it will be determined again by the ratios $s_\sigma/z_\sigma$, $s_\chi/z_\chi$, $s_{\sigma\sigma}/\D$ and $s_{\chi\chi}/z_{\chi\chi}$.
In order to get also a significant non-Gaussianity, we look for parameter values such that
\begin{equation}
z_\sigma, z_\chi \sim {\cal O}(\epsilon), \qquad s_\sigma, s_\chi \lesssim {\cal O}(\alpha\, \epsilon).
\end{equation}
These constraints can be satisfied by fine-tuning the parameters. Solving $s_\sigma\simeq 0$ and $s_\chi\simeq 0$ for the two parameters $f_{c1}$ and $f_{c2}$ yields
\begin{equation}
f_{c1}\simeq \frac{(x_{r1}-x_{c1})(1-f_{\cs 1})-x_{\cs 1}}{1-f_{\cs 1}-x_{\cs 1}} \,, \qquad
f_{c2}\simeq x_{r2}-x_{c2}+ \frac{1-x_{r2}}{1-f_{\cs 1}}\, x_{\cs 1}\, .
\end{equation}
The observational constraint on the isocurvature power spectrum is satisfied if these two fine-tuning relations hold simultaneously, at the level ${\cal O}(\alpha \,\varepsilon)$. Using these relations, one finds
interesting non-Gaussianity for the following set of parameters: $x_{r1}={\cal O}(\epsilon)$, $x_{r2}={\cal O}(\epsilon)$, $x_{\cs 1}={\cal O}(\alpha\, \epsilon)$,
$f_{c1}=x_{r1}-x_{c1}+{\cal O}(\alpha\, \epsilon)$, $f_{c2}=x_{r2}+{\cal O}(\alpha\, \epsilon)$, with negligible values for $x_{c2}$. In this scenario, both curvatons are subdominant at their decay and the fraction of produced dark matter is fine-tuned.
\section{Conclusions}
In this work, we have introduced a systematic treatment in order to compute the evolution of linear and non-linear cosmological perturbations in a cosmological transition due to the decay of some particle species. Our main results can be summarized as follows.
At the linear level, the evolution of all individual curvature perturbations can be expressed in terms of a transfer matrix, whose coefficients depend on background quantities, such as the relative abundances of the fluids at the decay, their equation of state parameters and the relative decay branching ratios
[see Eqs (\ref{decay_linear}-\ref{f_A})]. At the non-linear level, the post-decay curvature perturbations can also be given in terms of the pre-decay perturbations quite generally, and we have presented explicitly these relations at second order [see Eqs (\ref{zeta_2ndorder}-\ref{U_ABC})].
We have then applied our general formalism to two specific examples.
The first example is the mixed curvaton-inflaton scenario in which we allow the dark matter to be created both before {\it and} during the curvaton decay.
We find, in particular, the remarkable result that it is possible to obtain {\it isocurvature dominated} non-Gaussianities with, as required by the CMB measurements, an adiabatic dominated power spectrum.
In the second example, we have studied scenarios with several curvaton-like fields and obtained results that generalize previous works on two-curvaton scenarios by taking into account the various decay products of the curvatons. We have explored the parameter space to see whether it is possible to find significant non-Gaussianity while satisfying the isocurvature bound in the power spectrum. We have found that several such regions exist, but often at the price of a fine-tuning between the parameters.
In the presence of isocurvature modes, which can be correlated with the adiabatic modes, non-Gaussianity of the local type is much richer than in the purely adiabatic case and we have shown that the angular bispectrum is the sum of six different contributions. As a consequence, in addition to the traditional $f_{NL}$ (adiabatic) parameter, we have identified five new non-linear parameters that must be taken into account: one purely isocurvature parameter and four correlated parameters. We have computed these parameters in the two models we have investigated.
Beyond the two examples considered in this work, our formalism can be used as a general toolbox to study systematically the cosmological constraints, arising from linear perturbations and from non-Gaussianities, for particle physics models and their associated cosmological scenarios.
\vspace{1cm}
{\bf \noindent Acknowledgements}
The work of D.L. is partially supported by the ANR (Agence Nationale de la
Recherche) grant "STR-COSMO" (ANR-09-BLAN-0157).
The work of A. L. is partially supported by the International Doctorate
on AstroParticle Physics (IDAPP) program.
\smallskip
| -55,271.605151 |
[
-3.380859375,
2.990234375
] | 34.191555 |
[
-3.20703125,
0.1121826171875,
-2.0546875,
-6.546875,
-0.7158203125,
8.8984375
] |
[
4.4609375,
8.6640625,
3.470703125,
6.27734375
] | 310 | 7,179 |
[
-3.36328125,
3.873046875
] | 31.683732 |
[
-6.04296875,
-4.515625,
-4.99609375,
-2.46875,
2.013671875,
12.78125
] | 0.896314 | 17.916407 | 22.524028 | 3.109815 |
[
2.7233803272247314
] | -34,153.510126 | 5.827831 | -54,248.020038 | 0.239017 | 6.084705 |
[
-2.51953125,
-3.68359375,
-3.953125,
-5.08203125,
2.244140625,
12.4453125
] |
[
-5.5234375,
-2.076171875,
-2.541015625,
-1.259765625,
3.408203125,
4.5546875
] | |
BkiUf13xK1fBGwCLPSMO
|
\chapter{Introduction}
\abstract{
One characteristic that defines us, human beings, is the curiosity of the unknown. Since our birth, we have been trying to use any methods that human brains can comprehend to explore the nature: to mimic, to understand and to utilize in a controlled and repeatable way. One of the most ancient means lies in the nature herself, experiments, leading to tremendous achievements from the creation of fire to the scissors of genes. Then comes mathematics, a new world we made by numbers and symbols, where the nature is reproduced by laws and theorems in an extremely simple, beautiful and unprecedentedly accurate manner. With the explosive development of digital sciences, computer was created. It provided us the third way to investigate the nature, a digital world whose laws can be ruled by ourselves with codes and \textit{algorithms} to numerically mimic the real universe. In this chapter, we briefly review the history of tensor network algorithms and the related progresses made recently. The organization of our lecture notes is also presented.}
\section{Numeric renormalization group in one dimension}
Numerical simulation is one of the most important approaches in science, in particular for the complicated problems beyond the reach of analytical solutions. One distinguished example of the algorithms in physics as well as in chemistry is \textit{ab-initio} principle calculation, which is based on density function theory (DFT\index{DFT}) \cite{S10DFTRev,B12DFT,B14DFT}. It provides a reliable solution to simulate a wide range of materials that can be described by the mean-field theories and/or single-particle approximations. Monte Carlo method \cite{R14MCbook}, named after a city famous of gambling in Monaco, is another example that appeared in almost every corner of science. In contemporary physics, however, there are still many ``hard nuts to crack''. Specifically in quantum physics, numerical simulation faces un-tackled issues for the systems with strong correlations, which might lead to exciting and exotic phenomena like high-temperature superconductivity \cite{LNW06HighSCRev,KKNUZ15highTcRev} and fractional excitations \cite{L99FracRev}.
Tensor network (TN) \index{TN} methods in the context of many-body quantum systems have been developed recently. One could however identify some precursors of them in the seminal works of Krammers and Wannier \cite{kramers1941statistics,kramers1941statistics2}, Baxter \cite{B68Prototype,NB86Baxterfomu}, Kelland \cite{kelland1976estimates}, Tsang \cite{tsang1979square}, Nighingale and Blote \cite{NB86Baxterfomu}, Derrida \cite{DE93MPS000,DEHP93MPS0}, as found by T. Nishino \cite{NO96CTMRG0, nishino1996numerical, nishino2001two,nishino2000self, nishino1998density, okunishi2000kramers, nishino1997numerical}. Here we start their history from the Wilson numerical renormalization group (NRG) \cite{W75NRGRev} \index{NRG}. The NRG aims at finding the ground state of a spin systems. The idea of the NRG is to start from a small system whose Hamiltonian can be easily diagonalized. The system is then projected on few low-energy states of the Hamiltonian. A new system is then constructed by adding several spins and a new low-energy effective Hamiltonian is obtained working only in the subspace spanned by the low-energy states of the previous step and the full Hilbert space of the new spins. In this way the low-energy effective Hamiltonian can be diagonalized again and its low energy states can be used to construct a new restricted Hilbert space. The procedure is then iterated. The original NRG has been improved for example by combining it with the expansion theory \cite{K90DMRGexp,XG92DMRGexpan,XG93DMRG}. As already shown in \cite{W75NRGRev} the NRG successfully tackles the Kondo problem in one dimension \cite{K64Kondo1D}, however, its accuracy is limited when applied to generic strongly-correlated systems such as Heisenberg chains.
In the nineties, White and Noack were able to related the poor NRG\index{NRG} accuracy with the fact that it fails to consider properly the boundary conditions \cite{WN92RG}. In 1992, White proposed the famous density matrix renormalization group (DMRG)\index{DMRG} that is as of today the most efficient and accurate algorithms for one-dimensional (1D) models \cite{W92DMRG,W93DMRG}. White used the largest eigenvectors of the reduced denisty matrix of a block as the states describing the relevant part of the low energy physics Hilbert space. The reduced density matrix is obtained by explicitly constructing the ground state of the system on a larger region. In other words, the space of one block is renormalized by taking the rest of the system as an \textit{environment}.
The simple idea of environment had revolutionary consequences in the RG-based algorithms. Important generalizations of DMRG\index{DMRG} were then developed, including the finite-temperature variants of matrix renormalization group \cite{BXG96TMRG, MC96FTDMRG, WX97TMRG, S97TMRG}, dynamic DMRG algorithms \cite{H95dDMRG, RPKSB97dDMRG, KW99dDMRG, J02dDMRG}, and corner transfer matrix renormalization group by Nishino and Okunishi \cite{NO96CTMRG0} \footnote{We recommend a web page built by Tomotoshi Nishino, \url{http://quattro.phys.sci.kobe-u.ac.jp/dmrg.html}, where one exhaustively can find the progresses related to DMRG.}.
About ten years later, TN\index{TN} was re-introduced in in simplest form of matrix-product states (MPS) \cite{FNW92TTN, FNW92MPS, DE93MPS000, DEHP93MPS0, klumper1991equivalence} in the the context of the theory of entanglement in quantum many-body systems; see, e.g., \cite{ON02DMRGent, V03EntState, VPC04DMRGQinfo, V04Qsimu} \footnote{For the general theory of entanglement and its role in the physics of quantum-many body systems, see for instance \cite{BD00EntRev, NC00EntBook, AFOV08EntRev, HHHH09EntRev}.}. In this context, the MPS encodes the coefficients of the wave functions in a product of matrices, and are thus defined as the contraction of a one dimensional TN. Each elementary tensor has three indexes: one physical index acting on the physical Hilbert space of the constituent, and two auxiliary indexes that will be contracted. The MPS\index{MPS} structure is chosen since it represents the states whose entanglement only scales with the boundary of a region rather than its volume, something called the ``area law'' of entanglement. Furthermore, an MPS gives only finite correlations, thus is well suited to represent the ground states of the gapped short-range Hamiltonians. The relation between these two facts was evinced in seminal contributions \cite{H04CorFunDecExp-1, H04CorFunDecExp-2, H07EntArea, H15TNthesis, B73EntAreaLaw, S93Sarea, LRV04GSEnt, CC04EntAreaLaw, PEDC05EntAreaLaw, ECP10AreaLawRev} and led Verstraete and Cirac to prove that MPS can provide faithful representations of the ground states of 1D gapped local Hamiltoinan \cite{VC06MPSFaithfully}.
These results together with the previous works that identified the outcome of converged DMRG\index{DMRG} simulations with an MPS\index{MPS} description of the ground states \cite{OR95MPS} allowed to better understand the impressive performances of DMRG in terms of the correct scaling of entanglement of its underlying TN ansatz. The connection between DMRG and MPS stands in the fact that the projector onto the effective Hilbert space built along the DMRG iterations can be seen as an MPS. Thus, the MPS in DMRG can be understood as not only a 1D state ansatz, but also a TN representation of the RG flows (\cite{FNW92MPS,OR95MPS,RO97MPS,DMNS98MPS,M07DMRGsymme,S11DMRGRev}, as recently reviewed in \cite{PVWC07MPSRev}).
These results from the quantum information community fueled the search for better algorithms allowing to optimize variationally the MPS tensors in order to target specific states \cite{VMC08MPSPEPSRev}. In this broader scenario, DMRG can be seen as an alternating-least-square optimization method. Alternative methods include the imaginary-time evolution from an initial state encoded as in an MPS base of the time-evolving block decimation (TEBD)\index{TEBD} \cite{V03TEBD,V04TEBD,V07iTEBD,OV08canonical}, and time-dependent variational principle of MPS \cite{HCOPVV11TDVP}. Note that these two schemes can be generalized to simulate also the short out-of-equilibrium evolution of a slightly entangled state. MPS have been used beyond ground states, for example in the context of finite-temperature and low-energy excitations based on MPS or its transfer matrix \cite{OR95MPS, BSZ03exciteMPS, CW09EntPurt, PHV12exciteMPS, HPWC+12MPSexcitations, ZVHMV18exciteMPS}.
MPS have further been used to characterize state violating the area law of entanglement, such as ground states of critical systems, and ground states of Hamiltonian with long-range interactions \cite{LRV04GSEnt,HLW94CFTRG, VLRK03CritEnt, TOIL08EntScaling, PMTM09EntScaling, PMJ2010entanglement, SHMTV15ScalingcMPS, RPLLS17Scaling2D, hauke2013spread, koffel2012entanglement}.
The relevance of MPS goes far beyond their use as a numerical ansatz. There have been numerous analytical studies that have led to MPS exact solutions such as the Affleck-Kennedy-Lieb-Tasaki (AKLT)\index{AKLT} state \cite{AKLT87AKLTState,AKLT88AKLTState}, as well as its higher-spin / higher-dimensional generalizations \cite{FNW92TTN,VPC04DMRGQinfo,NKZ97HLS3-2PT,NKZ00S2SLGS,VMC04EntLenDiv,KM08MPSVBS}. MPS has been crucial in understanding the classification of topological phases in 1D \cite{PT12SPT}. Here we will not talk about these important results, but we will focus on numerical applications even though the theory of MPS is still in full development and constantly new fields emerge such as the application of MPS to 1D quantum field theories \cite{VC10cMPS}.
\section{Tensor network states in two dimensions}
The simulations of two-dimensional (2D) systems, where analytical solutions are extremely rare and mean-field approximations often fail to capture the long-range fluctuations, are much more complicated and tricky. For numeric simulations, exact diagonalization can only access small systems; quantum Monte Carlo (QMC)\index{QMC} approaches are hindered by the notorious ``negative sign'' problem on frustrated spin models and fermionic models away from half-filling, causing an exponential increase of the computing time with the number of particles \cite{WSSLGS89SignQMC,TW05SignQMC}.
While very elegant and extremely powerful for 1D models, the 2D version of DMRG \cite{W96DMRG2D, WS98tjDMRG,XLS01DMRG2D,SW12DMRG2DRev}\index{DMRG} suffers several severe restrictions. The ground state obtained by DMRG is an MPS\index{MPS} that is essentially a 1D state representation, satisfying the 1D area law of entanglement entropy \cite{H07EntArea, H15TNthesis, S93Sarea, SWVC08MPSent}. However, due to the lack of alternative approaches, 2D DMRG is still one of the most important 2D algorithms, producing a large number of astonishing works including discovering the numeric evidence of quantum spin liquid \cite{M00QSLrev,B10QSLRev,SB17QSLRev} on kagom\'e lattice (see, e.g., \cite{JWS08DMRGKagome,YHW12DMRGkagome,JWB12TopoEntKagome,DMS12DMRGKagome,NSH12kagomemag,HZOP17kagomeDMRG}).
Besides directly using DMRG in 2D, another natural way is to extend the MPS representation, leading to the tensor product state \cite{NHOMAG01TPS0}, or projected entangled pair state (PEPS)\index{PEPS} \cite{VC04PEPS, VC06PEPSArxiv}. While an MPS is made up of tensors aligned in a 1D chain, a PEPS is formed by tensors located in a 2D lattice, forming a 2D TN. Thus, PEPS can be regarded as one type of 2D tensor network states (TNS)\index{TNS}. Note the work of Affleck \textit{et al} \cite{AKLT88PEPSorigin} can be considered as a prototype of PEPS.
The network structure of the PEPS allows us to construct 2D states that strictly fulfill the area law of entanglement entropy \cite{VWPC06PEPSfamous}. It indicates that PEPS can efficiently represents 2D gapped states, and even the critical and topological states, with only finite bond dimensions. Examples include resonating valence bond states \cite{VWPC06PEPSfamous,PSPC12PEPSRVB,SPCP12PEPSRVB,WPGWV13PEPSRVB,PCSC14PEPSRVB} originally proposed by Anderson \textit{et al} for super-conductivity \cite{A73RVB,A74RVB,A87RVB,BZA87RVBSC,ABZH87RVBSC}, string-net states \cite{GLSW09StringTPS,BAV09StringTPS,CZGCW10StringTPS} proposed by Wen \textit{et al} for gapped topological orders \cite{W89TopoOrder,W90TopoOrder,WN90TopoOrder,W95TopoOrder,LW05TopoOrder,LW05TopoOrderRev,W05TopoOrderRev}, and so on.
The network structure makes PEPS so powerful that it can encode difficult computational problems including non-deterministic polynomial (NP)\index{NP} hard ones \cite{VWPC06PEPSfamous,SWVC07PEPSAreaLaw,GL11TNNPhardarxiv}. What is even more important for physics is that PEPS provides an efficient representation as a variational ansatz for calculating ground states of 2D models. However, obeying the area law costs something else: the computational complexity rises \cite{VWPC06PEPSfamous,SWVC07PEPSAreaLaw,HWS13PEPSAreaLaw}. For instance, after having determined the ground state (either by construction or variation), one usually wants to extract the physical information by computing, e.g., energies, order parameters, or entanglement. For an MPS, most of the tasks are matrix manipulations and products which can be easily done by classical computers. For PEPS, one needs to contract a TN \index{TN} stretching in a 2D plain, unfortunately, most of which cannot be neither done exactly or nor even efficiently. The reason for this complexity is what brings the physical advantage to PEPS: the network structure. Thus, algorithms to compute the TN contractions need to be developed.
Other than dealing with the PEPS, TN provides a general way to different problems where the \textit{cost functions} is written as the contraction of a TN. A cost function is usually a scalar function, whose maximal or minimal point gives the solution of the targeted optimization problem. For example, the cost function of the ground-state simulation can be the energy (e.g., \cite{SV09TNQMC,VHCV16GrdPEPS}); for finite-temperature simulations, it can be the partition function or free energy (e.g., \cite{CCD12FTPEPS, RXLS13NCD}); for the dimension reduction problems, it can be the truncation error or the distance before and after the reduction (e.g., \cite{V04TEBD,LN07TRG,RLXZS12ODTNS}); for the supervised machine learning problems, it can be the accuracy (e.g., \cite{SS16TNML}). TN can then be generally considered as a specific mathematical structure of the parameters in the cost functions.
Before reaching the TN\index{TN} algorithms, there are a few more things worth mentioning. MPS\index{MPS} and PEPS\index{PEPS} are not the only TN representations in one or higher dimensions. As a generalization of PEPS, projected entangled simplex state was proposed, where certain redundancy from the local entanglement is integrated to reach a better efficiency \cite{XCYKNX14PESS,LXCLX+17kagome}. Except for a chain or 2D lattice, TN can be defined with some other geometries, such as trees or fractals. Tree TNS\index{TNS} is one example with non-trivial properties and applications \cite{FNW92TTN, NKZ97HLS3-2PT, F97TreeDMRG, LCP00TreeDMRG, MRS02DendrimersDMRG, SDV06TTN, NFGSS08TreeMPS, TEV09TTN, MVLN10TTN,LDX12TTN, NC13TTN, PVK13TTN, GSRF+14TTN, MVSNL15TTN}. Another example is multi-scale entanglement renormalization ansatz (MERA)\index{MERA} proposed by Vidal \cite{V07EntRenor,V08MERA,CDR08MERA,EV09EntRenor,AV08EntRenor,EV09EntRenorAlgor,CV09EntRenor,EV10EntRenorBoson,EV10EntRenorFermi}, which is a powerful tool especially for studying critical systems \cite{PEV09MERAcrit, MRGF09MERAcrit, ECV10EntRenorOpera, SGCSF10EntRenorCrit, EV13EntRenorCrit, BOBD15EntRenorCrit} and AdS/CFT theories (\cite{EV11AdS/CFT, S12AdS/CFT, B13AdS/CFT, Q13AdS/CFTArxiv,MNSTW15AdS/CFT,BCCCHP+15AdS/CFT,CLMS15AdS/CFTArxiv}, see \cite{N15AdS/CFTBook} for a general introduction of CFT\index{CFT}). TN has also been applied to compute exotic properties of the physical models on fractal lattices \cite{GGN16fractal,WRLZZS16Fractal}.
The second thing concerns the fact that some TN's can indeed be contracted exactly. Tree TN is one example, since there is no loop of a tree graph. This might be the reason that a tree TNS\index{TNS} can only have a finite correlation length \cite{NFGSS08TreeMPS}, thus cannot efficiently access criticality in two dimensions. MERA modifies the tree in a brilliant way, so that the criticality can be accessed without giving up the exactly contractible structure \cite{EV09EntRenorAlgor}. Some other exactly contractible examples have also been found, where exact contractibility is not due to the geometry, but due to some algebraic properties of the local tensors \cite{KRV09exactTRG, DBJC12TopoTNS}.
Thirdly, TN can represent operators, usually dubbed as TN operators. Generally speaking, a TN state can be considered as a linear mapping from the physical Hilbert space to a scalar given by the contraction of tensors. A TN operator is regarded as a mapping from the \textit{bra} to the \textit{ket} Hilbert space. Many algorithms explicitly employ the TN operator form, including the matrix product operator (MPO)\index{MPO} for representing 1D many-body operators and mixed states, and for simulating 1D systems in and out of equilibrium \cite{VGC04MPDO, ZV04MPO, PMCV10MPO, LRGZXY+11LTRG, BCL14MPOopen, MFS15MPOopen, CCB15MPOsteady, BKTMWH17FT1D, GIK17FT1D, HV17MPO, CPSV17MPO}, tensor product operator (also called projected entangled pair operators) in for higher-systems \cite{CCD12FTPEPS, RXLS13NCD, RLXZS12ODTNS, FND10PEPO, O12PEPO, CD15TPO, CD15FTPEPS, CDO16TPO, CRD16TPO,DSCBZ17FT2D, CDO17TPOQMC, KREO18PEPOthermal, CDC19TPO}, and multiscale entangled renormalization ansatz \cite{MIH13MERA, MNRT14cMERAft, GSW16FTmera}.
\section{Tensor renormalization group and tensor network algorithms}
Since most of TN's cannot be contracted exactly (with \#P-complete computational complexity \cite{GL11TNNPhardarxiv}), efficient algorithms are strongly desired. In 2007, Levin and Nave generalized the NRG\index{NRG} idea to TN and proposed tensor renormalization group (TRG)\index{TRG} approach \cite{LN07TRG}. TRG consists of two main steps in each RG iteration: contraction and truncation. In the contraction step, the TN is deformed by singular value decomposition (SVD)\index{SVD} of matrix in such a way that certain adjacent tensors can be contracted without changing the geometry of the TN graph. This procedure reduces the number of tensors $N$ to $N/\nu$, with $\nu$ an integer that depends on the way of contracting. After reaching the fixed point, one tensor represents in fact the contraction of infinite number of original tensors, which can be seen as the approximation of the whole TN.
After each contraction, the dimensions of local tensors increase exponentially, and then truncations are needed. To truncate in an optimized way, one should consider the ``environment'', a concept which appears in DMRG\index{DMRG} and is crucially important in TRG-based schemes to determine how optimal the truncations are. In the truncation step of Levin's TRG, one only keeps the basis corresponding to the $\chi$-largest singular values from the SVD\index{SVD} in the contraction step, with $\chi$ called dimension cut-off. In other words, the environment of the truncation here is the tensor that is decomposed by SVD. Such a local environment only permits local optimizations of the truncations, which hinders the accuracy of Levin's TRG on the systems with long-range fluctuations. Nevertheless, TRG is still one of the most important and computationally-cheap approaches for both classical (e.g., Ising and Potts models) and quantum (e.g., Heisenberg models) simulations in two and higher dimensions \cite{KRV09exactTRG, JWX08SimpleUpdate,GLW08TERG,GW09TERG, CY09TRGapp, ZXCWCX10TNR, HL10MultiEnt, LGZS10honeycomb, GH10TRG, GHB10TRG, WKS11TRG, CQCWN+11PottsTRG, XCQZYX12HOSRG, S12TRGboson, GL13TRG3D, QCXCY+14PottsTRG, WXCBX14TRG3D, RH15TRGdimer, ZXXI16finiteTRG}. It is worth mentioning that for 3D classical models, the accuracy of the TRG algorithms have surpassed other methods \cite{XCQZYX12HOSRG,WXCBX14TRG3D}, such as QMC\index{QMC}. Following the contraction-and-truncation idea, the further developments of the TN contraction algorithms concern mainly two aspects: more reasonable ways of contracting, and more optimized ways of truncating.
While Levin's TRG ``coarse-grains'' a TN in an exponential way (the number of tensors decreases exponentially with the renormalization steps), Vidal's TEBD\index{TEBD} scheme \cite{V03TEBD,V04TEBD,V07iTEBD,OV08canonical} implements the TN contraction with the help of MPS in a linearized way \cite{LRGZXY+11LTRG}. Then, instead of using the singular values of local tensors, one uses the entanglement of the MPS to find the optimal truncation, meaning the environment is a (non-local) MPS\index{MPS}, leading to a better precision than Levin's TRG. In this case, the MPS at the fixed point is the dominant eigenstate of the transfer matrix of the TN. Another group of TRG algorithms, called corner transfer matrix renormalization group (CTMRG)\index{CTMRG} \cite{OV09CTMRG}, are based on the corner transfer matrix idea originally proposed by Baxter in 1978 \cite{B78CTM}, and developed by Nishina and Okunishi in 1996 \cite{NO96CTMRG0}. In CTMRG, the contraction reduces the number of tensors in a polynomial way and the environment can be considered as a finite MPS defined on the boundary. CTMRG has a compatible accuracy compared with TEBD\index{TEBD}.
With a certain way of contracting, there is still high flexibility of choosing the environment, i.e. the reference to optimize the truncations. For example, Levin's TRG\index{TRG} and its variants \cite{LN07TRG, JWX08SimpleUpdate, GLW08TERG, GW09TERG,ZXCWCX10TNR, XCQZYX12HOSRG}, the truncations are optimized by local environments. The second renormalization group proposed by Xie \textit{et al} \cite{XCQZYX12HOSRG, XJCWX09SRG} employs TRG to consider the whole TN\index{TN} as the environments.
Besides the contractions of TN's, the concept of environment becomes more important for the TNS update algorithms, where the central task is to optimize the tensors for minimizing the cost function. According to the environment, the TNS\index{TNS} update algorithms are categorized as the simple \cite{RXLS13NCD, RLXZS12ODTNS, JWX08SimpleUpdate, XCQZYX12HOSRG, LCB14PEPScontract, JO18graphPEPS}, cluster \cite{ RXLS13NCD, LCB14PEPScontract, WV11PEPSclusterArxiv, RPPSL17AOP3D}, and full update \cite{XCQZYX12HOSRG, OV09CTMRG, XJCWX09SRG, JOVVC08PEPS, PWV11tPEPS, O12CTMRG, LCB14fPEPS, PBTCO15FastFullUpdate, C16vPEPS}. The simple update uses local environment, hence has the highest efficiency but limited accuracy. The full update considers the whole TN as the environment, thus has a high accuracy. Though with a better treatment of the environment, one drawback of the full update schemes is the expensive computational cost, which strongly limits the dimensions of the tensors one can keep. The cluster update is a compromise between simple and full update, where one considers a reasonable subsystem as the environment for a balance between the efficiency and precision.
It is worth mentioning that TN\index{TN} encoding schemes are found to bear close relations to the techniques in multi-linear algebra (MLA)\index{MLA} (also know as tensor decompositions or tensor algebra; see a review \cite{KB09MLA}). MLA was originally targeted on developing high-order generalization of the linear algebra (e.g., the higher-order version of singular value or eigenvalue decomposition \cite{LMV00Rank1,DDV00HOSVD,LV04MLA,L06MLA}), and now has been successfully used in a large number of fields, including data mining (e.g., \cite{ACKY05MLADM,LZYCLB+05MLADM,SZLLC05MLADM,ACY06MLADM,SPY06MLADM}), image processing (e.g., \cite{KBK05MLAIP,KB06MLAIP,BHK07MLAIP,DZZHT17MLAimage}), machine learning (e.g., \cite{SDLFHP+17TDML}), and so on. The interesting connections between the fields of TN and MLA (for example tensor-train decomposition \cite{O11TTD} and matrix product state representation) open new paradigm for the interdisciplinary researches that cover a huge range in sciences.
\section{Organization of lecture notes}
Our lectures are organized as following. In Chapter 2, we will introduce the basic concepts and definitions of tensor and TN states/operators, as well as their graphic representations. Several frequently used architectures of TN states will be introduced, including matrix product state, tree TN state, and PEPS\index{PEPS}. Then the general form of TN, the gauge degrees of freedom, and the relations to quantum entanglement will be discussed. Three special types of TN's that can be exactly contracted will be exemplified in the end of this chapter.
In Chapter 3, the contraction algorithms for 2D TN's will be reviewed. We will start with several physical problems that can be transformed to the 2D TN contractions, including the statistics of classical models, observation of TN states, and the ground-state/finite-temperature simulations of 1D quantum models. Three paradigm algorithms namely TRG\index{TRG}, TEBD\index{TEBD}, and CTMRG\index{CTMRG}, will be presented. These algorithms will be further discussed from the aspect of the exactly contractible TN's.
In Chapter 4, we will concentrate on the algorithms of PEPS\index{PEPS} for simulating the ground states of 2D quantum lattice models. Two general schemes will be explained, which are the variational approaches and the imaginary-time evolution. According to the choice of environment for updating the tensors, we will explain the simple, cluster, and full update algorithms. Particularly in the full update, the contraction algorithms of 2D TN's presented in Chapter 3 will play a key role to compute the non-local environments.
In Chapter 5, a special topic about the underlying relations between the TN methods and the MLA\index{MLA} will be given. We will start from the canonicalization of MPS\index{MPS} in one dimension, and then generalize to the super-orthogonalization of PEPS in higher dimensions. The super-orthogonalization that gives the optimal approximation of a tree PEPS\index{PEPS} in fact extends the Tucker decomposition from single tensor to tree TN\index{TN}. Then the relation between the contraction of tree TN's and the rank-1 decomposition will be discussed, which further leads to the ``zero-loop'' approximation of the PEPS on the regular lattice. Finally, we will revisit the infinite DMRG (iDMRG)\index{iDMRG}, infinite TEBD (iTEBD)\index{iTEBD}, and infinite CTMRG\index{CTMRG} in a unified picture indicated by the tensor ring decomposition, which is a higher-rank extension of the rank-1 decomposition.
In Chapter 6, we will revisit the TN simulations of quantum lattice models from the ideas explained in Sec. 5. Such a perspective, dubbed as quantum entanglement simulation (QES)\index{QES}, shows an unified picture for simulating one- and higher-dimensional quantum models at both zero \cite{RPPSL17AOP3D, R16AOP} and finite \cite{RXPS+18FTQES} temperatures. The QES implies an efficient way of investigating infinite-size many-body systems by simulating few-body models with classical computers or artificial quantum platforms. In Sec. 7, a brief summary is given.
As TN makes a fundamental language and efficient tool to a huge range of subjects, which has been advancing in an extremely fast speed, we cannot cover all the related progresses in this review. We will concentrate on the algorithms for TN contractions and the closely related applications. The topics that are not discussed or are only briefly mentioned in this review include: the hybridization of TN with other methods such as density functional theories and \textit{ab-initio} calculations in quantum chemistry \cite{WM99DMRGabini, MFOLP01DMRGDFT, MOMR08DMRGQchem, MR10DMRGQchem, CS11DMRGQchem, W14DMRGQchemy, SA15MPSchem, KVLE16fTNS, RLJC17DMRGinitio, YSLM18DMRGChem}, the dynamic mean-field theory \cite{GJMNN03DMRGDMFT, NGJ04DMRGDMFT, GHR04DMFTwithDMRG, K06DMRGDMFT, WMPS14MPSDMFT, GTVHE14MPSimpu, WMIS14MPSDMFT, WGMM+15MPSDMFT, GAET+15MPSDMFT, BZTAE17TNDMFT}, and the expansion/perturbation theories \cite{GTVHE14MPSimpu, HWHSV11ChMPS, WJMS15Chebyshev, HKM15ChMPS, CLCL17TNexpand, TRFML17DMRGPT, VMSVV19TNpert}; the TN algorithms that are less related to the contraction problems such as time-dependent variational principle \cite{HCOPVV11TDVP, HLOVV16TDVPuni}, the variational TN state methods \cite{HPWC+12MPSexcitations, C16vPEPS, MHOV13vMPS, HOV13MPStan, VMVH15PEPStan, ZVFVH18vuMPS, ZMV18uMPS, VHV19uMPS}, and so on; the TN methods for interacting fermions \cite{YWTSC15PEPSchiral, EV10EntRenorFermi, KVLE16fTNS, BPE09fTN, COBV10fPEPS, CJV10fPEPS, PV10fPEPS, KSVC10fPEPS, MBRTV10fTNS, CWVT11fPEPS, MR11TNSchem, G13GTN, CD14fPEPS, SK14GrassTN, ZBWC15fPEPS, WBE17fTN, BWHV17fPEPS}, quantum field theories \cite{TCL14TNgaugeF, RPDZM14GaugeF, HVSC15TNgauge, XA15TNgauge, PDRZM16TNgauge, BMHVV17TNgauge, ZO17TNgauge}, topological states and exotic phenomena in many-body systems (e.g., \cite{JWS08DMRGKagome, YHW12DMRGkagome, DMS12DMRGKagome, HZOP17kagomeDMRG, PSPC12PEPSRVB, SPCP12PEPSRVB, WPGWV13PEPSRVB, PCSC14PEPSRVB, GLSW09StringTPS,BAV09StringTPS,PCBA+10AnyonTN, KB10AnyonER, WTSC13PEPSchiral, DR15TNchiral, YWTSC15PEPSchiral, PCS15PEPSchiral, MOP16TNsymme, HW16SPTTN, GRSDM17QHETNarxiv, LXCLX+17kagomeQSL, PRLCS17octakagome, LRLYZS14Husimi, YLPVV+14boundary, WHTCS14cPEPS, RLGWDS15StarPEPS, WBMH15MPSSPT, JR17TNanyon}), the open/dissipative systems \cite{VGC04MPDO, BCL14MPOopen, MFS15MPOopen, CCB15MPOsteady, GIK17FT1D, PZ09MPSsteady, SC16MPSopen, WJSKC+16PTNS, KWO17MPSopen, JMC18MPSopen}, quantum information and quantum computation \cite{VPC04DMRGQinfo, J06TNcuicirt, GE07QCPEPS, AL10QCTN, GESP07MPSQC, MS08TNQcomp, GMF08TNchannel, FM12QCTN, JBCJ13TNccuit, FP14TNcorrect, DESB+18TNQsimu}, machine learning \cite{SS16TNML, B13MERAML, BPT15MPSML, NPOV15TTNN, LRWP+17MLTN, CCXWX17TNML, HM17TNML, HWFWZ17MPSML, LYCS17TNML, GO17TNML, GJLP18MPOML, CLOPZ+17TNML1rev, CLOPZ+17TNML2rev, GJLP18MPOML, GPC18gTNML, S18MERAML, CBKW18SVMTT, CWXZ19generateTTNML}, and other classical computational problems \cite{EHHS11TNMLA, C14TNbigdata, BMT15TNSAT, BSU16TNMLA, YKCMR18TNCC, KCMR18TNCC}; the TN theories/algorithms with nontrivial statistics and symmetries \cite{GLSW09StringTPS,BAV09StringTPS,CZGCW10StringTPS,ZBWC15fPEPS, HVSC15TNgauge, PCBA+10AnyonTN, MOP16TNsymme, PSGWC10symme, W12Qspace, SCP10PEPSsymme, SPV10TNsymme, SPV11symmeU1, O11advTN, BCOT11assyme, SV12SU2symme, TCL14TNsymme, O14TNadvRev, RDS15PEPSZ2, JR15TNsymme, LH16TNsymme, ZB16PEPSgauge}; several latest improvements of the TN algorithms for higher efficiency and accuracy \cite{PWV11tPEPS, PBTCO15FastFullUpdate, BHVC09folding, MCB12folding, HM15folding, YGW17LoopTN, XLHX+19TNcon}.
\chapter{Tensor Network: Basic Definitions and Properties}
\abstract{This chapter is to introduce some basic definitions and concepts of TN\index{TN}. We will show that the TN can be used to represent quantum many-body states, where we explain MPS\index{MPS} in 1D and PEPS\index{PEPS} in 2D systems, as well as the generalizations to thermal states and operators. The quantum entanglement properties of the TN states including the area law of entanglement entropy will also be discussed. Finally, we will present several special TN's that can be exactly contracted, and demonstrate the difficulty of contracting TN's in general cases.}
\section{Scalar, vector, matrix, and tensor}
Generally speaking, a tensor is defined as a series of numbers labeled by $N$ indexes, with $N$ called the \textit{order} of the tensor \footnote{Note that in some references, $N$ is called the tensor \textit{rank}. Here, the word \textit{rank} is used in another meaning, which will be explained later.}. In this context, a scalar, which is one number and labeled by zero index, is a 0th-order tensor. Many physical quantities are scalars, including energy, free energy, magnetization and so on. Graphically, we use a dot to represent a scalar (Fig. \ref{fig-1Tensor}).
\begin{figure}[tbp]
\centering
\includegraphics[angle=0,width=0.75\linewidth]{Fig1.eps}
\caption{(Color online) From left to right, the graphic representations of a scalar, vector, matrix and tensor.}
\label{fig-1Tensor}
\end{figure}
A $D$-component vector consists of $D$ numbers labeled by one index, and thus is a 1st-order tensor. For example, one can write the state vector of a spin-$1/2$ in a chosen basis (say the eigenstates of the spin operator $\hat{S}^{[z]}$) as
\begin{eqnarray}
|\psi\rangle =C_1|0 \rangle +C_2|1 \rangle
=\sum_{s=0,1}C_s|s\rangle,
\end{eqnarray}
with the coefficients $C$ a two-component vector. Here, we use $|0\rangle$ and $|1\rangle$ to represent spin up and down states. Graphically, we use a dot with one open bond to represent a vector (Fig.\ref{fig-1Tensor}).
A matrix is in fact a 2nd-order tensor. Considering two spins as an example, the state vector can be written under a irreducible representation as a four-dimension vector. Instead, under the local basis of each spin, we write it as
\begin{eqnarray}
|\psi\rangle =C_{00}|0\rangle|0\rangle +C_{01}|0\rangle|1\rangle
+C_{10}|1\rangle|0\rangle+C_{11}|1\rangle|1\rangle =\sum_{ss'=0}^{1}C_{ss'}|s\rangle|s'\rangle,
\end{eqnarray}
with $C_{ss'}$ a matrix with two indexes. Here, one can see that the difference between a ($D \times D$) matrix and a $D^2$-component vector in our context is just the way of labeling the tensor elements. Transferring among vector, matrix and tensor like this will be frequently used later. Graphically, we use a dot with two bonds to represent a matrix and its two indexes (Fig.\ref{fig-1Tensor}).
It is then natural to define an $N$-th order tensor. Considering, e.g., $N$ spins, the $2^N$ coefficients can be written as a $N$-th order tensor $C$ \footnote{If there is no confuse, we use the symbol without all its indexes to refer to a tensor for conciseness, e.g., use $C$ to represent $C_{s_1...s_N}$.}, satisfying
\begin{eqnarray}
|\psi\rangle =\sum_{s_1 \cdots s_N=0}^{1}C_{s_1...s_N}|s_1\rangle...|s_N\rangle.
\end{eqnarray}
Similarly, such a tensor can be \textit{reshaped} into a $2^N$-component vector. Graphically, an $N$-th order tensor is represented by a dot connected with $N$ open bonds (Fig.\ref{fig-1Tensor}).
In above, we use states of spin-$1/2$ as examples, where each index can take two values. For a spin-$S$ state, each index can take $d=2S+1$ values, with $d$ called the \textit{bond dimension}. Besides quantum states, operators can also be written as tensors. A spin-$1/2$ operator $\hat{S}^{\alpha}$ ($\alpha=x,y,z$) is a ($2\times 2$) matrix by fixing the basis, where we have $ S^{\alpha}_{s_1's_2's_1s_2} = \langle s_1's_2' |\hat{S}^{\alpha} |s_1s_2\rangle$. In the same way, an $N$-spin operator can be written as a $2N$-th order tensor, with $N$ \textit{bra} and $N$ \textit{ket} indexes \footnote{Note that here, we do not distinguish \textit{bra} and \textit{ket} indexes deliberately in a tensor, if not necessary}.
We would like to stress some conventions about the ``indexes'' of a tensor (including matrix) and those of an operator. A tensor is just a group of numbers, where their indexes are defined as the labels labeling the elements. Here, we always put all indexes as the lower symbols, and the upper ``indexes'' of a tensor (if exist) are just a part of the symbol to distinguish different tensors. For an operator which is defined in a Hilbert space, it is represented by a hatted letter, and there will be no ``true'' indexes, meaning that both upper and lower ``indexes'' are just parts of the symbol to distinguish different operators.
\section{Tensor network and tensor network states}
\subsection{A simple example of two spins and Schmidt decomposition}
\label{sec.2SVD}
\begin{figure}[tbp]
\centering
\includegraphics[angle=0,width=0.4\linewidth]{Fig2.eps}
\caption{(Color online) The graphic representation of the Schmidt decomposition (singular value decomposition of a matrix). The positive-defined diagonal matrix $\lambda$, which gives the entanglement spectrum (Schmidt numbers), is defined on a virtual bond (dumb index) generated by the decomposition.}
\label{fig-1SVD}
\end{figure}
After introducing tensor (and it diagram representation), now we are going to talk about TN, which is defined as the contraction of many tensors. Let us start with the simplest situation, two spins, and consider to study the quantum entanglement properties for instance. Quantum entanglement, mostly simplified as entanglement, can be defined by the \textit{Schmidt decomposition} \cite{S1907SD,EK95SD,P95SD} of the state (Fig. \ref{fig-1SVD}) as
\begin{eqnarray}
| \psi \rangle = \sum_{ss'=0}^{1} C_{ss'} |s \rangle |s' \rangle = \sum_{ss'=0}^{1} \sum_{a=1}^{\chi} U_{sa} \lambda_{a a'} V^{\ast}_{a s'} |s \rangle |s' \rangle,
\label{eq-1Schmidt}
\end{eqnarray}
where $U$ and $V$ are unitary matrices, $\lambda$ is a positive-defined diagonal matrix in descending order \footnote{Sometime, $\lambda$ is treated directly as a $\chi$-component vector.}, and $\chi$ is called the \textit{Schmidt rank}. $\lambda$ is called the \textit{Schmidt coefficients} since in the new basis after the decomposition, the state is written in a summation of $\chi$ product states as $| \psi \rangle = \sum_{a} \lambda_a | u \rangle_{a} | v \rangle_{a}$, with the new basis $| u \rangle_{a} = \sum_{s} U_{s a} |s \rangle$ and $| v \rangle_{a} = \sum_{s'} V^{\ast}_{s a} |s' \rangle$.
Graphically, we have a small TN\index{TN}, where we use green squares to represent the unitary matrices $U$ and $V$, and a red diamond to represent the diagonal matrix $\lambda$. There are two bonds in the graph shared by two objects, standing for the summations (contractions) of the two indexes in Eq. (\ref{eq-1Schmidt}), $a$ and $a'$. Unlike $s$ (or $s'$), The space of the index $a$ (or $a'$) is not from any physical Hilbert space. To distinguish these two kinds, we call the indexes like $s$ the \textit{physical indexes} and those like $a$ the \textit{geometrical} or \textit{virtual indexes}. Meanwhile, since each physical index is only connected to one tensor, it is also called an \textit{open bond}.
Some simple observations can be made from the Schmidt decomposition. Generally speaking, the index $a$ (also $a'$ since $\lambda$ is diagonal) contracted in a TN carry the quantum entanglement \cite{NC10EntQIS}. In quantum information sciences, entanglement is regarded as a quantum version of correlation \cite{NC10EntQIS}, which is crucially important to understand the physical implications of TN. One usually uses the \textit{entanglement entropy} to measure the strength of the entanglement, which is defined as $S = - 2 \sum_{a=1}^{\chi} \lambda_{a}^2 \ln \lambda_{a}$. Since the state should be normalized, we have $\sum_{a=1}^{\chi}\lambda_{a}^2 = 1$. For $\dim(a) = 1$, obviously $| \psi \rangle = \lambda_1 | u \rangle_{1} | v \rangle_{1}$ is a product state with zero entanglement $S=0$ between the two spins. For $\dim(a) = \chi$, the entanglement entropy $S \leq \ln \chi$, where $S$ takes its maximum if and only if $\lambda_1 = \cdots = \lambda_{\chi}$. In other words, the dimension of a geometrical index determines the upper bound of the entanglement.
Instead of Schmidt decomposition, it is more convenient to use another language to present later the algorithms: \textit{singular value decomposition} (SVD)\index{SVD}, a matrix decomposition in linear algebra. The Schmidt decomposition of a state is the SVD of the coefficient matrix $C$, where $\lambda$ is called the \textit{singular value spectrum} and its dimension $\chi$ is called the \textit{rank} of the matrix. In linear algebra, SVD gives the optimal lower-rank approximations of a matrix, which is more useful to the TN algorithms. Specifically speaking, with a given matrix $C$ of rank-$\chi$, the task is to find a rank-$\tilde{\chi}$ matrix $C'$ ($\tilde{\chi} \leq \chi$) that minimizes the norm
\begin{eqnarray}
\mathcal{D} = |M-M'| = \sqrt{\sum_{ss'} (M_{ss'} - M'_{ss'})^2}.
\end{eqnarray}
The optimal solution is given by the SVD as
\begin{eqnarray}
M'_{ss'} = \sum_{a=0}^{\chi'-1} U_{sa} \lambda_{a a} V^{\ast}_{s' a}.
\end{eqnarray}
In other words, $M'$ is the optimal rank-$\chi'$ approximation of $M$, and the error is given by
\begin{eqnarray}
\varepsilon = \sqrt{\sum_{a=\chi'}^{\chi-1} \lambda_{a}^2},
\label{eq-2trunerr}
\end{eqnarray}
which will be called the \textit{truncation error} in the TN\index{TN} algorithms.
\subsection{Matrix product state}
\begin{figure}[tbp]
\centering
\includegraphics[angle=0,width=0.8\linewidth]{MPSSVD.eps}
\caption{(Color online) An impractical way to obtain an MPS from a many-body wave-function is to repetitively use the SVD\index{SVD}.}
\label{fig-1MPSSVD}
\end{figure}
Now we take a $N$-spin state as an example to explain the MPS\index{MPS}, a simple but powerful 1D TN state. In an MPS, the coefficients are written as a TN given by the contraction of $N$ tensors. Schollw\"ock in his review \cite{S11DMRGRev} provides a straightforward way to obtain such a TN\index{TN} is by repetitively using SVD\index{SVD} or QR\index{QR} decomposition (Fig. \ref{fig-1MPSSVD}). First, we group the first $N-1$ indexes together as one large index, and write the coefficients as a $2^{N-1} \times 2$ matrix. Then implement SVD or any other decomposition (for example QR decomposition) as the contraction of $C^{[N-1]}$ and $A{[N]}$
\begin{eqnarray}
C_{s_1 \cdots s_{N-1}s_N} = \sum_{a_{N-1}} C^{[N-1]}_{s_1 \cdots s_{N-1},a_{N-1}} A^{[N]}_{s_{N}, a_{N-1} }.
\end{eqnarray}
Note that as a convention in this paper, we always put the physical indexes in front of geometrical indexes and use a comma to separate them. For the tensor $C^{[N-1]}$, one can do the similar thing by grouping the first $N-2$ indexes and decompose again as
\begin{eqnarray}
C_{s_1 \cdots s_{N-1} a_{N-1}} = \sum_{a_{N-2}} C^{[N-2]}_{s_1 \cdots s_{N-2},a_{N-2}} A^{[N-1]}_{s_{N-1}, a_{N-2} a_{N-1}}.
\end{eqnarray}
Then the total coefficients becomes the contraction of three tensors as
\begin{eqnarray}
C_{s_1 \cdots s_{N-1}s_N} = \sum_{a_{N-2} a_{N-1}} C^{[N-2]}_{s_1 \cdots s_{N-2},a_{N-2}} A^{[N-1]}_{s_{N-1}, a_{N-2} a_{N-1}} A^{[N]}_{s_{N}, a_{N-1}}.
\end{eqnarray}
Repeat decomposing in the above way until each tensor only contains one physical index, we have the MPS\index{MPS} representation of the state as
\begin{eqnarray}
C_{s_1 \cdots s_{N-1}s_N} = \sum_{a_{1} \cdots a_{N-1}} A^{[1]}_{s_1, a_{1}} A^{[2]}_{s_2, a_{1} a_{2}} \cdots A^{[N-1]}_{s_{N-1}, a_{N-2} a_{N-1}} A^{[N]}_{s_{N}, a_{N-1}}.
\label{eq-1MPS}
\end{eqnarray}
One can see that an MPS is a TN\index{TN} formed by the contraction of $N$ tensors. Graphically, MPS is represented by a 1D graph with $N$ open bonds. In fact, an MPS given by Eq. (\ref{eq-1MPS}) has open boundary condition, and can be generalized to periodic boundary condition (Fig. \ref{fig-1MPSfinite}) as
\begin{eqnarray}
C_{s_1 \cdots s_{N-1}s_N} = \sum_{a_{1} \cdots a_{N}} A^{[1]}_{s_1, a_{N} a_{1}} A^{[2]}_{s_2, a_{1} a_{2}} \cdots A^{[N-1]}_{s_{N-1}, a_{N-2} a_{N-1}} A^{[N]}_{s_{N}, a_{N-1} a_{N}},
\label{eq-1MPS_PBC}
\end{eqnarray}
where all tensors are 3rd-order. Moreover, one can introduce translational invariance to the MPS, i.e. $A^{[n]} = A$ for $n=1,2,\cdots,N$. We use $\chi$, dubbed as \textit{bond dimension} of the MPS, to represent the dimension of each geometrical index.
MPS is an efficient representation of a many-body quantum state. For a $N$-spin state, the number of the coefficients is $2^N$ which increases exponentially with $N$. For an MPS given by Eq. (\ref{eq-1MPS_PBC}), it is easy to count that the total number of the elements of all tensors is $Nd\chi^2$ which increases only linearly with $N$. The above way of obtaining MPS with decompositions is also known as tensor train decomposition (TTD)\index{TTD} in MLA\index{MLA}, and MPS is also called tensor-train form \cite{O11TTD}. The main aim of TTD is investigating the algorithms to obtain the optimal tensor train form of a given tensor, so that the number of parameter can be reduced with well-controlled errors.
In physics, the above procedure shows that any states can be written in an MPS\index{MPS}, as long as we do not limit the dimensions of the geometrical indexes. However, it is extremely impractical and inefficient, since in principle, the dimensions of the geometrical indexes $\{a\}$ increase exponentially with $N$. In the following sections, we will directly applying the mathematic form of the MPS without considering the above procedure.
\begin{figure}[tbp]
\centering
\includegraphics[angle=0,width=0.9\linewidth]{MPSfinite.eps}
\caption{(Color online) The graphic representations of the matrix product states with open (left) and periodic (right) boundary conditions.}
\label{fig-1MPSfinite}
\end{figure}
Now we introduce a simplified notation of MPS that has been widely used in the community of physics. In fact with fixed physical indexes, the contractions of geometrical indexes are just the inner products of matrices (this is how its name comes from). In this sense, we write a quantum state given by Eq. (\ref{eq-1MPS}) as
\begin{eqnarray}
|\psi \rangle =tTr A^{[1]} A^{[2]} \cdots A^{[N]} |s_1s_2 \cdots s_N \rangle = tTr \prod_{n=1}^{N} A^{[n]} |s_n \rangle.
\label{eq-1MPSsimple}
\end{eqnarray}
$tTr$ stands for summing over all shared indexes. The advantage of Eq. (\ref{eq-1MPSsimple}) is to give a general formula for an MPS of either finite or infinite size, with either periodic or open boundary condition.
\begin{figure}[tbp]
\centering
\includegraphics[angle=0,width=0.45\linewidth]{SparseAF.eps}
\caption{(Color online) One possible configuration of the sparse anti-ferromagnetic ordered state. A dot represents the $S=0$ state. Without looking at all the $S=0$ states, the spins are arranged in the anti-ferromagnetic way.}
\label{fig-1SparseAF}
\end{figure}
\subsection{Afflect-Kennedy-Lieb-Tasaki state}
MPS\index{MPS} is not just a mathematic form. It can represent nontrivial physical states. One important example can be found with AKLT\index{AKLT} model proposed in 1987, a generalization of spin-$1$ Heisenberg model \cite{AKLT87AKLTState}. For 1D systems, Mermin-Wagner theorem forbids any spontaneously breaking of continuous symmetries at finite temperature with sufficiently short-range interactions. For the ground state of AKLT model called AKLT state, it possesses the \textit{sparse anti-ferromagnetic order} (Fig. \ref{fig-1SparseAF}), which provides a non-zero excitation gap under the framework of Mermin-Wagner theorem. Moreover, AKTL state provides us a precious exactly-solvable example to understand edge states and (symmetry-protected) topological orders.
AKLT state can be exactly written in an MPS with $\chi=2$ (see \cite{PVWC07MPSRev} for example). Without losing generality, we assume periodic boundary condition. Let us begin with the AKLT Hamiltonian that can be given by spin-1 operators as
\begin{eqnarray}
\hat{H}=\sum_n\left[\frac{1}{2} \hat{S}_n\cdot \hat{S}_{n+1}+\frac{1}{6} (\hat{S}_n\cdot \hat{S}_{n+1})^2+\frac{1}{3}\right].
\label{eq-1AKLTH}
\end{eqnarray}
By introducing the non-negative-defined projector $\hat{P}_2(\hat{S}_n+\hat{S}_{n+1})$ that projects the neighboring spins to the subspace of $S=2$, Eq. (\ref{eq-1AKLTH}) can be rewritten in the summation of projectors as
\begin{eqnarray}
\hat{H}=\sum_n \hat{P}_2(\hat{S}_n+\hat{S}_{n+1}).
\end{eqnarray}
Thus, the AKLT Hamiltonian is non-negative-defined, and its ground state lies in its kernel space, satisfying $\hat{H}|\psi_{AKLT}\rangle = 0$ with a zero energy.
\begin{figure}[tbp]
\centering
\includegraphics[angle=0,width=0.75\linewidth]{AKLT.eps}
\caption{(Color online) An intuitive graphic representation of the AKLT\index{AKLT} state. The big circles representing $S=1$ spins, and the small ones are effective $S=\frac{1}{2}$ spins. Each pair of spin-$1/2$ connecting by a red bond forms a singlet state. The two ``free'' spin-$1/2$ on the boundary give the edge state.}
\label{fig-1AKLT}
\end{figure}
Now we construct a wave-function which has a zero energy. As shown in Fig. \ref{fig-1AKLT}, we put on each site a projector that maps two (effective) spins-$1/2$ to a \textit{triplet}, i.e. the physical spin-1, where the transformation of the basis obeys
\begin{eqnarray}
|+\rangle&=&|00\rangle \\
|\tilde{0}\rangle&=&\frac{1}{\sqrt{2}}(|01\rangle+|10\rangle),\\
|-\rangle&=&|11\rangle.
\end{eqnarray}
The corresponding projector is determined by the Clebsch-Gordan coefficients \cite{S63CG}, and is a ($3 \times 4$) matrix. Here, we rewrite it as a ($3 \times 2 \times 2$) tensor, whose three components (regarding to the first index) are the ascending, $z$-component and descending Pauli matrices of spin-$1/2$ \footnote{Here, one has some degrees of freedom to choose different projectors, which is only up to a gauge transformation. But once one projector is fixed, the other is also fixed.},
\begin{eqnarray}
\sigma^+ = \left[ {\begin{array}{*{30}c}
0 \hspace{0.4cm} 1 \\
0 \hspace{0.4cm} 0 \\
\end{array}} \right], \hspace{0.6cm}
\sigma^{z} = \left[ {\begin{array}{*{30}c}
1 \hspace{0.4cm} 0 \\
0 \hspace{0.08cm} -1 \\
\end{array}} \right], \hspace{0.6cm}
\sigma^- = \left[ {\begin{array}{*{30}c}
0 \hspace{0.4cm} 0 \\
1 \hspace{0.4cm} 0 \\
\end{array}} \right]. \hspace{0.6cm}
\label{eq-1sigma}
\end{eqnarray}
In the language of MPS\index{MPS}, we have the tensor $A$ satisfying
\begin{eqnarray}
A_{0,a a'} = \sigma^+_{aa'}, \hspace{0.4cm} A_{1,a a'} = \sigma^z_{aa'}, \hspace{0.4cm} A_{2,a a'} = \sigma^-_{aa'}.
\label{eq-1AKLT-A}
\end{eqnarray}
Then we put another projector to map two spin-$1/2$ to a singlet, i.e. a spin-0 with
\begin{eqnarray}
|\bar{0}\rangle = \frac{1}{\sqrt{2}}(|01\rangle-|10\rangle)
\end{eqnarray}
The projector is in fact a ($2 \times 2$) identity with the choice of Eq. (\ref{eq-1sigma}),
\begin{eqnarray}
I = \left[ {\begin{array}{*{30}c}
1 \hspace{0.4cm} 0 \\
0 \hspace{0.4cm} 1 \\
\end{array}} \right].
\label{eq-1Identity22}
\end{eqnarray}
Now, the MPS of the AKLT state with periodic boundary condition (up to a normalization factor) is obtained by Eq. (\ref{eq-1MPS_PBC}), with every tensor $A$ given by Eq. (\ref{eq-1AKLT-A}). For such an MPS, every projector operator $\hat{P}_2(\hat{S}_n+\hat{S}_{n+1})$ in the AKLT Hamiltonian is always acted on a singlet, then we have $\hat{H}|\psi_{AKLT}\rangle = 0$.
\subsection{Tree tensor network state (TTNS)\index{TTNS} and projected entangled pair state (PEPS)\index{PEPS}}
\label{sec.peps}
\begin{figure}[tbp]
\centering
\includegraphics[angle=0,width=\linewidth]{TTNS.eps}
\caption{(Color online) The illustration of (a) and (b) two different TTNS's and (c) MERA\index{MERA}.}
\label{fig-1TTNS}
\end{figure}
TTNS is a generalization of the MPS\index{MPS} that can code more general entanglement states. Unlike an MPS where the tensors are aligned in a 1D array, a TTNS is given by a tree graph. Figs. \ref{fig-1TTNS} (a) and (b) show two examples of TTNS with the coordination number $z=3$. The red bonds are the physical indexes and the black bonds are the geometrical indexes connecting two adjacent tensors. The physical ones may locate on each tensor or put on the boundary of the tree. A tree is a graph that has no loops, which leads to many simple mathematical properties that parallel to those of an MPS. For example, the partition function of a TTNS can be efficiently exactly computed. A similar but more power TN state called MERA\index{MERA} also has such a property [Figs. \ref{fig-1TTNS} (c)]. We will get back to this in Sec. \ref{Sec-exactTN}. Note an MPS can be treated as a tree with $z = 2$.
An important generalization to the TN's of loopy structures is known as projected entangled pair state (PEPS), proposed by Verstraete and Cirac \cite{VC04PEPS, VC06PEPSArxiv}. The tensors of a PEPS\index{PEPS} are located in, instead of a 1D chain or a tree graph, a d-dimensional lattice, thus graphically forming a $d$-dimensional TN\index{TN}. An intuitive picture of PEPS is given in Fig. \ref{fig-1PEPSgraph}, i.e., the tensors can be understood as projectors that map the physical spins into virtual ones. The virtual spins form the maximally entangled state in a way determined by the geometry of the TN. Note that such an intuitive picture was firstly proposed with PEPS \cite{VC04PEPS}, but it also applies to TTNS.
Similar to MPS\index{MPS}, a TTNS or PEPS can be formally written as
\begin{eqnarray}
|\Psi\rangle = tTr \prod_{n} P^{[n]} |s_n\rangle,
\label{eq-2PEPSsimple}
\end{eqnarray}
where $tTr$ means to sum over all geometrical indexes. Usually, we do not write the formula of a TTNS or PEPS, but give the graph instead to clearly show the contraction relations.
\begin{figure}[tbp]
\centering
\includegraphics[angle=0,width=0.8\linewidth]{Fig7.eps}
\caption{(Color online) (a) An intuitive picture of the projected entangled pair state. The physical spins (big circles) are projected to the virtual ones (small circles), which form the maximally entangled states (red bonds). (b)-(d) Three kinds of frequently used PEPS's.}
\label{fig-1PEPSgraph}
\end{figure}
Such a generalization makes a lot of senses in physics. One key factor regards the \textit{area law of entanglement entropy} \cite{B73EntAreaLaw,S93Sarea,LRV04GSEnt,CC04EntAreaLaw,PEDC05EntAreaLaw,ECP10AreaLawRev} which we will talk about later in this chapter. In the following as two straightforward examples, we show that PEPS can indeed represents non-trivial physical states including nearest-neighbor \textit{resonating valence bond} (RVB)\index{RVB} and \textit{Z$_2$ spin liquid} states. Note these two types of states on trees can be similarly defined by the corresponding TTNS.
\subsection{PEPS can represent non-trivial many-body states: examples}
RVB state was firstly proposed by Anderson to explain the possible disordered ground state of the Heisenberg model on triangular lattice \cite{A73RVB,A74RVB}. RVB state is defined as the superposition of macroscopic configurations where all spins are paired to form the singlet states (dimers). The strong fluctuations are expected to restore all symmetries and lead to a spin liquid state without any local orders. The distance between two spins in a dimer can be short range or long range. For nearest-neighbor RVB, the dimers are only be nearest neighbors (Fig. \ref{fig-1NNRVB}, also see \cite{B10QSLRev}). RVB state is supposed to relate to high-$T_c$ copper-oxide-based superconductor. By doping the singlet pairs, the insulating RVB state can translate to a charged superconductive state \cite{A87RVB,BZA87RVBSC,ABZH87RVBSC}.
\begin{figure}[tbp]
\centering
\includegraphics[angle=0,width=1\linewidth]{NNRVB.eps}
\caption{(Color online) The nearest-neighbor RVB\index{RVB} state is the superposition of all possible configurations of nearest-neighbor singlets.}
\label{fig-1NNRVB}
\end{figure}
For the nearest-neighbor situation, an RVB state (defined on an infinite square lattice, for example) can be exactly written in a PEPS of $\chi=3$. Without losing generality, we take the translational invariance, i.e., the TN\index{TN} is formed by infinite copies of several inequivalent tensors. Two different ways have been proposed to construct the nearest-neighbor RVB\index{RVB} PEPS\index{PEPS} \cite{VWPC06PEPSfamous, SPCP12PEPSRVB}. In addition, Wang \textit{et al} proposed a way to construct the PEPS with long-range dimers \cite{WPGWV13PEPSRVB}. In the following, we explain the way proposed by Verstraete \textit{et al} to construct the nearest-neighbor one \cite{VWPC06PEPSfamous}. There are two inequivalent tensors: the tensor defined on each site whose dimensions are ($2 \times 3 \times 3 \times 3 \times 3$) only has five nonzero elements,
\begin{eqnarray}
P_{0,0000}=1, \ \ P_{1,2111}=1, \ \ P_{1,1211}=1, \ \ P_{1,1121}=1, \ \ P_{1,1112}=1.
\end{eqnarray}
We use the language of strings to understand physical meaning of the projector: the spin-up state (with the physical index $s=0$) stands for the vacuum state and the spin-down ($s=1$) for the occupied state of a string. In this sense, the first element $P_{0,0000}$ means it is vacuum in the physical space, thus all the geometrical spaces are vacuum. For the rest four elements, the physical space is occupied by a string that is mapped to one of the geometrical space with the same amplitude, leaving the rest three to be vacuum. For example, $P_{1,1211}=1$ means one possibility, where the physical string is mapped to the second geometrical space while the rest three remain vacuum \footnote{Note that for a geometrical space, $0$ and $1$ are to distinguish the vacuum states with vacuum and occupied physical states, respectively.}. The rest elements are all zero, which means the corresponding configurations are forbidden.
The tensor $P$ only maps physical strings to geometrical spaces. Then a projector $B$ is put on each geometrical bond to form the singlets in the RVB picture. $B$ is a ($3 \times 3$) matrix with only three nonzero elements
\begin{eqnarray}
B_{00}=1, \ \ B_{12}=1, \ \ B_{21}=-1.
\end{eqnarray}
Similarly, the first one means a vacuum state on this geometrical bond, and the rest two simply give a singlet state ($|12\rangle - |21\rangle$).
Then the infinite PEPS (iPEPS)\index{iPEPS} of the nearest-neighbor RVB\index{RVB} is given by the contraction of infinite copies of $P$'s on the sites and $B$'s (Fig.\ref{fig-1PEPSgraph}) on the bonds as
\begin{eqnarray}
|\Psi\rangle = \sum_{\{s,a\}} \prod_{n \in sites} P_{s_n, a_n^1 a_n^2 a_n^3 a_n^4} \prod_{m \in bonds} B_{a^1_m a^2_m } \prod_{j \in sites} |s_j\rangle.
\label{eq-2RVBPEPS}
\end{eqnarray}
After the contraction of all geometrical indexes, the state is the super-position of all possible configurations consisting of nearest-neighbor dimers. This iPEPS looks different from the one given in Eq. (\ref{eq-2PEPSsimple}) but they are essentially the same, because one can contract the $B$'s into $P$'s so that the PEPS is only formed by tensors defined on the sites.
Another example is the Z$_2$ spin liquid state, which is one of simplest string-net states \cite{GLSW09StringTPS,BAV09StringTPS,CZGCW10StringTPS}, firstly proposed by Levin and Wen to characterize gapped topological orders \cite{LW05TopoOrder}. Similarly with the picture of strings, the Z$_2$ state is the super-position of all configurations of string loops. Writing such a state with TN\index{TN}, the tensor on each vertex is ($2 \times 2 \times 2 \times 2$) satisfying
\begin{eqnarray}
P_{a_1 \cdots a_N}&=&
\left\{
\begin{array}{lll}
1, \ \ a_1+\cdots + a_N=even, \\
0, \ \ otherwise.
\end{array}
\right.
\end{eqnarray}
The tensor $P$ forces the \textit{fusion rules} of the strings: the number of the strings connecting to a vertex must be even, so that there are no loose ends and all strings have to form loops. It is also called in some literatures the \textit{ice rule} \cite{BG01SpinIce,BF1933IceRule} or \textit{Gauss' law} \cite{FDWSPB+09SpinIce}. In addition, the square TN formed solely by the tensor $P$ gives the famous \textit{eight-vertex model}, where the number ``eight'' corresponds to the eight non-zero elements (i.e. allowed sting configurations) on a vertex \cite{B71EightVertex}.
The tensors $B$ are defined on each bond to project the strings to spins, whose non-zero elements are
\begin{eqnarray}
B_{0,00}=1, \ \ B_{1,11}=1.
\end{eqnarray}
The tensor $B$ is a projector that maps the spin-up (spin-down) state to the occupied (vacuum) state of a string.
\subsection{Tensor network operators}
\label{sec-MPO}
The MPS\index{MPS} or PEPS\index{PEPS} can be readily generalized from representations of states to those of operators called MPO\index{MPO} \cite{VGC04MPDO,ZV04MPO, PMCV10MPO, LRGZXY+11LTRG, BKTMWH17FT1D, GIK17FT1D, HV17MPO, CPSV17MPO} or projected entangled pair operator (PEPO)\index{PEPO} \footnote{Generally, a representation of an operator with a TN can be called tensor product operator (TPO)\index{TPO}. MPO and PEPO are two examples.} \cite{CCD12FTPEPS,RXLS13NCD,RLXZS12ODTNS,FND10PEPO,O12PEPO,CD15TPO,CDO16TPO,CRD16TPO,DSCBZ17FT2D,CDO17TPOQMC}. Let us begin with MPO, which is also formed by the contraction of local tensors as
\begin{eqnarray}
\hat{O} = \sum_{\{s,a\}} \prod_n W_{s_n s_n', a_n a_{n+1}}^{[n]}|s_n\rangle \langle s_n'|.
\label{eq-2MPO}
\end{eqnarray}
\begin{figure}
\begin{center}
\includegraphics[width=0.6\textwidth]{Fig12.eps}
\end{center}
\caption{(Color online) The graphic representation of a matrix product operator, where the upward and downward indexes represent the \textit{bra} and \textit{ket} space, respectively.}
\label{fig-2MPO}
\end{figure}
Different from MPS\index{MPS}, each tensor has two physical indexes, of which one is a \textit{bra} and the other is a \textit{ket} index (Fig. \ref{fig-2TPO}). An MPO may represent several non-trivial physical models, for example the Hamiltonian. Crosswhite and Bacon \cite{CB08automata} proposed a general way of constructing an MPO from called \textit{automata}. Now we show how to construct the MPO of an Hamiltonian using the properties of a \textit{triangular MPO}. Let us start from a general lower-triangular MPO satisfying $W^{[n]}_{::,00}=C^{[n]}$, $W^{[n]}_{::,01}=B^{[n]}$, and $W^{[n]}_{::,11}=A^{[n]}$ with $A^{[n]}$, $B^{[n]}$, and $C^{[n]}$ some $d \times d$ square matrices. We can write $W^{[n]}$ in a more explicit $2 \times 2$ block-wise form as
\begin{eqnarray}
W^{[n]}=
\begin{pmatrix}
C^{[n]} & 0 \\
B^{[n]} & A^{[n]}
\end{pmatrix}
\end{eqnarray}
If one puts such a $W^{[n]}$ in Eq. (\ref{eq-2MPO}), it will give the summation of all terms in the form of
\begin{eqnarray}
O &=& \sum_{n=1}^N A^{[1]} \otimes \cdots \otimes A^{[n-1]} \otimes B^{[n]} \otimes C^{[n+1]} \otimes \cdots \otimes C^{[N]} \nonumber \\
&=& \sum_{n=1}^N \prod_{\otimes i=1}^{n-1} A^{[i]} \otimes B^{[n]} \otimes \prod_{\otimes j=n+1}^{N} C^{[j]},
\label{eq-2UpTriMPO}
\end{eqnarray}
with $N$ the total number of tensors and $\prod_{\otimes}$ the tensor product \footnote{For $n=0$, $A^{[0]}$ (or $B^{[0]}$, $C^{[0]}$) does not exist but can be defined as a scalar 1, for simplicity of the formula.}. Such a property can be easily generalized to a $W$ formed by $D \times D$ blocks.
Imposing Eq. (\ref{eq-2UpTriMPO}), we can construct as an example the summation of one-site local terms, i.e., $\sum_n X^{[n]}$ \footnote{Note that $X^{[n_1]}$ and $X^{[n_2]}$ are not defined in a same space with $n_1 \neq n_2$, Thus, precisely speaking, $\sum$ here is the direct sum. We will not specify this when it causes no confuse}, with
\begin{eqnarray}
W^{[n]}=
\begin{pmatrix}
I & 0 \\
X^{[n]} & I
\end{pmatrix},
\end{eqnarray}
with $X^{[n]}$ a $d \times d$ matrix and $I$ the $d \times d$ identity.
If two-body terms are included, such as $\sum_m X^{[m]} + \sum_n Y^{[n]} Z^{[n+1]}$, we have
\begin{eqnarray}
W^{[n]}=
\begin{pmatrix}
I & 0 & 0 \\
Z^{[n]} & 0 & 0 \\
X^{[n]} & Y^{[n]} & I
\end{pmatrix}.
\end{eqnarray}
This can be obviously generalized to $L$-body terms. With open boundary conditions, the left and right tensors are
\begin{eqnarray}
W^{[1]}=
\begin{pmatrix}
I & 0 & 0
\end{pmatrix}, \ \ \\
W^{[N]}=
\begin{pmatrix}
0 \\ 0 \\ I
\end{pmatrix}.
\end{eqnarray}
Now we apply the above technique on a Hamiltonian of, e.g., the Ising model in a transverse field
\begin{eqnarray}
\hat{H} = \sum_n \hat{S}^{z}_n \hat{S}^{z}_{n+1} + h \sum_m \hat{S}^{x}_m.
\end{eqnarray}
Its MPO is given by
\begin{eqnarray}
W^{[n]}=
\begin{pmatrix}
I & 0 & 0 \\
\hat{S}^z & 0 & 0 \\
h \hat{S}^x & \hat{S}^z & I
\end{pmatrix}.
\end{eqnarray}
Such a way of constructing an MPO\index{MPO} is very useful. Another example is the Fourier transformation to the number operator of Hubbard model in momentum space $\hat{n}_k = \hat{b}_k^{\dagger} \hat{b}_{k}$. The Fourier transformation is written as
\begin{eqnarray}
\hat{n}_k = \sum_{m,n=1}^{N} e^{i(m-n)k} \hat{b}_m^{\dagger} \hat{b}_{n},
\end{eqnarray}
with $\hat{b}_n$ ($\hat{b}_n^{\dagger}$) the annihilation (creation) operator on the $n$-th site. The MPO representation of such a Fourier transformation is given by
\begin{eqnarray}
\hat{W}_n = \begin{pmatrix}
\hat{I} & 0 & 0 & 0 \\
\hat{b}^{\dagger} & e^{ik} \hat{I} & 0 & 0 \\
\hat{b} & 0 & e^{-ik} \hat{I} & 0 \\
\hat{b}^{\dagger} \hat{b} & e^{+ik} \hat{b}^{\dagger} & e^{-ik} \hat{b} & \hat{I}
\end{pmatrix}
\end{eqnarray}
with $\hat{I}$ the identical operator in the corresponding Hilbert space.
The MPO formulation also allows for a convenient and efficient representation of the Hamiltonians with longer range interactions \cite{CDV08MPOLR}. The geometrical bond dimensions will in principle increase with the interaction length. Surprisingly, a small dimension is needed to approximate the Hamiltonian with long-range interactions that decay polynomially \cite{FND10PEPO}.
Besides, MPO can be used to represent the time evolution operator $\hat{U}(\tau) = e^{-\tau \hat{H}}$ with \textit{Trotter-Suzuki decomposition}, where $\tau$ is a small positive number called \textit{Trotter-Suzuki step} \cite{V04TEBD,V07iTEBD}. Such an MPO is very useful in calculating real, imaginary, or even complex time evolutions, which we will present later in detail. An MPO can also give a mixed state.
Similarly, PEPS\index{PEPS} can also be generalized to projected entangled pair operator (PEPO\index{PEPO}, Fig. \ref{fig-2TPO}), which on a square lattice for instance can be written as
\begin{eqnarray}
\hat{O} = \sum_{\{s,a\}} \prod_n W_{s_n s_n', a_n^1 a_n^2 a_n^3 a_n^4}^{[n]} |s_n\rangle \langle s_n'|.
\label{eq-2PEPO}
\end{eqnarray}
Each tensor has two physical indexes (\textit{bra} and \textit{ket}) and four geometrical indexes. Each geometrical bond is shared by two adjacent tensors and will be contracted.
\begin{figure}
\begin{center}
\includegraphics[width=0.5\textwidth]{Fig10.eps}
\end{center}
\caption{(Color online) The graphic representation of a projected entangled pair operator, where the upward and downward indexes represent the \textit{bra} and \textit{ket} space, respectively.}
\label{fig-2TPO}
\end{figure}
\subsection{Tensor network for quantum circuits}
A special case of TN\index{TN} are quantum circuits \cite{MS08TNQcomp}.
Quantum circuits encode computations made on qubits (or qudits in
general). Fig. \ref{fig:quantum circuit} demonstrates
the TN representation of a quantum circuit made by unitary gates that act on a product state of many constituents initialized as $\prod_{\otimes} |0\rangle$.
\textbf{An example of quantum circuits}. In order
to make contact with TN, we will consider the specific
case of quantum circuits where all the gates act on at most two neighbors.
An example of such circuit is the Trotterized evolution of a system
described by a nearest-neighbor Hamiltonian $\hat{H}=\sum_{i,i+1} \hat{h}_{i,i+1}$,
where $i,i+1$ label the neighboring constituents of a one-dimensional
system. The evolution operator for a time $t$ is $\hat{U}(t)=exp(-i\hat{H}t)$, and
can be decomposed into a sequence of infinitesimal time evolution
steps \cite{V03TEBD} (more details will be given in Sec. \ref{sec.2.evolution})
\begin{equation}
\hat{U}(t)=lim_{N\to\text{\ensuremath{\infty}}} \exp(-i\frac{t}{N}\hat{H})^{N}.
\end{equation}
In the limit, we can decompose the evolution into a product of two-body evolution
\begin{equation}
\hat{U}(t)=lim_{\tau \to 0}\prod_{i,i+1} \hat{U}(\tau)_{i,i+1}.
\end{equation}
where $\hat{U}_{i,i+1}(\tau)=\exp(-i\tau \hat{h}_{i,i+1})$ and $\tau = t/N$. This is obviously
a quantum circuit made by two-qubit gates with depth $N$. Conversely,
any quantum circuit naturally possesses an arrow of time; it transforms a product state into an entangled state after
a sequence of two-body gates.
\textbf{Casual cone}. One interesting concept in a quantum circuit is that of the causal
cone illustrated in Fig. \ref{fig:past-casual}, which becomes explicit with the TN\index{TN} representations. Given a quantum circuit
that prepares (i.e., evolves the initial state to) the state $|\psi\rangle$, we can ask a question: which subset of the
gates affect the reduced density matrix of a certain sub-region $A$ of $|\psi\rangle$? This can be seen by constructing the reduced
density matrix of the sub-region $A$ $\rho_{A}=tr_{\bar{A}}|\psi\rangle\langle\psi|$ with $\bar{A}$ the rest part of the system besides $A$.
The TN of the reduced density matrix is formed by a set of unitaries that define the past causal cone of the region $A$ (see the area between the green lines in Fig. \ref{fig:past-casual}). The rest unitaries (for instance the $\hat{U}_5$ and its conjugate in the right sub-figure of Fig. \ref{fig:past-casual}) will be eliminated in the TN of the reduced density matrix. The contraction of
the causal cone can thus be rephrased in terms of the multiplication
of a set of transfer matrices, each performing the computation from
$t$ to $t-1$. The maximal width of these transfer matrices defines the width
of the causal cone, which can be used as a good measure of the complexity
of computing $\rho_{A}$ \cite{jozsarichard2003}.
The best computational strategy one can find to compute exactly $\rho_{A}$
will indeed always scale exponentially with the width of the cone \cite{MS08TNQcomp}.
\begin{figure}[t]
\begin{center}
\includegraphics[scale=0.55]{ET_Fig1.eps}
\end{center}
\caption{The TN \index{TN} representation of a quantum circuit. Two-body unitaries act on a product state of a given number of constituents $|0\rangle\otimes\cdots\otimes|0\rangle$ and transform it into a target entangled state $|\psi\rangle$.\label{fig:quantum circuit}}
\end{figure}
\begin{figure}[t]
\includegraphics[scale=0.37]{ET_Fig2}
\caption{The past casual cone of the red site ($A$). The unitary gate $U_{5}$ does not affect the reduced density matrix of the red site. This is verified by computing explicitly $\rho_{A}$ by tracing over all the others constituents. In the TN\index{TN} of $\rho_{A}$, $U_{5}$ is contracted with $U_{5}^{\dagger}$, which gives an identity. \label{fig:past-casual}}
\end{figure}
\textbf{Unitary tensor networks and quantum circuits.} The simplest TN\index{TN}, the MP\index{MPS} can be interpreted as a sequential quantum
circuit \cite{schon2005}. The idea is that one can think of the MPS
as a sequential interaction between each constituent (a $d$- level
system) an ancillary $D$-level system (the auxiliary qDit, red bonds). The first
constituent interacts (say the bottom one shown in Fig. \ref{fig:MPS-as-quantum-circuit})
and then sequentially all the constituents interact with the same
$D-$level system. With this choice, the past causal cone of a constituent
is made by all the MPS matrices below it. Interestingly in the MPS
case, the causal cone can be changed using the gauge transformations (see sec. \ref{sec.gauge}),
something very different to what happens in two dimensional TN's\index{TN}. This amounts to finding appropriate unitary transformations
acting on the auxiliary degrees of freedom that allow to re-order the interactions
between the $D-$level system and the constituents. In such a way, a desired constituent can be made to interact first, then followed
by the others. An example of the causal cone in the center gauge used
in iDMRG calculation \cite{M08iDMRGarxiv} is presented in Fig. \ref{fig:center-gauge}.
This idea allows to minimize the number of tensors in the
causal cone of a given region. However, the scaling of
the computational cost of the contraction is not affected by such a
temporal reordering of the TN, since in this case the width of
the cone is bounded by one unitary in any gauge. The gauge choice
just changes the number of computational steps required to construct
the desired $\rho_{A}$. In the case that $A$ includes non-consecutive
constituents, the width of the cone increases linearly with the number
of constituents, and the complexity of computing $\rho_{A}$ increases
exponentially with the number of constituents.
\begin{figure}[t]
\includegraphics[scale=0.55]{ET_Fig3}
\caption{The MPS\index{MPS} as a quantum circuit. Time flows from right to left so that the lowest constituent is the first to interact with the auxiliary $D$-level system. Here we show the past causal cone of a single constituent. Similarly, the past causal cone of $A$ made by adjacent constituent has the same form starting from the upper boundary of $A$.
\label{fig:MPS-as-quantum-circuit}}
\end{figure}
Again, the gauge degrees of freedom can be used to modify the structure of the past
causal cone of a certain spin. As an example, the iDMRG center gauge
is represented in Fig. \ref{fig:center-gauge}.
\begin{figure}[t]
\begin{center}
\includegraphics[scale=0.55]{ET_Fig4}
\end{center}
\caption{Using the gauge degrees of freedom of an MPS, we can modify its past causal cone structure to make its region as small as possible, in such a way decreasing the computational complexity of the actual computation of specific $\rho_{A}$.
A convenient choice is the center gauge used in
iDMRG\index{iDMRG} \label{fig:center-gauge}}
\end{figure}
An example of a TN with a larger past causal cone can be obtained
by using more than one layers of interactions. Now the support of the causal
cone becomes larger since it includes transfer matrices acting on
two $D$-level systems (red bonds shown in Fig. \ref{fig:larger-causal-cone}). Notice that this TN has loops but it still exactly contractible since
the width of the causal cone is still finite.
\begin{figure}[t]
\begin{center}
\includegraphics[scale=0.55]{ET_Fig5}
\end{center}
\caption{The width of the causal cone increases as we increase the depth of the quantum circuit generating the MPS state.\label{fig:larger-causal-cone}}
\end{figure}
\section{Tensor networks that can be contracted exactly}
\subsection{Definition of exactly contractible tensor network states}
The notion of the past causal cone can be used to classify TNS's\index{TNS} based on
the complexity of computing their contractions. It is important to remember that
the complexity strongly depends on the object that we want to compute, not just the TN. For example, the complexity of an MPS\index{MPS} for a $N$-qubit state scales only linearly with $N$. However, to compute the $n$-site reduced density matrix, the cost scales exponentially with $n$ since the matrix itself is an exponentially large object. Here we consider to compute scalar quantities, such as the observables of one- and two-site operators.
We define the a TNS\index{TNS} to be \emph{exactly contractible} when it is allowed to compute their contractions with a cost that is a polynomial
to the elementary tensor dimensions $D$. A more rigorous definition can be given in terms of their tree
width see e.g. \cite{MS08TNQcomp}. From the discussion of the previous
section, it is clear that such a TNS corresponds to a bounded causal
cone for the reduced density matrix of a local sub-region. In order to show this, we now focus on the cost of computing
the expectation value of local operators and their correlation functions on a few examples of TNS's\index{TNS}.
\begin{figure}[t]
\begin{center}
\includegraphics[scale=0.55]{ET_Fig6_1}
\end{center}
\caption{The MPS wave-function representation in left canonical form.}
\label{fig:wf_mps}
\end{figure}
The relevant objects are thus the reduced density matrix of a region
$A$ made of a few consecutive spins, and the reduced density matrix
of two disjoint blocks $A_{1}$ and $A_{2}$ of which each made of a few consecutive spins. Once we have the reduced density matrices of such regions, we can compute arbitrary expectation values of local operators by $\langle{\cal {\cal O}\rangle=}tr(\rho_{A}{\cal O})$ and $\langle{\cal {\cal O}}_{A_{1}}{\cal O}'_{A_{2}}\rangle=tr(\rho_{A_{1}\cup A_{2}}{\cal O}_{A_{1}}{\cal O}'_{A_{2}})$
with ${\cal O}_{A}$, ${\cal {\cal O}}_{A_{1}}$, ${\cal O}'_{A_{2}}$ arbitrary operators defined on the regions $A$, $A_{1}$, $A_{2}$.
\subsection{MPS\index{MPS} wave-functions}
The simplest example of the computation of the expectation value of
a local operator is obtained by considering MPS wave-functions \cite{FNW92MPS,PVWC07MPSRev}.
Fig. \ref{fig:wf_mps} shows an MPS in the left-canonical form (see \ref{sec5.canon} for more details).
Rather than putting the arrows of time, here we put the direction
in which the tensors in the TN are isometric. In other words, an identity is obtained by contracting the inward bonds of a tensor in $|\psi \rangle$ with the outward bonds of its conjugate in $\langle \psi |$ (Fig. \ref{fig:obs_mps}). Note that $|\psi \rangle$ and $\langle \psi |$ have opposite arrows, by definition. These arrows are directly on the legs of the tensors. The arrows in $| \psi \rangle$ are in the opposite direction than the time, by comparing Fig. \ref{fig:MPS-as-quantum-circuit} with Fig. \ref{fig:obs_mps}. The two figures indeed represent the MPS in the same gauge. This means that the causal cone of an observable is on the right of that observable, as shown on the second line of Fig. \ref{fig:obs_mps}, where all the tensors on the left side are annihilated as a consequence of the isometric constraints. We immediately have that the causal cone
has at most the width of two. The contraction becomes a power of the transfer operator of the MPS $E=\sum_{i}A^{i}\otimes A^{i\dagger}$, where $A_{i}$ and $A^{i\dagger}$ represent the MPS tensors and its complex conjugate. The MPS transfer matrix $E$ only acts on two auxiliary degrees of freedom. Using the property that $E$ is a completely positive map and thus has a fixed point \cite{PVWC07MPSRev}, we can substitute the transfer operator by its largest eigenvector $v$, leading to the final TN\index{TN} diagram that encodes the expectation value of a local operator.
\begin{figure}[t]
\includegraphics[scale=0.55]{ET_Fig6_2}
\caption{The expectation value of a single-site operator with an MPS\index{MPS} wave-function.
\label{fig:obs_mps}}
\end{figure}
In Fig. \ref{fig:two-point-MPS}, we show the TN\index{TN} representation of the expectation value
of the two-point correlation functions. Obviously, the past-causal cone width is bounded
by two auxiliary sites. Note that in the second line, the directions of the arrows on the right side are changed. This in general does not happen in more complicated
TN's as we will see in the next subsection. Before going
there, we would like to comment the properties of the two-point correlation
functions of MPS. From the calculation we have just performed, we see
that they are encoded in powers of the transfer matrix that evolve
the system in the real space. If that matrix can be diagonalized, we can immediately
see that the correlation functions naturally decay exponentially with the ratio of the first to the second eigenvalue. Related details can be found in Sec. \ref{sec5.extract}.
\begin{figure}[t]
\includegraphics[scale=0.6]{ET_Fig7}
\caption{Two-point correlation function of an MPS wave-function. \label{fig:two-point-MPS}}
\end{figure}
\subsection{Tree tensor network wave-functions}
An alternative kind of wave-functions are the TTNS's\index{TTNS} \cite{SDV06TTN,TEV09TTN,alba2010,alba2011,calabrese2013,ferris2012,gliozzi2009}. In a TTNS, one can add the physical bond on each of the tensor, and use it as a many-body state defined on a Caley-tree lattice \cite{SDV06TTN}. Here, we we will focus on the TTNS with physical bonds only on the outer leafs of the tree.
The calculations with a TTNS normally correspond to the contraction of tree TN's.
A specific case
of a two-to-one TTNS is illustrated in Fig. \ref{fig:tree-tensor}, named binary Caley tree. This TN can be interpreted
as a quantum state of multiple spins with different boundary
conditions. It can also be considered as a hierarchical TN, in which
each layer corresponds to a different level of coarse-graining renormalization group (RG)\index{RG} transformation \cite{TEV09TTN}.
In the figure, different layers are colored differently. In the first layer,
each tensor groups two spins into one and so on. The tree TN can thus be interpreted
a specific RG transformation. Once more, the arrows on the tensors indicate
the isometric property of each individual tensor that; the directions are opposite as the time, if we interpret the tree TN as a quantum circuit. Note again that $|\psi \rangle$ and $\langle \psi |$ have opposite arrows, by definition.
The expectation value of a one-site operator is in fact a tree TN\index{TN} shown in Fig. \ref{fig:one-site-ttn}. We see that many of the tensors are completely
contracted with their Hermitian conjugates, which simply give identities. What are left is again a bounded causal cone. If we now build an infinite TTNS\index{TTNS} made by infinitely many layers, and
assume the scale invariance, the multiplication of infinitely many power
of the scale transfer matrix can be substituted with the corresponding
fixed point, leading to a very simple expression for the TN that encodes the expectation value of a single site operator.
\begin{figure}[t]
\begin{center} \label{fig:tree-tensor}
\includegraphics[scale=0.4]{ET_Fig8}
\end{center}
\caption{A binary TTNS made of several layers of third-order tensors. Different layers are identified with different colors. The arrows flow in the opposite direction of the time while being interpreted as a quantum circuit.}
\end{figure}
\begin{figure}[t]
\includegraphics[scale=0.32]{ET_Fig9}
\caption{The expectation value of a local operator of a TTNS\index{TTNS}. We see that after applying the isometric properties of the tensors, the past causal cone of a single site has a bounded width. The calculation again boils down to a calculation of transfer matrices. This time the transfer matrices evolve between different layers of the tree.\label{fig:one-site-ttn}}
\end{figure}
Similarly, if we compute the correlation function of local operators
at a given distance, as shown in Fig. \ref{fig:two_poin_ttn}, we
can once more get rid of the tensors outside the casual cone.
Rigorously we see that the causal cone width now increases to four sites,
since it consists of two different two-site branches. However, if
we order the contraction as shown in the middle, we see that the
contractions boil down again to a two-site causal cone. Interestingly,
since the computation of two-point correlations at very large distance
involve the power of transfer matrices that translate in scale rather
than in space, one would expect that these matrices are
all the same (as a consequence of scale-invariance for example). Thus, we
would get polynomially decaying correlations \cite{silvi2010}.
\begin{figure}[t]
\includegraphics[scale=0.26]{ET_Fig10}
\caption{The computation of the correlation function of two operators separated by a given distance boils down to the computation of a certain power of a transfer matrices. The computation of the casual cone can be simplified in a sequential way, as depicted in the last two sub-figures. \label{fig:two_poin_ttn}}
\end{figure}
\subsection{MERA\index{MERA} wave-functions}
Until now, we have discussed with the TN's\index{TN} that, even if they
can be embedded in a 2D space, they contain no loops. In the context
of network complexity theory, they are called mean-field networks \cite{bapst2013}.
However, there are also TN's with loops that are exactly contractible \cite{MS08TNQcomp}. A particular case is that
of a 1D MERA (and its generalizations) \cite{V07EntRenor,V08MERA,EV09EntRenor,EV09EntRenorAlgor,tagliacozzo2011}.
The MERA is again a TN that can be embedded in a 2D plane,
and that is full of loops as seen in Fig. \ref{fig:MERA}. This TN
has a very peculiar structure, again, inspired from RG\index{RG} transformation\cite{vidal2009}.
MERA can also be interpreted as a quantum circuit where the time
evolves radially along the network, once more opposite to the arrows
that indicate the direction along which the tensors are unitary. The
MERA is a layered TN, with where layer (in different colors in the figure) is
composed by the appropriate contraction of some third-order tensors
(isometries) and some forth-order tensors (disentangler). The concrete form of the network is not really important \cite{tagliacozzo2011}.
In this specific case we are plotting a two-to-one MERA that was
discussed in the original version of Ref. \cite{EV09EntRenorAlgor}. Interestingly, an
operator defined on at most two sites gives a bounded past-causal cone as shown in Fig. \ref{fig:one_site_MERA} and .
\begin{figure}
\begin{center}
\includegraphics[scale=0.40]{ET_Fig11}
\end{center}
\caption{The TN\index{TN} of MERA\index{MERA}. The MERA has a hierarchical structure consisting of several layers of disentanglers and isometries. The computational time
flows from the center towards the edge radially, when considering MERA as a quantum circuit. The unitary and isometric tensors and the network
geometry are chosen in order to guarantee that the width of the causal
cone is bounded. \label{fig:MERA}}
\end{figure}
As in the case of the TTNS\index{TTNS}, we can indeed perform the explicit calculation
of the past causal cone of a single site operators (Fig.\ref{fig:one_site_MERA}). There we show that
the TN\index{TN} contraction of the required expectation
value, and then simplify it by taking into account the contractions of the unitary
and isometric tensors outside the casual cone with a bounded width involving at most four auxiliary constituents.
\begin{figure}[t]
\includegraphics[height=5.5cm, width=12cm]{ET_Fig12}
\caption{Past causal cone of a single-site operator for a MERA\index{MERA}. \label{fig:one_site_MERA}}
\end{figure}
The calculation of a two-point correlation function of local operators follows
a similar idea and leads to the contraction shown in Fig.\ref{fig:MERA_two_point}.
Once more, we see that the computation of the two-point correlation function
can be done exactly due to the bounded width of the corresponding casual cone.
\begin{figure}[t]
\begin{center}
\includegraphics[scale=0.37]{ET_Fig13}
\end{center}
\caption{Two-point correlation function in the MERA\index{MERA}.
\label{fig:MERA_two_point}}
\end{figure}
\subsection{Sequentially generated PEPS\index{PEPS} wave-functions}
The MERA\index{MERA} and TTNS\index{TTNS} can be generalized to two dimensional lattices \cite{TEV09TTN, EV09EntRenor}.
The generalization of MPS to 2D, on the other hand, gives rise to PEPS\index{PEPS}.
In general, it belongs to the 2D TN's\index{TN} that cannot be exactly contracted \cite{VWPC06PEPSfamous,nishio2004}.
However for a subclass
of PEPS, one can implement the contract exactly, which is called sequentially generated PEPS \cite{banuls2008}. Differently
from the MERA where the computation of the expectation value of any
sufficiently local operator leads to a bounded causal cone, sequentially
generated PEPS have a central site, and the local observables around the
central site can be computed easily. However, the local observables in other
regions of the TN give larger causal cones. For example, we represent in Fig.\ref{fig:sequentially-generated-peps} a sequentially
generated PEPS for a $3\times3$ lattice. The norm of the state is
computed in (b), where the TN boils down to the norm of
the central tensor. Some of the reduced density matrices of the system
are also easy to compute, in particular those of the central site and
its neighbors [Fig. \ref{fig:rdm_seq_peps} (a)].
Other reduced density matrices, such as those of spins close
to the corners, are much harder to compute. As illustrated in Fig. \ref{fig:rdm_seq_peps} (b), the causal cone of a corner site in a $3\times\text{3}$ PEPS has a width $2$. In general for an $L\times L$ PEPS, the casual cone would
have a width $L/2$.
Differently from MPS, the causal cone of a PEPS cannot be transformed
by performing a gauge transformation. However, as firstly observed by
F. Cucchietti (private communication), one can try to approximate
a PEPS of a given causal cone with another one of
a different causal cone, by for example moving the center site. This is not an exact operation, and the approximations
involved in such a transformation need to be addressed numerically.
The systematic study of the effect of these approximations have been
studied recently in \cite{zaletel2019, haghshenas2019}. In general,
we have to say that the contraction of a PEPS wave-function
can only be performed exactly with exponential resources. Therefore, efficient approximate contraction schemes are necessary to deal with PEPS.
\begin{figure}[t]
\begin{center}
\includegraphics[scale=0.55]{ET_Fig14}
\end{center}
\caption{(a) A sequentially generated PEPS. All tensors but the central
one (green in the figure) are isometries, from the in-going bonds (marked with ingoing arrows) to the outgoing ones. The central tensor
represents a normalized vector on the Hilbert space constructed by
the physical Hilbert space and the four copies of auxiliary spaces,
one for each of its legs. (b) The norm of such PEPS, after implementing the isometric constraints, boils down to the norm of its central
tensor. \label{fig:sequentially-generated-peps}}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[scale=0.55]{ET_Fig15}
\end{center}
\caption{(a) The reduced density matrices of a PEPS that is sequentially generated
containing two consecutive spins (one of them is the central
spin. (b) The reduced density matrix of a local
region far from the central site is generally hard to
compute, since it can give rise to an arbitrarily large causal cone.
For the reduced density
matrix of any of the corners with a $L\times L$ PEPS, which is the most consuming case, it leads to a causal cone with a width up to $L/2$. That means the
computation is exponentially expensive with the size of the system.\label{fig:rdm_seq_peps}}
\end{figure}
\subsection{Exactly contractible tensor networks} \label{Sec-exactTN}
\begin{figure}
\begin{center}
\includegraphics[width=\textwidth]{Fig14.eps}
\end{center}
\caption{(Color online) If one starts with contracting an arbitrary bond, there will be a tensor with six bonds. As the contraction goes on, the number of bonds increases linearly with the boundary $\partial$ of the contracted area, thus the memory increases exponentially as $O(\chi^{\partial})$ with $\chi$ the bond dimension.}
\label{fig-3TNcontract}
\end{figure}
We have considered above, from the perspective of quantum circuits, whether a TNS\index{TNS} can be contracted exactly by the width of the casual cones. Below, we re-consider this issue from the aspect of TN.
Normally, a TN cannot be contracted without approximation. Let us consider a square TN, as shown in Fig. \ref{fig-3TNcontract}. We start from contracting an arbitrary bond in the TN\index{TN} (yellow shadow). Consequently, we obtain a new tensor with six bonds that contains $\chi^6$ parameters ($\chi$ is the bond dimension). To proceed, the bonds adjacent to this tensor are probably a good choice to contract next. Then we will have to restore a new tensor with eight bonds. As the contraction goes on, the number of bonds increases linearly with the boundary $\partial$ of the contracted area, thus the memory increases exponentially as $O(\chi^{\partial})$. For this reason, it is impossible to exactly contract a TN, even if it only contains a small number of tensors. Thus, approximations are inevitable. This computational difficulty is closely related to the area law of entanglement entropy \cite{ECP10AreaLawRev} (also see Sec. \ref{sec.TNent}), or the width of the casual cone as in the case of PEPS. Below, we give three examples of TN's that can be exactly contracted.
\textbf{\textit{Tensor networks on tree graphs}}. We here consider a scalar tree TN [Fig. \ref{fig-3ExactTN} (a)] with $N_L$ layers of third-order tensors. Some vectors are put on the out-most boundary. An example that a tree TN may represent is an observable of a TTNS. A tree TN is written as
\begin{eqnarray}
Z = \sum_{\{a\}} \prod_{n=1}^{N_L} \prod_{m=1}^{M_n} T^{[n,m]}_{a_{n,m,1},a_{n,m,2},a_{n,m,3}} \prod_k v^{[k]}_{a_k},
\end{eqnarray}
with $T^{[n,m]}$ the $m$-th tensor on the $n$-th layer, $M_n$ the number of tensors of the $n$-th layer, and $v^{[k]}$ the $k$-th vectors on the boundary.
Now we contract each of the tensor on the $N_L$-th layer with the corresponding two vectors on the boundary as
\begin{eqnarray}
v_{a_3}' = \sum_{a_1a_2} T^{[N_L m]}_{a_1a_2a_3} v^{[k_1]}_{a_1} v^{[k_2]}_{a_2}.
\end{eqnarray}
After the vectors are updated by the equation above, and the number of layers of the tree TN becomes $N_L-1$. The whole tree TN can be exactly contracted by repeating this procedure.
We can see from the above contraction that if the graph does not contain any loops, i.e. has a tree like structure, the dimensions of the obtained tensors during the contraction will not increase unboundedly. Therefore, the TN defined on it can be exactly contracted. This is again related to the area law of entanglement entropy that a loop-free TN satisfies: to separate a tree-like TN into two disconnecting parts, the number of bonds that needs to be cut is only one. Thus the upper bond of the entanglement entropy between these two parts is constant, determined by the dimension of the bond that is cut. This is also consistent with the analyses based on the maximal width of the casual cones.
\begin{figure}
\begin{center}
\includegraphics[width=0.95\textwidth]{Fig15.eps}
\end{center}
\caption{(Color online) Two kinds of TN's that can be exactly contracted: (a) tree and (b) fractal TN's. In (b), the shadow shows the Sierpi\'nski gasket, where the tensors are defined in the triangles.}
\label{fig-3ExactTN}
\end{figure}
\textbf{\textit{Tensor networks on fractals}}. Another example that can be exactly contracted is the TN defined on the fractal called Sierpi\'nski gasket [Fig. \ref{fig-3ExactTN} (b)] (see, e.g. \cite{GGN16fractal,WRLZZS16Fractal}). The TN can represent the partition function of the statistical model defined on the Sierpi\'nski gasket, such as Ising and Potts model. As explained in Sec. II, the tensor is given by the probability distribution of the three spins in a triangle.
Such a TN can be exactly contracted by iteratively contracting each three of the tensors located in a same triangle as
\begin{eqnarray}
T_{a_1a_2a_3}' = \sum_{b_1b_2b_3} T_{a_1b_1b_2} T_{a_2b_2b_3} T_{a_3b_3b_1}.
\end{eqnarray}
After each round of contractions, the dimension of the tensors and the geometry of the network keep unchanged, but the number of the tensors in the TN decreases from $N$ to $N/3$. It means we can exactly contract the whole TN by repeating the above process.
\textbf{\textit{Algebraically contractible tensor networks}}. The third example is called algebraically contractible TN's\index{TN} \cite{KRV09exactTRG, DBJC12TopoTNS}. The tensors that form the TN possess some special algebraic properties, so that even the bond dimensions increase after each contraction, the rank of the bonds is kept unchanged. It means one can introduce some projectors to lower the bond dimension without causing any errors.
The simplest algebraically contractible TN is the one formed by the \textit{super-diagonal tensor} $I$ defined as
\begin{eqnarray}
I_{a_1,\cdots, a_N}&=&
\left\{
\begin{array}{lll}
1, \ \ a_1 = \cdots = a_N, \\
0, \ \ \text{otherwise}.
\end{array}
\right.
\end{eqnarray}
$I$ is also called \textit{copy tensor}, since it forces all its indexes to take a same value.
For a square TN of an arbitrary size formed by the fourth-order $I$'s, obviously we have its contraction $Z=d$ with $d$ the bond dimension. The reason is that the contraction is the summation of only $d$ non-zero values (each equals to 1).
To demonstrate its contraction, we will need one important property of the copy tensor (Fig. \ref{fig-3DeltaFusion}): if there are $n \geq 1$ bonds contracted between two copy tensors, the contraction gives a copy tensor,
\begin{eqnarray}
I_{a_1 \cdots b_1 \cdots} = \sum_{c_1 \cdots} I_{a_1\cdots c_1 \cdots} I_{b_1 \cdots c_1 \cdots}.
\end{eqnarray}
This property is called the \textit{fusion rule}, and can be understood in the opposite way: a copy tensor can be decomposed as the contraction of two copy tensors.
\begin{figure}[tbp]
\centering
\includegraphics[angle=0,width=0.55\linewidth]{DeltaFusion.eps}
\caption{(Color online) The fusion rule of the copy tensor: the contraction of two copy tensors of $N_1$-th and $N_2$-th order gives a copy tensor of $(N_1+N_2-N)$-th order, with $N$ the number of the contracted bonds. }
\label{fig-3DeltaFusion}
\end{figure}
With the fusion rule, one will readily have the property for the dimension reduction: if there are $n \geq 1$ bonds contracted between two copy tensors, the contraction is identical after replacing the $n$ bonds with one bond,
\begin{eqnarray}
\sum_{c_1 \cdots c_n} I_{a_1\cdots c_1 \cdots c_n} I_{b_1 \cdots c_1 \cdots} = \sum_{c} I_{a_1\cdots c} I_{b_1 \cdots c}.
\end{eqnarray}
In other words, the dimension of the contracting bonds can be exactly reduced from $\chi^n$ to $\chi$. Applying this property to TN contraction, it means each time when the bond dimension increases after contracting several tensors into one tensor, the dimension can be exactly reduced to $\chi$, so that the contraction can continue until all bonds are contracted.
From the TN\index{TN} of the copy tensors, a class of exactly contractible TN can be defined, where the local tensor is the multiplication of the copy tensor by several unitary tensors. Taking the square TN as example, we have
\begin{eqnarray}
T_{a_1a_2a_3a_4} = \sum_{b_1b_2b_3b_4} X_{b_1} I_{b_1b_2b_3b_4} U_{a_1b_1} V_{a_2b_2} U^{\ast}_{a_3b_3} V^{\ast}_{a_4b_4},
\end{eqnarray}
with $U$ and $V$ two unitary matrices. $X$ is an arbitrary $d$-dimensional vector that can be understood as the ``weights'' (not necessarily to be positive to define the tensor). After putting the tensors in the TN, all unitary matrices vanish to identities. Then one can use the fusion rule of the copy tensor to exactly contract the TN, and the contraction gives $Z=\prod_b (X_{b})^{N_T}$ with $N_T$ the total number of tensors.
The unitary matrices are not trivial in physics. If we take $d=2$ and
\begin{eqnarray}
U=V=
\begin{bmatrix}
\sqrt{2}/2 & \sqrt{2}/2 \\
\sqrt{2}/2 & -\sqrt{2}/2 \\
\end{bmatrix},
\end{eqnarray}
the TN is in fact the inner product of the $Z_2$ topological state (see the definition of $Z_2$ PEPS\index{PEPS} in Sec. 2.2.3). If one cuts the system into two sub-regions, all the unitary matrix vanish into identities inside the bulk. However, those on the boundary will survive, which could lead to exotic properties such as topological orders, edge states and so on. Note that $Z_2$ state is only a special case. One can refer to a systematic picture given by X. G. Wen called the string-net states \cite{GLSW09StringTPS,BAV09StringTPS,CZGCW10StringTPS}.
\section{Some discussions}
\subsection{General form of tensor network}
One can see that a TN\index{TN} (state or operator) is defined as the contraction of certain tensors $\{T^{[n]}\}$ with a general form as
\begin{eqnarray}
\mathcal{T}_{\{s\}} = \sum_{\{a\}} \prod_{n} T^{[n]}_{s^n_1s^n_2 \cdots,a^n_1 a^n_2 \cdots}.
\label{eq-2TN}
\end{eqnarray}
The indexes $\{a\}$ are geometrical indexes, each of which is shared normally two tensors and will be contracted. The indexes $\{s\}$ are open bonds, each of which only belongs to one tensor. After contracting all the geometrical indexes, the TN\index{TN} represents a $\mathcal{N}$-th order tensor, with $\mathcal{N}$ the total number of the open indexes $\{s\}$.
Each tensor in the TN can possess different number of open or geometrical indexes. For an MPS\index{MPS}, each tensor has one open index (called physical bond) and two geometrical indexes; for PEPS\index{PEPS} on square lattice, it has one open and four geometrical indexes. For the generalizations of operators, the number of open indexes are two for each tensor. It also allows hierarchical structure of the TN, such as TTNS\index{TTNS} and MERA\index{MERA}.
One special kind of the TN's is the scalar TN with no open bonds, denoted as
\begin{eqnarray}
Z = \sum_{\{a\}} \prod_{n} T^{[n]}_{a^n_1 a^n_2 \cdots}.
\label{eq-2TNscalar}
\end{eqnarray}
It is very important because many physical problems can be transformed to computing the contractions of scalar TN's. A scalar TN can be obtained from the TN's that has open bonds, such as $Z = \sum_{\{s\}} \mathcal{T}_{\{s\}}$ or $Z = \sum_{\{s\}} \mathcal{T}_{\{s\}}^{\dagger} \mathcal{T}_{\{s\}}$, where $Z$ can be the cost function (e.g., energy or fidelity) to be maximized or minimized. The TN contraction algorithms mainly deal with the scalar TN's.
\subsection{Gauge degrees of freedom}
\label{sec.gauge}
For a given state, its TN\index{TN} representation is not unique. Let us take translational invariant MPS as an example. One may insert a (full-rank) matrix $U$ and its inverse $U^{-1}$ on each of the virtual bonds and then contracted them, respectively, into the two neighboring tensors. The tensors of new MPS\index{MPS} becomes $\tilde{A}^{[n]}_{s, a a'} = \sum_{bb'} U_{ab} A^{[n]}_{s, b b'} U^{-1}_{a' b'}$. In fact, we only put an identity $I = UU^{-1}$, thus do not implement any changes to the MPS. However, the tensors the form the MPS changes, meaning the TN representation changes. It is also the case when inserting an matrix and its inverse on any of the virtual bonds of a TN state, which changes the tensors without changing the state itself. Such degrees of freedom is known as the \textit{gauge degrees of freedom}, and the transformations are called \textit{gauge transformations}.
The gauge degrees of on the one hand may cause instability to TN simulations. Algorithms for finite and infinite PEPS were proposed to fix the gauge to reach higher stability \cite{LCB14fPEPS, PBTCO15FastFullUpdate,PMV15gaugePEPS}. On the other hand, one may use gauge transformation to transform a TN state to a special form, so that, for instance, one can implement truncations of local basis while minimizing the error non-locally \cite{OV08canonical, RLXZS12ODTNS} (we will go back to this issue later). Moreover, gauge transformation is closely related to other theoretical properties such as the global symmetry of TN states, which has been used to derive more compact TN representations \cite{SV13Gsymme}, and to classify many-body phases \cite{CGW11phase, SPC11PEPS} and to characterize non-conventional orders \cite{PWSC08symme, PTBO10EStopo}, just to name a few.
\subsection{Tensor network and quantum entanglement}
\label{sec.TNent}
\begin{figure}[tbp]
\centering
\includegraphics[angle=0,width=0.8\linewidth]{AreaLaw.eps}
\caption{Bipartition of a 1D system into two half chains. Significant quantum correlations in gapped ground states occur only on short length scales.}
\label{fig-Area_Law1}
\end{figure}
The numerical methods based on TN face great challenges, primarily that the dimension of the Hilbert space increases exponentially with the size. Such an ``\textit{exponential wall}'' has been treated in different ways by many numeric algorithms, including the DFT\index{DFT} methods \cite{K99DFT} and QMC\index{QMC} approaches \cite{TW05SignQMC}.
The power of TN\index{TN} has been understood in the sense of quantum entanglement: the entanglement structure of low-lying energy states can be efficiently encoded in TNS's\index{TNS}. It takes advantage of the fact that not all quantum states in the total Hilbert space of a many-body system are equally relevant to the low-energy or low-temperature physics. It has been found that the low-lying eigenstates of a gapped Hamiltonian with local interactions obey the area law of the entanglement entropy \cite{PKL99DMSpectra}.
More precisely speaking, for a certain subregion $\mathcal{R}$ of the system, its reduced density matrix is defined as $\hat{\rho}_{\mathcal{R}}= \rm Tr_{\mathcal{E}} (\hat{\rho})$, with $\mathcal{E}$ denotes the spatial complement of $\mathcal{R}$. The entanglement entropy is defined as
\begin{eqnarray}
S(\rho_{\mathcal{R}}) = - \rm Tr \lbrace \rho_{\mathcal{R}} \rm
log (\rho_{\mathcal{R}} ) \rbrace .
\end{eqnarray}
Then the area law of the entanglement entropy \cite{ECP10AreaLawRev,H15TNthesis} reads
\begin{eqnarray}
S(\rho_{\mathcal{R}})= O(\vert \partial \mathcal{R} \vert) ,
\end{eqnarray}
with $\vert \partial \mathcal{R}\vert$ the size of the boundary. In particular, for a $D$-dimensional system, one has
\begin{eqnarray}
S=O(l^{D-1}),
\label{eq-2arealaw}
\end{eqnarray}
with $l$ the length scale. This means that for 1D systems, $S= \rm \textit{const}$. The area law suggests that the low-lying eigenstates stay in a ``small corner'' of the full Hilbert space of the many-body system, and that they can be described by a much smaller number of parameters. We shall stress that the locality of the interactions is not sufficient to the area law. Vitagliano, \textit{el al} show that simple 1D spin models can exhibit volume law, where the entanglement entropy scales with the bulk \cite{VRL10arealaw, MS16arealaw}.
\begin{figure}[tbp]
\centering
\includegraphics[angle=0,width=0.8\linewidth]{AreaLaw2.eps}
\caption{The argue the 1D area law, the chain is separated into three sub-systems denoted by $A$, $B$ and $C$. If the correlation length $\xi_{corr}$ is much larger than the size of $B$ (denoted by $l_{A C}$), the reduced density matrix by tracing $B$ approximately satisfies $\hat{\rho}_{AC} \simeq \hat{\rho}_{A} \otimes \hat{\rho}_{C}$.}
\label{fig-Area_Law2}
\end{figure}
The area law of entanglement entropy is intimately connected to another fact that a non-critical quantum system exhibits a finite correlation length. The correlation functions between two blocks in a gapped system decay exponentially as a function of the distance of the blocks \cite{H04CorFunDecExp-1}, which is argued to lead to the area law. An intuitive picture can be seen in Fig. \ref{fig-Area_Law1}. Let us consider a 1D gapped quantum system whose ground state $| \psi_{ABC} \rangle$ possesses a correlation length $\xi_{corr}$. By dividing into three subregions $A$, $B$ and $C$, the reduced density operator $\hat{\rho}_{AC}$ is obtained when tracing out the block $B$, i.e. $\hat{\rho}_{AC} = \rm Tr_{B} | \psi_{ABC} \rangle \langle \psi_{ABC} |$ (see Fig. \ref{fig-Area_Law2}). In the limit of large distance between $A$ and $C$ blocks with $l_{AC} \gg \xi_{corr}$, one has the reduced density matrix satisfying
\begin{eqnarray}
\hat{\rho}_{AC} \simeq \hat{\rho}_{A} \otimes \hat{\rho}_{C},
\end{eqnarray}
up to some exponentially small corrections. Then $| \psi_{ABC} \rangle$ is a purification \footnote{Purification: Let $\rho$ be a density matrix acting an a Hilbert space $\mathcal{H}_A$ of finite dimension $n$. Then there exist a Hilbert space $\mathcal{H}_B$ and a pure state $\vert \psi \rangle \in \mathcal{H}_A \otimes \mathcal{H}_B$ such that the partial trace of $\vert \psi \rangle \langle \psi \vert$ with respect to $\mathcal{H}_B$: $\rho = Tr_B \vert \psi \rangle \langle \psi \vert$. We say that $\vert \psi \rangle$ is the purification of $\hat{\rho}$.} of a mixed state with the form $| \psi_{A B_l} \rangle \otimes | \psi_{B_r C} \rangle$ that has no correlations between $A$ and $C$; here $B_l$ and $B_r$ sit at the two ends of the block $B$, which together span the original block.
It is well known that all possible purifications of a mixed state are equivalent to each other up to a local unitary transformation on the virtual Hilbert space. This naturally implies that there exists a unitary operation $\hat{U}_B$ on the block $B$ that completely disentangles the left from the right part as
\begin{eqnarray}
\hat{I}_A \otimes \hat{U}_B \otimes \hat{I}_C \vert \psi_{ABC} \rangle \rightarrow \vert \psi_{A B_l} \rangle \otimes \vert \psi_{B_rC} \rangle .
\end{eqnarray}
$\hat{U}_B$ implies that there exists a tensor $B_{s,a a'}$ with $0\leq a, a', s \leq \chi-1$ and basis $\{| \psi^A \rangle\}$, $\{| \psi^B \rangle\}$, $\{| \psi^C \rangle\}$ defined on the Hilbert spaces belonging to $A$, $B$, $C$ such that
\begin{eqnarray}
\vert \psi_{ABC} \rangle \simeq \sum_{s a a'} B_{s,a a'} \vert \psi^A_{a} \rangle \vert \psi^B_{s} \rangle \vert \psi^C_{a'} \rangle .
\label{eq-3Btensor}
\end{eqnarray}
This argument directly leads to the MPS\index{MPS} description and gives a strong hint that the ground states of a gapped Hamiltonian is well represented by an MPS of finite bond dimensions, where $B$ in Eq. (\ref{eq-3Btensor}) is analog to the tensor in an MPS. Let us remark that every state of $N$ spins has an exact MPS representation if we allow $\chi$ grow exponentially with the number of spins \cite{VPC04DMRGQinfo}. The whole point of MPS is that a ground state can typically be represented by an MPS where the dimension $\chi$ is small and scales at most polynomially with the number of spins: this is the reason why MPS-based methods are more efficient than exact diagonalization.
For the 2D PEPS\index{PEPS}, it is more difficult to strictly justify the area law of entanglement entropy. However, we can make some sense of it from the following aspects. One is the fact that PEPS can exactly represent some non-trivial 2D states that satisfies the area law, such as the nearest-neighbor RVB and Z$_2$ spin liquid mentioned above. Another is to count the dimension of the geometrical bonds $\mathcal{D}$ between two subsystems, from which the entanglement entropy satisfies an upper bound as $S \leq \log \mathcal{D}$ \footnote{One can see this with simply a flat entanglement spectrum, $\lambda_n = 1/ \mathcal{D}$ for any $n$.}.
After dividing a PEPS\index{PEPS} into two subregions, one can see that the number of geometrical bonds $N_b$ increase linearly with the length scale, i.e. $N_b \sim l$. It means the dimension $\mathcal{D}$ satisfies $\mathcal{D} \sim \chi^{l}$, and the upper bound of the entanglement entropy fulfills the area law given by Eq. (\ref{eq-2arealaw}), which is
\begin{eqnarray}
S \leq O(l).
\end{eqnarray}
However, as we will see later, such a property of PEPS\index{PEPS} is exactly the reason that makes it computationally difficult.
\chapter{Two-dimensional tensor networks and contraction algorithms}
\label{sec3}
\abstract{In this section, we will first demonstrate in Sec. \ref{sec-phys2TN} that many important physical problems can be transformed to 2D TN's\index{TN}, and the central tasks become to compute the corresponding TN contractions. From Sec. \ref{sec.2.TRG} to \ref{transverse}, we will then present several paradigm contraction algorithms of 2D TN's including TRG\index{TRG}, TEBD\index{TEBD}, and CTMRG\index{CTMRG}. Relations to other distinguished algorithms and the exactly contractible TN's will also be discussed.}
\section{From physical problems to two-dimensional tensor networks}
\label{sec-phys2TN}
\subsection{Classical partition functions}
\label{Classical partition functions}
\begin{figure}[tbp]
\centering
\includegraphics[angle=0,width=1\linewidth]{IsingSquare.eps}
\caption{(Color online) (a) Four Ising spins (blue balls with arrows) sitting on a single square, and the red lines represent the interactions. The blue block is the tensor $T$ [Eq. (\ref{eq-3IsingT})], with the black lines denoting the indexes of $T$. (b) The graphic representation of the TN on a larger lattice with more than one squares. (c) The TN\index{TN} construction of the partition function on infinite square lattice.}
\label{fig-3IsingSquare}
\end{figure}
Partition function, which is a function of the variables of a thermodynamic state such as temperature, volume, and etc., contains the statistical information of a thermodynamic equilibrium system. From its derivatives of different orders, we can calculate the energy, free energy, entropy, and so on. Levin and Nave pointed out in Ref. \cite{LN07TRG} that the partition functions of statistical lattice models (such as Ising and Potts models) with local interactions can be written in the form of TN. Without losing generality, we take square lattice as an example.
Let us start from the simplest case: the classical Ising model on a single square with only four sites. The four Ising spins denoted by $s_i$ ($i=1,2,3,4$) locate on the four corners of the square, as shown in Fig.\ref{fig-3IsingSquare} (a); each spin can be up or down, represented by $s_i=0$ and $1$, respectively. The classical Hamiltonian of such a system reads
\begin{eqnarray}
H_{s_1s_2s_3s_4}=J(s_1s_2 + s_2s_3 + s_3s_4 + s_4s_1) - h (s_1+s_2+s_3+s_4)
\end{eqnarray}
with $J$ the coupling constant and $h$ the magnetic field.
When the model reaches the equilibrium at temperature $\rm T$, the probability of each possible spin configuration is determined by the Maxwell-Boltzmann factor
\begin{eqnarray}
T_{s_1s_2s_3s_4}=e^{-\beta H_{s_1s_2s_3s_4}},
\label{eq-3IsingT}
\end{eqnarray}
with the inverse temperature $\beta=1/ \rm T$ \footnote{In this paper, we set Bolzmann constant $k_B = 1$ for convenience.}. Obviously, Eq. (\ref{eq-3IsingT}) is a forth-order tensor $T$, where each element gives the probability of the corresponding configuration.
The partition function is defined as the summation of the probability of all configurations. In the language of tensor, it is obtained by simply summing over all indexes as
\begin{eqnarray}
Z=\sum_{s_1s_2s_3s_4} T_{s_1s_2s_3s_4}.
\end{eqnarray}
Let us proceed a little bit further by considering four squares, whose partition function can be written in a TN\index{TN} with four tensors [Fig.\ref{fig-3IsingSquare}(b)] as
\begin{eqnarray}
Z = \sum_{\{ss'\}} T_{s_1s_2s_2's_1'} T_{s_2's_3s_4s_3'} T_{s_4's_3's_5s_6} T_{s_8s_1's_4's_7}.
\end{eqnarray}
Each of the indexes $\{s'\}$ inside the TN is shared by two tensors, representing the spin that appears in both of the squares. The partition function is obtained by summing over all indexes.
For the infinite square lattice, the probability of a certain spin configuration ($s_1, s_2, \cdots$) is given by the product of infinite number of tensor elements as
\begin{eqnarray}\label{eq1}
e^{-\beta H_{\{s\}}}=e^{-\beta H_{s_1s_2s_3s_4}} e^{-\beta H_{s_4s_5s_6s_7}} \cdots =T_{s_1s_2s_3s_4}T_{s_4s_5s_6s_7} \cdots
\end{eqnarray}
Then the partition function is given by the contraction of an infinite TN\index{TN} formed by the copies of $T$ [Eq. (\ref{eq-3IsingT})] as
\begin{eqnarray}\label{eq2}
Z=\sum_{\{s\}} \prod_{n} T_{s^n_1 s^n_2 s^n_3 s^n_4},
\label{eq-3IsingTN}
\end{eqnarray}
where two indexes satisfy $s^n_j = s^m_k$ if they refer to the same Ising spin. The graphic representation of Eq.\ref{eq-3IsingTN} is shown in Fig.\ref{fig-3IsingSquare} (c). One can see that on square lattice, the TN still has the geometry of a square lattice. In fact, such a way will give a TN that has a geometry of the dual lattice of the system, and the dual of the square lattice is itself.
For the $Q$-state Potts model on square lattice, the partition function has the same TN representation as that of the Ising model, except that the elements of the tensor are given by the Boltzmann wight of the Potts model and the dimension of each index is $Q$. Note that the Potts model with $q=2$ is equivalent to the Ising model.
Another example is the eight-vertex model proposed by Baxter in 1971 \cite{B71EightVertex}. It is one of the ``ice-type'' statistic lattice model, and can be considered as the classical correspondence of the Z$_2$ spin liquid state. The tensor that gives the TN of the partition function is also ($2 \times 2\times 2\times 2$), whose non-zero elements are
\begin{eqnarray}
T_{s_1,\cdots, s_N}&=&
\left\{
\begin{array}{lll}
1, \ \ s_1+\cdots + s_N=even, \\
0, \ \ \text{otherwise}.
\end{array}
\right.
\end{eqnarray}
We shall remark that there are more than one ways to define the TN of the partition function of a classical system. For example, when there only exist nearest-neighbor couplings, one can define a matrix $M_{ss'} = e^{-\beta H_{ss'}}$ on each bond and put on each site a \textit{super-digonal} tensor $I$ (or called copy tensor) defined as
\begin{eqnarray}
I_{s_1,\cdots, s_N}=
\begin{cases}
1, \ \ s_1 = \cdots = s_N;\\
0, \ \ \text{otherwise}.
\end{cases}
\end{eqnarray}
Then the TN of the partition function is the contraction of copies of $M$ and $I$, and possesses exactly the same geometry of the original lattice (instead of the dual one).
\subsection{Quantum observables}
\label{sec.QobTN}
With a TN\index{TN} state, the computations of quantum observables as $\langle \psi | \hat{O} | \psi \rangle$ and $\langle \psi | \psi \rangle$ is the contraction of a scalar TN, where $\hat{O}$ can be any operator. For a 1D MPS\index{MPS}, this can be easily calculated, since one only needs to deal with a 1D TN stripe. For 2D PEPS\index{PEPS}, such calculations become contractions of 2D TN's. Taking $\langle \psi | \psi \rangle$ as an example, the TN of such an inner product is the contraction of the copies of the local tensor [Fig. \ref{fig-3IsingSquare} (c)] defined as
\begin{eqnarray}
T_{a_1a_2a_3a_4} = \sum_{s} P_{s,a''_1a''_2a''_3a''_4}^{\ast} P_{s,a_1'a_2'a_3'a_4'},
\end{eqnarray}
with $P$ the tensor of the PEPS\index{PEPS} and $a_i = (a'_i, a''_i)$. There are no open indexes left and the TN\index{TN} gives the scalar $\langle \psi | \psi \rangle$. The TN for computing the observable $\langle \hat{O} \rangle$ is similar. The only difference is that we should substitute some small number of $T_{a_1a_2a_3a_4}$ in original TN of $\langle \psi |\psi \rangle$ with ``impurities'' at the sites where the operators locate. Taking one-body operator as an example, the ``impurity'' tensor on this site can be defined as
\begin{eqnarray}
\widetilde{T}_{a_1a_2a_3a_4}^{[i]} = \sum_{s,s'} P_{s,a''_1a''_2a''_3a''_4}^{\ast} \hat{O}_{s,s'}^{[i]} P_{s',a_1'a_2'a_3'a_4'},
\end{eqnarray}
In such a case, the single-site observables can be represented by the TN contraction of
\begin{eqnarray}
\frac{\langle \psi| \hat{O}^{[i]}|\psi \rangle}{\langle \psi| \psi \rangle} = \frac{\text{tTr }\ \widetilde{T}^{[i]} \prod_{n\neq i} T}{\text{tTr }\ \prod_{n=1}^{N} T},
\end{eqnarray}
For some non-local observables, e.g., the correlation function, the contraction of $\langle \psi| \hat{O}^{[i]} \hat{O}^{[j]} |\psi \rangle$ is nothing but adding another ``impurity'' by
\begin{eqnarray}
\langle \psi| \hat{O}^{[i]} \hat{O}^{[j]} |\psi \rangle = \text{tTr }\ \widetilde{T}^{[i]}\widetilde{T}^{[j]} \prod_{n\neq i,j}^{N} T,
\end{eqnarray}
\subsection{Ground-state and finite-temperature simulations}
\label{sec.2.evolution}
Ground-state simulations of 1D quantum models with short-range interactions can also be efficiently transferred to 2D TN contractions. When minimizing the energy
\begin{eqnarray}
E=\frac{\langle \psi | \hat{H} | \psi \rangle} {\langle \psi | \psi \rangle},
\end{eqnarray}
where we write $|\psi\rangle$ as an MPS\index{MPS}. Generally speaking, there are two ways to solve the minimization problem: (i) simply treat all the tensor elements as variational parameters; (ii) simulate the imaginary-time evolution
\begin{eqnarray}\label{eq3}
|\psi_{gs}\rangle = \lim_{\beta \rightarrow \infty} \frac{e^{-\beta \hat{H}}|\psi\rangle}{\parallel{e^{-\beta \hat{H}}|\psi\rangle}\parallel}.
\end{eqnarray}
The first way can be realized by, e.g., Monte Carlo methods where one could randomly change or choose the value of each tensor element to locate the minimal of energy. One can also use the Newton method and solve the partial-derivative equations $\partial E / \partial x_n = 0$ with $x_n$ standing for an arbitrary variational parameter. Anyway, it is inevitable to calculate $E$ (i.e., $\langle \psi | \hat{H} | \psi \rangle$ and $\langle \psi | \psi \rangle$) for most cases, which is to contraction the corresponding TN's as explained above.
We shall stress that without TN\index{TN}, the dimension of the ground state (i.e., the number of variational parameters) increases exponentially with the system size, which makes the ground-state simulations impossible for large systems.
The second way of computing the ground state with imaginary-time evolution is more or less like an ``annealing'' process. One starts from an arbitrarily chosen initial state and acts the imaginary-time evolution operator on it. The ``temperature'' is lowered a little for each step, until the state reaches a fixed point. Mathematically speaking, by using Trotter-Suzuki decomposition, such an evolution is written in a TN defined on ($D+1$)-dimensional lattice, with $D$ the dimension of the real space of the model.
Here, we take a 1D chain as an example. We assume that the Hamiltonian only contains at most nearest-neighbor couplings, which reads
\begin{eqnarray}\label{eq4}
\hat{H}=\sum_{n} \hat{h}_{n,n+1},
\end{eqnarray}
with $\hat{h}_{n,n+1}$ containing the on-site and two-body interactions of the $n$-th and $n+1$-th sites. It is useful to divide $\hat{H}$ into two groups, $\hat{H}=\hat{H}^e+\hat{H}^{o}$ as
\begin{eqnarray}\label{eq5}
\begin{split}
\hat{H}^e \equiv\sum_{even\ n}\hat{h}_{n,n+1}, \ \ \
\hat{H}^{o} \equiv\sum_{odd\ n}\hat{h}_{n,n+1}.
\end{split}
\end{eqnarray}
By doing so, each two terms in $\hat{H}^e$ or $\hat{H}^{o}$ commute with each other. Then the evolution operator $\hat{U}(\tau)$ for infinitesimal imaginary time $\tau \to 0$ can be written as
\begin{eqnarray}
\hat{U}(\tau) = e^{-\tau \hat{H}} = e^{-\tau \hat{H}^e} e^{-\tau \hat{H}^o} + O(\tau^2) [\hat{H}^e, \hat{H}^o]
\label{eq6}
\end{eqnarray}
If $\tau$ is small enough, the high-order terms are negligible, and the evolution operator becomes
\begin{eqnarray}
\hat{U}(\tau) \simeq \prod_{n} \hat{U}(\tau)_{n,n+1},
\label{eq-3trotter}
\end{eqnarray}
with the two-site evolution operator $\hat{U}(\tau)_{n,n+1} = e^{-\tau \hat{H}_{n,n+1}}$.
The above procedure is known as the first-order Trotter-Suzuki decomposition \cite{T59TrotterDecomp, SI87Trotter, IS88Trotter}. Note that higher-order decomposition can also be adopted. For example, one may use the second order Trotter-Suzuki decomposition that is written as
\begin{eqnarray}\label{eq7}
e^{-\tau \hat{H}} \simeq e^{-\frac{\tau}{2} \hat{H}^e} e^{-\tau \hat{H}^o} e^{-\frac{\tau}{2} \hat{H}^e}.
\end{eqnarray}
With Eq. (\ref{eq-3trotter}), the time evolution can be transferred to a TN, where the local tensor is actually the coefficients of $\hat{U}(\tau)_{n,n+1}$, satisfying
\begin{eqnarray}
T_{s_n s_{n+1} s'_n s'_{n+1}} = \langle s'_n s'_{n+1} | \hat{U}(\tau)_{n,n+1} | s_n s_{n+1}\rangle.
\label{eq-3evolveT}
\end{eqnarray}
Such a TN is defined in a plain of two dimensions that corresponds to the spatial and (real or imaginary) time, respectively. The initial state is located at the bottom of the TN\index{TN} ($\beta = 0$) and its evolution is to do the TN contraction which can efficient solved by TN algorithms (presented later).
In addition, one can readily see that the evolution of a 2D state leads to the contraction of a 3D TN. Such a TN scheme provides a straightforward picture to understand the equivalence between a ($d+1$)-dimensional classical and a $d$-dimensional quantum theory. Similarly, the finite-temperature simulations of a quantum system can be transferred to TN contractions with Trotter-Suzuki decomposition. For the density operator $\hat{\rho}(\beta) = e^{-\beta \hat{H}}$, the TN is formed by the same tensor given by Eq. (\ref{eq-3evolveT}).
\section{Tensor renormalization group}
\label{sec.2.TRG}
In 2007, Levin and Nave proposed TRG\index{TRG} approach \cite{LN07TRG} to contract the TN\index{TN} of 2D classical lattice models. In 2008, Gu \textit{et al} further developed TRG to handle 2D quantum topological phases \cite{GLW08TERG}. TRG can be considered as a coarse-graining contraction algorithm. To introduce the TRG algorithm, let us consider a square TN formed by infinite number of copies of a forth-order tensor $T_{a_1a_2a_3a_4}$ (see the left side of Fig. \ref{fig-3TRG}).
\begin{figure}[tbp]
\centering
\includegraphics[angle=0,width=\linewidth]{TRG.eps}
\caption{(Color online) For an infinite square TN with translational invariance, the renormalization in the TRG algorithm is realized by two local operations of the local tensor. After each iteration, the bond dimensions of the tensor and the geometry of the network keep unchanged.}
\label{fig-3TRG}
\end{figure}
\textbf{\textit{Contraction and truncation}}. The idea of TRG is to iteratively ``coarse-grain'' the TN without changing the bond dimensions, the geometry of the network, and the translational invariance. Such a process is realized by two local operations in each iteration. Let us denote the tensor in the $t$-th iteration as $T^{(t)}$ (we take $T^{(0)}=T$). For obtaining $T^{(t+1)}$, the first step is to decompose $T^{(t)}$ by SVD\index{SVD} in two different ways [Fig. \ref{fig-3TRG}] as
\begin{eqnarray}
T^{(t)}_{a_1a_2a_3a_4} = \sum_{b} U_{a_1a_2b}V_{a_3a_4b},\\
T^{(t)}_{a_1a_2a_3a_4} = \sum_{b} X_{a_4a_1b}Y_{a_2a_3b}.
\label{eq-3TRGdecomp}
\end{eqnarray}
Note that the singular value spectrum can be handled by multiplying it with the tensor(s), and the dimension of the new index satisfies $dim(b)=\chi^2$ with $\chi$ the dimension of each bond of $T^{(t)}$.
The purpose of the first step is to deform the TN, so that in the second step, a new tensor $T^{(t+1)}$ can be obtained by contracting the four tensors that form a square [Fig. \ref{fig-3TRG}] as
\begin{eqnarray}
T^{(t+1)}_{b_1b_2b_3b_4} \leftarrow \sum_{a_1a_2a_3a_4} V_{a_1a_2b_1} Y_{a_2a_3b_2} U_{a_3a_4b_3} X_{a_4a_1b_4}.
\label{eq-3TRGnewT}
\end{eqnarray}
We use an arrow instead of the equal sign, because one may need to divide the tensor by a proper number to keep the value of the elements from being divergent. The arrows will be used in the same way below.
These two steps define the contraction strategy of TRG. By the first step, the number of tensors in the TN (i.e., the size of the TN\index{TN}) increases from $N$ to $2N$, and by the second step, it decreases from $2N$ to $N/2$. Thus, after $t$ times of each iterations, the number of tensors decreases to the $\frac{1}{2^t}$ of its original number. For this reason, TRG\index{TRG} is an \textit{exponential contraction algorithm}.
\textbf{\textit{Error and environment}}. The dimension of the tensor at the $t$-th iteration becomes $\chi^{2^t}$, if no truncations are implemented. that means that truncations of the bond dimensions are necessary. In its original proposal, the dimension is truncated by only keeping the singular vectors of the $\chi$-largest singular values in Eq. (\ref{eq-3TRGdecomp}). Then the new tensor $T^{(t+1)}$ obtained by Eq. (\ref{eq-3TRGnewT}) has exactly the same dimension as $T^{(t)}$.
Each truncation will absolutely introduce some error, which is called the \textit{truncation error}. Consistent with Eq. (\ref{eq-2trunerr}), the truncation error is quantified by the discarded singular values $\lambda$ as
\begin{eqnarray}
\varepsilon = \frac{\sqrt{\sum_{b=\chi}^{\chi^2-1} \lambda_{b}^2}}{\sqrt{\sum_{b=0}^{\chi^2-1} \lambda_{b}^2}}.
\label{eq-3TRGerr}
\end{eqnarray}
According to the linear algebra, $\varepsilon$ in fact gives the error of the SVD\index{SVD} given in Eq. (\ref{eq-3TRGdecomp}), meaning that such a truncation minimizes the error of reducing the rank of $T^{(t)}$, which reads
\begin{eqnarray}
\varepsilon = |T^{(t)}_{a_1a_2a_3a_4} - \sum_{b=0}^{\chi-1} U_{a_1a_2b} V_{a_3a_4b}|
\label{eq-3TRGerrT}
\end{eqnarray}
One may repeat the contraction-and-truncation process until $T^{(t)}$ converges. It usually only takes $\sim 10$ steps, after which one in fact contract a TN\index{TN} of $2^t$ tensors to a single tensor.
The truncation is optimized according to the SVD\index{SVD} of $T^{(t)}$. Thus, $T^{(t)}$ is called the \textit{environment}. In general, the tensor(s) that determines the truncations is called the environment. It is a key factor to the accuracy and efficiency of the algorithm. For those that use local environments, like TRG, the efficiency is relatively high since the truncations are easy to compute. But, the accuracy is bounded since the truncations are only optimized according to some local information (like in TRG the local partitioning $T^{(t)}$).
One may choose other tensors or even the whole TN\index{TN} as the environment. In 2009, Xie \textit{et al} proposed the second renormalization group (SRG)\index{SRG} algorithm \cite{XJCWX09SRG}. The idea is in each truncation step of TRG\index{TRG}, they define the global environment that is a forth-order tensor $\mathcal{E}_{a_1^{\tilde{n}} a_2^{\tilde{n}} a_3^{\tilde{n}} a_4^{\tilde{n}}} = \sum_{\{a\}} \prod_{n \neq \tilde{n}} T^{(n, t)}_{a_1^na_2^na_3^na_4^n}$ with $T^{(n, t)}$ the $n$-th tensor in the $t$-th step and $\tilde{n}$ the tensor to be truncated. $\mathcal{E}$ is the contraction of the whole TN after getting rid of $T^{(\tilde{n}, t)}$, and is computed by TRG. Then the truncation is obtained not by the SVD\index{SVD} of $T^{(\tilde{n}, t)}$, but by the SVD of $\mathcal{E}$. The word ``second'' in the name of the algorithm comes from the fact that in each step of the original TRG, they use a second TRG to calculate the environment. SRG is obviously more consuming, but bears much higher accuracy than TRG. The balance between accuracy and efficiency, which can be controlled by the choice of environment, is one main factor top consider while developing or choosing the TN algorithms.
\section{Corner transfer-matrix renormalization group}
\label{sec.CTMRG}
\begin{figure}[tbp]
\centering
\includegraphics[angle=0,width=0.7\linewidth]{CTMRGcontract.eps}
\caption{(Color online) Overview of the CTMRG\index{CTMRG} contraction scheme. The tensors in the TN are contracted to the variational tensors defined on the edges and corners.}
\label{fig-3CTMRGcontract}
\end{figure}
\begin{figure}[tbp]
\centering
\includegraphics[angle=0,width=\linewidth]{Fig20.eps}
\caption{(Color online) The first arrow shows absorbing tensors $R^{[1]}$, $T$, and $R^{[3]}$ to renew tensors $C^{[1]}$, $R^{[2]}$, and $C^{[2]}$ in left operation. The second arrow shows the truncation of the enlarged bond of $\tilde{C}^{[1]}$, $\tilde{R}^{[2]}$ and $\tilde{C}^{[2]}$. Inset is the acquisition of the truncation matrix $Z$.}
\label{fig-3CTMRG}
\end{figure}
In the 1960s, the corner transfer matrix (CTM)\index{CTM} idea was developed originally by Baxter in Refs. \cite{B68Prototype, B78CTM} and a book \cite{B16statBook}. Such ideas and methods have been applied to various models, for example, the chiral Potts model \cite{B91CTMPotts, B93CTMPottsP, B93CTMPottsTL}, the 8-vertex model \cite{B71EightVertex, B76CTM8VertexI, B76CTM8VertexII}, and to the 3D Ising model \cite{BF84CTM3DIsing}. Combining CTM with DMRG, Nishino and Okunishi proposed the CTMRG\index{CTMRG} \cite{NO96CTMRG0} in 1996 and applied it to several models \cite{NO96CTMRG0,nishino2001two,nishino2000self,nishino1998density,okunishi2000kramers,nishino1997numerical,nishino1997corner,tsushima1998phase,okunishi1999universal,li2001critical,gendiar2002latent}. In 2009, Or\'us and Vidal further developed CTMRG to deal with TN's \cite{OV09CTMRG}. What they proposed to do is to put eight \textit{variational tensors} to be optimized in the algorithm, which are four corner transfer matrices $C^{[1]}, C^{[2]}, C^{[3]}, C^{[4]}$ and four row (column) tensors $R^{[1]}, R^{[2]}, R^{[3]}, R^{[4]}$, on the boundary, and then to contract the tensors in the TN to these variational tensors in a specific order shown in Fig. \ref{fig-3CTMRGcontract}. The TN contraction is considered to be solved with the variational tensors when they converge in this contraction process. Compared with the boundary-state methods in the last subsection, the tensors in CTMRG define the states on both the boundaries and corners.
\textbf{\textit{Contraction}}. In each iteration step of CTMRG\index{CTMRG}, one choses two corner matrices on the same side and the row tensor between them, e.g., $C^{[1]}$, $C^{[2]}$ and $R^{[2]}$. The update of these tensors (Fig.\ref{fig-3CTMRG}) follows
\begin{eqnarray}
\tilde{C}^{[1]}_{\tilde{b}_2b'_1} &\leftarrow& \sum_{b_1} C^{[1]}_{b_1b_2}R^{[1]}_{b_1a_1b'_1},\\
\tilde{R}^{[2]}_{\tilde{b}_2 a_4 \tilde{b_3}} &\leftarrow& \sum_{a_2}R^{[2]}_{b_2a_2b_3}T_{a_1a_2a_3a_4}, \label{eq-3CTMRGcontract}\\
\tilde{C}^{[2]}_{\tilde{b}_3b'_4} &\leftarrow& \sum_{b_4} C^{[2]}_{b_3b_4}R^{[3]}_{b_4a_3b'_4},
\end{eqnarray}
where $\tilde{b}_2=(b_2,a_1)$ and $\tilde{b}_3=(b_3,a_1)$.
After the contraction given above, it can be considered that one column of the TN\index{TN} (as well as the corresponding row tensors $R^{[1]}$ and $R^{[3]}$) are contracted. Then one chooses other corner matrices and row tensors (such as $\tilde{C}^{[1]}$, $C^{[4]}$ and $R^{[1]}$) and implement similar contractions. By iteratively doing so, the TN is contracted in the way shown in Fig. \ref{fig-3CTMRGcontract}.
Note that for a finite TN\index{TN}, the initial corner matrices and row tensors should be taken as the tensors locating on the boundary of the TN. For an infinite TN, they can be initialized randomly, and the contraction should be iterated until the preset convergence is reached.
CTMRG\index{CTMRG} can be regarded as a \textit{polynomial contraction scheme}. One can see that the number of tensors that are contracted at each step is determined by the length of the boundary of the TN\index{TN} at each iteration time. When contracting a 2D TN defined on a $(L\times L)$ square lattice as an example, the length of each side is $L-2t$ at the $t$-th step. The boundary length of the TN (i.e., the number of tensors contracted at the $t$-th step) bears a linear relation with $t$ as $4(L-2t)-4$. For a 3D TN such as cubic TN, the boundary length scales as $6(L-2t)^2-12(L-2t)+8$, thus the CTMRG for a 3D TN (if exists) gives a polynomial contraction.
\textbf{\textit{Truncation}}. One can see that after the contraction in each iteration step, the bond dimensions of the variational tensors increase. Truncations are then in need to prevent the excessive growth of the bond dimensions. In Ref. \cite{OV09CTMRG}, the truncation is obtained by inserting a pair of isometries $V$ and $V^{\dagger}$ in the enlarged bonds. A reasonable (but not the only choice) of $V$ for translational invariant TN is to consider the eigenvalue decomposition on the sum of corner transfer matrices as
\begin{eqnarray}
\sum_{b}\tilde{C}^{[1]\dagger}_{\tilde{b}b}{\tilde{C}^{[1]}}_{\tilde{b}'b} + \sum_{b}\tilde{C}^{[2]\dagger}_{\tilde{b}b}{\tilde{C}^{[1]}}_{\tilde{b}'b}
\simeq \sum_{b=0}^{\chi-1} V_{\tilde{b}b} \Lambda_{b} V^{*}_{\tilde{b}'b}.
\label{eq-3CTMRGtrun}
\end{eqnarray}
Only the $\chi$ largest eigenvalues are preserved. Therefore, $V$ is a matrix of the dimension $D\chi \times \chi$, where $D$ is the bond dimension of $T$ and $\chi$ is the dimension cut-off. We then truncate $\tilde{C}^{[1]}$, $\tilde{R}^{[2]}$, and $\tilde{C}^{[2]}$ using $V$ as
\begin{eqnarray}
C^{[1]}_{b'_1b_2} &=& \sum_{\tilde{b}_2}\tilde{C}^{[1]}_{\tilde{b}_2b'_1}V^{*}_{\tilde{b}_2b_2},\\
R^{[2]}_{b_2a_4b_3} &=& \sum_{\tilde{b}_2,\tilde{b}_3}\tilde{R}^{[2]}_{\tilde{b}_2a_4\tilde{b}_3} V_{\tilde{b}_2b_2} V^{*}_{\tilde{b}_3b_3},\label{eq-3CTMRGtruncate} \\
C^{[2]}_{b_3b'_4} &=&
\sum_{\tilde{b}_3}\tilde{C}^{[2]}_{\tilde{b}_3b'_4} V_{\tilde{b}_3b_3}.
\end{eqnarray}
\textbf{\textit{Error and environment}}. Same as TRG\index{TRG} or TEBD\index{TEBD}, the truncations are obtained by the matrix decompositions of certain tensors that define the environment. From Eq. (\ref{eq-3CTMRGtrun}), the environment in CTMRG is the loop formed by the corner matrices and row tensors. Note that symmetries might be considered to accelerate the computation. For example, one may take $C^{[1]}=C^{[2]}=C^{[3]}=C^{[4]}$ and $R^{[1]}=R^{[2]}=R^{[3]}=R^{[4]}$ when the TN has rotational and reflection symmetries ($T_{a_1a_2a_3a_4} = T_{a_1'a_2'a_3'a_4'}$ after any permutation of the indexes).
\section{Time-evolving block decimation: linearized contraction and boundary-state methods}
\label{iTEBD}
The TEBD\index{TEBD} algorithm by Vidal was developed originally for simulating the time evolution of 1D quantum models \cite{V03TEBD,V04TEBD,V07iTEBD}. The (finite and infinite) TEBD algorithm has been widely applied to varieties of issues, such as criticality in quantum many body systems (e.g., \cite{TOIL08EntScaling, PMTM09EntScaling, PM10EScrit}), the topological phases \cite{PT12SPT}. the many-body localization \cite{DSK2013mbl, BPF2012unbounded, PPH2015mbl} and the thermodynamic property of quantum many-body systems \cite{PMJ2010entanglement, PMW2014correlations, BPG2009relaxation, FCE2014relaxation, BPG2010quantum, EF2016quench, LRGZX+11LTRG}.
In the language of TN\index{TN}, TEBD solves the TN contraction problems in a linearized manner, and the truncation is calculated in the context of an MPS. In the following, let us explain the infinite TEBD (iTEBD)\index{iTEBD} algorithm \cite{V07iTEBD} (Fig. \ref{fig-3iTEBD}) by still taking the infinite square TN formed by the copies of a forth-order tensor $T$ as an example. In each step, a row of tensors (which can be regarded as an MPO\index{MPO}) are contracted to an MPS\index{MPS} $|\psi\rangle$. Inevitably, the bond dimensions of the tensors in the MPS will increase exponentially as the contractions proceed. Therefore, truncations are necessary to prevent the bond dimensions diverging. The truncations are determined by minimizing the distance between the MPS's before and after the truncation. After the MPS $|\psi\rangle$ converges, the TN contraction becomes $\langle\psi |\psi\rangle$, which can be exactly and easily computed.
\textbf{\textit{Contraction}}. We use is two-site translational invariant MPS, which is formed by the tensors $A$ and $B$ on the sites and the spectrum $\Lambda$ and $\Gamma$ on the bonds as
\begin{eqnarray}
\sum_{\{a\}} \cdots \Lambda_{a_{n-1}} A_{s_{n-1}, a_{n-1} a_{n}} \Gamma_{a_{n}} B_{s_{n}, a_{n} a_{n+1}} \Lambda_{a_{n+1}} \cdots.
\end{eqnarray}
In each step of iTEBD, the contraction is given by
\begin{eqnarray}
A_{s, \tilde{a} \tilde{a}'} \leftarrow \sum_{s'} T_{s b s' b'} A_{s', a a'}, \ \ B_{s, \tilde{a} \tilde{a}'} \leftarrow \sum_{s'} T_{s b s' b'} B_{s', a a'},
\label{eq-3iTEBDEvolve}
\end{eqnarray}
where the new virtual bonds are entangled, satisfying $\tilde{a} = (b,a)$ and $\tilde{a}' = (b',a')$. Meanwhile, the spectrum are also updated as
\begin{eqnarray}
\Lambda_{\tilde{a}} \leftarrow \Lambda_{a} \textbf{1}_{b}, \ \ \Gamma_{\tilde{a}'} \leftarrow \Gamma_{a'} \textbf{1}_{b'},
\end{eqnarray}
where $\textbf{1}$ is a vector with $\textbf{1}_b=1$ for any $b$.
It is readily to see that the number of tensors in iTEBD will be reduced linearly as $tN$, with $t$ the number of the contraction-and-truncation steps and $N \to \infty$ the number of the columns of the TN. Therefore, iTEBD (also finite TEBD) can be considered as a \textit{linearized contraction algorithm}, in contrast to the exponential contraction algorithm like TRG\index{TRG}.
\begin{figure}[tbp]
\centering
\includegraphics[angle=0,width=\linewidth]{iTEBD.eps}
\caption{(Color online) The illustration of the contraction and truncation of the iTEBD\index{iTEBD} algorithm. In each iteration step, a row of tensors in the TN are contracted to the MPS, and truncations by SVD\index{SVD} are implemented so that the bond dimensions of the MPS\index{MPS} keep unchanged.}
\label{fig-3iTEBD}
\end{figure}
\textbf{\textit{Truncation}}. Truncations are needed when the dimensions of the virtual bonds exceed the preset dimension cut-off $\chi$. In the original version of iTEBD \cite{V07iTEBD}\index{iTEBD}, the truncations are done by local SVD's\index{SVD}. To truncate the virtual bond $\tilde{a}$ for example, one defines a matrix by contracting the tensors and spectrum connected to the target bond as
\begin{eqnarray}
M_{s_1 \tilde{a}_1,s_{2} \tilde{a}_2} = \sum_{\tilde{a}} \Lambda_{\tilde{a}_1} A_{s_1, \tilde{a}_1 \tilde{a}} \Gamma_{\tilde{a}} B_{s_2, \tilde{a} \tilde{a}_2} \Lambda_{\tilde{a}_2}.
\end{eqnarray}
Then, perform SVD on $M$, keeping only the $\chi$-largest singular values and the corresponding basis as
\begin{eqnarray}
M_{s_1 \tilde{a}_1,s_2 \tilde{a}_2} = \sum_{a=0}^{\chi-1} U_{s_1, \tilde{a}_1 a} \Gamma_{a} V_{s_2, a \tilde{a}_2}.
\label{eq-3iTEBDenv}
\end{eqnarray}
The spectrum $\Gamma$ is updated by the singular values of the above SVD. The tensors $A$ and $B$ are also updated as
\begin{eqnarray}
A_{s_1, \tilde{a} a} = (\Lambda_{\tilde{a}})^{-1} U_{s_1, \tilde{a} a}, \ \
B_{s_2, a \tilde{a}} = V_{s_2, a \tilde{a}} (\Lambda_{\tilde{a}})^{-1}.
\label{eq-3iTEBDtruncate}
\end{eqnarray}
Till now, the truncation of the spectrum $\Gamma$ and the corresponding virtual bond have been completed. Any spectra and virtual bonds can be truncated similarly.
\textbf{\textit{Error and environment}}. Similar to TRG\index{TRG} and SRG\index{SRG}, the environment of the original iTEBD is $M$ in Eq. (\ref{eq-3iTEBDenv}), and the error is measured by the discarded singular values of $M$. Thus iTEBD\index{iTEBD} seems to only use local information to optimize the truncations. What is amazing is that when the MPO\index{MPO} is unitary or near unitary, the MPS converges to a so-called \textit{canonical form} \cite{PVWC07MPSRev, OV08canonical}. The truncations are then optimal by taking the whole MPS as the environment. If the MPO is far from being unitary, Or\'us and Vidal proposed the \textit{canonicalization} algorithm \cite{OV08canonical} to transform the MPS\index{MPS} into the canonical form before truncating. We will talk about this issue in detail in the next section.
\textbf{\textit{Boundary-state methods: density matrix renormalization group and variational matrix product state}}. The iTEBD\index{iTEBD} can be understood as a boundary-state method. One may consider one row of tensors in the TN\index{TN} as an MPO\index{MPO} (see Sec. \ref{sec-MPO} and Fig. \ref{fig-2MPO}), where the vertical bonds are the ``physical'' indexes and the bonds shared by two adjacent tensors are the geometrical indexes. This MPO is also called the \textit{transfer operator} or \textit{transfer MPO} of the TN. The converged MPS\index{MPS} is in fact the dominant eigenstate of the MPO \footnote{For simplicity, we assume the MPO gives an Hermitian operator so that its eigenstates and eigenvalues are well-defined.}. While the MPO represents a physical Hamiltonian or the imaginary-time evolution operator (see Sec. \ref{sec-phys2TN}), the MPS is the ground state. For more general situations, e.g., the TN represents a 2D partition function or the inner product of two 2D PEPS's, the MPS can be understood as the \textit{boundary state} of the TN (or the PEPS\index{PEPS}) \cite{CPSV17MPO, SPCP13boundary, CPSV11boundaryMPS}. The contraction of the 2D infinite TN becomes computing the boundary state, i.e., the dominant eigenstate (and eigenvalue) of the transfer MPO.
The boundary-state scheme gives several non-trivial physical and algorithmic implications \cite{ RPLLS17Scaling2D,CPSV17MPO, YLPVV+14boundary,SPCP13boundary, CPSV11boundaryMPS}, including the underlying resemblance between iTEBD\index{iTEBD} and the famous infinite DMRG (iDMRG)\index{iDMRG} \cite{M08iDMRGarxiv}. DMRG \cite{W92DMRG, W93DMRG} follows the idea of Wilson's NRG\index{NRG} \cite{W75NRGRev}, and solves the ground states and low-lying excitations of 1D or quasi-1D Hamiltonians (see several reviews \cite{S11DMRGRev, SW12DMRG2DRev, S05DMRGrev, CS11DMRGrevChem}); originally it has no direct relations to TN contraction problems. After the MPS and MPO become well understood, DMRG\index{DMRG} was re-interpreted in a manner that is more close to TN (see a review by Schollw\"ock \cite{S11DMRGRev}). In particular for simulating the ground states of infinite-size 1D systems, the underlying connections between the iDMRG and iTEBD were discussed by McCulloch \cite{M08iDMRGarxiv}. As argued above, the contraction of a TN can be computed by solving the the dominant eigenstate of its transfer MPO. The eigenstates reached by iDMRG and iTEBD are the same state up to a gauge transformation (note the gauge degrees of freedom of MPS will be discussed in Sec. \ref{sec.gauge}). Considering that DMRG mostly is not used to compute TN contractions and there are already several understanding reviews, we skip the technical details of the DMRG algorithms here. One may refer to the papers mentioned above if interested. However, later we will revisit iDMRG in the clue of multi-linear algebra.
Variational matrix-product-state (VMPS)\index{VMPS} method is a variational version of DMRG\index{DMRG} for (but not limited to) calculating the ground states of 1D systems with periodic boundary condition \cite{VPC04DMRGQinfo}. Compared with DMRG, VMPS is more directly related to TN\index{TN} contraction problems. In the following, we explain VMPS by solving the contraction of the infinite square TN. As discussed above, it is equivalent to solve the dominant eigenvector (denoted by $|\psi \rangle$) of the infinite MPO\index{MPO} (denoted by $\hat{rho}$) that is formed by a row of tensors in the TN. The task is to minimize $\langle \psi| \hat{\rho} |\psi \rangle$ under the constraint $\langle \psi| \psi \rangle = 1$. The eigenstate $|\psi \rangle$ written in the form of an MPS\index{MPS}.
The tensors in $|\psi \rangle$ are optimized on by one. For instance, to optimize the $n$-th tensor, all other tensors are kept unchanged and considered as constants. Such a local minimization problem becomes $\hat{H}^{eff} |T_n\rangle = \mathcal{E} \hat{N}^{eff} |T_n\rangle$ with $\mathcal{E}$ the eigenvalue. $\hat{H}^{eff}$ is given by a 6-th order tensor defined by contracting all tensors in $\langle \psi| \hat{\rho} |\psi \rangle$ except for the $n$-th tensor and its conjugate [Fig. \ref{fig-3vMPS} (a)]. Similarly, $\hat{N}^{eff}$ is also given by a 6-th order tensor defined by contracting all tensors in $\langle \psi| \psi \rangle$ except for the $n$-th tensor and its conjugate [Fig. \ref{fig-3vMPS} (b)]. Again, the VMPS is different from the MPS obtained by TEBD\index{TEBD} only up to a gauge transformation.
\begin{figure}[tbp]
\centering
\includegraphics[angle=0,width=\linewidth]{VMPS.eps}
\caption{(Color online) The illustration of (a) $\hat{H}^{eff}$ and (b) $\hat{N}^{eff}$ in the variational matrix product state method}
\label{fig-3vMPS}
\end{figure}
Note that the boundary-state methods are not limited to solving TN contractions. An example is the time-dependent variational principle (TDVP)\index{TDVP}. The basic idea of TDVP was proposed by Dirac in 1930 \cite{dirac1930TDVP}, and then it was cooperated with the formulation of Hamiltonian \cite{KERMAN1976TDVP} and action function \cite{JACKIW1979TDVP}. For more details, one could refer to a review by Langhoff \textit{et al} \cite{LEK72TDVP}. In 2011, TDVP was developed to simulate the time evolution of many-body systems with the help of MPS \cite{HCOPVV11TDVP}. Since TDVP (and some other algorithms) concerns directly a quantum Hamiltonian instead of the TN\index{TN} contraction, we skip giving more details of these methods in this paper.
\section{Transverse contraction and folding trick}
\label{transverse}
\begin{figure}[tbp]
\centering
\includegraphics[width=0.9\columnwidth]{pg43fig23a}
\caption{Transverse contraction of the TN\index{TN} for a local expectation value $\langle O(t) \rangle$.}
\label{fig:folding}
\end{figure}
For the boundary-state methods introduced above, the boundary states are defined in the real space. Taking iTEBD\index{iTEBD} for the real-time evolution as an example, the contraction is implemented along the time direction, which is to do the time evolution in an explicit way. It is quite natural to consider implementing the contraction along the other direction. In the following, we will introduce the transverse contraction and the folding trick proposed and investigated in Refs. \cite{BHVC09folding, MCB12folding, HM15folding}. The motivation of transverse contraction is to avoid the explicit simulation of the time-dependent state $|\psi(t)\rangle$ that might be difficult to capture due to the fast growth of its entanglement.
\textbf{\textit{Transverse contraction}}. Let us consider to calculate the average of a one-body operator $o(t) = \langle \psi(t)| \hat{o} |\psi(t)\rangle$ with $|\psi(t)\rangle$ that is a quantum state of infinite size evolved to the time $t$. The TN\index{TN} representing $o(t)$ is given in the left part of Fig. \ref{fig:folding}, where the green squares give the initial MPS $|\psi(0)\rangle$ and its conjugate, the yellow diamond is $\hat{o}$, and the TN formed by the green circles represents the evolution operator $e^{it\hat{H}}$ and its conjugate (see how to define the TN in Sec. \ref{sec.2.evolution}).
To perform the transverse contraction, we treat each column of the TN\index{TN} as an MPO\index{MPO} $\hat{\mathcal{T}}$. Then as shown in the right part of Fig. \ref{fig:folding}, the main task of computing $o(t)$ becomes to solve the dominant eigenstate $|\phi \rangle$ (normalized) of $\hat{\mathcal{T}}$, which is an MPS\index{MPS} illustrated by the purple squares. One may solve this eigenstate problems by any of the boundary-state methods (TEBD\index{TEBD}, DMRG\index{DMRG}, etc.). With $|\phi\rangle$, $o(t)$ can be exactly and efficiently calculated as
\begin{equation}
o(t) = \frac{\langle \psi (t) | \hat{o} | \psi (t) \rangle }{\langle \psi (t) | \psi (t) \rangle} = \frac{\langle \phi | \hat{\mathcal{T}}_o | \phi \rangle }{\langle \phi | \hat{\mathcal{T}} | \phi \rangle},
\end{equation}
with $\hat{\mathcal{T}}_o$ is the column that contains the operator $\hat{o}$. Note that the length of $|\phi\rangle$ (i.e., the number of tensors in the MPS\index{MPS}) is proportional to the time $t$, thus one should use the finite-size versions of the boundary-state methods. It should also be noted that $\hat{\mathcal{T}}$ may not be Hermitian. In this case, one should not use $|\phi \rangle$ and its conjugate, but compute the left and right eigenstates of $\hat{\mathcal{T}}$ instead.
Interestingly, similar ideas of the transverse contraction appeared long before the concept of TN emerged. For instance, transfer matrix renormalization group (TMRG)\index{TMRG} \cite{BXG96TMRG, WX97TMRG, S97TMRG, N95TMRG2Dclassic} can be used to simulate the finite-temperature properties of a 1D system. The idea of TMRG is to utilize DMRG\index{DMRG} to calculate the dominant eigenstate of the transfer matrix (similar to $\mathcal{T}$). In correspondence with the TN\index{TN} terminology, it is to use DMRG to compute $|\phi \rangle$ from the TN that defines the imaginary-time evolution. We will skip of the details of TMRG since it is not directly related to TN. One may refer the related references if interested.
\begin{figure}[tbp]
\centering
\includegraphics[width=1\linewidth]{pg43fig23b}
\caption{The illustration of the folding trick.}
\label{fig:folding2}
\end{figure}
\textbf{\textit{Folding trick}}. The main bottleneck of a boundary-state method concerns the entanglement of the boundary state. In other words, the methods will become inefficient when the entanglement of the boundary state grows too large. One example is the real-time simulation of a 1D chain, where the entanglement entropy increases linearly with time. Solely with the transverse contraction, it will not essentially solve this problem. Taking the imaginary-time evolution as an example, it has been shown that with the dual symmetry of space and time, the boundary states in the space and time directions possess the same entanglement \cite{HM15folding, TTLR18tMPS}.
In Ref. \cite{BHVC09folding}, the folding trick was proposed. The idea is to ``fold'' the TN\index{TN} before the transverse contraction (Fig. \ref{fig:folding2}). In the folded TN, each tensor is the tensor product of the original tensor and its conjugate. The length of the folded TN in the time direction is half of the original TN, and so is the length of the boundary state.
The previous work (Ref. \cite{BHVC09folding}) on the dynamic simulations of 1D spin chains showed that the entanglement of the boundary state is in fact reduced compared with that of the boundary state without folding. This suggests that the folding trick provides a more efficient representation of the entanglement structure of the boundary state. The authors of Ref. \cite{BHVC09folding} suggested an intuitive picture to understand the folding trick. Consider a product state as the initial state at $t-0$ and a single localized excitation at the position $x$ that propagates freely with velocity $v$. By evolving for a time $t$, only $(x \pm vt)$ sites will become entangled. With the folding trick, the evolutions (that are unitary) besides the $(x \pm vt)$ sites will not take effects since they are folded with the conjugates and become identities. Thus the spins outside $(x \pm vt)$ will remain product state and will not contribute entanglement to the boundary state. In short, one key factor to consider here is the entanglement structure, i.e., the fact that the TN is formed by unitaries. The transverse contraction with the folding trick is a convincing example to show that the efficiency of contracting a TN can be improved by properly designing the contraction way according to the entanglement structure of the TN.
\section{Relations to exactly contractible tensor networks and entanglement renormalization}
\begin{figure}[tbp]
\centering
\includegraphics[angle=0,width=0.85\linewidth]{Fig21.eps}
\caption{(Color online) The exactly contractible TN in the HOTRG algorithm.}
\label{fig-3HOTRGexact}
\end{figure}
The TN algorithms explained above are aimed at dealing with contracting optimally the TN's that cannot be exactly contracted. Then a question rises: is a classical computer really able to handle these TN's? In the following, we show that by explicitly putting the isometries for truncations inside, the TN's that are contracted in these algorithms become eventually exactly contractible, dubbed as exactly contractible TN (ECTN)\index{ECTN}. Different algorithms lead to different ECTN. That means the algorithm will show a high performance if the TN can be accurately approximated by the corresponding ETNC.
Fig. \ref{fig-3HOTRGexact} shows the ECTN emerging in the plaquette renormalization \cite{WKS11TRG} or higher-order TRG (HOTRG)\index{HOTRG}\index{TRG} algorithms \cite{XCQZYX12HOSRG}. Take the contraction of a TN\index{TN} (formed by the copies of tensor $T$) on square lattice as an example. In each iteration step, four nearest-neighbor $T$'s in a square are contracted together, which leads to a new square TN formed by tensors ($T^{(1)}$) with larger bond dimensions. Then, isometries (yellow triangles) are inserted in the TN to truncate the bond dimensions (the truncations are in the same spirit of those in CTMRG\index{CTMRG}, see Fig. \ref{fig-3CTMRG}). Let us not contract the isometries with the tensors, but leave them there inside the TN. Still, we can move on to the next iteration, where we contract four $T^{(1)}$'s (each of which is formed by four $T$ and the isometries, see the darked-red plaques in Fig. \ref{fig-3HOTRGexact}) and obtain more isometries for truncating the bond dimensions of $T^{(1)}$. By repeating this process for several times, one can see that tree TN's appear on the boundaries of the coarse-grained plaques. Inside the 4-by-4 plaques (light red shadow), we have the two-layer tree TN's formed by three isometries. In the 8-by-8 plaques, the tree TN has three layers with seven isometries. These tree TN's separate the original TN into different plaques, so that it can be exactly contracted, similar to the fractal TN's introduced in Sec. \ref{Sec-exactTN}.
\begin{figure}[tbp]
\centering
\includegraphics[angle=0,width=0.7\linewidth]{Fig22.eps}
\caption{(Color online) The exactly contractible TN in the iTEBD algorithm.}
\label{fig-3iTEBDexact}
\end{figure}
In the iTEBD\index{iTEBD} algorithm \cite{V03TEBD,V04TEBD,V07iTEBD,OV08canonical} (Fig. \ref{fig-3iTEBDexact}), one starts with an initial MPS\index{MPS} (dark blue squares). In each iteration, one tensor (light blue circles) in the TN\index{TN} is contracted with the tensor in the MPS and then the bonds are truncated by isometries (yellow triangles). Globally seeing, the isometries separate the TN into many ``tubes'' (red shadow) that are connected only at the top. The length of the tubes equals to the number of the iteration steps in iTEBD. Obviously, this TN is exactly contractible. Such a tube-like structure also appears in the contraction algorithms based on PEPS.
\begin{figure}[tbp]
\centering
\includegraphics[angle=0,width=0.7\linewidth]{Fig23.eps}
\caption{(Color online) A part of the exactly contractible TN in the CTMRG algorithm.}
\label{fig-3CTMRGexact}
\end{figure}
For the CTMRG\index{CTMRG} algorithm \cite{OV09CTMRG}, the corresponding ECTN\index{ECTN} is a little bit complicated (see one quarter of it in Fig. \ref{fig-3CTMRGexact}). The initial row (column) tensors and the corner transfer matrices are represented by the pink and green squares. In each iteration step, the tensors (light blue circles) located most outside are contracted to the row (column) tensors and the corner transfer matrices, and isometries are introduced to truncate the bond dimensions. Globally seeing the picture, the isometries separate the TN into a tree-like structure (red shadow), which is exactly contractible.
For these three algorithms, each of them gives an ECTN that is formed by two part: the tensors in the original TN and the isometries that make the TN exactly contractible. After optimizing the isometries, the original TN is approximated by the ECTN. The structure of the ECTN depends mainly on the contraction strategy and the way of optimizing the isometries depend on the chosen environment.
The ECTN picture shows us explicitly how the correlations and entanglement are approximated in different algorithms. Roughly speaking, the correlation properties can be read from the minimal distance of the path in the ECTN that connects two certain sites, and the (bipartite) entanglement can be read from the number of bonds that cross the boundary of the bipartition. How well the structure suits the correlations and entanglement should be a key factor of the performance of a TN contraction algorithm. Meanwhile, this picture can assist us to develop new algorithms by designing the ECTN and taking the whole ECTN as the environment for optimizing the isometries. These issues still need further investigations.
The unification of the TN contraction and the ECTN\index{ECTN} has been explicitly utilized in the TN renormalization (TNR)\index{TNR} algorithm \cite{EV15TNR,EV15TNRmera}, where both isometries and unitaries (called \textit{disentangler}) are put into the TN\index{TN} to make it exactly contractible. Then instead of tree TN's or MPS's\index{MPS}, one will have MERA's\index{MERA} (see Fig. \ref{fig-1TTNS} (c) for example) inside which can better capture the entanglement of critical systems.
\section{A shot summary}
In this section, we have discussed about several contraction approaches for dealing with 2D TN's. Applying these algorithms, many challenging problems can be efficiently solved, including the ground-state and finite-temperature simulations of 1D quantum systems, and the simulations of 2D classical statistic models. Such algorithms consist of two key ingredients: contractions (local operations of tensors) and truncations. The local contraction determines the way how the TN is contracted step by step, or in other words, how the entanglement information is kept according to the ECTN\index{ECTN} structure. Different (local or global) contractions may lead to different computational costs, thus optimizing the contraction sequence is necessary in many cases \cite{BHVC09folding,EP14TNalgo,PHV14TNCorder}. The truncation is the approximation to discard less important basis so that the computational costs are properly bounded. One essential concept in the truncations is ``environment'', which plays the role of the reference when determining the weights of the basis. Thus, the choice of environment concerns the balance between the accuracy and efficiency of a TN\index{TN} algorithm.
\chapter{Tensor network approaches for higher-dimensional quantum lattice models}
\label{sec4}
\abstract{In this section, we will show several representative TN\index{TN} approaches for simulating the quantum lattice models in ($d>1$) dimensions. We will mainly use the language of TN contractions. One may refer to several existing reviews \cite{S11DMRGRev, VMC08MPSPEPSRev, CV09TNSRev, O14TNSRev, HV17TMTNrev} for more exhaustive understanding on the TN simulations for quantum problems. We will focus on the algorithms based on PEPS\index{PEPS}, and show the key roles that the 2D TN contraction algorithms presented in Sec. \ref{sec3} play in the higher-dimensional cases.}
\section{Variational approaches of projected-entangled pair state}
Without losing generality, we consider a 2D quantum system with nearest-neighbor coupling on an infinite square lattice as an example. The ground state can be represented by an iPEPS\index{iPEPS} (see Sec. \ref{sec.peps}). Similar to MPS\index{MPS} (Sec. \ref{sec.2.evolution}), the central task is to minimize the energy
\begin{eqnarray}
E= \frac{\langle \psi | \hat{H} | \psi \rangle}{ \langle \psi | \psi \rangle}.
\label{eq-4Eg}
\end{eqnarray}
There are in general two ways to do the minimization. One way proposed firstly by Verstraete and Cirac \cite{VC06PEPSArxiv} is considering the elements in the tensors as variational parameters. The tensors in the TN are updated one by one. In a similar spirit as the boundary-state methods (see Sec. \ref{iTEBD}), the key of this approach is to transform the global minimization to local ones, where one tensor (say $P^{[i]}$, see the PEPS form in Eq. (\ref{eq-2PEPSsimple}), Sec. \ref{sec.peps}) is updated by a local minimization problem
\begin{eqnarray}
E = \frac{P^{[i]\dagger} \hat{H}^{eff} P^{[i]}}{P^{[i]\dagger} \hat{N}^{eff} P^{[i]}}.
\label{eq-4Heff}
\end{eqnarray}
$\hat{H}^{eff}$ is an ``effective'' Hamiltonian by computing $\langle \psi | \hat{H} | \psi \rangle$ but after taking $P^{[i]\dag}$ in $\langle \psi |$ and $P^{[i]}$ in $| \psi \rangle$ out. Fig. \ref{fig-variational_PEPS} depicts $\hat{H}^{eff}$ where $\hat{H}$ is written as an infinite PEPO\index{PEPO} (iPEPO\index{iPEPO}, also see Sec. \ref{sec-MPO} for PEPO) for a better illustration. Similarly, $\hat{N}^{eff}$ is defined by computing $\langle \psi | \psi \rangle$ but after taking $P^{[i]\dag}$ and $P^{[i]}$ out.
Obviously, the computations of both $\hat{H}^{eff}$ and $\hat{N}^{eff}$ are in fact to contract the corresponding 2D TN's where the 2D TN\index{TN} contraction algorithms are needed. In \cite{C16vPEPS}, Corboz used CTMRG\index{CTMRG} (see \cite{NO96CTMRG0} or Sec. \ref{sec.CTMRG}) to compute the contractions. In \cite{VHCV16GrdPEPS}, Vanderstraeten \textit{et al} further developed this idea to a gradient method, where the gradient is calculated by implementing similar 2D TN contractions. The gradient is given as
\begin{eqnarray}
\frac{\partial E}{\partial P^{[i]\dag}} = \frac{\partial \langle \psi | \hat{H} | \psi \rangle / \langle \psi | \psi \rangle}{\partial P^{[i]\dag}} \nonumber = 2 \frac{\partial_{P^{[i]\dag}} \langle \psi | \hat{H} | \psi \rangle}{\langle \psi | \psi \rangle} - 2 \frac{\langle \psi | \hat{H} | \psi \rangle}{\langle \psi | \psi \rangle^2} \partial_{P^{[i]\dag}} \langle \psi | \psi \rangle.
\label{eq-4gradient}
\end{eqnarray}
By imposing the normalization condition $\langle \psi | \psi \rangle = 1$ and shifting the ground-state energy to zero by $\hat{H} \leftarrow \hat{H} - \langle \psi | \hat{H} | \psi \rangle$, the gradient is simplified as
\begin{eqnarray}
\frac{\partial E}{\partial P^{[i]\dag}} = 2 \partial_{P^{[i]\dag}} \langle \psi | \hat{H} | \psi \rangle.
\label{eq-4gradient_simp}
\end{eqnarray}
Thus the gradient is computed by contracting the TN of $\langle \psi | \hat{H} | \psi \rangle$ after taking $P^{[i]\dag}$ out.
The gradient method is consistent with the effective Hamiltonian schemes. In fact, one has $\frac{\partial E}{\partial P^{[i]\dag}} = 2\hat{H}^{eff} P^{[i]}$. At the minimal point, the gradient should vanish $\frac{\partial E}{\partial P^{[i]\dag}} = 0$. It means $2\hat{H}^{eff} P^{[i]} = 0$, i.e., $P^{[i]}$ is the dominant eigenstate of $\hat{H}^{eff}$ with a zero eigenvalue. Considering the ground-state energy is shifted to zero, $P^{[i]}$ is the ground state of the effective Hamiltonian $\hat{H}^{eff}$.
Note that the infinite PEPO (iPEPO)\index{iPEPO} representation is not enforced to define $\hat{H}^{eff}$. In fact, it is not easy to obtain the iPEPO of an arbitrary 2D (or 3D) Hamiltonian. The usual way is to start from the summation form of the Hamiltonian $\hat{H} = \sum \hat{H}_{ij}$, and compute the contribution to $\hat{H}^{eff}$ from each $\hat{H}_{ij}$ separately \cite{C16vPEPS}. Each term is computed by contracting a 2D TN, where one can reuse the results to improve the efficiency.
\begin{figure}[tbp]
\centering
\includegraphics[angle=0,width=0.6\linewidth]{Variational_PEPS.eps}
\caption{(Color online) The illustration of $\hat{H}^{eff}$ in Eq. (\ref{eq-4Heff}).}
\label{fig-variational_PEPS}
\end{figure}
Following the same clue (minimizing $E$), algorithms were proposed to combine TN with the QMC\index{QMC} methods \cite{SV09TNQMC, SWVC08QMCTN, SCSC10QMCTN, WPV11TNQMC, LDHGH17TNQMC}. Still let us focus on those based on PEPS\index{PEPS}. One may transform Eq. (\ref{eq-4Eg}) as
\begin{eqnarray}
E = \frac{\sum_{S,S'} W(S') \langle S'|\hat{H}| S\rangle W(S)}{\sum_{S} W(S)^2},
\end{eqnarray}
where $S = (s_1, s_2, \cdots)$ goes through all spin configurations and $W(S) = \langle s_1 s_2 \cdots| \psi \rangle$ is the coefficient of the iPEPS for the given configuration. QMC sampling can be implemented by defining the weight function as $W(S)^2$ and the estimator $E(S)$ as
\begin{eqnarray}
E(S) = \sum_{S'} \frac{W(S')}{W(S)} \langle S'|\hat{H}| S\rangle,
\end{eqnarray}
so that the energy becomes
\begin{eqnarray}
E = \langle E(S) \rangle = \sum_{S} W(S)^2 E(S).
\end{eqnarray}
It is easy to see that the normalization condition of the weights $\sum_S W(S)^2 = 1$ is satisfied.
The task becomes to compute $W(S)$ and $\langle S'|\hat{H}| S\rangle$ with different configurations. The computation of $\langle S'|\hat{H}| S\rangle$ is relatively easy since $|S\rangle$ and $|S'\rangle$ are just two product states. The computation of $W(S)$ is more tricky. When $|\psi \rangle$ is a PEPS on a square lattice, $W(S)$ is a 2D scalar TN\index{TN} by fixing all the physical indexes of the PEPS\index{PEPS} as
\begin{eqnarray}
W(S) = \text{tTr} \prod_{n} P^{[n]}_{s_n},
\label{eq-4TNQMCW}
\end{eqnarray}
where $P^{[n]}_{s_n}$ is a forth-order tensor that only has the geometrical index \footnote{One may refer to Eq. (\ref{eq-2PEPSsimple}) to better understand Eq. (\ref{eq-4TNQMCW})}. The $n$-th physical index is taken as $s_n$. Considering that most of the configurations are not translationally invariant, such QMC-TN\index{QMC} methods are usually applied to finite-size models. One may use the finite-TN version of the algorithms reviewed in Sec. \ref{sec3}.
\section{Imaginary-time evolution methods}
\label{sec.evo2D}
Another way to compute the ground-state iPEPS is to do imaginary-time evolution, analog to the MPS methods presented in Sec. \ref{sec.2.evolution}. For a $d$-dimensional quantum model, its ground-state simulation can be considered as computing the contraction of a ($d+1$)-dimensional TN.
Firstly, let us show how the evolution operator for an infinitesimal imaginary-time step $\tau$ can be written as an iPEPO, which is in fact one layer of the 3D TN (Fig. \ref{fig-TPO_iEV}). The evolution of the iPEPS is put the iPEPS at the bottom and to contract the TN layer by layer to it.
\begin{figure}[tbp]
\centering
\includegraphics[angle=0,width=0.8\linewidth]{TPO_iTE.eps}
\caption{(Color online) The evolution of a PEPS\index{PEPS} can be mapped to the contraction of a 3D TN\index{TN}.}
\label{fig-TPO_iEV}
\end{figure}
To proceed, we divide the local Hamiltonians on the square lattice into four group: $\hat{H}_{e,e} = \sum_{\text{even i,j}} \hat{H}^{[i,j;i,j+1]} + \hat{H}^{[i.j;i+1,j]}$, $\hat{H}_{o,o} = \sum_{\text{odd i,j}} \hat{H}^{[i,j;i,j+1]} + \hat{H}^{[i.j;i+1,j]}$, $\hat{H}_{e,o} = \sum_{\text{even i, odd j}} \hat{H}^{[i,j;i,j+1]} + \hat{H}^{[i.j;i+1,j]}$ and $H_{o,e} = \sum_{\text{odd i, even j}} \hat{H}^{[i,j;i,j+1]} + \hat{H}^{[i.j;i+1,j]}$. One can see that each two terms in one group commute to each other. The evolution operator for an infinitesimal imaginary-time step ($\tau \to 0$) then can be written as
\begin{equation}
\hat{U} = \exp(-\tau \hat{H})
= \exp(-\tau \hat{H}^{[e,e]}) \exp(-\tau \hat{H}^{[o,o]}) \exp(-\tau \hat{H}^{[e,o]}) \exp(-\tau \hat{H}^{[o,e]}) + O(\tau^2).
\label{eq-4evolU}
\end{equation}
Let us assume translational invariance to the Hamiltonian, i.e., $\hat{H}^{[i,j]} = \hat{H}^{[two]}$. The element of two-body evolution operator is a forth-order tensor $U_{s_is_js_i's_j'} = \langle s_i's_j'| \exp(-\tau \hat{H}^{[two]}) |s_is_j \rangle$. Implement SVD\index{SVD} or QR\index{QR} decomposition on $U$ (\ref{fig-TPO_iEV}) as
\begin{eqnarray}
U_{s_is_js_i's_j'} = \sum_{\alpha} L_{s_is_i', a} R_{s_js_j',a}.
\end{eqnarray}
Then the two tensors $T^{[L]}$ and $T^{[R]}$ that form the iPEPO\index{iPEPO} of $\hat{U}$ is obtained as
\begin{eqnarray}
\begin{aligned}
T^{[L]}_{ss', a_1a_2a_3a_4} = \sum_{s_1s_2s_3} L_{ss_1,a_1} L_{s_1s_2,a_2} L_{s_2s_3, a_3} L_{s_3s',a_4}, \\
T^{[R]}_{ss', a_1a_2a_3a_4} = \sum_{s_1s_2s_3} R_{ss_1,a_1} R_{s_1s_2,a_2} R_{s_2s_3, a_3} R_{s_3s',a_4}.
\label{eq-4PEPSevolve}
\end{aligned}
\end{eqnarray}
The four $L$'s (or $R$'s) in $T^{[L]}$ (or $T^{[R]}$) correspond to the evolution operators of the two-body terms in $\hat{H}^{[e,e]}$, $\hat{H}^{[o,o]}$, $\hat{H}^{[e,o]}$, and $\hat{H}^{[o,e]}$ in Eq. (\ref{eq-4evolU}), respectively (see the left part of Fig. \ref{fig-TPO_iEV}).
While the TN for the imaginary-time evolution with the iPEPO\index{iPEPO} is a cubic TN\index{TN}, one may directly use the tensor $U$, which also gives a 3D but not cubic TN. Without losing generality, we in the following will use the iPEPO to present the algorithms for contraction a cubit TN. The algorithm can be readily applied to deal with the statistic models on cubic lattice or other problems that can be written as the contraction of a cubic TN.
The evolution $\hat{U} |\psi \rangle$ is to contract the iPEPO (one layer of the tensors) to the iPEPS\index{iPEPS}. In accordance to the translational invariance of the iPEPO, the iPEPS is also formed by two inequivalent tensors (denoted by $P^{[L]}$ and $P^{[R]}$). Locally, the tensors in the evolved iPEPS are given as
\begin{eqnarray}
\tilde{P}^{[L]}_{s, \tilde{\alpha}_1\tilde{\alpha}_2\tilde{\alpha}_3\tilde{\alpha}_4} = \sum_{s'} T^{[L]}_{ss', a_1a_2a_3a_4} P^{[L]}_{s', \alpha_1\alpha_2\alpha_3\alpha_4}, \\
\tilde{P}^{[R]}_{s, \tilde{\alpha}_1\tilde{\alpha}_2\tilde{\alpha}_3\tilde{\alpha}_4} = \sum_{s'} T^{[R]}_{ss', a_1a_2a_3a_4} P^{[R]}_{s', \alpha_1\alpha_2\alpha_3\alpha_4},
\end{eqnarray}
with the composite indexes $\tilde{\alpha}_x = (a_x, \alpha_x)$ ($x = 1, 2, 3, 4$). Obviously, the bond dimensions of the new tensors are increased by $\dim(a_x)$ times. It is necessary to preset a dimension cut-off $\chi$: when the bond dimensions become larger than $\chi$, approximations will be introduced to reduce the dimensions back to $\chi$. One then can iterate the evolution of the iPEPS with bounded computational cost. After the iPEPS converges, it is considered that the ground state is reached. Therefore, one key step in the imaginary-time schemes (as well as the similar contraction schemes of 3D TN's\index{TN}) is to find the optimal truncations of the enlarged bonds. In the following, we will concentrate on the truncation of bond dimensions, and present three kinds of scheme known as \textit{full}, \textit{simple}, and \textit{cluster} updates according to which environment the truncations are optimized \cite{LCB14PEPScontract} \footnote{The definition of the update schemes also apply to finite-size PEPS and the variational methods; for example, the variational methods which contract the whole TN to update the (i)PEPS are also called full update.}.
\section{Full, simple, and cluster update schemes}
\label{sec.updateschemes}
For truncating the dimensions of the geometrical bonds of an iPEPS\index{iPEPS}, the task is to minimize the distance between the iPEPS's before and after the truncation, i.e.,
\begin{equation}
\mathcal{\varepsilon} = || \tilde{\psi} \rangle - |\psi \rangle|.
\end{equation}
With the normalization condition of the iPEPS's, the problem can be reduced to the maximization of the fidelity
\begin{equation}
\mathcal{Z} = \langle \tilde{\psi} |\psi \rangle.
\label{eq-4fidtrun}
\end{equation}
As discussed in Sec. \ref{sec.QobTN}, $\mathcal{Z}$ is in fact a scalar TN\index{TN}.
\textbf{Full update.} Among the three kinds of update schemes, full update seems to be the most natural and reasonable, in which the truncation is optimized referring to the whole iPEPS \cite{XCQZYX12HOSRG, OV09CTMRG, XJCWX09SRG, LCB14PEPScontract, O12CTMRG, JOVVC08PEPS, PBTCO15FastFullUpdate}. Les us consider a translationally invariant iPEPS. For square lattice, the iPEPS is formed by the infinite copies of two tensors $P^{[L]}$ and $P^{[R]}$ located on the two sub-lattices, respectively. Their evolution is given by Eq. (\ref{eq-4PEPSevolve}). We use $\tilde{P}^{[L]}$ and $\tilde{P}^{[R]}$ to denote the tensors with enlarged bond dimensions. Below, we follow Ref. \cite{XJCWX09SRG} to explain the truncation process. To truncate the forth bond $\tilde{\alpha}_4$ of the tensor for example, one firstly defines the tensor $M$ by contracting a pair of $\tilde{P}^{[L]}$ and $\tilde{P}^{[R]}$ as
\begin{equation}
M_{s_1\tilde{\alpha}_1 \tilde{\alpha}_2 \tilde{\alpha}_3, s_2 \tilde{\alpha}_1' \tilde{\alpha}_2' \tilde{\alpha}_3'} = \sum_{\tilde{\alpha}_4} \tilde{P}^{[L]}_{s_1, \tilde{\alpha}_1 \tilde{\alpha}_2 \tilde{\alpha}_3 \tilde{\alpha}_4} \tilde{P}^{[R]}_{s_2, \tilde{\alpha}_1' \tilde{\alpha}_2' \tilde{\alpha}_3' \tilde{\alpha}_4}.
\end{equation}
Note that $P^{[L]}$ and $P^{[R]}$ share the bond $\tilde{\alpha}_4$ that is to be truncated. Compute the \textit{environment tensor} $M^e$ by contracting the TN of $\mathcal{Z}$ after taking a pair of $\tilde{P}^{[L]}$ and $\tilde{P}^{[R]}$ out from the TN. $\mathcal{M}$ is in fact an eighth-order tensor of the same dimensions as $M$. Decompose $M^e$ by SVD as
\begin{equation}
M^e_{s_1\tilde{\alpha}_1 \tilde{\alpha}_2 \tilde{\alpha}_3, s_2 \tilde{\alpha}_1' \tilde{\alpha}_2' \tilde{\alpha}_3'} = \sum_{\alpha} V^{[L]}_{s_1 \tilde{\alpha}_1 \tilde{\alpha}_2 \tilde{\alpha}_3, \alpha} \Lambda_{\alpha} V^{[R]}_{s_2 \tilde{\alpha}_1' \tilde{\alpha}_2' \tilde{\alpha}_3', \alpha}.
\label{eq-4MSVD}
\end{equation}
Define a new matrix as $\tilde{M} = \Lambda^{1/2} V^{[R]} M V^{[L]} \Lambda^{1/2}$ and decompose it by SVD as $\tilde{M} \simeq \tilde{V}^{[L]} \tilde{\Lambda} \tilde{V}^{[R]}$ by taking only the $\chi$-largest singular values and singular vectors. Finally, two tensors are updated by $P^{[L]} = \tilde{\Lambda}^{1/2} \tilde{V}^{[L]T} \Lambda^{-1/2} V^{[R]T}$ and $P^{[R]} = \tilde{\Lambda}^{1/2} \tilde{V}^{[R]} \Lambda^{-1/2} V^{[L]}$. One can check
\begin{equation}
M_{s_1\tilde{\alpha}_1 \tilde{\alpha}_2 \tilde{\alpha}_3, s_2 \tilde{\alpha}_1' \tilde{\alpha}_2' \tilde{\alpha}_3'} \simeq \sum_{\alpha_4} P^{[L]}_{s_1, \tilde{\alpha}_1 \tilde{\alpha}_2 \tilde{\alpha}_3 \alpha_4} P^{[R]}_{s_2, \tilde{\alpha}_1' \tilde{\alpha}_2' \tilde{\alpha}_3' \alpha_4},
\label{eq-4recoverM}
\end{equation}
with the dimension of the shared bond $\dim(\alpha_4) = \chi$. We shall stress that Eq. (\ref{eq-4recoverM}) is not the SVD\index{SVD} of $M$; the decomposition and truncation are optimized by the SVD of $M^e$, hence is a non-local optimization.
With the formula given above, the task is to compute the environment tensor $M^e$ by the contraction algorithms of 2D TN's\index{TN}. In Ref. \cite{XJCWX09SRG}, the authors developed the SRG\index{SRG}, where $M^e$ is computed by a modified version of TRG\index{TRG} algorithm \cite{LN07TRG}. Other options include iTEBD\index{iTEBD} \cite{JOVVC08PEPS}, CTMRG\index{CTMRG} \cite{OV09CTMRG} and etc. Note that how to define the environment as well as how to truncate by the environment may have subtle differences in different works. The spirit is the same, which is to minimize the fidelity in Eq. (\ref{eq-4fidtrun}) referring to the whole iPEPS\index{iPEPS}.
\textbf{Simple update.} A much more efficient way known as the simple update was proposed by Jiang \textit{et al} \cite{JWX08SimpleUpdate}; it uses local environment to determine the truncations, providing an extremely efficient algorithm to simulate the 2D ground states. As shown in Fig. \ref{fig-1PEPSgraph} (c), the iPEPS\index{iPEPS} used in the simple update is formed by the tensors on the site and the spectra on the bonds: two tensors $P^{[L]}$ and $P^{[R]}$ located on the two sub-lattices, and $\lambda^{[1]}$, $\lambda^{[2]}$, $\lambda^{[3]}$, and $\lambda^{[4]}$ on the four inequivalent geometrical bonds of each tensor. The evolution of the tensors in such an iPEPS is given by Eq. (\ref{eq-4PEPSevolve}). $\lambda^{[i]}$ should be simultaneously evolved as $\tilde{\lambda}^{[i]}_{(a_i, \alpha_i)} = I_{a_i} \lambda_{\alpha_i}$ with $I_{a_i} = 1$.
To truncate the forth geometrical bond of $P^{[L]}$ (and $P^{[R]}$), for example, we construct a new tensor by contracting $P^{[L]}$ and $P^{[R]}$ and the adjacent spectra as
\begin{equation}
M_{s_1\tilde{\alpha}_1 \tilde{\alpha}_2 \tilde{\alpha}_3, s_2 \tilde{\alpha}_1' \tilde{\alpha}_2' \tilde{\alpha}_3'} = \sum_{\tilde{\alpha}_4} \tilde{P}^{[L]}_{s_1, \tilde{\alpha}_1 \tilde{\alpha}_2 \tilde{\alpha}_3 \tilde{\alpha}_4} \tilde{P}^{[R]}_{s_2, \tilde{\alpha}_1' \tilde{\alpha}_2' \tilde{\alpha}_3' \tilde{\alpha}_4} \tilde{\lambda}^{[1]}_{\tilde{\alpha}_1} \tilde{\lambda}^{[2]}_{\tilde{\alpha}_2} \tilde{\lambda}^{[3]}_{\tilde{\alpha}_3} \tilde{\lambda}^{[1]\prime}_{\tilde{\alpha}_1'} \tilde{\lambda}^{[2]\prime}_{\tilde{\alpha}_2} \tilde{\lambda}^{[3]\prime}_{\tilde{\alpha}_3} \tilde{\lambda}^{[4]}_{\tilde{\alpha}_4}.
\end{equation}
Then implement SVD\index{SVD} on $M$ as
\begin{equation}
M_{s_1\tilde{\alpha}_1 \tilde{\alpha}_2 \tilde{\alpha}_3, s_2 \tilde{\alpha}_1' \tilde{\alpha}_2' \tilde{\alpha}_3'} \simeq \sum_{\alpha=1}^{\chi} U^{[L]}_{s_1 \tilde{\alpha}_1 \tilde{\alpha}_2 \tilde{\alpha}_3, \alpha} \lambda_{\alpha} U^{[R]}_{s_2 \tilde{\alpha}_1' \tilde{\alpha}_2' \tilde{\alpha}_3', \alpha}.
\end{equation}
where one takes only the $\chi$-largest singular values and the basis. $P^{[L]}$ and $P^{[R]}$ are updated as
\begin{equation}
\begin{aligned}
P^{[L]}_{s_1, \tilde{\alpha}_1 \tilde{\alpha}_2 \tilde{\alpha}_3 \alpha} = U^{[L]}_{s_1 \tilde{\alpha}_1 \tilde{\alpha}_2 \tilde{\alpha}_3, \alpha} (\tilde{\lambda}^{[1]}_{\tilde{\alpha}_1})^{-1} (\tilde{\lambda}^{[2]}_{\tilde{\alpha}_2})^{-1} (\tilde{\lambda}^{[3]}_{\tilde{\alpha}_3})^{-1}, \\
P^{[R]}_{s_2, \tilde{\alpha}_1 \tilde{\alpha}_2 \tilde{\alpha}_3 \alpha} = U^{[R]}_{s_2 \tilde{\alpha}_1 \tilde{\alpha}_2 \tilde{\alpha}_3, \alpha} (\tilde{\lambda}^{[1]}_{\tilde{\alpha}_1})^{-1} (\tilde{\lambda}^{[2]}_{\tilde{\alpha}_2})^{-1} (\tilde{\lambda}^{[3]}_{\tilde{\alpha}_3})^{-1}.
\end{aligned}
\end{equation}
The spectrum $\tilde{\lambda}_4$ is updated by $\lambda$ in the SVD.
The above procedure truncates $\dim(\tilde{\alpha}_4)$ to the dimension cut-off $\chi$, which can be readily applied to truncate any other bonds. According to the discussion about SVD in Sec. \ref{sec.2SVD}, the environment is the two tensors and the adjacent spectra $\lambda$'s in $M$, where the $\lambda$'s play the role of an ``effective'' environment that approximate the true environment ($M^e$ in the full update). From this viewpoint, the simple update uses local environment. Later by borrowing from the idea of the orthogonal form of the iPEPS on Bethe lattices \cite{SDV06TTN, NFGSS08TreeMPS, TEV09TTN, MVLN10TTN, LDX12TTN, NC13TTN, PVK13TTN, MVSNL15TTN}, it was realized that the environment of the simple update is the iPEPS on the infinite trees \cite{RXLS13NCD, RLXZS12ODTNS, RPPSL17AOP3D}, not just several tensors. We will talk about this in detail in the next chapter from the perspective of the multi-linear algebra.
\textbf{Cluster update.} By keeping the same dimension cut-off, the simple update is much more efficient than the full update. On the other hand, obviously, the full update possesses higher accuracy than the simple update by considering better the environment. The cluster update is between the simple and full updates, which is more flexible to balance between the efficiency and accuracy \cite{RXLS13NCD, LCB14PEPScontract, WV11PEPSclusterArxiv}.
One way is to choose a finite cluster of the infinite TN\index{TN} and define the environment tensor by contracting the finite TN after taking a pair of $\tilde{P}^{[L]}$ and $\tilde{P}^{[R]}$ out. One can consider to firstly use the simple update to obtain the spectra and put them on the boundary of the cluster \cite{WV11PEPSclusterArxiv}. This is equivalent to using a new boundary condition \cite{RXLS13NCD, RPPSL17AOP3D}, different from the open or periodic boundary conditions of a finite cluster. Surely, the bigger the cluster becomes, more accurate but more consuming the computation will be. One may also consider an infinite-size cluster, which is formed by a certain number of rows of the tensors in the TN \cite{LCB14PEPScontract}. Again, both the accuracy and computational cost will in general increase with the number of rows. With infinite rows, such a cluster update naturally becomes the full update. Despite the progresses, there are still many open questions, for example, how to best balance the efficiency and accuracy in the cluster update.
\section{Summary of the tensor network algorithms in higher dimensions}
In this section, we mainly focused on the iPEPS\index{iPEPS} algorithm that simulate the ground states of 2D lattice models. The key step is to compute the environment tensor, which is to contract the corresponding TN\index{TN}. For several special cases such as trees and fractal lattices, the environment tensor corresponds to an exactly contractible TN, and thus can be computed efficiently (see Sec. \ref{Sec-exactTN}). For the regular lattices such as square lattice, the environment tensor is computed by the TN contraction algorithms, which is normally the most consuming step in the iPEPS approaches.
The key concepts and ideas, such as environment, (simple, cluster, and full) update schemes, and the use of SVD\index{SVD}, can be similarly applied to finite-size cases \cite{PWV11tPEPS, LCB14fPEPS}, the finite-temperature simulations \cite{CCD12FTPEPS, RXLS13NCD, RLXZS12ODTNS, CD15TPO, CD15FTPEPS, CDO16TPO, CRD16TPO, CDO17TPOQMC, KREO18PEPOthermal}, and real-time simulations \cite{CDC19TPO, PWV11tPEPS} in two dimensions. The computational cost of the TN approaches is quite sensitive to the spatial dimensions of the system. The simulations of 3D quantum systems are much more consuming than the 2D cases, where the task become to contract the 4D TN. The 4D TN contraction is extremely consuming, one may consider to generalize the simple update \cite{RPPSL17AOP3D, JO18graphPEPS}, or to construct finite-size effective Hamiltonians that mimic the infinite 3D quantum models \cite{RPPSL17AOP3D, RXPS+18FTQES}
Many technical details of the approaches can be flexibly modified according to the problems under consideration. For example, the iPEPO\index{iPEPO} formulation is very useful when computing a 3D statistic model, which is to contract the corresponding 3D TN. To simulating the imaginary-time evolution, to directly use the two-body evolution operators (see, e.g., \cite{JWX08SimpleUpdate, OV09CTMRG}) is normally more efficient than to use the iPEPO. The environment is not necessarily defined by the tensors; it can be defined by contracting everything of the TN\index{TN} except for the aimed geometrical bond \cite{CCD12FTPEPS, RLXZS12ODTNS}. The contraction order also significantly affects the efficiency and accuracy. One may consider to use the ``single-layer'' picture \cite{LCB14PEPScontract, PWV11tPEPS}, or an ``intersected'' optimized contraction scheme \cite{XLHX+19TNcon}.
\chapter{Tensor network contraction and multi-linear algebra}
\abstract{This chapter is aimed at understanding TN algorithms from the perspective of MLA\index{MLA}. In Sec. \ref{sec5-eig}, we start from a simple example with a 1D TN\index{TN} stripe, which can be ``contracted'' by solving the eigenvalue decomposition of matrices. This relates to several important MPS techniques such as canonicalization \cite{OV08canonical} that enables to implement optimal truncations of the bond dimensions of MPS's\index{MPS} (Sec. \ref{sec5-canon}). In Sec. \ref{sec5-SO}, we discuss about super-orthogonalization \cite{RLXZS12ODTNS} inspired by Tucker decomposition \cite{DDV00HOSVD} in MLA, which is also a higher-dimensional generalization of canonicalization; it is proposed to implement optimal truncations of the iPEPS's defined on trees. In Sec. \ref{sec5-rank1}, we explain based on the rank-1 decomposition \cite{LMV00Rank1}, that super-orthogonalization in fact provides the ``loopless approximation'' of the iPEPS's\index{iPEPS} on regular lattices \cite{RXLS13NCD}; it explains how the approximations in the simple update algorithm works for the ground-state simulations on 2D regular lattices \cite{JWX08SimpleUpdate}. In Sec. \ref{sec5-TRD}, we will discuss tensor ring decomposition (TRD)\index{TRD} \cite{R16AOP}, which is a rank-$N$ generalization of the rank-1 decomposition. TRD naturally provides a unified description of iDMRG\cite{W92DMRG,W93DMRG,M08iDMRGarxiv}\index{iDMRG}, iTEBD\cite{V07iTEBD}\index{iTEBD}, and CTMRG\index{CTMRG} \cite{OV09CTMRG,FVZHV17fastCTMRG} when considering the contractions of 2D TN's\index{TN}.}
\section{A simple example of solving tensor network contraction by eigenvalue decomposition}
\label{sec5-eig}
As discussed in the previous sections, the TN algorithms are understood mostly based on the linear algebra, such as eigenvalue and singular-value decompositions. Since the elementary building block of a TN is a tensor, it is very natural to think about using the MLA to understand and develop TN algorithms. MLA is also known as tensor decompositions or tensor algebra \cite{KB09MLA}. It is a highly inter-disciplinary subject. One of its tasks is to generalize the techniques in the linear algebra to higher order tensors. For instance, one key question is how to define the rank of a tensor and how to determine its optimal lower-rank approximation. This is exactly what we need in the TN algorithms.
Let us begin with a trivial example by simply considering the trace of the product of $N$ number of ($\chi \times \chi$) matrices $M$ as
\begin{eqnarray}
\text{Tr} \mathcal{M} = \text{Tr}(M^{[1]} M^{[2]} \cdots M^{[N]}) = \text{Tr} \prod_{n=1}^{N} M^{[n]},
\label{eq-4Z1D}
\end{eqnarray}
with $M^{[n]} = M$. In the language of TN\index{TN}, this can be regarded as a 1D TN with periodic boundary condition. For simplicity, we assume that the dominant eigenstate of $M$ is unique.
Allow us to firstly use a clumsy way to do the calculation: contract the shared bonds one by one from left to right. For each contraction, the computational cost is $O(\chi^3)$, thus the total cost is $O(N\chi^3)$.
Now let us be smarter by using the eigenvalue decomposition (assume it exists for $M$) in the linear algebra, which reads
\begin{eqnarray}
M = U \Lambda U^{\dagger},
\label{eq-4eig}
\end{eqnarray}
where $\Lambda$ are diagonal and $U$ is unitary satisfying $UU^{\dagger} = U^{\dagger} U= I$. Substituting Eq. (\ref{eq-4eig}) into Eq. (\ref{eq-4Z1D}), we can readily have the contraction as
\begin{eqnarray}
\text{Tr} \mathcal{M} = \text{Tr} (U \Lambda U^{\dagger} U \Lambda U^{\dagger} \cdots U \Lambda U^{\dagger}) = \text{Tr} (U \Lambda^N U^{\dagger}) = \sum_{a=0}^{\chi-1} \Lambda_a^N.
\end{eqnarray}
The dominant computational cost is around $O(\chi^3)$.
In the limit of $N \to \infty$, things become even easier, where we have
\begin{eqnarray}
\text{Tr} \mathcal{M} = \lim_{N \to \infty} \Lambda_0^N \sum_{a=0}^{\chi-1} (\frac{\Lambda_a}{\Lambda_0})^N = \Lambda_0^N,
\end{eqnarray}
where $\Lambda_0$ is the largest eigenvalue, and we have $\lim_{N \to \infty} (\frac{\Lambda_a}{\Lambda_0})^N = 0$ for $a>0$. It means all the contributions except for the dominant eigenvalue vanish when the TN is infinitely long. What we should do is just to compute the dominant eigenvalue. The efficiency can be further improved by numerous more mature techniques (such as Lanczos algorithm).
\subsection{Canonicalization of matrix product state}
\label{sec5-canon}
Before considering a 2D TN, let us take some more advantages of the eigenvalue decomposition on the 1D TN's, which is closely related to the \textit{canonicalization} of MPS proposed by Or\'us and Vidal for non-unitary evolution of MPS\index{MPS} \cite{OV08canonical}. The utilization of canonicalization are mainly in two aspects: locating optimal truncations of the MPS, and fixing the gauge degrees of freedom of the MPS for better stability and efficiency.
\subsection{Canonical form and globally optimal truncations of MPS}
\begin{figure}[tbp]
\centering
\includegraphics[angle=0,width=0.9\linewidth]{MPSbipart.eps}
\caption{(Color online) An impractical scheme to get the global optimal truncation of the virtual bond (red). First, the MPS\index{MPS} is cut into two parts. All the indexes on each side of the cut are grouped into one big index. Then by contracting the virtual bond and doing the SVD\index{SVD}, the virtual bond dimension is optimally reduced to $\chi$ by only taking the $\chi$-largest singular values and the corresponding vectors.}
\label{fig-4MPSbipar}
\end{figure}
As discussed in the above chapter, when using iTEBD\index{iTEBD} to contract a TN\index{TN}, one needs to find the optimal truncations of the virtual bonds of the MPS\index{MPS}. In other words, the problem is how to optimally reduce the dimension of an MPS.
The globally optimal truncation can be down in the following expensive way. Let us divide the MPS into two parts by cutting the bond that is to be truncated (Fig. \ref{fig-4MPSbipar}). Then, if we contract all the virtual bonds on the left hand side and reshape all the physical indexes there into one index, we will obtain a large matrix denoted as $L_{\cdots s_n,\alpha_n}$ that has one big physical and one virtual index. Another matrix denoted as $R_{s_{n+1}\cdots,\alpha_n}^{\ast}$ can be obtained by doing the same thing on the right hand side. The conjugate of $R$ is taken there to obey some conventions.
Then, by contracting the virtual bond and doing SVD\index{SVD} as
\begin{eqnarray}
\sum_{a_n} L_{\cdots s_n,a_n} R_{s_{n+1}\cdots,a_n}^{\ast} = \sum_{a_n'} \tilde{L}_{\cdots s_n,a_n'} \lambda_{a_n'} \tilde{R}_{s_{n+1}\cdots,a_n'}^{\ast},
\label{eq-4wholeMPSsvd}
\end{eqnarray}
the virtual bond dimension is optimally reduced to $\chi$ by only taking the $\chi$-largest singular values and the corresponding vectors. The truncation error that is minimized is the distance between the MPS before and after the truncation. Therefore, the truncation is optimal globally concerning the whole MPS as the environment.
In practice, we do not implement the SVD above. It is actually the decomposition of the whole wave function, which is exponentially expensive. Canonicalization provides an efficient way to realize the SVD through only local operations.
\begin{figure}[tbp]
\centering
\includegraphics[angle=0,width=0.75\linewidth]{Fig25.eps}
\caption{(Color online) The MPS\index{MPS} with two-site translational invariance.}
\label{fig-4MPS2}
\end{figure}
Considering an infinite MPS with two-site translational invariance (Fig. \ref{fig-4MPS2}); it is formed by the tensors $A$ and $B$ as well as the diagonal matrices $\Lambda$ and $\Gamma$ as
\begin{eqnarray}
\sum_{\{a\}} \cdots \Lambda_{a_{n-1}} A_{s_{n-1}, a_{n-1} a_{n}} \Gamma_{a_{n}} B_{s_{n}, a_{n} a_{n+1}} \Lambda_{a_{n+1}} \cdots = \text{tTr} (\cdots \Lambda A \Gamma B \Lambda \cdots).
\end{eqnarray}
This is the MPS\index{MPS} used in the iTEBD\index{iTEBD} algorithm (see Chap.\ref{iTEBD} and Fig.\ref{fig-3iTEBD}). Note that all argument can be readily generalized to the infinite MPS's with $n$-site translational invariance, or even to the finite MPS's.
\begin{figure}[tbp]
\centering
\includegraphics[angle=0,width=0.8\linewidth]{Fig26.eps}
\caption{(Color online) Four canonical conditions of an MPS.}
\label{fig-4CanonCond}
\end{figure}
An MPS is in the \textit{canonical form} if the tensors satisfy
\begin{eqnarray}
\sum_{sa} \Lambda_{a} A_{s, a a'} \Lambda^{\ast}_{a} A^{\ast}_{s, a a''} = I_{a' a''}, \label{eq-4Canonical1} \\
\sum_{sa} A_{s, a' a} \Gamma_{a} A^{\ast}_{s, a'' a} \Gamma^{\ast}_{a} = I_{a' a''}, \label{eq-4Canonical2} \\
\sum_{sa} \Gamma_{a} B_{s, a a'} \Gamma^{\ast}_{a} B^{\ast}_{s, a a''} = I_{a' a''}, \label{eq-4Canonical3} \\
\sum_{sa} B_{s, a' a} \Lambda_{a} B^{\ast}_{s, a'' a} \Lambda^{\ast}_{a} = I_{a' a''}, \label{eq-4Canonical4}
\end{eqnarray}
where $\Lambda$ and $\Gamma$ are positive-defined (Fig. \ref{fig-4CanonCond}). Eqs. (\ref{eq-4Canonical1}) - (\ref{eq-4Canonical4}) are called the \textit{canonical conditions} of the MPS\index{MPS}. Note there will be $2n$ equations with $n$-site translational invariance, meaning that each inequivalent tensor will obey to two (left and right) conditions.
In the canonical form, $\Lambda$ or $\Gamma$ directly give the singular values by cutting the MPS on the corresponding bond. To see this, let us calculate Eq. (\ref{eq-4wholeMPSsvd}) from a canonical MPS. From the canonical conditions, matrices $L$ and $R$ are unitary, satisfying $L^{\dagger}L=I$ and $R^{\dagger}R=I$ (the physical indexes are contracted). Meanwhile, $\Lambda$ (or $\Gamma$) is positive-defined, thus $L$, $\Lambda$ (or $\Gamma$) and $R$ of a canonical MPS directly define the SVD, and $\Lambda$ or $\Gamma$ is indeed the singular value spectrum. Then the optimal truncations of the virtual bonds are reached by simply keeping $\chi$-largest values of $\Lambda$ and the corresponding basis of the neighboring tensors. This is true when cutting any one of the bonds of the MPS. From the uniqueness of SVD\index{SVD}, Eqs. (\ref{eq-4Canonical1}) and (\ref{eq-4Canonical2}) leads to a unique MPS representation, thus such a form is called ``\textit{canonical}''. In other words, the canonicalization fixes the gauge degrees of freedom of the MPS.
For any finite MPS\index{MPS}, the uniqueness is robust. For an infinite MPS, there will be some additional complexity. Let us define the left and right \textit{transfer matrices} $M^L$ and $M^R$ of as
\begin{eqnarray}
M^L_{a_1 a_1' a_2 a_2'} = \sum_{s} \Lambda_{a_1} A_{s, a_1 a_2} \Lambda^{\ast}_{a_1'} A^{\ast}_{s, a_1' a_2'}, \\
M^R_{a_1 a_1' a_2 a_2'} = \sum_{s} A_{s, a_1 a_2} \Gamma_{a_1} A^{\ast}_{s, a_1' a_2'} \Gamma_{a_1'}^{\ast}.
\label{eq-4LTM}
\end{eqnarray}
Then the canonical conditions [Eq. (\ref{eq-4Canonical1})] say that the identity is the left (right) eigenvector of $M^L$ ($M^R$), satisfying
\begin{eqnarray}
\sum_{a_1 a_1'} I_{a_1 a_1'} M^L_{a_1 a_1' a_2 a_2'} = \lambda^L I_{a_2 a_2'}, \\
\sum_{a_1 a_1'} I_{a_2 a_2'} M^R_{a_1 a_1' a_2 a_2'} = \lambda^R I_{a_1 a_1'},
\end{eqnarray}
with $\lambda^L$ ($\lambda^R$) the eigenvalue.
Similar eigenvalue equations can be obtained from the canonical conditions associated to the tensor $B$, where we have the transfer matrices as
\begin{eqnarray}
N^L_{a_1 a_1' a_2 a_2'} = \sum_{s} \Gamma_{a_1} B_{s, a_1 a_2} \Gamma^{\ast}_{a_1'} B^{\ast}_{s, a_1' a_2'}, \\
N^R_{a_1 a_1' a_2 a_2'} = \sum_{s} B_{s, a_1 a_2} \Lambda_{a_1} B^{\ast}_{s, a_1' a_2'} \Lambda_{a_1'}^{\ast}.
\label{eq-4RTM}
\end{eqnarray}
Now the canonical conditions are given by four eigenvalue equations and can be reinterpreted as the following: with an infinite MPS formed by $A$, $B$, $\Lambda$ and $\Gamma$, it is canonical when the identity is the eigenvector of its transfer matrices.
Simply from the canonical conditions, it does not require the ``identity'' to be dominant eigenvector. However, if the identity is not the dominant one, the canonical conditions will become unstable under an arbitrarily small noise. Below, we will show that the canonicalization algorithm assures that the identity is the leading eigenvector, since it transforms the leading eigenvector to an identity. In addition, if the dominant eigenvector of $M^L$ and $M^R$ (also $N^L$ and $N^R$) is degenerate, the canonical form will not be unique. See Ref. \cite{OV08canonical} for more details.
\begin{figure}[tbp]
\centering
\includegraphics[angle=0,width=1\linewidth]{Fig27.eps}
\caption{(Color online) The illustration of the canonical transformations.}
\label{fig-4Canon}
\end{figure}
\subsection{Canonicalization algorithm and some related topics}
\label{sec5.canon}
Considering the iTEBD\index{iTEBD} algorithm \cite{V07iTEBD} (see Sec. \ref{iTEBD}), while the MPO represents an unitary operator, the canonical form of the MPS\index{MPS} will be reserved by the evolution (contraction). For the imaginary-time evolution, the MPO is near-unitary. For the Trotter step $\tau \to 0$, the MPO\index{MPO} approaches to be an identity. It turns out that in this case, the MPS will be canonicalized by the evolution in the standard iTEBD algorithm. When the MPO is non-unitary (e.g., when contracting the TN of a 2D statistic model) \cite{OV08canonical}, the MPS will not be canonical, and the canonicalization might be needed to better truncate the bond dimensions of the MPS.
\textbf{Canonicalization algorithm}. An algorithm to canonicalize an arbitrary MPS was proposed by Or\'us and Vidal \cite{OV08canonical}. The idea is to compute the first eigenvectors of the transfer matrices, and introduce proper gauge transformations on the virtual bonds that map the leading eigenvector to identity.
Let us take the gauge transformations on the virtual bonds between $A$ and $B$ as an example. Firstly, compute the dominant left eigenvector $v^L$ of the matrix $N^L M^L$, and similarly the dominant right eigenvector $v^R$ of the matrix $N^R M^R$. Then, reshape $v^L$ and $v^R$ as two matrices and decompose them symmetrically as
\begin{eqnarray}
v^R_{a_1 a_1'} &=& \sum_{a_1''} X_{a_1 a_1''}X^{\ast}_{a_1' a_1''}, \\
v^L_{a_1 a_1'} &=& \sum_{a_1''} Y_{a_1 a_1''}Y^{\ast}_{a_1' a_1''}.
\end{eqnarray}
$X$ and $Y$ can be calculated using eigenvalue decomposition, i.e., $v^R=WDW^{\dagger}$ with $X=W\sqrt{D}$.
Insert the identities $X^{-1}X$ and $Y Y^{-1}$ on the virtual bond as shown in Fig. \ref{fig-4Canon}, then we get a new matrix $\mathcal{M} = X \Gamma Y$ on this bond. Apply SVD\index{SVD} on on $\mathcal{M}$ as $\mathcal{M} = U \tilde{\Gamma} V^{\dagger}$, where we have the updated spectrum $\tilde{\Gamma}$ on this bond. Meanwhile, we obtain the gauge transformations to update $A$ and $B$ as $\mathcal{U}=X^{-1}U$ and $\mathcal{V}=V^{\dagger} Y^{-1}$, where the transformations are implemented as
\begin{eqnarray}
A_{s_1,a_1 a_2} \leftarrow \sum_{a} A_{s_1,a_1 a} \mathcal{U}_{a a_2},\\
B_{s_1,a_1 a_2} \leftarrow \sum_{a} B_{s_1,a a_2} \mathcal{V}_{a_1 a}.
\end{eqnarray}
Implement the same steps given above on the virtual bonds between $B$ and $A$, then the MPS is transformed to the canonical form.
\textbf{\textit{Variants of the canonical form}}. From the canonical form of an MPS, one can define the \textit{left or right canonical forms}. Define the follow tensors
\begin{eqnarray}
A^L_{s, a a'} &=& \Lambda_{a} A_{s, a a'}, \\
A^R_{s, a a'} &=& A_{s, a a'} \Gamma_{a'}, \\
B^L_{s, a a'} &=& \Gamma_{a} B_{s, a a'}, \\
B^R_{s, a a'} &=& B_{s, a a'} \Lambda_{a'}, \\
A^M_{s, a a'} &=& \Lambda_{a} A_{s, a a'} \Gamma_{a'}.
\end{eqnarray}
The left-canonical MPS is defined by $A^L$ and $B^L$ as
\begin{eqnarray}
\text{tTr} (\cdots A^L B^L A^L B^L \cdots).
\end{eqnarray}
Similarly, the right-canonical MPS\index{MPS} is defined by $A^R$ and $B^R$ as
\begin{eqnarray}
\text{tTr} (\cdots A^R B^R A^R B^R \cdots).
\end{eqnarray}
The central orthogonal MPS is defined as
\begin{eqnarray}
\text{tTr} (\cdots A^L B^L A^M B^R A^R \cdots).
\end{eqnarray}
One can easily check that these MPS's and the canonical MPS can be transformed to each other by gauge transformations.
From the canonical conditions, $A^L$, $A^R$, $B^L$, and $B^R$ are non-square orthogonal matrices (e.g., $\sum_{s a} A^L_{s, a a'} A^{L\ast}_{s, a a''} = I_{a' a''}$), called \textit{isometries}. $A^M$ is called the \textit{central tensor} of the central-orthogonal MPS. This MPS\index{MPS} form is the state ansatz behind the DMRG\index{DMRG} algorithm \cite{W92DMRG,W93DMRG}, and is very useful in TN-based methods (see for example the works of McCulloch \cite{M07DMRGsymme,M08iDMRGarxiv}). For instance, when applying DMRG to solve 1D quantum model, the tensors $A^L$ and $B^L$ define a left-to-right RG\index{RG} flow that optimally compresses the Hilbert space of the left part of the chain. $A^R$ and $B^R$ define a right-to-left RG flow similarly. The central tensor between these two RG flows is in fact the ground state of the effective Hamiltonian given by the RG flows of DMRG. Note that the canonical MPS is also called the \textit{central canonical form}, where the directions of the RG flows can be switched arbitrarily by gauge transformations, thus there is no need to define the directions of the flows or a specific center.
\textbf{\textit{Relations to tensor train decomposition}}. It is worth mentioning the TTD\index{TTD} \cite{O11TTD} proposed in the field of MLA\index{MLA}. As argued in Chap. 2, one advantage of MPS\index{MPS} is it lowers the number of parameters from an exponential size dependence to a polynomial one. Let us consider a similar problem: for a $N$-th order tensor that has $d^N$ parameters, how to find its optimal MPS\index{MPS} representation, where there are only $[2d\chi+(N-2)d\chi^2]$ parameters? TTD was proposed for this aim: by decomposing a tensor into a tensor-train form that is similar to a finite open MPS, the number of parameters becomes linearly relying to the order of the original tensor. The TTD algorithm shares many similar ideas with MPS and the related algorithms (especially DMRG\index{DMRG} which was proposed about two decades earlier). The aim of TTD\index{TTD} is also similar to the truncation tasks in the TN\index{TN} algorithms, which is to compress the number of parameters.
\section{Super-orthogonalization and Tucker decomposition}
\label{sec5-SO}
As discussed in the above section, the canonical form of an MPS brings a lot of advantages, such as determining the entanglement and the optimal truncations of the virtual bond dimensions by local transformations. The canonical form can be readily generalized to the iPEPS's\index{iPEPS} on trees. Can we also define the canonical form for the iPEPS's in higher-dimensional regular lattices, such as square lattice (Fig. \ref{fig-4SO})? If this can be done, we would known how to find the globally optimal transformations that reduces the bond dimensions of the iPEPS, just like what we can do with an MPS. Due to the complexity of 2d TN's\index{TN}, unfortunately, there is no such a form in general. In the following, we explain the \textit{super-orthogonal form} of iPEPS proposed in 2012 \cite{RLXZS12ODTNS}, which applies the canonical form of tree iPEPS to the iPEPS on regular lattices. The super-orthogonalization is a generalization of the Tucker decomposition (a higher-order generalization of matrix SVD\index{SVD}) \cite{DDV00HOSVD}, providing a zero-loop approximation scheme \cite{RXLS13NCD} to define the entanglement and truncate the bond dimensions.
\begin{figure}[tbp]
\centering
\includegraphics[angle=0,width=1\linewidth]{Fig28.eps}
\caption{(Color online) The first two figures show the iPEPS\index{iPEPS} on tree and square lattices, with two-site translational invariance. The last one shows the super-orthogonal conditions.}
\label{fig-4SO}
\end{figure}
\subsection{Super-orthogonalization}
Let us start from the iPEPS\index{iPEPS} on the (infinite) Bethe lattice with the coordination number $z=4$. It is formed by two tensors $P$ and $Q$ on the sites as well as four spectra $\Lambda^{(k)}$ ($k=1,2,3,4$) on the bonds, as illustrated in Fig. \ref{fig-4SO}. Here, we still take the two-site translational invariance for simplicity.
There are eight \textit{super-orthogonal conditions}, of which four associate to the tensor $P$ and four to $Q$. For $P$, the conditions are
\begin{eqnarray}
\sum_s \sum_{\cdots a_{k-1} a_{k+1} \cdots} P_{s,\cdots a_k \cdots} P_{s,\cdots a_k' \cdots}^{\ast} \prod_{n \neq k} \Lambda^{(n)}_{a_n} \Lambda^{(n)\ast}_{a_n} = I_{a_{k} a_{k}'}, \ \ (\forall \ k),
\label{eq-4SOconditions}
\end{eqnarray}
where all the bonds along with the corresponding spectra are contracted except for $a_k$. It means that by putting $a_k$ as one index and all the rest as another, the \textit{$k$-rectangular matrix} $S^{(k)}$ defined as
\begin{eqnarray}
S^{(k)}_{s\cdots a_{k-1} a_{k+1} \cdots, a_k} = P_{s,\cdots a_k \cdots} \prod_{n \neq k} \Lambda^{(n)}_{a_n},
\label{eq-4RCmatrix}
\end{eqnarray}
is an isometry, satisfying $S^{(k)\dagger} S^{(k)} = I$. The super-orthogonal conditions of the tensor $Q$ are defined in the same way. $\Lambda^{(k)}$ is dubbed \textit{super-orthogonal spectrum} when the super-orthogonal conditions are fulfilled.
\begin{figure}[tbp]
\centering
\includegraphics[angle=0,width=1\linewidth]{Fig29.eps}
\caption{(Color online) The illustrations of gauge transformations in the super-orthogonalization algorithm.}
\label{fig-4SOalgo}
\end{figure}
In the canonicalization of MPS\index{MPS}, the vectors on the virtual bonds give the bipartite entanglement defined by Eq. (\ref{eq-4wholeMPSsvd}). Meanwhile, the bond dimensions can be optimally reduced by discarding certain smallest elements of the spectrum. In the super-orthogonalization, this is not always true for iPEPS's. For example, given a translational invariant iPEPS\index{iPEPS} defined on a tree (or called Bethe lattice, see Fig. \ref{fig-4SO} (a)) \cite{SDV06TTN,NFGSS08TreeMPS,TEV09TTN,MVLN10TTN,LDX12TTN,NC13TTN,PVK13TTN,MVSNL15TTN}, the super-orthogonal spectrum indeed gives the bipartite entanglement spectrum by cutting the system at the corresponding place. However, when considering loopy lattices, such as the iPEPS defined on a square lattice (Fig. \ref{fig-4SO} (b)), this will no longer be true. Instead, the super-orthogonal spectrum provides an approximation of the entanglement of the iPEPS by optimally ignoring the loops. One can still truncate the bond dimensions according to the super-orthogonal spectrum, giving in fact the simple update (see Sec. \ref{sec.updateschemes}). We will discuss the loopless approximation in detail in Sec. \ref{sec.looplessTNrank-1} using the rank-1 decomposition.
\subsection{Super-orthogonalization algorithm}
Any PEPS\index{PEPS} can be transformed to the super-orthogonal form by iteratively implementing proper gauge transformations on the virtual bonds \cite{RLXZS12ODTNS}. The algorithm consists of two steps. Firstly, compute the reduced matrix $\mathcal{M}^{(k)}$ of the $k$-rectangular matrix of the tensor $P$ [Eq. (\ref{eq-4RCmatrix})] as
\begin{eqnarray}
\mathcal{M}^{(k)}_{a_{k} a_{k}'}=\sum_s \sum_{\cdots a_{k-1} a_{k+1} \cdots} S^{(k)}_{s\cdots a_{k-1} a_{k+1} \cdots, a_k} S^{(k)\ast}_{s\cdots a_{k-1} a_{k+1} \cdots, a_k'}.
\label{eq-4bondRM}
\end{eqnarray}
Compared with the super-orthogonal conditions in Eq. (\ref{eq-4SOconditions}), one can see that $\mathcal{M}^{(k)}=I$ when the PEPS is super-orthogonal. Similarly, we define the reduced matrix $\mathcal{N}^{(k)}$ of the tensor $Q$.
When the PEPS is not super-orthogonal, $\mathcal{M}^{(k)}$ and $\mathcal{N}^{(k)}$ are not identities but Hermitian matrices. Decompose them as $\mathcal{M}^{(k)}=X^{(k)}X^{(k)\dagger}$ and $\mathcal{N}^{(k)}=Y^{(k)} Y^{(k)\dagger}$. Then, insert the identities $X^{(k)} [X^{(k)}]^{-1}$ and $Y^{(k)} [Y^{(k)}]^{-1}$ on the virtual bonds to perform gauge transformations along four directions as shown in Fig. \ref{fig-4SOalgo}. Then, we can use SVD\index{SVD} to renew the four spectra by $X^{(k)}\Lambda^{(k)}Y^{(k)T} = U^{(k)}\tilde{\Lambda}^{(k)}V^{(k)\dagger}$. Meanwhile, we transform the tensors as
\begin{eqnarray}
P_{s,\cdots a_k \cdots} \leftarrow \sum_{a_k'a_k''} P_{s,\cdots a_k' \cdots} [X^{(k)}]^{-1}_{a_k'a_k''} U^{(k)}_{a_k'' a_k},\\
Q_{s,\cdots a_k \cdots} \leftarrow \sum_{a_k'a_k''} Q_{s,\cdots a_k' \cdots} [Y^{(k)}]^{-1}_{a_k'a_k''} V^{(k)\ast}_{a_k'' a_k}.
\end{eqnarray}
Compared with the canonicalization algorithm of MPS\index{MPS}, one can see that the gauge transformations in the super-orthogonalization algorithm are quite similar. The difference is that one cannot transform a PEPS\index{PEPS} into the super-orthogonal form by a single step, since the transformation on one bond might cause some deviation from obeying the super-orthogonal conditions on other bonds. Thus, the above procedure should be iterated until all the tensors and spectra converge.
\subsection{Super-orthogonalization and dimension reduction by Tucker decomposition}
Such an iterative scheme is closely related to the Tucker decomposition in MLA\index{MLA} \cite{DDV00HOSVD}. Tucker decomposition is considered as a generalization of (matrix) SVD to higher-order tensors, thus it is also called higher-order or multi-linear SVD. The aim is to find the optimal reductions of the bond dimensions for a single tensor.
\begin{figure}[tbp]
\centering
\includegraphics[angle=0,width=0.6\linewidth]{Fig30.eps}
\caption{(Color online) The illustrations of Tucker decomposition [Eq. (\ref{eq-4tucker})].}
\label{fig-4tucker}
\end{figure}
Let us define the $k$-reduced matrix of a tensor $T$ as
\begin{eqnarray}
M^{(k)}_{a_k a_k'} = \sum_{a_1 \cdots a_{k-1} a_{k+1} \cdots} T_{a_1 \cdots a_{k-1} a_k a_{k+1} \cdots} T_{a_1 \cdots a_{k-1} a_k' a_{k+1} \cdots}^{\ast},
\end{eqnarray}
where all except the $k$-th indexes are contracted. The Tucker decomposition (Fig. \ref{fig-4tucker}) of a tensor $T$ has the form as
\begin{eqnarray}
T_{a_1 a_2 \cdots} = \sum_{a_1a_2 \cdots} S_{b_1b_2 \cdots} \prod_{k} U^{(k)}_{a_kb_k},
\label{eq-4tucker}
\end{eqnarray}
where the following conditions should be satisfied:
\begin{itemize}
\item \textit{Unitarity}. $U^{(k)}$ are unitary matrices satisfying $U^{(k)} U^{(k)\dagger} = I$.
\item \textit{All-orthogonality}. For any $k$, the $k$-reduced matrix $M^{(k)}$ of the tensor $S$ is diagonal, satisfying
\begin{eqnarray}
M^{(k)}_{a_k a_k'} = \Gamma^{(k)}_{a_k} I_{a_k a_k'}.
\end{eqnarray}
\item \textit{Ordering}. For any $k$, the elements of $\Gamma^{(k)}$ in the $k$-reduced matrix are positive-defined and in the descending order, satisfying $\Gamma_0 > \Gamma_1 > \cdots$.
\end{itemize}
From these conditions, one can see that the tensor $T$ is decomposed as the contraction of another tensor $S$ with several unitary matrices. $S$ is called the \textit{core tensor}. In other words, the optimal lower-rank approximation of the tensor can be simply obtained by
\begin{eqnarray}
T_{a_1 a_2 \cdots} \simeq \sum_{a_1a_2 \cdots = 0}^{\chi-1} S_{b_1b_2 \cdots} \prod_{k} U^{(k)}_{a_kb_k},
\end{eqnarray}
where we only take the first $\chi$ terms in the summation of each index.
Such an approximations can be understood in terms of the SVD\index{SVD} of matrices. Applying the conditions of the $k$-reduced matrix of $T$, we have
\begin{eqnarray}
M^{(k)}_{a_k a_k'} = \sum_{b_k} U^{(k)}_{a_kb_k} \Gamma^{(k)}_{b_k} U^{(k)\dagger}_{a_k'b_k}.
\end{eqnarray}
Since $U^{(k)}$ is unitary and $\Gamma^{(k)}$ is positive-defined and in the descending order, the above equation is exactly the eigenvalue decomposition of $M^{(k)}$. From the relation between the SVD of a matrix and the eigenvalue decomposition of its reduced matrix, we can see that $U^{(k)}$ and $\Gamma^{(k)}$ in fact give the SVD of the matrix $T_{a_1 \cdots a_{k-1} a_{k+1} \cdots, a_{k}}$ as
\begin{eqnarray}
T_{a_1 \cdots a_{k-1} a_{k+1} \cdots, a_{k}} = \sum_{b_k} \mathcal{S}_{a_1 \cdots a_{k-1} a_{k+1} \cdots, b_{k}} \sqrt{\Gamma^{(k)}_{b_k}} U^{(k)}_{a_kb_k}.
\end{eqnarray}
Then, The optimal truncation of the rank of each index is reached by the corresponding SVD\index{SVD}. The truncation error is obviously the distance defined as
\begin{eqnarray}
\varepsilon^{(k)} = |T_{a_1 \cdots a_{k-1} a_{k+1} \cdots, a_{k}} - \sum_{b_k=1}^{\chi} \mathcal{S}_{a_1 \cdots a_{k-1} a_{k+1} \cdots} \Gamma^{(k)}_{b_k} U^{(k)}_{a_kb_k}|,
\end{eqnarray}
which is minimized in this SVD.
For the algorithms of Tucker decomposition, one simple way is to do the eigenvalue decomposition of each $k$-reduced matrix, or the SVD of each $k$-rectangular. Then for a $K$-th ordered tensor, $K$ SVD's will give us the Tucker decomposition and a lower-rank approximation. This algorithm is often called \textit{higher-order SVD} (HOSVD)\index{HOSVD}, which has been successfully applied to implement truncations in the TRG\index{TRG} algorithm \cite{XCQZYX12HOSRG}. The accuracy of HOSVD can be improved. Since the truncation on one index will definitely affect the truncations on other indexes, there will be some ``interactions'' among different indexes (modes) of the tensor. The truncations in HOSVD are calculated independently, thus such ``interactions''are ignored. One improved way is the \textit{high-order orthogonal iteration} (HOOI)\index{HOOI}, where the interactions among different modes are considered by iteratively doing SVD's until reaching the convergence. See more details in Ref. \cite{DDV00HOSVD}.
Compared with the conditions of Tucker decomposition, let us redefine the super-orthogonal conditions of a PEPS\index{PEPS} as
\begin{itemize}
\item \textit{Super-orthogonality}. For any $k$, the reduced matrix of the $k$-rectangular matrix $\mathcal{M}^{(k)}$ [Eq. (\ref{eq-4bondRM})] is diagonal, satisfying
\begin{eqnarray}
\mathcal{M}^{(k)}_{a_k a_k'} = \Gamma^{(k)}_{a_k} I_{a_k a_k'}.
\end{eqnarray}
\item \textit{Ordering}. For any $k$, the elements of $\Gamma^{(k)}$ are positive-defined and in the descending order, satisfying $\Gamma_0 > \Gamma_1 > \cdots$.
\end{itemize}
Note that the condition ``unitary'' (first one in Tucker decomposition) is hidden in the fact that we use gauge transformations to transform the PEPS into the super-orthogonal form. Therefore, the super-orthogonalization is also called \textit{network Tucker decomposition} (NTD)\index{NTD}.
In the Tucker decomposition, the ``all-orthogonality'' and ``ordering'' lead to an SVD\index{SVD} associated to a single tensor, which explains how the optimal truncations work from the decompositions in linear algebra. In the NTD, the SVD picture is generalized from a single tensor to a non-local PEPS. Thus, the truncations are optimized in a non-local way.
Let us consider a finite-size PEPS\index{PEPS} and arbitrarily choose one geometrical bond (say $a$). If the PEPS is on a tree, we can cut the bond and separate the TN\index{TN} into three disconnecting parts: the spectrum ($\Lambda$) on this bond and two tree brunches stretching to the two sides of the bond. Specifically speaking, each brunch contains one virtual bond and all the physical bonds on the corresponding side, formally denoted as $\Psi^L_{i_1i_2 \cdots,a}$ (and $\Psi^R_{j_1j_2 \cdots,a}$ on the other side). Then the state given by the iPEPS can be written as
\begin{eqnarray}
\sum_a \Psi^L_{i_1i_2 \cdots,a} \Lambda_a \Psi^R_{j_1j_2 \cdots,a}.
\label{eq-4TreeSVD}
\end{eqnarray}
To get the SVD\index{SVD} picture, we need to prove that $\Psi^L$ and $\Psi^R$ in the above equation are isometries, satisfying the orthogonal conditions as
\begin{eqnarray}
\begin{aligned}
\sum_{i_1i_2 \cdots} \Psi^L_{i_1i_2 \cdots,a} \Psi^L_{i_1i_2 \cdots,a'} = I_{aa'},\\
\sum_{j_1j_2 \cdots} \Psi^R_{j_1j_2 \cdots,a} \Psi^R_{j_1j_2 \cdots,a'} = I_{aa'}.
\end{aligned}
\label{eq-4OrtPhiLR}
\end{eqnarray}
Note that the spectrum $\Lambda$ is already positive-defined according to the algorithm. To this end, we construct the TN of $\sum_{i_1i_2 \cdots} \Psi^{L(R)}_{i_1i_2 \cdots,a} \Psi^{L(R)}_{i_1i_2 \cdots,a'}$ from its boundary. If the PEPS\index{PEPS} is super-orthogonal, the spectra must be on the boundary of the TN\index{TN} because the super-orthogonal conditions are satisfied everywhere \footnote{With the open boundary condition, one may consider the geometrical bond dimensions as one, and define the spectra by the one-dimensional vector $[1]$.}. Then the contractions of the tensors on the boundary satisfy Eq. (\ref{eq-4SOconditions}), which gives identities. Then we have on the new boundary again the spectra to iterate the contractions. All tensors can be contracted by iteratively using the super-orthogonal conditions, which in the end gives identities as Eq. (\ref{eq-4OrtPhiLR}). Thus, $\Psi^L$ and $\Psi^R$ are indeed isometries and Eq. (\ref{eq-4TreeSVD}) indeed gives the SVD\index{SVD} of the whole wavefunction. The truncations of the bond dimensions is globally optimized by taking the whole tree PEPS as the environment.
For an iPEPS\index{iPEPS}, it can be similarly proven that $\Psi^L$ and $\Psi^R$ are isometries. One way is to put any non-zero spectra on the boundary and iterate the contraction by Eq. (\ref{eq-4bondRM}). While the spectra on the boundary can be arbitrary, the results of the contractions by Eq. (\ref{eq-4bondRM}) converge to identities quickly \cite{RLXZS12ODTNS}. Then the rest of the contractions are exactly given by the super-orthogonal conditions [Eq. (\ref{eq-4SOconditions})]. In other words, the identity is a stable fixed point of the above iterations. Once the fixed point is reached, it can be considered that the contraction is from infinitely far away, meaning from the ``boundary'' of the iPEPS. In this way, one proves $\Psi^L$ and $\Psi^R$ are isometries, i.e., $\Psi^{L\dagger} \Psi^L = I$ and $\Psi^{R\dagger} \Psi^R = I$.
\section{Zero-loop approximation on regular lattices and rank-1 decomposition}
\label{sec.looplessTNrank-1}
\subsection{Super-orthogonalization works well for truncating the PEPS on regular lattice: some intuitive discussions}
\label{sec5-rank1}
From the discussions above, we can see that the ``canonical'' form of a TN\index{TN} state is strongly desired, because it is expected to give the entanglement and the optimal truncations of the bond dimensions. Recall that to contract a TN that cannot be contracted exactly, truncations are inevitable, and locating the optimal truncations is one of the main tasks in the computations. The super-orthogonal form provides a robust way to optimally truncate the bond dimensions of the PEPS\index{PEPS} defined on a tree, analog to the canonicalization of MPS\index{MPS}.
Interestingly, the super-orthogonal form does not require the tree structure. For an iPEPS\index{iPEPS} defined on a regular lattice, for example the square lattice, one can still super-orthogonalize it using the same algorithm. What is different is that the SVD\index{SVD} picture of the wave function (generally, see Eq. (\ref{eq-4wholeMPSsvd})) will not rigorously hold, as well as the robustness of the optimal truncations. In other words, the super-orthogonal spectrum does not exactly give the entanglement. A question rises: can we still truncate iPEPS defined on a square lattice according to the super-orthogonal spectrum?
Surprisingly, numeric simulations show that the accuracy by truncating according to the super-orthogonal spectrum is still good in many cases. Let us take the ground-state simulation of a 2D system by imaginary-time evolution as an example. As discussed in Sec. \ref{sec.evo2D}, the simulation becomes the contraction of a 3D TN\index{TN}. One usual way to compute this contraction is to contract layer by layer to an iPEPS\index{iPEPS} (see, e.g., \cite{JOVVC08PEPS,JWX08SimpleUpdate}). The contraction will enlarge the virtual bond dimensions, and truncations are needed. When the ground state is gapped (see, e.g., \cite{RLXZS12ODTNS,JWX08SimpleUpdate}), the truncations produce accurate results, which means the super-orthogonal spectrum approximates the true entanglement quite well.
It has been realized that using the simple update algorithm \cite{JWX08SimpleUpdate}, the iPEPS will converge to the super-orthogonal form for a vanishing Trotter step $\tau \to 0$. The success of the simple update suggests that the optimal truncation method on trees still works well for regular lattices. Intuitively, this can be understood in the following way. Comparing a regular lattice with a tree, if it has the same coordination number, the two lattices look exactly the same if we only inspect locally on one site and its nearest neighbors. The difference appears when one goes round the closed loops on the regular lattice, since there are no loop in the tree. Thus, the error applying the optimal truncation schemes (such as super-orthogonalization) of a tree to a regular lattice should be characterized by some non-local features associated to the loops. This explains in a descriptive way why the simple update works well for gapped states, where the physics is dominated by short-range correlations. For the systems that possess small gaps or are gapless, simple update is not sufficiently accurate\cite{XCYKNX14PESS}, particularly for the non-local physical properties such as the correlation functions.
\subsection{Rank-1 decomposition and algorithm}
\label{sec.rank1algo}
\begin{figure}[tbp]
\centering
\includegraphics[angle=0,width=0.5\linewidth]{Fig31.eps}
\caption{(Color online) The illustrations of rank-1 decomposition [Eq. (\ref{eq-4rank1})].}
\label{fig-4rank1}
\end{figure}
\textit{Rank-1 decomposition} in MLA\index{MLA} \cite{LMV00Rank1} provides a more mathematic and rigorous way to understand the approximation by super-orthogonalization (simple update) to truncate PEPS\index{PEPS} on regular lattices \cite{RXLS13NCD}. For a tensor $T$, its rank-1 decomposition (Fig. \ref{fig-4rank1}) is defined as
\begin{eqnarray}
T_{a_1a_2\cdots a_K} \simeq \Omega \prod_{k=1}^K v^{(k)}_{a_k},
\label{eq-4rank1}
\end{eqnarray}
where $v^{(k)}$ are normalized vectors and $\Omega$ is a constant that satisfies
\begin{eqnarray}
\Omega = \sum_{a_1a_2\cdots a_K} T_{a_1a_2\cdots a_K} \prod_{k=1}^K v^{(k)\ast}_{a_k}.
\label{eq-4rank1Const}
\end{eqnarray}
Rank-1 decomposition provides an approximation of $T$, where the distance between $T$ and its rank-1 approximation is minimized, i.e.,
\begin{eqnarray}
\min_{|v^{(k)}_{a_k}|=1} |T_{a_1a_2\cdots a_K} - \Omega \prod_{k=1}^K v^{(k)}_{a_k}|.
\label{eq-4rank1Min}
\end{eqnarray}
The rank-1 decomposition is given by the fixed point of a set of self-consistent equations (Fig. \ref{fig-4rank1cond}), which are
\begin{eqnarray}
\sum_{\text{all except } a_k} T_{a_1a_2\cdots a_K} \prod_{j \neq k} v^{(j)}_{a_j} = \Omega v^{(k)}_{a_k} \ \ (\forall \ k).
\label{eq-4rank1Self}
\end{eqnarray}
It means that $v^{(k)}$ is obtained by contracting all other vectors with the tensor. This property provides us an algorithm to compute rank-1 decomposition: one arbitrarily initialize the vectors $\{v^{(k)}\}$ of norm-1 and recursively update each vector by the tensor and the rest vectors using Eq. (\ref{eq-4rank1Self}) until all vectors converge.
\begin{figure}[tbp]
\centering
\includegraphics[angle=0,width=0.8\linewidth]{Fig32.eps}
\caption{(Color online) The illustrations of self-consistent conditions for the rank-1 decomposition [Eq. (\ref{eq-4rank1Self})].}
\label{fig-4rank1cond}
\end{figure}
Apart from some very special cases, such an optimization problem is concave, thus rank-1 decomposition is unique {\footnote{In fact, the uniqueness of rank-1 decomposition has not been rigorously proven.}}. Furthermore, if one arbitrarily chooses a set of norm-1 vectors, they will converge to the fixed point exponentially fast with the iterations. To the best of our knowledge, the exponential convergence has not been proved rigorously, but observed in most cases.
\subsection{Rank-1 decomposition, super-orthogonalization, and zero-loop approximation}
\label{sec.zeroloop}
Let us still consider an translational invariant square TN that is formed by infinite copies of the 4th-order tensor $T$ (Fig. \ref{fig-3TNcontract}). The rank-1 decomposition of $T$ provides an approximative scheme to compute the contraction of the TN, which is called the \textit{theory of network contractor dynamics} (NCD)\index{NCD} \cite{RXLS13NCD}.
The picture of NCD\index{NCD} can be understood in an opposite way to contraction, but by iteratively using the self-consistent conditions [Eq. (\ref{eq-4rank1Self})] to ``grow'' a tree TN\index{TN} that covers the whole square lattice (Fig. \ref{fig-4rank1encoding}). Let us start from Eq. (\ref{eq-4rank1Const}) of $\Omega$. Using Eq. (\ref{eq-4rank1Self}), we substitute each of the four vectors by the contraction of $T$ with the other three vectors. After doing so, Eq. (\ref{eq-4rank1Const}) becomes the contraction of more than one $T$'s with the vectors on the boundary. In other words, we ``grow'' the local TN contraction from one tensor plus four vectors to that with more tensors and vectors.
\begin{figure}[tbp]
\centering
\includegraphics[angle=0,width=0.8\linewidth]{Fig33.eps}
\caption{(Color online) Using the self-consistent conditions of the rank-1 decomposition, a tree TN\index{TN} with no loops can grow to cover the infinite square lattice. The four vectors gathering in a same site give the rank-1 approximation of the original tensor.}
\label{fig-4rank1encoding}
\end{figure}
By repeating the substitution, the TN\index{TN} can be grown to cover the whole square lattice, where each site is allowed to put maximally one $T$. Inevitably, some sites will not have $T$, but four vectors instead. These vectors (also called \textit{contractors}) give the rank-1 decomposition of $T$ as Eq. (\ref{eq-4rank1}). This is to say that some tensors in the square TN are replaced by its rank-1 approximation, so that all loops are destructed and the TN becomes a loopless tree covering the square lattice. In this way, the square TN is approximated by such a tree TN on square lattice, so that its contraction is simply computed by Eq. (\ref{eq-4rank1Const}).
The growing process as well as the optimal tree TN is only to understand the zero-loop approximation with rank-1 decomposition. There is no need to practically implement such a process. Thus, it does not matter how the TN is grown or where the rank-1 tensors are put to destroy the loops. All information we need is given by the rank-1 decomposition. In other words, the zero-loop approximation of the TN is encoded in the rank-1 decomposition.
For growing the TN\index{TN}, we shall remark that using the contraction of one $T$ with several vectors to substitute one vector is certainly not unique. However, the aim of ``growing'' is to reconstruct the TN formed by $T$. Thus, if $T$ has to appear in the substitution, the vectors should be uniquely chosen as those given in the rank-1 decomposition due to the uniqueness of rank-1 decomposition. Secondly, there are hidden conditions when covering the lattice by ``growing''. A stronger version is
\begin{eqnarray}
T_{a_1a_2a_3a_4} = T_{a_3a_2a_1a_4}^{\ast} = T_{a_1a_4a_3a_2}^{\ast} = T_{a_3a_4a_1a_2}.
\label{eq-4NCDcond1}
\end{eqnarray}
And a weaker one only requires the vectors to be conjugate to each other as
\begin{eqnarray}
v^{(1)} = v^{(3)\dagger}, \ \ v^{(2)} = v^{(4)\dagger}.
\label{eq-4NCDcond2}
\end{eqnarray}
These conditions assure that the self-consistent equations encodes the correct tree that optimally in the rank-1 sense approximates the square TN.
Comparing with Eqs. (\ref{eq-4SOconditions}) and (\ref{eq-4rank1Self}), the super-orthogonal conditions are actually equivalent to the above self-consistent equations of rank-1 decomposition by defining the tensor $T$ and vector $v$ as
\begin{eqnarray}
T_{a_1a_2 \cdots a_K} &=& \sum_s P_{s,a_1' a_2' \cdots a_K'} P_{s,a_1'' a_2'' \cdots a_K''}^{\ast} \prod_{k=1}^K \sqrt{\Lambda^{(k)}_{a_k'} \Lambda^{(k)\ast}_{a_k''}}, \label{eq-4DoubleT}\\
v^{(k)}_{a_k} &=& \sqrt{\Lambda^{(k)}_{a_k'} \Lambda^{(k)\ast}_{a_k''}},
\label{eq-4SOtensor}
\end{eqnarray}
with $a_k= (a_k',a_k'')$. Thus, the super-orthogonal spectrum provides an optimal approximation for the truncations of the bond dimensions in the zero-loop level. This provides a direct connection between the simple update scheme and rank-1 decomposition.
\subsection{Error of zero-loop approximation and tree-expansion theory based on rank-decomposition.}
The error of NCD\index{NCD} (and simple update) is an important issue. From the first glance, the error seems to be the error of rank-1 decomposition $\varepsilon = |T - \prod_k v^{(k)}|$. This would be true if we replaced all tensors in the square TN\index{TN} by the rank-1 version. In this case, the PEPS\index{PEPS} is approximated by a product state with zero entanglement. In the NCD\index{NCD} scheme, however, we only replace a part of the tensors to destruct the loops. The corresponding approximative PEPS is an entanglement state with a tree structure. Therefore, the error of rank-1 decomposition cannot properly characterize the error of simple update.
\begin{figure}[tbp]
\centering
\includegraphics[angle=0,width=0.6\linewidth]{Fig34.eps}
\caption{(Color online) The illustrations of rank-1 decomposition [Eq. (\ref{eq-4rank})].}
\label{fig-4rank}
\end{figure}
To control the error, let us introduce the \textit{rank decomposition} (also called CANDECOMP/PARAFAC decomposition)\index{CANDECOMP/PARAFAC} of $T$ in MLA\index{MLA} (Fig. \ref{fig-4rank}) that reads
\begin{eqnarray}
T_{a_1a_2 \cdots} = \sum_{r=0}^{R-1} \Omega_r \prod_k v^{(k,r)}_{a_{k}},
\label{eq-4rank}
\end{eqnarray}
where $v^{(k,r)}$ are normalized vectors. The idea of rank decomposition \cite{H1927rankorigin,H1928rankorigin1} is to expand $T$ into the summation of $R$ number of rank-1 tensors with $R$ called the \textit{tensor rank}. The elements of the vector $\Omega$ can always be in the descending order according to the absolute values. Then the leading term $\Omega_0 \prod_k v^{(k,0)}$ gives exactly the rank-1 decomposition of $T$, and the error of the rank-1 decomposition becomes $|\sum_{r=1}^{R-1} \Omega_r \prod_k v^{(k,r)}_{a_{k}}|$.
In the optimal tree TN\index{TN}, let us replace the rank-1 tensors back by the full rank tensor in Eq. (\ref{eq-4rank}). We suppose the rank decomposition is exact, thus we will recover the original TN by doing so. The TN contraction becomes the summation of $R^{\tilde{N}}$ terms with $\tilde{N}$ the number of rank-1 tensors in the zero-loop TN. Each term is the contraction of a tree TN, which is the same as the optimal tree TN except that certain vectors are changed to $v^{(k,r)}$ instead of the rank-1 term $v^{(k,0)}$. Note that in all terms, we use the same tree structure; the leading term in the summation is the zero-loop TN in the NCD\index{NCD} scheme. It means with rank decomposition, we expand the contraction of the square TN by the summation of the contractions of many tree TN's.
Let us order the summation referring to the contributions of different terms. For simplicity, we assume $R=2$, meaning $T$ can be exactly decomposed as the summation of two rank-1 tensors, which are the leading term given by the rank-1 decomposition, and the next-leading term denoted as $T_1 = \Omega_1 \prod_k v^{(k,1)}$. We dub as the next-leading term as the \textit{impurity tensor}. Defining $\tilde{n}$ as the number of the impurity tensors appearing in one of the tree TN in the summation, the expansion can be written as
\begin{eqnarray}
Z = \Omega_0^{\tilde{N}} \sum_{\tilde{n}=0}^{\tilde{N}} (\frac{\Omega_1}{\Omega_0})^{\tilde{n}} \sum_{\mathcal{C} \in \mathcal{C}(\tilde{n})} Z_{\mathcal{C}}.
\label{eq-4RankExpansion}
\end{eqnarray}
We use $\mathcal{C}(\tilde{n})$ to denote the set of all possible \textit{configurations} of $\tilde{n}$ number of the impurity tensors, where there are $\tilde{n}$ of $T_1$'s located in different positions in the tree. Then $Z_{\mathcal{C}}$ denotes the contraction of such a tree TN with a specific configuration of $T_1$'s. In general, the contribution is determined by the order of $|\Omega_1/\Omega_0|$ since $|\Omega_1/\Omega_0|<1$.
\begin{figure}[tbp]
\centering
\includegraphics[angle=0,width=1\linewidth]{Fig35.eps}
\caption{(Color online) The illustrations of the expansion with rank decomposition. The yellow and red circles stand for $v^{(k,0)}_{a_k}$ (zeroth order terms in the rank decomposition) and $v^{(k,1)}_{a_k}$ (first order terms), respectively. Here, we consider the tensor rank $R=2$ for simplicity.}
\label{fig-4rankexpansion}
\end{figure}
To proceed, we choose one tensor in the tree as the original point, and always contract the tree TN\index{TN} by ending at this tensor. Then the distance $\mathcal{D}$ of a vector is defined as the number of tensors in the path that connects this vector to the original point. Note that one impurity tensor is the tensor product of several vectors, and each vector may have different distance to the original point. For simplicity, we take the shortest one to define the distance of an impurity tensor.
Now, let us utilize the exponential convergence of the rank-1 decomposition. After contracting any vectors with the tensor in the tree, the resulting vector approaches to the fixed point (the vectors in the rank-1 decomposition) in an exponential speed. Define $\mathcal{D}_0$ as the average number of the contractions that will project any vectors to the fixed point with a tolerable difference. Consider any impurity tensors with the distance $\mathcal{D}>\mathcal{D}_0$, their contributions to the contraction are approximately the same, since after $\mathcal{D}_0$ contractions, the vectors have already been projected to the fixed point.
From the above argument, we can see that the error is related not only to the error of the rank-1 decomposition, but also to the speed of the convergence to the rank-1 component. The smaller $\mathcal{D}_0$ is, the smaller the error (the total contribution from the non-dominant terms) will be. Calculations show that the convergence speed is related to the correlation length (or gap) of the physical system, but their rigorous relations have not been established yet. Meanwhile, the expansion theory of the TN\index{TN} contraction given above requires the rank decomposition, which, however is not uniquely defined of an arbitrarily given tensor.
\section{iDMRG, iTEBD, and CTMRG revisited by tensor ring decomposition}
\label{sec5-TRD}
We have shown that the rank-1 decomposition solves the contraction of infinite-size tree TN\index{TN} and provides a mathematic explanation of the approximation made in the simple update. Then, it is natural to think: can we generalize this scheme beyond being only rank-1, in order to have better update schemes? In the following, we will show that besides the rank decomposition, the \textit{tensor ring decomposition} (TRD)\index{TRD} \cite{R16AOP} was suggested as another rank-N generalization for solving TN contraction problems.
TRD is defined by a set of self-consistent eigenvalue equations (SEE's)\index{SEE} with certain constraints. The original proposal of TRD requires all eigenvalue equations to be Hermitian \cite{R16AOP}. Later, a generalize version was proposed \cite{TLR16tMPSArxiv} that provides an unified description of the iDMRG\index{iDMRG} \cite{W92DMRG,W93DMRG,M08iDMRGarxiv}, iTEBD\index{iTEBD} \cite{V07iTEBD}, and CTMRG\index{CTMRG} \cite{OV09CTMRG} algorithms. We will concentrate on this version in the following.
\subsection{Revisiting iDMRG, iTEBD, and CTMRG: a unified description with tensor ring decomposition}
Let us start from the iDMRG\index{iDMRG} algorithm. The TN contraction can be solved using the iDMRG \cite{W92DMRG,W93DMRG,M08iDMRGarxiv} by considering an infinite-size row of tensors in the TN as an MPO \cite{VGC04MPDO,ZV04MPO,LRGZXY+11LTRG,BKTMWH17FT1D,GIK17FT1D} (also see some related discussions in Sec. \ref{iTEBD}). We introduce three third-order variational tensors denoted by $v^L$, $v^R$ (dubbed as the \textit{boundary} or \textit{environmental tensors}) and $\Psi$ (dubbed as the \textit{central tensor}). These tensors are the fixed-point solution of the a set of eigenvalue equations. $v^L$ and $v^R$ are, respectively, the left and right dominant eigenvector of the following matrices (Fig. \ref{fig-EigEqsiDMRG} (a) and (b))
\begin{eqnarray}
M^L_{c'b_1'b_1,cb_2'b_2} = \sum_{aa'} T_{a'c'ac} A^{\ast}_{a'b_1'b_2'} A_{a b_1b_2}, \label{eqs-MLiDMRG} \label{eq-4vL} \\
M^R_{c'b_1'b_1,cb_2'b_2} = \sum_{aa'} T_{a'c'ac} B^{\ast}_{a'b_1'b_2'} B_{a b_1b_2}, \label{eq-4vR}
\end{eqnarray}
where $A$ and $B$ are the left and right orthogonal parts obtained by the QR\index{QR} decomposition (or SVD\index{SVD}) of $\Psi$ (Fig. \ref{fig-EigEqsiDMRG} (d)) as
\begin{eqnarray}
\Psi_{abb'} = \sum_{b''} A_{abb''} \tilde{\Psi}_{b''b'} = \sum_{b''} \tilde{\Psi}^{\dagger}_{bb''} B_{ab''b'}. \label{eqs-QRPsi}
\end{eqnarray}
$\Psi$ is the dominant eigenvector of the Hermitian matrix (Fig. \ref{fig-EigEqsiDMRG} (c)) that satisfies
\begin{eqnarray}
\mathcal{H}_{a'b_1'b_2',ab_1b_2} = \sum_{cc'} T_{a'c'ac} v^{L}_{c'b_1'b_1} v^R_{cb_2'b_2}. \label{eq-4HeffDMRG}
\end{eqnarray}
\begin{figure}[tbp]
\centering
\includegraphics[angle=0,width=\linewidth]{EigEqsiDMRG_2.eps}
\caption{(Color online) The (a), (b) and (c) show the three local eigenvalue equations given by Eqs. (\ref{eq-4vR}) and (\ref{eq-4HeffDMRG}). The isometries $A$ and $B$ are obtained by the QR decompositions of $\Psi$ in two different ways in Eq. (\ref{eqs-QRPsi}), as shown in (d).}
\label{fig-EigEqsiDMRG}
\end{figure}
One can see that each of the eigenvalue problems is parametrized by the solutions of others, thus we solve them in a recursive way. First, we initialize arbitrarily the central tensors $\Psi$ and get $A$ and $B$ by Eq. (\ref{eqs-QRPsi}). Note that a good initial guess can make the simulations faster and more stable. Then we update $v^L$ and $v^R$ by multiplying with $M^L$ and $M^R$ as Eqs. (\ref{eq-4vL}) and (\ref{eq-4vR}). Then we have the new $\Psi$ by solving the dominant eigenvector of $\mathcal{H}$ in Eq. (\ref{eq-4HeffDMRG}) that is defined by the new $v^L$ and $v^R$. We iterate such a process until all variational tensors converge.
\begin{figure}[tbp]
\centering
\includegraphics[angle=0,width=0.55\linewidth]{Fig37.eps}
\caption{(Color online) The eigenvalue equations as illustrated ``encode'' the infinite TN.}
\label{fig-4TNE}
\end{figure}
Let us rephrase the iDMRG\index{iDMRG} algorithm given above in the language of TN\index{TN} contraction/reconstruction. When the variational tensors give the fixed point, the eigenvalue equations ``encodes'' the infinite TN, i.e., the TN can be reconstructed from the equations. To do so, we start from a local representation of $Z$ (Fig. \ref{fig-4TNE}) written as
\begin{eqnarray}
Z \leftrightharpoons \sum T_{a'c'ac} \Psi^{\ast}_{a'b_1b_2} \Psi_{ab_3b_4} v^{L}_{c'b_1b_3} v^R_{c b_2b_4},
\label{eq-4LocalTNE}
\end{eqnarray}
where the summation goes through all indexes. According to the fact that $\Psi$ is the leading eigenvector of Eq. (\ref{eq-4HeffDMRG}), $Z$ is maximized with fixed $v^L$ and $v^R$. We here and below use the symbol ``$\leftrightharpoons$'' to represent the contraction relation up to a difference of a constant factor.
Then, we use the eigenvalue equations of $v^L$ and $v^R$ to add one $M^L$ and one $M^R$ [Eq. (\ref{eq-4vL}) and (\ref{eq-4vR})] in the contraction, i.e., we substitute $v^L$ by $v^LM^L$ and $v^R$ by $M^Rv^R$. After doing so for one time, a finite central orthogonal MPS\index{MPS} appears, formed by $A$, $B$ and $\Psi$. Such substitutions can be repeated for infinite times, and then we will have an infinite central-orthogonal MPS formed by $\Psi$, $A$ and $B$ as
\begin{equation}
\Phi_{\cdots a_n \cdots} = \sum_{\{b\}} \cdots A_{a_{n-2}b_{n-2}b_{n-1}} A_{a_{n-1}b_{n-1}b_{n}} \Psi_{a_n b_nb_{n+1}} B_{a_{n+1}b_{n+1}b_{n+2}} B_{a_{n+2}b_{n+2}b_{n+3}} \cdots.
\label{eq-4iMPS}
\end{equation}
One can see that the bond dimension of $b_n$ is in fact the dimension cut-off of the MPS.
Now, we have
\begin{equation}
Z \leftrightharpoons \Phi^{\dagger} \rho \Phi,
\label{eq-4ZMPO}
\end{equation}
where $\rho$ is an infinite-dimensional matrix that has the form of an MPO (middle of Fig. \ref{fig-4TNE}) as
\begin{eqnarray}
\rho_{\cdots a_n' \cdots, \cdots a_n \cdots} = \sum_{\{c\}} \cdots T_{a_{n}'c_{n} a_{n} c_{n+1}} T_{a_{n+1}'c_{n+1} a_{n+1} c_{n+2}} \cdots.
\label{eq-4iMPO}
\end{eqnarray}
$\rho$ is in fact one raw of the TN\index{TN}. Compared with Eq. (\ref{eq-4LocalTNE}), the difference of $Z$ is only a constant factor that can be given by the dominant eigenvalues of $M^L$ and $M^R$.
After the substitutions from Eq. (\ref{eq-4LocalTNE}) to (\ref{eq-4ZMPO}), $Z$ is still maximized by the given $\Phi$, since $v^L$ and $v^R$ are the dominant eigenvectors. Note that such a maximization is reached under the assumption that the dominant eigenvector $\Phi$ can be well represented in an MPS\index{MPS} with finite bond dimensions. Meanwhile, one can easily see that the MPS is normalized $|\Phi_{\cdots a_n \cdots}|=1$, thanks to the orthogonality of $A$ and $B$. Then we come to a conclusion that $\Phi$ is the optimal MPS that gives the dominant eigenvector of $\rho$, satisfying $\Phi \leftrightharpoons \rho \Phi$ \footnote{For an Hermitian matrix $M$, $v$ is its dominant eigenvector if $|v| = 1$ and $v^{\dagger} M v$ is maximized.}. Then, we can rewrite the TN\index{TN} contraction as $Z \leftrightharpoons \lim_{K \to \infty} \Phi^{\dagger} \rho^K \Phi$, where the infinite TN appears as $\rho^K$ (Fig. \ref{fig-4TNE}).
\begin{figure}[tbp]
\centering
\includegraphics[angle=0,width=0.5\linewidth]{Fig38.eps}
\caption{(Color online) The illustrations of the TRD\index{TRD} in Eq. (\ref{eq-4TRD}).}
\label{fig-4TRD}
\end{figure}
Now we define the \textit{tensor ring decomposition} (TRD)\index{TRD} \footnote{The definition of TRD given here is from Ref. \cite{R16AOP}, which is completely different from the tensor ring decomposition proposed in Ref. \cite{ZZXZ+16tensorring}. While their TRD provides an approximation of a single tensor, the TRD discussed in this paper is more like an encoding scheme of an infinite-size TN\index{TN}.}: with the following conditions
\begin{itemize}
\item $Z$ [Eq. (\ref{eq-4LocalTNE})] is maximized under the constraint that $v^L$ and $v^R$ are normalized,
\item $\Phi^{\dagger} \rho \Phi$ is maximized under the constraint that $\Phi$ is normalized,
\end{itemize}
the TRD (Fig. \ref{fig-4TRD}) of $T$ is defined by $\tilde{T}$ as
\begin{eqnarray}
\tilde{T}_{a'c'ac} = \sum_{b_1b_2b_3b_4} \Psi^{\ast}_{a'b_1b_2} \Psi_{ab_3b_4} v^{L}_{c'b_1b_3} v^R_{c b_2b_4},
\label{eq-4TRD}
\end{eqnarray}
so that the TN contraction $Z = \sum T_{a'c'ac} \tilde{T}_{a'c'ac}$ [Eq. (\ref{eq-4LocalTNE})] is maximized. Like the NTD\index{NTD} and rank-1 decomposition, TRD belongs to the decompositions that encode infinite TN's, i.e., an infinite TN can be reconstructed from the self-consistent equations (note the rank decomposition does not encode any TN's). Comparing with Eq. (\ref{eq-4rank1Self}), TRD is reduced to the rank-1 decomposition by taking the dimensions of $\{b\}$ as one.
It was shown that for the same system, the ground state obtained by iDMRG\index{iDMRG} is equivalent to the ground state by iTEBD\index{iTEBD}, up to a gauge transformation \cite{M08iDMRGarxiv, HLOVV16TDVPMPS}. Different from this connection, TRD further unifies iDMRG and iTEBD. For iTEBD, after combining the contraction and truncation given by Eqs. (\ref{eq-3iTEBDEvolve}) and (\ref{eq-3iTEBDtruncate}), we have the equation for updating the tensor there as
\begin{eqnarray}
A_{s,cc'} \leftrightharpoons \sum_{s'aba'b'} T_{sbs'b'} A_{s',aa'} X_{ab,c} Y_{a'b',c'}.
\label{eq-4iTEBDupdate}
\end{eqnarray}
Looking at Eqs. (\ref{eq-4vR}) and (\ref{eq-4vR}), Eq. (\ref{eq-4iTEBDupdate}) is just the eigenvalue equation for updating $v^{[L(R)]}$ by $v^{[L(R)]} \leftrightharpoons M^{[L(R)]} v^{[L(R)]}$; the QR decomposition in Eq. (\ref{eqs-QRPsi}) guaranties that the ``truncations in iTEBD'' are implemented by isometries. In other words, one can consider another MPS\index{MPS} defined in the vertical direction, which is formed by $v^{[L(R)]}$ and updated by the iTEBD algorithm. It means that while implementing iDMRG in the parallel direction of the TN, one is in fact simultaneously implementing iTEBD to update another MPS along the vertical direction.
Particularly, when one uses iDMRG to solve the ground state of a 1D system, the MPS formed by $v^{[L(R)]}$ in the imaginary-time direction satisfies the continuous structure \cite{HM15folding, TLR16tMPSArxiv} that was originally proposed for continuous field theories \cite{VC10cMPS}. Such an iTEBD calculation can also be considered as the transverse contraction of the TN \cite{BHVC09folding, MCB12folding, HM15folding}.
CTMRG\index{CTMRG} \cite{OV09CTMRG, FVZHV17fastCTMRG} is also closely related to the scheme given above, which leads to the CTMRG without the corners. The tensors $\Psi$, $v^L$ and $v^R$ correspond to the row and column tensors, and the equations for updating these tensors are the same to the equations of updating the row and column tensors in CTMRG (see Eqs. (\ref{eq-3CTMRGcontract}) and (\ref{eq-3CTMRGtruncate})). Such a relation becomes more explicit in the rank-1 case, when corners become simply scalars. The difference is that in the original CTMRG by Or\'us \textit{et al} \cite{OV09CTMRG}, the tensors are updated with a power method, i.e., $\Psi \leftarrow \mathcal{H} \Psi$ and $v^{[L(R)]} \leftarrow M^{[L(R)]} v^{[L(R)]}$. Recently, eigen-solvers instead of power method were suggested in CTMRG (\cite{FVZHV17fastCTMRG} and a related review \cite{HV17TMTNrev}), where the eigenvalue equations of the row and column tensors are the same to those given in TRD\index{TRD}. The efficiency was shown to be largely improved with this modification.
\subsection{Extracting the information of tensor networks from eigenvalue equations: two examples}
\label{sec5.extract}
In the following, we present how to extract the properties of the TN\index{TN} by taking the \textit{free energy} and \textit{correlation length} as two example related to the eigenvalue equations. Note that these quantities correspond to the properties of the physical model and have been employed in many places (see, e.g., a review \cite{O14TNSRev}). In the following, we treated these two quantities as the properties of the TN itself. When the TN is used to represent different physical models, these quantities will be interpreted accordingly to different physical properties.
For an infinite TN, the contraction usually gives a divergent or vanishing value. The \textit{free energy} per tensor of the TN is defined to measure the contraction as
\begin{eqnarray}
f = -\lim_{N \to \infty} \frac{\ln \mathcal{Z}}{N},
\label{eq-4freeE}
\end{eqnarray}
with $\mathcal{Z}$ the value of the contraction in theory and $N$ denoting the number of tensors. Such a definition is closely related to some physical quantities, such as the free energy of classical models and the average fidelity of TN states \cite{ZOV08fidTN}. Meanwhile, $f$ can enable us to compare the values of the contractions of two TN's without actually computing $\mathcal{Z}$.
The free energy is given by the dominant eigenvalues of $M^L$ and $M^R$. Let us reverse the above reconstructing process to show this. Firstly, we use the MPS in Eq. (\ref{eq-4iMPS}) to contract the TN\index{TN} in one direction, and have $\mathcal{Z} = (\lim_{K \to \infty} \eta^K) \Phi^{\dagger} \Phi = \lim_{K \to \infty} \eta^K$ with $\eta$ the dominant eigenvalue of $\rho$. The problem becomes getting $\eta$. By going from $\Phi^{\dagger} \rho \Phi$ to Eq. (\ref{eq-4LocalTNE}), we can see that the eigenvalue problem of $\Phi$ is transformed to that of $\mathcal{H}$ in Eq. (\ref{eq-4HeffDMRG}) multiplied by a constant $\lim_{\tilde{K} \to \infty} \kappa_0^{\tilde{K}}$ with $\kappa_0$ the dominant eigenvalue of $M^L$ and $M^R$ and $\tilde{K}$ the number of tensors in $\rho$. Thus, we have $\eta = \eta_0 \kappa_0^{\tilde{K}}$ with $\eta_0$ the dominant eigenvalue of $\mathcal{H}$. Finally, we have the TN contraction $\mathcal{Z} = [\eta_0 \kappa_0^{\tilde{K}}]^K = \eta_0^K \kappa_0^{N}$ with $K\tilde{K}=N$. By substituting into Eq. (\ref{eq-4freeE}), we have $f = -\ln \kappa_0 - \lim_{\tilde{K} \to \infty} (\ln \eta_0) /K_1 = -\ln \kappa_0$.
The second issue is about the correlations of the TN\index{TN}. The \textit{correlation function} of a TN can be defined as
\begin{eqnarray}
F(\tilde{T}^{[\bf{r}_1]}, \tilde{T}^{[\bf{r}_2]}) = \mathcal{Z}(\tilde{T}^{[\bf{r}_1]}, \tilde{T}^{[\bf{r}_2]})/\mathcal{Z} - \mathcal{Z}(\tilde{T}^{[\bf{r}_1]}, T^{[\bf{r}_2]}) \mathcal{Z}(T^{[\bf{r}_1]}, \tilde{T}^{[\bf{r}_2]}) / \mathcal{Z}^2,
\label{eq-4CorrF}
\end{eqnarray}
where $\mathcal{Z}(\tilde{T}^{[\bf{r}_1]}, \tilde{T}^{[\bf{r}_2]})$ denotes the contraction of the TN\index{TN} after substituting the original tensors in the positions $\bf{r_1}$ and $\bf{r_2}$ by two different tensors $\tilde{T}^{[\bf{r}_1]}$ and $\tilde{T}^{[\bf{r}_2]}$. $T^{[\bf{r}]}$ denotes the original tensor at the position $\bf{r}$.
Though the correlation functions depend on the tensors that are substituted with, and can be defined in many different ways, the long-range behavior share some universal properties. For a sufficiently large distance ($|\mathbf{r_1}-\mathbf{r_2}| \gg 1$), if $\tilde{T}^{[\bf{r}_1]}$ and $\tilde{T}^{[\bf{r}_2]}$ are in a same column, $F$ satisfies
\begin{eqnarray}
F \sim e^{-|\mathbf{r_1}-\mathbf{r_2}|/\xi}.
\label{eq-4CorrelationExp}
\end{eqnarray}
One has the correlation length
\begin{eqnarray}
\xi = 1/(\ln \eta_0 - \ln \eta_1),
\label{eq-5Xiv}
\end{eqnarray}
with $\eta_0$ and $\eta_1$ the two dominant eigenvalues of $\mathcal{H}$. If $\tilde{T}^{[\bf{r}_1]}$ and $\tilde{T}^{[\bf{r}_2]}$ are in a same row, one has
\begin{eqnarray}
\xi = 1/(\ln \kappa_0 - \ln \kappa_1),
\label{eq-4CorrelationL}
\end{eqnarray}
with $\kappa_0$ and $\kappa_1$ the two dominant eigenvalues of $M^{L(R)}$.
To prove Eq. (\ref{eq-5Xiv}), we rewrite $\mathcal{Z}(\tilde{T}^{[\bf{r}_1]}, \tilde{T}^{[\bf{r}_2]})/\mathcal{Z}$ as
\begin{eqnarray}
\mathcal{Z}(\tilde{T}^{[\bf{r}_1]}, \tilde{T}^{[\bf{r}_2]})/\mathcal{Z} = [\Phi^{\dagger} \rho(\tilde{T}^{[\bf{r}_1]}, \tilde{T}^{[\bf{r}_2]}) \Phi]/(\Phi^{\dagger} \rho \Phi).
\end{eqnarray}
Then, introduce the transfer matrix $M$ of $\Phi^{\dagger} \rho \Phi$, i.e., $\Phi^{\dagger} \rho \Phi = {\bf Tr} M^{\tilde{K}}$ with $\tilde{K} \to \infty$. With the eigenvalue decomposition of $\mathcal{M} = \sum_{j=0}^{D-1} \eta_j v_j v_j^{\dagger}$ with $D$ the matrix dimension and $v_j$ the $j$-th eigenvectors, one can further simply the equation as
\begin{eqnarray}
\mathcal{Z}(\tilde{T}^{[\bf{r}_1]}, \tilde{T}^{[\bf{r}_2]})/\mathcal{Z} =
\sum_{j=0}^{D-1} (\eta_j/\eta_0)^{|\bf{r}_1 - \bf{r}_2|} v_0^{\dagger} \mathcal{M}(\tilde{T}^{[\bf{r}_1]}) v_{j} v_{j}^{\dagger} \mathcal{M}(\tilde{T}^{[\bf{r}_1]}) v_0,
\end{eqnarray}
with $\mathcal{M}(\tilde{T}^{[\bf{r}]})$ the transfer matrix after substituting the original tensor at $\bf{r}$ with $\tilde{T}^{[\bf{r}]}$. Similarly, one has
\begin{eqnarray}
\mathcal{Z}(\tilde{T}^{[\bf{r}_1]}, T)/\mathcal{Z} = v_0^{\dagger} \mathcal{M}(\tilde{T}^{[\bf{r}_1]}) v_0, \\
\mathcal{Z}(T,\tilde{T}^{[\bf{r}_2]})/\mathcal{Z} = v_0^{\dagger} \mathcal{M}(\tilde{T}^{[\bf{r}_2]}) v_0.
\end{eqnarray}
Note that one could transform the MPS into a translationally invariant form (e.g., the canonical form) to uniquely define the transfer matrix of $\Phi^{\dagger} \rho \Phi$. Substituting the equations above in Eq. (\ref{eq-4CorrF}), one has
\begin{eqnarray}
F(\tilde{T}^{[\bf{r}_1]}, \tilde{T}^{[\bf{r}_2]}) =
\sum_{j=1}^{D-1} (\eta_j/\eta_0)^{|\bf{r}_1 - \bf{r}_2|} v_0^{\dagger} \mathcal{M}(\tilde{T}^{[\bf{r}_1]}) v_{j} v_{j}^{\dagger} \mathcal{M}(\tilde{T}^{[\bf{r}_1]}) v_0.
\end{eqnarray}
When the distance is sufficiently large, i.e., $|\mathbf{r}_1 - \mathbf{r}_2| \gg 1$, only the dominant term takes effects, which is
\begin{eqnarray}
F(\tilde{T}^{[\bf{r}_1]}, \tilde{T}^{[\bf{r}_2]}) \simeq
(\eta_1/\eta_0)^{|\bf{r}_1 - \bf{r}_2|} v_0^{\dagger} \mathcal{M}(\tilde{T}^{[\bf{r}_1]}) v_{1} v_{1}^{\dagger} \mathcal{M}(\tilde{T}^{[\bf{r}_1]}) v_0.
\end{eqnarray}
Compared with Eq. (\ref{eq-4CorrelationExp}), one has $\xi = 1/(\ln \eta_0 - \ln \eta_1)$. The second case can be proven similarly.
These two quantities are defined independently on specific physical models that the TN\index{TN} might represent, thus they can be considered as the mathematical properties of the TN. By introducing physical models, these quantities are closely related to the physical properties. For example, when the TN represents the the partition function of a classical lattice model, Eq. (\ref{eq-4freeE}) multiplied by the temperature is exactly the free energy. And the correlation lengths of the TN are also the physical correlation lengths of the model in two spatial directions. When the TN gives the imaginary time evolution of an infinite 1D quantum chain, the correlation lengths of the TN are the spatial and dynamical correlation length of the ground state.
It is a huge topic to investigate the properties of the TN's or TN states. Paradigm examples include injectivity and symmetries \cite{ZBWC15fPEPS,HVSC15TNgauge, MOP16TNsymme,PSGWC10symme, W12Qspace, SCP10PEPSsymme, SPV10TNsymme, SPV11symmeU1, O11advTN, BCOT11assyme, SV12SU2symme, TCL14TNsymme, RDS15PEPSZ2, JR15TNsymme, LH16TNsymme, O14inject}, statistics and fusion rules \cite{GLSW09StringTPS, BAV09StringTPS, DBJC12TopoTNS, PCBA+10AnyonTN}. These issues are beyond the scope of this lecture notes. One may refer to the related works if interested.
\chapter{Quantum entanglement simulation inspired by tensor network}
\abstract{This chapter is focused the quantum entanglement simulation approach \cite{RPPSL17AOP3D}. The idea is to use few-body models embedded in the ``entanglement bath'' to mimic the properties of large and infinite-size quantum systems. The few-body models are dubbed as quantum entanglement simulators. Generally speaking, the QES\index{QES} approach consists of three stages: first, determine the effective interactions that give the infinite boundary condition \cite{RPPSL17AOP3D, PVM12InfBound} by the MPS/PEPS\index{MPS}\index{PEPS} methods, such as iDMRG\index{iDMRG} in one dimension or zero-loop scheme in two and three dimensions; second, construct the simulators by surrounding a finite-size cluster of the targeted model with the effective interactions; third, simulate the properties of the quantum entanglement simulator by the finite-size algorithms or quantum hardware, and extract the properties of the targeted model within the bulk.}
\section{Motivation and general ideas}
An impression one may have for the TN\index{TN} approaches of the quantum lattice models is that the algorithm (i.e., how to contract and/or truncate) will dramatically change when considering different models or lattices. This motivates us to look for more unified approaches. Considering that a huge number of problems can be transformed to TN contractions, one general question we may ask is: how can we reduce a nondeterministic-polynomial-hard TN contraction problem approximately to an effective one that can be computed exactly and efficiently by classical computers? We shall put some principles while considering this question: the effective problem should be as simple as possible, containing as few parameters to solve as possible. We would like to coin this principle for TN contractions as the \textit{ab-initio} optimization principle (AOP)\index{AOP} of TN \cite{R16AOP}. The term ``\textit{ab-inito}'' is taken here to pay respect to the famous \textit{ab-inito} principle approaches in condensed matter physics and quantum chemistry (see several recent reviews in \cite{S10DFTRev,B12DFT,B14DFT}). Here, ``\textit{ab-inito}'' means to think from the very beginning, with least prior knowledge of or assumptions to the problems.
One progress achieved in the spirit of AOP is the TRD\index{TRD} introduced in Sec. \ref{sec5-TRD}. Considering the TN on an infinite square lattice, its contraction is reduced to a set of self-consistent eigenvalue equations that can be efficiently solved by classical computers. The variational parameters are just two tensors. One advantage of TRD is that it connects the TN algorithms (iDMRG\index{iDMRG}, iTEBD\index{iTEBD}, CTMRG\index{CTMRG}), which are previously considered to be quite different, in a unified picture.
Another progress made in the AOP\index{AOP} spirit is called QES\index{QES} for simulating infinite-size physical models \cite{RPPSL17AOP3D, R16AOP, RXPS+18FTQES}. It is less dependent on the specific models; it also provides a natural way for designing quantum simulators and for hybridized-quantum-classical simulations of many-body systems. Hopefully in the future when people are aboe to readily realize the designed Hamiltonians on artificial quantum platforms, QES will enable us to design the Hamiltonians that will realized quantum many-body phenomena
\section{Simulating one-dimensional quantum lattice models}
Let us firstly take the ground-state simulation of the infinite-size 1D quantum system as an example. The Hamiltonian is the summation of two-body nearest-neighbor terms, which reads $\hat{H}_{Inf} = \sum_{n} \hat{H}_{n,n+1}$. The translational invariance is imposed. The first step is to choose a supercell (e.g., a finite part of the chain with $\tilde{N}$ sites). Then the Hamiltonian of the supercell is $\hat{H}_{B} = \sum_{n=1}^{\tilde{N}} \hat{H}_{n,n+1}$, and the Hamiltonian connecting the supercell to the rest part is $\hat{H}_{\partial} = \hat{H}_{n',n'+1}$ (note the interactions are nearest-neighbor).
\begin{figure}[tbp]
\centering
\includegraphics[angle=0,width=0.75\linewidth]{OperatorF.eps}
\caption{(Color online) Graphical representations of Eq.(\ref{eqs-Fpartial})-(\ref{eqs-cellT}).}
\label{fig-4OperatorF}
\end{figure}
Define the operator $\hat{F}^{\partial}$ as
\begin{eqnarray}
\hat{F}_{\partial} = \hat{I} - \tau \hat{H}_{\partial},
\label{eqs-Fpartial}
\end{eqnarray}
with $\tau$ the Trotter-Suzuki step. This definition is to construct the Trotter-Suzuki decomposition \cite{SI87Trotter,IS88Trotter}. Instead of using the exponential form $e^{-\tau \hat{H}}$, we equivalently chose to shift $\hat{H}_{\partial}$ for algorithmic consideration. The errors of these two ways concerning the ground state are at the same level ($\mathcal{O}(\tau^2)$). Introduce an ancillary index $a$ and rewrite $\hat{F}_{\partial}$ as a sum of operators as
\begin{eqnarray}
\hat{F}_{\partial} = \sum_a \hat{F}_{L}(s)_a \hat{F}_{R}(s')_a,
\label{eqs-Fboundary}
\end{eqnarray}
where $\hat{F}_{L}(s)_a$ and $\hat{F}_{R}(s')_a$ are two sets of one-body operators (labeled by $a$) acting on the left and right one of the two spins ($s$ and $s'$) associated with $\hat{H}_{\partial}$, respectively (Fig. \ref{fig-4OperatorF}). Eq. (\ref{eqs-Fboundary}) can be easily achieved directly from the Hamiltonian or using eigenvalue decomposition. For example for the Heisenberg interaction with $\hat{H}_{\partial} = \sum_{\alpha=x,y,z} J_{\alpha} \hat{S}^{\alpha}(s)\hat{S}^{\alpha}(s')$ with $\hat{S}^{\alpha}(s)$ the spin operators. We have $\hat{F}_{\partial} = \sum_{a = 0,x,y,z} \tilde{J}_{a} \hat{S}^{a}(s) \hat{S}^{a}(s')$ with $\hat{S}^{0} = I$, $\tilde{J}_0 = 1$, $\tilde{J}_{a} = -\tau J_{a}$ ($\alpha = x, y, z$), hence we can define $\hat{F}_{L}(s)_a = \sqrt{|\tilde{J}_{a}|} \hat{S}^{\alpha}(s)$ and $\hat{F}_{R}(s)_a = \text{sign}(\tilde{J}_{a}) \sqrt{|\tilde{J}_{a}|} \hat{S}^{\alpha}(s')$.
Construct the operator $\hat{\mathcal{F}}(S)_{a'a}$, with $S=(s_1,\cdots,s_{\tilde{N}})$ representing the physical spins inside the super-cell, as
\begin{eqnarray}
\hat{\mathcal{F}}(S)_{a'a}= \hat{F}_{R}(s_1)_{a'}^{\dagger} \tilde{H}_B \hat{F}_{L}(s_{\tilde{N}})_a,
\label{eqs-getF}
\end{eqnarray}
with $\tilde{H}_B = \hat{I} - \tau \hat{H}_B$. $\hat{F}_{R}(s_1)_{a'}^{\dagger}$ and $\hat{F}_{L}(s_{\tilde{N}})_{a}$ act on the first and last sites of the super-cell, respectively. One can see that $\hat{\mathcal{F}}(S)_{a'a}$ represents a set of operators labeled by two indexes ($a'$ and $a$) that act on the supercell.
In the language of TN\index{TN}, the co-efficients of $\hat{\mathcal{F}}(S)_{a'a}$ in the local basis ($|S\rangle = |s_1\rangle \cdots |s_{\tilde{N}}\rangle$) is a forth-order cell tensor (Fig. \ref{fig-4OperatorF}) as
\begin{eqnarray}
T_{S'a'Sa} = \langle S'| \hat{\mathcal{F}}(S)_{a'a} | S\rangle.
\label{eqs-cellT}
\end{eqnarray}
On the left-hand-side, the order of the indexes are arranged to be consistent with the definition in the TN algorithm introduced in the precious sections. $T$ is the cell tensor, whose infinite copies form the TN of the imaginary-time evolution up to the first Trotter-Suzuki order. One may consider the second Trotter-Suzuki order by defining $\hat{\mathcal{F}}(S)_{a'a}$ as $\hat{\mathcal{F}}(S)_{a'a} = (\hat{I} - \tau \hat{H}_B /2) \hat{F}_{R}(s_1)_{a'}^{\dagger} \hat{F}_{L}(s_{\tilde{N}})_a (\hat{I} - \tau \hat{H}_B /2)$. With the cell tensor $T$, the ground-state properties can be solved using the TN algorithms (e.g., TRD) introduced above. The ground state is given by the MPS given by Eq. (\ref{eq-4iMPS}).
Let us consider one of the eigenvalue equations [also see Eq. (\ref{eq-4HeffDMRG})] in the TRD\index{TRD}
\begin{eqnarray}
\mathcal{H}_{S'b_1'b_2',Sb_1b_2} = \sum_{aa'} T_{S'a'Sa} v^{L}_{a'b_1'b_1} v^R_{ab_2'b_2}. \label{eq-4HeffDMRG1}
\end{eqnarray}
We define a new Hamiltonian $\hat{\mathcal{H}}$ by using $\mathcal{H}_{S'b_1'b_2',Sb_1b_2}$ as the coefficients
\begin{eqnarray}
\hat{\mathcal{H}} =\sum_{SS'} \sum_{b_1b_2b_1'b_2'} \mathcal{H}_{S'b_1'b_2',Sb_1b_2} |S'b_1'b_2'\rangle\langle Sb_1b_2|.
\label{eq-4HeffDMRGop}
\end{eqnarray}
$\hat{\mathcal{H}}$ is the effective Hamiltonian in iDMRG\index{iDMRG} \cite{W92DMRG,W93DMRG,M07DMRGsymme} or the methods which represent the RG\index{RG} of Hilbert space by MPS \index{MPS} \cite{ZVFVH18vuMPS, HLOVV16TDVPMPS}. The indexes $\{b\}$ are considered as virtual spins with basis $\{|b\rangle\}$. The virtual spins are called the \textit{entanglement bath sites} in the QES\index{QES}.
By substituting with the cell tensor $T$ [Eqs. (\ref{eqs-getF}) and (\ref{eqs-cellT})] inside the above equation, we have
\begin{eqnarray}
\mathcal{\hat{H}} = \mathcal{\hat{H}}_L \tilde{H}_B \mathcal{\hat{H}}_R,
\label{eqs-Heffect}
\end{eqnarray}
where the Hamiltonians $\mathcal{\hat{H}}_L$ and $\mathcal{\hat{H}}_R$ locate on the boundaries of $\mathcal{\hat{H}}$, whose coefficients satisfy
\begin{eqnarray}
\begin{aligned}
\langle b_1's_{1}'| \mathcal{\hat{H}}_L |b_1 s_{1}\rangle = \sum_{a} v^{L}_{ab_1'b_1} \langle s_1' |\hat{F}_{R}(s_1)_{a}^{\dagger} | s_1\rangle, \\
\langle s_{\tilde{N}}' b_2'| \mathcal{\hat{H}}_R |s_{\tilde{N}} b_2\rangle = \sum_a \langle s_{\tilde{N}}'| \hat{F}_{L}(s_{\tilde{N}})_a^{\dagger} | s_{\tilde{N}}\rangle v^{R}_{ab_2b_2'}.
\end{aligned}
\label{eqs-HBtwobody}
\end{eqnarray}
$\mathcal{\hat{H}}_L$ and $\mathcal{\hat{H}}_R$ are just two-body Hamiltonians, of which each acts on the bath site and its neighboring physical site on the boundary of the bulk; they define the infinite boundary condition for simulating the time evolution of 1D quantum systems \cite{PVM12InfBound}.
$\mathcal{\hat{H}}_L$ and $\mathcal{\hat{H}}_R$ can also be written in a shifted form as
\begin{equation}
\mathcal{\hat{H}}_{L(R)} = I-\tau \hat{H}_{L(R)}.
\label{eqs-HfbLR}
\end{equation}
This is because the tensor $v^L$ (and also $v^R$) satisfies a special form \cite{TLR16tMPSArxiv} as
\begin{eqnarray}
v^{L}_{0,bb'} = I_{bb'} - \tau Q_{bb'}, \\
v^{L}_{a,bb'} = \tau^{a/2} (R^a)_{bb'} \ \ (a > 0), \\
\end{eqnarray}
with $Q$ and $R$ two Hermitian matrices independent on $\tau$. In other words, the MPS\index{MPS} formed by infinite copies of $v^{L}$ or $v^{R}$ is a continuous MPS \cite{VC10cMPS}, which is known as the temporal MPS \cite{HM15folding}. Therefore, $\hat{H}_{L(R)}$ is independent on $\tau$, called the \textit{physical-bath Hamiltonian}. Then $\mathcal{\hat{H}}$ can be written as the shift of a few-body Hamiltonian as $\mathcal{\hat{H}} = I- \tau \hat{H}_{FB}$, where $\hat{H}_{FB}$ has the standard summation form as
\begin{eqnarray}
\hat{H}_{FB} = \hat{H}_L + \sum_{n=1}^{L} \hat{H}_{n,n+1} + \hat{H}_R.
\label{eqs-Hfewbody}
\end{eqnarray}
For $\hat{H}_{L}$ and $\hat{H}_{R}$ with the bath dimension$\chi$, the coefficient matrix of $\hat{H}_{L(R)}$ is ($2\chi \times 2\chi$). Then $\hat{H}_{L(R)}$ can be generally expanded by $\hat{S}^{\alpha_1} \otimes \hat{\mathcal{S}}^{\alpha_2}$ with $\{\hat{\mathcal{S}}\}$ the generators of the SU($\chi$) group, and define the magnetic field and coupling constants associated to the entanglement bath
\begin{eqnarray}
\hat{H}_{L(R)} = \sum_{\alpha_1,\alpha_2} J^{\alpha_1\alpha_2}_{L(R)} \hat{\mathcal{S}}^{\alpha_1} \otimes \hat{S}^{\alpha_2},
\label{eqs-Constants}
\end{eqnarray}
with $\mathcal{S}$ denoting the SU($\chi$) spin operators and $\hat{S}$ the operators of the physical spin (with the identity included).
Let us take the bond dimension $\chi=2$ as an example, and $\hat{H}_{L(R)}$ just gives the Hamiltonian between two spin-$1/2$'s. Thus, it can be expanded by the spin (or Pauli) operators $\hat{S}^{\alpha_1} \otimes \hat{S}^{\alpha_2}$ as
\begin{eqnarray}
\hat{H}_{L(R)} = \sum_{\alpha_1,\alpha_2=0}^3 J^{\alpha_1\alpha_2}_{L(R)} \hat{S}^{\alpha_1} \otimes \hat{S}^{\alpha_2},
\label{eqs-ConstantsSU2}
\end{eqnarray}
where the spin-$1/2$ operators are labeled as $\hat{S}^0 = I$, $\hat{S}^1 = \hat{S}^x$, $\hat{S}^2 = \hat{S}^y$, and $\hat{S}^3 = \hat{S}^z$. Then with $\alpha_1 \neq 0$ and $\alpha_2 \neq 0$, we have $J^{\alpha_1\alpha_2}_{L(R)}$ as the coupling constants, and $J^{\alpha_1 0}_{L(R)}$ and $J^{0 \alpha_2}_{L(R)}$ the magnetic fields on the first and second sites, respectively. $J^{0 0}_{L(R)}$ only provides a constant shift of the Hamiltonian which does not change the eigenstates.
\begin{figure}[tbp]
\centering
\includegraphics[angle=0,width=1\linewidth]{Hbath.eps}
\caption{(Color online) The $\alpha$-dependence \cite{RPSL17CSW} of the coupling constants (left) and magnetic fields (right) of the few-body Hamiltonians [Eq. (\ref{eq-4HLR})]. Reused from \cite{RPSL17CSW} with permission.}
\label{fig-4Hbath}
\end{figure}
As an example, we show the $\hat{H}_{L}$ and $\hat{H}_{R}$ for the infinite quantum Ising chain in a transverse field \cite{RPSL17CSW}. The original Hamiltonian reads $\hat{H}_{Ising} = \sum_n \hat{S}^z_n \hat{S}^z_{n+1} -\alpha \sum_{n} \hat{S}^x_n$, and $\hat{H}_{L}$ and $\hat{H}_{R}$ satisfies
\begin{eqnarray}
\begin{aligned}
&\hat{H}_{L} = J^L_{xz} \hat{S}^x_1 \hat{S}^z_2 + J^L_{zz} \hat{S}^z_1 \hat{S}^z_2 - h^L_{x} \hat{S}^x_1 - h^L_{z} \hat{S}^z_1 - \tilde{h}^L_{x} \hat{S}^x_2,\\
&\hat{H}_{R} = J^R_{zx} \hat{S}^z_{N-1} \hat{S}^x_N + J^R_{zz} \hat{S}^z_{N-1} \hat{S}^z_{N} - h^R_{x} \hat{S}^x_{N} - h^R_{z} \hat{S}^z_{N} - \tilde{h}^R_{x} \hat{S}^x_{N-1}.
\end{aligned}
\label{eq-4HLR}
\end{eqnarray}
The coupling constants and magnetic fields depend on the transverse field $\alpha$, as shown in Fig. \ref{fig-4Hbath}. The calculation shows that except the Ising interactions and the transverse field that originally appear in the infinite model, the $\hat{S}^x \hat{S}^z$ coupling and a vertical field emerge in $\hat{H}_{L}$ and $\hat{H}_{R}$. This is interesting, because the $\hat{S}^x \hat{S}^z$ interaction is the stabilizer on the open boundaries of the cluster state, a highly entangled state that has been widely used in quantum information sciences \cite{BR01EntgleState,RB01EntgleState}. More relations with the cluster state are to be further explored.
The physical information of the infinite-size model can be extracted from the ground state of $\hat{H}_{FB}$ (denoted by $|\Psi(Sb_1b_2)\rangle$) by tracing over the entanglement-bath degrees of freedom. To this aim, we calculate the reduced density matrix of the bulk as
\begin{eqnarray}
\hat{\rho}(S) = \text{Tr}_{b_1b_2} |\Psi(Sb_1b_2) \rangle \langle \Psi(Sb_1b_2)|.
\label{eq-4RDM}
\end{eqnarray}
Note $|\Psi(Sb_1b_2)\rangle = \sum_{Sb_1b_2} \Psi_{Sb_1b_2} |Sb_1b_2\rangle$ with $\Psi_{Sb_1b_2}$ the eigenvector of Eq. (\ref{eq-4HeffDMRG1}) or (\ref{eq-4HeffDMRG}). It is easy to see that $\Psi_{Sb_1b_2}$ is the central tensor in the central-orthogonal MPS\index{MPS} [Eq. (\ref{eq-4iMPS})], thus the $\hat{\rho}(S)$ is actually the reduced density matrix of the MPS. Since the MPS optimally gives the ground state of the infinite model, therefore, $\hat{\rho}(S)$ of the few-body ground state optimally gives the reduced density matrix of the original model.
In Eq. (\ref{eqs-Hfewbody}), the summation of the physical interactions are within the supercell that we choose to construct the cell tensor. To improve the accuracy to, e.g., capture longer correlations inside the bulk, one just needs to increase the supercell in $\hat{H}_{FB}$. In other words, $\hat{H}_L$ and $\hat{H}_R$ are obtained by TRD\index{TRD} from the supercell of a tolerable size $\tilde{N}$, and $\hat{H}_{FB}$ is constructed with a larger bulk as $\hat{H}_{FB} = \hat{H}_L + \sum_{n=1}^{\tilde{N}'} \hat{H}_{n,n+1} + \hat{H}_R$ with $\tilde{N}'>\tilde{N}$. Though $\hat{H}_{FB}$ becomes more expensive to solve, we have any well-established finite-size algorithms to compute its dominant eigenvector. We will show below that this way is extremely useful in higher dimensions.
\section{Simulating higher-dimensional quantum systems}
For ($D>1$)-dimensional quantum systems on, e.g., square lattice, one can use different update schemes to calculate the ground state. Here, we explain an alternative way by generalizing the above 1D simulation to higher dimensions \cite{RPPSL17AOP3D}. The idea is to optimize the physical-bath Hamiltonians by the zero-loop approximation (simple update, see Sec. \ref{sec.looplessTNrank-1}), e.g., iDMRG\index{iDMRG} on tree lattices \cite{LCP00TreeDMRG,NC13TTN}, and then construct the few-body Hamiltonian $\hat{H}_{FB}$ with larger bulks. The loops inside the bulk will be fully considered when solving the ground state of $\hat{H}_{FB}$, thus the precision will be significantly improved compared with the zero-loop approximation.
\begin{figure}[tbp]
\centering
\includegraphics[angle=0,width=0.6\linewidth]{getF.eps}
\caption{(Color online) Graphical representation of the cell tensor for 2D quantum systems [Eq. (\ref{eq-getF})].}
\label{fig-4getF}
\end{figure}
The procedures are similar to those for 1D models. The first step is to contract the cell tensor, so that the ground-state simulation is transformed to a TN\index{TN} contraction problem. We choose the two sites connected by a parallel bond as the supercell, and construct the cell tensor that parametrizes the eigenvalue equations. The bulk interaction is simply the coupling between these two spins, i.e. $\hat{H}_B = \hat{H}_{i,j}$, and the interaction between two neighboring supercells is the same, i.e., $\hat{H}_{\partial} = \hat{H}_{i,j}$. By shifting $\hat{H}_{\partial}$, we define $\hat{F}_{\partial} = I-\tau \hat{H}_{\partial}$ and decompose it as
\begin{eqnarray}
\hat{F}_{\partial} = \sum_{a} \hat{F}_{L}(s)_a \otimes \hat{F}_{R}(s')_a.
\label{eq-Fsvd}
\end{eqnarray}
$\hat{F}_{L}(s)_a$ and $\hat{F}_{R}(s')_a$ are two sets of operators labeled by $a$ that act on the two spins ($s$ and $s'$) in the supercell, respectively (see the texts below Eq. (\ref{eqs-Fboundary}) for more detail).
Define a set of operators by the product of the (shifted) bulk Hamiltonian with $\hat{F}_{L}(s)_a$ and $\hat{F}_{R}(s)_a$ (Fig. \ref{fig-4getF}) as
\begin{equation}
\hat{\mathcal{F}}(S)_{a_1a_2a_3a_4}= \hat{F}_{R}(s)_{a_1} \hat{F}_{R}(s)_{a_2} \hat{F}_{L}(s')_{a_3} \hat{F}_{L}(s')_{a_4} \tilde{H}^B,
\label{eq-getF}
\end{equation}
with $S=(s,s')$ and $\tilde{H}^B = I- \tau \hat{H}^B$. The cell tensor that defines the TN\index{TN} is given by the coefficients of $\hat{\mathcal{F}}(S)_{a_1a_2a_3a_4}$ as
\begin{eqnarray}
T_{S'Sa_1a_2a_3a_4} = \langle S'| \hat{\mathcal{F}}(S)_{a_1a_2a_3a_4} |S\rangle.
\label{eq-4Tcell}
\end{eqnarray}
One can see that $T$ has six bonds, of which two ($S$ and $S'$) are physical and four ($a_1$, $a_2$, $a_3$, and $a_4$) are non-physical. For comparison, the tensor in the 1D quantum case has four bonds, where two are physical and two are non-physical [see Eq. (\ref{eqs-cellT})]. As discussed above in Sec. \ref{sec.evo2D}, the ground-state simulation becomes the contraction of a cubic TN formed by infinite copies of $T$. Each layer of the cubic TN gives the operator $\hat{\rho}(\tau) = I-\tau \hat{H}$, which is a PEPO\index{PEPO} defined on a square lattice. Infinite layers of the PEPO $\lim_{K \to \infty} \hat{\rho}(\tau)^K$ give the cubic TN\index{TN}.
The next step is to solve the SEE's\index{SEE} of the zero-loop approximation. For the same model defined on the loopless Bethe lattice, the 3D TN is formed by infinite layers of PEPO $\hat{\rho}_{Bethe}(\tau)$ that is defined on the Bethe lattice. The cell tensor is defined exactly in the same way as Eq. (\ref{eq-4Tcell}). With the Bethe approximation, there are five variational tensors, which are $\Psi$ (central tensor) and $v^{[x]}$ ($x=1,2,3,4$, boundary tensors). Meanwhile, we have five self-consistent equations that encodes the 3D TN $\lim_{K \to \infty} \hat{\rho}_{Bethe}(\tau)^K$, which are given by five matrices as
\begin{eqnarray}
\mathcal{H}_{S'b_1'b_2'b_3'b_4',Sb_1b_2b_3b_4} = \sum_{a_1a_2a_3a_4} T_{S'Sa_1a_2a_3a_4} v^{[1]}_{a_1b_1 b_1'} v^{[2]}_{a_2b_2 b_2'} v^{[3]}_{a_3b_3 b_3'} v^{[4]}_{a_4b_4 b_4'}, \label{eq-effectH}\\
M^{[1]}_{a_1b_1b_1',a_3b_3b_3'} = \sum_{S'Sa_2a_4b_2b_2'b_4b_4'} T_{S'Sa_1a_2a_3a_4} A^{[1]\ast}_{S'b_1'b_2'b_3'b_4'} v^{[2]}_{a_2b_2 b_2'} A^{[1]}_{Sb_1b_2b_3b_4} v^{[4]}_{a_4b_4 b_4'},\label{eq-M1}\\
M^{[2]}_{a_2b_2b_2',a_4b_4b_4'} = \sum_{S'Sa_1a_3b_1b_1'b_3b_3'} T_{S'Sa_1a_2a_3a_4} A^{[2]\ast}_{S'b_1'b_2'b_3'b_4'} v^{[1]}_{a_1b_1 b_1'} A^{[2]}_{Sb_1b_2b_3b_4} v^{[3]}_{a_3b_3 b_3'},\label{eq-M2}\\
M^{[3]}_{a_1b_1b_1',a_3b_3b_3'} = \sum_{S'Sa_2a_4b_2b_2'b_4b_4'} T_{S'Sa_1a_2a_3a_4} A^{[3]\ast}_{S'b_1'b_2'b_3'b_4'} v^{[2]}_{a_2b_2 b_2'} A^{[3]}_{Sb_1b_2b_3b_4} v^{[4]}_{a_4b_4 b_4'},\label{eq-M3}\\
M^{[4]}_{a_2b_2b_2',a_4b_4b_4'} = \sum_{S'Sa_1a_3b_1b_1'b_3b_3'} T_{S'Sa_1a_2a_3a_4} A^{[4]\ast}_{S'b_1'b_2'b_3'b_4'} v^{[1]}_{a_1b_1 b_1'} A^{[4]}_{Sb_1b_2b_3b_4} v^{[3]}_{a_3b_3 b_3'}.\label{eq-M4}
\end{eqnarray}
Eqs. (\ref{eq-effectH}) and (\ref{eq-M2}) are illustrated in Fig. \ref{fig-4SelfConsisEq} as two examples. $A^{[x]}$ is an isometry obtained by the QR\index{QR} decomposition (or SVD\index{SVD}) of the central tensor $\Psi$ referring to the $x$-th virtual bond $b_x$. For example for $x=2$, we have (Fig. \ref{fig-4SelfConsisEq})
\begin{eqnarray}
\Psi_{Sb_1b_2b_3b_4} = \sum_{b} A^{[2]}_{Sb_1bb_3b_4} R^{[2]}_{b b_2}.
\label{eq-4QRPsi}
\end{eqnarray}
$A^{[2]}$ is orthogonal, satisfying
\begin{eqnarray}
\sum_{Sb_1 b_3 b_4} A^{[2] \ast}_{Sb_1bb_3b_4} A^{[2]}_{Sb_1b'b_3b_4} = I_{bb'}.
\label{eq-OrtA3}
\end{eqnarray}
\begin{figure}[tbp] \includegraphics[angle=0,width=0.8\linewidth]{Fig43.eps}
\centering
\caption{(Color online) The left figure is the graphic representations of $\mathcal{H}_{S'b_1'b_2'b_3'b_4',Sb_1b_2b_3b_4}$ in Eq.(\ref{eq-effectH}), and we take Eq.(\ref{eq-M2}) from the self-consistent equations as an example shown in the middle. The QR\index{QR} decomposition in Eq.(\ref{eq-4QRPsi}) is shown in the right figure, where the arrows indicate the direction of orthogonality of $A^{[3]}$ in Eq.(\ref{eq-OrtA3}).}
\label{fig-4SelfConsisEq}
\end{figure}
The self-consistent equations can be solved recursively. By solving the leading eigenvector of $\mathcal{H}$ given by Eq. (\ref{eq-effectH}), we update the central tensor $\Psi$. Then according to Eq. (\ref{eq-4QRPsi}), we decompose $\Psi$ to obtain $A^{[x]}$, then update $M^{[x]}$ in Eqs. (\ref{eq-M1})-(\ref{eq-M4}), and update each $v^{[x]}$ by $M^{[x]} v^{[x]}$. Repeat this process until all the five variational tensors converge. The algorithm is the generalized DMRG\index{DMRG} based on infinite tree PEPS\index{PEPS} \cite{LCP00TreeDMRG,NC13TTN}. Each boundary tensor can be understood as the infinite environment of a tree branch, thus the original model is actually approximated at this stage by that defined on an Bethe lattice. Note that when only looking at the tree locally (from one site and its nearest neighbors), it looks the same to the original lattice. Thus, the loss of information is mainly long-range, i.e., from the destruction of loops.
\begin{figure}[tbp]
\centering
\includegraphics[angle=0,width=0.7\linewidth]{ZBethe.eps}
\caption{(Color online) The left figure shows the local contraction the encodes the infinite TN\index{TN} for simulating the 2D ground state. By substituting with the self-consistent equations, the TN representing $\tilde{Z} = \langle \tilde{\Phi}| \hat{\rho}_{Bethe}(\tau) | \tilde{\Phi} \rangle$ can be reconstructed, with $\hat{\rho}_{Bethe}(\tau)$ the tree PEPO\index{PEPO} of the Bethe model and $| \tilde{\Phi} \rangle$ a PEPS\index{PEPS}.}
\label{fig-4ZBethe}
\end{figure}
The Bethe approximation can be understood better from the rank-1 decomposition (see Sec. 4.5). Eqs. (\ref{eq-M1})-(\ref{eq-M4}) encodes a Bethe TN, whose contraction is written as $Z_{Bethe} = \langle \tilde{\Phi}| \hat{\rho}_{Bethe}(\tau) | \tilde{\Phi} \rangle$ with $\hat{\rho}_{Bethe}(\tau)$ the PEPO\index{PEPO} of the Bethe model and $| \tilde{\Phi} \rangle$ a tree iPEPS\index{iPEPS} (Fig.\ref{fig-4ZBethe}). To see this, let us start with the local contraction [Fig.\ref{fig-4ZBethe} (a)] as
\begin{eqnarray}
Z_{Bethe} = \sum \Psi^{\ast}_{S'b_1'b_2'b_3'b_4'} \Psi_{Sb_1b_2b_3b_4} T_{S'Sa_1a_2a_3a_4} v^{[1]}_{a_1b_1 b_1'} v^{[2]}_{a_2b_2 b_2'} v^{[3]}_{a_3b_3 b_3'} v^{[4]}_{a_4b_4 b_4'}.
\label{eq-4Zlocal}
\end{eqnarray}
Then, each $v^{[x]}$ can be replaced by $M^{[x]} v^{[x]}$ because we are at the fixed point of the eigenvalue equations. By repeating this substitution in a similar way as the rank-1 decomposition in Sec. \ref{sec.zeroloop}, we will have the TN\index{TN} for $Z_{Bethe} $, which is maximized at the fixed point [Fig.\ref{fig-4ZBethe} (b)]. With the constraint $\langle \tilde{\Phi}| \tilde{\Phi} \rangle =1$ satisfied, $| \tilde{\Phi} \rangle$ is the ground state of $\hat{\rho}_{Bethe}(\tau)$.
Now, we constrain the growth so that the TN\index{TN} covers the infinite square lattice. Inevitably, some $v^{[x]}$'s will gather at the same site. The tensor product of these $v^{[x]}$'s in fact gives the optimal rank-1 approximation of the ``correct'' full-rank tensor here (Sec. \ref{sec.zeroloop}). Suppose that one uses the full-rank tensor to replace its rank-1 version (the tensor product of four $v^{[x]}$'s), one will have the PEPO\index{PEPO} of $I- \tau \hat{H}$ (with $H$ the Hamiltonian on square lattice), and the tree iPEPS\index{iPEPS} becomes the iPEPS defined on the square lattice. Compared with the NCD\index{NCD} scheme that employs rank-1 decomposition explicitly to solve TN contraction, one difference here for updating iPEPS is that the ``correct'' tensor to be decomposed by rank-1 decomposition contains the variational tensor, thus is in fact unknown before the equations are solved. For this reason, we cannot use rank-1 decomposition directly. Another difference is that the constraint, i.e., the normalization of the tree iPEPS, should be fulfilled. By utilizing the iDMRG\index{iDMRG} algorithm with the tree iPEPS, the rank-1 tensor is obtained without knowing the ``correct'' tensor, and meanwhile the constraints are satisfied. The zero-loop approximation of the ground state is thus given by the tree iPEPS.
\begin{figure}[tbp]
\centering
\includegraphics[angle=0,width=1\linewidth]{Hbathx.eps}
\caption{(Color online) The left figure shows the few-body Hamiltonian $\mathcal{\hat{H}}$ in Eq. (\ref{eq-4Heffect2D}). The middle one shows the physical-bath Hamiltonian $\hat{\mathcal{H}}_{\partial}$ that gives the interaction between the corresponding physical and bath site. The right one illustrates the state ansatz for the infinite system. Note that the boundary of the cluster should be surrounded by $\hat{\mathcal{H}}_{\partial}$'s, and each $\hat{\mathcal{H}}_{\partial}$ corresponds to an infinite tree brunch in the state ansatz. For simplicity, we only illustrate four of the $\hat{\mathcal{H}}_{\partial}$'s and the corresponding brunches.}
\label{fig-4Hbathx}
\end{figure}
The few-body Hamiltonian is constructed in a larger cluster, so that the error brought by zero-loop approximation can be reduced. Similar to the 1D case, we embed a larger cluster in the middle of the entanglement bath. The few-body Hamiltonian (Fig. \ref{fig-4Hbathx}) is written as
\begin{eqnarray}
\mathcal{\hat{H}} = \prod_{\langle n \in cluster, \alpha \in bath \rangle} \hat{\mathcal{H}}_{\partial}(n,\alpha) \prod_{\langle i,j \rangle \in cluster} [I-\tau \hat{H}(s_i,s_j)].
\label{eq-4Heffect2D}
\end{eqnarray}
$\hat{\mathcal{H}}_{\partial}(n,\alpha)$ is defined as the physical-bath Hamiltonian between the $\alpha$-th bath site and the neighboring $n$-th physical site, and it is obtained by the corresponding boundary tensor $v^{[x(\alpha)]}$ and $\hat{F}_{L(R)}(s_n)$ (Fig. \ref{fig-4Hbathx}) as
\begin{eqnarray}
\langle b_{\alpha}'s_n'| \mathcal{\hat{H}}_{\partial}(n,\alpha) |b_{\alpha}s_n\rangle = \sum_{a} v^{[x(\alpha)]}_{ab_{\alpha}' b_{\alpha}} \langle s_n' |\hat{F}_{L(R)}(s_n)_{a} | s_n\rangle.
\label{eq-4Hbathx}
\end{eqnarray}
Here, $\hat{F}_{L(R)}(s_n)_{a}$ is the operator defined in Eq. (\ref{eq-Fsvd}), and $v^{[x(\alpha)]}_{ab_{\alpha}' b_{\alpha}}$ are the solutions of the SEE's\index{SEE} given in Eqs. (\ref{eq-effectH})-(\ref{eq-M4}).
$\hat{\mathcal{H}}$ in Eq. (\ref{eq-4Heffect2D}) can also be rewritten as the shift of a few-body Hamiltonian $\hat{H}_{FB}$, i.e. $\mathcal{\hat{H}} = I-\tau \hat{H}_{FB}$. We have $\hat{H}_{FB}$ possessing the standard summation form as
\begin{eqnarray}
\hat{H}_{FB} = \sum_{\langle i,j \rangle \in cluster} \hat{H}(s_i,s_j) + \sum_{\langle n \in cluster,\alpha \in bath \rangle} \hat{H}_{PB}(n,\alpha),
\label{eq-4Hfb2D}
\end{eqnarray}
with $\mathcal{\hat{H}}_{\partial}(n,\alpha) = I-\tau \hat{H}_{PB}(s_n,b_{\alpha})$. This equations gives a general form of the few-body Hamiltonian: the first term contains all the physical interactions inside the cluster, and the second contains all physical-bath interactions $\hat{H}_{PB}(s_n,b_{\alpha})$. $\hat{\mathcal{H}}$ can be solved by any finite-size algorithms, such as exact diagonalization, QMC\index{QMC}, DMRG\index{DMRG} \cite{W92DMRG, WS98tjDMRG, XLS01DMRG2D} or finite-size PEPS\index{PEPS} \cite{SV09TNQMC, LCB14fPEPS, LDHGH17TNQMC} algorithms. The error from the rank-1 decomposition will be reduced since the loops inside the cluster will be fully considered.
Similar to the 1D cases, the ground-state properties can be extracted by the reduced density matrix $\hat{\rho}(S)$ after tracing over the entanglement-bath degrees of freedom. We have $\hat{\rho}(S) = \text{Tr}_{/(S)} |\Phi \rangle \langle \Phi |$ (with $|\Phi \rangle$ the ground state of the infinite model) that well approximate by
\begin{eqnarray}
\hat{\rho}(S) \simeq \sum_{SS'b_1b_2 \cdots} \Psi^{\ast}_{S'b_1b_2 \cdots} \Psi_{Sb_1b_2 \cdots} |S \rangle \langle S'|.
\label{eq-rhoBath}
\end{eqnarray}
with $\Psi_{Sb_1b_2 \cdots} $ the coefficients of the ground state of $\hat{H}_{FB}$.
Fig. \ref{fig-4Hbathx} illustrates the ground state ansatz behind the few-body model. The cluster in the center is entangled with the surrounding infinite-tree brunches through the entanglement-bath degrees of freedom. Note that solving Eq. (\ref{eq-effectH}) in Stage one is equivalent to solving Eq. (\ref{eq-4Heffect2D}) by choose the cluster as one supercell.
Some benchmark results of simulating 2D and 3D spin models can be found in Ref. \cite{RPPSL17AOP3D}. For the ground state of Heisenberg model on honeycomb lattice, results of the magnetization and bond energy show that the few-body model of 18 physical and 12 bath sites suffers only a small finite-effect of $O(10^{-3})$. For the ground state of 3D Heisenberg model on cubic lattice, the discrepancy of the energy per site is $O(10^{-3})$ between the few-body model of 8 physical plus 24 bath sites and the model of 1000 sites by QMC\index{QMC}. The quantum phase transition of the quantum Ising model on cubic lattice can also be accurately captured by such a few-body model, including determining the critical field and the critical exponent of the magnetization.
\section{Quantum entanglement simulation by tensor network: summary}
Below, we summarize the QES\index{QES} approach for quantum many-body systems with few-body models \cite{RPPSL17AOP3D, R16AOP, RXPS+18FTQES}. The QES contains three stages (Fig. \ref{fig-4Protocol}) in general. The first stage is to optimize the physical-bath interactions by classical computations. The algorithm can be iDMRG\index{iDMRG} in one dimension or the zero-loop schemes in higher dimensions. The second stage is to construct the few-body model by embedding a finite-size cluster in the entanglement bath, and simulate the ground state of this few-body model. One can employ any well-established finite-size algorithms by classical computations, or build the quantum simulators according to the few-body Hamiltonian. The third stage is to extract physical information by tracing over all bath degrees of freedom. The QES approach has been generalized to finite-temperature simulations for one-, two-, and three-dimensional quantum lattice models \cite{RXPS+18FTQES}.
\begin{figure}[tbp]
\centering
\includegraphics[angle=0,width=1\linewidth]{Protocol.eps}
\caption{(Color online) The ``\textit{ab-initio} optimization principle'' to simulate quantum many-body systems.}
\label{fig-4Protocol}
\end{figure}
As to the classical computations, one will have a high flexibility to balance between the computational complexity and accuracy, according to the required precision and the computational resources at hand. On the one hand, thanks to the zero-loop approximation, one can avoid the conventional finite-size effects faced by the previous exact diagonalization, QMC\index{QMC}, or DMRG\index{DMRG} algorithms with the standard finite-size models. In the QES\index{QES}, the size of the few-body model is finite, but the actual size is infinite as the size of the defective TN\index{TN} (see Sec. \ref{sec.zeroloop}). The approximation is that the loops beyond the supercell are destroyed in the manner of the rank-1 approximation, so that the TN can be computed efficiently by classical computation. On the other hand, the error from the destruction of the loops can be reduced in the second stage by considering a cluster larger than the supercell. It is important that the second stage would introduce no improvement if no larger loops are contained in the enlarged cluster. From this point of view, we have no ``finite-size'' but ``finite-loop'' effects. In addition, this ``loop'' scheme explains why we can flexibly change the size of the cluster in stage two: which is just to restore the rank-1 tensors inside the chosen cluster with the full tensors.
The relations among other algorithms are illustrated in Fig. \ref{fig-4AOPrelation} by taking certain limits of the computational parameters. The simplest situation is to take the dimension of the bath sites $\rm dim (b) =1$, and then $\hat{\mathcal{H}}_{\partial}$ can be written as a linear combination of spin operators (and identity). Thus in this case, $v^{[x]}$ simply plays the role of a classical mean field. If one only uses the bath calculation of the first stage to obtain the ground-state properties, the algorithm will be reduced to the zero-loop schemes such as tree DMRG\index{DMRG} and simple update of iPEPS\index{iPEPS}. By choosing a large cluster and $\rm dim (b) =1$, the DMRG simulation in stage two becomes equivalent to the standard DMRG for solving the cluster in a mean field. By taking proper supercell, cluster, algorithms and other computational parameters, the QES\index{QES} approach can outperform others.
The QES approach with classical computations can be categorized as a cluster update scheme (see Sec. \ref{sec.updateschemes}) in the sense of classical computations. Compared with the ``traditional'' cluster update schemes \cite{LDX12TTN, LCB14PEPScontract, WV11PEPSclusterArxiv, LCB14fPEPS}, there exist some essential differences. The``traditional'' cluster update schemes use the super-orthogonal spectra to approximate the environment of the iPEPS. The central idea of QES is different, which is to give an effective finite-size Hamiltonian; the environment is mimicked by the physical-bath Hamiltonians instead of some spectra.
\begin{figure}[tbp]
\centering
\includegraphics[angle=0,width=1\linewidth]{aoprelation.eps}
\caption{(Color online) Relations to the algorithms (PEPS, DMRG and ED) for the ground-state simulations of 2D and 3D Hamiltonian. The corresponding computational set-ups in the first (bath calculation) and second (solving the few-body Hamiltonian) stages are given above and under the arrows, respectively. Reused from \cite{RPPSL17AOP3D} with permission.}
\label{fig-4AOPrelation}
\end{figure}
In addition, it is possible to use full update in the first stage to optimize the interactions related to the entanglement bath. For example, one may use TRD\index{TRD} (iDMRG\index{iDMRG}, iTEBD\index{iTEBD} or CTMRG\index{CTMRG}) to compute the environment tensors, instead of the zero-loop schemes. This idea has not been realized yet, but it can be foreseen that the interactions among the bath sites will appear in $\hat{H}_{FB}$. Surely the computation will become much more expensive. It is not clear yet how the performance would be.
The idea of ``bath'' has been utilized in many approaches and gained tremendous successes. The general idea is to mimic the target model of high complexity by a simpler model embedded in a bath. The physics of the target model can be extracted by integrating over the bath degrees of freedom. The approximations are reflected by the underlying effective model. Table \ref{tab-methods} shows the effective models of two recognized methods (DFT\index{DFT} and dynamic mean-field theory (DMFT)\index{DMFT} \cite{AGPTW10DMF}) and the QES\index{QES}. An essential difference is that the effective models of the former two methods are of single-particle or mean-field approximations, and the effective model of the QES is strongly correlated.
\begin{table}[tbp]
\caption{The effective models under several bath-related methods: density functional theory (DFT\index{DFT}, also known as the \textit{ab-initio} calculations), dynamical mean-field theory (DMFT)\index{DMFT} and QES\index{QES}.}
\begin{tabular*}{12cm}{@{\extracolsep{\fill}}rccc}
\\ \hline\hline
\textbf{Methods} & DFT & DMFT & QES \\ \hline
\textbf{Effective models} & Tight binding model & Single impurity model & Interacting few-body model
\\ \hline\hline
\label{tab-methods}
\end{tabular*}
\end{table}
The QES\index{QES} allow for quantum simulations of infinite-size many-body systems by realizing the few-body models on the quantum platforms. There are several unique advantages. The first one concerns the size. One of the main challenges to build a quantum simulator is to access a large size. In this scheme, a few-body model of only $O(10)$ sites already shows a high accuracy with the error $\sim O(10^{-3})$ \cite{RPPSL17AOP3D, R16AOP}. Such sizes are accessible by the current platforms. Secondly, the interactions in the few-body model are simple. The bulk just contains the interactions of the original physical model. The physical-bath interactions are only two-body and nearest-neighbor. But there exist several challenges. Firstly, the physical-bath interaction for simulating, e.g., spin-$1/2$ models, is between a spin-$1/2$ and a higher spin. This may require the realization of the interactions between SU(N) spins, which is difficult but possible with current experimental techniques \cite{GHGX+10SUNsimu,BBDRS+13SUNsimu,SHHDB14SUNsimu,ZBBK+14SUNsimu}. The second challenge concerns the non-standard form in the physical-bath interaction, such as the $\hat{S}^x\hat{S}^z$ coupling in $\hat{H}_{FB}$ for simulating quantum Ising chain [see Eq. (\ref{eq-4HLR})] \cite{RPSL17CSW}. With the experimental realization of the few-body models, the numerical simulations of many-body systems will not only be useful to study natural materials. It would become possible to firstly study the many-body phenomena by numerics, and then realize, control, and even utilize these many-body phenomena in the bulk of small quantum devices.
The QES Hamiltonian was shown to also mimics the thermodynamics \cite{RXPS+18FTQES}. The finite-temperature information is extracted from the reduced density matrix
\begin{equation}
\hat{\rho}_R = \text{Tr}_{\text{bath}} \hat{\rho},
\label{eq-6RDMFT}
\end{equation}
with $\hat{\rho}=e^{- \mathcal{\hat{H}_{FB}}/\rm{T}}$ the density matrix of the QES\index{QES} at the temperature $\rm{T}$ and $\text{Tr}_{\text{bath}}$ the trace over the degrees of freedom of the bath sites. $\hat{\rho}_R$ mimics the reduced density matrix of infinite-size system that traces over everything except the bulk. This idea has been used to simulate the quantum models in one, two, and three dimensions. The QES shows good accuracy at all temperatures, where relatively large error appears near the critical/crossover temperature.
One can readily check the consistency with the ground-state QES\index{QES}. When the ground state is unique, the density matrix is defined as $\hat{\rho} = |\Psi \rangle \langle \Psi|$ with $|\Psi \rangle$ the ground state of the QES. In this case, Eqs. (\ref{eq-6RDMFT}) and (\ref{eq-4RDM}) are equivalent. With degenerate ground states, the equivalence should still hold when the spontaneous symmetry breaking occurs. With the symmetry preserved, it is an open question how the ground-state degeneracy affects the QES, where at zero temperature we have $\hat{\rho} = \sum_a^{\mathcal{D}} |\Psi_a \rangle \langle \Psi_a|/ \mathcal{D}$ with $\{|\Psi_a \rangle\}$ the degenerate ground states and $\mathcal{D}$ the degeneracy.
\chapter{Summary}
The explosive progresses of TN\index{TN} that have been made in the recent years opened an interdisciplinary diagram for studying varieties of subjects. What is more, the theories and techniques in the TN algorithms are now evolving into a new numerical field, forming a systematic framework for numerical simulations. Our lecture notes are aimed at presenting this framework from the perspective of the TN contraction algorithms for quantum many-body physics.
The basic steps of the TN contraction algorithms are to contract the tensors and to truncate the bond dimensions to bound the computational cost. For the contraction procedure, the key is the contraction order, which leads to the exponential, linearized, and polynomial contraction algorithms according to how the size of the TN decreases. For the truncation, the key is the environment, which plays the role of the reference for determining the importance of the basis. We have the simple, cluster, and full decimation schemes, where the environment is chosen to be a local tensor, a local but larger cluster, and the whole TN, respectively. When the environment becomes larger, the accuracy increases, but so do the computational costs. Thus, it is important to balance between the efficiency and accuracy. Then, we show that by explicitly writing the truncations in the TN, we are essentially dealing with exactly contractible TN's.
Compared with the existing reviews of TN, a unique perspective that our notes discuss about is the underlying relations between the TN approaches and the multi-linear algebra (MLA)\index{MLA}. Instead of iterating the contraction-and-truncation process, the idea is to build a set of local self-consistent eigenvalue equations that could reconstruct the target TN. These self-consistent equations in fact coincide with or generalize the tensor decompositions in MLA, including Tucker decomposition, rank-1 decomposition and its higher-rank version. The equations are parameterized by both the tensor(s) that define the TN and the variational tensors (the solution of the equations), thus can be solved in a recursive manner. This MLA perspective provides a unified scheme to understand the established TN methods including iDMRG\index{iDMRG}, iTEBD\index{iTEBD}, and CTMRG\index{CTMRG}. In the end, we explain how the eigenvalue equations lead to the quantum entanglement simulation (QES)\index{QES} of the lattice models. The central idea of QES is to construct an effective few-body model surrounded by the entanglement bath, where its bulk mimics the properties of the infinite-size model at both zero and finite temperatures. The interactions between the bulk and the bath are optimized by the TN methods. The QES provides an efficient way for simulating one-, two-, and even three-dimensional infinite-size many-body models by classical computation and/or quantum simulation.
With the lecture notes, we expect that the readers could use the existing TN algorithms to solve their problems. Moreover, we hope that those who are interested in TN itself could get the ideas and the connections behind the algorithms to develop novel TN schemes.
\printindex
\extrachap{Acronyms}
\begin{description}[CABR]
\item[AKLT state]{Affleck-Kennedy-Lieb-Tasaki state}
\item[AOP]{\textit{ab-initio optimization principle}}
\item[CANDECOMP/PARAFAC]{canonical decomposition / parallel factorization}
\item[CFT]{conformal field theory}
\item[CTM]{corner transfer matrix}
\item[CTMRG]{corner transfer matrix renormalization group}
\item[DFT]{density functional theory}
\item[DMFT]{dynamical mean-field theory}
\item[DMRG]{density matrix renormalization group}
\item[ECTN]{exactly-contractible tensor network}
\item[HOOI]{higher-order orthogonal iteration}
\item[HOSVD]{higher-order singular value decomposition}
\item[HOTRG]{higher-order tensor renormalization group}
\item[iDMRG]{infinite density matrix renormalization group}
\item[iPEPO]{infinite projected entangled pair operator}
\item[iPEPS]{infinite projected entangled pair state}
\item[iTEBD]{infinite time-evolving block decimation}
\item[MERA]{multi-scale entanglement renormalization ansatz}
\item[MLA]{multi-linear algebra}
\item[MPO]{matrix product operator}
\item[MPS]{matrix product state}
\item[NCD]{network contractor dynamics}
\item[NP hard]{non-deterministic polynomial hard}
\item[NRG]{numerical renormalization group}
\item[NTD]{network Tucker decomposition}
\item[PEPO]{projected entangled pair operator}
\item[PEPS]{projected entangled pair state}
\item[QES]{quantum entanglement simulation/simulator}
\item[QMC]{quantum Monte Carlo}
\item[RG]{renormalization group}
\item[RVB]{resonating valence bond}
\item[SEE's]{self-consistent eigenvalue equations}
\item[SRG]{second renormalization group}
\item[SVD]{singular value decomposition}
\item[TDVP]{time-dependent variational principle}
\item[TEBD]{time-evolving block decimation}
\item[TMRG]{transfer matrix renormalization group}
\item[TN]{tensor network}
\item[TNR]{tensor network renormalization}
\item[TNS]{tensor network state}
\item[TPO]{tensor product operator}
\item[TRD]{tensor ring decomposition}
\item[TRG]{tensor renormalization group}
\item[TTD]{tensor-train decomposition}
\item[TTNS]{tree tensor network state}
\item[VMPS]{variational matrix product state}
\end{description}
\bibliographystyle{unsrt}
\input{Manuscript.bbl}
\end{document}
| -179,023.573779 |
[
-1.3447265625,
1.53125
] | 37.048666 |
[
-2.646484375,
0.919921875,
-1.7529296875,
-4.8046875,
0.1304931640625,
6.5390625
] |
[
4.23828125,
6.11328125,
2.345703125,
7.48828125
] | 2,362 | 39,456 |
[
-2.828125,
3.1015625
] | 27.221123 |
[
-5.9375,
-3.23046875,
-4.0625,
-2.01953125,
2.02734375,
11.5390625
] | 1.192441 | 22.966401 | 12.472121 | 2.899302 |
[
3.1521356105804443
] | -101,654.082469 | 5.696193 | -176,882.422081 | 0.320357 | 6.44006 |
[
-2.947265625,
-3.841796875,
-3.83984375,
-4.57421875,
2.49609375,
12.15625
] |
[
-5.8046875,
-1.478515625,
-2.044921875,
-0.9765625,
3.443359375,
4.4765625
] | |
BkiUbEHxK7ICUubsx6Ic
|
\section{Introduction}
Repetitive\blfootnote{Accepted for presentation as a regular paper in the Intelligent Short Video workshop, organized in conjunction with ICCV'19} patterns are ubiquitous in both natural and man-made environments.
Common human motions and activities such as walking, running, hand waving, breathing, etc, form temporally repetitive patterns.
Detecting such patterns is both effortless and useful for humans~\cite{Johansson1973}
who use visual, aural and tactile signals as the primary sources of sensory information to solve this task.
This work deals with the problem of {\em temporal localization of repetitive activities}, that is, the identification of all repetitive segments in a video. More specifically, given an input video, our goal is to identify all the frames of the video where a repetitive, periodic motion is observed. Consider, as an example, the scenario shown in Figure~\ref{fig:concept}.
In this example, a man starts by talking to the camera (non-repetitive activity), followed by doing crunches (repetitive activity), then stands up (non-repetitive), performs some kicks (repetitive), and finally again talks to the camera. The desired output is color-coded as red for non-repetitive video segments, and green for repetitive ones. Given this identification of repetitive segments, a potential next goal is {\em repetition counting}, also called {\em periodicity characterization} in the relevant literature: the estimation of the length of the period of the repetitive motion in each segment. Repetition counting is out of the scope of this work. However, a solution to the problem of localizing repetitive segments simplifies the problem of repetition counting.
\begin{figure}[t]
\centering
\includegraphics[width=\columnwidth]{figures/fig1}
\caption{We address the problem of classifying parts of an input video as belonging to repetitive activities or not. Still frames of an example video are shown, with the time running from left to right and from top to bottom. In the video, a man starts by talking to the camera (non-repetitive activity), followed by doing crunches (repetitive activity). Then he stands up (non-repetitive), performs some kicks (repetitive) and finally again talks to the camera (non-repetitive). The output of the proposed method is color-coded as red for non-repetitive parts, and green for repetitive ones.
\label{fig:concept}}
\end{figure}
Repetitive activity detection and repetition counting using visual input form a category of problems that has been widely studied~\cite{Liu1998,Vlachos2005}, and is still of significant interest to the community of computer vision \cite{Panagiotakis,Runia2018a}. The relevant methods need to address a number of challenges. To begin with, the amount of data contained in a regular video stream is big, even for the standards of modern hardware. Further complicating factors include camera motion, sensor noise and the non-stationarity of the observed motion.
Real-world signals deviate from being stationary and truly periodic.
Repetitions of an activity may differ in their duration (thus, the period of the activity is not constant) and/or in their actual execution.
Therefore, successful repetition detection methods that deal with these types of non-stationarity have to address significant theoretical challenges.
From a practical point of view, many real-world applications such as human action and motion detection, analysis and classification~\cite{Lu2004,Goldenberg2005,Albu2008,Ran2007}, 3D reconstruction~\cite{Belongie2006}, camera calibration~\cite{Huang}, medical assessment \cite{ID1} and rehabilitation, industrial inspection with minimal prior knowledge~\cite{Karvounas}, etc, can benefit from the development of robust solutions to periodicity-related problems.
\section{Related Work}
The detection and the analysis of repetitive patterns have received the attention of researchers in various computer science fields~\cite{Agrawal1995,polana1997detection,Elfeky2005,Pogalin2008} for, e.g., the detection of repeated actions within a video~\cite{polana1997detection,Pogalin2008}, of repeated patterns in a database~\cite{Agrawal1995,Elfeky2005,Exarchos2008}, etc.
\vspace*{0.2cm}
{\noindent \bf Repetition detection and characterization:}
The problems of repetition detection and repetition counting can be defined in the temporal and spatial domains~\cite{Cutler2000} but also in several others that involve either 1D signals~\cite{Lu2004,Elfeky2005,Burghouts2006} or multidimensional signal representations.
Naturally, Fourier transform and, more generally, spectral analysis tools have been employed by many methods~\cite{polana1997detection,Cutler2000,ID1}. This yields satisfactory results in clean, 1D signals, but fails in the presence of large amounts of noise or other sources of signal corruption that invalidate the assumption of signal stationarity. Special care must then be taken to preprocess the input appropriately so that the resulting signal(s) meet the requirements imposed by the selected spectrum analysis tool.
The work by Lu et al.~\cite{Lu2004} tackles the problem of segmenting and classifying repetitive motion events using visual input. The method decomposes the input video into motion primitives using a multidimensional signal segmentation algorithm and then uses these primitives to segment and classify repetitive motion.
Burghouts and Geusebroek~\cite{Burghouts2006} propose a method to detect and characterize periodic motion. A spatiotemporal filtering approach is adopted, yielding localization of the observed periodic events.
The method of Tong et al.~\cite{Tong2005} operates on local motion (optical flow) information~\cite{Black1996} computed in videos. Repetition detection and characterization is performed by estimating statistics of the autocorrelation computed over this motion representation. Albu et al.~\cite{Albu2008} use silhouettes as the main visual cue. They focus on human silhouettes to obtain 1D signals representing motion trajectories of body parts. These signals are then processed to detect their repetitve parts, with a final step of the method fusing these detections into a single coherent estimation of the repetitive parts of the input. The authors show experimental results of their method on repetitive activities such as aerobic exercises and walking.
Briassouli and Ahuja~\cite{Briassouli2007} propose a method to extract multiple periodic motions in a video sequence. They propose the use of Short Term Fourier Transform (STFT) on the volume of the input video. This approach allows for the simultaneous detection and characterization of repetitive activities.
Pogalin et al.~\cite{Pogalin2008} define $10$ different classes of repetitive motions, and build a system to detect and classify them. To do so, they begin by tracking all the objects in the input stream. This is followed by probabilistic PCA of the resulting tracks, and spectral analysis for detecting and characterizing periodic motion. Azy and Ahuja~\cite{Azy2008} propose a method to detect and segment objects that move in a repetitive manner using a maximum likelihood estimation of the period to characterize the motion. Image segmentation is performed using correlation of image segments over the estimated period. Taking the inverse approach, Gaojian Li et al.~\cite{Li2012} first localize the target object in a region of interest and then characterize its periodic motion. Motions with mild non-stationary components are shown to be handled effectively.
Karvounas et al.~\cite{Karvounas} detect and characterize the periodic part of an input video by formulating the task as an optimization problem. This work assumes that only one periodic activity exists per video. In the case that this assumption does not hold, only one of the periodic activities will be detected, or the method may completely fail to detect and characterize a periodic action. Panagiotakis et al.~\cite{Panagiotakis} treats the detection of multiple periodic actions as a problem of video co-segmentation. More specifically, they capitalize on a method~\cite{panagPR} that co-segments all common actions of two videos, in an unsupervised manner: by co-segmenting actions in a single video, the proposed work manages to identify repeated motions.
\vspace*{0.2cm}
{\noindent \bf Exploiting periodicity:}
The cue of repetitive motion is very strong and can be used to detect events in a video~\cite{Lu2004,Laptev2005}. In a work demonstrating the power of the repetition cue using visual input~\cite{Sarel2005}, Sarel and Irani show that it is possible to separate two superimposed layers in a video, under the assumption that one of the layers exhibits repetitive motion dynamics. As an example, a man performing jumping jacks is filmed through a glass door that reflects moving people on the other side of the door. The proposed method separates correctly the two scenes by assuming that repetitive motion occurs in one of the two layers. Another work where repetition serves as a strong cue to detect and segment motions is presented by Laptev et al.~\cite{Laptev2005}. In that work, the authors start by observing that corresponding time instances in successive repetitions of a periodic activity serve as approximate stereo pairs. This observation is then exploited to segment and characterize repetitive activities using space-time correspondences across different repetitions of the observed motion.
Goldenberg et al.~\cite{Goldenberg2005} propose the use of the sequence of silhouettes of a periodically moving object such as a walking dog as a basis for the representation and analysis of the motion towards action classification. The authors propose the eigen-decomposition of this set of silhouettes as a means to extract an intermediate representation. The authors proceed to show how this representation can efficiently solve the task of object and action classification.
Xiu Li et al.~\cite{Li} exploit periodicity to develop a method for non-rigid structure from motion. The input to the method is a monocular video of a non-rigid object undergoing a possibly repetitive dynamic motion. The authors observe that many deforming shapes tend to repeat themselves in time, a property termed ``shape recurrency''. Based on this property, the authors apply standard, rigid structure-from-motion tools on this problem.
\vspace*{0.2cm}
{\noindent \bf Repetition counting:}
Levy and Wolf~\cite{Levy2015} propose a method that tackles the problem of repetition counting. They use a convolutional neural network on windows of the input stream to detect and characterize repetitive activities. Based on this output, they proceed to estimate the number of repetitions in the input video stream. The recent method by Runia et al.~\cite{Runia2018a} for repetition counting analyzes the different possible viewpoints of a periodic motion with respect to the dominant orientation of motion in 3D. Then, the method selects the viewpoint with the best signal, estimating the length and period of motion. This, in turn, yields an estimation of the repetition count, the end goal of this work. This method only tackles the problem of counting the repetitions of a single action, so it cannot handle two or more different periodic activities in a video.
\vspace*{0.2cm}
{\noindent \bf Our contribution:}
The overview of the relevant literature indicates that, with the exception of~\cite{Panagiotakis}, no method can tackle the problem of detecting the repetitive segments of a real world input video in its generality. Existing methods make restrictive assumptions regarding the stationarity of the observed signals. For example, they consider strictly periodic signals of almost constant period and very similar execution of actions in the different repetitions. Others, assume that a single periodic activity is executed in a video and, therefore, cannot detect several of them.
Finally, others are tied to specific (and thus, limiting) settings as, for example, that the camera observing the scene is static. Others operate on representations of the human body, thus they may deal only with repetitive patterns of human motion. In our work, we present a deep learning method that overcomes all aforementioned limitations.
We first employ a generic video-content representation on which we show that temporal localization of repetitive activities can be learned. We also propose ReActNet, a lightweight deep learning architecture to solve this problem.
We show that, based on the employed representation, ReActNet learns to localize repetition on the basis of a relatively small training dataset.
Finally, we evaluate ReActNet quantitatively on existing public datasets and in comparison with existing approaches. The experimental results show that the proposed approach outperforms the state of the art.
\section{Localizing Repetitive Segments}
Given an input video, each frame is represented as a vector of deep features that encode high-level information for that frame.
Based on this frame representation, we compute a matrix $M$ of the pairwise distances of all video frames. The existence of repetitive activities in a video gives rise to distance matrices with a particular block diagonal structure. Based on this observation, we proceed to train a neural network that classifies square blocks on the main diagonal of $M$ into two classes (repetitive / non-repetitive). Figure~\ref{fig:DM} gives an example of a distance matrix of a video showing two repetitive activities among other, non-repetitive parts.
\subsection{Data Representation}
We approach the problem of temporally localizing repetitive activities building upon the idea of a distance matrix which has been adopted also by other works in video analysis~\cite{Cutler2000, Karvounas, papoutsakis2017a, Panagiotakis}. Specifically, assuming that the input is a video of $N$ frames, we construct an $N \times N$ matrix $M$. Each entry $m_{ij}$ of $M$ quantifies the distance between frames $i$ and $j$. Assuming that some temporally repetitive pattern is present in the observed video, the matrix exhibits particular structures. For example, it may contain sub-diagonals that are parallel to the main diagonal with low distance values, capturing the periodic nature of the observations.
An intermediate representation for each frame of the input video is useful for the task of computing the entries of matrix $M$.
The representation of a frame can, in principle, be anything that captures visual content, disentangling it from signal-specific and repetition-irrelevant details. Panagiotakis et al.~\cite{Panagiotakis} opted for the use of tracklets, short sequences of optical flow trajectories. This approach can successfully separate appearance from motion, but is prone to tracking failures. In this work, we propose the use of features that are computed by a deep convolutional neural network.
Specifically, the activation features of the VGG19 network~\cite{Krizhevsky} at the $15$th layer are computed for each frame of the input video. This particular layer has been selected as it strikes a good balance between the required level of feature abstraction and the need to keep features in spatial relation with the input. Then, for each pair $i$ and $j$ of frames of the video, the Euclidean distance of their precomputed feature vectors is computed as the value $m_{ij}$ of $M$.
An example of such a matrix is visualized in Figure~\ref{fig:DM}.
\subsection{Ground Truth Annotation}
\label{sec:annotation}
Having computed the distance matrix $M$ based on the input video, our next task is to denote each frame of the video as part of a repetitive activity or not.
We assume that there is ground truth annotation for a number of input videos. Specifically, the annotation for a video is a list of pairs of frames of the form $(s_i, e_i)$, where $s_i$ is the frame count of the start of the $i$-th repetitive segment, and $e_i$ is the ending frame of that segment.
\begin{figure}[t]
\centering
\includegraphics[width=.48\columnwidth]{figures/D_slw}
\includegraphics[width=.48\columnwidth]{figures/gn_slw}
\caption{A distance matrix $M$ (left) and the corresponding binary ground truth matrix $A$ (right). Two different sub-blocks are also shown in purple color (see text for details).
\label{fig:GT_subblocks}}
\end{figure}
For a number of reasons, it is not practical to train a neural network directly on the ground truth represented in the form of repetitive segments $(s_i, e_i)$.
Firstly, the input size of the image sequence (and of a repetitive part of it) can be arbitrarily large. This would make the resulting network prohibitively large for practical use on current hardware. A solution to the arbitrary input size is the use of recurrent neural networks. We don't adopt this design choice because of the additional complexity to fine-tune a recurrent neural network.
Additionally, this strategy has no clear way of representing multiple repetitive segments that may co-exist in a video, occurring one after the other.
To address these issues, we adopt a representation that is efficient with respect to input size, and represents the target output in a way that is compatible with a fully convolutional architecture.
\subsection{Learnable Repetition Representation}
To tackle the issues above, we resort to a binary, square matrix $A$ with the same dimensions as $M$. We adopt the convention that a value $a_{ij}$ of $A$ is equal to $1$ if both frames $i$ and $j$ belong to the same repetitive segment. Otherwise, $a_{ij} = 0$.
Figure~\ref{fig:GT_subblocks} shows a distance matrix $M$ (left) and the corresponding binary matrix $A$ (right).
Given the annotation information as described is Section~\ref{sec:annotation}, it is straightforward to compute the respective matrix $A$.
Representing the input video content using the matrix $M$, and the ground truth annotation as the matrix $A$, we propose to train a network on fixed-size sub-blocks of these matrices.
Since the information regarding repetitions is encoded around the main diagonal of the matrix $M$, any square block of the matrix $M$ that is centered on the main diagonal of $M$ can be used as a training sample. The corresponding block of the matrix $A$ can then serve as the ground truth annotation for this training sample.
Given this information, we proceed to train a deep neural network on such training samples.
Overall, the training samples are corresponding pairs of sub-blocks of the matrix $M$ and $A$.
It is fine to select sub-blocks of $M$ that have large overlap.
This process is repeated for all matrices $M_i$ coming from input videos $V_i$ in the training set, resulting in thousands of training sub-blocks given a few input videos.
Figure~\ref{fig:GT_subblocks} shows two example training sub-blocks (purple squares) superimposed on a distance matrix $M$ (left) and on the respective ground truth annotation matrix $A$ (right).
\begin{figure*}[t]
\centering
\includegraphics[width=0.6\textwidth]{figures/PerLNet}
\caption{The building blocks of the architecture of ReActNet, the proposed CNN for repetition detection. The colors denote layer types. Orange: convolution and non-linearity, red: max-pooling, blue: upsampling. The count of feature maps is shown under each convolution layer, and the size of the resulting feature maps is shown diagonally to the right of the layer (also visually as the size of the block). Finally, the blue sphere denotes addition.
\label{fig:architecture}}
\end{figure*}
\subsection{Data Augmentation}
\label{sec:augmentation}
In the case of a network that accepts visual input such as a regular color image, it is common practice to apply data augmentation in the form of geometric and intensity transformations~\cite{Krizhevsky} so as to force the neural network to become invariant to such transformations. Our case is different, since the goal is to learn the patterns generated from repetitive, periodic motion. The only useful form of data augmentation applicable to our input data would be one that achieves temporal scale invariance. This is effectively achieved by varying the size of the sub-blocks used during training.
The highlighted sub-blocks in Figure~\ref{fig:GT_subblocks} are of different size, following this data augmentation approach.
\subsection{Repetitive Activities Detection Network}
\label{sec:architecture}
{\noindent \bf ReActNet Architecture:}
Given the defined input and target output, we propose, train, and evaluate ReActNet, a custom, lightweight convolutional neural network to learn this mapping. The employed architecture is a stack of hourglass modules~\cite{Newell2016}, that are essentially autoencoders~\cite{Hinton2011} with skip connections~\cite{He2016}.
The architecture consists of three stacked autoencoders. The reason we adopted an autoencoder-like network is because we are targeting an output with the same size as the input.
Furthermore, the autoencoder architecture offers good generalization by squeezing the information through the (spatially) low-resolution intermediate layers.
The proposed neural network is built using copies of two building blocks, an encoder and a decoder. This architecture is shown in Figure~\ref{fig:architecture}. The encoder first applies $16$ convolution filters of dimension $11 \times 11$. This is a rather large spatial dimension, however it is justified because the patterns we are looking for are rather large-scale and noisy.
The encoder then contains a ReLu activation layer, followed by a batch normalization layer~\cite{ioffe2015batch}. Another set of convolution, activation, and batch-norm layers follow in the original input dimension, and then a max pooling layer halves the resolution in each of the two spatial dimensions of the input. Two more sets of convolution, activation, and batch normalization follow, and in parallel to these layers, an identity residual connection~\cite{He2016} is also used. The skip connection is added to the result of the other branch, and a last max-pooling layer halves again the spatial resolution. In total, the encoder applies $5$ different convolution layers and two max pooling operations, resulting in output spatial resolution that is $1/4$ of the input resolution in each spatial dimension.
The decoder follows essentially the mirrored connectivity pattern of the encoder, upsampling its input by a factor of $4$ for each input spatial dimension. There are again $5$ layers applying a convolution operation, and a skip connection runs parallel to a group of two such convolutions.
\vspace*{0.2cm}
{\noindent \bf Loss Function:}
We build a classifier with two output classes so it is natural to use binary cross entropy (BCE)~\cite{Janocha2017} as the loss function for training:
\begin{equation}
E(y, p) = -(y \log{p} + (1-y)\log(1-p)).
\label{eq:BCE}
\end{equation}
In Eq.(\ref{eq:BCE}), $y\in\{0,1\}$ is the ground truth class annotation, and $p\in[0,1]$ is the real-valued prediction of the network for this frame. In practice, we noticed that there is an imbalance between positive and negative examples in our training data. Specifically, our training set contains more segments with repetitive activities than non-repetitive ones.
This had an impact on the discriminative power of the trained models. To alleviate this problem, we used the weighted binary cross entropy (WBCE), a straightforward extension of the BCE loss:
\begin{equation}
E(y, p) = -(w y \log{p} + (1-w)(1-y)\log(1-p)).
\label{eq:WBCE}
\end{equation}
In Eq.(\ref{eq:WBCE}), $y$ and $p$ are as in Eq.(\ref{eq:BCE}), and $w$ is a weight term balancing the two classes.
Using the BCE loss proved sufficient for our problem: given that we are essentially tackling a classification problem with only two classes, it was not needed to resort to more complex loss functions.
In practice it proved beneficial to apply intermediate supervision in each stage of the proposed architecture, anf to vary the weight parameter $w$ of WBCE in each stage. Specifically, the first stage used a small value for $w$, and the next two stages used increasingly larger values. This technique has the effect that both positive and negative examples are sufficiently weighted at some stage during training.
\vspace*{0.2cm}
{\noindent \bf Testing:}
At run-time, we adopt a sliding window approach. Given an input video, the distance matrix $M$ is formed. Then, a square window is centered around each point of its diagonal. This is fed to ReActNet which returns a square window of the same size with real-valued predictions.
This sliding window approach results in multiple ($n_f$) predictions $p_i$, $1 \leq i \leq n_f$ per frame $f$, each in the range $[0,1]$, for the overlapping frames of consecutive windows. A frame $f$ is declared as being repetitive if:
\begin{equation}
P_f = \sum_{i=1}^{n_f}\frac{p_i}{n_f} > T.
\label{eq:classify}
\end{equation}
In Eq.(\ref{eq:classify}), $T$ was set to $0.5$ in all experiments.
\section{Experimental Evaluation}
The experimental evaluation of the proposed method was performed on two recent relevant datasets. A first category of experiments assessed the adopted design choices in an ablation study. Furthermore, the localization accuracy of ReActNet was compared to the current state of the art, for different choices of employed features. We also assessed the cross-dataset generalization of ReActNet by training it on a dataset and testing it on another.
\subsection{Training Details}
We implemented ReActNet using the Keras framework~\cite{chollet2015keras} on top of tensorflow~\cite{tensorflow2015-whitepaper}. The Adam optimizer was used to train it for $3$ epochs, with a learning rate value of $0.002$. For training, we employed an Nvidia GTX 1070 Ti GPU. On that machine, each epoch took $70$ seconds. The chosen range of sub-block sizes for data augmentation (see Section~\ref{sec:augmentation}) was between $100$ and $200$, with all the sub-blocks resized to $140 \times 140$ using Lanczos resampling.
\begin{figure*}[t]
\centering
\includegraphics[height=5cm]{figures/pertube}
\hspace*{0.2cm}
\includegraphics[height=5cm]{figures/quva}
\caption{Video frames from the PERTUBE (left) and QUVA (right) datasets.
\label{fig:datasets}}
\end{figure*}
\subsection{Datasets}
For the evaluation of the proposed methodology, we use the PERTUBE~\cite{Panagiotakis} and the QUVA~\cite{Runia2018a} datasets (see Fig.~\ref{fig:datasets}).
\vspace*{0.2cm}
{\noindent \bf The PERTUBE dataset~\cite{Panagiotakis}:} This dataset\footnote{Available online at \url{https://www.ics.forth.gr/cvrl/pd/}} contains a set of videos depicting repetitive motions (human activities, object motions, etc) that were obtained from YouTube. There is a total of $50$ annotated videos in the dataset. Each frame of every video is annotated with information that denotes whether it belongs or not to a repetitive segment.
\vspace*{0.2cm}
{\noindent \bf The QUVA dataset~\cite{Runia2018a}:}
This dataset\footnote{Available online at \url{http://tomrunia.github.io/projects/repetition/}} is compiled for the related problem of repetition counting. It contains $100$ videos depicting human activities. Each video is appropriately annotated for repetition detection.
Additionally to this information, specific frames are denoted as the starting frames of a new repetition, making the annotation useful for the task of repetition counting. We disregard this information, retaining only the repetitive/non-repetitive annotation per frame.
\subsection{Evaluation Metrics}
We view repetition detection as a classification problem. Thus, to evaluate the obtained results, we use the standard metrics of recall $\cal{R}$, precision $\cal{P}$, $F_1$ score and Overlap $\cal{O}$.
In our problem, precision quantifies how many of the frames that were classified as belonging to a repetitive segment are truly such. Recall quantifies the percentage of frames that actually belong to repetitive segments and were correctly classified by a method.
\subsection{Ablative Study}
In a first set of experiments, we justified specific design choices of ReActNet based on the PERTUBE dataset. The results of these experiments are presented in Table~\ref{table:ablation}. In each column, the best result is highlighted with bold.
The results were obtained by performing five randomized repetitions with half the videos of the dataset used as training and the other half as test. Then, the reported value is the average of the performance on the test set in the five randomized repetitions.
The first line of Table~\ref{table:ablation} refers to the architecture as this has been described in Section~\ref{sec:architecture}.
The experiment ``Filter size 11 $\times$ 11'' tests a larger filter size (11 $\times$ 11) for the first convolutional layer of the autoencoders, compared to the 5 $\times$ 5 of the proposed method.
The ``Single autoencoder'' experiment uses a single autoencoder instead of the three stacked autoencoders of the proposed method. The experiment ``Without skip connection'' disables the ResNet-like skip connections of the proposed method. Finally, the ``Intermediate supervision'' experiment explores the idea of adding two extra terms in the loss function of the network, at the end of each autoencoder in the stack. The target is the same for all three loss terms, the ground truth classification of the sub-block. Evidently, the proposed method outperforms all other network variants.
\begin{table}[t]
\caption{Meta-parameter study of the proposed neural network. Along with the values of the metrics, the standard deviations over the repeated experiments are also shown.}
\begin{center}
\scalebox{0.65}{
\vspace*{-10mm}
\begin{tabular}{|l|c|c|c|c|}
\hline
\cal{Configurations} & $\cal{R} (\%)$ & $\cal{P} (\%)$ & $F_1 (\%)$ & $\cal{O} (\%)$ \tabularnewline
\hline\hline
\hline
ReActNet (proposed) & $\textbf{87.6}\pm 3.4$ & $ 85.1\pm1 $ & $ \textbf{85.8}\pm2.2 $ & $ \textbf{75.5}\pm3.9$ \tabularnewline
\hline
Filter size 11$\times$11 & $ 86.6\pm9.2 $ & $ 85.2\pm3.9 $ & $ 85.3\pm3.2 $ & $ 75.6\pm3.5 $ \tabularnewline
\hline
Single autoencoder & $80.7\pm4 $ & $ 84.0\pm3.1 $ & $ 81.6\pm3.1 $ & $ 69.5\pm3.4 $ \tabularnewline
\hline
Without skip conn. & $ 84.6\pm7.6 $ & $ 83.7\pm4.7 $ & $ 83.6\pm2.6 $ & $ 73.0\pm3.3 $ \tabularnewline
\hline
Without interm. supervision & $ 84.2\pm6.2 $ & $\textbf{86.5}\pm1.9 $ & $ 85.3\pm3.7 $ & $ 75.0\pm4.7 $ \tabularnewline
\hline
\hline
\end{tabular}
}
\label{table:ablation}
\end{center}
\end{table}
In the following, unless otherwise stated, ReActNet and the term ``proposed method" refer to the architecture presented in Section~\ref{sec:architecture} and evaluated in the first row of Table~\ref{table:ablation}.
\begin{table}[t]
\caption{Comparative evaluation of ReActNet with~\cite{Panagiotakis} on the PERTUBE dataset with CNN-based (VGG19) and hand-crafted (IDT) features. Training of the proposed method has been performed on the PERTUBE dataset.}
\begin{center}
\scalebox{0.75}{
\vspace*{-10mm}
\begin{tabular}{|l|l|c|c|c|c|}
\hline
\cal{Method} & \cal{Features} & $\cal{R} (\%)$ & $\cal{P} (\%)$ & $F_1 (\%)$ & $\cal{O} (\%)$ \tabularnewline
\hline\hline
\cite{Panagiotakis} & IDT & 84.1 & 75.7 & 77.0 & 67.7\tabularnewline
\hline
ReActNet & IDT & $87.1\pm8.4$ & $78.6\pm5.7$ & $82.0\pm4.6$ & $471.9\pm4$ \tabularnewline
\hline
\cite{Panagiotakis} & VGG19 & 79.2 & 68.1 & 71.1 & 61.8\tabularnewline
\hline
ReActNet & VGG19 & $\textbf{87.6}\pm 3.4$ & $\textbf{85.1}\pm1$ & $\textbf{85.8}\pm2.2$ & $\textbf{75.5}\pm3.9$ \tabularnewline
\hline
\hline
\end{tabular}}
\label{table:pancomparison}
\end{center}
\end{table}
\subsection{Comparison to SoA \& the Impact of Features}
The recent method by Panagiotakis et al.~\cite{Panagiotakis} solves the problem of repetition localization by using a distance matrix computed on an Improved Dense Trajectories (IDT)~\cite{Wang2013} representation of each frame of the sequence. This distance matrix is then appropriately manipulated to extract the repetitive parts of the video. We note that both the method in~\cite{Panagiotakis} and the proposed method consist of a first phase that builds a distance matrix based on some feature representation of the frames of a video (IDT features for~\cite{Panagiotakis}, VGG19 features for us) and a second phase that localizes repetitions on this distance matrix (hand-crafted filtering process followed by Discrete Time Warping for~\cite{Panagiotakis}, ReActNet for us). It is therefore possible to interchange parts of the two methodologies, assessing the impact of deep neural networks on the proposed approach in a total of $4$ different experiments.
Table~\ref{table:pancomparison} presents the results we obtained with the $4$ different variants on the PERTUBE dataset. The first column regards the employed method (\cite{Panagiotakis} and ReActNet). The second column regards the employed features (Improved Dense Trajectories~\cite{Wang2013} suggested by~\cite{Panagiotakis}, VGG19 features~\cite{Krizhevsky} suggested in this work).
For the approach in~\cite{Panagiotakis}, the results were obtained by a single run over the whole dataset, using the publicly available implementation provided by the authors of that work\footnote{The implementation of the method in~\cite{Panagiotakis} is available at \url{https://sites.google.com/site/costaspanagiotakis/research/pd}}.
For our approach, the values were estimated with same methodology as above, using five repetitions with randomized training and test sets.
For each of the employed metrics, the best result is highlighted in bold. It can be verified that the proposed approach achieves the best results compared to all the other possible configurations.
Interestingly, the worst of our two variants performs better than the best of the variants of~\cite{Panagiotakis}, that is, regardless of whether we use IDT or VGG19 features. Thus, the quality of the obtained results is attributed mostly to the method itself and to a lesser extend to the employed features.
\subsection{Cross-dataset Validation}
We performed a set of experiments to investigate how ReActNet generalizes across datasets. In a first experiment we trained the proposed method on the PERTUBE dataset, and assessed its performance on the QUVA dataset. In a second experiment, we swapped the training and test sets.
\begin{table}[t]
\caption{Cross-dataset evaluation: The proposed method, ReActNet, was trained on QUVA and evaluated on PERTUBE.}
\begin{center}
\scalebox{0.95}{
\begin{tabular}{|l|c|c|c|c|c|}
\hline
\cal{Methods} & \cal{Features} & $\cal{R} (\%)$ & $\cal{P} (\%)$ & $F_1 (\%)$ & $\cal{O} (\%)$ \tabularnewline
\hline\hline
ReActNet &VGG19 & 82.8 & 76.8 & 79.7 & 67.8\tabularnewline
\hline
\hline
\end{tabular}}
\label{table:quva2per}
\end{center}
\end{table}
\begin{table}[t]
\caption{Cross-dataset evaluation: The proposed method was trained on PERTUBE and evaluated on
QUVA. The method in~\cite{Panagiotakis} requires no training.}
\begin{center}
\scalebox{0.95}{
\begin{tabular}{|l|c|c|c|c|c|}
\hline
\cal{Methods} & \cal{Features} & $\cal{R} (\%)$ & $\cal{P} (\%)$ & $F_1 (\%)$ & $\cal{O} (\%)$ \tabularnewline
\hline\hline
\hline
ReActNet & VGG19 & \textbf{89.5} & \textbf{87.4} & \textbf{88.5} & \textbf{80.3} \tabularnewline
\hline
\cite{Panagiotakis} & VGG19 & 83.8 & 80.6 & 83.1 & 72.8 \tabularnewline
\hline
\hline
\end{tabular}}
\label{table:per2quva}
\end{center}
\end{table}
\vspace*{0.2cm}
{\noindent \bf Training on QUVA, testing on PERTUBE:}
Table \ref{table:quva2per} shows the performance of ReActNet when trained on QUVA and tested on PERTUBE. The obtained results demonstrate that there is a small performance drop compared to training on PERTUBE and testing on PERTUBE (comparison with the fourth line of Table~\ref{table:pancomparison}). The performance drop is attributed to the smaller diversity of QUVA compared to PERTUBE. Still, the proposed approach generalizes well.
\vspace*{0.2cm}
{\noindent \bf Training on PERTUBE, testing on QUVA:}
The results of this experiment are shown in Table~\ref{table:per2quva}.
When training on PERTUBE, the results of testing on QUVA are better than those of testing on PERTUBE (Table~\ref{table:pancomparison}, fourth row). Given that training involved the same subset of PERTUBE in both cases, this serves as a further indication that PERTUBE is more diverse than QUVA (see previous paragraph). For comparison, the second row of Table~\ref{table:per2quva} shows the performance of~\cite{Panagiotakis} on the QUVA dataset using our proposed deep features. It can be verified the the proposed approach outperforms~\cite{Panagiotakis} in all performance metrics and that there is a $18\%$ increase in overlap ${\cal O}$.
\subsection{Qualitative Results}
Figure~\ref{fig:results} shows the distance matrices, the repetition localization estimation (1st green/red bar) and the ground truth (2nd green/red bar) for six sequences of the PERTUBE (top three rows) and the QUVA (bottom row) datasets. The repetition localization results are better illustrated in the supplementary material\footnote{Available online at \url{https://youtu.be/pPqg1lMkuaQ}} of this paper. More sequences have been selected from PERTUBE because it contains more complex activities of multiple repetitive segments per video.
\begin{figure}[t]
\centering
\includegraphics[width=0.44\columnwidth]{figures/qualitative/legD}
\hspace*{0.2cm}
\includegraphics[width=0.44\columnwidth]{figures/qualitative/runningD}\\
\includegraphics[width=0.44\columnwidth,height=0.5cm]{figures/qualitative/leg}
\hspace*{0.2cm}
\includegraphics[width=0.44\columnwidth,height=0.5cm]{figures/qualitative/running}\\
\includegraphics[width=0.44\columnwidth,height=0.5cm]{figures/qualitative/legPerL}
\hspace*{0.2cm}
\includegraphics[width=0.44\columnwidth,height=0.5cm]{figures/qualitative/runningPerl}\\
\vspace*{0.3cm}
\includegraphics[width=0.44\columnwidth]{figures/qualitative/howtoD}
\hspace*{0.2cm}
\includegraphics[width=0.44\columnwidth]{figures/qualitative/cardioD}\\
\includegraphics[width=0.44\columnwidth,height=0.5cm]{figures/qualitative/howto}
\hspace*{0.2cm}
\includegraphics[width=0.44\columnwidth,height=0.5cm]{figures/qualitative/cardio}\\
\includegraphics[width=0.44\columnwidth,height=0.5cm]{figures/qualitative/howtoPerl}
\hspace*{0.2cm}
\includegraphics[width=0.44\columnwidth,height=0.5cm]{figures/qualitative/cardioPerl}\\
\vspace*{0.3cm}
\includegraphics[width=0.44\columnwidth]{figures/qualitative/forgingD}
\hspace*{0.2cm}
\includegraphics[width=0.44\columnwidth]{figures/qualitative/martialD}\\
\includegraphics[width=0.44\columnwidth,height=0.5cm]{figures/qualitative/forging}
\hspace*{0.2cm}
\includegraphics[width=0.44\columnwidth,height=0.5cm]{figures/qualitative/martial}\\
\includegraphics[width=0.44\columnwidth,height=0.5cm]{figures/qualitative/forgingPerl}
\hspace*{0.2cm}
\includegraphics[width=0.44\columnwidth,height=0.5cm]{figures/qualitative/martialPerl}\\
\vspace*{0.3cm}
\includegraphics[width=0.44\columnwidth]{figures/qualitative/hometrainerD}
\hspace*{0.2cm}
\includegraphics[width=0.44\columnwidth]{figures/qualitative/trampolineD}\\
\includegraphics[width=0.44\columnwidth,height=0.5cm]{figures/qualitative/hometrainer}
\hspace*{0.2cm}
\includegraphics[width=0.44\columnwidth,height=0.5cm]{figures/qualitative/trampoline}\\
\includegraphics[width=0.44\columnwidth,height=0.5cm]{figures/qualitative/hometrainerPerl}
\hspace*{0.2cm}
\includegraphics[width=0.44\columnwidth,height=0.5cm]{figures/qualitative/trampolinePerl}
\caption{The distance matrices, the repetition localization estimation (1st green/red bar) and the ground truth (2nd green/red bar) for six sequences of the PERTUBE (top three rows) and the QUVA (last row) datasets.
\label{fig:results}}
\end{figure}
\section{Conclusions}
We presented a novel method for the temporal localization of repetitive activities within a video. In each frame of the video, the method extracts features using a deep neural network and then estimates a matrix of pairwise distances between video frames. Then, ReActNet, a specially designed convolutional neural network classifies the frames of the video as belonging to repetitive segments or not. Extensive experiments backed the design choices behind the proposed method, verified the use of the employed features, compared the approach to a state of the art method and assessed the generalization potential across datasets.
From a computational point of view, a shortcoming of the proposed method is that the current feature extraction method prevents its use in real-time or online applications.
Future plans include the investigation of alternatives that may allow online or even real-time operation, an extension of the existing framework to tackle the problem of repetition counting, and mechanisms for performing localization of the video periodicity also in the spatial domain.
\section*{Acknowledgments}
This work was partially supported by the H2020-ICT-2016-1-731869 project Co4Robots.
{\small
\bibliographystyle{ieee}
| -22,606.320585 |
[
-2.787109375,
2.640625
] | 45.6 |
[
-2.693359375,
1.142578125,
-2.623046875,
-5.8046875,
-0.85009765625,
8.609375
] |
[
2.875,
4.921875,
2.65234375,
7.62890625
] | 432 | 5,365 |
[
-3.109375,
3.673828125
] | 23.169114 |
[
-6.5546875,
-3.94140625,
-4.54296875,
-2.142578125,
2.322265625,
12.828125
] | 1.442027 | 19.91437 | 25.107176 | 4.399764 |
[
2.5844428539276123
] | -16,670.417329 | 6.075116 | -21,802.127339 | 0.730218 | 5.925074 |
[
-3.208984375,
-3.767578125,
-2.904296875,
-3.78515625,
2.75,
10.4921875
] |
[
-6.19140625,
-3.451171875,
-3.083984375,
-2.3125,
4.56640625,
7.59765625
] | |
BkiUbVQ5qhLACAkw_leP
|
\section{Supplementary Material}
\label{Sec:appendix}
\subsection{Normal and principal curvature estimation performance}
\subsubsection{Performance on real data.}
We qualitatively evaluate the performance of our method on the NYU Depth V2 dataset \cite{SilbermanECCV12}. This dataset was captured using a Kinect v1 RGBD camera and contains indoor scene environment and includes missing data and a noise pattern that is significantly different than the PCPNet dataset. Specifically, the noise often has the same magnitude as some of the features. Most importantly, this dataset, much like other real-world datasets, does not have ground truth normals. Fig. \ref{fig:appx:results:nyu_v2} and Fig. \ref{fig:appx:results:nyu_v2_curv} show the performance of DeepFit's normal and principal curvature estimation respectively compared to Jet. DeepFit was trained with 256 points, however, since the network's weights are shared between the points it can be used with any neighborhood size. In these results we show the performance for $128, 256, 512, 1024$ neighboring points. It shows that DeepFit is less sensitive to noise and is able to overcome the over-smoothing affect commonly attributed to using a large neighborhood while also preserving fine details.
\begin{figure}
\centering
\includegraphics[width=0.95\linewidth]{nyu_results.pdf}
\caption{Normal estimation results for DeepFit and Jet on NYU Depth V2 dataset for different neighborhoods sizes ($128, 256, 512, 1024$). The colors of the points are normal vectors mapped to RGB and projected to the image plane.}
\label{fig:appx:results:nyu_v2}
\vspace*{\floatsep}
\centering
\includegraphics[width=0.95\linewidth]{real_world_curvatures.pdf}
\caption{Principal curvature estimation results for DeepFit and Jet on NYU Depth V2 dataset for different neighborhoods sizes ($128, 256, 512, 1024$). The colors of the points correspond to their principal curvature values using the colormap in the bottom-left corner and projected to the image plane.}
\label{fig:appx:results:nyu_v2_curv}
\end{figure}
\subsubsection{Additional normal estimation results.}
We evaluate the normal estimation performance on the PCPNet datast using the percentage of good points (PGP $\alpha$) metric. Fig \ref{figure:results:baselines_pgp} shows the results of different learning based methods for increasing $\alpha$ values . It shows that for low and medium noise levels, DeepFit is comparable to Lenssen et.al. \cite{lenssen2019differentiable} while in all other categories their performance is better. This is most likely attributed to the dataset bias towards flat and low curvature surfaces, in which case, our method does not pose an advantage. DeepFit main advantage is in curvy surfaces where an $n$-jet yields a better fit than a plane.
\input{Normal_estimation_SOTA_pgp.tex}
We evaluate DeepFit's normal estimation performance using RMSE for different $n$-jet orders and number of points in the neighborhood. The results are shown in Fig. \ref{figure:appx:results:rmse_ablations_all}. It shows that the increase in the number of neighboring points slightly decreases the performance in the no noise augmentation however it significantly improves the performance in high noise. This is mainly attributed to the weight estimation network that softly selects the most relevant points for the fit. It also shows that 1-jet (planes) perform well, however higher order jets have an advantage in the low and medium noise augmentation categories. In theory, the higher order jets have the capacity to fit planes, however in practice it is not always the case.
\input{Normal_estimation_rmse_ablations_all.tex}
Fig. \ref{fig:appx:results:pcpnet_all} depicts a visualization of DeepFit's results on PCPNet point clouds. Here the normal vectors are mapped to the RGB cube.
Fig. \ref{fig:appx:results:pcpnet_all_err} depicts a visualization of the angular error in each point for the PCPNet dataset. Here, the points' color correspond to angular difference, mapped to a heatmap ranging from 0-60 degrees. It shows that for complex shapes with high noise levels, the general direction of the normal vector is predicted correctly, but, the fine details and exact normal vector are not obtained. For basic shapes the added noise does not affect the results substantially. Most notably, DeepFit shows robustness to point density corruptions.
\subsubsection{Additional principal curvature estimation results.} Fig. \ref{fig:appx:results:curv_pcpnet_all_err} qualitatively depicts DeepFit's results on the PCPNet dataset. For visualization, the principal curvatures are mapped to RGB values according to the commonly used mapping given in Fig. \ref{fig:curvature_result_visualization} i.e. both positive (dome) are red, both negative (bowl) are blue, one positive and one negative (saddle) are green, both zero (plane) are white, and one zero and one positive/negative (cylinder) are yellow/cyan. For consistency in color saturation we map each model differently according to the mean and standard deviation of the principal curvatures. Note that the curvature sign is determined by the ground truth normal orientation.
DeepFit's normalized RMSE metric is visualized in Fig. \ref{fig:appx:results:curv_err_pcpnet_all_err} as the magnitude of the error vector mapped to a heatmap. It can be seen that more errors occur near edges, corners and small regions with a lot of detail and high curvature.
Moreover, these visualizations show that for low noise levels, the principal curvature estimation is reliable, as expected, the reliability declines with the insertion of high magnitude noise.
\begin{figure}
\centering
\includegraphics[width=0.98\linewidth]{normal_visualization_appx.pdf}
\caption{DeepFit's normal estimation results for different noise levels (columns 1-4), and density distortions (columns 5-6). The colors of the points are normal vectors mapped to RGB.}
\label{fig:appx:results:pcpnet_all}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.98\linewidth]{normal_err_visualization_appx.pdf}
\caption{Normal estimation error visualization for different noise levels (columns 1-4), and density distortions (columns 5-6). The colors of the points correspond to angular difference, mapped to a heatmap ranging from 0-60 degrees. The number above each point cloud is the RMSE.}
\label{fig:appx:results:pcpnet_all_err}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.98\linewidth]{curv_visualization_appx.pdf}
\caption{Curvature estimation results visualization. The colors of the points corresponds to the mapping of $k_1, k_2$ to the color map given in Fig \ref{fig:curvature_result_visualization}. Values in the range $[-(\mu(|k_i|)+\sigma(|k_i|)),\mu(|k_i|)+\sigma(|k_i|)] |_{i=1,2}$.}
\label{fig:appx:results:curv_pcpnet_all_err}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.98\linewidth]{curv_err_visualization_appx.pdf}
\caption{Curvature estimation error results. The numbers under each point cloud are its RMSE and normalized RMSE. The color corresponds to the L2 norm of the error vector mapped to a heatmap ranging from 0-5.}
\label{fig:appx:results:curv_err_pcpnet_all_err}
\end{figure}
\subsection{Efficiency}
Table \ref{table:parameter_comparisong} shows a comparison between the number of parameters and run time between different deep learning based normal estimation methods. It can be seen that DeepFit has a significantly lower number of parameters compared to Nesti-Net and PCPNet and more parameters than Lenssen et. al.. This gap in the number of parameters can be explained by the lack of point structure in our method while Lessen et. al. construct a graph. Constructing a graph introduces a limitation with respects to the number of neighboring points, i.e. training and testing has to be done on the same neighborhood size, using a PointNet architecture allows to train and test on different sizes of neighborhoods. DeepFit's , number of parameters is mainly attributed to the PointNet transformation sub-networks. The reported run time is the average run time for a batch of size 64 (i.e. computing normals for 64 points simultaneously). We chose a batch of 64 in order to fairly compare to the more resource intensive methods (Nesti-Net). Most methods, including ours, can compute in larger batches for faster performance, particularly Lenssen et. al. that are able to fit a full size point cloud (100k points) in a single batch on the GPU.
\begin{table}[]
\centering
\begin{tabular}{M{0.2\textwidth} M{0.15\textwidth} M{0.15\textwidth} M{0.15\textwidth} M{0.15\textwidth}}
\toprule
Method & Our DeepFit & Nesti-Net \cite{ben2019nesti}& PCPNet \cite{guerrero2018pcpnet}& Lenssen et. al. \cite{lenssen2019differentiable}\\
\hline
Parameters & 3.5M&179M& 22M& 7981\\
Exec time (per point) & 0.35ms & 266ms & 0.61ms & 0.13ms\\
\bottomrule
\end{tabular}
\caption{Number of parameters and execution time performance for deep learning normals estimation methods. Run time is averaged for batches of size 64. }
\label{table:parameter_comparisong}
\end{table}
\section{DeepFit}
\label{Sec:DeepFit}
\subsection{Learning point-wise weights}
\label{subSec:learning_weights}
The full pipeline for our method is illustrated in Fig.~\ref{fig:approach}.
Given a 3D point cloud $S$ and a query point $q_i \in S$ we first extract a local subset of points $S_i$ using k-nearest neighbors. We then use a neural network to estimate the weight of each point in the neighborhood, which will subsequently be used for weighted least squares surface fitting.
Specifically, we feed $S_i$ into a PointNet~\cite{Qi_2017_CVPR} network, which outputs a global point cloud representation $G(S_i)$. Additionally, we extract local representations from an intermediate layer for each of the points $p_{j} \in S_i$ separately to give $g(p_{j})$. These representations are then concatenated and fed into a multi-layer perceptron $h(\cdot)$ followed by a sigmoid activation function. We choose a sigmoid in order to limit the output values to be between 0 and 1. The output of this network is a weight per point that is used to construct the diagonal point-weight matrix, $W = \text{diag}(w_j)$ with
\begin{align}
w_j &= \text{sigmoid}(h(G(S_i), g(p_{ij}))) + \epsilon
\end{align}
\\
For numerical stability, we add a constant small $\epsilon$ in order to avoid the degenrate case of a zero or poorly conditioned matrix. This weight matrix is then used to solve the WLS problem of Eq.~\ref{eq:wlstsqr_sol} and approximate the $n$-jet coefficients $\beta$. All parts of the network are differentiable and therefore it is trained end-to-end.
\subsection{Geometric quantities estimation}
\label{subSec:approach:normal_estimation}
Given the $n$-jet coefficients $\beta$ several geometric quantities can be easily extracted:
\subsubsection{Normal estimation.}
The estimated normal vector is given by:
\begin{equation}
N_i=\frac{(-\beta_{1}, -\beta_{2}, 1)}{\norm{(-\beta_{1}, -\beta_{2}, 1)}_2}
\end{equation}
\subsubsection{Shape operator and principal curvatures.}
For the second order information we compute the Weingarten map of the surface by multiplying the inverse of the first fundamental form and the second fundamental form. Its eigenvalues are the principal curvatures $(k_1, k_2)$, and its eigenvectors are the principal directions. The computation is done in the tangent space associated with the parametrization.
\begin{align}
M_{\text{Weingarten}} &= -\frac{1}{\sqrt{\beta_1^2+\beta_2^2+1}}
\begin{bmatrix}
1+\beta_1^2 & \beta_1\beta_2 \\
\beta_1\beta_2 & 1+\beta_2^2
\end{bmatrix}^{-1}\begin{bmatrix}
2\beta_3 & \beta_4 \\
\beta_4 & 2\beta_5
\end{bmatrix}
\end{align}
Generally, the principal curvatures can be used as ground truth in training, however, due to the eigenvalue decomposition, with the high probability of outputting two zero principal curvatures (planes) it suffers from numerical issues when computing the gradients for backpropagation \cite{dang2018eigendecomposition}. Therefore, we compute the curvatures only at test time. Note that Monge basis and higher order Monge coefficients can also be computed, similar to \cite{cazals2005estimating}.
\subsection{Consistency loss}
\label{subSec:con_loss}
In order to learn point-wise weights, we introduce a local consistency loss $L_{con}$. This loss is composed of two terms, the weighted normal difference term and a regularization term. The weighted normal difference term computes a weighted average of the sine of the angle between the ground truth normal and the estimated normal at every local neighborhood point. These normals are computed analytically by converting the $n$-jet to the implicit surface form of $F(x,y,z)=0$. Therefore, for every query point $q_i$ and its local neighborhood $S_i$ we can compute the normal at each neighboring point $p_{j} \in S_i$ using:
\begin{equation}
N_{j} = \left.\frac{\nabla F}{\norm{\nabla F}}\right|_{p_{j}} =
\left.\frac{(-\beta_i \frac{\partial M}{\partial x}^T,
\beta_i \frac{ \partial M}{\partial y}^T, 1)}{\norm{\nabla F}}\right|_{p_{j}}
\end{equation}
Note that this formulation assumes all points to lie on the surface, for points that are not on the surface, the normal error will be large, therefore that points weight will be encouraged to be small. This term can easily converge to an undesired local minimum by setting all weights to zero. In order to avoid that, we add a regularization term which computes the negative average $log$ of all weights.
In summary, the consistency loss for a query point $q_i$ is then given by:
\begin{equation}
L_{con} =\frac{1}{N_{q_i}} \left[ -\sum_{j=1}^{N_{q_i}}log(w_{j}) + \sum_{j=1}^{N_{q_i}}w_{j} \abs{N_{GT}\times N_{j}} \right]
\end{equation}
In contrast to Lenssen et. al. \cite{lenssen2019differentiable}, this formulation allows us to avoid solving multiple linear systems iteratively for each point in the local neighborhood.
In total, to train the network, we sum several loss terms: The $sin$ loss between the estimated unoriented normal and the ground truth normal at the query point, the consistency loss, and PointNet's transformation matrix regularization terms $L_{reg}=\abs{I-AA^T}$.
\begin{equation}
L_{tot} = \abs{N_{GT}\times N_{i}} + \alpha_1L_{con} +\alpha_2L_{reg}
\end{equation}
\\
Here, $\alpha_1$, and $\alpha_2$ are weighting factors, chosen empirically.
\subsection{Implementation notes}
\label{subSec:impl_notes}
In our experiments we report results using DeepFit with the following configuration, unless otherwise stated. A four layer MLP with sizes 512, 256, 128, and 1; a neighborhood size of 256 points, and a 3-order jet.
In order to avoid numerical issues, simplify the notation, and reduce the linear algebra operations, we perform the following pre-processing stages on every local point cloud:
\begin{enumerate}
\item Normalization: we translate the point cloud to position the query point in the origin and scale the point cloud to fit a unit sphere.
\item Basis extraction: we perform principal component analysis (PCA) on the point cloud. We then use the resulting three orthonormal eigenvectors as the fitting basis so that the vector associated with the smallest eigenvalue is the last vector of the basis.
\item Coordinate frame transformation: We perform a change of coordinates to move the points into the coordinate system of the fitting basis.
\item Preconditioning: we precondition the Vandermonde matrix by performing column scaling. Each monomial $x_i^ky_i^l$ is divided by $h^{k+l}$. That is, $M' = MD^{-1}$ with $D$ the diagonal matrix $D=\text{diag}(1, h, h^2, ..., h^n)$. We use the mean of the norm $\norm{(x_i, y_i)}$ as $h$. The new system is then $M'(D\beta)=B$ and $\beta=D^{-1}(M'^TWM')^{-1}M'^TWB$.
\end{enumerate}
\noindent
Note that after the normal is estimated we apply the inverse transform to output the result in the original coordinate frame.
\section{Introduction}
\label{Sec:intro}
Commodity 3D sensors are rapidly becoming an integral component of autonomous systems. These sensors, e.g., RGB-D cameras or LiDAR, provide a 3D point cloud representing the geometry of the scanned objects and surroundings. This raw representation, however, is challenging to process since it lacks connectivity information or structure, and is often incomplete, noisy and contains point density variations.
In particular, processing it by means of convolutional neural networks (CNNs)---highly effective for images---is problematic because CNNs require structured, grid-like data as input.
When available, additional local geometric information, such as the surface normal and principal curvatures at each point, induces a partial local structure and improves performance of different tasks for interpreting the scene, such as over-segmentation \cite{ben2018graph}, classification \cite{qi2017pointnet++} and surface reconstruction~\cite{guerrero2018pcpnet}.
Estimating the normals and curvatures from a raw point cloud with no additional information is a challenging task due to difficulties associated with sampling density, noise, outliers, and detail level. The common approach is to specify a neighborhood around a point and then fit a local basic geometric surface (e.g., a plane) to the points in this neighborhood. The normal at the point under consideration is estimated from the fitted geometric surface. The chosen size (or scale) of the neighborhood introduces an unavoidable trade-off between robustness to noise and accuracy of fine details. A large neighborhood over-smooths sharp corners and small details but is otherwise robust to noise. A small neighborhood, on the other hand, may reproduce the normals more accurately around small details but is more sensitive to noise. Evidently, a robust, scale-independent, data-driven surface fitting approach should improve normal estimation performance.
We propose a surface fitting method for unstructured 3D point clouds. It features a neural network for point-wise weight prediction for weighted least squares fitting of polynomial surfaces. This approach removes the multi-scale requirement entirely and significantly increases robustness to different noise levels, outliers, and varying levels of detail. Moreover, the approach enables extracting normal vectors and additional geometric properties without the need for retraining or additional ground truth information.
The main contributions of this paper are:
\begin{itemize}
\item A method for per-point weight estimation for weighted least squares fitting using deep neural networks.
\item A scale-free method for robust surface fitting and normal estimation.
\item A method for principal curvature and geometric properties estimation without using ground truth labels.
\end{itemize}
\begin{figure*}[t]
\centering
\includegraphics[width=0.98\linewidth]{DeepFit_teaser.pdf}
\caption{DeepFit pipeline for normal and principal curvature estimation. For each point in a given point cloud, we compute a global and local representation and estimate a point-wise wight. Then, we fit an $n$-jet by solving a weighted least squares problem.}
\label{fig:approach}
\end{figure*}
\section{Background and Related Work}
\label{Sec:related-work}
\subsection{Deep learning for unstructured 3D point clouds}
\label{SubSec:rel_work:DL_3D}
The point cloud representation of a 3D scene is challenging for deep learning methods because it is both unstructured and unordered. In addition, the number of points in the point cloud varies for different scenes. Several methods have been proposed to overcome these challenges. Voxel-based methods embed the point cloud into a voxel grid but suffer from several accuracy-complexity tradeoffs~\cite{maturana2015voxnet}. The PointNet approach \cite{Qi_2017_CVPR,qi2017pointnet++} applies a symmetric, order-insensitive, function on a high-dimensional representation of individual points. The Kd-Network \cite{klokov2017escape} imposes a kd-tree structure on the points and uses it to learn shared weights for nodes in the tree. The recently proposed 3D modified fisher vectors (3DmFV) \cite{ben20183dmfv} represents the points by their deviation from a Gaussian Mixture Model (GMM) whose Gaussians are uniformly positioned on a coarse grid.
In this paper we use a PoinNet architecture for estimating point-wise weights for weighted least squares surface fitting. We chose PointNet since it operates directly on the point cloud, does not require preprocessing, representation conversion or structure, and contains a relatively low number of parameters,
\subsection{Normal Estimation}
A classic method for estimating normals uses principal component analysis (PCA)~\cite{hoppe1992surface}. Here a neighborhood of points within some fixed scale is chosen and PCA regression used to estimate a tangent plane. Variants that fit local spherical surfaces \cite{guennebaud2007algebraic} or Jets \cite{cazals2005estimating} (truncated Taylor expansion) have also been proposed. Further detail on Jet fitting is given in Section \ref{subsec:jet_fitting_math}. To be robust to noise, these methods usually choose a large-scale neighborhood, leading them to smooth sharp features and fail to estimate normals near 3D edges. Computing the optimal neighborhood size can decrease the estimation error \cite{mitra2003estimating} but requires the (usually unknown) noise standard deviation value and a costly iterative process to estimate the local curvature and additional density parameters.
A few deep learning approaches have been proposed to estimate normal vectors from unstructured point clouds. Boulch and Marlet proposed to transform local point cloud patches into a 2D Hough space accumulator by randomly selecting point triplets and voting for that plane's normal. Then, the normal is estimated from the accumulator by designing explicit criteria \cite{boulch2012fast} for bin selection or, more recently, by training a 2D CNN \cite{boulch2016deep} to estimate it continuously as a regression problem. This method does not fully utilize available 3D information since it loses information during the transformation stage. Another method, named PCPNnet \cite{guerrero2018pcpnet}, uses a PointNet \cite{Qi_2017_CVPR} architecture over local neighborhoods at multiple scales. It achieves good normal estimation performance and has been extended to estimating principal curvatures. However, it processes the multi-scale point clouds jointly and requires selecting a predefined set of scales.
A more recent work, Nesti-Net \cite{ben2019nesti} tries to predict the appropriate scale using a mixture of experts network and a local representation for different scales. It achieves high accuracy but suffers from high computation time due to the multiple scale computations. Nesti-Net shares PCPNet's drawback of requiring a predefined set of scales.
A contemporary work \cite{lenssen2019differentiable} uses an iterative plane fitting approach which tries to predict all normals of a local neighborhood and iteratively adjusts the point weights to best fit the plane.
In this paper we propose a novel approach for normal estimation by learning to fit an $n$-order Jet while predicting informative points' weights. Our approach removes the need of predefined scales and optimal scale selection since the informative points are extracted at any given scale. Our method generalizes the contemporary method proposed by Lenssen et. al.~\cite{lenssen2019differentiable}, avoids the iterative process, and enables the computation of additional geometric properties.
\subsection{Jet fitting using least squares and weighted least squares}
\label{subsec:jet_fitting_math}
We now provide background and mathematical notation for truncated Taylor expansion surface fitting using least-squares (LS) and weighted least-squares (WLS). We refer the interested reader to Cazals and Pouget~\cite{cazals2005estimating} for further detail.
Any regular embedded smooth surface can be locally written as the graph of a bi-variate ``height function'' with respect to any z-direction that does not belong to the tangent space~\cite{spivak1970comprehensive}. We adopt the naming convention of Cazals and Pouget~\cite{cazals2005estimating} and refer to the truncated Taylor expansion as a degree $n$ jet or $n$-jet for short. An $n$-jet of the height function over a surface is given by:
\begin{equation}
\label{eq:jet}
f(x,y)=J_{\beta,n}(x,y)= \sum_{k=0}^{n}\sum_{j=0}^{k}\beta_{k-j,j}x^{k-j}y^j
\end{equation}
\\
Here $\beta$ is the jet coefficients vector that consists of $N_n=(n+1)(n+2)/2$ terms.
In this work we wish to fit a surface to a set of $N_p$ 3D points. For clarity, we move to the matrix notation and specify the Vandermonde matrix $M=(1, x_i, y_i, ..., x_i y_i^{n-1}, y_i^n)_{i=1,...,N_p}\in \mathbb{R}^{N_p \times N_n}$ and the height function vector $B=(z_1, z_2,...z_{N_p})^T \in \mathbb{R}^N_p$ representing the sampled points. We require that every point satisfy Eq. \ref{eq:jet}, yielding the system of linear equations:
\begin{equation}
M\beta = B
\end{equation}
When $ N_n > N_p$ the system is over-determined and an exact solution may not exist. Therefore we use an LS approximation that minimizes the sum of square errors between the value of the jet and the height function over all points:
\begin{equation}
\beta = \arg\min_{z \in \mathbb{R}^{N_n}} \norm{Mz - B}^2
\end{equation}
It is well known that the solution can be expressed in closed-form as:
\begin{equation}
\label{eq:lstsqr_sol}
\beta = (M^TM)^{-1}M^TB
\end{equation}
\\
Typically the sampled points include noise and outliers that heavily reduce the fitting accuracy. To overcome this, the formulation given in Eq.~\ref{eq:lstsqr_sol} can be extended to a weighted least square problem. In this setting, some points have more influence on the fitted model than others. Let $W\in\mathbb{R}^{N_p \times N_p}$ be a diagonal weight matrix $W=\textrm{diag}(w_1, w_2, ..., w_{N_p})$. Each element in the matrix's diagonal $w_i$ corresponds to the weight of that point.
\\
The optimization problem becomes:
\begin{align*}
\beta &= \arg\min_{z \in \mathbb{R}^{N_n}} \norm{W^{1/2}(Mz - B)}^2\\
& = \arg\min_{z \in \mathbb{R}^{N_n}}\sum_{i=1}^{N_p} w_i \left(\sum_{j=1}^{N_n} M_{ij}z_j - B_i\right)^2
\end{align*}
and its solution:
\begin{equation}
\label{eq:wlstsqr_sol}
\beta = (M^TWM)^{-1}M^TWB
\end{equation}
In this work, we choose to focus on $n$-jet fitting because any order $n$ differential quantity can be computed from the $n$-jet. This is one of the main advantages of our method. That is, our method is trained for estimating normal vectors but is then able to estimate other differential quantities, e.g., principal curvatures, depending on the jet order.
\section{Results}
\label{Sec:results}
\subsection{Dataset and training details}
For training and testing we used the PCPNet shape dataset \cite{guerrero2018pcpnet}. The training set consists of eight shapes: four CAD objects (fandisk, boxunion, flower, cup) and four high quality scans of figurines (bunny, armadillo, dragon and turtle). All shapes are given as triangle meshes and densely sampled with 100k points. The data is augmented by introducing i.i.d. Gaussian noise for each point's spacial location with a standard deviation of 0.012, 0.006, 0.00125 w.r.t the bounding box size. This yields a set with 3.2M training examples. The test set consists of 22 shapes, including figurines, CAD objects, and analytic shapes. For evaluation we use the same 5000 point subset per shape as in Guerrero et al.~\cite{guerrero2018pcpnet}.
All variations of our method were trained using 32,768 (1024 samples by 32 shapes) random subsets of the 3.2M training samples at each epoch. We used a batch size of $256$, the Adam optimizer and a learning rate of $10^{-3}$. The implementation was done in PyTorch and trained on a single Nvidia RTX 2080 GPU.
\subsection{Normal estimation performance}
\label{SubSec:results:baseline_n_est}
We use the RMSE metric for comparing the proposed DeepFit to other deep learning based methods \cite{guerrero2018pcpnet,ben2019nesti,lenssen2019differentiable} and classical geometric methods \cite{hoppe1992surface,cazals2005estimating}.
Additionally, we analyze robustness for two types of data corruption:
\begin{itemize}
\item Point density---applying two sampling regimes for point subset selection: gradient, simulating effects of distance from the sensor, and stripes, simulating local occlusions.
\item Point perturbations--adding Gaussian noise to the points coordinates with three levels of magnitude specified by $\sigma$, given as a percentage of the bounding box.
\end{itemize}
For the geometric methods, we show results for three different scales: small, medium and large, which correspond to 18, 112, 450 nearest neighbors. For the deep learning based methods we show the results for the single-scale (ss) and multi-scale (ms) versions.
Table \ref{table:results:baselines} shows the unoriented normal RMSE results for the methods detailed above. It can be seen that our method slightly outperforms all other methods for low, medium and no noise augmentation and for gradient density augmentation. For high noise, and striped occlusion augmentation we are a close second to the contemporary work of Lenssen et al. \cite{lenssen2019differentiable} which only estimates the normal vectors while DeepFit also estimates other geometric properties, e.g., principal curvatures.
The results also show that all method's performance deteriorate as the noise level rises. In this context, both PCA and Jet perform well for specific noise-scale pairs. In addition, for PCPNet, using a multiple scales only mildly improves performance. Nesti-Net's mixture of experts mitigate the scale-accuracy tradeoff well at the cost of computational complexity. DeepFit's soft point selection process overcomes this tradeoff. In the supplemental materials we perform additional evaluation using the percentage of good points (PGP$\alpha$) metric.
\input{Normal_estimation_SOTA_comparison.tex}
Figure \ref{fig:results_normals_visualiztion:a} depicts a visualization of DeepFit's results on three point clouds. Here the normal vectors are mapped to the RGB cube.
It shows that for complex shapes (pillar, liberty) with high noise levels, the general direction of the normal vector is predicted correctly, but, the fine details and exact normal vector are not obtained. For a basic shape (Boxysmooth) the added noise does not affect the results substantially. Most notably, DeepFit shows robustness to point density corruptions.
Figure \ref{fig:results_normals_visualiztion:b} depicts a visualization of the angular error in each point for the different methods using a heat map. For the Jet method~\cite{cazals2005estimating} we display the results for medium scale. For all methods, it can be seen that more errors occur in regions with small details, high curvature e.g. edges and corners, and complex geometry. DeepFit suffers the least from this effect due to its point-wise weight estimation, which allows it to adapt to the different local geometryand disregard irrelevant points in the fitting process.
\begin{figure}
\centering
\begin{subfigure}{.48\textwidth}
\centering
\includegraphics[width=0.98\linewidth]{normal_visualization.pdf}
\caption{}
\label{fig:results_normals_visualiztion:a}
\end{subfigure}
\unskip\ \vrule\
\begin{subfigure}{.48\textwidth}
\centering
\includegraphics[width=0.98\linewidth]{"error_comparison".pdf}
\caption{}
\label{fig:results_normals_visualiztion:b}
\end{subfigure}
\caption{(a) DeepFit's normal estimation results for different noise levels (columns 1-4), and density distortions (columns 5-6). The colors of the points are normal vectors mapped to RGB. (b) Normal estimation error visualization results of DeepFit compared to other methods for three types of point clouds without noise. The colors of the points correspond to angular difference, mapped to a heatmap ranging from 0-60 degrees.}
\end{figure}
Figure \ref{figure:results_weights_visualiztion} qualitatively visualizes the performance of DeepFit's point-wise weight prediction network. The colors of the points correspond to weight magnitude, mapped to a heatmap ranging from 0 to 1 i.e. red points highly affect the fit while blue points have low influence. It shows that the network learns to adapt well to corner regions (column $n=1$), assigning high weights to points on one plane and excluding points on the perpendicular one. Additionally, it shows how the network adapted the weight to achieve a good fit for complex geometries (column $n=2,3,4$).
\begin{figure}
\centering
\includegraphics[width=0.98\linewidth]{point_weight_visualization.pdf}
\caption{DeepFit point-wise weight prediction. Three views of different $n$-jet surface fits. The colors of the points correspond to weight magnitude , mapped to a heatmap ranging from 0 to 1; see color bar on the right i.e. red points highly affect the fit while blue points have low influence.}.
\label{figure:results_weights_visualiztion}
\end{figure}
Fig. \ref{fig:results:ablation} shows the unoriented normal RMSE results for different parameter choices of our method. We explore different Jet orders $n=1, 2, 3, 4$, and a different number of neighboring points $k=64, 128, 256$, It shows that using a large neighborhood size highly improves the performance in high noise cases while only minimally affecting the performance in low noise. It also shows that all jet orders are comparable with a small advantage for order 1-jet (plane) and order 3-jet which is an indication for a bias in the dataset towards low curvature geometry. Additional ablation results, including more augmentations and the PGP$\alpha$ metric are provided in the supplemental material.
Timing and efficiency performance are provided in the supplemental material. DeepFit is faster and has fewer parameters than PCPNet and Nesti-Net and has the potential of only being slightly slower than CGAL implementation of Jet fitting because the forward pass for weight estimation is linear with respect to the number of points and the network weights.
\begin{figure}[t]
\centering
\begin{subfigure}{.98\textwidth}
\centering
\includegraphics[width=0.4\linewidth]{ablations_legend_rmse.pdf}
\label{fig:results:ablations:b}
\end{subfigure}
\begin{subfigure}{.49\textwidth}
\centering
\includegraphics[width=0.98\linewidth]{ablations_angle_rmse_results_no_noise.pdf}
\caption{}
\label{fig:results:ablations:a}
\end{subfigure}
\begin{subfigure}{.49\textwidth}
\centering
\includegraphics[width=0.98\linewidth]{ablations_angle_rmse_results_high_noise.pdf}
\caption{}
\label{fig:results:ablations:b}
\end{subfigure}
\caption{Normal estimation RMSE results for DeepFit ablations for (a) no noise and (b) high noise augmentations. Comparing the effect of number of neighboring points and jet order.}
\label{fig:results:ablation}
\end{figure}
\subsection{Principal curvature estimation performance}
\label{SubSec:results:baseline_c_est}
Figure \ref{fig:curvature_result_visualization} qualitatively depicts DeepFit's results on five point clouds. For visualization, the principal curvatures are mapped to RGB values according to the commonly used mapping given in its bottom right corner i.e. both positive (dome) are red, both negative (bowl) are blue, one positive and one negative (saddle) are green, both zero (plane) are white, and one zero and one positive/negative (cylinder) are yellow/cyan. For consistency in color saturation we map each model differently according to the mean and standard deviation of the principal curvatures. Note that the curvature sign is determined by the ground truth normal orientation.
\begin{figure}
\centering
\includegraphics[width=0.80\linewidth]{curvature_visualization.pdf}
\caption{Curvature estimation results visualization. The colors of the points corresponds to the mapping of $k_1, k_2$ to the color map given in the bottom right. Values in the range $[-(\mu(|k_i|)+\sigma(|k_i|)),\mu(|k_i|)+\sigma(|k_i|)] |_{i=1,2}$.}
\label{fig:curvature_result_visualization}
\end{figure}
For quantitative evaluation we use the normalized RMSE metric curvature estimation evaluation proposed in Guerrero et. al.~\cite{guerrero2018pcpnet} and given in Eq. \ref{eq:D_k}, for comparing the proposed method to other deep learning based \cite{guerrero2018pcpnet} and geometric methods \cite{cazals2005estimating}. Table \ref{table:results:curvature_baselines} summarizes the results and shows an average error reduction of 35\% and 13.7\% for maximum and minimum curvatures respectively. We analyze robustness for the same types of data corruptions as in normal estimation i.e. point perturbation and density. DeepFit significantly outperforms all other methods for maximum principal curvature $k_1$. For the minimum principal curvature $k_2$ DeepFit outperforms all methods for low and no noise augmentation in addition to gradient and striped density augmentation, however PCPNet has a small advantage for medium and high noise levels. The results for the minimum curvature are very sensitive since most values are close to zero.
\begin{equation}
\label{eq:D_k}
D_{k_j} = \left\lvert \frac{k_j-k_{GT}}{ \max\{|k_{GT}|, 1\}} \right\rvert, \quad \text{for } j=1,2.
\end{equation}
\input{Curvature_results_table.tex}
The normalized RMSE metric is visualized in Fig. \ref{fig:results_curvature_error_comparison} for DeepFit and PCPNet as the magnitude of the error vector mapped to a heatmap. It can be seen that more errors occur near edges, corners and small regions with a lot of detail and high curvature.
These figures show that for both simple and complex geometric shapes DeepFit is able to predict the principal curvatures reliably.
\begin{figure}
\centering
\includegraphics[width=0.98\linewidth]{curvature_error_comparison.pdf}
\caption{Curvature estimation error results for DeepFit compared PCPNet. The numbers under each point cloud are its normalized RMSE errors in the format ($k_1$, $k_2$). The color corresponds to the L2 norm of the error vector mapped to a heatmap ranging from 0-5.}
\label{fig:results_curvature_error_comparison}
\end{figure}
\subsection{Surface reconstruction and noise removal}
We further investigate the effectiveness of our surface fitting in the context of two subsequent applications---Poisson surface reconstruction~\cite{kazhdan2006poisson} and noise removal.
\subsubsection{Surface reconstruction.}
Fig. \ref{fig:results:poisson_recon} shows the results for the classical Jet fitting and our DeepFit approach. Since the reconstruction requires oriented normals, we orient the normals, in both methods, according to the ground truth normal. It shows that using DeepFit, the poisson reconstruction is moderately more satisfactory by being smoother overall, and crispier near corners. It also retains small details (liberty crown, cup rim).
\subsubsection{Noise removal.}
The point-wise weight prediction network enables a better fit by reducing the influence of neighboring points. This weight can also be interpreted as the network's confidence of that point to lie on the object's surface. Therefore, we can use the weight to remove points with low confidence. We first aggregate the weights by summing all of its weight prediction from all of its neighbors. Then we compute the mean and standard deviation of the aggregateed weights and remove points under a threshold of $\mu(\sum w_i)-\sigma(\sum w_i)$. The output point cloud contains less points than the original one and the removed points are mostly attributed to outliers or noise. The results are depicted in Fig. \ref{fig:results:noise_removal}.
\begin{figure}[t]
\centering
\begin{subfigure}{.48\textwidth}
\centering
\includegraphics[width=0.98\linewidth]{poisson.pdf}
\caption{}
\label{fig:results:poisson_recon}
\end{subfigure}
\unskip\ \vrule\
\begin{subfigure}{.48\textwidth}
\centering
\includegraphics[width=0.98\linewidth]{noise_Removal.pdf}
\caption{}
\label{fig:results:noise_removal}
\end{subfigure}
\caption{DeepFit performance in two subsequent application pipelines: (a) Poisson surface reconstruction using estimated normal vectors from the classical Jet fitting and the proposed DeepFit. (b) Noise removal results using DeepFit predicted weights.}
\end{figure}
\section{Summary }
\label{Sec:summary}
In this paper we presented a novel method for deep surface fitting for unstructured 3D point clouds. The method consists of estimating point-wise weights for solving a weighted least square fitting of an $n$-jet surface.
Our model is fully differentiable and can be trained end-to-end. The estimated weights (at test time) can be interpreted as the a confidence measure for every point in the point cloud and used for noise removal. Moreover, the formulation enables the computation of normal vectors and higher order geometric quantities like principal curvatures. The approach demonstrates high accuracy, robustness and efficiency compared to state-of-the-art methods.
This is attributed to it's ability to adaptively select the neighborhood of points through a learned model while leveraging classic robust surface fitting approaches, allowing the network to achieve high accuracy with a low number of parameters and computation time.
| -22,626.523978 |
[
-2.078125,
1.9765625
] | 37.75 |
[
-2.765625,
1.18359375,
-1.130859375,
-4.55078125,
-1.447265625,
6.45703125
] |
[
0.5478515625,
4.90234375,
0.974609375,
4.44140625
] | 392 | 5,398 |
[
-2.359375,
2.40234375
] | 23.339142 |
[
-6.20703125,
-3.8203125,
-3.27734375,
-1.6494140625,
2.099609375,
10.6875
] | 0.581484 | 21.675112 | 24.490552 | 1.808736 |
[
1.6961500644683838
] | -16,042.399559 | 6.053168 | -22,291.126167 | 0.370314 | 6.014909 |
[
-2.837890625,
-3.708984375,
-3.337890625,
-4.078125,
2.736328125,
11.1875
] |
[
-5.7109375,
-1.40234375,
-2.15625,
-1.6240234375,
3.48046875,
4.6953125
] | |
BkiUd9zxK2li-F6JzCsS
|
\section{Introduction}\label{sec:introductionandresults}
The {\em characteristic rank} of a smooth closed connected $d$-dimensional manifold $M$ was
introduced, and in some cases also calculated, in \cite{korbas2010} as the largest integer
$k$, $0\leq k\leq d$, such that each cohomology class in $H^j(M;\mathbb Z_2)$ with $j\leq k$
can be expressed as a polynomial in the Stiefel-Whitney classes of $M$; further results can
be found in \cite{balko-korbasAMH2012}. Recently, A. Naolekar and A. Thakur
(\cite{naolekarthakur}) have adapted this homotopy invariant of smooth closed connected
manifolds to vector bundles. Given a path-connected topological space $X$ and a real vector
bundle $\alpha$ over $X$, by the {\em characteristic rank of} $\alpha$, denoted
$\rm{charrank}(\alpha)$, we understand the largest $k$, $0 \leq k \leq \mathrm{dim}_{\mathbb
Z_2}(X)$, such that every cohomology class in $H^j(X;\mathbb Z_2)$, $0 \leq j \leq k$, can be
expressed as a polynomial in the Stiefel-Whitney classes $w_i(\alpha)\in H^i(X;\mathbb Z_2)$;
$\mathrm{dim}_{\mathbb Z_2}(X)$ is the supremum of all $q$ such that the cohomology groups
$H^q(X;\mathbb Z_2)$ do not vanish. We shall always use $\mathbb Z_2$ as the coefficient
group for cohomology, and so we shall write $H^i(X)$ instead of $H^i(X;\mathbb Z_2)$ and
$\mathrm{dim}(X)$ instead of $\mathrm{dim}_{\mathbb Z_2}(X)$. Of course, if $TM$ denotes the
tangent bundle of $M$, then we have $\mathrm{charrank}(TM)=\mathrm{charrank}(M)$. Results on
the characteristic rank of vector bundles over the Stiefel manifolds can be found in
\cite{knt2012}. In addition to being an interesting question in its own right, there are
other reasons for investigating the characteristic rank; one of them is its close relation to
the cup-length of a given space (\cite{korbas2010}, \cite{naolekarthakur}).
This note presents a result on the characteristic rank that may also prove useful in some
non-topological questions (for instance, in the theory of partitions; see \cite{andrewstop}).
More precisely, we quantify (Theorem \ref{th:uboundbnum}), via a sharp inequality, an
interplay between (a) the characteristic rank of a vector bundle, (b) the
$\mathbb{Z}_2$-Betti numbers of base space of this vector bundle, and (c) sums of the numbers
of certain partitions of integers. In a particular context, (c) is transformed into a sum of
the readily calculable Betti numbers of the real Grassmann manifolds $G_{n,k}$ of all
$k$-dimensional vector subspaces in $\mathbb R^n$.
\section{The result and its proof}
\label{relationscharrankversusBettinumbers}
Let $p(a,b,c)$ be the number of partitions of $c$ into \emph{at most} $b$ parts, each $\leq
a$. At the same time, given a subset $A$ of the positive integers, let $p(A,b,c)$ denote the
number of partitions of $c$ into $b$ parts each taken from the set $A$. We recall that
(\cite[6.7]{milnorstasheff}) $p(a,b,c)$ is the same as the number of cells of dimension $c$
in the Schubert cell decomposition of the Grassmann manifold $G_{a+b,b}$, or the same as the
$\mathbb Z_2$-Betti number $b_c(G_{a+b,b})=b_c(G_{a+b,a})$. Of course, for dimensional
reasons, $b_c(G_{a+b,b})=p(a,b,c)=0$ for $c > ab = \mathrm{dim}(G_{a+b,b})$. In addition, we
denote by $p(c)$ the total number of partitions of $c$. Even if not explicitly stated, $X$
will always mean a path-connected topological space.
For obvious reasons, we confine our considerations to those vector bundles having total
Stiefel-Whitney class non-trivial. For a given space $X$, an ordered subset in $\{1, 2,
\dots, \mathrm{dim}(X)\}$, with the least element (denoted by) $\nu$ and the greatest element
(denoted by) $\kappa$, will be denoted by $S_{\nu}^{\kappa}$. If, in addition, $\alpha$ is a
real vector bundle over $X$, then $S_{\nu}^{\kappa}(\alpha)$ will denote any
$S_{\nu}^{\kappa}\neq \emptyset$ such that $w_i(\alpha)=0$ for every positive $i\notin
S_{\nu}^{\kappa}$. In general, there are several possible choices of
$S_{\nu}^{\kappa}(\alpha)$: one can always take $S_{\nu}^{\kappa}(\alpha)=\{1=\nu, 2, \dots,
\mathrm{dim}(X)=\kappa\}$, but if we know, for instance, that $w_3(\alpha)=0$, then we can
also take the set $\{1=\nu, 2, 4, \dots, \mathrm{dim}(X)=\kappa\}$ in the role of
$S_{\nu}^{\kappa}(\alpha)$. Now we can state and prove our result.
\begin{theorem} \label{th:uboundbnum}
Let $\alpha$ be a real vector bundle over a path-connected topological space $X$ such that
$w(\alpha)\neq 1$, and let $\mathrm{charrank}(\alpha)\geq t$ for some $t$. Then, for every
$j\in \{1, \dots, t\}$, we have an inequality,
\begin{align}
b_j(X)\leq \sum\limits_{s=1}^{\lfloor\frac{j}{\nu}\rfloor} p(\{x\in
S_{\nu}^{\kappa}(\alpha)\vert x\leq \mu\},s,j), \label{ineq:uboundbettinum}
\end{align}
where $\mu=\mathrm{min}\{j,\kappa\}$.
In particular, if the set $\{x\in S_{\nu}^{\kappa}(\alpha)\vert x\leq \mu\}$ in
\eqref{ineq:uboundbettinum} is gapless, then \eqref{ineq:uboundbettinum} turns into
\begin{align}
b_j(X)\leq \sum_{\frac{j}{\mu}\leq s \leq \frac{j}{\nu}} b_{j-\nu s}(G_{\mu-\nu+s,s}).
\label{ineq:uboundviaGrassmannmanif}
\end{align}
\end{theorem}
\begin{proof}
Let us fix one of the sets $S_{\nu}^{\kappa}(\alpha)$. If $\mathrm{charrank}(\alpha)\geq t$
for some $t$ then, for every $j\in \{1,2, \dots, t\}$, the $\mathbb Z_2$-vector space
$H^j(X)$ is spanned by all the products of the form
\begin{equation}
w_{\nu}^{i_\nu}(\alpha)\cup \cdots \cup w_{\mu}^{i_{\mu}}(\alpha)\in H^j(X).
\label{generatorsofH^j(X)}
\end{equation}
Since $\nu(i_\nu + \dots + i_{\mu}) \leq \nu i_{\nu}+ \dots + \mu i_{\mu} = j$ (thus $i_\nu +
\dots + i_{\mu}\leq \frac{j}{\nu}$), the number of generators of the form
\eqref{generatorsofH^j(X)}, that is, an upper bound for $b_j(X)$, is the number of partitions
of $j$ into at most $\lfloor\frac{j}{\nu}\rfloor$ positive parts each taken from the set
$\{x\in S_{\nu}^{\kappa}(\alpha)\vert x\leq \mu\}$. In other words, this upper bound is
$\sum\limits_{s=1}^{\lfloor\frac{j}{\nu}\rfloor} p(\{x\in S_{\nu}^{\kappa}(\alpha)\vert x\leq
\mu\},s,j)$ as was asserted.
In order to transform \eqref{ineq:uboundbettinum} into \eqref{ineq:uboundviaGrassmannmanif}
when the set $\{x\in S_{\nu}^{\kappa}(\alpha)\vert x\leq \mu\}$ in
\eqref{ineq:uboundbettinum} is gapless, it suffices to know that
\begin{equation}
\sum\limits_{s=1}^{\lfloor\frac{j}{\nu}\rfloor} p(\{\nu, \nu +1, \nu +2, \dots, \mu\},s,j) =
\sum\limits_{s=1}^{\lfloor\frac{j}{\nu}\rfloor} p(\mu-\nu,s,j-\nu s);
\label{formulaviarestrictedpartit} \end{equation} this equality is verified by the following
elementary considerations.
Let $P(j)_{\{\nu,\nu+1,\ldots,\mu\}}^x$ be the set of partitions of $j$ into $x$ positive
parts each taken from the set $\{\nu,\nu+1,\ldots,\mu\}$ and let
$P(l)_{\{1,2,\ldots,\mu-\nu\}}^{\leq x}$ be the set of partitions of $l$ $(l\geq 0)$ into at
most $x$ parts each taken from the set $\{1,2,\ldots,\mu-\nu\}$ (the set
$P(0)_{\{1,2,\ldots,\mu-\nu\}}^{\leq x}$ for all $x>0$ and $\mu-\nu > 0$ contains just one
element, namely the empty partition). Of course, the total number of elements in
$P(l)_{\{1,2,\ldots,\mu-\nu\}}^{\leq x}$ is $p(\mu-\nu,x,l)$.
Each element of $P(j)_{\{\nu,\nu+1,\ldots,\mu\}}^x$ has the form
\begin{align*}
a_1+\cdots+a_{i(\nu)}+a_{i(\nu)+1}+\cdots+a_x,
\end{align*}
where $i(\nu)\geq 0$, $a_1=\cdots =a_{i(\nu)}=\nu<a_{i(\nu)+1}\leq a_{i(\nu)+2}\leq \cdots
\leq a_x$ and $a_1+\cdots+a_{i(\nu)}+a_{i(\nu)+1}+\cdots+a_x=j$. The map
\[
P(j)_{\{\nu,\nu+1,\ldots,\mu\}}^x\longrightarrow P(j-\nu x)_{\{1,2,\ldots,\mu-\nu\}}^{\leq
x},
\]
\[
a_1+\cdots+a_{i(\nu)}+a_{i(\nu)+1}+\cdots+a_x\mapsto
(a_{i(\nu)+1}-\nu)+(a_{i(\nu)+2}-\nu)+\cdots+(a_x-\nu)
\]
is bijective; indeed, the inverse map is readily seen to be
\[
P(j-\nu x)_{\{1,2,\ldots,\mu-\nu\}}^{\leq x}\longrightarrow
P(j)_{\{\nu,\nu+1,\ldots,\mu\}}^x,
\]
\[
b_1+b_2+\cdots+b_{x-i} \mapsto \underbrace{\nu+\cdots+\nu}_{i\text{
times}}+(b_1+\nu)+\cdots+(b_{x-i}+\nu).
\]
Thus \eqref{formulaviarestrictedpartit} is verified, and the proof of Theorem
\ref{th:uboundbnum} is finished.
\end{proof}
\begin{example}\label{illustrationofthmandsharpness}
We recall (\cite[Theorem 7.1]{milnorstasheff}) that the cohomology algebra of the infinite
Grassmannian $G_{\infty,k}$ can be identified with a polynomial algebra,
$$H^\ast(G_{\infty,k})=\mathbb Z_2[w_1, \dots, w_k],$$
where $w_i\in H^i(G_{\infty,k})$ is the $i$th Stiefel-Whitney class of the universal
$k$-plane bundle $\gamma_{\infty,k}$. Thus $\mathrm{charrank}(\gamma_{\infty,k})=\infty$ and
we may take $S_{\nu}^{\kappa}(\gamma_{\infty,k})=\{1=\nu,2,\dots, k=\kappa\}$. Since for
$X=G_{\infty,k}$ there are no relations among the generators of the form
\eqref{generatorsofH^j(X)}, inequalities \eqref{ineq:uboundbettinum} and
\eqref{ineq:uboundviaGrassmannmanif} turn into one of the following equalities for any
positive integer $j$:
\begin{equation} b_j(G_{\infty,k})= p(j)=\sum_{s=1}^{j}
b_{j-s}(G_{j-1+s,s}) \textrm{\,\,if\,\,} j\leq k, \label{equalityforjleqk}\end{equation}
\begin{equation} b_j(G_{\infty,k})= \sum\limits_{s=1}^{j} p(\{1,2,\dots,k\},s,j)
=\sum_{s=\lceil\frac{j}{k}\rceil}^{j} b_{j-s}(G_{k-1+s,s}) \textrm{\,\,if\,\,} j > k.
\label{equalityforj>k}\end{equation} In a similar way, one can see that
\eqref{ineq:uboundbettinum} and \eqref{ineq:uboundviaGrassmannmanif} are also sharp for
$X=\tilde G_{\infty,k}$, the infinite oriented Grassmannian.
\end{example}
\begin{ack}The authors thank Professor James Stasheff for useful comments on a version of this
paper. Part of this research was carried out while J. Korba\v s was a member of the research
teams 1/0330/13 and 2/0029/13 supported in part by the grant agency VEGA (Slovakia).\end{ack}
| -13,165.366385 |
[
-2.58984375,
2.201171875
] | 20.731707 |
[
-6.43359375,
-4.04296875,
-3.578125,
-9.8203125,
0.65283203125,
15.1640625
] |
[
2.6328125,
9.0703125,
0.77587890625,
5.45703125
] | 46 | 1,132 |
[
-3.56640625,
3.984375
] | 33.367769 |
[
-5.109375,
-4.109375,
-3.328125,
-1.42578125,
1.64453125,
9.5625
] | 0.989948 | 13.059516 | 38.515901 | 1.601017 |
[
1.725799798965454
] | -9,415.962729 | 5.800353 | -13,221.703279 | 1.279318 | 5.440112 |
[
-1.5126953125,
-2.66796875,
-3.935546875,
-5.5625,
1.7490234375,
11.8984375
] |
[
-6.12890625,
-3.435546875,
-2.625,
-1.9580078125,
4.0234375,
6.2578125
] | |
BkiUbck4uBhhxKZ5uaCF
|
\section{Introduction}
Throughout this paper $\mathbb{N}$ denotes the set of the positive integers, further $\mathbb{Z}, \mathbb{Q}$, and
$\mathbb{R}$ stand for the set of the integer, the set of the rational and the set of the real numbers, respectively.
The aim of this work is to prove characterization theorems on derivations as well as on linear functions.
Therefore, firstly we have to recall some definitions and auxiliary results.
A function $f:\mathbb{R}\rightarrow\mathbb{R}$ is called an \emph{additive} function
if,
\[
f(x+y)=f(x)+f(y)
\]
holds for all $x, y\in\mathbb{R}$.
We say that an additive
function $f:\mathbb{R}\rightarrow\mathbb{R}$ is a \emph{derivation} if
\[
f(xy)=xf(y)+yf(x)
\]
is fulfilled for all $x, y\in\mathbb{R}$.
Clearly, the identically zero function is a real derivation.
It is rather difficult to give another example, since the following statements are valid concerning real derivations.
If $f\colon \mathbb{R}\to \mathbb{R}$ is a real derivation, then $f(x)=0$ holds for all
$x\in \mathrm{algcl}(\mathbb{Q})$ (the algebraic closure of the rationals). Further,
if $f\colon \mathbb{R}\to \mathbb{R}$ is a real derivation and $f$ is measurable or bounded (above or below) on a set of positive Lebesgue
measure, then $f$ is identically zero.
Despite of this very pathological behavior, there exist non identically zero derivations in $\mathbb{R}$,
see Kuczma \cite[Theorem 14.2.2.]{Kuc09}.
The additive function $f\colon\mathbb{R}\to\mathbb{R}$ is termed to be a \emph{linear function} if
$f$ is of the form
\[
f(x)=f(1)\cdot x
\qquad
\left(x\in\mathbb{R}\right).
\]
It is easy to see from the above definition that every derivation
$f:\mathbb{R}\rightarrow\mathbb{R}$ satisfies equation
\[
\tag{$\ast$}\label{ast} f(x^{k})=kx^{k-1}f(x)
\quad
\left(x\in\mathbb{R}\setminus\left\{0\right\}\right)
\]
for arbitrarily fixed $k\in\mathbb{Z}\setminus\left\{0\right\}$.
Furthermore, the converse is also true, in the following sense:
if $k\in\mathbb{Z}\setminus\left\{0, 1\right\}$ is fixed and
an additive function $f:\mathbb{R}\rightarrow\mathbb{R}$ satisfies (\ref{ast}),
then $f$ is a derivation, see e.g.,
Jurkat \cite{Jur65}, Kurepa \cite{Kur64}, and
Kannappan--Kurepa \cite{KanKur70}.
Concerning linear functions, Jurkat \cite{Jur65} and,
independently, Kurepa \cite{Kur64} proved that every additive function
$f:\mathbb{R}\rightarrow\mathbb{R}$ satisfying
\[
f\left(\frac{1}{x}\right)=\frac{1}{x^{2}}f(x)
\quad
\left(x\in\mathbb{R}\setminus\left\{0\right\}\right)
\]
has to be linear.
In \cite{NisHor68} A.~Nishiyama and S.~Horinouchi investigated
additive functions $f:\mathbb{R}\rightarrow\mathbb{R}$
satisfying the additional equation
\[
f(x^{n})=cx^{k}f(x^{m})
\quad
\left(x\in\mathbb{R}\setminus\left\{0\right\}\right),
\]
where $c\in\mathbb{R}$ and $n, m, k\in\mathbb{Z}$ are arbitrarily
fixed.
Henceforth we will say that the
function in question is \emph{regular} on its domain, if at least
one of the following statements are fulfilled.
\begin{enumerate}[(i)]
\item locally bounded;
\item continuous;
\item measurable in the sense of Lebesgue.
\end{enumerate}
Concerning rational functions F.~Halter-Koch and L.~Reich proved similar result for derivations as well as linear functions,
see \cite{HalRei01},\cite{HalRei00}. These results were strengthened in \cite{Gse13} in the following way.
\begin{thm}\label{thm1}
Let $n\in\mathbb{Z}\setminus\left\{0\right\}$ and
$\left(\begin{array}{cc}
a&b\\
c&d
\end{array}
\right)\in\mathbf{GL}_{2}(\mathbb{Q})$ be such that
\begin{enumerate}[--]
\item if $c=0$, then $n\neq 1$;
\item if $d=0$, then $n\neq -1$.
\end{enumerate}
Let further $f, g\colon\mathbb{R}\to\mathbb{R}$ be additive functions and define the function
$\phi$ by
\[
\phi(x)=f\left(\frac{ax^{n}+b}{cx^{n}+d}\right)-\frac{x^{n-1}g(x)}{\left(cx^{n}+d\right)^{2}}
\qquad
\left(x\in\mathbb{R}, \,
cx^{n}+d\neq 0\right).
\]
Let us assume $\phi$ to be regular.
Then, the functions $F, G\colon\mathbb{R}\to\mathbb{R}$ defined by
\[
F(x)=f(x)-f(1)x \quad \text{and}
\quad
G(x)=g(x)-g(1)x
\quad
\left(x\in\mathbb{R}\right)
\]
are derivations.
\end{thm}
Roughly speaking the above cited papers dealt with a special case of the following problem.
Assume that $\xi\colon \mathbb{R}\to \mathbb{R}$
is a given differentiable function and for the additive function $d\colon \mathbb{R}\to \mathbb{R}$, the mapping
\[
x\longmapsto d\left(\xi(x)\right)-\xi'(x)d(x)
\]
is regular on its domain. It is true that in case
$d$ admits a representation
\[
d(x)=\chi(x)+d(1)\cdot x
\quad
\left(x\in \mathbb{R}\right),
\]
where $\chi\colon \mathbb{R}\to \mathbb{R}$ is a real derivation?
In view of the above results, in case $n\in\mathbb{Z}\setminus\left\{0\right\}$ and
$\left(\begin{array}{cc}
a&b\\
c&d
\end{array}
\right)\in\mathbf{GL}_{2}(\mathbb{Q})$
and the function $\xi$ is
\[
\xi(x)=\dfrac{ax^{n}+b}{cx^{n}+d}
\quad
\left(x\in \mathbb{R}, cx^{n}+d\neq 0\right),
\]
then the answer is \emph{affirmative}.
The main aim of this note is to extend this result to other classes of elementary functions such as
the exponential function, the logarithm function, the trigonometric functions and the hyperbolic functions.
Concerning such type of investigations, we have to remark the paper of Gy.~Maksa (see \cite{Mak13}), where the previous
problem was investigated under the supposition that the mapping
\[
x\longmapsto d\left(\xi(x)\right)-\xi'(x)d(x)
\]
is identically zero.
\section{The main result}
Our main result is contained in the following.
\begin{thm}\label{thm2}
Assume that for the additive function
$d\colon \mathbb{R}\to \mathbb{R}$ the mapping $\varphi$ defined by
\[
\varphi(x)=d\left(\xi(x)\right)-\xi'(x)d(x)
\]
is regular. Then the function $d$ can be represented as
\[
d(x)=\chi(x)+d(1)\cdot x
\quad
\left(x\in \mathbb{R}\right),
\]
where $\chi\colon \mathbb{R}\to \mathbb{R}$ is a derivation,
in any of the following cases
\begin{multicols}{2}
\begin{enumerate}[(a)]
\item \[\xi(x)=a^{x}\]
\item \[\xi(x)=\cos(x)\]
\item \[\xi(x)=\sin(x)\]
\item \[\xi(x)=\cosh(x)\]
\item \[\xi(x)=\sinh(x). \]
\end{enumerate}
\end{multicols}
\end{thm}
\begin{proof}
\begin{enumerate}[{Case} (a)]
\item Let $a\in \mathbb{R}\setminus\left\{1\right\}$ be an arbitrary positive real number and suppose that the
mapping $\varphi$ defined by
\[
\varphi(x)=d\left(a^{x}\right)-a^{x}\ln(a)d(x)
\quad
\left(x\in \mathbb{R}\right)
\]
is regular.
A easy calculation shows that
\[
\varphi(2x)-2a^{x}\varphi(x)=d\left((a^{x})^{2}\right)-2a^{x}d\left(x\right)
\quad
\left(x\in \mathbb{R}\right),
\]
that is
\[
\varphi\left(2\log_{a}(u)\right)-2u\varphi\left(\log_{a}(u)\right)=
d(u^{2})-2ud(u)
\quad
\left(u\in ]0, +\infty[\right).
\]
Due to the regularity of the function $\varphi$, the mapping
\[
]0, +\infty[\ni u\longmapsto \varphi\left(2\log_{a}(u)\right)-2u\varphi\left(\log_{a}(u)\right)
\]
is regular, too. Thus by Theorem \ref{thm1},
\[
d(x)=\chi(x)+d(1)\cdot x
\qquad
\left(x\in \mathbb{R}\right),
\]
where the function $\chi\colon \mathbb{R}\to \mathbb{R}$ is a derivation.
\item Assume now that for the additive function $d\colon \mathbb{R}\to \mathbb{R}$, the mapping
$\varphi$ defined on $\mathbb{R}$ by
\[
\varphi(x)=d\left(\cos(x)\right)+\sin(x)d(x)
\qquad
\left(x\in \mathbb{R}\right)
\]
is regular.
If so, then
\[
\dfrac{\varphi(2x)-4\cos(x)\varphi(x)+d(1)}{2}=
d\left(\cos^{2}(x)\right)-2\cos(x)f\left(x\right)
\]
holds for all $x\in \mathbb{R}$.
Let now $u\in ]-1, 1[$ and write $\mathrm{arccos}(u)$ in place of $x$ to get
\[
\dfrac{\varphi(2\mathrm{arccos}(u))-4u\varphi(\mathrm{arccos}(u))+d(1)}{2}=
d(u^{2})-2ud(u).
\]
Again, due to the regularity of the function $\varphi$, the mapping
\[
]-1, 1[ \ni u\longmapsto \dfrac{\varphi(2\mathrm{arccos}(u))-4u\varphi(\mathrm{arccos}(u))+d(1)}{2}
\]
is regular, as well. Therefore, Theorem \ref{thm1} again implies that
\[
d(x)=\chi(x)+d(1)\cdot x
\qquad
\left(x\in \mathbb{R}\right),
\]
is fulfilled with a certain real derivation $\chi\colon \mathbb{R}\to \mathbb{R}$.
\item Suppose that for the additive function $d$, the mapping
\[
\varphi(x)=d\left(\sin(x)\right)-\cos(x)d(x)
\qquad
\left(x\in \mathbb{R}\right)
\]
is regular.
In this case
\begin{multline*}
\varphi\left(x-\frac{\pi}{2}\right)
=
d\left(\sin \left(x-\frac{\pi}{2}\right)\right)-\cos\left(x-\frac{\pi}{2}\right)d\left(x-\frac{\pi}{2}\right)
\\
-d\left(\cos(x)\right)-\sin(x)d(x)+\sin(x)d\left(\frac{\pi}{2}\right),
\end{multline*}
that is,
\[
-\varphi\left(x-\frac{\pi}{2}\right)+\sin(x)d\left(\frac{\pi}{2}\right)=
d\left(\cos(x)\right)+\sin(x)d(x)
\qquad
\left(x\in \mathbb{R}\right).
\]
In view of Case (b) this yields that the function $d$ has the desired representation as stated.
\item Assume the $d\colon \mathbb{R}\to \mathbb{R}$ is an additive function and the mapping
\[
\varphi(x)=d\left(\cosh(x)\right)-\sinh(x)d(x)
\qquad
\left(x\in \mathbb{R}\right)
\]
is regular. The additivity of $d$ and some addition formula of the $\cosh$ function furnish
\[
\dfrac{\varphi(2x)-4\cosh(x)\varphi(x)+d(1)}{2}=
d\left(\cosh^{2}(x)\right)-2\cosh(x)d\left(\cosh(x)\right)
\qquad
\left(x\in \mathbb{R}\right).
\]
Let now $u\in ]1, +\infty[$ arbitrary and put $x=\mathrm{arcosh}(u)$ into the previous identity to get
\[
\dfrac{\varphi(2\mathrm{arcosh}(u))-4u\varphi(\mathrm{arcosh}(u))+d(1)}{2}=
d(u^{2})-2ud(u).
\]
Since the function $\varphi$ is regular, the mapping
\[
]1, +\infty[\ni u\longmapsto \dfrac{\varphi(2\mathrm{arcosh}(u))-4u\varphi(\mathrm{arcosh}(u))+d(1)}{2}
\]
will also be regular. Therefore, Theorem \ref{thm1} implies again the desired decomposition of the function
$d$.
\item Finally, assume the $d\colon \mathbb{R}\to \mathbb{R}$ is an additive function so that
\[
\varphi(x)=d\left(\sinh(x)\right)-\cosh(x)d(x)
\qquad
\left(x\in \mathbb{R}\right)
\]
is regular.
Let $x, y\in \mathbb{R}$ be arbitrary, then
\begin{multline*}
\varphi(x+y)= d\left(\sinh(x+y)\right)-\cosh(x+y)d(x+y)
\\=
d\left(\sinh(x)\cosh(y)\right)+d\left(\sinh(y)\cosh(x)\right)
\\-\left[\sinh(x)\sinh(y)+\cosh(x)\cosh(y)\right]d(x+y)
\\=
d\left(\sinh(x)\cosh(y)\right)+d\left(\sinh(y)\cosh(x)\right)
-\sinh(x)\sinh(y)d(x+y)
\\-\cosh(x)d(x)\cosh(y)
-\cosh(x)\cosh(y)d(y)
\end{multline*}
If we use the definition of the function $\varphi$, after some rearrangement, we arrive at
\begin{multline*}
\varphi(x+y)-\varphi(x)\cosh(y)-\varphi(y)\cosh(x)
\\
=
d\left(\sinh(x)\cosh(y)\right)+d\left(\sinh(y)\cosh(x)\right)
-\sinh(x)\sinh(y)d(x+y)\\
-\cosh(y)d\left(\sinh(x)\right)-\cosh(x)d\left(\sinh(y)\right)
\end{multline*}
for all $x, y\in \mathbb{R}$. If we replace here $y$ by $-y$,
\begin{multline*}
\varphi(x-y)-\varphi(x)\cosh(y)-\varphi(-y)\cosh(x)
\\
=
d\left(\sinh(x)\cosh(y)\right)-d\left(\sinh(y)\cosh(x)\right)
+\sinh(x)\sinh(y)d(x-y)\\
-\cosh(y)d\left(\sinh(x)\right)+\cosh(x)d\left(\sinh(y)\right)
\end{multline*}
can be concluded, where we have also used that the function $\cosh$ is even and the function
$\sinh$ is odd.
Adding this two identities side by side,
\begin{multline*}
\Phi(x, y)= 2d\left(\sinh(x)\cosh(y)\right)
\\+\sinh(x)\sinh(x)\left[d(x-y)-d(x+y)\right]
-2\cosh(y)d\left(\sinh(x)\right)
\end{multline*}
for any $x, y\in \mathbb{R}$, where
\begin{multline*}
\Phi(x, y)=
\varphi(x+y)-\varphi(x)\cosh(y)-\varphi(y)\cosh(x)
\\+\varphi(x-y)-\varphi(x)\cosh(y)-\varphi(-y)\cosh(x)
\quad
\left(x, y\in \mathbb{R}\right).
\end{multline*}
If we put $x=\mathrm{arsinh}(1)$, we get that
\[
\dfrac{\Phi\left(\mathrm{arsinh}(1), y\right)+2\cosh(y)d(1)}{2}
=
d\left(\cosh(y)\right)-\sinh(y)d(y)
\quad
\left(y\in \mathbb{R}\right).
\]
Due to the regularity of the function $\varphi$, the mapping
\[
\mathbb{R}\ni y\longmapsto \dfrac{\Phi\left(\mathrm{arsinh}(1), y\right)+2\cosh(y)d(1)}{2}
\]
is regular, too. Hence, Case (d) yields the desired form of the function $d$.
\end{enumerate}
\end{proof}
In what follows, we would like to extend the list of the functions appearing in the previous statement.
Therefore we prove the following.
\begin{lem}\label{lem3}
Let $d\colon \mathbb{R}\to \mathbb{R}$ be an additive function,
$I\subset \mathbb{R}$ be a nonvoid open interval and
$\xi\colon I\to \mathbb{R}$ be a continuously differentiable function so that
the derivative of the function $\xi^{-1}\colon \xi(I)\to \mathbb{R}$ is nowhere zero.
The mapping
\[
I \ni x\longmapsto d(\xi(x))-\xi'(x)d(x)
\]
is regular if and only if the mapping
\[
\xi(I)\ni u\longmapsto d(\eta(u))-\eta'(u)d(u)
\]
is regular, where $\eta=\xi^{-1}$.
\end{lem}
\begin{proof}
Assume that for the additive function $d$, we have that the mapping
\[
\varphi(x)=d(\xi(x))-\xi'(x)d(x)
\qquad
\left(x\in I\right)
\]
is regular.
Let now $u\in \xi(I)$ and put $\xi^{-1}(u)$ in place of $x$ to get
\[
-\left(\xi^{-1}\right)'(u)\varphi(\xi^{-1}(u))=
d\left(\xi^{-1}(u)\right)-\left(\xi^{-1}\right)'(u)d(u).
\]
Due to the regularity of $\varphi$, the mapping appearing in the left hand side is also regular, as stated.
\end{proof}
In view of Theorem \ref{thm2} and Lemma \ref{lem3}, we immediately obtain the following theorem.
\begin{cor}\label{cor4}
Assume that for the additive function
$d\colon \mathbb{R}\to \mathbb{R}$ the mapping $\varphi$ defined by
\[
\varphi(x)=d\left(\xi(x)\right)-\xi'(x)d(x)
\]
is regular. Then the function $d$ can be represented as
\[
d(x)=\chi(x)+d(1)\cdot x
\quad
\left(x\in \mathbb{R}\right),
\]
where $\chi\colon \mathbb{R}\to \mathbb{R}$ is a derivation,
in any of the following cases
\begin{multicols}{2}
\begin{enumerate}[(a)]
\item \[\xi(x)=\ln(x)\]
\item \[\xi(x)=\mathrm{arccos}(x)\]
\item \[\xi(x)=\mathrm{arcsin}(x)\]
\item \[\xi(x)=\mathrm{arcosh}(x)\]
\item \[\xi(x)=\mathrm{arsinh}(x). \]
\end{enumerate}
\end{multicols}
\end{cor}
\section{Stability of derivations}
As a starting point of the proof of the main result of this section
the theorem of Hyers will be used. Originally this
statement was formulated in terms of functions that
are acting between Banach spaces, see Hyers \cite{Hye41}.
However, we will use this theorem only in the particular case
when the domain and
the range are the set of reals.
In this setting we have the following.
\begin{thm}
Let $\varepsilon\geq 0$ and suppose that the function
$f:\mathbb{R}\rightarrow\mathbb{R}$ fulfills the inequality
\[
\left|f(x+y)-f(x)-f(y)\right|\leq \varepsilon
\]
for all $x,y\in\mathbb{R}$. Then there exists an additive
function $a:\mathbb{R}\rightarrow\mathbb{R}$ such that
\[
\left|f(x)-a(x)\right|\leq \varepsilon
\]
holds for arbitrary $x\in\mathbb{R}$.
\end{thm}
In other words, Hyers' theorem states that if a function $f\colon \mathbb{R}\to \mathbb{R}$ fulfills
the inequality appearing above, then it can be represented as
\[
f(x)=a(x)+b(x)
\qquad
\left(x\in \mathbb{R}\right),
\]
where $a\colon \mathbb{R}\to \mathbb{R}$ is an additive and
$b\colon \mathbb{R}\to \mathbb{R}$ is a bounded function. Moreover, for all
$x \in \mathbb{R}$, we also have $\left|b(x)\right|\leq \varepsilon$.
With the aid of Hyers' theorem and the results of the previous section, the following
stability type result can be proved. Concerning stability properties of derivations
the interested reader may consult Badora \cite{Bad06} and Boros--Gselmann \cite{BorGse10}.
\begin{thm}
Let $\varepsilon>0$ be arbitrarily fixed,
$f\colon \mathbb{R}\to \mathbb{R}$ be a function and suppose that
\begin{enumerate}[(A)]
\item for all $x, y\in \mathbb{R}$ we have
\[
\left|f(x+y)-f(x)-f(y)\right|\leq \varepsilon.
\]
\item the mapping
\[
x\longmapsto f\left(\xi(x)\right)-\xi'(x)f(x)
\]
is locally bounded on its domain, where the function $\xi$ is one of the functions
\begin{multicols}{2}
\begin{enumerate}[(a)]
\item \[a^{x}\]
\item \[\cos(x)\]
\item \[\sin(x)\]
\item \[\cosh(x)\]
\item \[\sinh(x)\]
\item \[\ln(x)\]
\item \[\mathrm{arccos}(x)\]
\item \[\mathrm{arcsin}(x)\]
\item \[\mathrm{arcosh}(x)\]
\item \[\mathrm{arsinh}(x). \]
\end{enumerate}
\end{multicols}
\end{enumerate}
Then there exist $\lambda\in \mathbb{R}$ and a real derivation $\chi\colon \mathbb{R}\to \mathbb{R}$
such that
\[
\left|f(x)-\left[\chi(x)+\lambda\cdot x\right]\right|\leq \varepsilon
\]
holds for all $x\in \mathbb{R}$.
\end{thm}
\begin{proof}
Due to assumption (A), we immediately have that
\[
f(x)=a(x)+b(x)
\qquad
\left(x\in \mathbb{R}\right),
\]
where $a\colon \mathbb{R}\to \mathbb{R}$ is an additive and
$b\colon \mathbb{R}\to \mathbb{R}$ is a bounded function.
If we use supposition (B), from this we get that the mapping
\[
x\longmapsto \left[a(\xi(x))-\xi'(x)a(x)\right] +\left[b(\xi(x))-\xi'(x)b(x)\right]
\]
is locally bounded. From this however the local boundedness of the function
\[
x\longmapsto a(\xi(x))-\xi'(x)a(x)
\]
can be deduced. In view of the previous statements (see Theorem \ref{thm2} and Corollary \ref{cor4}),
\[
a(x)=\chi(x)+a(1)\cdot x
\qquad
\left(x\in \mathbb{R}\right)
\]
is fulfilled for any $x\in \mathbb{R}$, where $\chi\colon \mathbb{R}\to \mathbb{R}$ is a certain real derivation.
For the function $f$ this means that there exists $\lambda \in \mathbb{R}$ and a real derivation
$\chi\colon \mathbb{R}\to \mathbb{R}$ so that
\[
f(x)=\chi(x)+\lambda\cdot x+b(x)
\qquad
\left(x\in \mathbb{R}\right),
\]
or equivalently
\[
\left|f(x)-\left[\chi(x)+\lambda\cdot x\right]\right|\leq \varepsilon
\]
is staisfied for any $x\in \mathbb{R}$.
\end{proof}
\subsubsection*{Acknowledgements}
This note is dedicated to two important men in my life, to Fr\'{e}di and to P\'{e}ter.
| -30,845.914098 |
[
-2.076171875,
1.8076171875
] | 19.928187 |
[
-3.171875,
0.90185546875,
-2.365234375,
-6.44140625,
-0.65869140625,
9.0625
] |
[
1.2275390625,
6.890625,
-1.1220703125,
5.08984375
] | 95 | 1,801 |
[
-3.501953125,
4.125
] | 33.986594 |
[
-5.265625,
-3.484375,
-4.11328125,
-2.603515625,
1.6298828125,
11.71875
] | 1.103846 | 9.081654 | 28.928373 | 1.309172 |
[
2.8444855213165283
] | -20,601.899801 | 6.539145 | -30,759.216296 | 2.139764 | 5.344707 |
[
-2.107421875,
-3.02734375,
-3.787109375,
-5.69921875,
2.189453125,
12.6171875
] |
[
-5.390625,
-2.009765625,
-2.021484375,
-1.3291015625,
3.2734375,
4.1015625
] | |
BkiUd5s25V5jBFwBuKof
|
\section{Introduction}
\label{Sec:Intro}
The advent of high-throughput technology has generated tremendous amounts of large and high-dimensional classification data. This allows classification at unprecedented
scales with hundreds or even more classes. The standard approach to classification in the multiclass setting is to use a classification rule for assigning a single label. More specifically, it consists in assigning a single label $Y \in \mathcal{Y}$, with $ \mathcal{Y}= \{1, \ldots, K\}$, to a given input example $X \in \mathcal{X}$ among a
collection of labels. However, while a large number of classes can lead to precise characterizations of data points, similarities among classes also bear the risk of
confusion and misclassification. Hence, assigning a single label can lead to wrong or ambiguous results.
In this paper, we address this problem by introducing an approach that yields sets of labels as outputs, namely, confidence sets.
A confidence set $\Gamma$ is a function that maps $\mathcal{X}$ onto $2^{\mathcal{Y}}$.
A natural way to obtain a set of labels is to use
ranked outputs of the classification rule. For example, one could take the classes that correspond to the top-level conditional probabilities $\mathbb{P}(Y=\cdot|X =x)$.
Here, we provide a more general approach where we control the {\it expected} size of the confidence sets. For a confidence set $\Gamma$, the expected size of $\Gamma$ is defined as
$\mathbb{E}[|\Gamma(X)|]$, where $|\cdot|$ stands for the cardinality.
For a sample $X$ and given an expected set size $\beta$,
we provide an algorithm that outputs a set $\hat{\Gamma}(X)$ such that $\mathbb{E}[|\hat{\Gamma}(X)|] \approx \beta$.
Furthermore, the procedure aims at minimizing the classification error given by
\begin{equation*}
\mathbb{P}\left(Y \notin \hat{\Gamma}(X)\right) \approx \min_{{\scriptscriptstyle\Gamma \, : \, \mathbb{E}[|\Gamma(X)|] = \beta}} \mathbb{P}\left(Y \notin
\Gamma(X)\right) = \mathcal{R}^{*}_{\beta}.
\end{equation*}
We establish a close formula of the oracle confidence set~$\Gamma^* = \mathop{\rm argmin}_{{\scriptscriptstyle\Gamma \, : \, \mathbb{E}[|\Gamma(X)|] = \beta}} \mathbb{P}\left(Y \notin \Gamma(X)\right)$ that involves the cumulative distribution functions of the conditional probabilities.
Besides, we formulate a data-driven counterpart of $\Gamma^{*}$ based on the minimization of the empirical risk.
However, the natural risk function in the multiclass setting is non convex, and then minimizing it is proving to be a computational issue.
As a remedy, convex surrogates are often used in machine learning, and more specifically in classification as
reflected by the popularity of methods such as Boosting~\cite{Freund_Schapire97}, logistic regression~\cite{Friedman_Hastie_Tibshirani00}
and support vector machine~\cite{VapnikLivre}.
In our problem this translates by considering some convex surrogate of the $0/1$-loss in $\mathbb{P}\left(Y \notin \Gamma(X)\right)$;
we introduce a new convex risk function which enables us to emphasize specific aspects of confidence sets.
That convex risk function is partly inspired by the works in~\cite{Zhang04, BartlettJordanMcauliffe06, YW10} that deal with binary classification.
Our approach displays two main features.
First, our method can be implemented in a semi-supervised way [Vap98].
More precisely, the method is a two-steps procedure that requires the estimation of score functions (as minimizers of some convex risk function) in a first step and the estimation of the cumulative distribution of these scores in a second one. Hence, the first step requires labeled data whereas the second one involves only unlabeled samples.
Notably, the unlabeled sample does not necessary consists of examples coming from the whole classes. This aspects is fondamental when we deal with a large number of classes, some of which been sparsely represented.
Second, from the theoretical point of view, we provide an oracle inequality satisfied by $\hat{\Gamma}$,
the empirical confidence set that results from the minimization of an empirical convex risk.
The obtained oracle inequality shall enable us to derive rates of convergence
on a particular class of confidence sets under the Tsybakov noise assumption on the data generating distribution.
These rates are linear in the number of classes $K$ and also depend on the regularity of the cumulative distribution functions of the conditional probabilities.
An obvious benefit of considering convex risk minimization is when we deal with aggregation. However, aggregating confidence sets is no simple matter. Another contribution of the present paper is about applying the above methodology to aggregation of confidence sets. More specifically, we provide a generalization of the superlearning algorithm~\cite{vanderLaanSL}
initially introduced in the context of regression and binary classification.
This algorithm relies on the $V$-fold cross-validation principle. We prove the consistency of this aggregation procedure and illustrate
its relevance on real datasets.
Let us point out that any arbitrary library of machine learning algorithms may be used in this aggregation procedure; we propose in the present paper to exploit support vector machines, random forest procedures or softmax regression since these methods are popular in machine learning and that each of them is associated to a different shape.
We end up this discussion by highlighting in two words what we perceive as being the main contributions of the present paper. We describe an optimal strategy for building confidence sets in multiclass classification setting,
and we derive a new aggregation procedure for confidence sets that still allows controlling the expected size of the resulting confidence set.
\bigskip
\noindent {\it Related works:}
The closest learning task to the present work is {\it classification with reject option} which is a particular setting in binary classification.
Several papers fall within the scope of this area~\cite{Ch70,HW06,YW10,WY11,LeiConfTheori,DH_ConfSet_15} and differ from each other by the goal they consider.
Among these references, our procedure is partially inspired by the paper~\cite{DH_ConfSet_15} that also considers a semi-supervised approach to build confidence sets invoking some cumulative distribution functions (of the conditional probabilities themselves in their case).
The similarity is however limited to the definition of oracle confidence sets, the oracle confidence in the present paper being an extension of the one defined in~\cite{DH_ConfSet_15} to the multiclass setting. On the other hand, all the data-driven considerations are completely different (in particular, \cite{DH_ConfSet_15} focuses on plug-in rules) and importantly, we develop here new probabilistic results on sums of cumulative distribution functions of random variables, that are of own interest.
Assigning a set of labels instead of a single one for an input example is not new~\cite{Vovk_livre,WLW04,CDB09,LRW13,CCB_16}.
One of the most popular methods is based on {\it Conformal Prediction} approach~\cite{Vovk99IntroCP,Vovk02AsymptVal,Vovk_livre}.
In the multiclass classification framework, the goal of this algorithm is to build the smallest set
of labels such that its classification error is below a pre-specified level.
Since our procedure aims at minimizing the classification error while keeping under control the size of the set, {\it Conformal Prediction} can be seen as a dual of our method.
It is worth mentioning that conformal predictors approaches need two labeled datasets where we only need one labeled dataset, the second being unlabeled.
We refer to the very interesting statistical study of {\it Conformal Predictors} in the binary case in the paper~\cite{LeiConfTheori}.
\bigskip
\noindent {\it Notation:} First, we state general notation. Let $\mathcal{Y} = \{1,\ldots,K\}$,
with $K \geq 2$ being an integer.
Let $(X ,Y)$ be the generic data-structure taking its values in $\mathcal{X} \times \mathcal{Y}$
with distribution $\mathbb{P}$.
The goal in classification is to predict the label $Y$ given an observation of $X$.
This is performed based on a classifier (or classification rule) $s$ which is a function mapping $\mathcal{X}$ onto
$\mathcal{Y}$. Let $\mathcal{S}$ be the set of all classifiers.
The misclassification risk $R$ associated with $s \in \mathcal{S}$ is defined as
\begin{equation*}
R(s) = \mathbb{P}(s(X) \neq Y).
\end{equation*}
Moreover, the minimizer of $R$ over $\mathcal{S}$ is the Bayes classifier, denoted by $s^*$, and is characterized by
\begin{equation*}
s^{*}(\cdot) = \argmax_{k \in \mathcal{Y}} p_{k}(\cdot),
\end{equation*}
where $p_{k}(x) = \mathbb{P}(Y = k | X = x)$ for $x \in \mathcal{X}$ and $k \in \mathcal{Y}$.
\\
Let us now consider more specific notation related to the multiclass confidence set setting.
Let a confidence set be any measurable function that maps $\mathcal{X}$
onto $2^{\mathcal{Y}}$.
Let $\Gamma$ be a confidence set. This confidence set is characterized by two attributes.
The first one is the risk associated to the confidence set
\begin{equation}
\label{Sec1:RiskConfidenceSet}
\mathcal{R}\left(\Gamma\right) = \mathbb{P}\left(Y \notin \Gamma(X)\right),
\end{equation}
and is related to its accuracy.
The second attribute is linked to the information given by the confidence set.
It is defined as
\begin{equation}
\label{Sec1:ExpConfidenceSet}
\mathcal{I}(\Gamma) = \mathbb{E}\left(|\Gamma(X)|\right),
\end{equation}
where $|\cdot|$ stands for the cardinality.
Moreover, for some $\beta \in [1,K]$, we say that, for two confidence sets $\Gamma$ and $\Gamma '$ such
that
$\mathcal{I}\left(\Gamma \right)=\mathcal{I}\left(\Gamma '\right) = \beta$, the confidence set $\Gamma$
is ``better'' than $\Gamma '$
if $\mathcal{R}\left(\Gamma \right) \leq \mathcal{R}\left(\Gamma '\right)$.
\bigskip
\noindent {\it Organization of the paper:}
The rest of the paper is organized as follows.
Next section is devoted to the definition and the main properties
of the oracle confidence set for multiclass classification.
The empirical risk minimization procedure is provided in
Section~\ref{Sec:EmpRisk}.
Rates of convergence for the confidence set that results from this minimization can also be found in this section.
We present an application of our procedure to aggregation of confidence sets in Section~\ref{Sec:numRes}.
We finally draw some conclusions and present perspectives of our work in Section~5. Proofs of
our results are postponed to the Appendix.
\section{Confidence set for multiclass classification}
\label{Sec:confSet}
In the present section, we define a class of confidence sets that are suitable for multiclass classification and referred as {\it Oracle $\beta$-sets}.
For some $\beta \in (0,K)$, these sets are shown to be optimal according to the risk~\eqref{Sec1:RiskConfidenceSet}
with an information~\eqref{Sec1:ExpConfidenceSet} equal to $\beta$.
Moreover, basic but fondamental properties of Oracle $\beta$-sets can be found in Proposition~\ref{prop:propG},
while Proposition~\ref{prop:prop2} provides another interpretation of these sets.
\subsection{Notation and definition}
\label{subsec:confSetDef}
First of all, we introduce in this section a class of confidence sets that specifies oracle confidence sets.
Let $\beta\in(0,K)$ be a desired information level.
The so-called {\it Oracle $\beta$-sets} are optimal according to the risk~\eqref{Sec1:RiskConfidenceSet} among all the
confidence sets $\Gamma $ such that $\mathcal{I}(\Gamma) = \beta$.
Throughout the paper we make the following assumption
{\it
\begin{itemize}
\item[{\bf (A1)}] For all $k \in \{1,\ldots,K\}$, the cumulative distribution function $F_{p_k}$ of $p_k(X)$ is continuous.
\end{itemize}
}
The definition of the {\it Oracle $\beta$-set} relies on the continuous and decreasing function $G$ defined for any $t\in
[0,1]$ by
\begin{equation*}
G(t) = \sum_{k=1}^K \bar{F}_{p_k}(t),
\end{equation*}
where for any $k\in \{1,\ldots, K\}$, we denote by $ \bar{F}_{p_k}$ the tail distribution function of $p_k(X)$, that
is, $ \bar{F}_{p_k}= 1- F_{p_k}$ with $ F_{p_k}$ being the cumulative distribution function (c.d.f.) of $p_k(X)$.
The generalized inverse $G^{-1}$ of $G$ is given by (see \cite{VanderVaartLivre}):
\begin{equation*}
G^{-1}(\beta) = \inf\{t\in [0,1] : \ G(t) \leq \beta \}, \;\; \forall \beta \in (0,K).
\end{equation*}
The functions $G$ and $G^{-1}$ are central in the construction of the Oracle $\beta$-sets. We then provide some of their useful properties in the following proposition.
\begin{proposition}
The following properties on $G$ hold
\label{prop:propG}
\begin{itemize}
\item[$i)$] For every $t \in (0,1)$ and $\beta \in (0,K)$, $G^{-1}(\beta) \leq t \Leftrightarrow \beta \geq G(t)$.
\item[$ii)$]For every $\beta \in (0,K)$, $G(G^{-1}(\beta)) = \beta$.
\item[$iii)$] Let $\varepsilon$ be a random variable, independent of $X$, and distributed from a uniform distribution on
$\{1, \ldots, K\}$
and let $U$ be uniformly distributed on $[0,K]$.
Define
\begin{equation*}
Z = \sum_{k = 1}^K p_k(X) {\bf 1}_{\{\varepsilon = k\}}.
\end{equation*}
If the function $G$ is continuous,
then $G(Z) \overset{\mathcal{L}}{=} U$ and $G^{-1}(U) \overset{\mathcal{L}}{=} Z$.
\end{itemize}
\end{proposition}
The proof of Proposition~\ref{prop:propG} relies on Lemma~\ref{lemme:lm1} in the Appendix.
Now, we are able to defined the Oracle $\beta$-set:
\begin{definition}\label{def:OracleSet}
Let $\beta \in (0,K)$, the Oracle $\beta$-set is given by
\begin{eqnarray*}
\Gamma_\beta^* (X) & = & \left\{ k \in \{1,\ldots, K\} : \ G(p_{k}(X)) \leq \beta \right\} \\
& = & \left\{ k \in \{1,\ldots, K\} : \ p_{k}(X)\geq G^{-1}(\beta) \right\}.
\end{eqnarray*}
\end{definition}
This definition of the Oracle $\beta$-set is intuitive and can be related to the binary classification with reject
option setting~\cite{Ch70,HW06,DH_ConfSet_15} in the following way: a label $k$ is assigned to the Oracle $\beta$-set if the probability $p_k(X)$ is
large enough. It is worth noting that the function $G$ plays the same role as the c.d.f. of the score
function in~\cite{DH_ConfSet_15}.
As emphasized by Proposition~\ref{prop:propG}, their introduction allows to control exactly the information~\eqref{Sec1:ExpConfidenceSet}.
Indeed, it follows from the definition of the Oracle $\beta$-set that for each $\beta \in (0,K)$
\begin{equation*}
|\Gamma_{\beta}^*(X)| = \sum_{k=1}^K {\bf 1}_{\{p_k(X) \geq G^{-1}(\beta)\}},
\end{equation*}
and then $\mathcal{I}(\Gamma_\beta^*) = \mathbb{E}\left[ |\Gamma_{\beta}^*(X)| \right]=G(G^{-1}{(\beta)})$.
Therefore, Proposition~\ref{prop:propG} ensures that
\begin{equation*}
\mathcal{I}(\Gamma_\beta^*) = \beta.
\end{equation*}
This last display points out that the Oracle $\beta$-sets are indeed $\beta$-level (that is, its information equals $\beta$).
In the next section, we focus on the study of the risk of these oracle confidence sets.
\begin{remark}
Naturally, the definition of Oracle $\beta$-sets can be extended to any $\beta \in [0,K]$.
However, the limit cases $\beta=0$ and $\beta=K$ are of limited interest and are completely trivial.
We then exclude these two limit cases from the present study.
\end{remark}
\subsection{Properties of the oracle confidence sets}
\label{subsec:confSetProp}
Let us first state the optimality of the Oracle $\beta$-set:
\begin{proposition}
\label{prop:prop1}
Let Assumption (A1) be satisfied. For any $\beta\in (0,K)$, we have both:
\begin{enumerate}
\item The Oracle $\beta$-set $\Gamma_\beta^*$ satisfies the following property:
\begin{equation*}
\mathcal{R}\left( \Gamma_\beta^* \right) = \inf_{\Gamma } \mathcal{R}\left(\Gamma\right),
\end{equation*}
where the infimum is taken over all $\beta$-level confidence sets.
\item For any $\beta$-level confidence set $\Gamma$, the following holds
\begin{equation}
\label{eq:decompRisk}
0\leq \mathcal{R}\left(\Gamma\right) - \mathcal{R}\left(\Gamma_\beta^*\right)
=
\mathbb{E}\left[ \sum_{k\in (\Gamma_\beta^*(X) \, \Delta \, \Gamma(X))}
\left| p_{k}(X) - G^{-1}(\beta) \right| \right],
\end{equation}
where the symbol $\Delta$ stands for the symmetric difference of two sets, that is, for two subsets $A$ and $B$
of $\{1,\ldots,K\}$, we write $A \, \Delta \, B = (A \setminus B) \cup (B\setminus A)$.
\end{enumerate}
\end{proposition}
Several remarks can be made from Proposition~\ref{prop:prop1}. First, for $\beta \in (0,K)$, the
Oracle $\beta$-set is optimal for the misclassification risk, over the class of $\beta$-level confidence sets.
Moreover, the excess risk of any $\beta$-level confidence set relies on the behavior of the score functions $p_k$ around
the threshold $G^{-1}(\beta)$.
Finally, we can note that if $K = 2$ and $\beta = 1$, which implies that $G^{-1}(\beta) = 1/2$, Equation~\eqref{eq:decompRisk}
reduces to the misclassification excess risk in binary classification.
\begin{remark}
\label{rmk:max}
One way to build a confidence set $\Gamma$ with information $\beta$ is to set $\Gamma$ as the $\beta$ top levels conditional probabilities.
In the sequel this method is referred as the $\texttt{max}$ procedure.
This strategy is natural but actually suboptimal as shown by the first point of Proposition~\ref{prop:prop1}. As an illustration,
we consider a simulation
scheme with $K = 10$ classes.
We generate $(X,Y)$ according to a mixture model. More precisely,
\begin{itemize}
\item[i)] the label $Y$ is distributed from a uniform distribution on $\{1, \ldots, K\}$;
\item[ii)] conditional on $Y = k$, the feature $X$ is generated according to a multivariate gaussian distribution
with mean parameter $\mu_k \in \mathbb{R}^{10}$ and identity covariance matrix. For each $k =1, \ldots, K$, the vectors $\mu_k$ are i.i.d realizations of uniform distribution on
$[0,4]$.
\end{itemize}
For $\beta = 2$ we evaluate the risks of the Oracle $\beta$-set and the $\texttt{max}$ procedure and obtain respectively
$0.05$ and $0.09$ (with very small variance).
\end{remark}
\begin{remark}
An important motivation behind the introduction of confidence sets and in particular of Oracle $\beta$-sets is that they might outperform the Bayes rule which can be seen as the Oracle $\beta$-set associated to $\beta=1$.
This gap in performance is even larger when the number of classes $K$ is large and there is a big confusion between classes.
Such improvement will be illustrated in the numerical study (see Section~\ref{subsec:realData}).
\end{remark}
We end up this section by providing another characterization of the Oracle $\beta$-set.
\begin{proposition}
\label{prop:prop2}
For $t \in [0,1]$, and $\Gamma$ a confidence set, we define
\begin{equation*}
L_t(\Gamma) = \mathbb{P}\left(Y \notin \Gamma(X)\right) + t \mathcal{I}(\Gamma).
\end{equation*}
For $\beta \in [0,K]$, the following equality holds:
\begin{equation*}
L_{G^{-1}(\beta)}( \Gamma_\beta^*) = \min_{\Gamma} L_{G^{-1}(\beta)}(\Gamma).
\end{equation*}
\end{proposition}
The proof of this proposition relies on the same arguments as those of Proposition~\ref{prop:prop1}.
It is then omitted.
Proposition~\ref{prop:prop2} states that the Oracle $\beta$-set is defined as the minimizer, over all confidence sets $\Gamma$, of the risk function $L_{t}$ when the tuning
parameter $t$ is set equal to $G^{-1}(\beta)$.
Note moreover that the risk function $L_t$ is a trade-off, controlled by the parameter $t$, between the risk of a confidence set on the one hand, and the information provided
by this confidence set on the other hand.
Hence, the risk function $L_t$ can be viewed as a generalization to the multiclass case of the risk function
provided in~\cite{Ch70, HW06} for binary classification with reject option setting.
\section{Empirical risk minimization}
\label{Sec:EmpRisk}
In this section we introduce and study confidence sets which rely on the
minimization of convex risks. Their definitions and main properties are given in Sections~\ref{subsec:phiRisk}-\ref{subsec:confSetCalib}.
As a consequence, we deduce a data-driven procedure described in Section~\ref{subsec:dataDriven} with
several theoretical properties, such as rates of convergence, that we demonstrate in Section~\ref{subsec:rateOfConvergence}.
\subsection{$\phi$-risk}
\label{subsec:phiRisk}
Let $f = (f_1, \ldots, f_K):\mathcal{X} \rightarrow \mathbb{R}^{K}$ be a score function and $G_{f}(.) = \sum_{k = 1}^K \bar{F}_{f_{k}}(.)$.
Assuming that the function $G_{f}$ is continuous and given an information level $\beta \in (0,K)$,
there exists $\delta \in \mathbb{R}$, such that $G_{f}(-\delta) = \beta$. Given this simple but important fact, we define
the confidence set $\Gamma_{f,\delta}$ associated to $f$ and $\delta$ as
\begin{equation}
\label{eq:fconfSet}
\Gamma_{f,\delta}(X) = \{k \; : \; f_k(X) \geq -\delta\}.
\end{equation}
In this way, the confidence set $\Gamma_{f,\delta}$ consists of top scores, and the threshold $\delta$ is fixed so that $\mathcal{I}\left(\Gamma_{f,\delta}\right) = \beta$.
As a consequence, we naturally aim at solving the problem
\begin{equation*}
\min_{f \in \mathcal{F}}\mathcal{R}(\Gamma_{f,\delta}),
\end{equation*}
where $\mathcal{F}$ is a class of functions.
Due to computational issues, it is convenient to focus on a convex surrogate of the previous minimization problem.
To this end, let $\phi: \mathbb{R} \rightarrow \mathbb{R}$ be a convex function.
We define the $\phi$-risk of $f$ by
\begin{equation}
\label{eq:phirisk}
R_{\phi}\left(f\right) = \mathbb{E}\left[\sum_{k = 1}^K \phi(Z_k f_k(X))\right],
\end{equation}
where $Z_k = 2 \; {\bf 1}_{\{Y = k\}}-1$ for all $k = 1, \ldots, K$.
Therefore, our target score becomes
\begin{equation*}
\bar{f} \in \mathop{\rm argmin}_{f \in \mathcal{F}} R_{\phi}\left(f\right),
\end{equation*}
for the purpose of building the confidence $\Gamma_{\bar{f}, \delta}$.
In the sequel, we also introduce $f^*$, the minimizer over the class of all measurable functions, of the $\phi$-risk.
The notation suppresses the dependence on $\phi$.
It is worth mentioning at this point that the definition of the risk function $R_{\phi}$ is dictated by Equation~\eqref{eq:decompRisk} and suits for confidence sets. Moreover, this function differs from the classical risk function used in the multiclass setting (see~\cite{BartlettTewari}). The reason behind this is that building a confidence set is closer to $K$ binary classification problems.
\subsection{Classification calibration for confidence sets}
\label{subsec:confSetCalib}
Convexification of the risk in classification is a standard technique.
In this section we adapt classical results and tools to confidence sets in the multiclass setting.
We refer the reader to earlier papers as~\cite{Zhang04,BartlettJordanMcauliffe06,YW10} for interesting developments.
One of the main important concept when we deal with convexification of the risk is the notion of calibration of the loss function $\phi$.
This property permits to connect confidence sets deduced from the convex risk and from the classification risk.
\begin{definition}
\label{def:calibration}
We say that the function $\phi$ is confidence set calibrated if for all $\beta > 0$, there exists $\delta^{*} \in
\mathbb{R}$
such that
\begin{equation*}
\Gamma_{f^{*},\delta^{*}} = \Gamma_{\beta}^{*},
\end{equation*}
with $f^*$ being the minimizer of the $\phi$-risk
\begin{equation*}
f^* \in \mathop{\rm argmin}_{f} R_{\phi}\left(f\right),
\end{equation*}
where the infimum is taken over the class of all measurable functions.
Hence, the confidence set based on $f^*$
coincides with the Bayes confidence set.
\end{definition}
Given this, we can state the following proposition that gives a characterization of the confidence set calibration property in terms of the function $G$.
\begin{proposition}
\label{prop:classCal}
The function $\phi$ is confidence set calibrated if and only if for all $\beta \in (0,K)$,
there exists $\delta^{*} \in \mathbb{R}$
such that
$\phi'(\delta^{*})$ and $\phi'(-\delta^{*})$ both exists, $\phi'(\delta^{*}) <0$, $\phi'(-\delta^{*}) <0$ and
\begin{equation*}
G^{-1}(\beta) = \dfrac{\phi'(\delta^{*})}{\phi'(\delta^{*})+\phi'(-\delta^{*})} \; ,
\end{equation*}
where $\phi'$ denotes the derivative of $\phi$.
\end{proposition}
The proof of the proposition follows the lines of Theorem~1 in~\cite{YW10} with minor modifications.
These characterizations of calibration for confidence sets generalize the notion of calibration in the classification setting as well as the necessary and sufficient condition for the minimizer of the $\phi$-risk to be calibrated. Indeed, if we pick $\delta=0$ in Definition~\ref{def:calibration} and Proposition~\ref{prop:classCal} we exactly come back to the calibration property in the classical classification setting (see~\cite{BartlettJordanMcauliffe06}).
Note that commonly used loss functions as boosting ($x\mapsto \exp(-x)$), least squares ($x \mapsto (x-1)^2$) and logistic ($x \mapsto \log(1+\exp(-x))$)
are examples of calibrated losses (see for instance~\cite{BartlettJordanMcauliffe06,WY11}).
Now, for some score function $f$ and some real number $\delta$ such that $G_{f}(-\delta) = \beta$, we define the excess risk
\begin{equation*}
\Delta\mathcal{R}(\Gamma_{f,\delta}) = \mathcal{R}(\Gamma_{f,\delta}) - \mathcal{R}(\Gamma^{*}_{\beta}),
\end{equation*}
and the excess $\phi$-risk
\begin{equation*}
\Delta R_{\phi}(f) = R_{\phi}\left(f\right)-R_{\phi}\left(f^{*}\right).
\end{equation*}
We also introduce the marginal conditional excess $\phi$-risk on $f=(f_1,\ldots,f_K)$ as
\begin{equation*}
\Delta R^{k}_{\phi}(f(X)) = p_k(X)(\phi(f_k(X))-\phi(f_k^*(X))) + (1-p_k(X))(\phi(-f_k(X))-\phi(-f_k^*(X))),
\end{equation*}
for $k=1,\ldots,K$.
The following theorem shows that the consistency in terms of $\phi$-risk implies the consistency in terms of classification risk $\mathcal{R}$.
\begin{theorem}
\label{theo:theo1}
Assume that $\phi$ is confidence set calibrated and assume that there exists constants $C > 0$ and $s \geq 1$ such
that\footnote{With abuse of notation, we write $ \Delta R_{\phi}^k(-\delta^{*})$ instead of $\Delta R_{\phi}^k((-\delta^{*},\ldots,-\delta^*))$ since no confusion can occur.}
\begin{equation}
\label{eq:Zhansgg}
|p_k(X) - G^{-1}(\beta)|^s \leq C \Delta R_{\phi}^k(-\delta^{*}).
\end{equation}
Let $\hat{f}_n$ be a sequence of scores. We assume that for each $n$, the function $G_{\hat{f}_n}$ is continuous.
Let $\delta_n\in \mathbb{R}$ be such that $G_{\hat{f}_n}(-\delta_n) =\beta$, then
\begin{equation*}
\Delta R_{\phi}(\hat{f}_n) \overset{P}{\rightarrow} 0 \qquad \Rightarrow \qquad \Delta \mathcal{R}(\Gamma_{\hat{f}_n,\delta_n})
\overset{P}{\rightarrow} 0.
\end{equation*}
\end{theorem}
The theorem ensures that the convergence in terms of $\phi$ risk implies
the convergence in terms of risk $\mathcal{R}$.
This convergence is made possible since we manage, in the proof, to bound the excess risk by (a power of) the excess $\phi$-risk. The assumption needed in this theorem is also standard and is for instance satisfied for the boosting, least square and logistic losses with the parameter $s$ being equal to $2$ (see~\cite{BartlettJordanMcauliffe06}).
\subsection{Data-driven procedure}
\label{subsec:dataDriven}
In this section we provide the steps of the construction of our empirical confidence set that is deduced from the empirical risk minimization. Before going into further details, let us first mention that our procedure is semi-supervised in the sense that it requires two datasets, one of which being unlabeled. Hence
we introduce a first data set $\mathcal{D}_n = \{(X_i,Y_i), \; i = 1, \ldots, n\}$,
which consists of independent copies of $(X,Y)$.
We define the empirical $\phi$-risk associated to a score function~$f$ (which is the empirical counterpart of $R_\phi$ given in~\eqref{eq:phirisk}):
\begin{equation}
\label{eq:eqEmpPhiRisk}
\hat{R}_{\phi}(f) = \dfrac{1}{n} \sum_{i = 1}^n \sum_{k = 1}^K \phi(Z^i_k f_k(X_i)),
\end{equation}
where $Z^i_k = 2 \; {\bf 1}_{\{Y_i = k\}}-1$ for all $k = 1, \ldots, K$.
We also define the empirical risk minimizer over $\mathcal{F}$, a convex set of score functions, as
\begin{equation*}
\hat{f} = \arg\min_{f \in \mathcal{F}} \hat{R}_{\phi}(f).
\end{equation*}
At this point, we have in hands the optimal score function $\hat{f}$ and need to build the corresponding confidence set with the right expected size.
However, let us before introduce an intermediate confidence set that would help comprehension since it mimics the oracle $\beta$-set $\Gamma^*$ with regard to its construction using $\hat{f}$ instead of $f^*$.
For this purpose, we define
\begin{equation*}
F_{\hat{f}_k}(t) = \mathbb{P}_{X}\left(\hat{f}_k(X) \leq t | \mathcal{D}_n\right),
\end{equation*}
for $t \in \mathbb{R}$, where $\mathbb{P}_{X}$ is the marginal distribution of $X$.
As for the c.d.f. of $p_k$ and $f_k$, we make the following assumption:
{\it
\begin{itemize}
\item[{\bf (A2)}] The cumulative distribution functions $F_{\hat{f}_k}$ with $k=1,\ldots,K$ are continuous.
\end{itemize}
}
At this point, we are able to define an empirical approximation of the Oracle $\beta$-set for $\beta \in (0,K)$:
\begin{equation}
\label{eq:eqFirstEstim}
\widetilde{\Gamma}_{\beta}(X) = \left\{ k \in \{1,\ldots, K\} : \ \widetilde{G}(\hat{f}_{k}(X)) \leq \beta \right\},
\end{equation}
where for $t \in \mathbb{R}$
\begin{equation}
\label{eq:tildG}
\widetilde{G}(t) = \sum_{k=1}^K \bar{F}_{\hat{f}_k}(t),
\end{equation}
with $ \bar{F}_{\hat{f}_k}= 1 - F_{\hat{f}_k}$.
Since the function $\widetilde{G}$ depends on the unknown distribution of $X$, we consider a second but interestingly {\it unlabeled} dataset
$\mathcal{D}_N = \{X_i, i = 1, \ldots, N\}$, independent of $\mathcal{D}_n$ in order to compute the empirical versions of
the $\bar{F}_{\hat{f}_k}$'s.
By now, we can define the empirical $\beta$-set based on $\hat{f}$:
\begin{definition}
\label{def:empirConfSet}
Let $\hat{f}$ be the minimizer of the empirical $\phi$-risk given in~\eqref{eq:eqEmpPhiRisk} based on $\mathcal{D}_n$, and consider the unlabeled dataset $\mathcal{D}_N$. Let $\beta \in (0,K)$.
The empirical $\beta$-set is given by
\begin{equation}
\label{eq:eqEmpConfSet}
\hat{\Gamma}_{\beta}(X) = \left\{ k \in \{1,\ldots, K\} : \ \hat{G}(\hat{f}_{k}(X)) \leq \beta \right\},
\end{equation}
where
\begin{equation*}
\hat{G}(.) = \frac{1}{N} \sum_{i = 1}^N \sum_{k = 1}^K {\bf 1}_{\{\hat{f}_k(X_i) \geq .\}}.
\end{equation*}
\end{definition}
The most important remark about the construction of this data-driven confidence set is that it is made in a semi-supervised way.
Indeed, the estimation of the tail distribution functions of $\hat{f}_k$ requires only a set of {\it unlabeled} observations.
This is particularly attractive in applications where the number of label observations is small (because labelling examples is time consuming or may be costly) and where one has in hand several unlabeled features that can be used to make prediction more accurate.
As an important consequence is that the estimation of the tail distribution functions of $\hat{f}_k$ does not depend on the number of observations in each class label. That is to say, this unlabeled dataset can be unbalanced with respect to the classes, that can often occur in multiclass classification settings where the number of classes $K$ is quite large.
Next section deals with the theoretical performance of the empirical $\beta$-sets.
\subsection{Rates of convergence}
\label{subsec:rateOfConvergence}
In this Section, we provide rates of convergence for the empirical confidence sets defined in
Section~\ref{subsec:dataDriven}.
First, we state some additional notation.
In the sequel, the symbols $\mathbf{P}$ and $\mathbf{E}$ stand for generic probability and expectation, respectively.
Moreover, given the empirical $\beta$-set $\hat{\Gamma}_{\beta}$ from Definition~\ref{def:empirConfSet}, we consider the risk
$\mathbf{R}\left(\hat{\Gamma}_{\beta}\right) = \mathbf{P}\left(Y \notin \hat{\Gamma}_{\beta}(X)\right)$ and the
information $\mathbf{I}\left(\hat{\Gamma}_{\beta}\right) = \mathbf{E}\left[ | \hat{\Gamma}_{\beta}(X) | \right]$.
Our first result ensures that $\hat{\Gamma}_{\beta}$ is of level $\beta$ up to a term of order~$K/\sqrt{N}$.
\begin{proposition}
\label{prop:confSetSize}
For each $\beta \in (0,K)$, the following equality holds
\begin{equation*}
\mathbf{I}\left(\hat{\Gamma}_{\beta}\right) = \beta + O\left(\dfrac{K}{\sqrt{N}}\right).
\end{equation*}
\end{proposition}
In order to precise rates of convergence for the risk, we need to formulate several assumptions. First, we consider loss functions $\phi$ which have in common that the modulus of convexity of their underlying risk function $R_{\phi}$, defined by
\begin{equation}
\label{eq:deltaMod}
\delta(\varepsilon) = \inf\left\{\frac{R_{\phi}(f)+R_{\phi}(g)}{2} - R_{\phi}\left(\frac{f+g}{2}\right), \;\; \sum_{k=1}^K\mathbb{E}_{X}\left[(f_k-g_k)^2(X)\right] \geq \varepsilon^2\right\},
\end{equation}
satisfies $\delta(\varepsilon)\geq c_1\epsilon^2$ for some $c_1>0$ (we refer to~\cite{BartlettJordanMcauliffe06, MendelsonBartlett} for more details on modulus of convexity for classification risk).
Moreover, we assume that $\phi$ is classification calibrated and $L$-Lipschitz for $L>0$.
On the other hand, for $\alpha > 0$, we assume a margin condition ${\rm M_{\alpha}^k}$ on each $p_k, k = 1,\ldots,K$.
\begin{equation*}
{\rm M_{\alpha}^k}: \;\;\;\mathbb{P}_{X}\left(0<|p_k(X)-G^{-1}(\beta)|\leq t\right)\leq c_2t^{\alpha}, \;\;\; \text{for some constant} \;\; c_2>0 \;\; \text{and for all} \;\; t>0.
\end{equation*}
It is important to note that since we assume that the distribution functions of $p_k(X)$ are continuous for each $k$,
we have $\mathbb{P}_{X}\left(0<|p_k(X)-G^{-1}(\beta)|\leq t\right) \rightarrow 0$ with $t \rightarrow 0$. Therefore,
the margin condition only precise the rate of this decay to $0$.
Now, we provide the rate of convergence that we can expect for the empirical confidence sets defined by~\eqref{eq:eqEmpConfSet}.
\begin{theorem}
\label{theo:vitesse1}
Assume that $\|f\|_{\infty} \leq B$ for all $f \in \mathcal{F}$.
Let $M_n = \mathcal{N}(1/n, L_{\infty}, \mathcal{F})$ be the
cardinality of the set of closed balls with radius $1/n$ in $L_{\infty}$-norm needed to cover $\mathcal{F}$.
Under the assumptions of Theorem~\ref{theo:theo1}, with the modulos of convexity $\delta(\varepsilon)\geq c_1\epsilon^2$ with $c_1>0$, if $\phi$ is $L$-Lipschitz and if the margin assumptions ${\rm M_{\alpha}^k}$ are satisfied with $\alpha > 0$, then
\begin{equation*}
\mathbf{E}\left[|\Delta \mathcal R(\hat{\Gamma}_{\beta})|\right] \leq C(B,L,s, \alpha) K^{1-\alpha/(\alpha+s)}
\left\{\inf_{f\in \mathcal{F}} \Delta R_{\phi}(f) + \frac{K\log(M_n)}{n} \right\}^{\alpha/(\alpha+s)} + C^{'}\frac{K}{\sqrt{N}},
\end{equation*}
where $C^{'} >0$ is an absolute constant and $C(B,L,s,\alpha) > 0$ is a constant that depends only on $L,B,s$ and $\alpha$.
\end{theorem}
From this theorem we obtain the following rate of convergence:
$K\left(\frac{\log(M_n)}{n}\right)^{\alpha/(\alpha+s)} + \frac{K}{\sqrt{N}}$.
This is the first bound for confidence sets in multiclass setting. The regularity parameter $\alpha$ plays a crucial role and governs the rate; larger values of $\alpha$ lead to faster rates. Moreover, this rate is linear in K, the number of classes, that seems to be the characteristic of multiclass classification as compared to binary classification.
Note that the exponent $\alpha/(\alpha+s)$ is not classical and is due to the estimation of quantiles.
The second part of the rates which is of order $K/ \sqrt{N}$ relies on the estimation of the function $\tilde{G}$ given in~\eqref{eq:tildG}.
This part of the estimation is established under the mild Assumptions (A1) and (A2). For this term the proof of the linearity on $K$
is tricky and is obtained thanks to a new technical probabilistic results on sums of cumulative distribution functions (see Lemma~\ref{lemme:lm1}).
Let us conclude this paragraph by mentioning that Theorem~\ref{theo:vitesse1} applies for instance for $\mathcal{F}$ being the convex hull of a finite family of a score functions which is the scope of the next section.
\section{Application to confidence sets aggregation}
\label{Sec:numRes}
This Section is devoted to an aggregation procedure
which relies on the superlearning principle. The specifics of our superlearning procedure is given
in Section~\ref{subsec:superlearning}. In Section~\ref{subsec:algoCve}, we show the consistency of our procedure and finally
we provide a numerical illustration in Section~\ref{subsec:realData} of our algorithm.
\subsection{Description of the procedure: superlearning}
\label{subsec:superlearning}
In this section, we describe the superlearning procedure for confidence sets.
The superlearning procedure is based on the $V$-fold cross-validation procedure (see~\cite{vanderLaanSL}).
Initially, this aggregation procedure has been introduced in the context of regression and binary classification.
Let $V \geq 2$ be an integer and let $(B_v)_{1 \leq v \leq V}$ be a regular partition of $\{1, \ldots, n \}$, {\em i.e.}, a partition such that, for each $v= 1, \ldots, V$,
${\rm card} (B_v) \in \{ \lfloor n/V \rfloor, \lfloor n/V \rfloor + 1 \}$, where we write $\lfloor x \rfloor$ for the floor of any real number $x$ (that is, $x-1 \leq \lfloor x \rfloor \leq x$).
For each $v \in \{1, \ldots, V\}$, we denote by $D_n^{(v)}$ (respectively $D_n^{(-v)}$) the dataset $\{(X_i,Y_i), i \in B_v\}$ (respectively $\{(X_i,Y_i), i \not\in B_v\}$),
and define the
corresponding empirical measures
\begin{eqnarray*}
P_{n}^{(v)} & = & \frac{1}{{\rm card}(B_v)} \sum_{i \in B_v} \text{Dirac}(X_i,Y_i), \quad\text{and}\\
P_{n}^{(-v)} & = & \frac{1}{n - {\rm card}(B_v)} \sum_{i \not\in B_v} \text{Dirac}(X_i,Y_i). \\
\end{eqnarray*}
For a score algorithm $f$, that is to say a function which maps the empirical distribution to a score function.
We define the cross-validated risk of $f$ as
\begin{equation}
\label{eq:eqCrossVRisk}
\widehat{R}_{\phi}^n(f) = \frac{1}{V} \sum_{v = 1}^V R_{\phi}^{(P_n^{(v)})} \left( {f}(P_n^{(-v)}) \right),
\end{equation}
where
for each $v \in \{1, \ldots, V\}$,
$R_{\phi}^{(P_n^{(v)})}(f(P_{n}^{(-v)}))$ is the empirical estimator
of $R_{\phi} (f(P_n^{(-v)})$, based on $D_n^{(v)}$ and conditionally on $D_n^{(-v)}$:
\begin{equation*}
R_{\phi}^{(P_n^{(v)})} \left(f(P_n^{(-v)}) \right) = \dfrac{1}{{\rm card}(B_v)} \sum_{i
\in I_v} \sum_{k=1}^K \phi(Z_k^i f_k(P_n^{(-v)})(X_i)).
\end{equation*}
Next, we consider $\mathcal{F} = \left(f^1, \ldots,f^M\right)$ a family of $M$ score algorithms.
We define the cross-validated score by
\begin{equation}
\label{eq:eqCrossVScore}
\widehat{f} \in \mathop{\mathop{\rm argmin}_{f \in \rm{conv}(\mathcal{F})}} \widehat{R}_{\phi}^n (f).
\end{equation}
Finally, we consider the resulting cross-validated confidence set defined by
\begin{equation*}
\hat{\Gamma}^{{\rm CV}}_{\beta}(X) = \left\{ k \in \{1,\ldots, K\} : \ \hat{G}(\hat{f}_{k}(X)) \leq \beta \right\},
\end{equation*}
that we analyse in the next section.
\subsection{Consistency of the algorithm}
\label{subsec:algoCve}
In this Section, we show some results that illustrate the consistency
of the superlearning procedure described above.
To this end, we introduce the oracle counterpart of the cross-validated risk defined in~\eqref{eq:eqCrossVRisk}.
For a score $f$, we define
\begin{equation*}
\tilde{R}^n_{\phi}(f) = \frac{1}{V} \sum_{v= 1}^V R_{\phi} (f(P_n^{(-v)}),
\end{equation*}
that yields to the oracle counterpart of the cross-validated score defined in~\eqref{eq:eqCrossVScore}
\begin{equation*}
\tilde{f} \in \mathop{\rm argmin}_{f \in \rm{conv}(\mathcal{F})} \tilde{R}^n_{\phi} (f).
\end{equation*}
Here again, we assume that the loss function $\phi$ satisfies the properties described in Section~\ref{subsec:rateOfConvergence} so that we can state the following result:
\begin{proposition}
\label{prp:crossVal}
We assume that for each $f \in {\rm conv}(\mathcal{F})$ and $v \in \{1, \ldots,V\}$, $\|f(P_n^{(-v)})\|_{\infty} \leq B$.
Then the following holds
\begin{equation*}
\mathbf{E}\left[\tilde{R}^n_{\phi}(\hat{f}) - \tilde{R}^n_{\phi}(\tilde{f})\right] \leq C(B,L) \dfrac{KM\log(n)}{\lfloor n/V \rfloor},
\end{equation*}
where $C(B,L)>0$ is a constant that depends only on $B$ and $L$.
\end{proposition}
As usual when one deals with cross-validated estimators, the theorem compares $\hat{f}$ to the oracle counterpart
$\tilde{f}$ in terms of the oracle cross-validated $\phi$-risk.
The theorem teaches us that, for sufficiently large $n$, $\hat{f}$
perform as well as $\tilde{f}$.
However our main goal remains confidence sets. Therefore the next step consists in showing that the confidence set associated to the cross-validated score $\hat f$ has good properties in terms of the classification risk. Let $\Gamma_{f,\delta}$ be the confidence set that results from a choice of $\beta \in (0,K)$ and a score function $f \in {\rm conv}(\mathcal{F})$.
We introduce the following excess risks
\begin{eqnarray*}
\Delta \tilde{\mathcal{R}}^n(\Gamma_{f,\delta}) & = & \frac{1}{V} \sum_{v = 1}^V \mathcal{R}\left(\Gamma_{f\left(P_n^{(-v)}\right),\delta}\right) - \mathcal{R}^*,\\
\Delta \tilde{R}^n_{\phi}(f) & = & \frac{1}{V} \sum_{v = 1}^V R_{\phi}(f(P_n^{(-v)})) - R_{\phi}(f^*).
\end{eqnarray*}
These quantities can be view as cross-validated counterparts of the excess risks introduced in Section~\ref{subsec:confSetCalib}. Hereafter,
we provide a result which can be view as the cross-validation counterpart of Theorem~\ref{theo:vitesse1}.
It is interpreted in a same way.
\begin{proposition}
\label{prp:OracleCrossVal}
We assume that for each $v \in \{1, \ldots, V\}$, $t \mapsto \mathbb{P}_{X}\left(\hat{f}\left(P_n^{(-v)}\right) \leq t|\mathcal{D}_n\right)$ is continuous.
Under the assumptions of Proposition~\ref{prp:crossVal} and if the margin assumptions ${\rm M_{\alpha}^k}$ are satisfied with $\alpha > 0$,
\begin{equation*}
\mathbf{E}\left[|\Delta \tilde{\mathcal{R}}^n(\hat{\Gamma}_{\beta}^{{\rm CV}})|\right] \leq C(B,L,\alpha,s) K^{1-\alpha/(\alpha+s)}
\left\{\mathbf{E}\left[\Delta \tilde{R}^n_{\phi}(\tilde{f})\right] + \frac{KM\log(n)}{\lfloor n/V \rfloor} \right\}^{\alpha/(\alpha+s)} + C^{'}\frac{K}{\sqrt{N}}.
\end{equation*}
\end{proposition}
The proof of Proposition~\ref{prp:OracleCrossVal} relies on Proposition~\ref{prp:crossVal} and similar arguments as in Theorem~\ref{theo:vitesse1}.
\subsection{Application to real datasets}
\label{subsec:realData}
\begin{table}
\begin{center}
\footnotesize{
\begin{tabular}{l | c || ccccc}
\multicolumn{2}{c}{} & \multicolumn{5}{c}{{$\texttt{Forest}$ ($K=4$)}}\\
\hline
\multicolumn{2}{c}{} & \multicolumn{5}{c}{$\beta$-set} \\
\hline\noalign{\smallskip}
$\beta$ & & \;\; \texttt{rforest} \;\;\; & \texttt{softmax reg} \;\;\;& \texttt{svm} \;\;\; & \texttt{kknn} \;\;\; & \texttt{CV} \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
2 & $\mathbf{R}$ & 0.02 (0.02) & 0.06 (0.02) & 0.02 (0.01) & 0.05 (0.03) & 0.02 (0.01) \\
& $\mathbf{I}$ & 2.00 (0.09) & 2.00 (0.08) & 2.00 (0.09) & 2.00 (0.08) & 2.00 (0.08) \\ \hline
\end{tabular}
}
\vspace*{0.25cm}
\footnotesize{
\begin{tabular}{l | c || ccccc}
\multicolumn{2}{c}{} & \multicolumn{5}{c}{{$\texttt{Plant}$ ($K=100$)}}\\
\hline
\multicolumn{2}{c}{} & \multicolumn{5}{c}{$\beta$-set} \\
\hline\noalign{\smallskip}
$\beta$ & & \;\; \texttt{rforest} \;\;\; & \texttt{softmax reg} \;\;\;& \texttt{svm} \;\;\; & \texttt{kknn} \;\;\; & \texttt{CV} \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
2 & $\mathbf{R}$ & 0.18 (0.03) & 0.77 (0.02) & 0.32 (0.04) & 0.20 (0.03) & 0.17 (0.03)\\
& $\mathbf{I}$ & 2.00 (0.09) & 2.02 (0.18) & 1.99 (0.10) & 2.00 (0.08) & 2.00 (0.08) \\
10 & $\mathbf{R}$ & 0.02 (0.01) & 0.42 (0.04) & 0.03 (0.02) & 0.08 (0.03) & 0.02 (0.01) \\
& $\mathbf{I}$ & 9.95 (0.38) & 10.06 (0.58) & 9.98 (0.22) & 9.98 (0.23) & 9.96 (0.37)\\
\hline
\end{tabular}
}
\caption{\label{table:tableRealData}
For each of the $B = 100$ repetitions and for each dataset, we derive the estimated
risks~$\mathbf{R}$ and information~$\mathbf{I}$ of the different $\beta$-sets w.r.t. $\beta$.
We compute the means and standard deviations (between parentheses) over the $B = 100$ repetitions.
For each $\beta$, the $\beta$-sets are based on, from left to right,
\texttt{rforest}, \texttt{softmax reg} and \texttt{svm}, \texttt{kknn} and \texttt{CV}
which are respectively the random forest, the softmax regression, support vector machines, $k$ nearest neighbors and the superlearning procedure.
Top: the dataset is the $\texttt{Forest}$ -- the dataset is the $\texttt{Plant}$.}
\end{center}
\end{table}
In this section, we provide an application of our aggregation procedure described in Section~\ref{subsec:superlearning}.
For the numerical experiment we focus on the boosting loss and consider the library of algorithms constituted
by the random forest, the softmax regression, the support vector machines and the $k$ nearest neighbors (with $k = 11$) procedures.
To be more specifics, we respectively exploit the \texttt{R} packages \texttt{randomForest}, \texttt{polspline}, \texttt{e1071} and \texttt{kknn}.
All the \texttt{R} functions are used with standard tuning parameters. Finally the parameter $V$ of the aggregation procedure is fixed to $5$.
We evaluate the performance of the procedure on two real datasets: the {\it Forest type mapping} dataset and the {\it one-hundred plant species leaves} dataset coming from the UCI database.
In the sequel we refer to these two datasets as $\texttt{Forest}$ and $\texttt{Plant}$ respectively.
The $\texttt{Forest}$ dataset consists of $K=4$ classes and $523$ labeled observations (we gather the train and test sets) with $27$ features.
Here the classes are unbalanced.
In the $\texttt{Plant}$ dataset, there are $K=100$ classes and $1600$ labeled observations.
This dataset is balanced so that each class consists of $16$ observations.
The original dataset contains $3$ covariates (each covariate consists of $64$ features).
In order to make the problem more challenging, we drop $2$ covariates.
To get an indication of the statistical significance, it makes sense to compare our aggregated confidence set (referred as \texttt{CV}) to the confidence sets that result from each component of the library.
Roughly speaking, we evaluate risks (and informations) of these confidence sets on each dataset.
To do so, we use the {\it cross validation} principle.
In particular, we run $B=100$ times the procedure where we split the data each time in three:
a sample of size $n$ to build the scores $\hat{f}$;
a sample of size $N$ to estimate the function $G$ and to get the confidence sets;
and a sample of size $M$ to evaluate the risk and the information.
For both datasets, we make sure that in the sample of size $n$, there is the same number of observations in each class.
As a benchmark, we notify that the misclassification risks of the best classifier from the library in the
$\texttt{Forest}$ dataset is evaluated at $0.15$ , whereas in the $\texttt{Plant}$ dataset, it is evaluated at $0.40$.
As planned, the performance of the classical methods are quite bad in this last dataset.
We set the sizes of the samples as $n=200$, $N=100$ and $M=223$ for the $\texttt{Forest}$ dataset, and $n=1000$, $N=200$ and $M=400$ for the $\texttt{Plant}$ one.
The results are reported in Table~\ref{table:tableRealData}, and confirm our expectations.
In particular, our main observation is that the aggregated confidence set (\texttt{CV}) outperforms all components of the library in the sense that
it is at least as good as the best component in all of the experiments.
Second, let us state some remarks that hold for all of the confidence sets and in particular our aggregated confidence set. First, we note that the information $\mathbf{I}(\Gamma)$ has the good level $\beta$ which is supported by our theory. Moreover, we see that the risk gets drastically better with moderate $\beta$ as compared to the {\it best} misclassification risk.
For instance, for the $\texttt{Plant}$, the error rate of the confidence set with $\beta =2$ based on random forests is $0.18$ whereas the misclassification error rate
of the best component is $0.40$.
\section{Conclusion}
\label{Sec:Concl}
In multiclass classification setting, the present paper propose a new procedure that assigns a set of labels instead of a single label to each instance.
This set has a controlled expected size (or information) and its construction relies on cumulative distribution functions and on an empirical risk minimization procedure.
Theoretical guarantees, especially rates of convergence are also provided and rely on the regularity of these cumulative distribution functions.
The obtained rates of convergence highlight a linear dependence w.r.t the number of classes $K$.
The procedure described in Section~\ref{Sec:EmpRisk} is defined as a two steps algorithm whose second step consists in the estimation of the function $G$ (which is a sum of tail distribution functions).
Interestingly, this step does not require a set of labeled data and neither to explore the whole classes, that is suitable for
semi-supervised learning.
Moreover, we apply our methodology to derive an aggregation algorithm which is based on the $V$-fold cross-validation principle.
Future works will focus on the optimality in the minimax sense with respect to the classification error.
In particular, we will investigate whether the rates of convergence are optimal in terms of their dependence on $K$.
We believe that this dependence (linearity on $K$) is the correct one.
However we will instigate whether this dependence can be reduced under some sparsity assumption.
\section{Appendix}
\label{Sec:prof}
This section gathers the proofs of our results.
Let us first add a notation that will be used throughout the Appendix:
for any random variable (or vector) $Z$, we denote by $\mathbb{P}_Z$ the probability w.r.t. $Z$ and by
$\mathbb{E}_Z$, the corresponding expectation.
\subsection{Technical Lemmas}
\label{subsec:lemma}
We first lay out key lemmata, which are crucial to establish the main theory.
We consider $K \geq 2$ be an integer, and $Z_1, \ldots, Z_K$, $K$ random variables.
Moreover we define function $H$ by:
\begin{equation*}
H(t) = \dfrac{1}{K} \sum_{k = 1}^K F_k(t), \;\; \forall t \in [0,1],
\end{equation*}
where for all $k = 1, \ldots , K$, $F_k$ is the cumulative distribution function of $Z_k$.
Finally, let us define the generalized inverse $H^{-1}$ of $H$:
\begin{equation*}
H^{-1}(p) = \inf\{t: H(t) \geq p \}, \;\; \forall p \in (0,1).
\end{equation*}
\begin{lm}
\label{lemme:lm1}
Let $\varepsilon$ distributed from a uniform distribution on $\{1, \ldots, K\}$
and independent of $Z_k$, $k = 1, \ldots, K$. Let $U$ distributed from a uniform distribution
on $[0,1]$.
We consider
\begin{equation*}
Z = \sum_{k = 1}^K Z_k {\bf 1}_{\{\varepsilon = k\}}.
\end{equation*}
If $H$ is continuous then
\begin{equation*}
H(Z) \overset{\mathcal{L}}{=} U \;\; {\rm and} \;\; H^{-1}(U) \overset{\mathcal{L}}{=} Z
\end{equation*}
\end{lm}
\begin{proof}
First we note that for every $t \in [0,1]$,
$\mathbb{P}\left(H(Z) \leq t\right) = \mathbb{P}\left(Z \leq H^{-1}(t)\right)$.
Moreover, we have
\begin{eqnarray*}
\mathbb{P}\left(H( Z)\leq t\right) & = & \sum_{k = 1}^K \mathbb{P}(Z \leq H^{-1}(t), \varepsilon = k) \\
& = & \dfrac{1}{K} \sum_{k = 1}^K \mathbb{P}(Z_k \leq H^{-1}(t)) \;\; {\rm with} \;\;
\varepsilon \;\; {\rm independent \;\; of} \;\; X\\
& = & H(H^{-1}(t)) \\
& = & t \;\; {\rm with} \;\; H \;\; {\rm continuous}.
\end{eqnarray*}
To conclude the proof, we observe that
\begin{eqnarray*}
\mathbb{P}\left(H^{-1}(U) \leq t\right) & = & \mathbb{P}\left(U \leq H(t)\right) \\
& = & \dfrac{1}{K} \sum_{k = 1}^K F_k(t) \\
& = & \sum_{k = 1}^K \mathbb{P}\left(Z_k \leq t, \varepsilon = k\right)\\
& = & \mathbb{P}(Z\leq t).
\end{eqnarray*}
\end{proof}
\begin{lm}
\label{lem:lemFondamental}
There exists an absolute constant $C' > 0$ such that
\begin{equation*}
\sum_{k = 1}^K \mathbf{P}\left(|\hat{G}(\hat{f}_k(X) - \widetilde{G}(\hat{f}_k(X))| \geq |\widetilde{G}(\hat{f}_k(X)) -\beta|
\right) \leq \dfrac{C'K}{\sqrt{N}}.
\end{equation*}
\end{lm}
\begin{proof}
We define, for $\gamma > 0$ and $k \in \{1, \ldots K\}$
\begin{eqnarray*}
{A}^k_0 & = & \left\lbrace\left|\widetilde{G}(\hat{f}_k(X)) -\beta\right| \leq \gamma\right\rbrace\\
{A}^k_j & = & \left\{2^{j-1}\gamma < |\widetilde{G}(\hat{f}_k(X)) -\beta| \leq 2^{j} \gamma \right\} , \;\; j \geq 1.
\end{eqnarray*}
Since, for every $k$, the events $({A}^k_j)_{j\geq 0}$ are mutually exclusive, we deduce
\begin{multline}
\label{eq:eqDec}
\sum_{k = 1}^K \mathbf{P}\left(|\hat{G}(\hat{f}_k(X) - \widetilde{G}(\hat{f}_k(X))| \geq |\widetilde{G}(\hat{f}_k(X)) -\beta|
\right) = \\
\sum_{k = 1}^K \sum_{j\geq 0} \mathbf{P}\left(|\hat{G}(\hat{f}_k(X) - \widetilde{G}(\hat{f}_k(X))| \geq |
\widetilde{G} (\hat{f}_k(X)) -\beta| , A_j^k\right).
\end{multline}
Now, we consider a random variable $\varepsilon$ uniformly distributed on $\{1, \ldots, K\}$ and independent of $\mathcal{D}_n$ and $X$.
Conditional on $\mathcal{D}_n$ and under Assumption $(A2)$, we apply Lemma~\ref{lemme:lm1} with
$Z_k = \hat{f}_k(X)$, $Z = \sum_{k = 1}^K Z_k {\bf 1}_{\{\varepsilon = k\}}$ and then obtain that
$\widetilde{G}(Z)$ is uniformly distributed on $[0,K]$. Therefore, for all $j \geq 0$ and $\gamma > 0$, we deduce
\begin{eqnarray*}
\dfrac{1}{K} \sum_{k=1}^K \mathbb{P}_X\left(|\widetilde{G}(\hat{f}_k(X)) - \beta | \leq 2^j \gamma | \mathcal{D}_n\right) & = &
\mathbb{P}_X\left(|\widetilde{G}(Z) - \beta | \leq 2^j \gamma | \mathcal{D}_n\right) \\
& \leq & \dfrac{2^{j+1} \gamma}{K}.
\end{eqnarray*}
Hence, for all $j \geq 0$, we obtain
\begin{equation}
\label{eq:eqEns}
\sum_{k = 1}^K \mathbf{P}(A_j^k) \leq 2^{j+1} \gamma.
\end{equation}
Next, we observe that for all $j \geq 1$
\begin{multline}
\label{eq:eqAux}
\sum_{k = 1}^K \mathbf{P}\left(|\hat{G}(\hat{f}_k(X) - \widetilde{G}(\hat{f}_k(X))| \geq |
\widetilde{G} (\hat{f}_k(X)) -\beta| , A_j^k\right) \leq \\
\sum_{k = 1}^K \mathbb{E}_{(\mathcal{D}_n, X)}\left[\mathbb{P}_{\mathcal{D}_N}\left(|\hat{G}(\hat{f}_k(X)) - \widetilde{G}(\hat{f}_k(X))|
\geq 2^{j-1}\gamma |\mathcal{D}_n, X\right){\bf 1}_{A_j^k} \right].
\end{multline}
Now, since conditional on $(\mathcal{D}_n, X)$, $\hat{G}(\hat{f}_k(X))$ is an empirical mean of i.i.d random variables
of common mean $\widetilde{G}(\hat{f}_k(X)) \in [0,K]$, we deduce from Hoeffding's inequality that
\begin{equation*}
\mathbb{P}_{\mathcal{D}_N}\left(|\hat{G}(\hat{f}_k(X)) - \widetilde{G}(\hat{f}_k(X))|
\geq 2^{j-1}\gamma |\mathcal{D}_n, X\right) \leq 2\exp\left(-\dfrac{N\gamma^2 2^{2j-1}}{K^2}\right).
\end{equation*}
Therefore, from Inequalities~\eqref{eq:eqDec},~\eqref{eq:eqEns} and~\eqref{eq:eqAux}, we get
\begin{multline*}
\sum_{k = 1}^K \mathbf{P}\left(|\hat{G}(\hat{f}_k(X) - \widetilde{G}(\hat{f}_k(X))| \geq |\widetilde{G}(\hat{f}_k(X)) -\beta|\right)\\
\leq \sum_{k = 1}^K \mathbf{P}\left(A_0^k\right) + \sum_{j \geq 1}
2\exp\left(-\dfrac{N\gamma^2 2^{2j-1}}{K^2}\right)\left(\sum_{k=1}^K\mathbf{P}\left(A_j^k\right)\right) \\
\leq 2\gamma + \gamma \sum_{j \geq 1} 2^{j+2} \exp\left(-\dfrac{N\gamma^2 2^{2j-1}}{K^2}\right).\\
\end{multline*}
Finally, choosing $\gamma = \dfrac{K}{\sqrt{N}}$ in the above inequality, we finish the proof of the lemma.
\end{proof}
\subsection{Proof of Proposition~\ref{prop:prop1}}
\label{subsec:proofpropOracle}
Let $\beta > 0$ and $\Gamma$ be a confidence set such that $\mathcal{I}(\Gamma) = \beta$.
First, we note that the following decomposition holds
\begin{multline*}
\mathcal{R}(\Gamma) - \mathcal{R}(\Gamma_{\beta}^{*}) = \sum_{j = 1}^K \sum_{k = 1}^K\mathbb{E}\left[ \sum_{l = 1}^{K} ({\bf 1}_{\{Y \notin \Gamma(X)\}}-
{\bf 1}_{\{Y \notin \Gamma_{\beta}^*(X)\}}){\bf 1}_{\{Y = l\}} {\bf 1}_{\{|\Gamma(X)| = k\}}
{\bf 1}_{\{| \Gamma_{\beta}^*(X)| = j\}}\right]\\
= \sum_{j = 1}^K \sum_{k = 1}^K \mathbb{E}\left[\sum_{l=1}^{K} ({\bf 1}_{\{l \notin \Gamma(X)\}}- {\bf 1}_{\{l \notin
\Gamma_{\beta}^*(X)\}}) p_{l}(X) {\bf 1}_{\{|\Gamma(X)| = k\}} {\bf 1}_{\{|\Gamma_{\beta}^*(X)| = j\}} \right]\\
= \sum_{j = 1}^K \sum_{k = 1}^K \mathbb{E}\left[\sum_{l=1}^K ({\bf 1}_{\{l \in \Gamma_{\beta}^*(X)
\setminus \Gamma(X)\}}
- {\bf 1}_{\{l \in \Gamma(X) \setminus \Gamma^*_{\beta}(X)\}}) p_{l}(X) {\bf 1}_{\{|\Gamma(X)| = k,
| \Gamma_{\beta}^*(X)| = j\}} \right],
\end{multline*}
where we conditioned by $X$ to get the second equality.
From the last decomposition and with
\begin{eqnarray*}
\mathbb{E}\left[|\Gamma(X)|\right] & = & \mathbb{E}\left[|\Gamma_{\beta}^*(X)|\right] \\
& = & \mathbb{E}\left[ \sum_{j = 1}^K \sum_{k = 1}^K k {\bf 1}_{\{|\Gamma(X)| = k\}} {\bf 1}_{\{|\Gamma_{\beta}^*(X)| = j\}} \right] \\
& = & \mathbb{E}\left[ \sum_{j = 1}^K \sum_{k = 1}^K j {\bf 1}_{\{|\Gamma(X)| = k\}} {\bf 1}_{\{|\Gamma_{\beta}^*(X)| = j\}}\right] ,
\end{eqnarray*}
we can express the excess risk as the sum of two terms:
\begin{multline}
\label{eq:eqprop1}
\mathcal{R}(\Gamma) - \mathcal{R}(\Gamma_{\beta}^{*}) = \sum_{j = 1}^K \sum_{k = 1}^K \mathbb{E}\left[\left(\sum_{l=1}^K {\bf 1}_{\{l \in \Gamma_{\beta}^*(X) \setminus
\Gamma(X)\}}
p_{l}(X)- j G^{-1}(\beta)\right) {\bf 1}_{\{|\Gamma(X)| = k\}}
{\bf 1}_{\{|\Gamma_{\beta}^*(X)| = j\}} \right] \\
+ \sum_{j = 1}^K \sum_{k = 1}^K \mathbb{E}\left[\left(k G^{-1}(\beta) - \sum_{l=1}^K {\bf 1}_{\{l \in \Gamma(X) \setminus \Gamma_{\beta}^*(X)\}}
p_{l}(X)\right) {\bf 1}_{\{|\Gamma(X)| = k\}}
{\bf 1}_{\{|\Gamma_{\beta}^*(X)| = j\}} \right].
\end{multline}
Now, for $j,k \in \{1, \ldots, K\}$ on the event $\{|\Gamma(X)| = k, |\Gamma_{\beta}^*(X)| = j\}$, we have
\begin{equation*}
k = \sum_{l = 1}^K {\bf 1}_{\{l \in \Gamma(X) \setminus \Gamma_{\beta}^*(X)\}} + \sum_{l = 1}^K {\bf 1}_{\{l \in \Gamma(X) \cap \Gamma_{\beta}^*(X)\}},
\end{equation*}
and
\begin{equation*}
j = \sum_{l = 1}^K {\bf 1}_{\{l \in \Gamma_{\beta}^*(X) \setminus \Gamma(X)\}} + \sum_{l = 1}^K {\bf 1}_{\{l \in \Gamma(X) \cap \Gamma_{\beta}^*(X)\}}.
\end{equation*}
Therefore, since
\begin{equation*}
l \in \Gamma_{\beta}^* \Leftrightarrow p_{l}(X) \geq G^{-1}(\beta),
\end{equation*}
Equality~\eqref{eq:eqprop1} yields the result.
\subsection{Proof of Theorem~\ref{theo:theo1}}
\label{subsec:proofTheo1}
First we recall that $\hat{f}_n = (\hat{f}_{n,1}, \ldots, \hat{f}_{n,K})$ is a sequence of score functions and $\delta_n\in \mathbb{R}$ is such that $G_{\hat{f}_n}(-\delta_n) =\beta$.
We suppress the dependence on $n$ to simplify notation and write $\hat{f}= (\hat{f}_{1}, \ldots, \hat{f}_{K})$ and $\delta$ for $\hat{f}$ and $\delta_n$ respectively.
Moreover, since there is no doubt, we also suppress everywhere the dependence on $X$.
We also define the events
\begin{equation}
\label{eq:proofSetBk}
B_{k} = \{\hat{f}_{k} \in (-\delta,-\delta^{*}) \;\; {\rm or} \;\; \hat{f}_{k} \in
(-\delta^{*},-\delta)\},
\end{equation}
for $k = 1, \ldots, K$.
We aim at controlling the excess risk $\Delta \mathcal{R}(\Gamma_{\hat{f},\delta})$.
Since the risk $\mathcal{R}$ is decomposable, it is convenient to introduce ``marginal excess risks'':
\begin{equation*}
\Delta \mathcal{R}^{k}(\Gamma_{f,\delta}) = {\bf 1}_{\{k\in (\Gamma_{\hat{f}, \delta} \, \Delta \, \Gamma_\beta^{*})\}} |p_k - G^{-1}(\beta)|.
\end{equation*}
Recall also that by convexity of the loss function $\phi$, we have that for all $x,y\in \mathbb{R}$,
\begin{equation}
\label{eq:eqUtile}
\phi(y)-\phi(x) \geq \phi'(x)(y-x).
\end{equation}
Assume that $\hat{f}_{k} \leq -\delta$ and $\hat{f}_{k} \leq -\delta^{*}\leq f_{k}^{*}$, which translates as $p_k-G^{-1}(\beta) \geq 0$, we get thanks to~\eqref{eq:eqUtile}
\begin{equation*}
p_k(\phi(\hat{f}_{k})-\phi(-\delta)) - (1-p_k)(\phi(-\hat{f}_{k})-\phi(\delta))\geq (\phi'(\delta^*)-\phi'(-\delta^*))(p_k-
G^{-1}(\beta))(\hat{f}_{k} + \delta^{*}) \geq 0 .
\end{equation*}
Similarly,
if $\hat{f}_{k} \geq -\delta$ and $\hat{f}_{k} \geq -\delta^{*}\geq f_{k}^{*}$, that is, $p_k-G^{-1}(\beta) \leq 0$ we have
\begin{equation*}
p_k(\phi(\hat{f}_{k})-\phi(-\delta)) - (1-p_k)(\phi(-\hat{f}_{k})-\phi(\delta)) \geq 0.
\end{equation*}
Note that in the following two cases
\begin{eqnarray*}
\bullet \quad \hat{f}_{k} \leq \ - \delta & {\rm and} & f_{k}^{*} \leq -\delta^{*};\\
\bullet \quad \hat{f}_{k} \geq \ - \delta & {\rm and} & f_{k}^{*} \geq -\delta^{*},
\end{eqnarray*}
we have $\Delta \mathcal{R}^{k}(\Gamma_{\hat{f},\delta}) = 0$.
Therefore, from the above inequalities, on $B_{k}^{c}$ and by assumption~\eqref{eq:Zhansgg} we get
\begin{equation}
\label{eq:defout}
{\bf 1}_{B_k^c}{\bf 1}_{\{k\in (\Gamma_{\hat{f}, \delta} \, \Delta \, \Gamma_\beta^{*})\}} |p_k - G^{-1}(\beta)|^s \leq C(\Delta R_{\phi}^k(\hat{f})).
\end{equation}
Therefore, since $s \geq 1$, we have
\begin{eqnarray*}
\left(\mathbb{E}\left[\sum_{k = 1}^K {\bf 1}_{B_k^c \cap \{k\in (\Gamma_{\hat{f}, \delta} \, \Delta \, \Gamma_\beta^{*})\}} |p_k - G^{-1}(\beta)|\right]\right)^s &\leq & \dfrac{1}{K} \sum_{k = 1}^K \mathbb{E}\left[{\bf 1}_{B_k^c \cap \{k \in \Gamma_{\hat{f},\delta} \Delta
\Gamma_{\beta}^{*} \}} K^s |p_k-G^{-1}(\beta)|^s \right] \\
& \leq &
C K^{s-1} \Delta R_{\phi}(\hat{f}).
\end{eqnarray*}
Moreover
\begin{eqnarray*}
\mathbb{E} \left[\sum_{k = 1}^K {\bf 1}_{B_k}{\bf 1}_{\{k\in (\Gamma_{\hat{f}, \delta} \, \Delta \, \Gamma_\beta^{*})\}} |p_k - G^{-1}(\beta)| \right] & \leq &
\sum_{k = 1}^{K} \mathbb{P}(B_k) \\
& \leq & \mathbb{E}\left[|G_{\hat{f}}(-\delta) -G_{\hat{f}}(-\delta^{*})|\right] \\
& \leq & \mathbb{E}\left[|G_{f^{*}}(-\delta^{*}) -G_{\hat{f}}(-\delta^{*})|\right].
\end{eqnarray*}
Finally, we get the following bound
\begin{equation*}
\Delta \mathcal{R} (\Gamma_{\hat{f},\delta}) \leq K^{\frac{s-1}{s}} \Delta R_{\phi}(\hat{f})^{1/s} + \mathbb{E}\left[|G_{f^{*}}(-\delta^{*}) -G_{\hat{f}}(-\delta^{*})|\right].
\end{equation*}
Now we observe that
\begin{equation}
\label{eq:eqDecompRisk1}
|{\bf 1}_{\{\hat{f}_{k} \geq -\delta^{*}\}} - {\bf 1}_{\{f_k^{*} \geq -\delta^{*}\}} | \leq {\bf 1}_{\{|p_k - G^{-1}(\beta)|^s \leq
C \Delta R_{\phi}^{k}(\hat{f})\}}.
\end{equation}
Therefore, for each $\gamma > 0$, we have
\begin{eqnarray*}
\mathbb{E}_{X}\left[|G_{f^{*}}(-\delta^{*}) -G_{\hat{f}}(-\delta^{*})|\right] & \leq &
\sum_{k = 1}^K \mathbb{P}_{X}\left(|p_k(X) - G^{-1}(\beta)|^s \leq C \Delta R_{\phi}^{k}(\hat{f})\right)\\
& \leq & \sum_{k=1}^K \mathbb{P}_{X}\left(|p_k(X) - G^{-1}(\beta)| \leq \gamma^{1/s} \right)
+ \mathbb{P}_{X}\left(\gamma \leq C \Delta R_{\phi}^{k}(\hat{f})\right).\\
\end{eqnarray*}
Now using Markov Inequality, we have that
\begin{eqnarray*}
\sum_{k=1}^K \mathbb{P}_{X}\left(\gamma \leq C \Delta R_{\phi}^{k}(\hat{f})\right) & \leq &
\dfrac{C}{\gamma} \sum_{k =1}^K \mathbb{E}_{X} \left[ \Delta R_{\phi}^{k}(\hat{f})\right]\\
& \leq & \dfrac{C \Delta R_{\phi}(\hat{f})}{\gamma}.\\
\end{eqnarray*}
The above inequality yields
\begin{equation}
\label{eq:eqMajoration}
\mathbb{E}_{X}\left[|G_{f^{*}}(-\delta^{*}) -G_{\hat{f}}(-\delta^{*})|\right] \leq
\dfrac{C \Delta R_{\phi}(\hat{f})}{\gamma} + \sum_{k=1}^K \mathbb{P}_{X}\left(|p_k(X) - G^{-1}(\beta)| \leq
\gamma^{1/s} \right).
\end{equation}
Hence, with Equation~\eqref{eq:eqDecompRisk1}, we get
\begin{equation*}
\Delta \mathcal{R} (\Gamma_{\hat{f},\delta}) \leq K^{\frac{s-1}{s}} \Delta R_{\phi}(\hat{f})^{1/s} + \dfrac{C \Delta R_{\phi}(\hat{f})}{\gamma} + \sum_{k=1}^K \mathbb{P}_{X}\left(|p_k(X) -
G^{-1}(\beta)| \leq \gamma^{1/s} \right).
\end{equation*}
The term $\sum_{k=1}^K \mathbb{P}_{X}\left(|p_k(X) - G^{-1}(\beta)| \leq \gamma^{1/s} \right) \rightarrow 0$ when $\gamma \rightarrow 0$, given that the distribution function of the $p_k's$ are continuous. Then using the convergence in distribution of $\Delta R_{\phi}(\hat{f})$ to zero, the last inequality ensures the desired result.
\subsection{Proof of Proposition~\ref{prop:confSetSize}}
\label{subsec:proofPropSize}
For any $\beta \in (0,K)$, and conditional on $\mathcal{D}_n$ we define
\begin{equation}
\label{eq:eqtldeSeuil}
\widetilde{G}^{-1}(\beta) = \inf\{t\in \mathbb{R} : \ \widetilde{G}(t) \leq \beta \}.
\end{equation}
We note that Assumption $(A2)$ ensures that $t \mapsto \widetilde{G}(t)$ is continuous and then
\begin{equation*}
\widetilde{G}(\widetilde{G}^{-1}(\beta)) = \beta.
\end{equation*}
Now, we have
\begin{eqnarray*}
|\widetilde{\Gamma}_{\beta}(X)| & = & \sum_{k=1}^K {\bf 1}_{\{\widetilde{G}(\hat{f}_k(X)) \leq \beta\}}\\
& = & \sum_{k=1}^K {\bf 1}_{\{\hat{f}_k(X) \geq \widetilde{G}^{-1}(\beta)\}}.
\end{eqnarray*}
Hence, the last equation implies that
\begin{equation}
\label{eq:eqtldeCardSeuil}
\mathbb{E}_{X}\left[|\widetilde{\Gamma}_{\beta}(X)||\mathcal{D}_n\right] = \sum_{k = 1}^K \mathbb{P}_X\left(\hat{f}_k(X) \geq \widetilde{G}^{-1}(\beta)|\mathcal{D}_n\right) = \widetilde{G}
(\widetilde{G}^{-1}
(\beta)) = \beta.
\end{equation}
Therefore, we obtain $\mathbf{E}\left[|\widetilde{\Gamma}_{\beta}(X)|\right] = \beta$.
Also, we can write
\begin{eqnarray*}
\left|\mathbf{E}\left[\left|\hat{\Gamma}_{\beta}(X)\right|\right] - \beta \right| & \leq & \left|\mathbf{E}\left[|\hat{\Gamma}_{\beta}(X)| - |\widetilde{\Gamma}_{\beta}(X)|\right]\right|\\
& \leq & \left|\mathbf{E}\left[\sum_{k = 1}^K \left({\bf 1}_{\{\hat{G}(\hat{f}_k(X)) \leq \beta\}}
- {\bf 1}_{\{\widetilde{G}(\hat{f}_k(X)) \leq \beta\}}\right)\right] \right|\\
& \leq & \mathbf{E}\left[|\hat{\Gamma}_{\beta}(X) \; \Delta \; \widetilde{\Gamma}_{\beta}(X)|\right] \\
& \leq & \sum_{k = 1}^K \mathbf{E}\left[|{\bf 1}_{\{\hat{G}(\hat{f}_k(X)) \leq \beta\}}
- {\bf 1}_{\{\widetilde{G}(\hat{f}_k(X)) \leq \beta\}}|\right]\\
&\leq & \sum_{k=1}^K \mathbf{P}\left(|\hat{G}(\hat{f}_k(X) - \widetilde{G}(\hat{f}_k(X))| \geq |\widetilde{G}
(\hat{f}_k(X))
-\beta|\right).
\end{eqnarray*}
Hence, applying Lemma~\ref{lem:lemFondamental} in the above inequality, we obtain the desired result.
\subsection{Proof of Theorem~\ref{theo:vitesse1}}
When there is no doubt, we suppress the dependence on $X$.
First, let us state a intermediate result that is also needed to prove the theorem.
\begin{lm}
\label{lm:decompRisk}
Consider $\Gamma_{\hat f,\delta}$ the confidence set based on the score function $\hat f$ with information $\beta$ (that is, $G_{\hat{f}}(-\delta) = \beta$).
Under assumptions $M_{\alpha}^k$, the following holds
\begin{equation*}
\Delta \mathcal R(\Gamma_{\hat{f},\delta}) \leq C(\alpha,s)\left\{ K^{1-1/(s +\lambda -\lambda s)} \Delta R_{\phi}(\hat{f})^{1/(s+\lambda -\lambda s)}
+ K^{1-\lambda/(s+\lambda -\lambda s)} \Delta R_{\phi}(\hat{f})^{\lambda/(s+\lambda -\lambda s)} \right\},
\end{equation*}
where $\lambda = \frac{\alpha}{\alpha+1}$ and $C(\alpha,s)$ is non negative constant which depends only on $\alpha$ and $s$.
\end{lm}
\begin{proof}
For each $k = 1, \ldots, K$, we define the following events
$S_k = B_k^c \cap \{k\in (\Gamma_{\hat{f}, \delta} \, \Delta \, \Gamma_\beta^{*})\}$ and $T_k = B_k \cap \{k\in (\Gamma_{\hat{f}, \delta} \, \Delta \, \Gamma_\beta^{*})\}$, where the $B_k$'s are the events given in Eq.~\eqref{eq:proofSetBk}.
Now we observe that
\begin{equation*}
\Delta \mathcal R(\Gamma_{\hat{f},\delta}) =\mathbb{E}\left[\sum_{k =1}^K {\bf 1}_{S_k} |p_k - G^{-1}(\beta)|\right] + \mathbb{E}\left[\sum_{k =1}^K {\bf 1}_{T_k} |p_k - G^{-1}(\beta)|\right]
= \Delta \mathcal R_1(\Gamma_{\hat{f},\delta}) + \Delta \mathcal R_2(\Gamma_{\hat{f},\delta}),
\end{equation*}
where
\begin{eqnarray*}
\Delta \mathcal R_1(\Gamma_{\hat{f},\delta}) & = & \mathbb{E}\left[\sum_{k =1}^K {\bf 1}_{S_k} |p_k - G^{-1}(\beta)|\right], \\
\Delta \mathcal R_2(\Gamma_{\hat{f},\delta}) & = & \mathbb{E}\left[\sum_{k =1}^K {\bf 1}_{T_k} |p_k - G^{-1}(\beta)|\right].
\end{eqnarray*}
The end of the proof consists in controlling each of these two terms.
Let us first consider $\Delta \mathcal R_1(\Gamma_{\hat{f},\delta})$. For $\varepsilon > 0$, we have
\begin{eqnarray}
\label{eq:proofFinde}
\Delta \mathcal R_1(\Gamma_{\hat{f},\delta}) & = & \mathbb{E}\left[\sum_{k = 1}^K {\bf 1}_{S_k} |p_k - G^{-1}(\beta)| \left( {\bf 1}_{\{|p_k - G^{-1}(\beta)| \geq \varepsilon\}}
+ {\bf 1}_{\{|p_k - G^{-1}(\beta)| \leq \varepsilon\}} \right)\right] \nonumber \\
& \leq & \mathbb{E}\left[\sum_{k = 1}^K {\bf 1}_{S_k} \varepsilon^{1-s} |p_k - G^{-1}(\beta)|^s\right] + \varepsilon \sum_{k=1}^K \mathbb{P}(S_k) \nonumber \\
& \leq & C \varepsilon^{1-s} \Delta R_{\phi}(\hat{f}) + \varepsilon \sum_{k=1}^K \mathbb{P}(S_k),
\end{eqnarray}
where we used the assumption~\eqref{eq:Zhansgg} and more precisely~\eqref{eq:defout} to deduce the last inequality.
To control $\sum_{k=1}^K \mathbb{P}(S_k)$,
we require the following result that is a direct application of Lemma~5 in~\cite{BartlettJordanMcauliffe06}.
\begin{lm}
\label{lm:margin}
Under the assumptions $M_{\alpha}^k$ we have
\begin{equation*}
\sum_{k =1}^K \mathbb{P}\left(S_k\right) \leq
C(\alpha) \left(K^{1/\alpha} \Delta \mathcal R_1(\Gamma_{\hat f,\delta}) \right)^{\frac{\alpha}{\alpha+1}},
\end{equation*}
where $ C(\alpha) >0$ is a constant that depends only on $\alpha$.
\end{lm}
\begin{proof}
The proof of this result relies on the following simple fact: for all $\varepsilon > 0$
\begin{eqnarray*}
\mathbb{E}\left[{\bf 1}_{S_k} |p_k - G^{-1}(\beta)|\right] & \geq &
\varepsilon \mathbb{E}\left[{\bf 1}_{S_k} {\bf 1}_{\{|p_k - G^{-1}(\beta)| \geq \varepsilon\}} \right]\\
& \geq & \varepsilon \left[\mathbb{P}\left(S_k\right) - c_2 \varepsilon^{\alpha} \right].
\end{eqnarray*}
Choosing $\varepsilon = \left(\frac{1}{c_2K(\alpha + 1)}\sum_{k =1}^K \mathbb{P}\left(S_k)\right) \right)^{1/\alpha}$ we get the lemma.
\end{proof}
We go back to the proof of Lemma~\ref{lm:decompRisk}.
Applying Lemma~\ref{lm:margin} to~\eqref{eq:proofFinde}, we get
\begin{equation*}
\Delta \mathcal R_1(\Gamma_{\hat f,\delta}) \leq C\varepsilon^{1-s} \Delta R_{\phi}(\hat f) + \varepsilon C(\alpha) \left(K^{1/\alpha} \Delta \mathcal R_1(\Gamma_{\hat f,\delta}) \right)^{\frac{\alpha}
{\alpha+1}}.
\end{equation*}
Choosing $\varepsilon = \frac{s-1}{s C(\alpha)} K^{(\lambda - 1)} \Delta \mathcal R_1(\Gamma_{\hat f,\delta})^{(1-\lambda)}$, we obtain
\begin{equation}
\label{eq:eqlmMajoration}
\Delta \mathcal R_1(\Gamma_{\hat f,\delta}) \leq C_1(\alpha, s ) K^{1-1/(s-\lambda s +\lambda)} \Delta R_{\phi}(\hat f)^{1/(s+\lambda -\lambda s)},
\end{equation}
for a non negative constant $C_1(\alpha, s )$ that depends only on $\alpha$ and $ s$.
Let us now focus on the second term, $ \Delta \mathcal R_2(\Gamma_{\hat f,\delta})$.
Since the assumptions of Theorem~\ref{theo:theo1} are satisfied, we can use Eq.~\eqref{eq:eqMajoration} for any $\gamma > 0$.
Combined with the Margin assumptions $M_\alpha^k$, we obtain
\begin{eqnarray*}
\Delta \mathcal R_2(\Gamma_{\hat f,\delta}) & \leq & \mathbb{E}_{X}\left[|G_{f^{*}}(-\delta^{*}) -G_{{\hat f}}(-\delta^{*})|\right]\\
& \leq & \frac{C\Delta R_{\phi}({\hat f})}{\gamma} + c_2K\gamma^{\alpha/s}.
\end{eqnarray*}
Therefore, optimizing in $\gamma$, we have
\begin{equation*}
\Delta \mathcal R_2(\Gamma_{\hat f,\delta}) \leq C_2(\alpha,s) K^{1-\lambda/(s+\lambda -\lambda s)} \Delta R_{\phi}(\hat f)^{\lambda/(s+\lambda - \lambda s)},
\end{equation*}
for a non negative constant $C_2(\alpha, s )$ that depends only on $\alpha$ and $ s$.
The result stated in the lemma is deduced by combining the last equation with Eq.~\eqref{eq:eqlmMajoration} and by setting $C(\alpha,s)= \max\{ C_1(\alpha,s) ;C_2(\alpha,s)\}$.
\end{proof}
We now state another important lemma that describes the behavior of the empirical minimizer of the $\phi$-risk on the class $\mathcal F$:
\begin{lm}
\label{lm:estimPhiRisk}
Let $\bar{f} \in \mathcal{F}$ be the minimizer of $R_{\phi}(f)$ over $\mathcal{F}$.
Under the assumptions of Theorem~\ref{theo:vitesse1}, we have that
\begin{equation*}
\mathbf{E}\left[R_{\phi}(\hat{f_n}) -R_{\phi}(\bar{f})\right] \leq \frac{3KL}{n} + \frac{K C(B,L) \log(N_n)}{n},
\end{equation*}
where $C(B,L)>0$ is a constant that only depends on the constant $B$ given in the Proposition~\ref{prp:crossVal} and on the Lipschitz constant $L$.
\end{lm}
\begin{proof}
First, according to Eq.~\eqref{eq:deltaMod}, for each $f \in \mathcal{F}$, we can write
\begin{equation*}
\frac{R_{\phi}(f)+R_{\phi}(\bar{f})}{2} - R_{\phi}\left(\frac{f+\bar{f}}{2}\right) \geq \delta\left(\sqrt{\sum_{k=1}^K\mathbb{E}_{X}\left[(f_k-\bar{f}_k)^2(X)\right]}\right),
\end{equation*}
Hence, by assumption on the modulus of convexity, we deduce
\begin{equation*}
\frac{R_{\phi}(f)+R_{\phi}(\bar{f})}{2} - R_{\phi}\left(\frac{f+\bar{f}}{2}\right) \geq c_1 \sum_{k=1}^K\mathbb{E}_{X}\left[(f_k-\bar{f}_k)^2(X)\right].
\end{equation*}
Since $R_{\phi}\left(\frac{f+\bar{f}}{2}\right) \geq R_{\phi}(\bar{f})$, we obtain
\begin{equation*}
\sum_{k=1}^K\mathbb{E}_{X}\left[(f_k-\bar{f}_k)^2(X)\right] \leq \frac{1}{2c_1} R_{\phi}(f) - R_{\phi}(\bar{f}).
\end{equation*}
Now, denoting by $h(z,f(x)) = \sum_{k = 1}^ K \phi(z_kf_k(x)) - \phi(z_k\bar{f}_k(x))$, we get the following bound
\begin{eqnarray}
\label{eq:eqvarB}
\mathbb{E}_X\left[h^2(Zf(X))\right] & \leq & K L^2 \sum_{k=1}^K\mathbb{E}_{X}\left[(f_k-\bar{f}_k)^2(X)\right] \nonumber \\
& \leq & \frac{K L^2}{2c_1} \mathbb{E}_X\left[h(Zf(X))\right],
\end{eqnarray}
where $L$ is the Lipschitz constant $L$ for $\phi$.
On the other hand, we have the following decomposition
\begin{equation*}
R_{\phi}(\hat{f}) - R_{\phi}(\bar{f}) = R_{\phi}(\hat{f}) + 2(\hat{R}_{\phi}(\hat{f}) - \hat{R}_{\phi}(\bar{f})) - 2 (\hat{R}_{\phi}(\hat{f}) - \hat{R}_{\phi}(\bar{f}))
- R_{\phi}(\bar{f}).
\end{equation*}
Also, since $\hat{R}_{\phi}(\hat{f}) - \hat{R}_{\phi}(\bar{f}) \leq 0$, we get
\begin{eqnarray*}
R_{\phi}(\hat{f}) - R_{\phi}(\bar{f}) & \leq & (R_{\phi}(\hat{f}) - R_{\phi}(\bar{f})) - 2 (\hat{R}_{\phi}(\hat{f}) - \hat{R}_{\phi}(\bar{f})) \\
& \leq & \frac{3KL}{n} + \sup_{f \in \mathcal{F}_n} (R_{\phi}({f}) - R_{\phi}(\bar{f})) - 2 (\hat{R}_{\phi}({f}) - \hat{R}_{\phi}(\bar{f})),
\end{eqnarray*}
where $\mathcal{F}_n$ is the $\epsilon$-net of $\mathcal{F}$ w.r.t the $L_\infty$-norm and with $\epsilon = 1/n$.
Now, using Bernstein's Inequality, we have that for all $f \in \mathcal{F}_n$ and $t > 0$
\begin{multline*}
\mathbf{P}\left((R_{\phi}({f}) - R_{\phi}(\bar{f})) - 2 (\hat{R}_{\phi}({f}) - \hat{R}_{\phi}(\bar{f})) \geq t\right) \leq \\
\mathbf{P}\left(2((R_{\phi}({f}) - R_{\phi}(\bar{f})) - (\hat{R}_{\phi}({f}) -\hat{R}_{\phi}(\bar{f})) \geq t + R_{\phi}({f}) - R_{\phi}(\bar{f})\right)\\
\leq \exp\left(-\dfrac{n(t+\mathbb{E}\left[h(Z,f(X))\right])^2/8}{\mathbb{E}\left[h^2(Z,f(X))\right])+(2KLB/3)(t+\mathbb{E}\left[h(Z,f(X))\right]))}\right).
\end{multline*}
Using Eq.~\eqref{eq:eqvarB}, we get for all $f \in \mathcal{F}_n$
\begin{equation*}
\mathbf{P}\left((R_{\phi}({f}) - R_{\phi}(\bar{f})) - 2 (\hat{R}_{\phi}({f}) - \hat{R}_{\phi}(\bar{f})) \geq t\right)
\leq \exp\left(-\frac{nt}{8(KL^2/(2c_1) + KLB/3)}\right),
\end{equation*}
Therefore, using a union bound argument, and then integrating we deduce that
\begin{eqnarray*}
\mathbf{E}\left[R_{\phi}(\hat{f}) - R_{\phi}(\bar{f})\right] &\leq & \frac{3KL}{n} + \mathbf{E}\left[\sup_{f \in \mathcal{F}_n} (R_{\phi}({f}) - R_{\phi}(\bar{f})) - 2
(\hat{R}_{\phi}({f}) - \hat{R}_{\phi}(\bar{f}))\right]\\
& \leq & \frac{3KL}{n} + \frac{K C( B,L) \log(M_n)}{n}.
\end{eqnarray*}
\end{proof}
We are now ready to conclude the proof of the theorem.
We have the following inequality
\begin{equation}
\label{eq:eqDecomp}
| \mathcal R(\hat{\Gamma}_{\beta}) - \mathcal R_{\beta}^* |\leq \Delta \mathcal R(\widetilde{\Gamma}_{\beta})
+ |\mathcal R(\hat{\Gamma}_{\beta})-\mathcal R(\widetilde{\Gamma}_{\beta})|.
\end{equation}
We deal which each terms in the r.h.s separately.
First, we have from Jensen's Inequality that
\begin{equation*}
\left(\mathbf{E}\left[\Delta \mathcal R(\widetilde{\Gamma}_{\beta})\right]\right)^{\frac{s+\lambda-\lambda s}{\lambda}}
\leq \mathbf{E}\left[\Delta\mathcal R(\widetilde{\Gamma}_{\beta})^{\frac{s+\lambda-\lambda s}{\lambda}}\right].
\end{equation*}
Hence, from Lemma~\ref{lm:decompRisk}, we deduce
\begin{equation*}
\left(\mathbf{E}\left[\Delta \mathcal R(\widetilde{\Gamma}_{\beta})\right]\right)^{\frac{s+\lambda-\lambda s}{\lambda}}
\leq C(\alpha,s)^{\frac{s+\lambda-\lambda s}{\lambda}}K^{\frac{s+\lambda-\lambda s}{\lambda}-1}
\mathbf{E}\left[\Delta R_{\phi}(\hat{f})\right].
\end{equation*}
Moreover, from Lemma~\ref{lm:estimPhiRisk}, we have that
\begin{equation*}
\mathbf{E}\left[\Delta R_{\phi}(\hat{f})\right] \leq \inf_{f \in \mathcal{F}}\Delta R_{\phi}(f) +
\frac{3KL}{n} + \frac{K C( B,L) \log(M_n)}{n}.
\end{equation*}
Therefore, we can write
\begin{equation}
\label{eq:eqIneq1}
\mathbf{E}\left[\Delta \mathcal{R}(\widetilde{\Gamma}_{\beta})\right] \leq C(\alpha,s) K^{1-\lambda/(s+\lambda -\lambda s)}
\left\{ \inf_{f \in \mathcal{F}}\Delta R_{\phi}(f) + \frac{3KL}{n} + \frac{K C( B,L) \log(N_n)}{n} \right\}^{\lambda/(s+\lambda -\lambda s)}.
\end{equation}
For the second term $|\mathcal R(\hat{\Gamma}_{\beta})-\mathcal R(\widetilde{\Gamma}_{\beta})|$ in~\eqref{eq:eqDecomp}, we observe that
\begin{equation*}
{\bf 1}_{\{Y \notin \hat{\Gamma}_\beta(X) \}} - {\bf 1}_{\{Y \notin \widetilde{\Gamma}_\beta(X) \}}
=
\sum_{k =1}^{K} {\bf 1}_{ \left\{ Y = k\right\} } {\bf 1}_{ \left\{k \notin \hat{\Gamma}_\beta (X)\right\} } - \sum_{k =1}^{K}
{\bf 1}_{ \left\{ Y = k \right\} } {\bf 1}_{ \left\{ k \notin \widetilde{\Gamma}_\beta(X) \right\}}.
\end{equation*}
Therefore, we can write
\begin{eqnarray*}
\mathbf{E}\left[{\bf 1}_{Y \notin \hat{\Gamma}_\beta(X) \}} - {\bf 1}_{\{Y \notin \widetilde{\Gamma}_\beta(X) \}} \right]
& = & \sum_{k = 1}^K \mathbf{E}\left[{p}_k(X)\left( {\bf 1}_{ \left\{k \notin \hat{\Gamma}_\beta(X)\right\} } - {\bf 1}_{\{k \notin \widetilde{\Gamma}_\beta(X)\}}\right)\right]\\
& = & \sum_{k = 1}^K \mathbf{E}\left[{p}_k(X)\left( {\bf 1}_{ \left\{\hat{G}(\hat{f}_k(X)) > \beta\right\} } - {\bf 1}_{\{\tilde{G}(\hat{f}_k(X)) > \beta\}}\right)\right].
\end{eqnarray*}
Since $0\leq {p}_k (X) \leq 1$ for all $k \in \{1,\ldots,K\}$, the last equality implies
\begin{multline*}
\left|\mathbf{R}\left(\hat{\Gamma}_{\beta}\right) - \mathbf{R}\left(\tilde{\Gamma}_{\beta}\right)\right| = \left|\mathbf{E}\left[{\bf 1}_{Y \notin \hat{\Gamma}_\beta(X) \}} -
{\bf 1}_{\{Y \notin \widetilde{\Gamma}_\beta(X) \}} \right]\right| \\ \leq
\sum_{k = 1}^K \mathbf{E}\left[\left|{\bf 1}_{ \left\{\hat{G}(\hat{f}_k(X)) > \beta\right\} } - {\bf 1}_{\{\tilde{G}(\hat{f}_k(X)) > \beta\}}\right|\right] \\ \leq
\sum_{k=1}^K \mathbf{P}\left(|\hat{G}(\hat{f}_k(X) - \widetilde{G}(\hat{f}_k(X))| \geq |\widetilde{G}(\hat{f}_k(X)) -\beta|\right).
\end{multline*}
Therefore, Lemma~\ref{lem:lemFondamental} implies
\begin{equation}
\label{eq:eqIneq2}
\left|\mathbf{R}\left(\hat{\Gamma}_{\beta}\right) - \mathbf{R}\left(\tilde{\Gamma}_{\beta}\right)\right| \leq \dfrac{C' K}{\sqrt{N}}.
\end{equation}
Injecting Eqs.~\eqref{eq:eqIneq1} and~\eqref{eq:eqIneq2} to Eq.~\eqref{eq:eqDecomp}
we conclude the proof of the theorem.
\subsection{Proof of Proposition~\ref{prp:crossVal}}
We begin with the following decomposition
\begin{equation*}
\tilde{R}^n_{\phi}(\hat{f}) - \tilde{R}^n_{\phi}(\tilde{f}) = \tilde{R}^n_{\phi}(\hat{f}) + 2(\hat{R}^n_{\phi}(\hat{f}) - \hat{R}^n_{\phi}(\tilde{f})) - 2 (\hat{R}^n_{\phi}(\hat{f}) - \hat{R}^n_{\phi}(\tilde{f}))
- \tilde{R}^n_{\phi}(\tilde{f}),
\end{equation*}
since $\hat{R}^n_{\phi}(\hat{f}) - \hat{R}^n_{\phi}(\tilde{f}) \leq 0$, we get
\begin{equation}
\label{eq:eqCrossValDecomp}
\tilde{R}^n_{\phi}(\hat{f}) - \tilde{R}^n_{\phi}(\tilde{f}) \leq (\tilde{R}^n_{\phi}(\hat{f}) - \tilde{R}^n_{\phi}(\tilde{f})) - 2 (\hat{R}^n_{\phi}(\hat{f}) - \hat{R}^n_{\phi}(\tilde{f})).
\end{equation}
Now, we denote by $\mathcal{C}_n = \{(i_1/n, \ldots, i_M/n), (i_1,\ldots ,i_M) \in \{0, \ldots ,n\}\} \cap \mathcal{C}_M$,\\
and $\mathcal{F}_n = \{ f = \sum_{i = 1}^M m_i f_i, \;\; m_1,\ldots m_M) \in \mathcal{C}_n\}$.
For each $f \in {\rm conv}(\mathcal{F})$, there exists $f_n \in \mathcal{F}_n$ such that
\begin{eqnarray*}
|\tilde{R}^n_{\phi}(f)- \tilde{R}^n_{\phi}(f_n)| & \leq & \frac{2KLBM}{n} \\
|\hat{R}^n_{\phi}(f)- \hat{R}^n_{\phi}(f_n)| & \leq & \frac{2KLBM}{n}.
\end{eqnarray*}
Therefore, with Equation~\eqref{eq:eqCrossValDecomp}, we obtain
\begin{equation*}
\tilde{R}^n_{\phi}(\hat{f}) - \tilde{R}^n_{\phi}(\tilde{f}) \leq \frac{6LKB}{n} + \sup_{f \in \mathcal{F}_n} (\tilde{R}^n_{\phi}({f}) - \tilde{R}^n_{\phi}(\bar{f})) - 2 (\hat{R}^n_{\phi}({f}) - \hat{R}^n_{\phi}(\bar{f})).
\end{equation*}
Now, Similar arguments as in~\cite{DudoitvdLaan} and Lemma~\ref{lm:estimPhiRisk} yield the proposition.
\bibliographystyle{alpha}
| -93,948.829671 |
[
-3.125,
2.90234375
] | 35.697941 |
[
-2.744140625,
0.59375,
-2.111328125,
-5.23046875,
-0.568359375,
7.48046875
] |
[
2.4140625,
6.1875,
0.1749267578125,
7.25
] | 492 | 9,044 |
[
-3.357421875,
4.00390625
] | 37.347052 |
[
-5.99609375,
-4.75390625,
-5.1640625,
-2.109375,
2.5390625,
13.4140625
] | 0.83227 | 12.718681 | 19.648386 | 1.658254 |
[
2.207533836364746
] | -55,624.144674 | 5.978439 | -93,848.671363 | 0.236735 | 6.114688 |
[
-2.1796875,
-3.4375,
-3.6015625,
-4.9296875,
2.291015625,
12.03125
] |
[
-5.61328125,
-2.369140625,
-1.947265625,
-1.3837890625,
3.8046875,
5.03125
] | |
BkiUdeo5qX_BnmUhN6EM
|
\section*{Introduction}
Ever since their inception, Cuntz-Krieger algebras \cite{CuntzKrieger1980} and graph \cstar-algebras \cite{KumjianPaskRaeburn1998} form natural classes of nuclear \cstar-algebras satisfying the UCT \cite{RosenbergSchochet1987}, which keep attracting much attention.
One compelling feature is that many of these \cstar-algebras' structural properties have natural characterizations in terms of their matrices and graphs, respectively.
This feature makes them particularly approachable from the point of view of classification, as is evidenced for example by R\o rdam's work \cite{Rordam1995} on simple Cuntz-Krieger algebras.
More recently, the classification of unital graph \cstar-algebras in purely graph theoretic terms has been completed by Eilers-Restorff-Ruiz-S{\o}rensen; see \cite{EilersRestorffRuizSorensen2016}.
This paper deals with actions of finite abelian groups on Cuntz-Krieger algebras by quasi-free automorphisms in the sense of \cite{Zacharias2000} or \cite{Evans1980}.
This class of automorphisms contains the ones that permute the canonical generating partial isometries, as well as automorphisms for which each of these partial isometries is an eigenvector.
The latter ones fall into the class of diagonal quasi-free automorphisms, as they fix all range projections of the generating partial isometries.
Every action by diagonal quasi-free automorphisms is conjugate to one that is given by automorphisms for which the canonical partial isometries are eigenvectors; see Proposition~\ref{prop:charac quasi-free diagonal autos}.
In particular, diagonal quasi-free automorphisms are always homotopic to the identity map.
Therefore they are approximately inner, provided the Cuntz-Krieger algebra is simple, in which case it must be a unital Kirchberg algebra; see \cite{Rordam1995a, Phillips2000}.
This is in contrast to automorphisms induced by permutations on the set of the canonical generators.
In general, such automorphisms act non-trivially on the $K_0$-group of the Cuntz-Krieger algebra, as it is generated by the classes of domain projections of the generating partial isometries; see \cite{Cuntz1981a}.
One might now ask the following question.
\begin{questionoz} \label{ques:intro}
Is every outer action of a finite abelian group by outer diagonal quasi-free automorphisms on a simple, purely infinite Cuntz-Krieger algebra strongly approximately inner in the sense of Izumi \cite{Izumi2004}?
\end{questionoz}
Strong approximate innerness requires that unitaries witnessing approximate innerness can be chosen to lie in the fixed point algebra of the action. If the unitaries can in addition be chosen to give rise to (approximate) group representations, then the action is called approximately representable.
Although in general different, these notions coincide if the action absorbs the trivial action on a suitable UHF algebra; see \cite{Izumi2004} and \cite{Szabo2017}.
On the one hand, the key feature of these technically looking properties is the duality between approximate representability and the Rokhlin property.
As illustrated beautifully by Izumi \cite{Izumi2004,Izumi2004a}, this relationship paired with the rigid nature of Rokhlin actions make approximately representable actions particularly accessible to classification via their duals.
For strongly approximately inner actions of cyclic groups of prime power order on $\CO_2$, this perspective was successfully exploited by Izumi, and was even accompanied by a complete range result.
On the other hand, models for actions with the Rokhlin property such as in \cite{BarlakSzabo2017} together with Takai duality \cite{Takai1975} can be used to infer models or structural properties for approximately representable actions.
Along these lines, a characterization of the UCT for the fixed point algebra associated to a $M_{n^\infty}$-absorbing action $\alpha: \IZ_n \curvearrowright A$ on a unital UCT Kirchberg algebra has recently been shown by the first author in joint work with Xin Li; see \cite{BarlakLi2017}. More concretely, $A^\alpha$ satisfies the UCT if and only if there exists an inverse semigroup of $\alpha$-homogeneous partial isometries $\CS \subset A$\footnote{This means that $\CS$ is closed under multiplication and the $*$-operation.} such that $\CS$ generates $A$ as a \cstar-algebra and the projections in $\CS$ generate a Cartan subalgebra of $A$ in the sense of Renault \cite{Renault2008}.
Restricted to Cuntz-Krieger algebras, our question above therefore raises the issue to which extend the latter condition already implies approximate representabilty.
Indeed, the canonical partial isometries of a Cuntz-Krieger algebra generate an inverse semigroup whose projections in turn generate an abelian \cstar-subalgebra.
It is a Cartan subalgebra precisely when the matrix satisfies Condition (I) (see \cite{Renault2008} and \cite{CuntzKrieger1980}), which is equivalent to its spectrum being the Cantor set.
Our main result asserts that the above question indeed has an affirmatively answer if the matrix is aperiodic, that is, if some power of the matrix has only strictly positive entries.
\begin{theoremoz} [see Corollary {\ref{cor:main result}}]
Let $n \geq 2$ be an integer.
Let $A$ be an aperiodic $n \times n$-matrix with values in $\set{0,1}$.
Let $G$ be a finite abelian group and let $\sigma:G \curvearrowright \CO_A$ be an action by diagonal quasi-free automorphisms.
If $\sigma$ is outer, then it is strongly approximately inner.
\end{theoremoz}
As a consequence, one gets an analogous result for unital graph \cstar-algebras. The conditions we have to impose on the finite graphs are strong connectedness and aperiodicity. Here, a graph is said to be strongly connected aperiodic if there is some $k \geq 1$ such that for any two of its edges $v,w$ there is some path of length $k$ from $v$ to $w$. The notion of (diagonal) quasi-free automorphisms transfers to the setting of unital graph \cstar-algebras in a straightforward manner.
\begin{theoremoz} [see Corollary {\ref{cor:main result graphs}}]
Let $E$ be a finite graph that contains at least two edges.
Let $G$ be a finite abelian group and let $\sigma:G \curvearrowright \cstar(E)$ be an action by diagonal quasi-free automorphisms on the associated graph \cstar-algebra.
Suppose that $E$ is strongly connected aperiodic.
If $\sigma$ is outer, then it is strongly approximately inner.
\end{theoremoz}
Our main result generalizes Izumi's \cite[Proposition~5.6(2)]{Izumi2004} for actions on the Cuntz algebras by quasi-free automorphisms.
The argument in his work is a variant of R{\o}rdam's proof \cite{Rordam1993} that unital endomorphisms of Cuntz algebras are approximately inner.
A crucial ingredient therein is the Rokhlin property of the one-sided tensorial shift on the CAR algebra; see \cite{BratteliKishimotoRordamStormer1993}.
In the proof, this is used to derive an approximate cohomology vanishing-type result for the canonical endomorphism restricted to the fixed point algebra of the canonical UHF subalgebra.
Strong approximate innerness of the quasi-free action is then derived from the well-known bijective correspondence between unitaries and unital endomorphisms of Cuntz algebras; see \cite{Cuntz1980}.
Our approach is of a similar nature.
Although less striking than it is for Cuntz algebras, there is still a close connection between certain unitaries and unital endomorphisms of Cuntz-Krieger algebras; see \cite{Zacharias2000} and \cite{ContiHongSzymanski2012}.
In particular, one can associate to each diagonal quasi-free automorphism a unique unitary so that the automorphism is given on the canonical partial isometries by left multiplication with this unitary.
The shift map of a Cuntz-Krieger algebra commutes with any given action by diagonal quasi-free automorphisms for which the canonical partial isometries are eigenvectors.
In general, this map may only be a unital completely positive map, but it restricts to a $*$-homomorphism on the relative commutant of the domain projections of the canonical generators inside the fixed point algebra of the action.
This resulting endomorphism is injective.
As it is a priori unclear whether this endomorphism has the Rokhlin property (in the sense of \cite{BrownHirshberg2014} or \cite{Rordam1995a}), our approach deviates at this point from Izumi's original approach and instead considers the dilation of the shift map to an automorphism in the sense of \cite{Laca2000}.
Aperiodicity of the matrix enters the game here to make sure that the dilated \cstar-algebra is a unital Kirchberg algebra and the automorphism is aperiodic, provided that the action is outer; see Lemma~\ref{outerness dilated auto} and \cite{Kishimoto1981, Jeong1995, TomsWinter2007}.
A classical theorem of Nakamura \cite{Nakamura2000} then yields that this automorphism has the Rokhlin property.
As the dilated system embeds into the ultrapower of the shift endomorphism, a modified cohomology vanishing-type technique for the unitaries associated with the action can be performed.
Strong approximate innerness of the involved automorphisms then follows by going back from unitaries to unital endomorphisms.
In fact, a technically more involved argument yields strong approximate innerness for an a priori larger class of actions; see Theorem~\ref{main theorem}.
A consequence of our main result and Izumi's classification result \cite{Izumi2004a} is that outer actions of cyclic groups with prime power order by diagonal quasi-free automorphisms on Cuntz-Krieger algebras isomorphic to $\CO_2$ are classified in terms of their fixed point algebras and some additional information about their dual actions, provided the associated matrices are aperiodic.
In particular this holds for any (possibly non-standard) identification of $\CO_2$ with a Cuntz-Krieger algebra.
Using Kirchberg-Phillips classification \cite{Kirchberg, Phillips2000}, this simplifies as follows in the case of order two automorphisms.
\begin{coroz}[see Corollary {\ref{cor:classification}}]
Let $m, n \geq 2$.
Let $A$ be an $m \times m$-matrix and $B$ an $n \times n$-matrix with entries in $\set{0,1}$ such that $\CO_A\cong\CO_B\cong\CO_2$ as abstract \cstar-algebras.
Let $\alpha:\IZ_2 \curvearrowright \CO_A$ and $\beta:\IZ_2 \curvearrowright \CO_B$ be two actions by diagonal quasi-free automorphisms.
Suppose that that $A$ and $B$ are aperiodic and that both $\alpha$ and $\beta$ are outer.
Then $\alpha$ and $\beta$ are (cocycle) conjugate if and only if $\CO_A^\alpha$ and $\CO_B^\beta$ are (stably) isomorphic
\end{coroz}
\bigskip
\textbf{Acknowledgement.}
Substantial parts of this work were carried out during a research visit of the second author to the University of Southern Denmark in January 2017 and during a research visit of the first author to the University of Aberdeen in July 2017.
The authors are grateful to both institutions for their hospitality and support.
The first author would like to thank Wojciech Szyma{\'n}ski for inspiring discussions on endomorphisms of Cuntz-Krieger algebras and Rokhlin-type properties for endomorphisms. The authors are also grateful to Joachim Cuntz for valuable comments. The first author was supported by Villum Fonden project grant `Local and global structures of groups and their algebras' (2014-2018). The second author was supported by EPSRC grant EP/N00874X/1, the Danish National Research Foundation through the \emph{Centre for Symmetry and Deformation} (DNRF92), and the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement 746272.
\section{Preliminaries}
We start by recalling the definition of a Cuntz-Krieger algebra.
\begin{defi}[\cite{CuntzKrieger1980}] \label{defi CK algebras}
Let $n \geq 2$ be an integer. Let $A$ be an $n \times n$ matrix with entries in $\set{0,1}$ and no zero rows or columns. The Cuntz-Krieger algebra $\CO_A$ associated with $A$ is the universal \cstar-algebra generated by a family of partial isometries $\set{s_1,\ldots,s_n}$ subject to the relations
\begin{enumerate}[label=\textup{\arabic*)}, leftmargin=*]
\item $s_i^*s_j = 0$ \quad for $i \neq j$,\label{label:ck:1}
\item $s_i^*s_i = \sum_{j = 1}^n A(i,j) s_js_j^*$ \quad for $i=1,\ldots,n$.\label{label:ck:2}
\end{enumerate}
\end{defi}
\begin{nota} \label{nota: s_mu}
For $i \in \set{1,\ldots, n}$, we set $q_i = s_i^*s_i$ and $p_i = s_is_i^*$. One can check that $\CO_A$ is unital with $1 = \sum_{i = 1}^n p_i$.
If $\mu = (\mu_1,\ldots,\mu_k) \in \set{0,1}^k$ is a multi-index, then we set $s_\mu = s_{\mu_1}\ldots s_{\mu_k} \in \CO_A$. It holds that $s_\mu \neq 0$ if and only if $A(\mu_i,\mu_{i+1}) = 1$ for all $i\in \set{1,\ldots, k-1}$. We denote by $W^k$ the set of all multi-indices $\mu \in \set{0,1}^k$ such that $s_\mu \neq 0$.
\end{nota}
\begin{defi}
An $n \times n$-matrix $A$ with entries in $\set{0,1}$ is said to be aperiodic if there exists some $m \geq 1$ such that $A^m(i,j) > 0$ for all $i,j \in \set{1,\ldots,n}$.
\end{defi}
We shall later need the following well known result for Cuntz-Krieger algebras associated with aperiodic matrices over $\set{0,1}$.
\begin{theorem} \label{thm: Kirchberg aper mat}
Let $n\geq 2$.
Let $A$ be an aperiodic $n \times n$-matrix with entries in $\set{0,1}$.
Then $\CO_A$ is a unital Kirchberg algebra.
\end{theorem}
\begin{proof}
Clearly, $\CO_A$ is unital and separable. It follows from \cite[Proposition~1.6]{Cuntz1981} that $\CO_A$ is simple and purely infinite; see also \cite[Theorem~2.14]{CuntzKrieger1980}.
Furthermore, $\CO_A$ is nuclear, as it is stably isomorphic to a crossed product of an AF algebra by an automorphism; see \cite[Example~2.5]{Rordam1995a}.
\end{proof}
As shown in \cite{Cuntz1980}, there exists a canonical bijective correspondence between unital endomorphisms and unitaries of the Cuntz algebra $\CO_n$ for finite $n$.
A similar result was obtained in \cite{Zacharias2000} for Cuntz-Krieger-Pimsner algebras. In the case of Cuntz-Krieger algebras, such a correspondence exists for unital endomorphisms that fix all domain projections of the canonical generators and unitaries commuting with these projections; see also \cite{ContiHongSzymanski2012}.
\begin{prop} \label{prop:correspond graph alg endos}
Let $A$ be an $n \times n$-matrix with entries in $\set{0,1}$ and no zero rows or columns. For any unitary $u \in \CU(\CO_A) \cap \set{q_i \mid 1 \leq i \leq n}'$, there exists a unique unital $*$-endomorphism $\lambda_u$ of $\CO_A$ such that $\lambda_u(q_i) = q_i$ and $\lambda_u(s_i) = us_i$ for all $i \in \set{1,\ldots,n}$. Conversely, if $\sigma: \CO_A \to \CO_A$ is a unital $*$-endomorphism such that $\sigma(q_i) = q_i$ for all $i$, then $u_\sigma = \sum_{i = 1}^n \sigma(s_i)s_i^*$ is a unitary in $\CU(\CO_A) \cap \set{q_i \mid 1 \leq i \leq n}'$. These assignments are inverse to each other and continuous with respect to the norm and the point-norm topology, respectively.
\end{prop}
\begin{rem} \label{rem:conv formula}
It follows from Proposition~\ref{prop:correspond graph alg endos} that, given two unital endomorphisms $\rho,\sigma:\CO_A \to \CO_A$ fixing all domain projections of the canonical generators, the associated unitaries satisfy the convolution formula $u_{\sigma \circ \rho} = \sigma(u_\rho)u_\sigma$. In particular, for any $n\geq 1$, one has $\sigma^n = \id$ precisely when $\sigma^{n-1}(u_\sigma)\ldots\sigma(u_\sigma)u_\sigma = 1$.
\end{rem}
\begin{defi}[cf.\ {\cite{Zacharias2000}}] \label{defi:quasi-free}
Let $A$ be an $n \times n$-matrix with entries in $\set{0,1}$ and no zero rows or columns. An automorphism $\sigma \in \Aut(\CO_A)$ is said to be diagonal quasi-free if
\[
\begin{array}{c}
\sigma(q_i) = q_i \text{ for all } i \in \set{1,\ldots,n} \text{ and}\\
\sigma(\operatorname{span}\set{s_i \mid 1 \leq i \leq n}) = \operatorname{span}\set{s_i \mid 1 \leq i \leq n}.
\end{array}
\]
\end{defi}
We note that the above definition is slightly more general than the one in \cite{Zacharias2000}. In fact, Zacharias requires a diagonal quasi-free automorphism to restrict to the identity on the set of range projections of the canonical generators.
However, the two notions of diagonal quasi-free automorphisms lead to the same conjugacy classes of automorphisms, as the next result shows.
It is certainly known to experts, and partly contained in \cite{ContiHongSzymanski2012}. For the reader's convenience, we give a proof here.
\begin{prop} \label{prop:charac quasi-free diagonal autos}
Let $A$ be an $n \times n$-matrix with entries in $\set{0,1}$ and no zero rows or columns.
Consider the finite dimensional \cstar-subalgebra
\[
B = \operatorname{span}\big\{ s_is_j^* \mid 1 \leq i,j\leq n \big\}
\]
of $\CO_A$.
Then the following assertions hold:
\begin{enumerate}[label=\textup{\arabic*)}, leftmargin=*]
\item The bijective map in Proposition~\ref{prop:correspond graph alg endos} restricts to a group isomorphism between the set of diagonal quasi-free automorphisms and $\CU(B) \cap \set{q_i \mid 1 \leq i \leq n}'$. \label{label:cqf:1}
\item Let $G$ be an abelian group. Then every action $\sigma:G \curvearrowright \CO_A$ by diagonal quasi-free automorphisms is conjugate to an action $\rho:G \curvearrowright \CO_A$ with the property that for all $g \in G$ and $i \in \set{1,\ldots,n}$ there exists some $\eta_{g,i} \in \IT$ such that $\rho_g(s_i) = \eta_{g,i} s_i$. \label{label:cqf:2}
\end{enumerate}
\end{prop}
\begin{proof}
\ref{label:cqf:1}: Let $u,w \in \CU(B) \cap \set{q_i \mid 1 \leq i \leq n}'$. Using that $\lambda_u(w) = uwu^*$, one computes for $i \in \set{1,\ldots,n}$ that
\[
\lambda_u \lambda_w(s_i) = \lambda_u(w)us_i = uwu^*us_i = uws_i.
\]
Hence, $\lambda_{uw} = \lambda_u \lambda_w$. By taking $w = u^*$, it follows in particular that $\lambda_u$ is an automorphism. Moreover, if $u = \sum_{i,j =1}^n\eta_{i,j} s_i s_j^*$, then
\[
\lambda_u(s_j) = \sum_{i =1}^n \eta_{i,j} s_i.
\]
This shows that $\lambda_u$ is diagonal quasi-free. On the other hand, it is clear that $u_\sigma \in \CU(B) \cap \set{q_i \mid 1 \leq i \leq n}'$ if $\sigma \in \Aut(\CO_A)$ is diagonal quasi-free.
\ref{label:cqf:2}: Let $\sigma:G \curvearrowright \CO_A$ be an action of an abelian group by diagonal quasi-free automorphisms.
By \ref{label:cqf:1}, we find a unitary representation $u:G \to \CU(B) \cap \set{q_i \mid 1 \leq i \leq n}'$ such that $\sigma_g = \lambda_{u(g)}$ for all $g \in G$.
As $G$ is abelian, $u$ maps into some MASA of the finite dimensional \cstar-algebra $B \cap \set{q_i \mid 1 \leq i \leq n}'$.
Note that $\operatorname{span}\set{p_i \mid 1\leq i \leq n}$ is a MASA in this \cstar-algebra.
Using that every two MASAs in a finite dimensional \cstar-algebra are unitarily conjugate, we find some $w \in \CU(B) \cap \set{q_i \mid 1 \leq i \leq n}'$ such that
\[
wu(g)w^* \in \operatorname{span}\set{p_i \mid 1\leq i \leq n} \quad \text{for all } g \in G.
\]
It follows from \ref{label:cqf:1} that $\lambda_{wu(g)w^*} = \lambda_w \sigma_g \lambda_w^{-1}$ for all $g \in G$. Hence, the action $\rho:G \curvearrowright \CO_A$ given by $\rho_g = \lambda_{wu(g)w^*}$ is conjugate to $\sigma$. Furthermore, it is straighthforward to check that for $g \in G$ and $i \in \set{1,\ldots,n}$ there exists $\eta_{g,i} \in \IT$ such that $\rho_g(s_i) = \eta_{g,i} s_i$. This finishes the proof.
\end{proof}
\section{Main results}
\begin{nota}
We call an automorphism $\alpha$ on a unital \cstar-algebra $A$ aperiodic if $\alpha^k$ is outer for all $k \neq 0$.
For a free filter $\omega$ on $\IN$, we denote the $(\omega)$-sequence algebra of $A$ by
\[
A_\omega = \ell^\infty(\IN,A) \big / \set{(x_n)_n \in \ell^\infty(\IN,A) \mid \lim_{n \to \omega} \|x_n\| = 0}.
\]
If $\omega_\infty$ is the Fr{\'e}chet filter, we simply write $A_\infty = A_{\omega_\infty}$. There is a canonical embedding of $A$ into $A_\omega$ by (representatives of) constant sequences. Any endomorphism $\psi:A \to A$ induces endomorphisms $\psi_\omega:A_\omega \to A_\omega$ and $\psi_\infty:A_\infty \to A_\infty$. These assignments are functorial, so that any discrete group action $\alpha:G \curvearrowright A$ induces group actions $\alpha_\infty$ and $\alpha_\omega$ of $G$ on $A_\infty$ and $A_\omega$, respectively.
\end{nota}
\begin{prop} \label{outerness dilated auto}
Let $A$ be a separable, unital \cstar-algebra and $\psi: A \to A$ be a unital and injective $*$-homomorphism. Let
\[
(B, \bar{\psi}) = \lim_{\longrightarrow} \set{(A,\psi),\psi}
\]
denote its dilation to a $*$-automorphism; see \cite{Laca2000}. If $\psi$ is not an asymptotically inner $*$-automorphism\footnote{This means $\psi$ is a $*$-automorphism and there is no norm-continuous path $(u_t)_{t \in [0,\infty)}$ of unitaries in $A$ such that $\psi = \lim\limits_{t \to \infty} \ad(u_t)$.}, then $\bar{\psi}$ is outer.
In particular, for every $k \geq 1$ one has that if $\psi^k$ is not an asymptotically inner $*$-automorphism, then ${\bar{\psi}}^k$ is outer.
\end{prop}
\begin{proof}
By the standard construction of the inductive limit, $B$ arises as the closure of the $*$-algebra
\[
\set{[(a,\psi(a),\psi^2(a),\ldots)_{n \geq \ell}] \in A_\infty \mid a \in A,\ \ell \geq 1} \ \subset \ A_\infty,
\]
and on this dense subset the $*$-endomorphism $\bar{\psi}$ coincides with $\psi_\infty$.
Here the notation indicates that we only consider the tail of a representing sequence, which makes sense by definition of $A_\infty$.
Suppose $\bar{\psi}=\ad(v)$ for some unitary $v\in B$. By the definition of $B$, we can find a unitary\footnote{Note that the dense $*$-subalgebra of $B$ is closed under functional calculus.} $u \in A$ and $\ell \geq 1$ such that $\|v-\bar{u}\|<1/2$, where
\[
\bar{u} = [(u,\psi(u),\psi^2(u),\ldots)_{n \geq \ell}] \in B.
\]
Without loss of generality, let us assume $\ell=1$ here.
Let $a \in A$ with $\|a\|\leq 1$ and write $\bar{a} = [(\psi^{n-1}(a))_{n \geq 1}] \in A_\infty$.
We use that $\psi$ is injective and calculate
\[
\renewcommand\arraystretch{1.25}
\begin{array}{lcl}
\| \psi(a) - uau^* \| & = & \limsup\limits_{n \to \infty} \| \psi^n(\psi(a) - uau^*) \| \\
& = & \limsup\limits_{n \to \infty} \| \psi(\psi^n(a)) - \psi^n(u) \psi^n(a) \psi^n(u)^* \| \\
& = & \|\bar{\psi}(\bar{a})-\ad(\bar{u})(\bar{a})\| \\
& \leq & 2\|v-\bar{u}\| \ < \ 1.
\end{array}
\]
Hence, $\ad(u^*) \circ \psi$ has distance less than one in the operator norm to the identity operator on $A$. It thus follows from \cite[Theorem~8.7.7]{Pedersen1979} and \cite[Lemma~2.14 with proof]{DadarlatEilers2001} that $\psi$ is an asymptotically inner $*$-automorphism.
If $k \geq 1$ and ${\bar{\psi}}^k$ is inner, it now follows from the canonical isomorphism
\[
(B,{\bar{\psi}}^k) \cong \lim_{\longrightarrow} \set{(A,\psi^k),\psi^k}
\]
that $\psi^k$ is an asymptotically inner $*$-automorphism.
\end{proof}
\begin{rem} \label{rem:shift}
Let $A$ be an $n \times n$-matrix with entries in $\set{0,1}$ and no zero rows or columns. There exists a canonical unital completely positive map $\phi:\CO_A \to \CO_A$ given by $\phi(x) = \sum_{i =1}^n s_i x s_i^*$. Furthermore, it restricts to a $*$-endomorphism of $\CO_A \cap \set{q_i \mid 1 \leq i \leq n}'$. It is injective, as one finds for any $0\neq x \in \CO_A \cap \set{q_i \mid 1 \leq i \leq n}'$ some $j \in \set{1,\ldots,n}$ such that
\[
0 \neq xq_j = q_jxq_j,
\]
and consequently $0 \neq s_jx s_j^* = p_j \phi(x)$.
\end{rem}
\begin{lemma} \label{simplicity dilated system}
Let $n \geq 2$.
Let $A$ be an aperiodic $n \times n$-matrix with entries in $\set{0,1}$ and no zero rows or columns.
Let $C \subset \mathrm \CO_A$ be a simple \cstar-subalgebra containing $\set{q_i \mid 1\leq i \leq n}$.
Suppose that $\phi(C) \subset C$ and that the restricted $*$-endomorphism
\[
\phi:C \cap \set{q_i \mid 1\leq i \leq n}' \to C \cap \set{q_i \mid 1\leq i \leq n}'
\]
is not surjective. Denote its automorphic dilation by
\[
(B, \bar{\phi}) = \lim_{\longrightarrow} \set{(C \cap \set{q_i \mid 1\leq i \leq n}',\phi),\phi}.
\]
Then $B$ is a unital simple \cstar-algebra and $\bar{\phi}$ an aperiodic automorphism.
\end{lemma}
\begin{proof}
Observe that $\cstar(\set{q_i \mid 1\leq i \leq n}) \subset C$ is a commutative \cstar-sub\-algebra containing the unit of $\CO_A$. Hence, $C$ is unital and by Remark~\ref{rem:shift} it follows that
\[
\phi:C \cap \set{q_i \mid 1\leq i \leq n}' \to C \cap \set{q_i \mid 1\leq i \leq n}'
\]
is unital and injective, yielding that $B$ is unital as well. Moreover, aperiodicity of $\bar{\phi}$ follows immediately from Proposition~\ref{outerness dilated auto}.
It remains to show that $B$ is simple. Using Notation~\ref{nota: s_mu}, one computes for $k \geq 1$ and $x \in \CO_A$, that
\[
\phi^k(x) = \sum\limits_{\nu \in W^k} s_\nu x s_\nu^*.
\]
Let $\set{r_i \mid 1 \leq i \leq m}$ be the set of minimal projections in $\cstar(\set{q_i \mid 1\leq i \leq n})$. One has that
\[
C \cap \set{q_i \mid 1\leq i \leq n}' = \bigoplus_{i = 1}^m r_iCr_i.
\]
Now let $x \in C \cap \set{q_i \mid 1\leq i \leq n}'$ be a non-zero element.
Find $i \in \set{1,\ldots,m}$ such that $0 \neq xr_i = r_ixr_i$.
For any $k \geq 1$ and $j \in \set{1,\ldots,m}$ we have that
\begin{equation} \label{eq:shift^k}
r_j \phi^k(r_ixr_i)r_j = \sum_{ \nu \in W^k } r_j s_\nu r_i x r_i s_\nu^* r_j
= \sum_{ \nu \in W_{j,i}^k } s_\nu x s_\nu^*,
\end{equation}
where
\[
W_{j,i}^k = \set{\nu \in W^k \mid p_{\nu_1} \leq r_j,\ r_i \leq q_{\nu_k}}.
\]
Find $s,t \in \set{1,\ldots,n}$ such that $p_s \leq r_j$ and $r_i \leq q_t$.
As $A$ is aperiodic, there exists some $m_0 \geq 1$ such that $A^m(u,v) > 0$ for all $u,v \in \set{1,\ldots,n}$ and $m \geq m_0$.
In particular, we find for each $m \geq m_0$ some $\mu \in W^m$ with $\mu_1 = s$ and $\mu_m = t$.
It now follows from \eqref{eq:shift^k} that $0 \neq r_j\phi^m(r_ixr_i)r_j \in r_jCr_j $ for $m \geq m_0$, as
\[
s_\mu(r_j \phi^m(r_ixr_i)r_j)s_\mu^* = \sum_{\nu \in W_{j,i}^m } s^*_\mu s_\nu x s_\nu^*s_\mu = s_\mu^* s_\mu x s_\mu^* s_\mu = q_t x q_t \neq 0.
\]
As $C$ is simple, $r_j\phi^m(r_ixr_i)r_j \in r_jCr_j$ is a full element for all $j\in \set{1,\ldots,n}$ and $m \geq m_0$.
Hence, $\phi^m(x)$ is full in $C \cap \set{q_i \mid 1\leq i \leq n}'$ for any non-zero $x \in C \cap \set{q_i \mid 1\leq i \leq n}'$ and $m \geq m_0$.
We conclude that $B$ is simple. This finishes the proof.
\end{proof}
Next comes the main technical result of this paper. Before we state it, we need the following definition due to Izumi.
\begin{defi}[see {\cite[Definition~3.6]{Izumi2004}}]
Let $G$ be a finite abelian group. An action $\alpha: G \curvearrowright A$ on a separable, unital \cstar-algebra is said to be strongly approximately inner if for each $g \in G$ there exists a sequence of unitaries $\set{u_{g,n} \mid n \in \IN} \subset \CU(A^\alpha)$ such that $\alpha_g = \lim_{n \to \infty} \ad(u_{g,n})$.
\end{defi}
\begin{theorem} \label{main theorem}
Let $n \geq 2$.
Let $A$ be an aperiodic $n \times n$-matrix with values in $\set{0,1}$.
Let $G$ be a finite abelian group and let $\sigma: G \curvearrowright \CO_A$ be an action such that $\sigma_g(q_i) = q_i$ for all $g \in G$ and $i \in \set{1,\ldots,n}$.
For $g \in G$, denote by $u_{\sigma_g} \in \CO_A \cap \set{q_i \mid 1 \leq i \leq n}'$ the unitary associated with $\sigma_g$.
Assume furthermore that
\begin{enumerate}[label=\textup{\arabic*)}]
\item $\phi^k(u_{\sigma_g}) \in \CO_A^\sigma$ for all $g \in G$ and $k \geq 0$;
\item there exists a simple, nuclear \cstar-subalgebra $C \subset \CO_A^\sigma$ containing the set of domain projections $\set{q_i \mid 1 \leq i \leq n}$ such that $\phi(C) \subset C$ and the restricted $*$-endomorphism
\[
\phi:C \cap \set{q_i \mid 1 \leq i \leq n}' \to C \cap \set{q_i \mid 1 \leq i \leq n}'
\]
is not surjective.
\end{enumerate}
If $\sigma$ is outer, then it is strongly approximately inner.
\end{theorem}
\begin{proof}
Let $g \in G$. As $u_{\sigma_g}$ is fixed by $\sigma_g$ and $\sigma_g^n = \id$ for some $n \geq 1$, it follows from Remark~\ref{rem:conv formula} that
\[
u_{\sigma_g}^n = \sigma_g^{n-1}(u_{\sigma_g})\ldots\sigma_g(u_{\sigma_g})u_{\sigma_g} = 1.
\]
For $k \geq 0$, define unitaries $u_{g,k} \in \CO_A^\sigma \cap \set{q_i \mid 1 \leq i \leq n}'$ by
\[
u_{g,0} = 1,\ u_{g,1} = u_{\sigma_g} \text{ and } u_{g,k+1} = u_{g,\sigma}\phi(u_{g,\sigma}) \ldots \phi^k(u_{g,\sigma}).
\]
From the equality $\sigma_g \circ \phi = \ad(u_{\sigma_g}) \circ \phi \circ \sigma_g$, it follows that
\[
\sigma_g \circ \phi^k = \ad(u_{g,k}) \circ \phi^k \circ \sigma_g \quad \text{for all } k \geq 0.
\]
If $x \in \CO_A^{\sigma_g}$ and $\phi^k(x)$ are both fixed by $\sigma_g$, then this yields $[\phi^k(x),u_{g,k}] = 0$.
Using $\phi(C) \subset C\subset\CO_A^\sigma$ this shows that for $x \in C$ and $0 \leq i \leq k$,
\[
[\phi^k(x),u_{g,i}] = [\phi^i(\phi^{k-i}(x)),u_{g,i}] = 0.
\]
Thus, as $u_{g,i}^*u_{g,i + j} = \phi^i(u_{g,j})$ for all $i,j \in \IN$, it follows that
\[
\phi^k(x) \in C \cap \set{\phi^i(u_{g,j}),\ q_l \mid g \in G,\ 0 \leq j \leq k,\ 0 \leq i \leq k - j,\ 1 \leq l \leq n}'
\]
for all $x \in C \cap \set{q_i \mid 1 \leq i \leq n}'$ and $k \geq 1$.
In particular, $[\phi^k(u_{\sigma_g}),\phi^l(u_{\sigma_g})]= 0$ for all $k,l \geq 0$.
Hence, $u_{g,k}^n = 1$ for all $k \geq 0$ as $u_{\sigma_g}^n = 1$. Thus, each scalar in the spectrum of $u_{g,k}$ is an $n$-th root of unity.
Consider the automorphic dilation of $\phi$ on $C \cap \set{q_i \mid 1 \leq i \leq n}'$
\[
(B, \bar{\phi}) = \lim_{\longrightarrow} \set{(C \cap \set{q_i \mid 1 \leq i \leq n}',\phi),\phi}.
\]
By Lemma \ref{simplicity dilated system}, $B$ is a unital, simple, nuclear \cstar-algebra and $\bar{\phi}$ an aperiodic automorphism on $B$.
By the standard construction of the inductive limit, there exists a unital, injective and equivariant $*$-homomorphism
\[
\tilde{\rho}:(B,\bar{\phi}) \to \big( (C \cap \set{q_i \mid 1 \leq i \leq n}')_\infty,\phi_\infty \big)
\]
with the image being the closure of
\[
\set{[(x,\phi(x),\phi^2(x),\ldots)_{n \geq k}] \in C_\infty \mid x \in C \cap \set{q_i \mid 1 \leq i \leq n}',\ k\geq 1}.
\]
Observe that
\[
\tilde{\rho}(B) \subset (\CO_A^\sigma)_\infty \cap \set{\phi^i(u_{g,j}),\ q_k \mid g \in G,\ i,j \geq 0,\ 1\leq k \leq n}'.
\]
Let $\omega$ be a free ultrafilter on $\IN$.
As $\CO_A$ is a unital Kirchberg algebra by Theorem~\ref{thm: Kirchberg aper mat}, it follows from \cite[Proposition~3.4]{KirchbergPhillips2000} that $(\CO_A)_\omega \cap \CO_A'$ is unital, simple and purely infinite.
As $\sigma$ is a pointwise outer action, so is $\sigma_\omega$; see \cite[proof of Lemma~2]{Nakamura2000}. Hence, the fixed point algebra $((\CO_A)_\omega \cap \CO_A')^{\sigma_\omega}$ is unital, simple and purely infinite by \cite[Theorem 3.1]{Kishimoto1981} and \cite[Theorem~3]{Jeong1995}, and therefore admits a unital embedding of $\CO_\infty$. Note that $\phi_\omega$ acts trivially on $(\CO_A)_\omega \cap \CO_A'$.
A reindexation argument now shows the existence of a unital embedding
\[
\iota:\CO_\infty \to \big( (\CO_A)_\omega \cap (\rho(B)\cup \CO_A)' \big)^{\sigma_\omega},
\]
where $\rho(B) \subset (\CO_A)_\omega$ denotes the image of $\tilde{\rho}(B)$ under the canonical surjection $(\CO_A)_\infty \to (\CO_A)_\omega$.
Set $D = \mathrm C^*(\rho(B),\iota(\CO_\infty))$, which is a $\phi_\omega$-invariant \cstar-subalgebra of
\[
(\CO_A^\sigma)_\omega \cap \set{\phi^i(u_{g,j}),\ q_k \mid g\in G,\ i,j \geq 0,\ 1 \leq k \leq n}'
\]
containing the unit of $(\CO_A)_\omega$. In fact, the assignment $b\otimes c\mapsto \rho(b)\cdot\iota(c)$ yields an equivariant $*$-isomorphism
\[
(B \otimes \CO_\infty, \bar{\phi} \otimes \id) \to (D,\phi_\omega).
\]
Now $B \otimes \CO_\infty$ is a unital Kirchberg algebra and $\bar{\phi} \otimes \id$ is an aperiodic automorphism, so that $\bar{\phi} \otimes \id$ has the Rokhlin property by \cite[Theorem~1]{Nakamura2000}.
For given $r \geq 1$, it is thus possible to find projections $e_0,f_0,\ldots,e_{r-1},f_{r-1},f_r \in D$ such that
\begin{equation}\label{eq:Rok:1}
\sum\limits_{i=1}^{r-1}e_i + \sum\limits_{j=1}^r f_j = 1;
\end{equation}
\begin{equation} \label{eq:Rok:2}
\begin{split}
\phi_\omega(e_i) \approx e_{i+1} \text{ and } \phi_\omega(f_j) \approx f_{j+1} \text{ for all } i=0,\ldots,r-1 \\ \text{ and } j=1,\ldots,r,
\text{ where we set }e_r = e_0 \text{ and } f_{r+1} = f_0.
\end{split}
\end{equation}
By a reindexation trick, we may actually find such elements in
\[
(\CO_A^\sigma)_\omega \cap \set{\phi^i(u_{g,j}),\ q_k \mid g \in G,\ i,j \geq 0,\ 1 \leq k \leq n}'
\]
satisfying the second property \eqref{eq:Rok:2} on the nose.
Now fix $g \in G$. As $u_{g,r}$ and $u_{g,r+1}$ have finite spectrum, we find continuous maps
\[
z_g^{(i)}:[0,1] \to \CU\Big( (\CO_A^\sigma)_\omega \cap \set{e_0,f_0,\ldots,e_{r-1},f_{r-1},f_r,q_j \mid 1 \leq j \leq n}' \Big)
\]
for $i=0,1$ such that
\begin{equation} \label{eq:longPath:1}
z_g^{(0)}(0) = u_{g,r},\ z_g^{(1)}(0) = u_{g,r+1} \text{ and } z_g^{(0)}(1) = z_g^{(1)}(1) = 1;
\end{equation}
\begin{equation} \label{eq:longPath:2}
\begin{split}
\phi_\omega^k(z_g^{(i)}(s)) \in (\CO_A^\sigma)_\omega \cap \set{e_0,f_0,\ldots,e_{r-1},f_{r-1},f_r,q_j \mid 1 \leq j \leq n}' \\
\text{ for all } s \in [0,1],\ i=0,1 \text{ and } k \geq 0;
\end{split}
\end{equation}
\begin{equation} \label{eq:longPath:3}
\| z_g^{(i)}(s) - z_g^{(i)}(t) \| \leq 2\pi|s - t| \text{ for all } s,t \in [0,1] \text{ and } i=0,1.
\end{equation}
Define unitaries
\[
z^{(i)}_{g,j_i} = z_g^{(i)}(j_i/(r+i)) \quad \text{for }0 \leq j_i \leq r - 1 + i,\ i = 0,1.
\]
Then it follows from \eqref{eq:longPath:3} that
\begin{equation} \label{eq:longPath:4}
\| z^{(i)}_{g,j_i} - z^{(i)}_{g,j_i + 1} \| \leq 2\pi/r \text{ for all } 0 \leq j_i \leq r - 2 + i \text{ and } i = 0,1.
\end{equation}
Define a unitary $z_g \in \CU\Big( (\CO_A^\sigma)_\omega \cap \set{q_i \mid 1 \leq i \leq n}' \Big)$ by
\[
z_g = \sum_{k = 0}^{r - 1} u_{g,k} \phi_\omega^k (z^{(0)}_{g,k}) e_k + \sum_{l = 0}^r u_{g,l} \phi_\omega^l (z^{(1)}_{g,l}) f_l.
\]
One computes that
\[
\renewcommand\arraystretch{1.5}
\begin{array}{cl}
\multicolumn{2}{l}{ z_g\phi_\omega (z_g)^* } \\
\stackrel{\eqref{eq:Rok:1},\eqref{eq:Rok:2},\eqref{eq:longPath:1}}{=} & u_{g,r} e_0\phi_\omega(e_{r-1})\phi(u_{g,r-1})^* + u_{g,r+1} f_0\phi_\omega(f_r)\phi(u_{g,r})^* \\
& \dst + \sum_{k=1}^{r-1} u_{g,k} \phi_\omega^k (z^{(0)}_{g,k}) e_k \phi_\omega(e_{k-1}) \phi_\omega^k(z^{(0)}_{g,k-1})^*\phi(u_{g,k-1})^* \\
& \dst + \sum_{l=1}^{r} u_{g,l} \phi_\omega^l (z^{(1)}_{g,l}) f_l \phi_\omega(f_{l-1}) \phi_\omega^l(z^{(1)}_{g,l-1})^*\phi(u_{g,l-1})^* \\
\stackrel{\eqref{eq:Rok:2},\eqref{eq:longPath:2}}{=} & u_{g,r} \phi(u_{r-1})^* e_0 + u_{g,r+1} \phi(u_{g,r})^* f_0 \\
& \dst + \sum_{k=1}^{r-1} u_{g,k} \phi_\omega^k (z^{(0)}_{g,k}) \phi_\omega^k(z^{(0)}_{g,k-1})^*\phi(u_{g,k-1})^* e_k \\
& \dst + \sum_{l=1}^{r} u_{g,l} \phi_\omega^l (z^{(1)}_{g,l}) \phi_\omega^l(z^{(1)}_{g,l-1})^*\phi(u_{g,l-1})^* f_l \\
\stackrel{\eqref{eq:longPath:4}}{=}_{\makebox[0pt]{\footnotesize \hspace{2mm} $2\pi/r$}} & u_{g,r}\phi(u_{g,r-1})^*e_0 + u_{g,r+1}\phi(u_{g,r})^*f_0 \\
& \dst + \sum_{k=1}^{r-1} u_{g,k} \phi(u_{g,k-1})^*e_k \\
& \dst + \sum_{l=1}^{r} u_{g,l} \phi(u_{g,l-1})^*f_l \\
= & \dst \sum_{k=0}^{r-1} u_{\sigma_g} e_k + \sum_{l=0}^r u_{\sigma_g} f_l \\
\stackrel{\eqref{eq:Rok:1}}{=} & u_{\sigma_g}.
\end{array}
\]
Now let $\eps>0$ be given.
By choosing $r \in \IN$ so that $2\pi/r < \eps$, we may represent $z_g$ via unitaries and thus find a unitary
\[
w_g \in \CO_A^\sigma \cap \set{q_i \mid 1 \leq i \leq n}' \quad\text{with}\quad \|u_{\sigma_g} - w_g\phi(w_g)^* \| \leq \eps.
\]
It is easy to check that $\ad(w_g) = \lambda_{w_g\phi(w_g)^*}$, so that
\[
\|\sigma(s_i) - \ad(w_g)(s_i)\| \leq \eps \quad \text{for all } i \in \set{1,\ldots,n}.
\]
This yields a sequence of unitaries $\set{w_{g,n}}_{n\in\IN} \subset \CO_A^\sigma \cap \set{q_i \mid 1 \leq i \leq n}'$ such that $\sigma_g = \lim_{n \to \infty} \ad(w_{g,n})$.
The proof is complete.
\end{proof}
\begin{cor} \label{cor:main result}
Let $n \geq 2$.
Let $A$ be an aperiodic $n \times n$-matrix with values in $\set{0,1}$.
Let $G$ be a finite abelian group and $\sigma:G \curvearrowright \CO_A$ an action by diagonal quasi-free automorphisms.
If $\sigma$ is outer, then it is strongly approximately inner.
\end{cor}
\begin{proof}
By Proposition~\ref{prop:charac quasi-free diagonal autos}\ref{label:cqf:2}, we may assume that for each $g \in G$ and $i \in \set{1,\ldots,n}$, there exists some $\eta_{g,i} \in \IT$ such that $\sigma_g(s_i) = \eta_{g,i} s_i$.
Then
\[
u_{\sigma_g} = \sum_{i = 1}^n \eta_{g,i} p_i \ \in \ \CO_A^\sigma \cap \set{q_i \mid 1 \leq i \leq n}'.
\]
One checks that for each $g \in G$, $\sigma_g \circ \phi = \phi \circ \sigma_g$ so that $\phi(\CO_A^\sigma) \subset \CO_A^\sigma$. As $\sigma$ acts by diagonal quasi-free automorphisms, $\set{q_i \mid 1 \leq i \leq n}$ is contained in $\CO_A^\sigma$. Furthermore, as $\sigma$ is outer, $\CO_A^\sigma$ is a unital Kirchberg algebra.
The assumptions of Theorem~\ref{main theorem} are thus satisfied and prove the claim, provided that
\[
\phi: \CO_A^\sigma \cap \set{q_i \mid 1 \leq i \leq n}' \to \CO_A^\sigma \cap \set{q_i \mid 1 \leq i \leq n}'.
\]
is not surjective.
Suppose by way of contradiction that $\phi$ restricts to a surjective endomorphism of $\CO_A^\sigma \cap \set{q_i \mid 1 \leq i \leq n}'$. For $i \in \set{1,\ldots,n}$, find $x_i \in \CO_A^\sigma \cap \set{q_i \mid 1 \leq i \leq n}'$ such that $\phi(x_i) = p_i$. One computes that
\[
x_iq_i = q_ix_iq_i = s_i^* \phi(x_i) s_i = s_i^* p_i s_i = q_i.
\]
A similar computation shows that $x_i q_j = 0$ if $j \neq i$.
Hence, $x_i = q_i$ and it follows that
\[
\phi\left(\sum_{i=1}^n q_i \right) = \sum_{i = 1}^n p_i = 1.
\]
As $\phi$ is injective on $\CO_A^\sigma \cap \set{q_i \mid 1 \leq i \leq n}'$, we conclude that $\sum_{i=1}^n q_i = 1$. This in turn implies that $A$ is a permutation matrix, which contradicts the assumption that $A$ is aperiodic. The proof is complete.
\end{proof}
Our main result also applies to unital graph \cstar-algebras; see \cite{KumjianPaskRaeburn1998} for a definition. The conditions we have to impose on the finite graphs are strong connectedness and aperiodicity. Here, a graph is said to be strongly connected aperiodic if there is some $k \geq 1$ such that for any two of its edges $v,w$ there is some path of length $k$ from $v$ to $w$. The notion of quasi-free automorphisms transfers to the setting of unital graph \cstar-algebras in a straightforward manner. In particular, we call an automorphism of a unital graph \cstar-algebra diagonal quasi-free if it preserves the span of the canonical generating partial isometries and fixes all vertex projections.
\begin{cor} \label{cor:main result graphs}
Let $E$ be a finite graph that contains at least two edges.
Let $G$ be a finite abelian group and $\sigma:G \curvearrowright \cstar(E)$ be an action by diagonal quasi-free automorphisms on the associated graph \cstar-algebra.
Suppose that $E$ is strongly connected aperiodic.
If $\sigma$ is outer, then it is strongly approximately inner.
\end{cor}
\begin{proof}
Denote by $E^1$ the set of edges of $E$. The edge matrix of $E$ is the $E^1 \times E^1$ matrix $A_E$ defined by
\[
A_E(e,f) =
\begin{cases}
1 &,\ \text{if } r(e) = s(f), \\
0 &, \ \text{otherwise}.
\end{cases}
\]
The canonical partial isometries $\set{s_e \mid e \in E^1} \subset \cstar(E)$ define a Cuntz-Krieger family for $A_E$; see \cite[Section~1]{KumjianPaskRaeburn1998}. It follows that $\cstar(E)$ and $\CO_{A_E}$ are canonically isomorphic. In particular, this isomorphism intertwines (diagonal) quasi-free automorphisms. By Corollary~\ref{cor:main result}, it is thus enough to check that $A_E$ is aperiodic. However, this holds true as $A_E$ is the adjacency matrix of a graph, which in turn is strongly connected aperiodic since $E$ has this property.
\end{proof}
In combination with Izumi's classification theorem \cite[Theorem~4.6]{Izumi2004}, we obtain from Corollary~\ref{cor:main result} that outer actions of cyclic groups with prime power order by diagonal quasi-free automorphisms on Cuntz-Krieger algebras isomorphic to $\CO_2$ are classifiable in terms of their fixed point algebras and some additional information about their dual actions, provided the associated matrices are aperiodic. In the case of $\IZ_2$-actions, the latter information is not needed. We thus derive the following result.
\begin{cor} \label{cor:classification}
Let $m, n \geq 2$.
Let $A$ be an $m \times m$-matrix and $B$ be an $n \times n$-matrix with entries in $\set{0,1}$ such that $\CO_A\cong\CO_B\cong\CO_2$ as abstract \cstar-algebras.
Let $\alpha:\IZ_2 \curvearrowright \CO_A$ and $\beta:\IZ_2 \curvearrowright \CO_B$ be two actions by diagonal quasi-free automorphisms.
Suppose that $A$ and $B$ are aperiodic and that both $\alpha$ and $\beta$ are outer.
Then $\alpha$ and $\beta$ are (cocycle) conjugate if and only if $\CO_A^\alpha$ and $\CO_B^\beta$ are (stably) isomorphic
\end{cor}
\begin{proof}
The claim follows by combining Corollary~\ref{cor:main result}, \cite[Theorem~4.8]{Izumi2004} and Kirchberg-Phillips classification \cite{Kirchberg,Phillips2000}.
\end{proof}
\bibliographystyle{plain}
| -46,870.733583 |
[
-3.001953125,
2.712890625
] | 47.191011 |
[
-1.9560546875,
0.71875,
-2.193359375,
-5.1796875,
-0.94970703125,
7.9921875
] |
[
3.302734375,
9.53125,
2.560546875,
5.69140625
] | 280 | 5,564 |
[
-3.111328125,
3.765625
] | 32.712386 |
[
-5.10546875,
-3.56640625,
-5.484375,
-2.43359375,
1.6484375,
13.3203125
] | 0.688928 | 13.998053 | 21.24371 | 2.058579 |
[
1.3189260959625244
] | -29,288.854597 | 5.113228 | -46,605.594981 | 0.755712 | 5.867794 |
[
-1.4638671875,
-3.1640625,
-3.91796875,
-5.24609375,
1.84375,
12.25
] |
[
-5.8828125,
-2.6484375,
-2.3515625,
-1.6005859375,
4.375,
5.65625
] | |
BkiUcvY4eIXhokFxRhFy
|
\subsection{Derivation of the exact expressions of the first and second moments ($\tau_1$ and $\tau_2$) of the first-passage time from Eq.(3):}
\indent
Eq.(3) in the main text is a single variable, recursive, second-order differential equation in $\tau_n$, the nth moment of FPT. Utilizing the Einstein relation $D=(\beta\zeta)^{-1}$ in Eq.(3), where $\beta$ is the thermodynamic $\textit beta$, we get
\begin{align}
\dfrac{d^2 \tau_n}{d x_0^{2}}-\beta U^{\prime}(x_0)\dfrac{d \tau_n}{d x_0}+\left[\frac{n}{D}\right]\tau_{n-1}=0,
\label{eq:stnde}
\end{align}
\noindent
Setting $n=1$ in \eref{eq:stnde} and taking into account the fact that $\tau_0\coloneqq-\int_0^{\infty}dt\frac{\partial Q(t|x_0)}{\partial t}\equiv \left.Q(t|x_0)\right|_{t\to0}-\left.Q(t|x_0)\right|_{t\to\infty}=1$, we see that the resulting second order differential equation in $\tau_1$ can be reduced to a first order differential equation in $\tau_1^{\prime}\equiv d \tau_1/d x_0$ that reads
\begin{align}
\dfrac{d \tau_1^{\prime}}{d x_0}-\beta U^{\prime}(x_0)\tau_1^{\prime}+\frac{1}{D}=0.
\label{eq:st1de}
\end{align}
\noindent
With an integrating factor $e^{-\int dx_0\beta U^{\prime}(x_0)}\equiv e^{-\beta U(x_0)}$, \eref{eq:st1de} leads to the general solution
\begin{align}
\tau_1^{\prime}=\frac{1}{D}e^{+\beta U(x_0)} \left[C-\int_{x_a}^{x_0}dz~e^{-\beta U(z)}\right],
\label{eq:stp1_gsol}
\end{align}
\noindent
where $C$ is the arbitrary integration constant. Putting the
boundary condition $\lim_{x_0\to\infty} e^{-\beta U(x_0)}\tau_1^{\prime}(x_0)=0$ in \eref{eq:stp1_gsol}, we find $C=\int_{x_a}^{\infty}dz~e^{-\beta U(z)}$, and that in turn leads to the specific solution of \eref{eq:st1de}
\begin{align}
\tau_1^{\prime}=\frac{1}{D}e^{+\beta U(x_0)}\int_{x_0}^{\infty}dz~e^{-\beta U(z)}.
\label{eq:stp1_sol}
\end{align}
\noindent
Recalling that $x_a<x_0$, we now integrate \eref{eq:stp1_sol} within the interval $[x_a,x_0]$ and utilize the boundary condition $\tau_1({x_a})=0$ to obtain\cite{gardiner}
\begin{align}
\tau_1=\frac{1}{D}\int_{x_a}^{x_0}dy~e^{+\beta U(y)}\int_{y}^{\infty}dz~e^{-\beta U(z)}.
\label{eq:st1_sol}
\end{align}
\noindent
\eref{eq:st1_sol} presents an explicit expression for the mean FPT of a particle that diffuses in a potential $U(x)$ from its initial position $x_0$ to the absorbing boundary at $x_a$.\\
\indent
In a similar spirit, we can obtain an exact expression for $\tau_2$. For this, we set $n=2$ in \eref{eq:stnde} to obtain a second-order differential equation in $\tau_2$
\begin{eqnarray}
\dfrac{d^2 \tau_2}{d x_0^{2}}-\beta U^{\prime}(x_0)\dfrac{d \tau_2}{d x_0}=-\frac{2}{D^2}\left[\int_{x_a}^{x_0}dy~e^{+\beta U(y)}\int_{y}^{\infty}dz~e^{-\beta U(z)}\right],
\label{eq:st2de}
\end{eqnarray}
\noindent
where we utilized \eref{eq:st1_sol} to substitute for $\tau_1$. Solving for $\tau_2$ in the exact manner as before, we get
\begin{eqnarray}
\tau_2=\frac{2}{D^2}\int_{x_a}^{x_0}dw~e^{+\beta U(w)}
\int_{w}^{\infty}dv~e^{-\beta U(v)}
\int_{x_a}^{w}dy~e^{+\beta U(y)}\int_{y}^{\infty}dz~e^{-\beta U(z)}.
\label{eq:st2_sol}
\end{eqnarray}
\noindent
\eref{eq:st2_sol} presents an explicit expression for the second moment of the FPT.\\ \\
\subsection{Transformation of variable $x\to \phi(x)\equiv \beta U(x)$ for the power-law potential $U(x)=U_0|x|^{\alpha}$: Derivation of Eq. (8):}
As discussed in the main text before Eq. (7), we consider a general transformation of variable $x\to\phi(x)\coloneqq\beta U(x)$, which leads to $\phi(x)=\beta U_0|x|^{\alpha}$ for the power-law potential. Therefore, for $0\le x<\infty$ we get
\begin{align}
x\equiv\left[\frac{\phi(x)}{\beta U_0}\right]^{\frac{1}{\alpha}}.
\label{eq:svbt}
\end{align}
\noindent
Differentiating \eref{eq:svbt} with respect to $\phi(x)$
thus gives
\begin{align}
dx\equiv\frac{1}{\alpha(\beta U_0)^{\frac{1}{\alpha}}}\left[\frac{d\phi(x)}{\phi(x)^{1-\frac{1}{\alpha}}}\right].
\label{eq:sdvbt}
\end{align}
\noindent
Setting $x_a=0$, we can write \eref{eq:st1_sol} in terms of the transformed variable as
\begin{align}
\tau_1=\frac{1}{D\alpha^2(\beta U_0)^{\frac{2}{\alpha}}}\int_{0}^{\phi(x_0)}d\phi(y)~\frac{e^{+\phi(y)}}{\phi(y)^{1-\frac{1}{\alpha}}}\int_{\phi(y)}^{\infty}d\phi(z)~\frac{e^{-\phi(z)}}{\phi(z)^{1-\frac{1}{\alpha}}}.
\label{eq:st1_t}
\end{align}
\noindent
Note that $\phi(y)$ and $\phi(z)$ are integration variables in \eref{eq:st1_t}, and hence the value of the above double integral depends only on $\phi_0\equiv \phi(x_0)=\beta U_0|x_0|^{\alpha}$. To simplify notation, we use $\phi_z=\phi(z)$ and $\phi_y=\phi(y)$ hereafter. In a similar spirit as in the case of the mean FPT, \eref{eq:st2_sol} in terms of the transformed variable reads
\begin{align}
\tau_2=\frac{2}{D^2\alpha^4(\beta U_0)^{\frac{4}{\alpha}}}\int_{0}^{\phi_0}d\phi_w\frac{{ e}^{+\phi_w}}{\phi_w^{1-\frac{1}{\alpha}}}\int_{\phi_w}^{\infty}d\phi_v\frac{{ e}^{-\phi_v}}{\phi_v^{1-\frac{1}{\alpha}}}\int_{0}^{\phi_v}d\phi_y\frac{{ e}^{+\phi_y}}{\phi_y^{1-\frac{1}{\alpha}}}\int_{\phi_y}^{\infty}d\phi_z\frac{{ e}^{-\phi_z}}{\phi_z^{1-\frac{1}{\alpha}}},
\label{eq:st2_t}
\end{align}
\noindent
where $\phi_w=\phi(w)$ and $\phi_v=\phi(v)$. From Eqs.~(\ref{eq:st1_t}) and (\ref{eq:st2_t}), the condition for resetting transition, $\tau_2/2\tau_1^2=1$, can be obtained, which is presented by Eq. (8) in the main text.
\end{widetext}
\end{document}
| -9,936.780844 |
[
-1.4833984375,
1.330078125
] | 14.457831 |
[
-6.11328125,
-3.99609375,
-3.201171875,
-9.71875,
-0.302978515625,
14.1875
] |
[
1.2890625,
8.8359375,
0.08477783203125,
5.125
] | 28 | 496 |
[
-3.54296875,
3.9765625
] | 37.373168 |
[
-5.09375,
-4.140625,
-2.384765625,
-1.23046875,
1.7314453125,
7.625
] | 4.140311 | 5.591874 | 49.798387 | 2.590805 |
[
1.955967903137207
] | -7,228.040002 | 7.012097 | -9,926.0253 | 2.53019 | 5.043933 |
[
-1.9130859375,
-3.41015625,
-4.515625,
-5.7265625,
2.298828125,
12.4375
] |
[
-5.61328125,
-3.150390625,
-2.724609375,
-2.06640625,
3.58203125,
5.203125
] | |
BkiUdGM4ubnjow-6Mjh5
|
\section{Introduction}
The Yang-Mills gradient flow~\cite{Luscher:2010iy,Luscher:2011bx} is a new
powerful tool to investigate non-perturbative aspects of non abelian
quantum field theories. In the context of $SU(N)$ Yang-Mills theories
one introduces an extra coordinate $t$ (called
flow time, not the same as Euclidean time, denoted $x_4$), and defines
a flow gauge field $B_\mu(x,t)$ according to the equation
\begin{eqnarray}
\label{eq:flow}
\frac{ {\rm d}B_\mu(x,t)}{{\rm d}t} &=& D_\nu G_{\nu\mu}(x,t)\,, \\
G_{\mu\nu} &=& \partial_\mu B_\nu - \partial_\nu B_\mu +
[B_\mu,B_\nu] \,,
\end{eqnarray}
with initial condition
\begin{equation}
B_\mu(x,0) = A_\mu(x) \,.
\end{equation}
Since $D_\nu G_{\nu\mu}(x,t) \sim -\delta S_{YM}/\delta B_{\mu}$ the
flow drives the gauge field towards a classical solution (i.e. the
flow smooths the field). Due to this smoothing property ultra-violet
fluctuations are suppressed and composite operators do not need any
renormalization at positive flow time. In particular the energy
density
\begin{equation}
\langle E(t) \rangle = \frac{1}{4}\langle G_{\mu\nu}G_{\mu\nu}\rangle
\end{equation}
has the following perturbative expansion in $\mathbb
R^4$\cite{Luscher:2010iy}
\begin{equation}
\langle E(t)\rangle =
\frac{3g^2_{\overline{MS}}}{16\pi^2t^2}(1+c_1g^2_{\overline{MS}}+\mathcal O(g^4_{\overline{MS}}))
\end{equation}
and therefore can be used for a non-perturbative definition of the
coupling at a scale $\mu=1/\sqrt{8t}$
\begin{equation}
\alpha(\mu) = \frac{4\pi}{3}t^2\langle E(t)\rangle =
\alpha_{\overline{MS}}(\mu) + \dots\,.
\end{equation}
Moreover, in a finite volume one can identify the renormalization
scale $\mu$ with the size of the system and use the energy density to
define a running coupling.
In these proceedings we would like to explore an alternative to the
previously studied running coupling schemes based on the
gradient flow that use either periodic~\cite{Fodor:2012td} or Schr\"odinger
Functional~\cite{Fritzsch:2013je} boundary conditions. It
is based on twisted boundary conditions for the gauge fields and
has several practical advantages. The twisted boundary conditions, if
chosen appropriately,
guarantee that the only zero modes of the action are the gauge modes,
and therefore this coupling definition is analytic in
$g^2_{\overline{\rm MS}}$ and has an universal 2-loop beta
function. Moreover in
this setup the fields still live on a torus and therefore
there are no boundary counterterms: $\mathcal O(a)$ improvement is
guaranteed without any tuning. The weak
point of this running coupling scheme shows up when one considers
Yang-Mills fields coupled to matter. Fermions in the fundamental
representation are naively incompatible with the twisted boundary
conditions. On the other hand
fermions in multi-index representations (like adjoint fermions) do not
suffer from this obstruction.
\section{Twisted boundary conditions}
We have no space here to review the basics of twisted boundary
conditions. Instead we will say a few basic things to clarify the
notation and refer the reader to the
literature. In~\cite{Perez:2013dra} the reader can find a nice summary
on twisted boundary conditions with a similar
notation and in a similar setup than the ones used here. This setup is
basically the same as the one used in the Twisted Polyakov Loop (TPL)
scheme~\cite{deDivitiis:1994yp}. Finally the review~\cite{ga:torus}
contains more in-depth information and proofs about some of the
results used here.
We are going to work in a four dimensional torus $\mathcal T^4$ of
sides $L\times L\times L\times L$. The basic idea is that on a torus
physical quantities have to be
periodic, but this does not imply that the gauge potential $A_\mu(x)$
has to be periodic. It is enough if it is periodic modulo a gauge
transformation
\begin{equation}
A_\mu(x+L\hat \nu) = \Omega_\nu(x)A_\mu(x)\Omega^+_\nu(x) +
\Omega_\nu(x)\partial_\mu \Omega_\nu^+(x).
\end{equation}
The matrices $\Omega_{\mu}(x)$ are called twist matrices, and they
have to obey the consistency relation
\begin{equation}
\Omega_\mu(x+L\hat\nu)\Omega_\nu(x) = z_{\mu\nu}
\Omega_\nu(x+L\hat\mu)\Omega_\mu(x)
\end{equation}
where $z_{\mu\nu}$ are elements of the center of $SU(N)$. The
particular choice of twist matrices is irrelevant, since they change
under gauge transformations, but it is easy to check that $z_{\mu\nu}$
is gauge invariant, and therefore it encodes the physical part of the
twisted boundary conditions. We are going to use a particular setup of
this general scheme: we twist the plane $x_1,x_2$, while
the gauge potential will be periodic in the directions $x_3$ and
$x_4$. Moreover we are going to choose
\begin{equation}
z_{12} = z = e^{2\pi\imath/N},
\end{equation}
and the twist matrices to be constant $\Omega_{1,2}(x) = \Omega_{1,2}$
obeying the relation
\begin{equation}
\Omega_1\Omega_2 = e^{2\pi\imath /N}
\Omega_2\Omega_1.
\end{equation}
We will define the usual space momentum
\begin{equation}
p_\mu = \frac{2\pi n_\mu}{L}\,, \quad
\mu = 1,2,3,4;\; n_\mu = 0,\dots
\end{equation}
and what is usually called the color momentum
\begin{equation}
\tilde p_i = \frac{2\pi \tilde n_i}{NL}\,, \quad
i = 1,2;\; \tilde n_i = 0,\dots,N-1.
\end{equation}
The total momentum is the sum of both
\begin{eqnarray}
P_{i} &=& p_i + \tilde p_i \quad (i=1,2)\,, \\
P_{3,4} &=& p_{3,4} \,.
\end{eqnarray}
It can be proved~\cite{ga:torus} that any gauge connection compatible
with these particular boundary conditions can be written as
\begin{equation}
A_\mu^a(x)T^a = \frac{1}{L^4} \sum_{p,\tilde p\ne 0}\tilde
A_\mu(P)e^{\imath
Px}\hat\Gamma(P),
\end{equation}
where $\tilde A_\mu(P)$ are complex coefficients (\emph{not}
matrices) and $\hat\Gamma(P)$ are matrices given by
\begin{equation}
\hat\Gamma(P) = \Omega_1^{-k\tilde
n_2}\Omega_2^{k\tilde n_1}.
\end{equation}
Note that the only constant gauge connection compatible with our choice
of boundary conditions is $A_\mu = 0$, and this is the
only configuration (up to gauge transformations) with zero action.
\section{$\langle E(t) \rangle$ to leading order in a twisted box and
running coupling definition}
In perturbation theory one scales the gauge potential with the bare
coupling $A_\mu \rightarrow g_0 A_\mu$. The flow
field of equation~(\ref{eq:flow}) becomes a function of $g_0$ with
an asymptotic expansion
\begin{equation}
B_\mu(x,t) = \sum_{n=1}^{\infty} B_{\mu,n}(x,t)g_0^n.
\end{equation}
After gauge fixing and inserting this expansion in~(\ref{eq:flow}), we
find that to leading order the gradient flow equation reads
\begin{equation}
\frac{ {\rm d}B_{\mu,1}(x,t)}{{\rm d}t} =
\partial_\nu^2 B_{\mu,1}(x,t)\,, \quad
B_{\mu,1}(x,0) = A_\mu(x).
\end{equation}
The solution to this linear equation compatible with our twisted
boundary conditions can be written as
\begin{equation}
\label{eq:sol}
B_{\mu,1}(x,t) = \frac{1}{L^4}
\sum_{p,\tilde p\ne 0} e^{-P^2t} \tilde A_\mu(P) e^{\imath Px}
\hat\Gamma(P).
\end{equation}
We will expand our observable of interest $\langle E(t) \rangle$ in
powers of $g_0$
\begin{equation}
\langle E(t)\rangle = \frac{1}{4}\langle
G_{\mu\nu}(x, t)G_{\mu\nu}(x,t)\rangle =
\mathcal E(t) + \mathcal O(g_0^4)\,,
\end{equation}
where the leading order contribution is given by
\begin{equation}
\mathcal E(t) = \frac{g_0^2}{2}\langle
\partial_\mu B_{\nu,1}\partial_\mu B_{\nu,1} - \partial_\mu
B_{\nu,1}\partial_\nu B_{\mu,1}
\rangle.
\end{equation}
A short computation gives as result
\begin{equation}
\mathcal E(t) =
\frac{3g_0^2}{2L^4}\sum_{p,\tilde p\ne 0} e^{-P^2t}\,.
\end{equation}
To define a running coupling we simply identify the renormalization scale
$\mu = 1/\sqrt{8t}$ with the linear size of the finite volume box
\begin{equation}
\sqrt{8t} = cL\,.
\end{equation}
The parameter $c$ identify the scheme. The definition of the
twisted gradient flow running coupling reads
\begin{equation}
g_{TGF}^2(L) = \mathcal N^{-1}_T(c)t^2\langle E(t) \rangle \Big|_{t=c^2L^2/8}
= g_{\overline{\rm MS}}^2 + \mathcal O(g_{\overline{\rm MS}}^4)
\end{equation}
with
\begin{equation}
\mathcal N_T(c) =
\frac{3g_0^2c^4}{128}\sum_{p,\tilde p\ne 0} e^{-\frac{c^2L^2}{4}P^2}
= \frac{3g_0^2c^4}{128}\sum_{n_\mu=-\infty}^{\infty}
{\sum_{\tilde n_i=0}^{N-1}}'
e^{-{\pi^2c^2}(n^2 + \tilde n^2/N^2 + 2\tilde n_i n_i/N)}\,.
\end{equation}
The prime over the sum recalls that the term $\tilde n_1 =
\tilde n_2 = 0$ has to be dropped.
\section{$SU(2)$ running coupling}
As a test we have performed a non-perturbative running of the
$SU(2)$ coupling. Here we will only sketch the simulations and
results. A detailed description of the simulations, algorithms and
parameters will be presented later~\cite{toappear}. We have performed
simulations with the Wilson action for $\beta \in [2.75,12.0]$ on
lattices of sizes $L/a = 10, 12, 15, 18$, and to
perform the step-scaling, also on lattices of sizes $L/a = 20, 24, 30,
36$. We collect 2048 measurements of twisted gradient flow
coupling for $c=0.3$. The measurements are well spaced in simulation
time and autocorrelations are negligible. With this statistics we
achieve a precision between a $0.15-0.3\%$, independently of the value
of $L/a$ (see table~\ref{tab:gsq} for some representative values of
the coupling).
\begin{table}
\centering
\input{data.tex}
\caption{Values of the twisted gradient flow coupling for different
values of $\beta$ in lattices of sizes $L/a=20, 24, 30, 36$. These
values were determined with 2048 measurements well spaced in Monte
Carlo time so that autocorrelations were negligible.}
\label{tab:gsq}
\end{table}
For each value of $L/a$ we fit the data using a Pad\'e ansatz of the
form
\begin{equation}
g^2_{TGF}(a/L,\beta) = \frac{4}{\beta} \frac{\sum_{n=0}^{N-1} a_n(a/L)\beta^n
+ \beta^N}{\sum_{n=0}^{N-1} b_n(a/L)\beta^n+\beta^N}.
\end{equation}
We obtain fits with good quality ($\chi^2/{\rm ndof}\sim 1$)
with 4 parameters in all our cases. An example of such a fit, for the
case of the $L/a=36$ lattice can be seen in the figure~\ref{fig:fit}.
\begin{figure}
\centering
\subfloat[][]{
\includegraphics[width=0.3\textwidth]{fig/ss_1}
}
\subfloat[][]{
\includegraphics[width=0.3\textwidth]{fig/ss_2}
}
\subfloat[][]{
\includegraphics[width=0.3\textwidth]{fig/ss_3}
}
\caption{Continuum extrapolation of the step-scaling function. We
start the recursion in a volume $L_{\rm max}$ where $g^2_{\rm
TGF}(L_{\rm max})=7.5$.}
\label{fig:ss}
\end{figure}
We apply the usual step-scaling technique, starting the recursion in a
volume $L_{\rm max}$ where $g^2_{\rm TGF}(L_{\rm max})=7.5$. The
continuum extrapolations of the step-scaling function are very flat,
as can be seen in figure~\ref{fig:ss}.
\begin{figure}
\centering
\subfloat[][]{
\includegraphics[width=0.45\textwidth]{fig/gsq_c0_30}
\label{fig:fit}
}
\subfloat[][]{
\includegraphics[width=0.45\textwidth]{fig/gvsL}
\label{fig:gvsL}
}
\caption{Left: The twisted gradient flow coupling for $L/a=36$ as a
function of $\beta$. The points are fitted to a Pad\'e functional
form (grey band in the figure) with 4 parameters and the
$\chi^2/{\rm ndof} = 5.9/7$. Right: Non-perturbative running of
the twisted gradient flow coupling.}
\end{figure}
The non-perturbative
running of the coupling is recursively carried over a factor $2^{25}$
change in the scale down to a volume $L_{\rm min}$ where
$g^2_{\rm TGF}(L_{\rm min})=0.5323(83)$. Figure~\ref{fig:gvsL} shows
the scale dependence of the coupling and a comparison with
the universal 2-loop $\beta-$function.
\section{Conclusions and comments}
In this proceedings we have studied perturbatively the gradient flow
in a four dimensional torus with twisted boundary conditions. The
energy density of the flow field can be used to define a running
coupling. We have presented some preliminary results on a $SU(2)$
running coupling where we have shown that the twisted gradient flow
coupling is a convenient choice. The observable is precise and 2048
independent measurements are enough to reach a per mile
precision. Cutoff effects of the step-scaling functions are mild.
Some comments about the inclusion of matter
fields are in order. In principle our coupling definition is perfectly
valid with
any kind of matter fields, but fermions in the fundamental
representation are incompatible with the twisted boundary
conditions. A fermion in the fundamental representation transforms as
\begin{equation}
\psi(x+L\hat\mu) = \Omega_\mu\psi(x)\,,
\end{equation}
but this transformation is not consistent due to
\begin{eqnarray}
\psi(x+L\hat 1+L\hat 2) &=& \Omega_1\Omega_2\psi(x)\\
\psi(x+L\hat 2+L\hat 1) &=& \Omega_2\Omega_1\psi(x)
= e^{2\pi\imath/N} \Omega_1\Omega_2\psi(x)\,.
\end{eqnarray}
This obstruction can be overcome if the number of fermions is an
integer multiple of the degree of the gauge group
$N$~\cite{Parisi:1984cy}.
On the other hand fermions in multi-index representations (like
adjoint fermions) do not suffer from any kind of restriction.
\section*{Acknowledgments}
I want to thank M. Garc\'ia P\'erez and A. Gonz\'alez-Arroyo
for sharing some of their results before publication and A. Sastre for
the help in testing some parts of the codes. The continuous help of
R. Sommer in each of the steps of this work was invaluable.
I also want to thank P. Fritzsch, P. Korcyl, H. Simma, S. Sint and
U. Wolff for the many useful discussions on this work.
| -15,169.190084 |
[
-2.955078125,
2.689453125
] | 18.128655 |
[
-2.908203125,
0.0755615234375,
-2.03125,
-6.12890625,
-1.091796875,
8.59375
] |
[
2.8125,
9.1328125,
0.80908203125,
4.43359375
] | 99 | 1,683 |
[
-3.5,
3.84765625
] | 30.884304 |
[
-5.609375,
-4.1953125,
-4.46484375,
-2.37890625,
1.734375,
11.796875
] | 1.34492 | 11.266004 | 38.621509 | 2.965649 |
[
2.1009764671325684
] | -9,984.46039 | 5.743316 | -15,022.938186 | 2.327747 | 5.655537 |
[
-2.26171875,
-3.544921875,
-3.83984375,
-5.19921875,
2.189453125,
12.578125
] |
[
-5.41796875,
-2.294921875,
-2.806640625,
-1.6796875,
3.4921875,
5.015625
] | |
BkiUcnnxK4sA-46xNddg
|
\section{Introduction}\label{S:I}
In 1872, in his very influential ``Erlanger Programm'', Felix Klein suggested to base the study of (certain) geometric spaces on the
analysis of their transformation groups, thus assigning groups a fundamental role in geometry. In the late 1950s and early 1960s,
Jacques Tits reversed this idea. He invented geometric structures that he called ``buildings'' in order to interpret (certain) groups of Lie
type as symmetry groups. Until then, some of these groups, particularly those of exceptional type, had been studied in a purely algebraic
manner. Through the use of buildings, important parts of modern group theory have become accessible to geometric methods and ideas, and
this has continued to be a crucial link between group theory and geometry, with many applications ever since.
The goal of this paper
is to present several results on the structure and Galois cohomology of the groups of points of absolutely almost simple algebraic groups over polynomial rings that rely in a critical way on the analysis of the actions of these groups on affine Bruhat-Tits buildings and the Fixed Point Theorem for actions of finite groups on these buildings and their products. We hope that this approach can be developed further to treat similar issues for the groups of points over the coordinate rings of more general affine curves.
First, one of the notable consequences of reduction theory for arithmetic groups is that every arithmetic subgroup has finitely many conjugacy classes of finite subgroups (cf. \cite{BoHC}, \cite[Ch. 4, Theorem 4.3]{PlRa}) --- we will refer to this property as (FC). Later, using a combination of various techniques, including the Fixed Point Theorem for group actions on CAT(0) spaces, it was shown in \cite[Theorem 1.4]{GrPl} that (FC) remains valid for \emph{all} finite extensions of arithmetic groups.
We will use the Fixed Point Theorem to establish (FC) for groups of the form $G(k[t])$, where $G$ is a reductive algebraic group over a local field $k$ of characteristic zero, i.e. over a finite extension of some $p$-adic field $\mathbb{Q}_p$ (see Theorem \ref{T:FC-1}). This result can be viewed
as a contribution to the theory of algebraic groups over function fields of $p$-adic curves, where very significant progress has been achieved over the last decade on the cohomological Hasse principle and related issues --- see \cite{Par1}, \cite{Par2}, and references therein for the most recent installments of this work. Nevertheless, the question about (FC) for the groups of points over the coordinate rings of such curves has not been previously addressed. We plan to investigate this question for curves other than the affine line in future work. It should be pointed out that our finiteness result is derived from the following statement that is of independent interest.
\begin{thm}\label{T:I1}
Let $G$ be a reductive algebraic group over a field $k$ of characteristic 0. Then every finite subgroup of $G(k[t])$ is conjugate to a subgroup contained in $G(k)$.
\end{thm}
We note that if $p = \mathrm{char}\: k > 0$ and the derived group $[G , G]$ of $G$ is $k$-isotropic, then the group $G(k[t])$ contains finite abelian $p$-subgroups of unbounded orders (even when $k = \mathbb{F}_p$ is the field with $p$ elements). This implies that the above theorem fails in this situation and no finiteness theorem can be established unless we limit ourselves to finite subgroups of order prime to $p$.
Second, the result on (FC) for arithmetic groups was used in \cite{BoSe} to establish the finiteness of the cohomology set $H^1(\mathfrak{g} , \Gamma)$ for {\it certain} actions of a finite group $\mathfrak{g}$ on an arithmetic group $\Gamma$, which was then applied to prove the properness of the global-to-local map in Galois cohomology in some situations. Subsequently, the finiteness of $H^1(\mathfrak{g} , \Gamma)$ was established in \cite[Theorem 1.5]{GrPl} for {\it all} actions of a finite group $\mathfrak{g}$ on an arithmetic group $\Gamma$ as a consequence of the result on (FC) for all finite extensions of arithmetic groups. We will utilize this approach, in conjunction with the techniques employed in the proof of Theorem \ref{T:I1}, to give a short proof of the theorem of Raghunathan-Ramanathan \cite{RagRam} (see Theorem \ref{T:RagRam}). The proof quickly reduces to the following statement, which we then establish.
\begin{thm}\label{T:I2}
Let $G$ be a reductive algebraic group over a field $k$ of characteristic 0. Then for any finite Galois extension $\ell/k$, the natural map
$$
H^1(\mathrm{Gal}(\ell/k) , G(\ell)) \longrightarrow H^1(\mathrm{Gal}(\ell/k) , G(\ell[t]))
$$
is a bijection.
\end{thm}
It follows from this result that if $k$ is a local field of characteristic zero, then the set $H^1(\mathrm{Gal}(\bar{k}/k) , G(\bar{k}[t]))$ is finite. The result of \cite{CGP} on torsors over the punctured affine line implies the finiteness of the set $H^1(\mathrm{Gal}(\bar{k}/k) , G(\bar{k}[t , t^{-1}])$
over such $k$. It would be interesting to see if the finiteness result remains valid over the coordinate rings of more general $p$-adic curves.
\section{The action of $G(k[t])$ on the relevant building}\label{S:building}
We begin this section with a summary of the results of B.~Margaux \cite{Marg} that generalize results established by C.~Soul\'e \cite{Soule} in the split case. It should be noted that for anisotropic groups, the relevant building degenerates into a point, but the results we need remain valid in this case due to Proposition \ref{P:anis} below.
\vskip1mm
\noindent {\bf 2.1. Results of B.~Margaux \cite{Marg}.} Let $G$ be an absolutely almost simple simply connected algebraic group over a field $k$. Fix a maximal $k$-split torus $S$ of $G$, and let $T$ be a maximal $k$-torus containing $S$. Take $\ell/k$ to be a finite Galois extension with Galois group $\mathscr{G} = \mathrm{Gal}(\ell/k)$ such that $T$ split over $\ell$ (we do not assume that $\ell$ is necessarily the minimal splitting field of $T$).
Let $\Phi = \Phi(G , T)$ (resp., $\Phi_r = \Phi(G , S)$) denote the corresponding absolute (resp., relative) root system. We choose compatible orderings on $\Phi$ and $\Phi_r$, and denote by $\Delta$ and $\Delta_r$ the corresponding systems of simple roots.
Next, we set $K = k((1/t))$ and $L = \ell((1/t))$, and let $\mathscr{B}$ (resp., $\mathscr{B}_{\ell}$) denote the (affine) Bruhat-Tits building associated with the group $G_K := G \times_k K$ (resp., $G_L = G \times_k L$). Both $\mathscr{B}$ and $\mathscr{B}_{\ell}$ possess natural simplicial complex structures, and there is an isometric embedding $j \colon \mathscr{B} \to \mathscr{B}_{\ell}$ that identifies $\mathscr{B}$ with the fixed set $(\mathscr{B}_{\ell})^{\mathscr{G}}$ under the natural action of $\mathscr{G}$. The hyperspecial subgroup $G(\ell[[1/t]])$ of $G(L)$ fixes a unique vertex $p^0_{\ell} \in \mathscr{B}_{\ell}$. Since the subgroup $G(\ell[[1/t]])$ is $\mathscr{G}$-invariant, the point $p^0_{\ell}$ is $\mathscr{G}$-fixed, hence $p^0_{\ell} = j(p^0)$ for $p^0 \in \mathscr{B}$. Let $\mathscr{A}$ (resp., $\mathscr{A}_{\ell}$) be the standard apartment of $\mathscr{B}$ (resp., $\mathscr{B}_{\ell}$) associated with $S$ (resp., $T_L$). Then
\begin{equation}\label{E:apart}
\mathscr{A} = p^0 + \mathrm{Hom}_{k-\mathrm{gr}}(\mathbb{G}_m , S) \otimes_{\mathbb{Z}} \mathbb{R} \ \ \text{and} \ \ \mathscr{A}_{\ell} = p^0_{\ell} + \mathrm{Hom}_{\ell-\mathrm{gr}}(\mathbb{G}_m , T) \otimes_{\mathbb{Z}} \mathbb{R},
\end{equation}
hence $j(\mathscr{A}) = (\mathscr{A}_{\ell})^{\mathscr{G}}$. Let
$$
\langle \cdot , \cdot \rangle \colon \mathrm{Hom}_{k-\mathrm{gr}}(S , \mathbb{G}_m) \times \mathrm{Hom}_{k-\mathrm{gr}}(\mathbb{G}_m , S) \to \mathbb{Z}
$$
be the canonical pairing. We then define the {\it sector} (quartier) by
$$
\mathscr{Q} = p^0 + D, \ \ \text{where} \ \ D:= \{ \lambda \in \mathrm{Hom}_{k-\mathrm{gr}}(\mathbb{G}_m , S) \otimes_{\mathbb{Z}} \mathbb{R} \ \vert \ \langle \alpha , \lambda \rangle \geq 0 \ \ \forall \alpha \in \Delta \}.
$$
The following theorem, which is Theorem 2.1 in \cite{Marg}, generalizes the result that was established by Soul\'e \cite{Soule} for split groups.
\begin{thm}\label{T:Marg}
The set $\mathscr{Q}$ is a simplicial fundamental domain for the action of $G(k[t])$ on $\mathscr{B}$. In other words, any simplex of $\mathscr{B}$ is equivalent under the action of $G(k[t])$ to a unique simplex of $\mathscr{Q}$.
\end{thm}
We note that in this paper, the uniqueness will be used only in the split case. It follows from the description that if $G$ is $k$-split, then $S = T$ and consequently ${\mathscr{A}}_{\ell} = \mathscr{A}$. Therefore, every point of $\mathscr{B}_{\ell}$ is $G(\ell[t])$-equivalent to a point of $\mathscr{A}$.
Next, set $\Gamma = G(k[t])$. We will now recall the description of the stabilizers $\Gamma_x$ of points $x \in \mathscr{Q}$ obtained in \cite[\S 2]{Marg}.
Clearly, for $p^0 \in \mathscr{Q}$ we have $\Gamma_{p^0} = G(k)$. Let now $x \in \mathscr{Q} \setminus \{ p^0 \}$. Set
$$
I_x = \{ \alpha \in \Delta \ \vert \ \alpha(x) = 0 \}
$$
(we obviously have $I_{p^0} = \Delta$). As usual, for $I \subset \Delta$, we define
$$
S_I = \left( \bigcap_{\alpha \in I} \ker \alpha \right)^{\circ} \ \ \text{and} \ \ L_I = Z_G(S_I).
$$
Let $P_I$ be the standard parabolic subgroup of $G$ of type $I$ (cf. \cite[21.11]{Borel}). Then $P_I$ has $L_I$ as its (standard) Levi subgroup, i.e. $P_I$ is the semi-direct $U_I \rtimes L_I$, where $U_I$ is the unipotent radical of $P_I$, which is a connected $k$-defined $k$-split unipotent subgroup of $P_I$. Recall that the root system $\Phi(L_I , S_I)$ coincides with $[I]$, the subsystem of $\Phi = \Phi(G , S)$ consisting of roots that are linear combinations of elements of $I$. Then $U_I$ is generated by the $U_{\alpha}$ for $\alpha \in \Phi^+ \setminus [I]$. The following result is a consequence of (\cite[Proposition 2.5(1)]{Marg}).
\begin{prop}\label{P:stab1}
We have
\begin{equation}\label{E:stab1}
\Gamma_x = (\Gamma_x \cap U_{I_x}(K)) \rtimes L_{I_x}(k),
\end{equation}
and $\Gamma_x \cap U_{I_x}(K)$ is the group of $k$-points of a $k$-split unipotent subgroup.
\end{prop}
Now, given a subset $\Omega \subset \mathscr{Q}$, we deduce from the proposition that the pointwise stabilizer $\Gamma_{\Omega} := \bigcap_{x \in \Omega} \Gamma_x$ has the following description
\begin{equation}\label{E:stab2}
\Gamma_{\Omega} = (\Gamma_{\Omega} \cap U_{\Omega}(K)) \rtimes L_{I_{\Omega}}(k), \ \ \text{where} \ \ U_{\Omega} = \bigcap_{x \in \Omega} U_{I_x} \ \ \text{and} \ \ I_{\Omega} = \bigcap_{x \in \Omega} I_x.
\end{equation}
Thus, we obtain the following.
\begin{cor}\label{C:stab1}
For any subset $\Omega \subset \mathscr{Q}$, the pointwise stabilizer $\Gamma_{\Omega}$ is of the form $\mathscr{U} \rtimes L_{I_{\Omega}}(k)$, where $\mathscr{U}$ is the group of $k$-points of some $k$-defined $k$-split unipotent subgroup.
\end{cor}
\vskip.5mm
\noindent {\bf 2.2. A result concerning $G(k[t])$.} In order to extend some results from subsection 2.1 to arbitrary reductive groups, we need the following fact.
\begin{thm}\label{T:Stavr1}
Let $G$ be reductive algebraic $k$-group. Then
$$
G(k[t]) = G(k) \cdot G(k[t])^+,
$$
where $G(k[t])^+$ is the subgroup of $G(k[t])$ generated by the subgroups $U_P(k[t])$ for all minimal $k$-defined parabolic subgroups $P$ of $G$, with $U_P$ being the unipotent radical of $P$.
\end{thm}
This was established in \cite[Theorem 3.1]{Stav} under the additional assumption that every normal semi-simple $k$-subgroup of $[G , G]$ is $k$-isotropic. The argument in {\it loc. cit.} gives a reduction to the case where $G$ is semi-simple and simply connected, which was considered in \cite{Marg} by generalizing the techniques introduced in \cite{Soule}. To handle the general case in the theorem, we will need the fact that if $G$ is $k$-anisotropic, then $G(k[t]) = G(k)$. In fact, we have the following more general statement.
\begin{prop}\label{P:anis}
Let $\tilde{C}$ be a smooth absolutely irreducible projective curve over a field $k$ of characteristic 0, $P \in \tilde{C}(k)$ be a $k$-rational point, and $C = \tilde{C} \setminus \{ P \}$ be the corresponding affine curve. Then for any connected reductive algebraic $k$-group $G$ whose semi-simple part $H = [G , G]$ is $k$-anisotropic, we have $G(k[C]) = G(k)$.
\end{prop}
\begin{proof}
The claim is almost immediate if $G = T$ is a torus. Indeed, let $\ell$ be the splitting field of $T$. Then $\ell[C]^{\times} = \ell^{\times}$, and consequently
$$
T(k[C]) \subset T(\ell[C]) = T(\ell).
$$
So, $T(k[C]) \subset T(k[C] \cap \ell) = T(k)$.
We will reduce the proof to the case of a semi-simple $k$-anisotropic group. There is a central $k$-defined isogeny $\pi \colon G \to \bar{T} \times \bar{H}$ to the direct product of a torus and a semi-simple group. Clearly, $\pi$ yields a central isogeny $H \to \bar{H}$, so $\bar{H}$ is $k$-anisotropic. Assuming that the assertion of the proposition is valid for semi-simple $k$-anisotropic groups, we will have
$$
\pi(G(k[C]) \subset \bar{T}(k[C])) \times \bar{H}(k[C]) = \bar{T}(k) \times \bar{H}(k).
$$
Since $\ker \pi \subset G(\bar{k})$, we obtain $G(k[C]) \subset G(k[C] \cap \bar{k}) = G(k)$.
Now we will treat the main case where $G$ is a semi-simple $k$-anisotropic group. Let $K = k(C)$, and set $\mathcal{O}_P$, $\mathfrak{m}_P$, and $v = V_P$ to be the local ring of $P$, its maximal ideal, and the discrete valuation of $K$ associated with the point $P$, respectively. Then the completion $K_v$ can be identified with the field of formal power series $k((t))$, so one can easily see (say, by considering nilpotent elements in the Lie algebra --- here we use that ${\rm char}~k = 0$) that $G$ remains anisotropic over $K_v$, hence also over $K$. Fix a faithful $k$-defined representation $G \hookrightarrow \mathrm{GL}_n$. The fact that $G$ is $K_v$-anisotropic implies that $G(K_v)$, hence also $G(k[C])$, is a bounded subgroup of $\mathrm{GL}_n(K_v)$. Assume now that $\Gamma := G(k[C])$ is different from $G(k)$. The quotient map $\mathcal{O}_P \to \mathcal{O}_P/\mathfrak{m}_P$ gives rise to a split-surjective group homomorphism $\phi \colon \Gamma \to G(k)$. The assumption $\Gamma \neq G(k)$ implies the existence of a nontrivial element $g \in \ker \phi$. Let $f(t)$ be the characteristic polynomial of $g$ calculated in terms of our fixed embedding. We claim that $f(t)$ has a coefficient that is not in $k$. Indeed, since $g$ is congruent to the identity matrix modulo $\mathfrak{m}_P$, the polynomial $f(t)$ is congruent to $(t - 1)^n$. If all the coefficients of $f$ were in $k$, we would have $f(t) = (t - 1)^n$. Then $g$ would be a nontrivial unipotent matrix in $\Gamma$, which is impossible since $G$ is $K$-anisotropic. Next, the fact that not all coefficients of $f$ are in $k$ implies that $g$ has an eigenvalue $\lambda \in \bar{K} \setminus \bar{k}$. Then there exists an extension $w$ of $v$ to $L = K(\lambda)$ such that $w(\lambda) < 0$, which implies that the powers $g^n$, for $n = 1, 2, \ldots$, form an unbounded set in $\mathrm{GL}_n(K_v)$, a contradiction.\end{proof}
\vskip1mm
{\it Proof of Theorem \ref{T:Stavr1}.} We can write $G$ as an almost direct product $G = G_1 \cdot G_2$, where $G_1$ has the property that every semi-simple normal $k$-subgroup is $k$-isotropic and $G_2$ is $k$-anisotropic. Let $E = G_1 \cap G_2$, and set $\overline{G}_i = G_i/E$ for $i = 1, 2$ so that we have a $k$-isogeny
$$
\theta \colon G \to \overline{G}_1 \times \overline{G}_2,
$$
with $\ker \theta = E$. We have $\overline{G}_1(k[t]) = \overline{G}_1(k) \cdot \overline{G}_1(k[t])^+$ by \cite[Theorem 3.1]{Stav} and $\overline{G}_2(k[t]) = \overline{G}_2(k)$ by Proposition \ref{P:anis}. Since $\theta(G_1(k[t])^+) = \overline{G}_1(k[t])^+$, we see that
$$
G(k[t]) = (\theta^{-1}(\overline{G}_1(k) \times \overline{G}_2(k)) \cap G(k[t])) \cdot G(k[t])^+.
$$
However, $\theta^{-1}(\overline{G}_1(k) \times \overline{G}_2(k)) \subset G(\bar{k})$, and since $G(\bar{k}) \cap G(k[t]) = G(k)$, our claim follows. $\Box$
\vskip3mm
Now suppose that $G$ is a reductive $k$-group, and let $S$ be a maximal $k$-split torus, and $M = Z_G(S)$ be its centralizer. Furthermore, let $Z$ be the central torus (i.e. the connected center $Z(G)^{\circ}$) of $G$, let $H = [G , G]$ be the semi-simple part of $G$, and let $\pi_0 \colon \widetilde{H} \to H$ be a $k$-defined universal cover. Set $\widetilde{G} = \widetilde{H} \times Z$ and denote by $\pi \colon \widetilde{G} \to G$ the composition of the morphism $\widetilde{G} \to H \times Z$ induced by $\pi_0$ followed by the product map.
\begin{cor}\label{L:isog}
With the preceding notations, we have $G(k[t]) = M(k) \cdot \pi_{k[t]}(\widetilde{G}(k[t]))$.
\end{cor}
\begin{proof}
According to Theorem \ref{T:Stavr1}, we have $G(k[t]) = G(k) \cdot G(k[t])^+$. On the other hand, $G(k) = M(k) \cdot G(k)^+$ by \cite[Proposition 6.11]{BorelTits} So,
$$
G(k[t]) = M(k) \cdot G(k[t])^+.
$$
Since $G(k[t])^+ \subset \pi_{k[t]}(\widetilde{G}(k[t]))$, we obtain our claim.
\end{proof}
\vskip1mm
\noindent {\bf 2.3. The case of $G$ simple, but not necessarily simply connected.} Let $G$ be an absolutely almost simple, but not necessarily simply connected, algebraic $k$-group, and let $\pi \colon \widetilde{G} \to G$ be a $k$-defined universal cover. Let $\mathscr{B}$ be the building associated with the simply connected group $\widetilde{G}_K$ as above. Then there is an action of the group $G(K)$ on $\mathscr{B}$ which, if composed with the group homomorphism $\pi(K) \colon \widetilde{G}(K) \to G(K)$, yields the canonical action of $\widetilde{G}(K)$ on $\mathscr{B}$. (In other words, the buildings associated with the groups $\widetilde{G}_K$ and $G_K$ are canonically isomorphic.) Let $\mathscr{A} \subset \mathscr{B}$ be the apartment constructed above, and let $\mathscr{Q} \subset \mathscr{A}$ be the corresponding sector. We claim that Theorem \ref{T:Marg}, proved for the group $\widetilde{G}(k[t])$, remains valid for $G(k[t])$, i.e. $\mathscr{Q}$ is a simplicial fundamental domain for the group $G(k[t])$. To see this, we will use Corollary \ref{L:isog} that $G(k[t]) = M(k) \cdot \pi(\widetilde{G}(k[t]))$. Then the mere inclusion $\pi(\widetilde{G}(k[t])) \subset G(k[t])$ yields the fact that $\mathscr{B} = G(k[t]) \cdot \mathscr{Q}$. To continue the argument, we observe that $M(k)$ acts trivially on the entire apartment $\mathscr{A}$, which follows, for example from \cite[Proposition 9.3.16]{KaPr}.
So, if two simplices $\mathscr{S}_1 , \mathscr{S}_2 \subset \mathscr{Q}$ are related by $\mathscr{S}_2 = \gamma(\mathscr{S}_1)$ with $\gamma \in G(k[t])$, then writing $\gamma = m \cdot \delta$ with $m \in M(k)$ and $\delta \in \pi(\widetilde{G}(k[t]))$, we will have $\delta(\mathscr{S}_1) = \mathscr{S}_2$. Then $\mathscr{S}_1 = \mathscr{S}_2$ by Theorem \ref{T:Marg}, completing the argument.
\vskip.5mm
\vskip1mm
We will now show that the description of pointwise stabilizers of subsets of $\mathscr{Q}$ obtained in Corollary \ref{C:stab1} for simply connected groups remains valid in the general case. We will use $\widetilde{\ }$ to denote the objects associated with $\widetilde{G}$, dropping the tilde to denote the objects associated with $G$; in particular, we set $\widetilde{\Gamma} = \widetilde{G}(k[t])$ and $\Gamma = G(k[t])$. Let $\Omega \subset \mathscr{Q}$ be an arbitrary subset. According to Corollary \ref{C:stab1}, we have $\widetilde{\Gamma} = \widetilde{\mathscr{U}} \rtimes \widetilde{L}_{I_{\Omega}}(k)$, where $\widetilde{\mathscr{U}}$ is the group of $k$-points of a $k$-defined unipotent group $\widetilde{U}$. Moreover, it follows from the results of Margaux \cite{Marg} that $\widetilde{U}$ is normalized by $\widetilde{M}$. On the other hand, by Corollary \ref{L:isog}, we have $\Gamma = M(k) \cdot \pi(\widetilde{\Gamma})$. Since $M(k)$ acts on $\mathscr{A}$ trivially, we conclude that $\Gamma_{\Omega} = M(k) \cdot \pi(\widetilde{\Gamma}_{\Omega})$. Furthermore, $\mathscr{U} := \pi(\widetilde{\mathscr{U}})$ coincides with $U(k)$, where $U = \pi(\widetilde{U})$, and $\pi(\widetilde{\Gamma}_{\Omega}) = \mathscr{U} \rtimes \pi(\widetilde{L}_{I_{\Omega}}(k))$. As we observed above, $M(k)$ normalizes $\mathscr{U}$ and $M(k) \cdot \pi(\widetilde{L}_{I_{\Omega}}(k)) = L_{I_{\Omega}}(k)$, we conclude that
\begin{equation}\label{E:stab10}
\Gamma_{\Omega} = \mathscr{U} \rtimes L_{I_{\Omega}}(k),
\end{equation}
as required.
\vskip1mm
To close this subsection, we now consider one special situation that will come up in the proof of the Raghunathan-Ramanathan theorem in \S \ref{S:RR}. Let $G$ be an absolutely almost simple $k$-group that is {\it quasi-split} over $k$. As usual, we denote by $S$ a maximal $k$-split torus of $G$. Then the centralizer $M = Z_G(S)$ is a maximal $k$-torus $T$ of $G$ (cf., e.g., \cite[Lemma 15.3.2, Proposition 16.2.2]{Springer}). Let $\ell/k$ be a finite Galois extension with Galois group $\mathscr{G} = \mathrm{Gal}(\ell/k)$ that contains the splitting field of $T$. Let $\Delta \subset \Phi(G , T)$ be a $\mathscr{G}$-invariant system of simple roots that corresponds to a $k$-defined Borel subgroup of $G$ containing $T$. Let $\mathscr{Q}_{\ell}$ be the sector in the apartment $\mathscr{A}_{\ell}$ defined using this $\Delta$; clearly, $\mathscr{Q}_{\ell}$ is $\mathscr{G}$-invariant.
\begin{prop}\label{P:stab2}
With assumptions and notations as above, let $\Gamma_{\ell} = G(\ell[t])$. Then for any $\mathscr{G}$-invariant subset $\Omega \subset \mathscr{Q}_{\ell}$, the pointwise stabilizer $(\Gamma_{\ell})_{\Omega}$ is of the form $U(\ell) \rtimes L_{I_{\Omega}}(\ell)$, where $U$ and $L_{I_{\Omega}}$ are $k$-defined and $U$ is unipotent.
\end{prop}
This follows from the above discussion combined with the observation that $U$ and $I_{\Omega}$ are $\mathscr{G}$-invariant.
\vskip1mm
\noindent {\bf 2.4. Arbitrary reductive groups.} Let $G$ be a reductive $k$-group, $Z = Z(G)^{\circ}$ be the central torus, and $H = [G , G]$ be the semi-simple part. Let $H_1, \ldots , H_r$ be the $k$-simple components of $H$. For each $i = 1, \ldots , r$, we let $\widetilde{H}_i$ (resp., $\overline{H}_i$) denote the corresponding simply connected (resp., adjoint) group. Also, let $F = Z \cap H$ and $\overline{Z} = Z/F$. Set
$$
\widetilde{G} = Z \times \widetilde{H}_1 \times \cdots \times \widetilde{H}_r \ \ \text{and} \ \ \overline{G} = \overline{Z} \times \overline{H}_1 \times \cdots \times \overline{H}_r.
$$
We then have obvious $k$-isogenies $\pi_1 \colon \widetilde{G} \to G$ and $\pi_2 \colon G \to \overline{G}$. Next, for each $i = 1, \ldots , r$, we can write
$$
\widetilde{H}_i = \mathrm{R}_{\ell_i/k}(\widetilde{\mathscr{H}}_i) \ \ \text{and} \ \ \overline{H}_i = \mathrm{R}_{\ell_i/k}(\overline{\mathscr{H}}_i)
$$
for absolutely almost simple simply connected and adjoint groups $\widetilde{\mathscr{H}}_i$ and $\overline{\mathscr{H}}_i$. Set $L_i = \ell_i((1/t))$ and let $\mathscr{B}_i$ be the Bruhat-Tits building associated with $\widetilde{\mathscr{H}}_i$ and $\overline{\mathscr{H}}_i$ over $L_i$. The natural identifications
$$
\overline{H}_i(K) \simeq \overline{\mathscr{H}}_i(L_i)
$$
enable us to define an action of $\overline{G}(K)$ on the product of buildings $\mathscr{B} = \mathscr{B}_1 \times \cdots \mathscr{B}_r$, and hence an action of $G(K)$ that factors through $\pi_2$. (We observe that then the natural action of $\widetilde{G}(K)$ on $\mathscr{B}$ factors through $\pi_1$.) Furthermore, for each $i = 1, \ldots , r$, one can fix a maximal $\ell_i$-split torus $\mathscr{S}_i$ of $\mathscr{H}_i$ and a system of simple roots $\Delta_i \subset \Phi(\widetilde{\mathscr{H}}_i , \mathscr{S}_i)$, and then consider the corresponding apartments $\mathscr{A}_i$ of $\mathscr{B}_i$ and the sectors $\mathscr{Q}_i \subset \mathscr{A}_i$. Set
$$
\mathscr{A} = \mathscr{A}_1 \times \cdots \times \mathscr{A}_r \ \ \text{and} \ \ \mathscr{Q} = \mathscr{Q}_1 \times \cdots \times \mathscr{Q}_r.
$$
\begin{prop}\label{P:genred}
{\rm (1)} Every point of $\mathscr{B}$ is equivalent under the group $\Gamma = G(k[t])$ to a point of $\mathscr{Q}$.
\vskip1mm
\noindent {\rm (2)} For any $x \in \mathscr{Q}$, the stabilizer $\Gamma_x$ is of the form $U(k) \rtimes L(k)$ for some reductive $k$-subgroup $L \subset G$ and some unipotent $k$-group $U$.
\end{prop}
\begin{proof}
Part (1) follows from the fact that every point of $\mathscr{B}$ is equivalent to a point of $\mathscr{Q}$ under the action of $\widetilde{\mathscr{H}}_1(\ell_1[t]) \times \cdots \times \widetilde{\mathscr{H}}_r(\ell_r[t])$, which is a consequence of Theorem \ref{T:Marg}. Arguing as in subsection 2.3, one reduces the proof of part (2) to the simply connected case, where the required statement directly follows from the results of \cite{Marg} --- cf. subsection 2.1.
\end{proof}
We note that the result remains valid when some of the $k$-simple components are $k$-anisotropic; the argument for that is similar to the one used in the proof of Theorem \ref{T:Stavr1}.
\vskip1mm
\noindent {\bf 2.5. Type-preserving automorphisms.} We refer the reader to \cite{AbBr} for general background on buildings. We recall that given a building $\mathscr{B}$, every apartment $\mathscr{A}$ of $\mathscr{B}$ is isomorphic to a standard Coxeter complex $\Sigma(W , S)$, where the Coxeter system $(W , S)$ is uniquely determined by $\mathscr{B}$ and called the {\it type} of $\mathscr{B}$. There exists a {\it type function} $\tau$ defined on the set of vertices of $\mathscr{B}$ with values in $S$ such that for every chamber $\mathscr{C}$ of $\mathscr{B}$, the restriction of $\tau$ to the set of vertices of $\mathscr{C}$ is a bijection onto $S$. Since $\mathscr{B}$ is a chamber complex, $\tau$ is unquely determined by its restriction to the set of vertices of a single chamber. To every simplicial automorphism $\phi$ of $\mathscr{B}$ one can associate a permutation $\pi = \pi(\phi)$ of $S$ such that
$$
\tau(\phi(v)) = \pi(\tau(v)) \ \ \text{for all vertices} \ \ v \ \ \text{of} \ \ \mathscr{B}.
$$
Then $\phi$ is called {\it type-preserving} if $\pi(\phi) = \mathrm{id}_S$. We note that in order to establish that $\phi$ is type-preserving, it is enough to show that $\tau(\phi(v)) = \tau(v)$ for all vertices $v$ of a single chamber $\mathscr{C}$ of $\mathscr{B}$. Here is one simple but important fact concerning type-preserving automorphisms.
\begin{lemma}\label{L:TP}
Let $\phi$ be a type-preserving automorphism of a building $\mathscr{B}$, let $X = \vert \mathscr{B} \vert$ be the geometric realization
of $\mathscr{B}$, and let $f \colon X \to X$ be the homeomorphism induced by $\phi$.
\vskip1mm
\noindent {\rm (1)} \parbox[t]{16cm}{If $\Sigma$ is a simplex of $\mathscr{B}$ such that $\phi(\Sigma) = \Sigma$, then $\phi(v) = v$ for every vertex $v$ of $\Sigma$, and hence $f$ acts trivially on $\vert \Sigma \vert$.}
\vskip1mm
\noindent {\rm (2)} \parbox[t]{16cm}{If $f$ fixes a point of $X$, then $\phi$ fixes a vertex of $\mathscr{B}$.}
\end{lemma}
\begin{proof}
(1): Let $\mathscr{C}$ be a chamber containing $\Sigma$. Since $\phi(\Sigma) = \Sigma$, for every vertex $v$ of $\Sigma$, the image $w = \phi(v)$ is also a vertex of $\Sigma$, hence of $\mathscr{C}$. But $\tau(w) = \tau(\phi(v)) = \tau(v)$ as $\phi$ is type-preserving. Since the restriction of $\tau$ to the set of vertices of $\mathscr{C}$ is injective, we conclude that $w = v$, i.e. $\phi(v) = v$.
\vskip1mm
(2): Assume that $f(x) = x$ for a point $x \in X$. Then there is a unique simplex $\Sigma_x$ in $\mathscr{B}$ such that $\vert \Sigma_x \vert$ is the carrier $F_x$ of $x$, i.e. the closed cell/simplex in $X$ that contains $x$ in its interior. We have $F_x = F_{f(x)} = f(F_x)$, which implies that $\phi(\Sigma_x) = \Sigma_x$. By part (1), $\phi$ (hence also $f$) fixes every vertex of $\Sigma_x$.
\end{proof}
We now return to the notations introduced in subsection 2.3. It is known that the group $\widetilde{G}(K)$ acts on $\mathscr{B}$ by type-preserving transformations (cf. \cite[Proposition 5.2.10, p. 165]{BT2}), but this may not be true for the action of $G(K)$. Nevertheless, we have the following.
\begin{prop}\label{P:TP}
Let $G$ be an absolutely almost simple algebraic $k$-group. Then $G(k[t])$ acts on $\mathscr{B}$ by type-preserving transformations.
\end{prop}
\begin{proof}
As we mentioned above, the action of $\tilde{G}(k[t])$ on $\mathscr{B}$ is type-preserving and factors through $\pi_{k[t]}$. So, it follows from Corollary \ref{L:isog} that it is enough to show that $M(k)$ acts on $\mathscr{B}$ by type-preserving transformations. We have already seen that $M(k)$ acts trivially on the apartment $\mathscr{A} \subset \mathscr{B}$ corresponding to the maximal $k$-split torus $S$. On the other hand, for any vertex $v$ of $\mathscr{B}$, there exists $g \in G(k[t])$ (and even in $\pi_{k[t]}(\tilde{G}(k[t]))$) such that $gv \in \mathscr{A}$. Then for any $m \in M(k)$ we have $m(gv) = gv$, so
$$
mv = (mg^{-1}m^{-1}g)v.
$$
Let $C = \ker \pi.$ Since $C$ is central, it follows that the coboundary map $G(k[t]) \to H^1(\mathcal{G}_k, C(\bar{k}[t]))$ is a group homomorphism (where $\mathcal{G}_k = \Ga(\bar{k}/k)$) and we have an exact sequence of groups
$$
\tilde{G}(k[t]) \stackrel{\pi_{k[t]}}{\longrightarrow} G(k[t]) \to H^1(\mathcal{G}_k, C(\bar{k}[t]))
$$
(see, e.g., \cite[Ch. 1, \S1.3.2]{PlRa}).
From this, we see that $\pi_{k[t]}(\widetilde{G}(k[t]))$ is a normal subgroup of $G(k[t])$, and the quotient $G(k[t])/\pi_{k[t]}(\widetilde{G}(k[t]))$ is abelian, so
$$
\pi_{k[t]}(\widetilde{G}(k[t])) \supset [G(k[t]) , G(k[t])].
$$
In particular, the element $mg^{-1}m^{-1}g$ lies in $\pi_{k[t]}(\tilde{G}(k[t]))$, and since the latter acts by type-preserving transformations, we conclude that $m$ also acts by type-preserving transformations, as required.
\end{proof}
Although we differentiated between a building and its geometric realization in the above discussion, we do not make this distinction elsewhere in the paper.
\vskip1mm
\noindent {\bf 2.6. The Fixed Point Theorem.} The proofs of the main results of this paper critically depend on the following
statement.
\begin{thm}\label{T:FPT}
{\rm (cf. \cite[Theorem 11.23]{AbBr})}
Let $\mathscr{B}$ be a finite product of affine Bruhat-Tits buildings. Then every finite group of isometries of $\mathscr{B}$ has a fixed point.
\end{thm}
\section{Some auxiliary results}
In this section, we establish several auxiliary statements needed for the proof of the main results. The reader is referred to \cite[Ch. I, \S 5]{Serre-GC} or \cite[1.3.2]{PlRa} for the basics of nonabelian cohomology. We begin with the following well-known lemma.
\begin{lemma}\label{L:2}
Let $G = N \rtimes H$ be a semi-direct product of abstract groups, and let $\Psi \subset H$ be a subgroup. Then a map
$f_{\xi} \colon \Psi \to G$ of the form $\delta \mapsto (\xi(\delta) , \delta)$ is a group homomorphism if and only if
the map $\xi \colon \Psi \to N$ is a 1-cocycle. Furthermore, two such homomorphisms $f_{\xi_1}$ and $f_{\xi_2}$ are conjugate by
an element $n \in N$ if and only if the corresponding cocycles $\xi_1$ and $\xi_2$ are equivalent.
\end{lemma}
\begin{proof}
We find by direct computation that for $\delta_1 , \delta_2 \in \Psi$, we have
$$
f_{\xi}(\delta_1)f_{\xi}(\delta_2) = (\xi(\delta_1) , \delta_1) (\xi(\delta_2) , \delta_2) = (\xi(\delta_1) \cdot (\delta_1 \xi(\delta_2)) , \delta_1 \delta_2), $$
and our first assertion follows. Furthermore,
$$
(n , 1)^{-1} f_{\xi}(\delta) (n , 1) = ( n^{-1} \cdot \xi(\delta) \cdot (\delta n) , \delta),
$$
yielding our second assertion.
\end{proof}
\vskip1mm
\begin{lemma}\label{L:3}
Let $G = N \rtimes H$ be a semi-direct product and $\pi \colon G \to H$ be the canonical projection. If $\Gamma \subset G$ is a subgroup such that
$\Gamma \cap N = \{ e \}$ and $H^1(\pi(\Gamma) , N) = 1$, then $\Gamma$ is conjugate in $G$ to a subgroup of $H$.
\end{lemma}
\begin{proof}
Set $\Psi = \pi(\Gamma)$. By our assumption, $\pi$ induces an isomorphism $\Gamma \to \Psi$; let $f \colon \Psi \to G$ be the
inverse of this isomorphism. It follows from Lemma \ref{L:2} that $f$ is of the form $f_{\xi}(\delta) = (\xi(\delta) , \delta)$ for some
1-cocycle $\xi \colon \Psi \to N$. Since $H^1(\Psi , N)$ is trivial, applying Lemma \ref{L:2} we see that $f$ is conjugate by an element of $N$
to the identity embedding $\Psi \to H$, $\delta \mapsto (1 , \delta)$, yielding our claim.
\end{proof}
\vskip1mm
\begin{lemma}\label{L:1}
Let $U$ be a unipotent group defined over a field $k$ of characteristic 0. Then for any finite group $\Gamma$ acting on $U(k)$, the cohomology set $H^1(\Gamma , U(k))$ is trivial.
\end{lemma}
\begin{proof}
If $U$ is commutative, then $U \simeq (\mathbb{G}_a)^d$ (cf. \cite[Ch. II, Remark 7.3]{Borel}). Then $U(k) \simeq k^d$ is a uniquely divisible abelian group, so $H^1(\Gamma , U(k)) = 0$.
In the general case, we argue by induction on $\dim U$. We may assume that $U$ is noncommutative, hence the center $V := Z(U)$ is a proper $k$-defined subgroup of positive dimension. We then have the exact sequence
$$
1 \to V \longrightarrow U \longrightarrow U/V \to 1
$$
of unipotent $k$-groups, where $V$ and $U/V$ are of dimension $< \dim U$. Since the Galois cohomology $H^1(\bar{k}/k , V)$ is trivial (cf., for example, \cite[Ch. 2, Lemma 2.7]{PlRa}), the sequence of $k$-points
$$
1 \to V(k) \longrightarrow U(k) \longrightarrow (U/V)(k) \to 1
$$
is also exact, i.e. $(U/V)(k)$ can be naturally identified with the quotient $U(k)/V(k)$. Moreover, the fact that $U(k)$ is Zariski-dense in $U$ implies that $V(k)$ is precisely the center of $U(k)$, hence is invariant under the action of $\Gamma$. Then the action of $\Gamma$ on $U(k)$ also descends to $(U/V)(k) = U(k)/V(k)$. We have the following exact sequence of pointed sets
$$
H^1(\Gamma , V(k)) \longrightarrow H^1(\Gamma , U(k)) \longrightarrow H^1(\Gamma , (U/V)(k)).
$$
By the induction hypothesis, the sets $H^1(\Gamma , V(k))$ and $H^1(\Gamma , (U/V)(k))$ are trivial, so it follows from the above sequence that the set $H^1(\Gamma , U(k))$ is also trivial.
\end{proof}
\vskip1mm
\noindent {\bf Remark.} The assertion of Lemma \ref{L:1} remains valid if $p = \mathrm{char}\: k > 0$ if one assumes that $U$ is connected and $k$-split and the order of $\Gamma$ is prime to $p$.
\vskip1mm
\begin{cor}\label{C:Conj}
Let $G$ be a semi-direct product $N \rtimes H$, where $N$ is the group of $k$-points $U(k)$ of a unipotent group $U$ over a field $k$ of characteristic 0. Then every finite subgroup of $G$ is conjugate to a subgroup of $H$.
\end{cor}
This follows from Lemmas \ref{L:3} and \ref{L:1}.
\section{Finite subgroups}
\noindent {\it Proof of Theorem \ref{T:I1}.} Let $\mathscr{B}$ be the product of buildings constructed for the reductive $k$-group $G$ over the field $K = k((1/t))$ in \S 2.4, and consider the natural action of the group $\Gamma = G(k[t])$ on $\mathscr{B}$. According to Theorem \ref{T:FPT}, any finite subgroup $\Psi$ of $\Gamma$ fixes a point $x \in \mathscr{B}$. Applying Proposition \ref{P:genred}(1), we see that replacing $\Psi$ by a $\Gamma$-conjugate subgroup, we may assume that $x$ lies in the sector $\mathscr{Q}$ in the product $\mathscr{A}$ of standard apartments (cf. \S 2.4). According to Proposition \ref{P:genred}(2), the stabilizer $\Gamma_x$ is of the form $U(k) \rtimes L(k)$ for some reductive subgroup $L$ of $G$ and some unipotent group $U$. Then by Corollary \ref{C:Conj}, the subgroup $\Psi$ is conjugate to a subgroup of $L(k) \subset G(k)$ within $\Gamma_x$, and the required fact follows. \hfill $\Box$
\vskip1mm
Before proving property (FC) for $G(k[t])$ over a $p$-adic field $k$, we first establish one finiteness result over a significantly broader class of fields. For this, we recall that according to Serre \cite[Ch. III, \S 4]{Serre-GC}, a profinite group $\mathscr{G}$ has {\it type} $(\mathrm{F})$ if it satisfies the following property:
\vskip2mm
\noindent $(\mathrm{F})$ {\it For every $n \geq 1$, the group $\mathscr{G}$ has only finitely many open subgroups of index $\leq n$.}
\vskip2mm
\noindent Furthermore, a field $k$ is of type $(\mathrm{F})$ if it is perfect and its absolute Galois group $\mathrm{Gal}(\bar{k}/k)$ is of type $(\mathrm{F})$. Examples of fields of type $(\mathrm{F})$ include $\mathbb{C}$, $\mathbb{R}$, and finite extensions of $\mathbb{Q}_p$ --- cf. {\it loc. cit.} For more examples of fields satisfying condition $(\mathrm{F})$ and its generalizations, we refer the reader to \cite{IR}.
\begin{prop}\label{P:F1}
Let $G$ be a connected linear algebraic group over a field $k$ of characteristic 0 that is of type $(\mathrm{F})$. Then for every finite group $\Gamma$, the group $G(k)$ has only finitely many conjugacy classes of subgroups isomorphic to $\Gamma$.
\end{prop}
\begin{proof}
Let $R = R(\Gamma , G)$ be the variety of representations of $\Gamma$ into $G$ (cf. \cite[2.4.7]{PlRa}). It is enough to show that the number of orbits of the adjoint action of $G(k)$ on $R(k)$ is finite. According to \cite[Theorem 2.17]{PlRa}, there are only finitely many orbits for the adjoint action of $G$ on $R$ and these orbits are Zariski-closed. Now, fix some $\rho \in R(k)$; then the corresponding orbit $V = G \cdot \rho$ is a $k$-defined homogeneous space of $G$. It is well-known that the orbits of $G(k)$ on $V(k)$ are in one-to-one correspondence with the elements of the kernel of the map $H^1(k , C) \to H^1(k , G)$, where $C$ is the centralizer of $\rho$ (cf. \cite[Ch. I, \S5.4, Corollary 1]{Serre-GC}). However, since $k$ is of type $(\mathrm{F})$, the set $H^1(k , C)$ is finite \cite[Ch. III, \S 4, Theorem 4]{Serre-GC}, and our claim follows.
\end{proof}
\vskip1mm
\begin{cor}\label{C:F1}
Let $G$ be an absolutely almost simple algebraic group defined over a field $k$ of characteristic 0 that is of type $(\mathrm{F})$. Then for every finite group $\Gamma$, the group $G(k[t])$ has only finitely many conjugacy classes of finite subgroups isomorphic to $\Gamma$.
\end{cor}
\begin{proof}
By Theorem \ref{T:I1} every finite subgroup of $G(k[t])$ is conjugate to a subgroup of $G(k)$. On the other hand, by Proposition \ref{P:F1}, the group $G(k)$ has only finitely many conjugacy subgroup isomorphic to $\Gamma$, so our assertion follows.
\end{proof}
\vskip1mm
\begin{prop}\label{P:F2}
Let $G$ be a reductive algebraic group over a finite extension $k$ of the $p$-adic field $\Q_p$. Then the group $G(k)$ has finitely many conjugacy classes of finite subgroups.
\end{prop}
\begin{proof}
As we already mentioned above, $k$ is of type $(\mathrm{F})$. So, it follows from Proposition \ref{P:F1} that it is enough to show that finite subgroups of $G(k)$ split into finitely many {\it isomorphism classes}. Considering a faithful $k$-defined representation $G \hookrightarrow \mathrm{GL}_n$, we see that it is enough to prove this fact for $G = \mathrm{GL}_n$. It is well-known that every finite subgroup of $\mathrm{GL}_n(k)$ is conjugate to a subgroup of $\mathrm{GL}_n(\mathcal{O})$, where $\mathcal{O}$ is the valuation ring of
$k$ (cf. \cite[Proposition 1.12]{PlRa}). On the other hand, the congruence subgroup $\mathrm{GL}_n(\mathcal{O} , \mathfrak{p}^d)$, where $\mathfrak{p} \subset \mathcal{O}$ is the valuation ideal, is torsion-free for $d$ large enough (``Minkowski's Lemma," cf. \cite[p. 234]{PlRa}). Then every finite subgroup of $\mathrm{GL}_n(k)$ is isomorphic to a subgroup of the finite group $\mathrm{GL}_n(\mathcal{O}) / \mathrm{GL}_n(\mathcal{O} , \mathfrak{p}^d)$, and the required fact follows.
\end{proof}
\vskip1mm
\begin{thm}\label{T:FC-1}
Let $G$ be a reductive algebraic group defined over a finite extension $k$ of the $p$-adic field $\mathbb{Q}_p$. Then the group $G(k[t])$ has finitely many conjugacy classes of finite subgroups.
\end{thm}
This follows from Theorem \ref{T:I1} and Proposition \ref{P:F2}.
\vskip2mm
\section{The Raghunathan-Ramanathan theorem}\label{S:RR}
\noindent {\bf \ref{S:RR}.1. The statement and preliminary remarks.} In \cite{RagRam}, Raghunathan and Ramanathan proved the following theorem.
\begin{thm}\label{T:RagRam}
Let $G$ be a connected reductive algebraic group over a field $k$, and let $\pi \colon B \to \mathbb{A}^1_k$
be a $G$-torsor over the affine line $\mathbb{A}^1_k = \mathrm{Spec}\: k[t]$. If $\pi$ is trivialized by the base change from $k$ to $\bar{k}$ (separable closure), then $\pi$ is obtained by the base change $\mathbb{A}^1_k \to \mathrm{Spec}\: k$ from a $G$-torsor $\pi_0 \colon B_0 \to \mathrm{Spec}\: k$.
\end{thm}
In this section, we present a simple proof of this theorem over fields of characteristic zero that is based on the Fixed Point Theorem; the argument in the general case requires some technical adjustments. First, we note that if $k$ has characteristic zero (or, more generally, is perfect) then every $G$-torsor over the affine line is trivialized by the base change $\bar{k}/k$. In fact, this remains true if $\mathbb{A}^1_k$ is replaced by a Zariski-open subset, and actually by any geometrically connected smooth affine curve if $G$ is semi-simple.
\begin{lemma}\label{L:Triv}
Let $K$ be an algebraically closed field of characteristic 0, $G$ a connected reductive algebraic $K$-group, and $C$ a smooth affine curve over $K$. In each of the following situations
\vskip1mm
\noindent {\rm (1)} \parbox[t]{15cm}{$C$ is a Zariski-open subset of $\mathbb{A}^1_K$;}
\vskip1mm
\noindent {\rm (2)} \parbox[t]{15cm}{$G$ is semi-simple,}
\vskip1mm
\noindent we have $H^1(C , G) = 1$.
\end{lemma}
\begin{proof}
(Cf. the proof of Corollary 7.5 in \cite{CRR-GR}) Let $\beta \colon H^1(C , G) \to H^1(K(C) , G)$ be the map given by specialization to the
generic point. By Tsen's theorem, the field $K(C)$ has cohomological dimension $\leq 1$ (cf. \cite[Ch. II, 3.3]{Serre-GC}), so applying a theorem due to Steinberg \cite[Ch. III, 2.3]{Serre-GC}, we see that $H^1(K(C) , G) = 1$. So, it is enough to show that $\ker \beta$ is trivial. On the other hand, according to \cite{Nisn2}, there is a bijection between $\ker \beta$ and the double coset space
$$
\mathrm{cl}(G, K(C), V) := G(\mathbf{A}^{\infty}(V)) \backslash G(\mathbf{A}(V)) / G(K(C))
$$
where $G(\mathbf{A}(V))$ is the group of {\it rational} adeles of $G$ associated with the set $V$ of discrete valuations corresponding to the closed points of $C$, and $G(\mathbf{A}^{\infty}(V))$ and $G(K(C))$ are the subgroups of integral and principal adeles, respectively (see \cite{CRR-Isr} for a discussion of adeles in this context). Fix a maximal $K$-torus $T$ of $G$, which of course splits over $K$. A standard argument (cf. \cite{CRR-Isr}) using strong approximation for the opposite maximal unipotent with respect to $V$ (which holds because $C$ is assumed to be affine) shows that every double coset in the class set $\mathrm{cl}(G, K(C), V)$ has a representative lying in $T(\mathbf{A}(V))$. We have a $K$-isomorphism $T = \mathbb{G}_m^d$, where $d = \dim T$, and the double coset space $\mathrm{cl}(\mathbb{G}_m, K(C), V)$ is a group that can be identified with the Picard group $\mathrm{Pic}\: C$.
Now, if $C$ is a Zariski-open subset of $\mathbb{A}^1_K$, then the group $P := \mathrm{Pic}\: C$ is trivial, and the triviality of $\ker \beta$ follows immediately. In the general case, we assume that $G$ is semi-simple, and let $\pi \colon \widetilde{G} \to G$ denote the universal cover. Since the curve $C$ is affine, there exists a surjective group homomorphism $\mathbf{J}(K) \to P$, where $\mathbf{J}$ is the Jacobian of projective curve $\mathbf{C}$ that contains $C$. It follows that the group $P$ is divisible, i.e. $nP = P$ for any $n \geq 1$. Consequently, for any $n \geq 1$, every coset in the class group $\mathrm{cl}(T, K(C), V)$ can be chosen in $T(\mathbf{A}(V))^n$. On the other hand, for $n = \vert \ker \pi \vert$, we have the inclusion
\begin{equation}\label{E:N}
G(\mathbf{A}(V))^n \subset \pi(\widetilde{G}(\mathbf{A}(V))).
\end{equation}
Since $\widetilde{G}$ is $K$-split, by considering unipotent subgroups, we conclude that $\tilde{G}$ has strong approximation and therefore $\widetilde{G}(\mathbf{A}(V)) = \widetilde{G}(\mathbf{A}^{\infty}(V))\widetilde{G}(K(C))$. It follows that $\pi(\tilde{G}(\mathbf{A}(V)) \subset G(\mathbf{A}^{\infty}(V)) G(K(C))$. Combining this with (\ref{E:N}), we see that
$$
G(\mathbf{A}(V))^n \subset G(\mathbf{A}^{\infty}(V)) G(K(C)),
$$
and, in particular, $T(\mathbf{A}(V))^n$ is contained in $G(\mathbf{A}^{\infty}(V)) G(K(C))$. Eventually, we conclude that $\mathrm{cl}(G, K(C), V)$ reduces to a single element, and hence $\ker \beta$ is trivial.
\end{proof}
\vskip1mm
Turning now to the proof of Theorem \ref{T:RagRam} in the case where $\mathrm{char}\: k = 0$, we recall the Hochschild-Serre sequence (cf. \cite[p. 96]{Milne-LEC} for the commutative case):
\begin{equation}\label{E:HS}
1 \to H^1(\mathrm{Gal}(\bar{k}/k) , G(\bar{k}[t])) \longrightarrow H^1(\mathbb{A}^1_k , G) \stackrel{\rho}{\longrightarrow} H^1(\mathbb{A}^1_{\bar{k}} , G),
\end{equation}
which is exact as a sequence of pointed sets. The $G$-torsors over $\mathbb{A}^1_k$ trivialized by the base change $\bar{k}/k$ are classified by the
the elements of $\ker \rho$. On the other hand, (\ref{E:HS}) yields a natural bijection
$$
\ker \rho \, \simeq \, H^1(\mathrm{Gal}(\bar{k}/k) , G(\bar{k}[t])).
$$
It follows that to complete the proof of Theorem \ref{T:RagRam}, it is enough to prove Theorem \ref{T:I2}. Furthermore, the map $H^1(\bar{k}/k , G(\bar{k})) \to H^1(\bar{k}/k , G(\bar{k}[t]))$ is injective by specialization argument (we can specialize at any $t \in k$, e.g. $t = 0$). So, we only need to prove that it is surjective.
\vskip1mm
\noindent {\bf \ref{S:RR}.2. Reduction to quasi-split simple adjoint groups.} We will now show that it is enough to prove Theorem \ref{T:I2} for quasi-split simple adjoint groups. First, we reduce to the case of adjoint groups. Given a reductive $k$-group $G$, there exists an exact sequence of algebraic groups
$$
1 \to F \longrightarrow G \stackrel{\pi}{\longrightarrow} \overline{G} \to 1
$$
in which $F$ is finite and $\overline{G}$ is the direct product $\overline{H} \times T$ of a semi-simple adjoint $k$-group $\overline{H}$ and a $k$-torus $T$. As in the proof of Corollary \ref{L:isog}, we see that Theorem \ref{T:Stavr1} implies the equality
$$
\overline{G}(\bar{k}[t]) = \overline{G}(\bar{k}) \cdot \pi_{\bar{k}[t]}(G(\bar{k}[t])),
$$
and since $\pi_{\bar{k}}(G(\bar{k})) = \overline{G}(\bar{k})$, we conclude that the sequence
\begin{equation}\label{E:ExSeq1}
1 \to F(\bar{k}) \longrightarrow G(\bar{k}[t]) \stackrel{\pi_{\bar{k}[t]}}\longrightarrow \overline{G}(\bar{k}[t]) \to 1
\end{equation}
is exact. We then have the following commutative diagram with exact rows
$$
\xymatrix{H^1(\bar{k}/k , G(\bar{k})) \ar[r]^{\gamma_1} \ar[d]_{\alpha} & H^1(\bar{k}/k , \overline{G}(\bar{k})) \ar[r] \ar[d]_{\beta} & H^2(\bar{k}/k , F(\bar{k})) \ar[d]_{=} \\ H^1(\bar{k}/k , G(\bar{k}[t])) \ar[r]^{\gamma_2} & H^1(\bar{k}/k , \overline{G}(\bar{k}[t])) \ar[r] & H^2(\bar{k}/k , F(\bar{k}))}
$$
If we assume that the map $H^1(\bar{k}/k , \overline{H}(\bar{k})) \to H^1(\bar{k}/k , \overline{H}(\bar{k}[t]))$ is surjective, then in view of the fact that $T(\bar{k}[t]) =
T(\bar{k})$ (which follows from the case of $\mathbb{G}_m$), the map $\beta$ will also be surjective, hence bijective. So, by a diagram chase, we find that $\alpha$ is also surjective (one needs to use the fact that $H^1(\bar{k}/k , F(\bar{k}))$ acts transitively on the fibers of $\gamma_2$ --- cf. \cite[Ch. I, \S 5, Proposition 42, p. 54]{Serre-GC}).
Assume now that $G$ is semi-simple and adjoint. Writing $G$ as a direct product of $k$-simple groups, each of which is obtained by restriction of scalars from an absolutely simple group, we see that it is enough to consider the case where $G$ is an absolutely simple adjoint group. Such a group is an inner twist of a quasi-split group $G_0$, and we have compatible bijections
$$
H^1(\bar{k}/k , G(\bar{k})) \simeq H^1(\bar{k}/k , G_0(\bar{k})) \ \ \text{and} \ \ H^1(\bar{k}/k , G(\bar{k}[t])) \simeq H^1(\bar{k}/k , G_0(\bar{k}[t]))
$$
(cf. \cite[Ch. I, 5.3]{Serre-GC}). So, without loss of generality we may assume that $G$ is quasi-split.
\vskip1mm
\noindent {\bf \ref{S:RR}.3. Conclusion of the argument.} Thus, it remains to be shown that if $k$ is a field of characteristic 0, then {\it for an absolutely simple adjoint quasi-split $k$-group $G$ and a finite Galois extension $\ell/k$ with Galois group $\mathscr{G} = \mathrm{Gal}(\ell/k)$, the natural map
\begin{equation}\label{E:surj10}
H^1(\mathscr{G} , G(\ell)) \longrightarrow H^1(\mathscr{G} , G(\ell[t]))
\end{equation}
is surjective}.
For this we will use the natural action of the group $\Omega = \Gamma_{\ell} \rtimes \mathscr{G}$, where $\Gamma_{\ell} = G(\ell[t])$, on the Bruhat-Tits building $\mathscr{B}_{\ell}$ associated with $G$ over $L = \ell((1/t))$ --- see \S \ref{S:building} for the relevant notations.
Let $S$ be a maximal $k$-split torus of $G$, and $T$ be the maximal $k$-torus of $G$ containing $S$. Then the splitting field $\ell_0$ of $T$ is the minimal Galois extension of $k$ over which $G$ becomes split. Without loss of generality, we may assume that $\ell \supset \ell_0$.
Since $G$ is $k$-quasi-split, there exists a $k$-defined Borel subgroup $B$ of $G$ containing $T$. Then the system of simple roots $\Delta$ in the root system $\Phi(G , T)$ corresponding to $B$ is invariant under the Galois group $\mathscr{G}$.
Let $\mathscr{A}_{\ell}$ be the apartment in $\mathscr{B}_{\ell}$ corresponding to $T$, and let $\mathscr{Q}_{\ell}$ be the sector in $\mathscr{A}_{\ell}$ corresponding to the system of simple roots $\Delta$ (see \S \ref{S:building}). Since $T$ is $k$-defined, the apartment $\mathscr{A}_{\ell}$ is $\mathscr{G}$-invariant, and furthermore, since $\Delta$ is $\mathscr{G}$-invariant, $\mathscr{Q}_{\ell}$ is also $\mathscr{G}$-invariant.
Now, let $\xi \colon \mathscr{G} \to \Gamma_{\ell}$ be a 1-cocycle and let $$\iota_{\xi} \colon \mathscr{G} \to \Omega = \Gamma \rtimes \mathscr{G}, \ \ \sigma \mapsto (\xi(\sigma) , \sigma)$$ be the corresponding embedding (see Lemma \ref{L:2}). Using Theorem \ref{T:FPT}, we can find a point $x \in \mathscr{B}_{\ell}$ fixed by the finite subgroup $\iota_{\xi}(\mathscr{G}) \subset \Omega$. As we have seen in \S \ref{S:building}.3, Theorem \ref{T:Marg} remains valid for not necessarily simply connected groups, so there exists $g \in \Gamma_{\ell}$ such that $y = gx \in \mathscr{Q}_{\ell}$. Thus, replacing $\iota_{\xi}$ with the conjugate embedding $g \iota_{\xi} g^{-1}$, which is of the form $\iota_{\xi'}$ for a cocycle $\xi'$ that is equivalent to $\xi$ (see the proof of Lemma \ref{L:2}), we may (and will) assume that $\iota_{\xi}(\mathscr{G})$ fixes a point $y \in \mathscr{Q}_{\ell}$. Let $F = F_y$ be the carrier of $y$ (closed simplex containing $y$ in its interior). Clearly, $F \subset \mathscr{Q}_{\ell}$ and $F$ is invariant under $\iota_{\xi}(\mathscr{G})$. This means that for any $\sigma \in \mathscr{G}$, we have
$$
\xi(\sigma)(\sigma(F)) = F,
$$
and since $\sigma(F) \subset \mathscr{Q}_{\ell}$, we conclude from the uniqueness in Theorem \ref{T:Marg} that $\sigma(F) = F$. Thus, $F$ is $\mathscr{G}$-invariant. But then $\xi(\sigma)(F) = F$, so since the action of $\Gamma$ is type-preserving (cf. Proposition \ref{P:TP}), we obtain from Lemma \ref{L:TP}
that $\xi(\sigma)$ fixes $F$ pointwise. Thus, $\xi$ is a cocycle with values in $H(F)$, the pointwise stabilizer of $F$.
According to Proposition \ref{P:stab2}, the stabilizer $H(F)$ is of the form $U(\ell) \rtimes L(\ell)$ where $L \subset G$ is a $k$-defined reductive subgroup and $U$ is a $k$-defined unipotent group. Since the map $H^1(\mathscr{G} , L(\ell)) \to H^1(\mathscr{G} , L(\ell) \rtimes U(\ell))$ is bijective (cf. \cite[Ch. II, Proposition 2.9]{PlRa}), we see that $\xi$ is equivalent to a cocycle with values values in $L(\ell)$, hence lies in the image of the map $H^1(\ell/k , G(\ell)) \to H^1(\ell/k , G(\ell[t]))$, as required.
\vskip5mm
\vskip3mm
\vskip2mm
\noindent {\small {\bf Acknowledgements.} We are grateful to Vladimir Chernousov and Gopal Prasad for useful comments. The third-named author was partially supported by NSF grant DMS-2154408.}
\vskip5mm
\bibliographystyle{amsplain}
| -58,367.642031 |
[
-2.73046875,
2.509765625
] | 49.300699 |
[
-2.615234375,
0.55078125,
-2.291015625,
-5.98046875,
-1.3369140625,
8.78125
] |
[
4.50390625,
8.8046875,
2.716796875,
7.09375
] | 470 | 7,159 |
[
-3.544921875,
4.0625
] | 32.905373 |
[
-5.46875,
-3.962890625,
-5.7265625,
-2.458984375,
1.837890625,
13.734375
] | 0.531781 | 33.687121 | 18.103087 | 2.62563 |
[
1.7319358587265015
] | -36,723.470086 | 4.990781 | -58,276.786707 | 0.235103 | 5.896253 |
[
-1.3388671875,
-3.212890625,
-4.1875,
-5.921875,
1.81640625,
12.9921875
] |
[
-5.6484375,
-1.90234375,
-2.1875,
-1.521484375,
3.994140625,
4.33203125
] | |
BkiUbIA5qWTD6fM1V7xu
|
\section{Introduction and main results}
Correlation is a fundamental notion of science and the Gaussian distribution, which is a building block of probability and statistics, has a nearly 300-year history dating back to Abraham de Moivre. When combined, these two concepts have proven essential to many branches of mathematics and physics. However, our understanding of them is far from complete.
Inequalities involving correlations between Gaussian random variables have important connections to a number of active research areas, such as small-ball probabilities (cf. \cite{Li} and \cite{Shao}) and the zeros of random polynomials (cf. \cite{LS}). In 2014, Royen's proof of the elusive Gaussian correlation inequality conjecture \cite{Royen,LM} was a significant breakthrough that inspired renewed hope in solving another long-standing and challenging problem, the Gaussian product inequality (GPI) conjecture, which (see, for example, \cite{MNPP16}) implies the `real linear polarization constant conjecture' \cite{BST98} and is deeply linked to Kagan, Linnik and Rao's famous $U$-conjecture \cite{KLR1973}.
Most generally, the GPI conjecture (see Li and Wei \cite{LW12}) states that for any non-negative real numbers $y_j$, $j=1,\ldots,{n}$, and any ${n}$-dimensional real-valued centered Gaussian random vector $(X_1,\dots,X_{n})$,
\begin{eqnarray}\label{LW-inequ}
E \left[\prod_{j=1}^{n}|X_j|^{y_j}\right]\geq \prod_{j=1}^{n}E[|X_j|^{y_j}].
\end{eqnarray}
The difficulty of proving the GPI lies in the subtlety of the inequality and thus, so far, only partial results have been obtained. Frenkel \cite{Fr08} proved (\ref{LW-inequ}) when all exponents equal $2$ using algebraic methods. By using the Gaussian hypergeometric function as a powerful tool, Lan et al. \cite{LHS} were successful in proving the following $3$-dimensional GPI: for any $m_1,m_2\in\mathbb{N}$ and any centered Gaussian random vector $(X_1,X_2,X_3)$,
$$
E[X_1^{2m_1}X_2^{2m_2}X_3^{2m_2}]\ge E[X_1^{2m_1}]E[X_2^{2m_2}]E[X_3^{2m_2}].
$$
Recently, we gave a complete characterization of the $2$-dimensional GPI in \cite{RusSun3} and designed a computational algorithm based on sums-of-squares that can potentially be used to rigorously prove any GPI with all but one exponent fixed in \cite{RusSun2}. Furthermore, in \cite[Lemma 2.3]{RusSun} we proved a stronger version of the GPI for even-integer exponents when all correlations are non-negative (see also \cite{GenOui} for a related result, and \cite{Edel} and \cite{GenOui2} for distributional generalizations). In the same paper, we used combinatorial methods to prove the following GPIs: for any $m_3\in\mathbb{N}$ and any centered Gaussian random vector $(X_1,X_2,X_3)$,
$$
E[X_1^{2}X_2^{4}X_3^{2m_3}]\ge E[X_1^{2}]E[X_2^{4}]E[X_3^{2m_3}],
$$
and
$$
E[X_1^{2}X_2^{6}X_3^{2m_3}]\ge E[X_1^{2}]E[X_2^{6}]E[X_3^{2m_3}],
$$
and left the remainder of Theorem \ref{thmm1m2m3} below as a conjecture.
Throughout this paper, any Gaussian random variable is assumed to be real-valued and nondegenerate,
i.e., has positive variance. We will prove the following 3D-GPI.
\begin{thm}\label{thmm1m2m3}
Let $m_2,m_3 \in \mathbb{N}$. For any centered Gaussian random vector $(X_1,X_2,X_3)$,
\begin{equation}\label{GPIk}
E[X_1^{2} X_2^{2m_2} X_3^{2m_3}]\ge E[X_1^{2}] E[X_2^{2m_2}] E[X_3^{2m_3}].
\end{equation}
The equality sign holds if and only if $X_1,X_2,X_3$ are independent.
\end{thm}
In \cite{LHS}, all 3 exponents were unbounded, but 2 were equal to each other, allowing for some symmetry and convexity to be exploited. The main contribution of this present paper lies in the fact that we consider $3$ \textit{distinct} exponents, $2$ of which are unbounded. Even this special case is extremely challenging since the subtle nature of the GPI resists even small modifications. For this reason, we used the computer (\textbf{Mathematica}) as a time-saving guide at key junctures to help us guess appropriate simplifications under which the GPI still holds. We emphasize that despite this guidance, the computer was less of a crutch and more of a shot in the arm, and all proofs contained in this paper remain rigorous and easily verifiable. We strongly believe that computing will be an essential tool used in the eventual complete proof of the GPI.
As mentioned before, there is still much to be discovered about the multivariate Gaussian distribution. In particular, we found comparisons of moments to be scarce in the literature even for the bivariate case. In an attempt to address this gap, we discovered a very close relationship between Gaussian moment ratio inequalities (MRIs) and the GPI. On one hand, in \cite[Theorem 4.3]{RusSun} we gave a novel bivariate MRI that was weaker than the GPI. On the other hand, in this paper, we prove a new MRI (see Theorem \ref{thm2} below) that is stronger than Theorem \ref{thmm1m2m3}.
Define
$$
{\mathcal S}:=\{(1,m_3):m_3\ge 5\}\bigcup\{(2,m_3):m_3\ge 3\}\bigcup\{(m_2,m_3):m_3\ge m_2\ge3\},
$$
$$
r_{m_2,m_3}=(2m_2+1)(2m_3+1)+1,\ \ \ \ t_{m_2,m_3}=\frac{1}{r_{m_2,m_3}+\left(1+\frac{1}{2m_2}\right)\left(1+\frac{1}{2m_3}\right)},
$$
and for $\frac{1}{r^2_{m_2,m_3}}<z\le 1$,
\begin{eqnarray*}
&&H_{m_2,m_3}(z)\\
&=&\frac{(m_2+m_3+1)(r_{m_2,m_3}z-1)+\sqrt{[(m_3-m_2)(r_{m_2,m_3}z-1)]^2+(2m_2+1)^3(2m_3+1)^3z}}{r_{m_2,m_3}^2z-1}.
\end{eqnarray*}
For random variables $X$ and $Y$, denote by ${\rm Cov}(X,Y)$ and ${\rm Corr}(X,Y)$ their covariance and correlation, respectively. We will show that the GPI (\ref{GPIk}) is implied by the following MRI.
\begin{thm}\label{thm2}
Let $(X_2,X_3)$ be a centered Gaussian random vector. If $(m_2,m_3)\in{\mathcal S}$, then
\begin{eqnarray}\label{MRI22}
&&\frac{\left|E[ X_2^{2m_2+1}X_3^{2m_3+1}]\right|}{(2m_2+1)(2m_3+1)E[ X_2^{2m_2}X_3^{2m_3}]}\nonumber\\
&\le&
\begin{cases}
|{\rm Cov}(X_2,X_3)|,\ \ &{\rm if}\ |{\rm Corr}(X_2,X_3)|\le \sqrt{t_{m_2,m_3}},\nonumber\\
H_{m_2,m_3}([{\rm Corr}(X_2,X_3)]^2)\cdot|{\rm Cov}(X_2,X_3)|,\ \ &{\rm if}\ \sqrt{t_{m_2,m_3}}<|{\rm Corr}(X_2,X_3)|.
\end{cases}\nonumber\\
&&
\end{eqnarray}
The equality sign holds if and only $X_2$ and $X_3$ are independent.
\end{thm}
\noindent Note that the MRI (\ref{MRI22}) does not hold if $m_2=1$ and $1\le m_3\le 4$, or $m_2=m_3=2$. However, for these particular cases, the GPI (\ref{GPIk}) has been proved by \cite{LHS}. These cases can also be easily handled by the SOS method developed in \cite{RusSun2}.
In addition to the intrinsic value of solving the GPI and the implications a proof will have on related problems and fields, we can see that the study of the GPI itself has already led to some interesting new results, including the MRIs discussed above, and highly non-trivial combinatorial identities such as \cite[Lemma 2.5 and Corollary 2.8]{LHS}. Therefore, the GPI certainly deserves careful consideration. Finally, we hope that the interplay between computing and hard analysis that we describe in this paper will inspire confidence in the research community to extend our results.
The remainder of this paper is structured as follows. In Section \ref{sec2}, we explain the relation between the GPI (\ref{GPIk}) and the MRI (\ref{MRI22}). First, we derive an inequality (see (\ref{Augsdyy}) below) which is slightly stronger than the GPI. Then, we show that this inequality is equivalent to a hypergeometric function ratio inequality (HFRI) (see (\ref{Aug18a}) below) and hence is equivalent to the MRI (\ref{MRI22}). In Sections \ref{sec3} and \ref{sec4}, we use different methods to prove the HFRI (\ref{Aug18a}) for cases $1\le m_2\le 7$ and $m_2\ge 8$ separately. In Section \ref{sec5}, we make some concluding remarks. To complete the proof of Section \ref{sec4}, we need the sums-of-squares (SOS) expansion of a difference function. This expansion is obtained by \textbf{Mathematica} and given in Section \ref{sec6}, the Appendix.
To simplify notation, in the sequel, we use $r$, $t$ and $H$ to denote $r_{m_2,m_3}$, $t_{m_2,m_3}$ and $H_{m_2,m_3}$, respectively, whenever no confusion is caused. Throughout this paper, we assume without loss of generality that $m_3\ge m_2$.
\section{Relation between GPI and MRI}\label{sec2}\setcounter{equation}{0}
In this section, we derive the MRI (\ref{MRI22}) and show that its validity implies the validity of the GPI (\ref{GPIk}).
Denote by $F(a, b; c; z)$ the hypergeometric function (cf. \cite{R}):
$$
F(a, b; c; z)=\sum_{j=0}^{\infty}\frac{(a)_{j}(b)_{j}}{(c)_{j}}\cdot\frac{z^j}{j!},\ \ \ \ |z|<1,
$$
where $(\alpha)_{j}:=\alpha(\alpha+1)\cdots(\alpha+j-1)$ for $j\ge 1$, and $(\alpha)_0=1$ for $\alpha\not=0$. Let $(X_2,X_3)$ be a centered Gaussian random vector and $x={\rm Corr}(X_2,X_3)$. By the moment formula (cf. \cite[Page 261]{KBJ}), we get
\begin{eqnarray}\label{we2}
&&E[ X_2^{2m_2}X_3^{2m_3}]\nonumber\\
&=&(2m_2-1)!!(2m_3-1)!![{\rm Var}(X_2)]^{m_2}[{\rm Var}(X_3)]^{m_3}F\left(-m_2,-m_3;\frac{1}{2};x^2\right),
\end{eqnarray}
and
\begin{eqnarray}\label{we1}
&&E[ X_2^{2m_2+1}X_3^{2m_3+1}]\nonumber\\
&=&(2m_2+1)!!(2m_3+1)!![{\rm Var}(X_2)]^{\frac{2m_2+1}{2}}[{\rm Var}(X_3)]^{\frac{2m_3+1}{2}}xF\left(-m_2,-m_3;\frac{3}{2};x^2\right).\ \ \ \
\end{eqnarray}
Consider the following functions contiguous to ${F}(a,b;c;z)$:
\begin{equation}\label{contiguous}
{F}(a\pm 1,b;c;z),\quad {F}(a,b\pm 1; c;z),\quad {F}(a,b;c\pm 1;z).
\end{equation}
To simplify notation, we denote ${F}(a,b;c;z)$ and
the six functions in (\ref{contiguous}) respectively by
$$
{F},\quad {F}(a\pm 1),\quad {F}(b\pm 1),\quad {F}(c\pm 1).
$$
We have the following relations of Gauss between contiguous functions (cf. \cite[2.8-(31), (37), (38), pages 102 and 103]{B}):
\begin{eqnarray}\label{Gauss3}
&&[c-2a-(b-a)z]F+a(1-z)F(a+1)-(c-a)F(a-1)=0,\nonumber\\
&&(1-z)F=F(a-1)-c^{-1}(c-b)zF(c+1),\nonumber\\
&&(b-a)(1-z)F-(c-a)F(a-1)+(c-b)F(b-1)=0.
\end{eqnarray}
\subsection{A stronger inequality}
By \cite[Lemma 2.1]{RusSun}, to prove the GPI (\ref{GPIk}), we may assume without loss of generality that $X_1 = X_2+aX_3$ for some $a \in \mathbb{R}$ and $E[X_2^2]=E[X_3^2]=1$.
Define
$$x=E[X_2 X_3].
$$
Then,
$$
E[X_1^2]=a^2+1+2ax.
$$
Consider the moment ratio:
\begin{eqnarray}\label{moment}
&&\frac{E[(X_2+aX_3)^{2}X_2^{2m_2} X_3^{2m_3} ]} {E[X_1^{2}] E[X_2^{2m_2}] E[X_3^{m_3}]}\nonumber\\
&=&\frac{E[a^2X_2^{2m_2} X_3^{2m_3+2} + X_2^{2m_2+2} X_3^{2m_3} + 2aX_2^{2m_2+1} X_3^{2m_2+1}]} {(a^2+1+2ax)(2m_2-1)!!(2m_3-1)!!}\nonumber\\
&=&\Bigg[a^2\left(2m_3+1\right)F\left(-m_3-1,-m_2;\frac{1}{2};x^2\right)+\left(2m_2+1\right)F\left(-m_3,-m_2-1;\frac{1}{2};x^2\right)\nonumber\\
&&+2ax\left(2m_3+1\right)\left(2m_2+1\right) F\left(-m_3,-m_2;\frac{3}{2};x^2\right)\Bigg]\cdot\frac{1}{a^2+1+2ax}.
\end{eqnarray}
Then, the proof of Theorem \ref{thmm1m2m3} is complete if we can show that for any $a\in\mathbb{R}$ and $x\in[-1,1]$,
\begin{eqnarray}\label{gen2}
&&a^2\left(2m_3+1\right)F\left(-m_3-1,-m_2;\frac{1}{2};x^2\right)+\left(2m_2+1\right)F\left(-m_3,-m_2-1;\frac{1}{2};x^2\right)\nonumber\\
&&+2ax\left(2m_3+1\right)\left(2m_2+1\right) F\left(-m_3,-m_2;\frac{3}{2};x^2\right)\nonumber\\
&>&a^2+1+2ax.
\end{eqnarray}
Equivalently, we need to show that for any $a\in(-\infty,0]$ and $x\in[0,1]$,
\begin{eqnarray}\label{Aug9}
F(a,x)&:=&a^2\left[\left(2m_3+1\right)F\left(-m_2,-m_3-1;\frac{1}{2};x^2\right)-1\right]\nonumber\\
&&+2ax\left[\left(2m_3+1\right)\left(2m_2+1\right) F\left(-m_2,-m_3;\frac{3}{2};x^2\right)-1\right]\nonumber\\
&&+\left[\left(2m_2+1\right)F\left(-m_2-1,-m_3;\frac{1}{2};x^2\right)-1\right]\nonumber\\
&>&0.
\end{eqnarray}
Note that (\ref{Aug9}) holds when either $x\in\{0,1\}$ or $a=0$, and
$$
\lim_{a\rightarrow-\infty}\min_{x\in[0,1]}F(a,x)=\infty.
$$
Thus, we need only show that (\ref{Aug9}) holds under the following condition:
$$
\frac{\partial F(a,x)}{\partial a}=0,\ \ \ \ a\in(-\infty,0),\, x\in(0,1).
$$
By (\ref{Gauss3}), we get
\begin{eqnarray*}
&&F\left(-m_2,-m_3-1;\frac{1}{2};x^2\right)\\
&=&\frac{1}{2m_3+1}\left[(2m_2+1)F\left(-m_2-1,-m_3;\frac{1}{2};x^2\right)+2(m_3-m_2)(1-x^2)F\left(-m_2,-m_3;\frac{1}{2};x^2\right)\right],
\end{eqnarray*}
and
\begin{eqnarray}\label{Aug2222}
&&x^2F\left(-m_2,-m_3;\frac{3}{2};x^2\right)\nonumber\\
&=&\frac{1}{2m_3+1}\left[F\left(-m_2-1,-m_3;\frac{1}{2};x^2\right)-(1-x^2)F\left(-m_2,-m_3;\frac{1}{2};x^2\right)\right].
\end{eqnarray}
Then,
\begin{eqnarray*}
&&\frac{\partial F(a,x)}{\partial a}=0\nonumber\\
&\Leftrightarrow&0=a\left[\left(2m_3+1\right)F\left(-m_2,-m_3-1;\frac{1}{2};x^2\right)-1\right]\nonumber\\
&&\ \ \ \ \ +x\left[\left(2m_3+1\right)\left(2m_2+1\right) F\left(-m_2,-m_3;\frac{3}{2};x^2\right)-1\right]\nonumber\\
&\Leftrightarrow&0=ax\left[(2m_2+1)F\left(-m_2-1,-m_3;\frac{1}{2};x^2\right)+2(m_3-m_2)(1-x^2)F\left(-m_2,-m_3;\frac{1}{2};x^2\right)-1\right]\nonumber\\
&&\ \ \ \ \ +\left\{\left(2m_2+1\right)\left[F\left(-m_2-1,-m_3;\frac{1}{2};x^2\right)-(1-x^2)F\left(-m_2,-m_3;\frac{1}{2};x^2\right)\right]-x^2\right\}\nonumber\\
&\Leftrightarrow&a=-\frac{\left(2m_2+1\right)\left[F\left(-m_2-1,-m_3;\frac{1}{2};x^2\right)-(1-x^2)F\left(-m_2,-m_3;\frac{1}{2};x^2\right)\right]-x^2}{x\left[(2m_2+1)F\left(-m_2-1,-m_3;\frac{1}{2};x^2\right)+2(m_3-m_2)(1-x^2)F\left(-m_2,-m_3;\frac{1}{2};x^2\right)-1\right]}.\nonumber\\
&&
\end{eqnarray*}
Hence, we have that
\begin{eqnarray*}
F(a,x)&=&a^2\left[(2m_2+1)F\left(-m_2-1,-m_3;\frac{1}{2};x^2\right)+2(m_3-m_2)(1-x^2)F\left(-m_2,-m_3;\frac{1}{2};x^2\right)-1\right]\nonumber\\
&&+\frac{2a}{x}\left\{\left(2m_2+1\right) \left[F\left(-m_2-1,-m_3;\frac{1}{2};x^2\right)-(1-x^2)F\left(-m_2,-m_3;\frac{1}{2};x^2\right)\right]-x^2\right\}\nonumber\\
&&+\left[\left(2m_2+1\right)F\left(-m_2-1,-m_3;\frac{1}{2};x^2\right)-1\right]\nonumber\\
&=&\frac{\{\left(2m_2+1\right)\left[F\left(-m_2-1,-m_3;\frac{1}{2};x^2\right)-(1-x^2)F\left(-m_2,-m_3;\frac{1}{2};x^2\right)\right]-x^2\}^2}{x^2\left[(2m_2+1)F\left(-m_2-1,-m_3;\frac{1}{2};x^2\right)+2(m_3-m_2)(1-x^2)F\left(-m_2,-m_3;\frac{1}{2};x^2\right)-1\right]}\nonumber\\
&&-\frac{2\{\left(2m_2+1\right)\left[F\left(-m_2-1,-m_3;\frac{1}{2};x^2\right)-(1-x^2)F\left(-m_2,-m_3;\frac{1}{2};x^2\right)\right]-x^2\}^2}{x^2\left[(2m_2+1)F\left(-m_2-1,-m_3;\frac{1}{2};x^2\right)+2(m_3-m_2)(1-x^2)F\left(-m_2,-m_3;\frac{1}{2};x^2\right)-1\right]}\nonumber\\
&&+\left[\left(2m_2+1\right)F\left(-m_2-1,-m_3;\frac{1}{2};x^2\right)-1\right].
\end{eqnarray*}
Thus, to prove the GPI (\ref{GPIk}), we need to show that
\begin{eqnarray}\label{Aug111}
&&\left\{\left(2m_2+1\right)\left[F\left(-m_2-1,-m_3;\frac{1}{2};x^2\right)-(1-x^2)F\left(-m_2,-m_3;\frac{1}{2};x^2\right)\right]-x^2\right\}^2\nonumber\\
&<&x^2\left[\left(2m_2+1\right)F\left(-m_2-1,-m_3;\frac{1}{2};x^2\right)-1\right]\nonumber\\
&&\cdot\left[(2m_2+1)F\left(-m_2-1,-m_3;\frac{1}{2};x^2\right)+2(m_3-m_2)(1-x^2)F\left(-m_2,-m_3;\frac{1}{2};x^2\right)-1\right].\nonumber\\
&&
\end{eqnarray}
It can be shown that (\ref{Aug111}) is equivalent to \cite[Claim C, page 14]{RusSun}.
By (\ref{Gauss3}), we get
\begin{eqnarray*}
&&(1-x^2)F\left(-m_2,-m_3;\frac{1}{2};x^2\right)\\
&=&F\left(-m_2-1,-m_3;\frac{1}{2};x^2\right)-(2m_3+1)x^2F\left(-m_2,-m_3;\frac{3}{2};x^2\right),
\end{eqnarray*}
and
\begin{eqnarray*}
0&=&-(m_3-m_2)(1-x^2)F\left(-m_2,-m_3;\frac{1}{2};x^2\right)\\
&&-\left(m_2+\frac{1}{2}\right)F\left(-m_2-1,-m_3;\frac{1}{2};x^2\right)+\left(m_3+\frac{1}{2}\right)F\left(-m_2,-m_3-1;\frac{1}{2};x^2\right).
\end{eqnarray*}
Then, (\ref{Aug111}) becomes
\begin{eqnarray}\label{we3}
&&x^2\left\{\left(2m_2+1\right)(2m_3+1)F\left(-m_2,-m_3;\frac{3}{2};x^2\right)-1\right\}^2\nonumber\\
&<&\left[\left(2m_2+1\right)F\left(-m_2-1,-m_3;\frac{1}{2};x^2\right)-1\right]\nonumber\\
&&\cdot\left[\left(2m_3+1\right)F\left(-m_2,-m_3-1;\frac{1}{2};x^2\right)-1\right].
\end{eqnarray}
Note that, different from (\ref{Aug111}), the roles played by $m_2$ and $m_3$ in (\ref{we3}) are symmetric. This key fact will lead us to derive the stronger inequality (\ref{Augsdyy}) given below.
\begin{rem}
Let $(X_2,X_3)$ be a centered Gaussian random vector with ${\rm Var}(X_2)={\rm Var}(X_3)=1$. By (\ref{we2}) and (\ref{we1}), we can rewrite (\ref{we3}) as the following interesting and delicate moment inequality:
\begin{eqnarray*}
&&\left\{E[ X_2^{2m_2+1}X_3^{2m_3+1}]-E[X_2^{2m_2}]E[X_3^{2m_3}]E[X_2X_3]\right\}^2\nonumber\\
&<&\left\{E[X_2^{2m_2+2}X_3^{2m_3}]-E[X_2^{2m_2}]E[X_3^{2m_3}]\right\}\\
&&\cdot\left\{E[X_2^{2m_2}X_3^{2m_3+2}]-E[X_2^{2m_2}]E[X_3^{2m_3}]\right\}.
\end{eqnarray*}
\end{rem}
Let $z=x^2$. Then, (\ref{we3}) becomes
\begin{eqnarray}\label{Aug333}
&&z\left\{\left(2m_2+1\right)(2m_3+1)F\left(-m_2,-m_3;\frac{3}{2};z\right)-1\right\}^2\nonumber\\
&<&\left[\left(2m_2+1\right)F\left(-m_2-1,-m_3;\frac{1}{2};z\right)-1\right]\nonumber\\
&&\cdot\left[\left(2m_3+1\right)F\left(-m_2,-m_3-1;\frac{1}{2};z\right)-1\right].
\end{eqnarray}
Further, by (\ref{Aug2222}), we find that (\ref{Aug333}) is equivalent to
\begin{eqnarray*}
&&z\left\{\left(2m_2+1\right)(2m_3+1)F\left(-m_2,-m_3;\frac{3}{2};z\right)-1\right\}^2\\
&<&\left[\left(2m_2+1\right)(1-z)F\left(-m_2,-m_3;\frac{1}{2};z\right)+\left(2m_2+1\right)(2m_3+1)zF\left(-m_2,-m_3;\frac{3}{2};z\right)-1\right]\\
&&\cdot\left[\left(2m_3+1\right)(1-z)F\left(-m_2,-m_3;\frac{1}{2};z\right)+\left(2m_2+1\right)(2m_3+1)zF\left(-m_2,-m_3;\frac{3}{2};z\right)-1\right],
\end{eqnarray*}
i.e.,
\begin{eqnarray*}
&&z\left\{\left(2m_2+1\right)(2m_3+1)F\left(-m_2,-m_3;\frac{3}{2};z\right)-1\right\}^2\\
&<&\left[\left\{\left(2m_2+1\right)F\left(-m_2,-m_3;\frac{1}{2};z\right)-1\right\}(1-z)+\left(2m_2+1\right)(2m_3+1)zF\left(-m_2,-m_3;\frac{3}{2};z\right)-z\right]\\
&&\cdot\left[\left\{\left(2m_3+1\right)F\left(-m_2,-m_3;\frac{1}{2};z\right)-1\right\}(1-z)+\left(2m_2+1\right)(2m_3+1)zF\left(-m_2,-m_3;\frac{3}{2};z\right)-z\right].
\end{eqnarray*}
Hence, we need only show that for any $z\in(0,1)$,
\begin{eqnarray}\label{Augsd}
z
&<&\left[\frac{\left\{\left(2m_2+1\right)F\left(-m_2,-m_3;\frac{1}{2};z\right)-1\right\}(1-z)}{\left(2m_2+1\right)(2m_3+1)F\left(-m_2,-m_3;\frac{3}{2};z\right)-1}+z\right]\nonumber\\
&&\cdot\left[\frac{\left\{\left(2m_3+1\right)F\left(-m_2,-m_3;\frac{1}{2};z\right)-1\right\}(1-z)}{\left(2m_2+1\right)(2m_3+1)F\left(-m_2,-m_3;\frac{3}{2};z\right)-1}+z\right].
\end{eqnarray}
Note that
\begin{eqnarray*}
&&\frac{\left\{\left(2m_2+1\right)F\left(-m_2,-m_3;\frac{1}{2};z\right)-1\right\}(1-z)}{\left(2m_2+1\right)(2m_3+1)F\left(-m_2,-m_3;\frac{3}{2};z\right)-1}\\
&\ge&\frac{\left(2m_2+1\right)F\left(-m_2,-m_3;\frac{1}{2};z\right)(1-z)}{\left(2m_2+1\right)(2m_3+1)F\left(-m_2,-m_3;\frac{3}{2};z\right)}+\frac{-(1-z)}{\left(2m_2+1\right)(2m_3+1)F\left(-m_2,-m_3;\frac{3}{2};z\right)}\\
&\ge&\frac{F\left(-m_2,-m_3;\frac{1}{2};z\right)(1-z)}{(2m_3+1)F\left(-m_2,-m_3;\frac{3}{2};z\right)}+\frac{-(1-z)}{\left(2m_2+1\right)(2m_3+1)}.
\end{eqnarray*}
Therefore, to prove (\ref{Augsd}), it suffices to show that the following stronger inequality holds for $0<z<1$:
\begin{eqnarray}\label{Augsdyy}
z
&<&\left[\frac{F\left(-m_2,-m_3;\frac{1}{2};z\right)(1-z)}{(2m_3+1)F\left(-m_2,-m_3;\frac{3}{2};z\right)}+\frac{[(2m_2+1)(2m_3+1)+1]z-1}{\left(2m_2+1\right)(2m_3+1)}\right]\nonumber\\
&&\cdot\left[\frac{F\left(-m_2,-m_3;\frac{1}{2};z\right)(1-z)}{(2m_2+1)F\left(-m_2,-m_3;\frac{3}{2};z\right)}+\frac{[(2m_2+1)(2m_3+1)+1]z-1}{\left(2m_2+1\right)(2m_3+1)}\right].
\end{eqnarray}
Using \textbf{Mathematica}, it appears that inequality (\ref{Augsdyy}) holds if $
(m_2,m_3)\in{\mathcal S}$. In Sections \ref{sec3} and \ref{sec4} below, we will give a rigorous proof for (\ref{Augsdyy}) by combining computing and hard analysis.
\subsection{MRI $\Leftrightarrow$ HFRI $\Rightarrow$ GPI}
For $0<z< 1$, define
$$
y=\frac{F\left(-m_2,-m_3;\frac{1}{2};z\right)}{F\left(-m_2,-m_3;\frac{3}{2};z\right)}.
$$
Then, $y>1$ and
\begin{eqnarray*}
&&(\ref{Augsdyy})\ {\rm holds}\nonumber\\
&\Leftrightarrow&0<\frac{(1-z)^2}{(2m_2+1)(2m_3+1)}\cdot y^2\nonumber\\
&&\ \ \ \ +\frac{2(m_2+m_3+1)\{[(2m_2+1)(2m_3+1)+1]z-1\}(1-z)}{(2m_2+1)^2(2m_3+1)^2}\cdot y\nonumber\\
&&\ \ \ \ +\frac{[(2m_2+1)(2m_3+1)+1]^2z^2-\{[(2m_2+1)(2m_3+1)+1]^2+1\}z+1}{(2m_2+1)^2(2m_3+1)^2}\nonumber\\
&\Leftrightarrow&0<(1-z)y^2+2\beta y+\gamma,
\end{eqnarray*}
where
\begin{eqnarray*}
&&\beta:=-\frac{(m_2+m_3+1)\{1-[(2m_2+1)(2m_3+1)+1]z\}}{(2m_2+1)(2m_3+1)},\nonumber\\
&&\gamma:=\frac{1-[(2m_2+1)(2m_3+1)+1]^2z}{(2m_2+1)(2m_3+1)}.
\end{eqnarray*}
By the identity
$$
(m_2+m_3+1)^2=(m_3-m_2)^2+(2m_2+1)(2m_3+1),
$$
we get
\begin{eqnarray}\label{KKJJGG}
&&\beta^2-(1-z)\gamma\nonumber\\
&=&\left(\frac{(m_3-m_2)\{1-[(2m_2+1)(2m_3+1)+1]z\}}{(2m_2+1)(2m_3+1)}\right)^2+(2m_2+1)(2m_3+1)z.
\end{eqnarray}
Then,
\begin{eqnarray*}
&&0<(1-z)y^2+2\beta y+\gamma\nonumber\\
&\Leftrightarrow&\left|y+\frac{\beta}{1-z}\right|>\frac{\sqrt{\beta^2-(1-z)\gamma}}{1-z}.
\end{eqnarray*}
For $0<z\le\frac{1}{r^2}$, we have
\begin{eqnarray*}
&&y+\frac{\beta}{1-z}-\frac{\sqrt{\beta^2-(1-z)\gamma}}{1-z}\nonumber\\
&>&1-\frac{m_2+m_3+1}{(2m_2+1)(2m_3+1)\left[1-\frac{1}{[(2m_2+1)(2m_3+1)+1]^2}\right]}\nonumber\\
&&-\frac{\sqrt{\left[\frac{m_3-m_2}{(2m_2+1)(2m_3+1)}\right]^2+\frac{(2m_2+1)(2m_3+1)}{[(2m_2+1)(2m_3+1)+1]^2}}}{1-\frac{1}{[(2m_2+1)(2m_3+1)+1]^2}}\nonumber\\
&>&1-\frac{1}{(2m_2+1)\left(1-\frac{1}{10^2}\right)}-\frac{\sqrt{\left[\frac{1}{2(2m_2+1)}\right]^2+\frac{1}{(2m_2+1)^2}}}{1-\frac{1}{10^2}}\nonumber\\
&>&0.
\end{eqnarray*}
Hence, to prove (\ref{Augsdyy}), it suffices to show that for $\frac{1}{r^2}< z<1$,
$$
y>\frac{-\beta+\sqrt{\beta^2-(1-z)\gamma}}{1-z}.
$$
Note that
\begin{eqnarray}\label{HGF}
\frac{1}{H(z)}&=&\frac{\gamma}{-\beta-\sqrt{\beta^2-(1-z)\gamma}}\nonumber\\
&=&\frac{-\beta+\sqrt{\beta^2-(1-z)\gamma}}{1-z},\ \ \ \ 0<z<1.
\end{eqnarray}
Hence we need show that the following HFRI holds:
\begin{eqnarray*}
\frac{F\left(-m_2,-m_3;\frac{1}{2};z\right)}{F\left(-m_2,-m_3;\frac{3}{2};z\right)}>\frac{1}{H(z)},\ \ \ \ \frac{1}{r^2}< z<1.
\end{eqnarray*}
For $\frac{1}{r^2}< z<\frac{1}{r}$, we have
\begin{eqnarray*}
&&H(z)\ge1\nonumber\\
&\Leftrightarrow&[(m_3-m_2)(rz-1)]^2+(r-1)^3z\ge[(r^2z-1)+(m_2+m_3+1)(1-rz)]^2\nonumber\\
&\Leftrightarrow&(r-1)^3z-(r^2z-1)^2-(r-1)(rz-1)^2\ge2(m_2+m_3+1)(1-rz)(r^2z-1)\nonumber\\
&\Leftrightarrow&(r-1)(1-z)(r^2z-1)-(r^2z-1)^2\ge2(m_2+m_3+1)(1-rz)(r^2z-1)\nonumber\\
&\Leftrightarrow&(r-1)(1-z)-(r^2z-1)\ge2(m_2+m_3+1)(1-rz)\nonumber\\
&\Leftrightarrow&r(1-rz)-(r-1)z\ge2(m_2+m_3+1)(1-rz)\nonumber\\
&\Leftrightarrow&(2m_2)(2m_3)(1-rz)\ge(r-1)z\nonumber\\
&\Leftrightarrow&z\le\frac{1}{t}.
\end{eqnarray*}
Then,
\begin{eqnarray*}
\frac{F\left(-m_2,-m_3;\frac{1}{2};z\right)}{F\left(-m_2,-m_3;\frac{3}{2};z\right)}>\frac{1}{H(z)},\ \ \ \ \frac{1}{r^2}<z\le\frac{1}{t}.
\end{eqnarray*}
Thus, to prove the GPI (\ref{GPIk}), we need only prove
\begin{eqnarray}\label{Aug18a}
\frac{F\left(-m_2,-m_3;\frac{1}{2};z\right)}{F\left(-m_2,-m_3;\frac{3}{2};z\right)}>\frac{1}{H(z)},\ \ \ \ \frac{1}{t}<z< 1.
\end{eqnarray}
Finally, we show that the HFRI (\ref{Aug18a}) is equivalent to the MRI (\ref{MRI22}) with the equality sign holding if and only if $X_2$ and $X_3$ are independent. We assume without loss of generality that ${\rm Var}(X_2)={\rm Var}(X_3)=1$. By (\ref{we2}) and (\ref{we1}), we get
\begin{eqnarray}\label{Aug27a}
\frac{|E[ X_2^{2m_2+1}X_3^{2m_3+1}]|}{E[ X_2^{2m_2}X_3^{2m_3}]}={(2m_2+1)(2m_3+1)|x|}\frac{F\left(-m_2,-m_3;\frac{3}{2};x^2\right)}{F\left(-m_2,-m_3;\frac{1}{2};x^2\right)}.
\end{eqnarray}
Note that
$$
\frac{F\left(-m_2,-m_3;\frac{1}{2};0\right)}{F\left(-m_2,-m_3;\frac{3}{2};0\right)}=1,
$$
and
$$
\frac{F\left(-m_2,-m_3;\frac{1}{2};z\right)}{F\left(-m_2,-m_3;\frac{3}{2};z\right)}>1,\ \ \ \ 0<z< 1.
$$
Therefore, by (\ref{Aug27a}) and (\ref{HGFD}) (see below), we conclude that (\ref{Aug18a}) is equivalent to (\ref{MRI22}) with the equality sign holding if and only if $X_2$ and $X_3$ are independent.
\begin{rem}
On one hand, the MRI (\ref{MRI22}) implies the GPI (\ref{GPIk}). On the other hand, \cite[Theorem 4.3 and Remark 4.4]{RusSun} shows that the following MRI holds and is implied by the GPI (\ref{GPIk}):
\begin{eqnarray}\label{MRI33}
\frac{\left|E[ X_2^{2m_2-1}X_3^{2m_3+1}]\right|}{E[ X_2^{2m_2}X_3^{2m_3}]}<\frac{2m_3+1}{2m_2},
\end{eqnarray}
where $(X_2,X_3)$ is a centered Gaussian random vector with ${\rm Var}(X_2)={\rm Var}(X_3)=1$. Hence the MRI (\ref{MRI22}) is stronger than the MRI (\ref{MRI33}). However, these two MRIs have independent interests.
\end{rem}
\section{Proof of HFRI (\ref{Aug18a}) for the case $1\le m_2\le 7$}\label{sec3}\setcounter{equation}{0}
In this section, we show that (\ref{Aug18a}) holds for $1\le m_2\le 7$.
\subsection{Positiveness of bivariate polynomials}
Note that
\begin{eqnarray}\label{Aug27HH}
F\left(-m_2,-m_3;\frac{1}{2};z\right)&=&m_2!m_3!\sum_{j=0}^{m_2}\frac{2^{2j}z^j}{(m_2-j)!(m_3-j)!(2j)!},\nonumber\\
F\left(-m_2,-m_3;\frac{3}{2};z\right)&=&m_2!m_3!\sum_{j=0}^{m_2}\frac{2^{2j}z^j}{(m_2-j)!(m_3-j)!(2j+1)!}.
\end{eqnarray}
To simplify notation, we denote
$$
f_1(z):=F\left(-m_2,-m_3;\frac{1}{2};z\right),\ \ \ \ f_2(z):=F\left(-m_2,-m_3;\frac{3}{2};z\right).
$$
Then, we have
\begin{eqnarray}\label{Aug22g}
&&(\ref{Aug18a})\ {\rm holds}\nonumber\\
&\Leftarrow&\ \ \left[(m_2+m_3+1)(rz-1)+\sqrt{[(m_3-m_2)(rz-1)]^2+(2m_2+1)^3(2m_3+1)^3z}\right]f_1(z)\nonumber\\
&&>(r^2z-1)f_2(z)\nonumber\\
&\Leftarrow&\ \ \left\{[(m_3-m_2)(rz-1)]^2+(r-1)^3z\right\}f^2_1(z)\nonumber\\
&&>\left\{(r^2z-1)f_2(z)-(m_2+m_3+1)(rz-1)f_1(z)\right\}^2\nonumber\\
&\Leftrightarrow&\ \ (r-1)^3zf^2_1(z)\nonumber\\
&&>(r^2z-1)^2f_2^2(z)+(r-1)(rz-1)^2f_1^2(z)-2(m_2+m_3+1)(rz-1)(r^2z-1)f_1(z)f_2(z)\nonumber\\
&\Leftrightarrow&\ \ (r-1)(1-z)(r^2z-1)f_1^2(z)\nonumber\\
&&>(r^2z-1)^2f_2^2(z)-2(m_2+m_3+1)(rz-1)(r^2z-1)f_1(z)f_2(z)\nonumber\\
&\Leftrightarrow&\ \ (r-1)(1-z)f_1^2(z)+2(m_2+m_3+1)(rz-1)f_1(z)f_2(z)\nonumber\\
&&>(r^2z-1)f_2^2(z)\nonumber\\
&\Leftrightarrow&S_{m_2, m_3}(z):=(2m_2+1)(2m_3+1)\left(1-z\right)\left[\sum_{j=0}^{m_2}\frac{2^{2j}z^jm_2!m_3!}{(m_2-j)!(m_3-j)!(2j)!}\right]^2\nonumber\\
&&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +2(m_2+m_3+1)\left\{{[(2m_2+1)(2m_3+1)+1]z}-1\right\}\nonumber\\
&&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \cdot\left[\sum_{j=0}^{m_2}\frac{2^{2j}z^jm_2!m_3!}{(m_2-j)!(m_3-j)!(2j)!}\right]\left[\sum_{j=0}^{m_2}\frac{2^{2j}z^jm_2!m_3!}{(m_2-j)!(m_3-j)!(2j+1)!}\right]\nonumber\\
&&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -\left\{{[(2m_2+1)(2m_3+1)+1]^2z}-1\right\}\left[\sum_{j=0}^{m_2}\frac{2^{2j}z^jm_2!m_3!}{(m_2-j)!(m_3-j)!(2j+1)!}\right]^2\nonumber\\
&&\ >0,\ \ \ \ \frac{1}{r^2}<z<1.
\end{eqnarray}
By using the transformations
$$
z=\frac{c^2}{1+c^2},
$$
and
\begin{eqnarray*}
m_3=
\begin{cases}
b^2+5,\ \ &{\rm if}\ m_2=1,\\
b^2+3,\ \ &{\rm if}\ m_2=2,\\
b^2+m_2,\ \ &{\rm if}\ m_2\ge3,
\end{cases}
\end{eqnarray*}
we define
$$
h_{m_2}(b,c):=(1+c^2)^{2m_2+1}\cdot S_{m_2, m_3}(z),\ \ \ \ b,c \in \mathbb{R}.
$$
Then, by (\ref{Aug22g}), we find that the HFRI (\ref{Aug18a}) holds for $1\le m_2\le 7$ if $h_{m_2}$ is a positive bivariate polynomial for each $1\le m_2\le 7$.
By virtue of \textbf{Mathematica}, we obtain the expansion of $h_{m_2}$, a polynomial of $b$ and $c$. See the next subsection. Note that all exponents of $b$ and $c$ are even. We sum all terms with negative coefficients with some of the terms with positive coefficients and then apply the package `\texttt{SumsOfSquares}' in \textbf{Macaulay2} \cite{SOS,PP} to obtain exact SOS decompositions. Hence, since $h_{m_2}$ is strictly positive, we conclude that
\begin{eqnarray*}
(\ref{Aug18a})\ {\rm holds\ for}\ 1\le m_2\le 7.
\end{eqnarray*}
\subsection{Expansion of functions $h_{m_2}$}
Let $1\le m_2\le 7$. We have the following expansions of the functions $h_{m_2}$ for any $b,c \in \mathbb{R}$ and the SOS decompositions corresponding to the sums of the 10 terms in parentheses:
\begin{dmath*}
h_1(b,c)=\frac{8 b^6 c^6}{3}+\frac{124 b^4 c^6}{3}+\frac{622 b^2
c^6}{3}+\frac{1}{9} \left(48 b^6 c^4+664 b^4 c^4-48 b^4 c^2+2884 b^2
c^4-546 b^2 c^2+36 b^2+3003 c^6+3766 c^4-1557 c^2+180\right).
\end{dmath*}
Using `\texttt{SumsOfSquares}' in \textbf{Macaulay2}, we get
\begin{dmath*}
48 b^6 c^4+664 b^4 c^4-48 b^4 c^2+2884 b^2 c^4-546 b^2 c^2+36 b^2+3003 c^6+3766 c^4-1557 c^2+180 = 4924\bigg(\frac{3377}{39392}\,b^{2}c^{2}+c^{2}-\frac{35173}{196960} \bigg)^2
+ 3003\bigg(\frac{151}{8008}\,b^{2}c+c^{3}-\frac{193}{1001}\,c \bigg)^2
+\frac{3853}{2}\bigg(\frac{557}{77060}\,b^{3}c^{2}+b\,c^{2}-\frac{16027}{200356}\,b \bigg)^2
+\frac{945348187}{1575680}\bigg(b^{2}c^{2}-\frac{1722648687}{12289526431} \bigg)^2
+\frac{1591}{13}\bigg(b\,c \bigg)^2
+\frac{1802093}{20020}\bigg(-\frac{193955}{1802093}\,b^{2}c+c \bigg)^2
+\frac{147644951}{3082400}\bigg(b^{3}c^{2}-\frac{1109415365}{1919384363}\,b \bigg)^2
+\frac{178655579224727}{15976384360300}\bigg(1 \bigg)^2
+\frac{191380511436}{24951996719}\bigg(b \bigg)^2
+\frac{56147093713}{7496706880}\bigg(b^{2}c \bigg)^2.
\end{dmath*}
\begin{dmath*}
h_2(b,c)=\frac{32 b^{10} c^{10}}{15}+\frac{128 b^{10} c^8}{45}+\frac{1936
b^8 c^{10}}{45}+\frac{13376 b^8 c^8}{225}+\frac{128 b^8
c^6}{15}+\frac{5104 b^6 c^{10}}{15}+\frac{7424 b^6
c^8}{15}+\frac{400 b^6 c^6}{3}+\frac{59192 b^4
c^{10}}{45}+\frac{454432 b^4 c^8}{225}+\frac{32872 b^4
c^6}{45}+\frac{12374 b^2 c^{10}}{5}+\frac{19896 b^2
c^8}{5}+\frac{4892 b^2 c^6}{3}+\frac{9009
c^{10}}{5}+\frac{74484 c^8}{25}+\frac{1}{45} \left(576 b^6
c^4+3920 b^4 c^4-480 b^4 c^2+5376 b^2 c^4-3210 b^2 c^2+360
b^2+53838 c^6-4500 c^4-5535 c^2+1080\right).
\end{dmath*}
Using `\texttt{SumsOfSquares}' in \textbf{Macaulay2}, we get
\begin{dmath*}
576 b^6 c^4+3920 b^4 c^4-480 b^4 c^2+5376 b^2 c^4-3210 b^2 c^2+360
b^2+53838 c^6-4500 c^4-5535 c^2+1080 = 53838 \bigg(\frac{24865}{3876336}\,b^{2}c+c^{3}-\frac{26099}{107676}\,c \bigg)^2
+ 21599 \bigg(-\frac{5209}{194391}\,b^{2}c^{2}+c^{2}-\frac{18043}{86396} \bigg)^2
+\frac{210343}{36} \bigg(\frac{5751}{60098}\,b^{3}c^{2}+b\,c^{2}-\frac{538218}{2734459}\,b \bigg)^2
+\frac{19498324709}{6998076} \bigg(b^{2}c^{2}-\frac{88115791563}{506956442434}\bigg)^2
+\frac{251207361}{480784} \bigg( b^{3}c^{2}-\frac{717899012}{1814275385}\,b\bigg)^2
+\frac{69666947}{215352} \bigg(-\frac{15907018825}{32604131196}\,b^{2}c+c \bigg)^2
+\frac{15599}{52} \bigg(b\,c \bigg)^2
+\frac{11349643727532583}{152587333997280} \bigg(b^{2}c \bigg)^2
+\frac{1418170065555879}{26361735006568} \bigg(1 \bigg)^2
+\frac{42786226455192}{825495300175} \bigg(b \bigg)^2.
\end{dmath*}
\begin{dmath*}
h_3(b,c)=\frac{128 b^{14} c^{14}}{315}+\frac{256 b^{14} c^{12}}{525}+\frac{20672
b^{12} c^{14}}{1575}+\frac{71552 b^{12} c^{12}}{3675}+\frac{2816
b^{12} c^{10}}{525}+\frac{11296 b^{10} c^{14}}{63}+\frac{1172672
b^{10} c^{12}}{3675}+\frac{16544 b^{10} c^{10}}{105}+\frac{2368 b^{10}
c^8}{105}+\frac{2112752 b^8 c^{14}}{1575}+\frac{3416288 b^8
c^{12}}{1225}+\frac{988688 b^8 c^{10}}{525}+\frac{252416 b^8
c^8}{525}+\frac{1152 b^8 c^6}{35}+\frac{373432 b^6
c^{14}}{63}+\frac{17231216 b^6 c^{12}}{1225}+\frac{1227568 b^6
c^{10}}{105}+\frac{426784 b^6 c^8}{105}+\frac{18136 b^6
c^6}{35}+\frac{24322148 b^4 c^{14}}{1575}+\frac{150657352 b^4
c^{12}}{3675}+\frac{2972504 b^4 c^{10}}{75}+\frac{8826112 b^4
c^8}{525}+\frac{100708 b^4 c^6}{35}+\frac{766942 b^2
c^{14}}{35}+\frac{78310956 b^2 c^{12}}{1225}+\frac{2425358 b^2
c^{10}}{35}+\frac{3529636 b^2 c^8}{105}+\frac{230266 b^2
c^6}{35}+\frac{325611 c^{14}}{25}+\frac{10067706
c^{12}}{245}+\frac{8501427 c^{10}}{175}+\frac{4494432 c^8}{175}+\frac{1}{35} \left(784 b^6 c^4+4984 b^4 c^4-560 b^4 c^2+4844
b^2 c^4-3430 b^2 c^2+420 b^2+177111 c^6-10626 c^4-5495
c^2+1260\right).
\end{dmath*}
Using `\texttt{SumsOfSquares}' in \textbf{Macaulay2}, we get
\begin{dmath*}
784 b^6 c^4+4984 b^4 c^4-560 b^4 c^2+4844 b^2 c^4-3430 b^2 c^2+420 b^2+177111 c^6-10626 c^4-5495 c^2+1260 = 177111\bigg(\frac{343}{1159272}\,b^{2}c+c^{3}-\frac{470689}{2361480}\,c \bigg)^2
+\frac{1199547}{20}\bigg(-\frac{1090975}{10795923}\,b^{2}c^{2}+c^{2}-\frac{285517}{2399094} \bigg)^2
+\frac{607001}{36}\bigg(-\frac{14247}{6070010}\,b^{3}c^{2}+b\,c^{2}-\frac{1390563}{15782026}\,b \bigg)^2
+\frac{2162205376529}{485816535}\bigg(b^{2}c^{2}-\frac{12112877256075}{56217339789754} \bigg)^2
+\frac{54929220719}{31486400}\bigg(-\frac{4099630565195}{19280156472369}\,b^{2}c+c \bigg)^2
+\frac{190332960599}{242800400}\bigg(b^{3}c^{2}-\frac{1314759474005}{2474328487787}\,b \bigg)^2
+\frac{40527}{52}\bigg(b\,c \bigg)^2
+\frac{5960322065628821313}{29233016690672080}\bigg(1 \bigg)^2
+\frac{703832116675509863}{3759630512111955}\bigg(b^{2}c \bigg)^2
+\frac{4359683320181945}{64332540682462}\bigg(b \bigg)^2.
\end{dmath*}
\begin{dmath*}
h_4(b,c)=\frac{512 b^{18} c^{18}}{14175}+\frac{4096 b^{18}
c^{16}}{99225}+\frac{7936 b^{16} c^{18}}{3969}+\frac{2725888 b^{16}
c^{16}}{893025}+\frac{94208 b^{16} c^{14}}{99225}+\frac{4844032 b^{14}
c^{18}}{99225}+\frac{83216384 b^{14} c^{16}}{893025}+\frac{574976
b^{14} c^{14}}{11025}+\frac{8192 b^{14} c^{12}}{945}+\frac{68441344
b^{12} c^{18}}{99225}+\frac{281571328 b^{12}
c^{16}}{178605}+\frac{8088832 b^{12} c^{14}}{6615}+\frac{4169216
b^{12} c^{12}}{11025}+\frac{1024 b^{12} c^{10}}{27}+\frac{616465216
b^{10} c^{18}}{99225}+\frac{14723216384 b^{10}
c^{16}}{893025}+\frac{529583744 b^{10} c^{14}}{33075}+\frac{231241216
b^{10} c^{12}}{33075}+\frac{751808 b^{10} c^{10}}{567}+\frac{15872
b^{10} c^8}{189}+\frac{733857056 b^8 c^{18}}{19845}+\frac{99414094592
b^8 c^{16}}{893025}+\frac{1413614528 b^8
c^{14}}{11025}+\frac{334630784 b^8 c^{12}}{4725}+\frac{270950752 b^8
c^{10}}{14175}+\frac{1192064 b^8 c^8}{525}+\frac{2816 b^8
c^6}{35}+\frac{14425055648 b^6 c^{18}}{99225}+\frac{435356600192 b^6
c^{16}}{893025}+\frac{21284903264 b^6 c^{14}}{33075}+\frac{519067904
b^6 c^{12}}{1225}+\frac{685399136 b^6 c^{10}}{4725}+\frac{115082624
b^6 c^8}{4725}+\frac{171616 b^6 c^6}{105}+\frac{12032192752 b^4
c^{18}}{33075}+\frac{1195774968832 b^4
c^{16}}{893025}+\frac{196261589392 b^4
c^{14}}{99225}+\frac{49457683168 b^4 c^{12}}{33075}+\frac{8642687696
b^4 c^{10}}{14175}+\frac{28971328 b^4 c^8}{225}+\frac{59888 b^4
c^6}{5}+\frac{1158550822 b^2 c^{18}}{2205}+\frac{624204543248 b^2
c^{16}}{297675}+\frac{22520600344 b^2 c^{14}}{6615}+\frac{19041305312
b^2 c^{12}}{6615}+\frac{2719211668 b^2 c^{10}}{2025}+\frac{1577198528
b^2 c^8}{4725}+\frac{1309688 b^2 c^6}{35}+\frac{16338751
c^{18}}{49}+\frac{28338551384 c^{16}}{19845}+\frac{1107876484
c^{14}}{441}+\frac{25678259464 c^{12}}{11025}+\frac{5707836706
c^{10}}{4725}+\frac{532380728 c^8}{1575}+\frac{1}{315} \left(10752
b^6 c^4+99680 b^4 c^4-6720 b^4 c^2+228256 b^2 c^4-51030 b^2 c^2+5040
b^2+13164492 c^6-16408 c^4-99435 c^2+20160\right).
\end{dmath*}
Using `\texttt{SumsOfSquares}' in \textbf{Macaulay2}, we get
\begin{dmath*}
10752 b^6 c^4+99680 b^4 c^4-6720 b^4 c^2+228256 b^2 c^4-51030 b^2 c^2+5040 b^2+13164492 c^6-16408 c^4-99435 c^2+20160 = 13164492\bigg(\frac{871111}{473921712}\,b^{2}c+c^{3}-\frac{20957139}{175526560}\,c \bigg)^2
+\frac{62543257}{20}\bigg(-\frac{96138395}{1125778626}\,b^{2}c^{2}+c^{2}-\frac{7591495}{125086514} \bigg)^2
+\frac{25702673}{36}\bigg(-\frac{2527101}{128513365}\,b^{3}c^{2}+b\,c^{2}-\frac{26211375}{668269498}\,b \bigg)^2
+\frac{42535575011281679}{405280305360}\bigg(b^{2}c^{2}-\frac{92045449553337135}{552962475146661827} \bigg)^2
+\frac{649273640274437}{7021062400}\bigg(-\frac{665810223932870}{8440557323567681}\,b^{2}c+c \bigg)^2
+\frac{724643}{26}\bigg(b\,c \bigg)^2
+\frac{26925931846911}{2570267300}\bigg(b^{3}c^{2}-\frac{180424110237905}{350037114009843}\,b \bigg)^2
+\frac{82433649033959276332395}{14377024353813207502}\bigg(1 \bigg)^2
+\frac{1257596798250638235887177}{533274411703006085580}\bigg(b^{2}c \bigg)^2
+\frac{84337814929612787557}{72807719714047344}\bigg(b \bigg)^2.
\end{dmath*}
\begin{dmath*}
h_5(b,c)=\frac{2048 c^{22} b^{22}}{1091475}+\frac{4096 c^{20}
b^{22}}{1964655}+\frac{31744 c^{22} b^{20}}{200475}+\frac{354304
c^{20} b^{20}}{1440747}+\frac{53248 c^{18}
b^{20}}{654885}+\frac{11869696 c^{22} b^{18}}{1964655}+\frac{28832768
c^{20} b^{18}}{2401245}+\frac{4709888 c^{18}
b^{18}}{654885}+\frac{31744 c^{16} b^{18}}{24255}+\frac{3336448 c^{22}
b^{16}}{24255}+\frac{796876288 c^{20} b^{16}}{2401245}+\frac{109120768
c^{18} b^{16}}{392931}+\frac{186889216 c^{16}
b^{16}}{1964655}+\frac{806912 c^{14} b^{16}}{72765}+\frac{6798532352
c^{22} b^{14}}{3274425}+\frac{126837833216 c^{20}
b^{14}}{21611205}+\frac{12176539136 c^{18}
b^{14}}{1964655}+\frac{5950041088 c^{16}
b^{14}}{1964655}+\frac{49040128 c^{14} b^{14}}{72765}+\frac{15872
c^{12} b^{14}}{297}+\frac{23805268352 c^{22}
b^{12}}{1091475}+\frac{1521934275328 c^{20}
b^{12}}{21611205}+\frac{174559620352 c^{18}
b^{12}}{1964655}+\frac{108932344832 c^{16}
b^{12}}{1964655}+\frac{775136128 c^{14} b^{12}}{43659}+\frac{5957888
c^{12} b^{12}}{2205}+\frac{101888 c^{10} b^{12}}{693}+\frac{1315203392
c^{22} b^{10}}{8085}+\frac{1413228931712 c^{20}
b^{10}}{2401245}+\frac{1683844485056 c^{18}
b^{10}}{1964655}+\frac{1264240990336 c^{16}
b^{10}}{1964655}+\frac{3849765056 c^{14}
b^{10}}{14553}+\frac{12690841216 c^{12} b^{10}}{218295}+\frac{38016448
c^{10} b^{10}}{6237}+\frac{6016 c^8 b^{10}}{27}+\frac{1692144108832
c^{22} b^8}{1964655}+\frac{10617754825024 c^{20}
b^8}{3087315}+\frac{2217715657952 c^{18}
b^8}{392931}+\frac{17901245056 c^{16} b^8}{3645}+\frac{531886300384
c^{14} b^8}{218295}+\frac{4296463040 c^{12} b^8}{6237}+\frac{651507424
c^{10} b^8}{6237}+\frac{461312 c^8 b^8}{63}+\frac{3328 c^6
b^8}{21}+\frac{31157070814312 c^{22}
b^6}{9823275}+\frac{298407089852528 c^{20}
b^6}{21611205}+\frac{4480724112544 c^{18}
b^6}{178605}+\frac{48440168876992 c^{16}
b^6}{1964655}+\frac{3099620236336 c^{14}
b^6}{218295}+\frac{70563714464 c^{12} b^6}{14553}+\frac{280871456
c^{10} b^6}{297}+\frac{18037312 c^8 b^6}{189}+\frac{27784 c^6
b^6}{7}+\frac{25327698773092 c^{22}
b^4}{3274425}+\frac{785433486144952 c^{20}
b^4}{21611205}+\frac{141656838089504 c^{18}
b^4}{1964655}+\frac{154288192539328 c^{16}
b^4}{1964655}+\frac{67615137704 c^{14} b^4}{1323}+\frac{4421343351248
c^{12} b^4}{218295}+\frac{29711229488 c^{10} b^4}{6237}+\frac{12907648
c^8 b^4}{21}+\frac{2280668 c^6 b^4}{63}+\frac{2453547830422 c^{22}
b^2}{218295}+\frac{81423678540412 c^{20}
b^2}{1440747}+\frac{15858321568934 c^{18}
b^2}{130977}+\frac{18863764678564 c^{16}
b^2}{130977}+\frac{4533671397004 c^{14}
b^2}{43659}+\frac{10140634746104 c^{12} b^2}{218295}+\frac{7169154292
c^{10} b^2}{567}+\frac{368840488 c^8 b^2}{189}+\frac{8985182 c^6
b^2}{63}+\frac{1392896479 c^{22}}{189}+\frac{18906653425790
c^{20}}{480249}+\frac{3940199652805 c^{18}}{43659}+\frac{5059215383300
c^{16}}{43659}+\frac{1326722429110 c^{14}}{14553}+\frac{656332750372
c^{12}}{14553}+\frac{28692391130 c^{10}}{2079}+\frac{51226240
c^8}{21}+\frac{1}{63} \left(3024 c^4 b^6+37576 c^4 b^4-1680 c^2
b^4+135100 c^4 b^2-15246 c^2 b^2+1260 b^2+12895025 c^6+107170
c^4-34923 c^2+6300\right).
\end{dmath*}
Using `\texttt{SumsOfSquares}' in \textbf{Macaulay2}, we get
\begin{dmath*}
3024 c^4 b^6+37576 c^4 b^4-1680 c^2 b^4+135100 c^4 b^2-15246 c^2 b^2+1260 b^2+12895025 c^6+107170 c^4-34923 c^2+6300 = 12895025\bigg(\frac{9158}{12895025}\,b^{2}c+c^{3}-\frac{3751207}{46891000}\,c \bigg)^2
+\frac{43406677}{20}\bigg(-\frac{2898425}{43406677}\,b^{2}c^{2}+c^{2}-\frac{1705331}{43406677} \bigg)^2
+\frac{813253}{2}\bigg(-\frac{51669}{3253012}\,b^{3}c^{2}+b\,c^{2}-\frac{69487}{3020654}\,b \bigg)^2
+\frac{99568634438661}{1875640000}\bigg(-\frac{2717579463920}{33189544812887}\,b^{2}c+c \bigg)^2
+\frac{1771700819399}{43406677}\bigg(b^{2}c^{2}-\frac{6844893001259}{46064221304374} \bigg)^2
+\frac{186727}{13}\bigg(b\,c \bigg)^2
+\frac{76027180743}{26024096}\bigg(b^{3}c^{2}-\frac{481698502754}{988353349659}\,b \bigg)^2
+\frac{3067336004076266963}{1497087192392155}\bigg(1 \bigg)^2
+\frac{4822899680844187461}{9492209816485682}\bigg(b^{2}c \bigg)^2
+\frac{4508374085874757}{12848593545567}\bigg(b \bigg)^2.
\end{dmath*}
\begin{footnotesize}\begin{dmath*}
h_6(b,c)=\frac{8192 c^{26} b^{26}}{127702575}+\frac{32768 c^{24}
b^{26}}{468242775}+\frac{10760192 c^{26}
b^{24}}{1404728325}+\frac{73252864 c^{24}
b^{24}}{6087156075}+\frac{1933312 c^{22}
b^{24}}{468242775}+\frac{590606336 c^{26}
b^{22}}{1404728325}+\frac{248741888 c^{24}
b^{22}}{289864575}+\frac{251260928 c^{22}
b^{22}}{468242775}+\frac{4407296 c^{20}
b^{22}}{42567525}+\frac{3945715712 c^{26}
b^{20}}{280945665}+\frac{213729304576 c^{24}
b^{20}}{6087156075}+\frac{14447581184 c^{22}
b^{20}}{468242775}+\frac{1759670272 c^{20}
b^{20}}{156080925}+\frac{581632 c^{18}
b^{20}}{405405}+\frac{149143711232 c^{26}
b^{18}}{468242775}+\frac{634940704768 c^{24}
b^{18}}{676350675}+\frac{32664988672 c^{22}
b^{18}}{31216185}+\frac{17120804864 c^{20}
b^{18}}{31216185}+\frac{42243584 c^{18} b^{18}}{315315}+\frac{108544
c^{16} b^{18}}{9009}+\frac{346414848256 c^{26}
b^{16}}{66891825}+\frac{35540563938304 c^{24}
b^{16}}{2029052025}+\frac{332493773312 c^{22}
b^{16}}{14189175}+\frac{492099224576 c^{20}
b^{16}}{31216185}+\frac{15809266432 c^{18}
b^{16}}{2837835}+\frac{549275648 c^{16} b^{16}}{567567}+\frac{2854912
c^{14} b^{16}}{45045}+\frac{87228518486528 c^{26}
b^{14}}{1404728325}+\frac{1438014162632704 c^{24}
b^{14}}{6087156075}+\frac{1465996959232 c^{22}
b^{14}}{4002075}+\frac{46494374803456 c^{20}
b^{14}}{156080925}+\frac{42837903872 c^{18}
b^{14}}{315315}+\frac{10812772352 c^{16}
b^{14}}{315315}+\frac{117697024 c^{14} b^{14}}{27027}+\frac{149504
c^{12} b^{14}}{715}+\frac{31236824525056 c^{26}
b^{12}}{56189133}+\frac{14257521908077568 c^{24}
b^{12}}{6087156075}+\frac{1923969255517952 c^{22}
b^{12}}{468242775}+\frac{55196549321216 c^{20}
b^{12}}{14189175}+\frac{6107609131264 c^{18}
b^{12}}{2837835}+\frac{1995434665984 c^{16}
b^{12}}{2837835}+\frac{123057842944 c^{14}
b^{12}}{945945}+\frac{3828476416 c^{12} b^{12}}{315315}+\frac{1457152
c^{10} b^{12}}{3465}+\frac{5219339619628192 c^{26}
b^{10}}{1404728325}+\frac{11581321577165696 c^{24}
b^{10}}{676350675}+\frac{240480639853184 c^{22}
b^{10}}{7203735}+\frac{671544878167040 c^{20}
b^{10}}{18729711}+\frac{7298447665216 c^{18}
b^{10}}{315315}+\frac{17565080320 c^{16}
b^{10}}{1911}+\frac{2085803000192 c^{14}
b^{10}}{945945}+\frac{94694944768 c^{12}
b^{10}}{315315}+\frac{70219552 c^{10} b^{10}}{3465}+\frac{30592 c^8
b^{10}}{63}+\frac{1978323914357264 c^{26}
b^8}{108056025}+\frac{556119878319132224 c^{24}
b^8}{6087156075}+\frac{91412424181657792 c^{22}
b^8}{468242775}+\frac{2428744315669888 c^{20}
b^8}{10405395}+\frac{161932407186016 c^{18}
b^8}{945945}+\frac{45073399934720 c^{16}
b^8}{567567}+\frac{21927261204928 c^{14}
b^8}{945945}+\frac{86134081408 c^{12} b^8}{21021}+\frac{279871952
c^{10} b^8}{693}+\frac{395072 c^8 b^8}{21}+\frac{1920 c^6
b^8}{7}+\frac{3363043725212368 c^{26}
b^6}{52026975}+\frac{18017954680636544 c^{24}
b^6}{52026975}+\frac{41738823811154992 c^{22}
b^6}{52026975}+\frac{164252072011062592 c^{20}
b^6}{156080925}+\frac{271117905758368 c^{18}
b^6}{315315}+\frac{142978544588288 c^{16}
b^6}{315315}+\frac{29267159829280 c^{14}
b^6}{189189}+\frac{10494171683456 c^{12} b^6}{315315}+\frac{2949785008
c^{10} b^6}{693}+\frac{18196352 c^8 b^6}{63}+\frac{57328 c^6
b^6}{7}+\frac{690479779159336 c^{26}
b^4}{4459455}+\frac{54371705319629344 c^{24}
b^4}{61486425}+\frac{114494495345469496 c^{22}
b^4}{52026975}+\frac{23269671261024112 c^{20}
b^4}{7432425}+\frac{295079874070736 c^{18}
b^4}{105105}+\frac{520193894301824 c^{16}
b^4}{315315}+\frac{605240132759056 c^{14}
b^4}{945945}+\frac{50714422114784 c^{12}
b^4}{315315}+\frac{86651069368 c^{10} b^4}{3465}+\frac{46056544 c^8
b^4}{21}+\frac{625944 c^6 b^4}{7}+\frac{3192312960134 c^{26}
b^2}{14157}+\frac{2929503725483144 c^{24}
b^2}{2147145}+\frac{4185541698679412 c^{22}
b^2}{1156155}+\frac{19193256200856064 c^{20}
b^2}{3468465}+\frac{37709362565806 c^{18}
b^2}{7007}+\frac{24315798069560 c^{16}
b^2}{7007}+\frac{157559074357192 c^{14}
b^2}{105105}+\frac{1363235388608 c^{12} b^2}{3185}+\frac{29878490722
c^{10} b^2}{385}+\frac{24684424 c^8 b^2}{3}+424156 c^6
b^2+\frac{164190371975 c^{26}}{1089}+\frac{5361237368884
c^{24}}{5577}+\frac{89260211464222
c^{22}}{33033}+\frac{145446654518708
c^{20}}{33033}+\frac{4604119725105 c^{18}}{1001}+\frac{22528817322120
c^{16}}{7007}+\frac{2910348074588 c^{14}}{1911}+\frac{3389265520952
c^{12}}{7007}+\frac{1092754339 c^{10}}{11}+\frac{85356220
c^8}{7}+\frac{1}{7} \left(448 c^4 b^6+7056 c^4 b^4-224 c^2 b^4+34496
c^4 b^2-2366 c^2 b^2+168 b^2+5169090 c^6+48636 c^4-6223
c^2+1008\right).
\end{dmath*}\end{footnotesize}
Using `\texttt{SumsOfSquares}' in \textbf{Macaulay2}, we get
\begin{dmath*}
448 c^4 b^6+7056 c^4 b^4-224 c^2 b^4+34496 c^4 b^2-2366 c^2 b^2+168 b^2+5169090 c^6+48636 c^4-6223 c^2+1008 = 5169090\bigg(-\frac{17579}{41352720}\,b^{2}c+c^{3}-\frac{160928}{2584545}\,c \bigg)^2
+692348\bigg(-\frac{79731}{1384696}\,b^{2}c^{2}+c^{2}-\frac{153597}{5538784} \bigg)^2
+\frac{474487}{4}\bigg(-\frac{61767}{4744870}\,b^{3}c^{2}+b\,c^{2}-\frac{200809}{12336662}\,b \bigg)^2
+\frac{125461294753}{10338180}\bigg(-\frac{218508674731}{3261993663578}\,b^{2}c+c \bigg)^2
+\frac{108683246871}{13846960}\bigg(b^{2}c^{2}-\frac{735071362015}{5651528837292} \bigg)^2
+\frac{139289}{52}\bigg(b\,c \bigg)^2
+\frac{81212908111}{189794800}\bigg(b^{3}c^{2}-\frac{555976793995}{1055767805443}\,b \bigg)^2
+\frac{402956990227706381}{1175517998156736}\bigg(1 \bigg)^2
+\frac{1642106627418391347}{13569893640484480}\bigg(b^{2}c \bigg)^2
+\frac{245774985506386}{13724981470759}\bigg(b \bigg)^2.
\end{dmath*}
\begin{scriptsize}\begin{dmath*}
h_7(b,c)=\frac{32768 c^{30} b^{30}}{21070924875}+\frac{65536 c^{28}
b^{30}}{39131717625}+\frac{68337664 c^{30}
b^{28}}{273922023375}+\frac{231964672 c^{28}
b^{28}}{586975764375}+\frac{5439488 c^{26}
b^{28}}{39131717625}+\frac{242819072 c^{30}
b^{26}}{13043905875}+\frac{22728261632 c^{28}
b^{26}}{586975764375}+\frac{108470272 c^{26}
b^{26}}{4347968625}+\frac{5029888 c^{24}
b^{26}}{1003377375}+\frac{33536528384 c^{30}
b^{24}}{39131717625}+\frac{430012915712 c^{28}
b^{24}}{195658588125}+\frac{646418432 c^{26}
b^{24}}{323402625}+\frac{1106870272 c^{24}
b^{24}}{1449322875}+\frac{20676608 c^{22}
b^{24}}{200675475}+\frac{152180918272 c^{30}
b^{22}}{5590245375}+\frac{4411522617344 c^{28}
b^{22}}{53361433125}+\frac{35678670848 c^{26}
b^{22}}{372683025}+\frac{686989582336 c^{24}
b^{22}}{13043905875}+\frac{329676800 c^{22}
b^{22}}{24081057}+\frac{4870144 c^{20}
b^{22}}{3648645}+\frac{1177566524416 c^{30}
b^{20}}{1863415125}+\frac{1297201921439744 c^{28}
b^{20}}{586975764375}+\frac{120590942124032 c^{26}
b^{20}}{39131717625}+\frac{3164546449408 c^{24}
b^{20}}{1449322875}+\frac{496066573312 c^{22}
b^{20}}{602026425}+\frac{128628736 c^{20}
b^{20}}{825825}+\frac{9900032 c^{18}
b^{20}}{868725}+\frac{3033593405456896 c^{30}
b^{18}}{273922023375}+\frac{8547949053494272 c^{28}
b^{18}}{195658588125}+\frac{923760806332928 c^{26}
b^{18}}{13043905875}+\frac{793678304392192 c^{24}
b^{18}}{13043905875}+\frac{17900065622528 c^{22}
b^{18}}{602026425}+\frac{1644496237568 c^{20}
b^{18}}{200675475}+\frac{460692992 c^{18}
b^{18}}{394875}+\frac{18885632 c^{16}
b^{18}}{289575}+\frac{40866338200811264 c^{30}
b^{16}}{273922023375}+\frac{29504266738674176 c^{28}
b^{16}}{45151981875}+\frac{47023328208361216 c^{26}
b^{16}}{39131717625}+\frac{5241665987442688 c^{24}
b^{16}}{4347968625}+\frac{15985056058624 c^{22}
b^{16}}{22297275}+\frac{2452215712256 c^{20}
b^{16}}{9555975}+\frac{231990900992 c^{18}
b^{16}}{4343625}+\frac{8173356032 c^{16}
b^{16}}{1403325}+\frac{72306688 c^{14}
b^{16}}{289575}+\frac{6772078994287744 c^{30}
b^{14}}{4347968625}+\frac{4390359989064600832 c^{28}
b^{14}}{586975764375}+\frac{13327300113976832 c^{26}
b^{14}}{869593725}+\frac{228173159255886848 c^{24}
b^{14}}{13043905875}+\frac{349050803730176 c^{22}
b^{14}}{28667925}+\frac{3906839916032 c^{20}
b^{14}}{735075}+\frac{6252159817216 c^{18}
b^{14}}{4343625}+\frac{4194160620544 c^{16}
b^{14}}{18243225}+\frac{206661248 c^{14} b^{14}}{10725}+\frac{12118784
c^{12} b^{14}}{19305}+\frac{493086967257621952 c^{30}
b^{12}}{39131717625}+\frac{12849033838793635712 c^{28}
b^{12}}{195658588125}+\frac{5786157093705459712 c^{26}
b^{12}}{39131717625}+\frac{91015857306318848 c^{24}
b^{12}}{483107625}+\frac{29987049724226944 c^{22}
b^{12}}{200675475}+\frac{48708220722944 c^{20}
b^{12}}{637065}+\frac{9984511641344 c^{18}
b^{12}}{394875}+\frac{2742910942208 c^{16}
b^{12}}{521235}+\frac{62455173824 c^{14} b^{12}}{96525}+\frac{53144704
c^{12} b^{12}}{1287}+\frac{1478912 c^{10}
b^{12}}{1485}+\frac{437984906344618208 c^{30}
b^{10}}{5590245375}+\frac{257911544235495186752 c^{28}
b^{10}}{586975764375}+\frac{182138381290570528 c^{26}
b^{10}}{169401375}+\frac{1787221027036076224 c^{24}
b^{10}}{1185809625}+\frac{14622490648806592 c^{22}
b^{10}}{10945935}+\frac{6258176637986432 c^{20}
b^{10}}{8027019}+\frac{1313921511295936 c^{18}
b^{10}}{4343625}+\frac{40108717234304 c^{16}
b^{10}}{521235}+\frac{1191295347488 c^{14}
b^{10}}{96525}+\frac{22338232768 c^{12} b^{10}}{19305}+\frac{244108192
c^{10} b^{10}}{4455}+\frac{125504 c^8
b^{10}}{135}+\frac{685089599910084784 c^{30}
b^8}{1863415125}+\frac{1293046639384693665376 c^{28}
b^8}{586975764375}+\frac{17475723595240975024 c^{26}
b^8}{3010132125}+\frac{12828101842262460032 c^{24}
b^8}{1449322875}+\frac{472692113777782048 c^{22}
b^8}{54729675}+\frac{125670015478511936 c^{20}
b^8}{22297275}+\frac{3613520248080416 c^{18}
b^8}{1447875}+\frac{1942340622708352 c^{16}
b^8}{2606175}+\frac{14112701617264 c^{14}
b^8}{96525}+\frac{345486271904 c^{12} b^8}{19305}+\frac{1110551824
c^{10} b^8}{891}+\frac{1037824 c^8 b^8}{25}+\frac{2176 c^6
b^8}{5}+\frac{38357652830898352024 c^{30}
b^6}{30435780375}+\frac{1568896744214797657648 c^{28}
b^6}{195658588125}+\frac{58930304512834223408 c^{26}
b^6}{2608781175}+\frac{37232233827217872032 c^{24}
b^6}{1003377375}+\frac{23716853781261738904 c^{22}
b^6}{602026425}+\frac{5681394540067554544 c^{20}
b^6}{200675475}+\frac{6767389044544096 c^{18}
b^6}{482625}+\frac{12458458699837888 c^{16}
b^6}{2606175}+\frac{106297404992888 c^{14}
b^6}{96525}+\frac{1061306887088 c^{12} b^6}{6435}+\frac{7424792752
c^{10} b^6}{495}+\frac{495362848 c^8 b^6}{675}+\frac{226648 c^6
b^6}{15}+\frac{6044717066970154732 c^{30}
b^4}{2029052025}+\frac{1305734628921976084072 c^{28}
b^4}{65219529375}+\frac{260462198364442750552 c^{26}
b^4}{4347968625}+\frac{457436751807728788544 c^{24}
b^4}{4347968625}+\frac{24154389256946167852 c^{22}
b^4}{200675475}+\frac{899441721348971752 c^{20}
b^4}{9555975}+\frac{223013428893905216 c^{18}
b^4}{4343625}+\frac{71435302008678208 c^{16}
b^4}{3648645}+\frac{1491052459583948 c^{14}
b^4}{289575}+\frac{17479572907528 c^{12}
b^4}{19305}+\frac{448607662984 c^{10} b^4}{4455}+\frac{1443895168 c^8
b^4}{225}+\frac{2882692 c^6 b^4}{15}+\frac{1864510962275218 c^{30}
b^2}{429429}+\frac{423442590899827348 c^{28}
b^2}{13803075}+\frac{267667308866139454 c^{26}
b^2}{2760615}+\frac{1497908249959569716 c^{24}
b^2}{8281845}+\frac{2114042008345258174 c^{22}
b^2}{9555975}+\frac{593790439445694364 c^{20}
b^2}{3185325}+\frac{14562325156401034 c^{18}
b^2}{131625}+\frac{282509860040384828 c^{16}
b^2}{6081075}+\frac{1318793421067142 c^{14}
b^2}{96525}+\frac{3528101498444 c^{12} b^2}{1287}+\frac{1592710244282
c^{10} b^2}{4455}+\frac{18733155076 c^8 b^2}{675}+\frac{3194002 c^6
b^2}{3}+\frac{60139207926675 c^{30}}{20449}+\frac{444912639060106
c^{28}}{20449}+\frac{1478290911554065
c^{26}}{20449}+\frac{8737529975507480
c^{24}}{61347}+\frac{238105585232221
c^{22}}{1287}+\frac{2359065426708998
c^{20}}{14157}+\frac{186978113252539
c^{18}}{1755}+\frac{563604806416076
c^{16}}{11583}+\frac{20269646811239 c^{14}}{1287}+\frac{7572883528966
c^{12}}{2145}+\frac{778679579839 c^{10}}{1485}+\frac{10684326464
c^8}{225}+\frac{1}{45} \left(3696 c^4 b^6+70952 c^4 b^4-1680 c^2
b^4+438004 c^4 b^2-20250 c^2 b^2+1260 b^2+97752669 c^6+847658
c^4-60105 c^2+8820\right).
\end{dmath*}\end{scriptsize}
Using `\texttt{SumsOfSquares}' in \textbf{Macaulay2}, we get
\begin{dmath*}
3696 c^4 b^6+70952 c^4 b^4-1680 c^2 b^4+438004 c^4 b^2-20250 c^2 b^2+1260 b^2+97752669 c^6+847658 c^4-60105 c^2+8820 = 97752669\bigg(-\frac{23981}{97752669}\,b^{2}c+c^{3}-\frac{49373207}{977526690}\,c \bigg)^2
+\frac{53611497}{5}\bigg(-\frac{11286025}{214445988}\,b^{2}c^{2}+c^{2}-\frac{1006625}{47654664} \bigg)^2
+\frac{3229137}{2}\bigg(-\frac{94043}{7175860}\,b^{3}c^{2}+b\,c^{2}-\frac{758857}{55971708}\,b \bigg)^2
+\frac{350689158239069}{2443816725}\bigg(-\frac{793505948202085}{18235836228431588}\,b^{2}c+c \bigg)^2
+\frac{358437370956251}{4288919760}\bigg(b^{2}c^{2}-\frac{1009088034241725}{9319371644862526} \bigg)^2
+\frac{1448441}{52}\bigg(b\,c \bigg)^2
+\frac{981282369759}{287034400}\bigg(b^{3}c^{2}-\frac{6562235249365}{12756670806867}\,b \bigg)^2
+\frac{2961857827865059555905}{969214651065702704}\bigg(1 \bigg)^2
+\frac{934901241025865976851}{948263483878442576}\bigg(b^{2}c \bigg)^2
+\frac{19419295884942205}{331673440978542}\bigg(b \bigg)^2.
\end{dmath*}
\section{Proof of HFRI (\ref{Aug18a}) for the case $m_2\ge 8$} \label{sec4} \setcounter{equation}{0}
We will show that (\ref{Aug18a}) holds for $m_2\ge 8$. To this end, we split the unit interval into two subintervals according to the bound
\begin{eqnarray}\label{BBB}
B=\frac{2.75}{m_2m_3}.
\end{eqnarray}
In the following two subsections, we use different techniques to prove that
\begin{eqnarray}\label{Aug26A}
\frac{F\left(-m_2,-m_3;\frac{1}{2};z\right)}{F\left(-m_2,-m_3;\frac{3}{2};z\right)}>\frac{1}{H(z)}
\end{eqnarray}
for the case $z\le \frac{2.75}{m_2m_3}$ and the case $z> \frac{2.75}{m_2m_3}$ separately.
Here we would like to point out that although we choose to use the bound (\ref{BBB}) corresponding to the truncation number ``4" (see (\ref{Aug26CC}) below), we may also use the following bound corresponding to the truncation number ``5":
\begin{eqnarray}\label{BBB2}
B=\frac{3.8}{m_2m_3}.
\end{eqnarray}
The advantage of using the bound (\ref{BBB2}) is that we can apply the same method of this section to establish the inequality (\ref{Aug26A}) for all $m_2\ge 6$. The price is that the Appendix would become much longer. To shorten the length of this paper, we have decided to use the truncation number ``4" and the bound (\ref{BBB}).
\subsection{The case that $z\le \frac{2.75}{m_2m_3}$ }
First, we present a useful lemma.
\begin{lem} For $m_2,m_3\in\mathbb{N}$, we have
\begin{eqnarray}\label{Aug199}
H(z)>
\begin{cases}
\frac{1}{2},\ \ &{\rm if}\ \frac{1}{r^2}< z\le\frac{1}{r},\\
\frac{1}{7},\ \ &{\rm if}\ \frac{1}{r}< z\le\frac{2.75}{m_2m_3}.
\end{cases}
\end{eqnarray}
\end{lem}
\noindent{\bf Proof.}\ \ For $\frac{1}{r^2}< z\le\frac{1}{r}$, we have
\begin{eqnarray*}
&&H(z)>\frac{1}{2}\\
&\Leftrightarrow&4[(m_3-m_2)(rz-1)]^2+4(r-1)^3z>[(r^2z-1)+2(m_2+m_3+1)(1-rz)]^2\\
&\Leftrightarrow&4(r-1)^3z-(r^2z-1)^2-4(r-1)(rz-1)^2>4(m_2+m_3+1)(1-rz)(r^2z-1)\\
&\Leftrightarrow&4(r-1)(1-z)(r^2z-1)-(r^2z-1)^2>4(m_2+m_3+1)(1-rz)(r^2z-1)\\
&\Leftrightarrow&4(r-1)(1-z)-(r^2z-1)>4(m_2+m_3+1)(1-rz)\\
&\Leftarrow&3r-7>4(m_2+m_3+1)(1-rz)\\
&\Leftarrow&3r-7>4(m_2+m_3+1).
\end{eqnarray*}
Then,
\begin{eqnarray*}
H(z)>\frac{1}{2},\ \ \ \ \frac{1}{r^2}< z\le\frac{1}{r}.
\end{eqnarray*}
For $\frac{1}{r}< z\le\frac{2.75}{m_2m_3}$, we have
\begin{eqnarray*}
&&H(z)>\frac{1}{7}\\
&\Leftrightarrow&49[(m_3-m_2)(rz-1)]^2+49(r-1)^3z>[(r^2z-1)-7(m_2+m_3+1)(rz-1)]^2\\
&\Leftrightarrow&49(r-1)^3z-(r^2z-1)^2-49(r-1)(rz-1)^2>-14(m_2+m_3+1)(rz-1)(r^2z-1)\\
&\Leftrightarrow&49(r-1)(1-z)(r^2z-1)-(r^2z-1)^2>-14(m_2+m_3+1)(rz-1)(r^2z-1)\\
&\Leftrightarrow&49(r-1)(1-z)-(r^2z-1)>-14(m_2+m_3+1)(rz-1)\\
&\Leftrightarrow&49r-r^2z+49z+[14(m_2+m_3+1)-49]rz>14(m_2+m_3+1)+48\\
&\Leftarrow&14.5r>14(m_2+m_3+1)+48.
\end{eqnarray*}
Then,
\begin{eqnarray*}
H(z)>\frac{1}{7},\ \ \ \ \frac{1}{r}< z\le\frac{2.75}{m_2m_3}.
\end{eqnarray*}\hfill\fbox
By (\ref{Aug27HH}), we obtain that for $\frac{1}{r^2}< z<1$,
\begin{eqnarray}\label{Aug26CC}
&&\frac{F\left(-m_2,-m_3;\frac{1}{2};z\right)}{F\left(-m_2,-m_3;\frac{3}{2};z\right)}-\frac{1}{H(z)}>0\nonumber\\
&\Leftrightarrow&\ \ \left[(m_2+m_3+1)(rz-1)+\sqrt{[(m_3-m_2)(rz-1)]^2+(2m_2+1)^3(2m_3+1)^3z}\right]\nonumber\\
&&\ \ \cdot\sum_{j=0}^{m_2}\frac{2^{2j}z^j}{(m_2-j)!(m_3-j)!(2j)!}\nonumber\\
&&>(r^2z-1)\sum_{j=0}^{m_2}\frac{2^{2j}z^j}{(m_2-j)!(m_3-j)!(2j+1)!}\nonumber\\
&\Leftarrow&\ \ \ \ \ \ \ \ \ \ \left[(m_2+m_3+1)(rz-1)+\sqrt{[(m_3-m_2)(rz-1)]^2+(2m_2+1)^3(2m_3+1)^3z}\right]\nonumber\\
&&\ \ \ \ \ \ \ \ \ \ \cdot\sum_{j=0}^{4}\frac{2^{2j}z^j}{(m_2-j)!(m_3-j)!(2j)!}\nonumber\\
&&\ \ \ \ \ \ \ >(r^2z-1)\sum_{j=0}^{4}\frac{2^{2j}z^j}{(m_2-j)!(m_3-j)!(2j+1)!},\nonumber\\
&&{\rm and}\ \ H(z)\sum_{j=5}^{m_2}\frac{2^{2j}z^j}{(m_2-j)!(m_3-j)!(2j)!}>\sum_{j=5}^{m_2}\frac{2^{2j}z^j}{(m_2-j)!(m_3-j)!(2j+1)!}.
\end{eqnarray}
Define
$$
u=m_2m_3z.
$$
Similar to (\ref{Aug22g}), we can show that
\begin{eqnarray}\label{Aug20aa}
&&\ \ \left[(m_2+m_3+1)(rz-1)+\sqrt{[(m_3-m_2)(rz-1)]^2+(2m_2+1)^3(2m_3+1)^3z}\right]\nonumber\\
&&\ \ \cdot\left[\sum_{j=0}^{4}\frac{2^{2j}z^j}{(m_2-j)!(m_3-j)!(2j)!}\right]\nonumber\\
&&>(r^2z-1)\left[\sum_{j=0}^{4}\frac{2^{2j}z^j}{(m_2-j)!(m_3-j)!(2j+1)!}\right]\nonumber\\
&\Leftarrow&\ \ (2m_2+1)(2m_3+1)\left(1-\frac{u}{m_2m_3}\right)\left[1+\sum_{j=1}^{4}\frac{2^{2j}u^j(m_2-1)!(m_3-1)!}{(m_2m_3)^{j-1}(m_2-j)!(m_3-j)!(2j)!}\right]^2\nonumber\\
&&\ \ +2(m_2+m_3+1)\left\{\frac{[(2m_2+1)(2m_3+1)+1]u}{m_2m_3}-1\right\}\nonumber\\
&&\ \ \ \ \cdot\left[1+\sum_{j=1}^{4}\frac{2^{2j}u^j(m_2-1)!(m_3-1)!}{(m_2m_3)^{j-1}(m_2-j)!(m_3-j)!(2j)!}\right]\nonumber\\
&&\ \ \ \ \cdot\left[1+\sum_{j=1}^{4}\frac{2^{2j}u^j(m_2-1)!(m_3-1)!}{(m_2m_3)^{j-1}(m_2-j)!(m_3-j)!(2j+1)!}\right]\nonumber\\
&&>\left\{\frac{[(2m_2+1)(2m_3+1)+1]^2u}{m_2m_3}-1\right\}\nonumber\\
&&\ \ \cdot\left[1+\sum_{j=1}^{4}\frac{2^{2j}u^j(m_2-1)!(m_3-1)!}{(m_2m_3)^{j-1}(m_2-j)!(m_3-j)!(2j+1)!}\right]^2.
\end{eqnarray}
Obviously, the second inequality of (\ref{Aug20aa}) is a direct consequence of the following inequality: for $u\in(0,2.75)$ and $x_2\ge8$, $x_3\ge x_2$,
\begin{eqnarray}\label{Aug29l}
&&f(x_2,x_3,u)\nonumber\\
&:=&(2x_2+1)(2x_3+1)\left({x_2x_3}-{u}\right)\left[17!(x_2x_3)^3+\sum_{j=1}^{4}\frac{2^{2j}u^j(x_2x_3)^{4-j}(x_2-1)!(x_3-1)!17!}{(x_2-j)!(x_3-j)!(2j)!}\right]^2\nonumber\\
&&+2(x_2+x_3+1)\left\{{[(2x_2+1)(2x_3+1)+1]u}-{x_2x_3}\right\}\nonumber\\
&&\ \ \cdot\left[17!(x_2x_3)^3+\sum_{j=1}^{4}\frac{2^{2j}u^j(x_2x_3)^{4-j}(x_2-1)!(x_3-1)!17!}{(x_2-j)!(x_3-j)!(2j)!}\right]\nonumber\\
&&\ \ \cdot\left[17!(x_2x_3)^3+\sum_{j=1}^{4}\frac{2^{2j}u^j(x_2x_3)^{4-j}(x_2-1)!(x_3-1)!17!}{(x_2-j)!(x_3-j)!(2j+1)!}\right]\nonumber\\
&&-\left\{{[(2x_2+1)(2x_3+1)+1]^2u}-{x_2x_3}\right\}\nonumber\\
&&\ \ \cdot\left[17!(x_2x_3)^3+\sum_{j=1}^{4}\frac{2^{2j}u^j(x_2x_3)^{4-j}(x_2-1)!(x_3-1)!17!}{(x_2-j)!(x_3-j)!(2j+1)!}\right]^2\nonumber\\
&>&0.
\end{eqnarray}
By using the transformations
$$
u=\frac{2.75c^2}{1+c^2},\ \ x_2=a^2+8,\ \ x_3=b^2+8,
$$
we define
$$
g(a,b,c):=(1+c^2)^{9}\cdot f(x_2,x_3,u),\ \ \ \ a,b,c \in \mathbb{R}.
$$
Then, (\ref{Aug29l}) holds for any $u\in(0,2.75)$ and $x_2\ge8$, $x_3\ge x_2$ if $g$ is positive on $\mathbb{R}^3$.
By virtue of \textbf{Mathematica}, we obtain the expansion of $g$. See the Appendix.
Note that all terms in the expansion are positive. Hence (\ref{Aug29l}) holds. Therefore, the first inequality of (\ref{Aug20aa}) holds, which together with (\ref{Aug199}) and (\ref{Aug26CC}) implies that
\begin{eqnarray*}
\frac{F\left(-m_2,-m_3;\frac{1}{2};z\right)}{F\left(-m_2,-m_3;\frac{3}{2};z\right)}>\frac{1}{H(z)},\ \ \ \ \frac{1}{t}<z\le \frac{2.75}{m_2m_3}.
\end{eqnarray*}
\subsection{The case that $z> \frac{2.75}{m_2m_3}$ }
In this subsection, we show that for $m_2\ge 8$,
\begin{eqnarray*}
\frac{F\left(-m_2,-m_3;\frac{1}{2};z\right)}{F\left(-m_2,-m_3;\frac{3}{2};z\right)}>\frac{1}{H(z)},\ \ \ \ \frac{2.75}{m_2m_3}<z< 1.
\end{eqnarray*}
Note that
\begin{eqnarray*}
&&\frac{F\left(-m_2,-m_3;\frac{1}{2};z\right)}{F\left(-m_2,-m_3;\frac{3}{2};z\right)}>\frac{1}{H(z)}\nonumber\\
&\Leftrightarrow&F\left(-m_2,-m_3;\frac{3}{2};z\right)-H(z)F\left(-m_2,-m_3;\frac{1}{2};z\right)<0\nonumber\\
&\Leftrightarrow&\frac{F\left(-m_2-1,-m_3;\frac{1}{2};z\right)-(1-z)F\left(-m_2,-m_3;\frac{1}{2};z\right)}{(2m_3+1)z}-H(z)F\left(-m_2,-m_3;\frac{1}{2};z\right)<0\nonumber\\
&\Leftrightarrow&F\left(-m_2-1,-m_3;\frac{1}{2};z\right)-[(1-z)+(2m_3+1)zH(z)]F\left(-m_2,-m_3;\frac{1}{2};z\right)<0.
\end{eqnarray*}
For $0<z\le 1$, define
\begin{eqnarray}\label{Aug15aaa}
G(z):=F\left(-m_2-1,-m_3;\frac{1}{2};z\right)-[(1-z)+(2m_3+1)zH(z)]F\left(-m_2,-m_3;\frac{1}{2};z\right).
\end{eqnarray}
We will show that
\begin{eqnarray}\label{Aug15a}
G(z)<0,\ \ \ \ \frac{2.75}{m_2m_3}< z<1.
\end{eqnarray}
By (\ref{KKJJGG}) and (\ref{HGF}), we get
{\small\begin{eqnarray}\label{HGFD}
&&H(1)\nonumber\\
&=&\lim_{z\rightarrow 1}\frac{1-z}{-\beta+\sqrt{\beta^2-(1-z)\gamma}}\nonumber\\
&=&\lim_{z\rightarrow 1}\frac{1-z}{(m_2+m_3+1)\left[
\frac{1-z}{(2m_2+1)(2m_3+1)}-z\right]+\sqrt{(m_3-m_2)^2\left[
\frac{1-z}{(2m_2+1)(2m_3+1)}-z\right]^2+(2m_2+1)(2m_3+1)z}}\nonumber\\
&=&\lim_{z\rightarrow 1}\{(1-z)\nonumber\\
&&\left.\ \ \cdot\frac{(m_2+m_3+1)\left[
\frac{1-z}{(2m_2+1)(2m_3+1)}-z\right]-\sqrt{(m_3-m_2)^2\left[
\frac{1-z}{(2m_2+1)(2m_3+1)}-z\right]^2+(2m_2+1)(2m_3+1)z}}{(2m_2+1)(2m_3+1)\left[
\frac{1-z}{(2m_2+1)(2m_3+1)}-z\right]^2-(2m_2+1)(2m_3+1)z}\right\}\nonumber\\
&=&\lim_{z\rightarrow 1}\frac{(m_2+m_3+1)\left[
\frac{1-z}{(2m_2+1)(2m_3+1)}-z\right]-\sqrt{(m_3-m_2)^2\left[
\frac{1-z}{(2m_2+1)(2m_3+1)}-z\right]^2+(2m_2+1)(2m_3+1)z}}{\frac{1-z}{(2m_2+1)(2m_3+1)}-2z-(2m_2+1)(2m_3+1)z}\nonumber\\
&=&\frac{2(m_2+m_3+1)}{2+(2m_2+1)(2m_3+1)}.
\end{eqnarray}}
\noindent Then,
\begin{eqnarray*}
G(1)&=&F\left(-m_2-1,-m_3;\frac{1}{2};1\right)-(2m_3+1)\cdot H(1)\cdot F\left(-m_2,-m_3;\frac{1}{2};1\right)\\
&=&\frac{\Gamma(\frac{1}{2})\Gamma(m_2+m_3+\frac{3}{2})}{\Gamma(m_2+\frac{3}{2})\Gamma(m_3+\frac{1}{2})}-\frac{2(2m_3+1)(m_2+m_3+1)}{2+(2m_2+1)(2m_3+1)}\cdot\frac{\Gamma(\frac{1}{2})\Gamma(m_2+m_3+\frac{1}{2})}{\Gamma(m_2+\frac{1}{2})\Gamma(m_3+\frac{1}{2})}\\
&=&-\frac{(2m_2-1)(2m_3-1)-2}{(2m_2+1)[2+(2m_2+1)(2m_3+1)]}\cdot\frac{\Gamma(\frac{1}{2})\Gamma(m_2+m_3+\frac{1}{2})}{\Gamma(m_2+\frac{1}{2})\Gamma(m_3+\frac{1}{2})}\\
&<&0.
\end{eqnarray*}
Hence, to prove (\ref{Aug15a}), we need only show that for $\frac{2.75}{m_2m_3}< z<1$,
$$
G'(z)=0\Rightarrow G(z)<0.
$$
By (\ref{Gauss3}), we get
\begin{eqnarray}\label{Aug15j}
&&G'(z)=0\nonumber\\
&\Leftrightarrow&\left\{F\left(-m_2-1,-m_3;\frac{1}{2};z\right)-[(1-z)+(2m_3+1)zH(z)]F\left(-m_2,-m_3;\frac{1}{2};z\right)\right\}'=0\nonumber\\
&\Leftrightarrow&0=\frac{(m_2+1)\left[F\left(-m_2-1,-m_3;\frac{1}{2};z\right)-F\left(-m_2,-m_3;\frac{1}{2};z\right)\right]}{z}\nonumber\\
&&\ \ \ \ \ -[(1-z)+(2m_3+1)zH(z)]'F\left(-m_2,-m_3;\frac{1}{2};z\right)\nonumber\\
&&\ \ \ \ \ -[(1-z)+(2m_3+1)zH(z)]\frac{m_2\left[F\left(-m_2,-m_3;\frac{1}{2};z\right)-F\left(-m_2+1,-m_3;\frac{1}{2};z\right)\right]}{z}\nonumber\\
&\Leftrightarrow&0=\frac{(m_2+1)\left[F\left(-m_2-1,-m_3;\frac{1}{2},z\right)-F\left(-m_2,-m_3;\frac{1}{2};z\right)\right]}{z}\nonumber\\
&&\ \ \ \ \ -[(1-z)+(2m_3+1)zH(z)]'F\left(-m_2,-m_3;\frac{1}{2};z\right)\nonumber\\
&&\ \ \ \ \ -[(1-z)+(2m_3+1)zH(z)]\nonumber\\
&&\ \ \ \ \ \cdot\frac{(2m_2+1)F\left(-m_2-1,-m_3;\frac{1}{2};z\right)-[(2m_2+1)+2m_3z]F\left(-m_2,-m_3;\frac{1}{2};z\right)}{2z(1-z)}\nonumber\\
&\Leftrightarrow&\ \ \ \{2(m_2+1)(1-z)-(2m_2+1)[(1-z)+(2m_3+1)zH(z)]\}F\left(-m_2-1,-m_3;\frac{1}{2};z\right)\nonumber\\
&&=\{2(m_2+1)(1-z)+2z(1-z)[(1-z)+(2m_3+1)zH(z)]'\nonumber\\
&&\ \ \ \ -[(2m_2+1)+2m_3z][(1-z)+(2m_3+1)zH(z)]\}F\left(-m_2,-m_3;\frac{1}{2};z\right)\nonumber\\
&\Leftrightarrow&\ \ \ \{(1-z)-(2m_2+1)(2m_3+1)zH(z)\}F\left(-m_2-1,-m_3;\frac{1}{2};z\right)\nonumber\\
&&=\{[1-2(m_3+1)z](1-z)+2(2m_3+1)z^2(1-z)H'(z)\nonumber\\
&&\ \ \ \ +(2m_3+1)z[1-2m_2-2(m_3+1)z]H(z)\}F\left(-m_2,-m_3;\frac{1}{2};z\right).
\end{eqnarray}
Note that for $\frac{2.75}{m_2m_3}< z<1$,
\begin{eqnarray*}
&&(1-z)-(2m_2+1)(2m_3+1)zH(z)\\
&<&1-\frac{[(2m_2+1)(2m_3+1)]^{5/2}z^{3/2}}{r^2z-1}\\
&<&1-\frac{[(2m_2+1)(2m_3+1)]^{5/2}z^{1/2}}{[(2m_2+1)(2m_3+1)+1]^2}\\
&<&0.
\end{eqnarray*}
Thus, by (\ref{Aug15aaa}) and (\ref{Aug15j}), to complete the proof of (\ref{Aug15a}) it suffices to show that for $\frac{2.75}{m_2m_3}< z<1$,
\begin{eqnarray*}
&&[1-2(m_3+1)z](1-z)+2(2m_3+1)z^2(1-z)H'(z)+(2m_3+1)z[1-2m_2-2(m_3+1)z]H(z)\nonumber\\
&>&[(1-z)-(2m_2+1)(2m_3+1)zH(z)]\cdot[(1-z)+(2m_3+1)zH(z)],
\end{eqnarray*}
i.e.,
\begin{eqnarray}\label{Aug13v}
(1-z)+[2(m_2+m_3+1)z-1]H(z)<(2m_2+1)(2m_3+1)zH^2(z)+2z(1-z)H'(z).
\end{eqnarray}
We have
\begin{eqnarray*}
H'(z)&=&\frac{1}{2(r^2z-1)^2\sqrt{[(m_3-m_2)(rz-1)]^2+(2m_2+1)^3(2m_3+1)^3z}}\\
&&\cdot\left\{2r(r-1)(m_2+m_3+1)\sqrt{[(m_3-m_2)(rz-1)]^2+(2m_2+1)^3(2m_3+1)^3z}\right.\\
&&\ \ -(1+r^2z)(2m_2+1)^3(2m_3+1)^3+2(m_3-m_2)^2r(r-1)(rz-1)\}.
\end{eqnarray*}
Then, for $\frac{2.75}{m_2m_3}< z<1$, we get
\begin{eqnarray*}
&&(\ref{Aug13v})\ {\rm holds}\\
&\Leftrightarrow&\ \ (r-1)z\left\{\frac{(m_2+m_3+1)(rz-1)+\sqrt{[(m_3-m_2)(rz-1)]^2+(r-1)^3z}}{r^2z-1}\right\}^2\nonumber\\
&&\ \ +\frac{z(1-z)}{(r^2z-1)^2\sqrt{[(m_3-m_2)(rz-1)]^2+(r-1)^3z}}\nonumber\\
&&\ \ \ \ \cdot\left\{2r(r-1)(m_2+m_3+1)\sqrt{[(m_3-m_2)(rz-1)]^2+(r-1)^3z}+2(m_3-m_2)^2r(r-1)(rz-1)\right\}\nonumber\\
&&>\frac{(r-1)^3z(1-z)(1+r^2z)}{(r^2z-1)^2\sqrt{[(m_3-m_2)(rz-1)]^2+(r-1)^3z}}
+(1-z)\nonumber\\
&&\ \ +[2(m_2+m_3+1)z-1]\cdot\frac{(m_2+m_3+1)(rz-1)+\sqrt{[(m_3-m_2)(rz-1)]^2+(r-1)^3z}}{r^2z-1}\nonumber\\
&\Leftrightarrow&\ \ \frac{ (r-1)z\{[(r-1)+2(m_3-m_2)^2](rz-1)^2+(r-1)^3z\}}{(r^2z-1)^2}\nonumber\\
&&\ \ +\frac{2(m_2+m_3+1)(r-1)z(rz-1)\sqrt{[(m_3-m_2)(rz-1)]^2+(r-1)^3z}}{(r^2z-1)^2}\nonumber\\
&&\ \ +\frac{2(m_2+m_3+1)r(r-1)z(1-z)}{(r^2z-1)^2}\nonumber\\
&&\ \ +\frac{2(m_3-m_2)^2r(r-1)(rz-1)z(1-z)}{(r^2z-1)^2\sqrt{[(m_3-m_2)(rz-1)]^2+(r-1)^3z}}\nonumber\\
&&>\frac{(r-1)^3z(1-z)(1+r^2z)}{(r^2z-1)^2\sqrt{[(m_3-m_2)(rz-1)]^2+(r-1)^3z}}
+(1-z)\nonumber\\
&&\ \ +\frac{[2(m_2+m_3+1)z-1]\left[(m_2+m_3+1)(rz-1)+\sqrt{[(m_3-m_2)(rz-1)]^2+(r-1)^3z}\right]}{r^2z-1}\nonumber\\
&\Leftrightarrow&\ \ \frac{(r-1)^4z^2+(m_2+m_3+1)[2r(r-1)z(1-z)+(rz-1)(r^2z-1)]}{(r^2z-1)^2}\nonumber\\
&&>\frac{(r-1)z(1-z)\left[(r-1)^2(1+r^2z)-2(m_3-m_2)^2r(rz-1)\right]}{(r^2z-1)^2\sqrt{[(m_3-m_2)(rz-1)]^2+(r-1)^3z}}
+(1-z)\nonumber\\
&&\ \ +\frac{[2(m_2+m_3+1)z(r+rz-2)-(r^2z-1)]\sqrt{[(m_3-m_2)(rz-1)]^2+(r-1)^3z}}{(r^2z-1)^2}\nonumber\\
&&\ \ +\frac{(r-1)z(rz-1)(r^2z+rz+r-3)+2(m_3-m_2)^2z(rz-1)(rz+r-2)}{(r^2z-1)^2}\\
&\Leftrightarrow&\ \ (r-1)^4z^2+(m_2+m_3+1)[2r(r-1)z(1-z)+(rz-1)(r^2z-1)]\nonumber\\
&&>\frac{(r-1)z(1-z)\left[(r-1)^2(1+r^2z)-2(m_3-m_2)^2r(rz-1)\right]}{\sqrt{[(m_3-m_2)(rz-1)]^2+(r-1)^3z}}
+(r^2z-1)^2(1-z)\nonumber\\
&&\ \ +[2(m_2+m_3+1)z(r+rz-2)-(r^2z-1)]\sqrt{[(m_3-m_2)(rz-1)]^2+(r-1)^3z}\nonumber\\
&&\ \ +(r-1)z(rz-1)(r^2z+rz+r-3)+2(m_3-m_2)^2z(rz-1)(rz+r-2)\nonumber
\end{eqnarray*}
\begin{eqnarray}\label{RRRR}
&\Leftrightarrow&\ \ -1 + 4 z - 4 r z + 3 r^2 z + z^2 - 8 r z^2 + 8 r^2 z^2 - 4 r^3 z^2 +
r^2 z^3\nonumber\\
&&\ \ +(m_2+m_3+1)(1 - 3 r z + r^2 z + 2 r z^2 - 2 r^2 z^2 + r^3 z^2)\nonumber\\
&&\ \ -2(m_3-m_2)^2(2 z - r z - 3 r z^2 + r^2 z^2 + r^2 z^3)\nonumber\\
&&>\frac{1}{\sqrt{[(m_3-m_2)(rz-1)]^2+(r-1)^3z}}\nonumber\\
&&\ \ \cdot \left\{(r-1)z(1-z)\left[(r-1)^2(1+r^2z)-2(m_3-m_2)^2r(rz-1)\right]\right.\nonumber\\
&&\ \ \left.+[2(m_2+m_3+1)z(r+rz-2)-(r^2z-1)]\{[(m_3-m_2)(rz-1)]^2+(r-1)^3z\}\right\}.\nonumber\\
&&
\end{eqnarray}
For $\frac{2.75}{m_2m_3}< z<1$ , we have
\begin{eqnarray*}
&&-1 + 4 z - 4 r z + 3 r^2 z + z^2 - 8 r z^2 + 8 r^2 z^2 - 4 r^3 z^2 +
r^2 z^3\nonumber\\
&&+(m_2+m_3+1)(1 - 3 r z + r^2 z + 2 r z^2 - 2 r^2 z^2 + r^3 z^2)\nonumber\\
&&-2(m_3-m_2)^2(2 z - r z - 3 r z^2 + r^2 z^2 + r^2 z^3)\nonumber\\
&>&- 4 r^3 z^2+\sqrt{(m_3-m_2)^2+(r-1)}(- 3 r^2 z^2 + r^3 z^2)-4(m_3-m_2)^2 r^2z^2\\
&\ge&- 4 r^3 z^2+\sqrt{(m_3-m_2)^2+(r-1)}\cdot\frac{287r^3 z^2}{290} -4(m_3-m_2)^2 r^2z^2\\
&>&\left(\frac{1}{2}\sqrt{(m_3-m_2)^2+(r-1)}-4\right)r^3 z^2+\frac{142(m_3-m_2)r^3 z^2}{290}-4(m_3-m_2)^2 r^2z^2\\
&>&0,
\end{eqnarray*}
and, for $z\ge \frac{2.1}{2m_2+1}$, we have
\begin{eqnarray*}
&&(r-1)z(1-z)\left[(r-1)^2(1+r^2z)-2(m_3-m_2)^2r(rz-1)\right]\\
&&+[2(m_2+m_3+1)z(r+rz-2)-(r^2z-1)]\{[(m_3-m_2)(rz-1)]^2+(r-1)^3z\}\\
&<&\left(1-\frac{2.1}{2m_2+1}\right)(r-1)^3z(r^2z+1)+\{4(m_2+m_3+1)rz-(r^2z-1)\}(r-1)^3z\\
&=&\left\{2\left(1-\frac{2.1}{2m_2+1}\right)+4(m_2+m_3+1)rz-\frac{2.1(r^2z-1)}{2m_2+1}\right\}(r-1)^3z\\
&=&\left\{\frac{2-\frac{2.1}{2m_2+1}}{\frac{2.1r^2z}{2m_2+1}}+\frac{4(m_2+m_3+1)rz}{\frac{2.1r^2z}{2m_2+1}}-1\right\}\frac{2.1r^2(r-1)^3z^2}{2m_2+1}\\
&<&\left\{\frac{2}{(2.1)^2(2m_3+1)^2}+\frac{2}{2.1}-1\right\}\frac{2.1r^2(r-1)^3z^2}{2m_2+1}\\
&<&0.
\end{eqnarray*}
Then, by (\ref{RRRR}), we conclude that (\ref{Aug13v}) holds for $\frac{2.1}{2m_2+1}\le z<1$.
Finally, we will show that (\ref{RRRR}) holds for $\frac{2.75}{m_2m_3}< z<\frac{2.1}{2m_2+1}$. To this end, we assume without loss of generality that the right hand side of (\ref{RRRR}) is positive. Then, for $\frac{2.75}{m_2m_3}< z<\frac{2.1}{2m_2+1}$, we have
\begin{eqnarray*}
&&(\ref{RRRR})\ {\rm holds}\\
&\Leftarrow&\ \ \{-1 + 4 z - 4 r z + 3 r^2 z + z^2 - 8 r z^2 + 8 r^2 z^2 - 4 r^3 z^2 +
r^2 z^3\\
&&\ \ +\sqrt{(m_3-m_2)^2+(r-1)}(1 - 3 r z + r^2 z + 2 r z^2 - 2 r^2 z^2 + r^3 z^2)\\
&&\ \ -2(m_3-m_2)^2(2 z - r z - 3 r z^2 + r^2 z^2 + r^2 z^3)\}^2\{[(m_3-m_2)(rz-1)]^2+(r-1)^3z\}\\
&&>\left\{(r-1)z(1-z)\left[(r-1)^2(1+r^2z)-2(m_3-m_2)^2r(rz-1)\right]\right.\\
&&\ \ \left.+[2\sqrt{(m_3-m_2)^2+(r-1)}z(r+rz-2)-(r^2z-1)]\{[(m_3-m_2)(rz-1)]^2+(r-1)^3z\}\right\}^2\\
&\Leftrightarrow&\ \ [(m_3-m_2)^2 (r z-1)^2+(
r-1)^3 z ]\\
&&\ \ \cdot \left\{-1 + 4 z - 4 r z + 3 r^2 z + z^2 -
8 r z^2 + 8 r^2 z^2 - 4 r^3 z^2 + r^2 z^3\right.\\
&&\ \ \ \ \ +
\sqrt{(m_3-m_2)^2+(r-1)
} [1 + r^3 z^2 + r z (-3 + 2 z) + r^2 (z - 2 z^2)]\\
&&\ \ \ \ \ \left.-
2 (m_3-m_2)^2 z [2 + r^2 z (1 + z) - r (1 + 3 z)]\right\}^2\\
&&\ \ -\left\{[(m_3-m_2)^2 ( r z-1)^2+(r-1 )^3 z ] [1+
2 \sqrt{(m_3-m_2)^2+(r-1) } z (r + r z-2 )]\right. \\
&&\ \ \ \ \ \ -(m_3-m_2)^2rz ( r z-1)[2(r-1)(1-z)+r(rz-1)] \\
&&\ \ \ \ \ \ \left.-(r-1)^3r^2z^3+(r-1)^3z(1-z)\right\}^2\\
&&>0.
\end{eqnarray*}
Therefore, the proof of (\ref{RRRR}) for $\frac{2.75}{m_2m_3}< z<\frac{2.1}{2m_2+1}$ is complete by
\begin{eqnarray*}
&& [(m_3-m_2)^2 (r z-1)^2+(
r-1)^3 z ]\\
&&\ \ \cdot \left\{-1 + 4 z - 4 r z + 3 r^2 z + z^2 -
8 r z^2 + 8 r^2 z^2 - 4 r^3 z^2 + r^2 z^3\right.\\
&&\ \ \ \ \ +
\sqrt{(m_3-m_2)^2+(r-1)
} [1 + r^3 z^2 + r z (-3 + 2 z) + r^2 (z - 2 z^2)]\\
&&\ \ \ \ \ \left.-
2 (m_3-m_2)^2 z [2 + r^2 z (1 + z) - r (1 + 3 z)]\right\}^2\\
&&\ \ -\left\{[(m_3-m_2)^2 ( r z-1)^2+(r-1 )^3 z ] [1+
2 \sqrt{(m_3-m_2)^2+(r-1) } z (r + r z-2 )]\right. \\
&&\ \ \ \ \ \ -(m_3-m_2)^2rz ( r z-1)[2(r-1)(1-z)+r(rz-1)] \\
&&\ \ \ \ \ \ \left.-(r-1)^3r^2z^3+(r-1)^3z(1-z)\right\}^2\\
&>&( r-1)^3 z \left\{ - 4 r^3 z^2 +
\sqrt{(m_3-m_2)^2+(r-1)
} (r^3 z^2 -3rz)-2 (m_3-m_2)^2 r^2 z^2(1+z)\right\}^2\\
&&-\left\{(r-1 )^3 z (2-z-r^2z^2)+
2 \sqrt{(m_3-m_2)^2+(r-1) }(r-1 )^3 rz^2 (1 + z)\right\}^2\\
&>&( r-1)^3 r^6z^5 [(m_3-m_2)^2+(r-1)
]\left\{1-\frac{4}{(r-1)^{1/2}} - \frac{3}{r^2 z}- \frac{2(m_3-m_2)(1+z)}{r}\right\}^2\\
&&-4[(m_3-m_2)^2+(r-1) ](r-1)^6r^2z^4 (1 + z)^2\\
&>& [(m_3-m_2)^2+(r-1)](r-1 )^6 r^2 z ^4\\
&&\cdot\left[\left\{1-\frac{4}{(r-1)^{1/2}} - \frac{3}{r^2 z}- \frac{1+\frac{2.1}{2m_2+1}}{2m_2+1}\right\}^2rz-4(1 + z)^2\right]\\
&>& [(m_3-m_2)^2+(r-1)](r-1 )^6 r^2 z ^4\\
&&\cdot\left[\left\{1-\frac{4}{17} - \frac{3}{(17^2)(4)(2.75)}- \frac{1+\frac{2.1}{17}}{17}\right\}^2( 4)( 2.75)-4\left(1 + \frac{2.1}{17}\right)^2\right]\\
&>&0.3[(m_3-m_2)^2+(r-1)](r-1 )^6 r^2 z ^4\\
&>&0.
\end{eqnarray*}
\section{Concluding remarks}\label{sec5}\setcounter{equation}{0}
{\bf Remark 1}\ \ In this paper, the exponents $m_2$ and $m_3$ are assumed to be natural numbers. However, Theorems \ref{thmm1m2m3} and \ref{thm2} can be extended to the case that $m_2$ and $m_3$ are positive real numbers.
Let $y_2,y_3 \in (0,\infty)$. We consider a centered Gaussian random vector $(X_1,X_2,X_3)$. Let $X_1 = X_2+aX_3$ for some $a \in \mathbb{R}$. Define $x=E[X_2 X_3]$. Assume without loss of generality that $E[X_2^2]=E[X_3^2]=1$. We have
\begin{equation}\label{1d}
E[|X_2|^{y_2}]=\frac{2^{y_2/2}\Gamma(\frac{y_2+1}{2})}{\sqrt{\pi}},\ \ \ \ E[|X_3|^{y_3}]=\frac{2^{y_3/2}\Gamma(\frac{y_3+1}{2})}{\sqrt{\pi}}.
\end{equation}
By Nabeya \cite{Nabeya1951}, we get
\begin{eqnarray}\label{2deven}
&&E[|X_2|^{y_2} |X_3|^{y_3+2}] = (y_3+1)\cdot\frac{2^{(y_2+y_3)/2}\Gamma(\frac{y_2+1}{2})\Gamma(\frac{y_3+1}{2})}{\pi}F\left(-\frac{y_3}{2}-1,-\frac{y_2}{2};\frac{1}{2};x^2\right),\nonumber\\
&&E[|X_2|^{y_2+2} |X_3|^{y_3}] = (y_2+1)\cdot\frac{2^{(y_2+y_3)/2}\Gamma(\frac{y_2+1}{2})\Gamma(\frac{y_3+1}{2})}{\pi}F\left(-\frac{y_2}{2}-1,-\frac{y_3}{2};\frac{1}{2};x^2\right).\ \ \ \
\end{eqnarray}
Let $p(x_2,x_3)$ be the probability density function of $(X_2,X_3)$. By Kamat \cite{Kamat1958}, we get
\begin{eqnarray}\label{2dodd}
&&E[|X_2|^{y_2}X_2 |X_3|^{y_3}X_3] \nonumber\\
&=& \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}|x_2|^{y_2} x_2 |x_3|^{y_3} x_3 p(x_2,x_3) dx_2 dx_3 \nonumber\\
&=&\left[\int_{0}^{\infty}\int_{0}^{\infty}+\int_{-\infty}^{0}\int_{-\infty}^{0}-\int_{0}^{\infty}\int_{-\infty}^{0}- \int_{-\infty}^{0}\int_{0}^{\infty}\right]|x_2|^{y_2+1} |x_3|^{y_3+1} p(x_2,x_3) dx_2 dx_3\nonumber\\
&=&2\cdot\frac{2^{(y_2+y_3-2)/2}}{\pi}\Bigg[\Gamma\left(\frac{y_2+2}{2}\right)\Gamma\left(\frac{y_3+2}{2}\right)F\left(-\frac{y_2+1}{2},-\frac{y_3+1}{2};\frac{1}{2};x^2\right)\nonumber\\
&&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +2x\Gamma\left(\frac{y_2+3}{2}\right)\Gamma\left(\frac{y_3+3}{2}\right)F\left(-\frac{y_2}{2},-\frac{y_3}{2};\frac{3}{2};x^2\right)\Bigg]\nonumber\\
&&-2\cdot\frac{2^{(y_2+y_3-2)/2}}{\pi}\Bigg[\Gamma\left(\frac{y_2+2}{2}\right)\Gamma\left(\frac{y_3+2}{2}\right)F\left(-\frac{y_2+1}{2},-\frac{y_3+1}{2};\frac{1}{2};x^2\right)\nonumber\\
&&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -2x\Gamma\left(\frac{y_2+3}{2}\right)\Gamma\left(\frac{y_3+3}{2}\right)F\left(-\frac{y_2}{2},-\frac{y_3}{2};\frac{3}{2};x^2\right)\Bigg]\nonumber\\
&&=x(y_2+1)(y_3+1)\cdot\frac{2^{(y_2+y_3)/2}\Gamma\left(\frac{y_2+1}{2}\right)\Gamma\left(\frac{y_3+1}{2}\
\right)}{\pi}F\left(-\frac{y_2}{2},-\frac{y_3}{2};\frac{3}{2};x^2\right).
\end{eqnarray}
Then,
\begin{eqnarray}\label{Ka}
&&\frac{E[(X_2+aX_3)^{2}|X_2|^{y_2} |X_3|^{y_3} ]} {E[X_1^{2}]E[|X_2|^{y_2}] E[|X_3|^{y_3}] }\nonumber\\
&=&\frac{E[a^2|X_2|^{y_2} |X_3|^{y_3+2}] + |X_2|^{y_2+2} |X_3|^{y_3} + 2a|X_2|^{y_2}X_2 |X_3|^{y_3}X_3]} {(a^2+1+2ax)E[|X_2|^{y_2}] E[|X_3|^{y_3}]}\nonumber\\
&=&\Bigg[a^2\left(y_3+1\right)F\left(-\frac{y_3}{2}-1,-\frac{y_2}{2};\frac{1}{2};x^2\right)+\left(y_2+1\right)F\left(-\frac{y_3}{2},-\frac{y_2}{2}-1;\frac{1}{2};x^2\right)\nonumber\\
&&+2ax\left(y_3+1\right)\left(y_2+1\right) F\left(-\frac{y_3}{2},-\frac{y_2}{2};\frac{3}{2};x^2\right)\Bigg]\cdot\frac{1}{a^2+1+2ax}
\end{eqnarray}
by (\ref{1d})--(\ref{2dodd}). Hence, by a modified version of \cite[Lemma 2.1]{RusSun}, verifying the corresponding GPIs is equivalent to showing that for any $a\in\mathbb{R}$ and $x\in[-1,1]$,
\begin{eqnarray}\label{gen1}
&&a^2\left(y_3+1\right)F\left(-\frac{y_3}{2}-1,-\frac{y_2}{2};\frac{1}{2};x^2\right)+\left(y_2+1\right)F\left(-\frac{y_3}{2},-\frac{y_2}{2}-1;\frac{1}{2};x^2\right)\nonumber\\
&&+2ax\left(y_3+1\right)\left(y_2+1\right) F\left(-\frac{y_3}{2},-\frac{y_2}{2};\frac{3}{2};x^2\right)\nonumber\\
&>&a^2+1+2ax.
\end{eqnarray}
Note that the inequality (\ref{gen1}) is exactly the same as the inequality (\ref{gen2}) except that integer-valued exponents $m_2,m_3$ are replaced with real-valued exponents $\frac{y_2}{2},\frac{y_3}{2}$. The arguments used in Sections \ref{sec2} and \ref{sec4} can be applied without any change. Additionally, as explained in the beginning of Section \ref{sec4}, we can cover the case that
$\frac{y_2}{2},\frac{y_3}{2}\ge 6$ by using the truncation number ``5" and the bound $B=\frac{3.8}{\frac{y_2}{2}\cdot\frac{y_3}{2}}$. Therefore, we have the following propositions.
\begin{pro}\label{thmm1m2m3U}
Let $y_2,y_3 \in [12,\infty)$. For any centered Gaussian random vector $(X_1,X_2,X_3)$,
\begin{eqnarray}\label{GPIjkl}
E[X_1^{2} |X_2|^{y_2} |X_3|^{y_3}]\ge E[X_1^{2}] E[|X_2|^{y_2}] E[|X_3|^{y_3}].
\end{eqnarray}
The equality sign holds if and only if $X_1,X_2,X_3$ are independent.
\end{pro}
Define
$$
r_{y_2,y_3}=(y_2+1)(y_3+1)+1,\ \ \ \ t_{y_2,y_3}=\frac{1}{r_{y_2,y_3}+\left(1+\frac{1}{y_2}\right)\left(1+\frac{1}{y_3}\right)},
$$
and for $\frac{1}{r^2_{y_2,y_3}}<z\le 1$,
\begin{eqnarray*}
H_{y_2,y_3}(z)=\frac{\frac{(y_2+y_3+2)(r_{y_2,y_3}z-1)}{2}+\sqrt{\frac{[(y_3-y_2)(r_{y_2,y_3}z-1)]^2}{4}+(y_2+1)^3(y_3+1)^3z}}{r_{y_2,y_3}^2z-1}.
\end{eqnarray*}
\begin{pro}\label{thm2V}
Let $(X_2,X_3)$ be a centered Gaussian random vector. If $y_2,y_3\in [12,\infty)$, then
\begin{eqnarray}\label{MRIjkl}
&&\frac{\left|E[ |X_2|^{y_2}X_2|X_3|^{y_3}X_3]\right|}{(y_2+1)(y_3+1)E[ |X_2|^{y_2}|X_3|^{y_3}]}\nonumber\\
&\le&
\begin{cases}
|{\rm Cov}(X_2,X_3)|,\ \ &{\rm if}\ |{\rm Corr}(X_2,X_3)|\le \sqrt{t_{y_2,y_3}},\nonumber\\
H_{y_2,y_3}([{\rm Corr}(X_2,X_3)]^2)\cdot|{\rm Cov}(X_2,X_3)|,\ \ &{\rm if}\ \sqrt{t_{y_2,y_3}}<|{\rm Corr}(X_2,X_3)|.
\end{cases}\nonumber\\
&&
\end{eqnarray}
The equality sign holds if and only if $X_2$ and $X_3$ are independent.
\end{pro}
Through more delicate analysis, the interval $[12,\infty)$ used in Propositions \ref{thmm1m2m3U} and \ref{thm2V} can be enlarged. However, Proposition \ref{thm2V} does not hold for all $y_2,y_3\in(0,\infty)$. For example, \textbf{Mathematica} has shown that the inequality (\ref{MRIjkl}) does not hold when $y_2=4$ and $y_3\in[4,4.58]$, or when $y_2=2$ and $y_3\in[2,8.15]$. Therefore, the validity of the GPI (\ref{GPIjkl}) for the most general case that $y_2,y_3\in (0,\infty)$ still remains open, although using \textbf{Mathematica} to check the inequality (\ref{gen1}), it appears to be true.
\vskip 0.5cm
\noindent {\bf Remark 2}\ \ It is natural to ask if the MRI method developed in this paper can be adapted to solve the following 3D-GPI.
\begin{con}\label{conmm}
Let $m_1,m_2,m_3\in\mathbb{N}$. For any centered Gaussian random vector $(X_1,X_2,X_3)$,
$$
E[X_1^{2m_1} X_2^{2m_2}X_3^{2m_3}]\ge E[X_1^{2m_1}] E[X_2^{2m_2}] E[X_3^{2m_3}].
$$
The equality sign holds if and only if $X_1,X_2,X_3$ are independent.
\end{con}
By a modified version of \cite[Lemma 2.1]{RusSun}, we may assume without loss of generality that $X_1= X_2+aX_3$ for some $a \in \mathbb{R}$ and $E[X_2^2]=E[X_3^2]=1$. Define $x=E[X_2 X_3]$. Similar to (\ref{moment}), we consider the moment ratio:
\begin{eqnarray*}
&&\frac{E[(X_2+aX_3)^{2m_1}X_2^{2m_2}X_3^{2m_3} ]} {E[X_1^{2m_1}]E[X_2^{2m_2}] E[X_3^{2m_3}] }\nonumber\\
&=&\frac{\sum_{k=0}^{2m_1}\binom{2m_1}{k}a^kE[X_2^{2m_2+2m_1-k}X_3^{2m_3+k} ]} {(a^2+1+2ax)^{m_1}(2m_1-1)!!(2m_2-1)!!(2m_3-1)!!}\nonumber\\
&=&\Bigg[\sum_{i=0}^{m_1}\binom{2m_1}{2i}a^{2i}(2m_3+2i-1)\cdots(2m_3+1)\cdot(2m_2+2m_1-2i-1)\cdots(2m_2+1)\nonumber\\
&&\ \ \cdot F\left(-m_3-i,-m_2-m_1+i;\frac{1}{2};x^2\right)\nonumber\\
&&+x\sum_{j=1}^{m_1}\binom{2m_1}{2j-1}a^{2j-1}(2m_3+2j-1)\cdots(2m_3+1)\cdot(2m_2+2m_1-2j+1)\cdots(2m_2+1)\nonumber\\
&&\ \ \cdot F\left(-m_3-j+1,-m_2-m_1+j;\frac{3}{2};x^2\right)\Bigg]\cdot\frac{1}{(a^2+1+2ax)^{m_1}(2m_1-1)!!}.
\end{eqnarray*}
The proof of Conjecture \ref{conmm} is complete if we can show that
\begin{eqnarray}\label{gen3}
&&\sum_{i=0}^{m_1}\binom{2m_1}{2i}a^{2i}(2m_3+2i-1)\cdots(2m_3+1)\cdot(2m_2+2m_1-2i-1)\cdots(2m_2+1)\nonumber\\
&&\ \ \cdot F\left(-m_3-i,-m_2-m_1+i;\frac{1}{2};x^2\right)\nonumber\\
&&+x\sum_{j=1}^{m_1}\binom{2m_1}{2j-1}a^{2j-1}(2m_3+2j-1)\cdots(2m_3+1)\cdot(2m_2+2m_1-2j+1)\cdots(2m_2+1)\nonumber\\
&&\ \ \cdot F\left(-m_3-j+1,-m_2-m_1+j;\frac{3}{2};x^2\right)\nonumber\\
&>&(a^2+1+2ax)^{m_1}(2m_1-1)!!.
\end{eqnarray}
Obviously, the inequality (\ref{gen3}) extends the inequality (\ref{gen2}). Further, by using the relations (\ref{Gauss3}), we may use only two hypergeometric functions, $F\left(-m_2,-m_3;\frac{1}{2};x^2\right)$ and $F\left(-m_2,-m_3;\frac{3}{2};x^2\right)$, to simplify (\ref{gen3}).
We hope the MRI method introduced in this paper can be improved so as to prove (\ref{gen3}) with all three exponents unbounded.
\section{Appendix: expansion of the function $g$}\label{sec6}\setcounter{equation}{0}
We have the following SOS expression of the function $g$ for any $a,b,c \in \mathbb{R}$:
\begin{verbatim}
g(a,b,c)=148260632637820250986905600 + 148260632637820250986905600 a^2 +
64864026779046359806771200 a^4 + 16216006694761589951692800 a^6 +
2533751046056498429952000 a^8 + 253375104605649842995200 a^10 +
15835944037853115187200 a^12 + 565569429923325542400 a^14 +
8837022342551961600 a^16 + 148260632637820250986905600 b^2 +
148260632637820250986905600 a^2 b^2 +
64864026779046359806771200 a^4 b^2 +
16216006694761589951692800 a^6 b^2 +
2533751046056498429952000 a^8 b^2 +
253375104605649842995200 a^10 b^2 +
15835944037853115187200 a^12 b^2 + 565569429923325542400 a^14 b^2 +
8837022342551961600 a^16 b^2 + 64864026779046359806771200 b^4 +
64864026779046359806771200 a^2 b^4 +
28378011715832782415462400 a^4 b^4 +
7094502928958195603865600 a^6 b^4 +
1108516082649718063104000 a^8 b^4 +
110851608264971806310400 a^10 b^4 +
6928225516560737894400 a^12 b^4 + 247436625591454924800 a^14 b^4 +
3866197274866483200 a^16 b^4 + 16216006694761589951692800 b^6 +
16216006694761589951692800 a^2 b^6 +
7094502928958195603865600 a^4 b^6 +
1773625732239548900966400 a^6 b^6 +
277129020662429515776000 a^8 b^6 +
27712902066242951577600 a^10 b^6 + 1732056379140184473600 a^12 b^6 +
61859156397863731200 a^14 b^6 + 966549318716620800 a^16 b^6 +
2533751046056498429952000 b^8 + 2533751046056498429952000 a^2 b^8 +
1108516082649718063104000 a^4 b^8 +
277129020662429515776000 a^6 b^8 + 43301409478504611840000 a^8 b^8 +
4330140947850461184000 a^10 b^8 + 270633809240653824000 a^12 b^8 +
9665493187166208000 a^14 b^8 + 151023331049472000 a^16 b^8 +
253375104605649842995200 b^10 + 253375104605649842995200 a^2 b^10 +
110851608264971806310400 a^4 b^10 +
27712902066242951577600 a^6 b^10 + 4330140947850461184000 a^8 b^10 +
433014094785046118400 a^10 b^10 + 27063380924065382400 a^12 b^10 +
966549318716620800 a^14 b^10 + 15102333104947200 a^16 b^10 +
15835944037853115187200 b^12 + 15835944037853115187200 a^2 b^12 +
6928225516560737894400 a^4 b^12 + 1732056379140184473600 a^6 b^12 +
270633809240653824000 a^8 b^12 + 27063380924065382400 a^10 b^12 +
1691461307754086400 a^12 b^12 + 60409332419788800 a^14 b^12 +
943895819059200 a^16 b^12 + 565569429923325542400 b^14 +
565569429923325542400 a^2 b^14 + 247436625591454924800 a^4 b^14 +
61859156397863731200 a^6 b^14 + 9665493187166208000 a^8 b^14 +
966549318716620800 a^10 b^14 + 60409332419788800 a^12 b^14 +
2157476157849600 a^14 b^14 + 33710564966400 a^16 b^14 +
8837022342551961600 b^16 + 8837022342551961600 a^2 b^16 +
3866197274866483200 a^4 b^16 + 966549318716620800 a^6 b^16 +
151023331049472000 a^8 b^16 + 15102333104947200 a^10 b^16 +
943895819059200 a^12 b^16 + 33710564966400 a^14 b^16 +
526727577600 a^16 b^16 + 1178507184834162676727808000 c^2 +
1189545780444262836889190400 a^2 c^2 +
525249054431071582342348800 a^4 c^2 +
132516304956266486169600000 a^6 c^2 +
20893545903372971802624000 a^8 c^2 +
2108116095058273370112000 a^10 c^2 +
132928236194037025996800 a^12 c^2 +
4789200094813067673600 a^14 c^2 + 75482899175964672000 a^16 c^2 +
1189545780444262836889190400 b^2 c^2 +
1200259315024514565695078400 a^2 b^2 c^2 +
529794886474584937621094400 a^4 b^2 c^2 +
133617646849611109131878400 a^6 b^2 c^2 +
21060177855265766375424000 a^8 b^2 c^2 +
2124237435795556584652800 a^10 b^2 c^2 +
133902167677007088844800 a^12 b^2 c^2 +
4822789121955908812800 a^14 b^2 c^2 +
75989186914340044800 a^16 b^2 c^2 +
525249054431071582342348800 b^4 c^2 +
529794886474584937621094400 a^2 b^4 c^2 +
233771931867713966073446400 a^4 b^4 c^2 +
58939117104710746216857600 a^6 b^4 c^2 +
9286674706485076819968000 a^8 b^4 c^2 +
936402906520464497049600 a^10 b^4 c^2 +
59008125773560990924800 a^12 b^4 c^2 +
2124662473700986060800 a^14 b^4 c^2 +
33466770160562995200 a^16 b^4 c^2 +
132516304956266486169600000 b^6 c^2 +
133617646849611109131878400 a^2 b^6 c^2 +
58939117104710746216857600 a^4 b^6 c^2 +
14855018310596061403545600 a^6 b^6 c^2 +
2339866754286146813952000 a^8 b^6 c^2 +
235861969166930018304000 a^10 b^6 c^2 +
14858471394875591884800 a^12 b^6 c^2 +
534837935004136243200 a^14 b^6 c^2 +
8422067761525555200 a^16 b^6 c^2 +
20893545903372971802624000 b^8 c^2 +
21060177855265766375424000 a^2 b^8 c^2 +
9286674706485076819968000 a^4 b^8 c^2 +
2339866754286146813952000 a^6 b^8 c^2 +
368445497799113441280000 a^8 b^8 c^2 +
37128468304778231808000 a^10 b^8 c^2 +
2338260850645106688000 a^12 b^8 c^2 +
84142110929780736000 a^14 b^8 c^2 +
1324600466079744000 a^16 b^8 c^2 +
2108116095058273370112000 b^10 c^2 +
2124237435795556584652800 a^2 b^10 c^2 +
936402906520464497049600 a^4 b^10 c^2 +
235861969166930018304000 a^6 b^10 c^2 +
37128468304778231808000 a^8 b^10 c^2 +
3740334540329189376000 a^10 b^10 c^2 +
235487899861711257600 a^12 b^10 c^2 +
8471567863509811200 a^14 b^10 c^2 +
133325284442112000 a^16 b^10 c^2 +
132928236194037025996800 b^12 c^2 +
133902167677007088844800 a^2 b^12 c^2 +
59008125773560990924800 a^4 b^12 c^2 +
14858471394875591884800 a^6 b^12 c^2 +
2338260850645106688000 a^8 b^12 c^2 +
235487899861711257600 a^10 b^12 c^2 +
14821816246033305600 a^12 b^12 c^2 +
533057065377177600 a^14 b^12 c^2 + 8386907642265600 a^16 b^12 c^2 +
4789200094813067673600 b^14 c^2 +
4822789121955908812800 a^2 b^14 c^2 +
2124662473700986060800 a^4 b^14 c^2 +
534837935004136243200 a^6 b^14 c^2 +
84142110929780736000 a^8 b^14 c^2 +
8471567863509811200 a^10 b^14 c^2 +
533057065377177600 a^12 b^14 c^2 + 19165729108377600 a^14 b^14 c^2 +
301463750246400 a^16 b^14 c^2 + 75482899175964672000 b^16 c^2 +
75989186914340044800 a^2 b^16 c^2 +
33466770160562995200 a^4 b^16 c^2 +
8422067761525555200 a^6 b^16 c^2 +
1324600466079744000 a^8 b^16 c^2 +
133325284442112000 a^10 b^16 c^2 + 8386907642265600 a^12 b^16 c^2 +
301463750246400 a^14 b^16 c^2 + 4740548198400 a^16 b^16 c^2 +
4205418585929895618690416640 c^4 +
4283789227629131992808816640 a^2 c^4 +
1908717408260566013957898240 a^4 c^4 +
485887373715697908622295040 a^6 c^4 +
77291173706813363611238400 a^8 c^4 +
7867308845138505608724480 a^10 c^4 +
500410289854955472814080 a^12 c^4 +
18185010598892147834880 a^14 c^4 + 289071888149124218880 a^16 c^4 +
4283789227629131992808816640 b^2 c^4 +
4359668495142875067894988800 a^2 b^2 c^4 +
1940827262109243233733181440 a^4 b^2 c^4 +
493643668385112855360307200 a^6 b^2 c^4 +
78460828160884078333132800 a^8 b^2 c^4 +
7980057985327289786695680 a^10 b^2 c^4 +
507194239139627139072000 a^12 b^2 c^4 +
18417929327697859706880 a^14 b^2 c^4 +
292565273543914291200 a^16 b^2 c^4 +
1908717408260566013957898240 b^4 c^4 +
1940827262109243233733181440 a^2 b^4 c^4 +
863281484732864376172707840 a^4 b^4 c^4 +
219393134698552253837475840 a^6 b^4 c^4 +
34843167553001087198822400 a^8 b^4 c^4 +
3541090376643440730439680 a^10 b^4 c^4 +
224896162105929687367680 a^12 b^4 c^4 +
8160851353692882862080 a^14 b^4 c^4 +
129543066926689812480 a^16 b^4 c^4 +
485887373715697908622295040 b^6 c^4 +
493643668385112855360307200 a^2 b^6 c^4 +
219393134698552253837475840 a^4 b^6 c^4 +
55711953835203493468569600 a^6 b^6 c^4 +
8841138548643438841036800 a^8 b^6 c^4 +
897848160984087637524480 a^10 b^6 c^4 +
56981387314642865356800 a^12 b^6 c^4 +
2066240988872630599680 a^14 b^6 c^4 +
32776557579730944000 a^16 b^6 c^4 +
77291173706813363611238400 b^8 c^4 +
78460828160884078333132800 a^2 b^8 c^4 +
34843167553001087198822400 a^4 b^8 c^4 +
8841138548643438841036800 a^6 b^8 c^4 +
1401983290824405811200000 a^8 b^8 c^4 +
142272741033571044556800 a^10 b^8 c^4 +
9022876604746570137600 a^12 b^8 c^4 +
326960229932059852800 a^14 b^8 c^4 +
5183078021568921600 a^16 b^8 c^4 +
7867308845138505608724480 b^10 c^4 +
7980057985327289786695680 a^2 b^10 c^4 +
3541090376643440730439680 a^4 b^10 c^4 +
897848160984087637524480 a^6 b^10 c^4 +
142272741033571044556800 a^8 b^10 c^4 +
14427574047705645711360 a^10 b^10 c^4 +
914361084276374568960 a^12 b^10 c^4 +
33111348226576220160 a^14 b^10 c^4 +
524549875103170560 a^16 b^10 c^4 +
500410289854955472814080 b^12 c^4 +
507194239139627139072000 a^2 b^12 c^4 +
224896162105929687367680 a^4 b^12 c^4 +
56981387314642865356800 a^6 b^12 c^4 +
9022876604746570137600 a^8 b^12 c^4 +
914361084276374568960 a^10 b^12 c^4 +
57909653012277964800 a^12 b^12 c^4 +
2095686946952847360 a^14 b^12 c^4 +
33178745688883200 a^16 b^12 c^4 + 18185010598892147834880 b^14 c^4 +
18417929327697859706880 a^2 b^14 c^4 +
8160851353692882862080 a^4 b^14 c^4 +
2066240988872630599680 a^6 b^14 c^4 +
326960229932059852800 a^8 b^14 c^4 +
33111348226576220160 a^10 b^14 c^4 +
2095686946952847360 a^12 b^14 c^4 +
75792303051816960 a^14 b^14 c^4 + 1199191897128960 a^16 b^14 c^4 +
289071888149124218880 b^16 c^4 +
292565273543914291200 a^2 b^16 c^4 +
129543066926689812480 a^4 b^16 c^4 +
32776557579730944000 a^6 b^16 c^4 +
5183078021568921600 a^8 b^16 c^4 +
524549875103170560 a^10 b^16 c^4 + 33178745688883200 a^12 b^16 c^4 +
1199191897128960 a^14 b^16 c^4 + 18962192793600 a^16 b^16 c^4 +
9099687640450155263180144640 c^6 +
9328545438479094763770347520 a^2 c^6 +
4183431927861844189359636480 a^4 c^6 +
1071928757187759243201085440 a^6 c^6 +
171644911034162330625638400 a^8 c^6 +
17588428904315521943470080 a^10 c^6 +
1126299084549332077117440 a^12 c^6 +
41209030285424811048960 a^14 c^6 + 659566638832004628480 a^16 c^6 +
9328545438479094763770347520 b^2 c^6 +
9548456362608757308433367040 a^2 b^2 c^6 +
4275723411923189828325212160 a^4 b^2 c^6 +
1094020420323366888705884160 a^6 b^2 c^6 +
174943213236931741404364800 a^8 b^2 c^6 +
17902886471689191334871040 a^10 b^2 c^6 +
1144990602311449423380480 a^12 b^2 c^6 +
41842178448700232171520 a^14 b^2 c^6 +
668921265389064683520 a^16 b^2 c^6 +
4183431927861844189359636480 b^4 c^6 +
4275723411923189828325212160 a^2 b^4 c^6 +
1911906715172373635530752000 a^4 b^4 c^6 +
488523541416471233581547520 a^6 b^4 c^6 +
78015659142786066048614400 a^8 b^4 c^6 +
7973579571090166209576960 a^10 b^4 c^6 +
509328658192274549637120 a^12 b^4 c^6 +
18590714330907672576000 a^14 b^4 c^6 +
296866673570390999040 a^16 b^4 c^6 +
1071928757187759243201085440 b^6 c^6 +
1094020420323366888705884160 a^2 b^6 c^6 +
488523541416471233581547520 a^4 b^6 c^6 +
124660008517518226567987200 a^6 b^6 c^6 +
19882257503035609763020800 a^8 b^6 c^6 +
2029543733065706728980480 a^10 b^6 c^6 +
129485992445516216401920 a^12 b^6 c^6 +
4720829210372364042240 a^14 b^6 c^6 +
75300636585413836800 a^16 b^6 c^6 +
171644911034162330625638400 b^8 c^6 +
174943213236931741404364800 a^2 b^8 c^6 +
78015659142786066048614400 a^4 b^8 c^6 +
19882257503035609763020800 a^6 b^8 c^6 +
3167114835118667489280000 a^8 b^8 c^6 +
322903749309701343436800 a^10 b^8 c^6 +
20577407598801401241600 a^12 b^8 c^6 +
749368863900931276800 a^14 b^8 c^6 +
11939920073677209600 a^16 b^8 c^6 +
17588428904315521943470080 b^10 c^6 +
17902886471689191334871040 a^2 b^10 c^6 +
7973579571090166209576960 a^4 b^10 c^6 +
2029543733065706728980480 a^6 b^10 c^6 +
322903749309701343436800 a^8 b^10 c^6 +
32883202299738670709760 a^10 b^10 c^6 +
2093140483135599098880 a^12 b^10 c^6 +
76142089323605329920 a^14 b^10 c^6 +
1211899133470556160 a^16 b^10 c^6 +
1126299084549332077117440 b^12 c^6 +
1144990602311449423380480 a^2 b^12 c^6 +
509328658192274549637120 a^4 b^12 c^6 +
129485992445516216401920 a^6 b^12 c^6 +
20577407598801401241600 a^8 b^12 c^6 +
2093140483135599098880 a^10 b^12 c^6 +
133089176877671669760 a^12 b^12 c^6 +
4836180421689200640 a^14 b^12 c^6 +
76893677039370240 a^16 b^12 c^6 + 41209030285424811048960 b^14 c^6 +
41842178448700232171520 a^2 b^14 c^6 +
18590714330907672576000 a^4 b^14 c^6 +
4720829210372364042240 a^6 b^14 c^6 +
749368863900931276800 a^8 b^14 c^6 +
76142089323605329920 a^10 b^14 c^6 +
4836180421689200640 a^12 b^14 c^6 +
175552679006208000 a^14 b^14 c^6 + 2788384179732480 a^16 b^14 c^6 +
659566638832004628480 b^16 c^6 +
668921265389064683520 a^2 b^16 c^6 +
296866673570390999040 a^4 b^16 c^6 +
75300636585413836800 a^6 b^16 c^6 +
11939920073677209600 a^8 b^16 c^6 +
1211899133470556160 a^10 b^16 c^6 +
76893677039370240 a^12 b^16 c^6 + 2788384179732480 a^14 b^16 c^6 +
44245116518400 a^16 b^16 c^6 + 13763892782333028209308729344 c^8 +
14103109556141872838023839744 a^2 c^8 +
6324673191404469319070908416 a^4 c^8 +
1621383551616502173007872000 a^6 c^8 +
259876470921531193389219840 a^8 c^8 +
26666910412308406377381888 a^10 c^8 +
1710780105051900475342848 a^12 c^8 +
62734240818043847442432 a^14 c^8 + 1006726746667147591680 a^16 c^8 +
14103109556141872838023839744 b^2 c^8 +
14422290827620942671903719424 a^2 b^2 c^8 +
6455495906337291520375259136 a^4 b^2 c^8 +
1651871032509760639504220160 a^6 b^2 c^8 +
264291445364500788632616960 a^8 b^2 c^8 +
27073328183710343126581248 a^10 b^2 c^8 +
1733975602447550710284288 a^12 b^2 c^8 +
63483411143987520602112 a^14 b^2 c^8 +
1017187102704449617920 a^16 b^2 c^8 +
6324673191404469319070908416 b^4 c^8 +
6455495906337291520375259136 a^2 b^4 c^8 +
2884161073564195292272656384 a^4 b^4 c^8 +
736687926794666290603622400 a^6 b^4 c^8 +
117660716361730324841103360 a^8 b^4 c^8 +
12032471032357082048888832 a^10 b^4 c^8 +
769387376904868668506112 a^12 b^4 c^8 +
28123897552066588704768 a^14 b^4 c^8 +
449939007070399365120 a^16 b^4 c^8 +
1621383551616502173007872000 b^6 c^8 +
1651871032509760639504220160 a^2 b^6 c^8 +
736687926794666290603622400 a^4 b^6 c^8 +
187838478195020353744404480 a^6 b^6 c^8 +
29949502278670934707077120 a^8 b^6 c^8 +
3057671297003307423989760 a^10 b^6 c^8 +
195200133636469041561600 a^12 b^6 c^8 +
7124120769221169315840 a^14 b^6 c^8 +
113802566745088327680 a^16 b^6 c^8 +
259876470921531193389219840 b^8 c^8 +
264291445364500788632616960 a^2 b^8 c^8 +
117660716361730324841103360 a^4 b^8 c^8 +
29949502278670934707077120 a^6 b^8 c^8 +
4767259308474131471923200 a^8 b^8 c^8 +
485916731439371525422080 a^10 b^8 c^8 +
30971447743694520222720 a^12 b^8 c^8 +
1128605181568794961920 a^14 b^8 c^8 +
18001625313332183040 a^16 b^8 c^8 +
26666910412308406377381888 b^10 c^8 +
27073328183710343126581248 a^2 b^10 c^8 +
12032471032357082048888832 a^4 b^10 c^8 +
3057671297003307423989760 a^6 b^10 c^8 +
485916731439371525422080 a^8 b^10 c^8 +
49449484262486839781376 a^10 b^10 c^8 +
3146907814019581602816 a^12 b^10 c^8 +
114499517472837310464 a^14 b^10 c^8 +
1823600021515130880 a^16 b^10 c^8 +
1710780105051900475342848 b^12 c^8 +
1733975602447550710284288 a^2 b^12 c^8 +
769387376904868668506112 a^4 b^12 c^8 +
195200133636469041561600 a^6 b^12 c^8 +
30971447743694520222720 a^8 b^12 c^8 +
3146907814019581602816 a^10 b^12 c^8 +
199959513321408454656 a^12 b^12 c^8 +
7264597477448392704 a^14 b^12 c^8 +
115531967580825600 a^16 b^12 c^8 +
62734240818043847442432 b^14 c^8 +
63483411143987520602112 a^2 b^14 c^8 +
28123897552066588704768 a^4 b^14 c^8 +
7124120769221169315840 a^6 b^14 c^8 +
1128605181568794961920 a^8 b^14 c^8 +
114499517472837310464 a^10 b^14 c^8 +
7264597477448392704 a^12 b^14 c^8 +
263537599896539136 a^14 b^14 c^8 + 4185114339409920 a^16 b^14 c^8 +
1006726746667147591680 b^16 c^8 +
1017187102704449617920 a^2 b^16 c^8 +
449939007070399365120 a^4 b^16 c^8 +
113802566745088327680 a^6 b^16 c^8 +
18001625313332183040 a^8 b^16 c^8 +
1823600021515130880 a^10 b^16 c^8 +
115531967580825600 a^12 b^16 c^8 + 4185114339409920 a^14 b^16 c^8 +
66367674777600 a^16 b^16 c^8 + 15756690096587353153351974912 c^10 +
15989729995385305254359728128 a^2 c^10 +
7107051537369899368354676736 a^4 c^10 +
1807142753962795067114913792 a^6 c^10 +
287517911047290105220104192 a^8 c^10 +
29309044796012575693209600 a^10 c^10 +
1869372934965939939508224 a^12 c^10 +
68205782668786658574336 a^14 c^10 +
1089891946405907398656 a^16 c^10 +
15989729995385305254359728128 b^2 c^10 +
16192078017544193251671539712 a^2 b^2 c^10 +
7181976429540331818063888384 a^4 b^2 c^10 +
1822429215693235188274249728 a^6 b^2 c^10 +
289361208254518886947872768 a^8 b^2 c^10 +
29438150900192895538298880 a^10 b^2 c^10 +
1873946422946120358076416 a^12 b^2 c^10 +
68242692896110318731264 a^14 b^2 c^10 +
1088468406916298440704 a^16 b^2 c^10 +
7107051537369899368354676736 b^4 c^10 +
7181976429540331818063888384 a^2 b^4 c^10 +
3178937462854661344852033536 a^4 b^4 c^10 +
804992498756807975455500288 a^6 b^4 c^10 +
127554011929964148274658304 a^8 b^4 c^10 +
12950585392186199663345664 a^10 b^4 c^10 +
822766020254035072717824 a^12 b^4 c^10 +
29904297569975913154560 a^14 b^4 c^10 +
476071402416271441920 a^16 b^4 c^10 +
1807142753962795067114913792 b^6 c^10 +
1822429215693235188274249728 a^2 b^6 c^10 +
804992498756807975455500288 a^4 b^6 c^10 +
203426372326458213545598720 a^6 b^6 c^10 +
32167738934047077133175424 a^8 b^6 c^10 +
3259381783834845650128896 a^10 b^6 c^10 +
206658376561719306306432 a^12 b^6 c^10 +
7496446165027505167104 a^14 b^6 c^10 +
119112187839853307904 a^16 b^6 c^10 +
287517911047290105220104192 b^8 c^10 +
289361208254518886947872768 a^2 b^8 c^10 +
127554011929964148274658304 a^4 b^8 c^10 +
32167738934047077133175424 a^6 b^8 c^10 +
5076286611527597478163200 a^8 b^8 c^10 +
513307396459161216428544 a^10 b^8 c^10 +
32480254759253125136256 a^12 b^8 c^10 +
1175862257498290053888 a^14 b^8 c^10 +
18646843045199943168 a^16 b^8 c^10 +
29309044796012575693209600 b^10 c^10 +
29438150900192895538298880 a^2 b^10 c^10 +
12950585392186199663345664 a^4 b^10 c^10 +
3259381783834845650128896 a^6 b^10 c^10 +
513307396459161216428544 a^8 b^10 c^10 +
51799537371501017102592 a^10 b^10 c^10 +
3271057129163369419776 a^12 b^10 c^10 +
118182179400091301376 a^14 b^10 c^10 +
1870412179680688896 a^16 b^10 c^10 +
1869372934965939939508224 b^12 c^10 +
1873946422946120358076416 a^2 b^12 c^10 +
822766020254035072717824 a^4 b^12 c^10 +
206658376561719306306432 a^6 b^12 c^10 +
32480254759253125136256 a^8 b^12 c^10 +
3271057129163369419776 a^10 b^12 c^10 +
206143663367851808256 a^12 b^12 c^10 +
7432858707630370560 a^14 b^12 c^10 +
117400198041907200 a^16 b^12 c^10 +
68205782668786658574336 b^14 c^10 +
68242692896110318731264 a^2 b^14 c^10 +
29904297569975913154560 a^4 b^14 c^10 +
7496446165027505167104 a^6 b^14 c^10 +
1175862257498290053888 a^8 b^14 c^10 +
118182179400091301376 a^10 b^14 c^10 +
7432858707630370560 a^12 b^14 c^10 +
267461983098971136 a^14 b^14 c^10 +
4215962893476096 a^16 b^14 c^10 + 1089891946405907398656 b^16 c^10 +
1088468406916298440704 a^2 b^16 c^10 +
476071402416271441920 a^4 b^16 c^10 +
119112187839853307904 a^6 b^16 c^10 +
18646843045199943168 a^8 b^16 c^10 +
1870412179680688896 a^10 b^16 c^10 +
117400198041907200 a^12 b^16 c^10 +
4215962893476096 a^14 b^16 c^10 + 66320921028096 a^16 b^16 c^10 +
13742780160706977462591946752 c^12 +
13734790680534247390247387136 a^2 c^12 +
6013639640497792371342704640 a^4 c^12 +
1506752364669814563319775232 a^6 c^12 +
236313937135391341691437056 a^8 c^12 +
23758195326683836579577856 a^10 c^12 +
1495358873048699714764800 a^12 c^12 +
53875998037836790824960 a^14 c^12 +
850753640021892268032 a^16 c^12 +
13734790680534247390247387136 b^2 c^12 +
13694237638249619259888500736 a^2 b^2 c^12 +
5981317532153586870237462528 a^4 b^2 c^12 +
1494941162577765259899838464 a^6 b^2 c^12 +
233871812075639679059865600 a^8 b^2 c^12 +
23452919693787134374871040 a^10 b^2 c^12 +
1472373465380941577932800 a^12 b^2 c^12 +
52912361496480338829312 a^14 b^2 c^12 +
833415323489358446592 a^16 b^2 c^12 +
6013639640497792371342704640 b^4 c^12 +
5981317532153586870237462528 a^2 b^4 c^12 +
2605962317595385182145634304 a^4 b^4 c^12 +
649650422392580175465194496 a^6 b^4 c^12 +
101366414409614879861761536 a^8 b^4 c^12 +
10138058018991001950179328 a^10 b^4 c^12 +
634751999643330763061760 a^12 b^4 c^12 +
22749099031565413493760 a^14 b^4 c^12 +
357345976703529074688 a^16 b^4 c^12 +
1506752364669814563319775232 b^6 c^12 +
1494941162577765259899838464 a^2 b^6 c^12 +
649650422392580175465194496 a^4 b^6 c^12 +
161525152356629937695561088 a^6 b^6 c^12 +
25134573894694186953665088 a^8 b^6 c^12 +
2506812633892787236867584 a^10 b^6 c^12 +
156508807363778253191616 a^12 b^6 c^12 +
5593073416898433759360 a^14 b^6 c^12 +
87602730332509550592 a^16 b^6 c^12 +
236313937135391341691437056 b^8 c^12 +
233871812075639679059865600 a^2 b^8 c^12 +
101366414409614879861761536 a^4 b^8 c^12 +
25134573894694186953665088 a^6 b^8 c^12 +
3900132403002524747212128 a^8 b^8 c^12 +
387855110346978839463936 a^10 b^8 c^12 +
24143179046172411717792 a^12 b^8 c^12 +
860177785265876288448 a^14 b^8 c^12 +
13431301077326598144 a^16 b^8 c^12 +
23758195326683836579577856 b^10 c^12 +
23452919693787134374871040 a^2 b^10 c^12 +
10138058018991001950179328 a^4 b^10 c^12 +
2506812633892787236867584 a^6 b^10 c^12 +
387855110346978839463936 a^8 b^10 c^12 +
38454996493316833239168 a^10 b^10 c^12 +
2386317482377876081152 a^12 b^10 c^12 +
84749111716850193024 a^14 b^10 c^12 +
1319007260073823488 a^16 b^10 c^12 +
1495358873048699714764800 b^12 c^12 +
1472373465380941577932800 a^2 b^12 c^12 +
634751999643330763061760 a^4 b^12 c^12 +
156508807363778253191616 a^6 b^12 c^12 +
24143179046172411717792 a^8 b^12 c^12 +
2386317482377876081152 a^10 b^12 c^12 +
147604573564618136544 a^12 b^12 c^12 +
5224598076192918336 a^14 b^12 c^12 +
81033891718211712 a^16 b^12 c^12 +
53875998037836790824960 b^14 c^12 +
52912361496480338829312 a^2 b^14 c^12 +
22749099031565413493760 a^4 b^14 c^12 +
5593073416898433759360 a^6 b^14 c^12 +
860177785265876288448 a^8 b^14 c^12 +
84749111716850193024 a^10 b^14 c^12 +
5224598076192918336 a^12 b^14 c^12 +
184284235532203776 a^14 b^14 c^12 +
2847900669804288 a^16 b^14 c^12 + 850753640021892268032 b^16 c^12 +
833415323489358446592 a^2 b^16 c^12 +
357345976703529074688 a^4 b^16 c^12 +
87602730332509550592 a^6 b^16 c^12 +
13431301077326598144 a^8 b^16 c^12 +
1319007260073823488 a^10 b^16 c^12 +
81033891718211712 a^12 b^16 c^12 + 2847900669804288 a^14 b^16 c^12 +
43843813501824 a^16 b^16 c^12 + 8484085605964595841947271168 c^14 +
8357492172097111418258817024 a^2 c^14 +
3604367987313881249192306688 a^4 c^14 +
889019442515791385838240768 a^6 c^14 +
137185534904462043701732352 a^8 c^14 +
13564160503520378338427904 a^10 c^14 +
839348202010490600666112 a^12 c^14 +
29724796948765719951360 a^14 c^14 +
461342207452546990080 a^16 c^14 +
8357492172097111418258817024 b^2 c^14 +
8207865452903159147004665856 a^2 b^2 c^14 +
3528439047268155462981385728 a^4 b^2 c^14 +
867318322578665412093278976 a^6 b^2 c^14 +
133353272377956693875952384 a^8 b^2 c^14 +
13135057959056583273179904 a^10 b^2 c^14 +
809549935322361155516160 a^12 b^2 c^14 +
28550030654631813917184 a^14 b^2 c^14 +
441192672894941614080 a^16 b^2 c^14 +
3604367987313881249192306688 b^4 c^14 +
3528439047268155462981385728 a^2 b^4 c^14 +
1511588064954239822516718240 a^4 b^4 c^14 +
370188705526510989452094864 a^6 b^4 c^14 +
56693693484796908192803856 a^8 b^4 c^14 +
5560845258313469790409104 a^10 b^4 c^14 +
341210205056727472731984 a^12 b^4 c^14 +
11977026851892969939168 a^14 b^4 c^14 +
184176695116255380480 a^16 b^4 c^14 +
889019442515791385838240768 b^6 c^14 +
867318322578665412093278976 a^2 b^6 c^14 +
370188705526510989452094864 a^4 b^6 c^14 +
90298321408963532583380520 a^6 b^6 c^14 +
13769710164930643317681480 a^8 b^6 c^14 +
1344392366280131844404712 a^10 b^6 c^14 +
82084398036284935679400 a^12 b^6 c^14 +
2866143529903402417968 a^14 b^6 c^14 +
43828068120342762816 a^16 b^6 c^14 +
137185534904462043701732352 b^8 c^14 +
133353272377956693875952384 a^2 b^8 c^14 +
56693693484796908192803856 a^4 b^8 c^14 +
13769710164930643317681480 a^6 b^8 c^14 +
2089967517339107717532312 a^8 b^8 c^14 +
203018703620992356544488 a^10 b^8 c^14 +
12327705559305018120024 a^12 b^8 c^14 +
427897719198467689392 a^14 b^8 c^14 +
6501573696280797216 a^16 b^8 c^14 +
13564160503520378338427904 b^10 c^14 +
13135057959056583273179904 a^2 b^10 c^14 +
5560845258313469790409104 a^4 b^10 c^14 +
1344392366280131844404712 a^6 b^10 c^14 +
203018703620992356544488 a^8 b^10 c^14 +
19611469461279780410472 a^10 b^10 c^14 +
1183582106217037189800 a^12 b^10 c^14 +
40808010742778049264 a^14 b^10 c^14 +
615523644745847712 a^16 b^10 c^14 +
839348202010490600666112 b^12 c^14 +
809549935322361155516160 a^2 b^12 c^14 +
341210205056727472731984 a^4 b^12 c^14 +
82084398036284935679400 a^6 b^12 c^14 +
12327705559305018120024 a^8 b^12 c^14 +
1183582106217037189800 a^10 b^12 c^14 +
70946214174613099224 a^12 b^12 c^14 +
2427659157847862160 a^14 b^12 c^14 +
36310692234720672 a^16 b^12 c^14 +
29724796948765719951360 b^14 c^14 +
28550030654631813917184 a^2 b^14 c^14 +
11977026851892969939168 a^4 b^14 c^14 +
2866143529903402417968 a^6 b^14 c^14 +
427897719198467689392 a^8 b^14 c^14 +
40808010742778049264 a^10 b^14 c^14 +
2427659157847862160 a^12 b^14 c^14 +
82362003967435680 a^14 b^14 c^14 + 1220012565228576 a^16 b^14 c^14 +
461342207452546990080 b^16 c^14 +
441192672894941614080 a^2 b^16 c^14 +
184176695116255380480 a^4 b^16 c^14 +
43828068120342762816 a^6 b^16 c^14 +
6501573696280797216 a^8 b^16 c^14 +
615523644745847712 a^10 b^16 c^14 +
36310692234720672 a^12 b^16 c^14 + 1220012565228576 a^14 b^16 c^14 +
17870437083456 a^16 b^16 c^14 + 3200751090373242102395424768 c^16 +
3120375242287438392916818432 a^2 c^16 +
1330225478940252748792458240 a^4 c^16 +
323911462321972767356731008 a^6 c^16 +
49280117828387104634338944 a^8 c^16 +
4797500490695802288228480 a^10 c^16 +
291891297134935206092160 a^12 c^16 +
10149623019693724379904 a^14 c^16 +
154457830041819429888 a^16 c^16 +
3120375242287438392916818432 b^2 c^16 +
3030161292050287147244138304 a^2 b^2 c^16 +
1286195435060172020126343552 a^4 b^2 c^16 +
311694922276325132699328720 a^6 b^2 c^16 +
47170960300917588749225808 a^8 b^2 c^16 +
4565343041180991743266320 a^10 b^2 c^16 +
275974042051168432822704 a^12 b^2 c^16 +
9527886139626428882784 a^14 b^2 c^16 +
143861832060274976256 a^16 b^2 c^16 +
1330225478940252748792458240 b^4 c^16 +
1286195435060172020126343552 a^2 b^4 c^16 +
543311146926388146093030016 a^4 b^4 c^16 +
130954385822258762865421376 a^6 b^4 c^16 +
19698318888574614042787984 a^8 b^4 c^16 +
1893525464710155261670976 a^10 b^4 c^16 +
113592215262719748299680 a^12 b^4 c^16 +
3888262248501440183552 a^14 b^4 c^16 +
58147768521809800272 a^16 b^4 c^16 +
323911462321972767356731008 b^6 c^16 +
311694922276325132699328720 a^2 b^6 c^16 +
130954385822258762865421376 a^4 b^6 c^16 +
31370706809371817312198788 a^6 b^6 c^16 +
4685981621296027144362068 a^8 b^6 c^16 +
446875008803611741823188 a^10 b^6 c^16 +
26565366409412000016476 a^12 b^6 c^16 +
899923145554946102200 a^14 b^6 c^16 +
13298564543045002416 a^16 b^6 c^16 +
49280117828387104634338944 b^8 c^16 +
47170960300917588749225808 a^2 b^8 c^16 +
19698318888574614042787984 a^4 b^8 c^16 +
4685981621296027144362068 a^6 b^8 c^16 +
694340356303195521055816 a^8 b^8 c^16 +
65598010412755520282948 a^10 b^8 c^16 +
3857254377684448422964 a^12 b^8 c^16 +
129006593320829705528 a^14 b^8 c^16 +
1877879811124852308 a^16 b^8 c^16 +
4797500490695802288228480 b^10 c^16 +
4565343041180991743266320 a^2 b^10 c^16 +
1893525464710155261670976 a^4 b^10 c^16 +
446875008803611741823188 a^6 b^10 c^16 +
65598010412755520282948 a^8 b^10 c^16 +
6128988029114484676228 a^10 b^10 c^16 +
355646362213282150124 a^12 b^10 c^16 +
11706081894379726840 a^14 b^10 c^16 +
167114780727137712 a^16 b^10 c^16 +
291891297134935206092160 b^12 c^16 +
275974042051168432822704 a^2 b^12 c^16 +
113592215262719748299680 a^4 b^12 c^16 +
26565366409412000016476 a^6 b^12 c^16 +
3857254377684448422964 a^8 b^12 c^16 +
355646362213282150124 a^10 b^12 c^16 +
20303379571582356340 a^12 b^12 c^16 +
654826493762846312 a^14 b^12 c^16 +
9109645290581976 a^16 b^12 c^16 +
10149623019693724379904 b^14 c^16 +
9527886139626428882784 a^2 b^14 c^16 +
3888262248501440183552 a^4 b^14 c^16 +
899923145554946102200 a^6 b^14 c^16 +
129006593320829705528 a^8 b^14 c^16 +
11706081894379726840 a^10 b^14 c^16 +
654826493762846312 a^12 b^14 c^16 +
20566861605859504 a^14 b^14 c^16 + 276101622567072 a^16 b^14 c^16 +
154457830041819429888 b^16 c^16 +
143861832060274976256 a^2 b^16 c^16 +
58147768521809800272 a^4 b^16 c^16 +
13298564543045002416 a^6 b^16 c^16 +
1877879811124852308 a^8 b^16 c^16 +
167114780727137712 a^10 b^16 c^16 +
9109645290581976 a^12 b^16 c^16 + 276101622567072 a^14 b^16 c^16 +
3519912391812 a^16 b^16 c^16 + 542111206611168649565476884 c^18 +
524982172319000746684642980 a^2 c^18 +
222001337560468479090155757 a^4 c^18 +
53537395403205272787323130 a^6 c^18 +
8052255681877967991413346 a^8 c^18 +
773366096017811726390520 a^10 c^18 +
46313544366395388438237 a^12 c^18 +
1580944718467944344970 a^14 c^18 + 23549293329040239264 a^16 c^18 +
524982172319000746684642980 b^2 c^18 +
505918511898221668363740756 a^2 b^2 c^18 +
212741600552688131718348849 a^4 b^2 c^18 +
50972731961891258709020250 a^6 b^2 c^18 +
7609135378345884022816386 a^8 b^2 c^18 +
724450448170978615578744 a^10 b^2 c^18 +
42944058987966317470401 a^12 b^2 c^18 +
1448513766095888967930 a^14 b^2 c^18 +
21275411352230094024 a^16 b^2 c^18 +
222001337560468479090155757 b^4 c^18 +
212741600552688131718348849 a^2 b^4 c^18 +
355499814394584303038404373/4 a^4 b^4 c^18 +
42262848142252246126068549/2 a^6 b^4 c^18 +
6252034377523966127855049/2 a^8 b^4 c^18 +
294438468687311891683326 a^10 b^4 c^18 +
68923472314160553238917/4 a^12 b^4 c^18 +
1144544048340707430381/2 a^14 b^4 c^18 +
8248665160417831936 a^16 b^4 c^18 +
53537395403205272787323130 b^6 c^18 +
50972731961891258709020250 a^2 b^6 c^18 +
42262848142252246126068549/2 a^4 b^6 c^18 +
4978460794046178164373355 a^6 b^6 c^18 +
728384443784834978229167 a^8 b^6 c^18 +
67690678695978097084444 a^10 b^6 c^18 +
7792759928012633416213/2 a^12 b^6 c^18 +
126764727225743994703 a^14 b^6 c^18 +
1779973287150248186 a^16 b^6 c^18 +
8052255681877967991413346 b^8 c^18 +
7609135378345884022816386 a^2 b^8 c^18 +
6252034377523966127855049/2 a^4 b^8 c^18 +
728384443784834978229167 a^6 b^8 c^18 +
105124445925818113419857 a^8 b^8 c^18 +
9603934732450167217580 a^10 b^8 c^18 +
1081774023312500243521/2 a^12 b^8 c^18 +
17102805310330816883 a^14 b^8 c^18 +
231114800310247544 a^16 b^8 c^18 +
773366096017811726390520 b^10 c^18 +
724450448170978615578744 a^2 b^10 c^18 +
294438468687311891683326 a^4 b^10 c^18 +
67690678695978097084444 a^6 b^10 c^18 +
9603934732450167217580 a^8 b^10 c^18 +
858101874542786873044 a^10 b^10 c^18 +
46910552481517533290 a^12 b^10 c^18 +
1423167304273451128 a^14 b^10 c^18 +
18098762270078228 a^16 b^10 c^18 +
46313544366395388438237 b^12 c^18 +
42944058987966317470401 a^2 b^12 c^18 +
68923472314160553238917/4 a^4 b^12 c^18 +
7792759928012633416213/2 a^6 b^12 c^18 +
1081774023312500243521/2 a^8 b^12 c^18 +
46910552481517533290 a^10 b^12 c^18 +
9831401448174012965/4 a^12 b^12 c^18 +
139790538825606949/2 a^14 b^12 c^18 +
797031849137168 a^16 b^12 c^18 + 1580944718467944344970 b^14 c^18 +
1448513766095888967930 a^2 b^14 c^18 +
1144544048340707430381/2 a^4 b^14 c^18 +
126764727225743994703 a^6 b^14 c^18 +
17102805310330816883 a^8 b^14 c^18 +
1423167304273451128 a^10 b^14 c^18 +
139790538825606949/2 a^12 b^14 c^18 +
1773890417556559 a^14 b^14 c^18 + 15738263341034 a^16 b^14 c^18 +
23549293329040239264 b^16 c^18 +
21275411352230094024 a^2 b^16 c^18 +
8248665160417831936 a^4 b^16 c^18 +
1779973287150248186 a^6 b^16 c^18 +
231114800310247544 a^8 b^16 c^18 +
18098762270078228 a^10 b^16 c^18 + 797031849137168 a^12 b^16 c^18 +
15738263341034 a^14 b^16 c^18 + 33866423320 a^16 b^16 c^18
\end{verbatim}
\vskip 0.5cm
\begin{large} \noindent\textbf{Acknowledgements} \end{large} This work was supported by the Natural Sciences and Engineering Research Council of Canada (Nos. 559668-2021 and
4394-2018).
| -204,809.974639 |
[
-3.302734375,
2.93359375
] | 10.880077 |
[
-3.451171875,
1.0595703125,
-2.087890625,
-6.0703125,
-1.0615234375,
8.296875
] |
[
5.02734375,
9.328125,
3.873046875,
8.0234375
] | 410 | 8,010 |
[
-3.681640625,
4.15234375
] | 69.620365 |
[
-5.71875,
-4.17578125,
-4.3671875,
-2.03125,
2.193359375,
11.78125
] | 0.179005 | 5.452718 | 28.938826 | 54.019312 |
[
3.0575318336486816
] | -134,997.66814 | 9.206117 | -203,653.987876 | 0.35801 | 6.341993 |
[
-2.14453125,
-3.2734375,
-3.75,
-5.1640625,
2.294921875,
12.1484375
] |
[
-6.24609375,
-2.087890625,
-2.69921875,
-1.796875,
3.84375,
5.0703125
] | |
BkiUeL04dbghYS8bBLKb
|
\section*{Introduction}
It is known that formal
$p$-divisible groups $G$ over a $p$-adic ring $R$
are classified by Zink's nilpotent displays \cite{Zink-Disp}.
The theory includes a description of the Dieudonn\'e crystal
of $G$ in terms of the display associated to $G$.
Recall that a display over $R$ is a projective module over
the ring of Witt vectors $W(R)$ equipped with a filtration
and with certain semi-linear operators.
Assume that $R$ is a local Artin ring with perfect residue
field $k$ of characteristic $p$.
Then $W(R)$ has a unique subring ${\mathbb{W}}(R)$,
called here the Zink ring of $R$, which is stable under
the Frobenius and which sits in an exact sequence
$$
0\to\hat W(\mathcal{N}_R)\to{\mathbb{W}}(R)\to W(k)\to 0.
$$
If $p$ is odd or if $p$ annihilates $R$,
Zink obtains in \cite{Zink-DDisp}
a classification of all $p$-divisible groups over
$R$ by Dieudonn\'e displays; these are displays for ${\mathbb{W}}(R)$
in place of $W(R)$. The restriction on $p$ is equivalent
to ${\mathbb{W}}(R)$ being stable under the Verschiebung endomorphism
$v$ of $W(R)$; this is needed since
$v$ appears in the definition of displays.
However, we observe here that with a small modification the
theory can be extended to the case $p=2$ in general:
The Zink ring is
always stable under the modified Verschiebung ${\mathbbm{v}}(x)=v(u_0x)$,
where $u_0$ is the unit of $W(R)$ defined by the relation
$v(u_0)=p-[p]$. This allows to define Dieudonn\'e displays
which classify $p$-divisible groups over $R$ without restriction
on $p$.
As an application, the classification of $p$-divisible
groups and finite flat group schemes over a complete regular local
ring with perfect residue field by Breuil-Kisin modules
derived in \cite{Vasiu-Zink} and \cite{Lau-Frames}
holds without restriction on $p$.
In the case of discrete valuation rings this completes the proof
of a conjecture of Breuil \cite{Breuil}, which was proved by
Kisin \cite{Kisin-crys} if $p$ is odd, and in \cite{Kisin-2adic}
for connected groups if $p=2$.
The main object of this article is to relate the crystals
associated to a Dieudonn\'e display and to the corresponding
$p$-divisible group. It turns out that the Zink ring
and Dieudonn\'e displays can be defined for the following
class of rings $R$, which we call admissible:
The order of nilpotence of nilpotent
elements in $R$ is bounded, and $R_{\red}$
is a perfect ring of characteristic $p$.
Let us call $R$ odd if $p$ is odd or if $p$ annihilates $R$.
\bigskip
One can work in two directions.
\bigskip
If $R$ is odd, the kernel of the augmentation ${\mathbb{W}}(R)\to R$
carries natural divided powers.
In this case, the evaluation at ${\mathbb{W}}(R)\to R$ of the
Dieu\-donn\'e crystal of a $p$-divisible group
over $R$ can be extended to a functor
$$
\Phi_R:(\text{$p$-divisible groups over $R$})
\to(\text{Dieudonn\'e displays over $R$}).
$$
This requires some work because the usual construction
gives only a filtered $F$-$V$-module over ${\mathbb{W}}(R)$,
which does not in general determine a Dieudonn\'e
display. But the definition of $\Phi_R$ can be reduced to the
case where $R$ is the universal deformation ring of a
$p$-divisible group over a perfect ring.
Then the Dieudonn\'e display
is determined by the filtered $F$-$V$-module
because $p$ is not a zero divisor
in the Zink ring of $R$.
We will show that the Dieudonn\'e crystals
of a $p$-divisible group $G$ and of the associated
Dieudonn\'e display $\Phi_R(G)$ coincide on divided
power extensions of admissible rings which are compatible
with the canonical divided powers of $p$.
This is a consequence of the following extension of $\Phi_R$.
There is a notion of Dieudonn\'e displays for a
divided power extension of admissible rings $S\to R$,
called triples in the work of Zink.
If $R$ is odd and the divided powers
are compatible with the canonical
divided powers of $p$, the evaluation of the Dieudonn\'e
crystal of a $p$-divisible group over $R$
at the divided power extension
${\mathbb{W}}(S)\to R$ can be extended to a functor
$$
\Phi_{S/R}:(\text{$p$-divisible groups over $R$})
\to(\text{Dieudonn\'e displays for $S/R$}).
$$
Again this requires some work; the proof comes down to the fact
that the Zink ring of the divided power envelope of the
diagonal of the universal deformation space considered
above has no $p$-torsion.
The comparison of crystals implies that the functor
$\Phi_R$ induces an equivalence on infinitesimal deformations
over odd admissible rings.
By a theorem of Gabber, $p$-divisible groups over
perfect rings are classified by Dieudonn\'e modules,
which can also be viewed as
displays or Dieudonn\'e displays.
These two properties imply that $\Phi_R$ is an equivalence of categories if $R$ is odd.
If $R$ is not odd, in addition to ${\mathbb{W}}(R)$
we also consider the $v$-stabilised Zink ring
${\mathbb{W}}^+(R)={\mathbb{W}}(R)[v(1)]$,
which gives rise to an obvious notion of $v$-stabilised
Dieudonn\'e displays.
The preceding constructions give a functor $\Phi_R^+$
from $p$-divisible groups
to $v$-stabilised Dieudonn\'e displays, which again
preserves the Dieudonn\'e crystals on divided power
extensions of admissible rings which are compatible
with the canonical divided powers of $p$.
But it does not follow that $\Phi^+_R$ preserves
infinitesimal deformations since the canonical
divided powers of $p$ are not nilpotent.
\bigskip
In the opposite direction, the initial object in display
theory is usually a functor from displays to groups.
Assume now that $R$ is an admissible ring such that
$R_{\red}$ is a (perfect) field, without
restriction on $p$. In this case, an explicit formula
for the equivalence functor
$$
\BT_R:(\text{Dieudonn\'e displays over $R$})
\to(\text{$p$-divisible groups over $R$})
$$
is given in \cite{Lau-Dual} assuming that $R$ is odd,
but this restriction is not
necessary once the general definition of Dieudonn\'e
displays is available. The formula is stated
in terms of the category $\mathcal{J}_R$ of admissible $R$-algebras $A$
such that $A_{\red}$ is a union of finite dimensional $k$-algebras.
If $R$ is odd, the functors $\BT_R$ and
$\Phi_R$ are mutually inverse simply because the
fibered category of $p$-divisible groups over local
Artin schemes has very few automorphisms.
We will show that the Dieudonn\'e crystals of a Dieudonn\'e
display $\mathscr{P}$ and of the associated $p$-divisible group
$\BT_R(\mathscr{P})$, restricted to arbitrary divided power extensions
of rings in $\mathcal{J}_R$, are naturally isomorphic. The main point
is two write down a map between the crystals, which is done
by a variant of the duality construction in \cite{Lau-Dual}.
Here one needs that the definition of the Dieudonn\'e crystal
of $p$-divisible groups
as a crystalline $\uExt^1$ can be carried out in the crystalline
site formed by spectra of rings in the category $\mathcal{J}_R$.
This comparison of crystals implies that in the case
when $R$ is not odd, the composition $\Phi^+_R\circ\BT_R$
from Dieudonn\'e displays to $v$-stabilised Dieudonn\'e
displays coincides with the natural base change by
${\mathbb{W}}(R)\to{\mathbb{W}}^+(R)$.
\bigskip
Finally let us consider a complete regular local ring $R$
with perfect residue field $k$ of characteristic $p$.
Choose a representation $R=\mathfrak{S}/E\mathfrak{S}$ where $\mathfrak{S}$ is
a power series ring over $W(k)$ and where $E$ has
constant term $p$.
Let $\sigma$ be a Frobenius lift on $\mathfrak{S}$
that preserves the ideal generated by the variables.
The results of \cite{Lau-Frames} for odd $p$ extend
to the general case:
If $\sigma$ satisfies a nilpotence condition,
there is a ring homomorphism $\varkappa:\mathfrak{S}\to{\mathbb{W}}(R)$
which induces an equivalence between Breuil windows
relative to $\mathfrak{S}\to R$ in the sense of \cite{Vasiu-Zink,Lau-Frames} and Dieudonn\'e displays over $R$.
Assume in addition that $R$ has characteristic zero.
Let $S$ be the $p$-adic completion of
the divided power envelope of $E\mathfrak{S}\subset\mathfrak{S}$.
There is an obvious notion of windows relative to $S\to R$.
If $p$ is odd let us write ${\mathbb{W}}^+(R)={\mathbb{W}}(R)$.
Then $\varkappa$ extends for all $p$ to a homomorphism
$\varkappa_S:S\to{\mathbb{W}}^+(R)$ which induces an equivalence
between windows relative to $S\to R$ and $v$-stabilised
Dieudonn\'e displays. This can be shown by a variant of
the proof that $\varkappa$ induces an equivalence as above;
if $R$ is one-dimesional with odd $p$, the result also
follows from \cite{Zink-Windows}.
As a consequence we deduce that for a $p$-divisible group over $R$,
the value of its Dieudonn\'e crystal at $S$ coincides
with the base change of its Breuil window to $S$.
\bigskip
All rings are commutative with unit unless the contrary is
stated. For a $p$-divisible group $G$ we denote by ${\mathbb{D}}(G)$
the \emph{covariant} Dieudonn\'e crystal.
\bigskip
The author thanks Xavier Caruso, Tyler Lawson, and
Thomas Zink for helpful conversations.
\setcounter{tocdepth}{1}
\tableofcontents
\section{The Zink ring}
In this section we study the Zink ring ${\mathbb{W}}(R)$,
which is introduced in \cite{Zink-DDisp} under
the notation $\hat W(R)$, and variants of ${\mathbb{W}}(R)$
in the presence of not necessarily nilpotent divided
powers, following \cite{Zink-Windows}.
The definitions are stated in more generality, allowing
arbitrary perfect rings instead of perfect fields.
The modified Verschiebung ${\mathbbm{v}}$ for $p=2$ is new.
\subsection{Preliminaries}
We fix a prime $p$.
A commutative ring without unit
$N$ is called bounded nilpotent if there is
a number $n$ such that $x^n=0$
for every $x\in N$.
We will consider the following base rings.
\begin{Defn}
\label{Def-admissible}
A ring $R$ is called \emph{admissible} if the nilradical
$\mathcal{N}_R$ is bounded nilpotent and if $R_{\red}=R/\mathcal{N}_R$
is a perfect ring of characteristic $p$. An \emph{admissible
local ring} is an admissible ring $R$ such that $R_{\red}$
is a field.
\end{Defn}
Examples of admissible local rings are local Artin rings with
perfect residue field.
We will also consider projective limits of admissible rings:
\begin{Defn}
\label{Def-top-admissible}
An \emph{admissible topological ring} is a complete and separated
topological ring $R$ with linear topology such that the
ideal $\mathcal{N}_R$ of topologically nilpotent elements is open,
the ring $R_{\red}=R/\mathcal{N}_R$ is perfect of characteristic $p$,
and for each open ideal $N$ of $R$ contained in $\mathcal{N}_R$
the quotient $\mathcal{N}_R/N$ is bounded nilpotent.
Thus $R$ is the projective limit of the admissible rings $R/N$.
\end{Defn}
Examples of admissible topological rings
are complete local rings with perfect
residue field. Admissible topological rings in which
$\mathcal{N}_R$ is not topologically nilpotent can arise from
divided power envelopes; see Lemma \ref{Le-pd-alg}.
\begin{Not}
For a not necessarily unitary ring $A$ let $W(A)$ be
the ring of $p$-typical Witt vectors of $A$.
We denote by $f$ and $v$ the Frobenius and
Verschiebung of $W(A)$. Let $I_A=v(W(A))$, let
$w_i:W(A)\to A$ be the $i$-th Witt polynomial,
and let $\hat W(A)$ be the group of all elements of
$W(A)$ with nilpotent coefficients which are almost
all zero.
\end{Not}
Let us recall two well-known facts.
\begin{Lemma}
\label{Le-lift-W}
Let $A$ be a perfect ring of characteristic $p$
and let $B$ be a ring with a bounded nilpotent ideal
$J\subset B$.
Every ring homomorphism $A\to B/J$ lifts to a
unique ring homomorphism $W_n(A)\to B$.
\end{Lemma}
\begin{proof}
See \cite[Chap.~IV, Prop.~4.3]{Grothendieck-Montreal},
where the ideal $J$ is assumed nilpotent,
but the proof applies here as well.
\end{proof}
\begin{Lemma}[{\cite[Lemma 2.2]{Zink-Windows}}]
\label{Le-W(N)}
Let $N$ be a non-unitary ring which is bounded
nilpotent and annihilated by a power of $p$.
Then $W(N)$ is bounded
nilpotent and annihilated by a power of $p$.
\end{Lemma}
\subsection{The Zink ring}
Let $R$ be an admissible ring.
By Lemma \ref{Le-lift-W} the exact sequence
$$
0\to W(\mathcal{N}_R)\to W(R)\to W(R_{\red})\to 0
$$
has a unique ring homomorphism section $s:W(R_{\red})\to W(R)$,
which is $f$-equivariant by its uniqueness. Let
$$
{\mathbb{W}}(R)=sW(R_{\red})\oplus\hat W(\mathcal{N}_R).
$$
Since $\hat W(\mathcal{N}_R)$ is an $f$-stable ideal of $W(R)$,
the group ${\mathbb{W}}(R)$ is an $f$-stable subring of $W(R)$,
which we call the Zink ring of $R$.
\begin{Lemma}
\label{Le-WW-v}
The ring ${\mathbb{W}}(R)$ is stable under the Verschiebung
homomorphism $v:W(R)\to W(R)$ if and only if
$p\ge 3$ or $pR=0$. In this case we have an
exact sequence
$$
0\to{\mathbb{W}}(R)\xrightarrow v{\mathbb{W}}(R)\xrightarrow{w_0} R\to 0.
$$
\end{Lemma}
\begin{proof}
See \cite[Lemma 2]{Zink-DDisp}.
For some $r$ the ring $R_0={\mathbb{Z}}/p^r{\mathbb{Z}}$ is a
subring of $R$, and ${\mathbb{W}}(R_0)=W(R_0)\cap{\mathbb{W}}(R)$.
The calculation in loc.\ cit.\ shows that the element
$v(1)\in W(R_0)$ lies in ${\mathbb{W}}(R_0)$ if and only if
$p\ge 3$ or $r=1$. For $a\in W(R_{\red})$ we have
$v(s(f(a)))=v(f(s(a)))=v(1)s(a)$.
Since $\hat W(\mathcal{N}_R)$ is stable under $v$ and since
$f$ is surjective on $W(R_{\red})$, the first assertion
of the lemma follows. The sequence is an extension of
$$
0\to W(R_{\red})\xrightarrow v W(R_{\red})\to R_{\red}\to 0
$$
and
$$
0\to\hat W(\mathcal{N}_R)\xrightarrow v\hat W(\mathcal{N}_R)\to\mathcal{N}_R\to 0,
$$
which are both exact.
\end{proof}
With a slight modification the exception at the prime $2$
can be removed. The element $p-[p]$ of $W({\mathbb{Z}}_p)$ lies
in the image of $v$ because it maps to zero in ${\mathbb{Z}}_p$.
Moreover $v^{-1}(p-[p])$ lies in $1+W(p{\mathbb{Z}}_p)$, so
this element is a unit in $W({\mathbb{Z}}_p)$. We define
$$
u_0=
\begin{cases}
v^{-1}(p-[p]) & \text{if $p=2$,} \\
1 & \text{if $p\ge 3$.}
\end{cases}
$$
The image of $u_0$ in $W(R)$ is also denoted $u_0$.
For $x\in W(R)$ let
$$
{\mathbbm{v}}(x)=v(u_0x).
$$
One could also take $u_0=v^{-1}(p-[p])$ for all $p$, which
would allow to state results in a uniform way, but for odd
primes this seems overcomplicated.
\begin{Lemma}
\label{Le-WW-vv}
The ring ${\mathbb{W}}(R)$ is stable under ${\mathbbm{v}}:W(R)\to W(R)$,
and there is an exact sequence
$$
0\to{\mathbb{W}}(R)\xrightarrow {{\mathbbm{v}}}{\mathbb{W}}(R)\xrightarrow{w_0} R\to 0.
$$
\end{Lemma}
\begin{proof}
By Lemma \ref{Le-WW-v} we can assume that $p=2$.
For $a\in W(R_{\red})$ we have
${\mathbbm{v}}(s(f(a)))=v(u_0f(s(a)))=v(u_0)s(a)=(p-[p])s(a)$,
which lies in ${\mathbb{W}}(R)$. Since $\hat W(\mathcal{N}_R)$ is stable
under ${\mathbbm{v}}$ and since $f$ is surjective on $W(R_{\red})$
it follows that ${\mathbb{W}}(R)$ is stable under ${\mathbbm{v}}$.
The sequence is an extension of
$$
0\to W(R_{\red})\xrightarrow{\mathbbm{v}} W(R_{\red})\to R_{\red}\to 0
$$
and
$$
0\to\hat W(\mathcal{N}_R)\xrightarrow{\mathbbm{v}}\hat W(\mathcal{N}_R)\to\mathcal{N}_R\to 0.
$$
They are exact because in both cases ${\mathbbm{v}}=v\circ u_0$
where $u_0$ acts bijectively.
\end{proof}
\subsection{The enlarged Zink ring}
\label{Subse-enlarged-Zink}
Let us recall the logarithm of the Witt ring:
For a divided power extension of rings $(B\to R,\delta)$
with kernel $\mathfrak{b}\subset B$,
the $\delta$-divided Witt polynomials define an
isomorphism of $W(B)$-modules
\begin{equation*}
\Log:W(\mathfrak{b})\cong\mathfrak{b}^\infty
\end{equation*}
where $x\in W(B)$ acts on $\mathfrak{b}^\infty$ by
$[b_0,b_1,\ldots]\mapsto[w_0(x)b_0,w_1(x)b_1,\ldots]$.
The Frobenius and Verschiebung of $W(\mathfrak{b})$ act on $\mathfrak{b}^\infty$
by
$$
f([b_0,b_1,\ldots])=[pb_1,pb_2,\ldots],
\qquad
v([b_0,b_1,\ldots])=[0,b_0,b_1,\ldots].
$$
Moreover $\Log$ induces an injective map
$\hat W(\mathfrak{b})\to\mathfrak{b}^{(\infty)}$,
which is bijective when the divided powers $\delta$
are nilpotent; see \cite{Zink-Disp}, Eq.\ (149)
and the subsequent discussion. In general let
$$
\tilde W(\mathfrak{b})=\Log^{-1}(\mathfrak{b}^{(\infty)}).
$$
This is an $f$-stable and $v$-stable ideal of $W(B)$
containing $\hat W(\mathfrak{b})$.
Assume now that $(B\to R,\delta)$ is a divided power
extension of admissible rings (it suffices to assume that
$R$ is admissible and that $p$ is nilpotent in $B$
because then $\mathfrak{b}$ is bounded nilpotent, so $B$ is
admissible as well). Let
$$
{\mathbb{W}}(B,\delta)={\mathbb{W}}(B)+\tilde W(\mathfrak{b}).
$$
This is an $f$-stable subring of $W(B)$, which we call
the enlarged Zink ring of $B$ with respect to the
divided power ideal $(\mathfrak{b},\delta)$.
If the divided powers $\delta$ are nilpotent then
${\mathbb{W}}(B,\delta)={\mathbb{W}}(B)$. We have the following analogues of
Lemmas \ref{Le-WW-vv} and \ref{Le-WW-v}.
\begin{Lemma}
\label{Le-WW-large-vv}
The ring ${\mathbb{W}}(B,\delta)$ is stable under ${\mathbbm{v}}:W(R)\to W(R)$,
and there is an exact sequence
$$
0\to{\mathbb{W}}(B,\delta)\xrightarrow{\mathbbm{v}}{\mathbb{W}}(B,\delta)
\xrightarrow{w_0} B\to 0.
$$
\end{Lemma}
\begin{proof}
The ring ${\mathbb{W}}(B,\delta)$ is stable under ${\mathbbm{v}}$ because
so are ${\mathbb{W}}(B)$ and $\tilde W(\mathfrak{b})$; see Lemma \ref{Le-WW-vv}.
We have ${\mathbb{W}}(B,\delta)/\tilde W(\mathfrak{b})={\mathbb{W}}(R)$.
Thus the exact sequence follows from the exactness of
$0\to\tilde W(\mathfrak{b})\xrightarrow{\mathbbm{v}}\tilde{\mathbb{W}}(\mathfrak{b})\to\mathfrak{b}\to 0$
together with the exact sequence of Lemma \ref{Le-WW-vv}.
\end{proof}
\begin{Lemma}
\label{Le-WW-large-v}
The ring ${\mathbb{W}}(B,\delta)$ is stable under $v:W(R)\to W(R)$
if $p\ge 3$, or if $p\in\mathfrak{b}$ and the divided powers
$\delta$ on $\mathfrak{b}$ induce the canonical divided powers
on $pB$. In this case we have an exact sequence
$$
0\to{\mathbb{W}}(B,\delta)\xrightarrow v{\mathbb{W}}(B,\delta)
\xrightarrow{w_0}B\to 0.
$$
\end{Lemma}
\begin{proof}
If $p\ge 3$ then ${\mathbb{W}}(B,\delta)$ is stable under $v$
because ${\mathbb{W}}(B)$ and $\tilde W(\mathfrak{b})$ are stable under $v$;
see Lemma \ref{Le-WW-v}.
Assume that $p\in\mathfrak{b}$ and that $\delta$ induces the canonical
divided powers on $pB$. Let $\xi=p-v(1)\in W(B)$.
This element lies in $W(pB)\subseteq W(\mathfrak{b})$ and
satisfies $\Log(\xi)=[p,0,0,\ldots]$.
Thus $\xi\in\tilde W(\mathfrak{b})$,
which implies that $v(1)\in{\mathbb{W}}(B,\delta)$.
Using this, the proof of Lemma \ref{Le-WW-v}
shows that ${\mathbb{W}}(B,\delta)$ is stable under $v$.
The exact sequence follows as usual.
\end{proof}
\subsection{The $v$-stabilised Zink ring}
\label{Subse-v-stab-Zink}
Assume that $p=2$. For an admissible ring $R$
let $\gamma$ be the canonical divided powers
on the ideal $pR$.
We denote the associated enlarged Zink ring by
$$
{\mathbb{W}}^+(R)={\mathbb{W}}(R,\gamma)={\mathbb{W}}(R)+\tilde W(pR)\subseteq W(R).
$$
The kernel of the projection ${\mathbb{W}}^+(R)\to W(R/\mathcal{N}_R)$
will be denoted $\hat W^+(\mathcal{N}_R)$.
In view of the following lemma we call ${\mathbb{W}}^+(R)$
the $v$-stabilised Zink ring.
\begin{Lemma}
\label{Le-WW+}
Let $p=2$. We have
$$
{\mathbb{W}}^+(R)={\mathbb{W}}(R)+{\mathbb{W}}(R)v(1).
$$
The ring ${\mathbb{W}}^+(R)$ is equal to ${\mathbb{W}}(R)$ if and only if $pR=0$.
The ${\mathbb{W}}(R)$-module ${\mathbb{W}}^+(R)/{\mathbb{W}}(R)$ is
an $R_{\red}$-module generated by $v(1)$.
\end{Lemma}
\begin{proof}
By Lemma \ref{Le-WW-large-v} we have $v(1)\in{\mathbb{W}}^+(R)$.
Clearly $pR=0$ implies that ${\mathbb{W}}^+(R)={\mathbb{W}}(R)$.
In general we consider the filtration
$$
W(p\mathcal{N}_R)\subseteq W(pR)\subseteq W(R)
$$
and the graded modules for the induced filtrations
on ${\mathbb{W}}(R)$ and on ${\mathbb{W}}^+(R)$. The restriction of the
divided powers $\gamma$ to the ideal $p\mathcal{N}_R$
are nilpotent, which implies that
$$
{\mathbb{W}}^+(R)\cap W(p\mathcal{N}_R)=\tilde W(p\mathcal{N}_R)=
\hat W(p\mathcal{N}_R)={\mathbb{W}}(R)\cap W(p\mathcal{N}_R).
$$
On the other hand, we have ${\mathbb{W}}^+(R/pR)={\mathbb{W}}(R/pR)$
and thus
$$
{\mathbb{W}}^+(R)/{\mathbb{W}}^+(R)\cap W(pR)={\mathbb{W}}(R)/{\mathbb{W}}(R)\cap W(pR).
$$
Let $\mathfrak{c}=pR/p\mathcal{N}_R$.
By the preceding remarks we have an isomorphism
$$
{\mathbb{W}}^+(R)/{\mathbb{W}}(R)\cong\tilde W(\mathfrak{c})/\hat W(\mathfrak{c}).
$$
This is an $R/\mathcal{N}_R$-module. Assume that $pR\ne 0$,
which implies $\mathfrak{c}\ne 0$.
For some ideal $\mathcal{N}_R\subseteq\mathfrak{b}\subset R$,
multiplication by $2$ is an isomorphism $R/\mathfrak{b}\cong\mathfrak{c}$.
Modulo $2$ the divided Witt polynomials are
$\tilde w_i(x)\equiv\gamma_2(x_{i-1})+x_i$,
so the isomorphism $\Log:W(\mathfrak{c})\to\mathfrak{c}^\infty$
takes the form
$$
\Log(2a_0,2a_1,\ldots)=2[a_0,a_0^2+a_1,a_1^2+a_2,a_2^2+a_3,\ldots]
$$
with $a_i\in R/\mathfrak{b}$.
It follows that $\tilde W(\mathfrak{c})/\hat W(\mathfrak{c})$ can be
identified with the direct limit of the Frobenius homomorphism
$R/\mathfrak{b}\to R/\mathfrak{b}\to\ldots\,$, which is isomorphic to
$R/\sqrt\mathfrak{b}$. Under this identification, the element
$\xi=p-v(1)$ of $\tilde W(\mathfrak{c})$ maps to $1$ in $R/\sqrt\mathfrak{b}$
because we have $\Log(\xi)=[2,0,\ldots]$.
Hence ${\mathbb{W}}^+(R)/{\mathbb{W}}(R)$ is generated by $v(1)$,
with annihilator $\sqrt\mathfrak{b}$.
\end{proof}
Assume again that $p=2$.
Let $(B\to R,\delta)$ be a divided power
extension of admissible rings with kernel $\mathfrak{b}\subset B$
such that $\delta$ is compatible with the canonical divided powers
$\gamma$ on $pB$. Let $\delta^+$
be the divided powers on $\mathfrak{b}^+=\mathfrak{b}+pB$ that extend $\delta$
and $\gamma$. In this case we write
$$
{\mathbb{W}}^+(B,\delta)={\mathbb{W}}(B,\delta^+)={\mathbb{W}}(B)+\tilde W(\mathfrak{b}^+).
$$
Clearly ${\mathbb{W}}(B,\delta)\subseteq{\mathbb{W}}^+(B,\delta)\supseteq{\mathbb{W}}^+(B)$.
If the divided powers on $\mathfrak{b}^+/pB$ induced by $\delta$
are nilpotent then ${\mathbb{W}}^+(B,\delta)={\mathbb{W}}^+(B)$.
\subsection{Passing to the limit}
All of the preceding definitions and lemmas carry over to the
topological case. For an admissible topological ring $R$ let
$$
{\mathbb{W}}(R)=\varprojlim_N{\mathbb{W}}(R/N),
$$
the limit taken over all open ideals $N$ of $R$ with
$N\subseteq\mathcal{N}_R$. Then Lemmas \ref{Le-WW-v} and
\ref{Le-WW-vv} hold for admissible topological rings.
The enlarged Zink ring can be defined for topological
divided power extensions in the following sense.
\begin{Defn}
\label{Def-pd-top-admissible}
Let $B$ and $R$ be admissible topological rings.
A topological divided power extension is a surjective ring
homomorphism $B\to R$ whose kernel $\mathfrak{b}$ is equipped with
divided powers $\delta$ such that $\mathfrak{b}$ is closed in $B$,
the topology of $R$ is the quotient topology of $B/\mathfrak{b}$,
and the linear topology of $B$ is induced by open ideals
$N$ for which $N\cap\mathfrak{b}$ is stable under $\delta$.
\end{Defn}
\begin{Remark}
The existence of divided powers on $\mathfrak{b}$ implies that
$\mathfrak{b}\subseteq\mathcal{N}_B$. If $B$ is a noetherian complete
local ring then every ideal $\mathfrak{b}$ of $B$ is closed,
and for each $n$ there is an open ideal $N\subset\mathfrak{m}_B^n$
such that $\mathfrak{b}\cap N$ is stable under any divided powers
$\delta$ on $\mathfrak{b}$. Indeed, there is an $l$ with
$\mathfrak{m}_B^n\mathfrak{b}\supseteq\mathfrak{m}_B^l\cap\mathfrak{b}$; then
take $N=\mathfrak{m}_B^n\mathfrak{b}+\mathfrak{m}_B^l$, which implies that
$\mathfrak{b}\cap N=\mathfrak{m}_B^n\mathfrak{b}$.
\end{Remark}
For a topological divided power extension of admissible
topological rings $(B\to R,\delta)$ with kernel
$\mathfrak{b}\subset B$ we define
$$
{\mathbb{W}}(B,\delta)=\varprojlim_N{\mathbb{W}}(B/N,\delta/N)
$$
where $N$ runs through the open ideals of $B$ contained
in $\mathcal{N}_B$ such that $N\cap\mathfrak{b}$ is stable under $\delta$,
and $\delta/N$ denotes the divided powers induced by
$\delta$ on the ideal $\mathfrak{b}/N\cap\mathfrak{b}$ of $B/N$. Lemmas
\ref{Le-WW-large-vv} and \ref{Le-WW-large-v} hold
in the topological case.
Assume that $p=2$. Then for an admissible topological ring
we can also consider
$$
{\mathbb{W}}^+(R)=\varprojlim_N{\mathbb{W}}^+(R/N),
$$
the limit taken over all open ideals $N$ of $R$ contained
in $\mathcal{N}_R$. Assume that $(B\to R,\delta)$ is a topological
divided power extension of admissible rings such that the
divided powers $\delta/N$ considered above are
compatible with the canonical divided powers of $p$ for
all possible $N$. Then we can define
$$
{\mathbb{W}}^+(B,\delta)=\varprojlim_N{\mathbb{W}}^+(B/N,\delta/N).
$$
In section \ref{Se-bt2disp} we need the following example.
\begin{Lemma}
\label{Le-pd-alg}
Let $R$ be an $I$-adic ring such that $K=R/I$
is a perfect ring of characteristic $p$.
For a projective $R$-module $t$ of finite type we consider
the complete symmetric algebra $R[[t]]=\prod_{n\ge 0}\Sym^n(t)$.
Let $S$ be the $I$-adic completion of the divided power
envelope of the ideal $tR[[t]]\subset R[[t]]$.
\begin{enumerate}
\renewcommand{\theenumi}{\roman{enumi}}
\item
\label{Le-pd-alg-1}
The topological ring $S$ is admissible, and $S\to R$
is naturally a topological divided power extension
of admissible topological rings.
\item
\label{Le-pd-alg-2}
If $R$ has no $p$-torsion, $S$ has no $p$-torsion.
\end{enumerate}
\end{Lemma}
\begin{proof}
Let $\mathfrak{a}'\subset S'$ be
the divided power envelope of $tR[[t]]\subset R[[t]]$.
We write $R_n=R/I^n$ and $S_n=S'/I^nS'$, thus $S=\varprojlim S_n$.
The divided powers on $\mathfrak{a}'$ induce
divided powers on $\mathfrak{a}_n=\mathfrak{a}'/I^n\mathfrak{a}'$, which is the
kernel of $S_n\to R_n$. In particular, we have $x^{p^n}=0$
for $x\in\mathfrak{a}_n$. Hence $S_n$ is admissible, $S\to R$
is the projective limit of the divided power extensions
$S_n\to R_n$, and \eqref{Le-pd-alg-1} is proved.
Let $\mathfrak{a}''\subset S''$ be the divided power envelope
of $tR[t]\subset R[t]$. We claim that the $I$-adic completion of $S''$
coincides with $S$. Indeed, since $t$ is finitely generated,
for each $n$ there is an $r$ such that every element of
$t^rR[[t]]$ can be written as a linear combination
of $p^n$-th powers. Thus the homomorphism
$R[[t]]\to S_n$ factors over $R[t]/t^n$,
and the claim follows easily.
Assume now that $R$ has no $p$-torsion.
If $t$ is a free $R$-module with basis $x_1,\ldots,x_s$,
then $S''$ is a free $R$-module with basis
$x^{\underline m}/\underline m!$ for
$\underline m\in{\mathbb{N}}^s$; see \cite[Remark 3.20.5]{Berthelot-Ogus};
and $S$ is isomorphic to the $R$-module $R^{<\infty>}$ of infinite
sequences which converge to zero $I$-adically,
which has no $p$-torsion.
In general, since the divided power envelope $S''$
is compatible with flat base change in $R$, it follows
that $S''$ is an infinite direct sum of finitely
generated projective $R$-modules.
Thus $S$ is a direct summand of $R^{<\infty>}$,
and \eqref{Le-pd-alg-2} is proved.
\end{proof}
\subsection{Completeness}
For an admissible ring $R$, the Zink ring ${\mathbb{W}}(R)$ is $p$-adic.
Indeed, $W(R_{\red})$ is $p$-adic, and
$\hat W(\mathcal{N}_R)$ is annihilated by a power of $p$
because this holds for $W(\mathcal{N}_R)$ by Lemma \ref{Le-W(N)}.
The following topological variant of this fact seems
to be less obvious.
\begin{Prop}
\label{Pr-WW-p-adic}
Let $R$ be an $I$-adic ring such that the ideal $I$ is
finitely generated and $K=R/I$ is a perfect ring of
characteristic $p$. Then the ring ${\mathbb{W}}(R)$ is $p$-adic.
If $p=2$, then ${\mathbb{W}}^+(R)$ is $p$-adic as well.
\end{Prop}
\begin{proof}
The ring $W(R)$ is $p$-adically separated because this
holds for each $W(R/I^n)$. Thus ${\mathbb{W}}(R)$ is
$p$-adically separated too.
Let $S=W(K)[[t_1,\ldots t_r]]$ and let $S\to R$
be a homomorphism which maps $t_1,\ldots t_r$
to a set of generators of $I/I^2$. Then $S\to R$ is
surjective, and so is ${\mathbb{W}}(S)\to{\mathbb{W}}(R)$. Since
${\mathbb{W}}(R)$ is $p$-adically separated, in order to
show that ${\mathbb{W}}(R)$ is $p$-adically complete
we may assume that $S=R$. Consider the ideals
$J_n=p^nW(R)+W(I^n)$
of $W(R)$ and ${\mathbb{J}}_n={\mathbb{W}}(R)\cap J_n$ of ${\mathbb{W}}(R)$.
Then
$$
W(R)/J_n=W_n(K)\oplus W(I/I^n),
$$
$$
{\mathbb{W}}(R)/{\mathbb{J}}_n=W_n(K)\oplus\hat W(I/I^n).
$$
It follows that $W(R)$ and ${\mathbb{W}}(R)$ are complete and separated
for the linear topologies generated by the ideals $J_n$ and
${\mathbb{J}}_n$, respectively; moreover ${\mathbb{W}}(R)$
is closed in $W(R)$.
The ring $W(R)$ is also complete and separated
for the linear topology generated by the ideals
$J_{n,m}'=\Ker(W(R)\to W_m(R/I^n))$. The $J$-topology is
finer than the $J'$-topology because $J_{2n}\subseteq J'_{n,n}$.
We claim that for each $r$ the ideal $p^rW(R)$ of $W(R)$
is closed in the $J'$-topology, which is a variant
of \cite[Lemma 6]{Zink-Disp}.
The proof of \cite[Lemma 4]{Zink-Disp} shows that
an element $x=(x_0,\ldots,x_m)$ of $W_{m+1}(R)$
satisfies $x_i\in I^s$ for all $i$ if and only if
$w_i(x)\in I^{i+s}$ for all $i$. Then the proof of
\cite[Lemma 5]{Zink-Disp} shows that an element
$x\in W_m(R)$ is divisible by $p^r$ if and only
if for each $s$ the image $\bar x\in W_m(R/I^s)$
is divisible by $p^r$. Using this, the claim follows
from the proof of \cite[Lemma 6]{Zink-Disp}.
Thus $p^rW(R)$ is closed in the finer $J$-topology
as well. Assume that we have $p{\mathbb{W}}(R)=pW(R)\cap{\mathbb{W}}(R)$.
Then $p^r{\mathbb{W}}(R)$ is closed in the ${\mathbb{J}}$-topology, which implies
that ${\mathbb{W}}(R)$ is $p$-adic; see \cite[Lemma 7]{Zink-Disp}.
Thus for $p\ge 3$ the proof is completed by
Lemma \ref{Le-WW-p-adic} below.
For $p=2$ the same reasoning shows that ${\mathbb{W}}^+(R)$ is
$p$-adic. Now ${\mathbb{W}}^+(R)/{\mathbb{W}}(R)$ is isomorphic to $K$ as
abelian groups by the proof of Lemma \ref{Le-WW+}.
We get exact sequences
$$
0\to K\to {\mathbb{W}}(R)/p^n{\mathbb{W}}(R)\to{\mathbb{W}}^+(R)/p^n{\mathbb{W}}^+(R)\to K\to 0,
$$
where the transition maps from $n+1$ to $n$ are zero on
the left hand $K$ and the identity on the right
hand $K$. It follows that ${\mathbb{W}}(R)$ is $p$-adic as well.
\end{proof}
\begin{Lemma}
\label{Le-WW-p-adic}
For a perfect ring $K$ of characteristic $p$ we consider
the ring $R=W(K)[[t_1,\ldots,t_r]]$ with the
$(p,t_1,\ldots,t_r)$-adic topology. If $p\ge3$ then
$$
pW(R)\cap{\mathbb{W}}(R)=p{\mathbb{W}}(R).
$$
If $p=2$ then
$$
pW(R)\cap{\mathbb{W}}^+(R)=p{\mathbb{W}}^+(R).
$$
\end{Lemma}
\begin{proof}
Assume $p=2$. Let $I$ be the kernel of $R\to K$
and let $\bar I=I/pR$.
The filtration $0\subset W(pR)\subset W(I)\subset W(R)$ induces
a filtration of ${\mathbb{W}}(R)$ with successive quotients
$\tilde W(pR):=\varprojlim_n\tilde W(pR/I^npR)$ and
$\hat W(\bar I):=\varprojlim_n\hat W(\bar I/\bar I^n)$
and $W(K)$. To prove the lemma it suffices to show that
$$
pW(\bar I)\cap\hat W(\bar I)=p\hat W(\bar I)
$$
and
$$
pW(pR)\cap\tilde W(pR)=p\tilde W(pR).
$$
The first equality holds because multiplication by $p$
on $W(\bar I)$ is given by
$(a_0,a_1,\ldots)\mapsto(0,a_0^p,a_1^p,\ldots)$,
and for $a\in\bar I$ with $a^p\in\bar I^{pn}$
we have $a\in\bar I^n$. The second quality holds
because the isomorphism $\Log:W(pR)\cong (pR)^\infty$
induces an isomorphism between $\tilde W(pR)$ and the
group of all sequences in $(pR)^\infty$
that converge to zero $I$-adically.
The proof for $p\ge 3$ is similar.
\end{proof}
\section{Dieudonn\'e displays}
In this section, Dieudonn\'e displays and a number of
variants related with divided power extensions are
defined. We use the formalism of frames introduced
in \cite{Lau-Frames}.
First of all, let us recall a well-known fact.
\begin{Lemma}
\label{Le-W-f}
Let $A$ be a commutative, not necessarily unitary ring.
For $x\in W(A)$ we have $f(x)\equiv x^p$ modulo $pW(A)$.
Similarly, for $x\in\hat W(A)$ we have $f(x)\equiv x^p$ modulo
$p\hat W(A)$.
\end{Lemma}
\begin{proof}
For $x\in W(R)$ write $x=[x_0]+v(y)$ with $x_0\in R$
and $y\in W(R)$. Then $f(x)\equiv[x_0^p]\equiv x^p$ modulo
$pW(R)$ because $fv=p$ and $v(y)^p=p^{p-1}v(y^p)$.
The same is true when $W$ is replaced by $\hat W$.
\end{proof}
\subsection{Frames and windows}
\label{Subse-frames}
We recall the notion of frames and windows
from \cite{Lau-Frames} with some additions.
A \emph{pre-frame} is a quintuple
$$
\mathcal{F}=(S,I,R,\sigma,\sigma_1)
$$
where $S$ and $R=S/I$ are rings,
where $\sigma:S\to S$ is a ring endomorphism with
$\sigma(a)\equiv a^p$ modulo $pS$,
and where $\sigma_1:I\to S$
is a $\sigma$-linear map of $S$-modules
the image of which generates $S$ as an $S$-module.
Then there is a unique element $\theta\in S$ with
$\sigma(a)=\theta\sigma_1(a)$ for $a\in I$.
The pre-frame $\mathcal{F}$ is called a \emph{frame} if
$$
I+pS\subseteq\Rad(S).
$$
If in addition all projective $R$-modules of finite type
can be lifted to projective $S$ modules then $\mathcal{F}$ is
called a \emph{lifting frame}.
A homomorphism of pre-frames or frames $\alpha:\mathcal{F}\to\mathcal{F}'$
is a ring homomorphism $\alpha:S\to S'$ with
$\alpha(I)\subseteq I'$ such that $\sigma'\alpha=\alpha\sigma$
and $\sigma_1'\alpha=u\cdot\alpha\sigma_1$
for a unit $u\in S'$, which is determined by $\alpha$.
Then $\alpha(\theta)=u\theta'$.
We say that $\alpha$ is a
$u$-homomorphism of pre-frames or frames.
Let now $\mathcal{F}$ be a frame. An $\mathcal{F}$-\emph{window} is a quadruple
$$
\mathscr{P}=(P,Q,F,F_1)
$$
where $P$ is a finitely generated projective $S$-module
with a submodule $Q$ such that there exists
a decomposition of $S$-modules
$P=L\oplus T$ with $Q=L\oplus IT$, called \emph{normal
decomposition}, and where $F:P\to P$
and $F_1:Q\to P$ are $\sigma$-linear maps
of $S$-modules with
$$
F_1(ax)=\sigma_1(a)F(x)
$$
for $a\in I$ and $x\in P$;
we also assume that $F_1(Q)$ generates $P$ as an $S$-module.
Then $F(x)=\theta F_1(x)$ for $x\in Q$. If $F$ is a lifting
frame, every pair $(P,Q)$ such that $P$ is a finitely
generated projective $S$-module and $P/Q$ a projective
$R$-module admits a normal decomposition. In general,
for given $(P,Q)$ together with a normal
decomposition $P=L\oplus T$, giving $\sigma$-linear
maps $(F,F_1)$ which make an $\mathcal{F}$-window $\mathscr{P}$
is equivalent to giving a $\sigma$-linear isomorphism
$$
\Psi:L\oplus T\to P
$$
defined by $F_1$ on $L$ and by $F$ on $T$.
The triple $(L,T,\Psi)$ is called a
\emph{normal representation} of $\mathscr{P}$.
A frame homomorphism $\alpha:\mathcal{F}\to\mathcal{F}'$
induces a base change functor $\alpha_*$
from $\mathcal{F}$-windows to $\mathcal{F}'$-windows.
In terms of normal representations it is given by
$$
(L,T,\Psi)\mapsto(S'\otimes_SL,S'\otimes_ST,\Psi')
$$
with $\Psi'(s'\otimes l)=u\sigma'(s')\otimes\Psi(l)$
and $\Psi'(s'\otimes t)=\sigma'(s')\otimes\Psi(t)$.
A frame homomorphism $\alpha:\mathcal{F}\to\mathcal{F}'$ is called
\emph{crystalline} if the functor
$\alpha_*$ is an equivalence of categories.
The crystalline nature of frames is expressed
as follows; see \cite[Theorem 3.2]{Lau-Frames}.
\begin{Thm}
\label{Th-frame-crys}
Let $\alpha:\mathcal{F}\to\mathcal{F}'$ be a homomorphism of frames
which induces an isomorphism $R\cong R'$ and a
surjection $S\to S'$ with kernel $\mathfrak{a}$.
We assume that there is a finite filtration of ideals
$\mathfrak{a}=\mathfrak{a}_0\supseteq\ldots\supseteq\mathfrak{a}_n=0$
with $\sigma_1(\mathfrak{a}_i)\subseteq\mathfrak{a}_i$ and
$\sigma(\mathfrak{a}_i)\subseteq\mathfrak{a}_{i+1}$,
that $\sigma_1$ is elementwise nilpotent on each
$\mathfrak{a}_i/\mathfrak{a}_{i+1}$,
and that all finitely generated projective $S'$-modules
lift to projective $S$-modules.
Then $\alpha$ is crystalline.
\end{Thm}
Let us recall the operator $V^\sharp$ of a window.
For an $S$-module $M$ we write $M^{(1)}=S\otimes_{\sigma,S}M$.
We call a filtered $F$-$V$-module over $\mathcal{F}$ a quadruple
$$
(P,Q,F^\sharp,V^\sharp)
$$
where $P$ is a projective $S$-module of finite type,
$Q$ is a submodule of $P$ such that $P/Q$ is projective over
$R$, and $F^\sharp:P^{(1)}\to P$ and $V^\sharp:P\to P^{(1)}$
are $S$-linear maps with $F^\sharp V^\sharp=\theta$
and $V^\sharp F^\sharp=\theta$.
\begin{Lemma}
\label{Le-F-V-mod}
There is a natural functor from $\mathcal{F}$-windows
to filtered $F$-$V$-modules over $\mathcal{F}$,
which is fully faithful if $\theta$ is not
a zero divisor in $S$.
\end{Lemma}
\begin{proof}
The functor is $(P,Q,F,F_1)\mapsto(P,Q,F^\sharp,V^\sharp)$
where $F^\sharp$ is the linearisation of $F$, and $V^\sharp$
is the unique $S$-linear map such that $V^\sharp(F_1(x))=1\otimes x$
for $x\in Q$. Clearly this determines $V^\sharp$ if it exists.
In terms of a normal representation $(L,T,\Psi)$ of $\mathscr{P}$,
thus $P=L\oplus T$, one can define
$V^\sharp=(1\oplus\theta)(\Psi^\sharp)^{(-1)}$.
The relation $F^\sharp V^\sharp=\theta$ on $P$
is equivalent to $F^\sharp V^\sharp F_1=\theta F_1$
on $Q$, which is clear since $\theta F_1=F$.
The relation $V^\sharp F^\sharp=\theta$ on $P^{(1)}$
holds if and only if it holds after multiplication with
$\sigma_1(a)$ for all $a\in I$. For $x\in P$ we calculate
$\sigma_1(a)V^\sharp F^\sharp(1\otimes x)=V^\sharp F_1(ax)
=\sigma(a)\otimes x=\theta\sigma_1(a)(1\otimes x)$.
Assume that $\theta$ is not a zero divisor in $S$.
It suffices to show that the forgetful functors
from windows to triples $(P,Q,F)$ and from
filtered $F$-$V$-modules to triples $(P,Q,F^\sharp)$
are fully faithful. In the first case this holds
because $\theta F_1=F$. In the
second case, for an endomorphism $\alpha$ of $P$ with
$\alpha F^\sharp=F^\sharp\alpha^{(1)}$ we calculate
$V^\sharp\alpha\theta=V^\sharp\alpha F^\sharp V^\sharp=
V^\sharp F^\sharp\alpha^{(1)}V^\sharp=\theta\alpha^{(1)}V^\sharp$.
\end{proof}
Finally, we recall the duality formalism.
Let $\mathcal{F}$ denote the $\mathcal{F}$-window $(S,I,\sigma,\sigma_1)$.
A bilinear form between $\mathcal{F}$-windows
$$
\beta:\mathscr{P}\times\mathscr{P}'\to\mathcal{F}
$$
is an $S$-bilinear map $\beta:P\times P'\to S$
such that $\beta(Q\times Q')\subseteq I$
and $\beta(F_1x,F_1'x')=\sigma_1(\beta(x,x'))$ for $x\in Q$
and $x'\in Q'$. For each $\mathscr{P}$, the functor
$\mathscr{P}'\mapsto\Bil(\mathscr{P}\times\mathscr{P}',\mathcal{F})$ is represented
by an $\mathcal{F}$-window $\mathscr{P}^t$, called the dual of $\mathscr{P}$.
The tautological bilinear form $\mathscr{P}\times\mathscr{P}^t\to S$ is perfect.
There is a bijection between normal representations
$P=L\oplus T$ and normal representation $P^t=L^t\oplus T^t$
determined by
$\left<L,L^t\right>=0=\left<T,T^t\right>$. The associated
operators $\Psi:P\to P$ and $\Psi^t:P^t\to P^t$ are
related by $\left<\Psi x,\Psi'x'\right>=\sigma\left<x,x'\right>$.
There is also an obvious duality of filtered $F$-$V$-modules
over $\mathcal{F}$. The dual of $\mathcal{M}=(P,Q,F^\sharp,V^\sharp)$
is $\mathcal{M}^t=(P^*,Q',V^{\sharp*},F^{\sharp*})$
where $P^*=\Hom_S(P,S)$, and $Q'$ is the submodule
of all $y$ in $P^*$ with $y(Q)\subseteq I$.
It is easy to see that the functor in Lemma \ref{Le-F-V-mod}
preserves duality.
\subsection{Pre-frames associated to the Witt ring}
For an arbitrary ring $R$ let $f_1:I_R\to W(R)$
be the inverse of the Verschiebung $v$. Then
$$
\mathscr{W}_R=(W(R),I_R,R,f,f_1)
$$
is a pre-frame with $\theta=p$. If $R$ is $p$-adic,
$\mathscr{W}_R$ is a lifting frame because $W(R)$ is
$p$-adic by \cite[Proposition 3]{Zink-Disp}, and windows
over $\mathscr{W}_R$ are displays over $R$. For a divided power
extension of rings $(B\to R,\delta)$ with kernel $\mathfrak{b}\in B$
one can define a pre-frame
$$
\mathscr{W}_{B/R}=(W(B),I_{B/R},R,f,\tilde f_1)
$$
with $I_{B/R}=I_B+W(\mathfrak{b})$ such that $\tilde f_1:I_{B/R}\to W(B)$
is the unique extension of $f_1$ whose restriction
to $W(\mathfrak{b})$ is given by
$[a_0,a_1,a_2,\ldots]\mapsto[a_1,a_2,\ldots]$
in logarithmic coordinates; see section \ref{Subse-enlarged-Zink}.
The projection $\mathscr{W}_B\to\mathscr{W}_R$ factors into strict
pre-frame homomorphisms $\mathscr{W}_B\to\mathscr{W}_{B/R}\to\mathscr{W}_R$.
As a special case, assume that $R$ is a perfect ring
of characteristic $p$.
Then $f$ is an automorphism of $W(R)$, and $I_R=pW(R)$.
Let us call a Dieudonn\'e module over $R$ a triple $(P,F,V)$
where $P$ is a projective $W(R)$-module of finite type
equipped with an $f$-linear endomorphism $F$ and an
$f^{-1}$-linear endomorphism $V$ such that $FV=p$,
or equivalently $VF=p$.
\begin{Lemma}
\label{Le-disp-perf}
Displays over a perfect ring $R$ are equivalent to
Dieudonn\'e modules over $R$.
\end{Lemma}
\begin{proof}
To a display $(P,Q,F,F_1)$ we associate the Dieudonn\'e
module $(P,F,V)$ where the linearisation of $V:P\to P$ is
the operator $V^\sharp$ defined in Lemma \ref{Le-F-V-mod}.
Then $VF_1:Q\to P$ is the inclusion.
Here $F_1$ is surjective since $f$ is bijective.
Thus $Q=V(P)$, and the functor is fully faithful;
see Lemma \ref{Le-F-V-mod}.
It remains to show that for each Dieudonn\'e
module $(P,F,V)$ the $R$-module $M=P/V(P)$ is projective.
For $\mathfrak{p}\in\Spec R$ let $\ell_\mathfrak{p}(M)$ be the dimension
of the fibre of $M$ at $\mathfrak{p}$. Let $N=P/F(P)$. Then
$\ell_M+\ell_N=\ell_{P/pP}$ as functions on $\Spec R$.
Since $M$ and $N$ are finitely presented and since $P/pP$ is
projective, the functions $\ell_M$ and $\ell_N$ are
upper semicontinuous, and $\ell_{P/pP}$ is locally constant.
It follows that $\ell_M$ is locally constant, which implies
that $M$ is projective because $R$ is reduced.
\end{proof}
\subsection{Dieudonn\'e frames}
\label{Subse-Dieud-frames}
For an admissible ring $R$ let ${\mathbb{I}}_R$ be the kernel of
$w_0:{\mathbb{W}}(R)\to R$, and let ${\mathbbm{f}}_1:{\mathbb{I}}_R\to{\mathbb{W}}(R)$ be the inverse
of ${\mathbbm{v}}$, which is well-defined by Lemma \ref{Le-WW-vv}.
If $p$ is odd, then ${\mathbbm{v}}=v$ and ${\mathbbm{f}}=f$.
\begin{Lemma}
\label{Le-DDD-frame}
The quintuple
$$
\mathscr{D}_R=({\mathbb{W}}_R,{\mathbb{I}}_R,R,f,{\mathbbm{f}}_1)
$$
is a lifting frame.
\end{Lemma}
We call $\mathscr{D}_R$ the Dieudonn\'e frame associated to $R$.
\begin{proof}
In order that $\mathscr{D}_R$ is a pre-frame we need that
$f(a)\equiv a^p$ modulo $p{\mathbb{W}}(R)$ for $a\in{\mathbb{W}}(R)$,
which follows from Lemma \ref{Le-W-f}
applied to $W(R_{\red})$ and to $\hat W(\mathcal{N}_R)$.
Since $\hat W(\mathcal{N}_R)$ is a nilideal by Lemma \ref{Le-W(N)}
and since the quotient ${\mathbb{W}}(R)/\hat W(\mathcal{N}_R)=W(R_{\red})$
is $p$-adic,
the kernel of ${\mathbb{W}}(R)\to R_{\red}$ lies in the radical
of ${\mathbb{W}}(R)$, and projective $R_{\red}$-modules of finite
type lift to projective ${\mathbb{W}}(R)$-modules.
It follows that $\mathscr{D}_R$ is a lifting frame.
\end{proof}
The inclusion ${\mathbb{W}}(R)\to W(R)$ is a $u_0$-homomorphism
of frames $\mathscr{D}_R\to\mathscr{W}_R$.
Thus for $\mathscr{D}_R$ we have $\theta=1$ if $p\ge 3$ and
$\theta=pu_0=p-[p^p]$ if $p=2$.
\begin{Defn}
A Dieudonn\'e display over $R$ is a window over $\mathscr{D}_R$.
\end{Defn}
Thus a Dieudonn\'e display is a quadruple $\mathscr{P}=(P,Q,F,F_1)$
where $P$ is a projective ${\mathbb{W}}(R)$-module of finite type with
a filtration ${\mathbb{I}}_RP\subseteq Q\subseteq P$ such that
$P/Q$ is a projective $R$-module, $F:P\to P$ and
$F_1:Q\to P$ are $f$-linear maps with
$F_1(ax)={\mathbbm{f}}_1(a)F(x)$ for $a\in{\mathbb{I}}_R$ and $x\in P$,
and $F_1(Q)$ generates $P$. We write
$$
\Lie(\mathscr{P})=P/Q.
$$
The \emph{height} of $\mathscr{P}$ is the rank of the
${\mathbb{W}}(R)$-module $P$, and the \emph{dimension} of $\mathscr{P}$
is the rank of the $R$-module $\Lie(\mathscr{P})$, both viewed
as locally constant functions on $\Spec R$.
As in the case of general frames,
we also denote by $\mathscr{D}_R$ the Dieudonn\'e display
$({\mathbb{W}}(R),{\mathbb{I}}_R,f,{\mathbbm{f}}_1)$ over $R$.
\subsection{Extended Dieudonn\'e frames}
Let $(B\to R,\delta)$ be a divided power extension of
admissible rings with kernel $\mathfrak{b}\subset B$.
Let ${\mathbb{W}}(B/R)={\mathbb{W}}(B,\delta)$ and let
${\mathbb{I}}_{B/R}$ be the kernel of the projection
${\mathbb{W}}({B/R})\to R$, thus
$$
{\mathbb{I}}_{B/R}={\mathbb{I}}_B+\tilde W(\mathfrak{b}).
$$
\begin{Lemma}
\label{Le-tilde-ff1}
There is a unique extension of\/ ${\mathbbm{f}}_1:{\mathbb{I}}_B\to{\mathbb{W}}(B)$ to an
$f$-linear map $\tilde{\mathbbm{f}}_1:{\mathbb{I}}_{B/R}\to{\mathbb{W}}(B/R)$
of\/ ${\mathbb{W}}(B/R)$-modules such that the restriction of $\tilde{\mathbbm{f}}_1$
to $\tilde W(\mathfrak{b})$ is given by
\begin{equation}
\label{Eq-tilde-ff1}
\tilde{\mathbbm{f}}_1([a_0,a_1,a_2,\ldots])=
[w_0(u_0^{-1})a_1,w_1(u_0^{-1})a_2,\ldots]
\end{equation}
in logarithmic coordinates. The quintuple
$$
\mathscr{D}_{B/R}=\mathscr{D}_{B/R,\delta}=({\mathbb{W}}(B/R),{\mathbb{I}}_{B/R},R,f,\tilde{\mathbbm{f}}_1)
$$
is a lifting frame.
\end{Lemma}
\begin{proof}
Clearly $\tilde{\mathbbm{f}}_1$ is determined uniquely by
\eqref{Eq-tilde-ff1}.
Let ${\mathbb{I}}_B'$ be the kernel of ${\mathbb{W}}(B/R)\to B$.
By Lemma \ref{Le-WW-large-vv}, the inverse of ${\mathbbm{v}}$ is
an $f$-linear map ${\mathbbm{f}}_1':{\mathbb{I}}_B'\to{\mathbb{W}}(B/R)$
which extends ${\mathbbm{f}}_1$. In logarithmic coordinates,
the restriction of ${\mathbbm{v}}$ to $W(\mathfrak{b})$ is given by
$[a_0,a_1,\ldots]\mapsto[0,w_0(u_0)a_0,w_1(u_0)a_1,\ldots]$.
Thus ${\mathbbm{f}}_1'$ extends to the desired $\tilde{\mathbbm{f}}_1$.
As in the proof of Lemma \ref{Le-DDD-frame}, the
kernel of ${\mathbb{W}}(B/R)\to R_{\red}$ lies in the radical
of ${\mathbb{W}}(B/R)$, and finitely generated projective modules
lift over this homomorphism. The lemma follows.
\end{proof}
We call $\mathscr{D}_{B/R}$ the (extended) Dieudonn\'e frame
associated to $(B/R,\delta)$, and $\mathscr{D}_{B/R}$-windows
are called Dieudonn\'e displays for $B/R$.
There are natural strict frame homomorphisms
$$
\mathscr{D}_B\to\mathscr{D}_{B/R}\to\mathscr{D}_R.
$$
If the divided powers $\delta$ are nilpotent, then
${\mathbb{W}}(B)={\mathbb{W}}(B/R)$. In this case, lifts of Dieudonn\'e
displays from $B/R$ to $B$ correspond bijectively to
lifts of the Hodge filtration. In any case, we have:
\begin{Prop}
\label{Pr-DDD-crys}
The homomorphism $\mathscr{D}_{B/R}\to\mathscr{D}_R$ is crystalline.
\end{Prop}
\begin{proof}
This follows from Theorem \ref{Th-frame-crys}.
Indeed, let $\mathfrak{a}$ denote the kernel of the projection
${\mathbb{W}}(B/R)\to{\mathbb{W}}(R)$, thus
$\mathfrak{a}=\tilde W(\mathfrak{b})\cong\mathfrak{b}^{(\infty)}$.
The endomorphism $\tilde{\mathbbm{f}}_1$ of $\mathfrak{a}$ is elementwise
nilpotent by \eqref{Eq-tilde-ff1}. The required filtration
of $\mathfrak{a}$ can be taken to be $\mathfrak{a}_i=p^i\mathfrak{a}$; this is a
finite filtration by Lemma \ref{Le-W(N)}.
We have $\tilde{\mathbbm{f}}_1(\mathfrak{a}_i)=\mathfrak{a}_i$ by \eqref{Eq-tilde-ff1},
and $f(\mathfrak{a}_i)=\mathfrak{a}_{i+1}$ because the endomorphism $f$
of $\mathfrak{a}$
is given by $[a_0,a_1,\ldots]\mapsto [pa_1,pa_2,\ldots]$
in logarithmic coordinates.
\end{proof}
\subsection{$v$-stabilised Dieudonn\'e frames}
Assume that $p=2$.
The preceding constructions can be repeated with
${\mathbb{W}}^+$ and $v$ in place of ${\mathbb{W}}$ and $v$.
More precisely, for an admissible ring $R$ let
${\mathbb{I}}_R^+$ be the kernel of ${\mathbb{W}}^+(R)\to R$ and
let $f_1:{\mathbb{I}}_R^+\to{\mathbb{W}}^+(R)$ be the inverse of $v$,
which is well-defined by Lemma \ref{Le-WW-large-v}.
The $v$-stabilised Dieudonn\'e frame associated to
$R$ is defined as
$$
\mathscr{D}^+_R=({\mathbb{W}}^+(R),{\mathbb{I}}^+_R,R,f,f_1).
$$
This is a lifting frame by the proof of Lemma \ref{Le-DDD-frame}.
The inclusion ${\mathbb{W}}(R)\to{\mathbb{W}}^+(R)$ is a $u_0$-homomorphism of
frames $\mathscr{D}_R\to\mathscr{D}^+_R$, which is invertible if and only
if $pR=0$. Windows over $\mathscr{D}^+_R$ are called $v$-stabilised
Dieudonn\'e displays over $R$.
Assume again that $p=2$, and let $(B\to R,\delta)$ be
a divided power extension
of admissible rings with kernel $\mathfrak{b}\subset B$ which is
compatible with the canonical divided powers of $p$.
Let ${\mathbb{I}}^+_{B/R}$ be the kernel of the projection
${\mathbb{W}}^+_{B/R}\to{\mathbb{W}}^+(R)$, thus
$$
{\mathbb{I}}^+_{B/R}={\mathbb{I}}^+_B+\tilde W(\mathfrak{b}).
$$
There is a unique extension of $f_1:{\mathbb{I}}^+_B\to{\mathbb{W}}^+(B)$ to an
$f$-linear map $\tilde f_1:{\mathbb{I}}^+_{B/R}\to{\mathbb{W}}^+(B/R)$
of ${\mathbb{W}}^+(B/R)$-modules such that its restriction
to $\tilde W(\mathfrak{b})$ is given by
$[a_0,a_1,a_2,\ldots]\mapsto[a_1,a_2,\ldots]$
in logarithmic coordinates, and the quintuple
$$
\mathscr{D}^+_{B/R}=({\mathbb{W}}^+(B/R),{\mathbb{I}}^+_{B/R},R,f,\tilde f_1)
$$
is a lifting frame.
This follows from the proof of Lemma \ref{Le-tilde-ff1}.
We have a $u_0$-homomorphism of frames $\mathscr{D}_{B/R}\to\mathscr{D}^+_{B/R}$,
which is invertible if and only if $pR=0$, and strict frame
homomorphisms
$$
\mathscr{D}^+_B\to\mathscr{D}^+_{B/R}\to\mathscr{D}^+_R.
$$
If the divided powers induced by $\delta$ on $\mathfrak{b}+pB/pB$
are nilpotent, ${\mathbb{W}}^+(B)$ is equal to ${\mathbb{W}}^+(B/R)$.
In this case,
lifts of windows from $\mathscr{D}^+_{B/R}$ to $\mathscr{D}^+_B$ are
bijective to lifts of the Hodge filtration.
In general, we have:
\begin{Cor}
\label{Co-DDD+-crys}
The homomorphism $\mathscr{D}^+_{B/R}\to\mathscr{D}^+_R$ is crystalline.
\end{Cor}
\begin{proof}
This follows from the proof of Proposition \ref{Pr-DDD-crys}.
\end{proof}
\subsection{The crystals associated to Dieudonn\'e displays}
Let $R$ be an admissible ring.
We denote by $\Cris_{\adm}(R)$ the category of divided
power extensions $(\Spec A\to\Spec B,\delta)$ where
$A$ is an $R$-algebra which is an admissible ring,
and where $p$ is nilpotent in $B$. Then the kernel of
$B\to A$ is bounded nilpotent, so $B$ is an admissible
ring as well.
Let $\mathscr{P}$ be a Dieudonn\'e display over $R$.
For $(\Spec B\to\Spec A,\delta)$ in $\Cris_{\adm}(R)$ we
denote the base change of $\mathscr{P}$ to $A$ by $\mathscr{P}_A$ and
the unique Dieudonn\'e display for $B/A$ which
lifts $\mathscr{P}_A$ by
$$
\mathscr{P}_{B/A}=(P_{B/A},Q_{B/A},F,F_1);
$$
see Proposition \ref{Pr-DDD-crys}. A homomorphism
of divided power extensions of admissible rings
$\alpha:(A\to B,\delta)\to(A'\to B',\delta')$ induces a
frame homomorphism $\mathscr{D}_\alpha:\mathscr{D}_{B/A}\to\mathscr{D}_{B'/A'}$,
and we have a natural isomorphism
$$
(\mathscr{D}_{\alpha})_*(\mathscr{P}_{B/A})\cong\mathscr{P}_{B'/A'}.
$$
In more sophisticated terms this can be expressed as follows:
The frames $\mathscr{D}_{B/A}$ form a presheaf of frames
$\mathscr{D}_{**}$ on $\Cris_{\adm}(R)$, and
Proposition \ref{Pr-DDD-crys} implies that
the category of
Dieudonn\'e displays over $R$ is equivalent to
the category of crystals in $\mathscr{D}_{**}$-windows
on $\Cris_{\adm}(R)$. Then $\mathscr{P}_{B/A}$ is the value in
$(\Spec A\to\Spec B,\delta)$ of the crystal associated to $\mathscr{P}$.
For a Dieudonn\'e display $\mathscr{P}=(P,Q,F,F_1)$ over $R$,
the Witt crystal ${\mathbb{K}}(\mathscr{P})$ on $\Cris_{\adm}(R)$ is defined by
$$
{\mathbb{K}}(\mathscr{P})_{B/A}=P_{B/A}.
$$
This is a projective ${\mathbb{W}}({B/A})$-module of finite type.
The Dieudonn\'e crystal ${\mathbb{D}}(\mathscr{P})$ on $\Cris_{\adm}(R)$
is defined by
$$
{\mathbb{D}}(\mathscr{P})_{B/A}=P_{B/A}\otimes_{{\mathbb{W}}(B/A)}B.
$$
This is a projective $B$-module of finite type.
The Hodge filtration of $\mathscr{P}$ is the submodule
$$
Q/{\mathbb{I}}_RP\subseteq P/{\mathbb{I}}_RP={\mathbb{D}}(\mathscr{P})_{R/R}.
$$
\begin{Cor}
\label{Co-deform-DDD}
Let $(B\to R,\delta)$ be a nilpotent divided power
extension of admissible rings. The category of Dieudonn\'e
displays over $B$ is equivalent to the category of
Diedonn\'e displays $\mathscr{P}$ over $R$ together with a lift
of the Hodge filtration of $\mathscr{P}$ to a direct summand of\/
${\mathbb{D}}(\mathscr{P})_{B/R}$.
\end{Cor}
\begin{proof}
If the divided powers are nilpotent, then ${\mathbb{W}}(B/R)={\mathbb{W}}(B)$,
and lifts of windows under the frame homomorphism
$\mathscr{D}_{B}\to\mathscr{D}_{B/R}$ are in bijection to lifts of the
Hodge filtration.
\end{proof}
Assume that $p=2$. Then the preceding definitions
have a $v$-stabilised variant. Let $\Cris_{\adm}(R/{\mathbb{Z}}_p)$
be the subcategory of $\Cris_{\adm}(R)$ where the divided
powers are compatible with the canonical divided powers
of $p$. Let $\mathscr{P}^+$ be a $v$-stabilised Dieudonn\'e display
over $R$, i.e.\ a window over $\mathscr{D}^+_R$. For
$(\Spec A\to\Spec B,\delta)$ in $\Cris_{\adm}(R/{\mathbb{Z}}_p)$
we denote by $\mathscr{P}^+_A$ the base change of
$\mathscr{P}^+$ to $\mathscr{D}^+_A$ and by
$$
\mathscr{P}^+_{B/A}=(P^+_{B/A},Q^+_{B/A},F,F_1)
$$
the unique lift of $\mathscr{P}^+_A$ to a $\mathscr{D}^+_{B/A}$-window,
which exists by Corollary \ref{Co-DDD+-crys}.
The $v$-stabilised Witt crystal ${\mathbb{K}}^+(\mathscr{P})$
and the $v$-stabilised Dieudonn\'e crystal ${\mathbb{D}}^+(\mathscr{P})$
on $\Cris_{\adm}(R/{\mathbb{Z}}_p)$ are defined by
${\mathbb{K}}^+(\mathscr{P})_{B/A}=P^+_{B/A}$ and
$$
{\mathbb{D}}^+(\mathscr{P})_{B/A}=P^+_{B/A}\otimes_{{\mathbb{W}}^+(B/A)}B.
$$
\begin{Cor}
\label{Co-deform-DDD+}
Assume that $(B\to R,\delta)$ is a divided power extension
of admissible rings with $p=2$ which is compatible with the
canonical divided powers of $p$ such that the divided powers
induced by $\delta$ on the kernel of $B/pB\to R/pR$ are
nilpotent. Then the category of $v$-stabilised Dieudonn\'e
displays over $B$ is equivalent to the category of
$v$-stabilised Dieudonn\'e displays $\mathscr{P}^+$ over $R$
together with a lift of the Hodge filtration of $\mathscr{P}^+$
to a direct summand of\/ ${\mathbb{D}}^+(\mathscr{P}^+)_{B/R}$.
\end{Cor}
\begin{proof}
This is analogous to Corollary \ref{Co-deform-DDD},
using that ${\mathbb{W}}^+(B/R)={\mathbb{W}}^+(B)$ under the given
assumptions on $\delta$; see the end of section
\ref{Subse-v-stab-Zink}.
\end{proof}
\begin{Remark}
\label{Re-crys-DDD+}
Assume that $\mathscr{P}^+$ is the base change of a Dieudonn\'e
display $\mathscr{P}$ over $R$ by the $u_0$-homomorphism
of frames $\mathscr{D}_R\to\mathscr{D}^+_R$. Then for each
$(\Spec B\to\Spec A,\delta)$ in $\Cris_{\adm}(R/{\mathbb{Z}}_p)$,
$\mathscr{P}^+_{B/A}$ is the base change of $\mathscr{P}_{B/A}$ by
the $u_0$-homomorphism $\mathscr{D}_{B/A}\to\mathscr{D}^+_{B/A}$.
It follows that ${\mathbb{D}}^+(\mathscr{P}^+)$ coincides
with the restriction of ${\mathbb{D}}(\mathscr{P})$ to $\Cris_{\adm}(R/{\mathbb{Z}}_p)$.
However, this does not imply that infinitesimal deformations
of $\mathscr{P}$ and of $\mathscr{P}^+$ coincide: Let $B$ be an admissible ring
with $p^2B=0$ and $pB\ne 0$ and let $R=B/pB$. The ideal
$pB$ carries the canonical divided powers
$\gamma$ and the trivial divided powers $\delta$.
Corollary \ref{Co-deform-DDD} applies to $(B\to R,\delta)$
but not to $(B\to R,\gamma)$, while Corollary \ref{Co-deform-DDD+}
applies to $(B\to R,\gamma)$ but not to $(B\to R,\delta)$.
\end{Remark}
\subsection{Divided powers}
\label{Subse-pd}
In section \ref{Se-bt2disp} we will use that
the augmentation ideals of the Zink ring and its variants
carry natural divided powers, with some exception when $p=2$.
Let us first recall the canonical divided powers on the
Witt ring. If $R$ is a ${\mathbb{Z}}_{(p)}$-algebra, then
$W(R)$ is a ${\mathbb{Z}}_{(p)}$-algebra as well, and the
ideal $I_R$ carries divided powers $\gamma$ which
are determined by $(p-1)!\gamma_p(v(x))=p^{p-2}v(x^p)$.
Assume that $(B\to R,\delta)$ is a divided power
extension of ${\mathbb{Z}}_{(p)}$-algebras with kernel $\mathfrak{b}\subset B$.
Let $i:\mathfrak{b}\to W(\mathfrak{b})$ be defined by $\Log(i(b))=[b,0,0,\ldots]$.
Then $i(\mathfrak{b})$ is the kernel of the operator $\tilde f_1$ of
$\mathscr{W}_{B/R}$, and we have $I_{B/R}=I_B\oplus i(\mathfrak{b})$.
The divided powers $\gamma$ on $I_B$ extend to divided
powers $\gamma'=\gamma\oplus\delta$ on $I_{B/R}$ such
that $\gamma'_n(i(b))=i(\delta_n(b))$ for $b\in\mathfrak{b}$.
If $p\in\mathfrak{b}$ and if $\delta$ extends the
canonical divided powers of $p$, then $\gamma\oplus\delta$
extends the canonical divided powers
of $p$, and $f$ preserves $\gamma\oplus\delta$.
This is clear when $R$ has no $p$-torsion;
the general case follows because $(B\to R)$ can be written
as the quotient of a divided power extension $(B'\to R')$,
where $B'$ is the divided power algebra of a free module over a polynomial ring $R''$ over ${\mathbb{Z}}_{(p)}$, and $R'=R''/pR''$.
These facts extend to the Zink ring as follows.
\begin{Lemma}
\label{Le-pd-II}
Let\/ ${\mathbb{I}}\subset{\mathbb{W}}$ be one of the following.
\begin{enumerate}
\renewcommand{\theenumi}{\roman{enumi}}
\item
${\mathbb{I}}={\mathbb{I}}_R$ and\/ ${\mathbb{W}}={\mathbb{W}}(R)$ for an admissible ring $R$
with $p\ge 3$,
\item
${\mathbb{I}}={\mathbb{I}}^+_R$ and\/ ${\mathbb{W}}={\mathbb{W}}^+(R)$ for an admissible ring
$R$ with $p=2$.
\end{enumerate}
Then the divided powers $\gamma$ on $I_R$ induce
divided powers on ${\mathbb{I}}$.
\end{Lemma}
\begin{proof}
Since ${\mathbb{W}}$ is a ${\mathbb{Z}}_{(p)}$-algebra it suffices to show
that ${\mathbb{I}}$ is stable under $\gamma_p:I_R\to I_R$, which
is true because ${\mathbb{I}}=v({\mathbb{W}})$ by Lemmas \ref{Le-WW-v} and
\ref{Le-WW-large-v}.
\end{proof}
\begin{Lemma}
\label{Le-pd-II**}
Let $(B\to R,\delta)$ be a divided power extension
of admissible rings with kernel $\mathfrak{b}\subset B$.
Assume that $p\ge 3$; or that $p=2$ and $p\in\mathfrak{b}$ and $\delta$
extends the canonical divided powers of $p$.
Then the divided powers $\gamma\oplus\delta$ on $I_{B/R}$
induce divided powers on ${\mathbb{I}}_{B/R}$.
If $p\in\mathfrak{b}$ and if $\delta$ extends the canonical
divided powers of $p$, the divided powers on
${\mathbb{I}}_{B/R}$ induced by $\gamma\oplus\delta$ extend the canonical
divided powers of $p$ and are preserved by $f$.
\end{Lemma}
\begin{proof}
Let ${\mathbb{I}}'_B$ be the kernel of ${\mathbb{W}}(B/R)\to B$.
Then ${\mathbb{I}}_{B/R}={\mathbb{I}}'_B\oplus i(\mathfrak{b})$, and we have
${\mathbb{I}}'_B=v({\mathbb{W}}(B/R))$ by Lemma \ref{Le-WW-large-v}.
Thus ${\mathbb{I}}'_{B}$ is stable under $\gamma$, and
${\mathbb{I}}_{B/R}$ is stable under $\gamma\oplus\delta$.
The second assertion follows from the corresponding
fact for the Witt ring.
\end{proof}
\subsection{Passing to the limit}
All of the preceding definitions and results can be
extended to the case of admissible topological rings $R$.
For example, let
$$
\mathscr{D}_R=\varprojlim_N\mathscr{D}_{R/N}
$$
where $N$ runs through the open ideals of $R$
contained in $\mathcal{N}_R$, and let us call $\mathscr{D}_R$-windows
Dieudonn\'e displays over $R$ as before.
Then Dieudonn\'e displays over $R$ are equivalent to
compatible systems of Dieudonn\'e displays over ${R/N}$
for each $N$; see \cite[Lemma 2.1]{Lau-Frames}.
\section{From $p$-divisible groups to Dieudonn\'e displays}
\label{Se-bt2disp}
In this section we define a functor from $p$-divisible
groups over admissible rings $R$ with $p\ge 3$ or $pR=0$
to Dieudonn\'e displays over $R$ and show that it is an equivalence
of categories. For $p=2$ there is a $v$-stabilised variant
of this functor, but it is not an equivalence of categories.
\subsection{Finiteness over admissible rings}
\label{Subse-finiteness}
As a preparation we verify that the categories of $p$-divisible groups or Dieudonn\'e displays over an admissible ring $R$ are the direct limit of the corresponding categories over the finitely generated $W(R_{\red})$-subalgebras of $R$, with fully faithful transition maps. Let us begin with the easier case:
\begin{Prop}
\label{Pr-finiteness-disp}
Every Dieudonn\'e display over an admissible ring $R$
is defined over a finitely generated $W(R_{\red})$-subalgebra
of $R$. For an injective homomorphism of admissible rings
$R\to S$ such that $R_{\red}\to S_{\red}$ is bijective,
the base change of Dieudonn\'e displays from $R$ to $S$
is fully faithful.
\end{Prop}
\begin{proof}
A Dieudonn\'e display over $R$ is given by projective ${\mathbb{W}}(R)$-modules of finite type $L,T$ together with an $f$-linear automorphism $\Psi$ of $L\oplus T$. These objects can be described by three matrices: the idempotents corresponding to representations of $L$ and $T$ as direct summands of free modules of finite type, and the associated matrix representation of $\Psi$. The coefficients of these matrices are finitely many elements of ${\mathbb{W}}(R)=W(R_{\red})\oplus\hat W(\mathcal{N}_R)$, which involve only finitely many non-zero coefficients in $\hat W(\mathcal{N}_R)$. This proves the first assertion. Similarly, every homomorphism of Dieudonn\'e displays over $R$ is defined over a finitely generated $W(R_{\red})$-subalgebra of $R$. Thus for the second assertion we may assume that $\mathcal{N}_S^r=0$. Let $\bar S=S/\mathcal{N}_S^{r-1}$ and $\bar R=R/R\cap\mathcal{N}_S^{r-1}$. Let $R'\subseteq S$ be the inverse image of $\bar R\subseteq\bar S$. By induction on $r$, the base change of Dieudonn\'e displays from $\bar R$ to $\bar S$ is fully faithful. It follows that the base change from $R'$ to $S$ is fully faithful as well. By Corollary \ref{Co-deform-DDD}, Dieudonn\'e displays over $R$ or over $R'$ are equivalent to Dieudonn\'e displays over $\bar R$ together with a lift of the Hodge filtration to $R$ or to $R'$, respectively. Since $R\to R'$ is injective, it follows that the base change of Dieudonn\'e displays from $R$ to $R'$ is fully faithful.
\end{proof}
\begin{Prop}
\label{Pr-finiteness-bt}
${}$
\renewcommand{\theenumi}{\roman{enumi}}
\begin{enumerate}
\item
\label{It-finiteness-bt-1}
Every $p$-divisible group over an admissible
ring $R$ is defined over a finitely generated
$W(R_{\red})$-subalgebra of $R$.
\item
\label{It-finiteness-bt-2}
For an injective homomorphism of admissible rings $R\to S$
such that $R_{\red}\to S_{\red}$ is bijective, the base change
of $p$-divisible groups from $R$ to $S$ is fully faithful.
\end{enumerate}
\end{Prop}
\begin{Lemma}
\label{Le-reduce-pd-ideal}
Let $R$ be a ring and let $\mathfrak{a}\subset R$ be an ideal which is equipped with nilpotent divided powers $\delta$. Assume that $R$ is the filtered union of subrings $R_i\subseteq R$ such that $\mathfrak{a}\cap R_i$ is stable under $\delta$. Let $G$ be a $p$-divisible group over $R$. If $G\otimes_RR/\mathfrak{a}$ is defined over $R_i/\mathfrak{a}\cap R_i$ for some $i$, then $G$ is defined over $R_j$ for some $j$.
\end{Lemma}
\begin{proof}
We write $\bar R=R/\mathfrak{a}$ and $\bar R_i=R_i/\mathfrak{a}\cap R_i$.
The ring $S_i=R\times_{\bar R}\bar R_i$ is
the filtered union of the rings
$S_{ij}=R_j\times_{\bar R_j}\bar R_i$ for $j\ge i$.
If $\bar G=G\otimes_R\bar R$ comes from
a $p$-divisible group $\bar G_i$ over $\bar R_i$,
then $G$ comes from a $p$-divisible group
$G'_i$ over $S_i$. By \cite{Messing-Crys} the group
$G'_i$ is determined by $\bar G_i$ and by its Hodge
filtration in ${\mathbb{D}}(\bar G_i)_{S_i/\bar R_i}=
{\mathbb{D}}(\bar G_i)_{R_i/\bar R_i}\otimes_{R_i}S_i$.
For some $j\ge i$
the Hodge filtration is defined over $S_{ij}$,
which implies that $G'_i$ is defined over $S_{ij}$,
and thus $G$ is defined over $R_j$.
\end{proof}
\begin{proof}[Proof of Proposition \ref{Pr-finiteness-bt}
\eqref{It-finiteness-bt-1}]
Assume that $p^nR=0$ and $x^{p^r}=0$ for
$x\in\mathcal{N}_R$. Let $\Lambda=W_n(R_{\red})$.
For a set $\mathcal{I}$ we form the $\Lambda$-algebra
$$
S=\Lambda[\{x_i\}_{i\in\mathcal{I}}]/(\{x_i^{p^r}\}_{i\in\mathcal{I}}).
$$
This is an admissible ring because $x^{p^r+n}=0$
for $x\in\mathcal{N}_S$. If $\mathcal{I}$ is sufficiently large,
we find a surjective homomorphism of $\Lambda$-algebras
$S\to R$. Since every $p$-divisible group over $R$
can be lifted to $S$ by \cite[th\'eor\`eme 4.4]{Illusie},
it suffices to show that \eqref{It-finiteness-bt-1} holds
for $S$. Let $\mathfrak{a}\subset S$ be the ideal generated by
the elements $y_i=x_i^{p^{r-1}}$ for $i\in\mathcal{I}$.
This ideal carries nilpotent divided powers $\delta$
defined by $\delta_p(y_i)=0$ for all $i$.
By induction on $r$ we can assume that
\eqref{It-finiteness-bt-1} holds for $S/\mathfrak{a}$. Then
\eqref{It-finiteness-bt-1} holds for $S$ by Lemma
\ref{Le-reduce-pd-ideal}.
\end{proof}
\begin{Lemma}
\label{Le-ker-trunc}
Let $B\to A$ be a surjective ring homomorphism
with kernel $I$ such that $pI=0$ and
$x^p=0$ for all $x\in I$.
For a finite flat group scheme $H$ over $B$, the kernel
of $H(B)\to H(A)$ is annihilated by $p$.
\end{Lemma}
\begin{proof}
Let $B_0=B/pB$ and $H_0=H\otimes_BB_0$.
The abelian group $B_0\oplus I$
becomes a ring with multiplication
$(a\oplus i)(a'\oplus i')=aa'\oplus(ai'+a'i+ii')$,
and one can identify
$B\times_{A}B$ with $B\times_{B_0}(B_0\oplus I)$.
Since the evaluation of $H$ commutes with fibered
products of rings, we obtain an isomorphism of
abelian groups
$$
\Ker(H(B)\to H(A))\cong\Ker(H_0(B_0\oplus I)\to H_0(B_0)).
$$
By the assumptions on $I$, the right hand side lies
in the kernel of the Frobenius of $H_0$, which lies
in $H_0[p]$. This proves the lemma.
\end{proof}
\begin{proof}[Proof of Proposition \ref{Pr-finiteness-bt}
\eqref{It-finiteness-bt-2}]
Let $G$ and $H$ be $p$-divisible groups over $R$.
By \eqref{It-finiteness-bt-1} they are both
defined over a finitely generated $W(R_{\red})$-subalgebra
$R_0$ of $R$. Thus to prove \eqref{It-finiteness-bt-2}
we may replace $R$ by $R_0$; then $\mathcal{N}_R$ is a nilpotent
ideal. We have torsion free ${\mathbb{Z}}_p$-modules
$$
\Hom(G,H)\to\Hom(G_S,H_S)\to\Hom(G_{R_{\red}},H_{R_{\red}}).
$$
The first map is injective because $R\to S$ is
injective. The second map is injective because
Lemma \ref{Le-ker-trunc} implies that its kernel
is annihilated by a power of $p$.
The cokernel of the composition is annihilated
by a power of $p$ because $\mathcal{N}_R$ is nilpotent.
Hence it suffices to show
that a homomorphism $f:G\to H$ is divisible by $p$
if and only if $f_S:G_S\to H_S$
is divisible by $p$.
But $f$ is divisible by $p$ if and only if
$f[p]:G[p]\to H[p]$ is zero, which can be detected
over $S$ because $R\to S$ is injective.
\end{proof}
\subsection{Deformation rings}
Let $\Lambda\to K$ be a surjective ring homomorphism
with finitely generated kernel $I\subseteq\Lambda$
such that $\Lambda$ is $I$-adic. The ring $K$ is not
assumed to be a field. Let
$\Nil_{\Lambda/K}$ be the category of $\Lambda$-algebras
$A$ together with a homomorphism of $\Lambda$-algebras
$A\to K$ with nilpotent kernel.
We consider covariant functors
$$
F:\Nil_{\Lambda/K}\to({\text{sets}})
$$
with the following properties.
\begin{enumerate}
\item
The set $F(K)$ has precisely one element.
\item
For a surjective homomorphism $A_1\to A$ in $\Nil_{\Lambda/K}$
the induced map $F(A_1)\to F(A)$ is surjective.
\item
For each pair of homomorphisms $A_1\to A\leftarrow A_2$ in
$\Nil_{\Lambda/K}$ such that one of them is surjective
the natural map
$F(A_1\times_AA_2)\to F(A_1)\times_{F(A)}F(A_2)$
is bijective. Then for each $K$-module $N$ the set
$F(K\oplus N)$ is naturally a $K$-module. In particular,
$t_F=F(K[\varepsilon])$ is a $K$-module, which is
called the tangent space of $F$.
\item
For each $K$-module $N$ the natural homomorphism of $K$-modules
$
t_F\otimes_KN\to F(K\oplus N)
$
is bijective.
\item
The $K$-module $t_F$ is finitely presented.
\end{enumerate}
\noindent
The first three conditions imply that $F$
preserves exact sequences of $K$-modules.
Thus (4) is automatic if $N$ is finitely presented.
Moreover (1)--(4) imply that the $K$-module
$t_F$ is flat, so (5) implies that $t_F$
is projective.
\begin{Prop}
Assume that $F$ satisfies (1)--(5).
Then $F$ is pro-repre\-sented
by a complete $\Lambda$-algebra $B$.
Let $\tilde t$ be a projective $\Lambda$-module
which lifts $t_F$. Then $B$ is isomorphic to the
complete symmetric algebra $\Lambda[[\tilde t^*]]$,
where ${}^*$ means dual.
This is a power series ring over $\Lambda$ if $t_F$
is a free $K$-module.
\end{Prop}
\begin{proof}
The $K$-module $t_F$ is projective because it is
flat and finitely presented. Thus $\tilde t$ exists.
Let $B=\Lambda[[\tilde t^*]]$ and let $\bar B=K\oplus t_F^*$.
We have an obvious projection $B\to\bar B$.
Let $\bar\xi\in F(\bar B)=t_F\otimes t_F^*=\End(t_F)$
correspond to the identity of $t_F$ and let
$\xi\in F(B)$ be a lift of $\bar\xi$.
We claim that the induced homomorphism of functors
$\xi:B\to F$ is bijective. Note that the functor
$B$ satisfies (1)--(5). By induction it suffices
to show that if $A\to\bar A$ is a surjection
in $\Nil_{\Lambda/K}$ whose kernel $N$
is a $K$-module of square zero and if
$B(\bar A)\to F(\bar A)$ is bijective, then
$B(A)\to F(A)$ is bijective as well. We have a natural
isomorphism $A\times_{\bar A}A\cong A\times_K(K\oplus N)$.
It follows that the fibres of $B(A)\to B(\bar A)$
and the fibres of $F(A)\to F(\bar A)$ are
principal homogeneous sets under the $K$-modules
$B(K\oplus N)$ and $F(K\oplus N)$, respectively.
The homomorphism $t_B\to t_F$ induced by $\xi$
is bijective by construction, so $B(K\oplus N)\to F(K\oplus N)$
is bijective, and the proposition follows.
\end{proof}
\begin{Cor}
A homomorphism of functors which satisfy (1)--(5) is an
isomorphism if and only if it induces an isomorphism
on tangent spaces.
\qed
\end{Cor}
\begin{Remark}
\label{Re-deform-functorial}
Let $\Lambda'\to K'$ be another pair as above
and let $g:\Lambda'\to\Lambda$ be a ring homomorphism
which induces a homomorphism $\bar g:K'\to K$.
For given functors $F$ on $\Nil_{\Lambda/K}$ and
$F'$ on $\Nil_{\Lambda'/K'}$, a homomorphism
$h:F\to F'$ over $g$ is a compatible system of maps
$$
h(A):F(A)\to F'(A\times_KK')
$$
for $A$ in $\Nil_{\Lambda/K}$; here $A\times_KK'$
is naturally on object of $\Nil_{\Lambda'/K'}$.
If $F$ and $F'$ satisfy (1)--(5) and if $B$ and $B'$
are the complete algebras which pro-represent $F$
and $F'$, respectively, then $h$ corresponds to a
homomorphism $B'\to B$ compatible with $g$ and $\bar g$.
If $h(A)$ is bijective for all $A$, the induced
homomorphism $B'\hat\otimes_{\Lambda'}\Lambda\to B$ is an
isomorphism.
\end{Remark}
\begin{Defn}
Assume that $p$ is nilpotent in $K$.
For a $p$-divisible group $G$ over $K$ let
$$
\Def_G:\Nil_{\Lambda/K}\to(\text{sets})
$$
be the deformation functor of $G$. This means that
$\Def_G(A)$ is the set of isomorphism classes
of $p$-di\-visible groups $G'$ over $A$
together with an isomorphism $G'\otimes_AK\cong G$.
Let $t_G=\Lie(G^\vee)\otimes_K\Lie(G)$.
\end{Defn}
\begin{Prop}
\label{Pr-Def-G}
The functor $\Def_G$ is pro-represented by a
complete $\Lambda$-algebra $B$.
Explicitly, if $\tilde t$ is a projective
$\Lambda$-module which lifts $t_G$, then
$B$ is isomorphic to the complete symmetric
algebra $\Lambda[[\tilde t^*]]$.
\end{Prop}
\begin{proof}
The functor $\Def_G$ satisfies (1)--(5) with tangent space
$t_G$ because for a surjective homomorphism $A'\to A$ in
$\Nil_{\Lambda/K}$ whose kernel $N$ is a $K$-module of
square zero and for $H\in\Def_G(A)$, the set of lifts of $H$
to $A'$ is a principally homogeneous set under
the $K$-module $\Hom_K(t_G,N)$ by \cite{Messing-Crys}.
\end{proof}
\begin{Remark}
\label{Re-Def-G-functorial}
Let $g:\Lambda'\to\Lambda$ over $\bar g:K'\to K$
be as in Remark \ref{Re-deform-functorial}
such that $p$ is nilpotent in $K'$. Let $G$ over $K$
be the base change of a $p$-divisible group $G'$
over $K'$. For $A$ in $\Nil_{\Lambda/K}$
we have a natural map
$$
\Def_{G'}(A\times_KK')\to\Def_G(A).
$$
This map is bijective, and its inverse is a homomorphism
$\Def_G\to\Def_{G'}$ over $g$ in the sense of
\ref{Re-deform-functorial}.
If $B$ and $B'$ pro-represent $\Def_G$ and
$\Def_{G'}$, respectively, we get an isomorphism
$B'\hat\otimes_{\Lambda'}\Lambda\cong B$.
\end{Remark}
\begin{Defn}
Assume that $K$ is an admissible ring.
For a Dieudonn\'e display $\mathscr{P}$ over $K$ we denote by
$$
\Def_{\mathscr{P}}:\Nil_{\Lambda/K}\to(\text{sets})
$$
the deformation functor of $\mathscr{P}$.
Let $t_\mathscr{P}=\Hom(Q/I_KP,P/Q)$.
\end{Defn}
We are mainly interested in the case where $K$ is perfect
and $\Lambda=W(K)$. Then Dieudonn\'e displays over $K$
are displays because ${\mathbb{W}}(K)=W(K)$.
\begin{Prop}
\label{Pr-Def-PPP}
The functor $\Def_\mathscr{P}$ is pro-represented by a complete
$\Lambda$-algebra $B$. Explicitly, if $\tilde t$ is
a projective $\Lambda$-module which lifts $t_\mathscr{P}$, then
$B$ is isomorphic to the complete symmetric algebra
$\Lambda[[\tilde t^*]]$.
\end{Prop}
\begin{proof}
The functor $\Def_\mathscr{P}$ satisfies (1)--(5) with tangent space
$t_\mathscr{P}$ because for a surjective homomorphism $A'\to A$ in $\Nil_{\Lambda/K}$ whose kernel $N$ is a $K$-module of square zero and for $\mathscr{P}'\in\Def_\mathscr{P}(A)$, the set of lifts of $\mathscr{P}'$ to $A'$ is a principally homogeneous set under the $K$-module $\Hom_K(t_\mathscr{P},N)$ by Corollary \ref{Co-deform-DDD}.
\end{proof}
\begin{Remark}
\label{Re-Def-PPP-funct}
Let $g:\Lambda'\to\Lambda$ over $\bar g:K'\to K$ be as in
Remark \ref{Re-deform-functorial} such that $K$ and $K'$
are admissible rings. Assume that $\mathscr{P}$ is the base change
of a Dieudonn\'e display $\mathscr{P}'$ over $K'$.
If $B$ and $B'$ represent $\Def_\mathscr{P}$ and $\Def_{\mathscr{P}'}$,
respectively, then $B'\hat\otimes_{\Lambda'}\Lambda\cong B$.
This is analogous to Remark \ref{Re-Def-G-functorial}.
\end{Remark}
\subsection{Crystals and frames}
\label{Subse-cryst-frame}
Let $\mathcal{F}=(S,I,R,\sigma,\sigma_1)$ be a frame such that
$S$ and $R$ are $p$-adic, $S$ has no $p$-torsion,
$I$ carries divided powers, and $\sigma=p\sigma_1$ on $I$.
Thus $(S,\sigma)$ is a frame for each $R/p^nR$ in the
sense of \cite{Zink-Windows}.
By a well-known construction, the crystalline
Dieudonn\'e functor allows to associate to a $p$-divisible
group over $R$ an $\mathcal{F}$-window; this is explained in
the proof of \cite[Theorem 1.6]{Zink-Windows}
for the Dieudonn\'e crystal of a nilpotent display,
and in \cite{Kisin-crys, Kisin-2adic} for $p$-divisible groups.
The construction goes as follows.
First, one can define a filtered $F$-$V$-module;
here is is not necessary to assume that $S$ has no $p$-torsion:
\begin{Constr}
\label{Const-fil-mod}
Let $\mathcal{F}$ be a frame such that $S$ and $R$ are $p$-adic,
$I$ is equipped with divided powers $\delta$ which are compatible
with the canonical divided powers of $p$,
and $\sigma=p\sigma_1$ on $I$.
Let $R_0=R/pR$ and let $\delta'$ be the divided powers
on $I'=I+pS$ which extend $\delta$ and the canonical divided
powers of $p$. We assume that $\sigma$ preserves $\delta'$.
This is automatic if $S$ has no $p$-torsion. Let $\sigma_0$
be the Frobenius endomorphism of $R_0$. There is a functor
$$
G\mapsto(P,Q,F^\sharp,V^\sharp)
$$
from $p$-divisible groups over $R$ to filtered
$F$-$V$-modules over $\mathcal{F}$ defined as follows.
We put $P={\mathbb{D}}(G)_{S/R}={\mathbb{D}}(G_0)_{S/R_0}$,
where ${\mathbb{D}}(G)$ is the \emph{covariant}%
\footnote{\label{Ft-DD-covariant}
This differs from the notation of \cite{BBM},
where ${\mathbb{D}}(G)$ is contravariant.
One can switch between the covariant and contravariant crystals
by passing to the dual of $G$ or of ${\mathbb{D}}(G)$, which amounts to
the same by the crystalline duality theorem \cite[5.3]{BBM}.
}
Dieudonn\'e crystal.
The kernel of the natural projection $P\to\Lie(G)$ is $Q$.
Since $\sigma$ preserves $\delta'$, there is an isomorphism
$P^{(1)}\cong{\mathbb{D}}(\sigma_0^*G_0)_{S/R_0}$. Thus we can define
$V^\sharp:P\to P^{(1)}$ to be induced by the Frobenius
$F:G_0\to\sigma_0^*G_0$ and $F^\sharp:P^{(1)}\to P$
to be induced by the Verschiebung $V:\sigma_0^*G_0\to G_0$.
\end{Constr}
In the second step one associates $F_1$:
\begin{Prop}
\label{Pr-fil-mod-win}
Let $\mathcal{F}$ be a frame as in the beginning
of section \ref{Subse-cryst-frame}.
Let $(P,Q,F^\sharp,V^\sharp)$ be the filtered $F$-$V$-module
over $\mathcal{F}$ associated to a $p$-divisible group $G$ over $R$
by Construction \ref{Const-fil-mod}.
There is a unique map $F_1:Q\to P$ such that
$(P,Q,F,F_1)$ is an $\mathcal{F}$-window, and it induces $V^\sharp$.
\end{Prop}
\begin{proof}
We have functors $(P,Q,F,F_1)\mapsto
(P,Q,F^\sharp,V^\sharp)\mapsto(P,Q,F^\sharp)$
which are fully faithful; see Lemma \ref{Le-F-V-mod}.
Thus we have to show that $F(Q)$ lies in $pP$, so that
$F_1=p^{-1}F$ is well-defined, that $F_1(Q)$ generates $P$,
and that the pair $(P,Q)$ admits a normal decomposition.
Since $R$ and $S$ are $p$-adic and since the kernel of
$S/pS\to R/pR$ is a nil-ideal due to its divided powers,
all finitely generated projective $R$-modules lift to $S$.
Thus a normal decomposition exists.
The existence of $F_1$ and the surjectivity of its
linearisation are proved in \cite[A.2]{Kisin-crys}
if $S$ is local with perfect residue field, but the
proof can be easily adapted to the general case:
To prove surjectivity, for each maximal ideal of $S$,
which necessarily comes from a maximal ideal $\mathfrak{m}$ of $R$,
we choose an embedding of $R/\mathfrak{m}$ into a perfect field $k$.
There is a ring homomorphism $\alpha:S\to W(k)$ which lifts
$R\to k$ such that $f\alpha=\alpha\sigma$; it can be
constructed as $S\to W(S)\to W(k)$.
Then $\alpha$ is a homomorphism of frames $\mathcal{F}\to\mathscr{W}_k$,
and the assertion is reduced to the case of $\mathscr{W}_k$,
which is classical.
\end{proof}
\begin{Remark}
The surjectivity of $F_1$ in the proof of Proposition
\ref{Pr-fil-mod-win} can
also be deduced from the crystalline duality theorem.
Let $P=L\oplus T$ be a normal decomposition and let
$\Psi:P\to P$ be given by $F_1$ on $L$ and by $F$ on $T$.
We have to show that the linearisation
$\Psi^\sharp:P^{(1)}\to P$ is an isomorphism.
Let $(P',Q',F',F_1')$ be the quadruple associated to the
Cartier dual $G^\vee$. The duality theorem gives
a perfect pairing $P\times P'\to S$ such that
$\left<F(x),F'(x')\right>=p\sigma{\left<x,x'\right>}$.
It follows that
$\left<F(x),F_1'(x')\right>=\sigma{\left<x,x'\right>}$
and
$\left<F_1(x),F'(x')\right>=\sigma{\left<x,x'\right>}$
whenever this makes sense.
The unique decomposition $P'=L'\oplus T'$ with
$\left<L,L'\right>=0=\left<T,T'\right>$ is a normal
decomposition of $P'$, and the dual of the associated
$\Psi^{\prime\sharp}$ is an inverse of $\Psi^\sharp$.
\end{Remark}
\subsection{The Dieudonn\'e display associated to
a $p$-divisible group}
For an admissible ring $R$ with $p\ge 3$
we consider the Dieudonn\'e frame $\mathscr{D}_R$
defined in Lemma \ref{Le-DDD-frame}.
The ring ${\mathbb{W}}(R)$ is $p$-adic by the remark
preceding Proposition \ref{Pr-WW-p-adic}.
By Lemma \ref{Le-pd-II**} the ideal ${\mathbb{I}}_R$ carries
natural divided powers compatible with the canonical
divided powers of $p$, and the induced divided powers on
the kernel of ${\mathbb{W}}(R)\to R/pR$ are preserved by $f$.
Thus Construction \ref{Const-fil-mod} associates
to each $p$-divisible group $G$ over $R$
a filtered $F$-$V$-module over $\mathscr{D}_R$,
which we denote by $\Phi^o_R(G)$.
\begin{Prop}
\label{Pr-bt2disp}
For each admissible ring $R$ with $p\ge 3$
there is a unique functor
$$
\Phi_R:(\text{$p$-divisible groups over $R$})
\to(\text{Dieudonn\'e displays over $R$})
$$
which is compatible with base change in $R$ such that
the filtered $F$-$V$-module over $R$ associated to
$\Phi_R(G)$ is equal to $\Phi^o_R(G)$.
\end{Prop}
\begin{proof}
Clearly $\Phi^o_R(G)=(P,Q,F^\sharp,V^\sharp)$
is functorial in $R$ and $G$.
We have to show that there is a unique operator
$F_1:Q\to P$ which is functorial in $R$ and $G$ such that
$\Phi_R(G)=(P,Q,F,F_1)$ is a Dieudonn\'e display over $R$.
Let $K=R_{\red}$ and $\Lambda=W(K)$. Let $\bar G=G\otimes_RK$
and let $B$ be the complete $\Lambda$-algebra which
pro-represents the functor $\Def_{\bar G}$ on
$\Nil_{\Lambda/K}$; see Proposition \ref{Pr-Def-G}.
Let $\mathscr{G}$ be the universal deformation of $G$ over $B$.
If $I$ denotes the kernel of $B\to K$, we can define
$$
\Phi^o_B(\mathscr{G})=\varprojlim_n\Phi^o_{B/I^n}(\mathscr{G}\otimes_BB/I^n).
$$
On the other hand, the ring ${\mathbb{W}}(B)$ is $p$-adic
by Proposition \ref{Pr-WW-p-adic}.
Therefore we can also define $\Phi^o_B(\mathscr{G})$
be a direct application of Construction \ref{Const-fil-mod},
and this agrees with the limit definition.
The ring ${\mathbb{W}}(B)$ has no $p$-torsion because
$B$ has no $p$-torsion. Thus by Proposition
\ref{Pr-fil-mod-win} there is a unique operator
$F_1$ which makes $\Phi_B^o(\mathscr{G})$ into a
Dieudon\'e display $\Phi_B(\mathscr{G})$ over $R$.
By Proposition \ref{Pr-finiteness-bt} there is a unique
homomorphism $B\to R$ of augmented algebras
such that $G=\mathscr{G}\otimes_BR$ as deformations of $\bar G$.
Necessarily we define $\Phi_R(G)$ as the base change
of $\Phi_B(\mathscr{G})$ under $B\to R$. It remains to
show that $\Phi_R(G)$ is functorial in $R$ and $G$.
Assume that $G$ is the base change of a $p$-divisible
group $G'$ over $R'$ under a homomorphism of admissible
rings $R'\to R$. Let $K'$, $\Lambda'$, $\bar G'$, $B'$, $\mathscr{G}'$
have the obvious meaning. We have a natural homomorphism
of $W(K')$-algebras $B'\to B$ together with an
isomorphism $\mathscr{G}'\otimes_{B'}B\cong\mathscr{G}$;
see Remark \ref{Re-Def-G-functorial}. By the uniqueness
of $F_1$ over $B$ we see that $\Phi_B(\mathscr{G})$
coincides with the base change of $\Phi_{B'}(\mathscr{G}')$.
It follows that $\Phi_R(G)$ is the base change of
$\Phi_{R'}(G')$.
Assume that $u:G\to G_1$ is a homomorphism of $p$-divisible
groups over $R$. Let $\bar G_1$, $B_1$, $\mathscr{G}_1$
have the obvious meaning. We have to show that
$\Phi^o_R(u)$ commutes with $F_1$. We may assume that $u$ is an
isomorphism because otherwise one can pass to the automorphism
$\left(\begin{smallmatrix}1&0\\u&1\end{smallmatrix}\right)$
of $G\oplus G_1$. This reasoning uses that the natural isomorphism
$\Phi^o_R(G\oplus G_1)=\Phi^o_R(G)\oplus\Phi^o_R(G_1)$
preserves the operators $F_1$ defined on the three modules,
which follows from the uniqueness of $F_1$ over the
ring which pro-represents $\Def_{\bar G}\times\Def_{\bar G_1}$.
An isomorphism $u:G\to G_1$ induces an isomorphism
$\bar u:\bar G\cong\bar G_1$ and an isomorphism
$B\cong B_1$ together with an isomorphism
$\tilde u:\mathscr{G}\otimes_BB_1\cong\mathscr{G}_1$
which lifts $\bar u$. By the uniqueness of $F_1$
over $B_1$ it follows that
$\Phi_{B_1}^o(\tilde u)$ preserves $F_1$. Since
$u$ is the base change of $\tilde u$ by the
homomorphism $B_1\to R$ defined by $G_1$,
it follows that $\Phi_R^o(u)$ preserves $F_1$ as well.
\end{proof}
In order to analyse the action of $\Phi_R$
on deformations we need the following
extension of Proposition \ref{Pr-bt2disp}.
Let $(R'\to R,\delta)$ be a divided power extension
of admissible rings with $p\ge 3$ which is compatible
with the canonical divided powers of $p$.
Again, the ring ${\mathbb{W}}(R'/R)$ is $p$-adic, and
${\mathbb{I}}_{R'/R}$ carries natural divided powers compatible
with the canonical divided powers of $p$ such that
$f$ preserves their extension to the kernel of
${\mathbb{W}}(R'/R)\to R/pR$. Thus
Construction \ref{Const-fil-mod} associates
to each $p$-divisible group $G$ over $R$
a filtered $F$-$V$-module over $\mathscr{D}_{R'/R}$,
which we denote by $\Phi_{R'/R}^o(G)$.
The construction is functorial in the
triple $(R'\to R,\delta)$ and in $G$.
\begin{Prop}
\label{Pr-bt2disp-pd}
Assume that $p\ge 3$.
For each divided power extension of admissible rings
$(R'\to R,\delta)$ compatible with the
canonical divided powers of $p$ there is a unique functor
$$
\Phi_{R'/R}:(\text{$p$-divisible groups over $R$})
\to(\text{Dieudonn\'e displays for $R'/R$})
$$
which is compatible with base change in the triple
$(R'\to R,\delta)$
such that the filtered $F$-$V$-module over $\mathscr{D}_{R'/R}$
associated to $\Phi_{R'/R}(G)$ is equal to $\Phi_{R'/R}^o(G)$.
\end{Prop}
\begin{proof}
For a given $p$-divisible group $G$ over $R$ we choose
a lift to a $p$-divisible group $G'$ over $R'$, which
exists by \cite[th\'eor\`eme 4.4]{Illusie}.
The Dieudonn\'e display $\Phi_{R'}(G')$ is well-defined
by Proposition \ref{Pr-bt2disp}, and necessarily
$\Phi_{R'/R}(G)$ is defined as the base change of
$\Phi_{R'}(G')$ by the frame homomorphism
$\mathscr{D}_{R'}\to\mathscr{D}_{R'/R}$. We have to show
that the operator $F_1$ on $\Phi^o_{R'/R}(G)$
defined in this way does not depend on the choice
of $G'$. If this is proved it follows easily that
$\Phi_{R'/R}(G)$ is functorial in $G$ and in $(R'\to R,\delta)$;
here instead of arbitrary homomorphisms of $p$-divisible
groups it suffices to treat isomorphisms.
Let $K$, $\Lambda$, $\bar G$, $B$, $\mathscr{G}$ be
as in the proof of Proposition \ref{Pr-bt2disp}.
We have an isomorphism $B\cong\Lambda[[t]]$ for
a finitely generated projective $\Lambda$-module $t$.
Let $C=B\hat\otimes_\Lambda B$. The automorphism $\tau=\left(\begin{smallmatrix}1&1\\0&1\end{smallmatrix}\right)$
of $t\oplus t$ defines an isomorphism
$$
C=\Lambda[[t\oplus t]]\xrightarrow\tau
\Lambda[[t\oplus t]]=B[[t_B]]
$$
under which the multiplication homomorphism
$\mu:C\to B$ corresponds to the augmentation
$B[[t_B]]\to B$ defined by $t_B\mapsto 0$.
Let $I$ be the kernel of $B\to K$ and let
$C'$ be the $I$-adic completion of the divided
power envelope of the ideal $t_BB[[t_B]]\subset B[[t_B]]$.
By Lemma \ref{Le-pd-alg}, $\mu$ extends
to a divided power extension of admissible topological
rings $\mu':C'\to B$.
Assume that $G_1$ and $G_2$ are two lifts of $G$
to $p$-divisible groups over $R'$.
Let $\mathscr{G}_1$ and $\mathscr{G}_2$ be the $p$-divisible
groups over $C$ which are the base change of $\mathscr{G}$
by the two natural homomorphisms $B\to C$.
By Proposition \ref{Pr-finiteness-bt} there are
well-defined homomorphisms $\bar\alpha:B\to R$
and $\alpha:C\to R'$ such that $G=\mathscr{G}\otimes_{B,\bar\alpha}R$
and $G_i=\mathscr{G}_i\otimes_{C,\alpha}R'$ as deformations of $\bar G$.
We have the following commutative diagram of rings.
$$
\xymatrix@M+0.2em@C+1em{
C \ar[r] \ar@/^3ex/[rr]^\alpha \ar[dr]_\mu &
C' \ar@{.>}[r]_{\alpha'} \ar[d]^{\mu'} &
R' \ar[d] \\
& B \ar[r]^{\bar\alpha} & R
}
$$
The arrow $\alpha'$ exists uniquely due to the divided
powers on the kernel of $R'\to R$.
We obtain the following commutative diagram of frames,
where $\iota$ is given by $C\to C'$, and $\iota'$ is
given by the identity of $R'$.
$$
\xymatrix@M+0.2em{
\mathscr{D}_C \ar[r]^-{\iota} \ar[d]_{\alpha} &
\mathscr{D}_{C'/B} \ar[d]^{\alpha'} \\
\mathscr{D}_{R'} \ar[r]^-{\iota'} & \mathscr{D}_{R'/R}
}
$$
We have to show that the isomorphism of filtered
$F$-$V$-modules over $\mathscr{D}_{R'/R}$
\begin{equation}
\label{Eq-first-isom}
\iota'_*(\Phi^o_{R'}(G_1))
\cong\Phi^o_{R'/R}(G)\cong
\iota'_*(\Phi^o_{R'}(G_2))
\end{equation}
commutes with the operator $F_1$ defined
on the outer terms by the functor $\Phi_{R'}$.
The construction of $\Phi^o$ can be
extended to topological divided power extensions
of admissible topological rings by passing to
the projective limit.
Then \eqref{Eq-first-isom} arises by $\alpha'_*$ from
the natural isomorphism of filtered $F$-$V$-modules
over $\mathscr{D}_{C'/B}$
\begin{equation}
\label{Eq-second-isom}
\iota_*(\Phi^o_{C}(\mathscr{G}_1))
\cong\Phi^o_{C'/B}(\mathscr{G})\cong
\iota_*(\Phi^o_{C}(\mathscr{G}_2)).
\end{equation}
Since $\alpha_*$ preserves $F_1$ it suffices to show
that \eqref{Eq-second-isom} commutes with the operators
$F_1$ defined on the outer terms by the functor $\Phi_C$.
This follows from the relation $pF_1=F$ because
${\mathbb{W}}(C'/B)$ has no $p$-torsion by Lemma \ref{Le-pd-alg}.
\end{proof}
\subsection{Variant for the prime $2$}
Let $R$ be an admissible ring with $p=2$. The ring
${\mathbb{W}}^+(R)$ is $p$-adic, and its ideal ${\mathbb{I}}^+_R$
carries natural divided powers which are compatible
with the canonical divided powers of $p$.
The proof of Proposition \ref{Pr-bt2disp} with
${\mathbb{W}}^+$ in place of ${\mathbb{W}}$ shows the following.
\begin{Prop}
\label{Pr-bt2disp+}
For each admissible ring $R$ with $p=2$ there is
a unique functor
$$
\Phi_R^+:(\text{$p$-divisible groups over $R$})
\to(\text{$\mathscr{D}^+_R$-windows})
$$
which is compatible with base change in $R$ such that
the filtered $F$-$V$-module over $R$ associated to
$\Phi^+_R(G)$ is given by Construction \ref{Const-fil-mod}.
\qed
\end{Prop}
If $pR=0$ then $\mathscr{D}^+_R=\mathscr{D}_R$,
so in this case $\Phi_R^+$ is actually a functor
$$
\Phi_R:(\text{$p$-divisible groups over $R$})
\to(\text{Dieudonn\'e displays over $R$}).
$$
Assume again that $p=2$.
Let $(R'\to R,\delta)$ be a divided power extension
of admissible rings which is compatible
with the canonical divided powers of $p$.
The ring ${\mathbb{W}}^+(R'/R)$ is $p$-adic, and its ideal
${\mathbb{I}}^+_{R'/R}$ carries natural divided powers
compatible with the canonical divided
powers of $p$. The proof of
Proposition \ref{Pr-bt2disp-pd} with
${\mathbb{W}}^+$ instead of ${\mathbb{W}}$ shows the following.
\begin{Prop}
\label{Pr-bt2disp-pd+}
For each divided power extension of admissible rings
$(R'\to R,\delta)$ with $p=2$ such that $\delta$ is compatible
with the canonical divided powers of $p$ there is
a unique functor
$$
\Phi^+_{R'/R}:(\text{$p$-divisible groups over $R$})
\to(\text{$\mathscr{D}^+_{R'/R}$-windows})
$$
which is functorial in the triple
$(R'\to R,\delta)$ such that the filtered $F$-$V$-module
over $\mathscr{D}^+_{R'/R}$ associated to $\Phi^+_{R'/R}(G)$
is given by Construction \ref{Const-fil-mod}.
\qed
\end{Prop}
\subsection{Conclusions}
\begin{Cor}
\label{Co-bt2disp-DD}
Let $G$ be a $p$-divisible group over an admissible ring $R$.
If $p\ge 3$, let $\mathscr{P}=\Phi_R(G)$ be the associated Dieudonn\'e
display over $R$; then there is a natural isomorphism
of crystals on $\Cris_{\adm}(R/{\mathbb{Z}}_p)$
$$
{\mathbb{D}}(G)\cong{\mathbb{D}}(\mathscr{P})
$$
which is compatible with the obvious isomorphism
$\Lie(G)\cong\Lie(\mathscr{P})$.
If $p=2$, let $\mathscr{P}^+=\Phi^+_R(G)$ by the associated
$v$-stabilised Dieudonn\'e display over $R$;
then there is a natural isomorphism
of crystals on $\Cris_{\adm}(R/{\mathbb{Z}}_p)$
$$
{\mathbb{D}}(G)\cong{\mathbb{D}}^+(\mathscr{P}^+)
$$
which is compatible with the obvious isomorphism
$\Lie(G)\cong\Lie(\mathscr{P}^+)$.
\end{Cor}
\begin{proof}
Let $(R'\to R,\gamma)$ be a divided power extension
of admissible rings compatible with the canonical
divided powers of $p$. Assume that $p\ge 3$.
The Dieudonn\'e display $\Phi_{R'/R}(G)$ for $R'/R$
given by Proposition \ref{Pr-bt2disp-pd} is the unique
lift of $\mathscr{P}$ under the crystalline frame homomorphism
$\mathscr{D}_{R'/R}\to\mathscr{D}_R$. By the construction of the
underlying filtered $F$-$V$-module $\Phi_{R'/R}^o(G)$
and by the definition of ${\mathbb{K}}(\mathscr{P})$ we get a natural
isomorphism of ${\mathbb{W}}(R'/R)$-modules
$$
{\mathbb{D}}(G)_{{\mathbb{W}}(R')/R}\cong{\mathbb{K}}(\mathscr{P})_{R'/R}.
$$
The tensor product with the projection ${\mathbb{W}}(R'/R)\to R'$,
which is a homomorphism of divided power extensions of $R$,
gives a natural isomorphism of $R'$-modules
${\mathbb{D}}(G)_{R'/R}\cong{\mathbb{D}}(\mathscr{P})_{R'/R}$.
If $p=2$ the same argument applies to the
$\mathscr{D}^+_{R'/R}$-window $\Phi^+_{R'/R}(G)$
given by Proposition \ref{Pr-bt2disp-pd+}.
\end{proof}
Note that Dieudonn\'e displays over $R_{\red}$ are displays,
and they are equivalent to Dieudonn\'e modules
over $R_{\red}$ by Lemma \ref{Le-disp-perf}.
\begin{Cor}
\label{Co-Phi-Cartesian}
Let $R$ be an admissible ring such that $p\ge 3$
or $pR=0$.
The following diagram of categories is Cartesian.
$$
\xymatrix@M+0.2em@C+1em{
(\text{$p$-divisible groups over $R$})
\ar[r]^-{\Phi_R} \ar[d] &
(\text{Dieudonn\'e displays over $R$})
\ar[d] \\
(\text{$p$-divisible groups over $R_{\red}$})
\ar[r]^-{\Phi_{R_{\red}}} &
(\text{Dieudonn\'e modules over $R_{\red}$})
}
$$
\end{Cor}
\begin{proof}
The category of $p$-divisible groups (resp.\ Dieudonn\'e
displays) over $R$ is the direct limit of the corresponding
categories over all finitely generated $W(R_{\red})$-algebras
contained in $R$; see section \ref{Subse-finiteness}.
Thus we may assume that $\mathcal{N}_R$ is a nilpotent ideal.
If $\mathfrak{a}\subset R$ is an ideal equipped with nilpotent
divided powers which are compatible with the canonical
divided powers of $p$ and if the corollary holds for
$R/\mathfrak{a}$ then it holds for $R$. This follows from
Corollary \ref{Co-bt2disp-DD}, using that lifts from
$R/\mathfrak{a}$ to $R$ of $p$-divisible groups and of Dieudonn\'e
displays are both classified by lifts if the Hodge filtration
by \cite{Messing-Crys} and by Corollary \ref{Co-deform-DDD}.
By taking $\mathfrak{a}=pR$ the corollary is reduced to
the case $pR=0$. Then $\mathfrak{a}$ can be any
ideal of square zero equipped with the trivial
divided powers, and the corollary follows by induction
on the order of nilpotence of $\mathcal{N}_R$.
\end{proof}
We note the following result of Gabber.
It is also proved in \cite[Cor.~6.5]{Lau-Smoothness}.
\begin{Thm}
\label{Th-Phi-equiv-red}
The functor\/ $\Phi_{R_{\red}}$ is an equivalence of categories.
\end{Thm}
In view of Corollary \ref{Co-Phi-Cartesian}
we get the following result.
If $R_{\red}$ is a field then Theorem \ref{Th-Phi-equiv-red}
is classical, so in this case Theorem \ref{Th-Phi-equiv}
is proved completely in this article.
\begin{Thm}
\label{Th-Phi-equiv}
For every admissible ring $R$ such that $p\ge3$ or $pR=0$
the functor $\Phi_R$ is an equivalence of categories.
\qed
\end{Thm}
\begin{Remark}
Assume that $p=2$. If $pR\ne 0$, the proof of Corollary
\ref{Co-Phi-Cartesian} does not apply to the functor
$\Phi^+_R$ because the divided powers on the ideal $pR$
are not nilpotent, and lifts of $p$-divisible groups from
$\bar R=R/pR$ to $R$ are not bijective to lifts
of the Hodge filtration, while lifts of
$\mathscr{D}_{\bar R}$-windows to $\mathscr{D}^+_R$-windows
are bijective to lifts of the Hodge filtration.
On the other hand, assume that for each $R$ the functor
$\Phi_{\bar R}$ can be extended to a functor
$\Phi_R$ from $p$-divisible groups over $R$ to
Dieudonn\'e displays compatible with base change
in $R$. Such a functor necessarily satisfies
Corollary \ref{Co-Phi-Cartesian} and thus
Theorem \ref{Th-Phi-equiv}. Indeed, for
a given $p$-divisible group $G$ over a perfect
ring $K$ with associated display $\mathscr{P}$ the functors
$\Phi_R$ induce a homomorphism of deformation functors
$\Phi:\Def_G\to\Def_{\mathscr{P}}$ on the category
$\Nil_{W(K)/K}$. These functors are
pro-represented by complete symmetric algebras
of projective $W(K)$-modules; see Propositions
\ref{Pr-Def-G} and \ref{Pr-Def-PPP}. Thus $\Phi$
is an isomorphism as soon as it is bijective on
tangent spaces, which can be verified over rings
of characteristic $p$ and thus follows from
Corollary \ref{Co-Phi-Cartesian}.
This reasoning does not apply
to the functors $\Phi^+_R$ because the deformation
functor of $\mathscr{D}^+$-windows is not pro-representable,
which corresponds to the fact that the functor ${\mathbb{W}}^+$
does not preserve fibered products
of admissible rings. Explicitly, the proof of
Proposition \ref{Pr-Def-PPP} does not apply
to the deformation functor of $\mathscr{D}^+$-windows because
there are no functorial divided powers on ideals of
square zero which are
compatible with the canonical divided powers
of $p$ and nilpotent modulo $p$.
\end{Remark}
We have the following consequence of the crystalline
duality theorem. The duality of windows has been recalled
in the end of section \ref{Subse-frames}.
\begin{Cor}
Let $G$ be a $p$-divisible group over an admissible
ring $R$ and let $G^\vee$ be its Cartier dual.
If $p\ge 3$, there is a natural isomorphism
$$
\Phi_R(G^\vee)\cong \Phi_R(G)^t.
$$
If $p=2$, there is a natural isomorphism
$\Phi_R^+(G^\vee)\cong \Phi_R^+(G)^t$.
\end{Cor}
\begin{proof}
Let $p\ge 3$. The crystalline duality theorem \cite[5.3]{BBM}
gives an isomorphism $\Phi_R^o(G^\vee)^t\cong\Phi_R^o(G)$.
Since the functor from windows to filtered $F$-$V$-modules
preserves duality, the uniqueness part of
Proposition \ref{Pr-bt2disp} implies that
this isomorphism preserves $F_1$, so it is
an isomorphism $\Phi_R(G^\vee)^t\cong\Phi_R(G)$.
The case $p=2$ is similar.
\end{proof}
\subsection{Passing to the limit}
The results of this section carry over to admissible
topological rings $R$ because Dieudonn\'e
displays or $p$-divisible groups over $R$ are
equivalent to compatible systems of such objects
over $R/N$ for each open ideal $N$ of $R$ contained
in $\mathcal{N}_R$. We leave out the details.
\section{Rigidity of $p$-divisible groups}
In this section we collect some rigidity properties
of the fibered category of $p$-divisible groups over
local Artin rings and of the associated crystals.
\subsection{Finite flat group schemes}
Let $\mathcal{F}$ be the additive category of commutative
finite flat $p$-group schemes over a fixed local ring $R$.
It is known that $\mathcal{F}$ is equivalent to the full subcategory
$\mathcal{F}^\bullet$ of the bounded derived category of the exact
category of $p$-divisible groups over $R$ such that the
objects of $\mathcal{F}^\bullet$ are the complexes of
length one which are isogenies; see \cite[2.3.5]{Kisin-crys}.
In elementary terms this can be expressed as follows.
\begin{Prop}
\label{Pr-finite-derived}
Let $\mathcal{I}$ be the category of isogenies of $p$-divisible
groups over $R$ with homomorphisms up to homotopy.
The set $S$ of quasi-isomor\-phisms in $\mathcal{I}$ allows a
calculus of right fractions. In particular,
the localised category $S^{-1}\mathcal{I}$ is additive.
It is equivalent to the additive category $\mathcal{F}$.
\end{Prop}
\begin{proof}
For completeness we sketch a proof.
Isogenies of $p$-divisible groups are denoted
by $X=[X^0\to X^1]$ etc., and $H^0X$ denotes the kernel
of $X^0\to X^1$.
\smallskip
1.
A homomorphism of isogenies $f:X\to Y$ is null-homotopic
if and only if $H^0f:H^0X\to H^0Y$ is zero; the homotopy
is unique if it exists.
\smallskip
2.
For each homomorphism of isogenies $f:X\to Y$ there are
a quasi-isomorphism of isogenies $t:Z\to X$ and a homomorphism
$g:Z\to Y$ which is an epimorphism in
both components such that $ft$ is homotopic to $g$.
Proof: Embed $H^0(X)$ into $Z^0=X^0\oplus Y^0$ by $(1,f)$
and put $Z^1=Z^0/H^0(X)$. Define $t$ and $g$ by the projections
$Z^0\to X^0$ and $Z^0\to Y^0$. There is a homotopy
between $ft$ and $g$ because $ft=g$ on $H^0(Z)$.
3.
For given homomorphisms of isogenies
$X\xrightarrow fY\xleftarrow sY'$, where $s$ is
a quasi-iso\-mor\-phism, one can find an isogeny
$X'$ with a homomorphism $g:X'\to Y'$ and a
quasi-isomorphism $t:X'\to X$ such that $ft$
is homotopic to $sg$.
Proof: By 2.\ we can assume that the components of $f$
are epimorphisms. Then take $X'=X\times_YY'$ componentwise.
\smallskip
4.
Thus $S$ allows a calculus of right
fractions, and $S^{-1}\mathcal{I}$ is additive.
\smallskip
5.
We have an additive
functor $H^0:S^{-1}\mathcal{I}\to\mathcal{F}$, which is surjective on
isomorphism classes by a theorem of Raynaud
\cite[3.1.1]{BBM}. Let $X$ and $Y$ be isogenies.
The functor $H^0$ is full because for a given homomorphism
$f:H^0(X)\to H^0(Y)$, the construction in 2.\
allows to represents $f$ as $H^0(gt^{-1})$.
The functor is faithful because if a right fraction
$gt^{-1}:X\to Y$ induces zero on $H^0$ then $g$ induces
zero on $H^0$ and thus $g$ is null-homotopic.
\end{proof}
\subsection{The fibered category of $p$-divisible groups}
\label{Subse-fib-pdiv}
Let $({\Art})$ be the category of local Artin schemes
with perfect residue field of characteristic $p$, and
let $({\Art})_0\subset({\Art})$ be the full subcategory of
all schemes where $p$ is zero.
We denote by $(p\text{-}{\divv})\to({\Art})$ the fibered category of
$p$-divisible groups over schemes in $({\Art})$ and by
$(p\text{-}{\divv})_0\to({\Art})_0$ the inverse image of $({\Art})_0$.
\begin{Lemma}
\label{Le-Aut-pdiv}
Assume that $u$ is an exact automorphism of the fibered
category $(p\text{-}{\divv})$ over $({\Art})$ such that for the group
$E={\mathbb{Q}}_p/{\mathbb{Z}}_p$ over $\Spec{\mathbb{F}}_p$ there is an isomorphism
$u(E)\cong E$. Then $u$ is isomorphic to the identity.
Similarly, an exact automorphism $u$ of the fibered
category $(p\text{-}{\divv})_0$ over $({\Art})_0$ such that $u(E)\cong E$
is isomorphic to the identity.
\end{Lemma}
\begin{proof}
We work with $(p\text{-}{\divv})$; the case of $(p\text{-}{\divv})_0$ is similar.
For each $U$ in $({\Art})$ we are given a functor
$G\mapsto G^u$ from the category of $p$-divisible groups
over $U$ to itself, which preserves short exact sequences,
compatible with base change in $U$,
such that $\Hom(G,H)\cong\Hom(G^u,H^u)$.
We have to show that there is an natural isomorphism
$G^u\cong G$ for all $G$, compatible with base change in $U$.
Let $(p\text{-}{\fin})\to({\Art})$ be the fibered category of commutative
finite flat $p$-group schemes over schemes in $({\Art})$.
By Proposition \ref{Pr-finite-derived}, $u$ induces
an automorphism of $(p\text{-}{\fin})$ over $({\Art})$.
Let $H\in(p\text{-}{\fin})$ over $U\in({\Art})$ be given.
Assume that $p^n$ annihilates $H$ and $H^u$. For
each $T\to U$ in $({\Art})$ there is a natural isomorphism
$$
H(T)\cong\Hom_T({\mathbb{Z}}/p^n{\mathbb{Z}},H_T)\cong
\Hom_T({\mathbb{Z}}/p^n{\mathbb{Z}},H^u_T)\cong H^u(T),
$$
using that $({\mathbb{Z}}/p^n{\mathbb{Z}})^u\cong{\mathbb{Z}}/p^n{\mathbb{Z}}$.
Since commutative finite flat group schemes over $U$ form a full
subcategory of the category of abelian presheaves on
$({\Art})/U$, we get a natural isomorphism $H\cong H^u$,
which induces a natural isomorphism $G\cong G^u$ for
all $p$-divisible groups $G$ over $U$.
\end{proof}
One can also study the freedom in
the choice of the isomorphism $u\cong\id$.
\begin{Lemma}
\label{Le-End-id-pdiv}
Let $\alpha$ be an endomorphism of the identity of
$(p\text{-}{\divv})$ over $({\Art})$,
i.e.\ for each $U$ in $({\Art})$ and each $p$-divisible
group $G$ over $U$ an element $\alpha_G\in\End_U(G)$
is given, which is functorial in $G$ and $U$.
Then $\alpha$ is an element of\/ ${\mathbb{Z}}_p$.
Similarly, every endomorphism of the identity of
$(p\text{-}{\divv})_0$ over $({\Art})_0$ is an element of\/ ${\mathbb{Z}}_p$.
\end{Lemma}
\begin{proof}
We work with $(p\text{-}{\divv})$; the case of $(p\text{-}{\divv})_0$ is similar.
Since $\End_U(G)$ has no $p$-torsion we may
replace $\alpha$ by $1+p\alpha$ and thus assume that
$\alpha$ is an isomorphism. Since $\End({\mathbb{Q}}_p/{\mathbb{Z}}_p)={\mathbb{Z}}_p$,
after multiplying $\alpha$ with a $p$-adic unit we may further
assume that $\alpha$ is the identity on ${\mathbb{Q}}_p/{\mathbb{Z}}_p$.
By Proposition \ref{Pr-finite-derived}, $\alpha$ induces
an endomorphism $\alpha'$ of the identity of $(p\text{-}{\fin})$ over
$({\Art})$. For $H\in(p\text{-}{\fin})$ over $U\in({\Art})$, an element
$x$ of $H(U)$ corresponds to a homomorphism ${\mathbb{Z}}/p^n{\mathbb{Z}}\to H$
over $U$. Since $\alpha'$ is the identity on ${\mathbb{Z}}/p^n{\mathbb{Z}}$
it follows that $\alpha'(x)=x$. Thus $\alpha'$ is the identity.
Since $\alpha'$ determines $\alpha$, the latter is the identity too.
\end{proof}
For completeness we note the following extension
of Lemma \ref{Le-Aut-pdiv}.
\begin{Lemma}
Let $u$ be an exact automorphism of the fibered category
$(p\text{-}{\divv})$ over $({\Art})$ or $(p\text{-}{\divv})_0$ over $({\Art})_0$.
There is a unique continuous character
$\chi:\Gal(\bar{\mathbb{F}}_p/{\mathbb{F}}_p)\to{\mathbb{Z}}_p^*$ such that
$G^u\cong G\otimes\chi$ for all $G$.
\end{Lemma}
\begin{proof}
Again an automorphism $u$ of $(p\text{-}{\divv})$ over $({\Art})$
induces an automorphism of $(p\text{-}{\fin})$ over $({\Art})$.
Let $k$ be an algebraically closed
field of characteristic $p$. Then $u$ permutes
the three simple objects ${\mathbb{Z}}/p{\mathbb{Z}}$, $\mu_p$, and
$\alpha_p$ of $(p\text{-}{\fin})_{\Spec k}$. This permutation must be the
identity because only $\alpha_p$ has infinitely many
endomorphisms, and $\uHom(\mu_p,{\mathbb{Z}}/p{\mathbb{Z}})$ is trivial but
$\uHom({\mathbb{Z}}/p{\mathbb{Z}},\mu_p)$ is not. Thus the group
${\mathbb{Q}}_p/{\mathbb{Z}}_p$ over ${\mathbb{F}}_p$ is mapped by $u$ to an
\'etale $p$-divisible group of height $1$, which
corresponds to a character $\chi$ as in the lemma.
Then Lemma \ref{Le-Aut-pdiv} gives an isomorphism
$G^u\otimes\chi^{-1}\cong G$.
\end{proof}
\subsection{Homomorphisms of crystals}
We fix a perfect field $k$ of characteristic $p$.
Let $({\Art_k})$ be the category of local Artin schemes whose
residue field is a finite extension of $k$. By a slight
change of notation we denote by $(p\text{-}{\divv})\to({\Art_k})$ the
fibered category of $p$-divisible groups over schemes in
$({\Art_k})$ and by $(\DDD\text{-}{\win})\to({\Art_k})$ the fibered category
of Dieudonn\'e displays over schemes in $({\Art_k})$.
For $U$ in $({\Art_k})$ and for a $p$-divisible group or
Dieudonn\'e display $X$ over $U$ we consider the augmented
covariant\footnote{See footnote \ref{Ft-DD-covariant}
on page \pageref{Ft-DD-covariant}.}
Dieudonn\'e crystal ${\mathbb{D}}(X)\to\underline{\Lie X}$
(defined over different crystalline sites).
Denote by ${\mathbb{D}}'(X)$ the restriction of ${\mathbb{D}}(X)$
to divided power extensions in $({\Art_k})$.
This means that for $T\leftarrow U'\to U$
in $({\Art_k})$ where the first arrow is a divided power
embedding, we have a surjective homomorphism of
$\mathcal{O}_T$-modules ${\mathbb{D}}'(X)_{U'/T}\to\Lie(X_{U'})$.
\begin{Lemma}
\label{Le-DD-isom}
Let $\Psi:(\DDD\text{-}{\win})\to(p\text{-}{\divv})$ be an additive functor of fibered
categories over $({\Art_k})$ which preserves height and dimension.
Let $\beta:{\mathbb{D}}'\to{\mathbb{D}}'\circ\Psi$ be a homomorphism between
functors on $(\DDD\text{-}{\win})$ which respects the Hodge filtration.
This means that
for each divided power embedding $(U\to T,\delta)$ in $({\Art_k})$
and each Dieudonn\'e display $\mathscr{P}$ over $U$ we are given
a commutative diagram of $\mathcal{O}_T$-modules
$$
\xymatrix@M+0.2em@C+2em{
{\mathbb{D}}'(\mathscr{P})_{U/T} \ar[r]^-{\beta(\mathscr{P})_{U/T}} \ar[d] &
{\mathbb{D}}'(\Psi(\mathscr{P}))_{U/T} \ar[d] \\
{\Lie(\mathscr{P})} \ar[r]^-{\beta^0(\mathscr{P})} &
{\Lie(\Psi(\mathscr{P}))},\!
}
$$
which is functorial in $\mathscr{P}$ and in $(U\to T,\delta)$.
Assume that $\beta^0(\mathscr{P})$ is an isomorphism for all
$\mathscr{P}$. Then $\beta(\mathscr{P})$ is an isomorphism for all $\mathscr{P}$.
\end{Lemma}
\begin{proof}
Let $U=\Spec k$ and $T=\Spec k[\varepsilon]$.
It suffices to show that $\beta(\mathscr{P})_{U/U}$ is an isomorphism
for all $\mathscr{P}$ defined over $U$.
Let $G=\Psi(\mathscr{P})$. We consider the following homomorphism of
exact sequences of $k$-modules, where $V(\mathscr{P})$ and $V(G)$
have equal dimension. The diagram defines $\beta^1(\mathscr{P})$.
$$
\xymatrix@M+0.2em{
0 \ar[r] & V(\mathscr{P}) \ar[r] \ar[d]^{\beta^1(\mathscr{P})} &
{\mathbb{D}}'(\mathscr{P})_{U/U} \ar[r] \ar[d]^{\beta(\mathscr{P})_{U/U}} &
\Lie(\mathscr{P}) \ar[r] \ar[d]^{\beta^0(\mathscr{P})} & 0 \\
0 \ar[r] & V(G) \ar[r] &
{\mathbb{D}}'(G)_{U/U} \ar[r] &
\Lie(G) \ar[r] & 0
}
$$
Lifts of $\mathscr{P}$ to $T$ are in bijection
to $\Hom_k(V(\mathscr{P}),\Lie(\mathscr{P}))$, and lifts of $G$ to $T$
are in bijection to $\Hom_k(V(G),\Lie(G))$. Thus $\Psi$
induces a map
$$
\psi:\Hom_k(V(\mathscr{P}),\Lie(\mathscr{P}))\to\Hom_k(V(G),\Lie(G)).
$$
Since for two lifts $\mathscr{P}_1$ and $\mathscr{P}_2$ of $\mathscr{P}$ to
$T$ the isomorphisms
${\mathbb{D}}'(\mathscr{P}_1)_{T/T}\cong{\mathbb{D}}'(\mathscr{P})_{U/T}\cong{\mathbb{D}}'(\mathscr{P}_2)_{T/T}$
and their analogues for $G_i=\Psi(\mathscr{P}_i)$ are compatible with
$\beta$, for each $\alpha\in\Hom_k(V(\mathscr{P}),\Lie(\mathscr{P}))$
the following diagram commutes.
$$
\xymatrix@M+0.2em{
V(\mathscr{P}) \ar[r]^-\alpha \ar[d]_{\beta^1(\mathscr{P})} &
\Lie(\mathscr{P}) \ar[d]^{\beta^0(\mathscr{P})} \\
V(G) \ar[r]^-{\psi(\alpha)} & \Lie(G)
}
$$
By replacing $\mathscr{P}$ with $\mathscr{P}\oplus\mathscr{D}_k$
if necessary, we may assume that $\beta^0(\mathscr{P})$ is a
\emph{non-zero} isomorphism.
Since $\alpha$ is arbitrary, the diagram implies that
$\beta^1(\mathscr{P})$ is injective, thus bijective.
Thus $\beta(\mathscr{P})_{U/U}$ is an isomorphism.
\end{proof}
\begin{Lemma}
\label{Le-DD-aut}
Assume that for each $U$ in $({\Art_k})$ and each $p$-divisible
group $G$ over $U$ an endomorphism $\beta(G)$ of\/ ${\mathbb{D}}'(G)$
is given, which is functorial in $G$ and $U$ and which respects
the Hodge filtration, such that $\beta({\mathbb{Q}}_p/{\mathbb{Z}}_p)=\id$.
Then $\beta(G)=\id$ for all $G$.
The lemma also holds if $\beta(G)$ is an endomorphism
of the restriction of\/ ${\mathbb{D}}'(G)$ to nilpotent divided power
embeddings in $({\Art})$, or to (nilpotent) divided power embeddings
which are compatible with the canonical divided powers of $p$.
All this remains true is $({\Art_k})$ is replaced by its full
subcategory of schemes where $p$ is zero.
\end{Lemma}
\begin{proof}
Let $G$ be a $p$-divisible group over $U\in({\Art_k})$.
As in the proof of Lemma \ref{Le-DD-isom},
for each homomorphism $\alpha:V(G)\to\Lie(G)$,
corresponding to a deformation of $G$ over $U[\varepsilon]$,
we have a commutative diagram
$$
\xymatrix@M+0.2em{
V(G) \ar[r]^-\alpha \ar[d]_{\beta^1(G)} &
\Lie(G) \ar[d]^{\beta^0(G)} \\
V(G) \ar[r]^-\alpha & \Lie(G).\!
}
$$
Since $\beta^1({\mathbb{Q}}_p/{\mathbb{Z}}_p)$ is the identity
it follows that $\beta^0(G)$ and $\beta^1(G)$ are the
identity for all $G$. Thus $\gamma(G)=\beta(G)-\id$ is
an endomorphism of ${\mathbb{D}}'(G)$ such that for every lift
$G'$ of $G$ to $U[\varepsilon]$ the endomorphism
$g=\gamma(G)_{U/U[\varepsilon]}$ of $M={\mathbb{D}}'(G)_{U/U[\varepsilon]}$
vanishes on $V(G')$ and has image in $V(G')$.
If $V(G)$ is zero, $\gamma(G)$ must be zero.
If $V(G)$ is non-zero, the sum of all
possible $V(G')$ contains $\varepsilon M$, so
$g$ vanishes on $\varepsilon M$, and thus
the image of $g$ lies in $\varepsilon M$.
Since the inverse image of $g$ under $U\to U[\varepsilon]$,
$\varepsilon\mapsto 0$ is $\gamma(G)_{U/U}$, we see that
$\beta(G)_{U/U}$ is the identity.
By functoriality in $U$ we get that $\beta(G)=\id$.
\end{proof}
\section{From Dieudonn\'e displays to $p$-divisible groups}
In this section we define a functor from Dieudonn\'e
displays over an admissible local ring to $p$-divisible
groups, without restriction on $p$.
\subsection{Preliminaries}
We fix an admissible local ring $R$ with (perfect) residue
field $k=R_{\red}$ of characteristic $p$.
\begin{Defn}
Let $\mathcal{J}_R$ be the category of $R$-algebras $A$ such that
$\mathcal{N}_A$ is bounded nilpotent and $A_{\red}$ is the union
of finite dimensional $k$-algebras.
\end{Defn}
We call a ring homomorphism $A\to B$ ind-\'etale if
$B$ is the filtered direct limit of \'etale $A$-algebras.
\begin{Lemma}
\label{Le-JJJR}
Every $A\in\mathcal{J}_R$ is admissible.
The category $\mathcal{J}_R$ is stable under tensor products.
If $A\to B$ is an ind-\'etale or a quasi-finite ring
homomorphism with $A\in\mathcal{J}_R$ then $B\in\mathcal{J}_R$.
\end{Lemma}
\begin{proof}
Since a reduced finite $k$-algebra is \'etale
and thus perfect, every $A$ in $\mathcal{J}_R$ is admissible.
Let $A\to B$ a ring homomorphism with $A\in\mathcal{J}_R$.
Then $\mathcal{N}_AB$ is bounded nilpotent, so $B$ is lies in
$\mathcal{J}_R$ if and only if $B/\mathcal{N}_AB$ lies in $\mathcal{J}_R$.
For given homomorphisms $B\leftarrow A\to C$ in $\mathcal{J}_R$
we have to show that $B\otimes_AC$ lies in $\mathcal{J}_R$.
By the preceding remark we may assume that $A,B,C$
are reduced. Then $B\otimes_AC$ is the direct limit
of \'etale $k$-algebras and thus lies in $\mathcal{J}_R$.
Let $g:A\to B$ be an ind-\'etale or quasi-finite
ring homomorphism with $A\in\mathcal{J}_R$. In order to show
that $B\in\mathcal{J}_R$ we may assume that $A$ is reduced.
Then every finitely generated $k$-algebra in $A$ is
\'etale. Thus each \'etale $A$-algebra is defined
over an \'etale $k$-algebra in $A$. If $g$ is ind-\'etale
it follows that $B$ lies in $\mathcal{J}_R$.
Assume that $g$ is quasi-finite. Then $B$
is defined over an \'etale $k$-algebra in $A$.
Since all finite $k$-algebras lie in $\mathcal{J}_R$ and since
$\mathcal{J}_R$ is stable under tensor products it follows that
$B\in\mathcal{J}_R$.
\end{proof}
Let $S=\Spec R$ and let $\mathcal{J}_S$ be the category of
affine $S$-schemes $\Spec A$ with $A\in\mathcal{J}_R$.
If $\tau$ is one of ind-\'etale or fpqc, we consider
the $\tau$-topology on $\mathcal{J}_S$ in which coverings
are finite families of morphisms $(\Spec B_i\to\Spec A)$
such that the associated homomorphism $A\to\prod_i B_i$ is
faithfully flat, and ind-\'etale if $\tau$ is ind-\'etale.
Let $S_{\mathcal{J},\tau}$ be the category of $\tau$-sheaves on $\mathcal{J}_S$.
\begin{Lemma}
\label{Le-WW-sheaf}
The presheaf of rings ${\mathbb{W}}$ on $\mathcal{J}_S$ is an fpqc sheaf.
\end{Lemma}
\begin{proof}
See \cite[Lemma 1.5]{Lau-Dual}.
Since the presheaf $W$ is an fpqc sheaf it suffices to
show that for an injective ring homomorphism $A\to B$ in
$\mathcal{J}_R$ we have ${\mathbb{W}}(A)={\mathbb{W}}(B)\cap W(A)$ in $W(B)$.
This is easily verified using that $A_{\red}\to B_{\red}$
is injective and $\hat W(\mathcal{N}_A)=\hat W(\mathcal{N}_B)\cap W(A)$
in $W(\mathcal{N}_B)$.
\end{proof}
\subsection{The functor \boldmath $\BT$}
Let $\mathscr{P}$ be a Dieudonn\'e display over $R$.
For $\Spec A$ in $\mathcal{J}_S$ let $\mathscr{P}_A=(P_A,Q_A,F,F_1)$
be the base change of $\mathscr{P}$ to $A$. We define two
complexes of presheaves of abelian groups on $\mathcal{J}_S$ by
\begin{gather}
\label{Eq-ZPPP}
Z(\mathscr{P})(\Spec A)=[Q_A\xrightarrow{F_1-1}P_A],\\
\label{Eq-ZprimePPP}
Z'(\mathscr{P})(\Spec A)=[Q_A\xrightarrow{F_1-1}P_A]\otimes[{\mathbb{Z}}\to{\mathbb{Z}}[1/p]],
\end{gather}
such that $Z(\mathscr{P})$ lies in degrees $0,1$ and
$Z'(\mathscr{P})$ lies in degrees $-1,0,1$, so the second
tensor factor lies in degrees $-1,0$.
\begin{Prop}
\label{Pr-BT(PPP)}
The components of $Z'(\mathscr{P})$ are fpqc sheaves.
The ind-\'etale (and thus the fpqc) cohomology
sheaves of $Z'(\mathscr{P})$ vanish outside degree zero,
and the cohomology sheaf in degree zero is represented
by a well-defined $p$-divisible group $\BT_R(\mathscr{P})$ over $R$.
This defines an additive and exact functor
$$
\BT_R:(\text{Dieudonn\'e displays over $R$})
\to(\text{$p$-divisible groups over $R$}).
$$
\end{Prop}
\noindent
One can also express the definition of the functor
$\BT_R$ by the formula
$$
\BT_R(\mathscr{P})=[Q\xrightarrow{F_1-1} P]\otimes^L{\mathbb{Q}}_p/{\mathbb{Z}}_p
$$
in the derived category of either ind-\'etale or fpqc
abelian sheaves on $\mathcal{J}_S$.
\begin{proof}[Proof of Proposition \ref{Pr-BT(PPP)}]
This is essentially proved in \cite{Lau-Dual},
but we repeat the arguments for completeness and
because there is a small modification when $p=2$.
To begin with, $p$-divisible groups over $R$ form
a full subcategory of the abelian presheaves on $\mathcal{J}_S$
because finite group schemes over $R$
lie in $\mathcal{J}_S$; see Lemma \ref{Le-JJJR}.
Hence $\BT_R$ is a well-defined additive functor if the
assertions on the cohomology of $Z'(\mathscr{P})$ hold.
Since an exact sequence of Dieudonn\'e displays over $R$
induces an exact sequence of the associated complexes
of presheaves $Z'$, the functor $\BT_R$ is exact
if it is defined.
The components of $Z(\mathscr{P})$ and $Z'(\mathscr{P})$ are
fpqc sheaves by Lemma \ref{Le-WW-sheaf}.
These complexes carry two independent filtrations.
First, a Dieudonn\'e display is called \'etale if
$Q=P$, and nilpotent if $V^\sharp$
is topologically nilpotent. Every Dieudonn\'e
display over $R$ is naturally an extension of an
\'etale by a nilpotent Dieudonn\'e display,
which induces exact sequences of the associated
complexes $Z(\ldots)$ and $Z'(\ldots)$. Thus
we may assume that $\mathscr{P}$ is \'etale or nilpotent.
Second, for every $\mathscr{P}$
we have an exact sequence of complexes of presheaves
$$
0\to\hat Z(\mathscr{P})\to Z(\mathscr{P})\to Z_{\red}(\mathscr{P})\to 0,
$$
defined by $Z_{\red}(\mathscr{P})(\Spec A)=Z(\mathscr{P})(\Spec A_{\red})$.
The same holds for $Z'$ instead of $Z$.
We write $\hat Z(\mathscr{P})=[\hat Q\to\hat P]$.
Assume that $\mathscr{P}$ is \'etale. Then $F_1:P\to P$ is an $f$-linear
isomorphism. Thus $F_1:\hat P\to\hat P$ is elementwise nilpotent,
and the complex $\hat Z(\mathscr{P})$ is acyclic.
It follows that $Z(\mathscr{P})$ is quasi-isomorphic to the complex
$Z_{\red}(\mathscr{P})=Z_{\red}$,
which is the projective limit of the complexes
$Z_{\red,n}=Z_{\red}/p^nZ_{\red}$.
In the \'etale topology, each $Z_{\red,n}$ is a
surjective homomorphism of
sheaves whose kernel is a locally constant sheaf
$G_n$ of free ${\mathbb{Z}}/p^n{\mathbb{Z}}$-modules of rank equal to the
rank of $P$. The system $(G_n)_n$ defines an \'etale
$p$-divisible group $G$ over $R$, and $Z_{\red}$
is quasi-isomorphic to $T_pG=\varprojlim G_n$ as ind-\'etale
sheaves. It follows that
$Z'(\mathscr{P})\simeq Z'_{\red}(\mathscr{P})$ is quasi-isomorphic to
$[T_pG\to T_pG\otimes{\mathbb{Z}}[1/p]]$ in degrees $-1,0$,
which is quasi-isomorphic to $G$.
Assume that $\mathscr{P}$ is nilpotent. Then the complex
$Z_{\red}(\mathscr{P})$ is acyclic because its value over
$\Spec A$ is isomorphic to
$[1-V:P_{A_{\red}}\to P_{A_{\red}}]$ where $V$ is a
topologically nilpotent $f^{-1}$-linear homomorphism.
Thus $Z(\mathscr{P})$ is quasi-isomorphic to $\hat Z(P)$.
To $\mathscr{P}$ we associate a nilpotent display by the
$u_0$-homomorphism of frames $\mathscr{D}_R\to\mathscr{W}_R$.
By \cite[Theorem 81 and Corollary 89]{Zink-Disp}
there is a formal $p$-divisible group $G$ over $R$
associated to this display such that for each
$A\in\mathcal{J}_R$ there is an exact sequence
$$
0\to\hat Q(A)\xrightarrow{u_0F_1-1}\hat P(A)\to G(A)\to 0.
$$
Since $u_0\in W({\mathbb{Z}}_p)$ maps to $1$ in $W({\mathbb{F}}_p)$,
there is a unique $c\in W({\mathbb{Z}}_p)$ which maps to $1$
in $W({\mathbb{F}}_p)$ such that $u_0=cf(c^{-1})$,
namely $c=u_0\sigma(u_0)\sigma^2(u_0)\cdots$.
Then multiplication by
$c$ in both components defines an isomorphism
$$
[\hat Q(A)\xrightarrow{F_1-1}\hat P(A)]
\cong[\hat Q(A)\xrightarrow{u_0F_1-1}\hat P(A)]
$$
It follows that $Z'(\mathscr{P})\simeq\hat Z'(\mathscr{P})$
is quasi-isomorphic to $G$ in degree zero.
\end{proof}
\begin{Remark}
Recall that $\mathscr{D}_R=({\mathbb{W}}(R),{\mathbb{I}}_R,f,{\mathbbm{f}}_1)$ is viewed
as a Dieudonn\'e display over $R$.
We have $\BT_R(\mathscr{D}_R)=\mu_{p^{\infty}}$ by \cite[(211)]{Zink-Disp}.
\end{Remark}
\begin{Lemma}
\label{Le-BT-change}
Let $R\to R'$ be a homomorphism of admissible local rings.
For each Dieudonn\'e display $\mathscr{P}$ over $R$ there is a
natural isomorphism
$$
\BT_R(\mathscr{P})_{R'}\cong\BT_{R'}(\mathscr{P}_{R'}).
$$
\end{Lemma}
\begin{proof}
If the residue field of $R'$ is an algebraic extension of $k$,
every ring in $\mathcal{J}_{R'}$ lies in $\mathcal{J}_R$, and the assertion
follows directly from the construction of $\BT_R$. In general,
let $\mathscr{E}_R$ be the category of all $R$-algebras which are
admissible rings,
and let $\mathscr{E}_S$ be the category of affine $S$-schemes $\Spec A$
with $A\in\mathscr{E}_R$, endowed with the ind-\'etale topology.
The complexes of presheaves $Z(\mathscr{P})$ and $Z'(\mathscr{P})$
on $\mathcal{J}_S$ defined in \eqref{Eq-ZPPP} and \eqref{Eq-ZprimePPP}
extend to complexes of presheaves on $\mathscr{E}_S$ defined by the
same formulas. It is easy to see that
for an ind-\'etale ring homomorphism $A\to B$ with $A\in\mathscr{E}_R$
we have $B\in\mathscr{E}_R$ as well; see Lemma \ref{Le-JJJR}.
Using this, the proof of Proposition \ref{Pr-BT(PPP)}
shows that the ind-\'etale cohomology sheaves of $Z'(\mathscr{P})$ on
$\mathscr{E}_R$ vanish outside degree zero, and $H^0(Z'(\mathscr{P}))$
is naturally isomorphic with $\BT_R(\mathscr{P})$ as a sheaf on
$\mathscr{E}_R$. Since every ring in $\mathscr{E}_{R'}$ lies in $\mathscr{E}_R$,
the lemma follows as in the first case.
\end{proof}
\subsection{Comments}
\label{Subse-Comments}
At this point there are different ways to proceed.
The results and methods of \cite{Zink-DDisp} and \cite{Lau-Dual}
show that the functor $\BT_R$ is an equivalence of categories.
But one can also directly relate the
crystals associated to $\mathscr{P}$ and $\BT_R(\mathscr{P})$, which
gives another proof of the equivalence.
Before doing so we will discuss in more detail what is known.
\begin{Prop}
\label{Pr-BT-equiv}
The functor $\BT_R$ is an equivalence of exact categories.
If $p\ge 3$ or $pR=0$, then $\BT_R$ is a quasi-inverse of the
functor $\Phi_R$ defined in Proposition \ref{Pr-bt2disp}.
\end{Prop}
Recall that $\Phi_R$ is an equivalence by
Theorem \ref{Th-Phi-equiv}.
If $p=2$ and $R$ is not annihilated by $p$,
the relation between
$\BT_R$ and the functor $\Phi^+_R$ defined in
Proposition \ref{Pr-bt2disp+}
is explained in Corollary \ref{Co-BT-equiv} below.
\begin{proof}
By section \ref{Subse-finiteness} we may assume that $R$
is a local Artin ring.
Since $p$-divisible groups and Diedonn\'e displays over
$k$ have universal deformation rings which are power series
rings over $W(k)$, once the functor $\BT_R$ is defined,
in order to show that it is an equivalence of categories
it suffices so consider $R=k$ and $R=k[\varepsilon]$.
In particular, if $p=2$, we may assume that $pR=0$,
so that the
results of \cite{Zink-DDisp} and \cite{Lau-Dual} can
be applied. The category $\mathcal{C}_R$ used in \cite{Lau-Dual}
is the category of all $A\in\mathcal{J}_R$ such that $\mathcal{N}_A$
is nilpotent. Since this subcategory is stable under
ind-\'etale extensions, it does not make a difference
whether the functor $\BT_R$ is defined in terms of
$\mathcal{C}_R$ or $\mathcal{J}_R$. Thus $\BT_R$ is an equivalence
by \cite[Theorem 1.7]{Lau-Dual},
which relies on the equivalence proved in \cite{Zink-DDisp}.
Assume that $p\ge 3$ or $pR=0$.
It is easily verified that $\BT_R(\Phi_R({\mathbb{Q}}_p/{\mathbb{Z}}_p))$
is isomorphic to ${\mathbb{Q}}_p/{\mathbb{Z}}_p$. Thus $\BT_R$ is a quasi-inverse
of $\Phi_R$ by Lemmas \ref{Le-Aut-pdiv} and \ref{Le-BT-change}.
This proves also that, for general $R$,
the inverse of $\BT_R$ preserves exact sequences, using that
a sequence of Dieudonn\'e displays over $R$ is exact if and
only if it is exact over $R/pR$.
\end{proof}
The Serre dual of a $p$-divisible group $G$ is denoted by $G^\vee$.
\begin{Prop}
\label{Pr-BT-dual}
For every Dieudonn\'e display $\mathscr{P}$ over $R$ there
is a natural duality isomorphism $\BT_R(\mathscr{P}^t)\cong\BT_R(\mathscr{P})^\vee$.
\end{Prop}
If $p\ge 3$ or $pR=0$ this is proved in \cite{Lau-Dual}
by constructing explicitly a biextension between
$\BT_R(\mathscr{P})\times\BT_R(\mathscr{P}^t)$ and $\BT_R(\mathscr{D}_R)=\mu_{p^\infty}$.
This works for $p=2$ without restriction,
but the following argument is much easier.
The Lie version of the construction
of the duality isomorphism is used in the next section for
a different purpose, so the idea of the
construction survives.
\begin{proof}[Proof of Proposition \ref{Pr-BT-dual}]
We can assume that $R$ is a local Artin ring
with perfect residue field.
Since $\BT_R$ is an equivalence of exact categories
which is compatible with base change,
the assignment $\alpha:G\mapsto\BT(\BT^{-1}(G)^t)^\vee$
is an exact automorphism of the fibered category $(p\text{-}{\divv})$
over $({\Art})$ considered in section \ref{Subse-fib-pdiv}.
It is easily verified that $\alpha({\mathbb{Q}}_p/{\mathbb{Z}}_p)$ is
isomorphic to ${\mathbb{Q}}_p/{\mathbb{Z}}_p$. Then the proposition
follows from Lemma \ref{Le-Aut-pdiv}.
\end{proof}
\subsection{Passing to the limit}
Assume that $R$ is an admissible topological ring such
that $R_{\red}$ is a field, for example a complete local
ring with perfect residue field.
Then the functors $\BT_{R/N}$, where $N$ runs through
the open ideals of $R$ contained in $\mathcal{N}_R$, induce
a functor
$$
\BT_R:(\text{Dieudonn\'e displays over $R$})
\to(\text{$p$-divisible groups over $R$})
$$
because both sides are the projective limit of
the corresponding categories over $R/N$.
The assertions of Propositions \ref{Pr-BT-equiv}
and \ref{Pr-BT-dual} carry over to the topological case.
\section{Relation of the crystals}
In this section we relate the Dieudonn\'e crystals of a
Dieudonn\'e display over an admissible local ring
and of the associated $p$-divisible
group, which is defined as a crystalline
$\uExt^1$ following \cite{BBM}.
The main point is to write down a homomorphism
between the crystals. Initially we have to verify that
$\uExt^1$ can be computed in the smaller
crystalline site formed by spectra of admissible rings
in the category $\mathcal{J}_R$.
The results of this section are not used in section \ref{Se-Breuil}.
\subsection{The Dieudonn\'e crystal over admissible rings}
Let $\Sigma$ be a scheme equipped with a divided power
ideal $(I,\gamma)$. Let $S$ be a scheme over $\Sigma$
such that $p$ is locally nilpotent in $S$ and the
divided powers $\gamma$ extend to $S$. We recall that
the large crystalline site $\CRIS(S/\Sigma)$ used
in \cite{BBM} consists of
$S$-schemes $U$ together with a divided power extension
of $\Sigma$-schemes $U\to T$ compatible with $\gamma$
such that $p$ is locally nilpotent on $T$.
We fix a perfect field $k$ of characteristic $p$.
Let $\mathcal{J}=\mathcal{J}_{W(k)}$ be the category of $W(k)$-algebras
$A$ which are admissible such that $A_{\red}$ is a union
of finite dimensional $k$-algebras. We assume that
$S=\Spec R$ with $R\in\mathcal{J}$ and denote by
$\JCRIS(S/\Sigma)$ the category of all $U\to T$
in $\CRIS(S/\Sigma)$ such that $U=\Spec A$ with $A\in\mathcal{J}$.
Then $T=\Spec B$ with $B\in\mathcal{J}$ as well.
Let $\tau$ be one of the three topologies Zariski,
ind-\'etale, or fpqc. We denote the restriction functor of
$\tau$-sheaves from $\CRIS(S/\Sigma)$ to $\JCRIS(S/\Sigma)$
by
$$
\res_\tau:(S/\Sigma)_{\CRIS,\tau}\to(S/\Sigma)_{\JCRIS,\tau}.
$$
This is the direct image functor of a morphism of topoi.
If $\tau$ is Zariski or ind-\'etale, all $\tau$-coverings
in $\CRIS(S/\Sigma)$ of an object of $\JCRIS(S/\Sigma)$
lie in $\JCRIS(S/\Sigma)$. In these cases $\res_\tau$ is
also the inverse image functor of a morphism of topoi
in the opposite direction.
Let $\mathcal{O}_{S/\Sigma,\mathcal{J}}=\res_\tau(\mathcal{O}_{S/\Sigma})$.
Let $S_{\gamma,\tau}$ be the category of $\tau$-sheaves on
the category of $S$-schemes to which the divided powers
$\gamma$ extend and let $S_{\mathcal{J},\gamma,\tau}$
be the category of $\tau$-sheaves on the category
of $S$-schemes to which the divided powers
$\gamma$ extend of the type $\Spec A$ for $A\in\mathcal{J}_R$.
As before, the restriction $\res_\tau:S_{\gamma,\tau}\to
S_{\mathcal{J},\gamma,\tau}$
is the direct image functor of a morphism of topoi, and if
$\tau$ is Zariski or ind-\'etale, it is also
the inverse image functor of a morphism of topoi.
In addition we have morphisms of topoi
$$
i_{S/\Sigma,\tau}:S_{\gamma,\tau}\to(S/\Sigma)_{\CRIS,\tau},
$$
$$
i_{S/\Sigma,\mathcal{J},\tau}:S_{\mathcal{J},\gamma,\tau}\to(S/\Sigma)_{\JCRIS,\tau},
$$
defined by $i_*(G)(U\to T)=G(U)$.
We write $i_*(G)=\underline G$.
If $\tau$ is Zariski or ind-\'etale, $i_*$
is also the inverse image functor of a morphism
of topoi in the opposite direction.
\begin{Lemma}
\label{Le-CRIS-JCRIS}
Let $G$ be a finite flat group scheme or a $p$-divisible
group over $S$. For $i\le 2$ we have a natural isomorphism
of $\mathcal{O}_{S/\Sigma,\mathcal{J}}$-modules
\begin{equation}
\label{Eq-CRIS-JCRIS}
\res_\tau\uExt^i_{S/\Sigma,\tau}(\underline G,\mathcal{O}_{S/\Sigma})
\cong\uExt^i_{S/\Sigma,\mathcal{J},\tau}(\underline G,\mathcal{O}_{S/\Sigma,\mathcal{J}}).
\end{equation}
These modules are independent of the choice of $\tau$
as presheaves.
\end{Lemma}
\begin{proof}
This is a routine verification using the results of \cite{BBM}.
Assume first that $G$ is finite.
By \cite[1.3.6]{BBM}, for all $i\ge 0$ the
left hand side of \eqref{Eq-CRIS-JCRIS} is $\res_{\tau}$
of the $\tau$-sheaf associated to the Zariski sheaf
$$
\underline E^i=\uExt^i_{S/\Sigma,\Zar}
(\underline G,\mathcal{O}_{S/\Sigma}).
$$
The proof of loc.\ cit.\ shows that the right hand side
is the $\tau$-sheaf associated to the Zariski sheaf
$\uExt^i_{S/\Sigma,\mathcal{J},\Zar}
(\underline G,\mathcal{O}_{S/\Sigma,\mathcal{J}})$.
By \cite[2.3.11]{BBM},
$\underline E^i$ is already a $\tau$-sheaf if $i\le 2$.
Thus it suffices to prove that the
functor $\res_{\Zar}$ induces an isomorphism
\eqref{Eq-CRIS-JCRIS} if $i\le 2$.
By \cite[2.1.5]{BBM}, there is an exact sequence
in $S_{\gamma,\Zar}$
$$
C^{(3)}\to C^{(2)}\to C^{(1)}\to C^{(0)}\to G\to 0
$$
where each $C^{(n)}$ is a finite direct sum of sheaves
of the type ${\mathbb{Z}}[G^m]$. We break it into four short
exact sequences $0\to G^{(1)}\to C^{(0)}\to G\to 0$
and $0\to G^{(2)}\to C^{(1)}\to G^{(1)}\to 0$ and
$0\to G^{(3)}\to C^{(2)}\to G^{(2)}\to 0$ and
$0\to G^{(4)}\to C^{(3)}\to G^{(3)}\to 0$. These
sequences remain exact under $i_*$,
and the restriction $\res_{\Zar}$ induces
homomorphisms of the associated long exact sequences
$\res\uExt^i_{S/\Sigma,\Zar}(\ldots,\mathcal{O}_{S/\Sigma})\to
\uExt^i_{S/\Sigma,\mathcal{J},\Zar}(\ldots,\mathcal{O}_{S/\Sigma,\mathcal{J}})$.
We claim that the involved homomorphisms
\begin{equation}
\label{Eq-involved-hom}
\res\uExt^i_{S/\Sigma,\Zar}(\underline C^{(n)},\mathcal{O}_{S/\Sigma})
\to\uExt^i_{S/\Sigma,\mathcal{J},\Zar}
(\underline C^{(n)},\mathcal{O}_{S/\Sigma,\mathcal{J}})
\end{equation}
are isomorphisms for $n\le 3$ and all $i$.
Indeed, let $X=G^m$ and let $U\to T$ be an object of
$\JCRIS(S/\Sigma)$. Since $G$ is finite over $R$,
$X$ is the spectrum of a ring in $\mathcal{J}$, and
$\JCRIS(X_U/T)$ is defined; see Lemma \ref{Le-JJJR}.
By \cite[1.3.4]{BBM} and its proof we have isomorphisms
\begin{gather}
\label{Eq-Ext-Hi}
\Ext^i_{U/T,\Zar}({\mathbb{Z}}[\underline{X_U}],\mathcal{O}_{U/T})
\cong H^i((X_U/T)_{\Zar},\mathcal{O}_{X_U/T}), \\
\label{Eq-Ext-Hi-JJJ}
\Ext^i_{U/T,\mathcal{J},\Zar}({\mathbb{Z}}[\underline{X_U}],\mathcal{O}_{U/T,\mathcal{J}})
\cong
H^i((X_U/T)_{\mathcal{J},\Zar},\mathcal{O}_{X_U/T,\mathcal{J}}).
\end{gather}
Let $(X_U/T)_{\zar}$ be the category of Zariski sheaves
on the category of all $U'\to T'$ in $\CRIS(X_U/T)$ such
that $U'$ is an affine open subscheme of $X_U$; this
implies that $U'\to T'$ lies in $\JCRIS(X_U/T)$ as well.
The cohomology groups on the right in \eqref{Eq-Ext-Hi}
and \eqref{Eq-Ext-Hi-JJJ} both coincide with
$H^i((X_U/T)_{\zar}, \mathcal{O}_{X_U/T})$, and hence
\eqref{Eq-involved-hom} is an isomorphism
for $n\le 3$ and all $i$ as claimed.
By the $5$-lemma it follows that $\res_{Zar}$
induces an isomorphism \eqref{Eq-CRIS-JCRIS} for $i\le 2$.
This proves Lemma \ref{Le-CRIS-JCRIS} if $G$ is finite.
Assume now that $G$ is a $p$-divisible group.
For $i\le 2$, the
left hand side of \eqref{Eq-CRIS-JCRIS} is equal to
$$
\res_\tau\varprojlim_n\uExt^i_{S/\Sigma,\tau}
(\underline G[p^n],\mathcal{O}_{S/\Sigma})
$$
according to \cite[2.4.5]{BBM}.
Since $\res_\tau$ commutes with projective limits,
the $p$-divisible case of \eqref{Eq-CRIS-JCRIS} is
reduced to the finite case if the right hand side of
\eqref{Eq-CRIS-JCRIS} is equal to
$\varprojlim_n\uExt^i_{S/\Sigma,\mathcal{J},\tau}
(\underline G[p^n],\mathcal{O}_{S/\Sigma,\mathcal{J}})$.
This follows from the proof of \cite[2.4.5]{BBM}.
Indeed, the properties of the sheaves
$\uExt^i_{S/\Sigma,\tau}(\underline G[p^n],\mathcal{O}_{S/\Sigma})$
used in the proof of \cite[2.4.4]{BBM}
carry over from $\CRIS(S/\Sigma)$ to $\JCRIS(S/\Sigma)$
by the finite case of \eqref{Eq-CRIS-JCRIS}.
The proof of \cite[2.4.3]{BBM} uses that the sequences
\begin{equation}
\label{Eq-1.1.7}
0\to\underline G[p^n]\to\underline G
\xrightarrow{p^n}\underline G\to 0
\end{equation}
are exact in $(S/\Sigma)_{\CRIS,\fpqc}$ by \cite[1.1.7]{BBM}.
Since for $A\in\mathcal{J}_R$ and for a quasi-finite ring
homomorphism $A\to B$ we have $B\in\mathcal{J}_R$
by Lemma \ref{Le-JJJR},
the proof of \cite[1.1.7]{BBM} shows that \eqref{Eq-1.1.7}
is exact in $(S/\Sigma)_{\JCRIS,\fpqc}$ as well.
The remaining arguments relevant for the proof of
\cite[2.4.5]{BBM} carry over from $\CRIS(S/\Sigma)$
to $\JCRIS(S/\Sigma)$ easily.
\end{proof}
\begin{Lemma}
Let $G$ be a finite flat group scheme or a $p$-divisible
group over $S$. For $i\le 2$ we have a natural isomorphisms
of $\mathcal{O}_{S/\Sigma,\mathcal{J}}$-modules
\begin{equation}
\label{Eq-S-SJ}
\res_\tau\uExt^i_{S_{\gamma,\tau}}(G,{\mathbb{G}}_a)
\cong\uExt^i_{S_{\mathcal{J},\gamma,\tau}}(G,{\mathbb{G}}_a),
\end{equation}
\begin{equation}
\label{Eq-SJ-JCRIS}
i_{S/\Sigma,\mathcal{J}*}\uExt^i_{S_{\mathcal{J},\gamma,\tau}}(G,{\mathbb{G}}_a)\cong
\uExt^i_{S/\Sigma,\mathcal{J},\tau}(\underline G,\underline{\mathbb{G}}_a),\end{equation}
and these modules are independent of $\tau$ as presheaves.
\end{Lemma}
Note that \eqref{Eq-SJ-JCRIS} without $\mathcal{J}$ holds as well
by \cite[2.3.12 and 2.4.6]{BBM}.
\begin{proof}
The proof of \eqref{Eq-S-SJ} and of the fact that the
involved modules are independent of $\tau$ is an elementary
analogue of the proof of Lemma \ref{Le-CRIS-JCRIS};
see the discussion preceding \cite[2.3.12]{BBM},
and \cite[2.4.6]{BBM}. Then \eqref{Eq-SJ-JCRIS} follows from
the arguments given in \cite[2.3.12 and 2.4.6]{BBM}.
\end{proof}
We recall that if $p^nR=0$, for a $p$-divisible group $G$ over $R$
the exact sequence
$0\to G[p^n]\to G\xrightarrow{p^n}G\to 0$ on $S_{\gamma,\fpqc}$
induces an isomorphism
$$
\uLie(G^\vee)\cong\uHom_{S_\gamma}(G[p^n],{\mathbb{G}}_a)
\cong\uExt^1_{S_{\gamma,\fpqc}}(G,{\mathbb{G}}_a).
$$
The same holds for $S_{\mathcal{J},\gamma,\fpqc}$ instead of
$S_{\gamma,\fpqc}$.
\subsection{Relation of the crystals}
Let now $S=\Spec R$ for an admissible local ring $R$ with
perfect residue field $k$. We define $\mathcal{J}=\mathcal{J}_{W(k)}$ as before.
Let $\mathscr{P}$ be a Dieudonn\'e display over $R$ and
let $G^t=\BT_R(\mathscr{P}^t)$. We want to construct a homomorphism
of $\mathcal{O}_{S/\Sigma,\mathcal{J}}$-modules
$$
\beta(\mathscr{P}):{\mathbb{D}}(\mathscr{P})\to\uExt^1_{S/\Sigma,\mathcal{J}}
(\underline G^t,\mathcal{O}_{S/\Sigma,\mathcal{J}})={\mathbb{D}}(G^{t\vee})
$$
(see footnote \ref{Ft-DD-covariant}
on page \pageref{Ft-DD-covariant})
which is compatible with a homomorphism of $R$-modules
$$
\beta^0(\mathscr{P}):\Lie(\mathscr{P})\to\Lie(G^{t\vee}).
$$
The construction of $\beta^0(\mathscr{P})$ is the Lie-version
of the construction of the duality homomorphism in
\cite{Lau-Dual}, and the construction of $\beta(\mathscr{P})$
is a crystalline extension of this construction.
Let us fix some notation.
For given $U\to T$ in $\JCRIS(S/\Sigma)$
we write $U=\Spec A$ and $T=\Spec B$
and $\mathfrak{a}=\Ker(B\to A)$.
Let
$$
\mathscr{P}_{B/A}=(P_{B/A},Q_{B/A},F,F_1)
$$
be the unique lift of $\mathscr{P}_A$ to a Dieudonn\'e display
for $B/A$ (Proposition \ref{Pr-DDD-crys})
and let $\mathscr{P}_\mathfrak{a}$ be the kernel of the
projection $\mathscr{P}_{B/A}\to\mathscr{P}_A$. Thus
$$
\mathscr{P}_{\mathfrak{a}}=(P_{\mathfrak{a}},P_{\mathfrak{a}},F,F_1)
$$
with $P_\mathfrak{a}=P_{B/A}\otimes_{{\mathbb{W}}(B/A)}\tilde W(\mathfrak{a})$,
and $F_1(x\otimes a)=F(x)\otimes{\mathbbm{f}}_1(a)$.
Consider the surjection $B[\varepsilon]\to A$.
Its kernel $\mathfrak{a}\oplus\varepsilon B$ carries divided
powers which extend the given divided powers on
$\mathfrak{a}\cdot B[\varepsilon]$ by the trivial divided powers on
$\varepsilon B$. Let $\mathscr{P}_{\varepsilon B/A}$ be the
kernel of the projection
$\mathscr{P}_{B[\varepsilon]/A}\to\mathscr{P}_{B/A}$. Thus
$$
\mathscr{P}_{\varepsilon B/A}=(P_{\varepsilon B},P_{\varepsilon B},F,F_1)
$$
with $P_{\varepsilon B}=P_{B/A}\otimes_{{\mathbb{W}}(B/A)}
\hat W(\varepsilon B)$, and
$F_1(x\otimes a)=F(x)\otimes{\mathbbm{f}}_1(a)$.
Let
$$
IP_{\varepsilon B}=P_{B/A}\otimes_{{\mathbb{W}}(B/A)}v\hat W(\varepsilon B).
$$
In the case $A=B$ we also consider $\mathscr{P}_{A[\varepsilon]}$,
the scalar extension of $\mathscr{P}$ under $A\to A[\varepsilon]$.
Let $\mathscr{P}_{\varepsilon A}$ be the kernel
of the projection $\mathscr{P}_{A[\varepsilon]}\to\mathscr{P}_A$. Thus
$$
\mathscr{P}_{\varepsilon A}=(P_{\varepsilon A},Q_{\varepsilon A},F,F_1)
$$
where $P_{\varepsilon A}/Q_{\varepsilon A}=P/Q\otimes_R\varepsilon A$.
\begin{Defn}
We define complexes of abelian groups in
$(S/\Sigma)_{\mathcal{J},\indet}$ in degrees $0,1$ by
\begin{align*}
\underline Z(\mathscr{P})(U\to T)
&{}=[Q_{A}\xrightarrow{F_1-1}P_{A}] \\
Z_{S/\Sigma}(\mathscr{P})(U\to T)
&{}=[Q_{B/A}\xrightarrow{F_1-1}P_{B/A}] \\
Y_{S/\Sigma}(\mathscr{P})_\varepsilon(U\to T)
&{}=[IP_{\varepsilon B}\xrightarrow{F_1-1}P_{\varepsilon B}]
\end{align*}
and complexes of abelian groups in
$(S/\Sigma)_{\mathcal{J},\indet}$ in degrees $-1,0,1$ by
\begin{align*}
\underline Z'(\mathscr{P})(U\to T)
&{}=[Q_{A}\xrightarrow{F_1-1}P_{A}]\otimes[{\mathbb{Z}}\to{\mathbb{Z}}[1/p]] \\
Z'_{S/\Sigma}(\mathscr{P})(U\to T)
&{}=[Q_{B/A}\xrightarrow{F_1-1}P_{B/A}]
\otimes[{\mathbb{Z}}\to{\mathbb{Z}}[1/p]]
\end{align*}
and a complex of abelian groups in $S_{\mathcal{J},\gamma,\indet}$
in degrees $0,1$ by
$$
Z(\mathscr{P})_\varepsilon(\Spec A)=
[Q_{\varepsilon A}\xrightarrow{F_1-1}P_{\varepsilon A}].
$$
\end{Defn}
\begin{Lemma}
\label{Le-complexes-DD}
Let $G=\BT_R(\mathscr{P})$.
Then:
\begin{enumerate}
\renewcommand{\theenumi}{\roman{enumi}}
\item
\label{It-complexes-DD-1}
The projection $Z_{S/\Sigma}(\mathscr{P})\to\underline Z(\mathscr{P})$
is a quasi-isomorphism.
\item
\label{It-complexes-DD-2}
The projection $Z'_{S/\Sigma}(\mathscr{P})\to\underline Z'(\mathscr{P})$
is a quasi-isomorphism.
\item
\label{It-complexes-DD-3}
The complex $\underline Z'(\mathscr{P})$ is quasi-isomorphic to
$\underline G$ in degree zero.
\item
\label{It-complexes-DD-4}
The complex $Y_{S/\Sigma}(\mathscr{P})_\varepsilon$
is quasi-isomorphic to ${\mathbb{D}}(\mathscr{P})$ in degree one.
\item
\label{It-complexes-DD-5}
The complex $Z(\mathscr{P})_\varepsilon$ is quasi-isomorphic to
the quasi-coherent module $\Lie(\mathscr{P})=P/Q$ in degree one.
\end{enumerate}
\end{Lemma}
\begin{proof}
The projection \eqref{It-complexes-DD-1} is surjective
on the level of presheaves with kernel equal to
$[F_1-1:P_\mathfrak{a}\to P_\mathfrak{a}]$. Since ${\mathbbm{f}}_1$ is elementwise
nilpotent on $\tilde W(\mathfrak{a})$, here $F_1-1$ is
bijective. This proves \eqref{It-complexes-DD-1}
and \eqref{It-complexes-DD-2}.
Similarly, the endomorphism $F_1-1$ of $P_{\varepsilon B}$
is bijective; thus $Y_{S/\Sigma}(\mathscr{P})_\varepsilon(U\to T)$
is isomorphic to the inclusion
$[IP_{\varepsilon B}\to P_{\varepsilon B}]$.
Its cokernel is isomorphic to
$P_{B/A}\otimes_{{\mathbb{W}}(B/A)}B={\mathbb{D}}(\mathscr{P})_{B/A}$,
which proves \eqref{It-complexes-DD-4}.
The proof of \eqref{It-complexes-DD-5} is similar.
Since the functor $(i_{S/\Sigma,\mathcal{J},\indet})_*$
is exact, \eqref{It-complexes-DD-3} follows from the
definition of $G=\BT_R(\mathscr{P})$ in Proposition \ref{Pr-BT(PPP)}.
\end{proof}
\begin{Constr}
\label{Const-beta}
We define $\beta(\mathscr{P})$.
Let us write $\mathscr{P}^t=(P',Q',F',F_1')$.
We start with the tautological bilinear form
$$
\alpha:\mathscr{P}\times\mathscr{P}^t\to\mathscr{D}_R.
$$
This is a perfect bilinear form
$\alpha:P\times P'\to{\mathbb{W}}(R)$ such that
$\alpha(F_1x,F_1'x')={\mathbbm{f}}_1\alpha(x,x'))$
for $x\in Q$ and $x'\in Q'$.
Since for $B/A$ as above the base change under
$\mathscr{D}_{B/A}\to\mathscr{D}_A$
is compatible with duality, we have also a tautological
bilinear form
$$
\alpha_{B/A}:\mathscr{P}_{B/A}\times\mathscr{P}^t_{B/A}\to\mathscr{D}_{B/A}.
$$
The restriction of $\alpha_{B[\varepsilon]/A}$ induces
a bilinear form
$$
\alpha_{\varepsilon B/A}:\mathscr{P}_{\varepsilon B/A}
\times\mathscr{P}^t_{B[\varepsilon]/A}\to\mathscr{D}_{\varepsilon B/A},
$$
which vanishes on $P_{\varepsilon B}\times
P^t_{\varepsilon B}$ and thus induces a bilinear form
$$
\bar\alpha_{\varepsilon B/A}:\mathscr{P}_{\varepsilon B/A}
\times\mathscr{P}^t_{B/A}\to\mathscr{D}_{\varepsilon B/A}.
$$
Explicitly this is a ${\mathbb{W}}(B/A)$-bilinear map
$$
\bar\alpha:P_{\varepsilon B}\times P'_{B/A}\to\hat W(\varepsilon B)
$$
such that $\bar\alpha(F_1y,F_1'y')={\mathbbm{f}}_1\alpha(y,y')$
for $y\in P_{\varepsilon B}$ and
$y'\in Q'_{B/A}$. Since $\bar\alpha$ comes from
the bilinear map $\alpha_{B[\varepsilon]/A}$,
it maps $IP_{\varepsilon B}\times P'_{B/A}$
into $v\hat W(\varepsilon B)$.
As in \cite[Sec.~3]{Lau-Dual} we use $\bar\alpha$ to define a
homomorphism of complexes
$$
\gamma:Y_{S/\Sigma}(\mathscr{P})_\varepsilon\otimes
Z_{S/\Sigma}(\mathscr{P}^t)\to
Y_{S/\Sigma}(\mathscr{D}_R)_\varepsilon
$$
which is well-defined up to homotopy.
It consists of a map in degree zero:
$$
\gamma_0:
IP_{\varepsilon B}\otimes Q'_{B/A}
\to v\hat W(\varepsilon B)
$$
defined by $y\otimes y'\mapsto\bar\alpha(y,y')$,
and a map in degree one:
$$
\gamma_1:
(IP_{\varepsilon B}\otimes P'_{B/A})
\oplus (P_{\varepsilon B}\otimes Q'_{B/A})
\to \hat W(\varepsilon B)
$$
defined by $(y\otimes x')\oplus(x\otimes y')\mapsto
\bar\alpha(F_1y,x')+\bar\alpha(x,y')$ or
$\bar\alpha(y,x')+\bar\alpha(x,F_1'y')$.
It is easily verified that both choices of $\gamma$ are
homomorphisms of complexes that differ by a chain homotopy.
We compose $\gamma$ with the quasi-isomorphism
$Y_{S/\Sigma}(\mathscr{D}_R)_\varepsilon\to\mathcal{O}_{S/\Sigma,\mathcal{J}}[-1]$
from Lemma \ref{Le-complexes-DD} \eqref{It-complexes-DD-4}
and take the tensor product with $[{\mathbb{Z}}\to{\mathbb{Z}}[1/p]]$
placed in degrees $-1,0$. The resulting homomorphism
$$
Y_{S/\Sigma}(\mathscr{P})_\varepsilon\otimes
Z'_{S/\Sigma}(\mathscr{P}^t)\to\mathcal{O}_{S/\Sigma,\mathcal{J}}
$$
is adjoint to an arrow
$$
\gamma':Y_{S/\Sigma}(\mathscr{P})_\varepsilon\to
\uHom_{S/\Sigma,\mathcal{J}}(Z'_{S/\Sigma}(\mathscr{P}^t),\mathcal{O}_{S/\Sigma,\mathcal{J}}).
$$
Here $Z'_{S/\Sigma}(\mathscr{P}^t)$ is quasi-isomorphic
to $\underline G^t$ by Lemma \ref{Le-complexes-DD}
\eqref{It-complexes-DD-2} and \eqref{It-complexes-DD-3}.
We compose $\gamma'$ with $\uHom\to R\uHom$ and take the
cohomology sheaf in degree one.
By Lemma \ref{Le-complexes-DD} \eqref{It-complexes-DD-4},
this gives the desired homomorphism
$$
\beta(\mathscr{P}):{\mathbb{D}}(\mathscr{P})\to\uExt^1_{S/\Sigma,\mathcal{J},\indet}
(\underline G^t,\mathcal{O}_{S/\Sigma,\mathcal{J}})={\mathbb{D}}(G^{t\vee}).
$$
\end{Constr}
\begin{Constr}
The definition of $\beta^0(\mathscr{P})$ is similar.
The scalar extension of the bilinear form $\alpha$
under $A\to A[\varepsilon]$ is a bilinear form
$$
\alpha_{A[\varepsilon]}:\mathscr{P}_{A[\varepsilon]}\times
\mathscr{P}^t_{A[\varepsilon]}\to\mathscr{D}_{A[\varepsilon]}.
$$
Its restriction
$$
\alpha_{\varepsilon A}:\mathscr{P}_{\varepsilon A}\times
\mathscr{P}^t_{A[\varepsilon]}\to\mathscr{D}_{\varepsilon A}
$$
vanishes on $P_{\varepsilon A}\times P^t_{\varepsilon A}$
and thus induces a bilinear form
$$
\bar\alpha_{\varepsilon A}:\mathscr{P}_{\varepsilon A}\times
\mathscr{P}^t\to\mathscr{D}_{\varepsilon A}.
$$
We use it to define a homomorphism of complexes
$$
\bar\gamma:Z(\mathscr{P})_\varepsilon\otimes Z(\mathscr{P}^t)
\to Z(\mathscr{D}_R)_\varepsilon
$$
well-defined up to homotopy, such that the involved maps
$$
\bar\gamma_0:Q_{\varepsilon A}\otimes Q'_A\to v\hat W(\varepsilon A)
$$
and
$$
\bar\gamma_1:(Q_{\varepsilon A}\otimes P'_A)\oplus
(P_{\varepsilon A}\otimes Q'_A)\to\hat W(\varepsilon A)
$$
are defined by the same formulas as $\gamma_0$ and $\gamma_1$.
We compose $\bar\gamma$ with the quasi-isomorphism
$Z(\mathscr{D})_\varepsilon\to{\mathbb{G}}_a[-1]$ from
Lemma \ref{Le-complexes-DD} \eqref{It-complexes-DD-5}
and take the tensor product with $[{\mathbb{Z}}\to{\mathbb{Z}}[1/p]]$.
This gives the adjoint of an arrow
$$
\bar\gamma':Z(\mathscr{P})_\varepsilon\to
\uHom_{S_\mathcal{J}}(Z'(\mathscr{P}^t),{\mathbb{G}}_a).
$$
If we compose with $\uHom\to R\uHom$ and take $H^1$
we get a homomorphism
$$
\beta^0(\mathscr{P}):\uLie(\mathscr{P})\to\uExt^1_{S_{\mathcal{J},\indet}}(G^t,{\mathbb{G}}_a).
$$
\end{Constr}
\begin{Lemma}
\label{Le-beta-compat}
The hommorphism $\beta(\mathscr{P})$ and $\beta^0(\mathscr{P})$ are compatible.
\end{Lemma}
\begin{proof}
Let $i=i_{S/\Sigma,\mathcal{J},\indet}$.
The construction of $\beta^0(\mathscr{P})$ is preserved under
the functor $i_*$ because $i_*$ commutes with
$\uHom$ and because for a nil-immersion $U\to T$ the
ind-\'etale coverings of $U$ and of $T$ are the same.
There are obvious homomorphisms of complexes:
$$
Y_{S/\Sigma}(\mathscr{P})_\varepsilon\to i_*(Z(\mathscr{P})_\varepsilon),
$$
which induces the projection ${\mathbb{D}}(\mathscr{P})\to i_*(\Lie(\mathscr{P}))$
in the first cohomology;
$$
Z_{S/\Sigma}(\mathscr{P}^t)\to i_*(Z(\mathscr{P}^t))=\underline Z(\mathscr{P}^t),
$$
which is a quasi-isomorphism by Lemma \ref{Le-complexes-DD}
\eqref{It-complexes-DD-1}; and
$$
Y_{S/\Sigma}(\mathscr{D})_\varepsilon\to i_*(Z(\mathscr{D})_\varepsilon),
$$
which induces the projection $\mathcal{O}_{S/\Sigma}\to\underline{\mathbb{G}}{}_a$
in the first cohomology.
Since these homomorphisms are compatible with $\gamma$ and
$\bar\gamma$ the lemma follows easily.
\end{proof}
\begin{Lemma}
\label{Le-beta-0}
The homomorphism $\beta^0(\mathscr{P})$ is an isomorphism.
\end{Lemma}
This is the Lie version of \cite[Theorem 3.4]{Lau-Dual}.
Before proving it let us recall some facts from Cartier theory.
For an arbitrary ${\mathbb{Z}}_{(p)}$-algebra $R$ we consider functors
on the category of all $R$-algebras $A$. By \cite{Cartier-Dual},
formulated as in \cite[Proposition 117]{Zink-Disp},
the bilinear maps
$$
W(R)\times\hat W(A)\xrightarrow{\mult}\hat W(A)
\xrightarrow{\hex}\hat{\mathbb{G}}{}_m(A)
$$
induce an isomorphism $W(R)\cong\Hom(\hat W,{\mathbb{G}_m})$,
where $\hex$ identifies $\hat{\mathbb{G}}{}_m$ with the
cokernel of $1-v:\hat W\to\hat W$. By taking the
kernel of the base change under $R[\varepsilon]\to R$
we get an additive version: The bilinear map
\begin{equation}
\label{Eq-W-W-Ga}
W(\varepsilon R)\times\hat W(A)\xrightarrow{\mult}
\hat W(\varepsilon A)\xrightarrow{\Sigma}\varepsilon A\cong A,
\end{equation}
where $\varepsilon A$ is considered as an ideal in $A[\varepsilon]$
and where $\Sigma$ is the sum of the components,
induces an isomorphism $W(\varepsilon R)\cong\Hom(\hat W,{\mathbb{G}_a})$.
Next, $\Ext^1(\hat W,{\mathbb{G}_a})=\Ext^1(\hat W,\hat{\mathbb{G}}{}_a)$
vanishes by \cite[Proposition 114]{Zink-Disp}.
It follows that the exact sequence of fppf sheaves
$$
0\to W[f]\to\hat W\xrightarrow f\hat W\to 0,
$$
remains exact under $\Hom(\ldots,{\mathbb{G}_a})$.
Here the dual of $f$ is the endomorphism $v$ of $W(\varepsilon R)$,
and we get an isomorphism $\Hom(W[f],{\mathbb{G}_a})\cong R$.
\begin{proof}[Proof of Lemma \ref{Le-beta-0}]
Since $\beta^0(\mathscr{P})$ is a homomorphism of projective
$R$-mod\-ules which is compatible with base change,
we can assume that $R=k$ is an algebraically closed field.
Then we can assume that $\mathscr{P}$ is $F$-nilpotent or
$\mathscr{P}=\mathscr{D}_k$. Assume that $\mathscr{P}=\mathscr{D}_k$.
For $A\in\mathcal{J}_k$ we have
$$
Z(\mathscr{D}_k^t)(A)=[{\mathbb{W}}(A)\xrightarrow{f-1}{\mathbb{W}}(A)].
$$
The inclusion ${\mathbb{Z}}_p\to{\mathbb{W}}(A)$
is a quasi-isomorphism ${\mathbb{Z}}_p\to Z(\mathscr{D}_k^t)$, under which
the homomorphism $\bar\gamma$ corresponds to the identity
$Z(\mathscr{D}_k)_\varepsilon\otimes {\mathbb{Z}}_p\to Z(\mathscr{D}_k)_\varepsilon$.
By functoriality with respect to the quasi-isomorphism
$Z(\mathscr{D}_k)_\varepsilon\to{\mathbb{G}_a}[-1]$ we see that
$\beta^0(\mathscr{D}_k)$ can be computed starting with the identity
${\mathbb{G}_a}[-1]\otimes{\mathbb{Z}}_p\to{\mathbb{G}_a}[-1]$ instead of $\bar\gamma$.
The associated homomorphism
$$
\beta^0(\mathscr{D}_k):k=\Hom({\mathbb{Z}}_p,{\mathbb{G}_a})\to
\Ext^1({\mathbb{Q}}_p/{\mathbb{Z}}_p,{\mathbb{G}_a})
$$
is $\pm$ the connecting homomorphism of
$0\to{\mathbb{Z}}_p\to{\mathbb{Q}}_p\to{\mathbb{Q}}_p/{\mathbb{Z}}_p\to 0$,
which is an isomorphism because $\Ext^i({\mathbb{Q}}_p,{\mathbb{G}_a})$
vanishes for all $i$.
Assume now that $\mathscr{P}$ is $F$-nilpotent, so that $\mathscr{P}^t$
is nilpotent. First, as explained in the proof of Proposition
\ref{Pr-BT(PPP)}, the inclusion $\hat Z(\mathscr{P}^t)\to Z(\mathscr{P}^t)$
is a quasi-isomorhpism. Second, let $\tilde Z(\mathscr{P})_\varepsilon$
be the analogue of $Z(\mathscr{P})_\varepsilon$ with
$W$ instead of ${\mathbb{W}}$, thus
$$
\tilde Z(\mathscr{P})_\varepsilon(\Spec A)=
[\tilde Q_{\varepsilon A}\xrightarrow{F_1-1}\tilde P_{\varepsilon A}]
$$
with $\tilde P_{\varepsilon A}=P\otimes_{W(k)}W(\varepsilon A)$ and
$\tilde P_{\varepsilon A}/\tilde Q_{\varepsilon A}=P/Q\otimes_k\varepsilon A$.
Since $F:P\to P$ is topologically nilpotent and since the
$W(k)$-module $W(\varepsilon A)$ is annihilated by $p$,
the endomorphism $F_1$ of $\tilde P_{\varepsilon A}$ is
nilpotent, and Lemma \ref{Le-complexes-DD}
\eqref{It-complexes-DD-5} extends to a quasi-isomorphism
$\tilde Z(\mathscr{P})_\varepsilon\to\Lie(\mathscr{P})[-1]$. In view of
these remarks, in the definition of $\beta^0(\mathscr{P})$ the
homomorphism $\bar\gamma$ can be replaced by its analogue
$$
\tilde\gamma:\tilde Z(\mathscr{P})_\varepsilon
\otimes\hat Z(\mathscr{P}^t)\to Z(\mathscr{D}_k)_\varepsilon,
$$
which exists because the product $W(\varepsilon A)\cdot
\hat W(\mathcal{N}_A)$ lies in $\hat W(\varepsilon A)$.
Since the components of $\hat Z(\mathscr{P}^t)$ are $p$-torsion sheaves,
the tensor product with $[{\mathbb{Z}}\to{\mathbb{Z}}[1/p]]$ is just a shift by one.
Now the category $\mathcal{J}_k$ can be replaced by the category
of all $k$-algebras $A$ because all terms that appear in
$\tilde\gamma$ are naturally functors on the larger category.
The pairing \eqref{Eq-W-W-Ga} gives an isomorphism
$u:P_{\varepsilon k}\cong\Hom_k(\hat P',{\mathbb{G}_a})$.
Let $\iota:Q_{\varepsilon k}\to P_{\varepsilon k}$ be the inclusion.
We claim that there is the
following commutative diagram with exact rows,
where $g_1=u\circ\iota$ and $g_0=(F_1')^*\circ u$;
this is analogous to \cite[(3.8)]{Lau-Dual}.
$$
\xymatrix@M+0.2em{
0 \ar[r] &
Q_{\varepsilon k} \ar[r]^{F_1-1} \ar[d]^{g_1} &
P_{\varepsilon k} \ar[r] \ar[d]^{g_0} &
\Lie(\mathscr{P}) \ar[r] \ar[d]^{\beta^0(\mathscr{P})} & 0 \\
0 \ar[r] &
\Hom(\hat P',{\mathbb{G}_a}) \ar[r]^{(F_1'-1)^*} &
\Hom(\hat Q',{\mathbb{G}_a}) \ar[r] &
\Ext^1(G^t,{\mathbb{G}_a}) \ar[r] & 0
}
$$
It is easy to see that the left hand square commutes.
The right hand square commutes because the construction
$\tilde\gamma\mapsto\beta^0(\mathscr{P})_k$ is functorial with
respect to the obvious homomorphisms $\hat Q'\to\hat Z(\mathscr{P}^t)$
and $\tilde Z(\mathscr{P})_\varepsilon(k)\to P_{\varepsilon k}[-1]$.
The exact lower line is induced by the exact sequence
$$
0\to\hat Q'\xrightarrow{F_1'-1}\hat P'\to G^t\to 0;
$$
here $\Ext^1(\hat P',{\mathbb{G}_a})$ vanishes because $\hat P'$ is
isomorphic to a multiple of $\hat W$. The exactness of the
upper line follows
from Lemma \ref{Le-complexes-DD} \eqref{It-complexes-DD-5}.
Now $g_1$ is injective with cokernel isomorphic
to $P_{\varepsilon k}/Q_{\varepsilon k}\cong P/Q$.
Choose a normal decomposition $P'=L'\oplus T'$.
The $k$-modules $P/Q$ and $L'/pL'$ are the duals of
each other. Since $\Hom(W[f],{\mathbb{G}_a})=k$, the inclusion
$L'[f]\to\hat P'$ induces an isomorphism
$\Coker(g_1)\cong\Hom(L'[f],{\mathbb{G}_a})$. On the other hand,
the argument on \cite[page 255]{Lau-Dual}
shows that there is an exact sequence of fppf sheaves
$$
0\to L'[f]\to\hat Q'\xrightarrow{F_1'}\hat P'\to 0.
$$
This sequence remains exact under $\Hom(\ldots,{\mathbb{G}_a})$.
Thus $g_0$ is injective with cokernel isomorphic to
$\Hom(L'[f],{\mathbb{G}_a})$. The composition
$$
L'[f]\subseteq\hat Q'\xrightarrow{F_1'-1}\hat P'
$$
is the negative of the inclusion because $F_1'$ is
$f$-linear. Thus the natural homomorphism
$\Coker(g_1)\to\Coker(g_0)$ is bijective,
which implies that $\beta^0(\mathscr{P})$ is an isomorphism.
\end{proof}
\begin{Thm}
\label{Th-BT-DD}
Let $R$ be an admissible local ring.
For each Dieudonn\'e display $\mathscr{P}$ over $R$
and each divided power extension $(B\to A,\delta)$ in
$\mathcal{J}_R$ there is a natural isomorphism
\begin{equation}
\label{Eq-DD-DD}
{\mathbb{D}}(\mathscr{P})_{B/A}\cong{\mathbb{D}}(\BT_R(\mathscr{P}))_{B/A},
\end{equation}
functorial in $\mathscr{P}$ and in $(B\to A,\delta)$,
which respects the Hodge filtration.
\end{Thm}
Again we use the covariant Dieudonn\'e crystal; see footnote
\ref{Ft-DD-covariant} on page \pageref{Ft-DD-covariant}.
\begin{proof}
Let $\Sigma=\Spec{\mathbb{Z}}_p$ with the divided power ideal $I=(0)$.
In the notation of Lemma \ref{Le-DD-isom},
let $\Psi:(\DDD\text{-}{\win})\to(p\text{-}{\divv})$ be the functor
$\mathscr{P}\mapsto\BT(\mathscr{P}^t)^\vee$. The homomorphism
$\beta(\mathscr{P})$ defined in Construction \ref{Const-beta}
satisfies the hypotheses of
Lemma \ref{Le-DD-isom} by Lemmas \ref{Le-beta-compat}
and \ref{Le-beta-0}. Thus $\beta(\mathscr{P})$ is an isomorphism
${\mathbb{D}}(\mathscr{P})\cong{\mathbb{D}}(\BT(\mathscr{P}^t)^\vee)$.
(Lemma \ref{Le-DD-isom} is stated only with respect to
local Artin rings, but whether $\beta(\mathscr{P})$ is an
isomorphism can be verified over the residue field.)
One obtains the desired isomorphism \eqref{Eq-DD-DD}
either by using the duality isomorphism
$\BT(\mathscr{P}^t)^\vee\cong\BT(\mathscr{P})$ of Proposition \ref{Pr-BT-dual},
or by taking the dual of $\beta(\mathscr{P}^t)$,
using that ${\mathbb{D}}(\mathscr{P}^t)$ is the dual of ${\mathbb{D}}(\mathscr{P})$
and that ${\mathbb{D}}(G^\vee)$ is the dual of ${\mathbb{D}}(G)$
by the crystalline duality theorem \cite[5.3]{BBM}.
\end{proof}
\begin{Remark}
The proof of Theorem \ref{Th-BT-DD}
does not necessarily use Propositions \ref{Pr-BT-equiv} and
\ref{Pr-BT-dual}. Thus we get another proof of the fact that
$\BT_R$ is an equivalence of categories,
as announced in section \ref{Subse-Comments}:
By section \ref{Subse-finiteness} we can assume that $R$ is
a local Artin ring; then Theorem \ref{Th-BT-DD} reduces the
assertion to the case where $R=k$ is a perfect field,
which is well-known.
\end{Remark}
\begin{Remark}
Using Lemma \ref{Le-DD-aut} one can verify that the
isomorphisms of Theorem \ref{Th-BT-DD} and
Corollary \ref{Co-bt2disp-DD} coincide on their
common domain of definition.
\end{Remark}
\begin{Cor}
\label{Co-BT-equiv}
Let $R$ be an admissible local ring with $p=2$.
Let $\mathscr{P}$ be a Dieudonn\'e display over $R$,
let $G=\BT_R(\mathscr{P})$ be the associated $p$-divisible group,
and let $\mathscr{P}^+$ be the base change of $\mathscr{P}$ under
$\mathscr{D}_R\to\mathscr{D}_R^+$.
There is a natural isomorphism of $\mathscr{D}^+_R$-windows
$\Phi_R^+(G)\cong\mathscr{P}^+$.
\end{Cor}
\begin{proof}
We write $\mathscr{P}=(P,Q,F,F_1)$ and $\mathscr{P}^+=(P^+,Q^+,F,F_1)$
and $\Phi^+(G)=\mathscr{P}'=(P',Q',F,F_1)$. The base change
to $R_0=R/pR$ is denoted by a subscript zero;
all functors which appear are compatible with base
change. We equip the ideal $pR$ with the canonical
divided powers.
Proposition \ref{Pr-BT-equiv} gives a natural
isomorphism $\alpha$ between $\mathscr{P}'_0$ and $\mathscr{P}_0=\mathscr{P}_0^+$.
By Corollary \ref{Co-deform-DDD+} we have to show that the
induced isomorphism ${\mathbb{D}}(\alpha)_{R/R_0}$ between
${\mathbb{D}}(\mathscr{P}'_0)_{R/R_0}$ and
${\mathbb{D}}(\mathscr{P}_0)_{R/R_0}={\mathbb{D}}^+(\mathscr{P}_0^+)_{R/R_0}$
(see Remark \ref{Re-crys-DDD+} for this equality)
maps the Hodge filtration of $\mathscr{P}'$ to the
common Hodge filtration of $\mathscr{P}$ and $\mathscr{P}^+$.
If we restrict all crystals to divided power extensions
which are compatible with the canonical divided powers of $p$,
by Corollary \ref{Co-bt2disp-DD} and Theorem \ref{Th-BT-DD}
we have a chain of isomorphisms compatible with Hodge filtrations
$$
\beta:{\mathbb{D}}^+(\mathscr{P}')\cong{\mathbb{D}}(G)\cong{\mathbb{D}}(\mathscr{P})={\mathbb{D}}^+(\mathscr{P}^+).
$$
The analogous isomorphism
$\beta_0:{\mathbb{D}}^+(\mathscr{P}'_0)\cong{\mathbb{D}}^+(\mathscr{P}^+_0)$
maps the Hodge filtration of $\mathscr{P}'$ to that of $\mathscr{P}^+$.
Thus suffices to show that ${\mathbb{D}}(\alpha)$ is a scalar multiple
of $\beta_0$, i.e.\ that the automorphism
$\beta_0^{-1}\circ{\mathbb{D}}(\alpha)$ is a given by a scalar.
Since ${\mathbb{D}}({\mathbb{Q}}_p/{\mathbb{Z}}_p)$ is free
of rank one, this can be deduced from Lemma \ref{Le-DD-aut},
using the finiteness results of section \ref{Subse-finiteness}.
\end{proof}
\section{Breuil-Kisin modules}
\label{Se-Breuil}
We repeat the main construction of \cite{Lau-Frames}
without restriction on $p$.
Let $R$ be a complete regular local ring with perfect
residue field $k$ of characteristic $p$ and with
maximal ideal $\mathfrak{m}_R$.
Choose a representation $R=\mathfrak{S}/E\mathfrak{S}$ with
$$
\mathfrak{S}=W(k)[[x_1,\ldots,x_r]]$$
such that $E$ is a power series with constant term $p$.
Let $J\subset\mathfrak{S}$ be the ideal generated by $x_1,\ldots,x_r$.
Choose a ring endomorphism $\sigma:\mathfrak{S}\to\mathfrak{S}$
which lifts the Frobenius of $\mathfrak{S}/p\mathfrak{S}$
such that $\sigma(J)\subseteq J$. Let
$\sigma_1:E\mathfrak{S}\to\mathfrak{S}$ be defined by $\sigma_1(Ex)=\sigma(x)$
for $x\in\mathfrak{S}$. These data define a frame
$$
\mathscr{B}=(\mathfrak{S},E\mathfrak{S},R,\sigma,\sigma_1).
$$
For each integer $a\ge 1$ let $R_a=R/\mathfrak{m}_R^a$
and $\mathfrak{S}_a=\mathfrak{S}/J^a$. We have frames
$$
\mathscr{B}_a=(\mathfrak{S}_a,E\mathfrak{S}_a,R_a,\sigma,\sigma_1)
$$
where $\sigma$ and $\sigma_1$ are induced by the
corresponding operators of $\mathscr{B}$.
The frames $\mathscr{B}$ and $\mathscr{B}_a$ are related with the Witt
and Dieudonn\'e frames of $R$ and of $R_a$ as follows.
Let $\delta:\mathfrak{S}\to W(\mathfrak{S})$ be
the unique lift of the identity
of $\mathfrak{S}$ such that $f\delta=\delta\sigma$,
or equivalently $w_n\delta=\sigma^n$ for $n\ge 0$;
see \cite[IX.1, proposition 2]{Bourbaki-Comm-Alg}.
The composition of $\delta$ with the
projection $W(\mathfrak{S})\to W(R)$ is a ring homomorphism
$$
\varkappa:\mathfrak{S}\to W(R)
$$
which lifts the projection $\mathfrak{S}\to R$ such that
$f\varkappa=\varkappa\sigma$.
The same construction gives compatible homomorphisms
$$
\varkappa_a:\mathfrak{S}_a\to W(R_a)
$$
for $a\ge 1$, which induce $\varkappa$ in the projective limit.
Since the element $\varkappa(E)$ maps to zero in $R$ it lies in
the image of $v:W(R)\to W(R)$. Let
$$
u=v^{-1}(\varkappa(E))=f_1(\varkappa(E)).
$$
We will denote the image of $u$ in $W(R_a)$ also by $u$.
\begin{Lemma}
\label{Le-varkappa-W}
The element $u\in W(R)$ is a unit.
The homomorphisms $\varkappa$ and $\varkappa_a$
are $u$-homomorphisms of frames $\varkappa:\mathscr{B}\to\mathscr{W}_R$
and $\varkappa_a:\mathscr{B}_a\to\mathscr{W}_{R_a}$.
\end{Lemma}
\begin{proof}
This follows from \cite[Proposition 6.1]{Lau-Frames}.
One can also argue directly: Since the projection
$W(R)\to W(k)$ is a local homomorphism, in order to
show that $u$ is a unit we can work with $\varkappa_1$,
i.e.\ consider the case where $R=k$ and $\mathfrak{S}=W(k)$.
Then $E=p$ and $u=1$. In order that $\varkappa$ and
$\varkappa_a$ are $u$-homomorphisms of frames
we need that $f_1\varkappa=u\cdot\varkappa\sigma_1$.
For $x\in\mathfrak{S}$ we calculate
$f_1(\varkappa(Ex))=f_1(\varkappa(E)\varkappa(x))=
f_1(\varkappa(E))\cdot f(\varkappa(x))=u\cdot\varkappa(\sigma(x))=
u\cdot\varkappa(\sigma_1(Ex))$ as required.
\end{proof}
Let $\bar\sigma$ be the semi-linear endomorphism of the
free $W(k)$-module $J/J^2$ induced by $\sigma$.
Since $\sigma$ induces the Frobenius modulo $p$,
$\bar\sigma$ is divisible by $p$.
\begin{Prop}
\label{Pr-varkappa-image}
The following conditions are equivalent.
\begin{enumerate}
\renewcommand{\theenumi}{\roman{enumi}}
\item
\label{Cond-varkappa}
The image of $\varkappa:\mathfrak{S}\to W(R)$ lies in ${\mathbb{W}}(R)$.
\item
\label{Cond-delta}
The image of $\delta:\mathfrak{S}\to W(\mathfrak{S})$ lies in ${\mathbb{W}}(\mathfrak{S})$.
\item
\label{Cond-sigma-1}
The endomorphism $p^{-1}\bar\sigma$ of $J/J^2$
is nilpotent modulo $p$.
\end{enumerate}
\end{Prop}
Clearly \eqref{Cond-delta} implies \eqref{Cond-varkappa}.
In the special case $\sigma(x_i)=x_i^p$ it is easy to
see that \eqref{Cond-delta} holds since $\delta(x_i)=[x_i]$;
in this case $\bar\sigma$ is zero.
\begin{proof}[Proof of Proposition \ref{Pr-varkappa-image}]
For $p\ge 3$ the equivalence between \eqref{Cond-varkappa}
and \eqref{Cond-sigma-1} is \cite[Proposition 9.1]{Lau-Frames};
its proof shows that
\eqref{Cond-varkappa}
$\Rightarrow$
\eqref{Cond-sigma-1}
$\Rightarrow$
\eqref{Cond-delta}
$\Rightarrow$
\eqref{Cond-varkappa}.
The proof also applies for $p=2$ if \cite[Lemma 9.2]{Lau-Frames}
is replaced by the following Lemma \ref{Le-gr-tau}.
\end{proof}
\begin{Lemma}
\label{Le-gr-tau}
For $x\in\mathfrak{S}$ let $\tau(x)=(\sigma(x)-x^p)/p$.
Let $\mathfrak{m}$ be the maximal ideal of $\mathfrak{S}$.
For $n\ge 0$
the map $\tau$ preserves $\mathfrak{m}^nJ$ and induces a
$\sigma$-linear endomorphism $gr_n(\tau)$ of the $k$-module
$gr_n(J)=\mathfrak{m}^nJ/\mathfrak{m}^{n+1}J$. The endomorphism
$gr_0(\tau)$ is equal to $p^{-1}\bar\sigma$ modulo $p$.
For $n\ge 1$ there is a surjective $k$-linear map
$$
\pi_n:gr_n(J)\to gr_0(J)
$$
such that $gr_0(\tau)\pi_n=\pi_ngr_n(\tau)$ and
such that $gr_n(\tau)$ vanishes on $\Ker(\pi_n)$.
In particular, $p^{-1}\bar\sigma$ is nilpotent
modulo $p$ if and only if $gr_0(\tau)$ is nilpotent,
which implies that $gr_n(\tau)$ is nilpotent for each $n$.
\end{Lemma}
\begin{proof}
We have $\sigma(J)\subseteq J^p+pJ\subseteq\mathfrak{m} J$
and thus $\sigma(\mathfrak{m}^nJ)\subseteq\mathfrak{m}^{n+1}J$.
It follows that
$p\tau(\mathfrak{m}^nJ)\subseteq p\mathfrak{S}\cap\mathfrak{m}^{n+1}J=p\mathfrak{m}^nJ$
and $\tau(\mathfrak{m}^nJ)\subseteq\mathfrak{m}^nJ$.
For $x,y\in\mathfrak{m}^nJ$ the element $\tau(x+y)-\tau(x)-\tau(y)$
is a multiple of $xy$ and thus lies in $\mathfrak{m}^{2n+1}J$.
Hence $\tau$ induces an additive
endomorphism $gr_n(\tau)$ of $gr_n(J)$. It
is $\sigma$-linear because for $a\in\mathfrak{S}$ and $x\in\mathfrak{m}^nJ$
the element $\tau(ax)-\sigma(a)\tau(x)=\tau(a)x^p$
lies in $\mathfrak{m}^{pn}J^p\subseteq\mathfrak{m}^{n+1}J$.
Let us write $\sigma(x_i)=x_i^p+py_i$ with $y_i\in J$.
We have $\tau(x_i)=y_i$ and $p^{-1}\bar\sigma(x_i)\equiv y_i$
modulo $J^2$. Thus $gr_0(\tau)$ coincides with
$p^{-1}\bar\sigma$ modulo $p$.
For each $n\ge 0$, a basis of $gr_n(J)$ is given by
all elements $p^b\underline x^{\underline c}$ with
$\underline c\in{\mathbb{N}}^r$ and
$1\le|\underline c|\le n+1$ and $b+|\underline c|=n+1$.
Assume that $n\ge 1$ and define $\pi_n$ to be the $k$-linear
map with $\pi_n(p^nx_i)=x_i$ and
$\pi_n(p^b\underline x^{\underline c})=0$ if $|\underline c|>1$.
Then $gr_n(\tau)$ vanishes on $\Ker(\pi_n)$
because $\sigma(J)\subseteq\mathfrak{m} J$, thus
$\sigma(J^2)\subseteq\mathfrak{m}^2 J^2$, and because for
$x\in\mathfrak{m}^n J$ we have $x^p\in\mathfrak{m}^{n+2}J$.
The relation $gr_0(\tau)\pi_n=\pi_ngr_n(\tau)$ holds
since $\tau(p^nx_i)\equiv p^{n-1}x_i^p+p^n y_i$
modulo $\mathfrak{m}^{n+1}(J)$.
The last assertion of the lemma is immediate.
\end{proof}
\begin{Lemma}
\label{Le-varkappa-WW}
If the equivalent conditions of Proposition \ref{Pr-varkappa-image}
hold, $\varkappa$ and $\varkappa_a$ are
${\mathbbm{u}}$-homomorphisms of frames
$$
\varkappa:\mathscr{B}\to\mathscr{D}_R,\qquad
\varkappa_a:\mathscr{B}_a\to\mathscr{D}_{R_a},
$$
where the unit ${\mathbbm{u}}\in{\mathbb{W}}(R)$ is given by
$$
{\mathbbm{u}}={\mathbbm{v}}^{-1}(\varkappa(E))={\mathbbm{f}}_1(\varkappa(E)).
$$
In $W(R)$ we have ${\mathbbm{u}}=u$ if $p$ is odd and
${\mathbbm{u}}=(v^{-1}(p-[p]))^{-1}u$ if $p=2$.
\end{Lemma}
\begin{proof}
The proof of Lemma \ref{Le-varkappa-W} with $f_1$
replaced by ${\mathbbm{f}}_1$ shows that ${\mathbbm{u}}$ is a unit
of ${\mathbb{W}}(R)$ and that $\varkappa$ and $\varkappa_a$
are ${\mathbbm{u}}$-homomorphisms of frames as indicated. The
relation between ${\mathbbm{u}}$ and $u$ follows from the fact
that $\mathscr{D}_R\to\mathscr{W}_R$ is a $u_0$-homomorphism where
$u_0=1$ if $p$ is odd and $v(u_0)=p-[p]$ if $p=2$.
\end{proof}
\begin{Thm}
\label{Th-varkappa-crys}
If the equivalent conditions of Proposition \ref{Pr-varkappa-image}
hold, the frame homomorphisms $\varkappa:\mathscr{B}\to\mathscr{D}_R$ and
$\varkappa_a:\mathscr{B}_a\to\mathscr{D}_{R_a}$ are crystalline.
\end{Thm}
\begin{proof}
The proof for odd $p$ in \cite[Theorem 9.3]{Lau-Frames}
works almost literally for $p=2$. Let us repeat the
essential parts of the
argument. Fix an integer $a\ge 1$. One can define
a factorisation of the projection $\mathscr{B}_{a+1}\to\mathscr{B}_a$
into strict frame homomorphisms
\begin{equation}
\label{Eq-BBB-factor}
\mathscr{B}_{a+1}\xrightarrow{\iota}\tilde\mathscr{B}_{a+1}
\xrightarrow{\pi}\mathscr{B}_a
\end{equation}
such that
$\mathscr{B}_{a+1}=(\mathfrak{S}_{a+1},\tilde I,R_a,\sigma,\tilde\sigma_1)$.
This determines $\tilde I$ and $\tilde\sigma_1$ uniquely:
Let $\bar J^a=J^a/J^{a+1}$.
We have $\tilde I=E\mathfrak{S}_{a+1}+\bar J^a$ and
$E\mathfrak{S}_{a+1}\cap\bar J^a=p\bar J^a$.
The endomorphism $\bar\sigma$ of $\bar J^a$
induced by $\sigma$ is divisible by $p^a$,
and the operator $\tilde\sigma_1:\tilde I\to\mathfrak{S}_{a+1}$
is the unique extension of $\sigma_1$ such that
$\tilde\sigma_1(x)=p^{-1}\bar\sigma(x)$
for $x\in\bar J^a$. On the other hand,
we consider the factorisation
\begin{equation}
\label{Eq-DDD-factor}
\mathscr{D}_{R_{a+1}}\xrightarrow{\iota'}
\mathscr{D}_{R_{a+1}/R_a}\xrightarrow{\pi'}\mathscr{D}_{R_a}
\end{equation}
with respect to the trivial divided powers on the
kernel $\mathfrak{m}_R^{a}/\mathfrak{m}_R^{a+1}$.
Then $\varkappa_{a+1}$ is a ${\mathbbm{u}}$-homomorphism
of frames $\tilde\mathscr{B}_{a+1}\to\mathscr{D}_{R_{a+1}/R_a}$.
Here the only condition to be verified is that
for $x\in\bar J^a$ we have
\begin{equation}
\label{Eq-non-triv}
\tilde{\mathbbm{f}}_1(\varkappa_{a+1}(x))=
{\mathbbm{u}}\cdot\varkappa_{a+1}(\tilde\sigma_1(x))
\end{equation}
in the $k$-vector space $\hat W(\mathfrak{m}_R^a/\mathfrak{m}_R^{a+1})$.
On this space ${\mathbbm{u}}$ acts as the identity.
Let $y=(y_0,y_1,\ldots)$ in $W(\bar J^a)$ be defined
by $y_n=\tilde\sigma_1^n(x)$. Then $\delta(x)=y$
because the Witt polynomials give
$w_n(y)=p^n\tilde\sigma_1^n(x)=\sigma^n(x)=w_n(\delta(x))$
as required. Thus $\varkappa_{a+1}(x)$ is the reduction of $y$.
Since $\tilde{\mathbbm{f}}_1$ acts on $\hat W(\mathfrak{m}_R^a/\mathfrak{m}_R^{a+1})$
by a shift to the left, the relation \eqref{Eq-non-triv}
follows. We obtain compatible ${\mathbbm{u}}$-homomorphisms of frames
$\varkappa_*:\eqref{Eq-BBB-factor}\to\eqref{Eq-DDD-factor}$.
The homomorphisms $\pi$ and $\pi'$ are crystalline;
see the proof of \cite[Theorem 9.3]{Lau-Frames}.
Lifts of windows under $\iota$ and under $\iota'$
are both classified by lifts of the Hodge filtration
from $R_a$ to $R_{a+1}$ in a compatible way.
Thus if $\varkappa_a$ is crystalline then so is
$\varkappa_{a+1}$, and Theorem \ref{Th-varkappa-crys}
follows by induction, using that $\varkappa_1$ is
an isomorphism.
\end{proof}
Following the \cite{Vasiu-Zink} terminology,
a Breuil window relative to $\mathfrak{S}\to R$ is
a pair $(Q,\phi)$ where $Q$ is a free $\mathfrak{S}$-module of
finite rank and where $\phi:Q\to Q^{(\sigma)}$ is an
$\mathfrak{S}$-linear map with cokernel
annihilated by $E$. The category of $\mathscr{B}$-windows
is equivalent to the category of Breuil windows relative
to $\mathfrak{S}\to R$ by the assignment $(P,Q,F,F_1)\mapsto
(Q,\phi)$, where $\phi$ is the composition of the inclusion
$Q\to P$ with the inverse of $F_1^\sharp:Q^{(\sigma)}\cong P$;
see \cite[Lemma 8.2]{Lau-Frames}.
\begin{Cor}
\label{Co-B-win}
If the equivalent conditions of Proposition \ref{Pr-varkappa-image}
hold, there is an equivalence of exact categories between
$p$-divisible groups over $R$ and Breuil windows relative
to $\mathfrak{S}\to R$.
\end{Cor}
\begin{proof}
This is analogous to \cite[Corollary 8.3]{Lau-Frames},
using Proposition \ref{Pr-BT-equiv}.
\end{proof}
Following \cite{Vasiu-Zink} again,
a Breuil module relative to $\mathfrak{S}\to R$
is a triple $(M,\phi,\psi)$ where $M$ is a finitely
generated $\mathfrak{S}$-module annihilated by a power of $p$
and of projective dimension at most one, and where
$\phi:M\to M^{(\sigma)}$ and $\psi:M^{(\sigma)}\to M$
are $\mathfrak{S}$-linear maps with $\phi\psi=E$ and
$\psi\phi=E$. If $R$ has characteristic zero, such
triples are equivalent to pairs $(M,\phi)$
such that the cokernel of $\phi$ is annihilated
by $p$; see \cite[Lemma 8.6]{Lau-Frames}.
\begin{Cor}
If the equivalent conditions of Proposition \ref{Pr-varkappa-image}
hold, there is an equivalence of exact categories between
commutative
finite flat group schemes of $p$-power order over $R$
and Breuil modules relative to $\mathfrak{S}\to R$.
\end{Cor}
\begin{proof}
This is analogous to \cite[Theorem 8.5]{Lau-Frames}.
\end{proof}
\begin{Example}
\label{Ex-motiv}
Let $R=W(k)$ and $\mathfrak{S}=W(k)[[t]]$ with $\sigma(t)=t^p$.
Define $\mathfrak{S}\to R$ by $t\mapsto p$, thus $E=p-t$. We
have $\varkappa(E)=p-[p]$ and thus $u=v^{-1}(p-[p])$.
Assume that $p=2$. Then $u=u_0$, and $\mathscr{B}\to\mathscr{D}_R$ is
a strict frame homomorphism. This example has motivated
the definition of Dieudonn\'e displays for $p=2$.
\end{Example}
\section{Breuil-Kisin modules and crystals}
\label{Se-Breuil-crys}
We keep the notation of section \ref{Se-Breuil}
and assume that the equivalent conditions of Proposition
\ref{Pr-varkappa-image} hold. Assume that
$R$ has characteristic zero. Let $S$ be the
$p$-adic completion of the divided power envelope of
the ideal $E\mathfrak{S}\subset\mathfrak{S}$ and let $I$ be the kernel
of $S\to R$.
Since $\sigma:\mathfrak{S}\to\mathfrak{S}$ preserves the ideal $(E,p)$
it extends to $\sigma:S\to S$. It is easy to see that
$\sigma(I)\subseteq pS$; thus $\sigma:S\to S$ is a
Frobenius lift again.
\begin{Prop}
\label{Pr-Breuil-crys}
Let $(Q,\phi)$ be a Breuil window relative to $\mathfrak{S}\to R$
and let $G$ be the associated $p$-divisible group over $R$;
see Corollary \ref{Co-B-win}. There is a natural isomorphism
$$
{\mathbb{D}}(G)_{S/R}\cong S\otimes_{\mathfrak{S}}Q^{(\sigma)}
$$
such that the Hodge filtration of\/ ${\mathbb{D}}(G)_{S/R}$
corresponds to the submodule generated by
$\phi(Q)+IQ^{(\sigma)}$, and the Frobenius
of\/ ${\mathbb{D}}(G)_{S/R}$ corresponds to the
$\sigma$-linear endomorphism of $Q^{(\sigma)}$
defined by $x\mapsto 1\otimes\phi^{-1}(Ex)$.
\end{Prop}
In Kisin's theory (when $R$ is one-dimensional)
the analogous result is an immediate by-product.
Here Proposition \ref{Pr-Breuil-crys} will be deduced from
Theorem \ref{Th-BT-DD} via Corollary \ref{Co-BT-equiv},
together with a variant of \cite{Zink-Windows}.
To begin with,
the element $u=\sigma(E)/p$ of $S$ is a unit.
Indeed, since the arrow $\mathfrak{S}\to R$ is mapped surjectively
onto $W(k)\to k$, we have a local homomorphism $S\to W(k)$,
which maps $u$ to $1$. Consider the frame
$$
\mathscr{S}=(S,I,R,\sigma,\sigma_1)
$$
with $\sigma_1(x)=\sigma(x)/p$ for $x\in I$.
The inclusion $\iota:\mathfrak{S}\to S$ is a $u$-homomorphism
of frames $\iota:\mathscr{B}\to\mathscr{S}$.
If $p=2$ we have defined rings ${\mathbb{W}}(R)\subset{\mathbb{W}}^+(R)$;
if $p$ is odd let us write ${\mathbb{W}}^+(R)={\mathbb{W}}(R)$.
Then ${\mathbb{W}}^+(R)\to R$ is a divided power extension
of $p$-adic rings, so the composition $\mathfrak{S}\xrightarrow\varkappa
{\mathbb{W}}(R)\to{\mathbb{W}}^+(R)$ extends to a ring homomorphism
$\varkappa_S:S\to{\mathbb{W}}^+(R)$, which is a strict frame
homomorphism $\varkappa_S:\mathscr{S}\to\mathscr{D}^+_R$.
Thus we have a commutative square of frames:
$$
\xymatrix@M+0.2em{
\mathscr{B} \ar[r]^\iota \ar[d]_\varkappa &
\mathscr{S} \ar[d]^{\varkappa_S} \\
\mathscr{D}_R \ar[r] &
\mathscr{D}_R^+
}
$$
Here $\varkappa$ is crystalline by Theorem \ref{Th-varkappa-crys}.
\begin{Thm}
\label{Th-varkappa-S}
The frame homomorphism $\varkappa_S$ is crystalline.
\end{Thm}
This is a variant of the main result of \cite{Zink-Windows}.
It is easy to see that $S$ is an admissible topological ring
in the sense
of Definition \ref{Def-top-admissible} if and only if $r=1$,
i.e.\ if $R$ is a discrete valuation ring.
In that case, the methods of \cite{Zink-Windows}
apply directly, but additional effort is needed to prove
Theorem \ref{Th-varkappa-S} in general.
The proof is postponed to the next section.
In fact, we will show more: For $m\ge 0$ let $S_{<m>}$
be the closure of the $\mathfrak{S}$-algebra in $S$
generated by $E^i/i!$ for $i\le p^m$.
For $m\ge 1$ this ring naturally defines a sub-frame
$\mathscr{S}_{\!<m>}$ of $\mathscr{S}$, and the frame homomorphisms
$\mathscr{S}_{\!<m>}\to\mathscr{S}\to\mathscr{D}_R^+$ are crystalline.
\begin{proof}[Proof of Proposition \ref{Pr-Breuil-crys}]
Let $\mathscr{P}_{\!\mathscr{B}}=(P,Q,F,F_1)$ be the $\mathscr{B}$-window associated to
$(Q,\phi)$;
in particular $P=Q^{(\sigma)}$, the inclusion map $Q\to P$
is $\phi$, and $F:P\to P$ corresponds to
the $\sigma$-linear endomorphism of $Q^{(\sigma)}$
defined by $x\mapsto 1\otimes\phi^{-1}(Ex)$.
We write $\mathscr{P}_{\!\mathscr{S}}=\iota_*(\mathscr{P}_{\!\mathscr{B}})$ and
$\mathscr{P}^+=\varkappa_{S*}(\mathscr{P}_{\!\mathscr{S}})$.
The frames $\mathscr{S}$ and $\mathscr{D}_R^+$ both satisfy the
hypotheses of the beginning of section \ref{Subse-cryst-frame}.
Thus Construction \ref{Const-fil-mod} and
Proposition \ref{Pr-fil-mod-win}
applied to ${\mathbb{D}}(G)$ give an $\mathscr{S}$-window $\mathscr{P}'_{\!\!\mathscr{S}}$ and
a $\mathscr{D}^+_R$-window $\mathscr{P}^{+\prime}$ with
$\mathscr{P}^{+\prime}=\varkappa_{S*}\mathscr{P}'_{\!\!\mathscr{S}}$.
By the definition of $\Phi^+_R$ in Proposition \ref{Pr-bt2disp+}
we have $\mathscr{P}^{+\prime}=\Phi^+_R(G)$,
and by Corollary \ref{Co-BT-equiv} the latter is
isomorphic to $\mathscr{P}^+$.
Since the base change functor $\varkappa_{S*}$ is
fully faithful by Theorem \ref{Th-varkappa-S},
the isomorphism $\mathscr{P}^{+\prime}\cong\mathscr{P}^+$ descends
to an isomorphism $\mathscr{P}_{\!\!\mathscr{S}}'\cong\mathscr{P}_{\!\mathscr{S}}$.
The associated isomorphism of filtered $F$-$V$-modules
gives the proposition.
\end{proof}
\section{Proof of Theorem \ref{Th-varkappa-S}}
We keep the notation of sections \ref{Se-Breuil}
and \ref{Se-Breuil-crys}.
Let us begin with a closer look on the $p$-adic ring $S$
and its subrings $S_{<m>}$ defined above.
\begin{Prop}
\label{Pr-S-m}
For $m\ge 1$ there is a surjective ring homomorphism
$\mathfrak{S}[[t_1,\ldots,t_m]]\to S_{<m>}$ defined by
$t_i\mapsto E^{p^i}/p^i!$.
\end{Prop}
In particular, $S_{<m>}$ is a noetherian complete local ring.
Proposition \ref{Pr-S-m} is a consequence of
the following well-known fact.
\begin{Lemma}
\label{Le-lim-surjective}
Let $A$ be a noetherian complete local ring with a descending
sequence of ideals
$A\supseteq\mathfrak{a}_0\supseteq\mathfrak{a}_1\supseteq\cdots$.
Then $A\to\varprojlim_i A/\mathfrak{a}_i$ is surjective.
\end{Lemma}
\begin{proof}
Let $\mathfrak{m}$ be the maximal ideal of $A$. For each $r$,
the images of $\mathfrak{a}_i\to A/\mathfrak{m}^r$ stabilise for $i\to\infty$
to an ideal $\bar\mathfrak{a}_r\subset A/\mathfrak{m}^r$.
We have
$$
\varprojlim_i A/\mathfrak{a}_i=\varprojlim_{i,r}A/(\mathfrak{a}_i+\mathfrak{m}^r)
=\varprojlim_r(A/\mathfrak{m}^r)/\bar\mathfrak{a}_r.
$$
Since the ideals $\bar\mathfrak{a}_r$ form a surjective system,
taking the limit over $r$ of the exact sequences
$0\to\bar\mathfrak{a}_r\to A/\mathfrak{m}^r\to(A/\mathfrak{m}^r)/\bar\mathfrak{a}_r\to 0$
proves the lemma.
\end{proof}
\begin{proof}[Proof of Proposition \ref{Pr-S-m}]
Since the image of $E^{p^i}/p^i!$ in $S/p^nS$ is nilpotent,
there is a well-defined homomorphism
$\pi_{m,n}:\mathfrak{S}[[t_1,\ldots,t_m]]\to S/p^nS$
with $t_i\mapsto E^{p^i}/p^i!$.
By definition, $S_{<m>}$ is the projective limit over $n$
of the image of $\pi_{n,m}$.
The proposition follows by Lemma \ref{Le-lim-surjective}.
\end{proof}
Let $K=W(k)\otimes{\mathbb{Q}}$ and $\mathfrak{S}_{\mathbb{Q}}=K[[x_1,\ldots,x_r]]$.
Since $\sigma:\mathfrak{S}\to\mathfrak{S}$ preserves the ideal $J=(x_1,\ldots,x_r)$
it extends to a homomorphism $\sigma:\mathfrak{S}_{\mathbb{Q}}\to\mathfrak{S}_{\mathbb{Q}}$.
For $r=1$ it is easy to describe $S$ and $S_{<m>}$ as
explicit subrings of $\mathfrak{S}_{\mathbb{Q}}$ since instead of the divided
powers of $E$ one can take the divided powers of $x_1^e$
where $e$ is defined by $pR=\mathfrak{m}_R^e$.
For $r\ge 2$ the situation is more complicated.
\begin{Prop}
\label{Pr-S}
The natural embedding $\mathfrak{S}\to\mathfrak{S}_{\mathbb{Q}}$
extends to an injective homomorphism $S\to\mathfrak{S}_{\mathbb{Q}}$
that commutes with $\sigma$.
\end{Prop}
Thus $S_{<m>}$ is the image of $\mathfrak{S}[[t_1,\ldots,t_m]]\to\mathfrak{S}_{\mathbb{Q}}$
as in Proposition \ref{Pr-S-m}.
\begin{proof}[Proof of Proposition \ref{Pr-S}]
Recall that $J=(x_1,\ldots,x_r)$ as an ideal of $\mathfrak{S}$.
Choose $E'\in J^e$ with $E-E'\in p\mathfrak{S}$ such that $e$ is maximal,
thus $p\in\mathfrak{m}_R^e\setminus\mathfrak{m}_R^{e+1}$.
Let us write $gr_{E'}^i(\mathfrak{S})=E^{\prime i}\mathfrak{S}/E^{\prime i+1}\mathfrak{S}$ etc.
\begin{Lemma}
\label{Le-graded-inj}
The map of graded rings
$gr_{E'}(\mathfrak{S})\to gr_{E'}(\mathfrak{S}_{{\mathbb{Q}}})$ is injective.
\end{Lemma}
\begin{proof}
It suffices to show that $\mathfrak{S}/E'\mathfrak{S}\to\mathfrak{S}_{\mathbb{Q}}/E'\mathfrak{S}_{\mathbb{Q}}$
is injective.
The choice of $E'$ implies that the image of $E'$
in the regular local rings $\mathfrak{S}/p\mathfrak{S}$ and $\mathfrak{S}_{{\mathbb{Q}}}$ lies
in the same power of the maximal ideals. Therefore
the $k$-dimension of $\mathfrak{S}/(p\mathfrak{S}+E'\mathfrak{S}+J^n)$
is equal to the $K$-dimension of $\mathfrak{S}_{{\mathbb{Q}}}/(E'\mathfrak{S}_Q+J^n\mathfrak{S}_{\mathbb{Q}})$.
Since the last module is isomorphic to $\mathfrak{S}/(E'\mathfrak{S}+J^n)\otimes{\mathbb{Q}}$
it follows that $\mathfrak{S}/(E'\mathfrak{S}+J^n)$ is a free $W(k)$-module and
injects into $\mathfrak{S}_{{\mathbb{Q}}}/(E'\mathfrak{S}_Q+J^n\mathfrak{S}_{\mathbb{Q}})$.
Since $\mathfrak{S}/E'\mathfrak{S}$ and $\mathfrak{S}_{\mathbb{Q}}/E'\mathfrak{S}_{\mathbb{Q}}$ are $J$-adic
the lemma follows.
\end{proof}
Let $S_0\subset\mathfrak{S}_{\mathbb{Q}}$
be the $\mathfrak{S}$-algebra generated by $E^{\prime i}/i!$
for $i\ge 1$, or equivalently by $E^i/i!$ for $i\ge 1$,
so $S$ is the $p$-adic completion of $S_0$.
Let $S_{0,n}$ be the image of
$S_{0}\to\mathfrak{S}_{\mathbb{Q}}/E^{\prime n}\mathfrak{S}_{\mathbb{Q}}$
and let $\tilde S=\varprojlim_nS_{0,n}$.
Each $S_{0,n}$ is a noetherian complete local ring with
residue field $k$ and thus a $p$-adic ring.
Since $S_{0,n}$ has no $p$-torsion it follows that
$\tilde S$ is $p$-adic.
We obtain a homomorphism $S\to\tilde S\subset\mathfrak{S}_{\mathbb{Q}}$
which extends $S_0\subset\tilde S\subset\mathfrak{S}_{\mathbb{Q}}$.
\begin{Lemma}
\label{Le-S0-tildeS}
We have $S_0\cap p\tilde S=pS_0$ inside $\tilde S$.
\end{Lemma}
\begin{proof}
Let $x\in S_0\cap p\tilde S$ be given.
We have to show that $x$ lies in $pS_0$.
Assume that $x\ne 0$ and choose an expression
$(\star)$ $x=\sum_{i=0}^sa_iE^{\prime n_i}/n_i!$
with $a_i\in\mathfrak{S}$ such that $n_0<\ldots<n_s$.
We use induction on $n_s-n_0$.
Suppose $E'$ divides $a_0$ in $\mathfrak{S}$.
Then $a_0E^{\prime n_0}/n_0!=a_0'E^{\prime n_0'}/(n_0')!$
with $n_0'=n_0+1$ and $a_0'=n_0'a_0/E'$.
If $s>0$ this allows to find a new expression of $x$
of the type $(\star)$ with smaller value of $n_s-n_0$,
and we are done by induction.
If $s=0$ we replace the expression $(\star)$
by $x=a_0'E^{\prime n_0'}/n_0'!$;
call this a modification of the first type.
Suppose $E'$ does not divide $a_0$ in $\mathfrak{S}$.
Lemma \ref{Le-graded-inj} implies that the image of $x$
in $gr^{n_0}_{E'}(\mathfrak{S}_{\mathbb{Q}})$ is non-zero.
In $S_{0,n_0+1}$ we have $x=py$.
Choose an expression $y=\sum_{i=\ell}^{n_0}c_iE^{\prime i}/i!$
with $c_i\in\mathfrak{S}$ such that $\ell$ is maximal.
Then $E'$ does not divide $c_\ell$ in $\mathfrak{S}$,
and Lemma \ref{Le-graded-inj} implies that $y$ has
non-zero image in $gr^\ell_{E'}(\mathfrak{S}_{\mathbb{Q}})$.
Thus $\ell=n_0$. Using Lemma \ref{Le-graded-inj} again,
it follows that the image of $a_0$ in $\mathfrak{S}/E'\mathfrak{S}$ is divisible
by $p$. Let $a_0=pb_0+b_1E'$ with $b_i\in\mathfrak{S}$ and let
$x'=x-pb_0E^{\prime n_0}/n_0!$. Then $x-x'\in pS_0$;
thus $x'\in S_0\cap p\tilde S$,
and we have to show that $x'\in pS_0$.
If $s>0$ we get an expression of $x'$ of the type $(\star)$
with smaller value of $n_s-n_0$, and we are done
by induction. If $s=0$ we replace $x$ by $x'$ and
take for $(\star)$
the expression $x'=a'_0E^{\prime n'_0}/n_0'!$
with $n_0'=n_0+1$ and $a_0'=n_0'b_1$;
call this a modification of the second type.
If $s>0$ the inductive step is already finished.
So we may assume that $s=0$.
We apply successively modifications of the first or
second type depending on whether $E'$ divides $a_0$.
After at most $p$ steps, the new value of $a_0$ becomes
divisible by $p$, and thus $x$ lies in $pS_0$.
\end{proof}
Since $S_0/p^nS_0\to\tilde S/p^n\tilde S$ is injective by
Lemma \ref{Le-S0-tildeS}, the projective limit of these maps
$S\to\tilde S$ is injective, and so $S\to\mathfrak{S}_{\mathbb{Q}}$ is injective.
In order that this map commutes with $\sigma$
it suffices to show that $S\to\mathfrak{S}_{\mathbb{Q}}/J^n\mathfrak{S}_{\mathbb{Q}}$ commutes
with $\sigma$ for each $n$;
this is true since $S_0\to\mathfrak{S}_{\mathbb{Q}}/J^n\mathfrak{S}_{\mathbb{Q}}$
commutes with $\sigma$, and the image of this map is $p$-adic.
Proposition \ref{Pr-S} is proved.
\end{proof}
Let us now consider the frames.
\begin{Lemma}
\label{Le-SSS-m}
For $m\ge 1$ we have a sub-frame of $\mathscr{S}=(S,I,R,\sigma,\sigma_1)$,
$$
\mathscr{S}_{\!<m>}=(S_{<m>},I_{<m>},R,\sigma,\sigma_1).
$$
\end{Lemma}
\begin{proof}
Necessarily $I_{<m>}=I\cap S_{<m>}$.
We have to show that $\sigma:S\to S$ stabilises $S_{<m>}$
and that $\sigma_1=p^{-1}\sigma:I\to S$ maps
$I_{<m>}$ into $S_{<m>}$. We will show that
$\sigma(S)$ and $\sigma_1(I)$ are contained in $S_{<1>}$.
Namely, we have $\sigma(E)=px$ with $x\in\mathfrak{S}[E^p/p]$.
Thus $\sigma_1(E^i/i!)=(p\cdot i!)^{-1}(px)^i$ lies in
$\mathfrak{S}[E^p/p]$, since $1+v_p(i!)\le i$ for $i\ge 1$.
Since $I/p^n I$ is the kernel of $S/p^nS\to R/p^n R$,
it is generated as an $\mathfrak{S}$-module by
the elements $E^i/i!$ for $i\ge 1$.
Thus the image of the map $I/p^{n+1}I\to S/p^nS$
induced by $\sigma_1$ lies in the image of $S_{<1>}$,
and it follows that $\sigma_1(I)\subseteq S_{<1>}$.
Since $S=\mathfrak{S}+I$ we get $\sigma(S)\subseteq S_{<1>}$.
\end{proof}
\begin{Prop}
\label{Pr-varkappa-S-1}
For $m\ge 1$ the inclusion $\mathscr{S}_{\!<m>}\to\mathscr{S}$ is crystalline.
\end{Prop}
\begin{proof}
This is a formal consequence of the relations
$\sigma(S)\subseteq S_{<m>}$ and
$\sigma_1(I)\subseteq S_{<m>}$
verified in the proof of Lemma \ref{Le-SSS-m}.
Indeed, let $\mathscr{P}=(P,Q,F,F_1)$ be an $\mathscr{S}$-window.
Choose a normal decomposition $P=L\oplus T$,
and let $\Psi:L\oplus T\to P$ be the $\sigma$-linear
isomorphism defined by $F_1$ on $L$ and by $F$ on $T$.
Then $P_{<m>}:=S_{<m>}\Psi(L\oplus T)$ is a free
$S_{<m>}$-module with $S\otimes_{S_{<m>}}P_{<m>}=P$.
Moreover $F_1(Q)\subseteq P_{<m>}$ and $F(P)\subseteq P_{<m>}$.
We set $Q_{<m>}=Q\cap P_{<m>}$.
Let $P_{<m>}=L_{<m>}\oplus T_{<m>}$ be a normal decomposition
and let $\Psi_{<m>}:L_{<m>}\oplus T_{<m>}\to P_{<m>}$
be the $\sigma$-linear map defined by $F_1$ on $L_{<m>}$
and by $F$ on $T_{<m>}$.
In order that the quadruple $\mathscr{P}_{\!<m>}=(P_{<m>},Q_{<m>},F,F_1)$
is an $\mathscr{S}_{\!<m>}$-window with base change $\mathscr{P}$ we need that
the determinant of $\Psi_{<m>}$ is invertible. But the
determinant of $\Psi_{<m>}$ becomes invertible in $S$ because
$\mathscr{P}$ is a window; moreover $S_{<m>}\to S$ is a local homomorphism.
Thus the base change functor from $\mathscr{S}_{\!<m>}$-windows
to $\mathscr{S}$-windows is essentially surjective.
In order that the functor is fully faithful it suffices to
show that it induces a bijection $\End(\mathscr{P}_{\!<m>})\to\End(\mathscr{P})$.
Clearly the map is injective. We have to show that every
$h\in\End(\mathscr{P})$ stabilises $P_{<m>}$. But
$h(F_1(Q))=F_1(h(Q))\subseteq F_1(Q)\subseteq P_{<m>}$,
and $F_1(Q)$ generates $P_{<m>}$ as an $S_{<m>}$-module.
This proves the proposition.
\end{proof}
\begin{Prop}
\label{Pr-varkappa-S-2}
For $m\ge 1$ the composition
$\mathscr{S}_{\!<m>}\to\mathscr{S}\xrightarrow{\varkappa_S}\mathscr{D}_R^+$
is crystalline.
\end{Prop}
This is the main step in the proof of Theorem \ref{Th-varkappa-S}.
The proof of Proposition \ref{Pr-varkappa-S-1} is a variant
of the proof of Theorem \ref{Th-varkappa-crys}.
\begin{proof}
We choose $e$ such that $p\in\mathfrak{m}_R^e\setminus\mathfrak{m}_R^{e+1}$
and consider the index set $N=\{1,2,\ldots\}\cup\{e^+\}$,
ordered by the natural order of ${\mathbb{Z}}$ and $e<e^+<e+1$.
For $n\in N$ let $n^+\in N$ be its successor.
Let $\mathfrak{m}_R^{e^+}=\mathfrak{m}_R^{e+1}+pR$.
For $n\in N$ let $R_n=R/\mathfrak{m}_R^{n}$. We equip the
ideal $\mathfrak{m}_R^n/\mathfrak{m}_R^{n^+}$ of $R_{n^+}$
with the trivial divided powers if
$n\ne e^+$ and with the canonical divided powers of $p$
if $n=e^+$; these are again trivial if $p$ is odd.
In all cases the divided powers are compatible with
the canonical divided powers of $p$, and we obtain frames
$$
\mathscr{D}^+_{R_{n^+}/R_n}=
({\mathbb{W}}^+(R_{n^+}),{\mathbb{I}}^+_{R_{n^+}/R_n},R_n,f,\tilde f_1).
$$
Let $T_n$ be the image of
$S_{<m>}\xrightarrow{\varkappa_S}{\mathbb{W}}^+(R)\to{\mathbb{W}}^+(R_n)$.
Since $\varkappa_S\sigma=f\varkappa_S$,
the ring $T_n$ is stable under $f$.
Let $K_n$ be the kernel of $T_n\to R_n$ and let
$\tilde K_{n}$ be the kernel of $T_{n^+}\to R_n$.
We claim that $\tilde f_1(\tilde K_n)\subseteq T_{n^+}$.
If this is proved, we obtain frames
$$
\mathscr{T}_n=(T_n,K_n,R_n,f,f_1)
$$
$$
\mathscr{T}_{n^+/n}=(T_{n^+},\tilde K_n,R_n,f,\tilde f_1)
$$
and a commutative diagram of frames with strict homomorphisms:
$$
\xymatrix@M+0.2em{
\mathscr{T}_{n^+} \ar[r]^-{\psi'} \ar[d]_{\iota_{n^+}} &
\mathscr{T}_{n^+/n} \ar[r]^-{\pi'} \ar[d] &
\mathscr{T}_n \ar[d]^{\iota_n} \\
\mathscr{D}^+_{R_{n^+}} \ar[r]^-\psi &
\mathscr{D}^+_{R_{n^+}/R_n} \ar[r]^-\pi &
\mathscr{D}^+_{R_n}
}
$$
Here $\pi$ is crystalline because the hypotheses of
Theorem \ref{Th-frame-crys} are satisfied; see the
proof of Corollary \ref{Co-DDD+-crys}.
Since the vertical arrows are injective, it follows
that $\pi'$ satisfies the hypotheses of Theorem
\ref{Th-frame-crys} as well, thus $\pi'$ is crystalline.
Moreover lifts of windows under $\psi$ and under $\psi'$
correspond to lifts of the Hodge filtration from $R_n$
to $R_{n^+}$ in the same way.
Since $\iota_1$ is bijective, it follows that
$\iota_n$ is crystalline for each $n$.
Consider the limit frame
$$
\mathscr{T}=\varprojlim_n\mathscr{T}_n=(T,K,R,f,f_1).
$$
The inclusion $\iota:\mathscr{T}\to\mathscr{D}^+_R$
is the projective limit over $n$ of $\iota_n$
and thus crystalline.
Since $S_{<m>}$ is noetherian by Proposition \ref{Pr-S-m},
Lemma \ref{Le-lim-surjective} implies that
$T=\varprojlim_nT_n$ is the image of
$\varkappa_S:S_{<m>}\to{\mathbb{W}}^+(R)$.
If $\varkappa_S$ is injective, we get
$\mathscr{S}_{\!<m>}=\mathscr{T}$, so $\mathscr{S}_{<m>}\to\mathscr{D}^+_R$
is crystalline as required.
Since we have not proved that $\varkappa_S$ is injective
we need an extra argument. Let $\mathfrak{a}$ be the kernel of
$\varkappa_S:S_{<m>}\to{\mathbb{W}}^+(R)$
and let $\mathfrak{a}_n=\mathfrak{a}\cap J^n\mathfrak{S}_{\mathbb{Q}}$ for $n\ge 1$;
here we use
that $S$ is a subring of $\mathfrak{S}_{\mathbb{Q}}$ by Proposition \ref{Pr-S}.
We have $\mathfrak{a}=\mathfrak{a}_1$.
The ideals $\mathfrak{a}_n$ of $S_{<m>}$ are stable under $\sigma$,
and they are also stable under $\sigma_1$ since
$S_{<m>}/\mathfrak{a}$ and $\mathfrak{a}_n/\mathfrak{a}_{n+1}$ have no $p$-torsion.
Thus we can define frames
$\mathscr{S}_{\!<m>,n}=(S_{<m>}/\mathfrak{a}_n,I_{<m>}/\mathfrak{a}_n,R,\sigma,\sigma_1)$.
We have $\mathscr{S}_{\!<m>,1}=\mathscr{T}$,
and the projective limit over $n$ of $\mathscr{S}_{\!<m>,n}$
is isomorphic to $\mathscr{S}_{\!<m>}$ by Lemma \ref{Le-lim-surjective}
and Proposition \ref{Pr-S}.
The ideal $\mathfrak{a}_n/\mathfrak{a}_{n+1}$ is a finitely generated
$W(k)$-submodule of $(J^n/J^{n+1})\otimes{\mathbb{Q}}$
that contains $J^n/J^{n+1}$.
Since the conditions of Proposition \ref{Pr-varkappa-image}
are satisfied, the endomorphism
$\sigma_1$ of $\mathfrak{a}_n/\mathfrak{a}_{n+1}$ is $p$-adically nilpotent.
Thus $\mathscr{S}_{\!<m>,n+1}\to\mathscr{S}_{\!<m>,n}$ is crystalline;
see the proof of \cite[Theorem 9.3]{Lau-Frames}.
It follows that $\mathscr{S}_{\!<m>}\to\mathscr{T}$ is crystalline,
so $\mathscr{S}_{<m>}\to\mathscr{D}^+_R$ is crystalline too.
To prove Proposition \ref{Pr-varkappa-S-2}
it remains to show that $\tilde f_1(\tilde K_n)$
is contained in $T_{n^+}$.
Let $M_n$ be the kernel of $S_{<m>}\to R_n$,
so $\tilde K_n$ is the image of $M_n\to{\mathbb{W}}^+(R_{n^+})$.
Since $\varkappa_S\sigma=f\varkappa_S$
and since $f_1$ is $f$-linear it suffices
to show that a set of generators $x_i$ of the ideal $M_n$
with images $\varkappa_S(x_i)=\bar x_i\in\tilde K_n$ satisfies
$f_1(\bar x_i)\in T_{n^+}$.
Since $\mathfrak{m}_R=JR$, for $n\ne e^+$ the ideal $M_n$ is generated
by $I_{<m>}$ and $J^n$, while $M_{e^+}$
is generated by $I_{<m>}$ and $J^{e+1}$ and $p$.
We check these generators case by case.
First, for $x\in I_{<m>}$ we have $f_1(\bar x)\in T_{n^+}$
because $\mathscr{S}_{\!<m>}\to\mathscr{D}^+_{R_{n^+}}$ is a frame homomorphism.
Assume that $n\ne e^+$.
The homomorphism $\delta:J^n/J^{n+1}\to W(J^n/J^{n+1})$
is given by $\delta(x)=(x,\sigma_1(x),(\sigma_1)^2(x),\ldots)$.
Indeed, applying the Witt polynomial $w_n$ to this equation
gives $\sigma^n(x)=p^n(\sigma_1)^n(x)$, which is true.
Since the divided powers on $\mathfrak{m}_R^{n}/\mathfrak{m}_R^{n^+}$ are trivial,
the endomorphism $\tilde f_1$ of $W(\mathfrak{m}_R^{n}/\mathfrak{m}_R^{n^+})$ is
given by a shift to the left. Thus the map
$\varkappa_S:J^n/J^{n+1}\to W(\mathfrak{m}_R^{n}/\mathfrak{m}_R^{n^+})$
satisfies $\varkappa_S\sigma_1=\tilde f_1\varkappa_S$,
and we see that $f_1(\bar x)\in T_{n^+}$ for
$x\in J^n$.
Assume now that $n=e^+$. Since $J^{e+1}$ maps to
zero in $W(R_{e+1})$ it remains to show that
$\tilde f_1(p)\in T_{n^+}$. Now $\Log(p-v(1))=[p,0,0,\ldots]$;
cf.\ the proof of Lemma \ref{Le-WW-large-v}.
Thus $\tilde f_1(p)=f_1(v(1))=1$.
\end{proof}
Theorem \ref{Th-varkappa-S} follows from
Propositions \ref{Pr-varkappa-S-1} and \ref{Pr-varkappa-S-2}.
\qed
\begin{Remark}
Assume that $r=1$, i.e.\ $R$ is a discrete valuation ring.
If $pR=\mathfrak{m}_R^e$, the ring $S$ is the $p$-adic completion
of $W(k)[[t]][\{t^{em}/m!\}_{m\ge 1}]$. It is easy
to see that each quotient $S/p^nS$ is admissible,
so the $p$-adic ring $S$ is an admissible topological ring.
In particular, ${\mathbb{W}}^+(S)$ is defined.
Since we assumed that the image of $\delta:\mathfrak{S}\to W(\mathfrak{S})$
lies in ${\mathbb{W}}(\mathfrak{S})$,
the image of $\delta:S\to W(S)$ lies in ${\mathbb{W}}^+(S)$;
here one uses that
${\mathbb{W}}^+(S)\to R$ is the projective limit of the divided
power extensions ${\mathbb{W}}^+(S/p^nS)\to R/p^nR$ and that each
${\mathbb{W}}^+(S/p^nS)$ is $p$-adic.
If $p\ge 3$ this means that $\mathscr{S}$ is a Dieudonn\'e frame
in the sense of \cite[Definition 3.1]{Zink-Windows}, and
Theorem \ref{Th-varkappa-S} becomes a special case of
\cite[Theorem 3.2]{Zink-Windows}.
For $p=2$ the proof of loc.cit.\ works as well.
The starting point is the construction of an inverse
functor of $\varkappa_{S*}$; it maps a $\mathscr{D}_R^+$-window
$\mathscr{P}$ to the value of its crystal ${\mathbb{D}}^+(\mathscr{P})_{S/R}$,
equipped with an appropriate $\mathscr{S}$-window structure.
If $r\ge 2$, the ring $S$ is not admissible and thus
the crystal of a $\mathscr{D}_R^+$-window can not be evaluated at $S/R$.
However, one can define by hand a sub-frame
$\mathscr{D}^+_{S/R}$ of $\mathscr{W}_{S/R}$ such that
$\mathscr{D}^+_{S/R}\to\mathscr{D}^+_R$ is crystalline.
This allows to evaluate the crystal at $S/R$ and
to define an inverse functor of $\varkappa_{S*}$ as before.
The underlying ring of $\mathscr{D}^+_{S/R}$ is defined as follows.
Let $S_{m,n}$ be the image of $S_{<m>}\to S/p^nS$
and let $I_{m,n}$ be the kernel of $S_{m,n}\to R/p^nR$.
The divided Witt polynomials define an isomorphism
$\Log:W(I/p^nI)\cong (I/p^nI)^\infty$, and our ring is
$\varprojlim_n\varinjlim_m$ of the rings
${\mathbb{W}}^+(S_{0,n})+\Log^{-1}((I_{m,n})^{<\infty>})$.
If one takes $\varprojlim_n$ of these rings for fixed $m\ge 1$,
one gets a frame $\mathscr{D}^+_{S_{<m>}/R}$ with a crystalline
homomorphism to $\mathscr{D}^+_R$.
This allows to construct the inverse functor from
$\mathscr{D}^+_R$-windows to $\mathscr{S}_{\!<m>}$-windows.
We leave out the details.
\end{Remark}
| -201,331.839239 |
[
-2.771484375,
2.4765625
] | 29.775397 |
[
-2.095703125,
1.0595703125,
-2.130859375,
-5.67578125,
-1.23046875,
8.2109375
] |
[
2.8671875,
9.1796875,
1.0390625,
5.15234375
] | 1,471 | 22,972 |
[
-3.0625,
3.775390625
] | 33.278922 |
[
-4.73046875,
-3.111328125,
-5.109375,
-2.4765625,
1.18359375,
12.671875
] | 0.423946 | 13.387624 | 13.990945 | 1.160908 |
[
1.284579873085022
] | -123,593.09343 | 5.698807 | -199,388.428371 | 0.305546 | 6.132034 |
[
-1.2490234375,
-2.91796875,
-4,
-5.3203125,
1.603515625,
11.890625
] |
[
-5.45703125,
-1.3740234375,
-1.8271484375,
-0.84521484375,
3.24609375,
3.42578125
] | |
BkiUc_Q5qoYAlE7mUWSI
|
\chapter{\LARGE \bf {#1}}\vspace{-40mm}}
\renewcommand{\sectionl}[1]{\section{\LARGE \bf {#1}}}
\renewcommand{\subsectionl}[1]{\subsection{\LARGE \bf {#1}}}
\renewcommand{\subsubsectionl}[1]{\subsubsection{\Large \bf {#1}}}
\renewcommand{\paragraphl}[1]{\paragraph{\LARGE \bf {#1}}}
\renewcommand{\subparagraphl}[1]{\subparagraph{\LARGE \bf {#1}}}
\renewcommand{\captionl}[1]{\caption{\LARGE \bf {#1}}}
\renewcommand{\Largel}{\Large}}{{\caption}}
} \fi
\newcommand\end{proof}{\hfill{$\blacksquare$}\medskip}
\newenvironment{proofdu}[1]{{\noindent \bf D\'emonstration #1.}\quad}{\hfill{$\blacksquare$}\medskip}
\newcommand\ML{{\rm M}}
\def\hat{a}{\hat{a}}
\def\hat{b}{\hat{b}}
\def\hat{c}{\hat{c}}
\def\hat{d}{\hat{d}}
\def\hat{e}{\hat{e}}
\def\hat{f}{\hat{f}}
\def\hat{g}{\hat{g}}
\def\hat{h}{\hat{h}}
\def\hat{i}{\hat{i}}
\def\hat{j}{\hat{j}}
\def\hat{k}{\hat{k}}
\def\hat{l}{\hat{l}}
\def\hat{m}{\hat{m}}
\def\hat{n}{\hat{n}}
\def\hat{o}{\hat{o}}
\def\hat{p}{\hat{p}}
\def\hat{q}{\hat{q}}
\def\hat{r}{\hat{r}}
\def\hat{s}{\hat{s}}
\def\hat{t}{\hat{t}}
\newcommand\tth{\hat{t}}
\def\hat{u}{\hat{u}}
\def\hat{v}{\hat{v}}
\def\hat{w}{\hat{w}}
\def\hat{x}{\hat{x}}
\def\hat{y}{\hat{y}}
\def\hat{z}{\hat{z}}
\newcommand\Uh{\hat{U}}
\def\bar{a}{\bar{a}}
\def\bar{b}{\bar{b}}
\def\bar{c}{\bar{c}}
\def\bar{d}{\bar{d}}
\def\bar{e}{\bar{e}}
\def\bar{f}{\bar{f}}
\def\bar{g}{\bar{g}}
\def\bar{h}{\bar{h}}
\def\bar{i}{\bar{i}}
\def\bar{j}{\bar{j}}
\def\bar{k}{\bar{k}}
\def\bar{l}{\bar{l}}
\def\bar{m}{\bar{m}}
\def\bar{n}{\bar{n}}
\def\bar{o}{\bar{o}}
\def\bar{p}{\bar{p}}
\def\bar{q}{\bar{q}}
\def\bar{r}{\bar{r}}
\def\bar{s}{\bar{s}}
\def\bar{t}{\bar{t}}
\def\bar{u}{\bar{u}}
\def\bar{v}{\bar{v}}
\def\bar{w}{\bar{w}}
\def\bar{x}{\bar{x}}
\def\bar{y}{\bar{y}}
\def\zb{\bar{z}}
\def\bar{A}{\bar{A}}
\def\bar{B}{\bar{B}}
\def\bar{C}{\bar{C}}
\def\bar{D}{\bar{D}}
\def\bar{E}{\bar{E}}
\def\bar{F}{\bar{F}}
\def\bar{G}{\bar{G}}
\def\bar{H}{\bar{H}}
\def\bar{I}{\bar{I}}
\def\bar{J}{\bar{J}}
\def\bar{K}{\bar{K}}
\def\bar{L}{\bar{L}}
\def\bar{M}{\bar{M}}
\def\bar{N}{\bar{N}}
\def\bar{O}{\bar{O}}
\def\bar{P}{\bar{P}}
\def\bar{Q}{\bar{Q}}
\def\bar{R}{\bar{R}}
\def\bar{S}{\bar{S}}
\def\bar{T}{\bar{T}}
\def\bar{U}{\bar{U}}
\def\bar{V}{\bar{V}}
\def\bar{W}{\bar{W}}
\def\bar{X}{\bar{X}}
\def\bar{Y}{\bar{Y}}
\def\bar{Z}{\bar{Z}}
\def\bar{C}{\bar{C}}
\def\bar{T}{\bar{T}}
\def\bar{Z}{\bar{Z}}
\def\tilde{a}{\tilde{a}}
\def\tilde{b}{\tilde{b}}
\def\tilde{c}{\tilde{c}}
\def\tilde{d}{\tilde{d}}
\def\tilde{e}{\tilde{e}}
\def\tilde{f}{\tilde{f}}
\def\tilde{g}{\tilde{g}}
\def\tilde{h}{\tilde{h}}
\def\tilde{i}{\tilde{i}}
\def\tilde{j}{\tilde{j}}
\def\tilde{k}{\tilde{k}}
\def\tilde{l}{\tilde{l}}
\def\tilde{m}{\tilde{m}}
\def\tilde{n}{\tilde{n}}
\def\tilde{o}{\tilde{o}}
\def\tilde{p}{\tilde{p}}
\def\tilde{q}{\tilde{q}}
\def\tilde{r}{\tilde{r}}
\def\tilde{s}{\tilde{s}}
\def\tilde{t}{\tilde{t}}
\def\tilde{u}{\tilde{u}}
\def\tilde{v}{\tilde{v}}
\def\tilde{w}{\tilde{w}}
\def\tilde{x}{\tilde{x}}
\def\tilde{y}{\tilde{y}}
\def\tilde{z}{\tilde{z}}
\def\tA{\tilde{A}}
\def\tL{\tilde{L}}
\def\tS{\tilde{S}}
\def\tV{\tilde{V}}
\def\tW{\tilde{W}}
\def\tX{\tilde{X}}
\def\tY{\tilde{Y}}
\def\tZ{\tilde{Z}}
\def\ubf { {\bf u}}
\def\vbf { {\bf v}}
\def\wbf { {\bf w}}
\def\xbf { {\bf x}}
\def\ybf { {\bf y}}
\def{\mathcal A}{{\mathcal A}}
\def{\mathcal B}{{\mathcal B}}
\def{\mathcal C}{{\mathcal C}}
\def{\mathcal D}{{\mathcal D}}
\def{\mathcal E}{{\mathcal E}}
\def{\mathcal F}{{\mathcal F}}
\def{\mathcal G}{{\mathcal G}}
\def{\mathcal H}{{\mathcal H}}
\def{\mathcal I}{{\mathcal I}}
\def{\mathcal J}{{\mathcal J}}
\def{\mathcal K}{{\mathcal K}}
\def{\mathcal L}{{\mathcal L}}
\def{\mathcal M}{{\mathcal M}}
\def{\mathcal N}{{\mathcal N}}
\def{\mathcal O}{{\mathcal O}}
\def{\mathcal P}{{\mathcal P}}
\def{\mathcal Q}{{\mathcal Q}}
\def{\mathcal R}{{\mathcal R}}
\def{\mathcal S}{{\mathcal S}}
\def{\mathcal T}{{\mathcal T}}
\def{\mathcal U}{{\mathcal U}}
\def{\mathcal V}{{\mathcal V}}
\def{\mathcal W}{{\mathcal W}}
\def{\mathcal X}{{\mathcal X}}
\def{\mathcal Y}{{\mathcal Y}}
\def{\mathcal Z}{{\mathcal Z}}
\def{\mathcal L}{{\mathcal L}}
\def\mathbb{P}{{\mathcal P}}
\def\widetilde{\cal P}{\widetilde{\cal P}}
\def\mathcal{A}{\mathcal{A}}
\def\mathcal{B}{\mathcal{B}}
\def\mathcal{C}{\mathcal{C}}
\def\mathcal{D}{\mathcal{D}}
\def\mathcal{E}{\mathcal{E}}
\def\mathcal{F}{\mathcal{F}}
\def\mathcal{G}{\mathcal{G}}
\def\mathcal{H}{\mathcal{H}}
\def\mathcal{I}{\mathcal{I}}
\def\mathcal{J}{\mathcal{J}}
\def\mathcal{K}{\mathcal{K}}
\def\mathcal{L}{\mathcal{L}}
\def\mathcal{M}{\mathcal{M}}
\def\mathcal{N}{\mathcal{N}}
\def\mathcal{P}{\mathcal{P}}
\def\mathcal{Q}{\mathcal{Q}}
\def\mathcal{R}{\mathcal{R}}
\def\mathcal{S}{\mathcal{S}}
\def\mathcal{T}{\mathcal{T}}
\def\mathcal{U}{\mathcal{U}}
\def\mathcal{V}{\mathcal{V}}
\def\mathcal{W}{\mathcal{W}}
\def\mathcal{X}{\mathcal{X}}
\def\mathcal{Y}{\mathcal{Y}}
\def\mathcal{Z}{\mathcal{Z}}
\newcommand{\mathcal{B}}{\mathcal{B}}
\newcommand{\mathcal{C}}{\mathcal{C}}
\newcommand{\mathcal{E}}{\mathcal{E}}
\newcommand{\mathcal{L}}{\mathcal{L}}
\newcommand{\mathcal{S}}{\mathcal{S}}
\newcommand{\mathcal{T}}{\mathcal{T}}
\newcommand{\mathcal{V}}{\mathcal{V}}
\def\overline{\bf H}{\overline{\bf H}}
\def\underline{a}{\underline{a}}
\def\underline{x}{\underline{x}}
\def\underline{V}{\underline{V}}
\def\varepsilon{\varepsilon}
\def\bar\varepsilon{\bar\varepsilon}
\def{\Omega}{{\Omega}}
\def{\omega}{{\omega}}
\def\lambda{\lambda}
\def\bar{\lambda}{\bar{\lambda}}
\def\bar{\mu}{\bar{\mu}}
\def\bar{\nu}{\bar{\nu}}
\def{\bar\tau}{{\bar\tau}}
\def{\bf 1}{{\bf 1}}
\newcommand{\ceil}[1]{\lceil #1 \rceil}
\newcommand{\floor}[1]{\lfloor #1 \rfloor}
\def\mathop{\stackrel{\rightharpoonup}{\subset}}{\mathop{\stackrel{\rightharpoonup}{\subset}}}
\def\mathop{\rm ad}{\mathop{\rm ad}}
\def\mathop{\rm affhull}{\mathop{\rm affhull}}
\def\mathop{\rm argmin}{\mathop{\rm argmin}}
\def\mathop{\rm argmax}{\mathop{\rm argmax}}
\def\mathop{\rm cl}{\mathop{\rm cl}}
\newcommand\Comp{\mathop{\rm Comp}}
\def\mathop{\rm \overline{affhull}}{\mathop{\rm \overline{affhull}}}
\def\mathop{\rm cone}{\mathop{\rm cone}}
\def\mathop{\rm core}{\mathop{\rm core}}
\newcommand\conebar{\mathop{\rm \overline{cone}}}
\def\mathop{\rm conv}{\mathop{\rm conv}}
\def\mathop{\rm \overline{conv}}{\mathop{\rm \overline{conv}}}
\def\mathop{\rm deg}{\mathop{\rm deg}}
\def\mathop{\rm det}{\mathop{\rm det}}
\def\mathop{\rm det}{\mathop{\rm det}}
\def{\mathop{\rm diag}}{{\mathop{\rm diag}}}
\newcommand\diam{\mathop{\rm diam}}
\def\mathop{\rm dist}{\mathop{\rm dist}}
\def\mathop{\rm div}{\mathop{\rm div}}
\def\mathop{{\rm dom}}{\mathop{{\rm dom}}}
\def\mathop{\text{epi}}{\mathop{\text{epi}}}
\newcommand{\mathop{\rm ess}}{\mathop{\rm ess}}
\def\mathop{\rm essinf}{\mathop{\rm essinf}}
\def\mathop{\rm esssup}{\mathop{\rm esssup}}
\def\mathop{\rm esssup}{\mathop{\rm esssup}}
\def\mathop{\rm int}{\mathop{\rm int}}
\def\mathop{\rm inv}{\mathop{\rm inv}}
\def\mathop{\rm Isom}{\mathop{\rm Isom}}
\def\mathop{\rm lin}{\mathop{\rm lin}}
\newcommand\Jac{{\mathop{\rm Jac}}}
\newcommand\Lip{\mathop{\rm Lip}}
\newcommand\snorm{\|}
\newcommand\lnlt{\mathop{\rm lnlt}}
\newcommand\psicirc{ \psi}
\def\mathop{\rm Im}{\mathop{\rm Im}}
\def\mathop{\rm rint}{\mathop{\rm rint}}
\def\mathop{\rm rint}{\mathop{\rm rint}}
\def\mathop{\rm sign}{\mathop{\rm sign}}
\def\mathop{\rm supp}{\mathop{\rm supp}}
\def\mathop{\rm trace}{\mathop{\rm trace}}
\def\mathop{\rm val}{\mathop{\rm val}}
\def\mathop{\rm var}{\mathop{\rm var}}
\def\mathop{{\rm width}}{\mathop{{\rm width}}}
\def\mathop{\rm View}{\mathop{\rm View}}
\def\mathop{\rm Inf}{\mathop{\rm Inf}}
\def\mathop{\rm Ker}{\mathop{\rm Ker}}
\def\mathop{\rm Min}{\mathop{\rm Min}}
\def\mathop{\rm Max}{\mathop{\rm Max}}
\def\mathop{\delta w}{\mathop{\delta w}}
\def\mathop{\underline{\lim}}{\mathop{\underline{\lim}}}
\def\mathop{\overline{\lim}}{\mathop{\overline{\lim}}}
\def\mbox{$\frac{1}{2}$}{\mbox{$\frac{1}{2}$}}
\def\mbox{$\frac{1}{6}$}{\mbox{$\frac{1}{6}$}}
\def{\bf 1}{{\bf 1}}
\def\sbdeux#1#2{\mbox{\scriptsize$#1$}\atop\mbox{\scriptsize$#2$}}
\def\sbtrois#1#2#3{\mbox{\scriptsize$#1$}\atop\mbox{\scriptsize$#2$}\atop\mbox{\scriptsize$#3$}}
\newcommand{\mathbb{N}}{\mathbb{N}}
\newcommand{\mathbb{Z}}{\mathbb{Z}}
\newcommand{\mathbb{R}}{\mathbb{R}}
\newcommand{\mathsf{A}}{\mathsf{A}}
\newcommand{\mathsf{G}}{\mathsf{G}}
\newcommand{\mathsf{N}}{\mathsf{N}}
\newcommand{\mathsf{T}}{\mathsf{T}}
\newcommand{\mathsf{V}}{\mathsf{V}}
\newcommand{\mathsf{E}}{\mathsf{E}}
\newcommand{\mathcal{G}}{\mathcal{G}}
\newcommand{\mathcal{N}}{\mathcal{N}}
\newcommand{\mathcal{A}}{\mathcal{A}}
\def\mathbb{C}{\mathbb{C}}
\def\mathbb{N}{\mathbb{N}}
\def\mathbb{N}{\mathbb{N}}
\def\mathbb{P}{\mathbb{P}}
\def\mathbb{Q}{\mathbb{Q}}
\def\mathbb{Q}{\mathbb{Q}}
\def\mathbb{R}{\mathbb{R}}
\def\mathbb{Z}{\mathbb{Z}}
\def\mathbb{Z}{\mathbb{Z}}
\newcommand{I\!\!E}{I\!\!E}
\newcommand{\mathbb{\bar{R}}}{\mathbb{\bar{R}}}
\newcommand{\overline{\mathbb{R}}}{\overline{\mathbb{R}}}
\newcommand{\rbar^{_{\scriptstyle\sN}}}{\overline{\mathbb{R}}^{_{\scriptstyle\mathcal{N}}}}
\newcommand\be{\begin{equation}}
\newcommand\ee{\end{equation}}
\newcommand\ba{\begin{array}}
\newcommand\ea{\end{array}}
\newcommand{\begin{eqnarray}}{\begin{eqnarray}}
\newcommand{\end{eqnarray}}{\end{eqnarray}}
\newcommand{\begin{eqnarray*}}{\begin{eqnarray*}}
\newcommand{\end{eqnarray*}}{\end{eqnarray*}}
\newenvironment{paoplist}{\vspace{-1ex}\begin{list}{-}
{\itemsep 0mm \leftmargin 2mm \labelwidth 0mm}}
{\end{list} \vspace{-1ex}}
\newenvironment{myenumerate}{
\renewcommand{\theenumi}{\roman{enumi}}
\renewcommand{\labelenumi}{(\theenumi)}
\begin{enumerate}}{\end{enumerate}}
\newcommand{\noindent}{\noindent}
\newcommand{\bigskip}{\bigskip}
\newcommand{\medskip}{\medskip}
\newcommand{{\bigskip \noindent}}{{\bigskip \noindent}}
\newcommand{\refeq}[1]{(\ref{#1})}
\def\marginpar{$\leftarrow$}{\marginpar{$\leftarrow$}}
\def\rightarrow{\rightarrow}
\def\rightarrow{\rightarrow}
\def\displaystyle{\displaystyle}
\def\displaystyle{\displaystyle}
\def\langle{\langle}
\def\langle{\langle}
\def\rangle{\rangle}
\newcommand{\mypsfig}[3]
{\begin{figure}[hbtp]
\centerline{\input #1}
\caption{\rm{#2}} \label{#3}
\end{figure}}
\section{Introduction}
In this article we deal with an optimal control problem governed by the dynamics
\begin{equation*}
\dot x_t = f_0(x_t) + u_t f_1(x_t)\quad \text{for a.a. } t\in [0,T],
\end{equation*}
subject to the control bounds
$$
u_{\rm min} \leq u_t \leq u_{\rm max},
$$
with $u_{\rm min} < u_{\rm max}$,
endpoint constraints like
$$
\Phi(x_0,x_T) = 0,
$$
and a scalar state constraint of the form
$$
g(x_t) \leq 0.
$$
For this class of problems, we propose a shooting-like numerical scheme and we show a sufficient condition for its local quadratic convergence, that is also a second order sufficient condition for optimality (in a particular sense to be specified later on). Additionally, we solve an example of practical interest, for which we also prove optimality by applying second order sufficient optimality conditions obtained in \cite{AronnaBonnansGoh2016}.
This investigation is strongly motivated by applications since we deal with both control and state constraints, which naturally appear in realistic models. Many practical examples that are covered by our chosen framework can be found in the existing literature. A non exhaustive list includes the prey-predator model \cite{GLV74}, the Goddard problem in presence of a dynamic pressure limit \cite{SeywaldCliff1993,GraichenPetit2008}, the optimal control of the atmospheric arc for re-entry of a space shuttle seen in \cite{Bonnard2003}, an optimal production and maintenance system studied in \cite{MaurerKimVossen2005}, and a recent optimization problem on running strategies \cite{AftalionBonnans2014}.
We refer also to \cite{dePinho2005}, \cite{MaurerDePinho2014}, \cite{Schaettler2006} and references therein.
As it is commonly known, the application of the necessary conditions provided by Pontryagin's Maximum Principle leads to an associated two-point boundary-value problem (TPBVP) for the optimal trajectory and its associated multiplier \cite{Vinter2000}. A natural way for solving TPBVPs numerically is the application of {\em shooting algorithms} \cite{MorRilZan62}. This type of algorithms has been used extensively for the resolution of optimal control problems (see {\em e.g.} \cite{Bul71,BockPlitt1984,Pes94} and references therein). In particular, shooting methods have been applied to control-affine problems both with and without state constraints. Some works in this direction are mentioned in the sequel. Maurer \cite{Mau76} proposed a shooting scheme for solving a problem with bang-singular solutions, which was generalized quite recently by Aronna, Bonnans and Martinon in \cite{AronnaBonnansMartinon2013}, where they provided a sufficient condition for its local convergence.
Both these articles \cite{Mau76} and \cite{AronnaBonnansMartinon2013} analyze the case with control bounds and no state constraints.
Practical control-affine problems with state constraints were solved numerically in several articles, a non extensive list includes Maurer and Gillessen \cite{MauGil75}, Oberle \cite{Obe79} and Fraser-Andrews \cite{Fra89}.
Up to our knowledge, there is no result in the existing literature concerning sufficient conditions for the convergence of shooting algorithms in the framework considered here.
The paper is organized as follows. In Sections \ref{SecProblem} and \ref{SecArcs} we introduce the problem and give the basic definitions.
A shooting-like method and a sufficient condition for its local quadratic convergence are given in Sections \ref{SecShooting} and \ref{SecSuff}, respectively. The algorithm is implemented in Section \ref{SecExamples} where we solve numerically a variation of the regulator problem and we prove the optimality of the solution analytically.
\vspace{4pt}
\noindent {\bf Notations.}
Let $\mathbb{R}^k$ denote the $k-$dimensional real space, {\em i.e.} the space of column real vectors of dimension $k,$ and by $\mathbb{R}^{k*}$ its corresponding dual space, which consists of $k-$dimensional row real vectors. With $\mathbb{R}^k_+$ and $\mathbb{R}^k_-$ we refer to the subsets of $\mathbb{R}^k$ consisting of vectors with nonnegative, respectively nonpositive, components.
We write $h_t$ for the value of function $h$ at time $t$ if $h$ is a function that depends only on $t,$ and by $h_{i,t}$ the $i$th component of $h$ evaluated at $t.$
Let $h(t+)$ and $h(t-)$ be, respectively, the right and left limits of $h$ at $t,$ if they exist.
Partial derivatives of a function $h$ of $(t,x)$ are referred as $D_th$ or $\dot{h}$ for the derivative in time, and $D_xh,$ $h_x$ or $h'$ for the differentiations with respect to space variables. The same convention is extended to higher order derivatives.
By $L^p(0,T)^k$ we mean the Lebesgue space with domain equal to the interval $[0,T]\subset \mathbb{R}$ and with values in $\mathbb{R}^k.$ The notations $W^{q,s}(0,T)^k$ and $H^1(0,T)^k$ refer to the Sobolev spaces (see Adams \cite{Ada75} for further details on Sobolev spaces).
We let $BV(0,T)$ be the set of functions with bounded total variation.
In general, when there is no place for confusion, we omit the argument $(0,T)$ when referring to a space of functions. For instance, we write $L^\infty(0,T)$ for $L^\infty(0,T),$ or $(W^{1,\infty})^{k*}$ for the space of $W^{1,\infty}-$functions from $[0,T]$ to $\mathbb{R}^{k*}.$
We say that a function $h: \mathbb{R}^k \to \mathbb{R}^d$ is of class $C^\ell$ if it is $\ell-$times continuously differentiable in its domain.
\if{
\bigskip
{\bf References to be included}.
\begin{itemize}
\item Second order necessary conditions: abstract results \cite{PAOP}
(Kawasaki, Cominetti).
\item {\em Geometric approach:} Bonnard and Kupka \cite{MR1441391}:
optimality for the minimal time problem in the absence of
conjugate point.
\item {\em Others to look at:} Hoehener \cite{Hoe12}, Fraser-Andrews \cite{Fra96}, ...
\end{itemize}
\bigskip
}\fi
\section{The problem}\label{SecProblem}
Let us consider $L^\infty(0,T)$ and $W^{1,\infty}(0,T;\mathbb{R}^n)$ as control and state spaces, respectively.
We say that a control-state pair $(u,x)\in L^\infty(0,T)\times W^{1,\infty}(0,T;\mathbb{R}^n)$
is a {\em trajectory} if it satisfies both the {\em state equation}
\be
\label{bsbstateeq}
\dot x_t = f_0(x_t) +u_{t} f_1(x_t) \quad \text{for a.a. } t\in [0,T],
\ee
and the finitely many {\em endpoint constraints}
of equality type given by
\be
\Phi(x_0,x_T) = 0.
\ee
Here $f_0$ and $f_1$ are assumed to be Lipschitz continuous and twice continuously differentiable vector fields over $\mathbb{R}^n$,
$\Phi$ is of class $C^2$ from $\mathbb{R}^n\times\mathbb{R}^n$ to $\mathbb{R}^{q}.$
Under these hypotheses, for any pair control-initial condition $(u,x_0)$ in $L^\infty(0,T)\times \mathbb{R}^n$, the state equation
\eqref{bsbstateeq} has a unique solution. Additionally, we consider
a {\em cost functional}
$$
\phi:\mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R},$$
the
{\em control bounds}
\be
\label{bsbcc}
u_{\rm min}\leq u_{t} \leq u_{\rm max} \quad \text{for a.a. } t\in [0,T],
\ee
where $ u_{\rm min} < u_{\rm max}$,
and a {\em scalar state constraint}
\be
\label{stateconstraint1}
g(x_t) \leq 0\quad \text{for all } t\in [0,T],
\ee
with the functions $\phi$ and $g:\mathbb{R}^n\rightarrow \mathbb{R}$ being of class $C^2.$
A trajectory $(u,x)$ is said to be {\em feasible} if it satisfies
\eqref{bsbcc}-\eqref{stateconstraint1}.
\begin{remark}[On the control bounds]
We allow $u_{\min}$ and $u_{\max}$ to be either finite real numbers, or to take the values $-\infty$ or $+\infty,$ respectively, meaning that we also consider problems with control constraints of the form $u_t \leq u_{\max}$ or $u_{\min} \leq u_t$, as well as problems in the absence of control constraints.
\end{remark}
Summarizing, this article deals with the optimal control problem in the Mayer form given by
\be
\label{P}\tag{P}
\min \phi(x_0,x_T);
\qquad \text{subject to \eqref{bsbstateeq}-\eqref{stateconstraint1}.}
\ee
\if{
\section{First order analysis}\label{SecFirst}
\subsection{Pontryagin's Maximum Principle and multipliers}
Set
\be\label{deff}
f(u,x) := f_0(x)+u f_1(x),
\ee
and define the {\em pre-Hamiltonian function}
and the {\em endpoint Lagrangian,} respectively, by
\begin{gather}
\label{preham} H(u,x,p) := p f(u,x) = p \big(f_0(x)+u f_1(x) \big),
\\
\label{endLag} \ell^{\Psi}(x_0,x_T) := \phi(x_0,x_T) + \Psi \Phi(x_0,x_T),
\end{gather}
where $p\in \mathbb{R}^{n*}$ and $\Psi \in \mathbb{R}^{q*}.$
Let us recall that any function $\mu$ in $BV(0,T)$ has left limits on $(0,T]$ and right limits on
$[0,T)$. Therefore, the values $\mu_{0+}$ and $\mu_{T-}$ are well-defined. Furthermore, $\mu$ has derivative in the distributional sense that belongs to the space $\mathcal{M}(0,T)$ of finite Radon measures. Conversely, any measure ${\rm d}\mu$ in $\mathcal{M}$ can be identified with the derivative of a function $\mu$ of bounded variation such that $\mu_{T}=0,$ i.e., $\mu$ belongs to the space $BV_0$ of {\em bounded variation functions vanishing $T.$ }
For the sequel, let us consider a feasible trajectory $(u,x)$.
A function $\mu$ in $BV(0,T)$ is
{\em complementary to the state constraint} if
\be
\label{mucomsc}
{\rm d} \mu \geq 0 \quad \text{and} \quad
\int_0^T g(x_t) {\rm d} \mu_{t} =0.
\ee
The {\em costate equation} associated with a pair $(\Psi,{\rm d}\mu) \in \mathbb{R}^{q*} \times \mathcal{M}(0,T)$ is given by
\be
\label{costateeq}
- {\rm d} p_t =
p_t f_x(u_t,x_t) {\rm d} t + g'(x_t) \, {\rm d} \mu_{t}
\quad \text{for a.a. } t\in [0,T],
\ee
with boundary conditions
\be
\label{lmscac7}
(- p_{0},p_{T}) = D\ell^{\Psi} (x_0,x_T).
\ee
For any $(\Psi, {\rm d}\mu) \in \mathbb{R}^{q*} \times \mathcal{M}(0,T),$ the boundary value problem \eqref{costateeq}-\eqref{lmscac7} has at most one solution $p$ in $BV^{n*}(0,T)$.
Finally, the {\em minimization of
the pre-Hamiltonian} $H$ guaranteed by Pontryagin's Maximum Principle
can be expressed through the following three conditions
\be
\label{lmscac4a}
\left\{ \ba{lll}
u_t = u_{\rm min}, & \text{if $ u_{\rm min} > -\infty$ and $p_t f_1(x_t) >0$,}
\\
u_t = u_{\rm max}, & \text{if $ u_{\rm max} < +\infty$ and $p_t f_1(x_t) <0,$}
\\
p_tf_1(x_t) = 0, &\text{if } u_{\rm min} < u_t < u_{\rm max},
\ea\right. \qquad \text{for a.a. } t \in [0,T].
\ee
We shall write $\lambda := (\Psi,p,{\rm d}\mu)$ for the triplets of dual variables, that are elements of the set
\be
E^\Lambda:=\mathbb{R}^{q*}\times BV^{n*}(0,T) \times
\mathcal{M}(0,T).
\ee
The {\em Lagrangian} of problem \eqref{P} is given by $\mathcal{L} \colon L^\infty(0,T) \times W^{1,\infty}(0,T;\mathbb{R}^n) \times E^\Lambda \to \mathbb{R},$ with
\be
\label{lmscac5}
\mathcal{L}(u,x,\lambda):=\ell^{\Psi}(x_0,x_T)
+ \int_0^T p_t \big(f(u_t,x_t) - \dot x_t \big) {\rm d} t
+ \int_0^T g(x_t) {\rm d} \mu_{t}.
\ee
For a feasible trajectory $(u,x) \in L^\infty(0,T) \times W^{1,\infty}(0,T;\mathbb{R}^n),$ define the {\em set of Lagrange multipliers} as
\be
\label{Lambda}
\Lambda(u,x):= \left\{ \ba{lll}
\lambda = (\Psi,p,{\rm d}\mu) \in E^\Lambda: \;
\\
\text{\eqref{bsbstateeq}-\eqref{stateconstraint1} and
\eqref{mucomsc}-\eqref{lmscac4a} hold}
\ea\right\}.
\ee
\subsection{Constraint qualification}
In the remainder of the article we discuss conditions around a nominal feasible trajectory $(\hat{u},\hat{x}) \in L^\infty(0,T) \times W^{1,\infty}(0,T;\mathbb{R}^n).$ As a notation convention we establish that when the argument of a function is omitted, it is evaluated
at the nominal pair $(\hat{u},\hat{x}).$
Set
\be
\label{AB}
A_t := D_x f(\hat{u}_t,\hat{x}_t)\quad \text{for a.a. }t\in [0,T].
\ee
For $(v,z^0) \in L^\infty(0,T)\times \mathbb{R}^n$, let $z[v,z^0] \in W^{1,\infty}(0,T;\mathbb{R}^n)$ denote the solution
of the {\em linearized state equation}
\be
\label{lineq}
\dot z_t = A_t z_t + v_t f_{1,t} \quad \text{for a.a. }t\in [0,T],
\ee
with initial condition
\be
\label{lineq0}
z_0 = z^0.
\ee
Let $\bar\Phi$ denote the function from $L^\infty(0,T) \times \mathbb{R}^n$ to $\mathbb{R}^{q}$ that,
to each
$(u,x_0) \in L^\infty(0,T) \times \mathbb{R}^n,$ assigns the value $\big(\Phi_1(x_0,x_T),\dots,\Phi_{q}(x_0,x_T) \big),$ where $x$ is the solution of \eqref{bsbstateeq} associated with $(u,x_0).$
For the remainder in this article, we assume the following {\em constraint qualification condition,} which corresponds to {\em Robinson condition \cite{Robinson1976}} (see also \cite{AronnaBonnansGoh2016}[Remark 10]):
\be
\label{tool_ocfo.qualifoc}
\left\{ \ba{ll}
{\rm (i)} & D \bar\Phi(\hat{u},\hat{x}_0)\text{ is onto } \text{from } L^\infty(0,T) \times \mathbb{R}^n \text{ to } \mathbb{R}^{q},
\\
{\rm (ii)} & \text{there exists } (\bar{v},\zb^0)\in L^\infty(0,T) \times \mathbb{R}^n \text{ in the kernel of
$D \bar\Phi(\hat{u},\hat{x}_0)$,}
\\ & \text{such that, for some } \varepsilon>0, \text{ setting } \zb=z[\bar{v},\zb^0], \text{ one has:}
\\ & \hat{u}_t + \bar{v}_t \in [ u_{\rm min}+\varepsilon, u_{\rm max}-\varepsilon], \,
\text{ for a.a. $t \in [0,T],$}
\\ &
g(\hat{x}_t) + g'(\hat{x}_t)\zb_t < 0, \text{ for all } t\in [0,T].
\ea\right.
\ee
In the latter display, the notation $\hat{u}_t+\bar{v}_t \in [ u_{\rm min}+\varepsilon, u_{\rm max}-\varepsilon]$ means that $\hat{u}_t+\bar{v}_t \in [ u_{\rm min}+\varepsilon,+\infty)$ a.e. if $ u_{\rm max}=+\infty$ and $ u_{\rm min}$ is finite; $\hat{u}_t+\bar{v}_t \in (-\infty, u_{\rm max}-\varepsilon)$ a.e. if $ u_{\rm min}=-\infty$ and $ u_{\rm max}$ is finite, and no restriction is imposed on $\hat{u}_t+\bar{v}_t$ if both control bounds $ u_{\rm max}$ and $ u_{\rm min}$ are infinite.
}\fi
\subsection{Types of minima}
Throughout this article, we make use of two notions of optimality that are {\em weak} and {\em Pontryagin minima} and are defined as follows.
\begin{definition}[Weak and Pontryagin minima]
\label{defminpoint}
A {\em weak minimum} for \eqref{P} is a feasible trajectory $(u,x)$ for which there exists $\varepsilon>0$ such that $\phi(x_0,x_T) \leq \phi(\tilde x_0,\tilde x_T)$ for any feasible $( \tilde u,\tilde x)$ verifying $\|(\tilde u,\tilde x)-(u,x)\|_{\infty} < \varepsilon.$
A feasible trajectory $(u,x)$ is called a {\em Pontryagin minimum} for \eqref{P} if for any $M>0,$ there exists $\varepsilon_M>0$ such that $\phi(x_0,x_T) \leq \phi(\tilde x_0,\tilde x_T)$ for any feasible $( \tilde u,\tilde x)$ satisfying
\be
\label{defminpont1}
\| \tilde x - x\|_\infty +\| \tilde u -u \|_1 < \varepsilon_M,\qquad \| \tilde u - u\|_\infty < M.
\ee
\end{definition}
Note that any Pontryagin minimum is also a weak minimum. Consequently, necessary conditions that hold for weak minima, also do it for Pontryagin one.
This article provides a numerical scheme for approximating Pontryagin minima of \eqref{P}.
In order to achieve this, we make use of the auxiliary unconstrained transformed problem (TP) given in equations \eqref{costTP}-\eqref{continuityxTP}, which possesses neither control bounds not state constraints and can be solved numerically in an efficient way.
In Lemma \ref{LemmaPontWeak} below we prove that transformed Pontryagin minima of \eqref{P} that verify certain structural hypotheses are weak minima of the unconstrained transformed problem (TP).
\section{Bang, constrained and singular arcs}\label{SecArcs}
The {\em contact set} associated with the state constraint is defined as
\be
\label{defC}
C := \{ t\in [0,T]:\; g(\hat{x}_t)=0 \}.
\ee
For $0 \leq a < b \leq T$, we say that $(a,b)$ is an
{\em active arc} for the state constraint
or, shortly, a {\em $C$ arc,}
if $(a,b)$ is a maximal open interval contained in $C.$
A point $\tau\in (0,T)$ is a {\em junction point of the state constraint}
if it is the extreme point of a $C$ arc.
Similar definitions hold for the control constraint, with the difference that the control variable is only almost everywhere defined.
The {\em contact sets for the control bounds} are given by
\begin{gather*}
B_- := \{ t\in [0,T]:\; \hat{u}_t = u_{\rm min} \},
\quad B_+ := \{ t\in [0,T]:\; \hat{u}_t = u_{\rm max} \},\\
B:=B_- \cup B_+.
\end{gather*}
Note that these sets are defined up to null measure sets. Additionally, observe that if $ u_{\rm min} = - \infty$ then $B_- = \emptyset$ and, analogously, if $ u_{\rm max}=+\infty$ then $B_+ = \emptyset.$
We say that
$(a,b)$ is a {\em $B_-$} (resp. {\em $B_+)$ arc}
if $(a,b)$ is included, up to a null measure set, in
$B_-$ (resp. in $B_+$),
but no open interval strictly containing $(a,b)$ is.
We say that $(a,b)$ is a {\em $B$ arc} if it is either a
$B_{-}$ or a $B_+$ arc.
Finally, let $S$ denote the {\em singular set} given by
\be
S:=\{ t\in [0,T]: u_{\min} < \hat{u}_t < u_{\max} \text{ and } g(\hat{x}_t) <0 \}.
\ee
We say that $(a,b)$ is an {\em $S$ arc} if $(a,b)$ is included,
up to a null measure set, in $S$,
but no open interval strictly containing $(a,b)$ is.
We call {\em junction} or {\em switching times} the points $\tau \in
(0,T)$ at which the trajectory $(\hat{x},\hat{u})$ switches from one type of
arc ($B_-,B_+,C$ or $S$) to another type. Junction/switching times are
denominated by the type of arcs they separate. One can have, for instance, {\em $CS$ junction, $B_-B_+$ switching time, etc. }
Throughout the remainder of the article, we assume that the {\em state constraint is of first order}, this is,
\be
\label{1order}
g'(\hat{x}_t)f_1(\hat{x}_t) \neq 0\quad \text{on } C,
\ee
and we impose the following {\em hypotheses on the control structure:}
\be
\label{chypradn}
\left\{ \ba{cl}
{\rm (i)} &\text{the interval $[0,T]$ is (up to a zero measure set)
the disjoint}
\\ & \text{union of finitely many arcs of type $B$, $C$ and $S,$
and the set $C$} \\
& \text{does not contain isolated points,}
\\
{\rm (ii)} &\text{the control $\hat{u}$ is at uniformly positive distance
of the bounds }
\\& u_{\min} \text{ and } u_{\max}, \text{ over $C$ and $S$ arcs,}
\\
{\rm (iii)} &
\text{the control $\hat{u}$ is discontinuous at CS and SC junctions.}
\ea \right.
\ee
The example of the regulator problem studied in Section
\ref{SecExamples} fullfils the above hypothesis \eqref{chypradn} (see
as well the example given in
\cite[Remark 2]{AronnaBonnansGoh2016}).
When a control satisfying hypothesis \eqref{chypradn}(i) is, for instance, a concatenation of a bang and a singular arc, we call it a {\em BS control.} This denomination is extended for any finite sequence of arc types.
In order to formulate our shooting algorithm, we express the control as a function of the state on $C$ arcs and we fix the control to its bounds on $B$ arcs.
\subsection{Expression of the control on constrained arcs}\label{SubsectionC}
From $g(\hat{x}_t)=0$ on $C,$ we get
\be
\label{dtg0}
0 = \frac{\rm d}{{\rm d} t} g(\hat{x}_t) = g'(\hat{x}_t) \big( f_0(\hat{x}_t) + \hat{u}_t f_1(\hat{x}_t) \big)\quad \text{on } C,
\ee
and, since \eqref{1order} holds, we have that
\be
\label{uinC}
\hat{u}_t= -\frac{g'(\hat{x}_t)f_0(\hat{x}_t)}{g'(\hat{x}_t)f_1(\hat{x}_t)}\quad \text{on } C.
\ee
\if{
\subsection{Expression of the control on singular arcs}\label{SubsectionS}
Over singular arcs, the inequality
$u_{\min} < \hat{u}_t < u_{\max}$ and the minimum condition
\eqref{lmscac4a} imply that
\be
\label{stationarity}
H_u=0.
\ee
Differentiating in time, (see {\em e.g.}
\cite{Mau76,AronnaBonnansMartinon2013}),
we obtain
\be
H_u = pf_1,\quad \dot{H}_u = p[f_1,f_0],\quad \ddot{H}_u
= p\big[[f_1,f_0],f_0 \big] + \hat{u} p\big[[f_1,f_0],f_1\big],
\ee
where $[X,Y]:= X' Y - Y' X$ denotes the {\em Lie bracket}
associated with a pair of vector fields $X,Y : \mathbb{R}^n \to \mathbb{R}^n.$
In this control-affine case, the control variable does appear
explicitly neither in the expression of $H_u$ nor in its time
derivative $\frac{\rm d}{{\rm d} t} H_u.$
So, if
$p\big[[f_1,f_0],f_1\big]$ takes only nonzero values along singular arcs,
we obtain an expression for the control on singular arcs, namely
\be
\label{uinS}
\hat{u}=-\frac{p\big[[f_1,f_0],f_0 \big](\hat{x})}{p\big[[f_1,f_0],f_1\big](\hat{x})}.
\ee
Besides, it is known that
$\frac{\partial}{\partial u } \ddot{H}_{u} = p\big[[f_1,f_0],f_1\big] $
must satisfy the REF
{\em strengthened generalized Legendre-Clebsch condition:}
\be
\label{LCu-x}
-\frac{\partial}{\partial u } \ddot{H}_{u} > 0\quad \text{ on } S,
\ee
So that the strengthened generalized Legendre-Clebsch condition gives
\be
\label{LCscalar}
-p\big[[f_1,f_0],f_1\big] > 0 \quad \text{ on } S,
\ee
At this point of the article, $-\frac{\partial}{\partial u }
\ddot{H}_{u} = -p\big[[f_1,f_0],f_1\big]$ is scalar, but in Section
\ref{SecShooting} where we present the algorithm and transform the
problem to an unconstrained one, the resulting singular control may be
multidimensional and the corresponding matrix
$-\frac{\partial}{\partial u } \ddot{H}_{u}$ will no longer be
scalar.
\subsection{Expression for ${\rm d}\mu$ on constrained arcs}\label{SubsectionCmu}
Let us note that along a $C$ arc one has, in view of hypothesis \eqref{chypradn}(ii), that $u_{\min} < \hat{u}_t < u_{\max}$ a.e. Then, due to the minimum condition \eqref{lmscac4a}, the stationarity condition \eqref{stationarity} holds and, consequently,
\be
\label{dHu0}
\begin{split}
0 & = \frac{{\rm d}}{{\rm d} t} H_u = \frac{{\rm d}}{{\rm d} t} (pf_1) =
p[f_1,f_0] {\rm d} t- g' f_1{\rm d}\mu,
\end{split}
\ee
Thus, since the state constraint is of first order, ${\rm d}\mu$ has a density $\nu\geq 0$ over $C$ given by the absolutely continuous function
\be
\label{nu}
\nu:[0,T] \to \mathbb{R},\qquad
t \mapsto \nu_t := \frac{p_t[f_1,f_0](\hat{x}_t)}{g'(\hat{x}_t)f_1(\hat{x}_t)}.
\ee
}\fi
\section{Shooting formulation}\label{SecShooting}
We now explain how to state a transformed problem with neither control bounds nor running state constraints that serves as an intermediate step to write a numerical scheme for problem \eqref{P}. Afterwards the optimality system of the transformed problem is reduced to a nonlinear equation in a finite dimensional space.
The starting point is to estimate the arc structure of the control, {\em i.e.} the sequence of its different types of arcs and the approximate values of its junction times.
This is done in practice by some {\em direct method} such as solving the nonlinear programming (NLP) associated to the discretization of the optimal control problem.
Then we formulate a {\em transformed problem} in which the control is fixed to its bounds on B arcs, and is expressed as a function of the state on C arcs. So the optimzation variables are now the control over singular arcs and the switching times. Subsequently, we express the optimality conditions of the transformed problem. Finally, by eliminating the control as a function of the state and costate, we reduce the optimality system to a finite dimensional equation.
So, let us assume for the remainder of the section that $(\hat{u},\hat{x})$
is a Pontryagin minimum for \eqref{P}.
Additionally, without loss of generality and for the sake of simplicity of notation,
we set $u_{\min} :=0$ and $u_{\max} :=1.$ Recall further that $(\hat{u},\hat{x})$ complies with
the structural hypotheses \eqref{chypradn} for the control $\hat{u},$
and that the state constraint is of first order, {\em i.e.} \eqref{1order} holds true.
\subsection{The transformed problem}\label{SubsecReformulation}
We now state the transformed problem corresponding to \eqref{P}, in the
spirit of \cite{AronnaBonnansMartinon2013},
and we prove that any Pontryagin minimum for the original problem \eqref{P}
is transformed into a weak minimum of the unconstrained transformed problem.
For the Pontryagin minimum $(\hat{u},\hat{x})$, let
\be
0=: \hat\tau_0 < \hat\tau_1 < \dots < \hat\tau_N := T
\ee
denote its associated switching times. Recall the definition of the sets $C,$ $B_-,$ $B_+$ and $S$ given in Section \ref{SecArcs} above. Set $\hat I_k := [ \hat\tau_{k-1}, \hat\tau_k]$ for $k=1,\dots,N,$ and
\be
\mathcal{I}(S) := \big\{ k=1,\dots, N: \hat I_k \text{ is a singular arc} \big\}.
\ee
Analogously, define $\mathcal{I}(C),$ $\mathcal{I}(B_-),$ and $\mathcal{I}(B_+).$ For each $k=1,\dots,N,$ consider a state variable $x^k \in W^{1,\infty}(0,T;\mathbb{R}^n),$ and for each singular arc $k \in \mathcal{I}(S),$ a control variable $u^k \in L^\infty(0,T).$
On the set $B,$ we fix the control to the corresponding bound. Additionally, recall that from formula \eqref{uinC} we have that $\hat{u}_t=\Gamma(\hat{x}_t)$ on $C,$ where $\Gamma$ is given by
$$
\Gamma(x):= -\frac{g'(x)f_0(x)}{g'(x)f_1(x)}.
$$
After these considerations, we are ready to state the transformed problem.
Define the optimal control problem (TP), on the time interval $[0,1],$ by
\begin{align}
& \label{costTP} \text{min } \phi(x^1_0,x^N_1), \\
& \label{eqxk1} \dot{x}^k = (\tau_k -\tau_{k-1}) \big(f_0(x^k) + u^k f_1(x^k)\big)\quad \text{for } k\in \mathcal{I}(S), \\
& \dot{x}^k = (\tau_k -\tau_{k-1}) f_0(x^k)\quad \text{for } k\in \mathcal{I}(B_-), \\
& \dot{x}^k = (\tau_k - \tau_{k-1}) \big( f_0(x^k) + f_1(x^k) \big)\quad \text{for } k\in \mathcal{I}(B_+), \\
& \label{eqxk4} \dot{x}^k = (\tau_k - \tau_{k-1}) \left(f_0(x^k) + \Gamma(x^k) f_1(x^k) \right)\quad \text{for } k\in \mathcal{I}(C), \\
& \dot\tau_k =0 \quad \text{for } k=1,\dots,N-1,\\
&\Phi(x_0^1,x_1^N) = 0,\\
& \label{gx0k} g(x_0^k) = 0 \quad \text{for } k\in \mathcal{I}(C), \\
& \label{continuityxTP} x_1^k = x_0^{k+1} \quad \text{for } k=1,\dots,N-1.
\end{align}
\begin{remark}
Since we use the expression \eqref{uinC} to eliminate the control from the expression \eqref{dtg0} of the derivative of the state constraint equal to zero, we impose the entry conditions \eqref{gx0k} in the formulation of (TP)
in order to guarantee that the state constraint is active along $x^k$ for every $k \in \mathcal{I}(C).$
\end{remark}
Set
\be
\label{changet}
\begin{split}
\hat{x}^k_s &:= \hat{x} \big( \hat\tau_{k-1} + ( \hat\tau_k - \hat\tau_{k-1})s \big) \quad \text{for}\ s\in [0,1] \text{ and } k=1,\dots,N,\\
\hat{u}^k_s &:= \hat{u} \big( \hat\tau_{k-1} + ( \hat\tau_k - \hat\tau_{k-1})s \big) \quad \text{for}\ s\in [0,1] \text{ and } k\in \mathcal{I}(S).
\end{split}
\ee
\begin{lemma}
\label{LemmaPontWeak}
Let $(\hat{u},\hat{x})$ be a Pontryagin minimum of problem \eqref{P}. Then
the triple
$$
\Big( (\hat{u}^k)_{k \in \mathcal{I}(S)} , (\hat{x}^k)_{k=1}^N, ( \hat\tau_k)_{k=1}^{N-1} \Big)
$$
is a weak solution of (TP).
\end{lemma}
\begin{proof}
Consider the feasible trajectories $\big( (u^k),(x^k),(\tau_k) \big)$ for (TP) satisfying
\be
\label{estuk}
\| u^k - \hat{u}^k \|_\infty < \bar\varepsilon \quad \text{and} \quad | \tau_k - \hat\tau_k| \leq \bar\delta \quad \text{for all } k=1,\dots,N,
\ee
for some $\bar\varepsilon,\bar\delta >0$ to be determined later.
Set $I_k := [\tau_{k-1}, \tau_k]$ and consider the functions
$s_k \colon I_k \to [0,1]$ given by $s_{k,t} := \displaystyle \frac{t-\tau_{k-1}}{\tau_k - \tau_{k-1}}.$ Define $u\colon[0,T] \to \mathbb{R}$ by
\be
u_t :=
\left\{
\ba{cl}
\vspace{4pt} 0 & \text{if } t\in I_k,\, k\in \mathcal{I}(B_-), \\
\vspace{4pt}1 & \text{if } t\in I_k,\, k\in \mathcal{I}(B_+), \\
\vspace{4pt} \Gamma \big( x^k ( s_{k,t} ) \big) & \text{if } t\in I_k,\, k\in \mathcal{I}(C), \\
u^k(s_{k,t} ) & \text{if } t\in I_k,\, k\in \mathcal{I}(S).
\ea
\right.
\ee
Let $x \colon [0,T] \to \mathbb{R}^n$ be the state corresponding to the control $u$ and the initial condition $x(0)=x_0^1.$
We next show that if $\bar\varepsilon>0$ and $\bar\delta >0$ are small enough, then $(u,x)$ is feasible for \eqref{P} and arbitrarily close to $(\hat{u},\hat{x})$ in the sense of
\eqref{defminpont1}.
Observe that $x(t) = x^k(s_{k,t})$ for all $k=1,\dots,N$ and $t\in I_k$. Hence, $x$ satisfies the endpoint constraints. Furthermore, due to Gronwall's Lemma, $(u,x)$ verifies the estimate
\be
\label{estux}
\|u-\hat{u}\|_1 + \|x-\hat{x}\|_\infty < \mathcal{O}(\bar\varepsilon + \bar\delta).
\ee
Let us analyze the control constraints. Take $k=1,\dots,N.$
If $k\in \mathcal{I}(B_-) \cup \mathcal{I}(B_+),$ then $u_t \in \{0,1\}$ for a.a. $ t\in I_k.$
On the other hand, by the hypothesis \eqref{chypradn} on the control structure, there exists $\rho_1 > 0$ such that
\be
\label{bounduh}
\rho_1 < \hat{u}_t < 1-\rho_1\quad \text{over $C$ and $S$ arcs.}
\ee
Suppose now that
$k\in \mathcal{I}(S).$ Then, in view of \eqref{estuk} and \eqref{bounduh}, we can see that
the control constraints hold on $I_k$
provided that $\bar\varepsilon \leq \rho_1.$ Finally, let
$k$ be in $\mathcal{I}(C).$ Notice that in this case \eqref{bounduh} is equivalent to
\be
\rho_1 < \Gamma(\hat{x}_t) < 1- \rho_1 \quad \text{ on } \hat{I}_k.
\ee
Hence, by standard continuity arguments and for $\bar\varepsilon,\bar\delta$ sufficiently small, we get that
\be
0 < \Gamma (x_t) <1\quad \text{ on } I_k.
\ee
We therefore confirm that $(u,x)$ verifies the control constraints.
Let us now consider the state constraint. Take first $k \in \mathcal{I}(C).$ Then $g(x_{\tau_k})= g(x_0^k)=0$ and, by definition of $(u,x),$ we have that $\frac{\rm d}{{\rm d} t} g(x_t) =0$ for all $t\in I_k.$
Therefore, $x$ satisfies the state constraint on $I_k$ for $k\in \mathcal{I}(C).$
Next, observe that, due to \eqref{chypradn}, for any $t\in [0,T]$ sufficiently far from a $C$ arc, one has $g(\hat{x}_t) \leq -\rho$ for some small $\rho>0.$
Thus, by \eqref{estux} we get that $g(x_t)<0$ for appropriate $\bar\varepsilon,\bar\delta.$ On the other hand, for $t\in [0,T]$ close to a $C$ arc, we reason as follows. Assume, without loss of generality, that $t$ is near an entry point $\tau_k$ of a C arc. In view of hypothesis \eqref{chypradn} and of the relation \eqref{dtg0}, we have that $\left. \frac{\rm d}{ {\rm d} s} \right|_{s=\hat\tau_k-} g(\hat{x}_s) > 0,$ therefore, $\left. \frac{\rm d}{ {\rm d} s} \right|_{s=\tau_k-} g(x_s) > 0$ as well, if $\bar\varepsilon,\bar\delta$ are sufficiently small. Consequently, $g(x_t) < 0.$ Hence, $x$ verifies the state constraint on $[0,T].$
With this, we conclude that $(u,x)$ is feasible for the original problem \eqref{P}.
Finally, given $M>0$ as in Definition \ref{defminpoint}, we can easily show that $\bar\delta$ and $\bar\varepsilon$ can be taken in such a way that $(u,x)$ satisfies \eqref{defminpont1} for such $M$ and the corresponding $\varepsilon_M$ provided by the Pontryagin optimality of $(\hat{u},\hat{x}).$ Consequently,
$
\phi(x_0,x_1) \geq \phi(\hat{x}_0,\hat{x}_1)
$
or, equivalently,
\be
\phi(x^1_0,x^N_1) \geq \phi(\hat{x}^1_0,\hat{x}^N_1),
\ee
which proves that $\big( (\hat{u}^k)_{k \in \mathcal{I}(S)},(\hat{x}^k)_{k=1}^N, ( \hat\tau_k)_{k=1}^{N-1} \big)$ is a weak solution of (TP), as desired. This concludes the proof.
\end{proof}
\if{
\subsection{Optimality conditions of (TP) and computation of the control}
Here we give a set of optimality conditions implied by Pontryagin's Maximum Principle, and we present the hypothesis (in addition to assumption \eqref{chypradn}) that we impose along the trajectory, which allows us to compute the control as a function of the state and multipliers.
On each arc we have to take into account the state and costate
dynamics
\be
\left\{
\begin{split}
\dot x &= f_0(x) + u f_1(x), \\
\dot p &= -p \big( f'_0(x)+u f'_1(x) \big) - \lambda g'(x),
\end{split}
\right.
\ee
where the algebraic variables $u$ and $\lambda$ are determined as explained next.
On a singular arc $S,$ we have that $\lambda_t=0,$ and we compute $u$
such that $\ddot H_u =0$, i.e. we use the equation
\be
\label{ddotHu}
p[[f_1,f_0],f_0](x) + u\, p[[f_1,f_0],f_1](x) = 0.
\ee
We need to impose the following condition in order to solve $u$ from \eqref{ddotHu} that is known as {\em strengthened generalized Legendre-Clebsch condition:}
\be
\label{LC}
p[[f_1,f_0],f_1](\hat{x}) >0,\quad \text{on } S.
\ee
Hence, from \eqref{LC} and equation \eqref{ddotHu} we get
\be
\hat{u}= - \frac{p[[f_1,f_0],f_0]}{p[[f_1,f_0],f_1]},\quad \text{on } S.
\ee
Apart from equation \eqref{ddotHu}, to ensure that the stationarity condition $H_u=0$ holds, we need to add the following {\em junction conditions}
\be
H_u(\hat\tau_k)= \dot{H}_u(\hat\tau_k) = 0,\quad \text{for any } \hat\tau_k \text{ entry point of a singular arc.}
\ee
Over a constrained arc $C,$ we compute $u$ and $\lambda$ by setting to 0 the derivatives of the state constraint and of the switching function, i.e. we use the equations
\be
0 = g'(\hat{x})\big(f_0(\hat{x}) + \hat{u} f_1(\hat{x}) \big), \quad
0 = \frac{\rm d}{{\rm d} t} (pf_1) = p[f_1,f_0](\hat{x}) - \lambda g'(\hat{x}) f_1(\hat{x}).
\ee
In view of \eqref{1order} we get
\be
\lambda = \frac{p[f_1,f_0](\hat{x})}{g'(\hat{x})f_1(\hat{x})},\quad \text{on } C.
\ee
In order to guarantee that $g=0$ and $H_u=0$ on C, we have to impose the additional junction condition
\be
H_u(\hat\tau_k) = g(x_{\hat\tau_k})=0,\quad \text{for any } \hat\tau_k \text{ entry point of a constrained arc.}
\ee
In addition, we have the following junction condition
\be
H_u(\hat\tau_k)= 0,\quad \text{for any } \hat\tau_k \text{ bang-bang junction.}
\ee
\begin{remark}
Note that there is some redundancy here: at a CS junction, the
condition $H_u(\hat\tau_k)= 0$
follows from the same one put at the entry point of the C arc.
\end{remark}
\begin{remark}
Note that, if $\hat\tau_k$ is an entry (resp. exit) point, then $\dot H_u (\hat\tau_k-)=0$ (resp. $\dot H_u(\hat\tau_k+)=0$)
means that $p[f_1,f_0]=0$. It follows that
\be
\lambda(\hat\tau_k) = 0, \quad \text{at } \hat\tau_k \text{ CS and SC junctions}.
\ee
This is different from the case when the strong Legendre-Clebsch condition holds and the control is continuous, since in that case the density $\lambda$ has typically a nonzero limit at junction points
(REF A VOIR APRES articles avec Audrey).
\end{remark}
}\fi
\subsection{The shooting function}
We shall start by rewriting the problem (TP) in the following compact form, in order to ease the notation,
\begin{align}
\label{TPcost} & \min\,\, \tilde \phi(X_0,X_1),\\
\label{TPdyn} & \dot{X} = \tilde f_0(X) + \sum_{k\in \mathcal{I}(S)} U^k \tilde f_k(X),\\
\label{TPfinal} & \tilde \Phi (X_0,X_1)=0,
\end{align}
where $X:= \big( (x^k)_{k=1}^N,(\tau_k)_{k=1}^{N-1} \big),$ $U:=(u^k)_{k\in \mathcal{I}(S)},$ the vector field $ \tilde f_0: \mathbb{R}^{Nn+N-1} \to \mathbb{R}^{Nn+N-1}$ is defined as follows,
\begin{equation*}
\begin{split}
\big( \tilde f_0(X) \big)&_{i=(k-1)n+1}^{kn}\\
&:=
\left\{
\ba{cl}
(\tau_k-\tau_{k-1})f_0(x^k) &\text{ for } k\in \mathcal{I}(S) \cup \mathcal{I}(B_-),\\
(\tau_k-\tau_{k-1}) \big(f_0(x^k)+f_1(x^k) \big) &\text{ for } k\in \mathcal{I}(B_+),\\
(\tau_k-\tau_{k-1}) \left( f_0(x^k)+\Gamma(x^k) f_1(x^k) \right) &\text{ for } k\in \mathcal{I}(C),
\ea
\right.
\end{split}
\end{equation*}
and $\big( \tilde f_0(X) \big)_{i=nN+1}^{Nn+N-1}:=0.$ Additionally,
for $k\in \mathcal{I}(S)$ the vector field $\tilde f_k: \mathbb{R}^{Nn+N-1} \to \mathbb{R}^{Nn+N-1}$ is given by
$$
\big( \tilde f_k (X) \big)_{i=(k-1)n+1}^{kn} := (\tau_k-\tau_{k-1})f_1(x^k),
$$
and $\big( \tilde f_k (X) \big)_{i}:=0$ for the remaining index $i,$
the new cost $\tilde\phi:\mathbb{R}^{2(Nn+N-1)} \to \mathbb{R}$ is
$$
\tilde\phi (X_0,X_1) :=\phi(x_0^1,x_1^N),
$$
and the function $\tilde \Phi: \mathbb{R}^{2(Nn+N-1)} \to \mathbb{R}^{ {\rm d}_{\tilde \Phi}}$ with ${\rm d}_{\tilde \Phi}:=q+|\mathcal{I}(C)|+n(N-1)$ is defined as
$$
\tilde\Phi(X_0,X_1):=
\begin{pmatrix}
\Phi(x_0^1,x^N_1) \\
\big( g(x_0^k) \big)_{k \in \mathcal{I}(C)} \\
\big( x_1^k-x_0^{k+1} \big)_{k=1}^{N-1}
\end{pmatrix}.
$$
The pre-Hamiltonian for problem (TP) is given by
\be
\tilde H = P \Big( \tilde{f}_0(X) + \sum_{k\in \mathcal{I}(S)} U^k \tilde{f}_k(X) \Big) = \sum_{k=1}^N (\tau_k - \tau_{k-1})H^k,
\ee
where $P$ denotes the costate associated to (TP), \be
\label{Hk}
H^k:= p^k \big( f_0(x^k)+w^k f_1(x^k) \big),
\ee
with the notation $w^k$ defined as
\be
\label{wk}
w^k := \left\{
\ba{cl}
u^k & \quad \text{if } k\in \mathcal{I}(S),\\
0 & \quad \text{if } k\in \mathcal{I}(B_-),\\
1 & \quad \text{if } k\in \mathcal{I}(B_+),\\
\Gamma(x^k) & \quad \text{if } k\in \mathcal{I}(C),
\ea
\right.
\ee
and $p^k$ denotes the $n$-dimensional vector of components $P_{(k-1)n+1},\dots,P_{kn}.$ Note that $w^k$ is a variable only for $k\in \mathcal{I}(S)$, in which case it represents the control $u^k$.
\subsection{Constraint qualification and first order optimality condition for (TP)}
Since problem (TP) has only endpoint equality constraints,
and the Hamiltonian is an affine function of the control, it is known that Pontryagin's Maximum Principle is equivalent to the first-order optimality conditions. So, the qualification condition is that the derivative of the constraint is onto at the nominal trajectory $(\hat U,\hat X)$, see {\em e.g.} \cite[Ch. 3]{PAOP}.
This means that
\be
\begin{split}
\bar\Phi : \,\mathbb{R}^{Nn+N-1} \times (L^\infty)^{|\mathcal{I}(S)|} \to \mathbb{R}^{{\rm d}_{\tilde \Phi}},
\quad
(X_0,U) \mapsto \tilde \Phi(X_0,X_1),
\end{split}
\ee
where $X_t$ is the solution of \eqref{TPdyn} associated to $(X_0,U)$ is such that
\be
\label{CQ2}
D\bar\Phi(\hat X_0,\hat U) \text{ is surjective.}
\ee
Under this hypothesis, the first-order optimality condition in normal form is as follows,
defining the endpoint Lagrangian associated to (TP) by:
\be
\tilde \ell^\Psi := \phi(x_0^1,x_1^N) + \sum_{j=1}^{q} \Psi_j \Phi_j(x_0^1,x_1^N)
+ \sum_{k\in \mathcal{I}(C)} \gamma_k g(x_0^k) + \sum_{k=1}^{N-1} \theta_k(x_1^k-x_0^{k+1}).
\ee
\begin{theorem}
Let $(\hat U,\hat X)$ be a weak solution for (TP) satisfying the qualification condition \eqref{CQ2}. Then there exists a unique $\tilde \lambda := (\tilde \Psi,P) \in \mathbb{R}^{{\rm d}_{\tilde\Phi}*}\times (W^{1,\infty})^{Nn+N-1*}$ such that $P$ is solution of
\be
-\dot{P}_t = D_X \tilde H(\hat U_t,\hat X_t,P_t) \quad \text{a.e. on } [0,T],
\ee
with {\em transversality conditions}
\be
\begin{split}
P_0 & = -D_{X_0} \tilde\ell^{\tilde \Psi} (\hat X_0,\hat X_1),\\
P_1 & = D_{X_T} \tilde \ell^{\tilde \Psi} (\hat X_0,\hat X_1),
\end{split}
\ee
and with
\be
\label{stationarity}
\tilde H_U (\hat U_t,\hat X_t,P_t)=0.
\ee
\end{theorem}
Since there is a unique associated multiplier, we omit from now on the dependence on $\tilde \lambda$ for the sake of simplicity of the presentation. Moreover, in some ocassions, we omit the dependence on the nominal solution $(\hat U,\hat X).$
\subsection{Expression of the singular controls in problem (TP)}\label{SubsectionSingular}
{ It is known that in this control-affine case, the control variable does not appear explicitly neither in the expression of $\tilde{H}_U$ nor in its time
derivative $\dot{\tilde{H}}_U$ (see {\em e.g.} \cite{Rob67,AronnaBonnansMartinon2013}).
The {\em strengthened generalized Legendre-Clebsch condition} \cite{Rob67} for (TP) reads
\be
\label{LC}
-\frac{\partial}{\partial U } \ddot{\tilde H}_{U} \succ 0.
\ee
Here $A \succ B$, where $A$ and $B$ are symmetric matrices of same size, means that
$A-B$ is positive semidefinite.
At this point, recall the definitions of $H^k$ and $p^k$ given in \eqref{Hk} and in the first line after \eqref{wk}, respectively.
Simple calculations show that the l.h.s. of \eqref{LC} is a $|\mathcal{I}(S)|\times |\mathcal{I}(S)|$-diagonal matrix with positive entries equal to
\be
-(\tau_k-\tau_{k-1}) \frac{\partial}{\partial {u^k} } \ddot{H}^k_{u^k}\quad \text{for } k\in \mathcal{I}(S).
\ee
Then condition \eqref{LC} becomes
\be
\label{LCa}
\frac{\partial}{\partial {u^k} }\ddot{H}^k_{u^k} < 0\quad \text{for } k\in \mathcal{I}(S).
\ee
Hence, thanks to \eqref{LCa}, for each $k\in \mathcal{I}(S)$ one can compute the control $u^k$ from the identity
\be
\label{ddotHuk}
\ddot{H}^k_{u^k}=0.
\ee
Apart from the previous equation \eqref{ddotHuk}, in order to ensure the stationarity $H^k_{u^k}=0,$ we add the following endpoint conditions:
\be
0 = H^k_{u^k} (0) = p_0^k f_1(x_0^k),\quad 0=\dot{H}^k_{u^k}(0) = p_0^k [f_1,f_0](x_0^k)\quad \text{for } k\in \mathcal{I}(S).
\ee
\subsection{Lagrangians and costate equation}
The costate equation for $p^k$ is
\be
\label{eqpk}
\dot{p}^k = -(\tau_k-\tau_{k+1}) D_{x^k}H^k,
\ee
with endpoint conditions
\be
p_0^1=-D_{x_0^1} \tilde \ell^\Psi = -D_{x_0^1}\phi - \sum_{j=1}^{q} \Psi_j D_{x_0^1} \Phi_j - \chi_{\mathcal{I}(C)}(1) \gamma_1 g'(x_0^1),
\ee
\begin{gather}
\label{pk1} p_1^k = \theta^k \quad \text{for } k=1,\dots,N-1,\\
\label{pk0} p_0^k = \theta^{k-1} - \chi_{\mathcal{I}(C)}(k) \gamma_k g'(x_0^k) \quad \text{for } k=2,\dots,N,
\end{gather}
\be
p_1^N = D_{x_1^N} \phi + \sum_{j=1}^{q} \Psi_j D_{x_1^N} \Phi_j,
\ee
where $\chi_{\mathcal{I}(C)}$ denotes the {\em characteristic function} associated to the set $\mathcal{I}(C).$
For the costate $p^{\tau_k}$ we have the dynamics
\be
\label{pk}
\dot{p}^{\tau_k} = -H^k+H^{k+1},\quad p_0^{\tau_k}=0,\ p_1^{\tau_k}=0\qquad \text{for } k=1,\dots,N-1.
\ee
It is known that the pre-Hamiltonian of autonomous problems has a constant value along an optimal solution (see {\em e.g.}
\cite{Vinter2000}). By similar arguments it is easily seen that each $H^k$ is a constant function of time along an optimal solution.
Consequently, from \eqref{pk} we get that $p^{\tau_k}$ vanishes identically and that
\be
H_1^k = H_0^{k+1} \quad \text{for } k=1,\dots,N-1.
\ee
\subsection{The shooting function and method}
The shooting function associated with (TP) that we propose here is
\be
\begin{split}
\mathcal{S} : \mathbb{R}^{Nn+N-1} \times \mathbb{R}^{Nn+q+| \mathcal{I}(C)|,*} &\to \mathbb{R}^{(N-1)n+N-1 +q+| \mathcal{I}(C)|+ 2|\mathcal{I}(S)|} \times \mathbb{R}^{(N+1)n,*},\\
\big( (x_0^k),(\tau_k),(p_0^k),\Psi,\gamma \big) &\mapsto
\begin{pmatrix}
\Phi(x_0^1,x_1^N) \\
\big( g(x_0^k) \big)_{k\in \mathcal{I}(C)} \\
(x^k_1 - x^{k+1}_0)_{k=1,\dots,N-1} \\
p^1_0 + D_{x_0^1} \tilde\ell^\Psi \\
p_1^k-p_0^{k+1}-\chi_{\mathcal{I}(C)}(k) \gamma_k g'(x_0^k) \\
p^q-D_{x^N_1} \tilde \ell^\Psi \\
(H^k_1-H^{k+1}_0)_{k=1,\dots,N-1} \\
\big( p^k_0 f_1(x^k_0) \big)_{k\in \mathcal{I}(S)} \\
\big( p_0^k [f_1,f_0](x^k_0) \big)_{k \in \mathcal{I}(S)}
\end{pmatrix},
\end{split}
\ee
where $\big( (x^k),(p^k) \big)$ is the solution of the state and costate equations \eqref{eqxk1}-\eqref{eqxk4}, \eqref{eqpk} with initial values $(x^k_0),(p^k_0),$ and control $(u^k)_{k\in \mathcal{I}(S)},$ given by the stationarity condition \eqref{ddotHuk}. Note that we removed the variable $\theta$ by combining equations \eqref{pk1} and \eqref{pk0}.
The key feature of this procedure is that $\omega := \big( (x_0^k),(\tau_k),(p_0^k),\Psi,\gamma\big)$ satisfies
\be
\label{S=0}
\mathcal{S}(\omega)=0,
\ee
if and only if the associated solution $\big( (x^k),(p^k),(u^k) \big)$ verifies the Pontryagin's Maximum Principle for (TP). Briefly speaking, in order to find the candidate solutions of (TP), we shall solve \eqref{S=0}.
Let us observe that the system \eqref{S=0} has $2Nn+N-1+ q+| \mathcal{I}(C)|$ unknowns and $2Nn+N-1+ q +| \mathcal{I}(C)|+ 2|\mathcal{I}(S)|$ equations. Hence, as soon as a singular arc occurs, \eqref{S=0} has more equations than unknowns, {\em i.e.} it is overdetermined.
We then follow \cite{AronnaBonnansMartinon2013}, where the authors suggested to solve the shooting equations by the Gauss-Newton method.
We recall the following convergence result for Gauss-Newton, see {\em e.g.} Fletcher \cite{fletcher2013practical}, or alternatively Bonnans \cite{bonnans2006numerical}.
If $F$ is a $C^1$ mapping from $\mathbb{R}^n$ to $\mathbb{R}^p$ with $p>n$, the Gauss-Newton method computes a sequence
$(y^j)$ in $\mathbb{R}^n$ satisfying $F(y^j)+DF(y^j)(y^{j+1}-y^j)=0$.
When $F$ has a zero at $\bar{y}$ and $DF(\bar{y})$ is onto, the sequence $(y^j)$ is well-defined provided that
the starting point $y^0$ is close enough to $\bar{y}$ and in that case, $(y^j)$ converges superlinearly to $\bar{y}$
(quadratically if $DF$ is Lipschitz near $\bar{y}$).
In view of the regularity hypotheses done in Section \ref{SecProblem}, we know that $\mathcal{S}'$ is Lipschitz continuous.
\section{Sufficient condition for the convergence of the shooting algorithm}\label{SecSuff}
The main result of this article is Theorem \ref{ThConvergence} of current section. It gives a sufficient condition for the local convergence of the shooting algorithm, that is also a sufficient condition for weak optimality of problem (TP), as stated in Theorem \ref{SSC} below.
\subsection{Second order optimality conditions for (TP)}
We now recall some second order optimality conditions for (TP). Let us consider the quadratic mapping on the space $ (L^\infty)^{|\mathcal{I}(S)|} \times (W^{1,\infty})^{Nn+N-1},$ defined as
\be
\tilde Q(V,Z) := \mbox{$\frac{1}{2}$} D^2 \tilde\ell(Z_0,Z_1)^2 + \mbox{$\frac{1}{2}$} \int_0^1 \big[ Z^\top \tilde{H}_{XX} Z + 2 V \tilde H_{UX} Z \big] \mathrm{d}t.
\ee
We next introduce the {\em critical cone} associated to (TP).
Since the problem has only qualified equality constraints, this critical cone coincides with the tangent space to the constraints.
Consider first the linearized state equation
\be
\label{LINEQ}
\dot Z = \tilde A Z + \tilde B V\quad \text{a.e. on } [0,1],
\ee
where $F(U,X):= \tilde f_0(X) + \sum_{k\in \mathcal{I}(S)} U^k \tilde f_k(X),$ $\tilde A := F_X,$ $\tilde B:= F_U;$
and let the linearization of the endpoint constraints be given by
\be
\label{LINCONS}
D\tilde \Phi (Z_0,Z_1) = 0.
\ee
The {\em critical cone} for (TP) is defined as
\be
\tilde\mathcal{C} := \Big\{(V,Z) \in (L^\infty)^{|\mathcal{I}(S)|} \times (W^{1,\infty})^{Nn+N-1} : \text{\eqref{LINEQ}-\eqref{LINCONS} hold} \Big\}.
\ee
The following result follows (see {\em e.g.} \cite{LMO,ABDL12} for a proof).
\begin{theorem}[Second order necessary condition]
If $(\hat U,\hat X)$ is a weak minimum for (TP) that verifies \eqref{CQ2}, then
\be
\label{SONCineq}
\tilde Q(V,Z) \geq 0\quad \text{for all } (V,Z) \in \tilde\mathcal{C}.
\ee
\end{theorem}
In the sequel we present some optimality conditions for (TP). The first one is a necessary condition due to Goh \cite{Goh66} and the second one, a sufficient condition from Dmitruk \cite{Dmi77,Dmi87}.
The idea behind these results lies on the following observation. Note that $\tilde{H}_{UU}$ vanishes and, therefore, the quadratic mapping $\tilde Q$ does not contain a quadratic term on the control variation $V.$
Consequently, the Legendre-Clebsch necessary optimality condition on the positive semidefiniteness of $\tilde{H}_{UU}$ holds trivially and a second order sufficient condition cannot be obtained by strengthening inequality \eqref{SONCineq}.
In order to overcome this issue and derive necessary conditions for this singular case, Goh introduced a change of variables in \cite{Goh66a} and
applied it to derive necessary conditions in \cite{Goh66}. Some years later, Dmitruk \cite{Dmi77} showed a second order sufficient condition in terms of the coercivity of the transformation $\tilde \Omega$ of $\tilde Q$ introduced below.
The {\em Goh transformation} for the linear system \eqref{LINEQ} is given by
\be
\label{GOH}
Y_t: = \int_0^t V_s {\rm d} s,\qquad \Xi_t:= Z_t - \tilde B_t Y_t,
\ee
where $\tilde E := \tilde A \tilde B - \frac{\rm d}{{\rm d} t} \tilde B.$
Notice that if $(V,Z) \in \tilde \mathcal{C},$ then $(Y,\Xi)$ defined by the above transformation \eqref{GOH} is solution of the {\em transformed linearized equation}
\be
\label{LINEQGOH}
\dot\Xi = \tilde A \Xi + \tilde{E} Y,
\ee
and satisfies the {\em transformed linearized endpoint constraints}
\be
\label{LINCONSGOH}
D\tilde \Phi(\Xi_0,\Xi_1+\tilde B_1 h)=0,
\ee
where we set $h:= Y_1.$
Consider the function
\be
\rho(\zeta_0,\zeta_1,h) := D^2 \tilde\ell (\zeta_0,\zeta_1+\tilde B_1 h)^2 + h\tilde H_{UX,1} (2\zeta_1+\tilde B_1 h),
\ee
and the quadratic mapping
\be
\tilde\Omega (Y,\tilde h,\Xi) := \mbox{$\frac{1}{2}$} \rho(\Xi_0,\Xi_1,\tilde h) + \mbox{$\frac{1}{2}$} \int_0^1 \left( \Xi^\top \tilde{H}_{XX} \Xi + 2Y \tilde M \Xi + Y \tilde{R} Y\right) \mathrm{d}t,
\ee
for $(Y,\tilde h,\Xi) \in (L^2)^{|\mathcal{I}(S)|} \times \mathbb{R} \times (H^1)^{Nn+N-1}$ and
\be
\tilde M:= \tilde{f}_1^\top \tilde{H}_{XX} - \frac{\rm d}{{\rm d} t} \tilde{H}_{UX} -\tilde{H}_{UX} \tilde{A},
\quad
\tilde R:= \tilde{f}_1^\top \tilde{H}_{XX} \tilde{f}_1-2 \tilde{H}_{UX}\tilde{E}- \frac{\rm d}{{\rm d} t} (\tilde{H}_{UX} \tilde{f}_1).
\ee
Let us recall that the second order necessary condition for optimality stated by Goh \cite{Goh66} (and nowadays known as {\em Goh condition}) implies that if $(\hat U,\hat X)$ is a weak minimum for (TP) verifying \eqref{CQ2}, then
\be
\label{CB}
\tilde{H}_{UX}\tilde B \text{ is symmetric,}
\ee
or, equivalently, $P\cdotp D\tilde f_i \tilde f_j= P \cdotp D \tilde f_j \tilde f_i$ for all $i,j=1,\dots,N.$ In the recent literature, this condition can be encountered as $P \cdotp [\tilde f_i,\tilde f_j]=0.$
We shall mention that this necessary condition was first stated by Goh in \cite{Goh66} for the case with neither control nor state constraints, and extended in \cite{ABDL12,FraTon13} for problems containing control constraints.
Notice that when the control variable of (TP) is scalar ({\em i.e.} $|\mathcal{I}(S)|=1$), then \eqref{CB} is trivially verified since $\tilde{H}_{UX}\tilde B$ is also a scalar. Furthermore, given the special structure of the dynamics of (TP), we can see that $D \tilde f_i\, \tilde f_j =0$ for all $i,j \in \{1,\dots,N\}$ with $ i\neq j $ and, consequently, the matrix in \eqref{CB} is diagonal and the Goh condition holds trivially for (TP) even when $|\mathcal{I}(S)|>1$.
From \eqref{CB} and \cite[Theorem 4.4]{ABDL12} we get the following result:
\begin{proposition}
For all $(V,Z) \in (L^\infty)^{|\mathcal{I}(S)|} \times (W^{1,\infty})^{Nn+N-1}$ solution of \eqref{LINEQ} and $(Y,\Xi)$ given by the Goh transform \eqref{GOH}, it holds
\begin{equation*}
\tilde Q (V,Z) = \tilde \Omega (Y,Y_T,\Xi).
\end{equation*}
\end{proposition}
Define, for $(\Xi_0,Y,\tilde h) \in \mathbb{R}^n \times (L^2)^{|\mathcal{I}(S)|} \times \mathbb{R},$ the {\em order function}
\be
\gamma(\Xi_0,Y,\tilde h) := |\Xi_0|^2 + \int_0^1 |Y_t|^2 \mathrm{d}t + |\tilde h|^2.
\ee
\begin{definition}[$\gamma$-growth]
A feasible trajectory $(\hat U,\hat X)$ of (TP) satisfies the {\em
$\gamma$-growth condition in the weak sense} if there exists a
positive constant $c$ such that, for every sequence of feasible
variations $\big\{ (\delta X^k_0,V^k) \big\}_k$ converging to 0 in
$\mathbb{R}^{Nn+N-1} \times ( L^\infty)^{|\mathcal{I}(S)|}$,
one has that
\be
\tilde \phi(X_0^k,X_1^k) - \tilde \phi(\hat X_0,\hat X_1) \geq c \gamma(\Xi^k_0,Y^k,Y^k_T),
\ee
for $k$ large enough, where $(Y^k,\Xi^k)$ are given by Goh transform \eqref{GOH} and $X^k$ is the solution of the state equation \eqref{TPdyn} associated to $(\hat{X}_0+\delta X_0^k,\hat{U}+V^k).$
\end{definition}
Consider the {\em transformed critical cone}
\be
\tilde\mathcal{P}^2_S := \left\{ (Y,\tilde h,\Xi) \in(L^2)^{|\mathcal{I}(S)|} \times \mathbb{R} \times (H^1)^{Nn+N-1}: \text{\eqref{LINEQGOH}-\eqref{LINCONSGOH} hold} \right\}.
\ee
The following characterization of $\gamma$-growth holds (see \cite{Dmi77} or \cite[Theorem 3.1]{Dmi87} for a proof).
\begin{theorem}
\label{SSC}
Let $(\hat U,\hat X)$ be such that the qualification condition \eqref{CQ2} holds. Then $(\hat U,\hat X)$ is a weak minimum of (TP) that satisfies $\gamma$-growth in the weak sense if and only if \eqref{CB} holds and there exists $c>0$ such that
\be
\label{POS}
\tilde \Omega(Y,\tilde h,\Xi) \geq c \gamma (\Xi_0,Y,\tilde h) \quad \text{on } \tilde\mathcal{P}_S^2.
\ee
\end{theorem}
We are now ready to state the following convergence result for the shooting algorithm.
\begin{theorem}
\label{ThConvergence}
If $(\hat U,\hat X)$ is a weak minimum of problem (TP) satisfying the constraint qualification \eqref{CQ2} and the uniform positivity condition \eqref{POS}, then the shooting algorithm is locally quadratically convergent.
\end{theorem}
\begin{proof}
This is a consequence of the convergence result in \cite[Theorem 5.4]{AronnaBonnansMartinon2013}.
\end{proof}
Note that in the proof of the above Theorem, it is established that the hypotheses imply that the derivative of the shooting function is injective.
\section{Application to a regulator problem}\label{SecExamples}
Consider the following regulator problem, where $T=5$:
\be
\begin{split}
&\min \mbox{$\frac{1}{2}$} \int_0^5 (x_{1,t}^2 + x_{2,t}^2) {\rm d} t + \mbox{$\frac{1}{2}$} x_{1,5}^2, \\
& \dot x_{1,t} = x_{2,t},\quad \dot x_{2,t} = u_t \in [-1,1],
\end{split}
\ee
subject to the state constraint and initial conditions
\be
x_{2,t} \geq -0.2, \quad x_{1,0}=0,\quad x_{2,0}=1.
\ee
To write the problem in the Mayer form, we introduce an auxiliary state variable given by the dynamics
\be
\dot x_{3,t} = \mbox{$\frac{1}{2}$} (x_{1,t}^2 + x_{2,t}^2),\quad x_{3,0}=0.
\ee
The resulting problem is then
\be
\label{Preg}
\begin{split}
& \min\, x_{3,5} + \mbox{$\frac{1}{2}$} x_{1,5}^2, \\
& \dot x_1 = x_2, \quad \dot x_2 = u, \quad \dot x_3 = \mbox{$\frac{1}{2}$} (x_1^2 + x_2^2), \\
&x_{1,0}=0,\quad x_{2,0}=1,\quad x_{3,0}=0,\\
&-1 \leq u \leq 1,\; \; x_2 \geq -0.2.
\end{split}
\ee
Using the optimal control solver BOCOP \cite{Bocop} we estimated that the optimal control $\hat{u}$ is a concatenation of a bang arc in the lower bound, followed by a constrained arc and ended with a singular one. Briefly, we can say that the optimal control has a $B_-CS$ structure.
\subsection{Checking local optimality}
In this subsection we compute analytically the optimal solution of \eqref{Preg} and check that it verifies the second order sufficient condition for state-constrained control-affine problems proved in \cite[Theorem 5]{AronnaBonnansGoh2016}.
While the problem is convex, and hence, satisfying the first order optimality conditions is enough for proving optimality, the quoted second order conditions are of interest since they imply the quadratic growth, see
\cite[Definition 3]{AronnaBonnansGoh2016}.
To problem \eqref{Preg} we associate the functions $g:\mathbb{R}^3 \to \mathbb{R},$ $f_0,f_1:\mathbb{R}^3 \to \mathbb{R}^3$ given by
\begin{equation*}
g(x):= -x_2-0.2,\quad f_0(x):= \begin{pmatrix} x_2\\ 0 \\ \mbox{$\frac{1}{2}$}( x_1^2+x_2^2 )\end{pmatrix},\quad
f_1(x) = \begin{pmatrix} 0\\ 1\\ 0\end{pmatrix}.
\end{equation*}
So that the optimal control $\hat{u}$ is equal to -1 on the $B_-$ arc, and to
$\displaystyle-\frac{g'(\hat{x})f_0(\hat{x})}{g'(\hat{x})f_1(\hat{x})} = 0$
on $C$ according to formula \eqref{uinC}, where $\hat{x}$ is the associated optimal state.
The pre-Hamiltonian of \eqref{Preg} reads
$$
H(u,x,p):= p_1 x_2 + p_2u +\mbox{$\frac{1}{2}$} p_3(x_1^2+x_2^2),
$$
where $p:=(p_1,p_2,p_3)$ is the costate.
Over the singular arc $S$, the inequality
$u_{\min} < \hat{u}_t < u_{\max}$ and the minimum condition
of Pontryagin's Maximum Principle (see {\em e.g.} \cite[equation (2.12)]{AronnaBonnansGoh2016}) imply that
\be
\label{stationarity}
H_u=0.
\ee
Differentiating in time, (see {\em e.g.}
\cite{Mau76,AronnaBonnansMartinon2013}),
we obtain
\be
H_u = pf_1,\quad \dot{H}_u = p[f_1,f_0],\quad \ddot{H}_u
= p\big[[f_1,f_0],f_0 \big] + \hat{u} p\big[[f_1,f_0],f_1\big],
\ee
where $[X,Y]:= X' Y - Y' X$ denotes the {\em Lie bracket}
associated with a pair of vector fields $X,Y : \mathbb{R}^n \to \mathbb{R}^n.$
In this control-affine case, the control variable does not appear
explicitly neither in the expression of $H_u$ nor in its time
derivative $\dot{H}_u.$
So, if
$p\big[[f_1,f_0],f_1\big]$ takes only nonzero values along singular arcs,
we obtain an expression for the control on singular arcs, namely
\be
\label{uinS}
\hat{u}=-\frac{p\big[[f_1,f_0],f_0 \big](\hat{x})}{p\big[[f_1,f_0],f_1\big](\hat{x})}.
\ee
The involved Lie brackets for this examples are
\be
\label{Lie}
[f_1,f_0]= \begin{pmatrix} -1 \\ 0 \\ -x_2 \end{pmatrix},\quad
\big[ [f_1,f_0],f_0 \big] = \begin{pmatrix} 0 \\ 0 \\ x_1 \end{pmatrix},\quad
\big[ [f_1,f_0],f_1 \big]= \begin{pmatrix} 0 \\ 0\\ -1 \end{pmatrix}.
\ee
On the other hand, the costate equation on the singular arc $S$ gives
\be
\label{examplep}
\begin{split}
\dot{p}_1 &= - p_3\hat{x}_1,\quad p_{1,5} = \hat{x}_{1,5}, \\
{\rm d}{p}_2 &= -(p_1+p_3 \hat{x}_2){\rm d} t+\nu, \quad p_{2,5}=0, \\
\dot{p}_3 &= 0,\quad p_{3,5}=1,
\end{split}
\ee
where $\nu$ is the density of $ \mu.$
Thus,
\be\label{examplep3}
p_3 \equiv 1,
\ee
and from \eqref{uinS} and \eqref{Lie} we get
\be
\label{uonS}
\hat{u}=\hat{x}_1\quad \text{on}\ S.
\ee
Moreover, the first order optimality conditions imply that
\be
\label{p2CS}
0=H_u=p_2\quad \text{on } C\cup S.
\ee
Let us write $\hat{\tau}_1,\hat{\tau}_2$ for the switching times, so that
$$
B_-=[0,\hat{\tau}_1],\quad C=[\hat{\tau}_1,\hat{\tau}_2],\quad S=[\hat{\tau}_2,5].
$$
Since the control $\hat{u}$ is constantly equal to $-1$ on $B_-,$ then
\be
\label{examplex2}
\hat{x}_{2,t} = 1-t \quad \text{on. } B_-,
\ee
until it saturates the state constraint at $\hat{\tau}_1.$ Hence $1- \hat{\tau}_1 = -0.2,$ so it follows that
\be
\label{hattau1}
\hat\tau_1=1.2
\ee
and $B_- = [0,1.2].$ Consequently,
\be\label{examplex1}
\hat{x}_{1,t} = t-\frac{t^2}{2} \quad \text{on } B_-.
\ee
On $C=[1.2,\hat{\tau}_2],$ necessarily $\hat{x}_2 \equiv -0.2,$ thus $u\equiv 0$ and
\be
\label{x1C}
\hat{x}_{1,t} = 0.48 - 0.2(t-1.2) = 0.72 - t/5 \quad \text{ on } C.
\ee
On the singular arc $S=[\hat{\tau}_2,5]$ we get, from the expression of $\hat{u}$ on $S$ \eqref{uonS}, that
$\ddot \hat{x}_1 = \dot \hat{x}_2 = \hat{u} = \hat{x}_1.
$
Thus
\be
x_{1,t} = c_1 e^{5-t} + c_2 e^{t-5}\quad \text{ on } S,
\ee
for some real constant values $c_1,c_2.$
Therefore,
\be
{\hat{x}}_{2,t} = \dot{\hat{x}}_{1,t} = -c_1 e^{5-t} + c_2 e^{t-5} \quad \text{ on } S.
\ee
The stationarity condition $0=H_u=p_2$ on $S$ yields $0=\dot{p}_2 = -p_1-\hat{x}_2,$ where the second equality of latter equation follows from \eqref{examplep} and since $\nu \equiv 0$ on $S.$
Thus $p_1=-\hat{x}_2$ on $S$, so
\be
p_{1,t} = c_1 e^{5-t} - c_2 e^{t-5} \quad \text{ on } S.
\ee
The transversality condition for $p_1$ (see \eqref{examplep}) implies
$c_1+c_2 = c_1-c_2.$
Thus, $c_2=0$ and
\be
\hat{x}_1=-\hat{x}_2 = p_1 = c_1 e^{5-t} \quad \text{ on } S.
\ee
Since $-0.2=x_{2,\hat{\tau}_2},$ then
$
x_{1,\hat{\tau}_2} = 0.2.
$
Hence, from the expression of $\hat{x}_1$ on $C$ given in \eqref{x1C}, we obtain $0.72-\hat{\tau}_2/5 = 0.2$ and, consequently
\be
\label{hattau2}
\hat{\tau}_2=2.6.
\ee
Additionally,
$-0.2 = x_{2,\hat{\tau}_2} = -c_1 e^{2.4}$ so that
$
c_1 = 0.2 e^{-2.4}.
$
At time $\hat{\tau}_2$ we have that
$p_1= c_1 e^{2.4} = 0.2$
and, from the costate equation \eqref{examplep} and from \eqref{x1C}, we have
\be
\label{regp1-dc}
\dot p_{1,t} = - \hat{x}_{1,t} = t/5 - 0.72
\leq 2.6/5 -0.72 < 0 \quad \text{on } C = [1.2,2.6].
\ee
So $p_1$ is decreasing on $C$ and, recalling \eqref{p2CS}, this implies that
\be
\nu_t = p_{1,t} + \hat{x}_{2,t} > p_{1,\hat{\tau}_2}- 0.2 =0\quad \text{for } t\in [\hat{\tau}_1,\hat{\tau}_2).
\ee
Thus the complementarity condition for the state constraint \cite[equation(3.4)(ii)]{AronnaBonnansGoh2016} holds true.
In order to check that the strict complementarity hypothesis (i) of \cite[Theorem 5]{AronnaBonnansGoh2016} is satisfied, we will prove that the corresponding strict complementarity condition for the control constraint of \cite[Definition 4]{AronnaBonnansGoh2016} holds.
In view of \eqref{regp1-dc}, we get
\be
p_{1,t} = 0.2 + (t-\hat{\tau}_2)^2/10 - 0.72(t-\hat{\tau}_2).
\ee
Therefore
$
p_{1,\hat{\tau}_1} = 0.2 + (1.4)^2/10 +0.72 \times 1.4 = 1.404.
$
On the $B_-$ arc,
we have seen that $\hat{x}_1>0$ due to \eqref{examplex1} and $x_{1,0}=0,$ so
that $\dot p_1 = - \hat{x}_1 <0$, and since
$p_{1,\hat{\tau}_1}>0,$ it follows that $p_1$ has values greater than
$p_{1,\hat{\tau}_1} = 1.404$.
Therefore, since $\hat{x}_2 >-0.2$ over $B_-$,
\be
\dot{p}_2=-p_1-\hat{x}_2 < 0\quad \text{ on } B_-.
\ee
Since $p_{2,\hat{\tau}_1}=0,$ we get $p_2 > 0$ on $[0,\hat{\tau}_1)$ or, equivalently, $H_u>0$ on $[0,\hat{\tau}_1).$ We conclude that the strict complementarity condition for the control constraint given in holds. This completes the verification of \cite[Theorem 5 - (i)]{AronnaBonnansGoh2016}.
Let us now verify the uniform positivity \cite[equation (4.6)]{AronnaBonnansGoh2016}.
The dynamics for the linearized state is
\be
\begin{split}
& \dot{z}_1 = z_2,\quad \dot{z}_2=v,\quad \dot{z}_3 = \hat{x}_1 z_1 + \hat{x}_2 z_2,\\
& z_{1,0} = z_{2,0} = z_{3,0} =0.
\end{split}
\ee
Let $\mathcal{C}_S$ and ${\mathcal{P}}^2_*$ denote the strict critical cone and the extended cone (invoking the notation used in \cite{AronnaBonnansGoh2016}) at the optimal trajectory $(\hat{u},\hat{x}),$ respectively. Since $\hat{u}$ is $B_-CS$ then, for any $(v,z) \in \mathcal{C}_S,$ $v=0$ on the initial interval $B_-.$ Consequently,
\be
\label{yB-}
y=0 \quad \text{on } B_-,\text{ for all } (y,\xi,h) \in \mathcal{P}^2_*.
\ee
On the other hand, the dynamics for the transformed linearized state is
\begin{gather}
\label{eqxi}
\dot\xi_1 = \xi_2+y,\quad \dot\xi_2=0,\quad \dot\xi_3 = \hat{x}_1 \xi_1 + \hat{x}_2\xi_2 + \hat{x}_2y,\\
\xi_{1,0} = \xi_{2,0} = \xi_{3,0}=0.
\end{gather}
Take $(y,\xi,h) \in \mathcal{P}^2_*.$ Then
\be
\label{xi2}
\xi_2\equiv 0\quad \text{on } [0,5].
\ee
In view of \cite[equation (3.15)(i)]{AronnaBonnansGoh2016}, we have that $-\xi_2-y=0$ on $C.$ Thus, due to \eqref{xi2}, we get that
\be
\label{y=0}
y=0\quad \text{on } B_- \cup C.
\ee
Thus, from \eqref{eqxi}-\eqref{y=0}, we get $\xi_1=\xi_3=0$ on $B_- \cup C.$
As for the last component $h$ we get, from the linearized
cost equation \cite[equation (3.16)]{AronnaBonnansGoh2016} and due to the fact that there no final constraints, that
\be
\label{h1}
\hat{x}_{1,5}\,\xi_{1,5} + \xi_{3,5} = 0.
\ee
Then, we deduce that there is no restriction on $h.$
We obtain
\be
\mathcal{P}^2_* =
\left\{
\begin{split}
& (y,\xi,h) \in L^2 \times (H^1)^n \times \mathbb{R} : \,\, y=\xi_1=\xi_2 = 0 \text{ on } B_- \cup C,\\
& \xi_2 = 0 \text{ on } [0,5],\ \dot\xi_1 = y \text{ and } \dot \xi_3 = \hat{x}_1\xi_1 + \hat{x}_2 y\ \text{ on } S, \, \eqref{h1} \text{ holds}
\end{split}
\right\}.
\ee
The quadratic forms $Q$ and $\Omega$ are given by
\be
Q(v,z):= \int_0^T( z^2_1 +z^2_2) {\rm d} t + z^2_{1,5};
\quad
{\Omega}(y,h,\xi):= \int_0^T( \xi^2_1 +
(\xi_2+y)^2) {\rm d} t + h^2.
\ee
Thus, ${\Omega}$ is a Legendre form on $\{(y,h,\xi) \in L^2\times \mathbb{R}
\times (H^1)^3:\text{\eqref{eqxi} holds}\}$ and is coercive on
$\mathcal{P}^2_*.$ Hence Theorem 5 in \cite{AronnaBonnansGoh2016}
holds. Consequently, $(\hat{u},\hat{x})$ is a Pontryagin minimum of problem
\eqref{Preg} satisfying the $\gamma$-growth condition.
\subsection{Transformed problem}
Here we transform the problem \eqref{Preg} to obtain a problem with neither control nor state constraints, as done in Section \ref{SubsecReformulation}.
The optimal control associated with \eqref{Preg} has a $B_-CS$ structure, as said above. Then we triplicate the number of state variables obtaining the new variables $X_1,\dots,X_9$ and we
consider two switching times that we write $X_{10},X_{11}.$ The new
problem has only one control variable that corresponds to the
singular arc and which we call $V.$ The reformulation of
\eqref{Preg} is as follows
\be
\label{Pregref}
\begin{split}
\min\ &X_{9,1} + \mbox{$\frac{1}{2}$} X^2_{7,1}, \\
& \dot{X}_1 = X_{10} X_2, \\
& \dot{X}_2 = - X_{10}, \\
& \dot{X}_3 = \mbox{$\frac{1}{2}$} X_{10} (X_1^2 + X_2^2), \\
& \dot{X}_4 = (X_{11}-X_{10})X_5, \\
& \dot{X}_5 = 0, \\
& \dot{X}_6 = \mbox{$\frac{1}{2}$} (X_{11}-X_{10}) (X_4^2 + X_5^2), \\
& \dot{X}_7 = (5-X_{11}) X_8,\\
& \dot{X}_8 =(5-X_{11}) V,\\
& \dot{X}_9 = \mbox{$\frac{1}{2}$} (5-X_{11}) (X_7^2 +X_8^2),\\
& \dot{X}_{10} = 0, \\
& \dot{X}_{11} = 0,\\
& X_{1,0}=0,\quad X_{2,0}=1,\quad X_{3,0}=0, \\
& X_{1,1}=X_{4,0},\quad X_{2,1} = X_{5,0},\quad X_{3,1} = X_{6,0},\\
& X_{4,1}=X_{7,0},\quad X_{5,1} = X_{8,0},\quad X_{6,1} = X_{9,0},\\
& X_{2,1}=0.2, \, {( \text{or } g(x_2(\tau_2))=0)}. \\
\end{split}
\ee
From \eqref{uonS} we deduce that $V=X_7.$
We solved problem \eqref{Pregref} by the shooting algorithm proposed in Section \ref{SecShooting}. The graphics of the optimal control and states are shown in Figure \ref{regulator} in the original variables $u$ and $x,$ and the corresponding costate variables $p_1,p_2,p_3$ are displayed in Figure \ref{regulator2}.
In our numerical tests we take 1000 time steps.
The optimal switching times obtained numerically are
$\hat{\tau}_1=1.2$ and $\hat{\tau}_2 = 2.6036023,$ so these values
agree with the ones founded analytically
in \eqref{hattau1} and \eqref{hattau2},
while the optimal cost is $0.3934884$ and the shooting function evaluated at the optimal trajectory is $5.\times 10^{-6}.$
\begin{figure}[!h]
\centering
\begin{center}
\subfigure{\includegraphics [width=5.5cm]{control.eps}}
\subfigure{\includegraphics [width=5.5cm]{state.eps}}
\end{center}
\caption{Optimal control and state variables}
\label{regulator}
\end{figure}
\begin{figure}[!h]
\centering
\begin{center}
\includegraphics [width=5.5cm]{costate.eps}
\end{center}
\caption{Costate variables}
\label{regulator2}
\end{figure}
\section{Conclusion}
We have shown that, for problems that are affine w.r.t. the control, and for which the time interval can be partitioned in a finite set of arcs, the shooting algorithm converges locally quadratically if a second order sufficient condition holds, and thus the method may be an efficient way to get a highly accurate solution.
The essential tool was to formulate a transformed problem. We think that this approach could be extended to more general problems with several state constraints and control variables, the latter possibly entering nonlinearly in the problem. It would be then important to take into account the possibility of having high-order state constraints.
\bibliographystyle{plain}
| -93,898.798585 |
[
-1.70703125,
1.4560546875
] | 27.563396 |
[
-2.08984375,
3.541015625,
0.58935546875,
-3.88671875,
-2.603515625,
3.32421875
] |
[
-3.837890625,
-0.01218414306640625,
-4.578125,
-2.4765625
] | 466 | 8,875 |
[
-2.34765625,
2.7734375
] | 36.63492 |
[
-2.73828125,
-0.45703125,
-1.7978515625,
-2.0390625,
-0.8017578125,
5.5703125
] | 0.54944 | 19.11657 | 24.585915 | 3.133055 |
[
1.1932601928710938
] | -53,832.188813 | 5.639549 | -93,038.492952 | 0.53945 | 6.373667 |
[
-1.7734375,
-2.3203125,
-3.7890625,
-5.12890625,
1.771484375,
11.125
] |
[
-1.4873046875,
1.794921875,
-0.72802734375,
-0.77978515625,
0.455322265625,
-1.3076171875
] | |
BkiUbKs5qU2ApuiOWGYO
|
\section{Introduction}
Deep learning or deep neural networks (DNNs) have become the fundamental element and core enabler of ubiquitous artificial intelligence.
After obtaining DNN models trained with
a huge amount of data,
they can be
deployed for inference, perception and control tasks in various autonomous systems and internet-of-things (IoT) applications.
Recently, along with the rapid emergence of high-end mobile devices\footnote{Modern mobile platforms become increasingly sophisticated, usually equipped with both CPUs and GPUs, e.g., Qualcomm Snapdragon 855 \cite{snapdragon855} has an octa-core Kryo 485 CPU and an Adreno 640 GPU.}, executing DNNs on mobile platforms gains popularity and is quickly becoming the mainstream~\cite{deng2019deep,lane2016deepx,lane2017squeezing,ota2017deep,zhang2019deep}
for broad applications such as sensor nodes, wireless access points, smartphones, wearable devices, video streaming, augmented reality, robotics, unmanned vehicles, smart health devices, etc.~\cite{philipp2011sensor,lane2015early,boticki2010quiet,rodgers2014recent,bhattacharya2016smart}.
Considering the nature of these applications,
achieving {\em real-time DNN inference} is an ideal
but yet a very challenging goal for mobile devices
due to the limited computing resources of embedded processors.
For example, consider VGG-16 \cite{simonyan2014very}, one of the key DNN models in transfer learning with broad application scenarios. For an embedded GPU (Adreno 640, with 16-bit floating-point for weights/intermediate results), it takes 242ms to
perform inference using TVM \cite{chen2018tvm}, and is not even supported in TensorFlow-Lite (TFLite) \cite{TensorFlow-Lite} --- these are two representative mobile-oriented, end-to-end DNN inference acceleration frameworks. It is clearly far from
real-time execution.
To achieve the goal, it is necessary to
consider algorithm-level innovations.
To this end, {\em DNN model compression} techniques, including \emph{weight pruning}~\cite{han2015learning,han2015deep,guo2016dynamic,dai2017nest,mao2017exploring,wen2016learning,he2017channel} and \emph{weight/activation quantization}~\cite{leng2017extremely,park2017weighted,zhou2017incremental,lin2016fixed,wu2016quantized,rastegari2016xnor,hubara2016binarized,courbariaux2015binaryconnect,courbariaux2016binarized,gupta2015deep,hubara2017quantized}, have been proposed and
studied intensively for model storage reduction and computation acceleration.
Early efforts on DNN model compression~\cite{han2015learning,han2015deep,guo2016dynamic,dai2017nest,mao2017exploring,wen2016learning,he2017channel} mainly rely on iterative and heuristic methods, with limited and non-uniform model compression rates.
Recently, a systematic DNN model compression framework (ADMM-NN) has been developed using the powerful mathematical optimization tool ADMM (Alternating Direction Methods of Multipliers) \cite{boyd2011distributed,hong2016convergence,liu2018zeroth}, currently achieving the best performance (in terms of model compression rate under the same accuracy) on weight pruning \cite{zhang2018systematic,ren2019ADMMNN} and one of the best on weight quantization \cite{leng2017extremely}.
Despite the high compression ratio,
there is a significant gap between algorithm-level innovations and hardware-level performance
optimizations for DNN inference acceleration.
Specifically, the general but
{\em non-structured} weight pruning
(i.e., arbitrary weight can be pruned)~\cite{han2015learning,guo2016dynamic}
can seriously affect processing throughput
because the indices for the compressed
weight representation prevent achieving
high parallelism~\cite{wen2016learning,he2017channel,mao2017exploring}.
While ADMM-NN achieves higher and more reliable
compression ratios, hardware implementation
obstacle due to the non-structured nature still stays the same.
Alternatively, the {\em structured} pruning \cite{wen2016learning,he2017channel,mao2017exploring},
e.g., filter and channel pruning, can generate
more hardware-friendly models but result
in relatively higher accuracy drop.
To achieve the real-time inference for
representative DNNs in mobile devices,
it is imperative to develop
an end-to-end DNN acceleration framework
that achieves {\em both high accuracy
and high hardware efficiency}.
We make a key observation
that the general non-structured pruning
and current structured pruning
represent two extremes in the design space.
In non-structured pruning, {\em any} weight can be
pruned, while in structured pruning, the pruning
is done for
the {\em whole filter or channel}.
Thus, non-structured pruning is
completely {\em fine-grained}, which achieves
high compression ratio but is not hardware or
software optimization friendly,
while structured pruning is {\em coarse-grained},
which generates hardware-efficient regular models
with higher accuracy loss.
In this paper, we advance the state-of-the-art
by naturally introducing a new dimension,
{\em fine-grained
pruning patterns inside the coarse-grained
structures}, revealing a previously {\em unknown}
point in design space.
This new dimension allows more flexible
exploration of the trade-off between
accuracy and hardware efficiency.
In this paradigm, the key question is
{\em how to ``recover'' the hardware efficiency
lost due to the fine-grained patterns}.
The unique insight of our solution is
to use {\em compiler} to seamlessly
close the gap between hardware efficiency
of fully structured pruning and the
pattern-based ``semi-structured'' pruning.
Specifically, we propose
{\em PatDNN\xspace}, a novel end-to-end mobile DNN acceleration framework that can generate highly accurate
DNN models using pattern-based pruning methods
and guarantee execution efficiency with
compiler optimizations.
PatDNN\xspace consists of two stages: (1) \emph{pattern-based training stage}, which performs kernel pattern and connectivity pruning (termed \emph{pattern-based pruning} in general) with a pattern set generation and an extended ADMM solution framework. (2) \emph{execution code generation stage}, which converts DNN models into computational graphs and applies multiple optimizations including: a high-level and fine-grained DNN layerwise representation, filter kernel reorder, load redundancy eliminations, and automatic parameter tuning. All design optimizations are general, and applicable to both mobile CPUs and GPUs.
In sum, this paper makes several major contributions:
\begin{itemize
\item
First, it proposes a novel {\em pattern-based} DNN pruning approach that achieves the benefits of both non-structured and structured pruning while avoiding their weaknesses.
\item
Second, it enhances the recent ADMM-NN framework~\cite{ren2019ADMMNN,ye2019progressive} with pattern selection capability to map a pattern to each kernel, and train non-zero weights.
\item
Third, it identifies the compatibility of the proposed pattern-based pruning scheme with compiler code generation, and develop multiple novel compiler optimizations for compressed DNN execution. These optimization opportunities are enabled only by our pattern-based design, and do not exist in any prior DNN execution frameworks.
\item
Fourth, it implements an end-to-end DNN acceleration framework {\em PatDNN} on mobile platforms, compatible with modern embedded CPU and GPU architectures, achieving real-time performance on representative DNNs without accuracy loss for the first time.
\end{itemize}
We compare PatDNN with three state-of-the-art end-to-end DNN frameworks on both mobile CPU and GPU, TensorFlow Lite~\cite{TensorFlow-Lite}, TVM~\cite{chen2018tvm}, and Alibaba Mobile Neural Networks~\cite{Ali-MNN} using three widely used DNNs, VGG-16, ResNet-50, and MobileNet-V2 and two benchmark datasets, ImageNet and CIFAR-10. Our evaluation results show that PatDNN achieves up to $44.5\times$ speedup without any accuracy compromise. Using Adreno 640 embedded GPU, PatDNN achieves 18.9ms inference time of VGG-16 on ImageNet dataset. To the best of our knowledge, it is the first time to achieve real-time execution of such representative large-scale DNNs on mobile devices.
\section{Evaluation}\label{sec:evaluation}
This section evaluates the execution performance of PatDNN by comparing it with three state-of-the-art DNN inference acceleration frameworks,
TFLite~\cite{TensorFlow-Lite}, TVM~\cite{chen2018tvm}, and MNN~\cite{Ali-MNN}.
All major optimizations of these frameworks (and our PatDNN) are summarized in Table~\ref{tab:dnn-frameworks}.
\begin{figure*}[t]
\centering
\subfloat[ImageNet-CPU]{
\includegraphics[width=0.235\textwidth]{fig/Overview_ImageNet_CPU.pdf}
}
\subfloat[CIFAR-10-CPU]{
\includegraphics[width=0.25\textwidth]{fig/Overview_CIFA_CPU.pdf}
}
\subfloat[ImageNet-GPU]{
\includegraphics[width=0.25\textwidth]{fig/Overview_ImageNet_GPU_16.pdf}
}
\subfloat[CIFAR-10-GPU]{
\includegraphics[width=0.25\textwidth]{fig/Overview_CIFA_GPU16.pdf}
}
\caption{Overall performance: x-axis: different trained DNN models; y-axis: average DNN inference execution time on a single input.}
\label{fig:eva_overview_performance}
\vspace{-1em}
\end{figure*}
\subsection{Methodology}
\noindent{\bf Evaluation Objective:} Our overall evaluation
demonstrates that achieving real-time inference of large-scale DNNs on modern mobile devices is possible with PatDNN. Specifically, the evaluation has five objectives:
(1) demonstrating that PatDNN outperforms existing state-of-the-art DNN frameworks
without any accuracy compromise;
(2) studying the performance effect of our key compiler optimizations
and explaining the reasons for performance improvement;
(3) further confirming the performance of PatDNN by comparing its pure GFLOPS with our optimized dense baseline;
(4) showing that PatDNN performs similarly on different mobile platforms, i.e., PatDNN has a good portability; and
(5) unveiling the impact of pattern count selections on both the accuracy and performance.
\noindent{\bf DNNs and Datasets:} PatDNN is evaluated on three mainstream DNNs, VGG-16 (VGG), ResNet-50 (RNT), and Mobile-Net-V2 (MBNT). They are trained on two datasets, ImageNet and CIFAR-10. Table~\ref{tab:datasets} characterizes these trained DNNs.
Some information is omitted due to the space constraint, e.g., a uniform CONV pruning rate for VGG and RNT is $8\times$, and $4.4\times$, respectively (with uniform 3.6$\times$ connectivity pruning rate).
VGG has 13 CONV layers, and 5 of them have identical structures to others. Table~\ref{tab:eva_vgg_shortname} lists the filter shape ([{\tt \#output channel, \#input channel, kernel height, and kernel width}]) of these 9 unique layers and gives them a short name each.
\noindent{\bf Evaluation Platforms and Running Configurations:} Our experiments are conducted on a Samsung Galaxy S10 cell phone with the latest Qualcomm Snapdragon 855 mobile platform that consists of a Qualcomm Kryo 485 Octa-core CPU and a Qualcomm Adreno 640 GPU.
Our portability tests are conducted on a Xiaomi POCOPHONE F1 phone with a Qualcomm Snapdragon 845 that consists of a Kryo 385 Octa-core CPU and an Adreno 630 GPU, and an Honor Magic 2 phone with a Kirin 980 that consists of an ARM Octa-core CPU and a Mali-G76 GPU.
All tests run 50 times on different input (images) with 8 threads on CPU, and all pipelines on GPU. Because multiple runs do not vary significantly, this section only reports the average time for readability. Because CONV layers are most time-consuming, accounting for more than
$95\%$ ($90\%$ for VGG)
of the total execution time, our evaluation focuses on the CONV layers.
All runs are tuned to their best configurations, e.g., Winograd optimization~\cite{lavin2016fast} is used for all dense runs, and 16-bit float point is used for all GPU runs.
\subsection{Overall Performance}
Figure~\ref{fig:eva_overview_performance} shows the overall CPU and GPU performance of PatDNN compared to TFLite, TVM, MNN on all six trained DNNs. PatDNN outperforms all other frameworks for all cases.
On CPU, PatDNN achieves $12.3\times$ to $44.5\times$ speedup over TFLite, $2.4\times$ to $5.1\times$ over TVM, and $1.9\times$ to $7.1\times$ over MNN, respectively.
On GPU, PatDNN achieves $2.5\times$ to $20\times$, $2.8\times$ to $11.4\times$, and $1.6\times$ to $6.2\times$ speedup over TFLite, TVM, and MNN, respectively\footnote{TFLite does not support executing VGG on ImageNet data set on GPU due to its too large memory footprint.}. For the largest DNN (VGG) and largest data set (ImageNet), PatDNN completes CONV layers on a single input within
18.9 ms on GPU.
Even including the other rest layers (like FC), PatDNN can still meet the real-time requirement (usually 30 frames/sec,
i.e., 33 ms/frame).
PatDNN outperforms other frameworks because of two major reasons. First, its dense version is already $1.1\times$ to $1.6\times$ faster than TVM and MNN on mobile platforms because of some extra optimizations (as shown in Table~\ref{tab:dnn-frameworks}). Figure~\ref{fig:eva_dense_pattern_compare}(a) shows that PatDNN's dense version is faster than MNN on VGG, our largest DNN. Second, the pattern-based pruning reduces the overall computation by $3\times$ to $8\times$. Such computation reduction unfortunately cannot transfer to performance gains directly.
We confirmed this by implementing an optimized sparse matrix version of PatDNN based on CSR~\cite{greathouse2016clsparse}, which shows almost the same speed to PatDNN's dense version. However, the subsequent compiler-level optimizations (filter kernel reorder, load redundancy elimination, auto-tuning, and compressed weight storage) successfully convert this computation reduction into real performance gains. We conduct a more detailed study on these optimizations in the next Section, and Figure~\ref{fig:eva_reorder_rle_tuning_speedup} shows a break-down of these optimizations' contributions. Figures~\ref{fig:eva_reorder_rle_detail} to~\ref{fig:eva_csr_tight_storage} provide a detailed analysis of the underlying reasons.
\begin{figure}[t]
\centering
\subfloat[CPU]{
\includegraphics[width=0.485 \columnwidth]{fig/Reorder_RLE_Tuning_CPU.pdf}
}
\subfloat[GPU]{
\includegraphics[width=0.5 \columnwidth]{fig/Reorder_RLE_Tuning_GPU.pdf}
}
\caption{Speedup of opt/no-opt on each unique CONV layer.}
\label{fig:eva_reorder_rle_tuning_speedup}
\end{figure}
\subsection{Optimization Evaluation}
This section studies the effect of our key compiler optimizations
and shows that our PatDNN's good performance mainly comes from these pattern-enabled optimizations.
This part also compares the extra structure overhead between FKW and CSR.
Constrained by space, we only report the results of
VGG, our most complex DNN, on the most widely accepted dataset (ImageNet). Experiments on other DNNs and datasets show the same trend. The rest parts also use VGG on ImageNet as a representative example
Figure~\ref{fig:eva_reorder_rle_tuning_speedup} reports the speedup of the versions with optimizations over the version without any optimization on each unique CONV layer of VGG on CPU and GPU, respectively.
On CPU, reorder brings $1.6\times$ to $3.0\times$ speedup, load redundancy eliminations bring additional $1.6\times$ to $2.8\times$ speedup, and parameter tuning brings additional $1.2\times$ to $1.9\times$ speedup. On GPU, these numbers are $2.7\times$ to $6.1\times$, $1.5\times$ to $3.3\times$ and $1.4\times$ to $3.8\times$. It is interesting that FKR brings more benefits on GPU than on CPU, because GPU's performance is more sensitive to the thread divergence and load balance due to its massive parallel nature.
We next study why these optimizations work.
\begin{figure}[t]
\centering
\subfloat[Filter length distribution before and after filter kernel reorder for L4]{
\includegraphics[width=0.484 \columnwidth]{fig/Reorder_Load_Balance_Quality.pdf}
}
\subfloat[Register load counts before and after elimination]{
\includegraphics[width=0.516 \columnwidth]{fig/RLE_Reduction.pdf}
}
\caption{Profiling result: reorder and redundancy elimination.}
\label{fig:eva_reorder_rle_detail}
\end{figure}
\begin{figure}[t]
\centering
\subfloat[ImageNet]{\includegraphics[width=0.5 \columnwidth]{fig/Four_Computation_Comparison_VGG16_ImageNet.pdf}}
\subfloat[CIFAR-10]{\includegraphics[width=0.5 \columnwidth]{fig/Four_Computation_Comparison_VGG16_CIFA.pdf}}
\caption{Effect of different loop permutations and loop tiling.}
\label{fig:eva_four_comp_tuning}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=0.45\textwidth]{fig/CSR_VS_Tight_Storage.pdf}
\caption{Extra data structure overhead: FKW over CSR on unique VGG CONV layers with different pruning rates.}
\label{fig:eva_csr_tight_storage}
\end{figure}
\noindent{\bf Filter Kernel Reorder:}
Figure~\ref{fig:eva_reorder_rle_detail} (a) reports the filter length distribution of VGG L4 before and after FKR. Before reorder, the filters with varied lengths are distributed randomly, resulting in significant load imbalance if assigning them to different threads. After reorder, the filters are grouped into three groups, and the filters within each group have identical lengths. Each group could be executed by CPU threads simultaneously, or mapped to the same GPU thread block.
\noindent{\bf Load Redundant Elimination:}
Figure~\ref{fig:eva_reorder_rle_detail} (b) reports the register load counts before and after LRE for each unique CONV of VGG. It shows that our register LRE can significantly reduce the number of register loads.
Note that even if register load has lower
latency than cache or memory load, the memory/cache
performance has nevertheless been aggressively
optimized by conventional tiling.
Thus, the significant performance gains
must have been achieved with the reduced
number of register loads.
\noindent{\bf Auto-tuning:}
Figure~\ref{fig:eva_four_comp_tuning} reports
the CPU performance (in GFLOPS) of each unique VGG CONV layer with varied loop permutations, and with or w/o
blocking on ImageNet and CIFAR-10, respectively. It shows that different inputs and layers may require different configurations.
Proper tuning will bring significant benefits. Constrained by space, we omit the GPU results and tuning results about GPU data placement.
\noindent{\bf Compressed Weight Storage:}
Figure~\ref{fig:eva_csr_tight_storage} shows the extra data structure overhead
(i.e., the size of data structures other than weights) of FKW over CSR on each unique VGG CONV layer with three kinds of pruning rates, $18\times$, $12\times$, and $8\times$ respectively. For each one, FKW saves 93.4\%, 91.6\%, and 87.9\% extra data structure overhead over CSR in total, resulting in 46.7\%, 45.8\%, and 43.9\% overall storage space saving.
\begin{figure}[t]
\centering
\subfloat[Dense w/o Wino]{
\includegraphics[width=0.267 \columnwidth]{fig/Dense_No_Winograd.pdf}
}
\subfloat[Performance in GFLOPS: pattern vs dense]{
\includegraphics[width=0.7 \columnwidth]{fig/Dense_VS_Sparse_VGG16_ImageNet.pdf}
}
\caption{GFLOPS performance study: PatDNN vs dense.}
\label{fig:eva_dense_pattern_compare}
\end{figure}
\subsection{PatDNN Performance Analysis in GFLOPS}
To further analyze the performance of PatDNN, this part compares its pure GFLOPS with our dense implementation. To conduct an apple-to-apple comparison, we turn off the Winograd optimization that transforms the convolution operation to matrix-multiplication for a trade-off between the computation reduction and operation conversion overhead.
Figure~\ref{fig:eva_dense_pattern_compare} (a) shows that our dense version can serve as an optimized baseline, because it is even faster than MNN.
Figure~\ref{fig:eva_dense_pattern_compare} (b) shows that our pattern-based (sparse) PatDNN achieves comparable GFLOPS to our optimized dense baseline on CPU, and outperforms it on GPU. It implies that the memory performance of PatDNN is comparable to the dense baseline on CPU and even better than it on GPU. This benefits from our model compression and memory load (and register load) reductions. Without pattern-based pruning, the input, output, and DNN model compete for the limited memory/cache resource; after pruning, only the input and output compete for it. PatDNN also reduces the overall computation; thus, it significantly outperforms all other mobile frameworks.
We cannot achieve this performance without our pattern-based design, and our other sparse implementation with conventional sparse matrix optimizations can only get either comparable or even slower speed than other mobile frameworks.
\begin{figure}[t]
\centering
\subfloat[Kirin 980]{
\includegraphics[width=0.485 \columnwidth]{fig/Portability_Evaluation_Mali76.pdf}
}
\subfloat[Snapdragon 845]{
\includegraphics[width=0.485 \columnwidth]{fig/Portability_Evaluation_845.pdf}
}
\caption{Portability study: performance on two other platforms.}
\label{fig:eva_portability}
\end{figure}
\begin{table}[t]
\vspace{-5mm}
\caption{Pattern counts impact (with 3.6$\times$ connectivity pruning): accuracy loss and exe time for VGG.}
\label{tab:table_patterns_accu_speed}
\centering
\includegraphics[width=1\linewidth]{fig/table_accu_patterns_speed.pdf}
\vspace{5mm}
\end{table}
\subsection{Portability Study}
PatDNN is also evaluated on two other platforms to confirm its portability. Figure~\ref{fig:eva_portability} shows the result. On these platforms, PatDNN also outperforms other frameworks. Particularly, other frameworks run much slower on Magic 2 than on Snapdragon 855; however, PatDNN performs more stably.
This is because our pattern-based pruning leads to fewer computations and fewer memory accesses thus reducing the memory bandwidth pressure.
\subsection{Impact of Pattern Counts}
Table~\ref{tab:table_patterns_accu_speed} reports the impact of the pattern count selection on both the accuracy and execution time, under 3.6$\times$ uniform connectivity pruning rate. As increasing pattern counts, the accuracy increases slightly, however, the performance drops quickly. Our evaluation selects 8 patterns that result in ideal performance with a negligible accuracy loss.
\section{Overview of PatDNN}
\subsection{Pattern-based Pruning}
In pattern-based pruning, the key consideration
is how to design and select the patterns.
To achieve high accuracy and execution efficiency,
we need to design the patterns considering the
implication for {\em theory and algorithm},
{\em compiler optimization}, and
{\em hardware execution}.
Good patterns should have two key properties:
flexibility and regularity.
The {\em Flexibility} is not only desirable at theory and algorithm level but also enables efficient
compiler code generation.
Specifically, it allows compilers to
maximize or maintain
both instruction-level and thread-level parallelism.
The {\em regularity} not only results in
highly efficient hardware execution but also
enables efficient compiler optimizations such as
{\em redundant load elimination} to
further improve performance.
Compared to irregular structures,
recent works also show from theory and algorithm
level that high accuracy or function approximation
capability can be achieved at the same time
with certain regularity.
Given these two key properties,
we propose two pattern-based pruning techniques:
kernel pattern pruning and connectivity pruning.
\begin{figure
\centering
\includegraphics[width=0.45 \textwidth]{fig/Fig_Pattern_Connectivity_Pruning.pdf}
\caption{Illustration of (a) kernel pattern pruning on CONV kernels, and (b) connectivity pruning by removing kernels.}
\label{fig:pattern_connectivity}
\end{figure}
\textbf{Kernel Pattern Pruning} is illustrated in {Figure \ref{fig:pattern_connectivity}}. For each kernel (in a CONV filter), a fixed number of weights are pruned, and the remaining weights (white cells) form specific ``kernel patterns''. We define the example in {Figure \ref{fig:pattern_connectivity}} as 4-entry pattern pruning, since every kernel reserves 4 non-zero weights out of the original $3\times 3$ kernel (the most commonly used kernel).
The same approach is also applicable to other kernel sizes and the FC layer. For each kernel, it possesses \emph{flexibility} in choosing among a number of pre-defined patterns.
At {\em theory and algorithm} level, it is shown in {\cite{li2017pruning,lebedev2016fast}} that the desirable kernel shape has certain patterns
to match the connection structure in human
visual systems,
instead of a square shape.
The selection of appropriate pattern for each kernel
can be naturally done by extending
ADMM-based framework.
In Section~\ref{sec:pattern_acc_result}, we achieve
accuracy enhancement in all representative
DNNs in our testing.
At {\em compiler} level,
the pre-defined pattern allows compiler to
{\em re-order and generate codes} at filter and kernel level so that kernels with the same pattern
can be grouped for consecutive executions to
maximize instruction-level parallelism.
At {\em hardware} level, the 4-entry patterns are extremely friendly to the SIMD architecture in embedded processors based on either
GPUs or CPUs.
Note that our approach is general and can be
applied to any pre-defined patterns, not just
the 4-entry considered in the paper.
\begin{figure
\centering
\includegraphics[width=0.45 \textwidth]{fig/Fig_Connectivity1.pdf}
\caption{Illustration of connectivity pruning.}
\label{fig:connectivity}
\end{figure}
\textbf{Connectivity Pruning} is illustrated in {Figure \ref{fig:connectivity}}.
The key insight is to {\em cut the connections}
between certain input and output channels, which is equivalent to removal of corresponding kernels.
In CONV layers, the correlation between input channel $i$ and output channel $j$ is represented by the $i$-th kernel of filter $j$.
This method is proposed for overcoming the limited weight pruning rate by kernel pattern pruning.
At {\em theory and algorithm} levels, the connectivity pruning matches the desirability of locality in layerwise computations inspired by human visual systems \cite{yamins2016using,yamins2014performance}. It is more flexible than the prior filter/channel pruning schemes that remove whole filters/channels, thereby achieving higher accuracy. At {\em compiler and hardware} level, removed kernels and associated computations can be grouped by compiler using the \emph{re-ordering} capability without affecting the other computations, thereby maintaining parallelism degree.
\begin{table
\vspace{-5mm}
\caption{Qualitative comparison of different pruning schemes on accuracy and speedup under the same pruning rate.}\label{tab:prunecompare}
\centering
\includegraphics[width = 1\linewidth]{fig/Table_PruneCompare_HIGH_RESOLUTION.pdf}
\end{table}
\subsection{Overview of PatDNN Acceleration Framework}
\begin{figure*
\centering
\includegraphics[width=1 \textwidth]{fig/system_overview.pdf}
\caption{Overview of PatDNN acceleration framework.}
\label{fig:system-overview}
\end{figure*}
Based on the above discussions, we propose
{\em PatDNN\xspace}, a novel end-to-end mobile DNN acceleration framework that can generate highly accurate
DNN models using pattern-based pruning methods
and guarantee execution efficiency with
compiler optimizations.
Compared to recent prior works \cite{ren2019ADMMNN,he2018amc,he2017channel,wen2016learning},
PatDNN\xspace uniquely enables
{\em cross-layer vertical integration},
making it desirable across theory/algorithm,
compiler and hardware.
Allowing compilers to treat
pruned kernels as special patterns,
our approach not only achieves high pruning rate
with high accuracy,
but also effectively converts into performance
improvements due to hardware friendly properties.
As shown in {Table \ref{tab:prunecompare}}, PatDNN\xspace can achieve the benefits of both non-structured and structured pruning.
The key enabler to achieving this goal
is to leverage compiler to maintain the
efficiency of structured pruning based on
kernel pattern and connectivity pruning.
Our approach is an excellent example of
hardware and software co-design, which can be
compared to an intuitive analogy:
the multi-level cache memory hierarchy provides
sufficient hardware supports to hide memory
access latency and explore locality, but
compiler and software optimizations are still needed
to fully realize effective
cache management policy.
Figure~\ref{fig:system-overview}
shows the overview of PatDNN\xspace which consists of
two stages:
(1) {\em pattern-based training stage} (Section~\ref{sec:training}), which
performs kernel pattern and connectivity pruning
with an extended ADMM solution framework.
(2) {\em execution code generation stage} (Section~\ref{sec:inference}), which
performs multiple effective optimizations based
on the patterns.
Similar to TVM \cite{chen2018tvm}, PatDNN\xspace converts DNN models into computational graphs and applies multiple graph-based optimizations.
Based on these optimizations, we focus on
layerwise design and optimization including
a high-level and fine-grained DNN layerwise representation (LR), filter kernel reorder, load redundancy eliminations, and automatic parameter tuning. All of these designs and optimizations are general,
and applicable to both mobile CPUs and GPUs.
The second stage generates optimized execution codes as well as DNN models with weights stored in a novel compact format.
\section{Background and Motivation}
\subsection{Layerwise Computation of DNNs}
\begin{figure
\centering
\includegraphics[width=0.45 \textwidth]{fig/Fig_FilterandChannel1.pdf}
\caption{DNN CONV layer computation.}
\label{fig:DNNlayer}
\end{figure}
DNN models can be viewed as cascaded connections of multiple functional layers, such as convolutional (CONV), fully-connected (FC), and pooling (POOL) layers, to extract features for classification or detection~\cite{lee2009convolutional,karpathy2014large,yu2011deep}.
Take the most computation-intensive CONV layer as an example,
as shown in Figure \ref{fig:DNNlayer}, the input feature map of the $k$-th layer has a size of $M_k \times N_k \times C_k$, where $C_k$ is the number of \emph{channels} of the input feature map.
This layer uses $C_{k+1}$ CONV filters, each with a size of $P_k \times Q_k \times C_k$.
Note that the number of \emph{kernels} $C_k$ in a CONV filter should match the number of channels $C_k$ in the input feature map to perform convolution.
Each $j$-th CONV filter performs convolution with the input feature map, using a stride of $S_k$, resulting in the $j$-th channel in the output feature map.
Therefore, the number of channels in the output feature map equals to the number of filters $C_{k+1}$, while the size of the output feature map i.e., $M_{k+1}$ and $N_{k+1}$ is determined by $M_k$, $N_k$, $P_k$, $Q_k$, and $S_k$.
The CONV layer is followed by an activation layer, which performs an activation operation, typically ReLU, on the output feature map.
Besides the functional layers in DNNs, {batch normalization} becomes an essential operation to increase the stability of DNN training by overcoming the gradient vanishing issue \cite{ioffe2015batch}.
\subsection{Mobile Acceleration of DNNs}
In recent years, there have been intensive efforts
on DNN inference acceleration frameworks targeting mobile devices, include DeepX~\cite{lane2016deepx}, TFLite~\cite{TensorFlow-Lite}, DeepEar~\cite{lane2015deepear}, TVM~\cite{chen2018tvm}, Alibaba Mobile Neural Network (MNN) \cite{Ali-MNN}, DeepCache~\cite{xu2018deepcache}, DeepMon~\cite{huynh2017deepmon}, DeepSense~\cite{yao2017deepsense}, and MCDNN~\cite{han2016mcdnn}.
Most of these prior works do not fully utilize model compression techniques.
Other efforts that explore model sparsity and model compression to accelerate the DNN execution include Liu et al.~\cite{liu2015sparse}, DeftNN~\cite{hill2017deftnn}, SCNN~\cite{parashar2017scnn}, AdaDeep~\cite{liu2018demand}. However, they either do not target mobile platforms, or require new hardware, or trade off compression rate and accuracy,
introducing various drawbacks compared to our work.
Table~\ref{tab:dnn-frameworks} compares the major optimization techniques offered by three state-of-the-art, end-to-end DNN inference frameworks (TFLite~\cite{TensorFlow-Lite}, TVM~\cite{chen2018tvm}, and MNN~\cite{Ali-MNN}).
We do not include other efforts, e.g., DeepCache~\cite{xu2018deepcache} and DeepMon~\cite{huynh2017deepmon}, since they mainly focus on specific DNN applications rather than general DNNs.
In this work, our goal is to find
the most appropriate weight pruning scheme for mobile DNN acceleration and the corresponding full-stack acceleration framework. We utilize 16-bit floating point representation on GPU for both weights and intermediate results which is supported in mobile devices and shown to incur no accuracy loss \cite{TensorFlow-Lite,Ali-MNN,chen2018tvm} for DNNs.
\begin{table}
\scriptsize
\caption{DNN acceleration frameworks on mobile devices.}\label{tab:dnn-frameworks}
\vspace{-1.5em}
{\setlength{\tabcolsep}{3.6pt}
\begin{tabular}{llcccc}\\
\toprule
DNNs
& Optimization Knobs & TFLite & TVM & MNN & {\bf Ours}\\
\midrule
& Parameters auto-tuning & N & Y & N & {\bf Y}\\
& CPU/GPU support & Y & Y & Y & {\bf Y}\\
Dense
& Half-floating support & Y & Y & Y & {\bf Y} \\
& Computation graph optimization & Y\tmark[$!$] & Y\tmark[*] & Y\tmark[$!$] & {\bf Y}\tmark[**] \\
& Tensor optimization & Y\tmark[$!$] & Y\tmark[$\dag$] & Y\tmark[$!$] & {\bf Y}\tmark[$\dag\dag$]\\
\midrule
& Sparse DNN model support & N & N & N & {\bf Y}\\
& Pattern-based pruning & N & N & N & {\bf Y}\\
Sparse
& Connectivity pruning & N & N & N & {\bf Y}\\
& Filter kernel reordering & N & N & N & {\bf Y}\\
& Opt. sparse kernel code generation & N & N & N & {\bf Y}\\
& Auto-tuning for sparse models & N & N & N & {\bf Y}\\
\bottomrule
\multicolumn{5}{l}{{\tiny * Operator fusion, constant folding, static memory plan, and data layout transform}} \\
\multicolumn{5}{l}{{\tiny ** Besides above in *, operation replacement}} \\
\multicolumn{5}{l}{{\tiny $\dag$ Scheduling, nested parallelism, tensorization, explicit memory latency hiding}} \\
\multicolumn{5}{l}{{\tiny $\dag\dag$ Besides above in $\dag$, dense kernel reordering, SIMD operation optimization}} \\
\multicolumn{5}{l}{{\tiny $!$ Similar optimizations as TVM, but less advanced}}
\end{tabular}
}
\end{table}
\subsection{DNN Model Compression and Challenges}
DNN model compression has been proposed for simultaneously reducing the storage/computation and accelerating inference with minor classification accuracy (or prediction quality) loss. Model compression is performed during DNN training. Two important categories of DNN model compression techniques are weight pruning \cite{han2015learning,guo2016dynamic,dai2017nest,mao2017exploring,wen2016learning,he2017channel} and weight quantization \cite{leng2017extremely,park2017weighted,zhou2017incremental,lin2016fixed,wu2016quantized,rastegari2016xnor,hubara2016binarized,courbariaux2015binaryconnect}.
\begin{figure
\centering
\includegraphics[width=0.45 \textwidth]{fig/Fig_StructuredPruning1.pdf}
\caption{(a) Non-structured weight pruning and {(b) two types of structured weight pruning.}}
\label{fig:structuredpruning}
\end{figure}
{\em Weight pruning} reduces the redundancy in the number of weights.
As shown in Figure~\ref{fig:structuredpruning},
two main approaches of weight pruning are
(1) the general and non-structured pruning; and
(2) structured pruning, which produces
irregular and regular compressed DNN models.
{\bf Non-Structured Pruning:}
In this method, arbitrary weight can be pruned.
It can result in a high pruning rate, i.e.,
reduction in the number of weights,
which can reduce the actual computation.
For compiler and code optimization,
non-structured pruning
incurs several challenges
due to the irregularity in computation and memory access.
First, the irregular and sparse kernel
weights require
{\em heavy control-flow instructions}, which
degrade instruction-level parallelism.
Second, it introduces
{\em thread divergence and load imbalance} due to the fact that kernels in different filters have divergent workloads and they are usually processed by multiple threads --- a key concern for efficient thread-level parallelism.
Third, it usually incurs {\em low memory performance} due to poor data locality and cache performance.
More importantly, it prohibits advanced memory optimizations such as eliminating redundant loads that widely exist in convolution operations.
Similarly, for hardware acceleration,
since the pruned models are stored in
some sparse matrix format with indices,
they often
lead to performance degradation in GPU and CPU implementations \cite{han2015learning,guo2016dynamic,dai2017nest}.
{\bf Structured Pruning: }
This method can produce regular,
but smaller weight matrices.
{Figure \ref{fig:structuredpruning} (b)} illustrates the representative structured pruning schemes: \emph{filter pruning} and \emph{channel pruning} \cite{wen2016learning}.
Filter and channel pruning can be considered as equivalent in that pruning a filter in the $k$-th layer is equivalent to pruning the corresponding channel in the $(k+1)$-th layer. Filter/channel pruning is compatible with Winograd algorithm \cite{winograd1980arithmetic,lavin2016fast}
that has been used to accelerate
computation of the original DNNs.
Due to the regular structure,
the GPU/CPU implementations typically lead to more significant acceleration \cite{mao2017exploring,wen2016learning,he2017channel}.
However, the structured pruning suffers from notable accuracy loss~\cite{wen2016learning,he2017channel}.
\subsection{ADMM-based DNN Model Compression Framework}
Recent work ADMM-NN~\cite{ren2019ADMMNN,ye2019progressive}
leverages Alternating Direction Methods of Multipliers (ADMM) method for joint DNN weight pruning and quantization.
ADMM is a powerful tool for optimization, by decomposing an original problem into two subproblems that can be solved separately and efficiently. For example, considering optimization problem $\min_{\bf{x}} f({\bf{x}})+g({\bf{x}}).$ In ADMM, this problem is decomposed into two subproblems on $\bf{x}$ and $\bf{z}$ (auxiliary variable), which will be solved iteratively until convergence. The first subproblem derives $\bf{x}$ given $\bf{z}$: $\min_{\bf{x}} f({\bf{x}})+q_1(\bf{x}|\bf{z})$. The second subproblem derives $\bf{z}$ given $\bf{x}$: $\min_{\bf{z}} g({\bf{z}})+q_2(\bf{z}|\bf{x})$. Both $q_1$ and $q_2$ are quadratic functions.
As a unique property, ADMM can effectively deal with a subset of combinatorial constraints and yield optimal (or at least high quality) solutions~\cite{hong2016convergence,liu2018zeroth}. Luckily, the necessary constraints in the DNN weight pruning and quantization belong to this subset of combinatorial constraints,
making ADMM applicable to DNN model compression.
Due to the unprecedented results on accuracy
and pruning rate, ADMM-NN~\cite{ren2019ADMMNN}
is considered as the state-of-art results for
non-structured weight pruning
and one of state-of-art methods for weight quantization.
For non-structured pruning, ADMM-NN achieves 167$\times$, 24$\times$, and 7$\times$ weight reductions on LeNet-5, AlexNet, and ResNet-50 models, respectively, without accuracy loss. However, the framework only focuses on non-structured weight pruning, in which the pruning rate does
not directly translate to performance improvements.
ADMM-NN can be extended to perform structured
pruning, i.e., filter/channel pruning, and
our results show that it
leads to 1.0\% Top-5 accuracy degradation with 3.8$\times$ weight reduction on VGG-16 CONV layers using ImageNet dataset.
Although better than prior work (1.7\% in {\cite{he2017channel}} and 1.4\% in AMC {\cite{he2018amc}}), this accuracy loss is not negligible for many applications.
\subsection{Motivation}
Based on the discussion of prior work
on weight pruning, we rethink the design space
and observe that non-structured and structured
represent two extremes in the design space.
In non-structured pruning, any weight can be
pruned, we consider it as a fine-grained method;
in structured pruning, the weights of
whole filter or channel are pruned together,
we consider it as a coarse-grained method.
Correspondingly, the two methods have different
implications on hardware acceleration and
software optimization: non-structured pruning
is not hardware or software optimization
friendly, so the higher pruning ratio cannot
fully translate to performance gain, while
structured pruning incurs higher accuracy loss.
The motivation of our study is to seek
an approach that can offer the best of both methods.
To achieve that,
we naturally introduce a new dimension,
{\em fine-grained
pruning patterns inside the coarse-grained
structures}, revealing a previously {\em unknown}
point in design space.
With the higher accuracy enabled by
fine-grained pruning pattern, the key question is
how to re-gain similar hardware efficiency
as coarse-gained structured pruning.
We take a unique approach and leverage
compiler optimizations to
close the performance
gap between full structured pruning and
pattern-based ``semi-structured'' pruning.
\section*{Acknowledgements}
The authors would like to thank the anonymous reviewers for their valuable and thorough comments. The authors are especially grateful to the shepherd
Yufei Ding for her extensive feedback and constructive suggestions that help improve this paper substantially. This work was
supported in part by the NSF awards CNS-1739748, CCF-1937500, CCF-1919117, CCF-1901378, and CCF-1919289.
\section{Further Implementation and Optimization}
\subsection{Parameter Auto-tuning}\label{sec:auto-tuning}
Many configuration parameters require careful tuning to guarantee the performance of the generated execution code. However, manual tuning is tedious, and hard to yield the optimal code.
Therefore, PatDNN also includes an auto-tuning component for selecting the best execution configuration.
It consists of two parts: first, an {\em explorer model} based on Genetic Algorithm to generate the configuration exploration space; and second, a {\em performance estimation model} created from our historical data to predict the possible best configuration and performance for a given hardware.
Compared with the simulated annealing in TVM, our explorer model supports better parallelism because it allows the initialization of an arbitrary number of chromosomes to start the search.
For a typical (large-scale) DNN like VGG-16, our exploration can complete in 3-5ms. During the exploration, history data is also collected for training the performance estimator (based on Multilayer Perceptron and least square regression loss). The advantage of this approach is that when deploying PatDNN on a new platform, it can give a quick prediction of the optimal configuration parameters as well as the possible execution time.
In addition, these tuning parameters are crucial to the performance of our PatDNN execution, thus need
to be carefully tuned by our auto-tuning module
including: data placement configurations on GPU, tiling sizes, loop permutations, and loop unrolling factors.
\section{PatDNN Inference Code Optimization}\label{sec:inference}
For DNN models with kernel pattern and connectivity pruning, PatDNN\xspace ensures hardware execution
efficiency of DNN inference
with optimized compiler and code generation.
As aforementioned, compiler optimizations
play the key role in ``recovering'' the performance
loss due to the fine-grained pattern-based
pruning compared to fully structured pruning.
This stage includes two-levels of optimizations: (1) optimizations on computational graphs that explore the potential opportunities among multiple DNN layers;
and (2) optimizations within each layer. PatDNN\xspace adopts an enhanced TVM~\cite{chen2018tvm}-like approach together with other innovations from the latest efforts in this direction (e.g., Tensor Comprehensions~\cite{vasilache2018tensor}) to implement the former (with major optimizations summarized in Table~\ref{tab:dnn-frameworks}).
Due to space limit, we do not elaborate each
as they are not the main research contribution
and not specific to DNN execution optimization
leveraging pattern-based pruning.
This section focuses on PatDNN\xspace's layerwise optimizations based on kernel pattern and connectivity pruning
that are specifically designed to address the challenges in DNN acceleration with non-structured weight pruning, i.e., {\em heavy control-flow instructions, thread divergence and load imbalance, and poor memory performance}. These optimizations are general, and applicable to both mobile CPUs and GPUs. Our framework can generate both optimized CPU (vectorized C++) code and GPU (OpenCL) code.
Figure~\ref{fig:compiler-flow} illustrates PatDNN's compiler-based optimization and code generation flow with a CONV layer example.
\subsection{Compiler-based PatDNN Inference Framework}\label{sec:ir}
\begin{figure*}[t
\centering
\includegraphics[width=0.9 \textwidth]{fig/code_transformation.pdf}
\vspace{2mm}
\caption{{\bf PatDNN's compiler-based optimization and code generation flow:} compiler takes both model codes with graph-based optimizations and a layerwise representation (as an example in Figure~\ref{fig:ir-example}) to generate low-level C/C++ and OpenCL codes (as {\tt No-opt}). This low-level code is further optimized with filter kernel reorder and our FKW compact model storage ({\tt +Reorder}), the register-level load redundancy elimination ({\tt +LRE}), and other optimizations like auto-tuning. Finally, the code is deployed on mobile devices.}
\label{fig:compiler-flow}
\end{figure*}
\begin{figure
\centering
\includegraphics[width=0.45 \textwidth]{fig/layer_ir_example_new.pdf}
\vspace{2mm}
\caption{An LR example for a CONV layer.}
\label{fig:ir-example}
\end{figure}
\noindent{\bf Layerwise Representation:}
The key feature of PatDNN\xspace
is its {\em sparsity- and pruning-aware} design. To support it, PatDNN\xspace proposes a high-level fine-grained Layerwise Representation (LR) to capture
the sparsity information.
This LR includes intensive DNN layer
specific information to enable aggressive layerwise optimizations.
In particular, it includes detailed {\em kernel pattern and connectivity-related information} (e.g., the pattern types presented in this layer, the pattern order in each filter, the connection between kernels and input/output channels, etc.); and {\em tuning-decided parameters} (e.g., the input and output tile sizes, unrolling factors, the loop permutation of this layer, etc.).
PatDNN\xspace extracts the pattern/connectivity information from DNN models with computational graph optimizations, and determines the tuning-related
parameters by the auto-tuning.
This LR is used for PatDNN\xspace's following optimizations:
(1) filter kernel reordering, which operates
on kernel pattern and connectivity-related
information, i.e., specifically the compressed weight storage structure;
and (2) load redundancy elimination, which
requires each kernel's pattern, the connectivity between kernels and input/output channels, and the exact input/output tile size and unroll factor.
After these optimizations, high-level LR
can generate compressed model and associated optimized model execution code by using the pattern-related information and other basic layer information extracted from DNN models, (e.g., the kernel size, computation stride, computation dilation, etc).
Figure~\ref{fig:compiler-flow} shows the optimization flow and two sample code skeletons ({\tt +Reorder} and {\tt +LRE}) for these two optimizations, respectively
Figure~\ref{fig:ir-example} shows a simplified LR example for a CONV layer (with 2-D kernels). This LR will generate execution code for CPU ({\tt device}). Two types of kernel patterns ({\tt [1, 2]}) present in this layer ({\tt patterns}) and the filter kernels' pattern layout is specified by our {\tt FKW} compressed weight storage format (clarified in Section~\ref{sec:format} in detail)\footnote{This LR is used after our filter kernel reorder, so the pattern information is stored in the optimized FKW format. Before reorder, a relatively loose data format is used, which is omitted due to the space limit.}. Its computation loop permutation is {\tt cohwci\_b}, i.e., in the order of output channel, output height, output width, and input channel, with blocking and unrolling. Their blocking sizes are specified in {\tt tile}. Their unrolling factors are specified in {\tt unroll}. Figure~\ref{fig:compiler-flow} ({\tt +Reorder}) also shows the execution code generated from this LR, in which the outer loops iterating on all tiles are omitted. The inner-most iteration processes kernels in each filter in the order of their pattern types, i.e., all kernels with pattern 1 in each filter will be processed at first, then kernels with pattern 2. This code optimization does not require any loop control-flows. This is guaranteed by our filter kernel reorder that is introduced in Section~\ref{sec:reorder} in details.
\subsection{Filter Kernel Reorder (FKR)}\label{sec:reorder}
\begin{figure
\centering
\includegraphics[width=0.48 \textwidth]{fig/filter_kernel_reordering.pdf}
\caption{An example of filter kernel reorder.}
\label{fig:kernel-reorder}
\end{figure}
Kernel pattern and connectivity pruning offer better opportunities to address the performance challenges in non-structured pruning thanks to its better regularity.
Specifically,
Filter kernel reorder (FKR) is designed to address two key challenges, i.e., heavy control-flow instructions, and thread divergence and load imbalance. Our basic insight is: for a specific DNN layer, the patterns of all kernels are already known after model training, so the inference computation pattern is also known before model deployment. FKR leverages this knowledge to organize the filters with similar kernels together to improve {\em inter-thread} parallelization and order the same kernels in a filter together to improve {\em intra-thread} parallelization.
Figure~\ref{fig:kernel-reorder} explains FKR with a simplified example. Here, a matrix represents a CONV layer of DNN and each cell is a kernel with pattern type denoted by the number on it. Empty kernels are the ones pruned by {\bf connectivity pruning}. The kernels in the same row belong to the same filter, and are marked with the same color.
Before the reorder, kernels with different patterns are distributed in this DNN layer. When performing the convolution operation directly, the execution code will contain many branches
as the {\tt +No-opt} code in Figure~\ref{fig:compiler-flow})
that incur significant instruction pipeline stalls and thread divergences, hurting both instruction- and thread-level parallelism. According to our experimental results in Section~\ref{sec:evaluation}, this version results in sub-optimal performance
FKR is composed of two steps: {\em filter reorder} and {\em kernel reorder}. The filter reorder organizes similar filters next to each other and the kernel reorder groups kernels with identical patterns in each filter together. Particularly, the {\em filter similarity} used in filter reorder is decided by two factors: first, the number of non-empty kernels in each filter (i.e., the length of each filter); and second, for filters with the same length, the number of kernels at identical positions with identical pattern IDs when the kernels in each filter are ordered according to these IDs.
After the reorder, the filters with the same length are grouped together, and in each group, the filters with the highest degree of similarity are ordered next to each other.
The code {\tt +Reorder} in figure~\ref{fig:compiler-flow}
is for the execution of this reordered layer.
This code shows much better instruction-level parallelism because it eliminates all branches. In addition, it also allows the better exploration of thread-level parallelism, because it results in large thread execution similarity and good load balance, particularly, considering the example of mapping the filters in the same group to the same GPU thread block.
\subsection{Compressed DNN Weight Storage (FKW Format)}\label{sec:format}
\begin{figure
\centering
\includegraphics[width=0.48 \textwidth]{fig/tight_storage_detail.pdf}
\caption{An example of FKW compressed weight storage.}
\label{fig:fkw-example}
\end{figure}
\begin{figure*}[th]
\includegraphics[width=0.45\textwidth]{fig/redundant-kernel.pdf}
\hfill
\includegraphics[width=0.45\textwidth]{fig/redundant-filter.pdf}
\caption{Load redundancy elimination (left: kernel-level; right: filter-level).}
\label{figure:load-redundancy-elimination}
\vspace{-0.5em}
\end{figure*}
After FKR, our LR stores the DNN's weights in a novel compact format (called FKW, standing for Filter-Kernel-Weight format). Compared with existing compact data formats (like CSR), FKW is higher-level and results in much less extra structure overhead (i.e.,
the total size of all index arrays that are used for weights data acces
).
In addition, FKW leverages the pattern information, and stores the kernels with the FKR information that will support later branch-less DNN execution. Other compact data format cannot support this.
Figure~\ref{fig:fkw-example} shows an example. This DNN layer consists of four filters, each with 2, 2, 2, and 3 (after FKR)
non-empty kernels, respectively. The two kernels in the first filter (marked as blue) have pattern 1 and 2, corresponding to the input channel 3 and 1, respectively. FKW uses five arrays to represent this DNN layer: offset array, reorder array, index array, stride array, and weight array. The offset array and reorder array store filter-level information, index array and stride array store kernel-level, and the weight array stores actual weights.
More specifically, the offset array stores the offset of each filter (in terms of the number of non-empty kernels). In Figure~\ref{fig:fkw-example}, the offset of filter 0 is 0, and the offset of filter 1 is 2 because there are two kernels in filter 0, and so on.
The reorder array shows the reorder information that is used for accumulating the computation output to the correct output channel. In Figure~\ref{fig:fkw-example}, the reorder array tells us that filter 2 and filter 3 have been switched and their computation results should also be switched to the corresponding output channel.
The index array represents the corresponding input channel for each non-empty kernel. In Figure~\ref{fig:fkw-example}, kernel 1 in filter 0
corresponds to the input channel 3, and kernel 2 corresponds to the input channel 1. So, the first two elements in the index array are 3 and 1, respectively.
The stride array denotes the number of kernels in each pattern within the same filter. In Figure~\ref{fig:fkw-example}, the filter 0 has the stride array values 0, 1, and 2, denoting that
the filter 0 has
1 kernel with pattern 1 ($1 = 1 - 0$), and 1 kernel with pattern 2 ($1 = 2 - 1$).
In this example, each kernel has four (non-zero) weights, so each filter has 8, 8, 8, and 12 weights (after FKR), respectively.
\subsection{Load Redundancy Elimination (LRE)}\label{sec:redundancy}
As discussed before, irregular memory access (in the form of array indirection) is also a major cause of inefficient execution of weight pruned DNNs.
PatDNN uses two techniques to address this issue:
(1) a conventional input tiling to improve the cache performance; and
(2) the optimized code generation
with the help of the pre-defined pattern information.
The first one, specifically the determination of the optimal tiling size will be introduced in Section~\ref{sec:auto-tuning}.
This section focuses on the second, specifically, introducing our novel redundant {\em register} load elimination optimization applied in code generation procedure.
Our key insight is: in DNN execution, such as a convolution operation, the data access pattern of the input and output is decided by the (none-zero elements) patterns of kernels that are already known after training. Therefore, it is possible to generate the optimized data access code with this information for each pattern of kernels and call them dynamically during the DNN execution. The generated codes consist of all statically determined data access instructions for the kernel-level computation with a careful instruction reorganization to 1) eliminate all indirect memory accesses; and 2) eliminate all redundant {\em register} load operations.
The elimination of all indirect memory accesses is relatively straightforward, because in all data access instructions, the index of input data can be directly calculated from kernel pattern. We next explain two novel register-level load redundancy elimination methods in details.
Figure~\ref{figure:load-redundancy-elimination} illustrates both register-level load redundancy eliminations: the left one is within each kernel, and the right one is among multiple kernels. Within each kernel, the load redundancy is caused by the convolution operation. In the example (shown on the left part of Figure~\ref{figure:load-redundancy-elimination}), the kernel value 1 requires the elements in the first two rows of the input matrix while value 2 requires the second and third rows. The elements in the second row $[7, 8, 9, 10]$ are loaded twice (from cache to register). PatDNN eliminates this load redundancy by explicitly reusing the (SIMD) registers that already hold the required data (like the second row in the above example).
Multiple kernels on the same position of different filters may share the same pattern and input channel.
The input data required by these kernels are exactly identical. The right-hand side of Figure~\ref{figure:load-redundancy-elimination} shows a concrete example. If the computation of these filters on identical data is packed together, the possible redundant load of this input can be eliminated. PatDNN explores this optimization when it generates the optimized memory access code. The FKR organizes the kernels (in different filters) with identical patterns together. Together with a filter-level (or output channel) loop unrolling when processing these kernels, the redundant register load is eliminated.
Figure~\ref{fig:compiler-flow} ({\tt +LRE}) shows an example of this unrolling code.
It is worth noting that the above two redundancy elimination opportunities are more straightforward to exploit for dense models where the memory accesses of kernel weights are continuous and the data reuse pattern is periodically repeated. However, it is very challenging (or even not possible) to exploit for pruned sparse models with irregular memory accesses, because it is hard to detect the data reuse pattern (or the data reuse pattern does not even exist). Our pattern-based pruning can preserve the data reuse patterns and help the compiler to detect them, thus re-enabling these two kinds of register-level load redundancy elimination.
\section{Related Work}
\item Escoin: Efficient Sparse Convolutional Neural Network
Inference on GPUs(Direct sparse convolution): https://arxiv.org/pdf/1802.10280.pdf
\item Sparse Convolutional Neural Networks: https://zpascal.net/cvpr2015/Liu_Sparse_Convolutional_Neural_2015_CVPR_paper.pdf
\item Guide sparse CNN: https://openreview.net/pdf?id=rJPcZ3txx
\item EFFICIENT SPARSE-WINOGRAD CONVOLUTIONAL NEURAL NETWORKS(Training sparse winograd-based model directly): https://arxiv.org/pdf/1802.06367.pdf
\section{Discussion}
\noindent{\bf Generality:} The techniques proposed in PatDNN are general enough to be applied to other platforms. Compared to laptops or servers, mobile platforms are more resource-constrained, making it is more challenging to achieve real-time execution. However, the need for real-time DNN execution
is crucial due to many important mobile applications.
In fact, in addition to the mobile platforms in our paper, we also tested PatDNN on the latest Raspberry Pi 4 platform. It shows a similar speedup over other frameworks like TVM. We believe that it is a promising research direction to improve PatDNN's portability by incorporating it with TVM that emphasizes the DNN execution on varied computing devices.
\noindent{\bf Dense vs. Sparse DNNs:}
General end-to-end DNN inference acceleration frameworks like TFLite, TVM, and MNN do not support sparse DNN execution. If we simply add sparse DNN support with random pruning and general compression storage (like CSR) to these frameworks, it is expected that their speed cannot be improved significantly as shown in
the results of PatDNN's CSR implementation. Although there is potential to improve the performance with coarse-grained structured pruning (that prunes whole filters/channels), the accuracy will be obviously degraded as we discussed before. From this perspective, PatDNN opens a new door to accelerate DNN execution with a compression/compiler-optimization co-design. With such co-design, sparse (or compressed) DNN execution becomes a more promising solution in resource-constraint environments than dense DNN.
\section{Conclusion}\label{sec:conclusion}
This paper presents PatDNN, an end-to-end framework to achieve real-time DNN execution on mobile devices. PatDNN consists of two stages, a pattern-based pruning stage based on extended ADMM solution framework, and an optimized execution code generation stage including a high-level, fine-grained DNN layerwise representation and a set of archi-tecture-aware optimizations.
This design allows PatDNN to benefit from both high accuracy and hardware efficiency. Our evaluation results demonstrate that PatDNN outperforms other state-of-the-art end-to-end DNN execution frameworks
with up to $44.5\times$ speedup and no accuracy compromise, and achieves real-time execution of large-scale DNNs on mobile devices.
\section{PatDNN Training w/ Pattern-based Pruning}\label{sec:training}
This section describes the methods to generate
compressed DNN models for PatDNN\xspace.
The procedure is composed of two steps:
(1) we design a set of desired patterns to be
selected for each kernel;
(2) assign a pattern for each kernel (kernel pattern pruning) or prune the whole kernel (connectivity pruning), and train the pattern-based weights for maintaining accuracy.
The overall flow is shown in Figure~\ref{fig:algorithm-overview}.
Essentially, it reflects the algorithm aspects of
PatDNN\xspace.
Our method can be applied to either
a pre-trained DNN or train a model
from scratch.
\begin{figure}[t
\centering
\includegraphics[width=0.45 \textwidth]{fig/algorithm_overview.pdf}
\caption{The algorithm-level overview of PatDNN training.}
\label{fig:algorithm-overview}
\end{figure}
\subsection{Designing the Pattern Set}
We need to determine the number of patterns, and design each specific candidate pattern in the pattern set. The number of patterns is an important hyperparameter that should be carefully considered.
If it is too large, it is more challenging to
generate efficient codes, thereby affecting
performance;
if it is too small, the lack of flexibility may lead to
accuracy degradation.
Through empirical study, we validate that 6-8 patterns in the set achieves as a desirable tradeoff for the most common $3\times 3$ kernel---ensuring low compiler overhead while maintaining high accuracy.
When the number of patterns is determined and 4-entry patterns are utilized, the compiler optimization
and hardware efficiency are oblivious to the
specific pattern shapes.
However, the specific patterns to use need to be
carefully optimized to maintain high accuracy
after kernel pattern pruning.
The key insights of pattern design are:
(1) both theory and empirical studies~\cite{yamins2014performance,yamins2016using} show that the central weight in a $3\times 3$ kernel is critical and shall not be pruned; and
(2) it is desirable that the distortion is small for each kernel before and after kernel pattern pruning. Hence, we propose the following heuristic.
First, for the pre-trained DNN, we scan all the kernels, and for each kernel, we find the four weights with largest magnitudes (including the central weight). These four weights form a 4-entry pattern, called the \emph{natural pattern} of the kernel. According to the definition of natural patterns, there are a total of $\binom{8}{3}=56$ number of possible patterns.
Suppose we aim at $k$ different patterns in the candidate set. We count and select the Top-$k$ most commonly appeared natural patterns across all kernels in the DNN, thereby forming the pattern candidate set (to select from in the subsequent step).
Our study on pattern number and pattern style selection is consistent with the pattern pruning theory work that is proposed in ~\cite{ma2019pconv}. Different from pattern theory derivation in~\cite{ma2019pconv}, our approach focuses on system-level design and compiler optimization of the pattern-based acceleration framework.
\subsection{Kernel Pattern and Connectivity Pruning Algorithm}
\textbf{Problem Formulation:} Consider an $N$-layer DNN, and we focus on the most computationally intensive CONV layers. The weights and biases of layer $k$ are respectively denoted by ${\bf{W}}_{k}$ and ${\bf{b}}_{k}$, and the loss function of DNN is denoted by $f \big( \{{\bf{W}}_{k}\}_{k=1}^N, \{{\bf{b}}_{k} \}_{k=1}^N \big)$, refer to \cite{zhang2018systematic}
for more details. In our
discussion, $\{{\bf{W}}_{k}\}_{k=1}^N$ and $\{{\bf{b}}_{k} \}_{k=1}^N$ respectively characterize the collection of weights and biases from layer $1$ to layer $N$. Then the pattern and connectivity pruning is formulated as an optimization problem:
\begin{equation}
\label{opt0}\small
\begin{aligned}
& \underset{ \{{\bf{W}}_{k}\},\{{\bf{b}}_{k} \}}{\text{minimize}}
& & f \big( \{{\bf{W}}_{k}\}_{k=1}^N, \{{\bf{b}}_{k} \}_{k=1}^N \big),
\\ & \text{subject to}
& & {\bf{W}}_{k}\in {\mathcal{S}}_{k},\ {\bf{W}}_{k}\in {\mathcal{S}}'_{k},\ \; k = 1, \ldots, N.
\end{aligned}
\end{equation}
The collection of weights in the $k$-th CONV layer forms a four-dimensional tensor, i.e., ${\bf{W}}_{k} \in R^{P_k \times Q_k \times C_k \times C_{k+1}}$, where $P_k, Q_k, C_k$, and $C_{k+1}$ are respectively the height of kernel, the width of kernel, the number of kernels, and the number of filters, in layer $k$. Suppose $\bf{X}$ denotes the weight tensor in a specific layer, then $({\bf{X}})_{:,:,a,b}$ denotes a specific kernel.
In \emph{kernel pattern pruning}, the constraint in the $k$-th CONV layer is ${\bf{W}}_{k}\in {\mathcal{S}}_{k}
:=
\{{\bf{X}}\mid$ {each kernel in} ${\bf{X}}$ {needs to satisfy one specific pattern shape in the pattern set (and non-zero weight values can be arbitrary)}$\}$.
In \emph{connectivity pruning}, the constraint in the $k$-th CONV layer is ${\bf{W}}_{k}\in {\mathcal{S}}'_{k}
:=
\{{\bf{X}}\mid$ {the number of nonzero kernels in} ${\bf{X}}$ {is less than or equal to} $\alpha_k \}$ ($\alpha_k$ is a predetermined hyperparameter with more discussions later).
Both constraints need to be simultaneously satisfied.
\textbf{Extended ADMM-based Solution Framework:} The constraint ${\bf{W}}_{k}\in {\mathcal{S}}_{k}$ in problem (\ref{opt0}) is different from the clustering-like constraints in ADMM-NN \cite{ren2019ADMMNN}, in that it is flexible to select a pattern for each kernel from the pattern set. As long as a pattern is assigned for each kernel, constraints in problem (\ref{opt0}) become clustering-like and ADMM compatible.
Similar to ADMM-NN \cite{ren2019ADMMNN}, the ADMM-based solution is an iterative process, starting from a pre-trained DNN model. We assign an appropriate pattern for each kernel based on the $L_2$-norm metric in each iteration, to achieve higher flexibility.
By incorporating auxiliary variables ${\bf{Z}}_{k}$'s and ${\bf{Y}}_{k}$'s, and dual variables ${\bf{U}}_{k}$'s and ${\bf{V}}_{k}$'s, we decompose (\ref{opt0}) into three subproblems, and iteratively solve until convergence.
In iteration $l$, after assigning patterns we solve the first subproblem\vspace{-0.1in}
\begin{align}\label{eqn:5}\small \nonumber
&\underset{ \{{\bf{W}}_{k}\},\{{\bf{b}}_{k} \}}{\text{minimize}}
\ f \big( \{{\bf{W}}_{k} \}_{k=1}^N, \{{\bf{b}}_{k} \}_{k=1}^N \big)+\sum_{k=1}^{N} \frac{\rho_{k}}{2} \| {\bf{W}}_{k}-{\bf{Z}}_{k}^{l}+{\bf{U}}_{k}^{l} \|_{F}^{2}\vspace{-0.1in}\\
&+ \sum_{k=1}^{N} \frac{\rho_{k}}{2} \| {\bf{W}}_{k}-{\bf{Y}}_{k}^{l}+{\bf{V}}_{k}^{l} \|_{F}^{2}.
\vspace{-0.5in}\end{align}
The first term is the loss function of the DNN, while the other quadratic terms are convex.
As a result, this subproblem can be solved by stochastic gradient descent (e.g., the ADAM algorithm \cite{kingma2014adam}) similar to training the original DNN.
The solution $\{{\bf{W}}_{k}\}$ of subproblem 1 is denoted by $\{{\bf{W}}^{l+1}_{k}\}$. Then we aim to derive $\{{\bf{Z}}^{l+1}_{k}\}$ and $\{{\bf{Y}}^{l+1}_{k}\}$ in subproblems 2 and 3. These subproblems have the same form as those in ADMM-NN \cite{ren2019ADMMNN}. Thanks to the characteristics in combinatorial constraints, the optimal, analytical solution of the two subproblems are Euclidean projections, and are polynomial time solvable. For example, for connectivity pruning, the projection is: keeping $\alpha_k$ kernels with largest $L_2$ norms and setting the rest of kernels to zero. For kernel pattern pruning it is similar.
Finally, we update dual variables ${\bf{U}}_{k}$ and ${\bf{V}}_{k}$ according to the ADMM rule \cite{boyd2011distributed} and thereby complete the $l$-th iteration in the ADMM-based solution.
The hyperparameter determination process is relatively straightforward for joint pattern and connectivity pruning. There is no additional hyperparameters for kernel pattern pruning when the pattern set has been developed. For connectivity pruning we need to determine the pruning rate $\alpha_k$ for each layer. In this paper, we adopt a heuristic method of uniform pruning rate for all layers except for the first layer (which is smaller, yet more sensitive to pruning).
\subsection{Accuracy Validation and Analysis}\label{sec:pattern_acc_result}
We validate the accuracy of ADMM-based joint kernel pattern and connectivity pruning, based on ImageNet ILSVRC-2012 and CIFAR-10 datasets, using VGG-16 \cite{simonyan2014very}, ResNet-50 \cite{he2016deep}, and MobileNet-V2 \cite{sandler2018mobilenetv2} DNN models.
Our implementations are based on PyTorch, and the baseline accuracy results are in many cases higher than prior work, which reflects the recent
progress in DNN training. With a pre-trained DNN model, we limit the number of epochs in kernel pattern and connectivity pruning to 120, similar to the original DNN training in PyTorch and much lower than iterative pruning \cite{han2015learning}.
\begin{table}[h]
\vspace{2mm}
\caption{Top-5 accuracy comparison on kernel pattern pruning.}
\label{tab:pattern_acc_result}
\centering
\includegraphics[width = 1\linewidth]{fig/table_acc_pattern.pdf}
\end{table}
Table~\ref{tab:pattern_acc_result} illustrates the Top-5 accuracy comparison on kernel pattern pruning only, applied on the CONV layers of VGG-16 and ResNet-50 using ImageNet dataset. The baseline is the original DNN without patterns, and we demonstrate the accuracy results with 6, 8, and 12 patterns (all 4-entry patterns) in the pattern set.
Our first observation is that \emph{the accuracy will improve when the number of candidate patterns is sufficient} --- typically 4 - 8 patterns are sufficient. This is attributed to the compatibility of kernel pattern pruning with human visual system and the ability to eliminate overfitting (compared with square kernel shape). This observation has been also
validated for other types of DNNs and data sets (e.g., CIFAR-10).
\begin{table}[h]
\vspace{2mm}
\caption{Top-5 accuracy and CONV weight reduction on joint kernel pattern pruning (8 patterns in the set) and connectivity pruning.}
\label{tab:8pattern_compare}
\centering
\includegraphics[width = 1\linewidth]{fig/table_8pattern_compare.pdf}
\vspace{1mm}
\end{table}
Table~\ref{tab:8pattern_compare} illustrates the Top-5 accuracy comparison on joint kernel pattern pruning (8 patterns in the set) and connectivity pruning, on VGG-16 and ResNet-50 using ImageNet dataset. For VGG-16, all kernels are $3\times 3$. After applying 4-entry patterns on all kernels and 3.6$\times$ uniform connectivity pruning, we achieve around 8$\times$ weight reduction on CONV layers of VGG-16. For ResNet-50, a portion of kernels are $1\times 1$ besides the majority of $3\times 3$ kernels. We apply kernel pattern pruning on all $3\times 3$ ones, and apply uniform 3.6$\times$ connectivity pruning on all kernels. We achieve 4.4$\times$ weight reduction on CONV layers. One can observe from the table that (1) \emph{no Top-5 accuracy drop with this setup}; (2) \emph{under the same accuracy, the weight reduction rate is close to ADMM-based (and outperforms prior heuristic based) non-structured pruning on CONV layers.}
For the CIFAR-10 dataset, we observe consistent accuracy improvements with 8 patterns on 3$\times$3 kernels and 3.6$\times$ connectivity pruning, with results shown in Section \ref{sec:evaluation}.
| -34,212.699525 |
[
-2.15234375,
2.38671875
] | 50.670241 |
[
-2.6796875,
1.216796875,
-1.5537109375,
-5.0234375,
-1.3759765625,
7.2890625
] |
[
2.099609375,
6.2421875,
1.3603515625,
5.26171875
] | 588 | 9,289 |
[
-1.8271484375,
1.685546875
] | 23.543813 |
[
-6.1875,
-4.35546875,
-4.01953125,
-1.611328125,
2.56640625,
11.75
] | 0.336201 | 30.288586 | 19.948326 | 1.37779 |
[
2.4079651832580566
] | -25,221.245088 | 6.244052 | -33,257.409388 | 0.379304 | 6.222584 |
[
-2.83984375,
-3.55078125,
-3.65625,
-4.33984375,
2.6328125,
11.34375
] |
[
-6.25390625,
-3.046875,
-2.421875,
-2.06640625,
4.1640625,
6.89453125
] | |
BkiUc2zxK6wB9k0iLNKt
|
\section{Introduction}
The \textbf{uniform spanning forests} (USFs) of an infinite graph $G$ are defined
as weak limits of the uniform spanning trees of finite subgraphs of $G$. These limits can be taken with respect to two extremal boundary conditions, yielding the \textbf{free uniform spanning forest} (FUSF) and \textbf{wired uniform spanning forest} (WUSF). For transitive amenable graphs such as the hypercubic lattice $\mathbb Z^d$, the free and wired forests coincide and we speak simply of the USF.
In this paper we shall be concerned exclusively with the wired forest.
Uniform spanning forests have played a central role in the development of probability theory over the last twenty years, and are closely related to several other topics in probability and statistical mechanics including electrical networks \cite{kirchhoff1847ueber,BurPe93,BLPS}, loop-erased random walk \cite{Lawler80,Wilson96,BLPS}, the random cluster model \cite{GrimFKbook,Hagg95}, domino tiling \cite{BurPe93,Ken00}, random interlacements \cite{Szni10,hutchcroft2015interlacements}, conformally invariant scaling limits \cite{Schramm00,LaSchWe04,MR3719057}, and the Abelian sandpile model \cite{Dhar90,MajDhar92,JarRed08,JarWer14}. Indeed, our results have important implications for the Abelian sandpile model, which we discuss in \cref{subsec:sandpileintro}.
Following the work of many authors, the basic qualitative features of the WUSF are firmly understood on a wide variety of graphs. In particular, it is known that every tree in the WUSF is recurrent almost surely on any graph \cite{Morris03}, that the WUSF is connected a.s.\ if and only if two independent random walks on $G$ intersect almost surely \cite{Pem91,BLPS}, and that every tree in the WUSF is one-ended almost surely whenever $G$ is in one of several large classes of graphs \cite{Pem91,BLPS,LMS08,H15,AL07,HutNach15b} including all transient transitive graphs. (An infinite tree is one-ended if it does not contain a simple bi-infinite path.)
The goal of this paper is to understand the geometry of trees in the WUSF at a more detailed, quantitative level, under the assumption that the underlying graph is high-dimensional in a certain sense.
Our results can be summarized informally as follows.
Let $G$ be a network with vertex conductances bounded above and below by two positive constants, and suppose that the $n$-step return probabilities $p_n(v,v)$ for random walk on $G$ satisfy $\sum_{n\geq 1} n \sup_{v\in V} p_n(v,v)<\infty$. In particular, this holds for $\mathbb Z^d$ if and only if $d \geq 5$, and more generally for any transitive graph of at least quintic volume growth.
Let $\mathfrak F$ be the WUSF of $G$, and let $v$ be a vertex of $G$. The \textbf{past} of $v$ in $\mathfrak F$ is the union of the finite connected components of $\mathfrak F \setminus \{v\}$. The following hold:
\begin{enumerate}[leftmargin=*]
\itemsep0.5em
\item
\emph{The
intrinsic geometry} of each tree in $\mathfrak F$ is similar at large scales to that of a critical Galton-Watson tree with finite variance offspring distribution, conditioned to survive forever. In particular, every tree has volume growth dimension $2$, spectral dimension $4/3$, and walk dimension $3$ almost surely. The latter two statements mean that the $n$-step return probabilities for simple random walk on the tree decay like $n^{-2/3+o(1)}$, and that the typical displacement of the walk (as measured by the intrinsic graph distance in the tree) is $n^{1/3+o(1)}$.
These are known as the \emph{Alexander-Orbach} values of these dimensions~\cite{AlexOrb1982,KozNach09}.
\item \emph{The intrinsic geometry of the past} of $v$ in $\mathfrak F$ is similar in law to that of an \emph{unconditioned} critical Galton-Watson tree with finite variance offspring distribution. In particular, the probability that the past contains a path of length at least $n$ is of order $n^{-1}$, and the probability that the past contains more than $n$ points is of order $n^{-1/2}$. That is, the \emph{intrinsic diameter exponent} and \emph{volume exponent} are $1$ and $1/2$ respectively.
\item[3.] \emph{The extrinsic geometry of the past} of $v$ in $\mathfrak F$ is similar in law to that of an \emph{unconditioned} critical branching random walk on $G$ with finite variance offspring distribution. In particular, the probability that the past of $v$ includes a vertex at extrinsic distance at least $n$ from $v$ depends on the rate of escape of the random walk on $G$. For example, it is of order $n^{-2}$ for $G=\mathbb Z^d$ for $d\geq 5$ and is of order $n^{-1}$ for $G$ a transitive nonamenable graph. This is related to the fact that the random walk on the ambient network $G$ is diffusive in the former case and ballistic in the latter case.
\end{enumerate}
In light of the connections between the WUSF and the Abelian sandpile model, these results imply related results for that model, to the effect that an avalanche in the Abelian sandpile model has a similar distribution to a critical branching random walk (see \cref{subsec:sandpileintro}).
Precise statements of our results and further background are given in the remainder of the introduction.
The fact that our results apply at such a high level of generality is a strong vindication of \emph{universality} for high-dimensional spanning trees and sandpiles, which predicts that the large-scale behaviour of these models should depend only on the dimension, and in particular should be insensitive to the microscopic structure of the lattice. In particular, our results apply not only to $\mathbb Z^d$ for $d\geq 5$, but also to non-transitive networks that are similar to $\mathbb Z^d$ such as the half-space $\mathbb Z^{d-1}\times \mathbb N$ or, say, $\mathbb Z^d$ with variable edge conductances bounded between two positive constants. Many of our results also apply to long-range spanning forest models on $\mathbb Z^d$ such as those associated to the fractional Laplacian $-(-\Delta)^\beta$ of $\mathbb Z^d$ for $d\geq 1$, $\beta\in(0,d/4 \wedge 1)$. Long-range models such as these are motivated physically as a route towards understanding low-dimensional models via the $\varepsilon$-expansion \cite{wilson1972critical}, for which it is desirable to think of the dimension as a continuous parameter. (See the introduction of \cite{Slade2017} for an account of the $\varepsilon$-expansion for mathematicians.)
\medskip
\noindent \textbf{About the proofs.}
Our proof relies on the interplay between two different ways of sampling the WUSF. The first of these is \emph{Wilson's algorithm}, a method of sampling the WUSF by joining together loop-erased random walks which was introduced by David Wilson \cite{Wilson96} and extended to infinite transient graphs by Benjamini, Lyons, Peres, and Schramm \cite{BLPS}. The second is the \emph{interlacement Aldous-Broder algorithm}, a method of sampling the WUSF as the set of first-entry edges of Sznitman's \emph{random interlacement process} \cite{Szni10}. This algorithm was introduced in the author's recent work \cite{hutchcroft2015interlacements} and extends the classical Aldous-Broder algorithm \cite{Aldous90,broder1989generating} to infinite transient graphs. Generally speaking, it seems that Wilson's algorithm is the better tool for estimating the moments of random variables associated to the WUSF, while the interlacement Aldous-Broder algorithm is the better tool for estimating tail probabilities.
A key feature of the interlacement Aldous-Broder algorithm is that it enables us to think of the WUSF as the stationary measure of a natural continuous-time Markov chain. Moreover, the past of the origin evolves in an easily-understood way under these Markovian dynamics. In particular, as we run time backwards, the past of the origin gets monotonically smaller except possibly for those times at which the origin is visited by an interlacement trajectory. Indeed, the central insight in the proof of our results is that \emph{static} tail events (on which the past of the origin is large) can be related to to \emph{dynamic} tail events (on which the origin is hit by an interlacement trajectory at a small time). Roughly speaking, we show that these two types of tail event tend to occur together, and consequently have comparable probabilities. We make this intuition precise using inductive inequalities similar to those used to analyze one-arm probabilities in high-dimensional percolation \cite{KozNach09,MR2748397,MR3418547}.
Once the critical exponent results are in place, the results concerning the simple random walk on the trees can be proven rather straightforwardly using the results and techniques of Barlow, J\'arai, Kumagai, and Slade \cite{BJKS08}.
\subsection{Relation to other work}
\begin{itemize}
\item When $G$ is a regular tree of degree $k\geq 3$, the components of the WUSF are distributed exactly as augmented critical binomial Galton-Watson trees conditioned to survive forever, and in this case all of our results are classical \cite{kestensubdiff,MR1349164,BarKum06}.
\item In the case of $\mathbb Z^d$ for $d\geq 5$, Barlow and J\'arai \cite{barlow2016geometry} established that the trees in the WUSF have quadratic volume growth almost surely. Our proof of quadratic volume growth uses similar methods to theirs, which were in turn inspired by related methods in percolation due to Aizenman and Newman \cite{MR762034}.
\item Also in the case of $\mathbb Z^d$ for $d\geq 5$, Bhupatiraju, Hanson, and J\'arai \cite{bhupatiraju2016inequalities} followed the strategy of an unpublished proof of Lyons, Morris, and Schramm \cite{LMS08} to prove that the probability that the past reaches extrinsic distance $n$ is $n^{-2} \log^{O(1)}n$ and that the probability that the past has volume $n$ is $n^{-1/2} \log^{O(1)} n$. Our results improve upon theirs in this case by reducing the error from polylogarithmic to constant order. Their proof relies heavily on transitivity and cannot be used to derive universal results of the kind we prove here.
\item Peres and Revelle \cite{peres2004scaling} proved that the USTs of large $d$-dimensional tori converge under rescaling (with respect to the Gromov-weak topology) to Aldous's continuum random tree when $d\geq 5$. They also proved that their result extends to other sequences of finite transitive graphs satisfying a heat-kernel upper bound similar to the one we assume here. Later, Schweinsberg \cite{schweinsberg2009loop} established a similar result for four-dimensional tori. While these results are closely related in spirit to those that we prove here, it does not seem that either can be deduced from the other.
\item For planar Euclidean lattices such as $\mathbb Z^2$, the UST is very well understood thanks in part to its connections to conformally invariant processes in the continuum \cite{Ken00,Schramm00,LaSchWe04,Masson09,MR3719057}. In particular, Barlow and Masson \cite{BarMass10,BarMass11} proved that the UST of $\mathbb Z^2$ has volume growth dimension $8/5$ and spectral dimension $16/13$ almost surely (see also \cite{BCK17}).
\item
In \cite{HutNach15b}, the author and Nachmias established that the WUSF of any transient proper plane graph with bounded degrees and codegrees has mean-field critical exponents provided that measurements are made using the hyperbolic geometry of the graph's circle packing rather than its usual combinatorial geometry. Our results recover those of \cite{HutNach15b} in the case that the graph in question is also uniformly transient, in which case it is nonamenable and the graph distances and hyperbolic distances are comparable.
\item A consequence of this paper is that several properties of the WUSF are insensitive to the geometry of the graph once the dimension is sufficiently large. In contrast, the theory developed in \cite{BeKePeSc04,hutchcroft2017component} shows that some other properties describing the adjacency structure of the trees in the forest continue to undergo qualitative changes every time the dimension increases.
\item In forthcoming work with Sousi, we build upon the methods of this paper to analyze related problems concerning the uniform spanning tree in $\mathbb Z^3$ and $\mathbb Z^4$.
\end{itemize}
\subsection{Basic definitions}
In this paper, a \textbf{network} will be a connected graph $G=(V,E)$ (possibly containing loops and multiple edges) together with a function $c:E \to (0,\infty)$ assigning a positive \textbf{conductance} $c(e)$ to each edge $e \in E$ such that for each vertex $v\in V$, the sum $c(v):= \sum c(e)< \infty$, taken over edges incident to $v$, is finite.
We say that the network $G$ has \textbf{controlled stationary measure} if there exists a positive constant $C$ such that $C^{-1} \leq c(v) \leq C$ for every vertex $v$ of $G$. Locally finite graphs can always be considered as networks by setting $c(e) \equiv 1$, and in this case have controlled stationary measure if and only if they have bounded degrees. We write $E^\rightarrow$ for the set of oriented edges of a network. An oriented edge $e$ is oriented from its tail $e^-$ to its head $e^+$.
The \textbf{random walk} on a network $G$ is the process that, at each time step, chooses an edge emanating from its current position with probability proportional to its conductance, independently of everything it has done so far.
We use $\mathbf{P}_v$ to denote the law of the random walk started at a vertex $v$, $\mathbf{E}_v$ to denote the associated expectation operator, and $P$ to denote the Markov operator $P:\mathbb R^V \to \mathbb R^V$, defined by
\[ (Pf)(v) = \mathbf{E}_v f(X_1) = \sum_{e^-=v} \frac{c(e)}{c(v)}f(e^+).\]
Finally, for each two vertices $u$ and $v$ of $G$ and $n\geq 0$, we write $p_n(u,v) = \mathbf{P}_u(X_n = v)$ for the probability that a random walk started at $u$ is at $v$ at time $n$.
Given a graph or network $G$, a \textbf{spanning tree} of $G$ is a connected subgraph of $G$ that contains every vertex of $G$ and does not contain any cycles.
The \textbf{uniform spanning tree} measure $\mathsf{UST}_G$ on a finite connected graph $G=(V,E)$ is the probability measure on subgraphs of $G$ (considered as elements of $E^{\{0,1\}}$) that assigns equal mass to each spanning tree of $G$. If $G$ is a finite connected \emph{network}, then the
uniform spanning tree measure $\mathsf{UST}_G$
of $G$
is defined so that the mass of each tree is proportional to the product of the conductances of the edges it contains. That is,
\[\mathsf{UST}_G\left(\{\omega\}\right) = Z^{-1}_G \prod_{e\in \omega} c(e) \mathbbm{1}\left(\omega \text{ is a spanning tree of $G$}\right)\]
for every $\omega \in \{0,1\}^E$,
where $Z_G$ is a normalizing constant.
Let $G$ be an infinite network and let $\langle V_n \rangle_{n \geq1}$ be an \textbf{exhaustion} of $G$, that is, an increasing sequence of finite sets $V_n \subset V$ such that $\bigcup_{n\geq1} V_n = V$.
For each $n \geq 1$, let $G^*_n$ be the network obtained from $G_n$ by contracting every vertex in $V \setminus V_n$ into a single vertex, denoted $\partial_n$, and deleting all the resulting self-loops from $\partial_n$. The \textbf{wired uniform spanning forest} measure on $G$, denoted $\mathsf{WUSF}_G$, is defined to be the weak limit of the uniform spanning tree measures on the finite networks $G_n^*$. That is,
\[\mathsf{WUSF}_G\left(\{\omega : S \subset \omega \}\right) = \lim_{n\to\infty}\mathsf{UST}_{G^*_n}\left(\{\omega : S \subset \omega\}\right)\]
for every finite set $S \subseteq E$. It follows from the work of Pemantle \cite{Pem91} that this limit exists and does not depend on the choice of exhaustion; See also \cite[Chapter 10]{LP:book}.
In the limiting construction above, one can also orient the uniform spanning tree of $G_n^*$ towards the boundary vertex $\partial_n$, so that every vertex other than $\partial_n$ has exactly one oriented edge emanating from it in the spanning tree. If $G$ is transient, then the sequence of laws of these random oriented spanning trees converge weakly to the law of a random oriented spanning forest of $G$, which is known as the \textbf{oriented wired uniform spanning forest}, and from which we can recover the usual (unoriented) WUSF by forgetting the orientation. (This assertion follows from the proof of \cite[Theorem 5.1]{BLPS}.) It is easily seen that the oriented wired uniform spanning forest of $G$ is almost surely an \textbf{oriented essential spanning forest} of $G$, that is, an oriented spanning forest of $G$ such that every vertex of $G$ has exactly one oriented edge emanating from it in the forest (from which it follows that every tree is infinite).
\subsection{Intrinsic exponents}
Let $\mathfrak F$ be an oriented essential spanning forest of an infinite graph $G$.
We define the \textbf{past} of a vertex $v$ in $\mathfrak F$, denoted $\mathfrak P(v)$,
to be the subgraph of $\mathfrak F$ spanned by the set of vertices $u$ of $\mathfrak F$ such that every edge in the geodesic from $u$ to $v$ in $\mathfrak F$ is oriented in the direction of $v$. Thus, a component of $\mathfrak F$ is one-ended if and only if the past of each of its vertices is finite.
In order to quantify the one-endedness of the WUSF, it is interesting to estimate the probability that the past of a vertex is large in various senses. Perhaps the three most natural such measures of largeness are given by the \emph{intrinsic diameter}, \emph{extrinsic diameter}, and \emph{volume} of the past.
Here, given a subgraph $K$ of a graph $G$, we define the \textbf{extrinsic diameter} of $K$, denoted $\mathrm{diam}_\mathrm{ext}(K)$, to be the supremal graph distance in $G$ between two points in $K$, and define the \textbf{intrinsic diameter} of $K$, denoted $\mathrm{diam}_\mathrm{int}(K)$, to be the diameter of $K$. The \textbf{volume} of $K$, denoted $|K|$, is defined to be the number of vertices in $K$.
Generally speaking, for critical statistical mechanics models defined on Euclidean lattices such as $\mathbb Z^d$, many natural random variables arising geometrically from the model are expected to have power law tails. The exponents governing these tails are referred to as \textbf{critical exponents}.
For example, if $\mathfrak F$ is the USF of $\mathbb Z^d$, we expect that for each $d\geq 2$ there exists $\alpha_d$ such that
\[ \P\left(\mathrm{diam}_\mathrm{ext}\left(\mathfrak P\left(0\right)\right) \geq R \right) = R^{-\alpha_d +o(1)},\]
in which case we call $\alpha_d$ the \emph{extrinsic diameter exponent} for the USF of $\mathbb Z^d$.
Calculating and proving the existence of critical exponents is considered a central problem in probability theory and mathematical statistical mechanics.
It is also expected that each model has an \textbf{upper-critical dimension}, denoted $d_c$, above which the critical exponents of the model stabilize at their so-called \emph{mean-field} values. For the uniform spanning forest, the upper critical dimension is believed to be four. Intuitively, above the upper critical dimension the lattice is spacious enough that different parts of the model do not interact with each other very much. This causes the model to behave similarly to how it behaves on, say, the $3$-regular tree or the complete graph, both of which have a rather trivial geometric structure.
Below the upper critical dimension, the geometry of the lattice affects the model in a non-trivial way, and the critical exponents are expected to differ from their mean-field values.
The upper critical dimension itself is often characterised by the mean-field exponents holding up to a polylogarithmic multiplicative correction, which is not expected to be present at other dimensions.
For example, we expect that there exist constants $\gamma_2,\gamma_3,\gamma_{\mathrm{mf}}$ and $\delta$ such that
\begin{align*}
\P\left(|\mathfrak P(0)| \geq R\right) \asymp
\begin{cases}
R^{-\gamma_2} & d=2\\
R^{-\gamma_3} & d=3\\
R^{-\gamma_{\mathrm{mf}}} \log^{-\delta} R & d=4\\
R^{-\gamma_{\mathrm{mf}}} & d\geq 5,
\end{cases}
\end{align*}
where $\mathfrak F$ is the uniform spanning forest of $\mathbb Z^d$ and $\asymp$ denotes an equality that holds up to positive multiplicative constants. Moreover, the values of the exponents $\gamma_2,\gamma_3,\gamma_{\mathrm{mf}},$ and $\delta$ should depend only on the dimension $d$, and not on the choice of lattice; this predicted insensitivity to the microscopic structure of the lattice is an instance of the phenomenon of \emph{universality}.
Our first main result verifies the high-dimensional part of this picture for the intrinsic exponents. The corresponding results for the extrinsic diameter exponent are given in \cref{subsec:introextrinsic}.
Before stating our results,
let us give some further definitions and motivation.
For many models, an important signifier of mean-field behaviour is that certain \emph{diagrammatic sums} associated to the model are convergent. Examples include the \emph{bubble diagram} for self-avoiding walk and the Ising model, the \emph{triangle diagram} for percolation, and the \emph{square diagram} for lattices trees and animals; see e.g.\ \cite{MR2239599} for an overview.
For the WUSF, the relevant signifier of mean-field behaviour on a network $G$ is the convergence of the \emph{random walk bubble diagram}
\[\sum_{x} \bigg(\sum_{n\geq 0} p_n(v,x)\bigg)^2,\]
where $p_n(\cdot,\cdot)$ denotes the $n$-step transition probabilities for simple random walk on $G$ and $v$ is a fixed root vertex.
Note that the value of the bubble diagram is exactly the expected number of times that two independent simple random walks started at the origin intersect.
Using time-reversal, the convergence of the bubble diagram of a network with controlled stationary measure is equivalent to the convergence of the sum
\begin{equation*}
\sum_{n \geq0} (n+1) p_n(v,v). \end{equation*}
It is characteristic of the upper-critical dimension $d_c$ that the bubble diagram converges for all $d>d_c$, while at the upper-critical dimension itself we expect that the bubble diagram diverges logarithmically.
Our condition for mean-field behaviour of the
WUSF will be that the bubble diagram converges uniformly in a certain sense.
Let $G=(V,E)$ be a network, let $P$ be the Markov operator of $G$, and
let $\|P^n\|_{1\to\infty}$ be the $1\to\infty$ norm of $P^n$, defined by
\[\|P^n\|_{1\to\infty} = \sup_{u,v\in V}p_n(u,v) .\]
We define the \textbf{bubble norm}
of $P$ to be
\[\bubnorm{P} := \sum_{n=0}^\infty (n+1) \pstar{n}. \]
Thus, for transitive networks, $\bubnorm{P}<\infty$ is equivalent to convergence of the random walk bubble diagram. Here, a network is said to be \textbf{transitive} if for every two vertices $u,v\in V$ there exists a conductance-preserving graph automorphism mapping $u$ to $v$. Throughout the paper, we use $\asymp, \preceq$ and $\succeq$ to denote equalities and inequalities that hold to within multiplication by two positive constants depending only on the choice of network.
\vspace{0.1em}
\begin{theorem}[Mean-field intrinsic exponents]
\label{thm:transitivemain}
Let $G$ be a transitive network such that $\bubnorm{P} < \infty$,
and let $\mathfrak F$ be the wired uniform spanning forest of $G$.
Then
\vspace{0.25em}
\[
\vspace{0.25em}
\P\left(\mathrm{diam}_{\mathrm{int}}(\mathfrak P(v)) \geq R \right) \asymp R^{-1}
\quad \text{ and } \quad
\P\left(|\mathfrak P(v)| \geq R \right) \asymp R^{-1/2}
\]
for every vertex $v$ and every $R\geq 1$. In particular, the critical exponents governing the intrinsic diameter and volume of the past are $1$ and $1/2$ respectively.
\end{theorem}
For transitive \emph{graphs}, it follows from work of Hebisch and Saloff-Coste \cite{HebSaCo93} that $\bubnorm{P}<\infty$ if and only if the graph has at least quintic volume growth, i.e., if and only if there exists a constant $c$ such that the number of points in every graph distance ball of radius $n$ is at least $cn^5$ for every $n\geq 1$.
Thus, by Gromov's theorem \cite{Gromov1981}, Trofimov's theorem \cite{Trofimov1984}, and the Bass-Guivarc'h formula \cite{Bass1972,Guiv1973}, the class of graphs treated by \cref{thm:transitivemain} includes all transitive graphs not rough-isometric to $\mathbb Z,\mathbb Z^2,\mathbb Z^3,\mathbb Z^4$, or the discrete Heisenberg group. As mentioned above, the theorem also applies for example to long-ranged transitive networks with vertex set $\mathbb Z^d$, a single edge between every two vertices, and with translation-invariant conductances given up to positive multiplicative constants by
\[ c\left(\{x,y\}\right) =c\left(x-y\right) \asymp \| x-y\|^{-d-\alpha}
\vspace{0.5em}
\]
provided that either $1 \leq d \leq 4$ and $\alpha \in (0,d/2)$ or $d\geq 5$ and $\alpha>0$. The canonical example of such a network is that associated to the fractional Laplacian $-(-\Delta)^\beta$ of $\mathbb Z^d$ for $\beta\in(0,d/4 \wedge 1)$. (See \cite[Section 2]{Slade2017}.)
\medskip
The general form of our result is similar, but has an additional technical complication owing to the need to avoid trivialities that may arise from the local geometry of the network.
Let $G$ be a network, let $v$ be a vertex of $G$, let $X$ and $Y$ be independent random walks started at $v$, and let $q(v)$ be the probability that $X$ and $Y$ never return to $v$ or intersect each other after time zero.
\begin{theorem}
\label{thm:generalexponents}
Let $G$ be a network with controlled stationary measure such that $\bubnorm{P}<\infty$, and let $\mathfrak F$ be the wired uniform spanning forest of $G$. Then
\vspace{0.2em}
\[
\vspace{0.2em}
q(v) R^{-1}
\preceq
\P\Bigl(\mathrm{diam}_\mathrm{int}\bigl(\mathfrak P(v)\bigr) \geq R \Bigr)
\preceq
R^{-1}
\]
and
\vspace{0.2em}
\[
\vspace{0.2em}
q(v)^2 R^{-1/2}
\preceq
\P\Bigl(\bigl|\mathfrak P(v)\bigr| \geq R \Bigr)
\preceq
R^{-1/2}
\]
for all $v\in V$ and $R\geq1$.
\end{theorem}
The presence of $q(v)$ in the theorem is required, for example, in the case that we attach a vertex by a single edge to the origin of $\mathbb Z^d$, so that the past of this vertex in the USF is necessarily trivial. (The precise nature of the dependence on $q(v)$ has not been optimized.)
However, in any network $G$ with controlled stationary measure and with $\bubnorm{P}<\infty$, there must exist positive constants $\varepsilon$ and $r$ such that for every vertex $v$ in $G$, there exists a vertex $u$ within distance $r$ of $G$ such that $q(u)>\varepsilon$. In particular, if $G$ is a transitive network with $\bubnorm{P}<\infty$ then $q(v)$ is a positive constant, so that \cref{thm:transitivemain} follows from \cref{thm:generalexponents}.
Let us note that \cref{thm:generalexponents} applies in particular to any bounded degree nonamenable graph, or more generally to any network with controlled stationary measures satisfying a $d$-dimensional isoperimetric inequality for some $d>4$, see \cite[Theorem 3.2.7]{KumagaiBook}. In particular, it applies to $\mathbb Z^d$, $d\geq 5$, with any specification of edge conductances bounded above and below by two positive constants (in which case it can also be shown that $q(v)$ is bounded below by a positive constant). A further example to which our results are applicable is given by taking $G=H^d$ where $d\geq 5$ and $H$ is \emph{any} infinite, bounded degree graph.
\subsection{Volume growth, spectral dimension, and anomalous diffusion}
The theorems concerning intrinsic exponents stated in the previous subsection also allow us to determine exponents describing the almost sure asymptotic geometry of the trees in the WUSF, and in particular allow us to compute the almost sure spectral dimension and walk dimension of the trees in the forest.
See e.g.\ \cite{KumagaiBook} for background on these and related concepts.
Here, we always consider the trees of the WUSF as \emph{graphs}.
One could instead consider the trees as networks with conductances inherited from $G$, and the same results would apply with minor modifications to the proofs.
Let $G$ be an infinite, connected network and let $v$ be a vertex of $G$.
We define the \textbf{volume growth dimension} (a.k.a.\ \textbf{fractal dimension}) of $G$ to be
\begin{align*}
d_f(G) &:= \lim_{n\to\infty} \frac{\log |B(v,n)|}{\log n} &\qquad \text{ when this limit exists,}
\intertext{define the \textbf{spectral dimension} of $G$ to be}
d_s(G) &:= \lim_{n\to\infty} \frac{-2\log p_{2n}(v,v)}{\log n} &\qquad \text{ when this limit exists,}
\intertext{and define the \textbf{walk dimension}
of $G$ to be}
d_w(G) &:= \lim_{n\to\infty} \frac{\log n}{\log \mathbf E_v \max_{1\leq m \leq n} d(v,X_n)} &\qquad \text{ when this limit exists.}
\end{align*}
In each case, the limit used to define the dimension
does not depend on the choice of root vertex $v$. Our next theorem establishes the values of $d_f,d_s,$ and $d_w$ for the trees in the WUSF under the assumption that $\bubnorm{P}<\infty$. The results concerning $d_s$ and $d_w$ are new even in the case of $\mathbb Z^d$, $d\geq 5$, while the result concerning the volume growth was established for $\mathbb Z^d$, $d\geq 5$, by Barlow and J\'arai \cite{barlow2016geometry}.
\begin{theorem}
\label{thm:AlexanderOrbach}
Let $G$ be a network with controlled stationary measure and with $\bubnorm{P}<\infty$,
and let $\mathfrak F$ be the wired uniform spanning forest of $G$. Then almost surely, for every component $T$ of $\mathfrak F$, the volume growth dimension, spectral dimension, and walk dimension of $T$ satisfy
\[
d_f(T)=2, \quad d_s(T)=\frac{4}{3}, \quad \text{and} \quad d_w(T)=3.
\]
In particular, the limits defining these quantities are well-defined almost surely.
\end{theorem}
The values $d_f=2,d_s=4/3,$ and $d_w=3$ are known as the \emph{Alexander-Orbach} values of these exponents, following the conjecture due to Alexander and Orbach \cite{AlexOrb1982} that they held for high-dimensional incipient infinite percolation clusters. The first rigorous proof of Alexander-Orbach behaviour was due to Kesten \cite{kestensubdiff}, who established it for critical Galton-Watson trees conditioned to survive. The first proof for a model in Euclidean space was due to Barlow, J\'arai, Kumagai, and Slade \cite{BJKS08}, who established it for high-dimensional incipient infinite clusters in \emph{oriented} percolation. Later, Kozma and Nachmias \cite{KozNach09} established the Alexander-Orbach conjecture for high-dimensional \emph{unoriented} percolation.
See \cite{MR3206998} for an extension to long-range percolation, \cite{KumagaiBook} for an overview, and \cite{1609.03980} for results regarding scaling limits of a related model.
As previously mentioned, Barlow and Masson \cite{BarMass11} have shown that in the \emph{two-dimensional} uniform spanning tree, $d_f=8/5,$ $d_s=16/13$, and $d_w=13/5$.
\subsection{Extrinsic exponents}
\label{subsec:introextrinsic}
We now describe our results concerning the \emph{extrinsic} diameter of the past. In comparison to the intrinsic diameter, our methods to study the extrinsic diameter are more delicate and require stronger assumptions on the graph in order to derive sharp estimates. Our first result on the extrinsic diameter concerns $\mathbb Z^d$, and improves upon the results of Bhupatiraju, Hanson, and J\'arai \cite{bhupatiraju2016inequalities} by removing the polylogarithmic errors present in their results.
\begin{theorem}[Mean-field Euclidean extrinsic diameter]
\label{thm:extrinsicZd}
Let $d\geq 5$, and
let $\mathfrak F$ be the wired uniform spanning forest of $\mathbb Z^d$.
Then
\vspace{0.25em}
\[
\vspace{0.25em}
\P\left(\mathrm{diam}_{\mathrm{ext}}(\mathfrak P(0)) \geq R \right) \asymp R^{-2}
\]
for every $R\geq 1$.
\end{theorem}
We expect that it should be possible to generalize the proof of \cref{thm:extrinsicZd} to other similar graphs, and to long-range models, but we do not pursue this here.
On the other hand, if we are unconcerned about polylogarithmic errors, it is rather straightforward to deduce various estimates on the extrinsic diameter from the analogous estimates on the intrinsic diameter, \cref{thm:transitivemain,thm:generalexponents}. The following is one such estimate of particular interest. For notational simplicity we will always work with the graph metric, although our methods easily adapt to various other metrics.
We say that a network $G$ with controlled stationary measure is \textbf{$d$-Ahlfors regular} if there exist positive constants $c$ and
$C$ such that $ cn^d \leq B(v,n) \leq Cn^d$ for every vertex $v$ and $n\geq 1$. We say that $G$ satisfies \textbf{Gaussian heat kernel estimates} if there exist positive constants $c,c'$ such that
\begin{align} \frac{c}{|B(x,n^{1/2})|}e^{- d(x,y)^2/(cn)} \leq p_n(x,y) + p_{n+1}(x,y) \leq \frac{c'}{|B(x,n^{1/2})|}e^{-d(x,y)^2/(c' n)}
\label{eq:GHKE}
\end{align}
for every $n\geq 0$ and every pair of vertices $x,y$ in $G$ with $d(x,y)\leq n$. It follows from the work of Hebisch and Saloff-Coste \cite{HebSaCo93} that every transitive graph of polynomial volume growth satisfies Gaussian heat-kernel estimates, as does every bounded degree network with edge conductances bounded between two positive constants that is rough-isometric to a transitive graph of polynomial growth.
\begin{theorem}
\label{thm:extrinsic}
Let $G$ be a network with controlled stationary measure that is $d$-Ahlfors regular for some $d>4$ and that satisfies Gaussian heat kernel estimates.
Then
\vspace{0.25em}
\[
\vspace{0.25em}
q(v) R^{-2} \preceq \P\left(\mathrm{diam}_{\mathrm{ext}}(\mathfrak P(0)) \geq R \right) \preceq R^{-2}\log R
\]
for every vertex $v$ and every $R\geq 1$.
\end{theorem}
Finally, we consider networks in which the random walk is ballistic rather than diffusive. We say that a network $G$ is \textbf{uniformly ballistic} if there exists a constant $C$ such that
\begin{equation}
\label{eq:speedcondition}
\sup_{v\in V}\mathbf E_v\left[
\sum_{n\geq 0} \mathbbm{1}\Bigl(X_n \in B(v,r)\Bigr)
\right] \leq C r
\end{equation}
for every $r\geq 1$.
It is easily seen that any nonamenable network with bounded degrees and edge conductances bounded above and below is uniformly ballistic.
\begin{theorem}[Extrinsic diameter in the positive speed case]
\label{thm:extrinsicspeed}
Let $G$ be a uniformly ballistic network with controlled stationary measure and $\bubnorm{P}<\infty$, and let $\mathfrak F$ be the wired uniform spanning forest of $G$.
Then
\vspace{0.25em}
\[
\vspace{0.25em}
q(v)R^{-1} \preceq \P\left(\mathrm{diam}_{\mathrm{ext}}(\mathfrak P(0)) \geq R \right) \preceq R^{-1}
\]
for every vertex $v$ and every $R\geq 1$.
\end{theorem}
Note that the \emph{upper} bound of \cref{thm:extrinsicspeed} is a trivial consequence of \cref{thm:generalexponents}.
\subsection{Applications to the Abelian sandpile model}
\label{subsec:sandpileintro}
The \textbf{Abelian sandpile model} was introduced by Dhar \cite{Dhar90} as an analytically tractable example of a system exhibiting \emph{self-organized criticality}. This is the phenomenon by which certain randomized dynamical systems tend to exhibit critical-like behaviour at equilibrium despite being defined without any parameters that can be varied to produce a phase transition in the traditional sense. The concept of self-organized criticality was first posited in the highly influential work of Bak, Tang, and Wiesenfeld \cite{bak1987self,bak1988self}, who proposed (somewhat controversially \cite{Watkins2016}) that it may account for the occurrence of complexity, fractals, and power laws in nature. See \cite{Jar14} for a detailed introduction to the Abelian sandpile model, and \cite{jensen1998self} for a discussion of self-organized criticality in applications.
We now define the Abelian sandpile model.
Let $G=(V,E)$ be a connected, locally finite graph and let $K \subseteq V$ be a set of vertices. A \textbf{sandpile} on $K$ is a function $\eta: K \to \{0,1,\ldots\}$, which we think of as a collection of indistinguishable particles (grains of sand) located on the vertices of $K$. We say that $\eta$ is \textbf{stable} at a vertex $x$ if $\eta(x)\leq \deg(x)$, and otherwise that $\eta$ is \textbf{unstable} at $x$. We say that $\eta$ is stable if it is stable at every $x$, and that it is unstable otherwise. If $\eta$ is unstable at $x$, we can \textbf{topple} $\eta$ at $x$ to obtain the sandpile $\eta'$ defined by
\[
\eta'(y) = \begin{cases} \eta(x) - \deg(x) & y=x\\
\eta(y)+\#\{\text{edges between $x$ and $y$}\} & y \neq x
\end{cases}
\]
for all $y\in K$.
That is, when $x$ topples, $\deg(x)$ of the grains of sand at $x$ are redistributed to its neighbours, and grains of sand redistributed to neighbours of $x$ in $V\setminus K$ are lost. Dhar~\cite{Dhar90} observed that if $K$ is finite and not equal to $V$ then carrying out successive topplings will eventually result in a stable configuration and, moreover, that the stable configuration obtained in this manner does not depend on the order in which the topplings are carried out.
(This property justifies the model's description as \emph{Abelian}.)
We define a Markov chain on the set of stable sandpile configurations on $K$ as follows: At each time step, a vertex of $K$ is chosen uniformly at random, an additional grain of sand is placed at that vertex, and the resulting configuration is stabilized. Although this Markov chain is \emph{not} irreducible, it can be shown that chain has a unique closed communicating class, consisting of the \emph{recurrent} configurations, and that the stationary measure of the Markov chain is simply the uniform measure on the set of recurrent configurations. In particular, the stationary measure for the Markov chain is also stationary if we add a grain of sand to a \emph{fixed} vertex and then stabilize.
The connection between sandpiles and spanning trees was first discovered by Majumdar and Dhar \cite{MajDhar92}, who described a bijection, known as the \emph{burning bijection}, between recurrent sandpile configurations and spanning trees. Using the burning bijection, Athreya and J\'arai \cite{MR2077255} showed that if $d\geq 2$ and $\langle V_n\rangle_{n\geq 1}$ is an exhaustion of $\mathbb Z^d$ by finite sets, then the uniform measure on recurrent sandpile configurations on $V_n$ converges weakly as $n\to\infty$ to a limiting measure on sandpile configurations on $\mathbb Z^d$. More generally, their result applies to any infinite, connected, locally finite graph $G$ for which every component of the WUSF of $G$ is one-ended almost surely. We call a random sandpile configuration on $G$ drawn from this measure a \textbf{uniform recurrent sandpile} on $G$, and typically denote such a random variable by $\mathrm{H}$ (capital $\eta$).
We are particularly interested in what happens during one step of the dynamics at equilibrium, in which one grain of sand is added to a vertex $v$ in a uniformly random recurrent configuration $\mathrm{H}$, and then topplings are performed in order to stabilize the resulting configuration. The multi-set of vertices counted according to the number of times they topple is called the \textbf{Avalanche}, and is denoted $\operatorname{Av}_v(\mathrm{H})$. The \emph{set} of vertices that topple at all is called the \textbf{Avalanche cluster} and is denoted by $\operatorname{AvC}_v(\mathrm{H})$.
J\'arai and Redig \cite{JarRed08} showed that the burning bijection allows one to relate avalanches to the past of the WUSF, which allowed them to prove that avalanches in $\mathbb Z^d$ satisfy $\P( v \in \operatorname{AvC}_0(\mathrm{H})) \asymp \|v\|^{-d+2}$ for $d\geq 5$. (The fact that the \emph{expected} number of times $v$ topples scales this way is an immediate consequence of Dhar's formula, see \cite[Section 3.3.1]{Jar14}.) Bhupatiraju, Hanson, and J\'arai \cite{bhupatiraju2016inequalities} built upon these methods to prove that, when $d\geq 5$, the probability that the diameter of the avalanche is at least $n$ scales as $n^{-2} \log^{O(1)} n$ and the probability that the total number of topplings in the avalanche is at least $n$ is between $c n^{-1/2}$ and $n^{-2/5+o(1)}$. Using the combinatorial tools that they developed, the following theorem, which improves upon theirs, follows straightforwardly from our results concerning the WUSF. (Strictly speaking, it also requires our results on the $v$-WUSF, see \cref{subsec:vWUSF}.)
\begin{theorem}
\label{thm:sandpile}
Let $d\geq 5$ and let $\mathrm{H}$ be a uniform recurrent sandpile on $\mathbb Z^d$. Then
\begin{align*}
\P\Bigl(\operatorname{diam}_\mathrm{ext}\left(\operatorname{AvC}_0(\mathrm{H})\right) \geq n\Bigr) &\asymp n^{-2} && n \nearrow\infty\\
\P\Bigl(|\operatorname{AvC}_0(\mathrm{H})| \geq n\Bigr) &\asymp n^{-1/2} && n \nearrow \infty\\
\P\Bigl(|\operatorname{Av}_0(\mathrm{H})| \geq n\Bigr)&\asymp n^{-1/2} && n \nearrow \infty.
\end{align*}
\end{theorem}
As with the WUSF, our methods also yield several variations on this theorem for other classes of graphs, the following of which are particularly notable. See \cref{subsec:introextrinsic} for the relevant definitions. With a little further work, it should be possible to remove the dependency on $v$ in the lower bounds of \cref{thm:sandpilepolynomial,thm:sandpilenonamenable}. The \emph{upper} bounds of \cref{thm:sandpilepolynomial} only require that $G$ has polynomial growth, see \cref{prop:polyupperext}.
\begin{theorem}
\label{thm:sandpilepolynomial}
Let $G$ be a bounded degree graph that is $d$-Ahlfors regular for some $d>4$ and that satisfies Gaussian heat kernel estimates, and let $\mathrm{H}$ be a uniform recurrent sandpile on $G$. Then
\hspace{1cm}
\[
\begin{array}{llcll}
q(v)^2n^{-2} &\preceq &\P\Bigl(\operatorname{diam}_\mathrm{ext}\left(\operatorname{AvC}_v(\mathrm{H})\right) \geq n\Bigr) &\preceq &n^{-2}\log n,\vspace{.25em}\\
q(v)^2 n^{-1/2} &\preceq &\P\Bigl(|\operatorname{AvC}_v(\mathrm{H})| \geq n\Bigr) &\preceq & n^{-1/2}\log^{1/2}n,\vspace{.25em}\\
q(v)^2 n^{-1/2} &\preceq &\P\Bigl(|\operatorname{Av}_v(\mathrm{H})| \geq n\Bigr) &\preceq & n^{-1/2}\log^{1/2}n
\end{array}
\]
for all $n\geq 1$.
\end{theorem}
Similarly, the following theorem concerning uniformly ballistic graphs can be deduced from \cref{thm:extrinsicspeed,thm:extrinsicspeedv}.
Again, we stress that this result applies in particular to any bounded degree nonamenable graph.
\begin{theorem}
\label{thm:sandpilenonamenable}
Let $G$ be a bounded degree, uniformly ballistic graph such that $\bubnorm{P}<\infty$, and let $\mathrm{H}$ be a uniform recurrent sandpile on $G$. Then
\[
\begin{array}{llcll}
q(v)^2n^{-1} &\preceq &\P\Bigl(\operatorname{diam}_\mathrm{ext}\left(\operatorname{AvC}_v(\mathrm{H})\right) \geq n\Bigr) &\preceq &n^{-1},\vspace{.25em}\\
q(v)^2 n^{-1/2} &\preceq &\P\Bigl(|\operatorname{AvC}_v(\mathrm{H})| \geq n\Bigr) &\preceq & n^{-1/2},\vspace{.25em}\\
q(v)^2 n^{-1/2} &\preceq &\P\Bigl(|\operatorname{Av}_v(\mathrm{H})| \geq n\Bigr) &\preceq & n^{-1/2}
\end{array}
\]
for all $n\geq 1$.
\end{theorem}
\subsection*{Notation}
As previously discussed, we use $\asymp, \preceq$ and $\succeq$ to denote equalities and inequalities that hold to within multiplication by two positive constants depending only on the choice of network.
Typically, but not always, these constants will only depend on a few important parameters such as $\inf_{v\in V} c(v),$ $\sup_{v\in V}c(v),$ and $\bubnorm{P}$.
For the reader's convenience, we gather here several notations that will be used throughout the paper. Each such piece of notation is also defined whenever it first appears within the body of the paper.
\begin{itemize}[leftmargin=4cm]
\item[$\mathfrak F,\mathfrak F_v$] A sample of the wired uniform spanning forest and $v$-wired uniform spanning forest respectively.
\item[$\mathfrak T_v$] The tree containing $v$ in $\mathfrak F_v$.
\item[$\mathfrak{B}(u,n),\mathfrak{B}_v(u,n)$] The intrinsic ball of radius $n$ around $u$ in $\mathfrak F$ and $\mathfrak F_v$ respectively.
\item[$\partial \mathfrak{B}(u,n),\partial \mathfrak{B}_v(u,n)$] The set of vertices at distance exactly $n$ from $u$ in $\mathfrak F$ and $\mathfrak F_v$ respectively.
\item[$\mathfrak P(u),\mathfrak P_v(u)$] The past of $u$ in $\mathfrak F$ and $\mathfrak F_v$ respectively.
\item[$\mathrm{past}_{F}(u)$] The past of $u$ in the oriented forest $F$ (which need not be spanning).
\item[$\mathfrak P(u,n),\mathfrak P_v(u,n)$] The intrinsic ball of radius $n$ around $u$ in the past of $u$ in $\mathfrak F$ and $\mathfrak F_v$ respectively.
\item[$\Gamma(u,w), \Gamma_v(u,w)$] The path from $u$ to $w$ in $\mathfrak F$ and $\mathfrak F_v$ respectively, should these vertices be in the same component.
\item[$\Gamma(u,\infty), \Gamma_v(u,\infty)$] The future of $u$ in $\mathfrak F$ and $\mathfrak F_v$ respectively.
\item[$\partial \mathfrak P(u,n),\partial \mathfrak P_v(u,n)$] The set of vertices with intrinsic distance exactly $n$ from $u$ in the past of $u$ in $\mathfrak F$ and $\mathfrak F_v$ respectively.
\item[$\mathscr I,\mathscr I_v$] The interlacement process and $v$-wired interlacement process respectively.
\item[$\mathcal I_{[a,b]},\mathcal I_{v,[a,b]}$] The set of vertices visited by the interlacement process and the $v$-wired interlacement process in the time interval $[a,b]$ respectively.
\end{itemize}
\section{Background}
\subsection{Loop-erased random walk and Wilson's algorithm}
Let $G$ be a network. For each $-\infty \leq n \leq m \leq \infty$ we define $L(n,m)$ to be the line graph with vertex set $\{i \in \mathbb Z : n \leq i\leq m\}$ and with edge set $\{(i,i+1): n\leq i \leq m-1\}$. A \textbf{path} in $G$ is a multigraph homomorphism from $L(n,m)$ to $G$. We can consider the random walk on $G$ as a path by keeping track of the edges it traverses as well as the vertices it visits. Given a path $w : L(n,m)\to G$ we will use $w(i)$ and $w_i$ interchangeably to denote the vertex visited by $w$ at time $i$, and use $w(i,i+1)$ and $w_{i,i+1}$ interchangeably to denote the oriented edge crossed by $w$ between times $i$ and $i+1$.
Given a path in $w : L(0,m) \to G$ for some $m\in [0,\infty]$ that is transient in the sense that it visits each vertex at most finitely many times, we define the sequence of times $\ell_n(w)$ by $\ell_0(w)=0$ and $\ell_{n+1}(w) = 1+\max\{ k : w_k = w_{\ell_n}\}$, terminating the sequence when $\ell_n(w)=m$ in the case that $m<\infty$. The \textbf{loop-erasure} $\mathsf{LE}(w)$ of $w$ is the path defined by
\[\mathsf{LE}(w)_i = w_{\ell_i(w)} \qquad \mathsf{LE}(w)_{i,i+1} = w_{\ell_{i+1}-1,\ell_{i+1}}.\]
In other words, $\mathsf{LE}(w)$ is the path formed by erasing cycles from $w$ chronologically as they are created. The loop-erasure of simple random walk is known as \textbf{loop-erased random walk} and was first studied by Lawler \cite{Lawler80}.
\textbf{Wilson's algorithm} \cite{Wilson96} is a method of sampling the UST of a finite graph by recursively joining together loop-erased random walk paths. It was extended to sample the WUSF of infinite transient graphs by Benjamini, Lyons, Peres, and Schramm \cite{BLPS}. See also \cite[Chapters 4 and 10]{LP:book} for an overview of the algorithm and its applications.
Wilson's algorithm can be described in the infinite transient case as follows. Let $G$ be an infinite transient network, and let $v_1,v_2,\ldots$ be an enumeration of the vertices of $G$. Let $\mathfrak F^{0}$ be the empty forest, which has no vertices or edges.
Given $\mathfrak F^{n}$ for some $n\geq 0$, start a random walk at $v_{n+1}$. Stop the random walk if and when it hits the set of vertices already included in $\mathfrak F^n$, running it forever otherwise. Let $\mathfrak F^{n+1}$ be the union of $\mathfrak F^n$ with the set of edges traversed by the loop-erasure of this stopped path.
Let $\mathfrak F=\bigcup_{n\geq 0} \mathfrak F^n$.
Then the random forest $\mathfrak F$ has the law of the wired uniform spanning forest of $G$. If we keep track of direction in which edges are crossed by the loop-erased random walks when performing Wilson's algorithm, we obtain the oriented wired uniform spanning forest.
The algorithm works similarly in the finite and recurrent cases, except that we start by taking $\mathfrak F^{0}$ to contain one vertex and no edges.
\subsection{The $v$-wired uniform spanning forest and stochastic domination}
\label{subsec:vWUSF}
In this section we introduce the $v$-wired uniform spanning forest ($v$-WUSF), which was originally defined by J\'arai and Redig \cite{JarRed08} in the context of their work on the sandpile model (where it was called the WSF$_o$). The $v$-WUSF is a variation of the WUSF of $G$ in which, roughly speaking, we consider $v$ to be `wired to infinity'. The $v$-WUSF serves two useful purposes in this paper: its stochastic domination properties allow us to ignore interactions between different parts of the WUSF, and the control of the $v$-WUSF that we obtain will be applied to prove our results concerning the Abelian sandpile model in \cref{sec:sandpile}.
Let $G$ be an infinite network and let $v$ be a vertex of $G$. Let $\langle V_n \rangle_{n\geq 1}$ be an exhaustion of $G$ and, for each $n\geq1$, let $G^{*v}_n$ be the graph obtained by identifying $v$ with $\partial_n$ in the graph $G^*_n$. The measure $\mathsf{WUSF}_v$ is defined to be the weak limit
\[\mathsf{WUSF}_v(S \subset \mathfrak F; G) = \lim_{n\to\infty}\mathsf{UST}(S \subset T ; G^{*v}_n).\]
The fact that this limit exists and does not depend on the choice of exhaustion of $G$ is proved similarly to the corresponding statement for the WUSF, see \cite{LMS08}. As with the WUSF, we can also define the \textbf{oriented $v$-wired uniform spanning forest} by orienting the uniform spanning tree of $G_n^{*v}$ towards $\partial_n$ (which is identified with $v$) at each step of the exhaustion before taking the weak limit.
It is possible to sample the $v$-WUSF by running Wilson's algorithm rooted at infinity, but starting with $\mathfrak F^{0}_v$ as the forest that has vertex set $\{v\}$ and no edges (as we usually would in the finite and recurrent cases). Moreover, if we orient each edge in the direction in which it is crossed by the loop-erased random walk when running Wilson's algorithm, we obtain a sample of the oriented $v$-WUSF.
The following lemma makes the $v$-WUSF extremely useful for studying the usual WUSF, particularly in the mean-field setting. It will be the primary means by which we ignore the interactions between different parts of the forest. We denote by $\mathrm{past}_F(v)$ the past of $v$ in the oriented forest $F$, which need not be spanning. We write $\mathfrak T_v$ for the tree containing $v$ in $\mathfrak F_v$, and write $\Gamma(u,\infty)$ and $\Gamma_v(u,\infty)$ for the future of $u$ in $\mathfrak F$ and $\mathfrak F_v$ respectively.
\begin{lemma}[Stochastic Domination]
\label{lem:domination}
Let $G$ be an infinite network, let $\mathfrak F$ be an oriented wired uniform spanning forest of $G$, and for each vertex $v$ of $G$ let $\mathfrak F_v$ be an oriented $v$-wired uniform spanning forest of $G$.
Let $K$ be a finite set of vertices of $G$, and define $F(K) =\bigcup_{u\in K} \Gamma(u,\infty)$ and $F_v(K) =\bigcup_{v\in K} \Gamma_v(u,\infty)$. Then for every $u\in K$ and every increasing event $\mathscr A \subseteq \{0,1\}^E$ we have that
\begin{align}\P\Bigl( \mathrm{past}_{\mathfrak F \setminus F(K)}(u) \in \mathscr A \mid F(K) \Bigr) &\leq \P\bigl(\mathfrak T_v \in \mathscr A \bigr),
\label{eq:dom1}
\intertext{and similarly}
\P\Bigl( \mathrm{past}_{\mathfrak F_v\setminus F_v(K)} \in \mathscr A \mid F_v(K) \Bigr) &\leq \P\bigl(\mathfrak T_u \in \mathscr A \bigr).
\label{eq:dom2}
\end{align}
\end{lemma}
Note that when $K$ is a singleton, \eqref{eq:dom1} follows implicitly from \cite[Lemma 2.3]{LMS08}. The proof in the general case is also very similar to theirs, but we include it for completeness. Given a network $G$ and a finite set of vertices $K$, we write $G/K$ for the network formed from $G$ by identifying all the vertices of $K$.
\begin{lemma}
\label{lem:domaux}
Let $G$ be a finite network, let $K_1 \subseteq K_2$ be sets of vertices of $G$. For each spanning tree $T$ of $G$, let $S(T,K_2)$ be the smallest subtree of $T$ containing all of $K_2$. Then the uniform spanning tree of $G/K_1$ stochastically dominates $T \setminus S(T,K_2)$, where $T$ is a uniform spanning tree of $G$.
\end{lemma}
\begin{proof}
It follows from the spatial Markov property of the UST that, conditional on $S(T,K_2)$, the complement $T\setminus S(T,K_2)$ is distributed as the UST of the network $G/S(T,K_2)$ constructed from $G$ by identifying all the vertices in the tree $S(T,K_2)$, see \cite[Section 2.2.1]{HutNach15b}. On the other hand, it follows from the negative association property of the UST \cite[Theorem 4.6]{LP:book} that if $A \subseteq B$ are two sets of vertices, then the UST of $G / A$ stochastically dominates the UST of $G / B$. This implies that the claim holds when we condition on $S(T,K_2)$, and we conclude by averaging over the possible choices of $S(T,K_2)$.
\end{proof}
\begin{proof}[Proof of \cref{lem:domination}]
The claim \eqref{eq:dom1}
follows from \cref{lem:domaux} by considering the finite networks $G_n^*$ used in the definition of the WUSF, taking $K_1 = \{v,\partial_n\}$ and $K_2 = K \cup \{\partial_n\}$, and taking the limit as $n\to\infty$. The proof of \eqref{eq:dom2} is similar.
\end{proof}
\section{Interlacements and the Aldous-Broder algorithm}
The \textbf{random interlacement process} is a Poissonian soup of doubly-infinite random walks that was introduced by Sznitman \cite{Szni10} and generalized to arbitrary transient graphs by Texeira \cite{Teix09}. Formally, the interlacement process $\mathscr I$ on the transient graph $G$ is a Poisson point process on $\mathcal W^* \times \mathbb R$, where $\mathcal W^*$ is the space of bi-infinite paths in $G$ modulo time-shift, and $\mathbb R$ is thought of as a time coordinate. In \cite{hutchcroft2015interlacements}, we showed that the random interlacement process can be used to generate the WUSF via a generalization of the Aldous-Broder algorithm. By shifting the time coordinate of the interlacement process, this sampling algorithm also allows us to view the WUSF as the stationary measure of a Markov process; this dynamical picture of the WUSF, or more precisely its generalization to the $v$-WUSF, is of central importance to the proofs of the main theorems of this paper.
We now begin to define these notions formally.
We must first define the space of trajectories $\mathcal W^*$.
Recall that for each $-\infty \leq n \leq m \leq \infty$ we define $L(n,m)$ to be the line graph with vertex set $\{i \in \mathbb Z : n \leq i\leq m\}$ and with edge set $\{(i,i+1): n\leq i \leq m-1\}$. Given a graph $G$, we define $\mathcal W(n,m)$ to be the set of multigraph homomorphisms from $L(n,m)$ to $G$ that are transient in the sense that the preimage of each vertex of $G$ is finite. We define the set $\mathcal W$ to be the union
\[ \mathcal W:= \bigcup\left\{\mathcal W(n,m): -\infty \leq n \leq m \leq \infty\right\}.\]
The set $\mathcal W$ can be made into a Polish space in such a way that local times at vertices and first and last hitting times of finite sets are continuous, see \cite[Section 3.2]{hutchcroft2015interlacements}.
We define the \textbf{time shift} $\theta_k:\mathcal W\to \mathcal W$ by
$\theta_k : \mathcal W(n,m) \longrightarrow \mathcal W(n-k,m-k)$,
\[ \theta_k(w)(i)=w(i+k), \quad \theta_k(w)(i,i+1)= w(i+k,i+k+1),\]
and define the space $\mathcal W^*$ to be the quotient
\[\mathcal W^* = \mathcal W / \sim \text{ where } w_1\sim w_2 \text{ if and only if } w_1 = \theta_k (w_2) \text{ for some k}.\]
Let $\pi : \mathcal W \to \mathcal W^*$ denote the associated quotient function. We equip the set $\mathcal W^*$ with the quotient topology (which is Polish) and associated Borel $\sigma$-algebra.
An element of $\mathcal W^*$ is called a \textbf{trajectory}.
We now define the intensity measure of the interlacement process.
Let $G$ be a transient network.
Given $w \in \mathcal W(n,m)$, let $w^\leftarrow \in \mathcal W(-n,-m)$ be the reversal of $w$, which is defined by setting $w^\leftarrow(i)=w(-i)$ for all $-m \leq i \leq -n$ and setting $w^\leftarrow(i,i+1)=w(-i-1,-i)$ for all $-m \leq i \leq -n-1$. For each subset $\mathscr A \subseteq \mathcal W$, let $\mathscr A^\leftarrow$ denote the set
\[\mathscr A^\leftarrow:= \{w \in \mathcal W: w^\leftarrow \in \mathscr A \}.\]
For each set $K \subseteq V$, we let $\mathcal W_K(n,m)$ be the set of $w\in \mathcal W(n,m)$ such that there exists $n\leq i\leq m$ such that $w(i)\in K$, and similarly define $\mathcal W_K$ to be the union $\mathcal W_K=\bigcup \{\mathcal W_K(n,m): -\infty \leq n \leq m \leq \infty\}$. We
define a measure $Q_K$ on $\mathcal W_K$ by setting \[Q_K(\{w\in \mathcal W: w(0)\notin K\})=0\] and, for each $u\in K$ and each two Borel subsets $\mathscr A,\mathscr B\subseteq \mathcal W$,
\begin{multline*}
Q_K\left( \{w \in \mathcal W: w|_{(-\infty,0]} \in \mathscr A,\, w(0) = u \text{ and } w|_{[0,\infty)} \in \mathscr B \}\right)\\=
c(u)\mathbf{P}_u\big( X \in \mathscr A^\leftarrow \text{ and } \tau_K^+ =\infty\big) \mathbf{P}_u\big(X \in \mathscr B \big).
\end{multline*}
For each set $K \subseteq V$, let $\mathcal W^*_K= \pi(W_K)$ be the set of trajectories that visit $K$.
It follows from the work of Sznitman \cite{Szni10} and Teixeira \cite{Teix09} that there exists a unique $\sigma$-finite measure $Q^\ast$ on $\mathcal W^*$ such that for every Borel set $\mathscr A \subseteq \mathcal W^*$ and every finite $K\subset V$,
\begin{equation}\label{eq:ildefn} Q^*(\mathscr A \cap \mathcal W^*_K) = Q_K\left(\pi^{-1}(\mathscr A)\right). \end{equation}
We refer to the unique such measure $Q^*$ as the \textbf{interlacement intensity measure}, and define the \textbf{random interlacement process} $\mathscr I$ to be the Poisson point process on $\mathcal W^*\times \mathbb R$ with intensity measure $Q^* \otimes \Lambda$, where $\Lambda$ is the Lebesgue measure on $\mathbb R$. For each $t\in\mathbb R$, we write $\mathscr I_t$ for the set of $w\in \mathcal W^*$ such that $(w,t)\in\mathscr I$, and write $\mathscr I_{[a,b]}$ for the intersection of $\mathscr I$ with $\mathcal W^* \times [a,b]$.
See \cite[Proposition 3.3]{hutchcroft2015interlacements} for a limiting construction of the interlacement process from the random walk on an exhaustion with wired boundary conditions.
In \cite{hutchcroft2015interlacements}, we proved that the WUSF can be generated from the random interlacement process in the following manner. Let $G$ be a transient network, and let $t\in \mathbb R$. For each vertex $v$ of $G$, let $\sigma_t(v)$ be the smallest time greater than $t$ such that there exists a trajectory $W_{\sigma_t(v)} \in \mathscr I_{\sigma_t(v)}$ that hits $v$, and note that the trajectory $W_{\sigma_t(v)}$ is unique for every $t\in \mathbb R$ and $v\in V$ almost surely. We define $e_t(v)$ to be the oriented edge of $G$ that is traversed by the trajectory $W_{\sigma_t(v)}$ as it enters $v$ for the first time, and define
\[
\mathsf{AB}_t(\mathscr I):=\Bigl\{ -e_t(v): v \in V \Bigr\}.
\]
\cite[Theorem 1.1]{hutchcroft2015interlacements} states that $\mathsf{AB}_t(\mathscr I)$ has the law of the oriented wired uniform spanning forest of $G$ for every $t\in \mathbb R$. Moreover, \cite[Proposition 4.2]{hutchcroft2015interlacements} states that $\langle \mathsf{AB}_t(\mathscr I)\rangle_{t\in \mathbb R}$ is a stationary, ergodic, mixing, stochastically continuous Markov process.
\subsection{$v$-wired variants}
In this section, we introduce a variation on the interlacement process in which a vertex $v$ is wired to infinity, which we call the \textbf{$v$-wired interlacement process}. We then show how the $v$-wired interlacement process can be used to generate the $v$-WUSF in the same way that the usual interlacement process generates the usual WUSF.
Let $G$ be a (not necessarily transient) network and let $v$ be a vertex of $G$. Recall that $\tau_v$ denotes the first time that the random walk visits $v$, and $\tau^+_K$ denotes the first positive time that the random walk visits $K$. For each finite set $K \subset V$ we define a measure $Q_{v,K}$ on $\mathcal W$ by $Q_{v,K}(\{w \in \mathcal W : w(0)\notin K\})=0$,
\begin{multline*}
Q_{v,K}\left( \{w \in \mathcal W: w|_{(-\infty,0]} \in \mathscr A,\, w(0) = u \text{ and } w|_{[0,\infty)} \in \mathscr B \}\right)\\=
c(u)\mathbf P_u\big( X^{\tau_v} \in \mathscr A^\leftarrow \text{ and } \tau_K^+ >\tau_v \big) \mathbf P_u\big(X^{\tau_v} \in \mathscr B \big)
\end{multline*}
for every $u\in K\setminus \{v\}$ and every two Borel sets $\mathscr A,\mathscr B \subseteq \mathcal W$, and
\begin{multline*}
Q_{v,K}\left( \{w \in \mathcal W: w|_{(-\infty,0]} \in \mathscr A,\, w(0) = v \text{ and } w|_{[0,\infty)} \in \mathscr B \}\right)\\=
c(v)\mathbbm{1}(w_0 \in \mathscr A^\leftarrow) \mathbf P_v\big(X^{\tau_v} \in \mathscr B \big)
+c(v)\mathbf P_v\big( X^{\tau_v} \in \mathscr A^\leftarrow \text{ and } \tau_K^+ = \infty \big) \mathbbm{1}(w_0 \in \mathscr B)
\end{multline*}
for every two Borel sets $\mathscr A,\mathscr B \subseteq \mathcal W$ if $v\in K$,
where we write $w_0\in \mathcal W(0,0)$ for the path of length zero at $v$.
As with the usual interlacement intensity measure, we wish to define a measure $Q^\star_v$ on $\mathcal W^*$ via the consistency condition
\begin{equation}
\label{eq:rootedintensitydef}
Q^*_v(\mathscr A \cap \mathcal W^*_K) = Q_{v,K}(\pi^{-1}(\mathscr A))
\end{equation}
for every finite set $K \subset V$ and every Borel set $\mathscr A \subseteq \mathcal W^*$, and define the $v$-rooted interlacement process to be the Poisson point process on $\mathcal W^*\times \mathbb R$ with intensity measure $Q_v^* \otimes \Lambda$.
We will deduce that such a measure exists via the following limiting procedure, which also gives a direct construction of the $v$-rooted interlacement process.
Let $N$ be a Poisson point process on $\mathbb R$ with intensity measure $(c(\partial_n)+c(v))\Lambda$. Conditional on $N$, for each $t\in N$, let $W_t$ be a random walk on $G_n^{*v}$ started at $\partial_n$ and stopped when it first returns to $\partial_n$, where we consider each $W_t$ to be an element of $\mathcal W^*$. We define $\mathscr I^n_v$ to be the point process
$\mathscr I^n_v:=\left\{(W_t,t) : t \in N\right\}$.
\begin{proposition}\label{Prop:intexhaust}
Let $G$ be an infinite network, let $v$ be a vertex of $G$, and let $\langle V_n \rangle_{n\geq 0}$ be an exhaustion of $G$. Then the Poisson point processes $\mathscr I^n_v$ converge in distribution as $n\to\infty$ to a Poisson point process $\mathscr I_v$ on $\mathcal W^* \times \mathbb R$ with intensity measure of the form
$Q^*_v \otimes \Lambda$,
where $Q^*_v$ is a $\sigma$-finite measure on $\mathcal W^*$ such that \eqref{eq:rootedintensitydef} is satisfied
for every finite set $K \subset V$ and every event $\mathscr A \subseteq \mathcal W^*$.
\end{proposition}
The proof is very similar to that of \cite[Proposition 3.3]{hutchcroft2015interlacements}, and is omitted.
\begin{corollary}
Let $G$ be a network and let $v$ be a vertex of $G$. Then there exists a unique $\sigma$-finite measure $Q^*_v$ on $\mathcal W^*$ such that \eqref{eq:rootedintensitydef} is satisfied
for every finite set $K \subset V$ and every event $\mathscr A \subseteq \mathcal W^*$.
\end{corollary}
\begin{proof}
The existence statement follows immediately from \cref{Prop:intexhaust}.
The uniqueness part of the proposition follows immediately from existence, since sets of the form $\mathscr A \cap \mathcal W^*_K$ are a $\pi$-system generating the Borel $\sigma$-algebra on $\mathcal W^*$.
\end{proof}
We call $\mathscr I_v$ the \textbf{$v$-wired interlacement process}. Note that it may include trajectories that are either doubly infinite, singly infinite and ending at $v$, singly infinite and starting at $v$, or finite and both starting and ending at $v$.
We have the following $v$-rooted analogue of \cite[Theorem 1.1 and Proposition 4.2]{hutchcroft2015interlacements}, whose proof is identical to that of that theorem.
\begin{proposition}
Let $G$ be an infinite network, let $v$ be a vertex of $G$, and let $\mathscr I_v$ be the $v$-rooted interlacement process on $G$. Then
\[\mathsf{AB}_{v,t}(\mathscr I_v):= \bigl\{-e_t(u):u \in V \setminus \{v\} \bigr\} \]has the law of the oriented $v$-wired uniform spanning forest of $G$ for every $t\in \mathbb R$. Moreover, the process $\langle \mathsf{AB}_{v,t}(\mathscr I_v) \rangle_{t\in \mathbb R}$ is a stationary, ergodic, stochastically continuous Markov process.
\end{proposition}
\subsection{Relation to capacity}
In this section, we record the well-known relationship between the interlacement intensity measure $Q^*$ and the \textbf{capacity} of a set, and extend this relationship to the $v$-rooted interlacement intensity measure $Q^*_v$.
Recall that the \textbf{capacity} of a finite set of vertices $K$ in a network $G$ is defined to be
\[
\Cap(K):= Q^*(\mathcal W^*_K)=\sum_{v\in K} c(v)\mathbf P_v(\tau^+_K=\infty),
\]
where $\tau^+_K$ is the first positive time that the random walk visits $K$ and the second equality follows by definition of $Q^*$.
Similarly, we define the \textbf{$v$-wired capacity} of a finite set $K$ to be
\[
\Cap_v(K) := Q_v^*(\mathcal W^*_K) = \sum_{u\in K } c(u)\mathbf P_u(\tau^+_K>\tau_v) + c(v)\mathbbm{1}(v\in K)\left[1+\mathbf{P}_v(\tau^+_K=\infty)\right],
\]
with the convention that $\tau^+_K>\tau_v$ in the case that both hitting times are equal to $\infty$. In other words, $\Cap_v(K)$ is the effective conductance between $K$ and $\{\infty,v\}$. Thus, the number of trajectories in $\mathscr I_{[a,b]}$ that hit $K$ is a Poisson random variable with parameter $|a-b| \Cap(K)$, while the number of trajectories in $\mathscr I_{v,[a,b]}$ that hit $K$ is a Poisson random variable with parameter $|a-b| \Cap_v(K)$.
In our setting the capacity and $v$-rooted capacity of a set will always be of the same order:
The inequality $\Cap(K)\leq \Cap_v(K)$ is immediate, and on the other hand we have that
\[
\Cap_v(K) \leq \Cap(K) + 2c(v) + \sum_{u\in K\setminus \{v\}} c(u) \mathbf P_u( \tau_v < \tau^+_K <\infty).
\]
By time-reversal we have that $c(u)\mathbf P_u( \tau_v < \tau^+_K <\infty)=c(v)\mathbf P_v(\tau_K<\tau^+_u, X_{\tau_K} =u)$, and summing over $u\in K$ we obtain that
\begin{equation}
\label{eq:CapvCapComparison}
\Cap_v(K) \leq \Cap(K) + c(v)\mathbf P_v(\tau_K<\tau^+_u) + 2c(v) \leq \Cap(K) + 3c(v).
\end{equation}
Thus, in networks with bounded vertex conductances, the capacity and and $v$-rooted capacity agree to within an additive constant. Furthermore, the assumption that $\bubnorm{P}<\infty$ implies that $G$ is uniformly transient and hence that $\Cap(K)$ is bounded below by a positive constant for every non-empty set $K$.
\subsection{Evolution of the past under the dynamics}
The reason that the dynamics induced by the interlacement Aldous-Broder algorithm are so useful for studying the past of the origin in the WUSF is that the past itself evolves in a very natural way under the dynamics. Indeed, if we run time backwards and compare the pasts $\mathfrak P_{-t}(v)$ and $\mathfrak P_0(v)$ of $v$ in $\mathfrak F_{-t}$ and $\mathfrak F_{0}$, we find that the past can become larger only at those times when a trajectory visits $v$. At all other times, $\mathfrak P_{-t}(v)$ decreases monotonically in $t$ as it is `cut up' by newly arriving trajectories. This behaviour is summarised in the following lemma, which is adapted from \cite[Lemma 5.1]{hutchcroft2015interlacements}. We write $\mathcal I_{[a,b]}$ for the set of vertices that are hit by some trajectory in $\mathscr I_{[a,b]}$, and write $\mathfrak P_t(v)$ for the past of $v$ in the forest $\mathfrak F_t$.
\begin{lemma}
\label{lem:PastDynamics}
Let $G$ be a transient network, let $\mathscr I$ be the interlacement process on $G$, and let $\langle \mathfrak F_t\rangle_{t\in \mathbb R}=\langle \mathsf{AB}_t(\mathscr I) \rangle_{t\in \mathbb R}$. Let $v$ be a vertex of $G$, and let $s<t$. If $v \notin \mathcal I_{[s,t)}$, then $\mathfrak P_s(v)$ is equal to the component of $v$ in the subgraph of $\mathfrak P_t(v)$ induced by $V \setminus \mathcal I_{[s,t)}$.
\end{lemma}
\begin{proof}
Suppose that $u$ is a vertex of $V$, and let $\Gamma_s(u,\infty)$ and $\Gamma_t(u,\infty)$ be the futures of $u$ in $\mathfrak F_s$ and $\mathfrak F_t$ respectively. Let $u=u_{0,s},u_{1,s},\ldots$ and $u=u_{0,t},u_{1,t},\ldots$ be, respectively, the vertices visited by $\Gamma_s(u,\infty)$ and $\Gamma_t(u,\infty)$ in order.
Let $i_0$ be the smallest $i$ such that $\sigma_s(u_{i,s}) <t$. Then it follows from the definitions that $\Gamma_s(u,\infty)$ and $\Gamma_t(u,\infty)$ coincide up until step $i_0$, and that $\sigma_s(u_{i,s}) <t$ for every $i\geq i_0$. (Indeed, $\sigma_s(u_{i,s})$ is decreasing in $i$.) On the other hand, if $v\notin \mathcal I_{[s,t)}$ then $\sigma_s(v)>t$, and the claim follows readily.
\end{proof}
Similarly, we have the following lemma in the $v$-wired case, whose proof is identical to that of \cref{lem:PastDynamics} above. We write $\mathcal I_{v,[a,b]}$ for the set of vertices that are hit by some trajectory in $\mathscr I_{v,[a,b]}$, and write $\mathfrak P_{v,t}(u)$ for the past of $u$ in the forest $\mathfrak F_{v,t}$.
\begin{lemma}
\label{lem:vPastDynamics}
Let $G$ be a transient network, let $v$ be a vertex of $G$, let $\mathscr I_v$ be the $v$-wired interlacement process on $G$, and let $\langle \mathfrak F_{v,t}\rangle_{t\in \mathbb R}=\langle \mathsf{AB}_{v,t}(\mathscr I_v) \rangle_{t\in \mathbb R}$. Let $u$ be a vertex of $G$, and let $s<t$. If $u \notin \mathcal I_{v,[s,t)}$, then $\mathfrak P_{v,s}(u)$ is equal to the component of $v$ in the subgraph of $\mathfrak P_{v,t}(u)$ induced by $V \setminus \mathcal I_{v,[s,t)}$.
\end{lemma}
\section{Lower bounds for the diameter}
In this section, we use the interlacement Aldous-Broder algorithm to derive the lower bounds on the tail of the intrinsic and extrinsic diameter of \cref{thm:transitivemain,thm:generalexponents,thm:generalexponentsv,thm:extrinsicZd,thm:extrinsicv}.
\subsection{Lower bounds for the intrinsic diameter}
\begin{proposition}
\label{prop:intlower}
Let $G$ be a transient network. Then for each vertex $v$ of $G$ we have that
\begin{equation}
\label{eq:intdiamlower}
\P\Bigl(\mathrm{diam}_\mathrm{int}\bigl(\mathfrak P(v)\bigr)\geq r\Bigr) \geq \left(\frac{q(v)\Cap(v) }{e \sup_{u\in V}\Cap(u)}\right)\frac{1}{r+1}
\end{equation}
and similarly
\begin{equation}
\label{eq:intdiamlowerv}
\P\Bigl(\mathrm{diam}_\mathrm{int}\bigl(\mathfrak T_v\bigr) \geq r\Bigr) \geq \left(\frac{\Cap(v)}{e \sup_{u\in V} \Cap(u)}\right) \frac{1}{r+1}
\end{equation}
for every $r\geq 0$.
\end{proposition}
Note that \eqref{eq:intdiamlowerv} gives a non-trivial lower bound for every transitive network, and can be thought of as a mean-field lower bound. (For recurrent networks, the tree $\mathfrak T_v$ contains every vertex of the network almost surely, so that the bound also holds degenerately in that case.)
\begin{proof}
We prove \eqref{eq:intdiamlower}, the proof of \eqref{eq:intdiamlowerv} being similar.
Let $\mathscr I$ be the interlacement process on $G$ and let $\mathfrak F=\mathsf{AB}_0(\mathscr I)$.
Given a path $W=\langle W_n \rangle_{n\geq0}$ in $G$ and a vertex $u$ of $G$ visited by the path after time zero, we define $e(W,u)$ to be the oriented edge pointing into $u$ that is traversed by $W$ as it enters $u$ for the first time, and define
\[\mathsf{AB}(W) = \{-e(W,u) : u \text{ is visited by $W$ after time zero}\}.\]
Suppose that $W$ is doubly infinite and hits $v$, and parameterize $W$ so that it hits $v$ for the first time at time zero. One may verify from the definitions that the past of $v$ in $\mathsf{AB}(W)$ contains the portion of the loop-erasure of $\langle W_n \rangle_{n\geq 0}$ up until the first positive time that this loop-erasure intersects $\langle W_n \rangle_{n \leq 0}$.
For each $r\geq 1$ and $\varepsilon>0$, let $\mathscr A_{\varepsilon,n}$ be the event that $v$ is hit by exactly one trajectory in $\mathscr I_{[0,\varepsilon]}$, this trajectory hits $v$ exactly once, and the parts of the trajectory before and after hitting $v$ do not intersect each other. It follows from the definition of the interlacement intensity measure that
\begin{equation}\label{eq:Aeps}\P(\mathscr A_\varepsilon) = \varepsilon q(v) \Cap(v) e^{-\varepsilon\Cap(v)}.\end{equation}
Given $\mathscr A_\varepsilon$, let $\langle W_n \rangle_{n \in \mathbb Z}$ be the unique representative of this trajectory that has $W_0=v$, let $\eta$ be the set of vertices visited by the first $r$ steps of the loop-erasure of $\langle W_n \rangle_{n\geq0}$, and let $\mathscr B_{r,\varepsilon}\subseteq\mathscr A_\varepsilon$ be the event that $\mathscr A_\varepsilon$ occurs and that
$\eta$ is not hit by any trajectories in $\mathscr I_{[0,\varepsilon]}$ other than $W$.
On the event $\mathscr B_{r,\varepsilon}$ we have by definition of $\mathsf{AB}_0(\mathscr I)$ that the reversals of the first $n$ oriented edges traversed by the loop-erasure of $\langle W_n \rangle_{n\geq0}$ are all contained in $\mathfrak F$, and consequently we have that
\[ \P\left(\operatorname{diam}_\textrm{int}(\mathfrak P(v)) \geq r\right) \geq \P(\mathscr B_{r,\varepsilon})\]
for every $r\geq 1$ and $\varepsilon>0$. On the other hand,
by the splitting property of Poisson processes, for every set $K\subset V$, the number of trajectories of $\mathscr I_{[0,\varepsilon]}$ that hit $K$ but not $v$ is independent of the set of trajectories of $\mathscr I_{[0,\varepsilon]}$ that hit $v$. We deduce that
\begin{equation}
\P(\mathscr B_{r,\varepsilon} \mid \mathscr A_{r,\varepsilon}) \geq \mathbb E\left[ e^{-\varepsilon \Cap(\eta)} \mid \mathscr A_{r,\varepsilon}\right].
\end{equation}
Let $M = \sup_{u \in V} \Cap(u)$, which is at most $\sup_{u \in V} c(u)$ and is therefore finite.
By the subadditivity of the capacity we have that
$\Cap(\eta) \leq Mr$,
so that $\P(\mathscr B_{r,\varepsilon} \mid \mathscr A_{r,\varepsilon}) \geq e^{-\varepsilon Mr}$
and hence
\begin{equation}\label{eq:Beps}
\P(\mathscr B_{r,\varepsilon}) \geq q(v)\, \Cap(v) \, \varepsilon e^{-\varepsilon\Cap(v)}e^{-\varepsilon M r} \geq q(v) \Cap(v) \varepsilon e^{-\varepsilon M(r+1)}.
\end{equation}
The claimed inequality now follows by taking $\varepsilon= 1/(M(r+1))$.
\end{proof}
\subsection{Lower bounds for the extrinsic diameter}
In this section we apply a similar method to that used in the previous subsection to prove a lower bound on the tail of the \emph{extrinsic} diameter. The method we use is very general and, as well as being used in the proof of \cref{thm:extrinsicZd,thm:extrinsic,thm:extrinsicv,thm:extrinsicspeed}, can also be used to deduce similar lower bounds for e.g.\ long-ranged models.
Let $G$ be a network. For each $r\geq 0$, we define
\[
L(r) =\sup_{v\in V} \mathbf E_v\left[ \sup \left\{n\geq 0 : X_n \in B(v,r) \right\}\right]
\]
to be the maximum expected final visit time to a ball of radius $r$.
It is easily seen that every transitive graph of polynomial growth has $L(r)\asymp r^2$, and the same holds for any Ahlfors regular network with controlled stationary measure satisfying Gaussian heat kernel estimates, see \cref{lem:polyLr} below. On the other hand, uniformly ballistic networks have by definition that $L(r)\asymp r$.
\begin{proposition}
\label{prop:extlower}
Let $G$ be a transient network. Then for each vertex $v$ of $G$ we have that
\begin{equation}
\label{eq:extlower}
\P\Bigl(\mathrm{diam}_\mathrm{ext}\bigl(\mathfrak P(v)\bigr)\geq r \Bigr) \geq \left(\frac{2q(v)^2 \Cap(v) }{e \sup_{u\in V}\Cap(u)}\right)\frac{1}{L(r)+4}
\end{equation}
for every $r\geq 1$, and similarly
\begin{equation}
\label{eq:extlowerv}
\P\Bigl(\mathrm{diam}_\mathrm{ext}\bigl(\mathfrak T_v\bigr) \geq r\Bigr)\geq
\left(\frac{2\Cap(v) }{e \sup_{u\in V}\Cap(u)}\right)\frac{1}{L(r)+4}
\end{equation}
for every $r\geq 1$.
\end{proposition}
\begin{proof}[Proof of \cref{prop:extlower}]
We prove \eqref{eq:extlower}, the proof of \eqref{eq:extlowerv} being similar.
We continue to use the notation $\mathscr A_{R,\varepsilon},\mathscr B_{R,\varepsilon},W,$, $\eta$, and $M$ defined in the proof of \cref{prop:intlower} (but with the variable $R$ replacing the variable $r$ there).
We also define $\mathscr C_{r,R,\varepsilon} \subseteq \mathscr A_{R,\varepsilon}$ to be the event in which $\mathscr A_{R,\varepsilon}$ occurs and the distance in $G$ between
$v$ and the endpoint of $\eta$ is at least $r$, and define $\mathscr D_{r,R,\varepsilon} = \mathscr C_{r,R,\varepsilon} \cap \mathscr B_{R,\varepsilon}$.
Since the endpoint of $\eta$ is visited by the walk $\langle W_n \rangle_{n\geq0}$ at a time greater than $R$, we have by the definition of the interlacement intensity measure, the union bound, and Markov's inequality that
\[
\P(\mathscr C_{r,R,\varepsilon}) \geq \varepsilon \Cap(v) e^{-\varepsilon \Cap(v)}\left[q(v)-\frac{L(r)}{R}\right],
\]
and hence, by the same reasoning as in the proof of \cref{prop:intlower}, that
\[
\P(\mathrm{diam}_\mathrm{ext}(\mathfrak P(v))\geq r)\geq
\P(\mathscr D_{r,R,\varepsilon}) \geq e^{-\varepsilon M R} \P(\mathscr C_{r,R,\varepsilon}) \geq \varepsilon \Cap(v) e^{-\varepsilon M(R+1)}\left[q(v)-\frac{L(r)}{R}\right]
\]
for every $R,r\geq 1$ and $\varepsilon >0$. We conclude by taking $\varepsilon=1/M(R+1)$ and $R= \lceil L(r)/2q(v) \rceil$.
\end{proof}
\begin{lemma}
\label{lem:polyLr}
Let $G=(V,E)$ be a network with controlled stationary measure that is $d$-Ahlfors regular for some $d>2$ and satisfies Gaussian heat kernel estimates. Then $L(r)\preceq r^2$.
\end{lemma}
\begin{proof}
Let $v$ be a vertex of $G$ and let $T_r= \sup\{n \geq 0: X_n \in B(v,r)\}$.
It follows from the definitions that
\begin{align*}
\mathbf{P}_v\bigl(X_n \in B(v,r)\bigr) \asymp \begin{cases} 1 & n\leq r^2\\
n^{-d/2}r^d &n > r^2.
\end{cases}
\end{align*}
for every $n\geq 1$ and $r\geq 1$, and hence that
\[
\mathbf E_v\left[ \sum_{m\geq n} \mathbbm{1}\bigl(X_m \in B(v,2r)\bigr) \geq n \right] \preceq r^d n^{-d/2+1}
\]
for $n\geq r^2$.
On the other hand, it follows by the strong Markov property that
\[
\mathbf E_v\left[ \sum_{m\geq n} \mathbbm{1}\bigl(X_m \in B(v,2r)\bigr) \mid T_r \geq n \right] \geq \inf_{u\in V} \mathbf{E}_u\left[ \sum_{m\geq n} \mathbbm{1}\bigl(X_m \in B(u,r)\bigr) \right] \succeq r^2,
\]
and we deduce that
\[\mathbf{P}_v(T_r \geq n) \preceq r^{d-2}n^{-d/2+1}\]
for every $n\geq r^2$. The claim follows by summing over $n$.
\end{proof}
\section{The length and capacity of the loop-erased random walk}
\label{sec:LERWCap}
In this section, we study the length and capacity of loop-erased random walk. In particular, we prove that in a network with controlled stationary measure and $\bubnorm{P}<\infty$, an $n$-step loop-erased random walk has capacity of order $n$ with high probability. The estimates we derive are used extensively throughout the remainder of the paper. In the case of $\mathbb Z^d$, these estimates are closely related to classical estimates of Lawler, see \cite{MR2985195} and references therein.
\subsection{The number of points erased}
Recall that when $X$ is a path in a network $G$, the times $\langle \ell_n(X) : 0 \leq n \leq |\mathsf{LE}(X)|\rangle$ are defined to be the times contributing to the loop-erasure of $X$, defined recursively by $\ell_0(X)=0$ and $\ell_{n+1}(X) = 1+ \max\{m : X_m = X_{\ell_n(X)}\}$. We also define
\[\rho_n(X) = \max\{ k : \ell_k \leq n\}\]
for each $0 \leq n \leq |X|$, so that $\rho_n(X)$ is the number of times between $0$ and $n$ that contribute to the loop-erasure of $X$. The purpose of this section is to study the growth of $\ell$ and $\rho$ when $X$ is a random walk on a network with controlled stationary measure satisfying $\bubnorm{P}<\infty$.
We write $X^T$ for the random walk ran up to the (possibly random) time $T$, and use similar notation for other paths such as $\mathsf{LE}(X)$.
\begin{lemma}
\label{lem:LoopLength2}
Let $G$ be a transient network, let $v$ be a vertex of $G$, and let $X$ be a random walk on $G$ started at $v$. Then the following hold.
\begin{enumerate}[leftmargin=*]
\itemsep0.5em
\item
The random variables $\langle \ell_{n+1}(X)-\ell_{n}(X) \rangle_{n\geq0}$ are independent conditional on $\mathsf{LE}(X)$, and the estimate
\begin{equation}
\label{eq:ellloops1}
\mathbf P_v\left[ \ell_{n+1}(X)-\ell_n(X) -1 =m \mid \mathsf{LE}(X) \right] \leq \pstar{m}
\end{equation}
holds for every $n\geq 0$ and $m \geq 0$.
\item
If $\bubnorm{P}<\infty$, then
\begin{equation}
\label{eq:MFCLoopLength2}
\mathbf E_v \left[ \ell_n(X) \mid \mathsf{LE}\left(X\right)
\right] \leq \bubnorm{P} \, n
\end{equation}
and
\begin{equation}
\label{eq:MFCLoopLength3}
\mathbf P_v \left[\rho_n(X) \leq \lambda^{-1} n \mid \mathsf{LE}(X) \right] \leq \bubnorm{P} \, \lambda^{-1}
\end{equation}
for every $n\geq 1$ and $\lambda > 0$.
\end{enumerate}
\end{lemma}
Note that, in the other direction, we have the trivial inequalities $\ell_n \geq n$ and $\rho_n \leq n$.
\begin{proof}
We first prove item 1. Observe that the conditional distribution of $\langle X_i \rangle_{i \geq \ell_n}$ given $X^{\ell_n}$ is equal to the distribution of a random walk started at $X_{\ell_n}$ and conditioned never to return to the set of vertices visited by $\mathsf{LE}(X)^{n-1}$. This is essentially the \textbf{Laplacian random walk} representation of loop-erased random walk, see \cite{Lawler87}.
Thus, the conditional distribution of $X$ given $\mathsf{LE}(X) = \gamma$ can be described as follows.
For each finite path $x$ in $G$, let $w(x)$ be its random walk weight
\[w(x) = \mathbf P_{x_0}( X^{|x|} = x) = \prod_{i=0}^{|x|-1} \frac{c(x_{i,i+1})}{c(x_i)}.\]
For each time $n \geq 0$, let
$L_n=L_n(\mathsf{LE}(X))$ be the set of finite loops in $G$ that start and end at $\mathsf{LE}(X)_n$ and do not hit the trace of $\mathsf{LE}(X)^{n-1}$ (which we consider to be the empty set if $n=0$). In particular, $L_n$ includes the loop of length zero at $\mathsf{LE}(X)_n$ for each $n \geq 0$. Then the random walk segments $\langle X_i \rangle_{i=\ell_n}^{\ell_{n+1}-1}$ are conditionally independent given $\mathsf{LE}(X)$, and have law given by
\begin{equation}
\label{eq:conditionalloops1}
\mathbf P_v(\langle X_i \rangle_{i=\ell_n}^{\ell_{n+1}-1} = x \mid \mathsf{LE}(X))
= \frac{w(x)}{\sum_{y \in L_n}w(y)}\mathbbm{1}(x \in L_n).
\end{equation}
The contribution of the loop of length zero ensures that the denominator in \eqref{eq:conditionalloops1} is at least one, so that
\begin{equation*}
\mathbf P_v\left(\langle X_i \rangle_{i=\ell_n}^{\ell_{n+1}-1} =x \mid \mathsf{LE}(X)\right) \leq w(x)\mathbbm{1}(x \in L_n)
\end{equation*}
and hence, summing over $x \in L_n$ of length $m$,
\begin{equation}
\label{eq:looplengthestimate}
\mathbf P_v\left(\ell_{n+1}-\ell_{n} -1 = m \mid \mathsf{LE}(X) \right) \leq p_{m}\left(\mathsf{LE}(X)_n,\mathsf{LE}(X)_n\right) \leq \pstar{m}
\end{equation}
for all $m \geq 0$, establishing item 1.
For item 2, \eqref{eq:MFCLoopLength2} follows immediately from \eqref{eq:ellloops1}. Furthermore, $\rho_n \leq \lambda^{-1} n$ if and only if $\ell_{\lfloor \lambda^{-1} n \rfloor} \geq n$, so that \eqref{eq:MFCLoopLength3} follows from \eqref{eq:MFCLoopLength2} and Markov's inequality.\qedhere
\end{proof}
We remark that \cref{lem:LoopLength2} together with the strong law of large numbers for independent, uniformly integrable random variables has the following easy corollary. Since we do not require the result for the remainder of the paper, the proof is omitted.
\begin{corollary}
\label{cor:quenchedrhogrowth}
Let $G$ be a transient network, and let $X$ be a random walk on $G$.
If $\bubnorm{P}<\infty$, then
\[
\limsup_{n\to\infty} \frac{\ell_n(X)}{n} \leq \bubnorm{P}
\]
almost surely.
\end{corollary}
The following variation of \cref{lem:LoopLength}, applying to the loop-erasure of a random walk stopped upon hitting a vertex $v$, is proved similarly.
\begin{lemma}
\label{lem:LoopLength}
Let $G$ be a network. Let $u$ and $v$ be distinct vertices of $G$, let $X$ be a random walk started at $u$, and let $\gamma$ be a simple path connecting $u$ to $v$. Then the following hold.
\begin{enumerate}[leftmargin=*]
\itemsep0.5em
\item
The random variables $\langle \ell_{n+1}(X^{\tau_v})-\ell_{n}(X^{\tau_v}) \rangle_{n=0}^{|\gamma|-1}$ are independent conditional on the event that $\tau_v<\infty$ and $\mathsf{LE}(X^{\tau_v})=\gamma$, and the estimate
\vspace{0.2em}
\begin{equation}
\vspace{0.2em}
\mathbf P_u\left(\ell_{n+1}(X^{\tau_v})-\ell_n(X^{\tau_v}) -1 =m \mid \tau_v < \infty,\, \mathsf{LE}(X^{\tau_v})=\gamma \right) \leq \pstar{m}
\end{equation}
holds for every vertex $1 \leq n \leq |\gamma|-1$ and every $m\geq 0$.
\item
If $\bubnorm{P}<\infty$, then
\vspace{0.2em}
\begin{equation}
\vspace{0.2em}
\label{eq:MFCLoopLength1}
\mathbf E_u \left[ \tau_v \mid \tau_v < \infty,\, \mathsf{LE}(X^{\tau_v})=\gamma
\right] \leq \bubnorm{P} |\gamma|.
\end{equation}
\end{enumerate}
\end{lemma}
\begin{proof}
Write $\ell_n=\ell_n(X^{\tau_v})$.
Observe that the conditional distribution of $\langle X_i \rangle_{i=\ell_n}^{\tau_v}$ given the random variable $X^{\ell_n}$ and the event $\ell_n<\tau_v<\infty$ is equal to the distribution of a simple random walk started at $X_{\ell_n}$ and conditioned to hit $v$ before hitting the set of vertices visited by $\mathsf{LE}(X)^{n-1}$. The rest of the proof is very similar to that of \cref{lem:LoopLength2}. \qedhere
\end{proof}
\subsection{The capacity of loop-erased random walk}
Given a transient path $X$ in a network, we define
$\eta_n(X) = \max\{\ell_k(X) : n \geq 0, \ell_k(X) \leq n\}$. The time $\eta_n(X)$ is defined so that $\mathsf{LE}(X^{\eta_n}) = \mathsf{LE}(X^n)^{\rho_n} = \mathsf{LE}(X)^{\rho_n}$, and in particular, every edge traversed by $\mathsf{LE}(X^{\eta_n})$ is also traversed by both $\mathsf{LE}(X)$ and $\mathsf{LE}(X^n)$.
The goal of this subsection is to prove the following estimate, which will play a fundamental role in the remainder of our analysis.
\begin{proposition}
\label{prop:capacities}
Let $G$ be a network with controlled stationary measure.
If $\bubnorm{P}<\infty$, then
\vspace{0.6em}
\begin{equation}
\vspace{0.3em}
\mathbf P_v \left( \Cap\left(\mathsf{LE}(X^n)\right) \leq \lambda^{-1} n \right) \leq
\mathbf P_v \left( \Cap\left(\mathsf{LE}(X^{\eta_n})\right) \leq \lambda^{-1} n \right) \preceq \lambda^{-1/3}
\end{equation}
and similarly
\vspace{0.4em}
\begin{equation}
\label{eq:smallcapLERWprob}
\vspace{1em}
\mathbf P_v \left( \Cap\left(\mathsf{LE}(X^{\ell_n})\right) \leq \lambda^{-1} n \right) \preceq \lambda^{-1/2}
\end{equation}
for every vertex $v$ of $G$, every $n \geq 1$, and every $\lambda \geq 1$.
\end{proposition}
We do not expect these bounds to be optimal.
Our primary means of estimating capacity will be the following lemma. Given a network $G$, we write
$|A|_c=\sum_{v\in A} c(v)$ for the total conductance of a set of vertices $A$, write
\[\mathbf{G}(u,v) = \mathbb E_u\left[ \sum_{n\geq0} \mathbbm{1}(X_n=v) \right]\]
for the Green function on $G$,
and define for each finite set of vertices $A$ of $G$ the quantity
\[\mathbf{I}(A) = \sum_{u,v\in A} c(u)c(v) \mathbf{G}(u,v)=\sum_{u\in A} c(u)c(v) \mathbf E_u\bigg[ \sum_{n \geq 0} \mathbbm{1}\left(X_n \in A\right)\bigg]. \]
Note that for any two sets of vertices $A \subseteq B$, we have $\mathbf{I}(A) \leq \mathbf{I}(B)$.
When $G$ has controlled stationary measure, the ratio $\mathbf{I}(A)/|A|_c$ is comparable to the expected number of steps a random walk spends in $A$ when started from a uniform point of $A$.
\begin{lemma}
\label{lem:maincap}
Let $G$ be a transient network. Then
\[
\Cap(A) \geq \frac{|A|_c^2}{ \mathbf{I}(A)}
\]
for every finite set of vertices $A$ in $G$.
\end{lemma}
\begin{proof} Recall the following variational formula for the capacity of a finite set $A$ \cite[Lemma 2.3]{JainOrey-properties}:
\[
\Cap(A)^{-1} = \inf\left\{ \sum_{u,v\in A} \mathbf{G}(u,v) \mu(u)\mu(v) : \mu \text{ is a probability measure on $A$} \right\}.
\]
The claim follows by taking $\mu$ to be the measure
$\mu(v)=c(v)/|A|_c$.
\end{proof}
A useful feature of \cref{lem:maincap} is that once one has an upper bound on $\mathbf{I}(A)$ for some set $A$, one also obtains lower bounds on the capacity of all \emph{subsets} of $A$ in terms of their size.
In particular, \cref{lem:maincap} yields that
\[\Cap\left( \mathsf{LE}(X^{\eta_n}) \right) \geq \left[\inf_{u \in V} c(u)\right]^2 \frac{(\rho_n+1)^2}{ \mathbf{I} ( \mathsf{LE}(X^{\eta_n}))} \geq \left[\inf_{u \in V} c(u)\right]^2 \frac{(\rho_n+1)^2}{ \mathbf{I} (X^{n})}, \]
so that to lower bound $\Cap( \mathsf{LE}(X^{\eta_n}) )$ it will suffice to lower bound $\rho_n$ and upper bound $\mathbf{I}(X^n)$. Moreover, for our purposes, it will suffice to control the expectation of $\mathbf{I}(X^n)$.
\begin{lemma}
\label{lem:Imean}
Let $G$ be a network. Then
\[
\mathbf E_v \left[ \mathbf{I}(X^n) \right]
\leq \left[ \sup_{u \in V} c(u)\right]^2 \cdot \left[ 3(n+1) \sum_{m=0}^n (m+1) \pstar{m} + 2(n+1)^2\sum_{m=n+1}^\infty \pstar{m} \right]
\]
for every $v\in V$ and $n \geq 0$.
\end{lemma}
\cref{lem:Imean} has the following immediate corollary.
\begin{corollary}
\label{cor:Imean2}
Let $G$ be a network with controlled stationary measure.
If $\bubnorm{P}<\infty$ then
\[ \mathbf E_v\left[\mathbf{I}(X^n)\right] \preceq n\]
for every vertex $v$ of $G$ and every $n \geq 1$.
\end{corollary}
Before proving \cref{lem:maincap,lem:Imean}, let us use them, together with \cref{lem:LoopLength2}, to deduce \cref{prop:capacities}.
\begin{proof}[Proof of \cref{prop:capacities} given \cref{cor:Imean2}]
It follows from \cref{lem:maincap} and a union bound that
\begin{equation}
\label{eq:Iunionbound1}
\mathbf P_v\left( \Cap( \mathsf{LE}(X^{\eta_n}) ) \leq \lambda^{-1} n \right)
\leq \mathbf P_v \left( (\rho_n+1) \leq \left[\inf_{u\in V} c(u)\right]^{-1} \lambda^{-1/3} n \right) + \mathbf P_v \left( \mathbf{I}(X^n) \geq \lambda^{1/3} n \right)
\end{equation}
and similarly
\begin{equation}
\label{eq:Iunionbound2}
\mathbf P_v\left( \Cap( \mathsf{LE}(X^{\ell_n}) ) \leq \lambda^{-1} n \right)
\leq
\mathbf P_v \left(\ell_n \geq \lambda^{1/2} n \right) + \mathbf P_v \left( \mathbf{I}\left(X^{\lfloor \lambda^{1/2} n \rfloor}\right) \geq \left[\inf_{u\in V} c(u)\right]^{-2}\lambda n \right).
\end{equation}
The claim now follows immediately from \cref{lem:LoopLength2}, \cref{cor:Imean2} and Markov's inequality.
\end{proof}
\begin{proof}[Proof of \cref{lem:Imean}]
Conditional on $X$, let $Y^i$ be a random walk started at $i$ for each $i\geq0$, writing $\P$ for the joint law of $X$ and the walks $\langle Y^i \rangle_{i \geq0}$. Then we have
\[
\mathbf E_v\left[ \mathbf{I}(X^n) \right]
\leq
\sum_{j = 0}^{n} \sum_{i=0}^n \sum_{k \geq0} \mathbb E\left[ c(X_i)c(X_j) \mathbbm{1}(Y^i_k = X_j) \right].
\]
(This is an inequality rather than an equality because the right-hand side counts vertices with multiplicity according to how often they are visited by $X$.)
We split this sum into two parts according to whether $i\leq j$ or $i > j$.
For the first sum, we have that
\begin{align*}
\sum_{i=0}^n \sum_{j = i}^{n} \sum_{k \geq0} \mathbb E\left[ c(X_i)c(X_j) \mathbbm{1}(Y^i_k = X_j) \right]
&= \sum_{i=0}^n \sum_{j = i}^{n} \sum_{k \geq0} \sum_{u,w \in V} p_i(v,u)p_{m}(u,w)c(u)p_{j-i}(u,w)c(w).
\end{align*}
Reversing time and rearranging yields that
\begin{align*}
\sum_{i=0}^n \sum_{j = i}^{n} \sum_{k \geq0} \mathbb E\left[ c(X_i)c(X_j) \mathbbm{1}(Y^i_k = X_j) \right]
&= \sum_{i=0}^n \sum_{j = i}^{n} \sum_{k \geq0} \sum_{u,w \in V} p_i(v,u)p_{k}(u,w)p_{j-i}(w,u) c(w)^2\\
&\leq \left[\sup_{u \in V} c(u)\right]^2 \sum_{i=0}^n \sum_{j = i}^{n} \sum_{k \geq0} \pstar{k+j-i}.
\end{align*}
Using the substitution $m=k+j-i$ and rearranging, we obtain that
\begin{align}
\left(\sup_{u \in V} c(u)\right)^{-2}
\sum_{i=0}^n \sum_{j = i}^{n} &\sum_{k \geq0} \mathbb E\left[ c(X_i)c(X_j) \mathbbm{1}(Y^i_k = X_j) \right]
\nonumber
\\
& \leq (n+1) \sum_{m = 0}^n \sum_{j=0}^{m}\pstar{m} + (n+1)\sum_{m=n+1}^\infty \sum_{j=0}^n \pstar{m}
\nonumber
\\
&= (n+1) \sum_{m=0}^n (m+1)\pstar{m} + (n+1)^2 \sum_{ m =n+1}^\infty \pstar{m}.
\label{eq:lem4dwalkcap1}
\end{align}
Similarly, for the second sum, we have that
\begin{align*}
\sum_{i=0}^n \sum_{j = 0}^{i-1} \sum_{k \geq0} \mathbb E\left[ c(X_i)c(X_j) \mathbbm{1}(Y^i_k = X_j) \right]
&\leq \left[\sup_{u \in V} c(u)\right]^2 \sum_{i=0}^n \sum_{j=0}^{i-1} \sum_{k \geq0} \pstar{i-j+k}
\end{align*}
so that using the change of variables $m=i-j+k$, we obtain that
\begin{multline}
\label{eq:4dwalkcap2}
\left( \sup_{u \in V} c(u) \right)^{-2} \sum_{i=0}^n \sum_{j = 0}^{i-1} \sum_{k \geq0} \mathbb E\left[ c(X_i)c(X_j) \mathbbm{1}(Y^i_k = X_j) \right]
\\\leq \sum_{m = 0}^n \sum_{i=0}^m \sum_{j=0}^i \pstar{m} + \sum_{m =0}^n \sum_{i=m}^n \sum_{j=i-m}^i \pstar{m} + \sum_{m \geq n+1}\sum_{i=0}^n \sum_{0}^n \pstar{m}\\
\leq 2(n+1)\sum_{m=0}^n(m+1)\pstar{m} + (n+1)^2\sum_{m \geq n+1} \pstar{m}.
\end{multline}
Adding \eqref{eq:lem4dwalkcap1} and \eqref{eq:4dwalkcap2} yields the claim.
\end{proof}
\section{Volume bounds}
\label{sec:volume}
In this section, we study the volume of balls in both the WUSF and $v$-WUSF. In \cref{subsec:volupper} we prove upper bounds on the moments of the volumes of balls, while in \cref{subsec:volume_lower} we prove lower bounds on moments and upper bounds on the probability that the volume is atypically small. Together, these estimates will imply that $d_f(T)=2$ for every component $T$ of $\mathfrak F$ almost surely. The estimates in this section will also be important in \cref{sec:exponents,sec:AlexanderOrbach}.
\subsection{Upper bounds}
\label{subsec:volupper}
The goal of this subsection is to obtain tail bounds on the probability that an intrinsic ball in the WUSF contains more than $n^2$ vertices.
The upper bounds we obtain are summarized by the following two propositions, which are generalisations of \cite[Theorem 4.1]{barlow2016geometry}.
Recall that $\mathfrak{B}(v,n)$ denotes the intrinsic ball of radius $n$ around $v$ in the WUSF $\mathfrak F$, and $\mathfrak{B}_v(v,n)$ denotes the intrinsic ball of radius $n$ around $v$ in the $v$-WUSF $\mathfrak F_v$.
We define the constant
\begin{equation}
\label{eq:alphadefn}
\alpha=\alpha(G) = 4 \frac{\sup_{u \in V} c(u)}{\inf_{u \in V} c(u)} \bubnorm{P}.
\end{equation}
\begin{proposition}
\label{thm:moments}
Let $G$ be a network with controlled stationary measure such that $\bubnorm{P}<\infty$, and let
$\mathfrak F$ be a wired uniform spanning forest of $G$.
Then the estimates
\begin{align}
\label{eq:thmmomentsunrooted}
\vspace{0.1em}
\mathbb E \left[|\mathfrak{B}(v,n)|^k \right] \leq e\, k\, k!\, \alpha^k \, (n+1)^{2k}
\end{align}
and
\vspace{0.3em}
\begin{align}
\label{eq:thmmomentsunrootedexp}
\mathbb E\left[\exp\left( \frac{t}{\alpha(n+1)^2} |\mathfrak{B}(v,n)| \right)\right] \leq \frac{1}{1-t}
\vspace{1em}
\end{align}
hold for every $v\in V, n\geq 0$, and $0 \leq t <1 $.
\end{proposition}
We also obtain the following variation of this proposition applying to the $v$-WUSF.
\begin{proposition}
\label{thm:vmoments}
Let $G$ be a network with controlled stationary measure such that $\bubnorm{P}<\infty$, let $v$ be a vertex of $G$ and let
$\mathfrak F_v$ be a $v$-wired uniform spanning forest of $G$.
Then the estimates
\begin{align}
\label{eq:thmmomentsrooted}
\vspace{0.1em}
\mathbb E \left[|\mathfrak{B}_v(v,n)|^k \right] \leq (k-1)!\, \alpha^k \, (n+1)^{2k-1}
\end{align}
and
\vspace{0.1em}
\begin{align}
\label{eq:thmmomentsrootedexp}
\vspace{0.1em}
\mathbb E \left[\exp\left( \frac{t}{\alpha(n+1)^2} |\mathfrak{B}_v(v,n)| \right)\right] \leq 1 - \frac{\log(1-t)}{n+1}
\end{align}
hold for every $v\in V, n\geq 0$, and $0 \leq t <1 $.
\end{proposition}
(In fact, to prove our main theorems it suffices to have just the first and second moment bounds of \cref{thm:moments,thm:vmoments}. We include the exponential moment bounds for future application since they are not much more work to derive.)
Before proving
\cref{thm:moments,thm:vmoments}, we note the following important corollaries.
\begin{corollary}
\label{cor:exponentialproblargevolume}
Let $G$ be a network with controlled stationary measure and with $\bubnorm{P}<\infty$, and let $\mathfrak F$ be a wired uniform spanning forest of $G$. Then
\vspace{0.3em}
\[
\vspace{0.3em}
\P\left(|\mathfrak{B}(v,n)| \geq \lambda \alpha (n+1)^2\right) \leq \lambda e^{-\lambda +1}
\]
for all $v\in V$, $n\geq0$, and $\lambda \geq 1$.
\end{corollary}
\begin{proof}
By Markov's inequality, we have that
\begin{align*}
\P\left(|\mathfrak{B}(v,n)| \geq \lambda \alpha (n+1)^2 \right)
\leq e^{-t\lambda} \mathbb E\left[\exp\left( \frac{t}{\alpha(n+1)^2}|\mathfrak{B}(v,n)| \right)\right]
\leq \frac{e^{-t\lambda}}{1-t}
\end{align*}
for every $v\in V$, $n\geq 0$, and $0 \leq t <1$. The claim follows by taking $t=1-\lambda^{-1}$.
\end{proof}
\begin{corollary}
\label{cor:quenchedvolumeupper}
Let $G$ be a network with controlled stationary measure and with $\bubnorm{P}<\infty$, and let $\mathfrak F$ be a wired uniform spanning forest of $G$. Then
\[
\vspace{0.3em}
\limsup_{n\to\infty} \frac{|\mathfrak{B}(v,n)|}{n^2 \, \log \log n}
\leq \alpha
\]
almost surely for every vertex $v$ of $G$.
\end{corollary}
\begin{proof}
Let $a>1$ and let $n_k= \lfloor a^k \rfloor$. Then for every $\varepsilon>0$, $v\in V$ and all $k$ sufficiently large we have that, by \cref{cor:quenchedvolumeupper},
\[
\P\left(|\mathfrak{B}(v,n_k)| \geq (1+\varepsilon) \alpha (n_k+1)^2 \log\log n_k \right) \leq \frac{e(1+\varepsilon)\log (k+\log a) }{(k+\log a)^{1+\varepsilon}}.
\]
The right hand side is summable in $k$ whenever $\varepsilon>0$, and it follows by Borel-Cantelli that
\[
\limsup_{k \to\infty} \frac{|\mathfrak{B}(v,n_k)|}{\alpha(n_k+1)^2 \log\log n_k}
\leq 1
\]
almost surely.
Since $|\mathfrak{B}(v,n)|$ is increasing and for every $n$ there exists $k$ such that $n_k \leq n \leq a n_k$, it follows that
\[
\limsup_{k\to\infty} \frac{|\mathfrak{B}(v,n)|}{\alpha(n+1)^2\log \log n}
\leq a^2 \limsup_{k \to\infty} \frac{|\mathfrak{B}(v,n_k)|}{\alpha(n_k+1)^2 \log\log n_k}
\leq a^2
\]
almost surely, and the claim follows since $a>1$ was arbitrary.
\end{proof}
\begin{remark}
\cite[Proposition 2.8]{BarKum06} shows that \cref{cor:quenchedvolumeupper} is sharp in the sense that, when $G$ is a $3$-regular tree, $\log \log n$ cannot be replaced with $(\log \log n)^{1-\varepsilon}$ for any $\varepsilon>0$.
\end{remark}
\medskip
We now begin working towards the proof of \cref{thm:moments,thm:vmoments}.
We begin with a first moment estimate.
\begin{lemma}
\label{lem:expectedballgrowth1}
Let $G$ be a network with controlled stationary measure and with $\bubnorm{P}<\infty$. Then
\[
\mathbb E |\mathfrak{B}_v(v,n)| \leq \alpha (n+1)
\]
for every $v\in V$ and $n\geq 0$.
\end{lemma}
\begin{proof}
Let $u\in V$, and consider sampling the $v$-rooted uniform spanning forest $\mathfrak F_v$ using Wilson's algorithm rooted at $v$, starting with a random walk $X$ with $X_0=u$. Then $u \in \partial \mathfrak{B}(v,n)$ if and only if the random walk started at $u$ hits $v$ and the loop-erasure of the random walk path stopped when it first hits $v$ has length at most $n$. Denote this event $\mathscr A_n(u,v)$, so that
\[
\mathbb E|\mathfrak{B}_v(v,n)| = \sum_{u \in V} \mathbf P_u(\mathscr A_n(u,v)).
\]
If $\bubnorm{P}<\infty$, then for every two vertices $u$ and $v$ in $G$ and every simple path $\gamma$ from $u$ to $v$ we have that, by the estimate \eqref{eq:MFCLoopLength1} of \cref{lem:LoopLength} and Markov's inequality,
\[
\mathbf P_u\left( \tau_v \geq 2 \bubnorm{P} \cdot |\gamma| \; \mid \; \tau_v < \infty,\, \mathsf{LE}(X^{\tau_v})=\gamma\right) \leq \frac{1}{2}.
\]
Taking expectations over $\mathsf{LE}(X)$ conditional on the event $\mathscr A_n(u,v)$ yields that
\[
\mathbf P_u\left(\tau_v \leq 2 \bubnorm{P} \, n \mid \mathscr A_n(u,v)\right) \geq 1/2
\]
and hence, by Bayes' rule
\[
\mathbf P_u\left(\mathscr A_n(u,v)\right) \leq 2 \mathbf P_u\left(\tau_v \leq 2 \bubnorm{P} \, n\right) \leq 2\sum_{k=0}^{\lfloor 2\bubnorm{P} \, n \rfloor}p_k(u,v).
\]
Reversing time we have
\[
\mathbf P_u(\mathscr A_n(u,v)) \leq \frac{2 c(v)}{c(u)} \sum_{k=0}^{\lfloor 2\bubnorm{P} \, n \rfloor}p_k(v,u),
\]
from which the claim may be immediately be derived by summing over $u\in V$.
\end{proof}
We next use an inductive argument to control the higher moments of $|\mathfrak{B}_v(v,n)|$.
\begin{lemma}
\label{lem:ball2ndmoment}
Let $G$ be a network. Then
\[
\vspace{0.4em}
\sup_{v \in V} \mathbb E \Bigl[ |\mathfrak{B}_v(v,n)|^k \Bigr]
\leq (k-1)!\, (n+1)^{k-1} \, \sup_{v \in V} \mathbb E \Bigl[ |\mathfrak{B}_v(v,n)| \Bigr]^k
\]
for every $n\geq 0$ and $k\geq 1$.
\end{lemma}
\begin{proof}
By induction, it suffices to prove that the inequality
\vspace{0.2em}
\begin{equation}
\vspace{0.2em}
\label{eq:momentinduction}
\mathbb E \left[|\mathfrak{B}_v(v,n)|^k\right] \leq (k-1)(n+1) \, \mathbb E \left[|\mathfrak{B}_v(v,n)|^{k-1}\right] \, \sup_{w \in V}\mathbb E \left[|\mathfrak{B}_w(w,n)|\right]
\end{equation}
holds for every vertex $v$ of $G$ and every $k \geq 1$.
To this end, let $v$ be a vertex of $G$ and let $u_1,\ldots,u_{k-1}$ be vertices of $G$ such that $\P(u_1,\ldots,u_{k-1} \in \mathfrak{B}_v(v,n))>0$.
It follows from Wilson's algorithm that, conditional on the event that $u_1,\ldots,u_{k-1} \in \mathfrak{B}_v(v,n)$ and on the paths $\Gamma_v(u_1,v),\ldots,\Gamma_v(u_{k-1},v)$ connecting each of the vertices $u_i$ to $v$ in $\mathfrak F_v$, the probability that a vertex $w$ is contained in $\mathfrak{B}_v(v,n)$ is at most the probability that a random walk started at $w$ hits one of the paths $\Gamma_v(u_i,v)$ and that the loop-erasure of this stopped path has length at most $n$. Thus, we obtain
\begin{multline*}
\P\left(u_k \in |\mathfrak{B}_v(v,n)| \mid u_1,\ldots,u_{k-1} \in \mathfrak{B}_v(v,n),\, \langle \Gamma_v(u_i,v)\rangle_{i=1}^{k-1}\right) \\
\leq \sum_{i=1}^{k-1} \sum_{u \in \Gamma_v(u_i,v)} \mathbf P_{u_k}(\tau_u < \infty,\; |\mathsf{LE}(X^{\tau_u})| \leq n)
= \sum_{i=1}^{k-1} \sum_{u \in \Gamma_v(u_i,v)} \P(u_k \in \mathfrak{B}_u(u,n))
\end{multline*}
and hence, summing over $u_k$ and taking expectations over $\langle \Gamma(u_i,v)\rangle_{i=1}^{k-1}$ we obtain that
\begin{equation*}
\mathbb E\left[|\mathfrak{B}_v(v,n)| \mid u_1,\ldots,u_{k-1} \in \mathfrak{B}_v(v,n)\right] \\
\leq (k-1)(n+1) \sup_{u \in V}\mathbb E \left[|\mathfrak{B}_u(u,n)|\right].
\end{equation*}
(This inequality can also be deduced using \cref{lem:domination}.) The inequality \eqref{eq:momentinduction} now follows from this together with the inequality
\begin{multline*}
\mathbb E \left[|\mathfrak{B}_v(v,n)|^k\right] \\
= \sum_{u_1,\ldots,u_k \in V} \P\bigl(u_k \in \mathfrak{B}_v(v,n) \mid u_1,\ldots,u_k \in \mathfrak{B}_v(v,n)\bigr) \P\bigl(u_1,\ldots,u_{k-1} \in \mathfrak{B}_v(v,n)\bigr) \\
\leq \mathbb E \left[|\mathfrak{B}_v(v,n)|^{k-1}\right] \sup_{u_1,\ldots,u_{k-1}} \mathbb E \Bigl[|\mathfrak{B}_v(v,n)| \mid u_1,\ldots,u_{k-1} \in \mathfrak{B}_v(v,n)\Bigr],
\end{multline*}
where the supremum is taken over all collections of vertices $u_1,\ldots,u_{k-1} \in V$ such that the probability $\P(u_1,\ldots,u_{k-1} \in \mathfrak{B}_v(v,n))$ is positive. \qedhere
\end{proof}
Next, we control the moments of the volume of balls in the WUSF in terms of the moments in the $v$-WUSF.
\begin{lemma}
\label{lem:unrootedmoments}
Let $G$ be a transient network. Then
\begin{equation}
\label{eq:unrootedmoment}
\sup_{v\in V} \mathbb E\left[|\mathfrak{B}(v,n)|^k\right] \leq
\sum_{\substack{k_0,\,\ldots,\,k_n \geq 0:\\ k_0+ \cdots + k_n=k}} \, \prod_{i=0}^n\; \sup_{v\in V}\mathbb E\left[|\mathfrak{B}_v(v,n)|^{k_i}\right]
\end{equation}
for every $n\geq 0$ and $k\geq 1$.
\end{lemma}
\begin{proof}
Let $v \in V$ and let $\Gamma(v,\infty)$ be the future of $v$ in $\mathfrak F$.
Let $v=u_0,\ldots,u_n$ be the first $n+1$ vertices in the path $\Gamma(v,\infty)$.
For each $0\leq i \leq n$, let $W_i=\{w_{i,1},\ldots,w_{i,m_i}\}$ be a finite (possibly empty) collection of vertices of $G$, and let $\mathscr A_i$ be the event that for every vertex $w \in W_i$, $w$ is in $\mathfrak{B}(v,n)$ and that the path connecting $w$ to $v$ first meets $\Gamma(v,\infty)$ at $u_i$.
Let $\{ X^{i,j} : 0 \leq i \leq n, m_i\neq 0, 1\leq j \leq n \}$ be a collection of independent random walks, independent of $\Gamma(v,\infty)$, such that $X^{i,j}_0=w_{i,j}$ for each $0 \leq i \leq n$ such that $m_i \neq 0$ and $1 \leq j \leq m_i$. For each $0\leq i \leq n$ such that $m_i \neq 0,$ let $\mathscr B_i$ be the event that, if we sample $\mathfrak F_{u_i}$ using Wilson's algorithm, starting with the random walks $X^{i,1},\ldots,X^{i,m_i}$, then every vertex in $W_i$ is connected to $u_i$ in $\mathfrak F_{u_i}$. Observe that if we sample $\mathfrak F$ conditional on $\Gamma(v,\infty)$ using Wilson's algorithm starting with $X^{0,1},\ldots,X^{0,m_0}$, then $X^{1,1},\ldots,X^{1,m_1}$, and so on, then we have the containment $\mathscr A_i \subseteq \mathscr B_i$. We deduce that
\[\P\big(\cap_{i=0}^n\mathscr A_i \mid \Gamma(v,\infty)\big) \leq \P\big(\cap_{i=1}^n \mathscr B_i \mid \Gamma(v,\infty) \big) = \prod_{i=0}^n \P(\mathscr B_i \mid \Gamma(v,\infty)).\]
Summing over the possible choices of the sets $W_i$ such that $\sum_{i=0}^n |W_i| =k$, we obtain that
\[\mathbb E \left[|\mathfrak{B}(v,n)|^k \mid \Gamma(v,\infty) \right] \leq \sum_{\substack{k_0,\,\ldots,\,k_n \geq 0:\\ k_0+ \cdots + k_n=k}} \prod_{i=0}^n \mathbb E_{u_i} \left[|\mathfrak{B}(u_i,n)|^{k_i}\right],\]
and the claim follows. \qedhere
\end{proof}
We now prove \cref{thm:moments}.
\begin{proof}[Proof of \cref{thm:moments,thm:vmoments}]
The moment estimate \eqref{eq:thmmomentsrooted} follows immediately from \cref{lem:expectedballgrowth1,lem:ball2ndmoment}.
In order to prove the moment generating function estimates \eqref{eq:thmmomentsrootedexp} and \eqref{eq:thmmomentsunrootedexp}, define
\begin{align*}
\Phi(n,t) &= \sum_{k = 0}^{\infty} \frac{1}{k!} \left(\frac{t}{\alpha(n+1)^2}\right)^k \sup_{v\in V} \mathbb E \left|\mathfrak{B}(v,n)\right|^k
\intertext{and}
\Psi(n,t) &= \sum_{k = 0}^{\infty} \frac{1}{k!}\left(\frac{t}{\alpha(n+1)^2}\right)^k \sup_{v\in V} \mathbb E \left|\mathfrak{B}_v(v,n)\right|^k.
\end{align*}
The moment estimate \eqref{eq:thmmomentsrooted} implies that
\[
\Psi(n,t) \leq 1+ \frac{1}{n+1} \sum_{n=k}^\infty \frac{t^k}{k} = 1 - \frac{\log(1-t)}{n+1}
\]
for every $n \geq 0$ and $t\in [0,1)$. The moment generating function estimate
\eqref{eq:thmmomentsrootedexp} follows immediately.
Next, \cref{lem:unrootedmoments} implies that
\[
\Phi(n,t) \leq \sum_{k=0}^\infty \frac{1}{k!} \left(\frac{t}{\alpha(n+1)^2}\right)^k \sum_{\substack{k_0,\,\ldots,\,k_n \geq 0:\\ k_0+ \cdots + k_n=k}} \, \prod_{i=0}^n\; \sup_{v\in V}\mathbb E \left[ |\mathfrak{B}_v(v,n)|^{k_i} \right].
\]
On the other hand, we have that
\[
\Psi(n,t)^{n+1}
= \sum_{k=0}^\infty \sum_{\substack{ k_0,\ldots,k_n \geq 0 \\ k_0+ \cdots + k_n=k}} \prod_{i=0}^n \frac{1}{k_i!}\left(\frac{t}{\alpha(n+1)^2}\right)^{k_i} \sup_{v\in V}\mathbb E\left[\left| \mathfrak{B}_v(v,n)\right|^{k_i} \right],
\]
and since $\prod_{i=0}^n(k_i)! \leq k!$ whenever $k_0,\ldots,k_n$ are non-negative integers summing to $k$, we deduce that
\[
\Phi(n,t) \leq \Psi(n,t)^{n+1}
\]
for every $n \geq 0$ and $t\geq 0$. Thus, it follows that
\[
\Phi(n,t) \leq \left( 1 - \frac{\log(1-t)}{n+1}\right)^{n+1} \leq \frac{1}{1-t}
\]
for every $n \geq 0$ and $t\in [0,1)$, where the final inequality is a simple calculus exercise.
This yields the moment generating function estimate \eqref{eq:thmmomentsunrootedexp}.
Finally, to deduce \eqref{eq:thmmomentsunrooted}, we use the fact that, since $|\mathfrak{B}(v,n)|$ is non-negative,
\[
\mathbb E\left[ \exp\left(\frac{t}{\alpha(n+1)^2} |\mathfrak{B}(v,n)| \right)\right] \geq \frac{t^k \, \mathbb E\left[|\mathfrak{B}(v,n)|^k\right]}{\alpha^k (n+1)^{2k}}
\]
and hence
\[
\mathbb E\left[|\mathfrak{B}(v,n)|^k\right] \leq \frac{k!\alpha^k (n+1)^{2k}}{t^k(1-t)}
\]
for all $k\geq 1$ and $t\in [0,1)$. Optimizing by taking $t=k/(k+1)$ yields \eqref{eq:thmmomentsunrooted}. \qedhere
\end{proof}
\subsection{Lower bounds}
\label{subsec:volume_lower}
In this section, we give lower bounds on the first moment of the volume of the past, and derive upper bounds on the probability that the volume of an intrinsic ball is atypically small.
We begin with the following simple lower bounds on the first moments.
We write $\mathfrak P(v,n)$ for the ball of radius $n$ around $v$ in the past of $v$ in $\mathfrak F$, and write $\partial \mathfrak P(v,n)$ for the set of points that are in the past of $v$ in $\mathfrak F$ and have intrinsic distance exactly $n$ from $v$.
\begin{lemma}
\label{lem:expectedballgrowth2}
Let $G$ be a transient network, and let $\mathfrak F$ be the wired uniform spanning forest of $G$. Then
\[\mathbb E |\partial \mathfrak P(v,n)| \geq \frac{q(v)c(v)}{\sup_{u\in V}c(u)} \]
for every $v\in V$ and $n\geq 0$. Similarly, if $\mathfrak F_v$ is the $v$-wired uniform spanning forest of $G$ then
\[
\mathbb E|\partial \mathfrak P_v(v,n)| \geq \frac{c(v)}{\sup_{u\in V}c(u)}.
\]
\end{lemma}
\begin{proof}
We prove the first claim, the proof of the second being similar.
Let $u\in V$, and let $X$ be a random walk started at $u$. Consider sampling $\mathfrak F$ using Wilson's algorithm, starting with the walk $X$.
Let $\mathscr A_n(u,v)$ be the event that $X$ hits $v$, that the sets $\{ X_m : 0\leq m < \tau_v \}$ and $\{ X_m : m \geq \tau_v \}$ are disjoint, and that $|\mathsf{LE}(X^{\tau_v})|=n$, so that $u\in \partial \mathfrak P(v,n)$ on the event $\mathscr A_n(u,v)$ and hence
\[
\mathbb E\Bigl[\left|\partial \mathfrak P(v,n)\right|\Bigr] \geq \sum_{u\in V}\P\left(\mathscr A_n(u,v)\right).
\]
Let $Y$ be an independent random walk started at $v$. By time-reversal, we have that
\[\mathbf P_u(\mathscr A_n(u,v)) = \frac{c(v)}{c(u)}\mathbf E_v\left[\#\{k: X_k=u,\, |\mathsf{LE}(X^k)| = n\} \mathbbm{1}(\{X_i : i > 0 \} \cap \{Y_i : i \geq 0\} =\emptyset)\right]
\]
and hence, summing over $u \in V$,
\begin{align*}
\mathbb E |\partial \mathfrak{B}(v,n)| &\geq \frac{c(v)q(v)}{\sup_{u\in V} c(u)}
\mathbf{E}_v\Bigl[\#\{k: |\mathsf{LE}(X^k)| = n\} \mid \{X_i : i > 0 \} \cap \{Y_i : i \geq 0\} =\emptyset\Bigr]
\\
&\geq \frac{c(v)q(v)}{\sup_{u\in V} c(u)}
\end{align*}
as claimed, where the second inequality follows since there must be at least one time $k$ such that $|\mathsf{LE}(X^k)|=n$, namely the time $\ell_n$.
\end{proof}
Our next goal is to prove the following lemma.
\begin{lemma}
\label{lem:fullvolumelower}
Let $G$ be a network with controlled stationary measure and $\bubnorm{P}<\infty$, and let $\mathfrak F$ be the wired uniform spanning forest of $G$.
Then
\[ \P(|\mathfrak{B}(v,n)| \leq \lambda^{-1}n^2) \preceq \lambda^{-1/8}\]
for every vertex $v$, every $n\geq 0$ and every $\lambda \geq 1$.
\end{lemma}
\cref{lem:fullvolumelower} has the following immediate corollary, which together with \cref{cor:quenchedvolumeupper} establishes that $d_f(T)=2$ for every component of $\mathfrak F$ almost surely.
\begin{corollary}
\label{cor:quenchedvolumelowerbound}
Let $G$ be a network with controlled stationary measure with $\bubnorm{P}<\infty$, and let $\mathfrak F$ be the wired uniform spanning forest of $G$. Then
\[\liminf_{n\to\infty} \frac{\log^{8+\varepsilon} n}{n^2} |\mathfrak{B}(v,n)| > 0 \qquad \]
almost surely for every $v\in V$ and $\varepsilon>0$.
\end{corollary}
We remark that Barlow and J\'arai \cite{barlow2016geometry} established much stronger versions of \cref{lem:fullvolumelower,cor:quenchedvolumelowerbound} in the case of $\mathbb Z^d$, $d\geq 5$.
In order to prove \cref{lem:fullvolumelower}, we first show that the volume of the tree can be lower bounded with high probability in terms of quantities related to the capacity of the spine, and then show that these quantities are large with high probability.
Given two sets of vertices $A \subseteq B$ in a transient graph and $k \in [0,\infty]$, we define $\Cap_k(A,B)$ to be
\[\Cap_k(A,B):=\sum_{v\in A} c(v) \mathbf P_v(\tau^+_B \geq k),\]
so that $\Cap(A)=\Cap_\infty(A,A)$.
\begin{lemma}
\label{lem:volumelowerboundquant}
Let $G$ be a network with controlled stationary measure and with $\bubnorm{P}<\infty$. Let $v$ be a vertex of $G$ and let $\Gamma=\Gamma(v,\infty)$ be the future of $v$ in $\mathfrak F$.
\begin{enumerate}[leftmargin=*]
\itemsep1em
\item
The estimate
\begin{equation}
\label{eq:expectedvolumelowerboundfulltree}
\mathbb E\Big[|\mathfrak{B}(v,2n)| \mid \Gamma\Big] \succeq \, k \, \Cap_k\bigl(\Gamma^{n},\Gamma\bigr)
\end{equation}
holds for every $n\geq 0$ and $ 0 \leq k \leq n$.
\item
If $\bubnorm{P}<\infty$, then
\begin{equation}
\label{eq:smallvolumeprobability}
\P\left(|\mathfrak{B}(v,2n)| \leq \frac{m \Cap_k(\Gamma^{n},\Gamma)}{2 \sup_{u\in V} c(u)} \; \mid \; \Gamma \right)
\preceq \frac{m(n+1)^2}{\Cap_k(\Gamma^{n},\Gamma)^2}
\end{equation}
for every $1\leq m\leq k \leq n$.
\end{enumerate}
\end{lemma}
\begin{proof}
Let $v=v_0,v_1,\ldots$ be the vertices visited by the path $\Gamma$.
Let $\mathfrak T_i(m)$ be the set of vertices that are connected to $v$ in $\mathfrak F$ by a path that first meets $\Gamma$ at $v_i$, and such that the path connecting them to $\Gamma$ has length at most $m$. Clearly
\[|\mathfrak{B}(v,2n)| \geq \sum_{i=0}^{n} |\mathfrak T_i(m)|\]
for every $n\geq 0$ and every $0\leq m \leq n$.
We claim that
\begin{equation}
\label{eq:Tbetamean}
\mathbb E \left[ \sum_{i=1}^{n} |\mathfrak T_i(m)| \; \mid \; \Gamma \right] \succeq m \Cap_k(\Gamma^{n},\Gamma)
\end{equation}
for all $v\in V$, and $0\leq m \leq k \leq n$.
Indeed, let $u$ be a vertex of $G$, and suppose that we sample $\mathfrak F$ using Wilson's algorithm, starting with the vertices $v$ and $u$. For $u$ to be included in $\mathfrak T_i(m)$, it suffices for the random walk started at $u$ to hit $\Gamma$ for the first time at $v_i$, and to do so within time $m$. By a time-reversal argument similar to that used in the proof of \cref{lem:expectedballgrowth1}, it follows that
\vspace{0.4em}
\begin{align*}
\vspace{0.4em}
\mathbb E\left[ \sum_{i=1}^{n} |\mathfrak T_i(m)| \; \mid \; \Gamma \right] \geq \frac{ m \sum_{i=0}^{n} c(v_i)\mathbf P_{v_i}(\tau_{\Gamma}^+ > m)}{\sup_{u\in V} c(u)} = \frac{m \Cap_k(\Gamma^{n},\Gamma)}{\sup_{u\in V} c(u)}
\end{align*}
as claimed. The estimate \eqref{eq:smallvolumeprobability} follows immediately from
\eqref{eq:Tbetamean} by taking $m=k$.
On the other hand, for every $u_1,u_2 \in V$ and $0 \leq i < j \leq n$ we have that
\[\P\left( u_1 \in \mathfrak T_i(m) \mid \Gamma,\, u_2 \in \mathfrak T_j(m)\right) \leq \P\left( u_1 \in \mathfrak T_i(m) \mid \Gamma\right), \]
since the probability that $u_1 \in \mathfrak T_i(m)$ given $\Gamma$ is equal to the probability that a random walk started at $u_1$ hits $\Gamma$ for the first time at $v_i$, and the loop-erasure of the walk stopped at this time has length at most $m$, while when we also condition on $u_2 \in \mathfrak T_j$ the walk must also avoid the path connecting $u_2$ to $\Gamma$.
Thus, we have that
\[\mathbb E\left[|\mathfrak T_i(m)| \cdot |\mathfrak T_j(m)| \mid \Gamma\right] \leq \mathbb E\left[|\mathfrak T_i(m)| \mid \Gamma\right] \mathbb E\left[|\mathfrak T_j(m)| \mid \Gamma\right]\]
for every $0\leq i < j \leq n$, and hence that
\[
\operatorname{Var}\left( \sum_{i=0}^{n} |\mathfrak T_i(m)| \mid \Gamma\right) \leq \sum_{i=0}^{n}\mathbb E\left[ |\mathfrak T_i(m)|^2\right].
\]
If $\bubnorm{P}<\infty$, we deduce from stochastic domination (\cref{lem:domination}) and \cref{thm:moments} that
\begin{align*}
\operatorname{Var}\left( \sum_{i=0}^{n} |\mathfrak T_i(m)| \mid \Gamma\right)
\preceq (m+1)^3 (n+1),
\end{align*}
and
applying Chebyshev's inequality yields that
\[
\P\left(\sum_{i=0}^{n} |\mathfrak T_i(m)| \leq \frac{m \Cap_k(\Gamma^{n},\Gamma)}{2 \sup_{u\in V} c(u)} \mid \Gamma\right)
\preceq \frac{m(n+1)^2}{\Cap_k(\Gamma^{n},\Gamma)^2}.
\]
for all $1\leq m\leq k \leq n$ as claimed.
\qedhere
\end{proof}
\begin{lemma}
\label{lem:relativecapLERW}
Let $G$ be a network with controlled stationary measure and with $\bubnorm{P}<\infty$. Let $v$ be a vertex of $G$, let $X$ be a random walk started at $v$ and let $\Gamma=\mathsf{LE}(X)$. Then for every $0 \leq k \leq n$ we have that
\begin{equation}
\label{eq:relativecapexpectation}
\mathbb E\Bigl[\Cap\bigl(\Gamma^{n}\bigr)-\Cap_k\bigl(\Gamma^{n},\Gamma\bigr)\Bigr] \preceq k^{1/2}n^{1/2}
\end{equation}
and for every $0 \leq k \leq n$ and $\lambda \geq 1$ we have that
\begin{equation}
\label{eq:relativecapprob}
\P\left(\Cap_k\bigl(\Gamma^{n},\Gamma\bigr) \leq \lambda^{-1} n\right) \preceq \lambda^{-1/2}+\lambda k^{1/2} n^{-1/2}.
\end{equation}
\end{lemma}
\begin{proof}
Let $m\leq n$. Considering the contribution to $\Cap\bigl(\Gamma^n\bigr)-\Cap_k\bigl(\Gamma^{n},\Gamma)$ of the last $m$ steps and the first $n-m$ steps of $\Gamma^n$ separately, we obtain the bound
\[\Cap\bigl(\Gamma^n\bigr)-\Cap_k\bigl(\Gamma^{n},\Gamma) \leq \left(\sup_{u\in V} c(u)\right)\left[m + \sum_{0\leq i \leq n-m} \sum_{j > n} \sum_{\ell=0}^k p_\ell(\Gamma_i,\Gamma_j)\right],
\]
and taking expectations we obtain that
\begin{multline*}\mathbb E\left[\Cap(\Gamma^n)-\Cap_k(\Gamma^{n},\Gamma)\right] \\\preceq m +
\sum_{u_1,u_2} \P\Bigl(u_1 \in \{\Gamma_i : 0\leq i \leq m\}, u_2 \in \{\Gamma_j : j \geq n+1\} \Bigr) \sum_{\ell=0}^k p_\ell(u_1,u_2).
\end{multline*}
Applying \cref{lem:LoopLength}, we obtain that $\tau_{u_1}\leq 2\bubnorm{P} (n-m) \leq 2 \bubnorm{P} n$ with probability at least $1/2$ on the event that $u_1 \in \{\Gamma_i : 0\leq i \leq n-m\}$ and $u_2 \in \{\Gamma_j : j \geq n+1\}$. Moreover, on this event we must have that $u_2$ is visited (not necessarily for the first time) by $X$ some time at least $m$ steps after $\tau_{u_1}$.
Thus, we obtain that
\begin{multline*}\mathbb E\left[\Cap(\Gamma^n)-\Cap_k(\Gamma^{n},\Gamma)\right] \\\preceq m +
\sum_{u_1\in V} \P\Bigl(\tau_{u_1} \leq 2 \bubnorm{P}n \Bigr) \sum_{u_2\in V}\sum_{r\geq m} \sum_{\ell=0}^k p_r(u_1,u_2) p_\ell(u_1,u_2)\\
\preceq m +
k n \sum_{r\geq m} \|P^r\|_{1\to\infty} \preceq m + knm^{-1}
\end{multline*}
Taking $m=\lceil k^{1/2}n^{1/2} \rceil$ concludes the proof of \eqref{eq:relativecapexpectation}.
To obtain \eqref{eq:relativecapprob}, simply take a union bound and apply \eqref{eq:relativecapexpectation} and \eqref{eq:smallcapLERWprob} to get that
\begin{align*}\P\left(\Cap_k\bigl(\Gamma^{n},\Gamma\bigr) \leq \lambda^{-1} n \right)&\leq \P\left(\Cap(\Gamma^n) \leq 2 \lambda^{-1} n \right) + \P\left(\Cap(\Gamma^n)-\Cap_k(\Gamma^n,\Gamma) \geq \lambda^{-1} n \right)
\\
&\preceq \lambda^{-1/2} + \lambda k^{1/2} n^{-1/2}. \qedhere
\end{align*}
\end{proof}
\begin{proof}[Proof of \cref{lem:fullvolumelower}]
Take $k=\lfloor \lambda ^{-3/4} n \rfloor$.
Then it follows from \cref{lem:volumelowerboundquant} and \cref{lem:relativecapLERW} that if $\lambda$ is sufficiently large then
\begin{multline*}
\P\Bigl(|\mathfrak{B}(v,2n)|\leq \lambda^{-1} n^2 \Bigr) \\\leq \P\left(\Cap_k(\Gamma^{n},\Gamma) \leq \frac{2n^2 \sup_{u\in V}c(u)}{\lambda k} \right) + \P\left(|\mathfrak{B}(v,n)|\leq \frac{k \Cap_k(\Gamma^{n},\Gamma)}{2\sup_{u\in V}c(u)} \mid \Cap_k(\Gamma^{n/2},\Gamma) \geq \frac{n^2}{\lambda k}\right)\\
\preceq
\lambda^{-1/2}k^{-1/2}n^{1/2} + \lambda k^2 n^{-2} + \lambda^2 k^3 n^{-2} \preceq \lambda^{-1/8}
\end{multline*}
as claimed.
\end{proof}
\section{Critical exponents}
\label{sec:exponents}
In this section we apply the estimates obtained in \cref{sec:LERWCap,sec:volume} to complete the proofs of \cref{thm:transitivemain,thm:generalexponents,thm:extrinsicZd,thm:extrinsic,thm:extrinsicspeed}.
We will also prove the following extensions of these theorems to the $v$-wired case.
\begin{theorem}
\label{thm:generalexponentsv}
Let $G$ be a network with controlled stationary measure such that $\bubnorm{P}<\infty$, let $v$ be a vertex of $G$ and let $\mathfrak F_v$ be the $v$-wired uniform spanning forest of $G$. Then
\vspace{0.2em}
\[
\vspace{0.2em}
\P\Bigl(\mathrm{diam}_\mathrm{int}\bigl(\mathfrak T_v\bigr) \geq R \Bigr)
\asymp R^{-1}
\quad\text{and}\quad
\P\Bigl(\bigl|\mathfrak T_v\bigr| \geq R \Bigr) \asymp R^{-1/2}
\]
for all $v\in V$ and $R\geq1$.
\end{theorem}
\begin{theorem}
\label{thm:extrinsicZdv}
Let $d\geq 5$, and let $\mathfrak F_0$ be the $0$-wired uniform spanning forest of $\mathbb Z^d$.
Then
\vspace{0.25em}
\[
\vspace{0.25em}
\P\left(\mathrm{diam}_{\mathrm{ext}}(\mathfrak T_0) \geq R \right) \asymp R^{-2}
\]
for every vertex $v$ and every $R\geq 1$.
\end{theorem}
\begin{theorem}
\label{thm:extrinsicv}
Let $G$ be a network with controlled stationary measure that is $d$-Ahlfors regular for some $d>4$ and that satisfies Gaussian heat kernel estimates.
Let $v\in V$ and let $\mathfrak F_v$ be the wired uniform spanning forest of $G$.
Then
\vspace{0.25em}
\[
\vspace{0.25em}
R^{-2} \preceq \P\left(\mathrm{diam}_{\mathrm{ext}}(\mathfrak P(0)) \geq R \right) \preceq R^{-2}\log R
\]
for every $R\geq 1$.
\end{theorem}
\begin{theorem}
\label{thm:extrinsicspeedv}
Let $G$ be a uniformly ballistic network with controlled stationary measure and $\bubnorm{P}<\infty$, let $v\in V$ and let $\mathfrak F_v$ be the wired uniform spanning forest of $G$.
Then
\vspace{0.25em}
\[
\vspace{0.25em}
\P\left(\mathrm{diam}_{\mathrm{ext}}(\mathfrak P(0)) \geq R \right) \asymp R^{-1}
\]
for every $R\geq 1$.
\end{theorem}
\subsection{The intrinsic diameter: upper bounds}
The key estimate is provided by the following lemma, which will allow us to prove the upper bound on the probability of a large intrinsic diameter by an inductive argument.
For non-negative integers $n$, we define
\[Q(n) =
\sup_{v\in V}\P \left( |\partial \mathfrak{B}_v(v,n)| \neq \emptyset\right),
\]
to be the supremal probability that the component of $v$ in the $v$-WUSF survives to intrinsic distance $n$, and more generally define $Q(t)=Q(\lfloor t \rfloor)$ for $t\geq 0$.
\begin{lemma}
\label{lem:exponentsinduction}
Let $G$ be a network with controlled stationary measure satisfying $\bubnorm{P}<\infty$. Then there exist positive constants $N$ and $C$ such that
\begin{equation}
\label{eq:exponentsinduction}
Q(n) \leq \frac{C}{n} + \frac{1}{6} Q\left(\frac{n}{3}\right).
\end{equation}
for all $n\geq N$.
\end{lemma}
Before proving this lemma, let us prove that \cref{thm:generalexponents,thm:generalexponentsv} follow from it.
\begin{corollary}
\label{cor:intupperbound}
Let $G$ be a network with controlled stationary measure satisfying $\bubnorm{P}<\infty$, and let $Q(n)$ be as above. Then
$Q(n) \preceq n^{-1}$
for all $n\geq 1$.
\end{corollary}
\begin{proof}
Let $N=N(G)$ and $C=C(G)$ be as in \cref{lem:exponentsinduction}. We may assume that $C,N\geq 1$. We prove by induction that
\begin{equation}
\label{eq:Qnbound}
Q(n) \leq 2C N n^{-1}
\end{equation}
for every $n\geq 1$. For $n\leq N$ this is trivial. If $n>N$ and the claim holds for all $m <n$, then we have by \eqref{eq:exponentsinduction} and the induction hypothesis that
\begin{align*}
Q(n) \leq
\left( C + CN\right)n^{-1} \leq 2CN n^{-1}
\end{align*}
as claimed.
\end{proof}
\medskip
We now turn to the proof of \cref{lem:exponentsinduction}. We will require the following estimate. We denote by $\Gamma_v(u,v)$ the path connecting $u$ to $v$ in $\mathfrak F_v$.
\begin{lemma}
\label{lem:ln}
Let $G$ be a network with controlled stationary measure such that $\bubnorm{P}<\infty$. Then
\begin{equation*}
E(n,\delta):= \sup_{v\in V}\mathbb E \left|\Bigl\{ u \in \mathfrak{B}_v(v,2n/3) \setminus \mathfrak{B}_v(v,n/3) : \Cap(\Gamma_{v}(u,v)) \leq \delta n \Bigr\}\right| \preceq \delta^{1/3} n
\end{equation*}
for every $n\geq 1$.
\end{lemma}
\begin{proof}
By Wilson's algorithm we have that, for each two vertices $u$ and $v$ of $G$,
\begin{align*}
&\P\left(u \in \mathfrak{B}_v(v,2n/3) \setminus \mathfrak{B}_v(v,n/3) \text{ and } \Cap(\Gamma(u,v)) \leq \delta n \right)
\\
&\hspace{1cm}= \mathbf P_u\left(\tau_v < \infty,\, n/3 \leq |\mathsf{LE} (X^{\tau_v})| \leq 2n/3,\, \text{ and } \Cap\left(|\mathsf{LE} (X^{\tau_v})|\right) \leq \delta n\right)
\\
&\hspace{1cm}= \frac{c(v)}{c(u)}\mathbf P_v\left(\tau_u < \infty,\, n/3 \leq |\mathsf{LE} (X^{\tau_v})| \leq 2n/3,\, \text{ and } \Cap\left(|\mathsf{LE} (X^{\tau_v})|\right) \leq \delta n\right),
\end{align*}
where the second equality follows from time-reversal.
It follows from the estimate \eqref{eq:MFCLoopLength1} of \cref{lem:LoopLength} that
\begin{multline*}
\P\left(u \in \mathfrak{B}_v(v,2n/3) \setminus \mathfrak{B}_v(v,n/3) \text{ and } \Cap(\Gamma(u,v)) \leq \delta n \right)
\\
\preceq \mathbf P_v\left( n/3 \leq \tau_u \leq \bubnorm{P}n,\, \text{ and } \Cap\left(|\mathsf{LE} (X^{\tau_v})|\right) \leq \delta n\right).
\end{multline*}
Summing over $u\in V$, we obtain that
\begin{align*}
E(n,\delta)
\preceq \sum_{m=n/3}^{\bubnorm{P}n} \mathbf P_v\left(\Cap\left(|\mathsf{LE} (X^m) |\right) \leq \delta n \right)
\preceq \left[\sup_{m\geq n/3} \sup_{v\in V} \mathbf P_v\left(\Cap\left(\mathsf{LE}(X^m)\right) \leq \bubnorm{P}^{-1} \delta m \right) \right] n,
\end{align*}
and the claim follows from \cref{prop:capacities}.
\end{proof}
\begin{proof}[Proof of \cref{lem:exponentsinduction}]
Fix $v\in V$. Let $\mathscr I_v$ be the $v$-wired interlacement process on $G$, let $\mathfrak F_{v,t} = \langle \mathsf{AB}_{v,t}(\mathscr I_v) \rangle_{t\in \mathbb R}$, and let $\mathfrak{B}_{v,t}(v,n)$ denote the ball of radius $n$ about $v$ in $\mathfrak F_{v,t}$ for each $t\in \mathbb R$ and $n\geq0$. Recall that for each $t\in \mathbb R$, $\sigma_t(v) =\sigma_t(v,\mathscr I_v)$ is the first time greater than or equal to $t$ such that $v$ is hit by a trajectory of $\mathscr I_v$ at time $\sigma_t(v)$.
For each two vertices $u$ and $v$ of $G$, every $t\in \mathbb R$ and $n \geq 0$, let $\mathscr B_{t,n}(u,v)$ be the event that $u \in \mathfrak{B}_{v,t}(v,2n/3) \setminus \mathfrak{B}_{v,t}(v,n/3)$, and let $\mathscr C_{t,n}(u,v)\subseteq \mathscr B_{t,n}(u,v)$ be the event that $\mathscr B_{t,n}(u,v)$ occurs and that $v$ is connected to $\partial \mathfrak{B}_{v,t}(v,n)$ by a simple path that passes through $u$.
Let $\varepsilon,\delta>0$.
If $\partial \mathfrak{B}_{v,0}\left(v,n\right) \neq \emptyset$ then we must have that $\mathscr C_{0,n}(u,v)$ occurs for at least $n/3$ vertices $u$, namely those vertices on the middle third of some path connecting $v$ to $\partial \mathfrak{B}_{v,0}(v,n)$ in $\mathfrak F_0$. Thus, it follows by Markov's inequality that
\[Q(n) \leq \P(\sigma_t(v) \leq \varepsilon ) + \frac{3}{n}\sum_{u \in V} \P\left(\mathscr C_{0,n}(u,v) \cap \{\sigma_v > \varepsilon \}\right).\]
Let $\Gamma_{v,0}(u,v)$ be the path connecting $u$ to $v$ in $\mathfrak F_{v,0}$. By stationarity and \cref{lem:vPastDynamics}, we have that
\[ \P\Bigl(\mathscr C_{0,n}(u,v) \cap \{\sigma_v > \varepsilon\}\Bigr) =
\P\left(\mathscr C_{0,n}(u,v) \cap \left\{ \Gamma_{v,0}(u,v) \cap \mathcal I_{v,[-\varepsilon,0]} =\emptyset\right\}\right).
\]
Next, observe that, by stochastic domination (\cref{lem:domination}), we have that
\begin{multline*}
\P\Bigl(\mathscr C_{0,n}(u,v) \mid u \in \mathfrak T_v,\, \Gamma_{v,0}(u,v)=\gamma\Bigr)
\\
\leq \P\Bigl(\partial \mathfrak{B}_{v,0}\left(u,\lfloor n/3 \rfloor \right) \neq \emptyset \mid u \in \mathfrak T_v,\, \Gamma_{v,0}(u,v)=\gamma\Bigr)
\\
\leq \P\Bigl(\partial \mathfrak{B}_{u,0}(u,\lfloor n/3 \rfloor) \neq \emptyset\Bigr) \leq Q\left(\frac{n}{3} \right).
\end{multline*}
for every simple path $\gamma$ from $u$ to $v$. Since the events $\mathscr C_{0,n}(u,v)$ and $\{\Gamma_{v,0}(u,v) \cap \mathcal I_{v,[-\varepsilon,0]} =\emptyset \}$ are conditionally independent conditional on the event $\mathscr B_{0,n}(u,v)$ and the random variable $\Gamma_{v,0}(u,v)$, we deduce that
\begin{multline*}
\P\left(\mathscr C_{0,n}(u,v) \cap \left\{ \Gamma_{v,0}(u,v) \cap \mathcal I_{v,[-\varepsilon,0]} =\emptyset\right\} \mid \mathscr B_{0,n}(u,v),\,\Gamma_{v,0}(u,v) \right)\\
\leq
Q\left( \frac{n}{3}\right) \P\left(\mathscr B_{0,n}(u,v) \cap \left\{ \Gamma_{v,0}(u,v) \cap \mathcal I_{v,[-\varepsilon,0]} =\emptyset\right\} \mid \mathscr B_{0,n}(u,v),\, \Gamma_{v,0}(u,v) \right).
\end{multline*}
Taking expectations over $\Gamma_{v,0}(u,v)$ and applying a union bound, we deduce that
\begin{multline*}
\P\left(\mathscr C_{0,n}(u,v) \cap \left\{ \Gamma_{v,0}(u,v) \cap \mathcal I_{v,[-\varepsilon,0]} =\emptyset\right\}\right)
\\
\leq
Q\left(\frac{n}{3}\right)\P\left(\mathscr B_{0,n}(u,v) \cap \left\{ \Cap(\Gamma_{v,0}(u,v)) \geq \delta n \right\} \cap \left\{\Gamma_{v,0}(u,v) \cap \mathcal I_{v,[-\varepsilon,0]} =\emptyset \right\}\right)\\
+
Q\left(\frac{n}{3}\right)\P\left(\mathscr B_{0,n}(u,v) \cap \left\{\Cap(\Gamma_{v,0}(u,v)) \leq \delta n \right\}\right).
\end{multline*}
Summing the second term over $u \in V$ yields $Q\left(\frac{n}{3}\right) E(n,\delta)$, where $E(n,\delta)$ is the quantity from \cref{lem:ln}.
To control the first term, we apply \eqref{eq:CapvCapComparison} to deduce that
\begin{multline*}
\P\left(\mathscr B_{0,n}(u,v) \cap \left\{ \Cap(\Gamma_{v,0}(u,v)) \geq \delta n \right\} \cap \left\{\Gamma_{v,0}(u,v) \cap \mathcal I_{v,[-\varepsilon,0]} =\emptyset \right\}\right)\\
\preceq e^{- \delta \varepsilon n}\P\Bigl(\mathscr B_{0,n}(u,v)\cap\{\Cap(\Gamma_{v,0}(u,v)) \geq \delta n)\} \Bigr)
\preceq e^{-\varepsilon \delta n}\P\Bigl(\mathscr B_{0,n}(u,v)\Bigr).
\end{multline*}
Thus, summing over $u$ we obtain that
\begin{multline*}
Q(n) \leq \P(\sigma_v \leq \varepsilon) + \frac{3}{n}Q\left(\frac{n}{3}\right) e^{-\varepsilon \delta n} \mathbb E |\mathfrak{B}_{v,0}(0,2n/3)|
+ \frac{3}{n}Q\left(\frac{n}{3}\right)E(n,\delta)\\ \preceq \varepsilon + \left[ e^{-\varepsilon\delta n} + \delta^{1/3}\right] Q\left(\frac{n}{3}\right).
\end{multline*}
The claim now follows by taking $\delta$ to be a small constant and taking $\varepsilon$ to be $C/n$, where $C$ is a large constant. \qedhere
\end{proof}
\subsection{The volume}
\begin{proof}[Proof of \cref{thm:generalexponents,thm:generalexponentsv}]
Due to the stochastic domination between the $v$-WUSF and the past of $v$ in the WUSF (\cref{lem:domination}), the desired upper and lower bounds on the probability of a large intrinsic radius follow from Proposition \ref{prop:intlower} and \cref{cor:intupperbound}. Thus, it remains to prove only the desired upper and lower bounds on the probability of a large volume. We begin with the upper bound.
Using stochastic domination and taking a union bound, we have that
\begin{align*}
\P\bigl(|\mathfrak P(v)|\geq n^2\bigr) \leq \P\bigl(|\mathfrak T_v| \geq n^2\bigr) \leq
\P\Bigl(\partial \mathfrak{B}_v(v,n) \neq \emptyset\Bigr)
+
\P\Bigl(\big|\mathfrak{B}_v (v,n)\big| \geq n^2\Bigr).
\end{align*}
Applying the upper bound on the probability of a large intrinsic diameter from \cref{thm:generalexponentsv} to control the first term, and \eqref{eq:rootedmomentsexponentsproof} together with Markov's inequality to control the second term, we obtain
that
\begin{align*}
\P\bigl(|\mathfrak P(v)|\geq n^2\bigr) \leq \P\bigl(|\mathfrak T_v| \geq n^2\bigr) \preceq n^{-1},
\end{align*}
from which the claimed upper bounds follow immediately.
We now turn to the lower bounds.
For this, we recall the Paley-Zigmund inequality, which states that
\[
\P(Z \geq \varepsilon \mathbb E[Z]) \geq (1-\varepsilon)^2 \frac{\mathbb E[Z]^2}{\mathbb E[Z^2]}
\]
for every real-valued random variable $Z$ such that $\mathbb E[Z] \geq 0$ and $\P(Z=0)\neq1$ and every $\varepsilon>0$. Applying the Paley-Zigmund inequality to the conditional distribution of the real-valued random variable $Z$ on the event that $Z>0$ readily yields that
\begin{equation}
\label{eq:PaleyZigconditional}
\P\Bigl(Z \geq \varepsilon \mathbb E[Z \mid Z>0] \Bigr) \geq (1-\varepsilon)^2\frac{\mathbb E[Z]^2}{\mathbb E[Z^2]}
\end{equation}
for every real-valued random variable $Z$ such that $\P(Z>0) >0$ and every $\varepsilon>0$.
To apply the Paley-Zigmund inequality in our setting, we define random variables
\[
Z(v,n) = \left| \mathfrak P(v, 2n ) \setminus \mathfrak P(v,n ) \right|
\quad\text{and}\quad
Z_v(v,n) = \left| \mathfrak{B}_v(v,2 n ) \setminus \mathfrak{B}_v(v,n) \right|
\]
for each $v\in V$ and $n\geq 1$.
\cref{thm:moments,lem:expectedballgrowth2} yield the estimates
\begin{equation}
\label{eq:unrootedmomentsexponentsproof}
q(v) n\, \preceq
\mathbb E\left[Z(v,n)\right]
\preceq n,\,
\text{ and }
\mathbb E \left[Z(v,n)^2\right]
\preceq n^{3}
\end{equation}
and similarly
\begin{equation}
\label{eq:rootedmomentsexponentsproof}
\mathbb E\left[Z_v(v,n)\right]
\asymp n,\,
\text{ and }
\mathbb E \left[Z_v(v,n)^2\right]
\preceq n^{3}.
\end{equation}
Thus, the Paley-Zygmund inequality \eqref{eq:PaleyZigconditional} implies that
\begin{align*}
\P\bigl(Z\left(v,n\right) \geq c q(v) n^2\bigr) \geq \frac{1}{4}\frac{\mathbb E[Z(v,n)]^2}{\mathbb E[Z(v,n)^2]} \succeq q(v)^2 n^{-1}
\end{align*}
and similarly
\begin{align*}
\P(Z_v\left(v,n\right) \geq c n^2) \geq \frac{1}{4}\frac{\mathbb E[Z_v(v,n)]^2}{\mathbb E[Z_v(v,n)^2]} \succeq n^{-1}.
\end{align*}
Since $|\mathfrak P(v)| \geq Z(v,n)$ and $|\mathfrak T_v| \geq Z_v(v,n)$,
it follows easily that
\[
\P(|\mathfrak P(v)| \geq n) \succeq q(v)^2 n^{-1/2}
\quad \text{ and } \quad
\P(|\mathfrak T_v| \geq n) \succeq n^{-1/2}
\]
for all $n\geq 1$ as claimed.
\end{proof}
\begin{remark}
With a small amount of extra work one can obtain that $\P(|\mathfrak P(v)|\geq n) \succeq q(v) n^{-1/2}$.
\end{remark}
\subsection{The extrinsic diameter: upper bounds}
In this section we prove our results concerning the tail of the extrinsic diameter of the past.
We begin with the proofs of \cref{thm:extrinsic,thm:extrinsicv,thm:extrinsicspeed,thm:extrinsicspeedv}, which are straightforward.
\begin{proof}[Proof of \cref{thm:extrinsicspeed,thm:extrinsicspeedv}]
The lower bounds follow immediately from \cref{prop:extlower}. Since the extrinsic diameter is bounded from above by the intrinsic diameter, the upper bounds are immediate from \cref{thm:generalexponents,thm:generalexponentsv}.
\end{proof}
The upper bounds of \cref{thm:extrinsic,thm:extrinsicv} follow from the following more general bound.
\begin{prop}
\label{prop:polyupperext}
Let $G$ be a network with controlled stationary measure and $\bubnorm{P}<\infty$, and suppose that there exist $C$ and $d$ such that $|B(v,r)|\leq Cr^d$ for every $r\geq 1$ and $v\in V$. Then
\[
\P\bigl(\mathrm{diam}_\mathrm{ext}(\mathfrak P(v)) \geq n\bigr) \leq \P\bigl(\mathrm{diam}_\mathrm{ext}(\mathfrak T_v) \geq n\bigr) \preceq n^{-2}\log n
\]
for every $n\geq 1$.
\end{prop}
\begin{proof}
By the union bound we have that
\begin{align*}
\P\bigl( \mathrm{diam}_\mathrm{ext} (\mathfrak T_v)\geq n\bigr) &\leq \P\bigl( \mathrm{diam}_\mathrm{int} (\mathfrak T_v)\geq m\bigr) + \P\left(\max\left\{ d(v,u) : u \in \mathfrak{B}_v(v, m)\right\} \geq n \right)
\\ &\preceq m^{-1} + \P\left(\max\left\{ d(v,u) : u \in \mathfrak{B}_v(v, m)\right\} \geq n \right).
\end{align*}
Recall that the \textbf{Varopoulos-Carne bound} \cite[Theorem 13.4]{LP:book} states that in any network,
\[
p_n(u,v) \leq 2\sqrt{\frac{c(v)}{c(u)}} e^{-d(u,v)^2/(2n)}.
\]
Using \cref{lem:LoopLength} and the Varopoulos-Carne bound, one can easily derive that, using a time-reversal argument similar to that of \cref{lem:expectedballgrowth1},
\[
\P\left(\max\left\{ d(v,u) : u \in \mathfrak{B}_v(v, m) \right\} \geq n \right) \preceq \sum_{i=0}^{2 \bubnorm{P} m} \mathbf P_v\left( d(v,X_m) \geq n \right) \preceq m \sum_{k\geq n} k^d e^{-k^2/2m},
\]
so that taking $m= \varepsilon n^2 \log^{-1} n$ for a suitably small constant $\varepsilon>0$ yields that
\[
\P\bigl( \mathrm{diam}_\mathrm{ext} (\mathfrak T_v)\geq n\bigr) \preceq n^{-2} \log n. \qedhere
\]
\end{proof}
\begin{proof}[Proof of \cref{thm:extrinsic,thm:extrinsicv}]
The lower bounds both follow immediately from Proposition \ref{prop:extlower} and \cref{lem:polyLr}, while the upper bounds are immediate from \cref{prop:polyupperext}.
\end{proof}
We now wish to improve this argument and remove the logarithmic correction in the case of $\mathbb Z^d$, $d\geq 5$. To this end, let $\mathfrak F_0$ be the $0$-wired uniform spanning forest of $\mathbb Z^d$, and let $\mathfrak T_0$ be the component of $0$ in $\mathfrak F_0$. (We do not consider any time parameterised forests in this section, so this notation should not cause confusion.) We write $\Lambda_m=[-m,m]^d$ and $\partial \Lambda_m = \Lambda_m \setminus \Lambda_{m-1}$. We say that a vertex $v\in \partial \Lambda_m$ is a \textbf{pioneer} if it is in $\mathfrak T_0$ and the future of $v$ (i.e. the unique path connecting $v$ to $0$ in $\mathfrak T_0$) is contained in $\Lambda_m$.
\begin{lemma}
\label{lem:pioneers}
Let $d\geq 5$, and let $\mathfrak F_0$ be the $0$-wired uniform spanning forest of $\mathbb Z^d$. Then there exists positive constants $c_d$ and $C_d$ such that
\begin{equation}
\label{eq:pioneerbound}
\mathbb E\left[ \#\left\{ \text{pioneers in $\partial\Lambda_m\cap \mathfrak{B}_0(0,n)$} \right\}\right] \leq C_d \exp\left[-c_d\frac{m^2}{n}\right].
\end{equation}
\end{lemma}
Note that the expectation of $|\partial \Lambda_m \cap \mathfrak{B}_0(0,n)|$ is of order $m e^{-O(m^2/n)}$. The point of the lemma is that by considering only pioneers we can reduce the expectation by at least a factor of $m$.
Before proving \cref{lem:pioneers}, let us see how it can be applied to deduce \cref{thm:extrinsicZd,thm:extrinsicZdv}.
\begin{proof}[Proof of \cref{thm:extrinsicZd,thm:extrinsicZdv}]
The lower bounds follow from \cref{prop:extlower}, and so by stochastic domination (\cref{lem:domination}) it suffices to prove the upper bound on the probability that $\mathfrak T_v$ has a large intrinsic diameter.
Let
$Q(n) = \P(\partial \mathfrak{B}_0(0,n) \neq \emptyset)$ and let
$\tilde Q(m) = \P( \mathfrak T_0 \cap \partial \Lambda_m \neq \emptyset)$. By the union bound we have that
\begin{align*}
\tilde Q(2m) \leq Q(n) + \P( \mathfrak{B}_0(0,n) \cap \partial \Lambda_{2m} \neq \emptyset).
\end{align*}
Suppose that $\mathfrak{B}_0(0,n) \cap \partial \Lambda_{2m} \neq \emptyset$, and consider a geodesic $\gamma$ in $\mathfrak T_0$ from $0$ to $\partial \Lambda_{2m}$. If $\gamma$ visits $\partial \Lambda_m$ for the first time at $v$, then we necessarily have that $v$ is a pioneer in $\partial \Lambda_m \cap \mathfrak{B}_0(0,n)$ and that there is a path from $v$ to $v+\partial \Lambda_m$ that is disjoint from the future of $v$. Thus, counting the expected number of such $v$ and applying the stochastic domination property, we have that, by Markov's inequality,
\[
\P\left( \mathfrak{B}_0(0,n) \cap \partial \Lambda_{2m} \neq \emptyset\right) \leq \tilde Q(m) \mathbb E\left[ \#\left\{ \text{pioneers in $\partial\Lambda_m\cap \mathfrak{B}_0(0,n)$} \right\}\right].
\]
Thus, if we take $n= \lceil \varepsilon m^2 \rceil$ for some sufficiently small constant $\varepsilon$, it follows from \cref{thm:generalexponentsv} and \cref{lem:pioneers} that
\begin{align*}
\tilde Q(2m) \leq \frac{C}{m^2} + \frac{1}{8} \tilde Q(m).
\end{align*}
for all sufficiently large $m$. The proof can now be concluded via an induction similar to that used in the proof of \cref{cor:intupperbound}. (The $1/8$ here could be replaced with any number strictly smaller than $1/4$.)
\end{proof}
We now turn to the proof of \cref{lem:pioneers}.
\begin{proof}[Proof of \cref{lem:pioneers}]
The proof will come down to a few mildly involved estimates of diagrammatic sums involving random walks on boxes with Dirichlet boundary conditions.
We write $\preceq$ for upper bounds depending only on the dimension $d$, and, to simplify notation, use the convention that $0^{\alpha}=1$ for every $\alpha \in \mathbb R$.
We also use $\Omega$ asymptotic notation, where again the implicit constants depend only on $d$.
Let us first record some basic estimates concerning the random walk on $\mathbb Z^d$.
Let $\mathbf{G}^m_i(v,u)$ be the expected number of times that a walk started at $v$ visits $u$ when it is stopped either at time $i$ or when it first leaves $\Lambda_m$, whichever is sooner. It follows by Gambler's ruin applied to each coordinate that
\begin{equation}
\label{eq:GamblersRuin}
\sum_{u \in \Lambda_m} \mathbf{G}^{m+r}_\infty(v,u) \leq 2d r
\end{equation}
for every vertex $v\in \mathbb Z^d$. Furthermore, it follows from the elliptic Harnack inequality that
\[
\max_{w \in \partial \Lambda_m} \mathbf{G}^m_\infty(v,w) \preceq \left(m-\|v\|_1\right)^{-d+1}
\]
for every $v \in \Lambda_m$. Since the expected number of times the walk spends in $\partial\Lambda_m$ before leaving $\Lambda_m$ is order 1, we deduce that
\begin{equation}
\label{eq:EHI}
\sum_{w\in \Lambda_m} \mathbf{G}^m_\infty(v,w)^2 \leq \max \max_{w \in \partial \Lambda_m} \mathbf{G}^m_\infty(v,w) \sum_{w\in \Lambda_m} \mathbf{G}^m_\infty(v,w) \preceq \left(m-\|v\|_1\right)^{-d+1}
\end{equation}
for every $v\in \Lambda_m$, and similarly
\begin{equation}
\label{eq:EHI2}
\sum_{w\in \Lambda_m} \mathbf{G}^m_\infty(v,w)\mathbf{G}^m_{\infty}(u,w) \preceq \left(m-\|v\|_1 \wedge \|u\|_1\right)^{-d+1}
\end{equation}
for every $u,v \in \Lambda_m$.
Finally, we recall the maximal version of Azuma's inequality, which implies that
\begin{equation}
\label{eq:Azuma}
\mathbf P_0( \tau_{\partial \Lambda_m} \leq n) \leq C_d \exp\left[ - c_d\frac{m^2}{n}\right]
\end{equation}
for some constants $c_d,C_d$.
We now turn to the proof of \eqref{eq:pioneerbound}.
We begin with a proof that works only for $d>5$, and then show how it can be modified to obtain a proof for $d>4$. The variable names we use will be somewhat idiosyncratic; this is to avoid renaming them in the proof for $d=5$.
Let $v \in \partial\Lambda_m$, let $X$ be a random walk started at $v$, and consider sampling $\mathfrak F_0$ using Wilson's algorithm, starting with the walk $X$. Write $\tau_5$ for the first time that $X$ hits the origin. In order for $v$ to be a pioneer and be in $\mathfrak{B}(0,n)$, $X$ must hit $0$, and the loop-erasure $\mathsf{LE}(X^{\tau_5})$ must be contained in $\Lambda_m$. Applying \cref{lem:LoopLength}, we have furthermore that $\tau_5\leq 2\bubnorm{P} n$ with probability at least $1/2$ conditional on this event. Let $n'=\lceil2\bubnorm{P}n\rceil$, and let $\mathscr A_{n,m}(v)$ be the event that $X$ hits $0$, that $\mathsf{LE}(X^{\tau_5})$ is contained in $\Lambda_m$ and that $\tau_5 \leq n'$. Thus, we have that
\[\sum_{v\in \partial \Lambda_m} \P\left(\text{$v$ a pioneer in $\partial\Lambda_m \cap \mathfrak{B}_0(0,n)$}\right) \leq 2\sum_{v\in \partial \Lambda_m} \mathbf P_v\left(\mathscr A_{n,m}(v)\right).
\]
Let $R_2\geq 0$ be maximal such that $X$ visits $\partial \Lambda_{m+R_2}$ before it first visits the origin, let $0 \leq \tau_4 \leq \tau_5$ be the first time that $X$ visits $\partial \Lambda_{m+R_2}$, and let $\tau_3$ be the first time $i$ such that $X_i \in \Lambda_m$ and $X_i = X_j$ for some $j\geq \tau_4$, and let $\tilde \tau_3$ be the first time after $\tau_4$ that $X$ visits $X_{\tau_3}$. Clearly the time $\tau_3$ and $\tilde \tau_3$ must exist and satisfy $0 \leq \tau_3 \leq \tau_4 \leq \tilde \tau_3 \leq \tau_5$ on the event $\mathscr A_{n,m}(v)$. (Note that it is possible that $R_2=\tau_3=\tau_4=\tilde\tau_3=0$.)
Considering the possible choices for $r_2=R_2$ and for the vertices $w= X_{\tau_3}$ and $u = X_{\tau_4}$ yields that
\begin{align*}
\P\left(\mathscr A_{n,m}(v)\right) &\leq
\sum_{u \in \Lambda_m} \sum_{r_2 \geq 0} \sum_{w\in \partial \Lambda_{m+r_2}} \\
&\hspace{1.5cm}\mathbf P_v\Bigl( R_2=r_2, X_{\tau_3}=w,X_{\tau_4}=v, \text{ and } |\tau_3|, |\tau_4-\tau_3|, |\tilde\tau_3-\tau_4|, |\tau_5-\tilde\tau_3| \leq n'\Bigr)
\end{align*}
\begin{align*}
\phantom{ \P\left(\mathscr A_{n,m}(v)\right)}
&\leq \sum_{k=0}^m \sum_{u \in \partial\Lambda_{m-k}} \sum_{r_2 \geq 0} \sum_{w\in \partial \Lambda_{m+r_2}} \mathbf{G}^{m+r}_{n'}(v,u)\mathbf{G}^{m+r}_{n'}(u,w)^2\mathbf{G}^{m+r}_{n'}(u,0).
\end{align*}
Applying \eqref{eq:EHI} and reversing time yields that
\begin{equation*}
\sum_{v\in \partial \Lambda_m} \P\left(\mathscr A_{n,m}(v)\right)
\preceq \sum_{k=0}^m \sum_{r\geq 0} (r+k)^{-d+1} \sum_{u \in \partial \Lambda_{m-k}} \mathbf{G}^{m+r}_{n'}(0,u) \sum_{v\in \partial \Lambda_m} \mathbf{G}^{m+r}_{n'}(u,v).
\end{equation*}
Applying \eqref{eq:EHI} and \eqref{eq:Azuma} we have that
\begin{align*}
\sum_{v\in \partial \Lambda_m}\mathbf{G}^{m+r}_{n'}(u,v) &\leq \mathbf P_u\Bigl(\tau_{\partial\Lambda_m} \leq n'\Bigr) \max_{w \in \Lambda_m} \sum_{v\in \partial \Lambda_m}\mathbf{G}^{m+r}_{\infty}(w,v) \preceq r \exp\left[ - \Omega(m^2/n)\right],
\end{align*}
and so we deduce that
\begin{multline*}
\sum_{v\in \partial \Lambda_m} \P\left(\mathscr A_{n,m}(v)\right)
\preceq \sum_{k=0}^m \sum_{r\geq 0} r(r+k)^{-d+2} \exp\left[-\Omega\left(\frac{(m-k)^2+k^2}{n}\right)\right]\\
\preceq \exp\left[-\Omega(m^2/n)\right]\sum_{k=0}^m k^{-d+4} \preceq \exp\left[-\Omega(m^2/n)\right]
\end{multline*}
as claimed. (In five dimensions we would obtain an unwanted logarithmic correction to this bound since $\sum_{k\geq 0} k^{-d+4}$ diverges.)
To obtain a corresponding bound in $d=5$ dimensions, we sum over two loops instead of one. Let $R_2, \tau_3,\tau_4,\tau_5$ be as above.
Let $R_1$ be maximal such that $X$ visits $\partial\Lambda_{m+R_1}$ before time $\tau_3$, let $\tau_2$ be the first time that $X$ visits $\partial\Lambda_{m+R_1}$, and let $\tau_1$ be the first time $i$ such that $X_i \in \Lambda_m$ and $X_i=X_j$ for some $j\geq \tau_2$.
On the event $\mathscr A_{n,m}$, we must have that $\tau_1$ exists and that $0\leq \tau_1\leq \tau_2$.
Let $\tilde \tau_1$ be the first time $X$ visits $X_{\tau_1}$ after time $\tau_2$. It follows from the definitions that \[0\leq \tau_1 \leq \tau_2 \leq \tilde \tau_1,\tau_3 \leq \tau_4 \leq \tilde \tau_3\leq \tau_5,\] but it is possible for $\tilde \tau_1 $ and $\tau_3$ to occur in either order. We bound the contribution of the two possibilities separately. First, we argue as above to deduce that
\begin{multline*}
\sum_{v\in \partial \Lambda_m}\P\left(\mathscr A_{n,m}(v) , \tilde \tau_1 \leq \tau_3 \right) \\
\preceq \sum_{0\leq k_1,k_2\leq m} \sum_{r_2 \geq r_1 \geq 0} \sum_{v\in \partial \Lambda_m} \sum_{u \in \partial\Lambda_{m-k_1}} \sum_{x \in \partial\Lambda_{m-k_2}} \sum_{y\in \partial \Lambda_{m+r_2}} \sum_{w\in \partial \Lambda_{m+r_1}}\\
\hspace{5.2cm}\mathbf{G}_{n'}^{m+r_2}(0,x)\mathbf{G}_{n'}^{m+r_2}(x,y)^2 \mathbf{G}^{m+r_1}_{n'}(x,u) \mathbf{G}^{m+r_1}_{n'}(u,w)^2\mathbf{G}^{m+r_1}_{n'}(u,v),
\end{multline*}
and applying \eqref{eq:EHI} and \eqref{eq:Azuma} as before we obtain that
\begin{multline}
\sum_{v\in \partial \Lambda_m}\P\left(\mathscr A_{n,m}(v) , \tilde \tau_1 \leq \tau_3 \right)\\ \preceq
\exp\left[-\Omega(m^2/n)\right]\sum_{k_1\geq 0} \sum_{k_2\geq 0} \sum_{r_2 \geq r_1 \geq 0}
(k_2+r_2)(k_2+r_2)^{-d+1}(k_1+r_1)(k_1+r_1)^{-d+1}r_1.
\label{eq:diagram1}
\end{multline}
Similarly, we have that
\begin{multline*}
\sum_{v\in \partial \Lambda_m }\P\left(\mathscr A_{n,m}(v) , \tau_3 \leq \tilde \tau_1\right) \preceq
\sum_{0\leq k_1,k_2\leq m} \sum_{r_2 \geq r_1 \geq 0} \sum_{v\in \partial \Lambda_m} \sum_{u \in \partial\Lambda_{m-k_1}} \sum_{x \in \partial\Lambda_{m-k_2}} \sum_{y\in \partial \Lambda_{m+r_2}} \sum_{w\in \partial \Lambda_{m+r_1}}\\
\hspace{1.6cm} \mathbf{G}_{n'}^{m+r_2}(0,x)\mathbf{G}_{n'}^{m+r_2}(x,y)\mathbf{G}_{n'}^{m+r_2}(y,u)\mathbf{G}^{m+r_2}_{n'}(u,x)\mathbf{G}^{m+r_1}_{n'}(x,w)\mathbf{G}^{m+r_1}_{n'}(w,x)\mathbf{G}^{m+r_1}_{n'}(u,v).
\end{multline*}
Once again, applying \eqref{eq:EHI2} and \eqref{eq:Azuma} as before we obtain that
\begin{multline}
\sum_{v\in \partial \Lambda_m }\P\left(\mathscr A_{n,m}(v) , \tau_3 \leq \tilde \tau_1\right)
\\
\preceq
\exp\left[-\Omega(m^2/n)\right] \sum_{k_1\geq 0} \sum_{k_2\geq 0} \sum_{r_2 \geq r_1 \geq 0}
(k_2+r_2)(k_1\vee k_2 +r_2)^{-d+1}(k_1+r_2)(k_1\vee k_2 +r_1)^{-d+1}r_1.
\label{eq:diagram2}
\end{multline}
It remains to show that the sums on the right of \eqref{eq:diagram1} and \eqref{eq:diagram2} are of order $1$ for $d>4$. For the first, it is straightforward to compute that
\begin{align*}
\sum_{k_1\geq 0} \sum_{k_2\geq 0} \sum_{r_2 \geq r_1 \geq 0}
(k_2+r_2)^{-d+2}(k_1+r_1)^{-d+2}r_1 \preceq
\sum_{r_2 \geq r_1 \geq 0}
r_2^{-d+3}r_1^{-d+4} \preceq \sum_{r_1\geq 0} r_1^{-2d+8} \preceq 1
\end{align*}
as desired. For the second, we rewrite the sum in terms of $a=k_1 \wedge k_2$ and $b=k_1 \vee k_2$ to obtain that
\begin{multline*}
\sum_{k_1\geq 0} \sum_{k_2\geq 0} \sum_{r_2 \geq r_1 \geq 0}
(k_2+r_2)(k_1\vee k_2 +r_2)^{-d+1}(k_1+r_2)(k_1\vee k_2 +r_1)^{-d+1}r_1
\\= \sum_{b\geq a \geq 0} \sum_{r_2 \geq r_1 \geq 0} (a+r_2)(b+r_2)^{-d+2}(b+r_1)^{-d+1}r_1 \preceq \sum_{b\geq 0} \sum_{r_2 \geq r_1 \geq 0} b(b+r_2)^{-d+3}(b+r_1)^{-d+1}r_1.
\end{multline*}
Considering the contribution to this sum from the three cases $b\leq r_1$, $r_1<b<r_2$, and $b\geq r_2$ yields that
\begin{multline*}
\sum_{b\geq 0} \sum_{r_2 \geq r_1 \geq 0} b(b+r_2)^{-d+3}(b+r_1)^{-d+1}r_1 \\\preceq \sum_{r_2\geq r_1} \left[\sum_{b=0}^{r_1} b r_2^{-d+2} r_1^{-d+2} + \sum_{b=r_1}^{r_2} r_2^{-d+2} b^{-d+2} r_1 + \sum_{b \geq r_2} b^{-2d+4}r_1 \right]
\\
\preceq
\sum_{r_2\geq r_1} \left[ r_2^{-d+2} r_1^{-d+4} + r_2^{-d+2} r_1^{-d+4} + r_2^{-2d+5}r_1 \right]\preceq \sum_{r_1\geq 0} r_1^{-2d+7} \preceq 1
\end{multline*}
as desired.
\end{proof}
\section{Spectral dimension, anomalous diffusion}
\label{sec:AlexanderOrbach}
In this section, we apply the estimates of \cref{sec:volume} together with the intrinsic diameter exponent \cref{thm:generalexponentsv} to deduce \cref{thm:AlexanderOrbach}. This will be done via an appeal to the following theorem of Barlow, J\'arai, Kumagai, and Slade \cite{BJKS08}, which gives a sufficient condition for Alexander-Orbach behaviour. See \cite{BJKS08} for quantitative versions of the theorem, and \cite{kumagai2008heat} for generalizations.
We recall that the \textbf{effective conductance} between two finite sets $A,B$ in a finite network $G$ is defined to be
\[
\sC_{\mathrm{eff}}(A\leftrightarrow B ;\, G) = \sum_{v\in A}c(v)\mathbf{P}_v\bigl(\tau_B<\tau_A^+\bigr).
\]
The \textbf{effective resistance} $\mathscr{R}_\mathrm{eff}(A\leftrightarrow B ;\, G):= \sC_{\mathrm{eff}}(A\leftrightarrow B ;\, G)^{-1}$ is defined to be the reciprocal of the effective conductance. We also define the effective resistance and conductance by the same formulas when $A$ and $B$ are finite subsets of an infinite network $G$ that are such that every transient path from $A$ must pass through $B$. For further background on effective conductances and resistances see e.g.\ \cite[Chapters 2 and 9]{LP:book}.
\begin{theorem}[Barlow, J\'arai, Kumagai, and Slade 2008]
\label{thm:BJKSquenched}
Let $(G,\rho)$ be a random rooted graph, and suppose that there exist positive constants $C$, $\gamma$ and $N$ such that
\begin{equation}
\label{eq:BJKSgrowth}
\P\left(\lambda^{-1} n^2 \leq |B(\rho,n)| \leq \lambda n^2 \right)
\geq 1 - C \lambda^{-\gamma}
\end{equation}
and
\begin{equation}
\label{eq:BJKSresistance}
\P\left( \mathscr{R}_\mathrm{eff}\left(v \leftrightarrow \partial B(\rho,n);\, G\right) \geq \lambda^{-1}n \right)
\geq 1 - C \lambda^{-\gamma}
\end{equation}
for all $n\geq N$.
Then $d_s(G)=4/3$ and $d_w(G)=3$ almost surely. In particular, the limits defining both dimensions exist almost surely.
\end{theorem}
The estimate \eqref{eq:BJKSgrowth} has already been established in \cref{cor:exponentialproblargevolume,lem:fullvolumelower}. Thus, to apply \cref{thm:BJKSquenched}, it remains only to prove an upper bound on the probability that the effective conductance is large. The following lemma will suffice.
\begin{lemma}
\label{lem:expectedconductance}
Let $G$ be a network with controlled stationary measure such that $\bubnorm{P}<\infty$. Then
\[\mathbb E\Bigl[ \sC_{\mathrm{eff}}(v \leftrightarrow \partial \mathfrak{B}(v,n) ;\, \mathfrak F) \Bigr] \preceq n^{-1} \]
for all $n \geq 1$.
\end{lemma}
We begin with the following deterministic lemma.
\begin{lemma}
\label{lem:treeresistance}
Let $T$ be a tree, let $v$ be a vertex of $T$, and let $N_v(n,k)$ be the number of vertices $u \in \partial B(v,k)$ such that $u$ lies on a geodesic in $T$ from $v$ to $\partial B(v,n)$. Then
\[\sC_{\mathrm{eff}}\left( v \leftrightarrow \partial B(v,n) ; \, T \right) \leq \frac{1}{k}N_v(n,k)\]
for every $1\leq k \leq n$.
\end{lemma}
\begin{proof}
We use the extremal length characterisation of the effective resistance \cite[Exercise 2.78]{LP:book}. Given a graph $G$ and a function $m:E\to[0,\infty)$, we define the $m$-length of a path in G by summing m over the edges in the path, and define the $m$-distance $d_m(A,Z)$ between two sets of vertices $A$ and $Z$ to be the minimal $m$-length of a path connecting $A$ and $Z$. If $G$ is finite, then we have that
\[
\sC_{\mathrm{eff}}( A \leftrightarrow B) = \inf\left\{ \sum_{e} m(e)^2 : d_m(A,Z) \geq 1 \right\}.
\]
We now apply this bound with $G$ equal to the (subgraph of $T$ induced by the) ball $B(v,n)$ in $T$.
If we set $m(e)=1/k$ if $e$ lies on the first $k$ steps of some geodesic from $v$ to $\partial B(v,n)$ and set $m(e)=0$ otherwise, then we clearly have that $d_m(v,\partial B(v,n))=1$ and, since $N_v(n,k)$ is decreasing in $k$,
\[\sum_{e}m(e)^2 = \frac{1}{k^2} \sum_{r=1}^k N_v(n,r) \leq \frac{1}{k} N_v(n,k)\]
as claimed.
\end{proof}
\begin{proof}[Proof of \cref{lem:expectedconductance}]
For each $0 \leq m \leq n$, let $Z_{m,n}$ be the number of vertices $u$ in $\partial \mathfrak{B}(v,m)$ such that there is a geodesic from $v$ to $\partial \mathfrak{B}(v,n)$ in $\mathfrak F$ that passes through $u$. Thus, by \cref{lem:treeresistance}, we have that
\[\sC_{\mathrm{eff}}\left(v\leftrightarrow \partial \mathfrak{B}(v,n) ;\mathfrak F \right) \leq \frac{1}{m} Z_{m,n}\]
for every $1\leq m \leq n$, and hence that
\begin{equation}
\label{eq:conductancelanes}
\sC_{\mathrm{eff}}(v\leftrightarrow \partial \mathfrak{B}(v,3n) ;\mathfrak F ) \preceq \frac{1}{n^2} \sum_{m=n}^{2n} Z_{m,3n}.
\end{equation}
Now, for each vertex $u$ of $G$, let $\mathscr A_{m,n}(v,u)$ be the event that $u\in \mathfrak{B}(v,m)$ and that $u$ lies on a geodesic connecting $v$ to $\partial \mathfrak{B}(v,n)$ in $\mathfrak F$. By
By \cref{lem:domination} and \cref{thm:generalexponentsv}, we have that
\[
\P(\mathscr A_{m,n}(v,u)) \preceq \frac{1}{n-m}\P\left(u \in \mathfrak{B}(v,m)\right),
\]
for $n \geq m+1$.
Summing over $v$, we obtain that
\begin{equation}
\label{eq:expectedlanes}
\mathbb E\Bigl[\sC_{\mathrm{eff}}(v\leftrightarrow \partial \mathfrak{B}(v,3n) ;\mathfrak F )\Bigr] \preceq
\frac{1}{n^2}\mathbb E\left[ \sum_{m=n}^{2n} Z_{m,3n} \right] \preceq \frac{1}{n^3} \mathbb E|\mathfrak{B}(v,2n)| \preceq \frac{1}{n},
\end{equation}
where we have used \cref{thm:vmoments} in the final inequality. The claim follows immediately.
\end{proof}
\begin{proof}[Proof of \cref{thm:AlexanderOrbach}]
This follows immediately by applying \cref{thm:BJKSquenched}, the hypotheses of which are met by \cref{cor:quenchedvolumeupper,lem:fullvolumelower,lem:expectedconductance}.
\end{proof}
\section{Applications to the Abelian sandpile model}
\label{sec:sandpile}
Let $G$ be a transient graph and let $\mathrm{H}$ be a uniform infinite recurrent sandpile on $G$, as defined in \cref{subsec:sandpileintro}. Let $\mathfrak F$ be the wired uniform spanning forest of $G$, let $\mathfrak F_v$ be the $v$-wired uniform spanning forest of $G$ for each vertex $v$ of $G$, let $\mathfrak T_v$ be the component of $v$ in $\mathfrak F_v$, and let $\mathbf{G}$ be the Greens function on $G$.
Given a recurrent sandpile configuration $\eta$ and a vertex $v$ of $G$, we write $\operatorname{Av}_v(\eta,u)$ for number of times $u$ topples if we add a grain of sand to $\eta$ at $v$ and then stabilize, so that $|\operatorname{Av}_v(\eta)|=\sum_{u\in V} \operatorname{Av}_v(\eta,u)$.
We recall the following relationships between these objects:
\begin{enumerate}
\item \textbf{Dhar's formula} states that the expected number of times $u$ topples when we add a grain of sand at $v$ is given by the Greens function. That is,
\begin{equation}
\label{eq:Dhar}
\mathbb E\left[ \operatorname{Av}_v(\mathrm{H},u) \right] = \mathbf{G}(v,u).
\end{equation}
\item The avalanche cluster at $v$ approximately stochastically dominates the past of $v$ in the WUSF. More precisely, for any increasing Borel set $\mathscr A \subseteq \{0,1\}^V$, we have that
\begin{equation}
\label{eq:sandpilelowerbound}
\P\left( \operatorname{AvC}_v(\mathrm{H}) \in \mathscr A \right)
\geq \frac{1}{\deg(v)} \P\left( \mathfrak P(v) \in \mathscr A \right).
\end{equation}
This follows from the discussion in \cite[Section 2.5]{bhupatiraju2016inequalities}, see also equation $(3.2)$ of that paper.
\item The diameter of the avalanche cluster at $v$ is approximately stochastically dominated by the diameter of the component of $v$ in the $v$-WUSF. More precisely, we have that
\begin{equation}
\label{eq:sandpileupperbound}
\P\left(\mathrm{diam}_\mathrm{ext}\left[ \operatorname{AvC}_v(\mathrm{H}) \right] \geq r\right) \leq \mathbf{G}(v,v) \P\left(\mathrm{diam}_\mathrm{ext}\left[ \mathfrak T_v \right] \geq r\right).
\end{equation}
This follows from \cite[Lemma 2.6]{bhupatiraju2016inequalities}.
\end{enumerate}
See \cite{bhupatiraju2016inequalities,Jar14} for detailed discussions of these properties.
We now apply these relations to deduce \cref{thm:sandpile,thm:sandpilepolynomial,thm:sandpilenonamenable} from the analogous results concerning the WUSF and $v$-WUSF.
\begin{proof}
The lower bounds all follow immediately from \eqref{eq:sandpilelowerbound} together with the corresponding statements for the WUSF, which are given in \cref{thm:generalexponents,thm:extrinsicZd,thm:extrinsic,thm:extrinsicspeed}. Similarly, the upper bounds on the extrinsic radius of the avalanche follow from \cref{eq:sandpileupperbound} and the corresponding statements for the $v$-WUSF. Thus, it remains only to prove the upper bound on the probability of a large number of topplings. For this, we apply a union bound and Dhar's formula to obtain that
\begin{align*}
\P\Bigl(|\operatorname{Av}_v(\mathrm{H})| \geq n\Bigr) &\leq \P\left( \mathrm{diam}_\mathrm{ext}\left[ \operatorname{AvC}_v(\mathrm{H}) \right]\geq m \right) + \frac{1}{n}\sum_{u\in B(v,m)} \mathbf{G}(v,u)\\
&\leq \P\left( \mathrm{diam}_\mathrm{ext}\left[ \operatorname{AvC}_v(\mathrm{H}) \right]\geq m \right) + \frac{1}{n}L(m).
\end{align*}
The claimed upper bounds follow by applying \cref{thm:extrinsicZdv,thm:extrinsicv,thm:extrinsicspeedv} as appropriate to bound the first term on the right, using \cref{lem:polyLr} to bound the second term on the right when necessary, taking $m=\lceil n^{1/4} \rceil$ in the case of $\mathbb Z^d$, taking $m =\lceil n^{1/4} \log^{1/4} n \rceil$ in the case of \cref{thm:sandpilepolynomial}, and taking $m=\lceil n^{1/2} \rceil$ in the uniformly ballistic case.
\end{proof}
\subsection*{Acknowledgments}
We thank Martin Barlow, Antal J\'arai, and Perla Sousi for helpful discussions, and thank Russ Lyons for catching some typos. Much of this work took place while the author was a PhD student at the University of British Columbia, during which time he was supported by a Microsoft Research PhD Fellowship.
\phantomsection
\addcontentsline{toc}{section}{References}
\footnotesize{
\bibliographystyle{abbrv}
| -158,220.950747 |
[
-2.849609375,
2.611328125
] | 45.036916 |
[
-2.25390625,
0.93310546875,
-2.1328125,
-5.734375,
-1.0849609375,
8.3046875
] |
[
4.359375,
8.7734375,
3.251953125,
7.5625
] | 879 | 20,696 |
[
-3.341796875,
4.03125
] | 32.513639 |
[
-5.7578125,
-4.23828125,
-5.59375,
-2.521484375,
2.005859375,
13.9765625
] | 0.5779 | 25.033704 | 15.186509 | 2.320571 |
[
2.0096254348754883
] | -87,297.997194 | 5.392878 | -157,816.066585 | 0.29567 | 6.274646 |
[
-1.857421875,
-3.470703125,
-4.01171875,
-5.2109375,
2.037109375,
12.328125
] |
[
-6.09765625,
-2.62109375,
-2.15625,
-1.3896484375,
4.0546875,
5.1875
] | |
BkiUdYM4ukPiEUwWDEcV
|
\section{Introduction}
Systems in which individual photons can interact strongly with each other constitute an exciting frontier for the fields of quantum and nonlinear optics \cite{NLO14}. Such systems enable the generation and manipulation of non-classical light, which are crucial ingredients for quantum information processing and quantum networks \cite{K08}. At the many-body level, it has been predicted that these systems can produce phenomena such as quantum phase transitions of light \cite{HAR06,GRE06} or photon crystallization \cite{CRY08,OTT13}. Early examples of such systems where strong interactions between photons could be observed consisted of individual atoms coupled to single modes of high-finesse optical cavities, within the context of cavity quantum electrodynamics~(QED) \cite{TUR95,BIR05}. More recently, a number of systems have emerged that produce strong optical nonlinearities between freely propagating photons, including cold atomic gases coupled to guided modes of tapered fibers \cite{VET10,GOB12}, photonic crystal fibers \cite{BAJ09} or waveguides \cite{GOB14}, cold Rydberg gases in free space \cite{rydberg_experiments}, and superconducting qubits coupled to microwave waveguides \cite{HOI13,LOO13}.
The paradigm of cavity QED admits exact analytical or numerical solutions at the few-photon~(and/or few-atom) level through the elegant ``input-output'' formalism \cite{GAD95}, as illustrated conceptually in Fig.~\ref{fig_1}(a). In this formalism, all of the properties of the field exiting the system (the output) can be determined based upon knowledge of the input field and the dynamics of the atom-cavity system alone. For a few excitations, the latter can be solved due to the small Hilbert space associated with a small number of excitations of a discrete cavity mode. In contrast, despite rapid development on the experimental front, theoretical techniques to treat the dynamics of freely propagating photons interacting with spatially distributed emitters are generally lacking. At first glance, the challenge compared to the cavity case arises from the fact that a two-level system is a nonlinear frequency mixer, which is capable of generating a continuum of new frequencies from an initial pulse, as schematically depicted in Fig.~\ref{fig_1}(b). A priori, keeping track of this continuum as it propagates and re-scatters from other emitters appears to be a difficult task. An exception is the weak excitation limit, in which atoms can be treated as linear scatterers and the powerful transfer matrix method of linear optics can be employed~\cite{DEC12,DEU95}.
The full quantum case has been solved exactly in a limited number of situations in which nonlinear systems are coupled to 1D waveguides, such as up to three photons scattering on a single atom~\cite{SHI09,FAN10}, one and two photons scattering on a cavity QED system \cite{SHI11}, and many photons scattering on many atoms coupled to a chiral (monodirectional) waveguide \cite{PLE12}. The formalism employed in Ref.~\cite{FAN10} is particularly elegant, because it establishes an input-output relation to determine the nonlinear scattering from a two-level atom. Here, we show that this technique can be efficiently generalized to many atoms, chiral or bi-directional waveguides, and arbitrary atomic configurations, providing a powerful tool to investigate nonlinear optical dynamics in all systems of interest.
This paper is organized in the following way: first, we present a generalized input-output formalism to treat few-photon propagation in waveguides coupled to many atoms. We show that the infinite degrees of freedom associated with the photonic modes can be effectively integrated out, yielding an open, interacting ``spin'' model that involves only the internal degrees of freedom of the atoms. This open system can be solved using a number of conventional, quantum optical techniques. Then, we show that the solution of the spin problem can be used to re-construct the optical fields. In particular, we provide a prescription to map spin correlations to S-matrix elements, which contain full information about the photon dynamics, and give explicit closed-form expressions for the one- and two-photon cases. Importantly, in analogy with the cavity QED case, our technique enables analytical solutions under some scenarios, but in general allows for simple numerical implementation under a wide variety of circumstances of interest, such as different level structures, external driving, atomic positions, atomic motion, etc. Finally, to illustrate the ease of usage, we apply our technique to the study of nonlinear field propagation through an optically dense ensemble of atoms with Rydberg-like interactions.
\section{Generalized input-output formalism}
In this section we consider a generic system composed of many atoms located at positions $z_i$ along a bidirectional waveguide. We assume that there is an optical transition between ground and excited-state levels $\ket{g}$ and $\ket{e}$ to which the waveguide couples, but otherwise we leave unspecified the atomic internal structure and the possible interactions between them (\textit{e.g.}, Rydberg interactions), as such terms do not affect the derivation presented here. The bare Hamiltonian of the system is composed of a term describing the energy levels of the atoms $H_\text{at}$, and a waveguide part $H_\text{ph} = \sum_{v = \pm}\int dk \,\omega_k b^\dagger_{v,k}b_{v,k}$, where $k$ is the wavevector and $v=\pm$ is an index for the direction of propagation, with the plus (minus) denoting propagation towards the right (left) direction. We assume that within the bandwidth of modes to which the atoms significantly couple, the dispersion relation for the guided modes can be linearized as $\omega_k=c|k|$. The interaction between atoms and photons is given by
\begin{equation}
H_\text{int} = g\sum_{v = \pm}\sum_{i=1}^N \int dk\,\big(b_{v,k}\sigma_{eg}^i\,e^{{\rm i}} \def\end{equation}{{\rm e}} \def\vb{{\bf v} vkz_i} + \text{h.c.}\big),
\end{equation}
which describes the process where excited atoms can emit photons into the waveguide, or ground-state atoms can become excited by absorbing a photon. The coupling amplitude $g$ is assumed to be identical for all atoms, while the coupling phase depends on the atomic position ($e^{{\rm i}} \def\end{equation}{{\rm e}} \def\vb{{\bf v} vkz_i}$). Here, we will explicitly treat the more complicated bidirectional case, although all of the results readily generalize to the case of a single direction of propagation.
Our immediate goal is to generalize the well-known input-output formalism of cavity QED~\cite{GAD95} and the more recent formalism for a single atom coupled to a waveguide \cite{FAN10}, to the present situation of many spatially distributed atoms coupled to a common waveguide. In short, we will eliminate the photonic degrees of freedom by formal integration, which reveals that the output field exiting the collection of atoms is completely describable in terms of the input field and atomic properties alone. This formal integration also provides a set of generalized Heisenberg-Langevin equations that governs the atomic evolution.
The Heisenberg equations of motion for $\sigma_{ge}^i$ and $b_{v,k}$ can be readily obtained by calculating the commutators with $H$. To simplify the presentation of the resulting equations, we replace the spin operators $\sigma_{ge}^i$ with the bosonic annihilation operator $a_i$. The two-level nature of the atomic transition can be retained by introducing an interaction energy $U_0$ for multiple excitations, through the term $(U_0/2)\sum_i a^\dagger_ia_i(a^\dagger_ia_i-1)$ in $H_\text{at}$, and by taking the limit $U_0 \rightarrow \infty$ at the end of calculations for observable quantities.
\begin{figure}
\includegraphics[width=85mm,angle=0,clip]{fig_1.pdf}
\caption{{\bf Cavity QED vs. many-atom waveguide} (a) Schematic representation of a cavity QED system. The input-output formalism enables the outgoing field to be calculated based on knowledge of the input field and the internal dynamics of the cavity system. (b) System composed of many atoms coupled to a common waveguide. The nonlinearity of an atom enables the generation of a continuum of new frequencies upon scattering of an incoming state. This property, combined with multiple scattering from other atoms, appears to make this system more complicated than the cavity QED case.}
\label{fig_1}
\end{figure}
The Heisenberg equations for $b_{v,k}$ can be formally integrated and Fourier transformed, to obtain the real-space wave equation
\begin{eqnarray}
b_{v}(z,t)&=&b_{v,\text{in}}(t-vz/c)\nonumber\\
&-&\frac{{\rm i}} \def\end{equation}{{\rm e}} \def\vb{{\bf v} \sqrt{2\pi}g}{c}\,\sum_{j=1}^N\,\theta\big(v(z-z_j)\big)a_j\bigg(t-{\frac{v(z-z_j)}{c}}\bigg).
\label{eq:b}
\end{eqnarray}
Here $b_{v,\text{in}}$ is the homogeneous solution, physically corresponding to the freely propagating field in the waveguide, while the second term on the right consists of the part of the field emitted by the atoms. Inserting Eq.~\eqref{eq:b} into the equation for $a_i$, we obtain
\begin{eqnarray}
\label{eq:a_2}
\dot{a}_i&=&{\rm i}} \def\end{equation}{{\rm e}} \def\vb{{\bf v}[H_\text{at}, a_i] - {\rm i}} \def\end{equation}{{\rm e}} \def\vb{{\bf v}\sqrt{2\pi}g\sum_{v = \pm}\,b_{v,\text{in}}(t-vz_i/c)\nonumber\\
&-&\frac{2\pi g^2}{c}\sum_{j=1}^N\,a_j(t-|z_i-z_j|/c)).
\end{eqnarray}
In realistic systems, time retardation can be neglected, resulting in the Markov approximation $a_j(t-|z_i-z_j|/c) \approx a_j(t)e^{{\rm i}} \def\end{equation}{{\rm e}} \def\vb{{\bf v} \omega_{\text{in}}|z_i-z_j|/c}$. Here, $\omega_{\text{in}}$ is a central frequency around which the atomic dynamics is centered (typically the atomic resonance frequency). This approximation is valid when the difference in free-space propagation phases $ \Delta\omega L/c \ll 1$ is small across the characteristic system size $L$ and over the bandwidth of photons $\Delta\omega$ involved in the dynamics. As a simple example, the characteristic bandwidth of an atomic system is given by its spontaneous emission rate, corresponding to a few MHz, which results in a significant free-space phase difference only over lengths $L \gtrsim 1$ m much longer than realistic atomic ensembles.
We have thus obtained the generalized Heisenberg-Langevin equation
\begin{eqnarray}
\label{eq:many_1}
\dot{a}_i&=&{\rm i}} \def\end{equation}{{\rm e}} \def\vb{{\bf v}[H_\text{at}, a_i] - {\rm i}} \def\end{equation}{{\rm e}} \def\vb{{\bf v}\sqrt{c\,\Gamma_{1D}/2}\sum_{v = \pm}\,b_{v,\text{in}}(t-vz_i/c)\nonumber\\
&-&\frac{\Gamma_{1D}}{2}\sum_{j=1}^N\,a_j\,e^{{\rm i}} \def\end{equation}{{\rm e}} \def\vb{{\bf v} \omega_\text{in}|z_i-z_j|/c},
\end{eqnarray}
where we have identified $\Gamma_{1D} = 4\pi g^2/c$ as the single-atom spontaneous emission rate into the waveguide modes. If we keep separated the terms proportional to $a_j$ coming from the right and left-going photonic fields, we can find easily that the Lindblad jump operators corresponding to the decay of the atoms into the waveguide are $O_\pm = \sqrt{\Gamma_{1D}/4}\sum_j\,a_j e^{\mp{\rm i}} \def\end{equation}{{\rm e}} \def\vb{{\bf v} \omega_\text{in}z_j/c}$, in terms of which we can write the master equation for the atomic density matrix $\dot{\rho} = \mathcal{L}[\rho] \equiv -{\rm i}} \def\end{equation}{{\rm e}} \def\vb{{\bf v}[H_\text{at}, \rho] + \sum_{v=\pm}\,2O_v\rho O^\dagger_v - O^\dagger_vO_v\rho - \rho O^\dagger_v O_v$.
We also see that we can derive Eq.~\eqref{eq:many_1} from a non-Hermitian effective Hamiltonian
\begin{equation}
H_\text{eff} = H_{\text{at}} - \frac{{\rm i}} \def\end{equation}{{\rm e}} \def\vb{{\bf v}\Gamma_{1D}}{2}\sum_{ij}\,a_i^\dagger a_j\,e^{{\rm i}} \def\end{equation}{{\rm e}} \def\vb{{\bf v} \omega_\text{in}|z_i-z_j|/c},
\label{eff_H:eq}
\end{equation}
which can be used for a quantum jump description of the atomic dynamics. The resulting infinite-range interaction between a pair of atoms $i$,$j$ intuitively results from the propagation of a mediating photon between that pair, with a phase factor proportional to the separation distance.
Within the same approximations employed above to derive the Heisenberg-Langevin equations we can obtain a generalized input-output relation of the form $b_{v,\text{out}}(z,t) = b_{v,\text{in}}(t-vz/c) - {\rm i}} \def\end{equation}{{\rm e}} \def\vb{{\bf v}\sqrt{\Gamma_{1D}/(2c)}\sum_{i=1}^N\,a_i(t)\,e^{{\rm i}} \def\end{equation}{{\rm e}} \def\vb{{\bf v} v\omega_\text{in}(z-z_i)/c}$, where the output field is defined for $z > z_{R} \equiv \max[z_i]$ ($z < z_L \equiv \min[z_i]$) for right(left)-going fields. However, since the right-going output field propagates freely after $z_R$, it is convenient to simply define $b_{+,\text{out}}(t) = b_{+,\text{out}}(z_R+\epsilon,t)$ as the field immediately past the right-most atom (where $\epsilon$ is an infinitesimal positive number), and similarly for the left-going output. The derived relation shows that the out-going field properties are obtainable from those of the atoms alone.
The emergence of infinite-range interactions between emitters mediated by guided photons, and input-output relationships between these emitters and the outgoing field, have been discussed before in a number of contexts \cite{DEC12,FAN10,LUM13}, but the idea that such concepts could be used to study quantum interactions of photons in extended systems has not been fully appreciated. In the remaining sections, we will demonstrate the effectiveness of this approach to quantum nonlinear optics. In particular, the infinite-dimensional continuum of the photons is effectively reduced to a Hilbert space of dimension $\text{dim}[\mathcal{H}] = \sum^n_{i=0}\binom {N} {i}$ where $n$ is the maximum number of atomic excitations (for $n = N$ we have $\text{dim}[\mathcal{H}] = 2^N$). The atomic dynamics, on the other hand, having been reduced to standard Heisenberg-Langevin equations, quantum jump, or master equations, are solvable by conventional prescriptions~\cite{meystre}. It is also possible to derive generalized master equations describing the atomic dynamics in response to arbitrary (\textit{e.g.}, non-classical) incident states of light, as detailed in Appendix C.
\section{Relation to S-matrix elements}
The S-matrix characterizes how an incoming state of monochromatic photons evolves via interaction with atoms into a superposition of outgoing monochromatic photons. Because monochromatic photons form a complete basis, the S-matrix thus contains all information about photon dynamics. Here, we will show that S-matrix elements can be calculated from the dynamics of the atoms evolving under the effective spin model.
Formally, the S-matrix for the interaction of an $n$-photon state with an arbitrary system is defined as
\begin{eqnarray}
\label{eq:s_1}
S^{(n)}_{\mathbf{p};\mathbf{k}}&=&\bra{\mathbf{p}} S\ket{\mathbf{k}}\nonumber \\
&=&\bra{0}b_\text{out}(p_1)..b_\text{out}(p_n)b^\dagger_\text{in}(k_1)..b^\dagger_\text{in}(k_n)\ket{0}\nonumber \\
&=&\mathcal{FT}^{(2n)}\bra{0}b_\text{out}(t_1)..b_\text{out}(t_n)b^\dagger_\text{in}(t_1')..b^\dagger_\text{in}(t_n')\ket{0},
\end{eqnarray}
where the input and output creation operators respectively create freely propagating incoming and outgoing photonic states. The vectors $\mathbf{p}$ and $\mathbf{k}$ denote the outgoing and incoming frequencies of the $n$ photons. The input and output operators can be any combination of + and - propagation directions (we have omitted this index here for simplicity). In the last line we have used a global Fourier transformation $\mathcal{FT}^{(2n)} = (2\pi)^{-n}\int\prod_{i=1}^n\,dt_idt_i'\,e^{{\rm i}} \def\end{equation}{{\rm e}} \def\vb{{\bf v} c(t_ip_i-t_i'k_i)}$, to express the S-matrix in time.
The S-matrix has been calculated exactly before in a limited number of situations \cite{SHI09,FAN10,SHI11,PLE12}. Here, we provide a general prescription to numerically obtain the S-matrix starting from the input-output formalism. First, it should be noted that the operators $b_\text{in}$, $b_\text{out}$ correspond to the input and output operators defined in the previous section \cite{FAN10}. On the other hand, the input-output relation enables the correlator of Eq. \eqref{eq:s_1} to be written purely in terms of atomic operators. For notational simplicity, we give a derivation for a single spin and a monodirectional waveguide, but its generalization to the bidirectional waveguide and many atoms is straightforward. For our purpose it is enough to have an input-output relation of the form
\begin{equation}
\label{eq:bound_1}
b_{\text{out}} = b_{\text{in}} - {\rm i}} \def\end{equation}{{\rm e}} \def\vb{{\bf v}\sqrt{\gamma}a,
\end{equation}
where $a$ is in our case the spin operator $\sigma_{ge}$.
We summarize the main idea of the derivation here, while the details can be found in Appendix A.1. We begin by noting that Eq.~(\ref{eq:bound_1}) enables one to replace output operators by a combination of system and input operators, or input operators by system and output operators. Selectively using these substitutions, one can exploit favorable properties of either the input or output field, in order to gradually time order all of the system operators~(where operators at later times appear to the left of those at earlier times), while removing input and output operators from the correlation. The favorable properties that can be used are that 1) system and input operators commute, $[a(t), b_\text{in}(t')] = 0$, when $t'>t$, and likewise $[a(t), b_\text{out}(t')] = 0$ for $t' < t$, 2) input annihilation operators at different times commute amongst themselves, as do output annihilation operators, and 3) the correlator can be reduced in size using $[b_\text{in}(t),b^{\dagger}_\text{in}(t')]=\delta(t-t')$, or made to vanish using $b_\text{in}(t)\ket{0}=0$ or $\bra{0}b^{\dagger}_\text{out}(t)=0$. Through this procedure, the S-matrix can be expressed as a Fourier transform of a sum of terms involving only time-ordered atomic operators~(indicated by the operator $T$). These functions generally have the form
\begin{equation}
\label{eq:s_7}
\bra{0}T\big[a(t_1)..a(t_m)a^\dagger(t_1')..a^\dagger(t_m')\big]\ket{0},
\end{equation}
with $m \leq n$, multiplied by $n-m$ delta functions in time.
Moreover, using the general expression for the Heisenberg-Langevin equation and the quantum regression theorem, it can be proven that when external fields driving the system do not generate waveguide photons, the correlation function of Eq. \eqref{eq:s_7} can be evaluated by evolving $a(t)$ as $e^{{\rm i}} \def\end{equation}{{\rm e}} \def\vb{{\bf v} H_\text{eff}t}\,a\,e^{-{\rm i}} \def\end{equation}{{\rm e}} \def\vb{{\bf v} H_\text{eff}t}$ (see Appendix A.2 for a formal derivation). Here, $H_\text{eff}$ is the effective Hamiltonian from Eq. \eqref{eff_H:eq} that contains only the spin operators. Although such a form for $a(t)$ is not true in general due to quantum noise, these noise terms have no influence on the correlation. A similar procedure as above enables one to express other important observables of the field, such as the second-order correlation function $g^{(2)}(t)$, in terms of correlation functions involving atoms alone.
While the discussion has thus far been completely general, the case of S-matrix elements involving only one or two photons can be formally reduced to particularly simple expressions. For example, in Appendix B, we show that the transmission coefficient $T_k$ for the many-atom, bi-directional waveguide case is related to the S-matrix by $S^{(1)}_{p+;k+}\equiv \bra{0}b_{+,\text{out}}(p)b^{\dagger}_{+,\text{in}}(k)\ket{0}\equiv T_k\delta_{pk}$. Furthermore, it can be expressed in terms of a known $\sim N\times N$ matrix corresponding to the single-excitation Green's function $G_0$~(whose form varies depending on the system details),
\begin{equation}
T_k=1-\frac{{\rm i}} \def\end{equation}{{\rm e}} \def\vb{{\bf v}\Gamma_{1D}}{2}\sum_{ij}\left[G_0(k)\right]_{ij}e^{-{\rm i}} \def\end{equation}{{\rm e}} \def\vb{{\bf v} k_\text{in}(z_i-z_j)}.
\end{equation}
Similarly, the two-photon S-matrix in transmission is generally given by
\begin{multline}
S^{(2)}_{p_1+,p_2+;k_1+,k_2+}=T_{k_1}T_{k_2}\delta_{p_1 k_1}\delta_{p_2 k_2} + \\ - {\rm i}} \def\end{equation}{{\rm e}} \def\vb{{\bf v}\frac{\Gamma^2_{1D}}{8\pi}\delta_{p_1+p_2,k_1+k_2}\sum_{iji'j'}W_{ij}T_{ij;i'j'}W_{i'j'}+(p_1\leftrightarrow p_2),\label{eq:S2}
\end{multline}
where the first and second terms on the right describe the linear and nonlinear contributions, respectively. The latter term can be expressed in terms of matrices $W$ related to the single-excitation Green's function, and a known $\sim N^2 \times N^2$ matrix $T$ characterizing atomic nonlinearities and interactions.
\section{Electromagnetic Induced Transparency}
\begin{figure}[t]
\setlength{\unitlength}{1cm}
\begin{picture}(8,4.5)
\put(-0.3,0){\raisebox{4ex}{
\includegraphics[width=25mm,angle=0,clip]{eit_level.pdf}}}
\put(0,4){(a)}
\put(3,0) {\includegraphics[width=53mm,angle=0,clip]{susc.pdf}}
\put(2.6,4){(b)}
\end{picture}
\caption{(a) EIT level scheme. The atomic ground~($\ket{g}$) and excited states~($\ket{e}$) interact with the quantum propagating field~$b$ of the waveguide. An additional classical field with Rabi frequency $\Omega$ couples state $\ket{e}$ to a metastable state $\ket{s}$. The total single-atom linewidth of the excited state is given by $\Gamma$. (b) The real ($\chi '$) and imaginary
($\chi ''$) parts of the linear susceptibility for a two-level atom (upper panel) and three-level atom
(lower panel), as a function of the dimensionless detuning $\delta/\Gamma$ of the field $b$ from the resonance frequency of the $\ket{g}$-$\ket{e}$ transition. For the three-level atom, the parameters used are $\delta_L=0$ and $\Omega/\Gamma=1/3$.}
\label{eit_level:fig}
\end{figure}
\begin{figure}[t]
\setlength{\unitlength}{1cm}
\begin{picture}(8,4.5)
\put(0.4,0){\includegraphics[width=65mm,angle=0,clip]{eitSinglePolariton.pdf}}
\put(1.5,4){(a)}
\put(1.5,2){(b)}
\end{picture}
\caption{EIT: single polariton propagation for $\sigma _p<\sigma _{EIT}$ (a)
and $\sigma _p >\sigma _{EIT}$ (b). Plotted is the population
$P_j=\langle\sigma_{ss}^j\rangle$
of atom $j$ in the state $|s\rangle$. The blue line corresponds to the initial state,
the green line to the state evolved over a time $t_f$, and the red dots to the
theoretically predicted evolution.
Other parameters: $t_f v_g=Nd/6$, $\Gamma=5$, $\Gamma _{1D}=10$, $\Omega =1$, $N=500$ and
$\sigma _{EIT}\sim 22d$, where $d$ is the lattice constant.}
\label{eitSinglePolariton:fig}
\end{figure}
In this section, we apply our formalism to a specific example involving three-level atoms under conditions of electromagnetically induced
transparency (EIT) and with Rydberg-like interactions between atoms ~\cite{rydberg_experiments}.
The linear susceptibility for a two-level atom with states $|g\rangle$ and $|e\rangle$,
in response to a weak probe field with detuning $\delta=\omega _p-\omega _{eg}$ from the atomic
resonance, is shown in Fig.~\ref{eit_level:fig}b. It can be seen that the response on resonance
is primarily absorptive, as characterized by the imaginary part of the susceptibility
($\chi''$, red curve). In contrast, the response can become primarily dispersive near resonance
if a third level $|s\rangle$ is added,
and if the transition $|e\rangle -|s\rangle$ is driven by a control
field~(characterized by Rabi frequency $\Omega$ and single photon detuning
$\delta _L =\omega _L-\omega _{es}$). Specifically, via interference between the probe and control
fields, the medium can become transparent to the probe field ($\chi ''=0$) when two-photon
resonance is achieved, $\delta-\delta_L=0$, realizing EIT~\cite{eit_papers}.
In this process, the incoming probe field strongly mixes with spin wave excitations $\sigma_{sg}$
to create ``dark-state polaritons."
The medium remains highly transparent within a characteristic bandwidth
$\Delta_{EIT}$ around the two-photon resonance, which reduces to
$\Delta _{EIT}\sim 2\Omega ^2/(\Gamma \sqrt{D}) $,
when $\delta_L=0$. Here we have introduced the total single-atom linewidth $\Gamma$ (see below) and the optical depth $D$, corresponding to the opacity of the medium in absence of EIT
and defined in terms of input and output intensity as
$I_{\rm out}(\Omega =0)/I_{\rm in}(\Omega =0)=\exp(-D)$. These polaritons propagate at a
strongly reduced group velocity $v_g \ll c$, as indicated by the steep slope of the
real part of the susceptibility $\chi'$ in Fig.~\ref{eit_level:fig}b, which is proportional to the control field
intensity~\cite{eit_papers}.
Taking $s_i$ to be the annihilation operator for the state
$|s_i\rangle$, EIT is described within our spin model by the effective spin Hamiltonian
\begin{eqnarray}
H&=&-(\delta_L + {\rm i}} \def\end{equation}{{\rm e}} \def\vb{{\bf v}\frac{\Gamma'}{2})\sum _j a^\dagger _j a _j
-\Omega\sum _j (a^\dagger _js_j +s^\dagger _j a _j)\nonumber\\
&-& {\rm i}} \def\end{equation}{{\rm e}} \def\vb{{\bf v}\frac{\Gamma _{1D}}{2}\sum _{j,l}e^{{\rm i}} \def\end{equation}{{\rm e}} \def\vb{{\bf v} k_\text{in}|z_j-z_l|}a^\dagger _j a _l,
\label{eit_ham:eq}
\end{eqnarray}
where the first line represents the explicit form of $H_{at}$ in Eq.~(\ref{eff_H:eq})
for the EIT three-level atomic structure.
In addition to waveguide coupling, here we have added an independent atomic decay rate $\Gamma'$ into other channels (\textit{e.g.}, unguided modes), yielding a total single-atom linewidth of $\Gamma=\Gamma'+\Gamma_{1D}$.
Our theoretical model nearly ideally describes experiments in which atoms are coupled
to one-dimensional waveguides such as nanofibers~\cite{VET10} or photonic crystals~\cite{GOB12},
in which the interaction probability of a single photon and atom is
$\Gamma_{1D}/\Gamma\sim 0.1$ is significant and up to $N\sim 10^3$ atoms are trapped.
One consequence of the large interaction probability is that even a single atom can yield a significant reflectance for single photons~\cite{GOB12}. In free-space experiments,
the atom-photon interaction probability is much smaller, while a much larger number of
atoms is used to induce a significant optical response. While this large number cannot
be implemented numerically with our model, our system can nonetheless be used to reproduce
macroscopic observables, thus making our approach an excellent description for atom-light
interfaces in general.
Intuitively (and as can be rigorously shown, see below), provided that free-space
experiments only involve a single transverse mode of light, one expects that the system response
to light only depends on the interaction probability and number of atoms via their product.
This product in fact defines the optical depth $D=N(2\Gamma_{1D}/\Gamma)$~\cite{opticalDepth}.
As $D\lesssim 10^2$ in free-space experiments, such systems can be evaluated using our
spin model with several hundred atoms and suitably large $\Gamma_{1D}/\Gamma$.
The only remaining issue is the potentially large reflection generated by atoms in the
mathematical model, compared to free-space ensembles where reflection is negligible.
Reflection can be suppressed in the model by giving the atoms a lattice constant $d$
where $k_\text{in}d=(2m+1)\pi/2$, where $m$ is a non-negative integer, such that the reflection from atoms destructively
interferes~\cite{Chang_NJP11,perfectMirror}. All subsequent numerical calculations are performed under this condition.
The spin model on a lattice describing EIT, Eq.~(\ref{eit_ham:eq}), can be exactly solved
in the linear regime using the transfer matrix
formalism~\cite{Chang_NJP11}, which correctly
reproduces the free-space result and dependence on optical depth for
group velocity $v_g=2\Omega^2 n/\Gamma_{1D}$ and transparency window $\Delta_{EIT}$, where $n$ is the (linear) atomic density. The corresponding minimum spatial extent of a pulse that can propagate inside
the medium with high transparency is given by
$\sigma_{EIT}={v}_g/{\Delta}_{EIT}$.
A single photon propagating inside an ensemble of atoms under
EIT conditions is coherently mapped onto a single dark polariton,
corresponding to a delocalized spin wave populating
the single excitation subspace of the atomic ensemble. The polariton dynamics
can be therefore visualized directly by monitoring
the excitation probability $\langle \sigma _{ss}^j\rangle$ of the atoms in the ensemble.
In Fig.~\ref{eitSinglePolariton:fig}, we initialize a single polariton inside the medium
with an atomic wave function of the form
$|\psi\rangle=\sum_{j} f_j \sigma_{sg}^j |g\rangle^{\otimes N}$,
and we determine numerically the time evolution under $H$ in Eq.~(\ref{eit_ham:eq})
up to a final time $t_f$. Choosing an initially Gaussian spin wave, $f_j=\exp({\rm i}} \def\end{equation}{{\rm e}} \def\vb{{\bf v} k_\text{in}dj)\exp(-(jd-\mu)^2/4\sigma _{p}^2)/(2\pi \sigma _p ^2)^{1/4}$,
with spatial extent
$\sigma _p$ (blue line), one sees that the wavepacket propagates a distance $v_g\cdot t_f$, and with
little loss provided that $\sigma _p>\sigma_{EIT}$. Numerics (green line) show perfect agreement with theoretical predictions
(red lines) obtained via the transfer matrix formalism~\cite{Chang_NJP11}.
\section{Infinite range interaction}
\begin{figure}[top]
\setlength{\unitlength}{1cm}
\begin{picture}(8,15)
\put(0.5,10) {\includegraphics[width=60mm,angle=0,clip]{comparisonT2_vs_delta.pdf}}
\put(0,4.6) {\includegraphics[width=75mm,angle=0,clip]{copyE4_vs_DDP_C.pdf}}
\put(0.3,0) {\includegraphics[width=62mm,angle=0,clip]{g2_vs_t_comparison.pdf}}
\put(-0.4,14){{(a)}}
\put(-0.4,9){{(b)}}
\put(-0.4,4){{(c)}}
\end{picture}
\caption{(a) single-photon (circles) and two-photon (triangles) transmission spectrum for a weak probe field, for selected values $C/2=0$ and $C/2=0.2$ of the infinite-range interaction strength. The linear transmission is independent of $C$, while the two-photon spectrum exhibits a shift in the maximum transmission by an amount $C/2$.
Other parameters: $N=200$, $\Gamma _{1D}=1, \Gamma ' =3$, $\Omega= 2$, $\delta _L= 0$,
$\mathcal{E}=10^{-6}$.
(b) Contour plot of the two-photon transmission spectrum $T_2=\langle {b}^\dagger _\text{+,out}(t){b}^\dagger _\text{+,out}(t){b} _\text{+,out}(t){b} _\text{+,out}(t)\rangle/\mathcal{E} ^4$, as functions of interaction strength $C/2$ and two photon detuning $\Delta =\delta$ ($\delta _L=0$). Cuts of the contour plot (illustrated by the dashed lines) are plotted in a).
(c) Comparison of $g^{(2)}(\tau)$ evaluated by numerical simulations
(red dashed line) and S-matrix theory (black line). For constant infinite range interactions
$C=1$ and $\Delta=0$, the interactions induce photon antibunching.
Other parameters: $N=20$, $\Gamma _{1D}=2, \Gamma ' =2$, $\Omega= 1$, $\delta _L= 0$, $\mathcal{E}=10^{-6}$.}
\label{I_g20_mapping:fig}
\end{figure}
The spin model formalism can be easily extended to include arbitrary atomic interactions,
providing a powerful tool to study quantum nonlinear optical effects.
As a concrete example, we consider a system in which atoms can interact directly
over a long range, such as via Rydberg states \cite{Gorshkov_PRL11,Bienias_preprint,OTT13}
or photonic crystal bandgaps~\cite{Douglas_preprint}.
The total Hamiltonian is given by
\begin{eqnarray}
H&=&-(\delta_L +{\rm i}} \def\end{equation}{{\rm e}} \def\vb{{\bf v}\frac{\Gamma '}{2})\sum _j a^\dagger _j a _j
-\Omega\sum _j (a^\dagger _js_j +s^\dagger _j a _j)\nonumber\\
&-& {\rm i}} \def\end{equation}{{\rm e}} \def\vb{{\bf v}\frac{\Gamma _{1D}}{2}\sum _{j,l}e^{{\rm i}} \def\end{equation}{{\rm e}} \def\vb{{\bf v} k_\text{in}|z_j-z_l|}a^\dagger _j a _l
+\frac{1}{2}\sum _{j,l}U_{ij}s^\dagger _j s^\dagger _l s _l s _j\nonumber\\
&+& H _\text{drive}
\label{ham:eq}
\end{eqnarray}
in which $U_{ij}$ represents a dispersive interaction between atoms $i$ and $j$
when they are simultaneously in state $ |s\rangle $.
As we are primarily interested
in demonstrating the use of our technique, we take here a ``toy model'' where atoms
experience a constant infinite-range interaction, $U_{ij}/2\equiv C$.
Such a case enables the numerical results to be intuitively understood,
although we note that other choices of $U_{ij}$ do not increase the numerical complexity.
In particular we are interested in studying the propagation of a constant
weak coherent input field through the atomic ensemble. The corresponding driving
then is given by
$H _\text{drive}=\mathcal{E} \sum _j(a^\dagger _j e^{{\rm i}} \def\end{equation}{{\rm e}} \def\vb{{\bf v} k_\text{in}z_j}e^{-{\rm i}} \def\end{equation}{{\rm e}} \def\vb{{\bf v} \Delta t}+a _j e^{-{\rm i}} \def\end{equation}{{\rm e}} \def\vb{{\bf v} k_\text{in}z_j}e^{{\rm i}} \def\end{equation}{{\rm e}} \def\vb{{\bf v} \Delta t})$,
where $\mathcal{E}\ll \Gamma$ is the amplitude of the constant driving field,
$\Delta=\delta-\delta _L$ the detuning from two photon resonance
condition, and the initial state is given by the global atomic ground state
$|\psi _{i}\rangle =|g \rangle^{\otimes N}$~\cite{Mollow_PRA75,Chang_NAT07}.
With infinite-range interaction, one spin flip to state $|s\rangle _{\tilde{j}}$ shifts the energies
of all other states $|s\rangle _{j}$ by an amount $C$. A second photon should then be able
to propagate with perfect transparency, provided it has a detuning compensating for
the energy shift $C$, thus ensuring the two-photon resonance condition is satisfied.
As we result, we expect to see a transparency window for two photons, whose central frequency
shifts linearly with $C$.
This predicted behavior can be confirmed by plotting the transmitted intensity
fraction,
$T_1\!\!\!\!\!\!=\!\!\!\!\! I/I_\text{in}\!\!\!\!\!=\!\!\!\!\!\langle b_\text{+,out}^\dagger(t) b_\text{+,out}(t)\rangle/\mathcal{E}^2 $,
and also the second-order correlation function
$T_2\!\!\!=\langle {b}^\dagger _\text{+,out}(t){b}^\dagger _\text{+,out}(t){b} _\text{+,out}(t){b} _\text{+,out}(t)\rangle/\mathcal{E}^4$,
which corresponds roughly to the two-photon transmission.
Fig.~\ref{I_g20_mapping:fig} shows the single-photon transmission $T_1$
and two-photon transmission $T_2$
as a function of the interaction strength $C$ and detuning from two photon resonance $\Delta$.
As expected, $T_1$ shows a peak at $\Delta =0$ independently of the interaction intensity
$C$; instead the peak in $T_2$ shifts towards $\Delta =C/2$ with increasing $C$.
The decay of $T_2$ for increasing $C$ can be intuitively understood by noting that we have a constant coherent state input, in which photons are randomly spaced,
causing two photons to enter the medium at different times.
Thus, until the second photon enters, the first photon propagates as a single polariton
detuned by $\Delta$ from the single-photon transparency condition, getting partially
absorbed in the process.
By increasing the interaction we increase the detuning for this single polariton
and consequently its absorption, explaining the trend observed for $T_2$ in
Fig.~\ref{I_g20_mapping:fig}. A quantitative description of this phenomenon is given in Appendix D.
Field correlation functions like intensity
$I=\langle b_\text{+,out}^\dagger(t) b_\text{+,out}(t)\rangle$
or $g_{2}(\tau)=\langle {b}^\dagger _\text{+,out}(t){b}^\dagger _\text{+,out}(t+\tau){b} _\text{+,out}(t+\tau){b} _\text{+,out}(t)\rangle/I^2$
can be computed according to the following strategy. First
we switch from Heisenberg representation to Schr\"odinger representation,
so that for the intensity we get:
$I=\langle b_\text{+,out}^\dagger(t) b_\text{+,out}(t)\rangle=\langle \psi(t)|{b}^\dagger _\text{+,out}{b} _\text{+,out}|\psi(t)\rangle$.
The time evolved wave function,
$|\psi (t)\rangle$, is determined by numerically evolving the initial spin state
$|\psi _i\rangle$ under $H$ for
a time $t$. Then, the state immediately after detection of one photon,
${b} _\text{+,out}|\psi(t)\rangle=|\phi\rangle$, is evaluated by expressing $b_\text{+,out}$
in terms of spin operators using the input-output formalism:
$b_\text{+,out}=\mathcal{E}e^{{\rm i}} \def\end{equation}{{\rm e}} \def\vb{{\bf v} k_\text{in}z_R}-{\rm i}} \def\end{equation}{{\rm e}} \def\vb{{\bf v}\Gamma _{1D}/2\sum _j\sigma _{ge}e^{{\rm i}} \def\end{equation}{{\rm e}} \def\vb{{\bf v} k_\text{in}(z_R-z_j)}$.
Finally we obtain the intensity by computing the probability
of the one-photon detected state, $I=\langle\phi|\phi\rangle$.\\
For $g_{2}(\tau)$ an extra step is needed. Its numerator describes the process of detecting two photons, the first at time $t$ and the second at
time $t+\tau$:
$f(t+\tau)=\langle {b}^\dagger _\text{+,out}(t){b}^\dagger _\text{+,out}(t+\tau){b} _\text{+,out}(t+\tau){b} _\text{+,out}(t)\rangle$. As before we can switch to
Schr\"odinger picture,
$f(t+\tau)=\langle \psi(t)|{b}^\dagger _\text{+,out}e^{{\rm i}} \def\end{equation}{{\rm e}} \def\vb{{\bf v} H\tau}{b}^\dagger _\text{+,out}{b} _\text{+,out}e^{-{\rm i}} \def\end{equation}{{\rm e}} \def\vb{{\bf v} H\tau}{b} _\text{+,out}|\psi(t)\rangle$, and evaluate the
state after dectection of the first photon, ${b} _\text{+,out}|\psi(t)\rangle=|\phi\rangle$.
Then detection of a second photon after a time $\tau$ entails performing
an extra evolution under $H$
and annihilating a photon, that is:
${b} _\text{+,out}e^{-{\rm i}} \def\end{equation}{{\rm e}} \def\vb{{\bf v} H\tau}|\phi\rangle={b} _\text{+,out}|\phi (\tau)\rangle$.
Finally, we evaluate the quantity
$f(t+\tau)=\langle \phi(\tau)|{b}^\dagger _\text{+,out}{b} _\text{+,out}|\phi (\tau)\rangle$,
by again expressing $b_\text{+,out}$ in terms of spin operators.
In Fig.~\ref{I_g20_mapping:fig}c, we plot the numerically obtained result for
$g^{(2)}(\tau)$, for the case where infinite-range interactions are turned on ($C=1$) and for a weak coherent input state with detuning $\Delta=0$. In such a situation, one expects for the single-photon component of the coherent state to transmit perfectly, while the two-photon component is detuned from its transparency window and becomes absorbed. This nonlinear absorption intuitively yields the strong anti-bunching dip $g^{(2)}(\tau=0)<1$. We also evaluate this second-order correlation function using the analytical
result for the two-photon S-matrix in Eq.~(\ref{eq:S2}), which shows perfect agreement as expected.
\section{Conclusion}
We have shown that the dynamics of a few photons, propagating under strong interactions mediated by quantum emitters, is fully and efficiently characterized by the dynamics of an open quantum spin model. As the spin model is solvable by standard quantum optical techniques for open systems, our approach provides an easily implementable recipe for the exact numerical study of a large class of quantum nonlinear optical systems. As an example, it provides an attractive alternative to numerical simulations of propagation through cold Rydberg gases, where typically the continuous field is finely discretized and its degrees of freedom are explicitly kept track of~\cite{Gorshkov_PRL11}. Our technique can also be used to treat a number of strongly nonlinear systems based upon atomic saturation~\cite{CRY08}, where only approximate effective theories for field propagation were previously available. We anticipate that the availability of exact results will greatly aid in the development of effective theories for the generally challenging problem of quantum many-body states of light, and that our model will shed new insight on conceptual links between such systems and quantum spin systems.
\acknowledgments
We thank H.J. Kimble, A.V. Gorshkov, and H. de Riedmatten for stimulating discussions.
MTM acknowledges support from a La Caixa - Severo Ochoa PhD Fellowship. DEC acknowledges support from Fundacio Cellex Privada Barcelona, the Ramon y Cajal program, and the Marie Curie CIG project ATOMNANO.
TC, MTM and TS contributed equally to the work.
| -28,905.956279 |
[
-3.560546875,
3.17578125
] | 30.882353 |
[
-3.212890625,
0.180908203125,
-2.26171875,
-6.50390625,
-1.1748046875,
9.2109375
] |
[
5.375,
9.3125,
3.326171875,
7.796875
] | 239 | 5,030 |
[
-3.5703125,
4.09765625
] | 27.109457 |
[
-6.30859375,
-4.578125,
-4.75,
-2.447265625,
2.1640625,
12.859375
] | 0.845394 | 13.154989 | 27.833002 | 2.719201 |
[
2.6402435302734375
] | -19,241.20037 | 5.879125 | -28,634.730344 | 2.373867 | 6.090521 |
[
-2.6015625,
-4.03515625,
-4.203125,
-5.2421875,
2.396484375,
13.0546875
] |
[
-5.44140625,
-2.26171875,
-2.591796875,
-1.8359375,
3.716796875,
5.51953125
] | |
BkiUd3U4eIOjSJ68LGD2
|
\section*{Introduction}
Let us consider an algebraic variety $V$ admitting a set of equations whose coefficients belong in a number field,
$E\supset \QQ$. Consider $\sigma:E\to E$ a Galois transformation of $E$, then $\sigma$ acts on the defining equations
of $V$ to produce another algebraic variety $V^\sigma$ which is classically known as a \emph{conjugate} variety of $V$.
The same concept can be extended to any scheme. Note that the extension of $\sigma$ to $\CC$ is not necessarily a
homeomorphism and hence $V$ and $V^\sigma$ are not necessarily homeomorphic.
In~\cite{Serre-Examples}, Serre gave the first such example for a smooth surface whose conjugate has a non-isomorphic
fundamental group. Abelson~\cite{Abelson-conjugate} also found such examples for smooth projective varieties with isomorphic
fundamental groups but different homotopy types. In addition, he also gave examples of conjugate, smooth
homotopy equivalent, quasiprojective varieties that are not homeomorphic. Other examples by different authors can be
found in~\cite{Artal-Carmona-ji-effective,Shimada-non-homeomorphic,Milne-Suh-Non-homeomorphic,Sh:18,Sh:19}.
In the previous examples either fundamental groups are non abelian or they had not been explicitly calculated.
In this paper we present examples of conjugate plane algebraic curves, or equivalently conjugate complements of curves,
whose fundamental groups are abelian (and hence isomorphic) yet they are not homeomorphic.
In particular, we construct a family of irreducible plane projective curves whose deformation space is parametrized by
the roots of unity of any given order $d$ up to complex conjugation. Associated with this equisingular space, another
family is constructed whose deformation space is also parametrized by the roots of unity. In this paper we focus on
the topology both of their embedding and their complements. We prove that curves from different strata have both different
embeddings and different topological types of their complements. This is in contrast to the simplicity of the fundamental
group of their complements, which is abelian.
Particular examples of birationally equivalent curves, such as the union of smooth curves with three lines at maximal
order flexes have already been considered in the literature~\cite{CC:08}, however not necessarily for their topological
properties. The topology of their embedding was also recently studied by T.~Shirane in~\cite{Sh:18}, where a splitting
graph is used (a generalization of an invariant introduced in~\cite{afg:17}). Shirane also extended this study to the
union of smooth curves and any three non-concurrent lines in~\cite{Sh:19}. Some of our computations will apply to these
curves as well showing that their complements are not homeomorphic and their fundamental groups are still abelian.
The invariant we are using is not original of this paper, see~\cite{ea:Toulouse,afg:17,benoit-meilhan,ben-shi}.
Up to now, this invariant has been considered as an invariant of the pair given by the projective plane and the curve.
In this work, we show that in good conditions it is an invariant of the complement.
An outline of the paper is as follows: in \S~\ref{sec:construction} we give a description of three families of curves
whose equisingular space is non-connected and their connected components are parametrized by roots of unity of their
degree up to complex conjugation. Next, \S~\ref{sec:Zariski-tuples} is devoted to showing when curves in these families
have different embeddings by means of a linking invariant (first defined in \cite{afg:17} by the first author and
Florens-Guerville) that is sensitive to the embedding of a curve rather than just the fundamental group of its complement.
The purpose of \S~\ref{sec:pi1} is to explicitly describe different members of the different equisingular strata and
calculate the fundamental groups of their complements. Finally, in \S~\ref{sec:consecuencias} we extend our techniques
to the family studied by Shirane who showed that curves in different connected components of the equisingular strata
have different embeddings. These families share a similar behavior, that is, their complements are also non-homeomorphic
yet their fundamental groups are abelian.
\subsection*{Acknowledgments}
The second and third authors want to thank the Fulbright Program (within the Jos\'e Castillejo and Salvador de Madariaga
grants by Ministerio de Educaci\'on, Cultura y Deporte) for their financial support while writing this paper. They also want
to thank the University of Illinois at Chicago, especially Anatoly Libgober and Lawrence Ein for their warmth welcome and
support in hosting them.
\section{Construction of an equisingular family of curves}\label{sec:construction}
The purpose of this section is to construct three families of curves whose equisingular strata are not connected.
Such families have the property that conjugate curves belong to different connected components.
\subsection{The equisingular family \texorpdfstring{$\Sigma^{(d)}$}{Sigmad} of irreducible cuspidal curves of type
\texorpdfstring{$(d,d+1)$}{(d,d+1)}}
\mbox{}
For any given $d\in\bn$, $d>1$ let us denote by $\Sigma^{(d)}$ the realization space of (irreducible)
plane projective curves in $\PP^2=\PP_\CC^2$ of degree~$2d$ having exactly three singular points whose local topological
type is $u^d+v^{d+1}=0$. Such curves have genus
$\binom{d-1}{2}$ and a classical example is the family $\Sigma^{(2)}$
of tricuspidal quartics, whose fundamental group of the complement is the braid group $\BB_3(\PP^1)$ on three strings on the
sphere~$\mathbb S^2$ as was already known to Zariski~\cite{zr:29,Zariski-rational}.
Before we go any further note that $\Sigma^{(d)}\subset \PP^{N_d}$ for $N_d=\frac{d(d+3)}{2}$ is a smooth quasi-projective variety
and that the group $\PGL(3;\bc)$ of projective automorphisms of $\PP^2$ acts on $\Sigma^{(d)}$, maybe not freely. However, since
$\PGL(3;\bc)$ is a connected topological group, the number of connected components of $\Sigma^{(d)}$ and its quotient
$\PGL(3;\bc)\backslash\Sigma^{(d)}$ coincide. Moreover, since $\Sigma^{(d)}$ is smooth and quasi-projective, this number is also
the number of irreducible components of these spaces.
The following are properties of any curve $\cC\in\Sigma^{(d)}$:
\begin{enumerate}
\item\label{prop:1}
Since the singular points are locally irreducible, the curve~$\cC$ is \emph{globally} irreducible.
\item\label{prop:2}
Let $P_x,P_y,P_z$ denote the singular points of $\cC$ and $L_z$ the line joining $P_x,P_y$.
Since the local intersection number of two curves at a point is at least the product of their multiplicities,
we have
\[
2d=\cC\cdot L_z\geq (\cC\cdot L_z)_{P_x}+(\cC\cdot L_z)_{P_y}
\geq d+d\Longrightarrow
\begin{cases}
\cC\cap L_z=\{P_x,P_y\}, \textrm{ and }\\
\cC\pitchfork_{P_x} L_z,\cC\pitchfork_{P_y} L_z.
\end{cases}
\]
Hence, $L_z$ is not tangent to $\cC$ at $P_x$ and $P_y$. Analogously occurs with $L_x$ (resp.~$L_y$)
the line joining $P_y,P_z$ (resp.~$P_x,P_z$). Moreover, since they intersect pairwise at three different points,
we deduce that $L_x$, $L_y$, and $L_z$ are three distinct and non-concurrent lines.
\item\label{prop:3}
By the previous comment, there is a projective transformation~$\Phi$ such that $\Phi(L_x)=\{x=0\}$, $\Phi(L_y)=\{y=0\}$,
and $\Phi(L_z)=\{z=0\}$ and thus $\Phi(P_x)=[1:0:0]$, $\Phi(P_y)=[0:1:0]$ and $\Phi(P_z)=[0:0:1]$. Hence, up to a projective
change of coordinates, one can assume that $\cC$ satisfies these properties. Moreover, the projective transformation is not
unique, that is, any diagonal projective automorphism can be further applied to $\cC$.
\end{enumerate}
\begin{figure}[ht]
\begin{center}
\begin{tikzpicture}[scale=.5,vertice/.style={draw,circle,fill,minimum size=0.1cm,inner sep=0}]
\draw[fill, color=blue!10!white] (0,5)--(5,5)--(5,0)--cycle;
\draw[help lines,step=.5,dotted] (0,0) grid (10,10);
\draw[very thick] (0,10)--(0,0)--(10,0);
\foreach \x in {1,...,10}
\node[vertice] at (\x/2,5) {};
\foreach \y in {1,...,10}
\node[vertice] at (5,\y/2) {};
\foreach \z in {1,...,10}
\node[vertice] at (5-\z/2,\z/2) {};
\node[anchor=north] at (3,-.5) {$\ldots$};
\node[anchor=north] at (5,-.1) {$(d,0)$};
\node[anchor=north] at (7.5,-.5) {$\ldots$};
\node[anchor=north] at (10,-.1) {$(2d,0)$};
\node[anchor=north east] at (0,0) {(0,0)};
\node[anchor=east] at (-.25,3) {$\vdots$};
\node[anchor=east] at (-.1,5) {$(0,d)$};
\node[anchor=east] at (-.25,7.5) {$\vdots$};
\node[anchor=east] at (-.1,10) {$(0,2d)$};
\draw[very thick,dashed] (10,0)--(0,10);
\node[vertice] (a05) at (0,5) {};
\node[vertice] (a55) at (5,5) {};
\node[vertice] (a50) at (5,0) {};
\draw[very thick] (0,5)--(5,5)--(5,0)--cycle;
\draw[thick] (1,4.5)--(4.5,4.5)--(4.5,1)--cycle;
\end{tikzpicture}
\caption{Newton polygon}
\label{fig:np}
\end{center}
\end{figure}
Let $F(x,y,z)=0$ be the equation of a curve~$\cC\in \Sigma^{(d)}$ satisfying these properties. Our purpose is
to give the conditions the homogeneous polynomial $F$ of degree $2d$ should satisfy. A very useful tool to do
this is to consider the Newton polygon of $f(x,y)=F(x,y,1)=\sum_{0\leq i+j\leq 2d} a_{ij}x^iy^j$,
$$
N(f)=\textrm{ConvexHull}(\{(i,j)\in \ZZ^2_{\geq 0}\mid a_{ij}\neq 0\}).
$$
As a general restriction, since $F$ is a homogeneous polynomial of degree $2d$, $N(f)\subset T((0,0),(2d,0),(0,2d))$,
the triangle, or the convex hull of the three points. Moreover, since $\cC$ has multiplicity $d$ at $P_x$, $P_y$,
$P_z$ and the axes are not tangent, the Newton polygon is the triangle $T((0,d),(d,d),(d,0))$ of Figure~\ref{fig:np}.
The transversality of the axes at the singular points implies that the vertices of this triangle correspond
to non-zero coefficients; we assume that the coefficient of $x^d y^d$ equals~$1$.
Moreover, since the Zariski tangent cone at the singularities has only one direction, each of the (quasi)-homogeneous
polynomials of $F$ associated with the sides of $N(f)$ must be a $d$-power of type $z^d(a_zx+b_zy)^d=(b_zz)^d(\lambda x+y)^d$,
$y^d(a_yz+b_yx)^d=(a_yy)^d(z+\mu x)^d$, and $x^d(a_xy+b_xz)^d=(b_xx)^d(\gamma y+z)^d$. According to property~\eqref{prop:3} above
one can further assume that $b_z=\mu=\gamma=1$. Moreover, since the coefficient of $x^dy^d$ is 1, one has $a_y^d=b_x^d=1$
and hence the polynomial associated with the horizontal edge is $y^d(x+z)^d$, the one associated with the vertical edge is
$x^d(y+z)^d$ and the one associated with the diagonal edge is $z^d(y+\zeta x)^d$, where $\zeta^d=1$ since $\lambda^d$ is the
coefficient of $x^dz^d$ which is 1 in $x^d(y+z)^d$.
Note that the isomorphism $[x:y:z]\mapsto[y:x:z]$ transforms $z^d(y+\zeta x)^d$ in $z^d(y+\bar{\zeta}x)^d$ after an appropriate
diagonal isomorphism. Hence one obtains the following result.
\begin{prop}
\label{prop:sigma}
The realization set~$\Sigma^{(d)}$ decomposes in $\left(\left\lfloor\frac{d}{2}\right\rfloor+1\right)$ irreducible
(and connected) components
$\Sigma^{(d)}_\zeta$ parametrized by
$\TT_d^+:=\{\zeta\in\bc\mid\zeta^d=1,\im(\zeta)\geq 0\}$.
\end{prop}
\begin{proof}
As mentioned at the beginning of this section, we will prove the statement for the quotient space
$\PGL(3;\bc)\backslash\Sigma^{(d)}$, that is, up to projective automorphisms.
The above discussion implies that if $\cC\in\Sigma^{(d)}$, then there exists a projective transformation $\Phi_1$ sending $\cC$
to a curve $\cC_1=\{F_\zeta(x,y,z)=0\}$ (in the same connected component of~$\Sigma^{(d)}$) such that the polynomials associated
with the edges of its Newton polygon are $(x+z)^d, (y+z)^d, (y+\zeta x)^d$, $\zeta^d=1$. Moreover, after applying the symmetry with
respect to $z=0$ we can assume that $\im(\zeta)\geq 0$.
Assume the existence of another $\Phi_2\in\PGL(3;\bc)$ such that $\Phi_2(\cC):=\cC_2=\{F_\omega(x,y,z)=0\}$, whose associated
polynomials to the edges of its Newton polygon are $(x+z)^d, (y+z)^d, (y+\omega x)^d$, $\omega^d=1$, $\im(\omega)\geq 0$.
We want to prove that $\omega=\zeta$. This implies the existence of a projective transformation sending
$\{x+z=0, y+z=0, y+\zeta x=0\}$ to $\{x+z=0, y+z=0, y+\omega x=0\}$ and globally preserving the lines~$xyz=0$.
This transformation is a composition of a diagonal transformation and a permutation, and the result follows easily.
To end the proof one needs to show that for each $\zeta$ a curve $\cC\in \Sigma^{(d)}$ exists. Let us consider a polynomial
$F$ whose Newton polygon is as in Figure~\ref{fig:np}, with the fixed polynomials in the edges of the triangle. This fact does
not guarantee the topological type of the points $P_x,P_y,P_z$. Let us check for $P_z$. In order to have this topological type
it is enough that the homogeneous part of degree~$d$ is coprime with the homogeneous part of degree~$d+1$. This is achieved
for a generic choice of the coefficients of the inner triangle in Figure~\ref{fig:np} for $d>2$; the coefficients inside this
inner triangle have no effect in the local type. For a generic choice no other singular points will arise
(a confirmation of this fact will come later in this section). The connectivity for each $\zeta$ follows easily.
The case $d=2$ is special since the inner triangle mentioned above is empty. In that case, for $\zeta=-1$, one can check
that $x^2 y^2+ y^2 z^2 + x^2 z^2 +2 x y z(x+y-z)$ gives an appropriate equation; however, for $\zeta=1$, a non-reduced equation
is obtained.
\end{proof}
\subsection{The equisingular family \texorpdfstring{$\tilde\Sigma^{(d)}$}{tildeSigmad} of triangular curves associated with \texorpdfstring{$\Sigma^{(d)}$}{Sigmad}}
\mbox{}
Let $\tilde{\Sigma}^{(d)}$ be the realization space of the reducible curves formed by an element of $\Sigma^{(d)}$
and the three lines joining the singular points, which we refer to as \emph{triangular}.
This space coincides with the realization space of curves of degree~$2d+3$
with three singular points having the topological type of $u v((u+v)^d+v^{d+1})$. The argument starts by calculating the
multiplicity of intersection of a line joining two such singular points with a curve $\tilde \cC$ in $\tilde{\Sigma}^{(d)}$.
Since each singular point has multiplicity $d+2$, B\'ezout's Theorem implies that the lines are irreducible components of
$\tilde \cC$ and hence $\tilde \cC=L_1\cup L_2\cup L_3\cup \cC$ with $\cC\in \Sigma^{(d)}$. Hence combining the previous
discussion and Proposition~\ref{prop:sigma} one obtains the following.
\begin{cor}
The realization space~$\tilde{\Sigma}^{(d)}$ decomposes in $\left(\left\lfloor\frac{d}{2}\right\rfloor+1\right)$ irreducible
(and connected) components $\tilde{\Sigma}^{(d)}_\zeta$, parametrized by $\zeta\in \TT_d^+$.
\end{cor}
\subsection{An equisingular family \texorpdfstring{$\hat{\Sigma}^{(d)}$}{hatSigmad} of triangular curves birationally equivalent to \texorpdfstring{$\tilde{\Sigma}^{(d)}$}{tildeSigmad}}
\mbox{}
Applying the canonical Cremona transformation we obtain a family $\hat{\Sigma}^{(d)}$, the realization space of curves formed
by a smooth curve of degree~$d$ and three non-concurrent lines which are tangent at $d$-inflection points, which will also be
referred to as \emph{triangular}.
\begin{cor}
The realization space~$\hat{\Sigma}^{(d)}$ decomposes in $\left(\left\lfloor\frac{d}{2}\right\rfloor+1\right)$ irreducible
(and connected)
components $\hat{\Sigma}_\zeta^{(d)}$ parametrized by $\zeta\in \TT_d^+$.
\end{cor}
This fact was already studied in~\cite{CC:08,Sh:18}.
\section{Zariski tuples of conjugate curves}\label{sec:Zariski-tuples}
\subsection{A topological invariant of the embedding: the linking number}\label{sec:link}
\mbox{}
We review the results in~\cite{Sh:18} for completeness. Let $\hat \cC=\cC\cup L_1\cup L_2\cup L_3\in\hat{\Sigma}^{(d)}_\zeta$
for some $\zeta\in\TT_d^+$ where $\cC$ is the component of degree~$d$ of~$\hat \cC$. For the sake of simplicity the triangle
$L_1\cup L_2\cup L_3$ is denoted by $\cL$. As mentioned above, after a suitable
projective automorphism we can assume that $L_1=\{x=0\}$, $L_2=\{y=0\}$, $L_3=\{z=0\}$, and $\cC=\{F(x,y,z)=0\}$, where
\[
F(x,0,z)=(x+z)^d,\quad F(0,y,z)=(y+z)^d,\quad F(x,y,0)=(x+\zeta y)^d.
\]
Let us consider a model $S_\cC$ of the $d$-cyclic covering of $\bp^2$ ramified along~$\cC$,
\[
S_\cC:=\{[x:y:z:t]\in\bp^3\mid t^d=F(x,y,z)\}.
\]
Any deck transformation of the covering should be identified with elements of $H_1(\bp^2\setminus\cC;\bz)=\ZZ/d\ZZ$,
or analogously, with $\Hom(H_1(\bp^2\setminus\cC;\bz);\CC^*)$. Consider a character $\xi:H_1(\bp^2\setminus\cC;\bz)\to\bc^*$
such that if $\mu$ is a meridian of~$\cC$, then $\xi(\mu)=\exp\left(\frac{2i\pi}{d}\right)=:\zeta_d$.
This character is associated with the monodromy automorphism $\sigma:S_\cC\to S_\cC$,
$\sigma([x:y:z:t]):=[x:y:z:\zeta_d t])$ as follows. If $\gamma$ is a cycle in $\bp^2\setminus\cC$ based at $[x_0:y_0:z_0]$ and
$\tilde{\gamma}$ is the lift of $\gamma$ starting at $[x_0:y_0:z_0:t_0]\in S_\cC$, then the end point is
$[x_0:y_0:z_0:\xi(\gamma) t_0]$.
Let us construct a cycle~$\gamma$ representing the homology class of a loop based at $[0:1:0]$
as a composition of paths $\gamma_1*\gamma_2*\gamma_3$ with $\gamma_i\subset L_i\setminus\cC$.
Note that $\gamma$ is supported on $\cL\setminus\cC$ and it is based at $[0:1:0]$ with the
following trajectory:
\[
[0:1:0]\xrightarrow{\ \gamma_1\ }[0:0:1]\xrightarrow{\ \gamma_2\ }
[1:0:0]\xrightarrow{\ \gamma_3\ }[0:1:0].
\]
The fact that the contact order is~$d$ ensures that the choice of $\gamma$ does not affect the value of the character as
will be checked below.
The lift $\tilde{\gamma}=\tilde\gamma_1*\tilde\gamma_2*\tilde\gamma_3$ of $\gamma$ starting at $[0:1:0:1]$ can be
constructed as follows. First the path $\gamma_1$ is lifted, since $\cC\cap L_1$ has equations $x=t^d-(y+z)^d=0$,
it has $d$ irreducible components, and since $\im(\gamma)\cap \cC=\emptyset$, the path $\gamma_1$ lifts to $\tilde\gamma_1$
in $S_1=\{x=t-(y+z)=0\}\subset S_\cC$. Note that the lift $\tilde{\gamma}_1$ ends at~$[0:0:1:1]$, which is the only
point in $S_1$ and the fiber of $\gamma_1(1)=[0:0:1]$. Analogously, since $\cC\cap L_2$ is given by $\{y=t^d-(x+z)^d=0\}$,
the lift $\tilde{\gamma}_2$ is supported on $\{y=t-(x+z)=0\}\subset S_\cC$ and it ends at~$[1:0:0:1]$.
Finally, the equations of $\cC\cap L_3$ are $\{z=t^d-(x+\zeta y)^d=0\}$, and thus $\tilde{\gamma}_3$ is contained in
$\{z=t-(x+\zeta y)=0\}\subset S_\cC$ whose end is~$[0:1:0:\zeta]$, i.e., $\xi(\gamma)=\zeta$, which agrees with the action
of the monodromy~$\sigma$ on~$\gamma$ as discussed above.
Following~\cite{afg:17}, we will sketch how to obtain a topological invariant from these data.
Let $\hat\xi:H_1(\bp^2\setminus\hat \cC;\bz)\to\bc^*$ be the restriction of a character $\xi$ on $\bp^2\setminus\cC$ obtained
as $\hat{\xi}:=\xi\circ i_*$, where $i:\bp^2\setminus\hat \cC\to\bp^2\setminus\cC$ is the inclusion.
Consider $\hat\gamma$ a cycle homologous to $\gamma$ in $H_1(\bp^2\setminus\cC;\bz)$ whose support is outside
$\cL=L_1\cup L_2\cup L_3$. Then $\hat\xi(\gamma):=\hat\xi(\hat \gamma)=\zeta$ (since the character $\hat\xi$ is trivial
on the meridians of the lines) and hence $\hat\xi(\gamma)$ is well defined since it does not depend on the representative
$\hat \gamma$. Then, the value of $\hat\xi(\gamma)$ is an oriented topological invariant of $(\bp^2,\hat \cC^{\text{ord}},\hat\xi)$,
where the irreducible components in~$\hat \cC$ are ordered. Analogously, the set
$\{\hat\xi(\gamma)^{\pm 1}\}$ is a topological invariant of
$(\bp^2,\hat \cC,\hat\xi)$ when the conditions on orientation and ordering are removed.
\subsection{Main results}
\mbox{}
For completeness, we include the following result, already proven by T.~Shirane, to be refine
in Theorem~\ref{thm:complement}.
\begin{thm}[\cite{Sh:18}]\label{thm:sh}
If $\hat\cC_{\zeta_i}\in\hat{\Sigma}^{(d)}_{\zeta_i}$, $\zeta_i\in \TT_d^+$, then the pairs
$(\bp^2,\hat\cC_{\zeta_1})$ and $(\bp^2,\hat\cC_{\zeta_2})$ are homeomorphic if and only if $\zeta_1=\zeta_2$.
\end{thm}
\begin{proof}
Consider $\xi$ the character defined by $\xi(\mu)=\zeta_d$ for a meridian $\mu$ of $\cC$ and $\gamma$ the cycle described above.
Note that the pair $(\xi(\mu),\{\hat\xi(\gamma)^{\pm 1}\})$ becomes $(\zeta_d,\{\zeta^{\pm 1}_i\})$ when applied to the curve $\hat\cC_{\zeta_i}$.
Since this is an invariant of $(\bp^2,\hat \cC_{\zeta_i},\hat\xi)$ and $\{\zeta_i,\bar\zeta_i\}\neq \{\zeta_j,\bar\zeta_j\}$,
$\zeta_i,\zeta_j\in \TT_d^+$ if and only if $i=j$, this proves the \emph{only if} part.
The \emph{if} part follows from the fact that two curves in the same equisingular connected component have homeomorphic
embeddings. The homeomorphism is obtained via the locally trivial fibration along a compact differentiable path joining both
curves in the equisingular stratum of~$\hat\Sigma^{(d)}$.
\end{proof}
\begin{thm}\label{thm:complement}
Given $\hat\cC_{\zeta_i}\in\hat{\Sigma}^{(d)}_{\zeta_i}$, $\zeta_i\in \TT_d^+$, then
$\bp^2\setminus\hat\cC_{\zeta_1}$ and $\bp^2\setminus\hat\cC_{\zeta_2}$ are homeomorphic
if and only if $\zeta_1=\zeta_2$.
\end{thm}
\begin{proof}
Let $U(\hat\cC_\zeta)$ be a compact regular neighborhood of $\hat\cC_\zeta$ and let $M_\zeta:=\partial U(\hat\cC_\zeta)$
denote its boundary manifold. This is a graph manifold associated with the plumbing graph of a minimal resolution
of~$\hat\cC_\zeta$ (that is, the preimage of $\hat \cC_\zeta$ is normal crossing, but that can have self-intersections).
In principle, since $M_\zeta$ is associated with a normal bundle on the regular part of the normal crossing divisor
associated with the minimal resolution of $\hat \cC_\zeta$, the inclusion
$M_\zeta\hookrightarrow \bp^2\setminus\hat\cC_{\zeta}$ induces only an isomorphism
$\pi_1(M_\zeta)\to\pi_1^\infty(\bp^2\setminus\hat\cC_{\zeta})$. Nevertheless, a homeomorphism
$\Psi:\bp^2\setminus\hat\cC_{\zeta_1}\to\bp^2\setminus\hat\cC_{\zeta_2}$ induces a homeomorphism of Milnor balls around
the singular points of the curves and hence an isomorphism $\Psi^\infty:\pi_1(M_{\zeta_1})\to\pi_1(M_{\zeta_2})$.
From the properties of sufficiently large graph manifolds (see~\cite{wald:68}), this implies the existence of a homeomorphism
$\Psi_M:M_{\zeta_1}\to M_{\zeta_2}$ such that ${\Psi_M}_*=\Psi^\infty$.
Since the boundary manifolds $M_{\zeta_i}$ come with an extra structure due to the Milnor fibration around the singular points
this homeomorphism preserves the graph structure (see~\cite{wal:67b} and also the Appendix in~\cite{Neumann-Acalculus}).
In particular, a meridian $\mu_{\zeta_1}$ of $\cC_{\zeta_1}$ must be sent to $\mu_{\zeta_2}^{\pm 1}$, $\pm 1$-power
of a meridian of $\cC_{\zeta_2}$. Moreover the above cycle $\gamma_{\zeta_1}$ must be sent to a $\pm 1$-power of
$\gamma_{\zeta_2}$. Hence $\zeta_1=\zeta_2$ if such a homeomorphism exists.
\end{proof}
\begin{cor}
If $\tilde\cC_{\zeta_i}\in\tilde{\Sigma}^{(d)}_{\zeta_i}$, $\zeta_i\in \TT_d^+$, for $i=1,2$, then
$\bp^2\setminus\tilde\cC_{\zeta_1}$ and $\bp^2\setminus\tilde\cC_{\zeta_2}$ are homeomorphic if and only if~$\zeta_1=\zeta_2$.
\end{cor}
\begin{proof}
Recall that $\tilde\cC_{\zeta}\in\tilde{\Sigma}^{(d)}_{\zeta}$ is obtained from $\hat\cC_{\zeta}\in\hat{\Sigma}^{(d)}_{\zeta}$
via a standard Cremona transformation. This birational morphism produces a homeomorphism on their complements.
Theorem~\ref{thm:complement} applies and the result follows.
\end{proof}
\section{Fundamental groups of complements of curves in \texorpdfstring{$\Sigma^{(d)}$}{Sigma d},
\texorpdfstring{$\tilde\Sigma^{(d)}$}{tilde Sigma d}, and \texorpdfstring{$\hat\Sigma^{(d)}$}{hat Sigma d}}
\label{sec:pi1}
Our goal in this section is to compute the fundamental group for any curve in $\tilde\cC\in\tilde{\Sigma}_{\zeta}^{(d)}$
and for any~$d$. We will do this using two different techniques.
\subsection{Lower degree cases}
\mbox{}
Fundamental groups of complements of curves in $\Sigma^{(d)}$ and $\tilde\Sigma^{(d)}$ or $\hat\Sigma^{(d)}$ are
only well understood in the simplest cases, $d=2,3$. Here is a brief account of what is known.
For the case $d=2$, as mentioned above, it is classically known (see~\cite{zr:29}) that the fundamental group of the
complement of the tricuspidal quartic $\cC\in\Sigma_{-1}^{(2)}$ is the finite metabelian group of order 12
$$\BB_3(\mathbb S^2)=
\langle \sigma_1,\sigma_2: \sigma_1\sigma_2\sigma_1=\sigma_2\sigma_1\sigma_2, \sigma_1\sigma_2^2\sigma_1=1\rangle
\cong \ZZ_3 \rtimes\ZZ_4.$$
It is an easy exercise to check that the fundamental group of the complement of a curve
$\tilde\cC\in\tilde{\Sigma}_{-1}^{(2)}$ (a conic with three tangent lines) is the triangle Artin group~$(2,4,4)$.
For the case $d=3$, the fundamental group of the complement of a sextic curve with $3\EE_6$ singularities depended
on the connected component in $\Sigma^{(3)}$ it belongs (see~\cite{ac:98})
$$\pi_1(\PP^2\setminus \cC)=
\begin{cases}
\ZZ/2\ZZ*\ZZ/3\ZZ & \textrm{ if } \cC\in \Sigma_{1}^{(3)}\\
\ZZ/2\ZZ\times\ZZ/3\ZZ & \textrm{ if } \cC\in \Sigma_{\zeta_3}^{(3)},
\end{cases}
$$
where $\TT_3^+=\{1,\zeta_3\}$.
As for $\tilde{\Sigma}_{\zeta_3}^{(3)}$ it is also shown in~\cite{ac:98} that $\pi_1(\PP^2\setminus \tilde\cC)=\ZZ^3$
for $\tilde\cC\in\tilde{\Sigma}_{\zeta_3}^{(3)}$.
\subsection{Fermat curves}
\label{sec:fermat}
\mbox{}
In this section we present a way to calculate fundamental groups via finite coverings ramified along simpler curves.
In this case we will use Fermat curves.
Let us consider the Kummer map $\Phi_d:\bp^2\to\bp^2$ given by $\Phi_d([x:y:z])=[x^d:y^d:z^d]$.
Note that if $L=\{x+y+z=0\}$ then $\Phi_d^*(L)$ is a Fermat curve. The preimage of $L_x=\{y+z=0\}$
are $d$ tangent lines to $d$-inflection points, denoted by $L_{x,\tau}=\{y-\tau z=0\}$,
where $\tau^d=-1$. In the same way we consider $L_y=\{x+z=0\}$, $L_{y,\tau}=\{z-\tau x=0\}$
and also $L_z=\{x+y=0\}$, $L_{z,\tau}=\{x-\tau y=0\}$.
\begin{prop}
\label{prop:construction}
Let $\bt:=(\tau_1,\tau_2,\tau_3)$ where $\tau_i$ is a $d$-root of $-1$ and consider the following curve constructions:
\begin{enumerate}[label=\rm(\arabic{enumi})]
\item\label{prop:uno}
$\hat\cC_{d,\bt,1}:=\Phi_d^*(L)\cup L_{x,\tau_1}\cup L_{y,\tau_2}\cup L_{z,\tau_3}$ with $\tau:=\tau_1\tau_2\tau_3$, $\zeta:=\tau^2$,
\item\label{prop:dos}
$\hat\cC_{d,\bt,2}:=\Phi_d^*(L)\cup L_{x,\tau_1}\cup L_{x,\tau_2}\cup L_{z,\tau_3}$ with $\zeta:=\tau_1\tau_2^{-1}$
and $\tau_1\neq \tau_2$,
\item\label{prop:tres}
$\hat\cC_{d,\bt,3}:=\Phi_d^*(L)\cup L_{x,\tau_1}\cup L_{x,\tau_2}\cup L_{x,\tau_3}$ with $\zeta:=1$ and $\tau_i\neq \tau_j$,
for $i\neq j$.
\end{enumerate}
Then $\hat\cC_{d,\bt,i}\in \hat{\Sigma}^{(d)}_{\zeta}$, $i=1,2$, where $\zeta\in \TT_{d}^+$.
For $i=3$, $\hat\cC_{d,\bt,3}\in \overline{\hat{\Sigma}^{(d)}_{1}}$ the closure of the component $\hat{\Sigma}^{(d)}_{1}$.
Moreover, one can find a continuous equisingular family of curves $\{\hat\cC_t\}_{t\in(0,\varepsilon]}$ in $\hat{\Sigma}^{(d)}_{1}$
such that $\{\hat\cC_t\}_{t\in(0,\varepsilon]}\to\hat\cC_{d,\bt,3}$
where the triangle degenerates onto an ordinary triple point.
\end{prop}
\begin{proof}
For the family of curves in~\ref{prop:uno} consider the following change of coordinates:
\[
x_1=\tau x+y+z,\quad
y_1=\tau_3^{-1}(\tau x+y+\tau z),\quad
z_1=\tau_2(\tau x+z+\tau^{-1} y).
\]
Let us consider the homogeneous polynomial
$F(x,y,z)=x_1^d+y_1^d+z_1^d$. Then
\[
F(0,y,z)=(y+z)^d,\quad
F(x,0,z)=(z+x)^d,\quad
F(x,y,0)=(\zeta x+y)^d,
\]
and the statement follows.
The proof for the family of curves in construction~\ref{prop:dos} is analogous considering the following change of coordinates:
\[
x_1= x+ y+z,\quad
y_1=\tau_3(x+y),\quad
z_1=\tau_3(\tau_1 x+\tau_2 y)
\]
and
$F(x,y,z)=x_1^d+y_1^d+z_1^d$. Then
\[
F(0,y,z)=(y+z)^d,\quad
F(x,0,z)=(x+z)^d,\quad
F(x,y,0)=(x+\zeta^{-1} y)^d.
\]
For construction~\ref{prop:tres} consider the following family
\[
\{(x-\tau_1 y)(x-\tau_2 y)(x-\tau_3 y-t z)\left(x^d+y^d+z^d-t z\frac{x^d+y^d}{x-\tau_3 y}\right)=0\}_{t\in(0,\varepsilon]}.
\]
One can check that for $0<t\leq\varepsilon$ these curves are union of a smooth curve of degree~$d$ and three tangent lines
at aligned $d$-flexes. The fact that these flexes are aligned only occurs if the curve is in $\hat{\Sigma}^{(d)}_{1}$.
\end{proof}
\begin{obs}
Note that construction~\ref{prop:uno} in Proposition~\ref{prop:construction} produces representatives in
$\hat{\Sigma}^{(d)}_{\zeta}$ for all $\zeta\in\TT_d^+$ in the odd case, whereas construction~\ref{prop:dos}
produces representatives in $\hat{\Sigma}^{(d)}_{\zeta}$ as long as $\zeta\neq 1$. The only representative not
produced directly this way is one in the component $\hat{\Sigma}^{(d)}_{1}$ for $d$ even.
\end{obs}
\subsection{Fundamental group computation}
\mbox{}
We start with the fundamental group of a simple line arrangement, whose computation
can be done using its real affine picture shown in Figure~\ref{fig:lines1}
where $\ell_x=\{y=0\}$, $\tilde\ell_x=\{y+z=0\}$, $\ell_y=\{x=0\}$, $\tilde\ell_y=\{x+z=0\}$, $\ell=\{x+y+z=0\}$,
and $\ell_\infty=\{z=0\}$.
\begin{figure}[ht]
\begin{center}
\begin{tikzpicture}[vertice/.style={draw,circle,fill,minimum size=0.2cm,inner sep=0}]
\tikzset{%
suma/.style args={#1 and #2}{to path={%
($(\tikztostart)!-#1!(\tikztotarget)$)--($(\tikztotarget)!-#2!(\tikztostart)$)%
\tikztonodes}}
}
\coordinate (XY) at (0,0);
\coordinate (X) at (-1,0);
\coordinate (Y) at (0,-1);
\coordinate (Z) at (-1,-1);
\coordinate (X1) at (-1,1);
\draw[suma=.5 and .5] (X) to (XY) node[right] {$\ell_x$};
\draw[suma=.5 and .5] (Y) to (XY) node[above] {$\ell_y$} ;
\draw[suma=.5 and .5,dashed] ($.5*(XY)+.5*(X)$) to ($.5*(Y)+.5*(Z)$);
\draw[suma=.5 and .5] (X) to (Y) node[right] {$\ell$};
\draw[suma=.5 and .5] (Z) to (X) node[above] {$\tilde\ell_{y}$};
\draw[suma=.5 and 1.5] (Z) to (Y) node[above] {$\tilde\ell_{x}$};
\end{tikzpicture}
\caption{$6$-line arrangement}
\label{fig:lines1}
\end{center}
\end{figure}
\begin{prop}
The fundamental group $G$ of the complement of the line arrangement
$\ell_x\cup \ell_y\cup \tilde\ell_x\cup \tilde\ell_y\cup \ell\cup \ell_\infty\subset \bp^2$ has generators
$\gamma_x,\gamma_y,\gamma_\ell,\tilde\gamma_{x},\tilde\gamma_{y}$ and relations
\begin{multicols}{3}
\begin{enumerate}[label=\rm(G\arabic*)]
\item\label{rel1-xy} $[\tilde\gamma_{x},\tilde\gamma_{y}]=1$,
\item\label{rel1-tilde-x1} $[\gamma_y\cdot\gamma_\ell,\tilde\gamma_{x}]=1$,
\item\label{rel1-tilde-x2} $[\tilde\gamma_{x}\cdot \gamma_y,\gamma_\ell]=1$,
\end{enumerate}
\end{multicols}
\vspace{-5mm}
\begin{multicols}{3}
\begin{enumerate}[label=\rm(G\arabic*)]
\setcounter{enumi}{3}
\item\label{rel1-tilde-l1} $[\gamma_y,\gamma_x]=1$,
\item\label{rel1-tilde-y1} $[\gamma_x\cdot\tilde\gamma_{y},\gamma_\ell]=1$,
\item\label{rel1-tilde-y2} $[\gamma_\ell\cdot \gamma_x,\tilde\gamma_{y}]=1$.
\end{enumerate}
\end{multicols}
Moreover, the orbifold fundamental group~$\tilde G$ of the complement of $\ell\cup \tilde\ell_x\cup \tilde\ell_y\subset \PP^2$
with orbifold locus of order~$d$ along $\ell_x\cup \ell_y\cup \ell_\infty$, is obtained as a quotient of $G$ by the
normal subgroup generated by the relations
\begin{multicols}{3}
\begin{enumerate}[label=\rm(G\arabic*)]
\setcounter{enumi}{6}
\item\label{rel1-x} $\gamma_x^d=1$,
\item\label{rel1-y} $\gamma_y^d=1$,
\item\label{rel1-z}
$\gamma_\infty^d=1$,
\end{enumerate}
\end{multicols}
where $\gamma_\infty=(\tilde\gamma_{x}\cdot\gamma_\ell\cdot \gamma_x\cdot \tilde\gamma_{y}\cdot \gamma_y)^{-1}$.
\end{prop}
It is important to stress that the previous presentations have a geometrical meaning which will be relevant in the
upcoming calculations. The generators $\gamma_x$, $\gamma_\ell$, and $\tilde\gamma_x$ on the dashed vertical line in
Figure~\ref{fig:lines1} are meridians around the lines $\ell_x$, $\ell$, and $\tilde\ell_x$ respectively. They are
chosen as a geometrical basis on the punctured line and their product $\tilde\gamma_{x}\cdot\gamma_\ell\cdot \gamma_x$
is the inverse of a meridian around the point at infinity of such a line. The remaining generators $\tilde\gamma_{y}$,
$\gamma_y$ are meridians around the lines $\tilde\ell_y$ and $\ell_y$ respectively, on a horizontal line in
Figure~\ref{fig:lines1}.
namely, $\gamma_x$ (resp.~$\gamma_y$) is a meridian of $\ell_x$ (resp.~$\ell_y$),
$\tilde\gamma_x$ (resp.~$\tilde\gamma_y$) is a meridian of $\tilde\ell_x$ (resp.~$\tilde\ell_y$),
and so are $\gamma_\ell$ with respect to $\ell$ and $\gamma_\infty$ with respect to $\ell_\infty$.
Following the construction given in section~\ref{sec:construction} one needs to study the Kummer cover
\[
\sigma:\PP^2\to\PP^2,\qquad [x:y:z]\mapsto[x^d:y^d:z^d].
\]
Note that $\sigma$ is an abelian cover of order $d^2$ ramified along
$\ell_x\cup \ell_y\cup \ell_\infty$ with ramification index $d$ whose group of deck transformations is
$\ZZ/d\ZZ\times \ZZ/d\ZZ$. A possible strategy is to decompose $\sigma$ in two cyclic covers.
The first one $\sigma_1:S\to\bp^2$ can be seen as an orbifold covering ramified along the lines $\ell_y$ and $\ell_\infty$.
The second one $\sigma_2:\bp^2\to S$ can be seen as an orbifold covering ramified along the preimages by~$\sigma_1$
of $\ell_x$ and $[1:0:0]$. If we remove the line arrangement, the covering $\sigma_1$ is defined by an epimorphism
$\rho_1:\tilde G\to\mu_d$, $\rho(\gamma_y):=\zeta$, $d$-root of unity, and
$\rho(\gamma_x)=\rho(\tilde\gamma_{x})=\rho(\tilde\gamma_{y})=\rho(\gamma_\ell)=1$.
Let $\hat{K}:=\ker\rho_1$. Note that $\sigma_1^{-1}(\tilde\ell_y)$ breaks in $d$ irreducible
components, whose meridians are $\gamma_y^j\cdot\tilde\gamma_{y}\cdot \gamma_y^{j-1}$,
$0\leq j<d$. By Proposition~\ref{prop:construction}\ref{prop:dos}, we are interested in
keeping only one of these lines, say the one associated with~$\tilde\gamma_{y}$. Let $\hat{K}_1$
be the quotient of $\hat{K}$ obtained by killing the remaining meridians. It is the
orbifold fundamental group of the complement in~$S$ of the line arrangement with orbifold
structure at the preimage of $\ell_x$ and $[1:0:0]$.
\begin{lema}
The group $\hat{K}_1$ is generated by $\gamma_x,\gamma_\ell,\tilde\gamma_{x},\tilde\gamma_{y}$
with relations:
\begin{multicols}{3}
\begin{enumerate}[label=\rm(K\arabic*)]
\item\label{rel1-x-k} $\gamma_x^d=1$,
\item\label{rel1-z-k}
$(\tilde\gamma_{x}\cdot\gamma_\ell\cdot \gamma_x)^d\cdot\tilde\gamma_{y}=1$,
\item\label{rel1-xy-k} $[\tilde\gamma_{x},\tilde\gamma_{y}]=1$,
\end{enumerate}
\end{multicols}
\vspace{-7mm}
\begin{multicols}{3}
\begin{enumerate}[label=\rm(K\arabic*)]
\setcounter{enumi}{3}
\item\label{rel1-tilde-x1-k} $\left[(\ell\cdot\tilde\gamma_{x})^d,\tilde\gamma_{x}\right]=1$,
\item\label{rel1-tilde-y1-k} $[\gamma_x\cdot\tilde\gamma_{y},\gamma_\ell]=1$,
\item\label{rel1-tilde-y2-k} $[\gamma_\ell\cdot \gamma_x,\tilde\gamma_{y}]=1$,
\end{enumerate}
\end{multicols}
\vspace{-3mm}
\begin{enumerate}[label=\rm(K\arabic*)]
\setcounter{enumi}{6}
\item \label{rel1-j-k} $\left[\gamma_x,(\ell\cdot\tilde\gamma_{x})^{-j}\cdot\gamma_\ell\cdot(\gamma_\ell\cdot\tilde\gamma_{x})^j\right]=1$, for $0<j<d$.
\end{enumerate}
\end{lema}
\begin{proof}
We use Reidemeister-Schreier method. For each generator $\alpha$ of $\tilde G$, different from $\gamma_y$,
the generators are $\alpha_j:=\gamma_y^j\cdot\alpha\cdot \gamma_y^{-j}$, $j=0,1,\dots,d-1$. The element
$\gamma_y^d$ should be also a generator, which can be avoided by~\ref{rel1-y}.
Using~\ref{rel1-tilde-l1}, we obtain
\begin{equation*}
\gamma_{x_j}:=\gamma_y^j\cdot \gamma_x\cdot \gamma_y^{-j}=\gamma_x,
\end{equation*}
and we keep only $\gamma_{x_0}=\gamma_x$ as generator from $\gamma_{x_0},\dots,\gamma_{x_{d-1}}$.
The relation~\ref{rel1-x} is kept (all its conjugates provide the same relation).
By the definition of $\hat{K}_1$, we keep only $\tilde\gamma_{y}=\tilde\gamma_{{y}_0}$ as
$\tilde\gamma_{{y}_j}=1$, for $0<j<d$.
The relation~\ref{rel1-xy} also remains and it is unique. We use relations~\ref{rel1-tilde-x1}
and~\ref{rel1-tilde-x2} to eliminate some generators since
\begin{equation*}
\gamma_{\ell_j}=\gamma_y^j\cdot\gamma_\ell\cdot \gamma_y^{-j}=
(\gamma_\ell\cdot\tilde\gamma_{x})^{-j}\cdot\gamma_\ell\cdot(\gamma_\ell\cdot\tilde\gamma_{x})^j,
\quad
\tilde\gamma_{x_j}=\gamma_y^j\cdot\tilde\gamma_{x}\cdot \gamma_y^{-j}=
(\gamma_\ell\cdot\tilde\gamma_{x})^{-j}\cdot\tilde\gamma_{x}\cdot(\gamma_\ell\cdot\tilde\gamma_{x})^j,
\end{equation*}
producing the relation
\begin{equation*}
\left[(\gamma_\ell\cdot\tilde\gamma_{x})^d,\tilde\gamma_{x}\right]=1.
\end{equation*}
The relations given by~\ref{rel1-tilde-y1}
and~\ref{rel1-tilde-y2} are kept and produce for $0<j<d$:
\begin{equation*}
\left[\gamma_x,(\gamma_\ell\cdot\tilde\gamma_{x})^{-j}\cdot\gamma_\ell\cdot(\gamma_\ell\cdot\tilde\gamma_{x})^j\right]=1.
\end{equation*}
Finally the relation~\ref{rel1-z} becomes
\begin{equation*}
(\tilde\gamma_{x}\cdot\gamma_\ell\cdot \gamma_x)^d\cdot\tilde\gamma_{y}=1.
\qedhere
\end{equation*}
\end{proof}
The covering $\rho_2$ is defined by the morphism $\rho_2:\hat{K}_1\to\mu_d$ such that
$\rho_2(\gamma_x)=\zeta$ and the other generators are sent to~$1$.
\begin{lema}
The group $\tilde{K}:=\ker\rho_2$ is generated by
$\gamma_\ell,\tilde\gamma_{y},\tilde\gamma_{x_i}:=\gamma_x^i\cdot\tilde\gamma_{x}\cdot \gamma_x^{-i}$, $0\leq i<d$
with relations:
\begin{enumerate}[label=\rm($\tilde{\text{K}}$\arabic*)]
\item $\left[(\tilde\gamma_{y}\cdot\gamma_\ell)^d,\tilde\gamma_{y}\right]=1$,
\item\label{tk2} $[\tilde\gamma_{x_i},\left(\tilde\gamma_{y}\cdot\gamma_\ell\right)^{-i}\cdot\tilde\gamma_{y}\cdot
\left(\tilde\gamma_{y}\cdot\gamma_\ell\right)^{i}]=1$ for $0\leq i<d$,
\item\label{tk3} $\prod_{j=0}^{d-1}(\tilde\gamma_{x_j}\cdot(\tilde\gamma_{y}\cdot\gamma_\ell)^{-j}\cdot\gamma_\ell
\cdot (\tilde\gamma_{y}\cdot\gamma_\ell)^{j})\cdot \tilde\gamma_{y}=1$,
\item\label{tk4} $\left[(\left(\tilde\gamma_{y}\cdot\gamma_\ell\right)^{-i}\cdot\gamma_\ell\cdot
\left(\tilde{y}\cdot\gamma_\ell\right)^{i}\cdot\tilde\gamma_{x_i})^d,\tilde\gamma_{x_i}\right]=1$
for $0\leq i<d$,
\item\label{tk5} $1=[\gamma_\ell,(\gamma_\ell\cdot\left(\tilde\gamma_{y}\cdot\gamma_\ell\right)^{i}
\cdot\tilde\gamma_{x_i}\cdot\left(\tilde\gamma_{y}\cdot\gamma_\ell\right)^{-i})^j\cdot \left(\tilde\gamma_{y}\cdot\gamma_\ell\right)^{-1}\cdot (\gamma_\ell\cdot\left(\tilde\gamma_{y}\cdot\gamma_\ell\right)^{i+1}\cdot\tilde\gamma_{x_{i+1}}\cdot
\left(\tilde\gamma_{y}\cdot\gamma_\ell\right)^{-i-1})^{-j}]$, for $0<j<d$ and $0\leq i<d$.
\end{enumerate}
\end{lema}
\begin{proof}
We apply again Reidemeister-Schreier algorithm. We should start
with generators $\gamma_{\ell_j}=\gamma_x^j\cdot\gamma_\ell \gamma_x^{-j}$, $\tilde\gamma_{y_j}=
\gamma_x^j\cdot\tilde\gamma_{y} \gamma_x^{-j}$,
$\tilde\gamma_{x_j}=\gamma_x^j\cdot\tilde\gamma_{x} \gamma_x^{-j}$, and $\gamma_x^d$. The generator
$\gamma_x^d$ drops needed because of~\ref{rel1-x-k}.
Using~\ref{rel1-tilde-y1-k} and~\ref{rel1-tilde-y2-k}, we eliminate some generators
\begin{equation*}
\gamma_{\ell_i}=\gamma_x^i\cdot\gamma_\ell\cdot \gamma_x^{-i}=
\left(\tilde\gamma_{y}\cdot\gamma_\ell\right)^{-i}\cdot\gamma_\ell\cdot \left(\tilde\gamma_{y}\cdot\gamma_\ell\right)^{i},\quad
\tilde\gamma_{y_i}=\gamma_x^i\cdot\tilde\gamma_{y}\cdot \gamma_x^{-i}=
\left(\tilde\gamma_{y}\cdot\gamma_\ell\right)^{-i}\cdot\tilde\gamma_{y}\cdot
\left(\tilde\gamma_{y}\cdot\gamma_\ell\right)^{i},
\end{equation*}
and obtain the relation
\begin{equation*}
\left[(\tilde\gamma_{y}\cdot\gamma_\ell)^d,\tilde\gamma_{y}\right]=1.
\end{equation*}
The generator statement has been proven.
The relation~\ref{rel1-xy-k} becomes $d$ relations ($0\leq i<d$)
\[
[\tilde\gamma_{x_i},\left(\tilde\gamma_{y}\cdot\gamma_\ell\right)^{-i}\cdot\tilde\gamma_{y}\cdot
\left(\tilde\gamma_{y}\cdot\gamma_\ell\right)^{i}]=1.
\]
Using the above commutation relations, the relation~\ref{rel1-z-k} becomes
\[
\prod_{j=0}^{d-1}(\tilde\gamma_{x_j}\cdot(\tilde\gamma_{y}\cdot\gamma_\ell)^{-j}\cdot\gamma_\ell\cdot
(\tilde\gamma_{y}\cdot\gamma_\ell)^{j})\cdot \tilde\gamma_{y}=1.
\]
The relation~\ref{rel1-tilde-x1-k} becomes $d$ relations ($0\leq i<d$)
\begin{equation*}
\left[(\left(\tilde\gamma_{y}\cdot\gamma_\ell\right)^{-i}\cdot\gamma_\ell\cdot
\left(\tilde\gamma_{y}\cdot\gamma_\ell\right)^{i}\cdot\tilde\gamma_{x_i})^d,\tilde\gamma_{x_i}\right]=1.
\end{equation*}
Finally the relations \ref{rel1-j-k} produce $d(d-1)$ relations for $0<j<d$ and $0\leq i<d$:
\begin{gather*}
\!(\left(\tilde\gamma_{y}\cdot\gamma_\ell\right)^{-i}\cdot\gamma_\ell\cdot
\left(\tilde\gamma_{y}\cdot\gamma_\ell\right)^{i}\cdot\tilde\gamma_{x_i})^{-j}\cdot
\left(\tilde\gamma_{y}\cdot\gamma_\ell\right)^{-i}\cdot\gamma_\ell\cdot
\left(\tilde\gamma_{y}\cdot\gamma_\ell\right)^{i}\cdot (\left(\tilde\gamma_{y}\cdot
\gamma_\ell\right)^{-i}\cdot\gamma_\ell\cdot\left(\tilde\gamma_{y}\cdot\gamma_\ell\right)^{i}\cdot \tilde\gamma_{x_i})^j=\\
(\left(\tilde\gamma_{y}\!\cdot\!\gamma_\ell\right)^{-i-1}\!\!\!\cdot\!\gamma_\ell\!\cdot\!\left(\tilde\gamma_{y}\cdot
\gamma_\ell\right)^{i+1}\!\!\cdot\tilde\gamma_{x_{i+1}})^{-j}\!\cdot\!\left(\tilde\gamma_{y}\!\!\cdot
\gamma_\ell\right)^{-i-1}\!\!\!\cdot\!\gamma_\ell\!\cdot\!\left(\tilde\gamma_{y}\!\cdot\!\gamma_\ell\right)^{i+1}\!\!\cdot\!
(\!\left(\tilde\gamma_{y}\!\cdot\!\gamma_\ell\right)^{-i-1}\!\!\!\cdot\!\gamma_\ell\!\cdot\!\left(\tilde\gamma_{y}\!\cdot\!
\gamma_\ell\right)^{i+1}\!\!\cdot\!\tilde\gamma_{x_{i+1}})^j,
\end{gather*}
that is,
\begin{gather*}
1=[\gamma_\ell,(\gamma_\ell\cdot\left(\tilde\gamma_{y}\cdot\gamma_\ell\right)^{i}\cdot\tilde\gamma_{x_i}\cdot
\left(\tilde\gamma_{y}\cdot\gamma_\ell\right)^{-i})^j\cdot \left(\tilde\gamma_{y}\cdot\gamma_\ell\right)^{-1}\cdot
(\gamma_\ell\cdot\left(\tilde\gamma_{y}\cdot\gamma_\ell\right)^{i+1}\cdot\tilde\gamma_{x_{i+1}}\cdot
\left(\tilde\gamma_{y}\cdot\gamma_\ell\right)^{-i-1})^{-j}].
\qedhere
\end{gather*}
\end{proof}
\subsection{Main results on fundamental groups}
\mbox{}
In this section we summarize the main consequences in terms of fundamental groups for the curves in the
equisingular stratum $\hat{\Sigma}^{(d)}_\zeta$.
\begin{thm}\label{thm:sigmazeta}
Let $\cC\in\hat{\Sigma}^{(d)}_\zeta$, $\zeta\neq 1$, $d>2$. Its fundamental group is abelian, in particular
$\pi_1(\PP^2\setminus \cC)=\ZZ^3$.
\end{thm}
\begin{proof}
It is enough to prove the statement for a curve as in
Proposition{\rm~\ref{prop:construction}\ref{prop:dos}}, for which its fundamental group is $K_h$, $0<h<d$, the quotient of $\hat{K}_1$ obtained by
\emph{killing} $\tilde\gamma_{x_j}$, if $j\neq 0,h$. We can assume $h\geq 2$.
The relation~\ref{tk2} produces $[\tilde\gamma_{x_0},\tilde\gamma_{y}]=[\tilde\gamma_{x_h},(\tilde\gamma_{y}\cdot
\gamma_\ell)^{-h}\cdot\tilde\gamma_{y}\cdot(\tilde\gamma_{y}\cdot\gamma_\ell)^{h}]=1$. The relation~\ref{tk3} becomes
\[
\tilde\gamma_{x_0}\cdot(\tilde\gamma_{y}\cdot\gamma_\ell)^{h}\cdot
\tilde\gamma_{x_h}\cdot(\tilde\gamma_{y}\cdot\gamma_\ell)^{-h}\cdot(\tilde\gamma_{y}\cdot\gamma_\ell)^{d}\cdot
\tilde\gamma_{y}^{1-d}=1
\]
and $\tilde\gamma_{x_h}$ can be taken out from the generators. The second relation above is equivalent to the commutator
$[(\tilde\gamma_{y}\cdot\gamma_\ell)^{d},\tilde\gamma_{x_0}]=1$.
For \ref{tk4} we obtain
\[
\left[(\gamma_\ell
\cdot\tilde\gamma_{x_0})^d,\tilde\gamma_{x_0}\right]=
\left[(\left(\tilde\gamma_{y}\cdot\gamma_\ell\right)^{-h}\cdot\gamma_\ell\cdot
\left(\tilde\gamma_{y}\cdot\gamma_\ell\right)^{h}\cdot\tilde\gamma_{x_h})^d,\tilde\gamma_{x_h}\right]=1.
\]
Let us consider \ref{tk5} for $j=1,2$:
\[
[\gamma_\ell,\left(\tilde\gamma_{y}\cdot\gamma_\ell\right)^{i}\cdot\tilde\gamma_{x_i}\cdot
\tilde\gamma_{x_{i+1}}^{-1}\cdot\left(\tilde\gamma_{y}\cdot\gamma_\ell\right)^{-i-1}]=1
\]
and
\[
1=[\gamma_\ell,
\left(\tilde\gamma_{y}\cdot\gamma_\ell\right)^{i}\cdot\tilde\gamma_{x_i}^2\cdot\tilde\gamma_{x_{i+1}}^{-2}\cdot
\left(\tilde\gamma_{y}\cdot\gamma_\ell\right)^{-i-1}].
\]
The above relations for $i=0$ become
\[
1=[\gamma_\ell,\tilde\gamma_{x_0}\cdot\gamma_\ell^{-1}\cdot\tilde\gamma_{y}^{-1}]=
[\gamma_\ell,\tilde\gamma_{x_0}^2\cdot\gamma_\ell^{-1}\cdot\tilde\gamma_{y}^{-1}].
\]
As a consequence $[\gamma_\ell,\tilde\gamma_{x_0}]=[\gamma_\ell,\tilde\gamma_{y}]$ and the group is abelian.
\end{proof}
\begin{prop}\label{prop:triple}
Let $\cC$ be a curve as in Proposition{\rm~\ref{prop:construction}\ref{prop:tres}}, $d>3$. Its fundamental group is
generated by $\gamma_\ell,\tilde\gamma_{x_0},\tilde\gamma_{x_1},\tilde\gamma_{x_2}$, where $\gamma_\ell$ is central
and $\tilde\gamma_{x_0}\cdot\tilde\gamma_{x_1}\cdot\tilde\gamma_{x_2}\cdot\gamma_\ell^d=1$.
The relations $\tilde\gamma_{x_0}\cdot\tilde\gamma_{x_1}\cdot\tilde\gamma_{x_2}=\tilde\gamma_{x_1}\cdot
\tilde\gamma_{x_2}\cdot\tilde\gamma_{x_0}=\tilde\gamma_{x_2}\cdot\tilde\gamma_{x_0}\cdot\tilde\gamma_{x_1}$ correspond
to the ordinary triple point of intersection of the three lines.
\end{prop}
\begin{proof}
This fundamental group is the quotient of $\tilde{K}$ obtained by \emph{killing} $\tilde\gamma_{y}$
and $\tilde\gamma_{x_j}$, $j\geq 3$.
The generators are $\gamma_\ell,\tilde\gamma_{x_0},\tilde\gamma_{x_1},\tilde\gamma_{x_2}$ and the relations are:
\begin{enumerate}[label=\rm($\hat{\text{K}}$\arabic*)]
\item $\prod_{j=0}^{2}(\tilde\gamma_{x_j}\cdot\gamma_\ell)\cdot\gamma_\ell^{d-3}=1$,
\item\label{k2} $\left[(\gamma_\ell\cdot\tilde\gamma_{x_i})^d,\tilde\gamma_{x_i}\right]=1$
for $0\leq i<3$,
\item\label{k3} $1=[\gamma_\ell,(\gamma_\ell^{i+1}\cdot\tilde\gamma_{x_i}\cdot\gamma_\ell^{-i})^j\cdot
\gamma_\ell^{-1}\cdot (\gamma_\ell^{i+2}\cdot\tilde\gamma_{x_{i+1}}\cdot\gamma_\ell^{-i-1})^{-j}]$
for $0<j<d$ and $0\leq i<2$,
\item $1=[\gamma_\ell,\tilde\gamma_{x_2}]$,
\item $1=[\gamma_\ell,\tilde\gamma_{x_0}]$.
\end{enumerate}
Let us study the relations \ref{k3}. For $i=0$ and $j=1$ we obtain $[\gamma_\ell,\tilde\gamma_{x_1}]=1$.
\end{proof}
\begin{cor}\label{cor:sigma1}
Let $\cC\in\hat{\Sigma}^{(d)}_1$, $d>3$. Its fundamental group is abelian, that is, $\pi_1(\PP^2\setminus \cC)=\ZZ^3$.
\end{cor}
\begin{proof}
Let us consider the family $\{\cC_t\}_{t\in[0,1]}$ of Proposition~\ref{prop:construction}\ref{prop:tres}.
\begin{figure}[ht]
\begin{center}
\begin{tikzpicture}
\draw (0,0) circle [radius=2cm] node[below=1.05cm] {$\mathcal{C}_t$};
\draw ($1.7*(0,1)-.7*({-sqrt(3)/2},-1/2)$)--($-.7*(0,1)+1.7*({-sqrt(3)/2},-1/2)$);
\draw ($1.7*(0,1)-.7*({sqrt(3)/2},-1/2)$)--($-.7*(0,1)+1.7*({sqrt(3)/2},-1/2)$);
\draw ($1.7*({sqrt(3)/2},-1/2)-.7*({-sqrt(3)/2},-1/2)$)--($-.7*({sqrt(3)/2},-1/2)+1.7*({-sqrt(3)/2},-1/2)$);
\draw[-{[scale=2]>}] (2.5,0) -- (3.5,0);
\draw (6,0) circle [radius=2cm] node[below=1.05cm] {$\mathcal{C}_0$};
\draw ($(6,0)+(60:2.1)$)--($(6,0)-(60:2.1)$);
\draw ($(6,0)+(-60:2.1)$)--($(6,0)-(-60:2.1)$);
\draw ($(6,0)+(0:2.1)$)--($(6,0)-(0:2.1)$);
\end{tikzpicture}
\caption{Degeneration around the ordinary triple point}
\label{fig:deg1}
\end{center}
\end{figure}
For $t$ small enough, these curves are equisingular outside a small ball centered at the ordinary triple point
of $\cC_0$, see Figure~\ref{fig:deg1}. The fundamental group of $\cC_t$ is obtained from the group in Proposition~\ref{prop:triple}
by replacing the relations $\tilde\gamma_{x_0}\cdot\tilde\gamma_{x_1}\cdot\tilde\gamma_{x_2}=
\tilde\gamma_{x_1}\cdot\tilde\gamma_{x_2}\cdot\tilde\gamma_{x_0}=\tilde\gamma_{x_2}\cdot\tilde\gamma_{x_0}\cdot\tilde\gamma_{x_1}$
by $[\tilde\gamma_{x_0},\tilde\gamma_{x_1}]=[\tilde\gamma_{x_1},\tilde\gamma_{x_2}]=[\tilde\gamma_{x_2}\cdot\tilde\gamma_{x_0}]=1$,
and the result follows.
\end{proof}
\section{An extended family of arithmetic Zariski tuples and an open question}
\label{sec:consecuencias}
It is worth mentioning that recently, the equisingular stratum $\hat\Sigma^{(d)}$ has been generalized as follows
(see~\cite{Sh:18,Sh:19}). Fix $d\geq 3$ and consider three ordered tuples of positive integers
$\mathbf{a}:=(a_1,\dots,a_{n_1})$, $\mathbf{b}:=(b_1,\dots,b_{n_2})$, and $\mathbf{c}:=(c_1,\dots,c_{n_3})$ such that
$d=\sum_i\mathbf{a}=\sum_j\mathbf{b}=\sum_k\mathbf{c}$.
An \emph{Artal-Shirane curve} (defined as Artal curve in~\cite{Sh:18}) of type $(d;\mathbf{a},\mathbf{b},\mathbf{c})$ is a union of a smooth curve of degree~$d$
and three lines $L_{\mathbf{a}},L_{\mathbf{b}},L_{\mathbf{c}}$ such that the smooth curve intersects the line
$L_{\mathbf{a}}$ (resp.~$L_{\mathbf{b}}$, $L_{\mathbf{c}}$) at $n_1$ (resp.~$n_2$, $n_3$) points with intersection
multiplicities $a_i$, (resp.~$b_j$, $c_k$).
For $s:=\gcd(\mathbf{a},\mathbf{b},\mathbf{c})>1$ Shirane proved that these curves provide Zariski tuples related
to $s$-roots of unity. He used an invariant called \emph{splitting graph} based on the linking invariant described
in \S\ref{sec:link}.
Moreover, the following generalization of Theorem~\ref{thm:complement} on the homeomorphism type of the complements
of such curves can be stated and proven with the same arguments.
\begin{thm}
Let $\cC_1,\cC_2$ be two Artal-Shirane curves of the same type, associated with distinct and non-conjugate roots of unity.
Then, $\mathbb{P}^2\setminus\cC_1$ and $\mathbb{P}^2\setminus\cC_2$ are not homeomorphic.
\end{thm}
The interesting fact about these curves lies on the fact that the fundamental groups of their complements is as
simple as it can be. This can be obtained as a consequence of our Theorem~\ref{thm:sigmazeta},
Corollary~\ref{cor:sigma1}, Theorem~\ref{thm:deg2} and an equisingular deformation via the following very
useful result by Zariski proved in Dimca's book.
\begin{thm}[{\cite[Corollary 3.2]{dimca:li}}]\label{thm:deg2}
Let $\{\cC_t\}_{t\in[0,1]}$ be a continuous family of projective plane curves such that $\cC_0$ is reduced
and the family is equisingular for $(0,1]$. Then, there is an epimorphism
$\pi_1(\bp^2\setminus\cC_0)\twoheadrightarrow\pi_1(\bp^2\setminus\cC_1)$.
\end{thm}
\begin{cor}
Let $\cC$ be an Artal-Shirane curve of type $(d;\mathbf{a},\mathbf{b},\mathbf{c})$ where either $d>3$ or
$(d;\mathbf{a},\mathbf{b},\mathbf{c})\neq(3;(3),(3),(3))$ (with non-aligned intersection points).
Then the fundamental group of $\cC$ is abelian.
\end{cor}
\begin{proof}
It is clear that we can deform in an equisingular curve such a curve to a curve of type
$(d;(d),(d),(d))$ which is in some $\hat{\Sigma}^{(d)}_\zeta$; if $d=3$, then $\zeta\neq 1$.
The result is a direct consequence of Theorem~\ref{thm:sigmazeta}, Corollary~\ref{cor:sigma1}
and Theorem~\ref{thm:deg2}.
\end{proof}
Turning out attention back to the original equisingular family of irreducible curves $\Sigma^{(d)}$
the following question remains open.
\begin{question}
Given two irreducible curves $\cC_i\in \Sigma^{(d)}_{\zeta_i}$, $i=1,2$ where $\zeta_i\in \TT^+_d$,
$\zeta_1\neq \zeta_2$.
Are the pairs $(\PP^2,\cC_1)$ and $(\PP^2,\cC_2)$ topologically equivalent?
Are their complements homeomorphic, or their fundamental groups isomorphic?
\end{question}
\bibliographystyle{amsplain}
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
\providecommand{\MRhref}[2]{%
\href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
}
\providecommand{\href}[2]{#2}
| -59,231.083574 |
[
-2.65234375,
2.494140625
] | 28.848347 |
[
-2.75390625,
0.49560546875,
-2.658203125,
-5.26953125,
-0.74169921875,
8.484375
] |
[
4.66796875,
9.9375,
2.517578125,
6.9453125
] | 355 | 5,041 |
[
-3.17578125,
3.7421875
] | 33.757912 |
[
-4.72265625,
-3.2265625,
-4.734375,
-2.353515625,
1.23828125,
11.8046875
] | 0.514727 | 13.134454 | 30.886729 | 3.721945 |
[
1.83015775680542
] | -37,862.892471 | 6.937116 | -59,114.543219 | 0.543323 | 6.188254 |
[
-1.62890625,
-3.015625,
-3.53515625,
-5.125,
1.9482421875,
11.828125
] |
[
-5.52734375,
-1.3828125,
-2.12109375,
-0.98828125,
3.595703125,
3.4921875
] | |
BkiUfHg5qsFCifBZHZLT
|
\section{Background}
The existing time series analysis literature has proposed several different approaches for seasonality detection.
Often the term periodicity detection is used in literature for the same task.
These methods can be split in three different categories: explicit season length, discretized time series, and single season length.
\paragraph{1. Explicit season length}
The first type of algorithms depends on an external specification of the period
length to extract periodic features. The previously mentioned data cube
concept~\cite{Han1} was a pioneer method for mining periodic features in
time series databases. Other examples are the chi-squared
test~\cite{Ma1} or binary vectors~\cite{Berberidis1}. The
advantage of such procedures is their low time complexity and their reliable
results. However, relying on an external specification of the period inherently prevents these methods from detecting an unknown period.
In many applications a parameter-free method is being highly desired.
\paragraph{2. Discretized time series}
The second type of algorithms for detecting periodic features test all possible
periods of a time series. This is typically performed on a discretized time
series, as otherwise such algorithms would quickly become computationally
infeasible. Examples for such methods are suffix trees~\cite{Rasheed1}, convolution~\cite{Elfeky1} and data sketches~\cite{Indyk1}. Unlike
the previous type, these algorithms typically find all potentially relevant
periodic patterns without previous specification of the period. However,
since they depend on data discretization, they can only be as
accurate as the underlying data discretization allows. Furthermore they may return more than
one possible result. While this is often an accurate reflection of a time series
periodic features, it is often advantageous in practice to provide a single
dominant period for further automated processing of a time series.
\paragraph{3. Single season length}
The third type of algorithms approximates a single season length from the raw data.
While such an approach might be too limited for highly complex data, it can be useful
in cases where time series have one dominant frequency. One popular algorithm~\cite{Hyndman2} of this category can be found in the R forecast
library and was derived from another method~\cite{Hyndman1} which is based on
autocorrelation.
The algorithm presented in this paper belongs to the third type and focuses on improving
the robustness of the season length detection.
\paragraph{Combination of methods}
All three categories consequently serve different purposes in time series
seasonality detection and also can be combined in useful ways. For
example, an algorithm of type 3 can be used for season length detection, which
can then be used as input for a type 1 algorithm to enable correct periodicity
mining. Further, type 2 and 3 can be combined on the same time series to assess
the effect discretization has on a specific time series.
\section{Conclusion}
Detecting season length in time series without human assistance is challenging. The method presented in this paper attempts to complete
this task by interpolating, filtering and detrending a time series and then
analyzing the distances between zeros in its autocorrelation function. The implementation
of this concept is still leaves room for improvement, as it still relies on several empirically
chosen constants. However, the results demonstrated sufficient robustness in our evaluation.
Future work concerning automated season length detection might attempt to
eliminate the algorithm's constant-dependency by inferring them from the data.
Further, the detrending can likely be improved by including high-pass filtering
into the procedure. Another interesting concept would be to develop a season
length detection algorithm based on a machine learning method like a neural
network.
To advance automated seasonality and periodicity mining in general, it is
important to develop automated procedures for a dynamic computation of
otherwise static constants in contemporary algorithms. Achieving this is an
interesting challenge for the future.
\section{Introduction}
In many areas of natural science and economics there exists an abundance data
which can be used beneficially once they are processed. Often these data are
collected at regular intervals to enable making statistical assumptions and
predictions for numerous applications.
Data collected repeatedly over a time span is referred to as \textit{time series}.
The in-depth analysis of time series has been a central topic of research in
recent years. Typically, time series are investigated to better understand their past
behavior and to make forecasts for future data. To achieve this, identifying
trends and regularities are particularly useful, as they allow one to generalize
from the given data to a larger context.
A very natural approach to finding these relevant patterns is the Exploratory
Data Analysis~\cite{Tukey}. This method relies on visualizing the data, typically
without making any prior assumptions about the data. An advantage of this
procedure is that humans usually analyze visual data very effectively and thus
identify many relevant features of the data.
\begin{figure}[!h]
\centering
\includegraphics[scale=0.45]{intro_1.jpeg}
\caption{A simple example of a time series with trend and periodic patterns.
Here the season length is 12 month.}
\label{fig:example}
\end{figure}
While data visualization works well with many time series, there are also
several cases where it is insufficient. When dealing with very complex data,
it is often difficult to infer meaningful features. Furthermore,
it is also frequently necessary to analyze dozens of time series, where manual
analysis quickly becomes tedious or even infeasible.
Therefore, an automated and robust approach that can deal with complex data is required.
While there are many established algorithms for trend estimation~\cite{Draper}~\cite{Vamos},
methods for
finding periodic patterns and features are much younger. One popular approach is
the data cube concept~\cite{Han1}, which relies on constructing and employing
a data cube for finding periodic features in a time series. This method would,
for instance, successfully identify a peak of ice cream sales every summer given
monthly sales data over several years. However, to achieve this, the algorithm
requires the user to input the time series' \textit{season length} (or \textit{frequency}). In
the example depicted in Figure~\ref{fig:example} the season length would be 12, as the sales pattern repeats
every 12 months.
Today, there already exist a few algorithms for approximating season length.
Most of them rely on data discretization before analysis, which is
a simplification of data as seen above to discrete numbers such as 2, 4, and 6.
For finding so-called \textit{symbol periodicity} in discretized data one can
for instance use suffix trees~\cite{Rasheed1} or convolution~\cite{Elfeky1}.
One downside of discretization is that it inevitably leads to a loss of
information, which might change the result in some cases. A more
significant disadvantage of discretization is that contemporary time series symbolization methods such as SAX~\cite{Lin1} require the user to define
the number of symbols to which the data is discretized. This is be problematic
since data mining algorithms should in general have as few parameters as
possible~\cite{Keogh1}.
A different approach that does not rely on data discretization is searching
for local peaks and troughs in a time series' autocorrelation function
\cite{Hyndman1}. This method works fully automated and efficiently computes
the correct season length of many time series.
Yet there are still several cases where this algorithm is not able to identify the correct frequencies - particularly in time series with noisy data or very long periods.
Due to the above mentioned algorithm's disadvantages, it should be beneficial to
find a more reliable method. Such an algorithm should not rely on user defined
parameters or
data discretization and still correctly identify a time series' season length.
A method that was designed to meets these requirements is presented in this paper.
The source code of our system, the test data set and detailed test results can be accessed online\footnote{\url{https://github.com/mtoller/autocorr_season_length_detection}}.
\section{Concepts}
Let $x = \{x_1,x_2,...,x_n\}$ be a series of real-valued observations and $\Delta_x$ the
interval at which these observations are made. If $x$ has a subsequence of
observations $Q_x$ which
occurs every $s$ observations after its first occurrence, $x$ is seasonal.
Moreover, if $Q_x$ is not part of a longer repeating
subsequence of $x$ and it cannot be divided into shorter equal subsequences, then
$x$ is perfectly seasonal with season length $s\Delta_x$.
$\Delta_x$ is negligible since
it is independent from $x$. The problem of finding season length $s\Delta_x$ can
therefore be reduced to finding the characteristic subsequence $Q_x$ which fulfills the following formal
criteria~\ref{eq:criterion1} and \ref{eq:criterion2}. The first criterion is
\begin{equation}
|Q_x| = s
\label{eq:criterion1}
\end{equation}
which means the number of observations in $Q_x$ must be
equal to the number of observations in $x$ from one occurrence of $Q_x$ to its
next. The second criterion is
\begin{equation}
\forall q \in \rho(Q_x)\setminus Q_x : q^k \neq Q_x
\label{eq:criterion2}
\end{equation}
with
$\rho(Q_x)$ being the power set (set of all subsets) of $Q_x$ and $q^{k} :=
q^\frown q^{k-1}$ ($q$ followed by $q^{k-1}$). This means that $Q_x$ must not contain a subsequence q
that can be repeated $k$ times to create $Q_x$, i.e. $Q_x \neq \{q,q,q,...\}.$
For example, given a time series
\begin{equation}
y = \{0,2,1,2,0,2,1,2,0,2,1,2,0,2,1,2\}
\label{eq:example}
\end{equation}
the sequence $a = \{0,2,1,2\}$ is repeated four times in $y$.
Sequence $a$ cannot be reduced to shorter equal sequences, as $a_{1,2} = \{0,2\} \neq
a_{3,4}=\{1,2\}$ and the number of observations in $a$ is equal to the number
of observations from one occurrence of $a$ to its next occurrence $|a| = s_a = 4$.
Therefore $a = Q_y$ and $y$ has a season length of $4\Delta_y$.
If one extends the sequence by one element to $b=\{0,2,1,2,0\}$ then $|b| = 5
\neq s_b = 8$, which means it cannot be $Q_y$. Now considering
sequence $c = \{0,2,1,2,0,2,1,2\}$ the first criterion is fulfilled: $|c| = s_c = 8$.
Yet the second condition is violated as $c$ can be split into the equal
subsequences
$c_{1,2,3,4} = c_{5,6,7,8} = \{0,2,1,2\}$, therefore $c$ can be divided into
shorter equal sequences and also cannot be $Q_y$.
\subsection{Detrending}
Many time series have a trend in addition to its seasonality, which means that
the seasonal influences revolve around a trend function. For instance, in the
example in Equation~\ref{eq:example} there could be a linear increase in the observations after every
season, e.g.
\[y'=\{0,2,1,2,0.1,2.1,1.1,2.1,0.2,2.2,1.2,2.2,...\}\]
In this case $Q_{y'}$ would be time-dependent:
\[Q_{y'}(t) = \{0,2,1,2\} + \frac{1}{10}*\lfloor\frac{t}{4}\rfloor\]
with $t \in \mathbb{N}_0$.
However, the unscaled season length $s_{y'} = |Q_{y'}|$ is identical for all $t$
and is therefore not time-dependent. Hence it is desirable to remove trend
influences from a time series before analyzing seasonality.
Approximating the trend components can be achieved with regression analysis~\cite{Draper}, which
is a procedure for finding a function which minimizes the mean square error
between the observations and the approximated function. Its cost function can
be written as
\begin{equation}
C(\theta) = \frac{1}{N} \sum_{i=1}^{N}((X_i)^T \theta - x_i)^2
\label{eq:mse}
\end{equation}
where $X$ is the design matrix, $x$ the time series and $\theta =
\{\theta_1,...\theta_n\}$ the parameters of the regression.
A design matrix is a matrix where each row describes one observation
and each column models an assumed feature onto the corresponding observation.
For instance, in polynomial regression $X$ can be written as
\[
X =
\begin{bmatrix}
1 & 1 & 1 & 1 & ...\\
1 & 2 & 4 & 8 & ...\\
1 & 3 & 9 & 27 & ...\\
1 & 4 & 16 & 64 & ...\\
\vdots & \vdots & \vdots & \vdots& \ddots
\end{bmatrix}
\]
To find the parameters which minimize the cost function, one can use the
analytical solution
\begin{equation}
\theta = (X^T X)^{-1} X^T x.
\label{eq:linreg}
\end{equation}
The trend of time series $x$ can then be removed with
\begin{equation}
x_{detrend} = x - X \theta.
\label{eq:detrend}
\end{equation}
\begin{figure}[!h]
\centering
\includegraphics[scale=0.45]{meth1.jpeg}
\caption{Examples of time series with (blue) and without (black) quadratic trend.}
\end{figure}
Removing all trend influences from a time series significantly facilitates
investigating seasonal effects, as the characteristic subsequences $Q_x^{(t)}$
then directly correlate with each other. This property allows one to investigate
the time series' autocorrelation instead of the time series itself. Such an
approach is advantageous since the original observations are frequently much
more difficult to analyze than autocorrelation.
\subsection{Removing Noise}
Before directly analysing the characteristic subsequence $Q_x$, another major component of many time series needs to be considered.
Most non-discretized time series contain random inaccuracies, which are often named noise or residuals.
Minimizing these random influences is desirable, since they tend to obscure the true nature of the
observations. This is typically achieved by means of filters.
As noise typically affects every observation in a time series, it tends to
have a shorter period than seasonality, which has a period of at least 2
observations. Such a shorter period leads to noise having a higher frequency
than seasonality. Therefore, it is advantageous to apply a low-pass filter on
the time series, which rejects any frequency higher than a given threshold. This
smooths the curve while maintaining the original season length.
However, using a low-pass filter also has a negative effect on the
observations. While removing white noise, filters likewise alter the
observations' correlation.
To lessen these unwanted side effects, one may interpolate the data before applying the low-pass filter.
\subsection{Analyzing Correlation}
Only after removing most noise from the observations, it is possible to meaningfully investigate autocorrelation of a time series.
A time series' autocorrelation is the correlation of its observations at different times.
Autocorrelation is formally defined as
\begin{equation}
A_x(\tau) = \int_{-\infty}^{\infty}x_t\overline{x}_{t+\tau}dt
\label{eq:acorf1}
\end{equation}
where $\tau$ is the lag at which the observations are compared, and
$\overline{x}$ is the complex conjugate of $x$. Since time series are
time-discrete and real-valued, this can be further simplified to
\begin{equation}
A_x(\tau) = \sum_{t}^{T}x_t x_{t+\tau} .
\label{eq:acorf2}
\end{equation}
After a further normalization the resulting values range from $-1$ to $+1$.
The former implies complete anti-correlation between the two observations,
while the latter means full correlation.
In general, two objects $a$ and $b$ correlate, if a change in
$a$ is likely to be linked with the same change in $b$ as well, whereas negative
correlation would be associated with a change in $b$ in the opposite direction. In case
of $0$ correlation, a change in $a$ is independent from $b$.
Autocorrelation describes the correlation of $a$ with itself at a different
point in time. If $a$ is seasonal, then its autocorrelation will be seasonal
with the same season length. This property ensures that investigating
autocorrelation leads to the same season length as analyzing the original
observations, which is usually more challenging.
\subsection{Interpolating the Observations}
Autocorrelation should make it possible to observe seasonality.
Yet when applied on time series being preprocessed via low-pass filters one will experience a negative side-effect:
Its characteristic subsequence's length $|Q_{A_x}|$ no longer matches the length original
subsequence $|Q_x|$. Instead, its unscaled season length is $|Q_{A_x}| = k|Q_x|$ with $k \in \mathbb{N}$
and consequently a multiple of the original length, which makes finding
the correct season length more difficult.
To avoid this side-effect, it is possible to linearly interpolate the
observations before applying a filter. This means that between every neighboring
observations, several intermediate observations are inserted. For example, the
sequence $y=\{0,2,1,2\}$ could be interpolated to
$y'=\{\mathbf{0},1,\mathbf{2},1.5,\mathbf{1},1.5,\mathbf{2}\}$. The linear
interpolation $\psi(t)$ of two observations $x_1$ and $x_2$ can be written as
\begin{equation}
\psi(\Delta_t) = x_1 + \frac{x_2 - x_1}{t_{x_2} - t_{x_1}}(\Delta_t-t_{x_1})
\label{eq:interpolation}
\end{equation}
where $t_{x_i}$ is the time at which $x_i$ was observed and $\Delta_t$ denotes
a chosen point in time between these two observations. By interpolating all
neighboring observations, $x$ can be expanded to its interpolated sequence
$x'$
Interpolating the observations is advantageous, since it usually lessens
the filters negative effect on the length of the characteristic subsequence.
Let $\chi$ be $x$ after interpolation, filtering and detrending.
The unscaled season length of its autocorrelation $A_\chi$ matches
the unscaled season length of the original
observations $x$, which means $|Q_{A_\chi}| = |Q_x|$
\begin{figure}[!h]
\centering
\includegraphics[scale=0.45]{meth2.jpeg}
\caption{Autocorrelation plot computed after three preprocessing steps - after all three steps have been applied, the autocorrelation will be further analyzed.}
\end{figure}
After interpolating, filtering and detrending the observations and calculating
their autocorrelation, it is finally
possible to directly compute the time series' unscaled season length $s =
|Q_{A_\chi}|$.
This can be done by analyzing the zeros of $A_\chi$. The distance between two
zeros $A_\chi(\tau_1) = 0$ and $A_\chi(\tau_2) = 0$ with $\tau_1 < \tau_2$
is equal to half the unscaled season
length $s$. Therefore, $s$ can be computed with
\begin{equation}
s = |Q_{x}| = |Q_{A_\chi}| = 2(\tau_2 - \tau_1).
\label{eq:seasonlength}
\end{equation}
To calculate the true, scaled season length, one has to simply form the product
$s\Delta_x$, which is negligible in practice.
\section{System}
While the above mentioned concepts work well in theory, there are several
challenges in practice which still need to be met:
\begin{itemize}
\item Trends may change over time
\item Noise may vary
\item Machines have limited numeric precision
\item Seasonality is not always perfect
\item Presence of outliers
\end{itemize}
Dealing with these and other problems is imperative for robust parameter-free
season length detection. An overview of the steps which our system takes to
meet these challenges can be seen in Figure~\ref{fig:flowchart}.
Another crucial practical aspect is
runtime, since time series tend to contain thousands if not millions of
observations. To be valuable in practice, it is desirable for the
implementation to have a
worst-case computational complexity of at the most $\mathcal{O}(n\log n)$, with $n$ being
the number of observations in the time series.
\begin{figure}[!h]
\centering
\includegraphics[scale=0.45]{FlowChart_2.png}
\caption{The individual steps of the season length detection system.}
\label{fig:flowchart}
\end{figure}
\subsection{Approximating the Trend}
The seasonality of a time series typically revolves along a certain trend.
However, this trend may follow an arbitrary function, which makes removing
all trend in any time series impossible. Therefore, it is necessary to only
consider trends which are particularly common in practice.
Linear trends occur frequently in mathematical functions, but are not very
common in practice. Therefore, it is insufficient to assume that all time series
will have a close to linear trend. However, as polynomials of higher order tend
to over-fit the present data, linear trends can at least be compared with other
assumptions without over-fitting the data. Consequently, it is useful to begin
trend approximation by solving the 1st order polynomial linear regression.
Unlike linear tendencies, quadratic trends are fairly common. Many relations
in nature follow the square - for instance the relation between speed and
acceleration of moving objects. Therefore it is advantageous to attempt
modeling the data with a quadratic trend. To compare such a model with a linear
trend estimation one can compare the resulting mean square errors, which can
be achieved with
\begin{equation}
\ln(C(\theta_1) - C(\theta_2)) > k
\label{eq:constant1}
\end{equation}
where $\theta_i$ are the parameters which minimize the mean square error
$C(\theta)$ with a polynomial of $i$th degree. The logarithm is applied since
the difference tends to be very large. The constant $k$ determines the decision
boundary between linear and quadratic trend estimation. In the context of this
implementation it will be assumed that $k = e^2$. This value was chosen
empirically rather than theoretically, which obviously will not be appropriate
in all cases. However, determining the correct value of this and a few other
empirically chosen values for all cases would be beyond the scope of this paper.
\subsection{Choosing the Correct Filter}
A promising approach for detrending time series is high-pass
filtering. An advantage of this approach is that it can effectively remove
non-stationary trends from time series, which is difficult to achieve with
linear regression. A disadvantage of high-pass filters is that they depend more
strongly on threshold values, which limits their value to that of the chosen
models' constants.
There is a variety of ways to construct a low-pass filter, yet none of them
can create the ideal low-pass filter, which rejects all frequencies higher than
the cutoff frequency. It is only possible to approximate a close-to ideal
filter, which removes most too high frequencies while changing the less frequent
signals as little as possible. This can be achieved by generating a Butterworth
low-pass filter, which can be obtained by choosing its order and cutoff
frequency and then applying the corresponding algorithm.
However, choosing the correct order and cutoff frequency is problematic, as
the amount and amplitude of noise vary in time series. In the context of our
implementation, the order $\eta=2$ and cutoff-frequency $\omega=0.001\pi$ were
chosen empirically in preliminary tests.
\subsection{Numeric Inaccuracies}
Machines only have a finite numeric precision.
Therefore, the theoretical method must be adjusted at several points.
Firstly,
there usually is no exact $0$ in the autocorrelation, rather the data moves from
positive to negative values or vice versa. Consequently, it is necessary to
either create a tolerance interval $\epsilon$ around $A_\chi(\tau)=0$, or to interpolate the data until
it is accurate enough. In case of our implementation, both strategies are applied.
Secondly, even after an almost ideal detrending, the autocorrelation will not
have a linear trend of $0$, but rather a tendency very close to $0$. To deal
with this, a second linear regression is performed after the autocorrelation was
computed. The autocorrelation is then not searched for $A_\chi(\tau)=0$, but
rather $A_\chi(\tau)-X\theta=0$, where $X\theta$ is the linear trend of the
$A_\chi$.
Thirdly, due to the consequential mentioned tolerance interval
\begin{equation}
A_\chi(\tau)-X\theta \pm \epsilon=0
\label{eq:toleranceInterval}
\end{equation}
there will be several solutions for $\tau$ which are in the
same tolerance interval. Therefore, it is necessary to discard all $\tau_1$ and
$\tau_2$ for which the following condition is true
\begin{equation}
|\tau_1 - \tau_2| \leq 1.
\label{eq:removeOnes}
\end{equation}
This is obvious, since no time series can have a season length shorter than two
observations and the difference between two zeros corresponds to half the
season length.
\subsection{Imperfect Seasonality}
Another problem that exists in practice is that seasonality is rarely
perfect, as assumed in theory. Frequently, seasonality outliers or a
systematic seasonal
change occur, which invalidate the above formula $s=2(\tau_2 - \tau_1)$. To
deal with these problems, it is possible to not only analyze $\tau_2$ and
$\tau_1$, but rather any pair of adjacent solutions for Equation~\ref{eq:seasonlength}.
Let $\alpha=\{\alpha_1,\alpha_2,...,\alpha_m\}$ be all values for $\tau$ so
that Equation~\ref{eq:seasonlength} is fulfilled and let
$\delta=\{\delta_1,...,\delta_{m-1}\} = \{\alpha_2-\alpha_1,\alpha_3-\alpha_2
,...,\alpha_m-\alpha_{m-1}\}$ be the distances between each pair of adjacent
zeros in Equation~\ref{eq:seasonlength}. By removing all pairs that violate Equation~\ref{eq:removeOnes} from $\delta$ one
can compute all true distances between seasonal repetitions
$\delta'=\{\delta'_1,...,\delta'_l\}$, which are further sorted in ascending
order.
These true distances $\delta'$ provide a better representation of the season
length than an arbitrary pair of solutions $\tau_1$ and $\tau_2$. However,
simply taking an average of these values does not suffice, since many time
series contain seasonality outliers. Further, numeric imprecision may
cause the implementation to miss correct zeros, or any remaining noise might
add incorrect zeros. Therefore, it is necessary to perform additional
operations on $\delta'$.
To separate the correct values in $\delta'$ from the incorrect, it is
necessary to assume that $\delta'$ contains a sufficiently large number of
correct values. Since seasonality is stationary in time series, the correctly
identified distances have a low variance in most cases, while erroneous
values tend have a much higher sample variance.
When considering Equation~\ref{fig:delta}, it can be observed that the intervals
$\delta'_{[3,8]}$ and $\delta'_{[9,10]}$ have a low variance when compared
to intervals including other values.
\begin{figure}[H]
\begin{equation}
\delta' = [281,546,697,703,704,705,706,706,1411,1411,2823]
\label{fig:delta}
\end{equation}
\caption*{A typical zero-to-zero distance vector}
\end{figure}
This suggests that the correct
zero-to-zero distance is either $\frac{1}{2}s\approx703$ or
$\frac{1}{2}s\approx1411$. By further considering that
$703 * 2 \approx 1411$, it is intuitive to assume that these intervals are
multiples of each other. Since Equation~\ref{eq:criterion2} must still be valid, it is safe to
assume that a zero was missed between two zeros and thus caused the
multiplied distance.
Therefore, the correct zero-to-zero distance is very likely $703$.
While the above example suggests taking the low-variance interval with
the smallest mean, it is more reliable to rather search for the
low-variance interval with the highest cardinality. The reason for this is the necessary assumption that
a sufficiently large amount of the found zero-to-zero distances is almost correct,
as otherwise reliably finding the true season length would be impossible.
To apply the above reasoning in a generalized way, it is necessary to
reliably identify these low-variance intervals. This can be done by
computing the quotients between the distances
\begin{equation}
\gamma =
\{\gamma_1,\gamma_2,...,\gamma_{l-1}\} = \{\frac{\delta'_2}{\delta'_1},
\frac{\delta'_3}{\delta'_2},...,
\frac{\delta'_l}{\delta'_{l-1}}\}.
\label{eq:gamma}
\end{equation}
These quotients describe the rate at which the distances between zeros
change. A series of low values for $\gamma$ implies a stable, low-variance
interval of distances, while high values suggest a jump between intervals.
These high and low values can be separated with
\begin{equation}
\Gamma_i =
\begin{cases}
1 & \text{if } i=1 \land |\gamma_1 - \gamma_2| \leq k\\
i & \text{if } i=l-1\\
i+1 & \text{if }|\gamma_i - \gamma_{i+1}| > k\\
0 & \text{else}
\end{cases}
\label{eq:Gamma}
\end{equation}
where $k$ is a constant which was chosen empirically in preliminary tests with $0<k<1$ and
$i={1,2,...,l-1}$.
This gives a series of integers where $0$ represents a low change of
distances, and all other numbers the index of a high change. By then
discarding all zeros with $\Gamma' = \Gamma\setminus\{0\}$, the longest
low-variance interval of distances can be found with
\begin{equation}
i^* = \underset{i}{argmax}(\Gamma'_{i+1} - \Gamma'_{i})
\label{eq:imax}
\end{equation}
\[
a = \Gamma'_{i^*}
\]
\[
b = \Gamma'_{i^* +1}
\]
where $a$ is the index of the first distance in the interval, and $b$ the
index of the last. If the resulting interval $\delta'_{[a,b]}$ is long enough, it can be advantageous do further
discard an upper and lower percentile of the values within the interval.
However, in practice, we found that this alters the result very little in our experiments.
With the longest low-variance interval, an approximation of the season
length can be computed by taking the average of these distances with
\begin{equation}
s = \frac{2}{b-a}
\sum_{j=a}^{b}\delta'_j.
\label{eq:result}
\end{equation}
\section{Evaluation}
\subsection{Data Set}
To evaluate the implementation, it is necessary to test it with several
different time series. For this purpose, two distinct time series databases
were gathered. The first database contained extensive variations of synthetic data, while the second database consisted of real financial and climate data.
The time series from these databases were tested in nine test runs, where our
implementation was tested against the existing algorithm in the R
forecast library. The test cases for runs 1 to 7 were taken from the synthetic
database, while the test cases for runs 8 and 9 were taken from the real
database and are briefly described below:
In the $Diverse$ test, both algorithms were confronted with 20 time series,
which strongly vary in trend, seasonality, noise and length. However, all
these time series had a consistent season length, which does not vary
within the series. This first testing was meant to compare the general
applicability of the algorithm, which is achieved by challenging them with
very different problems, which yet have a distinct solution.
In the $Complex$ test, the 20 test cases included many numeric outliers,
largely changing season amplitude, season outliers and also varying
noise. This had the purpose to assess the algorithms' error tolerance.
The $Ambiguous$ test included 20 examples with more than one correct solution.
This aimed at observing the choices made by the algorithms and thus identifying
their tendencies or preferences.
The $Variations$ test featured 4 time series, which were each presented in
5 variations. This was meant to assess the algorithms' consistency.
The $Noise$ test started with a simple time series without any noise, which
was then tested repeatedly with increasing amounts of noise. The goal of this
run was to compare the noise resilience of the two algorithms.
The $Length$ test also consisted of a single, simple time series, which was
presented with varying season length. The purpose of this run was to test both
algorithms in different result domains.
The $NoSeason$ test featured 10 examples with no seasonality. This had the
purpose to explore the algorithms' capability to distinguish between seasonal
and non-seasonal time series.
The $Econonmy$ test consisted of 20 time series of seasonal economic data.
These financial data were taken primarily from sectors which display seasonality, such as tourisms and restaurants.
Three of these test cases contained both annual and quarterly seasonality, and
several others only showed a very slight manifestation of seasonality.
The $Climate$ test contained 20 time series of seasonal climate data about
temperature, precipitation, sun hours or storm counts.
\subsection{Results}
During testing, the R library algorithm which uses spectral density estimation was referred to as \textit{Spectral} and our system using autocorrelation as \textit{Autocorr}. For each test, the
season length suggested by the algorithms was considered correct if it was
within an error margin of $\pm20\%$ of the reference value.
The accumulated test results are depicted in Table~\ref{tab:results}, while the results of the individual test runs can be seen in Figure~\ref{fig:results}.
In the first 6 tests algorithm $Autocorr$ performed better than algorithm $Spectral$. The most
significant difference was in the noise test, were $Spectral$ passed 3 tests
and $Autocorr$ 12 tests.
The $NoSeason$ test, which only consisted of time series without seasonality,
was the only test run were both algorithms passed the same number of tests. In
all other test runs, including $Economy$ and $Climate$, $Autocorr$ passed more
tests than $Spectral$.
\begin{table}[h!]
\begin{tabular}{c|c |c c}
Database&\#Test Cases&Spectral&Autocorr \\
\hline Synthetic&125&62&96\\
Real&40&21&26 \\
\hline$\sum$&165&83&122\\
\end{tabular}
\caption{Numeric Test Results with the number of correctly identified season lengths for both algorithms.}
\label{tab:results}
\end{table}
\begin{figure}[!h]
\centering
\begin{tikzpicture}[scale=1.0] \begin{axis}[x tick label style={rotate=90,anchor=east}, ylabel=\%Passed Tests, enlargelimits=0.1, legend style={at={(0.35,0.95),font=\scriptsize}, anchor=north,legend columns=-1}, ybar=0pt
bar width=9pt, point meta=y,
xtick=data,
xticklabels={Diverse,Complex,Ambiguous,Variations,Noise,Length,NoSeason,Economy,Climate}
] \addplot coordinates {(1,55) (2,40) (3,60) (4,65) (5,15) (6,60) (7,60) (8,35) (9,70)}; \addplot coordinates {(1,90) (2,65) (3,80) (4,80) (5,60) (6,100) (7,60) (8,50) (9,80)}; \legend{Spectral,Autocorr} \end{axis} \end{tikzpicture}
\caption{Test results of algorithms \textit{Spectral} and \textit{Autocorr} for all 9 test runs.}
\label{fig:results}
\end{figure}
Figure~\ref{fig:errors} displays the relative difference between the detected
season length and the reference value for all test cases. The area between the
curves and zero represents the accumulated test error of the algorithms.
\textit{Spectral} had an accumulated error of 172\%, while \textit{Autocorr}'s
relative errors add up to 57\%.
\begin{figure}
\includegraphics[scale=0.45]{errors.png}
\caption{Relative detection error in all test cases - \textit{Autocorr}'s curve was negated to contrast the deviations}
\label{fig:errors}
\end{figure}
\section{Discussion}
The purpose of this paper was to present an algorithm developed for identifying time
series' season length which is sufficiently reliable to be used in real-world applications.
The conducted tests tried to cover many different aspects of time series behavior and our system compares favorably against the reference algorithm.
However, the overall results alone do not necessarily suggest that the methods
applied in $Autocorr$ are preferable for identifying season length in all settings. Both
implementations still rely on empirically chosen constants, which may have an influence on the outcome.
For example, had the constant in $Autocorr$ for distinguishing
linear and quadratic trends been chosen slightly higher, then several time
series with quadratic trends may have been misclassified. This argument can
be made for every empiric constant in both implementations.
The $Noise$ test, which was a repeated test of the same time series while
increasing the noise component from test to test, has revealed a noise
susceptibility of $Spectral$. This is likely an inherent property of working with
spectral density estimates, as noise tends to hide the relevant frequencies.
This is also supported by the $Variations$ test, where $Spectral$ failed to disregard
outliers induced on otherwise unchanged time series.
Another noteworthy observation is that $Spectral$ frequently misclassified season
lengths longer than $100\Delta_x$. In fact, $Spectral$ never suggested a season
length longer than $998\Delta_x$, although there were 18 cases where this
would have been necessary. This suggests that the implementation of $Spectral$ may not be the optimal choice for
investigating seasonality in large time series with very long season lengths.
Additional evidence for this assumption is provided by the $Length$ test, which was
designed to test this particular aspect of seasonality.
A dependency that greatly affects both algorithms is that they both rely on an adequate
removal of trend from the time series. Algorithm $Autocorr$ always returned an
erroneous result if it failed to correctly identify the trend. Moreover,
neither $Spectral$ nor $Autocorr$ were capable of detrending any non-linear or non-square
trend. Correctly removing trends from time series is an active research field, fostering the hope that this issue will be addressed.
There is also a single case of a false positive in $Autocorr$ that is not directly apparent. While it did correctly
analyze the season length of test case 5 in the the $NoSeason$ test, it was only
coincidental that the result is correct. The reason for this is that
$Autocorr$ perfectly removed all trend from this purely quadratic time series
and then considered almost every not interpolated point in the
autocorrelation as potential zero. Computing the mean distance between
these obviously yields an average of 1, which is then discarded due to
Equation~\ref{eq:removeOnes}. However, with an only slightly altered time series, the
average distance could be rounded up to 2, which would not be discarded, yet
still incorrect for a purely quadratic time series.
The most expensive operation in both implementations is computing the
matrix-inverse required for detrending the data, which has a worst-case
computational complexity of $\mathcal{O}(n^{2.3727})$. This is far above the desirable
computational complexity of $\mathcal{O}(n\log n)$, as for an exemplar time series with
$1000$ observations, in the ideal case $k * 10^3$ operations would be required,
whereas both implementations need in the worst case $k * 10^6$ operations, which
is a thousand times higher.
| -19,611.848567 |
[
-3.41015625,
3.126953125
] | 37.423313 |
[
-2.958984375,
0.1014404296875,
-2.134765625,
-4.74609375,
-0.1158447265625,
7.0234375
] |
[
1.9736328125,
6.01171875,
1.85546875,
7.1640625
] | 310 | 5,355 |
[
-2.857421875,
3.271484375
] | 23.886231 |
[
-6.62890625,
-5.015625,
-4.703125,
-2.044921875,
2.97265625,
12.96875
] | 2.640639 | 20.593005 | 24.3324 | 2.317961 |
[
2.7838711738586426
] | -13,330.732657 | 5.586741 | -19,036.337801 | 0.551526 | 5.986146 |
[
-3.25,
-3.70703125,
-2.630859375,
-3.67578125,
2.783203125,
10.125
] |
[
-6.078125,
-2.912109375,
-2.6796875,
-1.673828125,
3.5546875,
5.953125
] | |
BkiUe-bxK4tBVhat5pTg
|
\section{Introduction}
Nuclei far from the stability valley open a new test ground for nuclear
models.
Recently, many experimental and theoretical efforts have been
paid to study the structure and the reaction mechanisms in
the nuclei near the drip
lines. Modern radioactive nuclear beam facilities and
experimental detector setups have revealed several
unexpected effects in nuclei, such as the
existence of haloes and skins~\cite{Tani},
the modifications of shell closures~\cite{Ozawa}
and the pygmy dipole resonances~\cite{Pygmy,Pygmy_Sn}.
One of the current topics
is the role of the tensor interactions on the shell evolution
of nuclei far from the stability line.
The role of the tensor interactions in the evolution
of the single-particle states
was first discussed within the Skyrme Hartree-Fock (HF)
framework by Fl. Stancu {\em et al.}, almost thirty
years ago~\cite{Stancu}.
However, serious attempts have never been devoted, until very recently,
to the study of its effects on the evolution of the shell
structure in heavy exotic nuclei. In fact, the Skyrme parameter
sets which are widely used in nuclear structure calculations do not
include the tensor contribution. This contribution was
included only in the so-called Skyrme-Landau parametrizations
of Ref.~\cite{Liu}.
In the present paper, we discuss the isospin dependence of the shell
structure (in particular, the spin-orbit splitting)
of the Z=50 isotopes and N=82 isotones. We use the HF plus
Bardeen-Cooper-Schrieffer (BCS) approach, by employing
a Skyrme parameter set plus
the triplet-even and triplet-odd tensor zero-range
tensor terms, which read
\begin{eqnarray}
v_T &=& {T\over 2} \{ [ (\sigma_1\cdot k^\prime)
(\sigma_2\cdot k^\prime) - {1\over 3}(\sigma_1\cdot\sigma_2)
k^{\prime 2} ] \delta(r_1-r_2)
\nonumber \\
&+& \delta(r_1-r_2)
[ (\sigma_1\cdot k)(\sigma_2\cdot k) - {1\over 3}
(\sigma_1\cdot\sigma_2) k^2 ] \}
\nonumber\\
&+& U \{ (\sigma_1\cdot k^\prime) \delta(r_1-r_2)
(\sigma_1\cdot k) - {1\over 3} (\sigma_1\cdot\sigma_2)
[k^\prime\cdot \delta(r_2-r_2) k] \}.
\label{eq:tensor}
\end{eqnarray}
In the above expression,
the operator ${k}=(\nabla_1-\nabla_2)/2i$ acts on the right and
${k}'=-(\nabla_1-\nabla_2)/2i$ acts on the left.
The coupling constants $T$ and $U$ denote the strength of the
triplet-even and triplet-odd tensor interactions, respectively;
we treat these coupling constants as free parameters in the following study.
The tensor interactions (\ref{eq:tensor}) give contributions both to the
binding energy and to the spin-orbit splitting,
which are, respectively, quadratic and linear in the
proton and neutron spin-orbit densities,
\begin{equation}
J_q(r)=\frac{1}{4\pi r^3}\sum_{i}v_{i}^2(2j_{i}+1)\left[j_i(j_i+1)
-l_i(l_i+1) -\frac{3}{4}\right]R_i^2(r).
\label{eq:sd}
\end{equation}
In this expression $q=0(1)$ labels neutrons (protons), while
where $i=n,l,j$ runs over all states having the given $q$.
The $v_{i}^2$ is the BCS
occupation probability of each orbital and $R_i(r)$ is
the radial part of the wavefunction. It should be noticed that
the exchange part of the central Skyrme interaction gives
the same kind of contributions to the total energy density and
spin-orbit splitting.
The central exchange and tensor contributions to the energy
density $H$ are
\begin{equation}
\Delta H = {1\over 2}\alpha(J_n^2+J_p^2) + \beta J_n J_p.
\label{eq:dE}
\end{equation}
The spin-orbit potential is given by
\begin{eqnarray}
U_{s.o.}^{(q)} = {W_0\over 2r} \left( 2{d\rho_q\over dr} +
{d\rho_{q^\prime}\over dr} \right) + \left( \alpha
{J_q\over r} + \beta {J_{q^\prime}\over r} \right),
\label{eq:dW}
\end{eqnarray}
where the first term on the r.h.s comes from the Skyrme
spin-orbit interaction whereas the second term includes both the central
exchange and the tensor contributions, that is, $\alpha= \alpha_C +\alpha_T$
and $\beta=\beta_C +\beta_T$. The central exchange contributions are
written in terms of the usual Skyrme parameters,
\begin{eqnarray}
\alpha_C & = & {1\over 8}(t_1-t_2) - {1\over 8}
(t_1x_1+t_2x_2) \nonumber\\
\beta_C & = & -{1\over 8}(t_1x_1+t_2x_2).
\label{eq:dWc}
\end{eqnarray}
Basic definitions of all quantities derived from the Skyrme
parameters can be found in Refs.~\cite{Skyrme_old,Chabanat}.
The tensor contributions are expressed as
\begin{eqnarray}
\alpha_T & = & {5\over 12}U \hfill\nonumber\\
\beta_T & = & {5\over 24}(T+U).
\label{eq:dWT}
\end{eqnarray}
The central exchange contributions (\ref{eq:dWc}) have been
neglected when fitting most of the Skyrme parameter sets,
and when performing most of the previous HF calculations.
In this work, we employ the SLy5 parameter set~\cite{Chabanat}
which has been fitted with the same protocol of the more
widely used SLy4 set and should consequently have similar
quality. In the case of SLy5, the central exchange
contributions are included in the fit and we take
them into account here.
Except for the double-magic systems, we perform HF-BCS
in order to take into account the pairing correlations.
Our pairing force is a zero-range, density-dependent one,
namely
\begin{equation}
V=V_0 \left( 1- \left( {\varrho \left( {\vec r_1+\vec r_2\over 2} \right)
\over \varrho_0} \right)^\gamma \right)\cdot\delta(\vec r_1-\vec r_2).
\label{ppforce}
\end{equation}
The parameters of this force
(that is, $V_0$=680 MeV$\cdot$fm$^{3}$, $\gamma$=1 and
$\varrho_0$=0.16 fm$^{-3}$)
have been fixed
along the Z=50 isotopic chain in
connection with the SLy4 set in Ref.~\cite{Fracasso}.
Therefore, we employ the same force here both for the neutron
and the proton pairing interactions in the 50-82 shell, neglecting small
readjustments which could be made to account for the Coulomb
anti-pairing effect in the case of protons.
Before coming to a detailed analysis of our results, let us
mention the important general features associated with the tensor
and the central exchange
contributions to the
spin-orbit splitting.
The first point concerns the A-dependence (or isospin dependence) of the
the first and second terms in the r.h.s. of Eq.~(\ref{eq:dW}).
Since the Skyrme spin-orbit contributuion (proportional to $W_0$)
gives a value of the spin-orbit splitting which is
linear in the derivatives of the proton and neutron densities,
the associated mass number and isospin dependence
is very moderate in heavy nuclei. On the other hand, the second term
in Eq. (\ref{eq:dW}) depends on the spin density $J_q$ which has
a more peculiar behavior. $J_q$ gives essentially no contribution
in the spin-saturated cases, but it increases
linearly with the number of particles if only one of the
spin-orbit partners is filled. The sign of the $J_q$ will change
depending upon the quantum numbers of the orbitals which are
progressively filled: that is, the orbital with $j_{>}=l+1/2$ gives a positive
contribution to $J_q$ while the orbital with $j_{<}=l-1/2$ gives a
negative contribution to $J_q$. This must be kept in mind to understand
the results which are discussed below.
According to Ref.~\cite{Stancu}, the optimal parameters
$\alpha_T$ and $\beta_T$ should be found in a triangle in the two dimensional
($\alpha_T$, $\beta_T$) plane, lying in the quadrant of
negative $\alpha_T$ and positive $\beta_T$. At that time,
the force SIII was used. As already mentioned, we wish to use
here the Lyon forces which have been fitted using a more
complete protocol (including the neutron matter equation of
state) and have some better features like a more realistic
value of the incompressibilty $K_\infty$. Therefore, we have
refitted the values of ($\alpha_T$, $\beta_T$) using the
recent experimental data~\cite{Schiffer} for the single-particle
states in the N=82 isotones and the Z=50 isotopes. We have not
tried to refit all the Skyrme parameters after including
the tensor terms. This can be left for future work.
However, we have checked that the binding energies and the r.m.s.
charge radii of $^{132}$Sn ($^{208}$Pb) change, respectively,
by +0.65\% and -0.17\% (+0.46\% and -0.11\%) when we include
the tensor force. The parameters we have chosen are
$\alpha_T$ = -170 MeV$\cdot$fm$^5$ and $\beta_T$ = 100
MeV$\cdot$fm$^5$. We should mention that for the force
SLy5, $\alpha_C$ = 80.2 MeV$\cdot$fm$^5$ and $\beta_C$ = -48.9
MeV$\cdot$fm$^5$. The fact that we need significantly larger values of
($\alpha_T$, $\beta_T$) as compared to Ref.~\cite{Stancu}
can be adscribed to the fact that the effect of the
central exchange terms, with ($\alpha_C$, $\beta_C$) having
opposite sign as the ($\alpha_T$, $\beta_T$) required by our fit, must
be counterbalanced.
In Fig.~\ref{fig:Z50}, the energy differences of
the proton single-particle states,
$\varepsilon(h_{11/2})-\varepsilon(g_{7/2})$, along the Z=50 isotopes
are shown
as a function of the neutron excess N-Z. The original SLy5
interaction fails to reproduce the experimental trend both qualitatively and
quantitatively. Firstly, the energy differences obtained within HF-BCS are
much larger than the empirical data. Secondly, starting from
the double-magic $^{132}$Sn isotope, the experimental data markedly decrease
as the neutron excess decreases and reach
about 0.5 MeV at the minimum value in $^{112}$Sn. On the other hand,
using the HF-BCS approach with SLy5 the result is qualitatively the
opposite: the energy differences slightly increase as the neutron excess decreases,
and there is a maximum around $^{120}$Sn. We have studied also
several other Skyrme parameter sets and found almost the same
trends as that of SLy5.
In the results displayed in Fig.~\ref{fig:Z50},
we can see a substantial improvement due to the introduction of the
tensor interaction with ($\alpha_T$, $\beta_T$)=(-170,100) MeV$\cdot$fm$^5$.
This choice gives a very nice agreement with
the experimental data in the range $20\le$ (N-Z) $\le 32$, both quantitatively
and qualitatively. The HF+tensor results can be qualitatively understood by
simple arguments, looking at Eq.~(\ref{eq:dW}).
In the Z=50 core, only the proton $g_{9/2}$ orbital dominates the proton spin
density $J_p$ (cf. Eq. (\ref{eq:sd})); consequently, with a negative value
of $\alpha_T$, the spin-orbit potential (\ref{eq:dW}) is enlarged in
absolute value (notice that $W_0$ is positive and the radial derivatives of
the densities are negative), the values
of the proton spin-orbit splittings are increased,
and the energy difference $\varepsilon(h_{11/2})-\varepsilon(g_{7/2})$
is reduced with respect to HF-BCS without tensor. This reduction is seen
better around N-Z=20: in fact, $^{120}$Sn is, to a good extent, spin-saturated
as far as the neutrons are concerned so that one gets no contribution from
$J_n$. It should be noticed that the term in $\alpha$ does not give any
isospin dependence to the spin-orbit potential for a fixed
proton number, but only the term in $\beta$ can be responsible for
the isospin dependence. In a pure HF description,
from N-Z=6 to 14, the $g_{7/2}$ neutron orbit is gradually filled and
$J_n$ is reduced. Then, the positive value of $\beta_T$ enlarges in absolute value
the spin-orbit potential and increases the spin-orbit splitting,
so that the energy difference $\varepsilon(h_{11/2})-\varepsilon(g_{7/2})$
becomes smaller. Because of pairing, this decrease is not so pronounced
in the results of Fig.~\ref{fig:Z50}. Moreover, from N-Z=14 to 20, the
$s_{1/2}$ and $d_{3/2}$ neutron orbits are occupied and
in this region the spin density is not so much changed since
the $s_{1/2}$ orbital does not provide any contribution. Instead,
for N-Z=20 to 32, the $h_{11/2}$ orbital is gradually filled.
This gives a positive contribution to the spin-orbit potential (\ref{eq:dW})
and the the spin-orbit splitting becomes smaller.
$\varepsilon(h_{11/2})-\varepsilon(g_{7/2})$ consequently increases, and
this effect is well pronounced in our theoretical results.
The magnitude of $\beta$ determines the slope of the isospin dependence,
so that a larger $\beta$ would give a steeper slope.
In Fig.~\ref{fig:N82}, the energy difference $\varepsilon(i_{13/2})-\varepsilon(h_{9/2})$
for neutrons ouside the closed N=82 core is plotted as a function of the neutron
excess. The notation is the same of the previous figure.
Essentially, the same arguments already made in the previous
paragraph can be applied in order to understand the results;
simply, we should remind that the proton number is increasing
from the right (where the last nucleus displayed is $^{132}$Sn) to the left
(where the first isotope plotted is $^{150}$Er). The 1$g_{7/2}$ and 2$d_{5/2}$
orbitals are rather close in energy, above the last
occupied proton state 1$g_{9/2}$ of the Z=50 core, and their
occupations are affected by the pairing correlations introduced by
the BCS approximation. These two proton
orbitals have opposite effect on the spin orbit
potential (\ref{eq:dW}). Because of its larger value of $j$, the
1$g_{7/2}$ orbital turns out to play a more important role
on the spin-orbit potential, when the tensor interaction is included, in the nuclei
with N-Z decreasing from 32 to 24. Accordingly, with positive $\beta_T$ the
neutron spin-orbit potential is enlarged in absolute value:
the spin-orbit splitting is made
larger for these isotones, so that the $i_{13/2}$ orbital is pushed down and
the $h_{9/2}$ is pushed up. These changes make the energy gap
$\varepsilon(i_{13/2})-\varepsilon(h_{9/2})$ smaller
for the nuclei from N-Z=32 ($^{132}$Sn) to N-Z=24 ($^{140}$Ce).
Then, the occupation of the 2$d_{5/2}$ orbital reverses the trend around
N-Z=22 ($^{142}$Nd). The theoretical trend remains the same
until N-Z=14, since the effect of the 2$d_{3/2}$ occupation is
counterbalanced by the occupation of the 1$h_{11/2}$ which is
not much higher and enters the active BCS space.
The role of the tensor interaction due to the $\beta_T$ term
is expected from the discussion made by J.M. Blatt and
V.F. Weisskopf \cite{BW} for the
deuteron. In Ref. \cite{Otsuka} the same argument was also
presented. The role of $\alpha_T$ has not yet been examined
in a quantitative way within the mean field calculations, as
this term comes from the triplet-odd tensor interaction.
The assessment of its role is new, since
the triplet-odd tensor interaction was
not included in the analysis of Refs. \cite{BW,Otsuka}.
Recently, Brown {\em et al.}~\cite{Brown} studied the tensor
interactions in $^{132}$Sn and $^{114}$Sn, based on the
parameter set Skx. They considered both positive and
negative values of $\alpha_T$ in the HF calculations and
they concluded that $\alpha_T<$ 0 gives a better agreement
with the experimental data. This result is consistent with
the present systematic study of the single-particle
states in the Z=50 isotopes and N=82 isotones, performed
within the HF+BCS model.
The effect of the tensor interactions can be
tested on other single-particle states which are empirically
known. In this work, we have also considered the relative
position of the 2$d_{3/2}$ and 1$h_{11/2}$ neutron states
in $^{132}$Sn and $^{100}$Sn. In the former case ($^{132}$Sn),
experimentally the two states are the last occupied,
with the 2$d_{3/2}$ being less bound than 1$h_{11/2}$ by
about 240 keV (see Fig. 8 of~\cite{GL}). Theoretically, all
the mean field calculations with Skyrme or Gogny forces, as well as
the relativistic mean field (RMF) ones, result with the
opposite ordering (see Fig. 7 of~\cite{RMP}). In particular,
with the SLy5 force employed here, the 1$h_{11/2}$ orbital
is less bound by 1.76 MeV. The contribution of the added
tensor force reduces this value to 0.67 MeV. It
has to be noted that the the right position of the 2$d_{3/2}$
level may be rather important for the proper description of
the low-lying dipole strength in $^{132}$Sn. In fact, the
results obtained using a Skyrme force in Ref.~\cite{Sarchi}
show that this strength has basically single-particle
character; but even in the calculation of Ref.~\cite{ll_rmf},
in which the low-lying dipole strength turns out to
present a certain degree of collectivity,
the energy position of the levels we have mentioned is relevant
since the configurations involving the
2$d_{3/2}$ hole contribute to
about 50\% of the low-lying collective state.
In the nucleus $^{100}$Sn, experimentally
the 2$d_{3/2}$ level is more bound by 0.9 MeV (see
Fig. 7 of~\cite{GL}). In our HF calculation with the SLy5
force, the two levels have the right order but their
energy difference is 2.13 MeV. It is quite satisfactory that
this difference becomes 1.33 MeV after including the
tensor interactions.
We have already mentioned that total binding energies and
charge radii of $^{132}$Sn and $^{208}$Pb are not
extremely sensitive to the tensor interactions.
We have also checked in our work the effect of the tensor
terms on the isotope shift of the Pb isotopes.
Actually, this effect is small (and does not even have
the correct sign to reproduce experiment).
Thus, the tensor interactions is unable to produce the
well-known empirical kink in the trend of the charge radii beyond
$^{208}$Pb, which instead results from the introduction
of generalized spin-orbit functionals~\cite{RF}.
Single-particle energies, and other ground-state properties,
are not the only observables which are affected by the
inclusion of tensor interactions. There exist excited
states which reflect very much the behavior of the
spin-orbit splittings. One of them is the well-known
Gamow-Teller resonance (GTR). We have made a simple
estimate of the effect of the tensor interactions
on the GTR centroid by using the sum rules.
In charge-exchange RPA calculations,
the following sum rules are satisfied~\cite{AK},
\begin{eqnarray}
m_-(0) - m_+(0) & = & \langle 0 \vert [F^\dagger,F] \vert 0\rangle,
\nonumber\\
m_-(1) + m_+(1) & = & \langle 0 \vert [F^\dagger, [H,F]] \vert 0\rangle.
\label{sum_rules}
\end{eqnarray}
In the above expressions, $m_\pm(k)$ denotes the $k$-th
moment of the strength in the $\Delta T_z=\pm 1$ channel:
in particular, we are considering the non energy-weighted sum
rule $m(0)$ and the energy-weighted sum rule $m(1)$.
The associated operators $F$ and $F^\dagger$ act, respectively,
in the $\Delta T_z=-1$ and $\Delta T_z=+1$ channels. In
the Gamow-Teller case, they read
\begin{eqnarray}
F & = & \sum_i \vec\sigma(i)t_-(i), \nonumber\\
F^\dagger & = & \sum_i \vec\sigma(i)t_+(i).
\end{eqnarray}
Moreover, in Eq. (\ref{sum_rules}), $H$ is the total
Skyrme Hamiltonian and $\vert 0\rangle$ is the HF ground
state. In nuclei with neutron excess, the contributions
associated with the $\Delta T_z=+1$ channel are negligible
and we can approximate the GT centroid as
\begin{equation}
E_{GT} = {m_-(1)\over m_-(0)} \sim
{\langle 0 \vert [F^\dagger, [H,F]] \vert 0\rangle \over
\langle 0 \vert [F^\dagger,F] \vert 0\rangle}.
\end{equation}
The contribution from the tensor interaction to the
GT centroid is obtained by replacing the total Hamiltonian
$H$ in the previous formula, with the two-body force
of Eq. (\ref{eq:tensor}). The calculation of the
ground-state expectation value of the
double commutator $[F^\dagger, [v_T,F]]$ has been
worked out and it gives the contribution of the
tensor force to the GT centroid,
\begin{equation}
\Delta E_{GT} = {4\pi\over 9(N-Z)}
\int dr r^2 [{24\over 5}(\beta-5\alpha)J_pJ_n -
12\alpha(J_p^2+J_n^2)].
\label{deltaE_GT}
\end{equation}
We have evaluated this latter expression, for the
Sn isotopes having N-Z larger than 20, by using
our
optimal ($\alpha_T$, $\beta_T$) values of (-170, 100).
The
results are reported in Table~\ref{table_GT}.
The numbers are not small, but this should not be
surprising since the shifts of the single-particle
states displayed in the Figures can also be of the
order of 1-2 MeV. In fact, the positive energy shift
can be expected in $^{208}$Pb as well and can be
understood as follows. The spin-orbit densities (\ref{eq:sd})
receive contribution only from the $i_{13/2}$ orbital
(in the case of neutrons) and from the
$h_{11/2}$ orbital (in the case of protons).
From Eq. (\ref{eq:dW}), one sees that, since
$\alpha_T$ is negative and $\vert\alpha_T\vert
> \vert\beta_T\vert$, the net effect is an increase
of the spin-orbit splitting. Consequently, the
excitation energy of the dominant unperturbed
Gamow-Teller configuration, that is, $\nu i_{13/2}
\rightarrow \pi i_{11/2}$, is shifted upwards by
the tensor correlations. With similar arguments we
can understand the numbers reported in Table~\ref{table_GT},
as we did for the values plotted in the previous
Figures. Actually, we should remind that the analysis in terms
of the sum rules is not able to tell whether
the peak energy is affected as much as the
centroid $m(1)/m(0)$ since the main peak does not
exhaust, as a rule, the whole strength.
Accurate QRPA calculations of the Gamow-Teller and
spin-dipole resonances are reported in
Ref.~\cite{Fracasso2}. The behavior of different Skyrme
parameter sets, without the tensor contribution, is
critically discussed. In fact, no RPA or QRPA calculations
including the two-body tensor force are presently
available, for any kind of vibrational mode;
accordingly, results obtained without the tensor
interactions should be still kept as reference
until a global refit of the Skyrme plus tensor
parameters is carefully accomplished. The tensor
force is not only expected to produce effect on the
Gamow-Teller states. Other vibrational
states (like the low-lying 2$^+$ which in many
systems, once more, reflects the spin-orbit
splitting~\cite{Peru})
will be certainly affected.
A further question for future work is the role of
correlations beyond mean field. As discussed at length
in Ref.~\cite{Mahaux}, the coupling of single-particle
states to vibrational states
has the net effect of increasing the level
density around the Fermi surface by about 30\%, by shifting occupied and
unoccupied states in opposite directions. Smaller effects are
expected for the energy differences we are considering here since
these differences involve pairs of states which are either
occupied or unoccupied. The net shift may be of the
order of few hundreds of keV as estimated from
$^{132}$Sn~\cite{unp}.
In conclusion, the present work has shed light on the necessity
to include the tensor component in the Skyrme framework. The
first attempts in this direction were focusing on the effect
of the tensor force in magic nuclei but, as we have
stressed, if the nuclei are spin-saturated the spin-orbit
splittings are not affected at all by the tensor force. The
experimental mesurement of the isospin dependence of single-particle
energies has opened the possibility to fit the parameters of
the zero-range effective tensor force we are employing.
Our results show that the introduction of the tensor force can
fairly well explain the isospin dependence of energy differences
between single-particle proton states outside the Z=50 core, and
neutron states outside the N=82 core. We have not attempted to
refit a Skyrme force by including the tensor contribution, but
we have discussed, by using the case of the Gamow-Teller
centroids, that excited state properties will also be affected
by the tensor. An ambitious refitting program of Skyrme forces
should therefore be undertaken and deformed systems should
be considered as well~\cite{priv}. This is left as a future prospect,
together with the role of particle-vibration coupling in this
context.
\begin{acknowledgments}
We would like to thank D. M. Brink for stimulating and enlightening
discussions. One of us (G.C.) gratefully acknowledges the
hospitality of the University of Aizu, where the present work
has started. The work is supported in part by the Japanese
Ministry of Education, Culture, Sports, Science and Technology
by Grant-in-Aid for Scientific Research under
the program number (C (2)) 16540259.
\end{acknowledgments}
| -15,620.469488 |
[
-3.490234375,
3.208984375
] | 16.735537 |
[
-3.212890625,
-0.15771484375,
-2.1171875,
-6.6484375,
-0.63720703125,
9.3515625
] |
[
2.884765625,
8.2890625,
2.384765625,
4.86328125
] | 198 | 3,430 |
[
-2.89453125,
3.1484375
] | 27.069957 |
[
-5.99609375,
-3.9140625,
-4.15625,
-2.423828125,
1.8525390625,
12.15625
] | 1.864426 | 6.458893 | 27.667638 | 2.65359 |
[
1.3582030534744263
] | -11,586.94895 | 5.238484 | -15,163.828204 | 0.612199 | 5.689008 |
[
-2.787109375,
-3.7109375,
-3.6171875,
-4.6796875,
2.400390625,
12.1015625
] |
[
-5.47265625,
-2.298828125,
-2.166015625,
-1.193359375,
3.771484375,
4.546875
] | |
BkiUdTA4uBhivVgojJ1E
|
\section{Introduction}
Systems described by first order actions, \textit{i.e.} Lagrangians linear in
the velocities \cite{Schwinger, Symanzik} appear in many branches in Physics.
In 1928 Dirac proposed a system with first order action \cite{Dirac-eq} which
has been used since then to describe fermion fields. Later in the 1930's, a
Lagrangian with a linear kinematic term was also proposed in order to describe
bosonic fields (DKP\ theory) \cite{DKP, DKP A, DKP B}. In gravitation, this
type of Lagrangian appeared for the first time in 1919 with Palatini's work
\cite{Palatini}, which came to be the basis of a new method of variation.
First order actions were also present in Schwinger's development of quantum
theory \cite{Schwinger-FT, Schwinger-FT 2}.
A significant feature of these systems is that they always have a null Hessian
matrix, \textit{i.e.} they are singular (constrained) systems, and hence they
must be properly treated in order to accomplish a hamiltonian formulation.
In the particular case of first-order actions, different approaches can be
applied. The most usual one is the formalism developed in 1950 by Dirac
\cite{Dirac0, Dirac1, Dirac2} to treat general constrained systems, in which
the hamiltonian structure is employed \cite{Ham-Struc, Ham-Struc1, Ham-Struc2,
Ham-Struc3, Ham-Struc4}. One of the main features of the Dirac formalism is
the fact that it allows one to introduce generalized brackets which conduct to
a consistent quantization of the system. The application of this formalism to
first-order action can be found in reference \cite{Govaerts}.
Another approach to deal with Lagrangians with linear velocities was developed
by Faddeev and Jackiw \cite{F-J} in 1988, where generalized brackets framed on
the symplectic structure of phase space were also introduced, leading to a
consistent quantization at least to purely bosonic variables.
A third approach that can also be applied to study such systems is the
Hamilton-Jacobi (HJ) formalism, which is based on the Carath\'{e}odory
Equivalent Lagrangian method \cite{Carat}. This method, developed by
Carath\'{e}odory to treat regular systems with first derivatives, is an
alternative way to obtain the Hamilton-Jacobi equation starting from
lagrangian formalism. In 1992 G\"{u}ler generalized Carath\'{e}odory's method
to treat singular systems \cite{Guler 1, Guler 2} and more recently Pimentel
and Teixeira \cite{Rand Pim1, Rand Pim2}, worked with Lagrangians with higher
order derivatives. In 1998 Pimentel, Teixeira and Tomazelli made an extension
to deal with Berezinian singular systems \cite{Rand Pim Jef}. Important
applications of this method can be found in literature \cite{REF HJ, REF HJ 1,
REF HJ 2, REF HJ 3,REF HJ 4}, including an application to Lagrangians linear
in the velocities \cite{Guler -lin-veloc}, where no generalized brackets are introduced.
In this work we intend to study systems with first-order actions \textit{via}
Hamilton-Jacobi formalism and show how generalized brackets and a symplectic
structure appear in a natural way. Accordingly, in the two next sections we
will make a review of first-order actions and HJ formalism, respectively.
Afterwards we will apply the HJ structure to the Lagrangians of interest and
see how generalized brackets are introduced. Then we will show some examples
and\ at last some concluding remarks will be made.
\section{First Order Actions\label{Sec1}}
We shall consider here a system whose dynamical evolution is described by some
variational principle from the action integral%
\begin{equation}
\mathcal{A}\left[ z_{A}\right] =\int_{i_{i}}^{t_{f}}dtL\left( z_{A},\dot
{z}_{A}\right) ,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,A=1,...,N. \label{acao}%
\end{equation}
In this expression $z_{A}$ are coordinates and $\dot{z}_{A}$ are the time
derivative of $z_{A}\,$. We will assume that the Lagrangian function has
linear dependence on the velocities $\dot{z}_{A}$, \textit{i.e.},%
\begin{equation}
L\left( z_{A},\dot{z}_{A}\right) =\dot{z}_{A}K^{A}\left( z_{B}\right)
-V\left( z_{B}\right) , \label{L}%
\end{equation}
where $K^{A}$ and $V$ are arbitrary functions of the coordinates $z_{B}$.
One can immediately verify that the variational problem remains the same if we
consider, instead of $L$, another Lagrangian function $\tilde{L}\left(
z_{A},\dot{z}_{A},t\right) $, which differs from the first by a total time
derivative:%
\begin{align}
\tilde{L}\left( z_{A},\dot{z}_{A},t\right) & =L\left( z_{A},\dot{z}%
_{A}\right) +\frac{d}{dt}Y\left( z_{A},t\right) =\nonumber\\
& =L+\frac{\partial Y}{\partial t}+\dot{z}_{A}\frac{\partial Y}{\partial
z_{A}}. \label{L_transf}%
\end{align}
An interesting property of this transformation in $L$ is that we can preserve
the structure of (\ref{L}) for $\tilde{L}$, $\tilde{L}\left( z,\dot
{z},t\right) =\dot{z}_{A}\tilde{K}^{A}\left( z,t\right) -\tilde{V}\left(
z,t\right) ,$ if $K^{A}$ and $V$ transform respectively as%
\begin{align}
\tilde{K}^{A}\left( z\right) & =K^{A}\left( z\right) +\frac{\partial
Y}{\partial z_{A}}\left( z,t\right) ,\nonumber\\
\tilde{V}\left( z\right) & =V\left( z\right) -\frac{\partial Y}{\partial
t}. \label{K_transf}%
\end{align}
Since the variational problem is unchanged by the transformations above, we
can expect that the equations of motion depend on quantities that are
invariant by these transformations. One example of a quantity that has such a
property is the {}``curl'' of $K^{A}$:%
\begin{equation}
M^{AB}\equiv\frac{\partial K^{B}}{\partial z_{A}}-\frac{\partial K^{A}%
}{\partial z_{B}}=-M^{BA}=\tilde{M}^{AB}, \label{M}%
\end{equation}
and as it will be seen in the next section, it is related to equations of motions.
\section{The Hamilton-Jacobi Formalism for an Arbitrary Lagrangian}
The Hamilton-Jacobi (HJ) equation is usually obtained from the hamiltonian
approach when a specific canonical transformation is considered. An
alternative path to reach HJ formalism was developed by
Carath\'{e}odory\cite{Carat} and starts from the lagrangian approach without
mentioning the hamiltonian one. According to Carath\'{e}odory Equivalent
Lagrangian method, a minimum of the action $\mathcal{A}$ can be found when we
consider a set of functions $\beta_{A}\left( z_{B},t\right) $ such that
\begin{equation}
\tilde{L}\left( z_{A},\dot{z}_{A}=\beta_{A}\left( z_{B},t\right) \right)
=L\left( z_{A},\dot{z}_{A}\right) +\frac{\partial}{\partial t}Y\left(
z_{A},t\right) +\frac{\partial Y\left( z_{A},t\right) }{\partial z_{A}}%
\dot{z}_{A}=0, \label{L'=0}%
\end{equation}
and in a neighbourhood of $\dot{z}_{A}=\beta_{A}\left( z_{B},t\right) $ the
condition $\tilde{L}\left( z_{A},\dot{z}_{A}\right) >0$ is satisfied. From
these conditions it follows that%
\begin{equation}
p^{A}\equiv\left. \frac{\partial L}{\partial\dot{z}_{A}}\right| _{\dot
{z}_{B}=\beta_{B}}=-\left. \frac{\partial Y}{\partial z_{A}}\right|
_{\dot{z}_{B}=\beta_{B}}. \label{momenta}%
\end{equation}
If $L$ is a singular Lagrangian then the Hessian matrix, $H^{AB}%
=\frac{\partial^{2}L}{\partial\dot{z}_{A}\partial\dot{z}_{B}}$, has a null
determinant, $\det H^{AB}=0$; and if this matrix has rank $P=N-R$, then we can
find a $P\mathsf{x}P$ submatrix such that%
\begin{equation}
\det H^{ab}=\det\frac{\partial p^{a}}{\partial\dot{z}_{b}}\neq
0,~\,\,\,\,\,\,\,\,\,\,\,\,a,b=R+1,...,N. \label{subHess}%
\end{equation}
For this case we can verify that $R$ momenta $p^{\alpha}$ ($\alpha=1,...R$)
have no dependence on any velocity, which means that $R$ velocities cannot be
written as functions of $z$ and $p$, as it happens to $\dot{z}_{b}$, $\dot
{z}_{b}=f_{b}\left( z_{A},p^{a}\right) $. We conclude that $R$ relations of
the type%
\begin{equation}
p^{\alpha}=-H^{\alpha}\left( t,z_{\beta}\equiv t_{\beta},z_{a},p^{a}\right)
\label{vinc1}%
\end{equation}
must be satisfied.
Moreover if we define $H_{0}\equiv p^{A}\dot{z}_{A}-L$ it follows from
(\ref{L'=0}) that%
\begin{equation}
p^{0}+H^{0}\left( t,t_{\alpha},z_{a},p^{a}\right) =0, \label{eqHJ}%
\end{equation}
where $p^{0}\equiv\frac{\partial S}{\partial t}$.
We see that equations (\ref{vinc1}) and (\ref{eqHJ}) lead us to define $R+1$
conditions%
\begin{equation}
\phi^{\alpha^{\prime}}\equiv p^{\alpha^{\prime}}+H^{\alpha^{\prime}}\left(
t_{\beta^{\prime}},z_{a},p^{a}\right) =0,~\,\,\,\,\,\,\,\,\alpha^{\prime
},\beta^{\prime}=0,1,...,R, \label{vinculos}%
\end{equation}
where $t^{0}\equiv t$. These conditions $\phi^{\alpha^{\prime}}=0$ are usually
called \textit{constraints}, and they constitute a set of first order partial
differential equations, called \textit{Hamilton-Jacobi Partial Differential
Equations} (HJPDE).
\subsection{Integrability Conditions}
In order to integrate the HJPDE (\ref{vinculos}) we can use the method of
characteristics\cite{Carat}, which conducts us to total differential equation%
\begin{equation}
\left\{
\begin{array}
[c]{c}%
d\eta^{I}=E^{IJ}\frac{\partial\phi^{\alpha}}{\partial\eta^{J}}dt_{\alpha
}=\left\{ \eta^{I},\phi^{\alpha}\right\} dt_{\alpha},\\
dS=\frac{\partial S}{\partial z_{A^{\prime}}}dz_{A^{\prime}}=p^{A^{\prime}%
}\frac{\partial\phi^{\alpha}}{\partial p^{A^{\prime}}}dt_{\alpha},
\end{array}
\right. \,\,\
\begin{array}
[c]{c}%
I,J=\left( \zeta;A^{\prime}\right) ,\,\zeta=1,2;\\
\alpha=0,...,R;~A^{\prime}=0,...,N;
\end{array}
\label{eq carac}%
\end{equation}
where $\left\{ \eta^{1A\prime}\right\} =\left\{ z_{A^{\prime}}\right\} $
and $\left\{ \eta^{2A^{\prime}}\right\} =\left\{ p^{A^{\prime}}\right\} $,
$E^{IJ}=\delta_{\,B^{\prime}}^{A^{\prime}}\left[ \delta_{\,1}^{\zeta}%
\delta_{\,\sigma}^{2}-\delta_{\,2}^{\zeta}\delta_{\,\sigma}^{1}\right] ,$
$(I=\left( \zeta,A^{\prime}\right) ,\,J=\left( \sigma,\,B^{\prime}\right)
)$. In this expression we use the definition of Poisson Brackets $\left\{
F,G\right\} =\frac{\partial F}{\partial\eta^{I}}E^{IJ}\frac{\partial
G}{\partial\eta^{J}}$. According to this method, if the characteristic
equations are integrable then the HJPDE will have a unique solution
(determined by initial conditions). To obtain (\ref{eq carac}) we assume that
the momenta and coordinates are independent quantities, and we can observe
that if $d\eta^{I}$ (whose equations will be called equations of motions)
constitute an integrable system, then $dS$ will be integrable as a consequence.
To assure the integrability of the equations of motion we must recall from the
theory of differential equations that, associated with a set of total
equations, $dx_{I}=b_{I}^{\alpha}\left( x_{J}\right) dt_{\alpha}$, there are
linear operators $X^{\alpha}$ such that%
\begin{equation}
X^{\alpha}F\left( x^{J}\right) =b_{I}^{\alpha}\frac{\partial F}{\partial
x_{I}}=0. \label{eq dif parc}%
\end{equation}
From this result it is obvious that%
\begin{equation}
\left[ X^{\alpha},X^{\beta}\right] F=\left( X^{\alpha}X^{\beta}-X^{\beta
}X^{\alpha}\right) F=0, \label{comut F}%
\end{equation}
if $F$ is at least twice differentiable.
The partial differential equations $X_{\alpha}F=0$ are said to be
\textit{complete} if
\[
\left[ X^{\alpha},X^{\beta}\right] F=C_{~\,\,\gamma}^{\alpha\beta}X^{\gamma
}F.
\]
If this condition is not satisfied we can define a new operator $X$ such that
$XF=0$, and it must be added to previous set $X_{\alpha}$, and we must verify
if this new set is complete. This procedure must be repeated until a complete
set is obtained. The total differential equations will be integrable when the
associated partial equations constitute a complete set.
We must notice that when we define a new operator $X$ we are imposing a
restriction to phase space. In fact we are searching a subspace of the
original phase space where the equations of motions can be integrated.
Considering now the specific case of the equations of motion (\ref{eq carac})
we have $X^{\alpha}F=\left\{ F,\phi^{\alpha}\right\} $, and using Jacobi
identity for Poisson Brackets it follows $\left[ X^{\alpha},X^{\beta}\right]
F=-\left\{ F,\left\{ \phi^{\alpha},\phi^{\beta}\right\} \right\} $. The
equations of motion will be integrable if%
\begin{equation}
\left\{ \phi^{\alpha},\phi^{\beta}\right\} =C_{\,\,\gamma}^{\alpha\beta}%
\phi^{\gamma}=0, \label{cond int0}%
\end{equation}
or considering the independence of $t_{\alpha}$,%
\begin{equation}
d\phi^{\alpha}=\left\{ \phi^{\alpha},\phi^{\beta}\right\} dt_{\beta}=0.
\label{cond int1}%
\end{equation}
If these conditions are not satisfied we must restrict our phase space with
new relations $\phi=0$ until a complete set of partial differential equations
is obtained.
\section{The HJ Formalism for First Order Actions}
Let us now consider the specific case of section (\ref{Sec1}). According to
the previous section when the condition $\tilde{L}=0$ and the action is a
minimum, the momenta canonically conjugated to $z_{A}$ and $z_{0}=t$ are
respectively%
\begin{align}
p^{A} & =-\frac{\partial Y}{\partial z_{A}}%
,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,A=1,...N,\nonumber\\
p^{0} & =-\frac{\partial Y}{\partial t}. \label{momentos}%
\end{align}
However when $\tilde{L}=0$, we have%
\begin{equation}
\tilde{L}=\dot{z}_{A}\left( K^{A}\left( z\right) -p^{A}\right) -\dot
{z}_{0}\left( V\left( z\right) +p^{0}\right) =\dot{z}_{A^{\prime}}%
\tilde{K}^{A^{\prime}}\left( z,t\right) =0,\,\,\,\,A^{\prime}=0,1,...,N,
\label{L_transf=0}%
\end{equation}
where%
\begin{equation}
\tilde{K}^{0}\equiv-\left( V\left( z\right) -\frac{\partial Y}{\partial
t}\left( z,t\right) \right) . \label{K0}%
\end{equation}
If we consider $\dot{z}_{A^{\prime}}$ as independent quantities then%
\begin{equation}
\tilde{K}^{A^{\prime}}\left( z,t\right) =0\Rightarrow\left\{
\begin{array}
[c]{c}%
K^{A}\left( z\right) -p^{A}=0,\\
V\left( z\right) +p^{0}=0.
\end{array}
\right. \label{EDPHJ_1ord}%
\end{equation}
This is the set of HJPDE, which leads us to define the constraints
\begin{equation}
\phi^{A^{\prime}}\equiv p^{A^{\prime}}-K^{A^{\prime}}\left( z\right) =0,
\label{Vinc_1ord}%
\end{equation}
with $K^{0}\equiv-V\left( z\right) $.\ From this result we see that all the
coordinates have the status of parameters and then, to be consistent with the
notation of the previous section, we define%
\begin{align*}
t_{A} & \equiv z_{A},\\
t_{0} & \equiv z_{0}\equiv t.
\end{align*}
\subsection{Integrability Conditions}
To test the integrability conditions we must calculate the total differential
of the constraints (\ref{Vinc_1ord}):%
\[
d\phi^{A^{\prime}}=\left\{ \phi^{A^{\prime}},\phi^{B^{\prime}}\right\}
dt_{B^{\prime}}=\left\{ \phi^{A^{\prime}},\phi^{0}\right\} dt_{0}+\left\{
\phi^{A^{\prime}},\phi^{B}\right\} dt_{B},
\]
or explicitly
\begin{equation}
d\phi^{0}=\left\{ \phi^{0},\phi^{B}\right\} dt_{B}, \label{d_fi_zero}%
\end{equation}%
\begin{align}
d\phi^{A} & =\left\{ \phi^{A},\phi^{0}\right\} dt_{0}+\left\{ \phi
^{A},\phi^{B}\right\} dt_{B}=\nonumber\\
& =\left\{ \phi^{A},\phi^{0}\right\} dt_{0}+M^{AB}dt_{B}. \label{de_fi_A}%
\end{align}
If we consider the independence of the parameters $dt_{B^{\prime}}$ then we
see that the equations of motion are integrable only if $\left\{ \phi
^{A},\phi^{0}\right\} =0,~M^{AB}=0$. If this is not the case we have some
problems, because all the coordinates are already parameters and no further
restriction can be done, \textit{i.e.} we cannot define a new constraint
$\phi=0$. And the obvious conclusion is: the system is not integrable.
Of course this analysis is valid when we consider $dt_{B^{\prime}}$ as
independent quantities. The question naturally arises: can we find a subspace
of the parameters space where the system becomes integrable? To answer this
question we admit that $M^{AB}$ is not nule and that such construction can be
done. Hence, in this subspace we have%
\begin{align*}
d\phi^{0} & =0,\\
d\phi^{A} & =0.
\end{align*}
The last expression shows
\begin{equation}
d\phi^{A}=\left\{ \phi^{A},\phi^{0}\right\} dt_{0}+M^{AB}dt_{B}=0\Rightarrow
M^{AB}dt_{B}=-\left\{ \phi^{A},\phi^{0}\right\} dt_{0}, \label{Mdt_B=PP_dt}%
\end{equation}
and the dependence among $dt_{B}$ and $dt_{0}$ becomes clear.
\subsubsection{The $M^{AB}$ Regular Case}
Let us now consider the case when $M^{AB}$ is a regular matrix. Hence
$\det\left( M^{AB}\right) \neq0$ and the matrix $M_{AB}^{-1}$ does exist. In
this case it is straightforward to verify that%
\begin{equation}
dt_{B}=-M_{BA}^{-1}\left\{ \phi^{A},\phi^{0}\right\} dt_{0}. \label{dt_B}%
\end{equation}
If we substitute this result in (\ref{d_fi_zero}) it follows%
\[
d\phi^{0}=-\left\{ \phi^{0},\phi^{B}\right\} M_{BA}^{-1}\left\{ \phi
^{A},\phi^{0}\right\} dt_{0}=\left\{ \phi^{B},\phi^{0}\right\} M_{BA}%
^{-1}\left\{ \phi^{A},\phi^{0}\right\} dt_{0},
\]
and since $M_{BA}^{-1}=-M_{AB}^{-1}$ it is immediate that $d\phi^{0}=0$,
because $\left\{ \phi^{B},\phi^{0}\right\} \left\{ \phi^{A},\phi
^{0}\right\} =\left\{ \phi^{A},\phi^{0}\right\} \left\{ \phi^{B},\phi
^{0}\right\} $.
Now considering the dependence stablished by (\ref{dt_B}), the differential of
any function $E=E\left( z,p\right) $ is given by%
\[
dE=\left[ \left\{ E,\phi^{0}\right\} -\left\{ E,\phi^{B}\right\}
M_{BA}^{-1}\left\{ \phi^{A},\phi^{0}\right\} \right] dt_{0}.
\]
We can now introduce new Brackets%
\begin{equation}
\left\{ F,G\right\} _{\ast}\equiv\left\{ F,G\right\} -\left\{ F,\phi
^{B}\right\} M_{BA}^{-1}\left\{ \phi^{A},G\right\} , \label{PD_HJ}%
\end{equation}
such that%
\begin{equation}
dE=\left\{ E,\phi^{0}\right\} _{\ast}dt_{0}. \label{dE}%
\end{equation}
In particular if we consider in (\ref{PD_HJ}) functions $F=F\left(
z_{A}\right) $ and $G=G\left( z_{B}\right) $, then%
\begin{equation}
\left\{ F,G\right\} _{\ast}=\frac{\partial F}{\partial z_{A}}M_{AB}%
^{-1}\frac{\partial G}{\partial z_{B}}, \label{Est_Simp}%
\end{equation}
and if $F=z_{A}$ and $G=z_{B}$,%
\begin{equation}
\left\{ z_{A},z_{B}\right\} _{\ast}=M_{AB}^{-1}. \label{Est_Simp1}%
\end{equation}
Equations (\ref{Est_Simp}) and (\ref{Est_Simp1}) show there is a symplectic
structure in phase space in HJ approach.
We can still verify the consistence of this construction by taking $E=z_{C}$
in (\ref{dE}) and see if (\ref{dt_B}) is obtained:%
\begin{align*}
dz_{C} & =\left\{ z_{C},\phi^{0}\right\} _{\ast}dt_{0}=\left[ \left\{
z_{C},\phi^{0}\right\} -\left\{ z_{C},\phi^{B}\right\} M_{BA}^{-1}\left\{
\phi^{A},\phi^{0}\right\} \right] dt_{0}=\\
& =-\delta_{C}^{B}M_{BA}^{-1}\left\{ \phi^{A},\phi^{0}\right\}
dt_{0}=-M_{CA}^{-1}\left\{ \phi^{A},\phi^{0}\right\} dt_{0}.
\end{align*}
If we consider that $z_{C}=t_{C}$ then the verification is straightforward.
At last, expliciting $\left\{ \phi^{A},\phi^{0}\right\} $, we see%
\begin{equation}
dz_{C}=M_{CA}^{-1}\frac{\partial V}{\partial z_{A}}dt_{0}. \label{eq_mov_reg}%
\end{equation}
This result is in agreement to that one presented in \cite{Govaerts}, where
the lagrangian and hamiltonian approach are considered.
\subsubsection{The $M^{AB}$ Singular Case}
In what follows we will consider the case when $M^{AB}$ is a singular matrix
(\textit{i.e.} $\det\left( M^{AB}\right) =0$) of rank $P=N-R$. Even in this
case the expression (\ref{Mdt_B=PP_dt}) holds, and from this result it is
quite simple to verify that if $\lambda^{^{(\alpha)}}$ are $R$ eigenvectors of
$M^{AB}$, $M^{AB}\lambda_{A}^{^{(\alpha)}}=0$ then
\[
\frac{\partial V}{\partial z_{A}}\lambda_{A}^{^{(\alpha)}}dt_{0}=0.
\]
Since $M^{AB}$ has rank $P=N-R$ then there is a submatrix $P\mathsf{x}P$ of
$M^{AB}$ such that%
\[
\det\left( M^{ab}\right) \neq
0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,a,b=1,...,P,
\]
which implies in the existence of $M_{ab}^{-1}$. Hence we can rewrite
(\ref{Mdt_B=PP_dt}) as (considering $a=1,...,P;\,\alpha=P+1,...,N$)
\[
-\left\{ \phi^{A},\phi^{0}\right\} dt_{0}=M^{AB}dt_{B}=M^{Ab}dt_{b}%
+M^{A\beta}dt_{\beta},\Rightarrow
\]%
\begin{equation}
\Rightarrow-\left\{ \phi^{A},\phi^{0}\right\} dt_{0}=\delta_{a}^{A}%
M^{ab}dt_{b}+\delta_{\alpha}^{A}M^{\alpha b}dt_{b}+\delta_{a}^{A}M^{a\beta
}dt_{\beta}+\delta_{\alpha}^{A}M^{\alpha\beta}dt_{\beta}. \label{PP_deltas}%
\end{equation}
Taking $A=a$ we see some $dt_{b}$ can be expressed as a linear combinations of
$dt_{\beta}$ and $dt$:%
\[
dt_{b}=-M_{ba}^{-1}\left[ \left\{ \phi^{a},\phi^{0}\right\} dt_{0}%
+M^{a\beta}dt_{\beta}\right] \Rightarrow
\]%
\begin{equation}
\Rightarrow dt_{b}=-M_{ba}^{-1}\left\{ \phi^{a},\phi^{\beta^{\prime}%
}\right\} dt_{\beta^{\prime}},\,\,\,\,\,\,\,\,\,\,\,\beta^{\prime}=\left\{
0,\beta\right\} . \label{d_t_bar}%
\end{equation}
If we consider now the case $A=\alpha$ it follows%
\[
M^{\alpha b}dt_{b}=-M^{\alpha\beta}dt_{\beta}-\left\{ \phi^{\alpha},\phi
^{0}\right\} dt,
\]
and if we use (\ref{d_t_bar}) and consider $dt_{\beta}$and $dt_{0}$ as
independent parameters, then%
\begin{equation}
\left\{
\begin{array}
[c]{c}%
\left\{ \phi^{\alpha},\phi^{0}\right\} =\left\{ \phi^{\alpha},\phi
^{b}\right\} M_{ba}^{-1}\left\{ \phi^{a},\phi^{0}\right\} \\
\left\{ \phi^{\alpha},\phi^{\beta}\right\} =\left\{ \phi^{\alpha},\phi
^{b}\right\} M_{ba}^{-1}\left\{ \phi^{a},\phi^{\beta}\right\}
\end{array}
\right. , \label{fix_sub_esp}%
\end{equation}
which tell us that, if (\ref{d_t_bar}) is satisfied, $\left\{ \phi^{\alpha
},\phi^{0}\right\} $ and the elements $\left\{ \phi^{\alpha},\phi^{\beta
}\right\} $ of the matrix $M^{AB}=\left\{ \phi^{A},\phi^{B}\right\} $ must
be related to $\left\{ \phi^{a},\phi^{0}\right\} ,\,\left\{ \phi^{a}%
,\phi^{\beta}\right\} ,$\thinspace$\left\{ \phi^{a},\phi^{b}\right\} $. The
second expression is in agreement with the fact that $M^{AB}$ is singular,
while the first one must be faced as conditions that actually fix the subspace
of the parameters where the system can be integrable. It becomes clear from
the results above that the case $A=a$ brings information about the dependence
among the $P$ parameters $dt_{b}$ and the $R$ parameters $dt_{\beta^{\prime}}%
$, while the case $A=\alpha$ stablishes $R$ conditions that determine the
subspace of integrability.
With (\ref{d_t_bar}) the differential of $E=E\left( z,p\right) $ becomes%
\begin{align}
dE & =\left[ \left\{ E,\phi^{\beta^{\prime}}\right\} -\left\{ E,\phi
^{b}\right\} M_{ba}^{-1}\left\{ \phi^{a},\phi^{\beta}\right\} \right]
dt_{\beta^{\prime}}\Rightarrow\nonumber\\
& \Rightarrow dE=\left\{ E,\phi^{\beta^{\prime}}\right\} _{\ast}%
dt_{\beta^{\prime}}, \label{dE_sing}%
\end{align}
where the Brackets $\left\{ F,G\right\} _{\ast}$ are introduced%
\begin{equation}
\left\{ F,G\right\} _{\ast}\equiv\left\{ F,G\right\} -\left\{ F,\phi
^{b}\right\} M_{ba}^{-1}\left\{ \phi^{a},G\right\} . \label{PD_HJ_sing}%
\end{equation}
And if we consider $F=F\left( z_{A}\right) $ and $G=G\left( z_{B}\right)
,$%
\[
\left\{ F,G\right\} _{\ast}=\frac{\partial F}{\partial z_{a}}M_{ab}%
^{-1}\frac{\partial G}{\partial z_{b}},
\]
and for $F=z_{A}$ and $G=z_{B}$%
\[
\left\{ z_{A},z_{B}\right\} _{\ast}=\delta_{A}^{a}M_{ab}^{-1}\delta_{B}%
^{b}\Rightarrow\left\{
\begin{array}
[c]{c}%
\left\{ z_{a},z_{b}\right\} _{\ast}=M_{ab}^{-1},\\
\left\{ z_{a},z_{\beta}\right\} _{\ast}=0,\\
\left\{ z_{\alpha},z_{\beta}\right\} _{\ast}=0.
\end{array}
\right. .
\]
These two last results show the existence of a reduced symplectic structure in
phase space of a singular $M^{AB}$.
Taking $E=z_{C}$ in (\ref{dE_sing}) it follows%
\begin{align*}
dz_{C} & =\left\{ F,\phi^{\beta^{\prime}}\right\} _{\ast}dt_{\beta
^{\prime}}=\left[ \left\{ z_{C},\phi^{\beta^{\prime}}\right\} -\left\{
z_{C},\phi^{b}\right\} M_{ba}^{-1}\left\{ \phi^{a},\phi^{\beta^{\prime}%
}\right\} \right] dt_{\beta^{\prime}}=\\
& =\left[ \delta_{C}^{\beta^{\prime}}-\left\{ z_{C},\phi^{b}\right\}
M_{ba}^{-1}\left\{ \phi^{a},\phi^{\beta^{\prime}}\right\} \right]
dt_{\beta^{\prime}}=\\
& =\left[ \delta_{C}^{\beta^{\prime}}-\delta_{C}^{b}M_{ba}^{-1}\left\{
\phi^{a},\phi^{\beta^{\prime}}\right\} \right] dt_{\beta^{\prime}}%
\end{align*}
and for $C=\beta^{\prime}$:%
\[
dz_{\beta^{\prime}}=dt_{\beta^{\prime}},
\]
that shows $z_{\beta^{\prime}}$ remain arbitrary parameters in this
construction. For $C=b$,%
\[
dz_{b}=-M_{ba}^{-1}\left\{ \phi^{a},\phi^{\beta^{\prime}}\right\}
dt_{\beta^{\prime}},
\]
which is consistent with (\ref{d_t_bar}) since $z_{b}=t_{b}$. This expression
can still be written as
\[
dz_{b}=M_{ba}^{-1}\left[ \frac{\partial K^{a}}{\partial z_{\beta^{\prime}}%
}-\frac{\partial K^{\beta^{\prime}}}{\partial z_{a}}\right] dt_{\beta
^{\prime}},
\]
when we explicit the Poisson Bracket $\left\{ \phi^{A},\phi^{\beta^{\prime}%
}\right\} $.
\section{Examples}
In order to ilustrate how the method works, let us consider the two examples
studied in \cite{Guler -lin-veloc}, where the HJ method was also applied.
\textbf{1}- Starting with the Lagrangian
\begin{gather*}
L=\left( z_{2}+z_{3}\right) \dot{z}_{1}+z_{4}\dot{z}_{3}+W\left(
z_{2},z_{3},z_{4}\right) ,\\
W\left( z_{2},z_{3},z_{4}\right) =\frac{1}{2}\left[ \left( z_{4}\right)
^{2}-2z_{2}z_{3}-\left( z_{3}\right) ^{2}\right] ,
\end{gather*}
in a four dimensional (coordinate) space, it is immediate to identify $K^{A}$
and the constraints $\phi^{A}$:%
\[
\left\{
\begin{array}
[c]{l}%
K^{1}=z_{2}+z_{3},\\
K^{2}=0,\\
K^{3}=z_{4},\\
K^{4}=0,\\
V=-W,
\end{array}
\right. \Rightarrow\left\{
\begin{array}
[c]{l}%
\phi^{1}=p^{1}-K^{1}=p^{1}-\left( z_{2}+z_{3}\right) =0,\\
\phi^{2}=p^{2}-K^{2}=p^{2}=0,\\
\phi^{3}=p^{3}-K^{3}=p^{3}-z_{4}=0,\\
\phi^{4}=p^{4}-K^{4}=p^{4}=0,\\
\phi^{0}=p^{0}+V=p^{0}-\frac{1}{2}\left[ \left( z_{4}\right) ^{2}%
-2z_{2}z_{3}-\left( z_{3}\right) ^{2}\right] =0.
\end{array}
\right.
\]
From here, the matrix $M^{AB}$ is given by%
\[
M^{AB}=\frac{\partial K^{B}}{\partial z_{A}}-\frac{\partial K^{A}}{\partial
z_{B}}=\left\{ \phi^{A},\phi^{B}\right\} \Rightarrow\left( M^{AB}\right)
=\left(
\begin{array}
[c]{cccc}%
0 & -1 & -1 & 0\\
1 & 0 & 0 & 0\\
1 & 0 & 0 & -1\\
0 & 0 & 1 & 0
\end{array}
\right) ,
\]
which is regular and whose inverse is%
\[
\left( M_{AB}^{-1}\right) =\left(
\begin{array}
[c]{cccc}%
0 & 1 & 0 & 0\\
-1 & 0 & 0 & -1\\
0 & 0 & 0 & 1\\
0 & 1 & -1 & 0
\end{array}
\right) .
\]
Now we can find the equations of motion by constructing the generalized
brackets and using (\ref{dE}), or using (\ref{eq_mov_reg}) in a
straightforward way:%
\[
\left\{
\begin{array}
[c]{l}%
dz_{1}=z_{3}dt,\\
dz_{2}=z_{4}dt,\\
dz_{3}=-z_{4}dt,\\
dz_{4}=-z_{2}dt,
\end{array}
\right. \Rightarrow\left\{
\begin{array}
[c]{l}%
\dot{z}_{1}=z_{3},\\
\dot{z}_{2}=z_{4},\\
\dot{z}_{3}=-z_{4},\\
\dot{z}_{4}=-z_{2}.
\end{array}
\right.
\]
Manipulating the second and fourth equations we see that%
\[
\ddot{z}_{2}+z_{2}=0\Rightarrow z_{2}=-A\cos t+B\sin t,
\]
and by direct substitution into the second equation it follows
\[
z_{4}=A\sin t+B\cos t.
\]
By integration it is verified that%
\begin{align*}
z_{3} & =A\cos t-B\sin t+C,\\
z_{1} & =A\sin t+B\cos t+Ct+D.
\end{align*}
Now, substituting these results in the constraints we can obtain the momenta:%
\[
\left\{
\begin{array}
[c]{l}%
p^{1}=C,\\
p^{2}=0,\\
p^{3}=A\sin t+B\cos t,\\
p^{4}=0,
\end{array}
\right.
\]
and the problem is completely solved since we know all phase space
variables.\ Comparing these results with those obtained in \cite{Guler
-lin-veloc} we see some differences, the main one being the linear dependence
of $z_{1}$ with $t$. In fact we verify that the result of \cite{Guler
-lin-veloc} is a particular case of the one obtained here.
\textbf{2}- Let us now consider the Lagrangian
\begin{gather*}
L=\left( z_{2}+z_{3}\right) \dot{z}_{1}+k\dot{z}_{3}+W\left( z_{2}%
,z_{3}\right) ,\\
W\left( z_{2},z_{3},z_{4}\right) =\frac{1}{2}\left[ k^{2}-2z_{2}%
z_{3}-\left( z_{3}\right) ^{2}\right] ,
\end{gather*}
in a three dimensional (coordinate) space. We identify%
\[
\left\{
\begin{array}
[c]{l}%
K^{1}=z_{2}+z_{3},\\
K^{2}=0,\\
K^{3}=k,\\
V=-W,
\end{array}
\right. \Rightarrow\left\{
\begin{array}
[c]{l}%
\phi^{1}=p^{1}-\left( z_{2}+z_{3}\right) =0,\\
\phi^{2}=p^{2}=0,\\
\phi^{3}=p^{3}-k=0,\\
\phi^{0}=p^{0}-\frac{1}{2}\left[ k^{2}-2z_{2}z_{3}-\left( z_{3}\right)
^{2}\right] =0,
\end{array}
\right.
\]
and construct $M^{AB}$:%
\[
\left( M^{AB}\right) =\left(
\begin{array}
[c]{ccc}%
0 & -1 & -1\\
1 & 0 & 0\\
1 & 0 & 0
\end{array}
\right) .
\]
This is a singular matrix and has rank 2, and then we must find an inversible
submatrix $M^{ab}$; this can be done by choosing%
\[
\left( M^{ab}\right) =\left(
\begin{array}
[c]{cc}%
M^{11} & M^{13}\\
M^{31} & M^{33}%
\end{array}
\right) =\left(
\begin{array}
[c]{cc}%
0 & -1\\
1 & 0
\end{array}
\right) \Rightarrow\left( M_{ab}^{-1}\right) =\left(
\begin{array}
[c]{cc}%
0 & 1\\
-1 & 0
\end{array}
\right) .
\]
This choice implies that $t_{\beta}=z_{2}$ ($\phi^{\alpha}=\phi^{2}$) and
$t_{b}=\left\{ z_{1},z_{3}\right\} $ ($\phi^{a}=\left\{ \phi^{1},\phi
^{3}\right\} $), and the construction of the generalized brackets leads us to
the following equations of motion:%
\[
\left\{
\begin{array}
[c]{l}%
dz_{1}=\left( z_{2}+z_{3}\right) dt,\\
dz_{3}=-dz_{2}.
\end{array}
\right.
\]
By direct integration of the second equation it follows%
\[
z_{3}=-z_{2}+C;
\]
which shows us that%
\[
z_{1}=Ct+D.
\]
Now we must look for the condition that fixes the subspace where the system is
integrable:%
\begin{align*}
\left\{ \phi^{2},\phi^{0}\right\} & =\left\{ \phi^{2},\phi^{b}\right\}
M_{ba}^{-1}\left\{ \phi^{a},\phi^{0}\right\} \Rightarrow\\
& \Rightarrow-z_{3}=\left(
\begin{array}
[c]{cc}%
1 & 0
\end{array}
\right) \left(
\begin{array}
[c]{cc}%
0 & 1\\
-1 & 0
\end{array}
\right) \left(
\begin{array}
[c]{c}%
0\\
-\left( z_{2}+z_{3}\right)
\end{array}
\right) \Rightarrow z_{2}=0.
\end{align*}
We see that this system is integrable in the subspace where $z_{2}=0$, which
leads to conclude that $z_{3}=C$. Substituting these results in the
constraints, it follows%
\[
\left\{
\begin{array}
[c]{l}%
p^{1}=C,\\
p^{2}=0,\\
p^{3}=k.
\end{array}
\right.
\]
The problem is then completely solved.\ If we now compare these results with
those obtained in \cite{Guler -lin-veloc} we see that they are in agreement if
we correctly fix the values of $C_{1}$,\ $C_{2}$, and $C_{3}$\ of reference
\cite{Guler -lin-veloc}.
\textbf{3} - Now we will consider a third example of a system of fields known
as Proca's model. The Lagrangian density considered here has the Palatini's
form, where the fields $A_{\mu}\left( x\right) $ and $F^{\mu\nu}\left(
x\right) (\mu,\nu=0,1,2,3;\,i,j=1,2,3)$\ are considered as independent fields
($z_{A}=A_{\mu},F^{\mu\nu}$):%
\begin{align*}
\mathcal{L} & =\frac{1}{4}A_{\nu}\partial_{\mu}\left( F^{\mu\nu}-F^{\nu\mu
}\right) -\frac{1}{4}F^{\mu\nu}\left( \partial_{\mu}A_{\nu}-\partial_{\nu
}A_{\mu}\right) +\frac{1}{4}F^{\mu\nu}F_{\mu\nu}+\frac{1}{2}m^{2}A_{\mu
}A^{\mu}\\
& =-\frac{1}{4}\left( F^{0\nu}-F^{\nu0}\right) \partial_{0}A_{\nu}+\frac
{1}{4}A_{i}\partial_{0}F^{0i}-\frac{1}{4}A_{i}\partial_{0}F^{i0}-\mathcal{H},
\end{align*}
where%
\[
\mathcal{H}=-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}+\frac{1}{4}\left( F^{i\nu
}-F^{\nu i}\right) \partial_{i}A_{\nu}-\frac{1}{4}A_{\nu}\partial_{i}\left(
F^{i\nu}-F^{\nu i}\right) -\frac{1}{2}m^{2}A_{\mu}A^{\mu}.
\]
We then identify%
\begin{gather*}
\left\{
\begin{array}
[c]{l}%
K^{\nu}\left( x\right) =-\frac{1}{4}\left( F^{0\nu}\left( x\right)
-F^{\nu0}\left( x\right) \right) ,\\
K_{00}\left( x\right) =0,\\
K_{0i}\left( x\right) =\frac{1}{4}A_{i}\left( x\right) ,\\
K_{i0}\left( x\right) =-\frac{1}{4}A_{i}\left( x\right) ,\\
K_{ij}\left( x\right) =0\\
V\left( x\right) =\mathcal{H}\left( x\right) ,
\end{array}
\right. \Rightarrow\\
\Rightarrow\left\{
\begin{array}
[c]{l}%
\phi^{\nu}\left( x\right) =\pi^{\nu}\left( x\right) +\frac{1}{4}\left(
F^{0\nu}\left( x\right) -F^{\nu0}\left( x\right) \right) =0,\\
\phi_{00}\left( x\right) =\Pi_{00}\left( x\right) =0,\\
\phi_{0i}\left( x\right) =\Pi_{0i}\left( x\right) -\frac{1}{4}A_{i}\left(
x\right) =0,\\
\phi_{i0}\left( x\right) =\Pi_{i0}\left( x\right) +\frac{1}{4}A_{i}\left(
x\right) =0,\\
\phi_{ij}\left( x\right) =\Pi_{ij}\left( x\right) =0,\\
\phi^{t}\left( x\right) =p^{0}+\mathcal{H}\left( x\right) =0.
\end{array}
\right.
\end{gather*}
The $M^{AB}$ matrix, now defined as
\[
M^{A_{x}B_{y}}=\frac{\delta K^{B}\left( y\right) }{\delta z_{A}\left(
x\right) }-\frac{\delta K^{A}\left( x\right) }{\delta z_{B}\left(
y\right) },
\]
is%
\[
\left( M^{A_{x}B_{y}}\right) =\left(
\begin{array}
[c]{ccccc}%
0 & 0 & \frac{1}{2}\delta_{j}^{\mu}\delta\left( x-y\right) & -\frac{1}%
{2}\delta_{j}^{\mu}\delta\left( x-y\right) & 0\\
0 & 0 & 0 & 0 & 0\\
-\frac{1}{2}\delta_{i}^{\nu}\delta\left( x-y\right) & 0 & 0 & 0 & 0\\
\frac{1}{2}\delta_{i}^{\nu}\delta\left( x-y\right) & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & 0
\end{array}
\right) ,
\]
and it is not inversible. So we must find an inversible submatrix $M^{ab}$,
what can be done by choosing $t_{\beta}\left( x\right) =\left\{
A_{0}\left( x\right) ,F^{00}\left( x\right) ,F^{j0}\left( x\right)
,F^{ij}\left( x\right) \right\} $ and $t_{b}=\left\{ A_{i}\left(
x\right) ,F^{0j}\left( x\right) \right\} $, such that%
\begin{align*}
\left( M^{a_{x}b_{y}}\right) & =\left(
\begin{array}
[c]{cc}%
\mathbf{0}_{3\times3} & \mathbf{1}_{3\times3}\\
-\mathbf{1}_{3\times3} & \mathbf{0}_{3\times3}%
\end{array}
\right) \frac{1}{2}\delta\left( x-y\right) \Rightarrow\\
& \Rightarrow\left( M_{c_{z}a_{x}}^{-1}\right) =\left(
\begin{array}
[c]{cc}%
\mathbf{0}_{3\times3} & -\mathbf{1}_{3\times3}\\
\mathbf{1}_{3\times3} & \mathbf{0}_{3\times3}%
\end{array}
\right) 2\delta\left( z-x\right)
\end{align*}
In order to construct the generalized brackets we must calculate $\left\{
\phi^{a_{y}},\phi^{\beta_{x}^{\prime}}\right\} $:%
\begin{align*}
&
\begin{array}
[c]{c}%
\left\{ \phi^{i}\left( y\right) ,\phi^{0}\left( x\right) \right\}
=M^{i_{y}\left( 00\right) _{x}}=0,\\
\left\{ \phi^{i}\left( y\right) ,\phi_{00}\left( x\right) \right\}
=M^{i_{y}\left( 00\right) _{x}}=0,\\
\left\{ \phi^{i}\left( y\right) ,\phi_{j0}\left( x\right) \right\}
=M^{i_{y}\left( j0\right) _{x}}=-\frac{1}{2}\delta_{j}^{i}\delta\left(
x-y\right) ,\\
\left\{ \phi^{i}\left( y\right) ,\phi_{mn}\left( x\right) \right\}
=M^{i_{y}\left( mn\right) _{x}}=0,
\end{array}
\\
&
\begin{array}
[c]{c}%
\left\{ \phi_{0i}\left( y\right) ,\phi^{0}\left( x\right) \right\}
=M^{\left( 0i\right) _{y}0_{y}}=0,\\
\left\{ \phi_{0i}\left( y\right) ,\phi_{00}\left( x\right) \right\}
=M^{\left( 0i\right) _{y}\left( 00\right) _{x}}=0,\\
\left\{ \phi_{0i}\left( y\right) ,\phi_{j0}\left( x\right) \right\}
=M^{\left( 0i\right) _{y}\left( j0\right) _{x}}=0,\\
\left\{ \phi_{0i}\left( y\right) ,\phi_{mn}\left( x\right) \right\}
=M^{\left( 0i\right) _{y}\left( mn\right) _{x}}=0,
\end{array}
\end{align*}%
\begin{align*}
\left\{ \phi^{i}\left( y\right) ,\phi^{t}\left( x\right) \right\} &
=-\frac{\partial\mathcal{H}\left( x\right) }{\partial A_{i}\left( y\right)
}=-\frac{1}{4}\left[ F^{ji}\left( x\right) -F^{ij}\left( x\right)
\right] \partial_{j}^{x}\delta\left( x-y\right) +\\
& +\frac{1}{4}\left[ \partial_{j}F^{ji}\left( x\right) -\partial_{j}%
F^{ij}\left( x\right) \right] \delta\left( x-y\right) +m^{2}A^{i}\left(
x\right) \delta\left( x-y\right) ,\\
\left\{ \phi_{0i}\left( y\right) ,\phi^{t}\left( x\right) \right\} &
=-\frac{\partial\mathcal{H}\left( x\right) }{\partial F^{0i}\left(
y\right) }=\left[ \frac{1}{2}F_{0i}\left( x\right) +\frac{1}{4}%
\partial_{i}A_{0}\left( x\right) \right] -\frac{1}{4}A_{0}\left( x\right)
\partial_{i}^{x}\delta\left( x-y\right) .
\end{align*}
From this results we see that for any function $G=G\left( z,p\right) $\
\begin{align}
dG & =\left\{ G,\phi^{\beta_{x}^{\prime}}\right\} dt_{\beta_{x}^{\prime}%
}+\nonumber\\
& -\left\{ G,\phi^{c_{z}}\right\} \left( M_{c_{z}a_{y}}^{-1}\right)
\left\{ \phi^{a_{y}},\phi^{t}\left( x\right) \right\} dt\left( x\right)
+\nonumber\\
& -\left\{ G,\phi^{c_{z}}\right\} \left( M_{c_{z}a_{y}}^{-1}\right)
\left\{ \phi^{a_{y}},\phi_{j0}\left( x\right) \right\} dF^{j0}\left(
x\right) .\label{dG_Proca}%
\end{align}
Before obtaining the field equations we will look for the conditions that fix
the subspace where the system is integrable,
\[
\left\{ \phi^{\alpha_{w}},\phi^{t}\left( x\right) \right\} =\left\{
\phi^{\alpha_{w}},\phi^{b_{z}}\right\} M_{b_{z}a_{y}}^{-1}\left\{
\phi^{a_{y}},\phi^{t}\left( x\right) \right\} .
\]
For $\phi^{\alpha_{w}}=\phi^{0}\left( w\right) $,
\begin{align*}
\left\{ \phi^{0}\left( w\right) ,\phi^{t}\left( x\right) \right\} &
=-\frac{1}{4}\left[ F^{j0}\left( x\right) -F^{0j}\left( x\right) \right]
\partial_{j}^{x}\delta\left( x-w\right) +\\
& +\frac{1}{4}\left[ \partial_{j}F^{j0}\left( x\right) -\partial_{j}%
F^{0j}\left( x\right) \right] \delta\left( x-w\right) +\\
& +m^{2}A^{0}\left( x\right) \delta\left( x-w\right) ,
\end{align*}
and%
\[
\left\{ \phi^{0}\left( w\right) ,\phi^{b_{z}}\right\} M_{b_{z}a_{y}}%
^{-1}\left\{ \phi^{a_{y}},\phi^{t}\left( x\right) \right\} =0,
\]
which lead us to conclude that%
\begin{equation}
\frac{1}{2}\left[ \partial_{j}F^{j0}\left( x\right) -\partial_{j}%
F^{0j}\left( x\right) \right] +m^{2}A^{0}\left( x\right) =0.\label{cond1}%
\end{equation}
Considering now $\phi^{\alpha_{w}}=\phi_{00}\left( w\right) $,%
\[
\left\{ \phi_{00}\left( w\right) ,\phi^{t}\left( x\right) \right\}
=\frac{1}{2}F_{00}\left( x\right) \delta\left( w-x\right) ,
\]%
\[
\left\{ \phi_{00}\left( w\right) ,\phi^{\bar{B}_{z}}\right\} M_{\bar
{B}_{z}\bar{A}_{y}}^{-1}\left\{ \phi^{\bar{A}_{y}},\phi^{t}\left( x\right)
\right\} =0,
\]%
\begin{equation}
\therefore F_{00}\left( x\right) =0.\label{cond2}%
\end{equation}
For $\phi^{\alpha_{w}}=\phi_{ij}\left( w\right) $,%
\[
\left\{ \phi_{ij}\left( w\right) ,\phi^{t}\left( x\right) \right\}
=\frac{1}{2}\left[ F_{ij}\left( x\right) -\left( \partial_{i}A_{j}\left(
x\right) -\partial_{j}A_{i}\left( x\right) \right) \right] \delta\left(
w-x\right) ,
\]%
\[
\left\{ \phi_{ij}\left( w\right) ,\phi^{\bar{B}_{z}}\right\} M_{\bar
{B}_{z}\bar{A}_{y}}^{-1}\left\{ \phi^{\bar{A}_{y}},\phi^{t}\left( x\right)
\right\} =0,
\]%
\begin{equation}
\therefore F_{ij}\left( x\right) =\partial_{i}A_{j}\left( x\right)
-\partial_{j}A_{i}\left( x\right) .\label{cond3}%
\end{equation}
At last, considering $\phi^{\alpha_{w}}=\phi_{j0}\left( w\right) $,%
\[
\left\{ \phi_{j0}\left( w\right) ,\phi^{t}\left( x\right) \right\}
=\frac{1}{2}\left[ F_{j0}\left( x\right) -\partial_{j}A_{0}\left(
x\right) \right] \delta\left( w-x\right) ,
\]%
\[
\left\{ \phi_{j0}\left( w\right) ,\phi^{\bar{B}_{z}}\right\} M_{\bar
{B}_{z}\bar{A}_{y}}^{-1}\left\{ \phi^{\bar{A}_{y}},\phi^{t}\left( x\right)
\right\} =-\frac{1}{2}\left[ F_{0j}\left( x\right) -\partial_{j}%
A_{0}\left( x\right) \right] \delta\left( w-x\right) ,
\]%
\begin{equation}
\therefore F_{j0}\left( x\right) =-F_{0j}\left( x\right) .\label{cond4}%
\end{equation}
Now we are able to obtain the field equations. Taking $G=A_{j}\left(
w\right) $ in (\ref{dG_Proca}) we have%
\begin{equation}
dA_{j}\left( w\right) =\left[ F_{0j}\left( w\right) +\partial_{j}%
A_{0}\left( w\right) \right] dt. \label{eq_mov_Proca1}%
\end{equation}
If we now consider $G=F^{0j}\left( w\right) $,%
\begin{equation}
d\left[ F^{0j}\left( w\right) -F^{j0}\left( w\right) \right] =\left[
\partial_{m}F^{jm}\left( w\right) -\partial_{m}F^{mj}\left( w\right)
-2m^{2}A^{k}\left( w\right) \right] dt. \label{eq_mov_Proca2}%
\end{equation}
The results above (eqs. (\ref{cond1}-\ref{eq_mov_Proca2}))can be summarized as%
\begin{gather*}
F_{\mu\nu}\left( x\right) =-F_{\nu\mu}\left( x\right) =\partial_{\mu
}A_{\nu}\left( x\right) -\partial_{\nu}A_{\mu}\left( x\right) ,\\
\partial_{\mu}F^{\mu\nu}\left( x\right) +m^{2}A^{\nu}\left( x\right) =0.
\end{gather*}
Moreover, when we take the divergence of this last result, as consequence of
the antisymmetry property of $F_{\mu\nu}$, it follows:%
\[
\partial_{\nu}A^{\nu}\left( x\right) =0.
\]
When we compare these results with those obtained in reference \cite{coreanos}%
, where an analysis of the Proca model is conducted with the usual
Hamilton-Jacobi formalism, the agreement is manifest.
\section{Final Remarks}
In this work we have studied how systems described by Lagrangians with linear
velocities can be treated in Hamilton-Jacobi formalism. Initially we observed
that all \emph{momenta} were constrained and therefore all coordinates had the
status of parameters. Then, after applying integrability conditions, we saw
that if the conditions $\left\{ \phi^{A},\phi^{0}\right\} =0,$ $M^{AB}=0$
are satisfied, then the system of total differential equation is integrable,
and all the parameters are independent.
But, if these conditions are not satisfied then the system of differential
equation can be integrable only in a subspace of the phase space. Two distinct
cases were analysed. Firstly we considered the $M^{AB}$regular case, where the
system is integrable in the subspace where all the parameters are
time-dependent (see (\ref{dt_B})). In this subspace we verified that new
generalized brackets could be introduced, which allowed us to recognize the
symplectic structure of phase space in HJ approach.
Secondly we considered the $M^{AB}$ singular case, and we saw that
integrability is achieved in a subspace where $t_{0}$ and $t_{\beta}$ are
independent parameters. We also verified that some extra conditions must be
satisfied in order that this subspace could be determined. In this subspace
generalized brackets were also introduced and the symplectic structure could
also be recognized. Here one interesting feature can be pointed out. In this
case we can separate the parameters $t_{B}$ in two distinct sets: one is
composed by the parameters $t_{\beta}$, and the other by $t_{b}$, which are
the real dynamical variables of the problem (in example 1, all $z_{A},$
$A=1,...,4,$ are dynamical variables, while in example 2 and 3, only $z_{1}$,
$z_{3}$ and $A_{i}\left( x\right) $, $F^{0j}\left( x\right) $ are
dynamical, respectively). In the same way, the associated constraints can also
be separated in two sets composed by $\phi^{b}$ and $\phi^{\beta}$,
respectively. The interesting feature of this separation is to identify the
constraints that allow one to construct the inversible submatrix $M^{ab}$,
used to define the generalized brackets.
Moreover we believe the introduction of generalized brackets can be done not
only in systems described by first order actions, but also in any system which
has a non-null matrix $M^{AB}=\left\{ \phi^{A},\phi^{B}\right\} $ with an
inversible submatrix $M^{ab}$. This case is still under consideration by the authors.
At last we must notice that, although we considered only usual variables in
this work, the extension to treat berezinian variables is quite immediate.
\bigskip
\textbf{Acknowledgements}
\bigskip
The authors would like to thanks the referee for his comments, which allowed
them to improve their work. M. C. Bertin thanks CNPq for full support; B.
M.\ Pimentel thanks CNPq and FAPESP,\ (grant number 02/00222-9) for partial
support; P. J. Pompeia thanks the staff of CTA for incentive and support.
| -67,035.747305 |
[
-3.19140625,
2.861328125
] | 14.012739 |
[
-3.552734375,
-0.216796875,
-2.146484375,
-6.203125,
-0.2042236328125,
8.6015625
] |
[
2.900390625,
8.96875,
2.12890625,
6.203125
] | 194 | 4,432 |
[
-3.478515625,
4.1015625
] | 39.320678 |
[
-5.71875,
-3.9375,
-4.23828125,
-1.90625,
2.048828125,
11.5
] | 1.417943 | 9.575279 | 26.737365 | 4.910198 |
[
2.6752400398254395
] | -42,988.18149 | 6.126354 | -66,818.717902 | 0.165734 | 6.004603 |
[
-2.541015625,
-3.552734375,
-3.71484375,
-4.75390625,
2.328125,
11.9765625
] |
[
-5.04296875,
-1.8359375,
-2.181640625,
-0.87744140625,
3.19921875,
3.521484375
] | |
BkiUbovxK4sA-9F9jBrS
|
\section{Introduction:}
Globular clusters (GCs) have now in general been established, both photometrically and spectroscopically,
to have multiple populations (MPs) with differing compositions and possibly different ages.
Photometric detections of MPs in GCs
include Omega Cen (Bedin et~al.\ 2004), NGC 2808 (Piotto et~al.\ 2007), NGC 1851 (Milone et~al.\ 2008,
hereafter M08; Lee et~al.\ 2009a, hereafter L09; Han et~al.\ 2009, hereafter H09; Cummings et~al.\
2014, hereafter Paper I), M22 (Lee et al. 2009b), M4 (Marino et~al.\ 2008), and M2 (Lardo et~al.\ 2013;
Milone et~al.\ 2015). In the recent Piotto et~al.\ (2015) investigation, photometric detections of MPs are
found in all 56 globular clusters observed with their special combination of filters (see below).
In all of these clusters two or more red giant branches (RBGs), sub giant branches (SGBs),
or even main sequences (MSs) have been observed.
NGC 1851 is a noteworthy cluster because it photometrically has two RGBs
(L09; H09) and SGBs (M08; H09) plus evidence for two MSs (Paper I). Additionally, it has both a blue
horizontal branch (BHB) and a red horizontal branch (RHB), where the Paper I analysis also suggested
the RHB has two sequences. A further advantage to photometrically studying NGC 1851 is its very low
reddening of E(B-V)=0.02 (Harris 1996), so there is no concern about the photometric effects of
a large variable reddening.
Spectroscopically, the two RGBs observed in NGC 1851 typically exhibit different abundances in a variety of
elements, most strikingly in sodium (Na)
and barium (Ba) (Villanova et~al.\ 2010, hereafter V10; Carretta et~al.\ 2011a, hereafter Ca11)
and in nitrogen (N) (Carretta et~al.\ 2014, hereafter Ca14). Spectroscopic observations of
molecular bands in NGC 1851 show its RGB has moderate variations in CH strength but large variations in
CN strengths (e.g., Lim et~al.\ 2015), with no CN-CH anti-correlation. There is also evidence for a
quadrimodal CN distribution (Campbell et~al.\ 2012; Simpson et~al.\ 2017). The two SGBs appear to correspond
to high and low [Ba/Fe] (Gratton et~al.\ 2012a, hereafter G12). Therefore, abundance
differences likely play a key role in creating the separate sequences.
Models to explain MPs of
differing abundances include 1) An initial population formed and soon after ($<$1 Gyr) a second
population formed from the gas contaminated by the ejecta of the high-mass stars of the first
generation (see M08; Joo \& Lee 2013; Ventura et~al.\ 2009). 2) There was a merger of two GCs of
slightly different age and composition (see Ca11). The merger explanation is based on
the Ca11 observation that between the two populations there is an apparent real spread in the
heavy elements (e.g., iron (Fe)), not just in the light s-process elements. This spread cannot simply be
explained by two distinct episodes of star formation within one cluster. 3) Enriched material
was ejected from interacting massive binaries and rapidly rotating stars, and this material
accreted onto the circumstellar discs of young pre-main sequence stars formed at the same time
as these massive stars (Bastian et~al.\ 2013).
Unlike the other models, this model only has a single star-formation burst that creates MPs
based on variations in the amount of enriched material accreted, if any at all, on each individual
cluster member. This avoids the mass budget problem that plagues multiple star-formation burst models.
However, a critical assessment of all current MP theories by Renzini et al. (2015) finds only the AGB
pollution scenario to be possibly viable.
The key to photometrically distinguishing these MPs has been ultraviolet (UV) filters, where Sbordone
et~al.\ (2011) and Carretta et~al.\ (2011b) have shown that realistic abundance differences in Carbon
(C), Nitrogen (N), and Oxygen (O) greatly affect the UV filter bandpasses because of the strong CN, NH,
and CH molecular bands present. The ground-based detections of multiple sequences
have used the Johnson U, Stromgren u, or sloan u filters, but these filters are narrow and inefficient and
require a significant amount of large telescope time to observe most GCs well. The Hubble Space Telescope (HST)
has also been a powerful tool to photometrically analyze MPs with the F336W filter, which is comparable to
Johnson U, and the F275W filter, which goes farther into the UV beyond what can be observed from the ground.
Building on these filters, Milone et~al.\ (2013) and Piotto et~al.\ (2015) have defined the pseudo-colors
C$_{\rm F275W,F336W,F410M}$ = (m$_{\rm F275W}$-m$_{\rm F336W}$)-(m$_{\rm F336W}$-m$_{\rm F410M}$)
and the similar C$_{\rm F275W,F336W,F438W}$, respectively, terming these three the ''magic trio". These pseudo-colors take advantage of the strong OH
features in F275W's bandpass in combination with F336W's strong sensitivity to N, and this provides a key
method to photometrically distinguish most MPs. Their team is carrying out a legacy survey of Galactic globulars
in the magic trio and uncovering MPs in all of them, with each GC exhibiting unique behavior (e.g., Piotto et al.
2015). F275W, however, requires the use of HST, and this far into the UV it is even more difficult to acquire the
appropriate signal on the typically cool GC stars, greatly increasing the need for very limited HST time.
Furthermore, given the limited lifetime of HST and because no other current or planned space facility will
be sensitive below the atmospheric cutoff, it is important to investigate other filters that can
uncover MPs from the ground.
In Paper I we were thus motivated to consider other available tools without these limitations. We demonstrated
that the broader and more efficient Washington C filter was quite adept at photometrically detecting
the NGC 1851 MPs with a ground-based 1-meter telescope and only moderate observation times.
Color distribution analysis of the Washington photometry illustrated that these two populations
are a dominant ($\sim$70\%) bluer population (in C-T1 and C-T2) that is narrow in color and a secondary
($\sim$30\%) redder but partially overlapping population that is broadly distributed in color.
While the narrower band UV filters like Johnson U (H09) and Stromgren u (L09) appear to more
cleanly separate the two populations in NGC 1851 into two distinct photometric branches, these two narrow
and farther UV filters required five and three times as much telescope time, respectively, to perform
these observations (see Paper I for more details).
For this second paper in our analysis of NGC 1851, we look at the photometric variations in the Washington
C filter and how they are connected to elemental abundance variations. This is based on the detailed NGC 1851
abundance analyses in Ca11, V10, Ca14, G12, Gratton et~al.\ (2012b, hereafter G12b), Yong et~al.\ (2015,
hereafter Y15), and Lim et~al.\ (2015, hereafter L15). We also performed photometric synthesis and
focused on the effects of variations in the three CNO abundances at constant C+N+O. Additionally, we briefly
looked at the photometric effects of variations in total C+N+O, metallicity, and in He. We compare
these effects in the C filter to the effects that identical variations have on the commonly used HST F336W
filter, which is also very similar to Johnson U.
In Section 2 we discuss the previous abundance results, including their trends and distributions. In
Section 3 we match previous abundances to out RGB photometry and analyze the abundance differences
between the two branches observed in Washington C. In Section 4 we match previous
abundances to the SGB and turnoff stars. In Section 5 we match previous abundances to HB stars.
In Section 6 we synthesize for a representative RGB star the photometric effects that abundance
variations have on C and F336W magnitudes. In Section 7 we synthesize for a representative MS star
the photometric effects that abundance variations have on C and F336W magnitudes.
Lastly, in Section 8 we summarize our results and conclusions.
\section{Spectroscopic Abundances}
There have been multiple spectroscopic studies of NGC 1851 that have focused on its RGB stars (e.g., Ca11; V10;
Campbell et~al.\ 2012; Ca14; Y15; L15), its turnoff and SGB stars (Pancino et~al.\ 2010; Lardo et~al.\
2012, hereafter L12; G12), and its HB stars (G12b). All of these studies have shown that there is a broad
spread of abundances (e.g., Na, O, Ba, Sr, Ni, and CN strengths) a possibly significant spread
in Fe (Ca11), several abundance correlations and anti-correlations (e.g., O-Na and Ba-Sr), but
unlike most globular clusters it has no CN-CH anti-correlation (Pancino et~al.\ 2010; L12; L15; Simpson et~al.\ 2017).
In our analysis we first looked at the abundance trends and distributions for each element and used these to
define the high and low abundance ranges for each element.
Matching these abundances to our Washington photometry allows us to analyze the abundance
variations in these stars in comparison to their placement on the photometrically observed MPs.
\subsection{Abundance Trends}
In each element we have searched for abundance trends with T$_{\rm eff}$ to account for either true variations
in surface abundances as stars undergo evolution or those caused by potential systematics in the analysis.
Therefore, the distributions across a broad range of giants can more appropriately be analyzed. Beginning
with the analysis of 124 RGB stars from Ca11,
we looked at the abundances of Fe, O, Na, Ca, Cr, and Ba. Across the $>$1000 K range in T$_{\rm eff}$
there is no significant correlation with T$_{\rm eff}$ for Fe, Ca, and Ba. For Na we find evidence for a
minor anti-correlation of significance at greater than 99\% confidence. Figure 1's Na panel shows this trend, which
is minor relative to the overall observed scatter. It is likely the result of systematics rather than a true
abundance trend, but we have used this trend to divide the high and low Na abundances.
\begin{figure}[htp]
\begin{center}
\includegraphics[clip, scale=0.45]{n1851ca11cor2top.eps}
\includegraphics[clip, scale=0.45]{n1851ca11cor2bottom.eps}
\vspace{-0.75cm}
\end{center}
\caption{The upper two panels plot [Na/Fe] and [O/Fe] versus T$_{\rm eff}$ for the RGB stars observed in Ca11.
Solid data points represent detections and open-inverted triangles represent O upper limits. For Na the weak
but statistically significant correlation shown is Na$_{\rm cor}$ = 1.66 -- T$_{\rm eff}$ $\times$ 3.17$\times$
10$^{-4}$. For O, in the cooler stars there is an apparent bimodal abundance distribution, but for the hotter
stars the abundance trend at first appears to significantly increase. However, this is because at these hotter
T$_{\rm eff}$ the oxygen spectroscopic feature becomes very weak and unmeasurable in the O-poor stars. This
is illustrated by the large number of upper limits in the hotter stars. Hence, there is no intrinsic
trend with T$_{\rm eff}$ and O possibly has a bimodal distribution. The lower three panels plot [Cr/Fe],
[Ba/Fe], and [C/Fe] versus T$_{\rm eff}$ for the SGB stars observed in G12. For Cr and Ba the minor but
statistically significant correlations shown are Cr$_{\rm cor}$ = --3.25 + T$_{\rm eff}$ $\times$ 5.27 $\times$ 10$^{-4}$
and Ba$_{\rm cor}$ = --1.17 + T$_{\rm eff}$ $\times$ 3.51 $\times$ 10$^{-4}$. Lastly, the [C/Fe] correlation is
the most significant with C$_{\rm cor}$ = --4.29 + T$_{\rm eff}$ $\times$ 7.28 $\times$ 10$^{-4}$. This may
represent an intrinsic T$_{\rm eff}$ correlation caused by the increasing depth of
the surface convection zone along the SGB. For these four abundance sets, these correlations define our rich and
poor populations for each element.}
\end{figure}
The O abundances have an apparent trend where at cooler T$_{\rm eff}$ O appears to have a bimodal distribution
while at higher T$_{\rm eff}$ there appears to be no O-poor stars. Figure 1's O panel illustrates
this, including upper limits for stars without detectable O shown as open-inverted triangles. All of these
upper limits are in the hotter RGB stars, and this is because the O spectral feature becomes increasingly
weak in the fainter (hotter) RGB stars and was unmeasurable in the faint stars that are also
O poor. Therefore, this strongly suggests that the O abundances have no significant trend with
T$_{\rm eff}$ and may have a bimodal distribution across the full T$_{\rm eff}$ range, but in the fainter
(hotter) giants this bimodality is washed out by errors and the inability to measure the weakest [O/Fe].
\begin{figure*}[htp]
\begin{center}
\includegraphics[scale=0.8]{abdisttotcor.eps}
\vspace{-0.43cm}
\end{center}
\caption{The abundance distributions for various elements are shown for the RGB (Ca11; black), the SGB (G12;
red), and the HB/RGB (G12b; blue). When available the typical abundance errors are shown above the distributions.
For display purposes, and to directly compare the characteristics of the abundance distributions, we have
applied systematic offsets to place the distributions in line with those of the RGB when possible. For [O/Fe]
we applied -0.29 (G12b); no [Na/Fe] offsets; for [Cr/Fe] we applied +0.16 (G12); for [Ba/Fe] we applied
-0.45 (G12) and +0.3 (G12b); for [Ca/Fe] we applied -0.09 (G12) and +0.09 (G12b); no [Fe/H] offsets.}
\end{figure*}
Carretta et~al.\ (2014, hereafter Ca14) built further on the RGB analysis in Ca11 and looked at the CN
strengths of 62 of the 124 RGB stars from Ca11. In their analysis they state these CN strengths in terms of
[N/Fe] with an assumed constant [C/Fe]=0. While the RGB abundances from NGC 1851 in V10
and Y15 suggest that [C/Fe] does not vary as significantly as [N/Fe] and [O/Fe] at a given magnitude, both find
that [C/Fe] still does have important variations. Additionally, surface abundance evolution due to
deep mixing along the RGB will cause decreasing [C/Fe] in more evolved stars. In the upper RGB, for example,
[C/Fe] is more appropriately defined as $\sim$-0.8 dex rather than scaled solar. Therefore, these [N/Fe] values
from Ca14 will be used as a CN strength index, but we acknowledge that these [N/Fe] values
likely trace true [N/Fe] variations. Testing these CN strengths versus T$_{\rm eff}$ finds that there are no trends.
Building on these RGB abundances, we looked for similar correlations with T$_{\rm eff}$ in Fe, C, Ca, Cr,
Sr, and Ba across the 77 SGB stars observed in G12 that span $\sim$800 K. As for the cases in the RGB, we
again did not find any trends for [Fe/H] and [Ca/Fe]. Sr was not analyzed in the RGB, but in the SGB we did
not find any meaningful trend in [Sr/Fe]. In contrast to the RGB, there is evidence for weak but statistically
significant trends in [Cr/Fe] and [Ba/Fe]. Both of these trends are shown in Figure 1. Again, these are
trends that are likely the result of minor systematics, and the inconsistency of the trends in the SGB and RGB
supports this. In any case, we have used these trends to help define our rich and poor abundances for these
elements. Lastly, the lower panel of Figure 1 shows [C/Fe] vs T$_{\rm eff}$ in the SGB from G12, which is the
most scientifically interesting and strongest observed abundance-T$_{\rm eff}$ trend. [C/Fe] steeply decreases
when moving from the hotter (less evolved) stars up the SGB. As G12 discuss, this trend is likely the result
of mixing as the surface convection zones increase in depth along the SGB.
\subsection{Abundance Distributions}
Figure 2 shows the abundance distributions of nine elements or molecules of interest, where when available we
have plotted the distributions derived from Ca11 (black; RGB stars), G12 (red; SGB stars), and G12b (blue;
HB stars). The median abundance errors (when published) are shown above the distributions. For
elements that we found
correlations of significance in the previous section, we have adjusted these distributions to reflect that.
Additionally, for elements that were analyzed in both Ca11 and G12, we have estimated and corrected any
observed systematic differences in the distribution relative to the RGB abundances. These applied
corrections are given in the caption of Figure 2. For the HB analysis, these systematics can be based
directly on the overlapping samples of Ca11 and
G12b, where G12b also looked at a number of RGB stars that included seven from Ca11. The calculated
systematics (Ca11 minus G12b) between these RGB stars found no meaningful difference in [Fe/H],
a 0.09 dex difference in [Ca/Fe], and a 0.3 dex difference in [Ba/Fe], which have been applied in Figure 2.
For [Na/Fe] and [O/Fe] the measured systematics are even larger at 0.31 and -0.56, respectively, but in
Figure 2 these resulted in clear offsets between the observed distributions. For display purposes we
instead matched up the distributions with no offset in [Na/Fe] and a -0.29 offset in [O/Fe]. While the
[Na/Fe] and [O/Fe] differences between the RGB and HB may be real, for the purposes of this paper the cause
of this systematic is not important.
\begin{figure*}[htp]
\begin{center}
\includegraphics[scale=0.75]{n1851CRab1finalcencut2new.eps}
\vspace{-0.43cm}
\end{center}
\caption{Matches of the Ca11 and V10 abundances to our Washington photometry for Ba, Ca, Si, and Fe. In each
panel the analyzed stars rich in the element labeled are shown in red while the analyzed stars poor in
that element are shown in blue. The abundance ranges are defined in each panel.
The photometric matches find that stars poor in Ba (and less so in Ca or Si) are consistently
found to fall along the blue branch of the RGB. In contrast, the stars that are rich in either Ba, Ca, or Si,
are broadly distributed
in color and fall along both the blue branch as well as create the sparse red branch of the RGB. The
photometric difference between the Fe-rich and Fe-poor populations are not significant.}
\end{figure*}
Looking at the distributions themselves, while errors and number statistics limit the significance of some
of these abundance variations, the distributions of many of the elements suggest a significant spread beyond
that caused by errors. Furthermore, some abundances even suggest a bimodal distribution (O, Na, Sr). The
significance of these bimodalities are further strengthened by being observed in more than one study. For
example, the smaller sample of 15 RGB stars from V10, not shown here, also shows a clear bimodality in Sr.
The distribution of [O/Fe] is of particular interest with (as seen in Figure 1) a small population of very
O-poor RGB stars from Ca11, but here we also see that this population exists in the HB and RGB analysis of
G12b, strengthening the significance of this bimodality.
\section{Matching Red Giant Branch Abundances to Photometry}
\subsection{Characterizing the Red Giant Branch Abundances}
Figure 3 matches the Ca11 abundances of Ba, Ca, Si, and Fe to our Paper I photometry of the two RGB
populations in NGC 1851. We have also supplemented these abundances with three RGB star abundances from V10,
which have been systematically adjusted to be consistent with the Ca11 abundances. This finds that the two
photometric RGB branches have different abundance characteristics in Ba and Ca, but any differences
are less clear in Si and Fe. The Ba-poor stars and the Ca-poor primarily
fall tightly along the blue RGB, while the Ba-rich and Ca-rich stars are more widely
distributed and cover both the blue and red RGB. This photometric abundance distribution, most clearly
observed in Ba, is remarkably similar to the two photometric populations observed in Paper I. Based
solely on photometric color distribution analysis, we found that the C filter does not distinctly separate
the MPs but creates a narrow blue population and an overlapping broader and redder second population.
A more robust statistical analysis of these population distributions can be performed with a
Kolmogorov-Smirnov test (KS-test). In Figure 4 we look at the photometric C-T1 colors these
stars relative to a mean RGB trend, and we present the cumulative distributions of the rich and poor populations
for each element. The KS-test looks at the measurement of the greatest separation (D) between each distribution,
and based on the population sizes this provides the significance (p-value) of whether or not these different
abundances correlate with distinct colors. As is common we adopt p-values of $<$0.05 as
representative of a statistically significant difference in color distributions between the two populations
(i.e., that we can reject the null hypothesis that these abundance groups are derived from the same color
distribution).
In Table 1 we give the KS-test statistics for each element and molecule analyzed (see Figures
3 to 6 for display of most of these elements and molecules). This finds that in Figures 3 and 4 the
distinct color distributions based on abundance in Ba and Ca are statistically significant. While Si
shows a weaker distinction, the differences are still significant. In contrast
to this, the comparison of the Fe-rich and Fe-poor populations from Ca11 in the RGB gives a D of 0.2379 and
p-value 0.078. Therefore, while this does not rule out the significance of this possible [Fe/H] spread
observed by Ca11, Fe does not correlate well with the different photometric branches. Ca11 also found this
when they matched the different [Fe/H] to the Stromgren photometry by Calamida et~al.\ (2007), where both the
metal-rich and metal-poor stars showed similar double RGBs. This is the foundation for their argument
that NGC 1851 was formed from the merger of two different clusters.
\vspace{-0.4cm}
\begin{center}
\tablefontsize{\footnotesize}
\begin{deluxetable}{c c c c c}
\multicolumn{5}{c}%
{{\bfseries \tablename\ \thetable{} - Abundance Distribution KS-Test Statistics}} \\
\hline
Element & Poor Count & Rich Count & D & p-value\\
\hline
\hline
Ba & 51 & 47 & 0.4293 & 0.000\\
Ca & 53 & 56 & 0.3133 & 0.007\\
Si & 54 & 54 & 0.2593 & 0.043\\
Fe & 56 & 53 & 0.2379 & 0.078\\
Cr & 53 & 45 & 0.1379 & 0.711\\
Mg & 48 & 57 & 0.2007 & 0.218\\
O & 36 & 36 & 0.3333 & 0.028\\
Na & 49 & 57 & 0.3863 & 0.000\\
CN & 47 & 42 & 0.5228 & 0.000\\
CH & 27 & 27 & 0.2222 & 0.466\\
\hline
\caption{For each element or molecule we matched to the RGB, we give the corresponding
number of rich and poor stars, (D) the greatest separation between the cumulative distributions of
the two populations, and the corresponding p-value where we adopt p$<$0.05 as significant.}
\end{deluxetable}
\end{center}
\begin{figure}[htp]
\begin{center}
\includegraphics[scale=0.44]{ksdist.comb1.eps}
\vspace{-0.43cm}
\end{center}
\caption{We analyze the statistical properties of the photometric abundance distributions in
Figure 3. Adopting the same abundance groups (and color schemes), we look at how the cumulative
distributions compare between the rich and poor populations for each element in C-T1 color space.
We derive relative C-T1 colors by fitting the mean RGB trend with the relation of
C-T1=507.974-T1$\times$129.2119+T1$^2\times$12.44539-T1$^3\times$0.5351378+T1$^4\times$0.00864987.}
\end{figure}
In the upper panels of Figure 5 we match the key elements of O and Na (from Ca11 supplemented
with V10) to the RGB.
Like with Ba, Na clearly shows that nearly all Na-poor stars are consistent with the narrow
blue branch while the Na-rich population is broadly spread covering the blue branch but also creating
the red branch. We similarly find a distinct distribution between the O-poor and O-rich stars, but
for this element the blue branch is primarily O-rich (not poor like with Ba and Na) while the red
branch is predominantly created by O-poor stars.
Looking in more detail at [O/Fe], as we noted in Section 2.1, its abundance may have a bimodal
distribution. Much of this is washed out, however, by the difficult measurement of O in the faint and
hotter RGB stars, which appears to have been unmeasurable in the O-poor stars in this regime. Qualitatively
consistent with this idea is that none of the faint red-branch RGB stars that were analyzed in Ca11
have O detections in Figure 3; only the faint blue-branch RGB stars have O detections. In the
upper panels of Figure 6 we look at the cumulative distributions of the O and Na abundance populations
from Figure 5.
Why do these different abundance populations have such different photometric characteristics?
As described, the ``red RGB branch" is composed of second population stars but many second population
stars also fall on the ``blue RGB branch".
On average the second population stars that fall on the red RGB branch are moderately
more Ba-rich, Na-rich, or O-poor than the second population stars that fall on the blue RGB branch.
However, several of the most Ba-rich, Na-rich, or O-poor stars still fall on the blue RGB branch. This
suggests it is not our adopted definitions of the abundance groups that create these overlapping distributions.
Ca11 similarly found photometrically overlapping populations in matches of their abundances to the Stromgren
photometry of L09, and this suggests that MPs can create similar photometric characteristics in the C
and Stromgren u filters.
\begin{figure*}[htp]
\begin{center}
\includegraphics[scale=0.75]{n1851CRab2finalcencut3.eps}
\vspace{-0.43cm}
\end{center}
\caption{Matches of the Ca11 and V10 abundances to our Washington photometry for Na, and O, and matches
of the CN and CH bands strengths from Ca14 and L15 to our Washington photometry. In each
panel the analyzed stars rich in the element labeled are shown in red while the analyzed stars poor in
that element are shown in blue. The O abundance ranges are defined in the panel. As seen
in Figure 1, we use the [Na/Fe] correlation with T$_{\rm eff}$ to define the Na-rich and Na-poor stars.
For CN the CN-poor stars are [N/Fe]$\leq$-0.05 for the Ca14 abundances or $\delta$CN$\leq$0 for the L15
abundances, and the CN-rich stars are [N/Fe]$>$-0.05 for the Ca14 abundances or $\delta$CN$>$0 for the
L15 abundances. For CH the CH-rich stars are $\delta$CH$>$0 and the CH-poor stars are $\delta$CH$<$0 from L15.
Like with Ba, the photometric matches find that stars poor in either Na or CN, or those that are rich
in O, are consistently found to fall along the blue branch of the RGB. In contrast, the stars that are
rich in either Na or CN, or those that are poor in O, are broadly distributed
in color and fall along both the blue branch as well as create the sparse red branch of the RGB. The photometric
difference between the CH-rich and CH-poor stars are not significant.}
\end{figure*}
Carretta et~al.\ (2011b, hereafter Ca11b), Ca11, and more recently Ca14 have analyzed why the population
that is typically poor in light s-process elements is photometrically very narrow and the population that is
typically rich in light s-process elements is very broad in color and centered redward in the Stromgren
filters. Ca11b originally suggested a model based purely on variations in CNO abundances where
the two stellar populations can be defined as C-normal and C-rich, which can recreate the general observed
photometric characteristics. However, in this paper we have focused instead on a photometrically similar
model of two populations that are distinctly N-normal and N-rich in abundance. This is based on four
recent abundance analyses: First, in Ca14 the redder RGB stars were found to be distinctly CN-rich
and the bluer RGB stars were primarily CN-weak. Supplementing this finding, the analysis by V10 suggested
that the most clear abundance distinction between the giants in NGC 1851 are a Ba-rich and a Ba-poor
population, but these Ba-rich and Ba-poor populations were also found to be N-rich/O-poor and
N-normal/O-rich, respectively. In V10 the two populations had a difference in [N/Fe] nearly as significant
as [Ba/Fe]. Additionally, while the total V10 sample has moderate variations in [C/Fe], the Ba-rich and
Ba-poor populations themselves show no significant difference in their mean [C/Fe]. The recent
analysis by Y15 also looked at CNO in the two RGBs of NGC 1851 for 11 stars, and they find the same
qualitative details as previous analyses. Lastly, L15 looked directly at CN and CH band strengths
in 62 RGB stars from NGC 1851. While they found a moderate spread in CH band indexes, they found a significant
spread in the observed CN band indexes. Based on this, N (and O) appears to be a key element in
distinguishing the populations, which is expected due to the strong molecular bands sensitive to N in
the UV.
\begin{figure}[htp]
\begin{center}
\includegraphics[scale=0.44]{ksdist.comb2.eps}
\vspace{-0.43cm}
\end{center}
\caption{We analyze the statistical properties of the photometric abundance distributions in
Figure 5. Adopting the same abundance groups (and color schemes) we look at how the cumulative
distributions compare between the rich and poor populations for each element or molecule in C-T1 color space,
adopting the same mean RGB trend as in Figure 4.}
\end{figure}
Another key factor to consider between the abundances of the two RGB branches is whether the observed
variations in C, N, and O still result in a constant total C+N+O abundance. The analysis in V10
found that both populations did have a consistent log $\epsilon$(CNO)$\sim$8.00\textsuperscript{3}.
Comparison of V10 to the results of Y15 finds encouraging consistency in the blue RGB,
but the Y15 analysis of the red RGB stars finds they are nearly a factor of 10 ($\sim$0.9 dex) richer
in N than the blue RGB. V10 only found them twice as rich in N ($\sim$0.35 dex) with respect to the blue
RGB. A comparison of the CN strengths in Ca14 similarly suggests the CN-rich stars are $\sim$0.4 dex
richer in N. This far more significant increase in N found
by Y15 also results in a distinct log $\epsilon$(CNO) abundance between the two branches,
with the red RGB branch being $\sim$0.5 dex richer. In our synthetic magnitude analysis (see Sections 6
and 7) we have adopted a constant log $\epsilon$(CNO) of 8.0, but consistent with Y15 we also briefly
consider the effects of a significant increase in [N/Fe] in the red branch that results in a distinct
log $\epsilon$(CNO) of 8.5 for the red.
To look more directly at the effects of CNO and the corresponding molecular bands, we first use
the CN strengths from Ca14 (represented by [N/Fe]) and L15 (represented by $\delta$CN). These CN band
strengths are useful because they (with the CH and NH bands) are the primary cause of the observed
photometric differences in these stars. The lower-left panel of Figure 5 shows the CN abundances matched
to our photometry with blue representing CN-poor ([N/Fe]$\leq$-0.05 for the Ca14 abundances or
$\delta$CN$\leq$0 for the L15 abundances) and red representing CN-rich ([N/Fe]$>$-0.05 for the Ca14 abundances or
$\delta$CN$>$0 for the L15 abundances).
Consistent with what we saw with Ba and Na abundances, we similarly see that the CN-poor stars are narrowly
distributed and are the primary component of the blue-RGB branch, while the CN-rich stars are
broadly distributed and create the red-RGB branch but also are heavily overlapping with the blue RGB.
In the lower-left panel of Figure 6 we show the cumulative
population distributions for these CN-rich versus CN-poor stars, and the KS test finds CN gives a
D of 0.5228 with a p-value of 0.000. This is the largest D given for a single abundance, but the two
populations are still heavily overlapping.
This suggests that while CN bands play a critical role, other factors, like the CH and NH bands,
also must play a role in NGC 1851 and the C filter.
The CH indices from L15 do not show the broad variations observed in CN indices, but they still have
meaningful variation. In the lower-right panel of Figure 5 we have matched these CH band strengths to
our RGB photometry with CH-rich ($\delta$CH$>$0) in red and CH-poor ($\delta$CH$<$0) in blue. This finds
that there is no clear matching of either CH population with the two RGB branches. In the lower-right panel
of Figure 6 the cumulative distributions of the CH-poor and CH-rich populations are not distinct. This is
consistent with the lack of anti-correlation between CH and CN observed in Pancino et~al.\ (2010), L12, and L15,
and with the two RGB populations observed in V10 showing no significant difference in [C/Fe].
Therefore, there is a moderate spread in [C/Fe] throughout NGC 1851, but the two populations are not meaningly
different in their [C/Fe] abundance distributions.
\footnotetext[3]{We adopt standard abundance notation where for a given element X,
log $\epsilon$ = log (N$_{\rm X}$/N$_{\rm H}$) and [X] = log $\epsilon$(X)$_{\rm star}$ -
log $\epsilon$(X)$_\odot$.}
\subsection{Characterizing the Two Photometric Red Giant Branches}
What could cause the large color range observed only in the second RGB population? A possible explanation is
that NGC 1851 has two populations with distinct [O/Fe], and when adopting no variation in C+N+O, at a
constant [O/Fe] the variations in [C/Fe] will be anti-correlated with [N/Fe].
Focusing on CN molecular bands, which dominate in the C filter bandpass, variations in [C/Fe] do
not greatly affect the CN strengths of the O-rich (blue RGB) stars because the strengths are limited
by their weaker N abundance. Conversely, for the O-poor (red RGB) stars these CN bands are more significantly
affected by [C/Fe] variations because they are N-rich and the N abundance is no longer a limiting factor.
The first population stars
are O-rich (and typically N-normal) and will cover a tight color range while the second population stars are
O-poor (and typically N-rich) and will be more broadly distributed in color. Therefore, the sparser and distinct
red RGB branch (which is only the reddest part of the second population) is composed of only the N-rich, C-rich,
and O-poor stars, which have the strongest CN bands but also the strongest CH and NH bands. In contrast, the
blue RGB is composed of all the O-rich (first population) stars in addition to the
O-poor stars that are also C-poor, which all have relatively weaker CN bands and either weak CH or NH
bands, if not both.
\begin{figure*}[htp]
\begin{center}
\includegraphics[scale=0.75]{n1851.CRCNCH.eps}
\vspace{-0.43cm}
\end{center}
\caption{In these four panels we look at various methods to distinctly differentiate the abundances of the
red and blue RGB branches in the Washington filters. In the upper-left panel we see that considering O and
Ba simultaneously has some success at differentiating the abundance characteristics of the two branches. This shows
that the blue RGB are stars that are either O-rich or Ba-poor (blue), and the red
RGB are stars that are both very O-poor and Ba-rich (red). In the upper-right
panel we consider the more complete abundances of Na with Ba. This more clearly differentiates
the blue RGB as stars that are either Na-poor or Ba-poor (blue), and the red RGB as
stars that are both very Na-rich and Ba-poor. In the lower-left panel we consider CN-poor or Na-poor stars (blue),
CN-rich and Na-rich (green), and CN-rich and very Na-rich (red). This most distinctly separates the two
photometric branches. Lastly, in the lower-right panel we consider CN together with CH, where CH by itself did not
show any meaningful photometric differences. Consistent with this, for CN-poor stars there is no significant difference
between CH-rich and CH-poor stars. However, this comparison finds that the CN-rich stars that are consistent with
the blue branch are primarily CH-poor and the reddest giants are both CN-rich and CH-rich.}
\end{figure*}
\begin{figure}[htp]
\begin{center}
\includegraphics[scale=0.44]{ksdist.comb3.eps}
\vspace{-0.43cm}
\end{center}
\caption{We analyze the statistical properties of the photometric abundance distributions in
Figure 7. Adopting the same abundance groups (and color schemes) we look at how the cumulative
distributions compare between the defined abundance populations in C-T1 color space,
adopting the same mean RGB trend as in Figure 4.}
\end{figure}
Another key element for differentiating the two populations is Ba. The Ba absorption itself
is not significant enough to affect the observed flux, but its differentiating power may be related to
a connection between Ba and N, where V10 found that the Ba-rich stars were also more N-rich.
This is also found by our matching of the [Ba/Fe] from the much larger RGB sample in Ca11 to their CN
abundances from Ca14; the Ba-rich stars are typically found to be more CN-rich by $\sim$0.15 dex.
However, there is significant dispersion in the comparison and only evidence for a weak but statistically
significant ($>$95\% confidence) [Ba/Fe] and CN correlation.
Whatever connection Ba may have, looking at the upper-left panel in Figure 3 shows that the Ba-poor
stars show little scatter and almost all of them fall directly on the blue RGB, while the Ba-rich stars
stars show a very large scatter covering both the red and the blue RGB.
\begin{figure*}[htp]
\begin{center}
\includegraphics[scale=0.85]{n1851SGBcencutabcorCT1C.eps}
\vspace{-0.43cm}
\end{center}
\caption{Similar to Figures 3, 5, and 7, we now match SGB abundances from G12 to our Washington photometry
of NGC 1851. In the upper-left panel we see that again, like in the RGB, Ba-poor (blue) stars nearly all
fall tightly along the brighter (bluer) SGB while the Ba-rich (red) stars are more broadly distributed
in color and create the sparser fainter (redder) SGB but still also fall on the bright SGB. Sr also creates a
similar distribution. Cr, however, does not create a clear distinction between the two SGB branches. In
the lower-right panel we look at the C abundances and show that the C-rich stars predominantly fall
along the bright SGB while the C-poor stars are more broadly distributed. (See Figure 1 for the definitions
of Ba$_{\rm cor}$, Cr$_{\rm cor}$, and C$_{\rm cor}$.)}
\end{figure*}
In Figure 7 we look at four approaches to distinctly characterizing the abundances of the red and
blue RGBs. The upper-left panel simultaneously looks at both [O/Fe] and [Ba/Fe]. We have colored all of the
Ba-poor (N-normal) stars blue and all of the O-rich stars blue because as discussed these stars should
be CN-poor and either CH-poor or NH-poor. We clarify that these blue stars are either Ba-poor or O-rich
and not necessarily both, and we include Ba-poor stars that do not have
an O abundance and vice-versa. We have used three colors to mark the stars that are both
Ba-rich and O-poor: green represents the Ba-rich and moderately O-poor stars, red represents the
Ba-rich and very O-poor stars, and orange represents the Ba-rich stars with O upper limits that define
them to be at least moderately O-poor if not very O-poor.
Quite remarkably, the blue data are in agreement with the well defined blue RGB, while the green data show
a moderately redder distribution typically falling on the red edge of the blue RGB and extending slightly
redder, and the red data are mostly consistent with the red RGB. As should be expected the orange data
are consistent with both the green and red data. This is promising but the
matches are not perfect because two of the stars that clearly belong to the red RGB are colored blue. We
also note the one colored red data point that lies well above the RGB. Its characteristics strongly suggest
that it is an asymptotic giant branch star. This, rather than its abundances, explains its photometric
deviation above the RGB. In the upper-left panel of Figure 8 we show the
cumulative population distributions of these four abundance groups.
The one limitation of the Ca11 O abundances is that O is a challenging measurement. This resulted in 52 of
the 111 stars we have matched to our photometry having only an upper limit or no O abundance information at
all. Therefore, because Na has a well established anti-correlation with O and Ca11 has 108 measured Na
abundances, we have analyzed a similar combination of Na and Ba abundances. This is shown in the upper-right
panel of Figure 7 with all of the Ba-poor or Na-poor stars colored blue. The Ba-rich
and Na-rich stars are grouped into Ba-rich and moderately Na-rich as green or Ba-rich and very Na-rich as
red. This provides the most striking plot we have seen so far, where the blue data are consistently in
agreement with the well defined blue RGB, while the green data show a moderately redder distribution
typically falling where the red and blue RGBs meet, and the red data are consistent with the red RGB.
Similarly, in the upper-right panel of Figure 8 we show the cumulative population distributions
of these three abundance groups.
Unlike with O, use of Na provides both a larger sample and abundance information extending to the
faintest observed red RGB stars, where we still see abundance patterns consistent with the bright RGB
stars. Overall, we have greatly increased the number of stars and find a result consistent with the proposed
model.
As with Ba, the O (Na) abundances in combination
with CN may tell us more. Again, the O abundances are limited in number, but the Na abundances will reliably
be indicative of O. The lower-left panel of Figure 7 shows the combination of the CN strengths
and Na abundances, where we color stars that are both CN-rich and very Na-rich (very O-poor) red, the
CN-rich and moderately Na-rich stars green, and all stars that are either CN-poor stars or Na-poor blue.
We now see that when considering the O abundances (represented by Na) in addition to the CN strengths,
the red data and the blue data agree very well with the red RGB and the blue RGB, respectively, and
the green data primarily falls where the red and the blue RGB meet. This is displayed more
clearly in the lower-left panel of Figure 8, where there is almost no color overlap between the red and
blue population distributions. This result implies that because of
the possible lack of large C+N+O variations, the CN-rich and Na-poor (O-rich) stars will not be both very C-rich
and N-rich, so their moderately strong CN strengths will be balanced out by either weak CH or NH lines.
Additionally, the CN-poor and Na-rich stars are O-poor but likely have band strengths limited by weak
N or weak C. Only the CN-rich and the Na-rich (O-poor) stars will be significantly rich enough in both C
and N to also be CH-rich and NH-rich, leading only these stars to be significantly fainter in the C magnitude.
Lastly, in the lower-right panels of Figures 7 and 8 we consider CH and
CN band strengths together. Here stars that are both CN-poor and CH-poor are blue, those that
are both CN-poor but CH-rich are cyan, those that are both CN-rich and CH-poor are magenta, and lastly those
that are both CN-rich and CH-rich are red. Remarkably consistent with our explanation given at the beginning
of this section, and most clearly shown in the lower-right panel of Figure 8, for the CN-poor stars the variations in CH strengths
(i.e., [C/Fe]) have no meaningful affect on the resulting color. This is not the case for the CN-rich stars,
where all CN-rich stars consistent with the blue RGB are also CH-poor, while the red RGB is composed of nearly
all of the stars that are both CN-rich and CH-rich.
\section{Matching Turnoff/Subgiant Branch Abundances to Photometry}
We have also matched the SGB and turnoff abundances of G12 to our photometry. While the lack of [O/Fe]
and [Na/Fe] measurements for these stars limit us from performing the more detailed analysis we did with the
RGB stars, we still detect that in several elements the two branches exhibit different abundance characteristics.
In Figure 9 both [Ba/Fe] and [Sr/Fe] show similar patterns as those observed for Ba in the RGB. The Sr-poor
stars and Ba-poor stars are both concentrated on the brighter and blue branch while the Sr-rich stars and the
Ba-rich stars are more broadly distributed and create the sparse fainter and red branch but also compose part of the bright
branch. Sr and Ba are the two elements that both G12 (in the SGB) and V10 (in the RGB) show to be strongly
correlated. The Cr-rich and Cr-poor stars do not show a strong photometric distinction; this illustrates the
importance of adopting the observed trends (see Figure 1) to define rich and poor populations because adopting
a constant separation for Cr-rich and Cr-poor does result in an apparent photometric distinction in the SGB.
These Ba and Sr distributions suggest that in the C filter, as on the RGB, these are not two
photometrically distinct SGB sequences representing two cleanly separated populations, but that photometrically
the two populations are a narrow and a broad population that overlap each other.
The broad second population extends significantly farther to the red and creates a distinctly redder branch, but again
this clearly redder group is not the entire second population.
\begin{figure*}[htp]
\begin{center}
\includegraphics[scale=0.75]{n1851HBtot.eps}
\vspace{-0.43cm}
\end{center}
\caption{Matches of the HB abundances from G12b to our Washington photometry
of NGC 1851. The left grid of three panels shows the BHB and the right grid of 6 panels shows the
RHB, with its two potential sequences: a faint (bluer) sequence and a bright (redder) sequence, which we
have divided by a solid grey line. O, Na,
and Mg have both BHB and RHB abundances and their abundances distributions are very broad. Therefore,
we have grouped them in three abundance groups with blue being poor, green being intermediate, and
red being rich. This clearly illustrates the abundance distinctions between the RHB and BHB. Looking
closely at the RHB itself finds that there are no clear abundance distinctions between the two
apparent RHB sequences in any of these six elements.}
\end{figure*}
G12 also have direct measurement of the C abundances from the CH bands for the SGB stars, which we
discussed a temperature trend for in Section 2 and Figure 1. Using the correlation from Figure 1 to define our
C-rich and C-poor populations and matching these abundances to our photometry shows that the faint
SGB is predominantly C-poor while the bright SGB is predominantly C-rich. This is in contrast to what
we would infer based on the RGB abundances and CH indices, but the bright SGB is on average only
$\sim$0.1 dex richer in C. This difference is relatively minor compared to the total range of
C abundances, when correcting for the trend with T$_{\rm eff}$, spanning $\sim$0.6 dex. To expand
on this further, we can define the two SGB populations based on their [Ba/Fe], and we find that the Ba-rich (second
population) and Ba-poor (first population) stars have on average no meaningful difference in C.
We also acknowledge the work in L12, which self-consistently added to the work of Pancinco et~al.\ (2010)
and in total analyzed 70 turnoff and SGB stars. They found C and N abundances from their
direct measurement of the CH and CN bands and matched them to both the HST photometry of Milone et~al.\
(2008) and the Stromgren photometry of L09. They similarly found that the bright (blue) SGB is
typically more C-rich and the faint (red) SGB is typically more C-poor, but also that the bright SGB
is poorer in N and the faint SGB is richer in N. They argue that there is a significant and possibly
bimodal spread in the distribution of C+N abundances, with the faint SGB typically being much richer.
\begin{figure}[htp]
\begin{center}
\includegraphics[scale=0.44]{HBdist.eps}
\vspace{-0.43cm}
\end{center}
\caption{Illustration of the bimodal RHB distribution perpendicular to the solid grey dividing line shown
in Figure 10.}
\end{figure}
G12 question these large variations in the L12 C+N abundances and
suggest that their T$_{\rm eff}$ were 500 K too cool. This would cause them to greatly underestimate the C
abundances, and because the N abundances are found from CN this would also cause the N abundances to be
overestimated. However, based on the idea that C+N+O is fairly constant, this variation in C+N would
at least be qualitatively in agreement with the variations in O abundance observed in the RGB. The
C+N-rich stars would
be O-poor and the C+N-poor stars would be O-rich. L12 also concluded this by comparing the average RGB
O-abundances from V10 to their average C+N abundance for the two branches and found no meaningful difference
between their average C+N+O abundances. We have also matched 65 of the abundances
from L12 to our photometry, but only one of these stars could reasonably be defined as belonging to our
faint SGB. Most of the limited number of faint-SGB stars from L12 were affected by crowding issues in
our ground-based photometry from Paper I. The one clear faint-SGB star
from Paper I with L12 abundances is both N-rich and C-rich, but this is too limited to draw any conclusions.
\section{Matching Horizontal Branch Abundances to Photometry}
The horizontal branch (HB) abundances provide a unique case to analyze the two populations
because here they create two photometrically distinct groups of stars in the blue and red
HB, unlike the overlapping RGBs and SGBs. It is believed that the BHB corresponds
to the red and broader population on the RGB and that the RHB corresponds to the blue and narrower
population on the RGB. The distinct color differences in the HB itself are believed to result from a
moderate He enhancement in the BHB population (e.g., the observations of G12b and the models of Joo
\& Lee 2013). This variation in He can also explain the observed variations of the pulsational properties
of RR Lyrae variables in NGC 1851 (Kunder et~al.\
2013). Another advantage of the HB is that these are bright stars with relatively small photometric
error. Our observations from Paper I were able to observe a broad range of both T1 and T2
magnitudes in the RHB, and in T1 there is an apparent split that creates two sequences of T1 magnitudes.
Does this suggest that there may be key differences for stars within the RHB itself?
Figure 10 shows the abundances from G12b matched to the RHB and BHB (when available). Similar to previous Figures for
Ca, Fe, and Ba the blue data represent poor stars and the red data represent rich stars for each element.
However, for the elements that also have BHB abundances measured we used three abundance groups because
they typically have very broad elemental distributions: for O the red data represent O-rich stars, the
green data represent moderately O-poor stars, and the blue data represent very O-poor stars. For Na and
Mg the blue data represent poor stars, the green data represent moderately rich stars, and the red data
represent very rich stars (see Figure 10 for the detailed abundance ranges).
The BHB is very O-poor, Na-rich, and very Mg-rich, consistent with it being the same population that
creates the red RGB.
The RHB overall does show a broader range of O, Na, and Mg abundances than the BHB, but there are
no consistent abundance differences between the two apparent RHB sequences. While the faint RHB may
on average be more O-poor, Na-rich, and Mg-rich, it still contains many O-rich, Na-poor,
and Mg-poor stars. For both [Fe/H] and [Ca/Fe] there are no meaningful abundance differences between the two
RHB sequences, but when considering errors the observed distribution spreads in [Fe/H] and
possibly [Ca/Fe] are not meaningful. Lastly, Ba appears to similarly show no meaningful difference between
the bright and faint RHB, but there is a meaningful sample of very
Ba-rich stars ([Ba/Fe]$>$0.5; see Figure 2) that nearly all fall on the faint RHB.
To look more closely at the double distribution in NGC 1851, we divide the two sequences with the
solid grey line shown in Figure 10 (T1=14.487+(C-T1)$\times$1.156). In Figure 11 we illustrate the
bimodal distribution perpendicular to this dividing line. This also provides a reference to statistically
test for differences in photometric distributions on the RHB abundances of Figure 10. Consistent with
expectations, there are no significant (p-value $<$ 0.05) differences in distribution of
any of the defined abundance groups within the RHB. Even the very-rich Na stars in Figure 10 ([Na/Fe]$>$0.23),
which in the RHB primarily fall on the faint RHB, do not have a statistically meaningful difference
in distribution in comparison to the Na-poorer ([Na/Fe]$\leq$0.23) stars. This is because while a KS-test provides a moderate
D of 0.4312, the small number limitations of only 11 very-rich Na RHB stars gives a p-value of
only 0.063. However, we again make note of the RHB stars richest in Ba ([Ba/Fe]$>$0.5) all primarily
fall on the faint RHB. They have a meaningfully different photometric distribution in comparison to all
RHB stars with weaker Ba, where a KS-test test provides a D of 0.4701 and a p-value of 0.033.
In comparisons to other cluster, this double sequence in the RHB of NGC 1851 appears similar to the double
sequence in the RHB of 47 Tuc (Milone et~al.\ 2012). But unlike NGC 1851, 47 Tuc does not also have a BHB.
Another distinction is that these two 47 Tuc RHB sequences are separated in the UV. Therefore, adopting
in 47 Tuc two populations with appropriate CNO variations that can create its observed double sequences in
the MS, SGB, and RGB
would also create this double sequence in the RHB. In NGC 1851 its two populations instead create its well
observed BHB and RHB. Is this the result of the NGC 1851 populations possibly having a more significant
difference in He than those of 47 Tuc? The two RHBs of NGC 1851 are further differentiated
because they are not defined by a difference in UV but by differences in the T1 and T2 magnitudes,
which are not meaningfully affected by variations in CNO.
Is there a possible age spread between these
two RHBs? Milone et~al.\ (2008) suggested an age difference of $\sim$1 Gyr between the two
NGC 1851 populations to explain the observed split SGB, but this corresponds to the RHB and the BHB.
Is there possibly a spread in age between the apparently single population that creates the RHB?
Is there some factor driving a difference in mass-loss rates between the two RHB sequences?
Lastly, is this related to the four, not just two, abundance group spectroscopically observed
in NGC 1851 by Campbell et~al.\ (2012) and Simpson et~al.\ (2017)?
The apparent double RHB remains a mystery, but the differences between the RHB and BHB have further
established the key abundance differences between the two populations of NGC 1851.
\section{Photometric Synthesis of Red Giant Branch Populations}
To create our synthetic magnitudes we applied the models of Kurucz et~al.\ (1992) with overshoot
and used the spectral synthesis program SPECTRUM v2.76\textsuperscript{4} and its linelist (Gray
\& Corbally 1994). For our standard input abundances we applied a [Fe/H] of -1.23, which is based
on that found by V10 (-1.23$\pm$0.01) and is similar to that found by Ca11 ([Fe/H]=-1.16$\pm$0.05).
We also applied a standard He abundance of Y=0.246. To analyze the effects of possible
variations in He, we have performed comparisons to a more He-rich (Y=0.30) population. For CNO
abundances we applied the result of V10 that for both populations C+N+O is constant at log
$\epsilon$(CNO)=8.00. Additionally, we considered the effects of distinct log $\epsilon$(CNO) in
the two populations that results from the more significant differences in [N/Fe] measured in Y15.
This gives the second population a significantly richer log $\epsilon$(CNO)=8.50. All other
abundances are scaled solar.
\begin{figure*}[htp]
\begin{center}
\includegraphics[scale=0.8]{UvsCrgbcomp3join2.eps}
\vspace{-0.43cm}
\end{center}
\caption{Relative magnitude synthesis of a representative RGB star for the Washington
C filter (black) and the F336W filter (green). Varying levels of [C/Fe], [N/Fe], and [O/Fe] are
considered but with an adopted constant log $\epsilon$(CNO)=8.0. The different curves represent constant
[O/Fe] (see key) with corresponding variations in [C/Fe] (illustrated in the left panel) and [N/Fe]
(illustrated in the right panel). Both panels show reflections of the same data curves in [C/Fe] and
[N/Fe] space, respectively. For both filters their magnitude at the most C-poor abundance synthesized
is defined as the zero point (brightest) and as standard an increase in magnitude (going down the y-axis) is
increasingly fainter. This demonstrates the strong dependence of C magnitudes on [C/Fe] across a large range
of abundances. F336W magnitudes are also strongly dependent on [C/Fe] but are more counterbalanced by an
important dependence on [N/Fe].}
\end{figure*}
Variations in [Fe/H] can also play an important role in MPs, but evidence for a meaningful spread in [Fe/H] in
NGC 1851 is limited to the analysis in the RGB by Ca11. V10 did not find any evidence for such a spread.
If there is a true variation it is
relatively minor ($\sim$0.1 dex). Additionally, matches of these [Fe/H] values to both Stromgren photometry (see
Ca11) and our own Washington photometry find that these apparent [Fe/H] variations do not match with
the two identified populations and are randomly distributed photometrically. Therefore, if this minor [Fe/H]
variation is real it does not play a role in the photometric differences of the two populations, and it would
only increase the photometric spread observed in each population. For completeness, however, we briefly
considered the photometric effects of a 0.1 dex metallicity increase.
\footnotetext[4]{http://www.appstate.edu/~grayro/spectrum/spectrum.html}
\subsection{Effects of CNO on Magnitude at Constant Log $\epsilon$(CNO)}
To thoroughly evaluate the effects of variations of C, N, and
O, we analyzed nearly the full range of abundances within the constraint of log $\epsilon$(CNO)=8.0.
This allows us to move beyond the constraints of the abundances of NGC 1851 and also make more general
conclusions about the photometric effects of CNO variations at constant log $\epsilon$(CNO) and how
this varies in differing filters.
For our representative RGB star we adopted T$_{\rm eff}$=4863 K, log g=2.18,
microturbulence=1.52 km/s, and [Fe/H]=-1.23. This places this giant right below the RGB bump in
NGC 1851. In addition to the C filter we also synthetically looked at the commonly adopted
F336W filter (which is very similar to Johnson U). In contrast to both the C and Stromgren filters,
Johnson U appears to
create two more cleanly separated photometric branches in the RGB of NGC 1851 (H09) instead of heavily
overlapping narrow and broad branches. However, based on published analyses of NGC 1851, it is unclear
if the U or F336W filters also distinctly separate, for example, the Ba- and Na-poor stars from the Ba- and
Na-rich stars. H09 showed that in U-I their red branch is Ca-rich and their blue branch is Ca-poor
with little meaningful overlap, but these Ca abundances were photometrically based, and Han et~al.\ subsequently
discovered that their Ca filter was old and degraded, leading to an increased sensitivity to CN variations
and a decreased sensitivity to Ca variations. In other clusters, however, the recent photometric and spectroscopic
analysis of the similar double RGB in M2 (Lardo et~al.\ 2013) also finds in U-V that the redder RGB is cleanly
separated from the bluer RGB. The abundance and spectral indices in Lardo et~al.\ (2013) also illustrate that
U magnitudes cleanly separate stars of differing Sr and Ba abundances and CH and CN indices. These
characteristic differences in how MPs are photometrically distributed in C and U or F336W filters
was briefly discussed in Paper I, but in this Section and the following we will analyze the reasons for this difference
in more detail.
The general relations between abundance and magnitude for the C and F336W filters can be defined by
looking at broad abundance ranges for [C/Fe], [N/Fe], and [O/Fe]. Figure 12 assumes a constant
log $\epsilon$(CNO)=8.00, and the left panel looks at several curves of constant [O/Fe] and how variations in [C/Fe]
affect both C magnitudes and F336W magnitudes.\footnotetext[5]{In our analysis we adopt for
the Sun a log $\epsilon$(C)=8.49, log $\epsilon$(N)=7.95, and log $\epsilon$(O)=8.83. This is
consistent with that adopted in V10 and agrees with the Grevesse \& Sauval (1998) solar abundances
within 0.03 dex.} We remind the reader that variations in [C/Fe] must have corresponding [N/Fe] variations
in order to keep total CNO constant, which are similarly illustrated in the right panel of Figure 12.
Throughout this paper we focus on the comparisons plotted relative to [C/Fe] because the
general characteristics are more clearly displayed in this manner. The magnitudes are placed on a
relative scale where the magnitude of the most C-poor and O-rich abundance we analyzed ([N/Fe]=-0.49,
[C/Fe]=-1.50, and [O/Fe]=0.39) is set as the zero-point for both filters. Each curve represents a constant
[O/Fe] with black curves for the C filter and green curves for the F336W filter. Lastly, we find that
these same very broad variations in CNO have only marginal effect on the synthetic R (T1) magnitudes
($\lesssim$0.02 mag, or only 2 to 5\%\, of the variations found in C magnitude) but they become more
important for the synthetic I (F814W) magnitudes ($\lesssim$0.04 mag, or 4 to 10\%\, of the variations found in
C magnitude or 2 to 7\%\, of the variations found in F336W). Therefore, these plotted variations in C and F336W
magnitude reliably trace variations in C-T1 and F336W-F814W, in particular when looking at realistic
(moderate) CNO variations observed within a single cluster, but for the highest precision
comparisons to cluster observations these variations in R (T1) and I (F814W) should be noted.
At constant [O/Fe], increasing [C/Fe] with the corresponding decrease in [N/Fe] leads to a
fainter C magnitude in all but the richest [C/Fe] (weakest [N/Fe]). Therefore, in the C magnitude
the [C/Fe] generally plays the dominant role, which was the filter's original purpose (note that the C
designation in fact refers to Carbon). [N/Fe] remains important but generally
plays a secondary role. In contrast, the F336W magnitude patterns are more complex and typically show at
constant [O/Fe] a relatively weaker dependence of F336W magnitude on [C/Fe], but followed by quickly brightening
F336W magnitude at the richest [C/Fe]. This pattern can be explained by the F336W filter being strongly dependent
on [C/Fe] but [N/Fe] plays nearly as important of a role, resulting from the prominent NH band at 3360 \AA\, near
the center of its bandpass and the lack of the CH feature at 4300 \AA\ (the G band). In contrast the G band
plays a more important role in the C filter while the strong NH band is at the far blue edge of its bandpass
where the transmission is relatively low.
Therefore, for the F336W filter in Figure 12 the increasing [C/Fe] at constant [O/Fe] is counterbalanced by a
decreasing [N/Fe], leading to a comparatively weaker magnitude change with increasing [C/Fe] followed by
a significant brightening at the C-rich extreme because so little N remains, greatly weakening the
NH line.
\begin{figure*}[htp]
\begin{center}
\includegraphics[scale=0.85]{UvsCrgbcompgrid.eps}
\vspace{-0.43cm}
\end{center}
\caption{The left panel applies the curves from Figure 12 to C-T1 and F336W-F814W color space for
abundances of the two populations in NGC 1851. An O-rich/N-normal population is circled in blue and
an O-poor/N-rich population is circled in red for both color curves. This illustrates
photometric variations similar to the corresponding observations of Paper I for C-T1 (a narrow blue
population and a heavily overlapping but broad red population) and H09 for U-I (which is comparable to
F336W-F814W with two more cleanly separated populations). The central panel looks at identical CNO variations
but illustrates the effects of
O-poor/N-rich population ([O/Fe]=-0.3) also being He-rich (Y=0.30). This causes this redder population to
be shifted slightly bluer, but the two populations are still distinct. The right panel looks at the significant
effect of the O-poor/N-rich population ([O/Fe]=-0.3) also having a richer log $\epsilon$(CNO)=8.5. The significant
increase in [N/Fe] shifts the redder populations $\sim$0.06 further to the red in C-T1 and $\sim$0.12 further
to the red in F336W-F814W.}
\end{figure*}
Comparing the relative shifts in the differing lines (representing changing [O/Fe]) indicates
another key difference between the two filters caused by the stronger [N/Fe] dependence of F336W.
With increasing [O/Fe], at constant [C/Fe], the F336W magnitudes become fainter more rapidly
than the corresponding C magnitudes. This is not directly the result of the changing [O/Fe], which
plays relatively little importance in the actual flux, but the corresponding increase in [N/Fe] with
the decreasing [O/Fe]. This also represents the advantage that the F336W (and the similar Johnson U)
filter has over the C filter in separating MPs that have a significant difference in [N/Fe]. It
is also important to note an anti-correlation (or correlation) between NH and CH strengths is not clearly
observed in NGC 1851, but they are observed in most globular clusters (e.g., Pancino et~al.\ 2010; L15). For
clusters with anti-correlations (e.g., 47 Tuc, M15, NGC 288) the weaker CH bands in the N-rich/O-poor
second population will weaken the observed magnitude shift in C more than it does in F336W, but for clusters
with a positive correlation (e.g., M22) this will increase the C magnitude sensitivity more than for F336W.
Despite these caveats, the C filter
is still highly sensitive to CNO abundance variations and retains many advantages with its significant
(4 to 5 times) increase in photometric throughput in cool RGB stars versus the F336W filter.
\subsection{Application to NGC 1851 Red Giant Branch Abundances}
Here we apply these curves more specifically to the photometric and spectroscopic observations of NGC 1851.
In Figure 13, because we look at a number of special cases where variations in R (T1) and I
(F814W) play a role, we plot these curves in color space with C-T1 in black and F336W-F814W in green.
In the left panel of Figure 13 we have selected two appropriate abundance ranges for the two observed
populations of NGC 1851. Both populations have a comparable average [C/Fe] of $\sim$-0.25 but with a
moderate variation, consistent with no CN-CH correlation. The dominant blue
population has a [O/Fe]$\sim$0.2 while the secondary red population has a [O/Fe]$\sim$-0.3. This is more
C-rich than the RGB abundances measured by V10 and Y15, but those are from upper RGB stars
where deep mixing has further depleted the [C/Fe] beyond the lower RGB and the SGB. For
both the C-T1 and F336W-F814W curves we illustrate the abundance groups with a blue and red ellipse plotted over
each curve set. As would be expected, the O-rich/N-normal population (blue ellipse) has both brighter C and F336W
magnitudes giving relatively narrow and blue populations in both colors. Also, as expected, the O-poor/N-rich
population (red ellipse) has both fainter C and F336W magnitudes, giving redder colors, but there are many key
differences. First, in C-T1 the weaker [N/Fe] sensitivity leads to a relatively weaker color
difference causing strong overlap between the primary and secondary population at the O-poor/C-poor edge versus
the O-rich/C-rich edge. However, the sensitivity of C-T1 to varying [C/Fe] is increased in the O-poor
stars giving a very broadly distributed color range with a distinctly redder subpopulation consisting of only
the O-poor/N-rich/C-rich stars. The second population (O-poor/N-rich) in F336W-F814W shows more significant
color separation from the primary blue population. Additionally, the weaker sensitivity to [C/Fe] variations
leads to a narrower population than observed in C-T1. These findings are strongly consistent with our
observations in Paper I of the two RGB branches in C-T1 and C-T2 and the comparable U-I used by H09.
\begin{figure*}[htp]
\begin{center}
\includegraphics[scale=0.85]{UvsCmscomp2.eps}
\vspace{-0.43cm}
\end{center}
\caption{The left panel shows the relative color synthesis of a representative upper MS star for
C-T1 (black) and for F336W-F814W filter (green). The format is the same as that described in Figure 13.
Additionally, the C-T1 and F336W-F814W curve sets are arbitrarily offset for clarity. While the [C/Fe]
sensitivity of these colors are weaker here than in the RGB, due to the hotter T$_{\rm eff}$, it is still
strong and the trends are similar to that found in the RGB. We note that in the MS these same two populations
should overall be more C-rich and correspondingly poorer in N than the RGB. Two photometrically separated
populations are predicted in both colors. A double MS, however, has not been observed in F336W-F814W for NGC
1851. In the right panel we consider that the redder (O-poor/N-rich) population is likely also
He-rich. Adjusting the relative colors for an He difference
greatly diminishes the predicted photometric difference in F336W-F814W, and with only moderate photometric
error these populations may appear as a somewhat broad but single MS. The C-T1 color difference
is also diminished, but the O-poor/N-rich population remains significantly broader in color and can be detected
more easily through the distribution analysis performed in Paper I.}
\end{figure*}
The consistency of the general photometric characteristics seen in observations is promising.
More quantitatively, the observations in both Paper I and H09 find an observed difference of
$\sim$0.1 in C-T1 and U-I colors,
respectively, between the centers of the two RGB branches. This is consistent with
the separation of the centers of the red and blue ellipses in F336W-F814W color. For C, while the
separation of the two ellipses in C-T1 color is $\sim$0.05, this is the separation of the two
complete and overlapping populations. The RGB ``red branch" in C-T1 is only the redder half
of the red ellipse, and again it has an extension of $\sim$0.1 beyond the dominant blue branch.
(For reference, the O-poor/N-rich population is only $\sim$0.005 magnitudes fainter in R (T1)
and $\sim$0.015 magnitudes fainter in I (F814W; T2) in comparison to the O-rich/N-poor population.
The lack of significant effect of CNO variations on these redder filters is also consistent with
the two RGB branches not being observed in Paper I's T1-T2.)
While our photometric matches do not find that redder RGB population is more metal-rich, it is important
to test what effects metallicity variations may have between the two populations. Adopting that the
O-poor/N-rich population is also 0.1 dex richer in all metals (besides CNO) finds that all filters are moderately
affected. Accounting for the effects in both filters, the metallicity increase would shift the O-poor/N-rich
population $\sim$0.02 redder in C-T1 and $\sim$0.025 redder in F336W-F814W. By itself, this metallicity increase
is not large enough to create the observed color differences between the two populations in NGC 1851, but
this shows that metallicity is important to consider in clusters with well defined metallicity differences
between their MPs. As suggested by the NGC 1851 observations in Ca11, however, if there is a true metallicity
spread of $\sim$0.1 dex within each population, this would further enhance the color spread within each population.
In the center panel of Figure 13 we also consider the effects of the second population (O-poor/N-rich)
being more He-rich (Y=0.3) than the first population. In an RGB star of these characteristics, this
has minor but measurable effects on its F336W and C magnitudes. However, we note that variations in He
also have smaller but meaningful effects on the R (T1) and
F814W (I) magnitudes. Therefore, in the center panel of Figure 13 we account for all filter variations
in our color curves. As expected, a richer He shifts the both C-T1 and F336W-F814W to bluer colors.
In the right panel of Figure 13 we look at the effects that an increase of log $\epsilon$(CNO) from 8.0
to 8.5 for the second population would have. This is driven by a significant increase in [N/Fe] but with
consistent [O/Fe] and [C/Fe] abundances and is in line with with the abundances from Y15. Unlike
at constant log $\epsilon$(CNO), this variation in total log $\epsilon$(CNO) leads to more important effects
on R (T1) at $\sim$0.03 and F814W (I) at $\sim$0.05. Therefore, we apply the effects of all filters on our
colors. We find that this large increase in [N/Fe] causes a significant shift
to the red for the second population that is far more than is found in either C-T1 or U-I observations. Even when
correcting for the second population also being richer in He, this remains inconsistent
with observations. One additional factor suggested by the abundances of V10 and G12 is that the second
population on average is slightly ($\sim$0.1 dex) more C-poor. This would help mitigate the effect of
such a large increase in [N/Fe] and suggests that the log $\epsilon$(CNO) may not be uniform across both
populations, but any differences are likely more moderate ($\lesssim$0.2-0.3 dex).
\section{Photometric Synthesis of Main Sequence Populations}
In Paper I, using Washington photometry we discovered evidence for NGC 1851 having two photometric branches
in the MS consistent with those observed in its RGB. Unlike in the other stellar
groups, there are no direct spectroscopic abundances available for MS stars in NGC 1851.
However, we can still infer general abundance information from the observed SGB and RGB abundances.
Therefore, we have looked specifically at an O-rich ([O/Fe]=+0.2) population
and an O-poor ([O/Fe]=-0.3) population, consistent with our RGB analysis and the abundances
of Ca11. At these [O/Fe] we again adopt a constant log $\epsilon$(CNO)=8.0 and at a range of
[C/Fe] near scaled solar. Our representative MS model has T$_{\rm eff}$=5800 K, log g=4.59,
microturbulence=0.74 km/s, and [Fe/H]=-1.23. This places our star in the upper MS below the turnoff.
Figure 14 looks at the color synthesis for an MS star in both the C and F336W filters. Both the general
C-T1 and F336W-F814W color trends are consistent with that found in the RGB (Section 6). Again,
these color variations are driven by magnitude variations in C and F336W because the synthetic R (T1) and
F814W (I) magnitudes variations ($<$0.01) are even weaker than those predicted in the RGB. Overall, in
comparison to the RGB, the variations in synthetic color are weaker but still significant. This is primarily
the result of the MS model star being $\sim$1000 K hotter than the RGB model star, giving weaker CN, CH,
and NH bands, rather than any other difference between MS and RGB stars.
Focusing on the expected abundance differences in the two MS populations of NGC 1851,
the [C/Fe] abundances of the SGB and upper turnoff stars from G12 (see Figure 1) suggest that the [C/Fe] in the MS
will have moderate variations and be richer than scaled solar. The MS is expected
to be more C-rich than the RGB because the mixing processes that dilute the surface C during evolution
have not yet begun. In Figure 14 we have adopted
abundance ranges consistent with this for the two MS populations and find that the O-rich/N-normal
population (blue ellipse) is bluer in both C-T1 and F336W-F814W and relatively narrow, but
at these very C-rich abundances the decreasing [N/Fe] makes the F336W-F814W colors increasingly bluer.
The O-poor/N-rich population (red ellipses) create in C-T1 a very broadly distributed
and slightly redder population with minor overlap with the blue population. The F336W-F814W creates a
significantly redder but still photometrically narrow population.
Milone et~al.\ (2008) did not find evidence in NGC 1851 for a
double MS in F336W-F814W, but the C-T1 observations from Paper I do find evidence for a heavily overlapping
but broadly distributed second redder MS. In the upper MS this second MS extends to the red with an apparent separation
of $\sim$0.05 magnitudes. In the lower MS this separation increases further, resulting from the increasing
strengths of the molecular bands in cooler stars.
Figure 14 suggests that the double MS should be cleanly separated in the F336W-F814W observations of Milone
et~al.\ (2008), but this is not the case. A possible explanation for this is that in the MS the effect of the
varying He abundances in the two populations becomes more important. The O-poor/N-rich population
is believed to be more He-rich. In the right panel of Figure 14 we analyze the effects of this by keeping
the O-rich/N-normal population at our standard Y=0.246 and the O-poor/N-rich population
at Y=0.30. Across the range of [C/Fe] and [N/Fe] at constant [O/Fe]=-0.3, the effect on the C and F336W
magnitudes are uniform at 0.043 brighter and 0.052 brighter, respectively. The He-rich population is also
0.028 magnitudes brighter in T1 and 0.022 magnitudes brighter in F814W. This shifts the He-rich population
0.03 magnitudes bluer in F336W-F814W and 0.015 magnitudes bluer in C-T1. This brings the populations in
C-T1 closer together but because of their distinct color distributions the heavily blended redder
population was still observed in Paper I using C-T1. However, the larger effect of He on the F336W-F814W
color nearly brings the two narrow MS populations together. With even small photometric error and
considering that this second population may be slightly more C-poor ($\sim$0.1 dex), this may appear
similar to a single MS that is broad in color but otherwise normal.
\section{Summary \& Conclusions}
Matching previously published spectroscopic abundances to our Washington photometry, we have analyzed many of the quantitative
differences between the two NGC 1851 populations. The large sample of RGB abundances from Ca11 have shown
that Ba, Na, and O are useful elements to distinguish between the two populations in the RGB. In particular,
Ba and Na clearly show that stars poor in either of these elements are consistently part of the (in C-T1) narrow
and blue RGB, while stars that are rich in either of these elements are more broadly distributed in color.
The reddest stars of this second population create the red RGB while the bluer stars of this second population
are consistent in color with the blue RGB. In agreement with the strong O and Na anti-correlation, the
O-rich stars are consistent with the narrow and blue RGB while the O-poor stars are more broadly distributed.
To better understand this broad color distribution of the second population, we considered two abundances
simultaneously. The stars that are both Ba-rich and O-poor (or similarly Ba-rich and Na-rich)
all fall primarily on the red RGB. All other stars are consistent with the
blue RGB. This may result from [Ba/Fe] acting as a tracer of [N/Fe]. Therefore, the red RGB is composed of
only the N-rich/O-poor stars. This is further supported because stars that are both CN-rich
and Na-rich (or CH-rich) all fall primarily on the red RGB.
Matching the G12 abundances to the two branches of the SGB shows many consistencies with what we found in
the RGB abundances, in particular for the correlated elements of Ba and Sr. While the lack of O and Na
abundances for the SGB limit our analysis tools, the C abundances from G12 are useful to analyze these two
branches. Most interestingly, the C abundances from G12 suggest that the faint SGB is more C-poor and the
bright SGB is more C-rich, but the difference is only $\sim$0.1 dex while the total range of C abundances
span $\sim$0.6 dex.
For the G12b abundances of the HB, as expected we find clear abundance differences between the RHB and BHB
where the BHB is relatively very O-poor and more Na-rich and Mg-rich. However, other than a group
of very Ba-rich stars being predominantly in the faint RHB, there are no clear abundance
differences between the observed bright and faint RHB sequences. The cause of these sequences in the T1 and
T2 magnitudes remains unclear, but it is likely different than the CNO and He variations that can explain the
two established populations, represented by the BHB and the RHB. It would be of interest to investigate what
other factors may cause two sequences of such characteristics in the ``single population" of the RHB.
To build on these abundance analyses and look at the role that CNO plays in the C and F336W filters, we performed
photometric synthesis for a broad range of [C/Fe], [N/Fe], and [O/Fe] at a constant log $\epsilon$(CNO)=8.0.
Comparisons of the effects on both the C and F336W filters find that at constant C+N+O the variations in
magnitude can be quite significant and adopting larger log $\epsilon$(CNO) values are not necessary
to explain the observed photometric differences between the two populations. This analysis also showed that
the C filter is highly sensitive to [C/Fe], as its name would suggest, with some sensitivity to [N/Fe].
In contrast, F336W and the similar Johnson U filter have more comparable sensitivities to both [C/Fe]
and [N/Fe].
To compare these synthetic magnitudes to observations of NGC 1851, we adopted abundance characteristics for
each population based on the RGB spectral analyses from Ca11, V10,
Ca14, and Y15. We define the two populations as an O-rich ([O/Fe]=0.2) and an O-poor ([O/Fe]=-0.3) population.
In our representative RGB star that is just below the RGB bump, we assume that both populations have comparable
but broadly distributed [C/Fe] near [C/Fe]$\sim$-0.25, which is consistent with NGC 1851 not having
a CN-CH correlation. Based on the constant C+N+O, and consistent with the CN observations of
Ca14, the O-rich and O-poor populations are also N-normal and N-rich, respectively. Application of this
to our photometric synthesis finds that the O-rich/N-normal population is blue and narrowly distributed
in color in both C-T1 and F336W-F814W. Similarly, the O-poor/N-rich population are both redder in
C-T1 and F336W-F814W. However, in C-T1 the second population's colors are very broadly distributed
and overlapping with the bluer population. In F336W-F814W the second population is redder and nearly
as narrow in color as the blue population. These color variations are driven almost completely by variations
in the C and F336W filters and are consistent with the photometric observations of both Paper I and H09.
In application to general globular cluster observations, it is important
to discuss the differences between clusters with a CN-CH anti-correlation or correlation in contrast to our
adoption of no correlation (as observed in NGC 1851).
In general, the photometric sensitivity to MPs will be decreased in clusters with CN-CH
anti-correlations, but more so in the C filter versus F336W. However, for clusters with CN-CH correlations the
increase in photometric sensitivity to MPs will be greater in the C filter versus F336W. Our broad abundance
analysis in Figure 12 helps illustrate this.
We also looked at additional differences that may play a role in the photometric differences of these two
populations. First, we considered the O-poor/N-rich population being He-rich (Y=0.3). This diminishes the
color difference between the two populations, but the effect is relatively minor and the predicted total
photometric differences are still significant and detectable. Second, we considered the
O-poor/N-rich population being significantly more N-rich than the primary population at log $\epsilon$(CNO)=8.5.
This greatly increases the predicted color differences between the two populations, strikingly
in F336W-F814W, and well beyond the differences that have been observed in NGC 1851. A more moderate difference
in log $\epsilon$(CNO) ($\lesssim$0.2 to 0.3) for the two populations, however, could be further mitigated
by the O-poor/N-rich population being both He-rich and slightly more C-poor ($\sim$0.1 dex). This could bring
two populations of a meaningfully different log $\epsilon$(CNO) into agreement with the photometric observations.
Applying the same [O/Fe] based populations to photometric synthesis in the MS finds further interesting
comparisons between the C and F336W filters. Because the MS will be more C-rich we adopted a moderately spread
[C/Fe] consistent with that found in the least unevolved stars from G12 ([C/Fe]$\sim$0.2).
In the hotter MS stars, compared to the cooler RGB stars, the weaker molecular bands lead to comparable
dependencies but overall less significant sensitivity to the CNO abundances in both the C and F336W filters.
At these richer [C/Fe] the second, redder population is still very broadly distributed in the C-T1 but quite narrow
in F336W-F814W. This is consistent with the observations of a heavily overlapping double MS from
Paper I, but how does it explain the lack of observed double MS in the F336W-F814W observations of Milone et~al. (2008)?
We believe this again relates to the increase in He abundance in the second O-poor/N-rich population.
In the MS the increase in He has a more important effect than in the RGB, and it is also stronger in
the F336W-F814W colors than the C-T1 colors. This He correction is significant enough that in F336W-F814W
the two narrowly distributed in color MS populations are nearly shifted on top of each other. Considering appropriate
photometric error and the second population being slightly more C-poor ($\sim$0.1 dex), this could create
a broad but apparently single MS. In C-T1, the significant difference between the color distribution widths of
the two populations leaves a comparably overlapping but more distinguishable second population.
The C filter is a very important filter in analyzing the MPs in GCs for four reasons. The first is that in
the typically very cool stars in GCs the C filter has $\sim$3 to 5 times the throughput of other UV filters.
This results from the C filter being both broader but also centered redward of the other UV filters, which
makes it less susceptible to reddening and interstellar extinction. Lastly, the C-filter's peak sensitivity
is also usually higher than its competitors. These factors combined allow MPs to be relatively quickly
identified using smaller telescopes, like in the analysis from Paper I of NGC 1851 using relatively little
time on a 1-meter telescope. This increased sensitivity also greatly improves our ability to observe distant
populations in both our own Galaxy and also in the Magellanic Clouds, as well as the Andromeda Galaxy with
the HST. Second, the C filter may provide much stronger sensitivity than the F336W filter for detecting
multiple MSs, at least for populations like that found in NGC 1851, but potentially for many other GCs as
well. Third, while both the C and F336W filters can detect MPs, we have shown several important differences
in how these two magnitudes are affected by the compositional variations expected in MPs. Analyzing GCs with
both filters and comparing how their MPs are distributed differently in both filters provides an efficient
method to help constrain the abundance differences between the two populations, in particular when
spectroscopic abundances may be limited. Finally, although the combined ``magic trio" of F275W,
F336W, and F438W filters on board HST are unmatched in their ability to distinguish MPs, we will be forced to
use a different technique once HST has stopped operations. We will be ''uv-blind" with no ability to observe
blueward of the atmospheric cutoff and ground-based filters like Washington C will be especially important.
Acknowledgements: J.C. acknowledges Christian Johnson for his help with spectral synthesis software
and support from the National Science Foundation (NSF) through grant AST-1211719.
D.G. and S.V. gratefully acknowledge support from the Chilean
BASAL Centro de Excelencia en Astrof\'isica y Tecnolog\'ias Afines (CATA) grant PFB-06/2007.
| -42,047.271489 |
[
-3.640625,
3.2890625
] | 22.694394 |
[
-2.50390625,
0.52685546875,
-1.609375,
-5.31640625,
-0.6904296875,
7.671875
] |
[
3.69140625,
6.02734375,
3.9375,
5.625
] | 826 | 13,937 |
[
-3.189453125,
3.6015625
] | 24.693151 |
[
-6.234375,
-3.685546875,
-3.525390625,
-2.0625,
1.8427734375,
11.5
] | 0.991704 | 13.135814 | 11.135826 | 3.533924 |
[
2.4531304836273193
] | -36,390.554086 | 4.956088 | -41,471.286027 | 0.639903 | 5.819145 |
[
-3.560546875,
-3.61328125,
-2.599609375,
-3.451171875,
2.626953125,
9.8125
] |
[
-5.7265625,
-2.806640625,
-2.513671875,
-2.21484375,
4.05078125,
6.4140625
] | |
BkiUdq45qoYA2zk0vrSH
|
\section{Introduction}
We study aperiodic tilings of Euclidean space arising from a substitution rule. Aperiodic tilings provide important examples of topological dynamical systems, over spaces of tilings called \emph{tiling spaces} \cite{Sadun}. Since the seminal work of Anderson and Putnam \cite{AP}, much is known about the topology of tiling spaces of substitution tilings, including calculations of their topological invariants \cite{BDHS,Sadun}. A beautiful result of Sadun and Williams \cite{SW} proves that the tiling space of an FLC tiling is a fibre bundle over the torus, with totally disconnected fibres (which are homeomorphic to the Cantor set, in the case of a repetitive, nonperiodic tiling). Kellendonk \cite{Kel1,Kel2} established a complementary algebraic approach, by constructing an inverse semigroup of partial translations for such a tiling. Tiling semigroups were extensively studied by Kellendonk and Lawson in \cite{KL} and their algebraic properties were determined by Zhu in \cite{Zhu}. In this paper, we show that tiling semigroups arising from a substitution are self-similar, and that the limit solenoid naturally associated to the tiling semigroup is homeomorphic to the tiling space.
Self-similarity of groups has been a hugely fruitful mechanism with which to construct groups enjoying certain properties, particularly growth properties. As such they have been key in solving several important problems in Group Theory and Geometric Group Theory, most notably Grigorchuk's famous group with intermediate growth \cite{nek_book, Gr, Gr2}. Self-similar inverse semigroups acting on topological Markov shifts were introduced by Bartholdi, Grigorchuk and Nekrashevych in \cite{BGN} and Nekrashevych went on to show they give rise to Smale spaces in \cite{Nek_SSIS}. It has already been noted that self-similar groups and semigroups can be associated to some substitution tilings. However, the general theory for substitution tilings is not worked out in either paper, which focus entirely on a quotient by rotations and reflections of the Penrose tilings, see \cite[p.13--15]{BGN} and \cite[p.859--861]{Nek_SSIS}. Moreover, we outline a different, more global approach, which makes it implicit that the full object constructed is Kellendonk's tiling semigroup, rather than starting with generators and (self-similar) relations. Our constructions are easily modified to construct the analogous object where translations are replaced with general rigid motions, but we consider it important to present the theory in the translational case, which has remained a major focus in Aperiodic Order owing to connections between translational dynamics of aperiodic patterns and their spectral properties, which finds application to the study of quasicrystals \cite{BG}.
For a substitution tiling satisfying standard conditions we show that Kellendonk's tiling semigroup acts self-similarly on a topological Markov chain that is naturally homeomorphic to the canonical transversal of the tiling space. We recall the substitution graph, whose path-space is a topological Markov chain. It is well-known that this is conjugate to Kellendonk's discrete hull of a tiling with dynamics arising from (the inverse of) substitution. The tiling semigroup acts on this space by translation. Indeed, an element of the tiling semigroup specifies a patch of tiles with two distinguished tiles that specify the domain and range of the translation between them. The self-similarity arises from the fact that translation across patches of tiles can be lifted to translations between supertiles, with these structures being analogous at all levels of the hierarchy in the topological Markov chain.
In \cite{AP}, Anderson and Putnam define a branched manifold from a substitution tiling, now called the Anderson--Putnam (AP) complex. Starting with a collection of prototiles, the AP complex is the quotient space defined by the relation that two prototiles are glued along their codimension-1 faces if those faces ever meet in a tiling. For a substitution tiling, one can define addresses of points of prototiles by a left-infinite topological Markov shift. We define the limit space of a general self-similar semigroup action, in an analogous way to the self-similar group case \cite{BGN}, as a quotient space of such left-infinite topological Markov shifts by asymptotic equivalence, which itself we modify for the semigroup case (Definition \ref{def:ae}). We prove that the limit space of a tiling semigroup is homeomorphic to the AP complex. The limit space naturally inherits a self-map from the shift map of the Markov shift, which corresponds to the usual self-map by substitution on the AP complex, which thus has inverse limit homeomorphic to the tiling space.
The paper is organised as follows. In Section \ref{sec: Nonperiodic tilings and their semigroups} we recall the basic facts required about aperiodic tilings, their tiling semigroups, tiling spaces and tiling substitutions. In Section \ref{sec: Self-similarity of substitution tiling semigroups} we prove that tiling semigroups of substitution tilings (with standard restrictions) are self-similar. We give a general notion of a self-similar semigroup being contracting (Definition \ref{def: contracting}) and show this property holds for substitution tiling semigroups. In Section \ref{sec: limit space} we introduce the asymptotic equivalence relation for self-similar semigroups (Definition \ref{def:ae}). The associated quotient space, called the limit space, is shown to be homeomorphic to the Anderson--Putnam complex (Theorem \ref{thm: limit spaces of sub tilings}). Finally, in Section \ref{sec: examples}, we see how the definitions given apply to various examples.
\section{Nonperiodic tilings and their semigroups} \label{sec: Nonperiodic tilings and their semigroups}
Tilings will be built from a finite set $\mathcal{P}$ of \textbf{prototiles}, labelled compact subsets of $\mathbb{R}^d$ that are equal to the closure of their interior. The support of $p$ is denoted $\mathrm{supp}(p) \subset \mathbb{R}^d$. A translate of a prototile is called a {\bf tile} and a finite, connected set of tiles which overlap only on their boundaries is called a {\bf patch}. By {\bf connected} here, we mean that any two tiles can be connected through a path of {\bf meeting tiles}. By tiles {\bf meeting} we allow two options: that the tiles are {\bf adjacent}, meaning that they intersect non-trivially (on their boundaries) or alternatively, if the tiles and patches have cell decompositions (for example, the tiles are polyhedra), then the tiles meet along a shared codimension-1 face. When cells have a cellular decomposition, the constructions to follow will hold for whichever of these two conventions the reader prefers. The set of all finite patches is denoted $\mathcal{P}^*$. A \textbf{tiling} $T$ is a covering of $\mathbb{R}^d$ by tiles which intersect only on their boundaries. Given a tiling $T$ and bounded subset $S \subseteq \mathbb{R}^d$ we define
\[
T \sqcap S \coloneqq \{t \in T \mid \mathrm{supp}(t) \cap S \neq \varnothing \}.
\]
If $S$ is a closed ball of radius $r$ then $T \sqcap S$ is called an {\bf $r$-patch}. If $T$ is a tiling and $x \in \mathbb{R}^d$, the translate of $T$ by $x$ is $T+x \coloneqq \{t+x \mid t \in T\}$ and the \textbf{orbit} of $T$ is $\mathcal{O}(T) \coloneqq \{T+x \mid x \in \mathbb{R}^d\}$. We say that $T$ is \textbf{nonperiodic} if $T+x=T$ implies that $x=0$.
\begin{definition}\label{tiling metric}
For tilings $T$, $T'$ we define their distance in the \textbf{tiling metric} as
\[
d(T,T') \coloneqq \inf \{\varepsilon, 1 \mid (T-x) \sqcap B_{1/\varepsilon} = (T'-x') \sqcap B_{1/\varepsilon},\ x, x' \in \mathbb{R}^d,\ |x|, |x'| < \varepsilon\},
\]
where $B_r$ denotes the closed ball of radius $r$ centred at $0 \in \mathbb{R}$.
\end{definition}
Two tilings $T$, $T'$ are close if $T$ and $T'$ have the same patch of tiles on a large ball centred about the origin, up to a small translation. The \textbf{continuous hull} (or {\bf tiling space}) of a tiling $T$ is the space of tilings whose finite patches all belong to $T$, up to translation, with topology induced by the tiling metric (that is, it is the space of tilings which are {\bf locally indistinguishable} from $T$). Equivalently, $\Omega_T$ may be regarded as the completion of $\mathcal{O}(T)$ under the tiling metric. We call $T$ {\bf repetitive} if, for every finite patch $P$, there exists some $r > 0$ so that a translated copy of $P$ can be found in every $r$-patch of $T$. In this case, every element of $\Omega_T$ has the same set of finite patches and thus $\Omega_T = \Omega_{T'}$ for all $T' \in \Omega_T$. In particular, if $T$ is nonperiodic and repetitive, then $T$ is {\bf strongly aperiodic}, that is, every element of $\Omega_T$ is nonperiodic. A tiling $T$ is said to have \textbf{finite local complexity (FLC)} if there are only a finite number of two-tile patches in $T$, up to translation (equivalently, there are finitely many $r$-patches for each $r > 0$). Of course, every repetitive tiling has FLC. With the topology above, FLC is equivalent to compactness of $\Omega_T$ \cite[Lemma 2]{RW}.
A \textbf{substitution} on a set of prototiles $\mathcal{P}$ is a map $\varphi \colon \mathcal{P} \to \mathcal{P}^*$ for which there is a scaling factor $\lambda>1$ with $\mathrm{supp}(\varphi(p)) = \lambda \cdot \mathrm{supp}(p)$ for each $p \in \mathcal{P}$. Since the support of a substituted tile is exactly equal (rather than just covering) its inflated tile, $\varphi$ is more specifically a {\bf stone inflation}. This property is necessary in what follows, but the inflation being a similarity $x \mapsto \lambda x$ is not and can instead be taken as an expansive linear map; we assume an expansion constant merely for expository convenience. A substitution $\varphi$ on a tile $t=p+x$ is defined to be $\varphi(t) \coloneqq \varphi(p) + \lambda x$. Then a substitution may be applied to a patch, which by a slight abuse of notation we also denote $\varphi \colon \mathcal{P}^* \to \mathcal{P}^*$; similarly, substitution may be applied to tilings. An \text{$n$-supertile} is a translate of the patch $\varphi^n(p)$ for some $p \in \mathcal{P}$. We will always assume that $\varphi$ generates FLC tilings, which is to say that it generates only finitely many two-tile patches (up to translation equivalence) under iteration. We call $\varphi$ {\bf primitive} if there exists some $k \in \mathbb{N}$ so that, for each $a$, $b \in \mathcal{P}$, we have that $\varphi^k(a)$ contains a translated copy of $b$.
A tiling $T$ is {\bf admitted} by the substitution if every finite sub-patch of $T$ is contained in some $n$-supertile. The set of such tilings is denoted $\Omega_\varphi$. It follows easily from FLC that $\Omega_\varphi$ is non-empty and, if $\varphi$ is primitive, then every admitted tiling is repetitive, so that $\Omega_\varphi = \Omega_T$ for any $T \in \Omega_\varphi$.
The induced map $\varphi \colon \Omega_\varphi \to \Omega_\varphi$ is always surjective, so that for each tiling $T$ there is a corresponding `supertiling', which decomposes under $\varphi$ to $T$. If $\varphi$ is additionally injective, then we say that $\varphi$ is {\bf recognisable}. This means that, for any $T \in \Omega_\varphi$, there is a unique way to group tiles into supertiles, whose associated tiling in $\Omega_\varphi$ decomposes to $T$ under substitution. By continuity and compactness, this may always be done by a locally defined rule for a recognisable substitution. Recognisability is equivalent to non-periodicity of the tilings of $\Omega_\varphi$ \cite{Sol2}. We will assume throughout that $\varphi$ is recognisable.
We always assume that $\varphi$ {\bf forces the border} \cite[p.24]{Kel1}. This means that there is some $k \in \mathbb{N}$ so that any $k$-supertile $\varphi^k(p)$, for $p \in \mathcal{P}$, extends uniquely to a valid patch containing $\varphi^k(p)$ and any tiles intersecting the boundary of $\varphi^k(p)$ (that is, $\varphi^k(p)$ uniquely extends to its $1$-corona). Border forcing may always be assumed by passing to a dynamically equivalent substitution by collaring tiles (see \cite{AP,Sadun}).
The \textbf{Anderson--Putnam complex} of a tiling is the compact Hausdorff topological space formed through taking the transitive closure of gluing together prototiles in all ways their translations can be adjacent in a tiling, see \cite[Section 4]{AP}.
In this paper we make use of the discrete hull of a tiling, a particular subset of the continuous hull. Let $T$ be a tiling with prototile set $\mathcal{P}$. Following Kellendonk \cite[Section 2.1]{Kel1}, for each $p \in \mathcal{P}$, choose a point in the interior of $\mathrm{supp}(p)$ called a \textbf{puncture} and denote it by $x(p)$. This naturally punctures tiles $t=p+y$ by $x(t) \coloneqq x(p)+y$ and defines sets of punctures for patches and tilings. The \textbf{discrete hull} of a tiling $T$ is given by
\[
\Omega_\mathrm{punc} \coloneqq \{T' \in \Omega_T \mid \text{there exists}\ t \in T'\ \text{with}\ x(t) = 0 \} \subset \Omega_T,
\]
i.e., the subset of tilings with a puncture over the origin of $\mathbb{R}^d$. If $T$ is repetitive, non-periodic and has FLC then $\Omega_\mathrm{punc}$ is a Cantor set. In particular, $\Omega_\mathrm{punc}$ is a compact metric space that has a basis of clopen sets. Indeed, for a patch $P$ and a tile $t$ in $P$, the set
\begin{equation}\label{disc_top}
U(P,t) \coloneqq \{ T' \in \Omega_\mathrm{punc} \mid P-x(t) \subset T' \}
\end{equation}
is clopen in $\Omega_\mathrm{punc}$, and the set of all such sets forms a basis for the metric topology on $\Omega_\mathrm{punc}$.
\subsection{The substitution graph and discrete hull}
For $p \in \mathcal{P}$ and $t \in \varphi(p)$ we call $(t,p)$ a {\bf supertile extension}, and denote the set of such supertile extensions by $\mathcal{S}$. If at most one copy of each prototile appears in each substituted prototile, then the elements of $\mathcal{S}$ can be identified with all pairs $(a,b) \in \mathcal{P}^2$ for which $a \in \varphi(b)$, but we do not need to assume this in general (see Example \ref{ex: supertile duplicates} below). Given a supertile extension $e = (t,p) \in \mathcal{S}$, we denote $r(e) \coloneqq t$ (considered as a prototile in $\mathcal{P}$) and $s(e) \coloneqq p$. We construct the {\bf substitution graph} $G$ with vertex set $\mathcal{P}$, edge set $\mathcal{S}$ and source and range maps $s$, $r \colon \mathcal{S} \to \mathcal{P}$. The associated set of right-infinite (left-pointing) paths is denoted
\[
\mathcal{F} \coloneqq \{e_0 e_1 e_2 \cdots \mid s(e_i) = r(e_{i+1})\},
\]
and comes equipped with the left shift map $\sigma: \mathcal{F} \to \mathcal{F}$ defined by $\sigma(e_0 e_1 e_2 \cdots)=e_1 e_2 \cdots$. Generally, given a finite graph $G$, the set $\mathcal{F}$ of right-infinite words as above is called a \emph{topological Markov chain}.
\begin{remark} \label{rem: composition order}
One could use the opposite convention to above, taking an arrow $e \colon t \to p$ as a `subtile inclusion' of a tile $t$ into a $p$ supertile. In that case, all arrows on graphs would be reversed and one could take $\mathcal{F}$ as right-infinite, right-pointing paths. The advantage of the convention we take here instead is that a string $e_0 e_1 \cdots e_n$ may be read analogously to function composition (with range on the left, source on the right), and when introducing semigroup elements, which also have domain/codomain or `in/out' tiles, a valid string has consistently matching adjacent tiles in the domain/codomain or source/range, whether the term is a semigroup element or supertile extension term. By this convention arrows point in the direction of substitution application, which is also similar to standard conventions on inverse limits defining the tiling space, as we shall see in Section \ref{sec: limit space}.
\end{remark}
\begin{figure}
\begin{center}
\begin{tikzpicture}
\begin{scope}[xshift=0cm,yshift=2cm]
\draw[|-|,thick] (0,0)-- node[above]{$a$} (1.61,0);
\draw[|-|,thick] (3,0)-- node[above]{$c$} (4.61,0);
\draw[-|,thick] (4.61,0)-- node[above]{$d$} (5.61,0);
\draw[|-|,thick] (0,-0.75)-- node[above]{$b$} (1.61,-0.75);
\draw[|-|,thick] (3,-0.75)-- node[above]{$a$} (4.61,-0.75);
\draw[-|,thick] (4.61,-0.75)-- node[above]{$d$} (5.61,-0.75);
\draw[|-|,thick] (0,-1.5)-- node[above]{$c$} (1.61,-1.5);
\draw[|-|,thick] (3,-1.5)-- node[above]{$a$} (4.61,-1.5);
\draw[-|,thick] (4.61,-1.5)-- node[above]{$d$} (5.61,-1.5);
\draw[|-|,thick] (0,-2.25)-- node[above]{$d$} (1,-2.25);
\draw[|-|,thick] (3,-2.25)-- node[above]{$b$} (4.61,-2.25);
\draw[->] (2,-0)-- (2.59,0);
\draw[->] (2,-.75)-- (2.59,-.75);
\draw[->] (2,-1.5)-- (2.59,-1.5);
\draw[->] (2,-2.25)-- (2.59,-2.25);
\end{scope}
\begin{scope}[xshift=7cm,yshift=0cm]
\node[vertex] (vert_b) at (0,0) {$b$};
\node[vertex] (vert_d) at (2,2) {$d$}
edge [<-,>=latex,out=215,in=55,thick] node[left,pos=0.5]{$(d,b) \ $} (vert_b)
edge [->,>=latex,out=235,in=35,thick] node[right,pos=0.65]{$\ (b,d)$} (vert_b);
\node[vertex] (vert_c) at (6,2) {$c$}
edge [->,>=latex,out=180,in=0,thick] node[above,pos=0.5]{$(d,c)$} (vert_d);
\node[vertex] (vert_a) at (4,0) {$a$}
edge [<-,>=latex,out=35,in=235,thick] node[right,pos=0.5]{$\ (a,c)$} (vert_c)
edge [->,>=latex,out=55,in=215,thick] node[left,pos=0.5]{$(c,a)\ $} (vert_c)
edge [->,>=latex,out=135,in=-45,thick] node[left,pos=0.5]{$(d,a)$} (vert_d)
edge [<-,>=latex,out=180,in=0,thick] node[below,pos=0.5]{$(a,b)$} (vert_b);
\end{scope}
\end{tikzpicture}
\end{center}
\caption{The border forcing Fibonacci substitution and its graph of supertile extensions.}
\label{fig:AP sub}
\end{figure}
\begin{figure}
\begin{center}
\begin{tikzpicture}
\begin{scope}[xshift=0cm,yshift=0cm,decoration={markings,mark=at position 0.52 with {\arrow{>}}}]
\draw[-,thick,postaction={decorate}] (-1,0)-- node[below] {$d$} (1,0);
\draw[-,thick,postaction={decorate}] (1,0) -- node[right] {$b$} (0,3.387);
\draw[-,thick,postaction={decorate}] (0,3.387) -- node[left] {$a$} (-1,0);
\draw[-,thick] (-1,0) arc (180:270:1) node[below] {$c$};
\draw[<-,thick] (0,-1) arc (270:360:1);
\node at (-1,0) {$\bullet$};
\node at (1,0) {$\bullet$};
\node at (0,3.387) {$\bullet$};
\end{scope}
\end{tikzpicture}
\end{center}
\caption{The AP complex for the border forcing Fibonacci tiling \cite{AP}.}
\label{fig:AP complex fib}
\end{figure}
\begin{example}\label{ex: fib}
We illustrate supertile extensions and the substitution graph by studying a border forcing version of the Fibonacci tiling, as defined in \cite[Section 10.1]{AP}. In particular, starting with the usual Fibonacci substitution $0 \mapsto 01$ and $1 \mapsto 0$ we define our tiles as a sliding block code, with $a=0[0]1$, $b=1[0]0$, $c=1[0]1$ and $d=0[1]0$ (the terms in brackets are the tiles being collared, and terms to the left and right denote collaring information). From this we obtain the substitution $\varphi: \mathcal{P} \to \mathcal{P}^*$ given by
\begin{equation}\label{eq: Fib substitutions}
\varphi(a)=cd, \, \varphi(b)=ad, \, \varphi(c)=ad \text{ and } \varphi(d)=b.
\end{equation}
It is routine to check that this substitution is recognisable. Using \eqref{eq: Fib substitutions} we immediately obtain the supertile extensions:
\[
(c,a), (d,a), (a,b), (d,b), (a,c), (d,c) \text{ and } (b,d),
\]
which can unambiguously be labelled by pairs in $\mathcal{P}^2$ since at most one copy of each prototile appears in each supertile. The substitution appears in Figure \ref{fig:AP sub}, along with the graph associated with this substitution. Note that a smaller substitution (on a 3-letter alphabet) could be used to force the border for this example, by only collaring tiles on the left, since every supertile is adjacent to a $0$-tile on the right. \qed
\end{example}
\begin{figure}
\begin{center}
\begin{tikzpicture}
\node (a) at (0,0) {$a$};
\node (b) at (4,0) {$b$};
\path[->,thick,>=latex]
(a) edge [out=30,in=150,"{$(b_2,a)$}"] (b)
(a) edge ["{$(b_3,a)$}"] (b)
(b) edge [out=210,in=-30] node[above] {$(a,b)$} (a)
(a) edge [out=210, in=150, looseness=10, "{$(a,a)$}"] (a)
(b) edge [out=30, in=-30, looseness=10, "{$(b,b)$}"] (b);
\end{tikzpicture}
\end{center}
\caption{The supertile extension graph of Example \ref{ex: supertile duplicates}.}
\label{fig: supertile duplicates}
\end{figure}
\begin{example} \label{ex: supertile duplicates}
Consider the substitution on $\mathcal{P} = \{a,b\}$ defined by
\begin{equation*}
\varphi(a)=abb \text{ and } \varphi(b)=ab.
\end{equation*}
It is easy to see that this substitution is recognisable and forces the border since every supertile is followed by an $a$ tile to its right and is preceded by a $b$ to its left. In this case there two ways tile $b$ is extended into an $a$ supertile: by the $b$ being included as either the second or third letter. These could be distinguished by denoting them as, say, $(b_2,a)$ and $(b_3,a)$. The substitution graph is given in Figure \ref{fig: supertile duplicates}. \qed
\end{example}
We have a natural bijection between a right infinite sequence of supertile extensions
\begin{equation}\label{string}
(t_0,t_1)(t_1,t_2)(t_2,t_3) \cdots \in \mathcal{F}
\end{equation}
and the discrete hull $\Omega_\mathrm{punc}$ of all substitution tilings $T$ generated by $\varphi$ (where here, and later, we allow a very minor abuse of notation by denoting a supertile extension by $(t_n,t_{n+1})$ with each $t_n$ simultaneously denoting a subtile of $\varphi(t_{n+1})$ and also its corresponding prototile in $\mathcal{P}$). Indeed, to such a string \eqref{string} the puncture of tile $t_0$ is placed on the origin and we obtain a sequence of inclusions $t_0 \subseteq \varphi(t_1) \subseteq \varphi^2(t_2) \subseteq \varphi^3(t_3) \subseteq \cdots$. The nested patches $\varphi^n(t_n)$ determine the entire tiling by the border forcing property. Conversely, a punctured tiling determines such a string by recognisability (which itself follows from FLC and aperiodicity).
There is a natural topology on $\mathcal{F}$ whose basis consists of clopen cylinder sets of all infinite strings starting with some given finite initial string. Under this topology, the above bijection induces a homeomorphism
\begin{equation} \label{eq: strings <-> transversal}
\tau \colon \mathcal{F} \xrightarrow{\cong} \Omega_\mathrm{punc}
\end{equation}
to the discrete hull $\Omega_\mathrm{punc}$ with the topology generated by \eqref{disc_top}. The map $\tau$ above constructs an infinite tiling with a puncture on the origin from an infinite string. We may also define $\tau$ on a finite string $s = (t_0,t_1)(t_1,t_2)\cdots (t_{n-1},t_{n})$ to obtain a finite marked patch. This is the patch $\varphi^n(t_n)$, where the location of $t_0$ is positioned in this supertile according to how the supertile extensions embed into each other.
The discrete hull $\Omega_\mathrm{punc}$ is the object space of the {\bf (discrete) translation groupoid}
\[
R_\mathrm{punc} \coloneqq \{ (T-x(t),T) \in \Omega_\mathrm{punc} \times \Omega_\mathrm{punc} \mid t \in T \},
\]
and the product topology coming from $\Omega_\mathrm{punc} \times \mathbb{R}^d$ makes $R_\mathrm{punc}$ a principal topological groupoid. Given a patch $P$ and tiles $t,t'$ in $P$, the sets
\begin{equation}\label{Rpunc_top}
V(t',P,t) \coloneqq \{ (T',T) \in R_\mathrm{punc} \mid P-x(t) \subset T \text{ and } P-x(t') \subset T' \}
\end{equation}
are open in $R_\mathrm{punc}$, and the set of all such sets forms a basis for the product topology. In this topology, $R_\mathrm{punc}$ is an \'etale equivalence relation. See \cite{Kel1} for further details.
The open set $V(t',P,t) \subset R_\mathrm{punc}$ defines a bijection from $U(P,t)$ to $U(P,t')$, which may be thought of as the `partial translation' which shifts the origin tile from $t$ to $t'$, within any tiling of $\Omega_\mathrm{punc}$ with $t \in P$ centred over the origin. For example, moving across a certain face from one prototile to an adjacent one may be interpreted as such an open subset of the discrete groupoid, or as a partial bijection within $\Omega_\mathrm{punc}$ (or, via the identification $\tau$, within the Markov shift $\mathcal{F}$ for a substitution tiling). This collection of partial translations naturally leads us to the tiling semigroup:
\subsection{The tiling semigroup}
We recall Kellendonk's construction of the inverse semigroup associated to an FLC tiling \cite{KL,KelPut}. We use notation similar to \cite{KL} for the elements of this semigroup, the doubly pointed patches.
A semigroup $S$ is an \textbf{inverse semigroup} if for each $s \in S$ there exists a unique element $s^* \in S$ such that $ss^*s=s$ and $s^*ss^*=s^*$. According to the Vagner--Preston Representation Theorem \cite[Theorem V.1.10]{Howie}, every inverse semigroup is isomorphic to a subsemigroup of $\mathcal{I}(X)$, the inverse semigroup of partial bijections on a set $X$. An \emph{action} of an inverse semigroup $S$ on a set $X$ is a homomorphism $\pi:S \to \mathcal{I}(X)$. If the homomorphism $\pi$ is fixed, we usually write $g \cdot x$ for $\pi_g(x)$. See \cite{Exel} for further details.
\begin{definition}
A {\bf doubly pointed patch} $[b,P,a]$ is given by a finite patch $P$ and tiles $a,b \in P$, where we take the tuple $(b,P,a)$ up to translation equivalence. Let $\mathcal{T}$ be the set of all doubly pointed patches along with a `zero element' $0 \in \mathcal{T}$.
Let $[d,Q,c]$, $[b,P,a] \in \mathcal{T}$ be two doubly pointed patches which, without loss of generality (by translating each, if necessary) have $x(b) = x(c)$. If $P$ and $Q$ agree on any tiles with intersecting interiors and $P \cup Q$ is a valid patch, then we define
\[
[d,Q,c] [b,P,a] = [d,P \cup Q,a].
\]
Otherwise, we define $[d,Q,c] [b,P,a] = 0$. Any product with $0$ is defined as $0$. We call $\mathcal{T} = (\mathcal{T},\cdot)$ the {\bf tiling semigroup}.
\end{definition}
\begin{remark}
One could also define the tiling semigroup for a space $\Omega$ of tilings, considering patches over all tilings in $\Omega$. For $\Omega = \Omega_T$ with $T$ a repetitive tiling, all tilings of $\Omega$ have the same finite patches, up to translation, so this does not affect the construction. However, if we consider $\Omega_\varphi$ with $\varphi$ non-primitive then the sets of finite patches can differ between orbits. We take the full collection of patches over all of $\Omega_\varphi$ in such a case. We note, typically we are most interested in the case where $\varphi$ is primitive and $\Omega_\varphi = \Omega_T$ for any $T \in \Omega_\varphi$, with each such being repetitive.
\end{remark}
Notice that, again, our notation here mirrors function composition: the element $[b,P,a]$ has `in tile' $a$ and `out tile' $b$, with a product of elements $[d,Q,c][b,P,a]$ interpreted as applying the right then the left-hand term, and requiring that the intermediate tiles $b$ and $c$ agree. There is a bijective correspondence between elements of $\mathcal{T}$ and basis elements of the \'etale topology for $R_\mathrm{punc}$ as described in \eqref{Rpunc_top} via $[b,P,a] \mapsto V(b,P,a)$ (and where $0$ has empty (co)domain). The product of semigroup elements is identified with the composition of partial bijections on the largest compatible domain. Thus, the tiling semigroup $\mathcal{T}$ naturally acts by partial bijections on the discrete hull $\Omega_\mathrm{punc}$, where a doubly pointed patch $g = [b,P,a]$ has domain $U(P,a) \subset \Omega_\mathrm{punc}$ and codomain $U(P,b) \subset \Omega_\mathrm{punc}$.
It is easy to establish that $\mathcal{T}$ is an inverse semigroup. In particular, note that for $s = [b,P,a] \in \mathcal{T}$, the unique $t \in \mathcal{T}$ with $s = sts$ and $t = tst$ is given by $t = [a,P,b]$. We interpret $[b,P,a]$ as a \emph{translation from $a$ to $b$, within $P$}, where the product of two such translations is allowed when the union is itself a valid patch. The idempotents (or `partial identities') are of the form $[a,P,a]$, along with the $0$ element. Since $\mathcal{T}$ is an inverse semigroup, it naturally inherits a partial ordering: we have that $[b,P,a] \preceq [d,Q,c]$ if and only if, up to translation, we have an inclusion $(d,Q,c) \subseteq (b,P,a)$ of doubly pointed patches; that is, $P$ extends $Q$ as a doubly pointed patch with $a=c$ and $b=d$. It is not hard to see that $\mathcal{T}$ is generated by idempotents $[p,\{p\},p]$ for $p \in \mathcal{P}$, where $\{p\}$ is a single-tile patch, and elements $[b,P,a]$, where $P$ is a connected two-tile patch containing distinct $a$ and $b$.
\section{Self-similarity of substitution tiling semigroups} \label{sec: Self-similarity of substitution tiling semigroups}
We now show that the above semigroup action of $\mathcal{T}$ on $\mathcal{F} \cong \Omega_\mathrm{punc}$ is self-similar.
\begin{definition}[{\cite[Definition 3.6]{BGN}}] \label{def: self-similar}
Let $\mathcal{F}$ be a topological Markov chain over an alphabet $X$. An inverse semigroup $G$ acting on $\mathcal{F}$ is called {\bf self-similar} if for every $g \in G$ and $x \in X$ there exist $y_1$, \ldots, $y_k \in X$ and $h_1$, \ldots, $h_k \in G$ such that the sets $\mathrm{dom}(h_i)$ are disjoint, $\bigcup_{i=1}^k x \mathrm{dom}(h_i) = x\mathcal{F} \cap \mathrm{dom}(g)$, and for every $xw \in \mathcal{F}$ we have
\begin{equation} \label{eq: self-sim}
g \cdot xw = y_i (h_i \cdot w),
\end{equation}
where $i$ is such that $w \in \mathrm{dom}(h_i)$.
\end{definition}
Often in the context of self-similar groups and semigroups the action of $g$ on $w$ is denoted by $w^g$, but here we choose to use $g \cdot w$. Note that the $h_i$ in \eqref{eq: self-sim} are not uniquely defined. Indeed, given a partial bijection $h_i$ one could partition its domain and use instead the restrictions of $h_i$ to each such subset. From the opposite perspective, one may always replace the expression $g \cdot w$ with $h \cdot w$ whenever $h$ is an extension of $g$ to a larger domain. This will be useful later in simplifying the semigroup elements $h_i$ generated by successive application of \eqref{eq: self-sim}.
\begin{remark}
We briefly explain here an equivalent description of self-similarity to highlight its algorithmic quality (and note that one may equivalently define self-similarity of inverse semigroup actions by automata, see \cite{Nek_SSIS}). We make the standing assumption throughout that all semigroup elements have clopen domains. For each $g \in G$, there is some $N(g) = N \in \mathbb{N}$, given by the distance required to `read forwards' in the sequence to evaluate the first letter of $g \cdot w$, for an infinite word $w \in \mathcal{F}$, as well as determining the necessary semigroup element to apply to the remainder of the string. Let $\mathcal{F}^N$ denote the set of words of length $N$. Self-similarity means that there exist letters $y_1$, \ldots, $y_\ell \in X$, elements $h_1$, \ldots, $h_\ell \in G$ and subsets $S_i \subseteq \mathcal{F}^N$ so that:
\begin{enumerate}
\item $\mathrm{dom}(g) = \bigcup_{i=1}^\ell S_i \mathcal{F}$, where $S_i \mathcal{F}$ denotes the set of infinite words with initial $N$-letter string in $S_i$;
\item $S_i \cap S_j = \emptyset$ for $i \neq j$ (so the above is a disjoint union);
\item for each $i = 1$, \ldots, $\ell$ we have $S_i \mathcal{F} = x_i\mathrm{dom}(h_i)$ for some $x_i \in X$;
\end{enumerate}
Then given an infinite word $w \in \mathcal{F}$, to evaluate $g \cdot w$ we first determine the initial $N$-letter string $s$ of $w$. We have that $s \in S_i$ for a unique $i$, and then we have the rule:
\[
g \cdot w = y_i (h_i \cdot \sigma(w)),
\]
where $\sigma \colon \mathcal{F} \to \mathcal{F}$ is the left shift (i.e., the map removing the initial letter $x_i$ from $w$). Thus, the first letter of $g \cdot w$ is $y_i$. To apply $h_i$ to the remainder, we look forward distance $N(h_i)$ in $\sigma(w)$ to determine the second letter of $g \cdot w$, as well as the next element of $G$ to apply to $\sigma^2(w)$. This may be repeated indefinitely. This is best demonstrated here through Example \ref{geometric intuition}, which may help the reader with the following proof.
\end{remark}
\begin{theorem}
The tiling semigroup $\mathcal{T}$ of a recognisable substitution tiling is self-similar.
\end{theorem}
\begin{proof}
Let $g = [b,P,a]$ be a doubly pointed patch and recall that the domain of $g$ is $U(P,a)$ corresponding to tilings with patch $P$ at the origin centred at the puncture $x(a)$ of tile $a$. Since $\varphi$ forces the border there is some $N \in \mathbb{N}$ so that any $w \in \mathcal{F}$, with $\tau(w) \in U(P,a)$, has initial string $s \in \mathcal{F}^N$ with $P \subseteq \tau(s)$. In particular, there is a finite set $S = \{s_1,\ldots,s_k\} \subseteq \mathcal{F}^N$ of all possible length-$N$ strings for which $P \subseteq \tau(s_i)$.
Fix $s_i \in \mathcal{F}^N$ and define $h_i \in \mathcal{T}$ by
\[
h_i = [b',\tau(s_i'),a'],
\]
where $s'_i \in \mathcal{F}^{N-1}$ is given by removing the first term of $s_i$, and $a'$ and $b'$ are the tiles of $\tau(s_i')$ whose substitutes contain the tiles $a$, $b \in P$. Let $y_i \in \mathcal{S}$ denote the unique supertile extension that includes $b \in P \subset \tau(s_i)$ into its 1-supertile $b' \in \tau(s_i')$.
We may now check that the above assignments fulfil the definition of self-similarity. Take any $w \in \mathrm{dom}(g)$. Then $w$ has initial $N$-letter string $s_i \in S$. We must check that
\begin{equation}\label{tiling_SS}
g \cdot w = y_i( h_i \cdot \sigma(w)).
\end{equation}
By definition, $\tau(w)$ is a tiling $T \in U(P,a)$ and $\tau(g\cdot w)=T-x(b)$ so that $T-x(b) \in U(P,b)$. Let $T' = \tau(\sigma(w))$ which is the 1-supertiling of $T$ with the puncture of tile $a'$ at the origin. Then our definition of $h_i$ implies that $\tau(h_i \cdot \sigma(w))$ is the 1-supertiling with the puncture of tile $b'$ at the origin. Pre-appending $y_i$ corresponds to substituting this tiling, translated appropriately to position the puncture of $b$ over the origin. Thus \eqref{tiling_SS} holds, as required.
\end{proof}
\begin{example}\label{geometric intuition}
We consider the Fibonacci substitution from Example \ref{ex: fib}. We denote by $P_{xy}$ the two-tile patch consisting of $x$, $y \in \mathcal{P}$, with $x$ on the left and $y$ on the right. Later, in Example \ref{ex: fib rules}, we give a complete set of rules on applying doubly-pointed two-tile patches to strings. To be applied, such elements need to look at either the next term, or the next two terms. However, we quickly give an example application to give the flavour of the definitions above, where one sees the group element working algorithmically through the string:
\begin{figure}
\begin{center}
\begin{tikzpicture}
\begin{scope}[xshift=0cm,yshift=0cm]
\draw[|-|,dashed] (-5.22,0) -- node[below,pos=0.5]{$a$} (-3.61,0);
\draw[|-|,dashed] (-3.61,0) -- node[below,pos=0.5]{$d$} (-2.61,0);
\draw[|-|] (-2.61,0) -- node[below,pos=0.5]{$c$} (-1,0);
\draw[|-|] (-1,0) -- node[below,pos=0.5]{$d$} (0,0);
\draw[|-|] (0,0) -- node[below,pos=0.5]{$b$} (1.61,0);
\draw[|-|,dashed] (1.61,0) -- node[below,pos=0.5]{$a$} (3.22,0);
\draw[|-|,dashed] (3.22,0) -- node[below,pos=0.5]{$d$} (4.22,0);
\draw[|-|,dashed] (4.22,0) -- node[below,pos=0.5]{$b$} (5.83,0);
\draw[|-|,dashed] (5.83,0) -- node[below,pos=0.5]{$a$} (7.44,0);
\draw[|-|,dashed] (7.44,0) -- node[below,pos=0.5]{$d$} (8.44,0);
\draw[|-|,dashed] (-5.22,2) -- node[below,pos=0.5]{$b$} (-2.61,2);
\draw[|-|] (-2.61,2) -- node[below,pos=0.5]{$a$} (0,2);
\draw[|-|] (0,2) -- node[below,pos=0.6]{$d$} (1.61,2);
\draw[|-|,dashed] (1.61,2) -- node[below,pos=0.6]{$c$} (4.22,2);
\draw[|-|,dashed] (4.22,2) -- node[below,pos=0.5]{$d$} (5.83,2);
\draw[|-|,dashed] (5.83,2) -- node[below,pos=0.5]{$b$} (8.44,2);
\draw[|-|,dashed] (-5.22,4) -- node[below,pos=0.5]{$d$} (-2.61,4);
\draw[|-|] (-2.61,4) -- node[below,pos=0.45]{$b$} (1.61,4);
\draw[|-|,dashed] (1.61,4) -- node[below,pos=0.6]{$a$} (5.83,4);
\draw[|-|,dashed] (5.83,4) -- node[below,pos=0.6]{$d$} (8.44,4);
\draw[|-|] (-2.61,6) -- node[below,pos=0.45]{$d$} (1.61,6);
\node[vertex] (vert_l) at (0.8,0) {};
\node[vertex] (vert_r) at (2.41,0) {}
edge [<-,>=latex,out=135,in=45,thick] node[above,pos=0.5]{$[a,P_{ba},b]$} (vert_l);
\node[vertex] (vert_ul) at (0.8,2) {}
edge [->,>=latex,out=270,in=90,thick] node[left,pos=0.5]{$(b,d)$} (vert_l);
\node[vertex] (vert_ur) at (3,2) {}
edge [<-,>=latex,out=135,in=45,thick] node[above,pos=0.5]{$[c,P_{dc},d]$} (vert_ul)
edge [->,>=latex,out=270,in=90,thick] node[right,pos=0.5]{$(a,c)$} (vert_r);
\node[vertex] (vert_uul) at (-0.4,4) {}
edge [->,>=latex,out=270,in=90,thick] node[left,pos=0.5]{$(d,b)$} (vert_ul);
\node[vertex] (vert_uur) at (3.8,4) {}
edge [<-,>=latex,out=135,in=45,thick] node[above,pos=0.5]{$[a,P_{ba},b]$} (vert_uul)
edge [->,>=latex,out=270,in=90,thick] node[right,pos=0.5]{$(c,a)$} (vert_ur);
\node[vertex] (vert_uuul) at (-0.4,6) {}
edge [->,>=latex,out=270,in=90,thick] node[left,pos=0.5]{$(b,d)$} (vert_uul);
\node at (0,0) {$\ddag$};
\node at (1.61,0) {$\ddag$};
\node at (4.22,0) {$\ddag$};
\node at (5.83,0) {$\ddag$};
\node at (8.44,0) {$\ddag$};
\node at (-5.22,0) {$\ddag$};
\node at (-2.61,0) {$\ddag$};
\node at (1.61,2) {$\ddag$};
\node at (-2.61,2) {$\ddag$};
\node at (-5.22,2) {$\ddag$};
\node at (5.83,2) {$\ddag$};
\node at (1.61,4) {$\ddag$};
\node at (8.44,4) {$\ddag$};
\node at (-2.61,4) {$\ddag$};
\node at (1.61,6) {$\ddag$};
\end{scope}
\end{tikzpicture}
\end{center}
\caption{A graphical representation of the patch formed by the prefix $(b,d)(d,b)(b,d)$ of the word $w$. Tile lengths are increased by the golden ratio at each increasing level. The vertical arrows on the left denote the tile inclusions $(b,d)(d,b)(b,d)$. The bottom horizontal arrow depicts the action of moving one tile to the right by a semigroup element and a non-idempotent element of $\mathcal{T}$ acts on the remainder of the string if we translate across a supertile boundary, denoted by double daggers. Thus, the vertical arrows on the right represent the output of applying the semigroup element $[a,P_{ba},b]$ to $(b,d)(d,b)(b,d)$ as shown in \eqref{fib_comp_2}. Note that the solid tiles are given directly from the tile inclusions and the dashed tiles are defined implicitly by the border forcing property.}
\label{fig:Fib_tiling_example}
\end{figure}
Take the infinite word
\[
w = (b,d)(d,b)(b,d) e f z \in \mathcal{F}
\]
where $(b,d)$, $(d,b)$ and $(b,d)$ are specified supertile extensions in $\mathcal{S}$, $e$ and $f$ are in $\mathcal{S}$ and $z \in \mathcal{F}$ is an infinite tail. We apply the element $[a,P_{ba},b]$ to $w$ according to the rules found in Example \ref{ex: fib}, which corresponds to moving one tile to the right in the tiling formed by $w$. It is important to note that all these relations can be deduced from the information given, as depicted in Figure \ref{fig:Fib_tiling_example}.
\begin{align}
\notag
[a,P_{ba},b] \cdot (b,d)(d,b)(b,d)e f z & =(a,c) \left( [c,P_{dc},d] \cdot (d,b) (b,d) efz \right) \\
\label{fib_comp_2}
&=(a,c)(c,a) \left( [a,P_{ba},b] \cdot (b,d)e f z \right)
\end{align}
At this stage, we cannot evaluate $[a,P_{ba},b]$ without knowing $e$. If, for example, $e=(d,a)$ and $f=(a,b)$ then one finds that \eqref{fib_comp_2} evaluates as
\begin{align*}
(a,c)(c,a) \left( [a,P_{ba},b] \cdot (b,d)(d,a)(a,b) z \right) &=(a,c)(c,a)(a,b) \left( [b,P_{db},d] \cdot (d,a)(a,b) z \right) \\
&=(a,c)(c,a)(a,b)(b,d) \left( [d,P_{ad},a] \cdot (a,b) z \right) \\
&=(a,c)(c,a)(a,b)(b,d)(d,b) \left( [b,P_{b},b] \cdot z \right) \\
&=(a,c)(c,a)(a,b)(b,d)(d,b)z \\
\end{align*}
where the last line is fully evaluated, since the idempotent $[b,P_b,b]$ acts as the identity. If instead we took $e=(d,b)$ then one finds that \eqref{fib_comp_2} evaluates as
\begin{align*}
(a,c)(c,a) \left( [a,P_{ba},b] \cdot (b,d)(d,b)f z \right) &=(a,c)(c,a)(a,c) \left( [c,P_{dc},d] \cdot (d,b)f z \right) \\
&=(a,c)(c,a)(a,c)(c,a) \left( [a,P_{ba},b] \cdot f z \right)
\end{align*}
and we cannot evaluate further without knowing both $f$ and the first letter of $z$.
If we translate left one tile instead, by applying $[d,P_{db},b]$ to $w$, then we find:
\begin{align*}
[d,P_{db},b] \cdot (b,d) (d,b)(b,d) e f z &=(d,a) \left( [a,P_{ad},d] \cdot (d,b) (b,d)e f z\right)\\
&= (d,a) (a,b) \left( [b,P_b,b] \cdot (b,d) e f z \right) \\
&= (d,a) (a,b)(b,d) e f z
\end{align*}
and the application is fully evaluated, again because $[b,P_b,b]$ is an idempotent. We encourage the reader to consider translating to the left in Figure \ref{fig:Fib_tiling_example} to work out similar equations.
Notice the necessary consistency in the above strings: there is agreement between adjacent tile types of both the supertile extension pairs $(y,x)$ and the `in/out' tiles of the doubly pointed patches $[y,P,x]$. As in Remark \ref{rem: composition order}, this follows from the convention of orientation in the substitution graph and writing strings in an order corresponding to function composition.
\qed
\end{example}
One should observe that, in terms of the action of $\mathcal{T}$ on $\mathcal{F}$, there is some degree of superfluous information in the semigroup elements: for $g \in \mathcal{T}$ and $T \in \mathrm{dom}(g)$, all that is required to evaluate $g(T)$ is the relative displacement of the `in and out' tiles. This fact is also reflected algebraically in terms of the inverse semi-group: For a general inverse semigroup one has the partial ordering defined by letting $x \preceq y$ if there is an idempotent $e$ for which $x = ey$. For the tiling semigroup $\mathcal{T}$, this says that $[b,P,a] \preceq [d,Q,c]$ if and only if, up to translation, we have an inclusion of doubly pointed patches $(d,Q,c) \subseteq (b,P,a)$ (i.e., $P$ extends $Q$ with $a=c$ and $b=d$). If $x \preceq y$ then $\mathrm{dom}(x) \subseteq \mathrm{dom}(y)$ and $x \cdot \tau^{-1}(T) = y \cdot \tau^{-1}(T)$ for all $T \in U(P,a)$.
A consequence of the above is that there is significant choice of semigroup elements satisfying the self-similarity rule \eqref{eq: self-sim}. This is easily dealt with in practice since, if we ignore the domain and range of elements, we may always replace a term such as $h_i \cdot w$ with $j \cdot w$ for any $j \succeq h_i$ in \eqref{eq: self-sim}. In fact, after enough iterations, we see that we may take $j$ to be a `small' patch. This is made precise via the following more general definition.
\begin{definition} \label{def: contracting}
Let $(G,\mathcal{F})$ be a self-similar inverse semigroup. We call $G$ {\bf contracting} if there exists some finite $N \subseteq G$ satisfying the following: For any $g \in G$ there exists some $k \in \mathbb{N}$ for which, for any $uw \in dom(g)$ with $u \in \mathcal{F}^k$, there exists some $v \in \mathcal{F}^k$ and $h \in N$ with
\begin{equation} \label{eq: contracting}
g \cdot (uw) = v (h \cdot w).
\end{equation}
\end{definition}
\begin{remark}
\label{rem: alt contracting}
We can write the above definition in the following alternative way. There exists some finite $N \subseteq G$ satisfying the following: For any $g \in G$ there exists some $k \in \mathbb{N}$ for which, for every $w \in \mathrm{dom}(g)$, there is some $v \in \mathcal{F}^k$ and $h \in N$ with
\begin{equation} \label{eq: alt contracting}
g \cdot w = v (h \cdot (\sigma^k w)).
\end{equation}
\end{remark}
In the standard language of self-similar group actions, the above says that after sufficiently many applications of the `restriction' of $g$ (the elements $h_i$ of \eqref{eq: self-sim}) the new semigroup element to apply to the remainder of the string may be taken in the finite set $N$, at least after an appropriate adjustment of its domain and range.
\begin{definition}
Let $(G,\mathcal{F})$ be a contracting self-similar inverse semigroup. Call $\mathcal{N}$ a \emph{semi-nucleus} if it satisfies the contracting condition above and is such that for all $g \in \mathcal{N}$ and $ew \in \mathrm{dom}(g)$ with $e \in \mathcal{S} = \mathcal{F}^1$, there is some $f \in \mathcal{S}$ and $h \in \mathcal{N}$ so that
\begin{equation} \label{eq: semi-nucleus}
g \cdot (ew) = f (h \cdot w).
\end{equation}
\end{definition}
The above says that not only do all semigroup elements eventually `restrict' to elements in $\mathcal{N}$ (possibly after extended the domain and range), we also have that a single iteration of restriction of an element of $\mathcal{N}$ can be chosen to remain in $\mathcal{N}$. That is, we may take $k=1$ for any $g \in N$ in Definition \ref{def: contracting}.
\begin{lemma} \label{lem: semi-nucleus}
A contractive self-similar inverse semigroup has a semi-nucleus.
\end{lemma}
\begin{proof}
Let $N \subseteq G$ be as required for the definition of contractivity and $g \in N$, satisfying \eqref{eq: alt contracting} for $k = k(g) \in \mathbb{N}$. Let $u = u_1 \cdots u_k \in \mathcal{F}^k$ and $w \in \mathcal{F}$ with $uw \in \mathrm{dom}(g)$.
Consider the elements $h_1^1$, $h_2^1$, \ldots, $h_n^1 \in G$ arising from \eqref{eq: self-sim} for $g$ and $x = u_1$. For each $h_i^1$, apply \eqref{eq: self-sim} again with $x = u_2$ to obtain elements $h_1^2$, $h_2^2$, \ldots, $h_m^2$. We continue this procedure to generate elements $h_i^j$ for $j = 1$, \ldots, $k$.
Iteratively applying self-similarity we have
\[
g \cdot (u_1 u_2 \cdots u_k w) = \cdots = y_1 \cdots y_{k-1} (h_\ell^{k-1} \cdot (u_k \cdot w)) = y_1 \cdots y_k (h_j^k \cdot w) = v(h \cdot w),
\]
for $v = y_1 \cdots y_k \in \mathcal{F}^k$ and $h \in N$, by \eqref{eq: contracting}. This shows, at least in the above expression, that we may replace $h_j^k$ with $h \in N$. In fact, again by repeated application of self-similarity and with more careful consideration of the domains, we have
\[
\mathrm{dom} g \cap u \mathcal{F} = \bigsqcup_\ell u (\mathrm{dom} h_\ell^k),
\]
where the above is a disjoint union and $g \cdot (uw) = v (h_\ell^k \cdot w)$ for all $w \in \mathrm{dom} h_\ell^k$. Thus, for all $w \in \mathrm{dom} h_\ell^k$, we have $h_\ell^k \cdot w = h \cdot w$ for some $h \in N$.
Thus, let $\mathcal{N}$ be the union of $N$ and all elements $h_i^j$, for $j < k(g)$, generated by $k(g)-1$ applications of the self-similar rule for each $g \in N$. Then \eqref{eq: semi-nucleus} holds for $g \in N$ using some $h = h_i^1 \in \mathcal{N}$, by construction. Similarly, for $j < k(g)-1$, each $h_i^j \in \mathcal{N}$ satisfies \eqref{eq: semi-nucleus} with $h = h_\ell^{j+1} \in \mathcal{N}$ for some $\ell$. Finally, for $j = k(g)-1$, \eqref{eq: semi-nucleus} is satisfied for each $h_\ell^{k(g)-1} \in \mathcal{N}$ using some $h \in N \subseteq \mathcal{N}$, as above, so that $\mathcal{N}$ is a semi-nucleus.
\end{proof}
\begin{remark} \label{rem: disconnected patches}
For the tiling semigroup, we could define a larger inverse semigroup which does not demand that patches are connected. Then for every (non-zero) $g \in \mathcal{T}$ we have $g \preceq z$ for some $z = [b,P,a]$ with $P$ a two-tile patch containing $a$ and $b$ (or a $1$-tile patch, if $a=b$). We will sometimes make temporary use of such partial translations not in $\mathcal{T}$, such as in the proof below.
\end{remark}
\begin{proposition} \label{prop: tiling semigroup contractive}
Let $(\mathcal{T},\mathcal{F})$ be the self-similar tiling semigroup of an aperiodic substitution tiling. Then for each $g \in \mathcal{T}$ there is some $k = k(g) \in \mathbb{N}$ so that, for all $uw \in \mathrm{dom}(g)$ with $u \in \mathcal{F}^k$, we have that
\[
g \cdot (uw) = v (h \cdot w)
\]
for some $v \in \mathcal{F}^k$ and $h = [b,P,a]$ for which $a=b$ or $a$ and $b$ are adjacent. Hence, $\mathcal{T}$ defines a contractive action on $\mathcal{F}$. We may choose $\mathcal{N}$ to be a semi-nucleus consisting of all doubly-pointed {\bf star patches} $[b,P,a]$, that is, with $P$ a patch of tiles all sharing a common point.
\end{proposition}
\begin{proof}
Without loss of generality, we can take $g=[b,P,a]$ where $P$ is a two-tile patch containing only tiles $a$ and $b$ (but with $P$ possibly disconnected, see Remark \ref{rem: disconnected patches}). Let $g=h_0$ and consider any element $w \in \mathrm{dom}(g)$ along with a sequence $\{h_i\} \subset \mathcal{T}$ arising from recursively applying the self-similar relation \eqref{eq: self-sim} to $g \cdot w$. We may choose each $h_i=[b_i,P_i,a_i]$ using a one or (possibly disconnected) two-tile patch.
Consider the sequence of tile pairs $\{(a_i,b_i)\}$ coming from the doubly pointed patches $h_i=[b_i,P_i,a_i]$. Let $r_i \coloneqq \inf\{|x-y| \mid x \in \mathrm{supp}(a_i) \text{ and } y \in \mathrm{supp}(b_i)\}$ be the distance between tiles $a_i$ and $b_i$; that is, the infimum of distances between points of $a_i$ and $b_i$. It follows from the proof of Theorem \ref{tiling_SS} that the supertile extensions (of $a_{i-1}$ into $a_i$ and $b_{i-1}$ into $b_i$) arising from the self-similar relation \eqref{eq: self-sim} geometrically embed the two-tile patch $(a_{i-1},b_{i-1})$ as a subpatch of the substitution of $(a_i,b_i)$. It follows immediately that $r_i \leq \lambda^{-1} r_{i-1}$ for all $i \in \mathbb{N}$ since pairs of supertile extensions within a patch uniformly scale tiles by $\lambda^{-1}$ between the range and source, and the support of the source covers the support of the range.
We claim the sequence $(r_i)$ is eventually zero. If not, then this would provide an infinite sequence of two-tile patches with arbitrarily small distance apart. But it follows from FLC that $r_i$ can only take finitely many values less than $r_0$, so this cannot happen. Hence, we have that $r_i = 0$ for $i > k$ where, by FLC, $k$ can be taken to only depend on $(a,b)$ and thus on $g$. Since $r_i = 0$ if and only if $a_i$ and $b_i$ are equal or adjacent, it follows that $h_i$ may be given by a doubly pointed one or two-tile patch of intersecting tiles for $i > k$.
Let $\mathcal{N}$ be the set of doubly pointed star-patches. By the above, for each $g \in \mathcal{T}$ there is some $k \in \mathbb{N}$ satisfying \eqref{eq: contracting} with $h \in \mathcal{N}$. Since the source and range tiles of the $h_i$ intersect also for $i > k$, further restrictions may be taken in $\mathcal{N}$, which is thus a semi-nucleus for $\mathcal{T}$.
\end{proof}
\begin{remark}
For self-similar groups one defines \emph{the} nucleus of a contracting group $G$ to be the minimal $\mathcal{N}$ so that all elements eventually restrict to $\mathcal{N}$. In \cite{Nek_SSIS}, a notion of contractivity and (minimal, uniquely defined) nucleus is given in the case of self-similar semigroups. However, in this setup self-similar semigroups are treated via automatons which are $\omega$-deterministic, which amounts to declaring fixed restrictions of partial bijections in \eqref{eq: self-sim} as part of the structure. In our setup we have allowed this to remain flexible, which is an alternative approach which we feel may be of further interest. Indeed, it was beneficial in the proof above that it was not necessary to manage the shapes of patches under restriction down to the semi-nucleus. It is also clear that, in this setting, it may be impossible and unnatural to have a unique and minimal semi-nucleus $\mathcal{N}$. For example, for a cellular $2$-dimensional tiling, if we define connected patches via meeting tiles merely being adjacent, then we only need $1$ and $2$-tile doubly pointed patches in $\mathcal{N}$. If we instead define tiles to be meeting when they meet over a shared edge, then star patches $[b,P,a]$ with $a$ and $b$ meeting at a shared vertex (but not over an edge) can be removed, and replaced with star patches $[b,P',a]$, with $P' \subset P$ connected, for which there is some degree of arbitrary choice.
\end{remark}
In the case of a $d$-dimensional cellular tiling, it is not hard to see that $\mathcal{T}$ is generated by idempotents (which may be identified with $\mathcal{P}$) together with $\mathcal{P}_2$, defined as the finite set of elements $[b,P,a]$ for $a \neq b$ and $P$ a two-tile patch consisting of $a$ and $b$ meeting over a particular shared $(d-1)$-dimensional face. Idempotents restrict to idempotents, and elements of $\mathcal{P}_2$ restrict to elements of $\mathcal{P} \cup \mathcal{P}_2$ (after possibly extending domains). So the action of $\mathcal{T}$ on $\mathcal{F}$ may be completely described by the action of the finite set $\mathcal{P}_2$ on strings of sufficiently large length, together with how they restrict to elements of $\mathcal{P} \cup \mathcal{P}_2$. However, the semi-nucleus still requires more elements for tilings of dimension greater than one, since the restriction of a `diagonally adjacent doubly pointed patch' can remain as such after arbitrarily many restrictions.
\section{The limit space}\label{sec: limit space}
Let $G$ be a finite graph with associated topological Markov shift $\mathcal{F}$ (the right-infinite, left-pointing paths). We define
\[
\mathcal{F}^- \coloneqq \{\cdots e_{-3}e_{-2}e_{-1} \mid r(e_i) = s(e_{i-1})\};
\]
that is, the space of left-infinite, left-pointing paths, which is equipped with the product topology. The following is a natural adaptation of the asymptotic equivalence relation from the case of self-similar groups to semigroups:
\begin{definition}\label{def:ae}
Two elements $x = \cdots e_{-3}e_{-2}e_{-1}$ and $y = \cdots f_{-3}f_{-2}f_{-1} \in \mathcal{F}^-$ are called {\bf asymptotically equivalent} with respect to the action of the semigroup $G$ if there is a sequence $(g_n)$ of $G$, with $\{g_n\} \subseteq G$ finite, and some $w \in \mathcal{F}$ so that for each $n \in \mathbb{N}$ the element
\begin{equation} \label{eq: ae relation}
g_n \cdot (e_{-n} \cdots e_{-3}e_{-2}e_{-1}w) \in \mathcal{F}
\end{equation}
has initial string of $n$ terms given by $f_{-n} \cdots f_{-3}f_{-2}f_{-1} \in \mathcal{F}^n$. In this case we write $x \sim_\mathrm{ae} y$. We define the {\bf asymptotic equivalence relation} $\sim$ on $\mathcal{F}^{-}$ to be the equivalence relation generated by $\sim_\mathrm{ae}$.
\end{definition}
The main difference between the above definition and the case of self-similar groups is that we need to append the infinite word $w$ to the right of the finite string $e_{-n} \cdots e_{-1}$ so that $g_n$ may be unambiguously applied to it. However, by self-similarity, it is in fact only necessary to append a finite string of sufficiently large length.
In the lemma below, and henceforth, we will always assume that for each $x \in \mathcal{F}$ there is some $g_x \in G$ with $x \in \mathrm{dom}(g_x)$ and $\mathrm{dom}(g_x)$ open.
\begin{lemma} \label{lem: aeq ref sym}
Let $G$ be a self-similar inverse semigroup acting on the Markov chain $\mathcal{F}$. Then $\sim_\mathrm{ae}$ is reflexive. Suppose that $G$ is contractive and $e \sim_\mathrm{ae} f$. Then we may take each $g_n \in \mathcal{N}$ in \eqref{eq: ae relation} for $\mathcal{N}$ some semi-nucleus, and there exists $w \in \mathcal{F}$ and $h \in \mathcal{N}$, not depending on $n$, such that
\begin{equation} \label{eq: aeq alt}
g_n \cdot (e_{-n} \cdots e_{-3}e_{-2}e_{-1}w) = f_{-n} \cdots f_{-3}f_{-2}f_{-1} (h \cdot w)
\end{equation}
In particular, $\sim_\mathrm{ae}$ is symmetric.
\end{lemma}
\begin{proof}
By compactness, one can choose a finite number of $x \in \mathcal{F}$ with the union of $\mathrm{dom}(g_x)$ covering $\mathcal{F}$. We have idempotents $h_i = g_x^{-1} g_x$, $i=1$, \ldots $k$, which still have domains $\mathrm{dom}(h_i) = \mathrm{dom}(g_x)$ covering $\mathcal{F}$. Given $e = \cdots e_{-2} e_{-1} \in \mathcal{F}^-$ take any $w \in \mathrm{ran}({e_{-1}})$. Then we may take each $g_n$ to be some $h_i$, with $e_{-n} \cdots e_{-1} w \in \mathrm{dom}(h_i)$. Since each $h_i$ is an idempotent, we have that $g_n (e_{-n} \cdots e_{-1} w) = e_{-n} \cdots e_{-1} w$, so $e \sim_\mathrm{ae} e$.
Suppose now that $G$ is contractive. By Lemma \ref{lem: semi-nucleus} we may choose a semi-nucleus $\mathcal{N}$ for $G$. Given $g_n$, we have some $k(g_n) \in \mathbb{N}$ as required from Definition \ref{def: contracting}. By finiteness of $\{g_n\}$, we may take $K = \max_n k(g_n) < \infty$. Then
\[
g_{n+K} \cdot (e_{-(n+K)} \cdots \cdots e_{-1} w) = f_{-(n+K)} \cdots f_{-(n+1)} h \cdot (e_{-n} \cdots e_{-1} w) = f_{-(n+K)} \cdots f_{-1} w',
\]
for some $w' \in \mathcal{F}$ and $h \in \mathcal{N}$, so we may suppose without loss of generality that $g_n = h \in \mathcal{N}$. Since this applies for each $n \in \mathbb{N}$, and $\mathcal{N}$ is finite, we see that we may take $\{g_n\}$ as a sequence in $\mathcal{N}$.
By repeated application of \eqref{eq: semi-nucleus}, for each $n \in \mathbb{N}$ we may write
\begin{equation} \label{eq: aeq sym}
g_n \cdot (e_{-n} \cdots e_{-3}e_{-2}e_{-1}w) = f_{-n} \cdots f_{-3}f_{-2} f_{-1} (h \cdot w),
\end{equation}
where $h \in \mathcal{N}$. By finiteness of $\mathcal{N}$, some $h \in \mathcal{N}$ as above occurs for infinitely many $n$. For each $n$ in this subsequence, we similarly have
\[
g_n \cdot (e_{-n} \cdots e_{-3}e_{-2}e_{-1}w) = f_{-n} \cdots f_{-3}f_{-2} h_1 \cdot (e_{-1} w)
\]
for some $h_1 \in \mathcal{N}$. Some such $h_1$ occurs infinitely often, and we may take $g_1 = h_1$. Repeating for each $n \in \mathbb{N}$, the resulting Cantor diagonalisation argument implies that we may take each $g_n$ so that application of \eqref{eq: semi-nucleus} restricts each $g_n$ to $g_{n-1}$, with the final restriction to the right-infinite tail $h \cdot w$ not depending on $n$, as required.
Finally, applying $g_n^{-1}$ to both sides of \eqref{eq: aeq alt}, we see that $f \sim_\mathrm{ae} e$.
\end{proof}
Whilst the above shows that $\sim_\mathrm{ae}$ is reflexive and symmetric in the contractive case, it need not be transitive, as we will see for the tiling semigroup. So $\sim$ is the transitive closure of $\sim_\mathrm{ae}$.
\begin{remark}
Lemma \ref{lem: aeq ref sym} implies that, in the contractive case, we may equivalently define $\sim_\mathrm{ae}$ by demanding that the right-infinite tail $w'$ of \eqref{eq: aeq alt} remains constant in $n$, and that each $g_n \in \mathcal{N}$.
\end{remark}
\begin{definition} \label{def: limit space}
The {\bf limit space} $\mathcal{J}$ of a self-similar semigroup action is defined as the quotient space $\mathcal{F}^- / \sim$. The shift map $\sigma \colon \mathcal{F}^{-} \to \mathcal{F}^{-}$, given by $\cdots e_{-3} e_{-2} e_{-1} \mapsto \cdots e_{-4} e_{-3} e_{-2}$ induces a map $\sigma \colon \mathcal{J} \to \mathcal{J}$. We denote its inverse limit by
\begin{equation}\label{omega_inv_limit}
\Omega \coloneqq \varprojlim (\mathcal{J} \xleftarrow{\sigma} \mathcal{J} \xleftarrow{\sigma} \mathcal{J} \xleftarrow{\sigma} \cdots).
\end{equation}
\end{definition}
We now return to the case of the tiling semigroup $\mathcal{T}$ acting on $\mathcal{F} \cong \Omega_\mathrm{punc}$. We construct a map
\begin{equation}\label{alpha_map}
\alpha \colon \mathcal{F}^- \hspace{-0.2cm} \longrightarrow Y \coloneqq \bigsqcup_{p \in \mathcal{P}} \mathrm{supp}(P),
\end{equation}
where the range of the map is the disjoint union of (supports of) prototiles. Let us recall the following elementary lemma.
\begin{lemma}\label{lem_compact_inclusion}
Let $\cdots \subset S_{-3} \subset S_{-2} \subset S_{-1} $ be a sequence of nested non-empty compact subsets of $\mathbb{R}^d$ such that the corresponding diameters $d_i \coloneqq sup_{x_1, x_2 \in S_i} |x_1-x_2| \to 0$ as $i \to -\infty$. Then $\cap_{i=-1}^{-\infty} S_i$ is a single point in $\mathbb{R}^d$.
\end{lemma}
We will construct such a nested sequence from elements of $\mathcal{F}^-$. This is done using successively finer partitions of $Y$ under substitution: Given $e = \cdots e_{-2}e_{-1} \in \mathcal{F}^-$, we let $S_{-1} \coloneqq \mathrm{supp}(s(e_{-1}))$. For $n > 1$, each supertile extension $e_{-n}$ embeds $\mathrm{supp}(r(e_{-n}))$ into $\mathrm{supp}(s(e_{-n}))$ as subtiles of scale $\lambda^{-1}$ of the original size. Thus, letting $S_{-n} \coloneqq \mathrm{supp}(r(e_{-n}))$ be the corresponding subset of $S_{-(n-1)}$, we get a nested sequence of subtile inclusions $\cdots \subset S_{-2} \subset S_{-1}$. By Lemma \ref{lem_compact_inclusion} their intersection is some point $x_{-\infty} \in S_{-n} \subset Y$ for each $n$, and we define a continuous map $\alpha \colon \mathcal{F}^- \hspace{-0.2cm} \longrightarrow Y$ by $\alpha(e) \coloneqq x_{-\infty}$.
\begin{lemma} \label{thm: aeq}
Suppose $(\mathcal{T},\mathcal{F})$ is a self-similar inverse semigroup associated with a recognisable substitution. We have that $e \sim_\mathrm{ae} f$ if and only if there is some tiling $T \in \Omega_\mathrm{punc}$, and tiles $t=p+x $ and $t'=q+y$ in $T$ with $p$, $q \in \mathcal{P}$ and $x$, $y \in \mathbb{R}^d$ such that $\alpha(e) \in \mathrm{supp}(p)$, $\alpha(f) \in \mathrm{supp}(q)$ and $\alpha(e)+x=\alpha(f)+y$. That is, $\alpha(e)$ and $\alpha(f)$ are identical points of a prototile, or points on prototile boundaries which can coincide in adjacent tiles.
\end{lemma}
\begin{proof}
Suppose that such a tiling $T$ exists with $\alpha(e)+x=\alpha(f)+y$ in the supports of $t$ and $t'$. Using the homeomorphism $\tau:\mathcal{F} \to \Omega_\mathrm{punc}$ from \eqref{eq: strings <-> transversal}, let $w_e = \tau^{-1}(T-x(t))$ and $w_f = \tau^{-1}(T-x(t'))$. For each $n \in \mathbb{N}$, consider the tilings $E_n=\tau(e_{-n} \cdots e_{-2}e_{-1}w_e)$ and $F_n=\tau(f_{-n} \cdots f_{-2} f_{-1}w_f)$, respectively. Notice that the sequences of tilings $(E_n)$ and $(F_n)$ are given by successive substitution. Moreover, since $\alpha(e)$ and $\alpha(f)$ correspond to a shared point, we have that the tilings $E_n$ and $F_n$ are equal, up to translating from the origin tile of $E_n$ to the adjacent origin tile of $F_n$. It follows that we may choose semigroup elements $g_n \in \mathcal{T}$ corresponding to translations between adjacent tiles and so that $g_n \cdot \tau^{-1}(E_n) = \tau^{-1}(F_n)$. This shows that \eqref{eq: aeq alt} is satisfied with $h = [q,P_{pq},p]$ where $P_{pq}$ may be taken as a star-patch with $p$, $q$ meeting analogously to $t$ and $t'$. By FLC, there are only finitely many such patches, so $e \sim_\mathrm{ae} f$.
Conversely, suppose that $e \sim_\mathrm{ae} f$, and take $w$, $(g_n)$ and $h$ as in \eqref{eq: aeq alt}. We have that
\begin{equation} \label{eq: limit space proof}
g_n\cdot (e_{-n} \cdots e_{-3}e_{-2}e_{-1}w) = f_{-n} \cdots f_{-1}(h \cdot w),
\end{equation}
where each $g_n \in \mathcal{N}$ and $h \in \mathcal{N}$. Hence, we have tilings $T = \tau(w)$ and $T' = \tau(h \cdot w)$ which are equal up to a translation between adjacent origin tiles $t = t_0 \in T$ and $t' = t'_0 \in T'$. Moreover, for each level $n$ of substitution, the tilings $E_n = \tau(e_{-n} \cdots e_{-2}e_{-1}w)$ and $F_n = \tau(f_{-n} \cdots f_{-2} f_{-1}w)$ remain equal up to translation between adjacent origin tiles $t_n \in \varphi(t_{n-1})$ and $t'_n \in \varphi(t'_{n-1})$. So we have $T-x(t')=T'$ and, letting $t=p+x$ and $t'=q+y$ for $p$, $q \in \mathcal{P}$, by definition of $\alpha$ we have that $\alpha(e)+x=\alpha(f)+y$, as required.
\end{proof}
\begin{theorem} \label{thm: limit spaces of sub tilings}
Suppose $(\mathcal{T},\mathcal{F})$ is a self-similar inverse semigroup associated with a recognisable substitution $\varphi$. The limit space $\mathcal{J}$ is homeomorphic to the Anderson--Putnam complex of the substitution, and the inverse limit $\Omega$ in \eqref{omega_inv_limit} is conjugate to the continuous hull $\Omega_\varphi$.
\end{theorem}
\begin{proof}
Let $x \sim_\mathrm{AP} y$ be the the relation on $Y$ that identifies points of prototiles that coincide in some tiling. The Anderson--Putnam complex $\Gamma_0$ \cite{AP} is defined as the quotient of $Y$ under the transitive closure of $\sim_\mathrm{AP}$. Let us denote the quotient map by $q_\mathrm{AP} \colon Y \to \Gamma_0$. We have that $\alpha \colon \mathcal{F}^- \to Y$ is also a quotient map, since it is a surjective map between compact Hausdorff spaces. By Lemma \ref{thm: aeq}, we have that $e \sim_\mathrm{ae} f$ in $\mathcal{F}^-$ if and only if $\alpha(e) \sim_\mathrm{AP} \alpha(f)$. It follows that the quotient map $q \colon \mathcal{F}^- \to \mathcal{J}$ may be identified with composition $\alpha \circ q_\mathrm{AP}$ and hence $\mathcal{J}$ is homeomorphic to the Anderson--Putnam complex $\Gamma_0$.
Given $e \in \mathcal{F}^-$, its shift in the limit space may be identified with $q_\mathrm{AP}(\alpha(\sigma (e)))$. By the definition of $\alpha$, the point $\alpha(\sigma (e)) \in Y$ is given by substituting $\alpha(e)$, considered as a point of a prototile in $Y$. Hence $\sigma \colon \mathcal{J} \to \mathcal{J}$ agrees with the map induced by substitution on the Anderson--Putnam complex, so $\Omega$ is the continuous hull by \cite[Theorem 4.3]{AP}.
\end{proof}
\begin{remark}
Definition \ref{def: limit space} generalises the notion of the limit space and associated inverse limit (the {\bf limit solenoid}) for self-similar groups. In the case of self-similar semigroups, a notion of the limit solenoid has already been defined without use of the intermediary limit space, as a quotient on the bi-infinite Markov shift $\mathcal{F}_\mathbb{Z}$ by an equivalence relation similar to the asymptotic equivalence relation above \cite[Definition 3.4]{Nek_SSIS}. In the case considered here, there is a natural map $\beta \colon \mathcal{F}_\mathbb{Z} \to \Omega_\varphi$, defined as follows. Given $w = w_- w_+ \in \mathcal{F}_\mathbb{Z}$, where $w_- = \cdots e_{-2} e_{-1} \in \mathcal{F}_-$ and $w_+ = e_0 e_1 \cdots \in \mathcal{F}$, we define $\beta(w)$ to be the tiling $\tau(w_+)$, translated with the origin over the point corresponding to $\alpha(w_-)$ in the origin tile $r(w_+) = s(w_-)$. This defines a quotient map to the tiling space, and it is easy to see that it identifies points of the Markov shift if and only if they correspond to addresses which are adjacent at all levels, that is, they may be related by a finite sequence of elements $g_n \in \mathcal{N}$.
\end{remark}
\section{examples} \label{sec: examples}
In this section we study several well-known 1- and 2-dimensional examples of tiling semigroups and their self-similar actions.
\begin{figure}
\begin{center}
\begin{tikzpicture}
\begin{scope}[xshift=0cm,yshift=0cm]
\draw[|-|] (1,2) -- node[above,pos=0.5]{$b$} (3,2);
\draw[|-|] (0,0) -- node[below,pos=0.5]{$a$} (2,0);
\draw[|-|] (2,0) -- node[below,pos=0.5]{$d$} (4,0);
\node[vertex] (vert_l) at (1,0) {};
\node[vertex] (vert_r) at (3,0) {}
edge [->,>=latex,out=135,in=45,thick] node[above,pos=0.5]{$[a,P_{ad},d]$} (vert_l);
\node[vertex] (vert_u) at (2,2) {}
edge [->,>=latex,out=225,in=90,thick] node[left,pos=0.5]{$(a,b)$} (vert_l)
edge [->,>=latex,out=315,in=90,thick] node[right,pos=0.5]{$(d,b)$} (vert_r);
\node at (0,0) {$\ddag$};
\node at (4,0) {$\ddag$};
\node at (1,2) {$\ddag$};
\node at (3,2) {$\ddag$};
\end{scope}
\begin{scope}[xshift=5cm,yshift=0cm]
\draw[|-|] (1,2) -- node[above,pos=0.5]{$c$} (3,2);
\draw[|-|] (0,0) -- node[below,pos=0.5]{$a$} (2,0);
\draw[|-|] (2,0) -- node[below,pos=0.5]{$d$} (4,0);
\node[vertex] (vert_l) at (1,0) {};
\node[vertex] (vert_r) at (3,0) {}
edge [->,>=latex,out=135,in=45,thick] node[above,pos=0.5]{$[a,P_{ad},d]$} (vert_l);
\node[vertex] (vert_u) at (2,2) {}
edge [->,>=latex,out=225,in=90,thick] node[left,pos=0.5]{$(a,c)$} (vert_l)
edge [->,>=latex,out=315,in=90,thick] node[right,pos=0.5]{$(d,c)$} (vert_r);
\node at (0,0) {$\ddag$};
\node at (4,0) {$\ddag$};
\node at (1,2) {$\ddag$};
\node at (3,2) {$\ddag$};
\end{scope}
\begin{scope}[xshift=10cm,yshift=0cm]
\draw[|-|] (1,2) -- node[above,pos=0.5]{$d$} (3,2);
\draw[|-|] (3,2) -- node[above,pos=0.5]{$b$} (5,2);
\draw[|-|] (0,0) -- node[below,pos=0.5]{$b$} (2,0);
\draw[|-|] (2,0) -- node[below,pos=0.5]{$a$} (4,0);
\draw[|-|] (4,0) -- node[below,pos=0.5]{$d$} (6,0);
\node[vertex] (vert_l) at (1,0) {};
\node[vertex] (vert_r) at (3,0) {}
edge [->,>=latex,out=135,in=45,thick] node[above,pos=0.5]{$[b,P_{ba},a]$} (vert_l);
\node[vertex] (vert_ul) at (2,2) {}
edge [->,>=latex,out=225,in=90,thick] node[left,pos=0.5]{$(b,d)$} (vert_l);
\node[vertex] (vert_ur) at (4,2) {}
edge [->,>=latex,out=135,in=45,thick] node[above,pos=0.5]{$[d,P_{db},b]$} (vert_ul)
edge [->,>=latex,out=225,in=90,thick] node[right,pos=0.5]{$(a,b)$} (vert_r);
\node at (0,0) {$\ddag$};
\node at (2,0) {$\ddag$};
\node at (6,0) {$\ddag$};
\node at (5,2) {$\ddag$};
\node at (3,2) {$\ddag$};
\end{scope}
\end{tikzpicture}
\end{center}
\caption{An illustration of how the first three formulae after \eqref{Fib_auto1} are deduced. The double daggers at tile edges denote supertile boundaries.}
\label{fig:Auto left}
\end{figure}
\begin{figure}
\begin{center}
\begin{tikzpicture}
\begin{scope}[xshift=0cm,yshift=0cm]
\draw[|-|] (0,4) -- node[above,pos=0.5]{$a$} (2,4);
\draw[|-|] (2,4) -- node[above,pos=0.5]{$d$} (4,4);
\draw[|-|] (-1,2) -- node[below,pos=0.5]{$a$} (1,2);
\draw[|-|] (1,2) -- node[below,pos=0.5]{$d$} (3,2);
\draw[|-|] (3,2) -- node[below,pos=0.5]{$b$} (5,2);
\draw[|-|] (0,0) -- node[below,pos=0.5]{$b$} (2,0);
\draw[|-|] (2,0) -- node[below,pos=0.5]{$a$} (4,0);
\draw[|-|] (4,0) -- node[below,pos=0.5]{$d$} (6,0);
\node[vertex] (vert_l) at (1,0) {};
\node[vertex] (vert_r) at (3,0) {}
edge [<-,>=latex,out=135,in=45,thick] node[above,pos=0.5]{$[a,P_{ba},b]$} (vert_l);
\node[vertex] (vert_ul) at (2,2) {}
edge [->,>=latex,out=225,in=90,thick] node[left,pos=0.5]{$(b,d)$} (vert_l);
\node[vertex] (vert_ur) at (4,2) {}
edge [<-,>=latex,out=135,in=45,thick] node[above,pos=0.5]{$[b,P_{db},d]$} (vert_ul)
edge [->,>=latex,out=225,in=90,thick] node[right,pos=0.5]{$(a,b)$} (vert_r);
\node[vertex] (vert_uul) at (1,4) {}
edge [->,>=latex,out=315,in=90,thick] node[left,pos=0.5]{$(d,a)$} (vert_ul);
\node[vertex] (vert_uur) at (3,4) {}
edge [->,>=latex,out=315,in=90,thick] node[right,pos=0.5]{$(b,d)$} (vert_ur);
\node at (0,0) {$\ddag$};
\node at (2,0) {$\ddag$};
\node at (-1,2) {$\ddag$};
\node at (6,0) {$\ddag$};
\node at (5,2) {$\ddag$};
\node at (3,2) {$\ddag$};
\node at (0,4) {$\ddag$};
\node at (4,4) {$\ddag$};
\end{scope}
\begin{scope}[xshift=8cm,yshift=0cm]
\draw[|-|] (0,4) -- node[above,pos=0.5]{$b$} (2,4);
\draw[|-|] (2,4) -- node[above,pos=0.5]{$a$} (4,4);
\draw[|-|] (-2,2) -- node[above,pos=0.5]{$a$} (0,2);
\draw[|-|] (0,2) -- node[above,pos=0.4]{$d$} (2,2);
\draw[|-|] (2,2) -- node[above,pos=0.6]{$c$} (4,2);
\draw[|-|] (4,2) -- node[above,pos=0.5]{$d$} (6,2);
\draw[|-|] (0,0) -- node[below,pos=0.5]{$b$} (2,0);
\draw[|-|] (2,0) -- node[below,pos=0.5]{$a$} (4,0);
\draw[|-|] (4,0) -- node[below,pos=0.5]{$d$} (6,0);
\node[vertex] (vert_l) at (1,0) {};
\node[vertex] (vert_r) at (3,0) {}
edge [<-,>=latex,out=135,in=45,thick] node[above,pos=0.5]{$[a,P_{ba},b]$} (vert_l);
\node[vertex] (vert_ul) at (1,2) {}
edge [->,>=latex,out=270,in=90,thick] node[left,pos=0.5]{$(b,d)$} (vert_l);
\node[vertex] (vert_ur) at (3,2) {}
edge [<-,>=latex,out=135,in=45,thick] node[above,pos=0.5]{$[c,P_{dc},d]$} (vert_ul)
edge [->,>=latex,out=270,in=90,thick] node[right,pos=0.5]{$(a,c)$} (vert_r);
\node[vertex] (vert_uul) at (1,4) {}
edge [->,>=latex,out=270,in=90,thick] node[left,pos=0.5]{$(d,b)$} (vert_ul);
\node[vertex] (vert_uur) at (3,4) {}
edge [->,>=latex,out=270,in=90,thick] node[right,pos=0.5]{$(c,a)$} (vert_ur);
\node at (0,0) {$\ddag$};
\node at (2,0) {$\ddag$};
\node at (-2,2) {$\ddag$};
\node at (6,0) {$\ddag$};
\node at (6,2) {$\ddag$};
\node at (2,2) {$\ddag$};
\node at (0,4) {$\ddag$};
\node at (4,4) {$\ddag$};
\end{scope}
\end{tikzpicture}
\end{center}
\caption{An illustration of how the first three formulae after \eqref{Fib_auto2} are deduced. The double daggers at tile edges denote supertile boundaries.}
\label{fig:Auto right}
\end{figure}
\begin{example}\label{ex: fib rules}
We return to the border forcing Fibonacci tiling of Examples \ref{ex: fib} and \ref{geometric intuition}. The self-similar inverse semigroup is generated by doubly pointed patches consisting of all possible single tile patches and connected pair patches appearing anywhere in a Fibonacci tiling. Note that Figures \ref{fig:Auto left} and \ref{fig:Auto right} show how we geometrically deduce the self-similar relation on a selection of generating elements. The following doubly pointed patches, represented here along with their self-similar action, generate the semigroup of the Fibonacci tiling.
\begingroup
\allowdisplaybreaks
\begin{align}
\label{Fib_auto1}
[a,P_{ad},d] \cdot (d,b)w&=(a,b)\ [b,P_b,b] \cdot w; \\
\notag
[a,P_{ad},d] \cdot (d,c)w&=(a,c) \ [c,P_c,c] \cdot w; \\
\notag
[b,P_{ba},a] \cdot (a,b)w&=(b,d) \ [d,P_{db},b] \cdot w; \\
\notag
[b,P_{ba},a] \cdot (a,c)w&=(b,d)\ [d,P_{dc},c] \cdot w; \\
\notag
[c,P_{cd},d] \cdot (d,a)w&=(c,a)\ [a,P_{a},a] \cdot w; \\
\notag
[d,P_{db},b] \cdot (b,d)(d,a)w&=(d,c)\ [c,P_{cd},d] \cdot (d,a)w; \\
\notag
[d,P_{db},b] \cdot (b,d)(d,b)w&=(d,a)\ [a,P_{ad},d] \cdot (d,b)w; \\
\notag
[d,P_{db},b] \cdot (b,d)(d,c)w&=(d,a)\ [a,P_{ad},d] \cdot (d,c)w; \\
\notag
[d,P_{dc},c] \cdot (c,a)w&=(d,b)\ [b,P_{ba},a] \cdot w; \\
\label{Fib_auto2}
[a,P_{ba},b] \cdot (b,d)(d,a)w&=(a,b)\ [b,P_{db},d] \cdot (d,a)w; \\
\notag
[a,P_{ba},b] \cdot (b,d)(d,b)w&=(a,c)\ [c,P_{dc},d] \cdot (d,b)w; \\
\notag
[a,P_{ba},b] \cdot (b,d)(d,c)w&=(a,b)\ [b,P_{db},d] \cdot (d,c)w; \\
\notag
[b,P_{db},d] \cdot (d,a)w&=(b,d)\ [d,P_{ad},a] \cdot w; \\
\notag
[b,P_{db},d] \cdot (d,c)w&=(b,d)\ [d,P_{cd},c] \cdot w; \\
\notag
[c,P_{dc},d] \cdot (d,b)w&=(c,a)\ [a,P_{ba},b] \cdot w; \\
\notag
[d,P_{ad},a] \cdot (a,b)w&=(d,b)\ [b,P_{b},b] \cdot w; \\
\notag
[d,P_{ad},a] \cdot (a,c)w&=(d,c)\ [c,P_{c},c] \cdot w; \\
\notag
[d,P_{cd},c] \cdot (c,a)w&=(d,a)\ [a,P_{a},a] \cdot w.
\end{align}
\qed
\end{example}
\endgroup
\begin{example}
The simplest border-forcing 2-dimensional example comes from the half-hex tiling. We note that there are six prototiles $\{p_0,p_1,p_2,p_3,p_4,p_5\}$, where the subscript denotes the number of rotations of $p_0$ by $\pi/3$. Similarly, the substitution of each prototile is equivalent up to rotations by $n\pi/3$, see Figure \ref{proto}. Thus, always taking addition to be mod 6, the tile inclusions can be written as:
\[
\{(p_i,p_i), (p_{i+2},p_i), (p_{i+3},p_i), (p_{i+4},p_i) \mid i=0,1,2,3,4,5\}.
\]
\begin{figure}
\begin{center}
\begin{tikzpicture}
\begin{scope}[yshift=1cm]
\HHex{0}{1.73}{180}{0};
\HHex{1.5}{0.866}{120}{0};
\HHex{-1.5}{0.866}{240}{0};
\HHex{0}{-1.73}{0}{0};
\HHex{1.5}{-0.866}{60}{0};
\HHex{-1.5}{-0.866}{-60}{0};
\node at (30:1.3) {$p_2$};
\node at (90:1.3) {$p_3$};
\node at (150:1.3) {$p_4$};
\node at (210:1.3) {$p_5$};
\node at (270:1.3) {$p_0$};
\node at (330:1.3) {$p_1$};
\end{scope}
\begin{scope}[xshift=10cm]
\HHex{-4.5}{0.5}{0}{0};
\node at (-4.5,0.85) {$p_0$};
\HHexI{0}{0}{0}{-1};
\node at (90:0.35) {$p_0$};
\node at (30:1.3) {$p_2$};
\node at (90:1.3) {$p_3$};
\node at (150:1.3) {$p_4$};
\draw[->, thick] (-3.25,1) -- node[above] {$\varphi$} (-2.15,1);
\end{scope}
\end{tikzpicture}
\end{center}
\caption{The half-hex prototiles are on the left and the substitution of $p_0$ is on the right. All other substitutions are rotations of $p_0$ by $n\pi/3$.}
\label{proto}
\end{figure}
The substitution of $p_0$ appears in Figure \ref{proto}. The graph associated with this substitution appears in Figure \ref{HH_sub}.
\begin{figure}
\begin{center}
\begin{tikzpicture}
\node[vertex] (vert_p0) at (270:2) {$p_0$}
edge [->,>=latex,out=250,in=290,thick,loop] node[below,pos=0.5]{} (vert_p0);
\node[vertex] (vert_p1) at (330:2) {$p_1$}
edge [->,>=latex,out=310,in=350,thick,loop] node[right,pos=0.5]{} (vert_p1);
\node[vertex] (vert_p2) at (30:2) {$p_2$}
edge [->,>=latex,out=10,in=50,thick,loop] node[right,pos=0.5]{} (vert_p2)
edge [<-,>=latex,out=250,in=50,thick] node[right,pos=0.5]{} (vert_p0)
edge [->,>=latex,out=230,in=70,thick] node[right,pos=0.5]{} (vert_p0);
\node[vertex] (vert_p3) at (90:2) {$p_3$}
edge [->,>=latex,out=70,in=110,thick,loop] node[above,pos=0.5]{} (vert_p3)
edge [<-,>=latex,out=260,in=100,thick] node[right,pos=0.5]{} (vert_p0)
edge [->,>=latex,out=280,in=80,thick] node[right,pos=0.5]{} (vert_p0)
edge [<-,>=latex,out=250+60,in=50+60,thick] node[right,pos=0.5]{} (vert_p1)
edge [->,>=latex,out=230+60,in=70+60,thick] node[right,pos=0.5]{} (vert_p1);
\node[vertex] (vert_p4) at (150:2) {$p_4$}
edge [->,>=latex,out=130,in=170,thick,loop] node[left,pos=0.5]{} (vert_p4)
edge [<-,>=latex,out=260+60,in=100+60,thick] node[right,pos=0.5]{} (vert_p1)
edge [->,>=latex,out=280+60,in=80+60,thick] node[right,pos=0.5]{} (vert_p1)
edge [<-,>=latex,out=250+60,in=50+60,thick] node[right,pos=0.5]{} (vert_p0)
edge [->,>=latex,out=230+60,in=70+60,thick] node[right,pos=0.5]{} (vert_p0)
edge [<-,>=latex,out=250+120,in=50+120,thick] node[right,pos=0.5]{} (vert_p2)
edge [->,>=latex,out=230+120,in=70+120,thick] node[right,pos=0.5]{} (vert_p2);
\node[vertex] (vert_p5) at (210:2) {$p_5$}
edge [->,>=latex,out=190,in=230,thick,loop] node[left,pos=0.5]{} (vert_p5)
edge [<-,>=latex,out=260+120,in=100+120,thick] node[right,pos=0.5]{} (vert_p2)
edge [->,>=latex,out=280+120,in=80+120,thick] node[right,pos=0.5]{} (vert_p2)
edge [<-,>=latex,out=10,in=170,thick] node[right,pos=0.5]{} (vert_p1)
edge [->,>=latex,out=-10,in=190,thick] node[right,pos=0.5]{} (vert_p1)
edge [<-,>=latex,out=250+180,in=50+180,thick] node[right,pos=0.5]{} (vert_p3)
edge [->,>=latex,out=230+180,in=70+180,thick] node[right,pos=0.5]{} (vert_p3);
\end{tikzpicture}
\end{center}
\caption{The substitution graph of the half-hex tiling.}
\label{HH_sub}
\end{figure}
The self-similar inverse semigroup is generated by the collection of doubly pointed patches consisting of all single tile patches and connected pair patches appearing in a half-hex tiling. All such tile pairs appear in Figure \ref{automaton_elements} up to rotation. For each connected pair of tiles, the generating doubly pointed patch $[q,X,p]$ represents a translation across an edge of tile $p$ where $X \in \{A,B,C,D\}$ represents the two-tile patch connected across one of the 4 edges of $p$. For $p_0$, we set edges $A$--$D$ to be the 4 edges starting from the bottom and rotating counterclockwise. See Figure \ref{automaton_elements} for complete clarity.
\begin{figure}
\begin{center}
\begin{tikzpicture}
\begin{scope}[xshift=0cm,yshift=0cm]
\HHex{0}{0}{0}{-1};
\HHex{0}{0}{180}{-1};
\node[vertex] (vert_p) at (90:1.1) {$(p_i,x)$};
\node[vertex] (vert_q) at (-90:1.1) {$(p_{i+3},x)$}
edge [<-,>=latex,out=132,in=228,thick] node[right,pos=0.55]{$\scriptstyle [p_{i+3},A,p_i]$} (vert_p);
\end{scope}
\begin{scope}[xshift=4.2cm,yshift=-1.8cm]
\HHex{0}{0}{0}{-1};
\HHex{3}{1.732}{60}{-1};
\node[vertex] (vert_p) at (90:0.6) {$(p_i,x)$};
\node[vertex] (vert_q) at (40:3.1) {$(p_{i+1},x)$}
edge [<-,>=latex,out=220,in=20,thick] node[left,pos=0.45]{$\scriptstyle [p_{i+1},B,p_i]$} (vert_p);
\end{scope}
\begin{scope}[xshift=9cm,yshift=-1.8cm]
\HHex{0}{0}{0}{-1};
\HHex{3}{1.732}{120}{-1};
\node[vertex] (vert_p) at (90:0.6) {$(p_i,x)$};
\node[vertex] (vert_q) at (35:2.8) {$(p_{i+2},x)$}
edge [<-,>=latex,out=220,in=0,thick] node[left,pos=0.15]{$\scriptstyle [p_{i+2},B,p_i]$} (vert_p);
\end{scope}
\begin{scope}[xshift=0cm,yshift=-7.5cm]
\HHex{0}{0}{0}{-1};
\HHex{0}{3.464}{180}{-1};
\node[vertex] (vert_p) at (90:0.6) {$(p_i,x)$};
\node[vertex] (vert_q) at (90:2.8) {$(p_{i+3},x)$}
edge [<-,>=latex,out=228,in=132,thick] node[right,pos=0.6]{$\scriptstyle [p_{i+3},C,p_i]$} (vert_p);
\end{scope}
\begin{scope}[xshift=4.7cm,yshift=-7.5cm]
\HHex{0}{0}{0}{-1};
\HHex{0}{3.464}{240}{-1};
\node[vertex] (vert_p) at (90:0.6) {$(p_i,x)$};
\node[vertex] (vert_q) at (80:3.2) {$(p_{i+4},x)$}
edge [<-,>=latex,out=228,in=132,thick] node[right,pos=0.65]{$\scriptstyle [p_{i+4},C,p_i]$} (vert_p);
\end{scope}
\begin{scope}[xshift=10cm,yshift=-7.5cm]
\HHex{0}{0}{0}{-1};
\HHex{0}{3.464}{120}{-1};
\node[vertex] (vert_p) at (90:0.6) {$(p_i,x)$};
\node[vertex] (vert_q) at (100:3.2) {$(p_{i+2},x)$}
edge [<-,>=latex,out=312,in=48,thick] node[left,pos=0.65]{$\scriptstyle [p_{i+2},C,p_i]$} (vert_p);
\end{scope}
\begin{scope}[xshift=3cm,yshift=-12cm]
\HHex{0}{0}{0}{-1};
\HHex{-3}{1.732}{300}{-1};
\node[vertex] (vert_p) at (90:0.6) {$(p_i,x)$};
\node[vertex] (vert_q) at (140:3.2) {$(p_{i+5},x)$}
edge [<-,>=latex,out=320,in=160,thick] node[right,pos=0.47]{$\scriptstyle [p_{i+5},D,p_i]$} (vert_p);
\end{scope}
\begin{scope}[xshift=10cm,yshift=-12cm]
\HHex{0}{0}{0}{-1};
\HHex{-3}{1.732}{-120}{-1};
\node[vertex] (vert_p) at (90:0.6) {$(p_i,x)$};
\node[vertex] (vert_q) at (145:2.8) {$(p_{i+4},x)$}
edge [<-,>=latex,out=320,in=180,thick] node[right,pos=0.10]{$\, \scriptstyle [p_{i+4},D,p_i]$} (vert_p);
\end{scope}
\end{tikzpicture}
\end{center}
\caption{The possible two-tile patches with respect to reference tile $p_i$.}
\label{automaton_elements}
\end{figure}
We begin by describing the self-similar relation for the generating doubly pointed patches across the long edge of tile $p_i$, labelled $A$. Note that all subscripts are treated mod$\ 6$ and $w \in \mathcal{F}$.
\begingroup
\allowdisplaybreaks
\begin{align}
\label{HH_auto_1}
[p_{i+3},A,p_i] \cdot (p_i,p_i)w&=(p_{i+3},p_{i+3}) \ [p_{i+3},A,p_i] \cdot w; \\
\notag
[p_{i+3},A,p_i] \cdot (p_i,p_{i+2})(p_{i+2},p_{i+5})w&=(p_{i+3},p_{i+1}) \ [p_{i+1},D,p_{i+2}] \cdot (p_{i+2},p_{i+5})w; \\
\notag
[p_{i+3},A,p_i] \cdot (p_i,p_{i+2})(p_{i+2},p_{i+2})w&=(p_{i+3},p_{i}) \ [p_{i},D,p_{i+2}] \cdot (p_{i+2},p_{i+2})w; \\
\notag
[p_{i+3},A,p_i] \cdot (p_i,p_{i+3})(p_{i+3},p_{i+5})w&=(p_{i+3},p_{i+5}) \ [p_{i+5},C,p_{i+3}] \cdot (p_{i+3},p_{i+5})w;\\
\notag
[p_{i+3},A,p_i] \cdot (p_i,p_{i+3})(p_{i+3},p_{i+1})w&=(p_{i+3},p_{i+1}) \ [p_{i+1},C,p_{i+3}] \cdot (p_{i+3},p_{i+1})w;\\
\notag
[p_{i+3},A,p_i] \cdot (p_i,p_{i+3})(p_{i+3},p_{i})w&=(p_{i+3},p_{i}) \ [p_{i},C,p_{i+3}] \cdot (p_{i+3},p_{i})w;\\
\notag
[p_{i+3},A,p_i] \cdot (p_i,p_{i+4})(p_{i+4},p_{i+1})w&=(p_{i+3},p_{i+5}) \ [p_{i+5},B,p_{i+4}] \cdot (p_{i+4},p_{i+1})w;\\
\notag
[p_{i+3},A,p_i] \cdot (p_i,p_{i+4})(p_{i+4},p_{i+4})w&=(p_{i+3},p_{i}) \ [p_{i},B,p_{i+4}] \cdot (p_{i+4},p_{i+4})w.
\end{align}
\endgroup
In order to geometrically understand these relations, we illustrate the first two self-similar relations from \eqref{HH_auto_1} in Figures \ref{fig:HH_auto_1} and \ref{fig:HH_auto_2}.
\begin{figure}
\begin{center}
\begin{tikzpicture}
\begin{scope}[xshift=0cm,yshift=0cm]
\HHexI{0}{0}{0}{-1};
\HHexI{0}{0}{180}{-1};
\node (vert_p) at (90:0.6) {$(p_0,p_0)$};
\node (vert_q) at (270:0.6) {$(p_{3},p_3)$}
edge [<-,>=latex,out=132,in=228,thick] node[right,pos=0.75]{$\scriptstyle [p_{3},A,p_0]$} (vert_p);
\end{scope}
\begin{scope}[xshift=6cm,yshift=0cm]
\HHex{0}{0}{0}{-1};
\HHex{0}{0}{180}{-1};
\node[vertex] (vert_p) at (90:1.1) {$(p_0,x)$};
\node[vertex] (vert_q) at (270:1.1) {$(p_{3},x)$}
edge [<-,>=latex,out=132,in=228,thick] node[right,pos=0.65]{$\scriptstyle [p_{3},A,p_0]$} (vert_p);
\end{scope}
\end{tikzpicture}
\end{center}
\caption{An illustration of the first formula $[p_{3},A,p_0] \cdot (p_0,p_0)w=(p_{3},p_{3}) \ [p_{3},A,p_0] \cdot w$ in \eqref{HH_auto_1} with $i=0$. The left hand side shows the action $[p_{3},A,p_0] \cdot (p_0,p_0)=(p_3,p_3)$ and the right hand side shows that the element $[p_{3},A,p_0]$ comes from the relationship between 1-supertiles.}
\label{fig:HH_auto_1}
\end{figure}
\begin{figure}
\begin{center}
\begin{tikzpicture}
\begin{scope}[xshift=0cm,yshift=0cm]
\HHexII{-3}{0}{-60}{-2};
\node (vert_p) at (90:0.6) {$(p_0,p_2)$};
\node (vert_q) at (270:0.6) {$(p_{3},p_1)$}
edge [<-,>=latex,out=132,in=228,thick] node[right,pos=0.75]{$\scriptstyle [p_{3},A,p_0]$} (vert_p);
\end{scope}
\begin{scope}[xshift=7cm,yshift=0cm]
\HHexI{-3}{0}{-60}{-2};
\node (vert_p) at (115:1.3) {$(p_2,x)$};
\node (vert_q) at (245:1.3) {$(p_{1},x)$}
edge [<-,>=latex,out=132,in=228,thick] node[right,pos=0.65]{$\scriptstyle [p_{1},D,p_2]$} (vert_p);
\end{scope}
\end{tikzpicture}
\end{center}
\caption{An illustration of the second formula $[p_{3},A,p_0] \cdot (p_0,p_{2})(p_{2},p_{5})w=(p_{3},p_{1}) \ [p_{1},D,p_{2}] \cdot (p_{2},p_{5})w$ in \eqref{HH_auto_1} with $i=0$. The left hand side shows the action $[p_{3},A,p_0] \cdot (p_0,p_2)=(p_3,p_1)$ and the right hand side shows that the element $[p_{1},D,p_{2}]$ comes from the relationship between 1-supertiles.}
\label{fig:HH_auto_2}
\end{figure}
We now describe the automaton elements $b_i$, $c_i$ and $d_i$ across the shorter edges of tile $p_i$. Again, we note that all subscripts are treated mod$\ 6$ and $w \in \mathcal{F}$.
\begingroup
\allowdisplaybreaks
\begin{align*}
[p_{i+1},B,p_i] \cdot (p_i,p_{i+2})w&=(p_{i+1},p_{i+5}) \ [p_{i+5},A,p_{i+2}] \cdot w; \\
[p_{i+1},B,p_i] \cdot (p_i,p_{i+3})w&=(p_{i+1},p_{i+3}) w; \\
[p_{i+1},B,p_i] \cdot (p_i,p_{i+4})w&=(p_{i+1},p_{i+4}) w; \\
[p_{i+2},B,p_i] \cdot (p_i,p_i)w&=(p_{i+2},p_{i}) w; \\
[p_{i+2},C,p_i] \cdot (p_i,p_{i+2})w&=(p_{i+2},p_{i+2}) w; \\
[p_{i+3},C,p_i] \cdot (p_i,p_i)w&=(p_{i+3},p_{i}) w; \\
[p_{i+3},C,p_i] \cdot (p_i,p_{i+3})w&=(p_{i+3},p_{i+3}) w; \\
[p_{i+4},C,p_i] \cdot (p_i,p_{i+4})w&=(p_{i+4},p_{i+4}) w; \\
[p_{i+4},D,p_i] \cdot (p_i,p_i)w&=(p_{i+4},p_{i}) w; \\
[p_{i+5},D,p_i] \cdot (p_i,p_{i+2})w&=(p_{i+5},p_{i+2}) w; \\
[p_{i+5},D,p_i] \cdot (p_i,p_{i+3})w&=(p_{i+5},p_{i+3}) w; \\
[p_{i+5},D,p_i] \cdot (p_i,p_{i+4})w&=(p_{i+5},p_{i+1}) \ [p_{i+1},A,p_{i+4}] \cdot w.
\end{align*}
\endgroup
Let us note that some relations here could have been omitted by also exploiting the rotational equivariance of substitution. We shall make use of this in the following example. \qed
\end{example}
\begin{figure}
\begin{center}
\includegraphics[scale=0.07]{PenroseStarPatch}
\end{center}
\caption{A patch of a Penrose tiling.}
\label{Penrose_patch}
\end{figure}
\begin{example}
The most well-known 2-dimensional example was given by Penrose \cite{pentaplexity}, represented here as Robinson triangles. We note that there are forty prototiles $\{a_i,b_i,ra_i,rb_i \mid i=0, \ldots,9\}$, where the subscript denotes the number of rotations by $\pi/5$. By $ra_i$, we mean the reflection of the tile $a_i$ across the vertical, followed by rotation by $i\pi/5$, and analogously for $rb_i$ (we emphasise that we reflect the tile $a_0$ first, and then rotate). The substitutions of $a_0$ and $b_0$ appear in Figure \ref{Penrose_proto}. We have not attempted to display the graph of the substitution. All other prototiles are rigid motions of these and substitution on them is determined by equivariance of the substitution $\varphi$. For example, we have that $\varphi(ra_4) = \varphi(\theta_4(\tau a_0)) = \theta_4 \circ \tau(\varphi(a_0))$, where $\theta_4$ is rotation by $4\pi/5$ and $\tau$ is reflection across the vertical. Thus, always taking addition to be (mod$\ 10$), the tile inclusions can be written as
\begin{align*}
(a_7,a_0),(b_3,a_0),(rb_0,b_0),(ra_6,b_0),(b_4,b_0)
\end{align*}
along with the required rigid motions of the above (thus there are $20 \times 5 = 100$ in total). A patch of the Penrose tiling appears in Figure \ref{Penrose_patch}.
\begin{figure}
\begin{center}
\begin{tikzpicture}
\node at (0,1.8) {\includegraphics[scale=0.2]{PenroseSubsNoRef}};
\node at (-6.15,0.8) {$a_0$};
\draw[->,thick] (-5.6,1.5) -- node[above] {$\varphi$} (-5.0,1.5);
\node at (-4.35,0.65) {$a_7$};
\node at (-4.0,1.7) {$b_3$};
\node at (-0.7,0.7) {$b_0$};
\draw[->,thick] (0.9,0.9) -- node[above] {$\varphi$} (1.5,0.9);
\node at (3.1,0.7) {$rb_0$};
\node at (5.2,0.8) {$b_4$};
\node at (3.9,1.4) {$ra_6$};
\end{tikzpicture}
\end{center}
\caption{Penrose substitution.}
\label{Penrose_proto}
\end{figure}
\begin{figure}
\begin{center}
\begin{tikzpicture}
\node at (0,0) {\includegraphics[scale=0.21]{PenroseAP_Complex}};
\node at (-2.35,1.8) {$a_0$};
\node at (-1.0,2.7) {$a_2$};
\node at (-1.5,4.3) {$a_4$};
\node at (-3.25,4.3) {$a_6$};
\node at (-3.75,2.7) {$a_8$};
\node at (-1.55,2.0) {$ra_1$};
\node at (-1.0,3.5) {$ra_3$};
\node at (-2.35,4.5) {$ra_5$};
\node at (-3.75,3.5) {$ra_7$};
\node at (-3.25,2) {$ra_9$};
\node at (-2.35+4.75,1.8) {$ra_0$};
\node at (-1.0+4.75,2.7) {$ra_2$};
\node at (-1.5+4.75,4.3) {$ra_4$};
\node at (-3.25+4.75,4.3) {$ra_6$};
\node at (-3.75+4.75,2.7) {$ra_8$};
\node at (-1.55+4.75,2.0) {$a_1$};
\node at (-1.0+4.75,3.5) {$a_3$};
\node at (-2.35+4.75,4.5) {$a_5$};
\node at (-3.75+4.75,3.5) {$a_7$};
\node at (-3.25+4.75,2) {$a_9$};
\node at (-4.6,-1.5) {$rb_0$};
\node at (-3.9,-3.4) {$rb_2$};
\node at (-1.8,-3.3) {$rb_4$};
\node at (-1.3,-1.4) {$rb_6$};
\node at (-2.9,-0.3) {$rb_8$};
\node at (-4.6,-2.5) {$b_5$};
\node at (-2.95,-3.6) {$b_7$};
\node at (-1.3,-2.5) {$b_9$};
\node at (-1.8,-0.6) {$b_1$};
\node at (-3.9,-0.6) {$b_3$};
\node at (-4.6+6,-1.5) {$b_4$};
\node at (-4.1+6,-3.3) {$b_6$};
\node at (-2.13+6,-3.3) {$b_8$};
\node at (-1.4+6,-1.55) {$b_0$};
\node at (-3.05+6,-0.3) {$b_2$};
\node at (-4.6+6,-2.5) {$rb_1$};
\node at (-3.05+6,-3.6) {$rb_3$};
\node at (-1.4+6,-2.4) {$rb_5$};
\node at (-2.1+6,-0.6) {$rb_7$};
\node at (-4.1+6,-0.7) {$rb_9$};
\end{tikzpicture}
\end{center}
\caption{The Anderson--Putnam complex of the Penrose tiling \cite{AP}.}
\label{Penrose_AP}
\end{figure}
The self-similar inverse semigroup is generated by the collection of doubly pointed patches consisting of all single tile patches and adjacent pair patches appearing anywhere in a Penrose tiling. The Anderson--Putnam complex \cite[Section 10.4]{AP}, copied in Figure \ref{Penrose_AP}, neatly illustrates each possible two-tile patch using the edge identifications. For each connected pair of tiles, the doubly pointed patch $[q,X,p]$ represents translation from $p$ to $q$ across a specific edge of tile $p$ where $X \in \{B,L,R\}$ denotes crossing the \emph{B}ottom, \emph{L}eft, or \emph{R}ight edges of prototile $p$ with orientation from Figure \ref{Penrose_proto}.
We now describe the generating two-tile patch elements associated with moving across an edge of $a_i$. Again, we note that all subscripts are treated mod$\ 10$ and $w \in \mathcal{F}$.
\begingroup
\allowdisplaybreaks
\begin{align*}
[ra_{i+5},B,a_i] \cdot (a_{i},a_{i+3})w&=(ra_{i+5},b_{i+9}) \ [b_{i+9},L,a_{i+3}] \cdot w; \\
[b_{i+6},L,a_i] \cdot (a_{i},a_{i+3})w&=(b_{i+6},a_{i+3})w; \\
[ra_{i+1},R,a_i] \cdot (a_{i},a_{i+3})w&=(ra_{i+1},ra_{i+8}) \ [ra_{i+8},B,a_{i+3}] \cdot w; \\
[ra_{i+5},B,a_i] \cdot (a_{i},rb_{i+6})(rb_{i+6},ra_{i+2})w&=(ra_{i+5},ra_{i+2}) \ [ra_{i+2},R,rb_{i+6}] \cdot (rb_{i+6},ra_{i+2})w; \\
[ra_{i+5},B,a_i] \cdot (a_{i},rb_{i+6})(rb_{i+6},b_{i+6})w&=(ra_{i+5},ra_{i+2}) \ [ra_{i+2},R,rb_{i+6}] \cdot (rb_{i+6},b_{i+6})w; \\
[ra_{i+5},B,a_i] \cdot (a_{i},rb_{i+6})(rb_{i+6},rb_{i})w&=(ra_{i+5},b_{i+9}) \ [b_{i+9},L,rb_{i+6}] \cdot (rb_{i+6},rb_{i})w; \\
[b_{i+6},L,a_i] \cdot (a_{i},rb_{i+6})w&=(b_{i+6},rb_{i+6})w; \\
[rb_{i+2},R,a_i] \cdot (a_{i},rb_{i+6})w&=(rb_{i+2},rb_{i+6})w.
\end{align*}
\endgroup
Note that reflection acts on tiles by $a_i \leftrightarrow ra_{-i}$, $b_i \leftrightarrow rb_{-i}$ and edge types by $B \leftrightarrow B$, $L \leftrightarrow R$. Therefore, the above relations determine also those for reflections. For example, the last row determines the relation $[b_{i+8},L,ra_i] \cdot (ra_{i},b_{i+4})w = (b_{i+8},b_{i+4})w$, where we apply the above conversions, write $-2 \equiv 8 \mod 10$ etc., and also substitute $-i$ with $i$.
The following relations describe the generating two-tile patch inverse semigroup elements associated with moving across an edge of $b_i$.
\begingroup
\allowdisplaybreaks
\begin{align*}
[rb_{i+5},B,b_i] \cdot (b_{i},a_{i+7})(a_{i+7},a_{i})w&=(rb_{i+5},ra_{i+8}) \ [ra_{i+8},R,a_{i+7}] \cdot (a_{i+7},a_{i})w; \\
[rb_{i+5},B,b_i] \cdot (b_{i},a_{i+7})(a_{i+7},rb_{i+3})w&=(rb_{i+5},rb_{i+9}) \ [rb_{i+9},R,a_{i+7}] \cdot (a_{i+7},rb_{i+3})w; \\
[a_{i+4},L,b_i] \cdot (b_{i},a_{i+7})w&=(a_{i+4},a_{i+7})w; \\
[rb_{i+3},R,b_i] \cdot (b_{i},a_{i+7})w&=(rb_{i+3},b_{i+3}) \ [b_{i+3},L,a_{i+7}] \cdot w; \\
[rb_{i+5},B,b_i] \cdot (b_{i},b_{i+6})(b_{i+6},a_{i+3})w&=(rb_{i+5},rb_{i+9}) \ [rb_{i+9},R,b_{i+6}] \cdot (b_{i+6},a_{i+3})w; \\
[rb_{i+5},B,b_i] \cdot (b_{i},b_{i+6})(b_{i+6},b_{i+2})w&=(rb_{i+5},ra_{i+8}) \ [ra_{i+8},R,b_{i+6}] \cdot (b_{i+6},b_{i+2})w; \\
[rb_{i+5},B,b_i] \cdot (b_{i},b_{i+6})(b_{i+6},rb_{i+6})w&=(rb_{i+5},rb_{i+9}) \ [rb_{i+9},R,b_{i+6}] \cdot (b_{i+6},rb_{i+6})w; \\
[rb_{i+7},L,b_i] \cdot (b_{i},b_{i+6})w&=(rb_{i+7},rb_{i+1}) \ [rb_{i+1},B,b_{i+6}] \cdot w; \\
[ra_{i+2},R,b_i] \cdot (b_{i},b_{i+6})w&=(ra_{i+2},b_{i+6})w; \\
[rb_{i+5},B,b_i] \cdot (b_{i},rb_{i})w&=(rb_{i+5},b_{i+5}) \ [b_{i+5},B,rb_{i}] \cdot w; \\
[a_{i+4},L,b_i] \cdot (b_{i},rb_{i})w&=(a_{i+4},rb_{i})w; \\
[rb_{i+3},R,b_i] \cdot (b_{i},rb_{i})(rb_i,ra_{i+3})w&=(rb_{i+3},ra_{i+6}) \ [ra_{i+6},R,rb_{i}] \cdot (rb_i,ra_{i+3})w; \\
[rb_{i+3},R,b_i] \cdot (b_{i},rb_{i})(rb_i,b_{i})w&=(rb_{i+3},ra_{i+6}) \ [ra_{i+6},R,rb_{i}] \cdot (rb_i,b_{i})w; \\
[rb_{i+3},R,b_i] \cdot (b_{i},rb_{i})(rb_i,rb_{i+4})w&=(rb_{i+3},b_{i+3}) \ [b_{i+3},R,rb_{i}] \cdot (rb_i,rb_{i+4})w. \qed
\end{align*}
\end{example}
\endgroup
| -101,909.761891 |
[
-3.29296875,
3.015625
] | 22.01087 |
[
-2.634765625,
0.2236328125,
-2.59765625,
-6.5390625,
-1.330078125,
9.890625
] |
[
4.6015625,
8.984375,
3.427734375,
7.01171875
] | 829 | 11,773 |
[
-3.375,
4.05078125
] | 36.795631 |
[
-5.484375,
-3.6171875,
-5.765625,
-2.66015625,
1.345703125,
13.9296875
] | 1.039188 | 13.82072 | 17.939353 | 10.805125 |
[
1.5917317867279053
] | -62,474.110411 | 5.247516 | -101,453.004857 | 0.615096 | 6.241837 |
[
-1.5322265625,
-3.3359375,
-4.0390625,
-5.55078125,
1.8779296875,
12.6171875
] |
[
-6.11328125,
-2.328125,
-2.224609375,
-1.046875,
4.56640625,
4.90625
] | |
BkiUdV45i7PA9OlIEcJP
|
\section{Introduction}
Unmanned aerial vehicles (UAVs) have gained much attention in advanced communication systems \cite{H18UAV, CG17, CE17, ZL16, MP15, BY16, LY18, MD16, MD18, CH17}. Due to the advantages such as mobility, flexibility, efficiency, and low-cost in deployment, cellular-connected UAVs have potentials in 5G beyond and IoT \cite{MD18, CB18, H18LIS, H18LIS1} systems. UAVs can be integrated into multi-tier relay networks as amplify-and-forward (AF) and decode-and-forward (DF) nodes to increase data-throughput and received signal-to-noise ration (SNR) \cite{CG79, WJ05, HA03, FB09, CK10}. This is particular helpful for sudden appearances of massive connections such as in a stadium or an outdoor concert, and also for cellular users that are far away from a base-transceiver station (BTS) or obstructed by surrounding objects. UAVs can boost the connections in these circumstances by means of amplifying and beaming the signals.
Compared to a traditional terrestrial relay system, a UAV-assisted relay system is very flexible and has the capability to adapt its deployment according to real-time situations to maximize the performance. Further, the UAVs can appear anywhere anytime when there is an assignment, which makes it a powerful assistance to traditional cellular systems. By exploiting the flexibility in deployment, in this paper we consider multiple UAV-tier assisted communication systems in cellular networks. The target is to optimize the deployment for a given total number of UAVs through maximizing the ergodic capacity under different practical scenarios.
Previous works on UAV assisted cellular networks can be referred to e.g., \cite{BY16, LY18, MD16, MD18, CH17}. The potential and challenges of using UAVs in cellular networks were discussed in \cite{BY16, MD16, MD18, CH17}, as well as initial performance evaluation and trade-offs. The energy-efficiency and power control of UAVs were considered in \cite{WW18} and \cite{AY17}. The channel modeling and measurement of the air-to-ground (A2G) and air-to-air (A2A) communication channels were carried out in \cite{KD18, MS17, AJ14}. One observation from \cite{MS17, AJ14} is that both A2G and A2A channels will not always contain a line-of-sight (LoS) component. Especially in urban environment and for low-altitude UAVs, there are rich reflections and diffractions by surface-based obstacles such as tall-buildings, terrain, trees, and the UAV itself. For this fact and also for analytical tractability, in this paper we model the channels of considered UAV-assisted relay systems as independent and identically distributed (i.i.d.) Rayleigh channel. Although we do not consider other channel models, the analysis in this paper can be applied as a basis for studies on other channels. For instance, under the cases that there is LoS which yields Rician fading \cite{MS17}, the communication property obtained based only on the i.i.d. assumption can still apply. Moreover, under the cases that UAVs are deployed in rural areas or with high altitude, the LoS component becomes a dominant factor and an optimal deployment of UAVs is to minimize the distance between two adjacent tiers \cite{WB03, KA06} by using a single UAV at each tier.
By modeling the multi-input and multi-output (MIMO) channels between different UAV-tiers as i.i.d. Rayleigh fading, the effective propagation channel between the transmitting users and the receiving BTS is modeled as a Rayleigh product channel. The Rayleigh product channel origins from a double-scattering model \cite{GP02}, which comprises two i.i.d. channel components. Literatures addressing the achievable rate and diversity-multiplexing trade-off under such channels can be seen e.g. in \cite{ZS04, JG08}. Latter, Rayleigh product channel is extended to comprise three channel components in \cite{FM10}, and then eventually to an ensemble of arbitrary $K$ i.i.d. Rayleigh MIMO channels as in \cite{YB07, AK13, RK14, F14}. Although eigenvalue statistics and ergodic capacity have been discussed in \cite{AK13, RK14, R02} for a Rayleigh product channel with $K$ tiers (in our case, $K\!-\!1$ UAV-tiers and one last BTS-tier), the results are based on hyper-geometric \textit{Meijer} $G$-function \cite{ET55} which is difficult to analyze. Capacity results for normal Rayleigh MIMO channels\footnote{By a normal Rayleigh MIMO channel, we refer to a direct MIMO channel between the users and the BTS without UAV, i.e., $K\!=\!1$.} can be reviewed as a special case with $K\!=\!1$ \cite{T99, AK13}.
To optimize the number of tiers for a given total number\footnote{To simplify the description, we assume that both users and UAVs are equipped with a single-antenna. The cases that a user or a UAV is equipped with multiple antennas following similar analysis by treating each antenna as a separate user or UAV, respectively.} of $M$ UAVs, in principle one needs to evaluate all the integer partition sets of $M$ to find a partition set that maximizes the ergodic capacity $\tilde{R}$. With the analytical-form of $\tilde{R}$ in \cite{AK13}, calculating it for all partition sets requires extensive numerical computations or look-up-table operations, which renders a high cost and processing latency in real-time applications\footnote{As what becomes clear latter, the optimal number of UAV-tiers changes under different settings such as the numbers of antennas of the users and the BTS, the transmitting power, and the power attenuation factor. Therefore, the UAVs may need the capability to adapt to different practical scenarios.}. To simply the expression of $\tilde{R}$, one direct approach is to approximate it with an upper-bound through Jessen\rq{}s inequality. Such an obtained upper-bound can be tight when dimensions of the MIMO channel are sufficiently large, such as with traditional massive MIMO systems~\cite{T99, HT04}. However, with Rayleigh product channel the upper-bound becomes loose, due to the fact that the approximation errors of the upper-bound increases when the total number of tiers (i.e., the number of component random matrices in the Rayleigh product channel) increases. Therefore, finding other tight bound of $\tilde{R}$ is of interest.
For a given setting of UAV-tiers, the UAV-based relay system is similar to a traditional terrestrial relay system with the same settings. However, to our best knowledge, there is little work on considering optimizing the ergodic capacity $\tilde{R}$ under Rayleigh product channels. Previous works considering approximating and asymptotic properties of $\tilde{R}$ in multi-tier terrestrial wireless relay systems can be found in e.g., \cite{FB09, HA03,CK10, LS12, LL10, M14, NL11, S04}. But these works either consider the case that each tier has only a single-antenna relay \cite{FB09, HA03,CK10}, or there is only a single intermediate tier (the case when $K\!=\!2$) \cite{LL10}. Therefore, our analysis on ergodic capacity in the first part is also meaningful for traditional relay systems. There are also works consider the optimal of number of tiers in multi-tier terrestrial relay systems from different perspective. The authors in \cite{LS12} and \cite{M14} consider optimal power allocation and relay placements for multi-tier systems. In \cite{NL11}, the authors consider the optimal number of hops in a linear multi-tier AF relay model with maximizing a random coding error exponent (RCEE) instead of achievable rates. However, the expression of RCEE is also complex which make a direct optimization difficult. In \cite{S04}, the authors consider the optimum number of hops with time division multiple access (TDMA) multi-tier transmissions, which is optimized to minimize the transmission power for a given end-to-end rate.
In \cite{H18UAV}, we have derived a tight lower-bound of $\tilde{R}$ for the product of two Rayleigh MIMO channels, i.e., a single UAV-tier assisted communication system. We analyze trade-offs between the number of antennas and the transmit power of the UAV-tire in order to have higher ergodic capacity than a direct connection between the uses and the BTS. Following \cite{H18UAV}, we consider tight approximation of the ergodic capacity for multi-tier UAV assisted communication systems, and the optimization of number of tiers that maximizes $\tilde{R}$ for a given $M$ UAVs. We point out that we only consider the cases that the distance between adjacent UAV-tiers are relatively far and the channel can be model as Rayleigh fading such as in \cite{FB09, HA03,CK10, LS12, LL10, M14, NL11, S04}. Further, although we consider uplink transmission from users to the BTS, the analysis also applies to downlink transmission due to the channel reciprocity.
Although TDMA transmission can be used to mitigate cross-talks among UAV-tiers, one drawback is that $\tilde{R}$ is linearly scaled down by $K$. This renders the outcome that as SNR increases, the optimal number of tiers quickly decreases to 1 \cite{S04, OS06}. In our considered UAV-assisted system, we assume that the communications among UAV-tiers use approaches such as frequency-division multiplexing access (FDMA) \cite{MD16} (i.e., different tiers transmit on different frequency bands) or code-division multiplexing access (CDMA) \cite{LD13} (i.e., different tiers use orthogonal codes to spread transmit data). Further, with a pipelined transmission scheme on top of that, the number of tiers $K$ has negligible impact on the ergodic capacity \cite{FM10, NK04}, at a cost of wider bandwidth which can use free WIFI frequency band such as at 2.4 GHz or other bandwidth dedicated for UAV communications.
Assuming there are $N_0$ users are connecting to a BTS with $N_K$ receiving antennas through $M$ UAVs, which are know aforehand. There are many integer partition sets of $M$ with
\begin{eqnarray} \label{partM} \sum\limits_{k=1}^{K-1}N_k\!=\!M.\end{eqnarray}
With each partition scheme in (\ref{partM}) and together with $N_0$ and $N_K$, we form a UAV-assisted communication system with $K$ tiers according to a parameter setting ($N_0,\,N_1,\,\cdots,\,N_K$), where $N_k$ denotes the number of UAVs at the $k$th tier. Since the spatial multiplexing gain is determined by the minimum value of $N_k$ ($0\!\leq\!k\!\leq\!K$), it is not always optimal to put all UAVs in a single UAV-tier, i.e., setting $K\!=\!2$. On the other hand, there can be multiple schemes in (\ref{partM}) that have the same spatial multiplexing gain, but render different gains in terms of power attenuation and information-rate. With the derived lower-bound, these trade-offs can be directly evaluated and based on which, the number of tiers can be optimized to maximize the ergodic capacity $\tilde{R}$.
The main contribution of this paper are as follows:
\begin{itemize}
\item We derive a lower-bound of the ergodic capacity $\tilde{R}$ for Rayleigh product channel that comprises arbitrary $K$ i.i.d. rectangular Rayleigh MIMO channels with arbitrary dimensions. We show that the lower-bound is asymptotically tight as SNR increases and has a much simpler closed-form than its original form.
\item We show that the approximation error $\Delta\epsilon$ of the trivial upper-bound by switching the order of expectation operation and \lq\lq{}$\ln\!\det$\rq\rq{} function, asymptotically satisfies
\begin{eqnarray} \label{eq2} \Delta\epsilon>N_0\sum_{k=1}^{K}\frac{1}{2N_k},\end{eqnarray}
which increases when $K$ increases, and $N_0$ is the minimum value of $N_k$ ($0\!\leq\!k\!\leq\!K$)
\item We show the differences between different settings of Rayleigh product channels such as with rectangular or square MIMO components. As a special case, adding an extra antenna\footnote{Note that, this result is only for Rayleigh product channel. For UAV-assisted cases, the impact on the received SNR also needs to be considered.} to the $k$th tier whose original number of antennas is $N_k$ can bring an increment to the ergodic capacity as
\begin{eqnarray} \label{eq3} \Delta\tilde{R}=\sum_{r=1}^{N_0}\frac{1}{N_k-\ell+1}.\end{eqnarray}
\item We analyze the optimal number of tiers for a given total $M$ UAVs with the derived bounds of $\tilde{R}$, and we show that the lower-bound based optimization is close-to-optimal and has much less computational-cost. Further, we propose an effective algorithm that significantly reduces the size of searching sets in the procedure, where we show that in general the optimal number of UAV-tier $K$ is
\begin{eqnarray} \label{Ksub} K=\max\left(1+\biggl\lfloor \frac{M}{\min\{N_0, N_K\}}\biggr\rfloor,\;2\right). \end{eqnarray}
\item We also analyze the asymptotic properties of the solutions and show that under low and high SNR cases, using a single UAV-tier ($K\!=\!2$) and setting each tier with a single UAV ($N_k\!=\!1$,$0\!<\!k\!<\!K$) is optimal for the two extreme cases, respectively.
\end{itemize}
The organization of the paper is as follows. In Section II, we briefly introduce the Rayleigh product channel model and the ergodic capacity. We also show a communication property between the tiers, and the high SNR property in Theorem 1. In Section III, we derive upper and lower bounds of the ergodic capacity and analyze the differences between them. We show the asymptotic properties of the lower-bound, and compare the ergodic capacity differences for different parameter settings of the Rayleigh product channel. In Section VI, we consider the number of tier optimization in the UAV-assisted systems, and propose a low-complexity algorithm which is shown to be effective. Simulation results are presented in Section V, and Section VI summarizes the paper.
\subsubsection*{Notation}
Throughout the paper, a capital bold letter such as $\vec{A}$ represents a matrix, a lower case bold letter $\vec{a}$ represents a vector, and matrix $\vec{I}$ represents an identity matrix. The superscripts $(\cdot)^\dag$ denotes the conjugate transpose of a matrix, and $(\cdot)^{-1}$ is the inverse. Further, $\ln(\cdot)$ is the natural logarithm function, $\det(\cdot)$ is the determinant, $\lfloor \cdot\rfloor$ and $\mod(\cdot)$ denotes the floor and modulo operations, respectively. In addition, $\mathbb{E[\cdot]}$ is the expectation operator, $\mathrm{Tr(\cdot)}$ takes the trace of a matrix, and $\min(\cdot)$ takes the minimum of inputs.
\begin{figure*}[t]
\vspace*{-25mm}
\begin{center}
\hspace*{-30mm}
\scalebox{0.36}{\includegraphics{RayleighProdChan_Model.pdf}}
\vspace*{-34mm}
\caption{\label{fig1}Rayleigh product channel model in a UAV-assisted communication system with $K$ tiers, where the first $K\!-\!1$ tiers are the UAVs and the last tier is the BTS. The channels $\vec{Q}_k$ between adjacent tiers are modeled as i.i.d. Rayleigh MIMO channels, multiplying with the factors of power attenuations.}
\vspace*{-6mm}
\end{center}
\end{figure*}
\section{Preliminaries}
\subsection{Rayleigh Product Channel}
Consider a MIMO received signal model
\begin{eqnarray} \label{md1} \vec{y}=\sqrt{q}\vec{H}{x}+\vec{n}, \end{eqnarray}
where $\vec{x}$ is the transmitted symbols from one or multiple users, and the transmit power\footnote{Although we call $q$ the transmit power, it however, denotes the combined impact of the transmit power, the power-amplifying in all UAV tiers, and the propagation losses.} is denoted as $q$. The channel $\vec{H}$ is of size $N_K\!\times\!N_0$, where $N_K$ denotes the number of receive antennas at a BTS, and $N_0$ is the total number of transmit antennas for users transmitting to the BTS simultaneously. For simplicity, we model $\vec{n}$ as additive Gaussian white noise (AWGN) with zero-mean and unit-variance.
In a UAV-assisted communication system such as depicted in Fig. 1, the channel $\vec{H}$ is modeled as a Rayleigh product model. That is, each tier of the UAVs as depicted is assumed to be an independent scatter that beams the received signal from the previous tier (and with possible power-amplifying) to the next tier until it reaches the BTS. In other words, we assume
\begin{eqnarray} \label{Hmd} \vec{H}=\vec{Q}_K\times\vec{Q}_{K-1}\times\cdots\times\vec{Q}_1=\prod\limits_{k=1}^K\vec{Q}_k, \end{eqnarray}
where $\vec{Q}_k$ are of size $N_k\!\times\!N_{k-1}$, and comprise of i.i.d. complex-valued Gaussian elements with zero-mean and unit-variance. Therefore, a parameter setting $(N_0,\,N_1,\,\cdots,\,N_K)$ uniquely determines the structure of $\vec{H}$. Note that in the rest of the paper, when we use the term $\prod\limits_{k=1}^K\vec{Q}_k$, it is always refereed to the multiplexing order in (\ref{Hmd}). Further, in order to model the UAV-assisted communication systems with the received signal model (\ref{md1}), we assume that the received noise is white, which can be due to the fact that the noise power at each UAV is negligible compared to the received signal.
\subsection{A Communication Property of the Ergodic Capacity}
The capacity (nats per channel use) corresponding to the received signal model (\ref{md1}) equals
{\setlength\arraycolsep{1pt} \begin{eqnarray} \label{R} R_{\left(\vec{Q}_1, \vec{Q}_2, \dots,\vec{Q}_K\right)}&=&\ln\!\det\!\left(\vec{I}\!+\!q\vec{H}^{\dag}\vec{H}\right) \notag \\
&=&\ln\!\det\!\left(\!\vec{I}\!+\!q\left(\prod\limits_{k=1}^K\vec{Q}_k\!\right)^{\dag}\!\left(\prod\limits_{k=1}^K\vec{Q}_k\right)\!\!\right)\!\!, \notag \\ \end{eqnarray}
\hspace{-1.2mm}and the ergodic capacity is
\begin{eqnarray} \label{tR} \tilde{R}= \mathbb{E}\!\left[R_{\left(\vec{Q}_1, \vec{Q}_2, \dots,\vec{Q}_K\right)}\right]\!,\end{eqnarray}
where the expectation is taken over the probability density function (pdf) of $\vec{Q}_k$. Since different UAVs tier are independent from the others, we assume that \cite{G63}
\begin{eqnarray} p\big(\vec{Q}_1, \vec{Q}_2, \dots,\vec{Q}_K\big)&=&\prod\limits_{k=1}^K p\big(\vec{Q}_k\big) \notag \\
&=&\prod\limits_{k=1}^K\frac{\exp\Big(\!\!-\!\text{Tr}\big\{Q_{k}^{\dag} Q_{k}\big\}\Big)}{\pi^{N_{K\!-\!1}N_K}}.\;\end{eqnarray}
We first state Property 1 that shows that permuting $N_k$ in a Rayleigh product channel will not change the ergodic capacity $\tilde{R}$, which is known as a weak commutation property for a product of i.i.d. random matrices in \cite{RK14, AK13}.
\begin{property}
The ergodic capacity $\tilde{R}$ of the Rayleigh product channel (\ref{md1}) is invariant under permutations of $(N_0, N_1, \cdots, N_K)$.
\end{property}
\begin{proof}
See Appendix A.
\end{proof}
With Property 1, the analysis of $\tilde{R}$ for Rayleigh product channel is significantly simplified, as the order of $N_k$ is independent from the achieved ergodic capacity. With proper permutations we can always assume $N_0\!\leq\!N_1\!\leq\cdots\leq\!N_K$ when analyzing the properties of ergodic capacity.
Letting $N_0\!=\!\min\limits_{0\leq k \leq K}\{N_k\}$, at high SNR\footnote{By high SNR we mean that either $q$ or the product $\prod\limits_{k=1}^K N_k$ is large, since the mean-value of the diagonal elements in $q\vec{H}^{\dag}\vec{H}$ equals $q\prod\limits_{k=1}^K N_k$. } it holds that
\begin{eqnarray} \label{tRh} \tilde{R}&\approx&N_0\ln q+\mathbb{E}\!\left[\ln\det\!\left(\vec{H}^{\dag}\vec{H}\right)\right]\!.\end{eqnarray}
\begin{lemma}
For a Rayleigh product channel $\vec{H}$ in (\ref{Hmd}), it holds that
\begin{eqnarray} \label{lemeq} \mathbb{E}\!\left[\ln\det\!\left(\!\vec{H}^{\dag}\vec{H}\right)\right]\!=\! \sum_{k=1}^{K}\mathbb{E}\!\left[\ln\det\!\left(\!\hat{\vec{Q}}_k^{\dag}\hat{\vec{Q}}_k\right)\!\right]\!, \end{eqnarray}
where $\hat{\vec{Q}}_k$ are i.i.d. random Rayleigh MIMO channels with dimensions $N_k\!\times\!N_0$.
\end{lemma}
\begin{proof}
See Appendix B.
\end{proof}
From Lemma 1, $\tilde{R}$ can be expressed as a summation over i.i.d. Rayleigh MIMO channels $\hat{\vec{Q}}_k$ at high SNR, but with reduced dimensions $N_k\!\times\!N_0$, instead of the original $N_k\!\times\!N_{k-1}$ of $\vec{Q}_k$. Since $\hat{\vec{Q}}_k^{\dag}\hat{\vec{Q}}_k$ is complex Wishart distributed, it can be readily seen from \cite{OP02, G63} that
\begin{eqnarray} \label{EHH} \mathbb{E}\!\left[\ln\det(\hat{\vec{Q}}_k^{\dag}\hat{\vec{Q}}_k)\right] &=&\sum_{\ell=1}^{N_0}\psi(N_k-\ell+1) \notag \\
&=&-N_0\gamma+\sum_{\ell=1}^{N_0}\sum_{r=1}^{N_k-\ell}\frac{1}{r}, \end{eqnarray}
where the \textit{digamma} function $\psi(n)$ is $$\psi(n)=-\gamma+\sum_{k=1}^{n-1}\frac{1}{k}$$
and $\gamma\!\approx\!0.5772$ is the \textit{Euler-Mascheroni} constant.
Inserting (\ref{EHH}) back into (\ref{lemeq}), we have the below Theorem 1, which will be useful in deriving a lower-bound for $\tilde{R}$.
\begin{theorem}
For a Rayleigh product channel $\vec{H}$, it holds that
\begin{eqnarray} \label{c2} \mathbb{E}\!\left[\ln\det(\vec{H}\vec{H}^{\dag})\right] &=&\sum_{k=1}^{K}\sum_{\ell=1}^{N_{0}}\psi(N_k-\ell+1) \notag \\
&=&-KN_0\gamma+\sum_{k=1}^{K}\sum_{\ell=1}^{N_0}\sum_{r=1}^{N_k-\ell}\frac{1}{r}. \end{eqnarray}
\end{theorem}
From Theorem 1, we see that $N_0$ (the minimum of $N_k$) and $K$ (the total number of tiers) play fundamental roles in the ergodic capacity that can be achieved for the Rayleigh product channel $\vec{H}$.
\section{Bounds of the Ergodic Capacity for Rayleigh Product Channel}
\subsection{Exact-Form of the Ergodic Capacity}
Given the Rayleigh product channel model (\ref{md1}), the ergodic capacity $\tilde{R}$ in (\ref{tR}) can be solved in an analytical-form stated in Lemma 2 \cite{AK13}.
\begin{lemma}
With Rayleigh product channel model (\ref{md1}), the ergodic capacity $\tilde{R}$ (nats per channel use) in (\ref{tR}) equals
\begin{eqnarray} \tilde{R}&=& \!\sum_{n=0}^{N_0-1}\sum_{m=0}^{n}\!\Bigg(\sum_{s=0}^{n}\frac{(-1)^{m+s}n!(n+\nu_1)!}{(n-m)!m!(n-s)!s!(s+\nu_1)!} \notag \\
&& \quad\; \times\, \MeijerG*{K+2,1}{2,K+2}{0,\,1 \\ m+1+\nu_K,\,\cdots,\,m+1+\nu_2,\,m+s+1+\nu_1,\,0,\,0 } {q^{-1}}\!\!\Bigg) \notag \\
&& \quad\; \times\Bigg(\prod_{k=1}^{K}\frac{1}{(m+\nu_k)!}\Bigg),
\end{eqnarray}
where the positive dimension differences
$$ \nu_k=N_k-N_0.$$
\end{lemma}
Although $\tilde{R}$ can be expressed in analytical-form, it is complex to evaluate with involving the \textit{Meijer} $G$-function and the \textit{gmama} function $\Gamma(\cdot)$, and the \textit{Meijer} $G$-function is defined as a line integral on the complex-plane as \cite{ET55}
\begin{eqnarray} \label{exacttR} &&\MeijerG*{m, n}{p, q}{a_1,\,a_2,\,\cdots,\,a_p \\ b_1,\,b_2,\,\cdots,\,b_q } {z}\notag \\
&& =\frac{1}{2\pi i}\bigintss_{L}\frac{\prod\limits_{j=1}^m\Gamma(b_j-s)\prod\limits_{j=1}^n\Gamma(1-a_j+s)}{\prod\limits_{j=m+1}^q\Gamma(1-b_j+s)\prod\limits_{j=n+1}^p\Gamma(a_j-s)}z^s \mathrm{d}z. \notag \end{eqnarray}
Further, it is also difficult to understand the connections between different parameter settings $(N_0, N_1, \cdots, N_K)$ and the attained ergodic capacity $\tilde{R}$. Hence, next we find bounds for $\tilde{R}$ with simpler forms.
\subsection{Upper and Lower Bounds}
By Jessen\rq{}s inequality, the ergodic capacity $\tilde{R}$ can be upper bounded as
{\setlength\arraycolsep{2pt} \begin{eqnarray} \label{ub}\tilde{R}&\leq&\ln\!\det\!\left(\vec{I}+q\mathbb{E}\big[\vec{H}^{\dag}\vec{H}\big]\right)\notag \\
&=&N_0\ln\!\left(\!1+q\prod\limits_{k=1}^{K}N_k\!\right)\!.\end{eqnarray}}
\hspace{-1.4mm}This bound is trivial and widely used to approximate the ergodic capacity for normal Rayleigh channels, such as in massive MIMO systems \cite{T99, HT04}. For Rayleigh product channel ($K\!\geq\!2$), however, this upper-bound becomes loose as what will be explained later. Therefore, seeking another bound that is tight is of interest. Following the similar idea in \cite{H18UAV}, we derive a lower-bound that is asymptotically tight, which is stated in Property 2 together with the aforementioned upper-bound (\ref{ub}).
\begin{property}
The ergodic capacity of the Rayleigh product channel model (\ref{md1}) is bounded as
\begin{equation} \label{tR3} N_0\ln\!\big(1+q\exp\Big(g-K\gamma\big)\Big)\leq \tilde{R}\leq N_0\ln\!\left(\!1+q\prod\limits_{k=1}^{K}N_k\!\right)\!, \end{equation}
where
\begin{eqnarray} \label{gK} g&=&K\gamma+\frac{1}{N_0}\sum_{k=1}^{K}\sum_{\ell=1}^{N_0}\psi(N_k-\ell+1)\notag \\
&=& \frac{1}{N_0}\sum_{k=1}^{K}\sum_{\ell=1}^{N_0}\sum_{r=1}^{N_k-\ell}\frac{1}{r}. \end{eqnarray}
\end{property}
\begin{proof}
See Appendix C.
\end{proof}
\subsection{Asymptotic Properties of the Bounds}
Under cases that $q\exp\Big(g-K\gamma\big)\!\gg\!1$, it holds that
\begin{equation} N_0\ln\!\big(1+q\exp\Big(g-K\gamma\big)\Big)\approx N_0\ln q+N_0\Big(g-K\gamma\big). \end{equation}
Then, from (\ref{tRh}) and Theorem 1 we have the below corollary.
\begin{corollary}
The lower-bound in (\ref{tR3}) for the ergodic capacity $\tilde{R}$ is asymptotically tight.
\end{corollary}
To show the gap between the derived upper and lower bounds, we notice that the difference between them is asymptotically equal to $\Delta\epsilon$ and
\begin{eqnarray} \frac{\Delta\epsilon}{N_0}=-\big(g-K\gamma\big)+ \sum\limits_{k=1}^{K}\ln N_k. \end{eqnarray}
Using the approximation of \textit{digamma} function \cite{ET55} that
$$\psi(x)\approx\ln x-\frac{1}{2x}, \;\; x\!>\!1,$$
and by the definition of $g$, $\Delta\epsilon$ can be approximated as
\begin{eqnarray} \frac{\Delta\epsilon}{N_0}&=&-\frac{1}{N_0}\sum_{k=1}^{K}\sum_{\ell=1}^{N_0}\psi(N_k-\ell+1)+ \sum\limits_{k=1}^{K}\ln N_k\notag \\
&\approx& \frac{1}{N_0}\sum_{k=1}^{K}\sum_{\ell=1}^{N_0}\!\left(\!\ln\!\left(\!\frac{N_k}{N_k-\ell+1}\!\right)\!+\frac{1}{2(N_k-\ell+1)}\!\right)\!\!.\qquad \end{eqnarray}
Therefore, the ergodic capacity difference $\Delta\epsilon$ satisfies
\begin{eqnarray} \label{deltae} \Delta\epsilon&>&\sum_{k=1}^{K}\sum_{\ell=1}^{N_0}\frac{1}{2(N_k-\ell+1)} \\
&\geq&N_0\sum_{k=1}^{K}\frac{1}{2N_k}. \notag \end{eqnarray}
As the lower-bound is asymptotically tight, the ergodic capacity difference in (\ref{deltae}) is asymptotically equal to the errors between the upper-bound and the exact value of $\tilde{R}$. As can be seen from (\ref{deltae}), the error $\Delta\epsilon$ increases as $K$ increases, and in order for $\Delta\epsilon$ to be close to zero, it must holds that $N_k\!\gg\!N_0$ for $k\!>\!0$.
Using Corollary 1 we can obtain a below corollary for a normal Rayleigh MIMO channel (i.e., $K\!=\!1$).
\begin{corollary}
The ergodic capacity $\tilde{R}$ when $q\!\to\!\infty$ for a normal Rayleigh channel $\vec{H}$ of sizes $N_0\!\times\!N_1$ ($N_1\!\geq\!N_0$) can be approximated as
\begin{eqnarray} \label{ntR}\tilde{R}&=&\mathbb{E}\big[\ln\!\det\!\left(\vec{I}+q\vec{H}^{\dag}\vec{H}\right)\!\big]\notag \\
&\approx& N_0\big(\ln\! q-\gamma\big)+\sum_{\ell=1}^{N_0}\sum_{r=1}^{N_1-\ell}\frac{1}{r}. \end{eqnarray}
\end{corollary}
Note that when $N_1\!\gg\!N_0$, the harmonic series
$$\sum_{r=1}^{N_1}\frac{1}{r}\approx\ln\! N_1+\gamma,$$
and (\ref{ntR}) becomes
\begin{eqnarray} \label{hsnrbd} \tilde{R}\approx N_0\ln\! \big(qN_1\big), \end{eqnarray}
which is aligned with the upper-bound in Property 2 for the case $K\!=\!1$. However, (\ref{hsnrbd}) only holds for cases $N_1\!\gg\!N_0$, but the derived (\ref{ntR}) holds for general settings of $N_0$ and $N_1$.
\subsection{The Connection between Rectangular and Square Random Matrices in Rayleigh Product Channel}
Based on Property 2, we have Property 3 that states the ergodic capacity difference between the Rayleigh product channel formed by a number of rectangular and square i.i.d. random matrices.
\begin{property}
At high SNR the ergodic capacity increment $\Delta\tilde{R}$, between Rayleigh product channels (for an identical $q$) with a parameter setting $(N_0,\, N_1,\, \cdots,\, N_K)$ and with square matrices $N_k\!=\!N_0$ ($1\!\leq\!k\!\leq\!K$), is
\begin{eqnarray} \Delta\tilde{R}&=&\sum_{k=1}^{K}\sum_{\ell=1}^{N_{0}}\psi(N_k-\ell+1)-\sum_{k=1}^{K}\sum_{\ell=1}^{N_{0}}\psi(N_0-\ell+1) \notag \\ &=& \sum_{k=1}^{K}\sum_{\ell=1}^{N_{0}}\sum_{s=N_0}^{N_k-1}\frac{1}{s-\ell+1}. \notag \end{eqnarray}
\end{property}
\begin{proof}
See Appendix D.
\end{proof}
To interpret Property 3, as a special case, we consider adding an extra antenna\footnote{Such an operation changes both the dimensions of $\vec{Q}_{k-1}$ and $\vec{Q}_k$. However, if $N_k\!<\!N_{k+1}$, it still holds $\tilde{N}_k\!\leq\!N_{k+1}$. If $N_k\!=\!N_{k+1}$, then adding an extra-antenna to $N_k$ is equivalent to add that antenna to $N_{k+1}$ which yields the same capacity increment.} in the UAV-assisted communication system by increasing $\tilde{N}_k\!=\!N_k\!+\!1$ can bring an increment $\Delta\tilde{R}$ that is asymptotically equal to (\ref{eq3}), that is,
\begin{eqnarray} \Delta\tilde{R}=\sum_{r=1}^{N_0}\frac{1}{N_k-\ell+1}. \notag \end{eqnarray}
\begin{figure*}[t]
\vspace*{-3mm}
\begin{center}
\hspace*{-0mm}
\scalebox{0.43}{\includegraphics{UAV_Relay_Mod_simple.pdf}}
\vspace*{-12mm}
\caption{\label{fig2}Partitioning the total number of $M$ UAVs into $K\!-\!1$ tiers with parameters $(N_1,\,N_2,\,\cdots,\,N_{K-1})$, and the values of $N_0$ and $N_K$ are fixed.}
\vspace*{-6mm}
\end{center}
\end{figure*}
\section{Number of Ties Optimization}
With the derived bounds, in this section we consider optimizing the number of tiers in a UAV-assisted communication system with a given $M$ UAVs. Such a UAV relay system is analogous to a linear multi-tier terrestrial relay system \cite{NL11, S04, OS06}, but the target now is to find an optimal partition of $M$ that splits it into $K\!-\!1$ integers that satisfy (\ref{partM}) according to different scenarios, which yield a deployment of UAV-assisted relay system that can have the highest ergodic capacity for each applied scenario.
There are obvious trade-offs between the number of tiers $K$ and the number of UAVs at each tier $N_k$. When $K$ is larger, the minimum value of $N_k$ becomes smaller. That is, the spatial multiplexing gain is reduced seen from (\ref{tR3}). On the other hand, when $K$ is smaller, the distance between two adjacent tiers are larger. The power attenuation factor can be modeled as \cite{S04}
\begin{eqnarray} \eta\propto\Big(\frac{d}{K}\Big)^{\!-\alpha}, \end{eqnarray}
where $d$ is the distance between the users to the BTS, and $\alpha$ is the path-loss exponent with typical values between 2 and 4.
Further, for fair comparisons we assume that the transmit-power at each antenna of the UAV is equal to $p$. That is, at the $k$th UAV-tier, the received signal at each antenna is scaled by a factor $p/N_{k-1}$, and the total transmit power of all UAVs is equal to $Mp$. Note that, the cases different UAVs with unequal transmit power follow the similar analysis since this will only impact the definition of parameter $q$ in the ergodic capacity of the modeled Rayleigh product channel.
With the above assumptions, the ergodic capacity for the UAV-assisted systems can be modeled as
\begin{eqnarray} \label{tRK}\tilde{R}&=&\mathbb{E}\big[\ln\det\!\left(\vec{I}+q\vec{H}^{\dag}\vec{H}\right)\!\big], \end{eqnarray}
where
\begin{eqnarray} \label{qcon} q=cp_0\frac{K^\alpha p^{K-1}}{\prod\limits_{k=0}^{K-2}N_k},\end{eqnarray}
and $c$ is a constant representing the power attenuation from the users to the BTS with respect to the distance $d$, and $p_0$ is the transmit power from each of the users. Without loss of generality, we let
$$\tilde{p}=(cp_0)^{\frac{1}{K-1}}p,$$
and (\ref{qcon}) can be rewritten as
\begin{eqnarray} \label{qcon1} q=\frac{K^\alpha \tilde{p}^{K-1}}{\prod\limits_{k=0}^{K-2}N_k}.\end{eqnarray}
With the modeled Rayleigh product channel $\vec{H}$ in (\ref{tRK}), the analysis on ergodic capacity in Sec. III can be used for optimizing the deployment of UAVs.
\subsection{Problem Formulation}
The optimization problem can be formulated as:
{\setlength\arraycolsep{2pt} \begin{eqnarray} \label{prbm} &&\underset{\begin{subarray}{c}
K\\
(N_1,N_2,\cdots,N_{K-1}) \end{subarray}}{\text{maximize}} \;\; \tilde{R} \text{\,\;in\;} (\ref{tRK})\notag \\
&&\quad\mathrm{subject\; to\;\;\;\;\;} (\ref{partM}) \text{\;and\;} (\ref{qcon1}).\end{eqnarray}}
\hspace{-1.5mm}where both $K$ and $(N_1,N_2,\cdots,N_{K-1})$ are yet to be optimized, and $N_0$ and $N_K$ are known parameters in computing $\tilde{R}$ that denotes the number of antennas of the users and the BTS, respectively.
An exhaust search over all possible partition sets yields a prohibitive complexity when $M$ is large\footnote{Under certain circumstances, the optimization can be simplified. For instance, if the values $q$ in (\ref{tRK}) are unaltered for different settings such as the UAVs are only used as scatters, then to optimize the ergodic capacity with the upper-bound is equivalent to maximum the product of the elements in the partition set of $M$. The optimal partition follows the rule that $M$ are partitioned only with 2 and 3, and with as many 3\rq{}s as possible \cite{D05, B93}.}. An asymptotic expression of the number\footnote{We use $\#M$ to denote the number of integer partitions of $M$.} of integer partitions for $M$ is \cite{HR18}
\begin{eqnarray} \label{nbset} \#M\approx\frac{1}{4\sqrt{3}M}\exp\left(\! \pi\sqrt{\frac{2M}{3}}\right)\!, \end{eqnarray}
which increases rapidly as $M$ increases. Due to the numerical calculations needed for evaluating hyper-geometric functions, directly solving (\ref{prbm}) with the exact-form of $\tilde{R}$ in Lemma 2 is also complex. Therefore, we consider to use the derived bounds.
If the upper-bound of $\tilde{R}$ in Property 2 is used in the optimization problem (\ref{prbm}), $\tilde{R}$ is approximated as
\begin{eqnarray} \label{optub} \tilde{R}\approx\tilde{N}_0\ln\!\left(\!1+\frac{K^\alpha \tilde{p}^{K-1} N_{K-1}N_K}{N_0}\!\right)\!. \end{eqnarray}
Note that, $\tilde{N}_0$ denotes the minimal values among all $N_k$ including $N_0$ and $N_K$, and $N_0$ is the number of users which is fixed in the optimizations.
Similarly, if we use the lower-bound in (\ref{prbm}), $\tilde{R}$ can be expressed as
\begin{eqnarray} \label{optlb} \tilde{R}\approx\tilde{N}_0\ln\!\left(\!1+\frac{K^\alpha \tilde{p}^{K-1}}{\prod\limits_{k=0}^{K-2}N_k}\exp\Big(g-K\gamma\big)\!\right)\!, \end{eqnarray}
where
$$g= \frac{1}{\tilde{N}_0}\sum_{k=1}^{K}\sum_{\ell=1}^{\tilde{N}_0}\sum_{r=1}^{N_k-\ell}\frac{1}{r}. $$
As can be seen, both optimizations with the expressions in (\ref{optub}) and (\ref{optlb}) need to search over all possible partition sets of $M$, despite that the optimization (\ref{optub}) is slightly simple since only $\tilde{N}_0$ and $N_{K-1}$ need to be considered.
\subsection{The Proposed Optimization Routine}
To further reduce the search-size in the optimizations, we notice that there are two main principles to maximize the ergodic capacity $\tilde{R}$ for a given $\tilde{p}$:
\begin{enumerate}
\item $\tilde{N}_0$, the minimum of all $N_k$, shall be maximized.
\item $K$, the total number of tiers, shall be maximized to reduce the power attenuations.
\end{enumerate}
With the above two principles, for a given pair $(N_0, N_K)$, the optimal value of $K$ can be determined via (\ref{Ksub}), that is,
\begin{eqnarray} K=\max\left(1+\biggl\lfloor \frac{M}{\min\{N_0, N_K\}}\biggr\rfloor,2\right)\!. \notag \end{eqnarray}
Then, what left is to find all possible partitions sets for the remainder
$$R=M-(K-1)\min\{N_0, N_K\},$$
if $R\!>\!0$. That is, denoting $(r_1, r_2,\cdots,r_t)$ as a partition set of $R$, the corresponding partition set of $M$ is set to
\begin{eqnarray} \label{subpart} N_k=\left\{\begin{array}{cc}\min\{N_0, N_K\}\quad & \quad 1\leq k\leq K-t-1,\\
\min\{N_0, N_K\}+r_t\quad & \quad K-t\leq k\leq K-1 . \end{array}\right. \quad \end{eqnarray}
In total only $\#R$ partition sets need to be evaluated, which yields great search-size reduction, due to the fact that
$$\#R\!\ll\!\#M.$$
For instance, letting $\min\{N_0, N_K\}\!=\!3$ and $M\!=\!16$, the number of partition sets $\#M\!=\!231$, while $\#R\!=\!1$ as $R\!=\!1$. That is, the optimal solution is directly given as the partition set $\{5, 5, 6\}$ of $M$, which is also aligned with the numerical simulation result in Fig. 7 shown later Sec. V.
To further reduce the complexity, a further simplification is to let $r_1\!=\!R$, which gives a suboptimal solution of (\ref{prbm}) directly as
\begin{eqnarray} \label{subpart1} N_k=\left\{\begin{array}{cc}\min\{N_0, N_K\}\qquad & \qquad 1\leq k\leq K-2,\\
\min\{N_0, N_K\}+R\qquad & \qquad k=K-1 . \end{array}\right.\end{eqnarray}
The idea is to maximize $q$ in (\ref{qcon}) for a given $K\!-\!1$, or equivalently, minimize the term $\prod\limits_{k=0}^{K-2}N_k$ by adding the reminder $R$ onto $N_{K-1}$.
As an example, in Table I we list the optimal $K$ for $M\!=\!20$ and different values of ($N_0$, $N_K$). One disadvantage of the suboptimal solution (\ref{subpart}) is that the impact of SNR is not taken into account. However, as shown later by simulation results, the suboptimal solution (\ref{subpart}) is close-to-optimal in a wide range of SNR values. At extreme low or high SNR values, the optimal solutions can also be derived as shown in the next. Therefore, a practical optimizing approach is to combine both (\ref{subpart}) and the asymptotic solutions.
\begin{table}[ht!]
\renewcommand{\arraystretch}{1.5}
\vspace{-0mm}
\centering
\caption{Optimal $K$ in (\ref{Ksub}) with $M\!=\!20$ and different ($N_0$, $N_K$).}
\label{tab1}
\vspace{-1mm}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline
\backslashbox{$N_0$}{$N_K$} & 8 &16& 32 & 48& 64 &96&128&256 \\ \hline
2&11&11 &11&11&11&11 &11&11\\ \hline
4&6&6&6&6&6&6&6&6\\ \hline
6&4&4&4&4&4&4&4&4\\ \hline
8&3&3&3&3&3&3&3&3 \\ \hline
10&3&3&3&3&3&3&3&3\\ \hline
12&3&2 &2&2&2&2&2&2 \\ \hline
14&3&2&2&2&2&2&2&2 \\ \hline
16&3&2 &2&2&2&2&2&2\\ \hline
\end{tabular}
\vspace{-2mm}
\end{table}
\subsection{Asymptotic Solutions and Practical Optimization Procedure}}
Under the case that $p\!\gg\! K$ is sufficiently large, it holds from both (\ref{optub}) and (\ref{optlb}) that
$$ \tilde{R}\approx\tilde{N}_0(K-1)\ln q.$$
In such case, the optimal number of tiers can be optimized through maximizing $\tilde{N}_0(K-1)$. For a given $\tilde{N}_0$, the maximal value of $K\!-\!1$ is $\bigl\lfloor M/\tilde{N}_0\bigr\rfloor$, and hence,
$$\max(\tilde{N}_0(K-1))\leq M,$$
which can be achieved by a partition set\footnote{However, a partition set with all 1\rq{}s is not a unique solution to achieve the maximum. For instance, $\tilde{N}_0\!=\!K\!-\!1\!=\!4$ is also optimal for $M\!=\!16$.} with all $N_k\!=\!1$. This is to say, when SNR increases, setting the number of tiers to $M$ with each UAV-tier only containing a single UAV is close to optimal.
On the other hand, under the case that $p$ is sufficiently small, $p^{K-1}$ decreases as $K$ increases, and the optimal number of tiers is $K\!=\!2$, that is, using a single UAV-tier comprising all $M$ UAVs is close-to-optimal.
Note that, these conclusions are different from the observations in \cite{S04, OS06}, due to the fact that they assume TDMA transmission schemes and the ergodic capacity $\tilde{R}$ linearly decreases in $K$. However, similar optimizations for the number of tiers with TDMA transmissions can follow the same analysis shown above.
Combing the above discussions, a practical optimization procedure for the number of UAV-tier with low-complexity is to find the partition set that maximizes the lower-bound derived in (\ref{optlb}), and with the partition sets defined in (\ref{subpart}) and the two asymptotic settings for low and high SNR cases. Such an optimizing approach yields a significantly reduced search-size of ($\#R+2$), and is more robust against SNR changes.
\section{Numerical Results}
In this section, we show simulations results with the consider UAV-assisted communication systems and the Rayleigh product channels. We use various settings such that the previous elaborated properties can be clearly explained.
\subsection{Tightness of the Lower-bound}
In Fig. 3 and 4, we show comparisons between the derived bounds and the numerical results of the ergodic capacity $\tilde{R}$. In both cases we test with a UAV-assisted system with three tiers, i.e., $K\!=\!3$. In Fig. 3, we set $N_0\!=\!N_1\!=\!N_2\!=\!4$, and $N_3\!=\!8$, while in Fig. 4 we set $N_0\!=\!4$, $N_1\!=\!N_2\!=\!8$, and $N_3\!=\!16$, respectively. As can be seen, in both cases, the derived lower-bounds are much tighter than the traditional upper-bounds. Further, as $q$ increases, the lower-bounds become tight and converge to the exact $\tilde{R}$. Moreover, with larger values of $N_k$ such as in Fig. 4, the lower-bound is also tight even with small values of $q$. These results are well aligned with the derivations in Sec. III-B.
\subsection{Asymptotic Properties}
In Fig. 5, we show asymptotic properties of $\tilde{R}$ with numerical simulations. We evaluate $\tilde{R}$ for three different scenarios, but all with $K\!=\!4$, $N_0\!=\!3$, and $N_4\!=\!8$. In the first case, we set $N_1\!=\!N_2\!=\!4$, while in the second case we only increase $N_2\!=\!5$ and the others remain unchanged. According to Property~3, adding one extra antenna the increment of $\tilde{R}$ under high SNR equals
$$ \sum_{r=1}^{3}\frac{1}{r+1}\approx 1.08. $$
In the third case, we further increase both $N_2\!=\!5$ and $N_3\!=\!6$, the increment of $\tilde{R}$ over the first case under high SNR according to Property~3 now is
$$ 2\sum_{r=1}^{3}\frac{1}{r+1}+\sum_{r=1}^{3}\frac{1}{r+2}\approx 2.95. $$
As can be seen, these two values are well aligned with the numerical results shown in the lower part of Fig. 5, where we use the ergodic capacity of the latter two cases and subtract them from the first case, respectively.
\begin{figure}[t]
\vspace*{-3mm}
\begin{center}
\hspace*{-5mm}
\scalebox{0.42}{\includegraphics{bounds_4448.pdf}}
\vspace*{-8mm}
\caption{\label{fig3}The ergodic capacity with $K\!=\!3$, $N_0\!=\!N_1\!=\!N_2\!=\!4$, and $N_3\!=\!8$.}
\vspace*{-4mm}
\end{center}
\end{figure}
\begin{figure}
\vspace*{-2mm}
\begin{center}
\hspace*{-5mm}
\scalebox{0.42}{\includegraphics{bounds_488_16.pdf}}
\vspace*{-8mm}
\caption{\label{fig4}The ergodic capacity with $K\!=\!3$, $N_0\!=\!4$, $N_1\!=\!N_2\!=\!8$, and $N_3\!=\!16$.}
\vspace*{-6mm}
\end{center}
\end{figure}
\begin{figure}[t]
\vspace*{0mm}
\begin{center}
\hspace*{-6mm}
\scalebox{0.33}{\includegraphics{ErgoCapIncs.pdf}}
\vspace*{-8mm}
\caption{\label{fig5}The ergodic capacity increments with increasing the number of antennas of the UAVs.}
\vspace*{-4mm}
\end{center}
\end{figure}
\begin{figure}
\vspace*{-2mm}
\begin{center}
\hspace*{-6mm}
\scalebox{0.42}{\includegraphics{New_OptInd_User1to8_eNB1to8Space1_UAV10_q20_alpha2.pdf}}
\vspace*{-8mm}
\caption{\label{fig6}The optimal partitions sets based on different formulas for the ergodic capacity.}
\vspace*{-6mm}
\end{center}
\end{figure}
\begin{figure}[t]
\vspace*{-2mm}
\begin{center}
\hspace*{-5mm}
\scalebox{0.42}{\includegraphics{OptInd_User1to8_eNB1to8Space1_UAV16_q10_alpha3.pdf}}
\vspace*{-8mm}
\caption{\label{fig7}The optimal partition sets based on different formulas for a large $M\!=\!16$.}
\vspace*{-6mm}
\end{center}
\end{figure}
\subsection{Number of Tiers Optimization}
In the remaining simulations followed, for simplicity we always assume $cp_0\!=\!1$ in (\ref{qcon}). In Fig.6, we test the number of tier optimizations using different formulas of $\tilde{R}$ with $p\!=\!20$ dB and an power-attenuation exponent $\alpha\!=\!2$. The total number of UAVs is $M\!=\!10$, and the all 42 partition sets are listed in Table~II (where the partition sets considered in the proposed scheme (\ref{subpart}) are marked in bold and italic font). We test the total 64 different combinations of $N_0$ and $N_K$, i.e., the number of antennas for the users and the BTS, respectively. For a given combination index $N$ ($1\!\leq\!N\!\leq\!64$) in the $x$-axis, the values of $N_0$ and $N_K$ are determined by
\begin{eqnarray} \label{N0} N_0=\Bigl\lfloor \frac{N-1}{8}\Bigr\rfloor+1,\end{eqnarray}
and
\begin{eqnarray} \label{NK} N_K\!=\!\!\!\!\!\mod(N-1,\;8)+1,\end{eqnarray}
respectively.
As can be seen, optimizing the number of tiers with the lower-bound yields identical outputs as the optimal case that uses the exact-form, while there are two discrepancies between the upper-bound and the exact-form based optimizations. Importantly, the optimal partition indexes only varies in 6 different indexes, 1, 6, 21, 30, 35, and 42, which correspond to the partition sets with all 1\rq{}s, all 2\rq{}s, \{3, 3, 4\}, \{5, 5\}, \{4, 6\}, and \{10\}, respectively. This is perfectly aligned with the proposed optimization routine stated in Sec. IV-B and (\ref{subpart}).
In Fig. 7, we test another case with $M\!=\!16$, $p\!=\!10$ dB, and $\alpha\!=\!3$. In this case, $\#M\!=\!231$ and we are not able to list all the partitioning sets. In this case the number of discrepancies between the upper-bound and the exact-form based optimization is 10, while that between the lower-bound and the exact-form is only 3. Moreover, as can be seen, the optimal index also only concentrated on 7 different indexes, 1, 9, 49, 64, 131, 186, and 201, which correspond to partition sets 16 1\rq{}s, 8 2\rq{}s, \{3, 3, 3, 3, 4\}, \{4, 4, 4, 4\}, \{5, 5, 6\}, \{8, 8\}, and \{7, 9\}, respectively, which are also well aligned with the proposed solution in (\ref{subpart}). This means that for the upper and lower bounds as well as exact-form based optimizations, checking the sets defined in (\ref{subpart}) provides the same results as evaluating all 231 possible sets, which has a significant complexity reduction. Furthermore, we see that the direct solution in (\ref{subpart1}) is suboptimal since the optimal partition set can be \{7, 9\} for both $N_0\!=\!6$ and 7, but (\ref{subpart1}) generates the solution for $N_0\!=\!6$ as \{6, 10\}.
In Fig. 8, we test the ergodic capacity $\tilde{R}$ with $M\!=\!8$, $N_0\!=\!4$, and $\alpha\!=\!2$. We compare the proposed solution in (\ref{subpart}), in this case, using two UAV-tiers with each containing 4 UAVs, with other heuristic schemes that use 8, 4, and 1 UAV-tiers, respectively. The values of $N_K$ changes from 8 to 64, and the power $q$ (in dB) is modeled as a random Gaussian variable with both mean and variance equal to 10. As can be seen, the partitioning set with the proposed solution quite close to the optimal scheme, with the latter one optimized individually for each SNR realization and with an exhaustive search over all 22 possible partitions. This verifies the validity of the proposed low-complexity optimization scheme with (\ref{subpart}) in practical applications.
\subsection{Asymptotic Solutions}
In Fig. 9 we test the ergodic capacity $\tilde{R}$ under different values of $p$, with $M\!=\!8$ and $\alpha\!=\!3$. We set $N_0\!=\!4$ and $N_K\!=\!8$. In this case, the number of total possible partition sets is 22. As can be seen, when $q\!=\!-15$ and 0 dB, the highest ergodic capacity is attained by a single UAV-tier, i.e., with the partition set \{8\}. When $p$ increases to 15 dB, the highest ergodic capacity is attained by partition set \{4,\,4\} with a two tiers. As $p$ further increases to 20 dB, the highest ergodic capacity now is attained by partition with all 1\rq{}s, i.e., 8 UAV-tiers and each tier contains only a single UAV. This is also well aligned with the analysis in Sec. IV-C.
\section{Summary}
In this paper, we have considered the ergodic capacity and tier optimization in UAV-assisted communication systems. With multiple tiers of UAVs, the channel between the users and the BTS can be modeled as a Rayleigh product channel. We then have derived a tight lower-bound for the ergodic capacity which is asymptotically tight in high SNR regime or with a large number of UAVs. Further, with the derived lower-bound, the ergodic capacity difference between different Rayleigh product channels can be easily computed. Furthermore, to maximize the ergodic capacity for a given total number of UAVs, we have proposed a low-complexity scheme to optimize the number of tiers in the UAV assisted systems with the derived lower-bound.
\begin{figure}[t]
\vspace*{-2mm}
\begin{center}
\hspace*{-6mm}
\scalebox{0.315}{\includegraphics{PerformanceComps.pdf}}
\vspace*{-8mm}
\caption{\label{fig8}The ergodic capacities obtained with different settings of UAV-tiers. The proposed solution in (\ref{subpart}) which is two UAV-tiers with each containing 4 UAVs, is close to the optimal scheme that is optimized individually for each SNR realization and over all possible partitions. }
\vspace*{-6mm}
\end{center}
\end{figure}
\begin{figure}[t]
\vspace*{-2mm}
\begin{center}
\hspace*{-5mm}
\scalebox{0.42}{\includegraphics{N_0_4_N_K_8_UAV8_new_alpha3.pdf}}
\vspace*{-8mm}
\caption{\label{fig9}The ergodic capacity with different partition sets for different values of $q$ with $N_0\!=\!4$, $N_K\!=\!8$, and $M\!=\!8$.}
\vspace*{-6mm}
\end{center}
\end{figure}
\begin{table*}[b]
\renewcommand{\arraystretch}{1.5}
\vspace{-0mm}
\centering
\caption{Partition Indexes and the correspondent combinations for $M\!=\!10$.}
\label{tab1}
\vspace{-1mm}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
Index & Combination &Index& Combination & Index &Combination \\ \hhline{|=|=|=|=|=|=|}
\bf{1}&\{ \bf{\emph{1 1 1 1 1 1 1 1 1 1}} \}&2 &\{ 1 1 1 1 1 1 1 1 2 \} &3 &\{1 1 1 1 1 1 2 2 \}\\ \hline
4&\{1 1 1 1 2 2 2\} &5 &\{ 1 1 2 2 2 2 \} &\bf{6} &\{ \bf{\emph{2 2 2 2 2}} \} \\ \hline
7 &\{ 1 1 1 1 1 1 1 3 \} &8 & \{ 1 1 1 1 1 2 3 \} &9 &\{ 1 1 1 2 2 3 \} \\ \hline
10&\{ 1 2 2 2 3 \}&11 &\{ 1 1 1 1 3 3 \} &12 &\{ 1 1 2 3 3 \} \\
\hline
13& \{ 2 2 3 3 \} & 14 &\{ 1 3 3 3 \} &15 &\{ 1 1 1 1 1 1 4 \} \\ \hline
16 &\{ 1 1 1 1 2 4 \} &17 &\{ 1 1 2 2 4 \}
&18& \{ 2 2 2 4 \} \\ \hline 19 &\{ 1 1 1 3 4 \} &20 &\{ 1 2 3 4 \}
&\bf{21}& \{ \bf{\emph{3 3 4 }} \} \\ \hline 22 &\{ 1 1 4 4 \} &23 &\{ 2 4 4 \}
&24& \{ 1 1 1 1 5 \} \\ \hline 25 &\{ 1 1 1 2 5 \} &26 &\{ 1 2 2 5 \}
&27& \{ 1 1 3 5 \} \\ \hline 28 &\{ 2 3 5 \} & 29 &\{ 1 4 5 \}
&\bf{30}& \{\bf{ \emph{5 5}} \} \\ \hline 31 &\{ 1 1 1 1 6 \} & 32 &\{ 1 1 2 6 \}
&33& \{ 2 2 6 \} \\ \hline 34 &\{ 1 3 6 \} & \bf{35} &\{ \bf{\emph{4 6}} \}
&36& \{ 1 1 1 7 \} \\ \hline 37 &\{ 1 2 7 \} &38 &\{ 3 7 \}
&39& \{ 1 1 8\} \\ \hline 40 &\{ 2 8 \} &41 &\{ 1 9 \} &\bf{42} &\{ \bf{\emph{10}} \} \\ \hline
\end{tabular}
\vspace{-2mm}
\end{table*}
\section*{Appendix A: Proof of Property 1}
Arguments leading to Property 1 can be found in previous work \cite{RK14, AK13}. Here we provide a simpler and more straightforward proof. We first prove that switching any two adjacent parameters $N_i$ and $N_{i+1}$ will not change the ergodic capacity. That is, the ergodic capacities are the same with Rayleigh product channel of settings ($N_0, N_1, \cdots,N_K$) and ($N_0, N_1, \cdots,N_{i-1}, N_{i+1}, N_i, N_{i+2},\cdots,N_K$), where the parameters $N_i$ and $N_{i+1}$ ($0\!<\!i\!<\!K$) are switched.
For the latter one, we let
\begin{eqnarray} \label{Hmd1} \tilde{\vec{H}}\!&=&\!\left(\prod\limits_{k=1}^{i-1}\vec{Q}_k\right) \left(\tilde{\vec{Q}}_i\tilde{\vec{Q}}_{i+1}\tilde{\vec{Q}}_{i+2}\right)\left(\prod\limits_{k=i+3}^{K}\vec{Q}_k\right)\notag \\
\!&=&\!\vec{A}\left(\tilde{\vec{Q}}_i\tilde{\vec{Q}}_{i+1}\tilde{\vec{Q}}_{i+2}\right)\vec{B}, \end{eqnarray}
where $\vec{A}\!=\!\prod\limits_{k=1}^{i-1}\vec{Q}_k$ and $\vec{B}\!=\!\prod\limits_{k=i+3}^{K}\vec{Q}_k$. Since the statistic properties of $\vec{A}$ and $\vec{B}$ remain the same for these two different settings, it is sufficient to show that
\begin{eqnarray} \label{W1} \tilde{\vec{W}}=\tilde{\vec{Q}}_i\tilde{\vec{Q}}_{i+1}\tilde{\vec{Q}}_{i+2} \end{eqnarray}
also has the same statistic properties as
\begin{eqnarray} \label{W2} \vec{W}=\vec{Q}_i\vec{Q}_{i+1}\vec{Q}_{i+2}. \end{eqnarray}
The difference between (\ref{W1}) and (\ref{W2}) is that $\tilde{\vec{Q}}_i$, $\tilde{\vec{Q}}_{i+1}$, and $\tilde{\vec{Q}}_{i+2}$ are with dimensions $N_{i-1}\!\times\!N_{i+1}$, $N_{i+1}\!\times\!N_i$, and $N_i\!\times\!N_{i+2}$; while $\vec{Q}_i$, $\vec{Q}_{i+1}$, and $\vec{Q}_{i+2}$ are with dimensions $N_{i-1}\!\times\!N_i$, $N_i\!\times\!N_{i+1}$, and $N_{i+1}\!\times\!N_{i+2}$, respectively.
Denoting ${\vec{a}}(m,n)$ as the element on the $m$th row and $n$th column of a matrix $\vec{A}$, the element $\tilde{\vec{w}}(m,n)$ of $\tilde{\vec{W}}$ equals
\begin{eqnarray} \label{w1} \tilde{\vec{w}}(m,n)=\!\!\sum_{s=0}^{N_{i+1}-1}\!\sum_{r=0}^{N_i-1}\tilde{\vec{q}}_i(m,s)\tilde{\vec{q}}_{i+1}(s,r)\tilde{\vec{q}}_{i+2}(r,n),\;\end{eqnarray}
and the element $\vec{w}(m,n)$ of $\vec{W}$ equals
\begin{eqnarray} \label{w2} \vec{w}(m,n)=\!\!\sum_{s=0}^{N_i-1}\!\sum_{r=0}^{N_{i+1}-1}\vec{q}_i(m,s)\vec{q}_{i+1}(s,r)\vec{q}_{i+2}(r,n),\;\end{eqnarray}
respectively. Since $\vec{Q}_k$ and $\tilde{Q}_k$ are Rayleigh MIMO channels, elements in $\vec{q}_k$ and $\tilde{\vec{q}}_k$ are complex Gaussian variables with zero-mean and unit-variance. Therefore, $\vec{w}(m,n)$ and $\tilde{\vec{w}}(m,n)$ have the same pdf. Further, since $\vec{W}$ and $\tilde{\vec{W}}$ have the same size $N_{i-1}\!\times\!N_{i+2}$, they also have identical pdf. That is to say, $\vec{H}$ and $\tilde{\vec{H}}$ have the same statistical properties and the ergodic capacities of them are the same.
Following similar discussions, we can also show that the above conclusion holds for the cases $i\!=\!0$ (switching $N_0$ and $N_1$) and $i\!=\!K-1$ (switching $N_{K-1}$ and $N_K$). Hence, we conclude that switching any two adjacent parameters in ($N_0, N_1, \cdots,N_K$) will not change the ergodic capacity. Since any permutation of ($N_0, N_1, \cdots,N_K$) can be decomposed as a constitution of operations that switching the order of two adjacent parameters, the Property 1 thusly holds.
One remark from the above proof is that Property 1 holds for a general condition that all elements in all matrices $\vec{Q}_k$ are i.i.d. (but not necessary Gaussian).
\section*{Appendix B: Proof of Lemma 1}
We prove Theorem 1 by deduction. Firstly, we assume the SVD decomposition
\begin{eqnarray} \label{svd1} \vec{Q}_1= \vec{U}^{\dag}\vec{\Lambda}\vec{V}^{\dag}, \end{eqnarray}
where $\vec{U}$ and $\vec{V}$ are unitary matrices with dimensions $N_1\!\times\!N_1$ and $N_0\!\times\!N_0$, respectively. The matrix $\vec{\Lambda}$ has dimensions $N_1\!\times\!N_0$ and the last $N_1\!-\!N_0$ diagonal elements are 0s. That is
\[
\vec{\Lambda} =
\begin{bmatrix}
\hat{\vec{\Lambda}} \\
\vec{0}_{(N_1-N_0)\times N_0}\\
\end{bmatrix}\!,
\]
where $\hat{\vec{\Lambda}}$ is diagonal and with dimensions $N_0\!\times\!N_0$.
Letting
$$ \vec{A}=\prod\limits_{k=3}^{K}\vec{Q}_k,$$
it holds that
\begin{eqnarray} \label{ab2} &&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\mathbb{E}\!\left[\ln\det\!\left(\!\vec{H}^{\dag}\vec{H}\right)\right]\!\qquad \qquad \notag \\
&&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!=\mathbb{E}_{\{\vec{Q}_1,\,\vec{Q}_2,\,\cdots,\,\vec{Q}_K\}}\!\!\left[\ln\det\!\left(\vec{Q}_1^{\dag}\vec{Q}_2^{\dag}\vec{A}^{\dag}\vec{A}\vec{Q}_2\vec{Q}_1\right)\!\right]\!. \end{eqnarray}
Inserting (\ref{svd1}) back into (\ref{ab2}) yields
\begin{eqnarray} \label{ab3} &&\!\!\!\!\!\!\!\!\mathbb{E}\!\left[\ln\det\!\left(\!\vec{H}^{\dag}\vec{H}\right)\right]\!\qquad \qquad \notag \\
&&\!\!\!\!\!\!\!=\mathbb{E}_{\{\vec{\Lambda},\,\vec{V},\,\tilde{\vec{Q}}_2,\,\cdots,\,\vec{Q}_K\}}\!\!\left[\ln\det\!\left(\!\vec{V}\vec{\Lambda}^{\dag}\tilde{\vec{Q}}_2^{\dag}\vec{A}^{\dag}\vec{A}\tilde{\vec{Q}}_2\vec{\Lambda}\vec{V}^{\dag}\right)\!\right]\! \notag \\
&&\!\!\!\!\!\!\!=\mathbb{E}_{\{\vec{\Lambda},\,\tilde{\vec{Q}}_2,\,\cdots,\,\vec{Q}_K\}}\!\!\left[\ln\det\!\left(\!\vec{\Lambda}^{\dag}\tilde{\vec{Q}}_2^{\dag}\vec{A}^{\dag}\vec{A}\tilde{\vec{Q}}_2\vec{\Lambda}\right)\!\right]\! \notag \\
&&\!\!\!\!\!\!\!\overset{(a)}{=}\mathbb{E}_{\{\vec{\Lambda},\,\vec{Q}_2,\,\cdots,\,\vec{Q}_K\}}\!\!\left[\ln\det\!\left(\!\vec{\Lambda}^{\dag}\vec{Q}_2^{\dag}\vec{A}^{\dag}\vec{A}\vec{Q}_2\vec{\Lambda}\right)\!\right]\! \notag \\
&&\!\!\!\!\!\!\!\overset{(b)}{=}\mathbb{E}_{\{\hat{\vec{\Lambda}},\,\hat{\vec{Q}}_2,\,\cdots,\,\vec{Q}_K\}}\!\!\left[\ln\det\!\left(\!\hat{\vec{\Lambda}}^{\dag}\hat{\vec{Q}}_2^{\dag}\vec{A}^{\dag}\vec{A}\hat{\vec{Q}}_2\hat{\vec{\Lambda}}\right)\!\right]\! \notag \\
&&\!\!\!\!\!\!\!=\mathbb{E}_{\{\hat{\vec{\Lambda}}\}}\!\!\left[\ln\det\!\left(\!\hat{\vec{\Lambda}}\hat{\vec{\Lambda}}^{\dag}\right)\!\right]\!+\!\mathbb{E}_{\{\hat{\vec{Q}}_2,\,\cdots,\,\vec{Q}_K\}}\!\!\left[\ln\det\!\left(\!\hat{\vec{Q}}_2^{\dag}\vec{A}^{\dag}\vec{A}\hat{\vec{Q}}_2\right)\!\right]\!, \notag \\
&&\!\!\!\!\!\!\!\overset{(c)}{=}\mathbb{E}_{\{\vec{Q}_1\}}\!\!\left[\ln\det\!\left(\!\vec{Q}_1^{\dag}\vec{Q}_1\right)\!\right] \notag \\
&&+\mathbb{E}_{\{\hat{\vec{Q}}_2,\,\cdots,\,\vec{Q}_K\}}\!\!\left[\ln\det\!\left(\!\hat{\vec{Q}}_2^{\dag}\vec{A}^{\dag}\vec{A}\hat{\vec{Q}}_2\right)\!\right]\!, \!\!\! \end{eqnarray}
where
$$\tilde{\vec{Q}}_2=\vec{U}^{\dag}\vec{Q}_2,$$
and $\hat{\vec{Q}}_2$ denotes the submatrix of $\tilde{\vec{Q}}_2$ obtained by by removing the last $N_1\!-\!N_0$ columns, which is a Rayleigh channel with dimensions dimensions $N_2\!\times\!N_0$.
The equation \lq{}(a)\rq{} holds because $\tilde{\vec{Q}}_2$ and $\vec{Q}_2$ has the same statistic properties since $\vec{U}$ is unitary, and \lq{}(b)\rq{} holds since the diagonal matrix $\vec{\Lambda}$ has the last $N_1\!-\!N_0$ diagonal elements as 0s. The equation \lq{}(c)\rq{} holds due to the fact that
$$\mathbb{E}\!\left[\ln\det\!\left(\!\vec{Q}_1^{\dag}\vec{Q}_1\right)\!\right]\!=\mathbb{E}\!\left[\ln\det\!\left(\!\hat{\vec{\Lambda}}\hat{\vec{\Lambda}}^{\dag}\right)\!\right]\!.$$
Since the last term in (\ref{ab3}) can be equivalently rewritten as
$$\mathbb{E}_{\{\hat{\vec{Q}}_2,\,\cdots,\,\vec{Q}_K\}}\!\!\left[\ln\det\!\left(\!\hat{\vec{Q}}_2^{\dag}\vec{A}^{\dag}\vec{A}\hat{\vec{Q}}_2\right)\!\right]\!=\mathbb{E}\!\left[\ln\det\!\left(\!\hat{\vec{H}}^{\dag}\hat{\vec{H}}\right)\right]\!,$$
where
$$\hat{\vec{H}}=\hat{\vec{Q}}_2\vec{A}$$
is a Rayleigh product channel with $K\!-1\!$ components and with parameter settings $(N_0,\,N_2,\,\cdots,\,N_K)$. Following the same analysis as in (\ref{ab3}) it holds that
\begin{eqnarray} \mathbb{E}\!\left[\ln\det\!\left(\!\hat{\vec{H}}^{\dag}\hat{\vec{H}}\right)\right]\!&=&\!\mathbb{E}_{\{\hat{\vec{Q}}_2\}}\!\!\left[\ln\det\!\left(\!\hat{\vec{Q}}_2^{\dag}\hat{\vec{Q}}_2\right)\!\right] \notag \\
&&+\mathbb{E}_{\{\hat{\vec{Q}}_3,\,\cdots,\,\vec{Q}_K\}}\!\!\left[\ln\det\!\left(\!\hat{\vec{Q}}_3^{\dag}\vec{B}^{\dag}\vec{B}\hat{\vec{Q}}_3\right)\!\right]\!, \notag \\ \end{eqnarray}
where $\hat{\vec{Q}}_3$ is a Rayleigh channel with dimensions $N_3\!\times\!N_0$ and
$$ \vec{B}=\prod\limits_{k=4}^{K}\vec{Q}_k.$$
Repeat such a process it can be shown that
$$\mathbb{E}\!\left[\ln\det\!\left(\!\vec{H}^{\dag}\vec{H}\right)\right]\!=\! \sum_{k=1}^{K}\mathbb{E}\!\left[\ln\det\!\left(\!\hat{\vec{Q}}_k^{\dag}\hat{\vec{Q}}_k\right)\!\right]\!, $$
where $\hat{\vec{Q}}_k$ are Rayleigh channel with dimensions $N_k\!\times\!N_0$, which proves Lemma 1.
\section*{Appendix C: Proof of Property 2}
Applying Minkowski\rq{}s inequality \cite{HJ85} for $N_0\!\times\!N_0$ positive definite matrix,
\begin{eqnarray} \big(\det(\vec{A}+\vec{B})\big)^{1/N_0}\geq \big(\det\vec{A}\big)^{1/N_0}+\big(\det\vec{B}\big)^{1/N_0}, \notag \end{eqnarray}
the ergodic capacity $\tilde{R}$ in (\ref{tR}) satisfies
{\setlength\arraycolsep{2pt} \begin{eqnarray} \label{tRab1} \tilde{R}&\geq&N_0\mathbb{E}\!\left[\ln\!\left(1+q\big(\!\det(\vec{H}\vec{H}^{\dag})\big)^{1/N_0}\right)\!\right]\notag \\
&=&N_0\mathbb{E}\!\left[\ln\!\left(1+q\exp\!\left(\frac{1}{N_0}\ln\det(\vec{H}\vec{H}^{\dag})\right)\!\right)\!\right].\end{eqnarray}}
\hspace{-1.2mm}By Jessen\rq{}s inequality, it holds from (\ref{tRab1}) that
\begin{eqnarray} \label{tRab2} \tilde{R}\geq N_0\ln\!\left(1+q\exp\!\left(\frac{1}{N_0}\mathbb{E}\!\left[\ln\det(\vec{H}\vec{H}^{\dag})\right]\right)\!\right).\end{eqnarray}
Noticing that with Rayleigh product channel $\vec{H}$, it holds from Theorem 1 that
\begin{eqnarray} \label{c3} \mathbb{E}\left[ \ln\det(\vec{H}\vec{H}^{\dag})\right]=N_0(g-\gamma K) , \end{eqnarray}
where $g$ is given in (\ref{gK}). Inserting (\ref{c3}) back into (\ref{tRab2}), the ergodic capacity is lower bounded as
\begin{eqnarray} \label{tRab3} \tilde{R}\geq N_0\ln\!\left(1+q\exp\big(g-\gamma K\big)\right)\!. \notag \end{eqnarray}
\section*{Appendix D: Proof of Property 3}
From Theorem 1, at high SNR the ergodic capacity difference, between the Rayleigh product channel with settings ($N_0,\,N_1,\,\cdots,N_K$) and $N_k\!=\!N_0$ for all $0\!\leq\!k\!\leq\!K$, equals
\begin{eqnarray} \Delta\tilde{R}=\sum_{k=1}^{K}\sum_{\ell=1}^{N_{0}}\psi(N_k-\ell+1)-\sum_{k=1}^{K}\sum_{\ell=1}^{N_{0}}\psi(N_0-\ell+1) . \notag \end{eqnarray}
Using the identity \cite{LN13}
$$ \psi(x+1)=\frac{1}{x}+ \psi(x),$$
it holds that
\begin{eqnarray} \Delta\tilde{R}= \sum_{k=1}^{K}\sum_{\ell=1}^{N_{0}}\sum_{s=N_0}^{N_k-1}\frac{1}{s-\ell+1}. \notag \end{eqnarray}
\iffalse
\section*{Appendix E: Proof of Property 4}
With the definition of $\vec{Z}$ in (\ref{vZ}), it holds that
$$ \vec{Z}^{\dag}\vec{Z}=\vec{A}+\vec{B},$$
where
\begin{eqnarray} \vec{A}\!\!\!&=&\!\!\!\sum_{k=0}^{K-1}\alpha_m^2\vec{H}_m^{\dag}\vec{H}_m, \notag \\
\vec{B}\!\!\!&=&\!\!\!\sum_{\substack{n,m=0\\n\neq m}}^{K-1}\!\alpha_m\alpha_n\vec{H}_m^{\dag}\vec{H}_n. \notag \end{eqnarray}
Further, when $m\!\neq\!n$ using the fact that
$$ \mathbb{E}\big[\vec{H}_m^{\dag}\vec{H}_n\big]=\vec{0},$$
we have
\begin{eqnarray} \label{appE1}\mathbb{E}\big[\ln\det\!\left(\!\vec{Z}^{\dag}\vec{Z}\right)\!\big]\!\!\!\!&=&\!\!\!\!\mathbb{E}\big[\ln\det \vec{A} \big]\!+\!\mathbb{E}\big[\ln\det\!\left(\vec{I}\!+\!\vec{A}^{-1}\vec{B}\right)\!\big]\notag \\
\!\!\!\!&\leq&\!\!\!\! \mathbb{E}\big[\ln\det \vec{A} \big]\!+\!\ln\det\!\left( \vec{I}\!+\!\mathbb{E}\big[\vec{A}^{-1}\vec{B}\big]\right)\!\notag \\
\!\!\!\!&=&\!\!\!\!\mathbb{E}\big[\ln\det \vec{A} \big]. \end{eqnarray}
Under the condition $K$ is sufficiently large, approximately it holds that
$$\vec{B} \approx\vec{0},$$
and the equality in (\ref{appE1}) holds.
On the other hand, as $\ln\!\det$ is concave \cite{B04}, it also holds that
\begin{eqnarray} \ln\det \vec{A}\geq N_0 \ln \eta+\frac{1}{\eta}\sum_{m=0}^{K-1}a_m^2\ln\det\!\left(\!\vec{H}_m^{\dag}\vec{H}_m\right)\!,\end{eqnarray}
where $\eta$ is in (\ref{eta}). From the definition of $\vec{H}_m$ in (\ref{vHm}), it is a Rayleigh product channel with parameters $(N_0,\,N_1,\,\cdots,\,N_{m-1},\,N_{m-2},\,\cdots,\,N_K)$. Using Theorem 1, it holds that
$$ \mathbb{E}\!\left[\ln\det\!\left(\!\vec{H}_m^{\dag}\vec{H}_m\right)\! \right]\!= \sum_{\substack{k=1\\k\neq m}}^{K}\sum_{\ell=1}^{N_{0}}a_m^2\psi(N_k-\ell+1).$$
Therefore, under high SNR and when $K$ is sufficiently large, it holds that
\begin{eqnarray} \tilde{R}\!\!\!\!&\approx&\!\!\!\! N_0\ln q+\mathbb{E}\big[\ln\det\!\left(\vec{Z}^{\dag}\vec{Z}\right)\!\big] \notag \\
\!\!\!\!&\approx&\!\!\!\!N_0 \ln q +\mathbb{E}\big[\ln\det \vec{A} \big] \notag \\
\!\!\!\!&\geq&\!\!\!\!N_0\big(\ln q + \ln \eta\big)+\frac{1}{\eta}\sum_{m=0}^{K-1} \sum_{\substack{k=1\\k\neq m}}^{K}\sum_{\ell=1}^{N_{0}}a_m^2\psi(N_k-\ell+1),\notag\end{eqnarray}
which completes the proof.
\fi
| -60,457.732405 |
[
-2.330078125,
2.53125
] | 34.943639 |
[
-3.427734375,
0.6787109375,
-2.423828125,
-5.9296875,
-0.6923828125,
8.9375
] |
[
2.16015625,
6.515625,
2.3046875,
4.84765625
] | 828 | 7,320 |
[
-1.2666015625,
1.078125
] | 34.437573 |
[
-6.203125,
-4.4453125,
-4.515625,
-2.11328125,
2.525390625,
12.5546875
] | 0.29428 | 25.404449 | 21.734973 | 5.986107 |
[
2.5227835178375244
] | -38,587.63339 | 5.942077 | -59,753.669099 | 1.98639 | 5.989566 |
[
-2.818359375,
-3.42578125,
-3.646484375,
-4.66796875,
2.56640625,
11.65625
] |
[
-5.2578125,
-2.005859375,
-2.24609375,
-1.521484375,
4.08984375,
5.25
] | |
BkiUa5w5qX_AYzDLroyZ
|
\section{Introduction}
Let $S = \{X_1, X_2, \ldots, X_n\}$ be a set of random variables. Let $ \sP(A) $ refer to the set of partitions of set $ A $ into nonempty subsets, which we call blocks. So, for $ \tau \in \sP(A) $, each $ \be \in \tau $ is a block. Then, the joint cumulant of the set of random variables $ S $ is defined to be
\[
\K(S) = \K(X_1, X_2, \ldots, X_n) = \sum_{ \tau \in \sP(S) } (-1)^{|\tau| - 1} (|\tau| - 1)! \prod_{\be \in \tau} \E \lp \prod_{X \in \be} X \rp.
\]
The joint cumulant is a measure of how far random variables are from independence. Notice that if $ n = 1 $ or $ n = 2 $, the joint cumulant reduces to the expected value and covariance, respectively:
\[
\K(X) = \E(X), \qquad \K(X,Y) = \E(XY) - \E(X) \E(Y).
\]
The identities we prove here are all known results for the joint cumulant. The cumulant was first introduced by Thiele in \cite{thiele} in 1899. Brillinger provides a history of the development of higher order moments and cumulants as well as their applications in \cite{history}. For another interpretation of most of these identities involving Mobius inversion by Speed, see \cite{speed}.
In this paper, we use the combinatorial method of Description, Involution, Exception - DIE - a technique for evaluating alternating sums, to prove properties of the joint cumulant. We learned this technique from Prof. Arthur Benjamin at Harvey Mudd College, who introduced it in \cite{benjamin}. Because the expressions we are evaluating are in terms of joint cumulants, which are in terms of expectation values, we use a version of DIE that proceeds as follows:
\begin{itemize}
\item D (Description): Describe a combinatorial interpretation of the expression. Define a set $ C $ of combinatorial objects, a sign function $s: C \to \{-1,1\}$, and a weight function $W: C \to \R$ such that the sum can be written as
$$ \sum_{\rho \in C}s(\rho) W(\rho). $$
\item I (Involution): Find a weight-preserving, sign-reversing involution on a subset $ C_0 \sube C $. That is, find $f: C_0 \to C_0 $, such that for any $\rho \in C_0$,
$$ f(f(\rho)) = \rho, \qquad W(f(\rho)) = W(\rho), \qquad s(f(\rho)) = -s(\rho). $$
\item E (Exception): Determine the elements of $ C - C_0 $, which we call exceptions. Then,
$$ \sum_{\rho \in C} s(\rho) W(\rho) = \sum_{\rho \in C - C_0 } s(\rho) W(\rho). $$
\end{itemize}
The reason DIE works is simple: If we find such an involution $ f $, then we can break up $ C_0 $ into pairs $ (x, f(x)) $ such that $ s(x) W(x) $ and $ s(f(x)) W(f(x)) $ cancel in the sum for all $ x \in C_0 $, leaving us with the exceptions. Usually we describe the involution first and then give the exceptions.
In order to use DIE to prove properties of the joint cumulant, we introduce a combinatorial interpretation of the joint cumulant. We define $ \sC(A) $ to be the set of cyclically arranged partitions of set $ A $. That is, first partition $ A $ and then arrange the blocks of the partition into a cycle. In general, such an object $ \sigma \in \sC(A) $ looks like:
\begin{center}
\begin{tikzpicture}
\tikzstyle{every node} = [draw, shape = circle, fill=gray!5]
\node (1) at (0:2) {$ \beta_1 $} ;
\node (2) at (60:2) {$ \beta_2 $} ;
\node (3) at (300:2) {$ \beta_{|\si|} $} ;
\draw (14:2) arc (15:46:2) ;
\draw (-46:2) arc (-46:-14:2) ;
\draw[style = dashed] (74:2) arc (74:286:2) ;
\end{tikzpicture}
\end{center}
where the $ \beta_i \in \si $ are the blocks of the partition and $ |\si| $ is the size of the partition, or the number of blocks. Next, we define a weight function for $\sC(S)$. Let $V: \sC(S) \to \R $ be given by
\[
V(\sigma) = \prod_{ \beta \in \sigma } \E\left( \prod_{X \in \beta} X \right).
\]
Note that $ V $ depends only on the partition, not the cyclic arrangment. For example,
\begin{center}
\begin{tikzpicture}
\tikzstyle{every node} = [draw, shape = circle, fill=gray!10]
\node (1) at (60:1.5) {$ X_1 X_4 $} ;
\node (2) at (180:1.5) {$ X_2 X_6 X_7 $} ;
\node (3) at (300:1.5) {$ X_3 X_5 $} ;
\draw (87:1.5) arc (87:143:1.5) ;
\draw (217:1.5) arc (217:273:1.5) ;
\draw (-33:1.5) arc (-33:33:1.5) ;
\end{tikzpicture}
\end{center}
represents the partition $ \{ \{ X_1,X_4\}, \{X_2,X_6,X_7\},\{X_3,X_5\} \} $ arranged in the cycle \\ $ ( \{ X_1,X_4\} \{X_2,X_6,X_7\} \{X_3,X_5\} ) $, and its weight is
\[
\E(X_1 X_4) \E(X_2X_6X_7) \E(X_3X_5).
\]
Because there are $ (|\tau| - 1)! $ ways to arrange the blocks of $ \tau $ into a cycle, all with value $ \prod_{ \beta \in \tau } \E\left( \prod_{X \in \beta} X \right) $, we readily observe that
\[
\K(S) = \sum_{ \si \in \sC(S) } (-1)^{ |\si| - 1 } V( \si ).
\]
This is our combinatorial interpretation of the joint cumulant. The set of combinatorial objects is $ \sC(S) $, the sign function is $ s(\si) = (-1)^{ |\si| - 1 } $, and the weight function is $ V(\si) = \prod_{ \beta \in \si } \E \lp \prod_{X \in \be} X \rp $. We will continue to use the above definition for $ \sC(S) $ and $ V $ throughout this paper. Furthermore, we will define more sets and functions which we will continue to use throught the paper.
\section{Joint Cumulant Identities}
\begin{thm}
\label{SumCoef}
For $ n \ge 2 $, the sum of the coefficients in the joint cumulant is 0.
\end{thm}
\begin{proof} (DIE)
\begin{itemize}
\item D (Description): By definition of the joint cumulant, the sum of the coefficients is
\[
\sum_{ \tau \in \sP(S) } (-1)^{|\tau| - 1} (|\tau| - 1)! = \sum_{\si \in \sC(S)} (-1)^{ |\si| - 1}.
\]
Here, our set of combinatorial objects is $ \sC(S) $, the sign function is $ s(\si) = (-1)^{ |\si| - 1} $, and the weight function is 1.
\item I (Involution): In any cyclically arranged partition of $ S $, either $ X_1 $ is or is not alone in its block. If $ X_1 $ is alone, there must exist a block after it as $ n \ge 2 $. Let $ f $ be the map that merges $ X_1 $ into the next block in the cycle if $ X_1 $ is alone and pulls $ X_1 $ out of its block and puts it before that block if $ X_1 $ is not alone. That is, let $ f $ map
\begin{center}
\begin{tikzpicture}
\tikzstyle{every node} = [draw, shape = circle, fill=gray!10]
\node (1) at (0,2) {$ X_1 $} ;
\node (2) at (0,0) {$ B_1 $} ;
\node (3) at (0, -2) {$ B_2 $} ;
\draw (1)--(2);
\draw (2)--(3);
\draw[-)] (-1,1)--(-1,-1);
\draw[red,<->] (1.5,0) -- (2.5,0);
\node (4) at (5,1.5) {$ X_1, B_1 $} ;
\node (5) at (5,-1.5) {$ B_2 $} ;
\draw (4)--(5);
\draw[-)] (4,1)--(4,-1);
\end{tikzpicture}
\end{center}
where $ B_1 $ and $ B_2 $ are nonempty sets of random variables. Clearly, $ f $ is an involution and is weight-preserving as all weights are 1. It is also sign-reversing as it changes the number of blocks by $ \pm 1 $.
\item E (Exceptions): None.
\end{itemize}
Therefore, the sum of the coefficients is 0.
\end{proof}
\begin{thm}
\label{(n - 1)-indep}
Suppose $ n \ge 2 $ and every proper subset of $ S $ is independent. Then,
\[
\K(S) = \E \lp \prod_{ X \in S } X \rp - \prod_{ X \in S } \E(X).
\]
\end{thm}
\begin{proof} (DIE)
\begin{itemize}
\item D (Description): As before,
\[
\K(S) = \sum_{ \si \in \sC(S) } (-1)^{ |\si| - 1 } V( \si ).
\]
\item I (Involution): Let $ f $ be the same involution as in Theorem \ref{SumCoef}. Notice that each cyclic arrangment of the blocks $ \si \ne \{ S \} $ has weight
\[
V( \si) = \prod_{\be \in \si} \E \lp \prod_{X \in \be} X \rp = \prod_{ X \in S } \E(X)
\]
because every proper subset of $ S $ is independent. Hence, $ f $ is weight-preserving for $ \si \ne \{ S \}, \{ \{X_1\}, \{ X_2, X_3 \cdots X_n \} \} $. It is also sign-reversing as it changes the number of blocks by $ \pm 1 $.
\item E (Exceptions): We have two exceptions - $ \{ S \} $, with value $ \E \lp \prod_{ X \in S } X \rp $ and sign $ (1) $, and its pair in the involution, $ \{ \{X_1\}, \{ X_2, X_3 \cdots X_n \} \} $, with value $ \prod_{ X \in S } \E(X) $ and sign $ (-1) $.
\end{itemize}
Therefore,
\[
\K(S) = \E \lp \prod_{ X \in S } X \rp - \prod_{ X \in S } \E(X).
\]
\end{proof}
For the next theorem, we will need the following defintion. We say two sets of random variables $ M = \{ M_1, M_2, \cdots M_p \} $ and $ N = \{ N_1, N_2, \cdots N_q \} $ are independent if
\[
P( M_i = a_i \fa i , N_j = b_j \fa j) = P( M_i = a_i \fa i ) P( N_j = R_j \fa j).
\]
Furthermore, if $ M $ and $ N $ are independent, then for all $ Q \sube M $, $ R \sube N $,
\[
\E \lp \prod_{X \in Q} X \prod_{Y \in R} Y \rp = \E \lp \prod_{X \in Q} X \rp \E \lp \prod_{Y \in R} Y \rp.
\]
\begin{thm}
If $ S $ can be partitioned into two nonempty, independent subsets, then
\[
\K(S) = 0.
\]
\end{thm}
\begin{proof} (DIE)
\begin{itemize}
\item D (Description): As before,
\[
\K(S) = \sum_{ \si \in \sC(S) } (-1)^{ |\si| - 1 } V( \si ).
\]
\item I (Involution): Let $ M = \{ M_1, M_2, \cdots M_p \} $ and $ N = \{ N_1, N_2, \cdots N_q \} $ be a partition of $ S $ into two relatively independent subsets. Consider the block containing $ M_1 $. Starting from this block in the cycle, find the first block containing an element of $ N $. Call this block $ B $. Let $ C = B \cap M $ and $ D = B \cap N $. Now, if $ C = \es $, the previous block must have contained only elements of $ M $, or else we would have found the desired block $ B $ sooner, as we started at a block containing $ M_1 $. Then, let $ f $ be the following map: If $ C \ne \es $, pull out $ C $ from $ B $ and make it its own block in front of $ B $. Otherwise, if $ C = \es $, move the previous block into $ B $. That is, let $ f $ map
\begin{center}
\begin{tikzpicture}
\tikzstyle{every node} = [draw, shape = circle, fill=gray!10]
\node (1) at (0,2) {$ C' $} ;
\node (2) at (0,0) {$ D $} ;
\node (3) at (0, -2) {$ E $} ;
\draw (1)--(2);
\draw (2)--(3);
\draw[-)] (-1,1)--(-1,-1);
\draw[red,<->] (1.5,0) -- (2.5,0);
\node (4) at (5,1.5) {$ C', D $} ;
\node (5) at (5,-1.5) {$ E $} ;
\draw (4)--(5);
\draw[-)] (4,1)--(4,-1);
\end{tikzpicture}
\end{center}
where $ C' \sube M, D \sube N $ are nonempty. Clearly, $ f $ is an involution. Because $M$ and $N$ are relatively independent and $ P \subseteq M, Q \subseteq N$, we have
\[
\E \lp \prod_{X \in Q} X \prod_{Y \in R} Y \rp = \E \lp \prod_{X \in Q} X \rp \E \lp \prod_{Y \in R} Y \rp.
\]
Also, $ f $ changes no other blocks besides those involving $ C' $ and $ D $. Hence, $ f $ is weight-preserving. It is also sign reversing since it changes the number of blocks by $ \pm 1 $.
\item E (Exceptions): None.
\end{itemize}
Hence,
\[
\K(S) = 0.
\]
\end{proof}
To shorten notation, we will make the following definition. For $ \tau \in \sP(S) $, define
\[
\K( \tau) = \prod_{ \be \in \tau } \K(\be).
\]
\begin{thm}
\label{CumulantSum}
\[
\E \lp \prod_{ X \in S } X \rp = \sum_{ \tau \in \sP(S) } \K(\tau).
\]
\end{thm}
\begin{proof} (DIE)
\begin{itemize}
\item D (Description): Consider the expression
\[
\sum_{ \tau \in \sP(S) } \K(\tau) = \sum_{ \tau \in \sP(S) } \prod_{ \be \in \tau } \K( \be) = \sum_{ \tau \in \sP(S) } \prod_{ \be \in \tau } \sum_{ \si \in \sC(\be) } (-1)^{ |\si| - 1 } V( \si ).
\]
Choose a partition $ \tau$ of $ S $. Then, for each $ \beta \in \tau $, we choose an element of $\sC(\beta)$. Let $\sD(S)$ be the set of these combinatorial objects. For each $ \rho \in \sD(S) $, we refer to each $ \be \in \rho $ as an outer block and each $ \g \in \be $ as an inner block. Such an object looks like
\begin{center}
\begin{tikzpicture}
\path (0,0) node {$\be$} ;
\path (3,0) node {$\be$} ;
\path (9,0) node {$\be$} ;
\tikzstyle{every node} = [draw, shape = circle, fill=gray!5]
\path (0,0) +(0:1) node {$\g$};
\path (0,0) +(60:1) node {$ \g$};
\path (0,0) +(300:1) node {$ \g$};
\draw[style = dashed] (0,0) +(90:1) arc (90:270:1);
\draw (0,0) +(25:1) arc (25:35:1);
\draw (0,0) +(-35:1) arc (-35:-25:1);
\path (3,0) +(0:1) node {$ \g $};
\path (3,0) +(60:1) node {$ \g $};
\path (3,0) +(300:1) node {$ \g $};
\draw[style = dashed] (3,0) +(90:1) arc (90:270:1);
\draw (3,0) +(25:1) arc (25:35:1);
\draw (3,0) +(-35:1) arc (-35:-25:1);
\draw[style = loosely dotted, line width = 2pt] (5, 0) -- (7.5, 0);
\path (9,0) +(0:1) node {$ \g $};
\path (9,0) +(60:1) node {$ \g $};
\path (9,0) +(300:1) node {$ \g $};
\draw[style = dashed] (9,0) +(90:1) arc (90:270:1);
\draw (9,0) +(25:1) arc (25:35:1);
\draw (9,0) +(-35:1) arc (-35:-25:1);
\end{tikzpicture}
\end{center}
where the $\g$s are the inner blocks and the $\be$s are the outer blocks. Rewriting the sum using these new objects,
\[
\sum_{ \tau \in \sP(S) } \K(\tau) = \sum_{ \rho \in \sD(S) } \prod_{ \be \in \rho } (-1)^{ |\be| - 1 } V( \be ) .
\]
Or, using our expression for the weight and distributing
\[
\sum_{ \tau \in \sP(S) } \K(\tau) = \sum_{ \rho \in \sD(S) } (-1)^{ \sum_{ \be \in \rho }|\be| - |\rho|} \prod_{ \be \in \rho } \prod_{ \g \in \be} \E \lp \prod_{ X \in \g } X \rp.
\]
Thus, our set of combinatorial objects is $ \sD(S) $, our sign function is
\[
s(\rho) = (-1)^{ \sum_{ \be \in \rho }|\be| - |\rho|},
\]
and our weight function is
\[
W(\rho) = \prod_{ \be \in \rho } \prod_{ \g \in \be} \E \lp \prod_{ X \in \g } X \rp.
\]
\item I (Involution): Consider any $\rho \in \sD(S)$. Let $\g_{1}, \g_2, \ldots, \g_{m}$ be the inner blocks of $\rho$ ordered by increasing order of their minimum index. (This order is arbitrary, but we are explicit about the order because we need the order to be the same for all elements of $\sD(S)$ with the same inner blocks). Then, we write each cycle in terms of the inner blocks using cycle notation. We will keep the inner blocks the same and apply our map to the cycles.
Now, either $ \g_1 $ and $ \g_2 $ do or do not lie in the same outer block or cycle. Let $ f $ be the map that merges the cycles containing $ \g_1 $ and $ \g_2 $ if they are not in the same cycle and splits the cycle containing both if they are in the same cycle. We split and merge these cycles by adding or removing parentheses in cycle notation, respectively. That is, let $ f $ map
\[
(\g_1 C_1)(\g_2 C_2)(C_3)(C_4) \cdots (C_k) \leftrightarrow (\g_1 C_1 \g_2 C_2) (C_3)(C_4) \cdots (C_k)
\]
where each $ C_i $ represents any sequence of inner blocks. Such merging and splitting is well defined because of the order of the cycle. Clearly, $ f $ is an involution. Notice that the weight function is
\[
W(\rho) = \prod_{i=1}^m \E\left(\prod_{X \in \g_i} X\right),
\]
which depends only on the inner blocks. As $ f $ does not change the inner blocks, $ f $ is weight-preserving. Furthermore, $ f $ changes the number of outer blocks by $ \pm 1 $ and maintains the number of inner blocks, and we have the sign function $ (-1)^{ \sum_{ \be \in \rho }|\be| - |\rho|} $. Thus, $f$ is sign-reversing.
\item E (Exceptions): This involution is defined for any $\rho$ with two or more inner blocks, so $ f $ is not defined for any $\rho$ with only one inner block, which also means that $\rho$ has one outer block. Thus the only exception is $ \rho = \{ \{ S \} \} $, with sign and weight
\[
s(\rho) = 1, \qquad W(\rho) = \E \lp \prod_{ X \in S} X \rp.
\]
\end{itemize}
Therefore,
\[
\E \lp \prod_{ X \in S } X \rp = \sum_{ \tau \in \sP(S) } \K(\tau).
\]
\end{proof}
For the next theorem, we will need to define a partial order, $ \preceq $ ,on $ \sP(S) $. For $\pi, \tau \in \sP(S)$, we say that $ \pi $ is finer than $\tau$, or $\pi \preceq \tau$, if for all $\beta \in \pi$, there exists $\alpha \in \tau $ such that $\beta \subseteq \al$. Or, every block of $ \pi $ is contained in a block of $ \tau $.
\begin{thm}
\label{Mobius} For all $ \tau \in \sP(S) $,
\[
\prod_{ \beta \in \tau} \E \lp \prod_{ X \in \beta } X \rp = \sum_{ \pi \preceq \tau } \K( \pi ).
\]
\end{thm}
\begin{proof} (DIE)
\begin{itemize}
\item D (Description): Similarly to Theorem \ref{CumulantSum},
\[
\sum_{ \pi \preceq \tau } \K( \pi ) = \sum_{ \pi \preceq \tau } \prod_{ \be \in \pi } \sum_{ \si \in \sC(\be) } (-1)^{ |\si| - 1 } V( \si ).
\]
Let $ \sD(S) $ be defined as in Theorem \ref{CumulantSum}. Choose a partition $ \pi $ of $ S $ such that $ \pi \preceq \tau $. Then, for each $ \beta \in \pi $, we choose an element of $\sC(\beta)$. Let $\sD_{\tau}(S) \sube \sD(S) $ be the set of these combinatorial objects. We continue to refer to inner and outer blocks as in Theorem \ref{CumulantSum}. We have the same objects as in \ref{CumulantSum} except the partition forming the outer blocks is finer than $ \tau $. Then let $W$ and $s$ be defined as in Theorem \ref{CumulantSum}. Then, similarly to Theorem \ref{CumulantSum},
\[
\sum_{ \pi \preceq \tau } \K( \pi ) = \sum_{ \rho \in \sD_\tau(S)} s(\rho) W(\rho).
\]
\item I (Involution): Let $\al_1, \ldots, \al_{|\tau|}$ be the blocks of $ \tau $. Let $\rho \in \sD_\tau(S)$. Now, each inner block is contained in an outer block, and each outer block is contained in a block of $ \tau $ by definition of $ \sD_{ \tau}(S) $. Thus, each inner block is contained in a block of $ \tau $. Let $\g_{i, 1}, \g_{i,2}, \ldots, \g_{i, m}$ be the inner blocks of $\rho$ that are contained in $\al_i$, ordered as in Theorem \ref{CumulantSum}. Find the first $i$ such that there is more than one $\g_{i,j}$. Then let $ f $ be the same map as in Theorem 4, but just acting on the inner blocks with first index $ i $. Let $ f $ merge the cycles containing $ \g_{i,1} $ and $ \g_{i,2} $ if they are not in the same cycle and split the cycle containing both if they are in the same cycle. That is, let $ f $ map
\[
(\g_{i,1} C_1)(\g_2 C_2)(C_3)(C_4) \cdots (C_k) \leftrightarrow (\g_{i,2} C_1 \g_2 C_2) (C_3)(C_4) \cdots (C_k)
\]
where each $ C_j $ represents any sequence of inner blocks. Clearly, $ f $ is an involution. As before, the weight function depends only on the inner blocks. So, as $ f $ does not change the inner blocks, $ f $ is weight-preserving. Furthermore, $ f $ changes the number of outer blocks by $ \pm 1 $, and we have the same sign function as in Theorem \ref{CumulantSum}, so $ f $ is sign-reversing.
\item E (Exceptions): This involution is defined as long as we have two or more inner blocks for some block of $ \tu $. Thus the exception is the single $ \rho \in \sD_\tau(S) $ for which each outer block contains one inner block, and the inner blocks are the blocks of $\tau$. For this exception,
\[
s(\rho) = 1, \qquad W(\rho) = \prod_{ \beta \in \tau} \E \lp \prod_{ X \in \beta } X \rp.
\]
\end{itemize}
Therefore,
\[
\sum_{ \pi \preceq \tau } \prod_{\beta \in \pi} \K( \beta ) = \prod_{ \beta \in \tau} \E \lp \prod_{ X \in \beta } X \rp.
\]
\end{proof}
For the next theorem, we will need the following definition. Let
\[
T = \{ X_{ i,j} \mid i \in [n], j \in [j_i] \}
\]
be a set of random variables, where $[n]$ denotes the set $\{1,2, \ldots, n\}$. Each $ \tau \in \sP([n]) $ induces a partition $ \tau^* \in \sP(T) $ defined by
\[
\tau^* = \{ \{ X_{i,j} \, | \, i \in \beta, j \in [j_i] \} \, | \, \beta \in \tau \}.
\]
Then we say that $ \pi \in \sP(T) $ is indecomposable if $\pi \preceq \tau^* $ implies $ \tau = \{ [n] \} $. The following result is due to Leonov and Shiryaev in \cite{leonov}.
\begin{thm}
\label{Indecomposable}
\[
\sum_{ \tau \in \sP(T) }^* \K(\tau) = \K \lp \prod_{ j = 1}^{j_1} X_{1, j},\prod_{ j = 1}^{j_2} X_{2, j}, \ldots, \prod_{ j = 1}^{j_n} X_{n , j} \rp ,
\]
where the $ * $ indicates the sum is over all indecomposable partitions of $ T $.
\end{thm}
\begin{proof} (DIE)
\begin{itemize}
\item D (Description): Let $ \sD(S) $ be defined as in Theorem \ref{CumulantSum}. Choose an indecomposable partition $ \tau $ of $ T $. Then, for each $ \beta \in \tau $, we choose an element of $\sC(\beta)$. Let $\sD^*(T) \sube \sD(T) $ be the set of these combinatorial objects. We continue to refer to inner and outer blocks as in Theorem \ref{CumulantSum}. We have the same objects as in \ref{CumulantSum} except the partition forming the outer blocks is indecomposable. Then let $W$ and $s$ be defined as in Theorem \ref{CumulantSum}. Then, similarly to Theorem \ref{CumulantSum},
\[
\sum_{ \tau \in \sP(T) }^* \K( \tau ) = \sum_{ \rho \in \sD^*\tau(S)} s(\rho) W(\rho).
\]
\item I (Involution): Consider $\rho \in \sD^*(T)$. We define an $ i $-split as a pair $ \{ X_{i,j}, X_{i,k} \} $ that belong to different inner blocks of $ \rho $. Find the smallest $ i $ such that there exists an $ i $-split, $ \{ X_{i,j}, X_{i,k} \} $, with $j < k$. Let $ \be_1 $ and $ \be_2 $ be the inner blocks containing $ X_{i,j} $ and $ X_{i,k} $, respectively. Now, we perform the same involution as in Theorem \ref{CumulantSum} using these blocks. Let $ f $ merge the cycles containing $ \be_1 $ and $ \be_2 $ if they are not in the same cycle and split the cycle containing both if they are in the same cycle. That is, let $ f $ map
\[
(\beta_1 C_1)(\beta_2 C_2)(C_3)(C_4) \cdots (C_m) \leftrightarrow (\beta_1 C_1 \beta_2 C_2) (C_3)(C_4) \cdots (C_m).
\]
where each $ C_r $ represents any sequence of inner blocks. First, we verify that if $ f $ is defined for $ \rho \in \sD^*(T) $, then $ f(\rho) \in \sD^{*}(T) $, or that $ f $ preserves indecomposability of the outer partition of $\rho$. We will consider the cases when $f$ merges cycles and splits cycles separately.
If $ f $ merges 2 cycles into a single cycle, then the new outer partition is larger under the refinement order. So clearly, indecomposability is preserved in this case.
Otherwise, $ f $ splits a single outer block, $ C $, into 2 outer blocks, say $ C_1 $ and $ C_2 $. Let $ \sigma $ be the outer partition of $ \rho $, and let $ \sigma' $ be the outer partition of $ f(\rho) $. We know $ \sigma $ is indecomposable. Suppose that $ \sigma' \preceq \tau^{*} $ for some $ \tau \in \sP([n]) $. So, $ C_1 $ and $ C_2 $ must be contained in some block of $ \tau^* $. By definition of $ f $, $ C_1 $ and $ C_2 $ both contain an element with first index $ i $. Thus, as $ \tau^* $ is an induced partition, $ C_1 $ and $ C_2 $ must be contained in the same block of $ \tau^{*} $. Hence, $ C $ is also contained in some block of $ \tau^* $. As all other blocks of $ \sigma $ are also blocks of $ \sigma' $, it follows that $ \sigma \preceq \tau^* $ as well. But, because $ \sigma $ is indecomposable, $ \tau = \{ [n] \} $. Thus, $ \sigma' $ is indecomposable as well. Therefore, $ f $ preserves indecomposability.
As before, $ f $ is a sign-reversing, weight-preserving involution.
\item E (Exceptions): This involution is defined for any $\rho \in \sD^*(T)$ with an $ i $-split for some $ i $, but it fails if we have no such splits. Note that $\rho \in \sD^*(T)$ cannot have more than one outer block and no splits, because then the outer partition would not be indecomposable. Thus, each exception $\rho \in \sD^*(T)$ has outer partition $ \{ T \} $, and for each index $ i $, all random variables with first index $ i $ must belong to the same inner block. So there exists a unique $\si \in \sC([n])$ for which the set of inner blocks of $\rho$, with the same cyclic order, is
\[
\{\{ X_{i,j} \, | \, i \in \beta, j \in [j_i] \} \, | \, \beta \in \si\},
\]
and an exception of this form exists uniquely for every $\si \in \sC([n])$.The sign and weight of each exception $\rho$ with corresponding $\si$ is
\[
s(\rho) = (-1)^{|\si|-1}, \qquad W(\rho) = \prod_{\beta \in \si} \E \left( \prod_{i \in \beta} \left( \prod_{j = 1}^{j_i} X_{i,j} \right) \right).
\]
Hence, the total sum reduces to
\[
\sum_{\si \in \sC([n])} (-1)^{|\si|-1} \prod_{\beta \in \si} \E \left( \prod_{i \in \beta} \left( \prod_{j = 1}^{j_i} X_{i,j} \right) \right) = \K \lp \prod_{ j = 1}^{j_1} X_{1, j},\prod_{ j = 1}^{j_2} X_{2, j}, \ldots, \prod_{ j = 1}^{j_n} X_{n , j} \rp
\]
by definition of the Joint Cumulant.
\end{itemize}
Therefore,
\[
\sum_{ \tau \in \sP(T) }^* \K(\tau) = \K \lp \prod_{ j = 1}^{j_1} X_{1, j},\prod_{ j = 1}^{j_2} X_{2, j}, \ldots, \prod_{ j = 1}^{j_n} X_{n , j} \rp.
\]
\end{proof}
Let $ Y $ be a random variable. Define the conditional joint cumulant as
\[
\kappa( S \, | \, Y ) = \sum_{ \tau \in \sP(S) } (-1)^{|\tau| - 1} (|\tau| - 1)! \prod_{\beta \in \tau}\E \lp \prod_{X \in \beta} X \, \bigg| \, Y \rp
\]
It is the same as the joint cumulant except the expectations are conditional. The following theorem comes from Brillinger in \cite{brillinger}.
\begin{thm}
\label{Conditional}
We can generalize the Tower Formula as follows
\[
\K(S) = \sum_{ \tau \in \sP(S) } \K(\K(\beta_1 \mid Y), \K(\beta_2 \mid Y ), \ldots, \K(\beta_{|\tau|} \mid Y) ),
\]
where the $\beta_i$ are the blocks of partition $\tau$.
\end{thm}
\begin{proof} (DIE)
\begin{itemize}
\item D (Description): First, we have
\[
\sum_{ \tau \in \sP(S) } \K(\K(\beta_1 \mid Y), \K(\beta_2 \mid Y ), \ldots, \K(\beta_{|\tau|} \mid Y) )
\]
\[
= \sum_{ \tau \in \sP(S) } \; \sum_{ \pi \in \sC( \K(\be_1 \mid Y), \K(\be_2 \mid Y ), \ldots \K(\be_{|\tau|} \mid Y)) } (-1)^{ |\pi| - 1} V(\pi).
\]
Here, each $ \K(\be_i | Y) $ is a sum over all cyclic arrangments of partitions of $ \be_i $. So, the sum represents partitioning $ S $ into blocks which we arrange into inner cycles, and then partitioning the set of cycles into outer blocks, which we arrange into an outer cycle. But this is equivalent to first partitioning $ S $ into outer blocks, which we arrange into a cycle, then partitioning the outer blocks into middle blocks, and then partitioning each middle block into inner blocks, which we arrange into a cycle.
Choose a cyclically arranged partition $ \si $ of $ S $. Then, for each $ \be \in \si $, choose an element of $ \sD(\be) $. Let $\sG(T) $ be the set of these combinatorial objects. Then, each object $ \rho \in \sG(S) $ looks like
\begin{center}
\begin{tikzpicture}
\path (0,0) +(160: 2) coordinate (a);
\draw[fill=gray!20] (a) circle (1.7);
\path (a) +(-1.05,0) coordinate (a1);
\path[fill=black] (a1) +(0:.5) circle (.2);
\path[fill=black] (a1) +(60:.5) circle (.2);
\path[fill=black] (a1) +(300:.5) circle (.2);
\draw[style = dashed] (a1) +(90:.5) arc (90:270:.5);
\draw (a1) +(25:.5) arc (25:35:.5);
\draw (a1) +(-35:.5) arc (-35:-25:.5);
\draw[style = loosely dotted, line width = 1.5pt] (a1) +(.8,0) -- +(1.3, 0);
\draw (a1) +(1.9,0) coordinate (a2);
\path[fill=black] (a2) +(0:.5) circle (.2);
\path[fill=black] (a2) +(60:.5) circle (.2);
\path[fill=black] (a2) +(300:.5) circle (.2);
\draw[style = dashed] (a2) +(90:.5) arc (90:270:.5);
\draw (a2) +(25:.5) arc (25:35:.5);
\draw (a2) +(-35:.5) arc (-35:-25:.5);
\path (0,0) +(20: 2) coordinate (a);
\draw[fill=gray!20] (a) circle (1.7);
\path (a) +(-1.05,0) coordinate (a1);
\path[fill=black] (a1) +(0:.5) circle (.2);
\path[fill=black] (a1) +(60:.5) circle (.2);
\path[fill=black] (a1) +(300:.5) circle (.2);
\draw[style = dashed] (a1) +(90:.5) arc (90:270:.5);
\draw (a1) +(25:.5) arc (25:35:.5);
\draw (a1) +(-35:.5) arc (-35:-25:.5);
\draw[style = loosely dotted, line width = 1.5pt] (a1) +(.8,0) -- +(1.3, 0);
\draw (a1) +(1.9,0) coordinate (a2);
\path[fill=black] (a2) +(0:.5) circle (.2);
\path[fill=black] (a2) +(60:.5) circle (.2);
\path[fill=black] (a2) +(300:.5) circle (.2);
\draw[style = dashed] (a2) +(90:.5) arc (90:270:.5);
\draw (a2) +(25:.5) arc (25:35:.5);
\draw (a2) +(-35:.5) arc (-35:-25:.5);
\draw (0,0) +(71: 2) arc (71: 109: 2);
\draw[style = dashed] (0,0) +(215: 2) arc (215: 325: 2);
\end{tikzpicture}
\end{center}
where the black circles are the inner blocks, the cycles of these are the middle blocks, and the larger two gray circles are the outer blocks. For $ \rho \in \sG(S) $, we refer to each $ \al \in \rho $ as an outer block, each $ \be \in \al $ as a middle block, and each $ \g \in \be $ as an inner block. Then, distributing, we see that the sum becomes
\[
\sum_{ \rho \in \sG(S) } (-1)^{|\rho| -1} (-1)^{ \sum_{\alpha \in \rho} \left( \left(\sum_{\beta \in \alpha}|\beta|\right) - |\alpha| \right) } \prod_{\alpha \in \rho} \E\left( \prod_{\beta \in \alpha} \prod_{\gamma \in \beta} \E \left( \prod_{X \in \gamma} X \, \bigg| \, Y \right) \right).
\]
So, our set of combinatorial objects is $ \sG(S) $, our sign function is
\[
s(\rho) = (-1)^{|\rho| -1} (-1)^{ \sum_{\alpha \in \rho} \left( \left(\sum_{\beta \in \alpha}|\beta|\right) - |\alpha| \right) },
\]
and our weight function is
\[
W(\rho) = \prod_{\alpha \in \rho} \E\left( \prod_{\beta \in \alpha} \prod_{\gamma \in \beta} \E \left( \prod_{X \in \gamma} X \, \bigg| \, Y \right) \right).
\]
\item I (Involution): Consider $\rho \in \sG(S)$. Let $\alpha_1, \ldots, \alpha_m$ be the outer blocks, ordered by increasing minimum index. Consider the smallest $ i $ such that $ \alpha_i $ contains more than one inner block. Let $\g_1, \g_2, \ldots, \g_k $ be the inner blocks of $\alpha_i$. Then perform the same merge- split involution as in the previous proofs on these inner blocks. That is, within $\alpha_i$, let $ f $ map
\[
(\gamma_1 C_1)(\gamma_2 C_2)(C_3)(C_4) \cdots (C_r) \leftrightarrow (\gamma_1 C_1 \gamma_2 C_2) (C_3)(C_4) \cdots (C_r)
\]
where each $ C_j $ represents any sequence of inner blocks. As before, $ f $ is an involution. Here, the weight function
\[
W(\rho) = \prod_{\alpha \in \rho} \E\left( \prod_{\beta \in \alpha} \prod_{\gamma \in \beta} \E \left( \prod_{X \in \gamma} X \, \bigg| \, Y \right) \right)
\]
depends only on the inner and outer blocks. Because $f$ does not change the inner or outer blocks, $ f $ is weight-preserving. Furthermore, $ f $ changes the number of middle blocks by $ \pm 1 $ but does not change the number of inner or outer blocks, and we have the sign function
\[
s(\rho) = (-1)^{|\rho| -1} (-1)^{ \sum_{\alpha \in \rho} \left( \left(\sum_{\beta \in \alpha}|\beta|\right) - |\alpha| \right) },
\]
so $f$ is sign-reversing.
\item E (Exceptions): This involution is defined for $\rho \in \sG(S)$ if and only if there exists an outer block containing more than one inner block. Therefore, $\rho$ is an exception if each outer blocks contains only one inner block. So, the set of exceptions has a one-to-one correspondence with $ \sC(S) $, corresponding to choosing the outer blocks and cyclically arranging them. Suppose that the outer blocks of $\rho$, considering also their cyclic order, form the cyclically ordered partition $ \sigma \in \sC(S)$. Then exception $\rho$ has weight and sign
\[
s(\rho) = (-1)^{ |\sigma| - 1 }, \qquad W(\rho) = \prod_{\beta \in \sigma} \E \left( \E\left( \prod_{X \in \beta} X \, \bigg| \, Y \right) \right) = \prod_{\beta \in \sigma} \E \left( \prod_{X \in \beta} X \right),
\]
by the tower formula. So, the sum totals
\[
\sum_{\sigma \in \sC(S)} (-1)^{|\sigma| - 1} \prod_{\beta \in \sigma} \E \left( \prod_{X \in \beta} X \right) = \K(S).
\]
\end{itemize}
Therefore,
\[
\sum_{ \tau \in \sP(S) } \K(\K(\beta_1 \mid Y), \K(\beta_2 \mid Y ), \ldots, \K(\beta_{|\tau|} \mid Y) ) = \kappa(S).
\]
\end{proof}
| -44,301.572541 |
[
-2.57421875,
2.30859375
] | 33.938294 |
[
-2.45703125,
0.71630859375,
-1.662109375,
-5.52734375,
-1.0390625,
7.9765625
] |
[
2.208984375,
8.234375,
1.427734375,
6.9140625
] | 303 | 4,608 |
[
-3.466796875,
3.69140625
] | 42.009504 |
[
-5.1640625,
-3.25390625,
-3.388671875,
-1.7822265625,
1.5458984375,
9.8828125
] | 1.533594 | 23.768133 | 14.561632 | 9.747107 |
[
2.4866063594818115
] | -29,417.510728 | 4.047092 | -43,664.52565 | 0.729262 | 5.546553 |
[
-2.42578125,
-2.6953125,
-3.1015625,
-4.76953125,
2.19921875,
10.953125
] |
[
-5.55078125,
-0.470458984375,
-1.5390625,
-0.89013671875,
3.130859375,
3.150390625
] | |
BkiUeGjxK7FjYEXSFtkH
|
\section{Introduction}
\label{intro}
Accurate human hand pose estimation forms an important task in Human Computer Interaction (HCI) and Augmented Reality (AR) systems and is one of the most highlighted problems of the Computer Vision community that is garnering a lot of attention recently \cite{Ref4, Ref5, Ref6, Ref10, Ref12, Ref18, Ref25, Ref27, Ref28, Ref41, Ref43}. Hand pose estimation forms the core of several interesting applications such as sign-language recognition \cite{Ref22, Ref29}, aerial keyboards \cite{Ref30, Ref35} and driver hand-gesture analyses \cite{Ref37, Ref39}.
With the advent of Convolutional Neural Networks (CNNs) to be used for object classification \cite{Ref1}, there has been a considerable boost in depth-based hand pose estimation methods using 2D CNNs \cite{Ref3, Ref5, Ref13, Ref28, Ref41, Ref44}. Most of these methods take input as a 2D depth image and regress 3D joint coordinates directly \cite{Ref4, Ref16, Ref27, Ref28, Ref38}. The main problem with methods using 2D CNNs for hand pose estimation is that they try to map a 2D depth image to regress 3D world coordinates directly. This causes significant loss of information as the depth images intrinsically depict 3D coordinates of the hand with respect to the depth camera and interpreting them in two-dimensions misses out on important spatial information. Methods \cite{Ref11, Ref43} use multi-view CNNs and a feedback-loop respectively to account for the lost information but this makes it a multistage approach which increases the time and compute taken for results. Most of these methods are limited to the dataset they are trained on and hence do not generalize well on real-world scenarios.
\begin{figure}[t]
\sidecaption[t]
\includegraphics[scale=.5]{figure1.png}
\caption{Overview of our proposed method. The 2D hand depth image after localization is projected into 3D world coordinates which are then voxelized as described in Section \ref{input}. This voxelized grid is then fed to our network after data augmentations which regresses joint locations. The ground truth joint locations on the depth image are shown in the above figure for the purpose of representation.}
\label{fig:1}
\end{figure}
In this paper, we propose a novel voxel-to-coordinate approach that can efficiently capture the spatial information of the depth image. An overview of the proposed method is shown in Figure \ref{fig:1}.The proposed method is end-to-end and reduces the time and compute power significantly opposed to previously used methods, as shown through the experiments performed in Section \ref{exp}. Voxelizing the input depth map retains the 3D nature of the input and provides the network with greater information as shown by \cite{Ref6}. Specifically, we segment the hand in the depth map from its surroundings using depth-thresholding and use reference points to localize the hand. We design a deep network to capture the 3D essence of the depth image and predict accurate joint locations. The depth image of the hand is cropped and fed to this network after performing 3D data augmentations to make our network robust to real-world scenarios. The network regresses 3D joint coordinates faster than previous methods. We evaluate our method on a publicly available benchmark dataset \cite{Ref19} and through experiments quantify the reduction in complexity by our network.
Our contributions can be summarized as follows:
\begin{enumerate}
\item{We propose a voxel-to-coordinate architecture(FastV2C - HandNet) which efficiently captures 3D information as opposed to 2D CNN methods and efficiently regresses 3D world joint coordinates. We convert the input depth map into a voxel grid which is then fed to the network.}
\item{We empirically show that our method reduces the time complexity and compute power significantly when compared to methods. This makes possible faster achievement of results while retaining the whole information of the input.}
\item{We evaluate our method on a publicly available benchmark dataset \cite{Ref19}. We also perform three-dimensional data augmentations to better generalize our model and make it more robust when used in real-time applications.}
\end{enumerate}
The remainder of the paper is organized as follows:
Previous related work has been discussed in Section \ref{relatedwork}. Section \ref{input} elaborates upon the steps used to localize the hand, voxelize it and perform 3D data augmentations. The network architecture of our model along with its implementation is discussed in Section \ref{method}. Experiments in Section \ref{exp} provides experimental results and Section \ref{comp} compares the proposed model with state-of-the-art methods. Section \ref{conc} concludes the paper.
\section{Related Work}
\label{relatedwork}
There is a significant work done on hand pose estimation. This section classifies the methods used into 3 classes - non-neural network methods, methods using 2D CNNs and methods using 3D CNNS. Methods in each of these classes are described in the following subsections.
\subsection{Non-neural Network Methods}
\label{rel:1}
Decision Forests played a major role in hand pose estimation from depth images in earlier methods \cite{Ref7, Ref17, Ref19, Ref23, Ref24, Ref25, Ref31}. These methods required manual feature selection and performed significantly lower than more recent methods. Approaches \cite{Ref8, Ref15} based on particle swarm optimization (PSO) require an increased compute because of the large number of particles involved. Tagliasacchi et al. \cite{Ref20} use iterative closest point (ICP) approach and Qian et al. \cite{Ref21} use a combination of ICP and PSO.
\subsection{2D Convolutional Neural Networks}
\label{rel:2}
A majority of methods \cite{Ref3, Ref4, Ref5, Ref11, Ref13, Ref16, Ref27, Ref28, Ref38, Ref40, Ref41} involve the use of 2D CNNs. Guo et al. \cite{Ref3, Ref4} partition the convolution feature maps into regions and integrate the results from multiple regressors on each region. Chen et al. \cite{Ref5} improve upon this method by using a guided initial hand pose. Ge et al. \cite{Ref11} project the 2D depth map onto three orthogonal planes and fuse the result of 2D CNNs trained on each plane. Tompson et al. \cite{Ref13} were first to predict heatmaps representing 2D joint positions in the depth image. Madadi et al. \cite{Ref16} use a hierarchical tree-structured CNN for estimation. Oberwerger et al. \cite{Ref28} bettered their work \cite{Ref27} by introducing data augmentations, improved localization and a newer network architecture. Sinha et al. \cite{Ref38} use a matrix completion method. Yang et al. \cite{Ref40} use a CNN to classify depth images into different types and later perform regression on them. Xu et al. \cite{Ref41} apply Lie group theory to estimate poses.
All of the above methods use 2D filters in 2D CNNs to extract features from a depth map. As a result of this 2D mapping, important spatial information is lost while predicting 3D joint locations. Our method avoids this loss of information by employing a 3D CNN which can efficiently capture spatial information to map 3D joint locations from depth images.
\subsection{3D Convolutional Neural Networks}
\label{rel:3}
Volumetric representations of depth maps in the form of binary variables representing a voxel grid were first proposed by Wu et al. \cite{Ref33}. They used a convolutional deep belief network to map the probability distribution for each binary variable to represent the 3D geometric shape. Inspired from Maturana et al. \cite{Ref36}, where they show different representations for representing a voxel grid, we use a binary voxel grid to represent the depth map input in our model.
The most recent methods \cite{Ref6, Ref10, Ref12} make use of 3D CNNs for joint estimations. Ge et al. \cite{Ref10} and Deng et al. \cite{Ref12} make use of truncated signed distance function (TSDF) to represent the depth map points in a volumetric shape. Moon et al. \cite{Ref6} use input similar to our method but output a heatmap showing per-voxel likelihood of each joint. The time and compute taken in their method is significantly reduced in our method as shown in Table \ref{tab:2}.
\section{Input to the Network}
\label{input}
This section discusses the steps involved in generating the input which is fed to our model. We localize the hand from its surroundings, voxelize it to capture the 3D spatial information and perform data augmentations to generalize our model.
\subsection{Hand Localization}
\label{in:1}
Segmenting the hand from its surrounding background is a prerequisite to hand pose estimation. Hand localization feeds the network the complete hand discarding much of the unnecessary background information that comes along with it when using our method on real-time images. Inspired from Obwerger et al. \cite{Ref27, Ref28} we segment the hand using depth-thresholding and calculate its center of mass. A 3D bounding box is built around the center of mass. We then train a 2D CNN, as shown in Figure \ref{fig:2} to refine the hand locations by regressing one reference point per frame based on the center of mass calculated for that frame. We use data augmentations to generalize the input as described in Section \ref{in:3}.
\begin{figure}[b]
\sidecaption
\includegraphics[scale=.5]{figure2.png}
\caption{ Hand localization network. An offset to correct an inaccurate hand localization is predicted by the network from the previous reference point to the new reference point.}
\label{fig:2}
\end{figure}
\subsection{Volumetric Representation}
\label{in:2}
The images captured by depth cameras are a function of the 2D depth image pixel coordinates $(p,q)$ represented as $D(p,q)$. We project these pixel coordinates $(p,q)$ to world coordinates $(x, y, z)$ using the camera intrinsic parameters $f_p$, $f_q$.
\begin{equation}
(x, y, z) = \left( \frac{p}{f_p}D(p,q), \frac{q}{f_q}D(p,q), D(p,q) \right)
\end{equation}
We discretize the 3D projected points into a voxel grid based on a predefined voxel size of $10 mm^3$ . Each voxel $V[x,y,z]=1$ if it is occupied by a depth point and is set to $0$ if it is not. Using the reference point obtained in Section \ref{in:1}, we set the voxel grid to a size of $G^3$ voxels where $G=200$ to cover the entire camera coordinate system around the reference point. A similar method to voxelize has also been used in \cite{Ref9}.
\subsection{Data Augmentation}
\label{in:3}
In order to make our model robust to different hand
sizes and global orientations we perform data augmentations on the 3D projected depth map points. We perform scaling by a random factor in the range [0.7, 1.2] to resize the input to a random size. We then perform translation on the voxel grid by a random number of voxels in the range [-7, 7] linearly. Finally, we perform rotation of the voxel grid in the XY plane by an angle chosen randomly from the range [-40, 40] degrees. For a voxel in the 3D space $v$ , we perform the following operations as described above to change it into a voxel $v''$ :
\begin{equation}
v' = v.s + t
\end{equation}
\begin{eqnarray}
\begin{avm}
\[ $v_x''$ \\
$v_y''$ \]
\end{avm}
=
\begin{avm}
\[ $\cos\theta$ & $-\sin\theta$ \\
$\sin\theta$ & $\sin\theta$ \]
\end{avm}
.
\begin{avm}
\[$v_x'$ \\
$v_y'$\]
\end{avm} ;v_z'' = v_z'
\end{eqnarray}
Here $s$ is the scale factor and $t$ is the number of voxels by which translation is performed. $v'$ is the voxel after performing translation and scaling with $v_x, v_y, v_z$ being its components along X, Y, Z axes respectively. We then perform rotation in the XY
plane by the angle $\theta$ resulting in the augmented voxel $v''$.
The entire training set is generated by augmenting the data in the original training set.
\begin{figure}[t]
\sidecaption[t]
\includegraphics[scale=.55]{figure3.png}
\caption{Overall architecture of FastV2C-HandNet. Each Conv3D layer comprises of 3D Convolutional Layer (3x3x3) + Batch Normalization + ReLU Activation. Each MaxPool3D layer consists of a 3D Max Pooling layer with size (2x2x2). The Conv3DTranspose layer has kernel size of (2x2x2) and stride 2.$F$ are the number of joints per hand pose. The network takes in a voxelized grid input and regresses 3D world joint coordinates.}
\label{fig:3}
\end{figure}
\section{FastV2C-HandNet (Our Method)}
\label{method}
In this section we describe the network architecture used along with the implementation details for training our model. The input passes through a series of 3D convolutional blocks with two residual connections \cite{Ref26} in between to facilitate increased depth at a reduced complexity.
\subsection{Network Architecture}
\label{method: 1}
As seen in Figure \ref{fig:3}, the input to our model is an 88x88x88 voxelized depth image. Our proposed model is composed of eleven 3D convolutional layers. We use three 3D max pooling layers to down-sample the input size so that the most important features are captured. Two residual connections \cite{Ref26} are adopted after the first and the second max pooling layers to increase the dimension size. The filter size of each pair of Conv3D layers increases from 16 to 64 with the last Conv3D layer having a filter size of $F$ , which is the number of joints in the hand pose.To up-sample the input we then introduce a 3D transpose convolutional layer with a filter size of 32 and a stride of 2. The output from this layer passes through a pair of convolutional layers with 32 filters. All
the Conv3D layers and the Conv3DTranspose layer are followed by Batch Normalization and Rectified Linear Unit (ReLU) activation. The Conv3D layers each have a kernel size of 3 except the last layer which has a kernel size of 1. The MaxPool3D layers have a stride of 2 each. The Conv3DTranspose layer has a filter size of 32 with kernel size 2 and stride 2. This completes the feature extraction part of our network.
The features extracted through the above network are then passed through three fully connected (FC) layers to regress the joint coordinates. For $F$ number of joints in a pose each fully connected layer has $F$ x $c$ units, where c = 44, 11 and 3 respectively. Each of the first two fully connected layers is followed by a DropOut \cite{Ref42} layer with a dropout rate of 0.5 to prevent the network from overfitting on the training set. We get a $F$ x $c$ dimensional output vector with each row corresponding to the 3D world coordinates for each joint.
\subsection{Implementation}
\label{method: 2}
We segment the hand from the depth image and
voxelize the depth map, on which we perform data augmentations as described in Sections \ref{in:1}, \ref{in:2} and \ref{in:3}. We input an 88x88x88 voxel grid obtained by cropping the 96x96x96 3D projection. We use the Mean Squared Error (MSE) loss function $L$ to calculate the loss between the estimated joint locations and the ground truth joint locations.
\begin{equation}
L = \frac{1}{N} \sum_{n=1}^{N} \sum_{i,j,k} (i-i_n)^2 + (j-j_n)^2 + (k-k_n)^2
\end{equation}
Here $N$ are the number of joints per frame and $i_n, j_n, k_n$ are the ground truth joint locations of the $n^{th}$ joint.
The weights are updated using the Adam Optimizer \cite{Ref14} with a learning rate of $3.0$x$10^{-4}$ . We use a mini-batch size of 4 and train the network for 3 epochs. The kernel weights are initialized from a zero-mean normal distribution with
$\sigma = 0.005$ and the biases are initialized with zeros. To demonstrate the speed of our network we trained it on a single NVIDIA Tesla P100 GPU and got state-of-the-art-results in terms of speed. (Section \ref{exp:3})
\begin{figure}
\sidecaption
\includegraphics[scale=.25]{figure4.png}
\caption{Mean error per joint for the 21 joints in the MSRA Hand Dataset.}
\label{fig:4}
\end{figure}
\begin{figure}
\sidecaption
\includegraphics[scale=.25]{figure5.png}
\caption{Fraction of success frames vs maximum distance from ground truth location in experiments performed on the MSRA Hand Dataset.}
\label{fig:5}
\end{figure}
\section{Experiments}
\label{exp}
We perform extensive experiments and provide results in this section. The experiments are performed using a publicly available benchmark dataset. The evaluation metrics are then discussed which are followed by the results of our experiments.
\subsection{Dataset}
\label{exp:1}
We evaluate our model on the MSRA Hand Pose Dataset \cite{Ref19} which is a publicly available benchmark dataset. The Dataset is composed of depth images captured from 9 different subjects depicting 17 hand pose gestures each. Each gesture has about 500 frames. The creators use Intel\textquotesingle s Creative Interactive Gesture Camera \cite{Ref32} to obtain more than 76K depth images with 21 hand joints per image. Since there are no explicitly defined train and test sets, we train on 8 subjects and test on the remaining one subject. This is repeated 9 times for 9 different test subjects.
\subsection{Evaluation Metrics}
\label{exp:2}
We use 2 evaluation metrics to evaluate the performance of our model:
\begin{itemize}
\item We use the 3D mean joint error to evaluate the accuracy of our model. This metric gives the average error per joint in all the samples.
\item As our second metric, we plot the fraction of frames having all predicted joints\textquotesingle Euclidean distance from the ground truth less than a maximum value \cite{Ref34}.
\end{itemize}
\subsection{Results}
\label{exp:3}
We train our network on the MSRA Hand Dataset \cite{Ref19} as described in Section \ref{exp:1}. We plot the mean error per joint as shown in Figure \ref{fig:4}. We also plot the fraction of samples having distance less than a maximum from the ground truth. As seen in Figure \ref{fig:5}, predicted joints in more than 50 percent of the samples fed to the network have less than 100mm distance from the ground truth joints. From Figure \ref{fig:4} we find that the lowest error across all samples is recorded for little\_icp joint and the highest error for thumb\_tip which can also be intuitively understood from Figure \ref{fig:5}. The total training time taken by our network is about 7 hours on the MSRA Hand Dataset which to the best of our knowledge, is significantly less compared to all previous hand pose estimation methods using 3D CNNs.
\section{Comparison with state-of-the-arts}
\label{comp}
We compare the running time of our method with
state-of-the art methods \cite{Ref6, Ref10, Ref12} on the MSRA Hand Dataset. As shown in Table \ref{tab:1}, our method significantly reduces the time complexity thereby achieving faster results than any of the previous methods. Our network is end-to-end which avoids the time-consuming multistage pipelines \cite{Ref7, Ref8, Ref15} and feedback loops \cite{Ref43} used in previous methods.
We also list a separate comparison with V2V-PoseNet \cite{Ref6} which gives the lowest error currently for this task to show that our method reduces the training and prediction time taken significantly while retaining the 3D spatial information of the depth image. This is shown in Table \ref{tab:2}.
\begin{table}
\caption{Table 1: Comparison of the proposed method (FastV2C-HandNet) with state-of-the-art methods based on time taken per frame for joint prediction. Our method outperforms all previous methods.}
\label{tab:1}
\begin{tabular}{p{6cm}p{3cm}}
\hline\noalign{\smallskip}
\textbf{Methods} & \textbf{Time per frame(ms)} \\
\noalign{\smallskip}\hline\noalign{\smallskip}
DeepPrior++ \cite{Ref28} & 33.33 \\
V2VPoseNet \cite{Ref6} & 28.5 \\
PointNet \cite{Ref45} & 23.92\\
HandPointNet \cite{Ref46} & 20.833 \\
Madadi \textit{et al.} \cite{Ref16} & 20.0 \\
CascadedPointNet \cite{Ref47} & 14.3 \\
CrossingNets \cite{Ref44} & 11 \\
Ge \textit{et al.} \cite{Ref12} & 7.9 \\
Latent2.5D \cite{Ref48} & 6.89 \\
REN \cite{Ref4} & 0.31 \\
\textbf{FastV2C-HandNet (Ours)} & \textbf{0.185} \\
\noalign{\smallskip}\hline
\end{tabular}
\end{table}
\begin{table}
\caption{Table 2: Comparison of the proposed method's (FastV2C-HandNet) total training time and model size on the MSRA Hand Dataset \cite{Ref19} with V2V-PoseNet \cite{Ref6}.}
\label{tab:2}
\begin{tabular}{p{5cm}p{3cm}p{2cm}}
\hline\noalign{\smallskip}
\textbf{Methods} & \textbf{Total Training Time} & \textbf{Model Size} \\
\noalign{\smallskip}\hline\noalign{\smallskip}
V2V-PoseNet \cite{Ref6} & 12 hours & 457 MB \\
\textbf{FastV2C-HandNet (Ours)} & \textbf{7 hours} & \textbf{42 MB} \\
\noalign{\smallskip}\hline
\end{tabular}
\end{table}
\section{Conclusion}
\label{conc}
In this paper, we presented a novel voxel-to-coordinate model for hand pose estimation. We segment the hand from its background and localize it by regressing reference points using a 2D CNN network. To overcome the drawbacks in methods using 2D CNNs and random forests we voxelize the depth map to preserve the spatial information. In order to make our network robust to real-world scenarios we perform data augmentations on our training set before feeding it to the network. The network is trained in an end-to-end manner which significantly reduces the time and compute power required. We evaluate our model on one publicly available hand dataset. Through experiments we show that our model performs faster than the state-of-the-art methods for hand pose estimation which makes it more suitable to be used in real-world scenarios.
A limitation of our method is a slight decrease in accuracy with the increase in speed. Due to the regression of the hand joints directly as 3D coordinates instead of a per-voxel likelihood heatmap as in \cite{Ref6}, a decrease in accuracy is observed. An extension of this work will present a modification of this algorithm that can maintain the speeds achieved by this method without affecting the accuracy. This forms a scope for future research.
\input{reference}
\end{document}
| -15,610.232144 |
[
-0.93505859375,
1.2333984375
] | 43.921569 |
[
-3.525390625,
0.82080078125,
-2.12109375,
-4.7421875,
-0.376220703125,
7.33203125
] |
[
2.47265625,
6.21484375,
1.193359375,
6.7421875
] | 202 | 3,202 |
[
-2.271484375,
2.392578125
] | 24.102049 |
[
-6.3359375,
-3.712890625,
-3.75390625,
-1.3232421875,
2.42578125,
10.9140625
] | 0.672239 | 33.466218 | 28.888195 | 3.613265 |
[
2.0509605407714844
] | -11,818.555161 | 5.342598 | -15,467.973038 | 1.052201 | 5.819122 |
[
-3.193359375,
-3.62109375,
-3.416015625,
-3.978515625,
2.703125,
10.90625
] |
[
-5.921875,
-2.30078125,
-2.982421875,
-2.326171875,
4.4765625,
6.6484375
] | |
BkiUdQs4eIXhqTbAZ6vi
|
\section{Introduction}
One of the central challenges of modern cosmology is still to shed light on the physical mechanism behind the cosmic acceleration. Current measurements have already sharply improved constraints on this phenomena. Several observations like Supernovas SNeIa \cite{Riess:1998cb}, Cosmic Microwave Background Radiation (CMBR) \cite{Spergel:2003cb}, Baryonic Acoustic Oscillations (BAO) \cite{Eisenstein:2005su}, among others \cite{Tegmark:2003ud,Jain:2003tba,Magana,Riess:2016jrr}, has been useful to constraints the cosmological parameters that define a specific model. Future observations are expected to do much better, especially for models that allow a time-evolving Equation of State (EoS).
One of the main candidate to explain this cosmic acceleration is Dark Energy (DE). This component also features baryonic matter, dark matter and radiation. The advantage of DE is that relax some tensions in the cosmological parameters measurements, which can explain in particular the fact the geometry of the universe is consistent with the flatness predicted by inflation. Despite the large observational progress in measuring DE properties, not fundamental insights into the physics behind this dark sector has been solved. Even thought, while the statistical error have shrunk dramatically, current constraints are still roughly consistent with 68.3\% \cite{Ade:2015xua} current energy budget with an EoS ratio $\omega\approx-1$. This had led to the idea in where a Cosmological Constant (CC) $\Lambda$ can explain the cosmic acceleration. Also, in agreement with the described observations, the $\Lambda$CDM or concordance model has the advantage to provide an accelerated behavior driven by $\Lambda$ and filled with Cold Dark Matter (CDM).
Despite its simplicity, there are fundamental problems if we assume that CC is related with the quantum vacuum fluctuations. Some theoretical efforts point out to a value of density energy $\sim120$ orders of magnitude of difference with the observational value or at least it is expected a strictly vanishing value under protective symmetry \cite{Weinberg}. For this reason, some research has turned to find new alternatives as: quintessence \cite{Qreferences}, phantom fields \cite{Caldwell:2003vq}, Chaplygin models \cite{Bento:2002ps}, brane models \cite{Aghababaie:2003wz}, just to mention a few.
Between the plethora of models, one of the most interesting alternatives comes from a brane model based in the idea of Dvali, Gabadadze and Porrati (DGP), where it is assumed a $5$D Minkowski space time, within a $4$D Minkowski brane embedded \cite{dgpmodels}. The region of transition between the four and five dimensional manifold is encoded in the crossover scale parameter $r_c$, which is a function of the five and four Planck masses. It is interesting to notice that this scenario allows to mimic the universe acceleration as a transition between the dimensions of spacetime mimicking the CC with the crossover region parameter. A natural extension of DGP models can be performed when the brane is generalized
by using a Friedmann-Lemaitre-Robertson-Walker (FLRW) metric. Therefore, this model offers an attractive explanation for the accelerated expansion of the universe without to invoke a dark energy component. From the DGP background evolution emerge two solution branches depending the choice of the sign: the self-accelerated branch (which corresponds to a negative sign) and the normal or stable branch (which correspond to a positive sign). In the first branch there is a late cosmic acceleration without the presence of DE. However, this branch is ruled out by supernovae data \cite{fang}. The normal/stable branch has the property of introducing a $\Lambda$ over the evolution and fixed a bidimensional model with two cosmological parameters: $\Omega_{rc}$ and $\Omega_{\Lambda}$. This latter characteristic allow us to perform a directly astrophysical test using $H(z)$ data set and Planck analysis \cite{Ade:2015xua} which can set constraints on the cosmological parameters of the stable DGP model. However, the impact of these cosmological parameters will became looser (stronger) depending of the weakness (strength) of the fifth force. Interesting results related to these cases are reported in \cite{Aguilera:2013tmp,Barreira:2016ovx}, studying a IR cutoff or the growth rate of structure or in \cite{Raccanelli} it was studied tests of gravity using large-scale redshift-space distortions.
In this paper, we will work with the stable DGP model for three different pipelines in where we can control the strength of the $r_c$ parameter and set the constraints over this parameter using direct $H(z)$ measurements: the Cosmic Chronometers (Cosmic-C) and the radial BAO scale in the galaxy distribution. In \cite{Wan:2007pm} was study a DGP universe using these observations, however the variation of the curvature in this analysis shows a DGP model with best fits that correspond to a/an closed/open universe using a WMAP prior.
This paper is organised as follows. In Sec. 2 we will present an overview of the equations related to the DGP background cosmology. In Sec. 3 we describe the astrophysical samples for $H(z)$. In Sec. 4 we present the constraints over the DGP cosmological parameters of our interest. In Sec. 5 with set a discussion of the results obtained.
\section{DGP cosmological background}
The DGP model \cite{Dvali:2000hr} suggest an universe on a brane which is embedded in a 5D Minkowski space-time with a infinite extra dimension.
This model give us two important reasons to consider it. First, it describe a 4D Newtonian gravity on the brane at short distances whereas on the bulk the gravity shows as 5D. Second, the short distances are fixed by a crossover scale $r_c$ denoted by $r_c \equiv M^2_P/2M^3$, where $M_P$ and $M$ are the five and four Planck masses, respectively. Only gravity is present in both the brane and the bulk but not the other force of the standard model.
Let us begin with the action that we have taken in 4D Einstein-Hilbert action for the bulk added:
\begin{equation}\label{eq:actionDGP}
S=M^3 \int d^5 X \sqrt{-g_{(5)}}(R_{(5)}-\mathcal{L}_{m}) +M^2_{P}\int d^4x\sqrt{-g}R,
\end{equation}
where $g_{(5)}$ and $g$ are the determinants of the metric of the five-dimensional bulk $g_{AB}^{(5)}$ and four-dimensional brane $g_{\mu\nu}$ respectively, t, and $R_{(5)}$ and $R$ are their corresponding Ricci scalars. Similarly, $\mathcal{L}_m$ is the Lagrangian associated with the fields confined on the brane, included if we consider the CC as a fluid.
Therefore, the induced metric is defined as usual from the bulk metric as $g_{\mu\nu}=\partial_{\mu}X^{A}\partial_{\nu}X^{B}g^{(5)}_{AB}$. Notice that the capital letters runs as $A,B=0,1,2,3,4$ and greeks letters runs as $\mu,\nu=0,1,2,3$.
Thus, the background expansion rate in the DGP model using a flat FRW metric can be written as (see \cite{Deffayet:2000uy} for details):
\begin{eqnarray}\label{eq:evol_DGP}
H(z)^2 &=& H_0^2 \left[\sqrt{\Omega_m (1+z)^3 +\Omega_{r}(1+z)^4 +\Omega_{\Lambda}+\Omega_{rc}}
\right. \nonumber \\ && \left.
\pm \sqrt{\Omega_{rc}}\right]^2,
\end{eqnarray}
where $H_0 =100h\text{km}/\text{s}\text{Mpc}^{-1}$ is the expansion rate today, $\Omega_m$ represents the fractional matter density today, $\Omega_\Lambda$ the CC term and $\Omega_{rc}=(4H_{0}^{2}r_{c}^{2})^{-1}$. Here, in addition to the matter and the crossover scale contributions we have included the radiation term.
We can compare (\ref{eq:evol_DGP}) with the standard flat Friedmann evolution equation with a dark energy component $\Omega_{DE}$:
\begin{eqnarray}
H^2(z) &=& H^{2}_{0} \left[\Omega_m (1+z)^3 +\Omega_r (1+z)^4
\right. \nonumber\\ && \left.
+ \Omega_{DE} (1+z)^{3(1+\omega_{DE})}\right],
\end{eqnarray}
where $\omega_{DE}$ is the EoS for the DE component. Comparing the latter with (\ref{eq:evol_DGP}) we observe that $(\Omega_{rc} +\Omega_{\Lambda})$ behaves similarly to an effective CC.
If we set the $z=0$ value in (\ref{eq:evol_DGP}) leads to the constraint condition:
\begin{equation}\label{eq:constraintC}
\sqrt{\Omega_m +\Omega_r +\Omega_{\Lambda} +\Omega_{rc}} \pm \sqrt{\Omega_{rc}} =1,
\end{equation}
which differs from the conventional $\Omega_m +\Omega_r +\Omega_{DE} =1$. Therefore, from (\ref{eq:constraintC}) we get
\begin{equation}
\Omega_{rc} = \frac{1}{4} \left(\Omega_m +\Omega_r +\Omega_{\Lambda} -1\right)^2.
\end{equation}
The latter shows that for a flat universe with radiation component, $\Omega_{rc}$ is \textit{always} smaller that $\Omega_{DE}$. Even more, at large scales ($\Omega_{\Lambda}\approx 0.7$, $\Omega_m \approx 0.3$, $\Omega_r=2.469\times10^{-5}h^{-2}(1+0.2271\times N_{eff}$, $h=H_0/100$kms$^{-1}$Mpc$^{-1}$, and $N_{eff}=3.04$) the $\Omega_{rc}$ vanishes and we obtain the standard cosmology with a CC.
We observe from the evolution equation (\ref{eq:evol_DGP}) that there are two branches:
considering the positive sign emerges the branch in where is necessary to introduce a CC (i.e $\Omega_{\Lambda}\neq 0$) to drive a late cosmic acceleration. Considering the negative sign it is not necessary to add a CC (i.e $\Omega_{\Lambda}= 0$) component to describe acceleration at late-time. This latter is however ruled out by supernovae data \cite{fang}.
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth,origin=c,angle=0]{evolution1.pdf}
\includegraphics[width=0.45\textwidth,origin=c,angle=0]{evolution2.pdf}
\caption{$H(z)^2$ ratio between DGP stable model and $\Lambda$CDM model. The curves represents the cases in where the strength (weakness) DGP stable model can be fixed. \textit{Left:} Evolution of the $H(z)_{\text{DGP}}^{2}/H(z)_{\Lambda \text{CDM}}^{2}$ with a $\Omega_\Lambda =0$. \textit{Right:} Evolution of the $H(z)_{\text{DGP}}^{2}/H(z)_{\Lambda \text{CDM}}^{2}$ with a $\Omega_\Lambda \neq 0$.
} \label{fig:evolution_dgp}
\end{figure}
Therefore we consider three values of the cross-over scale: $r_c H_0 =0.2$, $r_c H_0 =0.6$ and $r_c H_0 =1.9$, which we renamed as: DGP strong, DGP medium and DGP weak stable models, respectively. The advantage of these slightly changes over the values in comparison to \cite{Barreira:2016ovx} is that we can observe in Figure \ref{fig:evolution_dgp} a distinguishable difference between each DGP stable model and $\Lambda$CDM at early times.
\section{DGP stable cosmological analysis}
Since for both proposals of the DGP models we have cosmic acceleration, in order to perform the analysis of the DGP stable model (with positive sign in (\ref{eq:evol_DGP})) we require observational Hubble rate data. The basic assumption of this data is due that the differential age approach estimates the Hubble rate directly from the data without assuming a specific spatial geometry or any other cosmological model. These measurements has become an effective probe in cosmology comparison with SNeIa, BAO and CMB data. Following a similar methodology from \cite{Escamilla-Rivera:2015odt}, we use the cosmic chronometer (Cosmic-C) data and we complete the dataset with six measurements of $H(z)$ obtained from BAO. We summarize these data sets as:
\subsection{$H(z)$ observations}
Usually, it is has more precision to study the observational $H(z)$ data directly due that all these test use the distance scale measurement to determinate the values of the cosmological parameters, which needs the integral of $H(z)$ and therefore loses some important information of this quantity. As an independent approach of this measure we provide two samples:
\begin{enumerate}\renewcommand\labelenumi{(\theenumi)}
\item Cosmic Chronometers (Cosmic-C) data. This kind of sample gives a measurement of the expansion
rate without relying on the nature of the metric between the chronometer and us. We are going to employ
several data sets presented in \cite{Hsamples}. A full compilation of the latter, which includes 28 measurements
of $H(z)$ in the range $0.07 < z < 2.3$, are reported in \cite{Hsamples2}. The normalized {parameter $h(z)$ can be easily
determined by considering the value $H_0 = 67.31 \pm 0.96$ km s${}^{-1}$ M~pc${}^{-1}$ \cite{Ade:2015xua}.}
\item Data from BAO. Unlike the angular diameter $d_A$ measures given by the transverse BAO
scale, the $H(z)$ data can be extracted from the measurements of the line-of-sight of this BAO scale.
Because the BAO distance scale is embodied in the CMB, its measurements on DE parameters are strongest
at low redshift. The samples that we are going to consider consist of three data points from \cite{Blake:2012pj}
and three more from \cite{Gaztanaga:2008xz} measured at six redshifts in the range $0.24 < z < 0.73$.
This data set is shown in Table \ref{tab:dataBAO}.
\begin{table}
\caption{\textit{\textit{BAO}} sample data from
\cite{Blake:2012pj,Gaztanaga:2008xz}.
}\label{tab:dataBAO}
\centering
\begin{tabular}{ccccc}
\toprule
{\textbf{z}} & \boldmath{$H(z)$} \textbf{[km \boldmath{$\text{s}^{-1}$}$\bf\text{M pc}$\boldmath{$^{-1}$}]}
& \textbf{\boldmath{${\sigma_{H}}^2$}} \\
\hline
$0.24$ & $79.69$ & $2.32$ \\
$0.34$ & $83.80$ & $2.96$ \\
$0.43$ & $86.45$ & $3.27$ \\
$0.44$ & $82.6$ & $7.8$ \\
$0.6$ & $87.9$ & $6.1$ \\
$0.73$ & $97.3$ & $7.0$ \\
\hline
\end{tabular}
\end{table}
\end{enumerate}
To perform the statistical analysis we employ (\ref{eq:evol_DGP}), where
$(\Omega_{\Lambda}, r_c)$ are the free parameters of the model. We compute the best fits of these cosmological parameters
by minimizing the quantity
\begin{eqnarray} \label{eq:min}
\chi_{\text{H(z)}}^2
=\sum^{N}_{i=1}{\frac{\left[H_{\text{theo}}(z_i ,\Omega_m ;\Omega_{\Lambda}, r_c)-H_{\text{obs}}(z_i)\right]^2}{\sigma^{2}_{H,i}}},
\end{eqnarray}
where the $\sigma^{2}_{H,i}$ are the measurements variances and $N$ is the number of the total sample, which for our purpose will be consider three combinations between datasets.
\subsection{DGP stable model cosmological tests}
First we are going to study the case for a DGP stable model with a prior $H_0 = 67.31 \pm 0.96$ km s${}^{-1}$ M~pc${}^{-1}$ and $\Omega_m =0.315\pm 0.017$, where the set of cosmological parameters to constrains are $(\Omega_{rc},\Omega_{\Lambda})$. We perform the minimization of (\ref{eq:min}) to get the best fit values. The confidence regions in the $\Omega_m -\Omega_{rc}$ plane are show in Figure \ref{fig:cosmo_dgp1} and the statistical values are given in Table \ref{tab:datacosmo1}.
\begin{figure*}[ht]
\centering
\includegraphics[width=1.\textwidth,origin=c,angle=0]{Total.pdf}
\caption{DGP stable model confidence contours $(\Omega_\Lambda,\Omega_{rc})$ until $3$-$\sigma$. \textit{Left:} Using Cosmic-C dataset. \textit{Middle:} Using BAO dataset. \textit{Right:} Using Cosmic-C + BAO dataset.
} \label{fig:cosmo_dgp1}
\end{figure*}
We notice that for these priors, the cosmic acceleration at late-times is performed by the $\Omega_{\Lambda}$ term. Also, the $\Omega_{rc}$ shows a constant value for the three posible combinations of data sets.
For our second analysis, we consider the DGP strong stable model ($r_c H_0 =0.2$ ) with the same $H_0$ prior, where the set of cosmological parameters to constrains are $(\Omega_{m},\Omega_{\Lambda})$. The confidence regions in the $\Omega_m -\Omega_{\Lambda}$ plane are show in Figure \ref{fig:cosmo_dgp2} and the statistical values are given in Table \ref{tab:datacosmo2}.
\begin{figure*}[ht]
\centering
\includegraphics[width=1.\textwidth,origin=c,angle=0]{Total2.pdf}
\caption{ DGP strong model confidence contours $(\Omega_\Lambda,\Omega_m)$ until $3$-$\sigma$. \textit{Left:} Using Cosmic-C dataset. \textit{Middle:} Using BAO dataset. \textit{Right:} Using Cosmic-C + BAO dataset.
} \label{fig:cosmo_dgp2}
\end{figure*}
\begin{table}
\caption{Cosmological parameter constraints for a DGP stable model with a prior $H_0 = 67.31 \pm 0.96$ km s${}^{-1}$ M~pc${}^{-1}$ and $\Omega_m =0.31$. }
\label{tab:datacosmo1}
\centering
\begin{tabular}{ccccc}
\toprule
\textbf{Dataset} & $\chi^2$ &\boldmath{$\Omega_{\Lambda}$} & \boldmath{$\Omega_{rc}$} \\
\hline
Cosmo-C & $18.827$ &$0.427\pm 0.161$ & $0.01\pm 0.177$ \\
BAO & $4.652$ &$0.501\pm 0.235$ & $0.01\pm 0.534$ \\
Cosmo-C + BAO & $24.404$ & $0.472 \pm 0.021$ & $0.01\pm 0.178$ \\
\hline
\end{tabular}
\end{table}
\begin{table}
\caption{Cosmological parameter constraints for a DGP strong stable model with a prior $H_0 = 67.31 \pm 0.96$ km s${}^{-1}$ M~pc${}^{-1}$ and $r_c H_0 =0.2$. }
\label{tab:datacosmo2}
\centering
\begin{tabular}{ccccc}
\toprule
\textbf{Dataset} & $\chi^2$ &\boldmath{$\Omega_{\Lambda}$} & \boldmath{$\Omega_{m}$} \\
\hline
Cosmo-C & $16.984$ &$0.240\pm 0.131$ & $0.089\pm 0.221$ \\
BAO & $3.718$ &$0.401\pm 1.635$ & $0.131\pm 1.554$ \\
Cosmo-C + BAO & $21.329$ & $0.131 \pm 0.021$ & $0.231\pm 0.113$ \\
\hline
\end{tabular}
\end{table}
We notice in this case that $\Lambda$CDM model ($\Omega_m =0.3$ and $\Omega_\Lambda =0.7$) is discarded beyond $3$-$\sigma$. Also the best fits suggest that the cosmic acceleration in the DGP strong model is performed by only the $\Omega_{rc}$ component.
\section{Discussion}
We notice that DGP stable model with or non addition of $\Omega_{\Lambda}$ can be distinguishable from $\Lambda$CDM at early times. Also, as we see from the Figure 1, at large redshift it seems that each DGP models starts to loiters to $\Lambda$CDM case.
Therefore, we observed some important results about the contribution of the crossover scale tested by $H(z)$ data which is shown in Figure 2, where the values for the free parameters ($\Omega_{rc}, \Omega_{\Lambda}$) are almost constant in the redshift range given by the $H(z)$ measurements. For the three confidence regions these results indicate that for our Planck priors there is no tension between these two datasets.
In addition, the obtained value for the density parameter $\Omega_{r_c}$ is approximately equal to the value of $\Omega_{\Lambda}$, this result give us a prediction about the dominant term in the evolution equations (\ref{eq:evol_DGP}). Hence, the density of CC is the main responsible of the accelerated expansion of the universe at late times.
Indeed, the $\Lambda$CDM model is recovered for small contribution of the crossover scale density parameter.
As well, in Figure 3 we illustrate the obtained values for $\Omega_{\Lambda}$ and $\Omega_{m}$ for the DGP strong model with prior $r_c H_0 =0.2$, it is necessary to remark the difference between both values of the model. There is a tension at around 2-$\sigma$ between the two confidence contours $[\Omega_{\Lambda}-\Omega_{m}]$ using Cosmic-C and BAO. Furthermore, the obtained values from Cosmic-C, BAO and the joined dataset analysis for $\Omega_{m}$ are below the expected, the results of both densities are no consistent with the well known values for them.
Finally, we remark that for cosmological perturbations in DGP models, the main characteristics are that the integrated Sach-Wolfe (ISW) effect shows more suppression than in the standard paradigm \cite{Cardoso:2007xc} and the evolution of metric perturbations is no longer necessarily scale free \cite{Seahra:2010fj}. It is important to notice that these results could also be studied in this paper. However, to assess the impact of the brane perturbations, a full CMB analysis should be carried out, which is beyond of the scope of this article.
\begin{acknowledgments}
G.B.-E acknowledges support from CONACYT fellowship with number 785554. C. E.-R. acknowledges support from MCTP-UNACH and M.A.G.-A. acknowledges support from SNI-M\'exico and CONACyT research fellow. Instituto Avanzado de Cosmolog\'ia (IAC) collaborations.
\end{acknowledgments}
| -15,668.598414 |
[
-1.544921875,
1.552734375
] | 39.784946 |
[
-3.21875,
0.105712890625,
-1.970703125,
-5.26171875,
0.0672607421875,
7.484375
] |
[
3.640625,
8.0703125,
2.517578125,
5.65234375
] | 217 | 2,588 |
[
-1.9501953125,
2.0078125
] | 28.316067 |
[
-6.01171875,
-4.1953125,
-4.1875,
-2.298828125,
1.974609375,
11.796875
] | 1.230012 | 21.724657 | 32.341577 | 7.591401 |
[
2.5151445865631104
] | -10,819.70346 | 5.65456 | -15,215.174203 | 0.669673 | 5.764621 |
[
-2.9609375,
-3.603515625,
-3.224609375,
-4.1953125,
2.369140625,
10.8359375
] |
[
-5.55078125,
-1.640625,
-1.7744140625,
-0.91455078125,
3.537109375,
4.28515625
] | |
BkiUc-vxK6nrxpQc1A6q
|
\section{Introduction: Why study saturation at strong coupling ?}
\setcounter{equation}{0}
Since the idea of parton saturation has first emerged
\citep{Gribov:1984tu,Mueller:1985wy}, in the mid eighties, as a possible
solution to the unitarity problem in QCD at high energy, this phenomenon
has been generally associated with weak coupling. This was based on the
asymptotic freedom of QCD together with the following, self--consistent,
argument: in the kinematical domain for saturation, one expects parton
densities to be large, hence the relevant values of the QCD running
coupling should be weak; and indeed calculations in QCD at weak coupling
($\alpha_s\ll 1$) predict large gluon occupation numbers at saturation,
$n\sim 1/\alpha_s\gg 1$, thus closing the argument. Following this logic,
and also by lack of non--perturbative tools, all the subsequent studies
of this phenomenon from first principles were performed within
perturbative QCD, with increasingly higher degrees of sophistication
\citep{CGCreviews} (and Refs. therein). In particular, the observation
that a regime characterized by high occupation numbers and weak coupling
is semi--classical \citep{McLerran:1993ni} paved the way to the modern
effective theory for gluon saturation within pQCD, which is the Color
Glass Condensate (CGC) \citep{CGCreviews}. These studies demonstrated the
existence of an intrinsic scale associated with this phenomenon, the {\em
saturation momentum} $Q_s$ --- the transverse momentum below which
non-linear effects in the gluon distribution become important ---, which
increases quite fast with the energy, and thus eventually becomes `hard'
($Q_s \gg \Lam \simeq 200$ MeV). This scale also controls the gluon
density at and near saturation, and hence it sets the scale for the
running coupling in the approach towards saturation. All these results
have confirmed the original intuition that, {\em for sufficiently high
energies}, parton saturation is a weak coupling phenomenon which is
driven by the rapid evolution of the gluon distribution via
bremsstrahlung.
But what about the {\em current} energies, as attained in the present
days colliders ? These energies are relatively high, allowing to explore
values of the Bjorken's $x$ variable --- the longitudinal momentum
fraction of a parton inside the hadron wavefunction --- as small as
$x\sim 10^{-4}$ at RHIC, $10^{-5}$ at HERA and even $10^{-6}$ at the LHC.
Besides, in collisions involving large nuclei, the gluon density and thus
the saturation momentum are further enhanced by the atomic number $A \gg
1$. Yet, in spite of such favorable circonstances, the corresponding
values of $Q_s$ remain quite modest: this scale does not exceed $1.5$~GeV
at HERA and RHIC (with nuclei) and it should be around $2\div 3$~GeV in
the `forward' kinematics at the LHC. Moreover, the theoretical analyses
of the phenomenology initiate the high--energy evolution at some
intermediate value $x_0$ which must be small enough to justify the focus
on the evolution with increasing energy (as opposed, e.g., to the DGLAP
evolution), but large enough (with respect to the $x$ values of interest)
to minimize the effects of the uncertainties in the initial conditions at
$x_0$ (which must be taken from a model, so like the
McLerran--Venugopalan model \citep{McLerran:1993ni}). This value $x_0$
and the associated saturation momentum $Q_s(x_0)$ are generally chosen in
such a way to optimize the description of some set of data, so like the
HERA data for the DIS structure function $F_2$, and some typical values
(taken from Ref. \citep{Albacete:2009fh}) are $x_0\sim 0.01$ and
$Q_{s}^2(x_0)\,\sim\, 0.4$~GeV$^2$. For such `semihard' values $Q_s$, the
weak coupling techniques are only marginally applicable.
So, clearly, it would be very interesting to have some (at least,
qualitative) understanding of the phenomenon of parton saturation in the
transition region towards strong coupling. In general, that problem is
extremely complicated not only because the coupling is strong, but
especially because of the possible mixing with the physics of
confinement, for which there is no analytic understanding from first
principles. It is therefore both interesting and remarkable that there
exists a physical regime of QCD, in which one can isolate the physics of
(relatively) strong coupling from that of confinement, and which moreover
might have some relevance for the present day phenomenology, as suggested
by some of the data at RHIC. This refers to the deconfined phase of QCD,
the {\em quark--gluon plasma} (QGP), which in thermodynamical equilibrium
exists for temperatures larger than a critical value $T_c\simeq 170$~MeV.
This phase has been rather extensively studied (at least, in so far as
its thermodynamical properties are concerned) via lattice QCD
calculations. By now we have rather firm evidence that it has been also
experimentally produced at RHIC, in the intermediate stages of the heavy
ion collisions \citep{Gyulassy:2004zy}. There are moreover strong
indications that the partonic matter liberated by the collision
equilibrates quite fast, over a time $\tau_0\sim 1$~fm/c, at a
temperature $T=(2\div 3) T_c$, and then lives in the plasma phase for
about 5~fm/c, before eventually cooling down and hadronizing. (In
lead--lead collisions at LHC, one should reach $T\sim 5 T_c$ and a QGP
lifetime $\tau\sim 10$~fm/c.)
In the forthcoming two sections, I shall argue that: \texttt{(i)} there
are strong indications, notably from the heavy ion experiments at RHIC,
that this deconfined matter is effectively strongly coupled,
\texttt{(ii)} the physics of parton saturation in a strongly coupled
plasma is interesting not only at a conceptual level, but also for the
phenomenology at RHIC, and \texttt{(iii)} this physics can be reliably
studied, at least at a qualitative level, by using the gauge/gravity
duality. Then, in the remaining part of the discussion, I will explain
the consequences of this approach for parton evolution and saturation at
strong coupling.
\section{sQGP at RHIC}
\setcounter{equation}{0}
There are several arguments why the quark--gluon plasma produced at RHIC
or LHC can be considered as strongly coupled. At a formal level, one can
note that the relevant temperatures are relatively low (less than 1 GeV),
so the respective QCD coupling is quite high: $\alpha_s\equiv g^2/4\pi
\simeq 0.4$, or $g\simeq 2$. Moreover, unlike at zero temperature, where
the perturbative expansion is a series in powers of $\alpha_s$, at finite
temperature this is truly a series in powers of $g$ and it shows very bad
convergency unless $g\ll 1$ (which in QCD requires astronomically high
temperatures) \citep{Blaizot:2003tw}. But this formal argument is not
decisive by itself, as shown by the following fact: one has demonstrated
that appropriate resummations of the perturbative expansion are able to
cure the problem of the lack of convergency and thus yield results for
the QCD thermodynamics which agree very well with lattice QCD for all
temperatures $T\gtrsim 2.5 T_c$ \citep{Blaizot:2003tw}. Underlying such
resummation schemes, there is the picture of QGP as a gas of weakly
interacting `quasiparticles' --- quarks and gluons with energies and
momenta of order $T$ which are `dressed' by medium effects.
But this picture, which would be natural at weak coupling, has been
shaken by some of the experimental discoveries at RHIC
\citep{RHICglobal}, especially the unexpectedly large `elliptic flow' and
`jet quenching', which are rather suggestive of strong coupling
\citep{Gyulassy:2004zy,Muller:2007rs}.
The `elliptic flow' refers to an azimuthal anisotropy in the distribution
of the particles produced in a peripheral nucleus--nucleus collision.
Such a pattern is natural for a {\em fluid}, which is a system with
strong interactions, but it would be very difficult to explain for a
weakly coupled gas. The elliptic flow measured at RHIC \citep{RHICglobal}
not only is strong, but it is so even for the heavy quarks $c$ and $b$,
which appear to be dragged by the medium in spite of their large masses.
The RHIC data for elliptic flow can be well accommodated within
theoretical analyses using hydrodynamics, which assume early
thermalization ($\tau_0 \lesssim 1$~fm/c) and small viscosity --- more
precisely, a very small viscosity to entropy--density ratio $\eta/s$.
These features are signatures of a system with strong interactions:
indeed, when $g\ll 1$, both the equilibration time $\tau_0$ and the ratio
$\eta/s$ are parametrically large, since proportional to the mean free
path $\sim 1/g^4$. On the other hand, AdS/CFT calculations for gauge
theories with a gravity dual \citep{Policastro:2001yc} suggest that, in
the limit of an infinitely strong coupling, the ratio $\eta/s$ should
approach a universal lower bound which is $\hbar/4\pi$
\citep{Kovtun:2004de}. Interestingly, it appears that, within the error
bars, the ratio $\eta/s$ extracted (via the theoretical analyses) from
the RHIC data \citep{Luzum:2008cw} is rather close to this lower bound,
thus supporting the new paradigm of a {\em strongly coupled Quark--Gluon
Plasma} (sQGP).
Whereas the elliptic flow is a manifestation of long--range correlations
which could be indeed sensitive to larger values of the coupling, the
observation of strong--coupling aspects in relation with `jet quenching'
looks even more surprising, since it seems to be in conflict with
asymptotic freedom. The `jet quenching' refers to the energy loss and
transverse momentum broadening of an energetic parton (the `jet') which
interacts with the medium. The jet has a relatively large transverse
momentum $k_\perp\gg T$ and hence it explores the structure of the plasma
on relatively small space--time distances $\ll 1/T$ (`hard probe'). Some
typical values at RHIC are $k_\perp\sim 2\div 20$~GeV and $T\sim
0.5$~GeV. Because of this large separation in scales, one would expect
the medium to be relatively transparent for the jets, but the
measurements at RHIC show that this is actually not the case: the medium
appears to be opaque \citep{RHICglobal}.
This opaqueness is manifest e.g. in the RHIC measurements of the `nuclear
modification factor' --- the ratio $R_{AA}$ between the particle yield in
Au+Au collisions and the respective yield in proton--proton collisions
scaled up by $A^2$. This ratio would be one in the absence of medium
effects, but in reality one finds a much lower value, $R_{AA}\simeq
0.2\div 0.3$, which is interpreted as a sign of strong energy loss in the
medium. Another observable which points in the same direction is the
`away--jet suppression' observed in the azimuthal correlations of the
produced jets: unlike in p+p or d+Au collisions, where the hard particles
typically emerge from the collision region as pairs of back--to--back
jets, in the Au+Au collisions at RHIC one sees `mono--jet' events in
which the second jet is missing (see Fig.~\ref{fig:JETS} left). This has
the following natural interpretation (see Fig.~\ref{fig:JETS} right): the
hard scattering producing the jets has occurred near the edge of the
interaction region, so that one of the jets has escaped and triggered a
detector, while the other one has been deflected, or absorbed, via
interactions in the surrounding medium.
\begin{figure*}{\centerline{
\includegraphics[width=.6\textwidth]{JetsSTAR.eps}\
\includegraphics[width=.45\textwidth]{JetsAA.eps}}
\caption{\sl\small Left: Azimuthal correlations for jet measurements
($k_\perp(assoc) > 2$~GeV) at RHIC (STAR) in p+p, d+Au, and Au+Au
collisions. Right: Jet production in a nucleus--nucleus collision.}
\label{fig:JETS}}
\end{figure*}
Assuming weak coupling, it is possible to compute energy loss and
momentum broadening within perturbative QCD \citep{Baier:1996kr}. If the
medium is composed of weakly interacting quasiparticles (quarks and
gluons), then the deflection of the hard jet is due to its successive
scattering off these quasiparticles (see Fig.~\ref{fig:quench} left).
Also, energy loss at weak coupling is dominated by {\em medium induced
radiation}, that is, the emission of a hard gluon in the presence of
medium rescattering. Both phenomena are controlled by the same transport
coefficient, the `jet quenching parameter' $\hat q$, defined as the rate
of transverse momentum broadening. In pQCD $\hat q$ is estimated as the
cross--section for the scattering between the jet and the plasma
constituents `seen' by the jet on its hard resolution scale. At high
energy, these constituents are mostly gluons and $\hat q$ is estimated as
\citep{Baier:1996kr}
\begin{eqnarray}\label{qhat}
\hat q\,\equiv\, \frac{{\rm d} \langle k_\perp^2\rangle}{{\rm d} t}
\,\simeq\, \frac{\alpha_s N_c}{N_c^2-1}\,
\,{\mathcal G}(x,Q^2),
\end{eqnarray}
where ${\mathcal G}(x,Q^2)$ is the gluon distribution in the medium on
the resolution scale $Q^2\sim \langle k_\perp^2\rangle$, as produced via
the quantum evolution of the quasiparticles from their intrinsic energy
scale to the hard scale $Q$ (see Fig.~\ref{fig:RG}). For instance, if the
medium is a finite--temperature plasma with temperature $T$, then
${\mathcal G}\simeq n_q(T)\,{\mathcal G}_q\,+\,n_g(T)\,{\mathcal G}_g$,
where $n_{q,g}(T)\propto T^3$ are the quark and gluon densities in
thermal equilibrium and ${\mathcal G}_{q,g}(x,Q^2)$ are the gluon
distributions produced by a single quark, respectively gluon, on the
scale $Q\gg T$. $\hat q$ is also related to the saturation scale $Q_s$ in
the plasma, via $Q_s^2 \simeq \hat q L$ where $L$ is the longitudinal
extent of the medium. At weak coupling, one can evaluate all these
quantities within pQCD. By doing that, one finds an estimate $\hat q_{\rm
pQCD}\simeq (0.5\,\div\,1) {\rm GeV}^2/{\rm fm}$, while phenomenology
\citep{Abelev:2006db,Adare:2006nq} rather suggests that $\hat q$ should
be somehow larger, between 5 and 15 ${\rm GeV}^2/{\rm fm}$. One should
nevertheless keep in mind that this phenomenology is quite difficult and
not devoid of ambiguities: strong assumptions are necessary in order to
compute $\hat q$, and also to extract its value from the RHIC data (see,
e.g., the discussion in \citep{Baier:2006fr}).
\begin{figure*}{ \centerline{
\includegraphics[width=.46\textwidth]{Broad_QCD-QT.eps}\qquad
\includegraphics[width=.46\textwidth]{HeavyQ_BROAD.eps}}
\caption{\sl\small Transverse momentum broadening for a heavy quark which
propagates through a quark--gluon plasma. Left: weak coupling (successive
scattering off thermal quasiparticles). Right: strong coupling (medium
induced branching).} \label{fig:quench}}
\end{figure*}
This discrepancy suggests that the actual gluon distribution in the
plasma is significantly larger than expected in pQCD. A possible
explanation for that is a stronger value for the coupling, which would
enhance the quantum evolution from $T$ up to $Q$. Note that there is not
necessarily a conflict with asymptotic freedom: to get an enhanced gluon
distribution on the relatively hard scale $Q$, it is enough to have a
stronger coupling at the lower end of the evolution, that is, at the
relatively soft scale $T$ (where we know that $g\simeq 2$ is indeed quite
large). Actually, in Ref. \citep{Iancu:2009zz} we proposed a strategy for
numerically studying this evolution in lattice QCD at finite temperature
and thus directly test the hypothesis of strong coupling.
\begin{figure}[htb]{\centerline{
\includegraphics[width=.8\textwidth]{RGflow.eps}}
\caption{\small\sl Parton evolution from the thermal scale $T$ up to the
harder scale $Q\gg T$.} \label{fig:RG}}
\end{figure}
\section{The AdS/CFT correspondence}
The previous discussion invites us to a better understanding of parton
evolution and saturation in deconfined QCD matter at strong coupling,
that is, for $\alpha_s\equiv g^2/4\pi \simeq 1$. However, even without
the complications of confinement, the QCD calculations at strong coupling
remain notoriously difficult. (In particular, lattice QCD cannot be used
for real--time phenomena so like scattering.) So it has become common
practice to look to the ${\mathcal N}=4$ supersymmetric Yang--Mills (SYM)
theory, whose strong coupling regime can be addressed within the AdS/CFT
correspondence, for guidance as to general properties of strongly coupled
plasmas (see the review papers
\citep{Son:2007vk,Iancu:2008sp,Gubser:2009sn}).
${\mathcal N}=4$ SYM has the `color' gauge symmetry SU$(N_c)$, so like
QCD, but differs from the latter in some other aspects: it has conformal
symmetry (the coupling $g$ is fixed) and no confinement, and all the
fields in its Lagrangian (gluons, scalars, and fermions) transform in the
adjoint representation of SU$(N_c)$. But these differences are believed
not to be essential for a study of the quark--gluon plasma phase of QCD
in the temperature range of interest for heavy ion collisions at RHIC and
LHC ($2T_c\lesssim T \lesssim 5T_c$), where QCD itself is known (e.g.,
from lattice studies \citep{Cheng:2007jq}) to be nearly conformal.
The AdS/CFT correspondence
\citep{Maldacena:1997re,Gubser:1998bc,Witten:1998qj} is the statement
that the conformal field theory (CFT) ${\mathcal N}=4$ SYM is `dual'
({\em i.e.}, equivalent) to a specific string theory (`type IIB') living
in a $(9+1)-$dimensional space time with AdS$_5\times S^5$ geometry. The
5--dimensional Anti-de-Sitter space--time AdS$_5$ is a space with Lorentz
signature and uniform negative curvature and can be roughly imagined as
the direct product between our $(3+1)-$dimensional Minkowski world and a
radial, or `fifth', dimension $\chi$, with $0\le \chi < \infty$. Our
physical world is the boundary of AdS$_5$ at $\chi=0$ (see the sketch in
Fig.~\ref{fig:WAVE}). The radial dimension is, roughly speaking, dual to
the virtual momenta of the quantum fluctuations that we implicitly
integrate out in the boundary gauge theory (see the discussion in
Sect.~\ref{DIS}).
This gauge/string equivalence is conjectured to hold for arbitrary values
of the parameters $g$ and $N_c$, but in practice this is mostly useful in
the strong `t Hooft coupling limit $\lambda\equiv g^2N_c\to \infty$ with
$g\ll 1$, where the string theory becomes tractable --- it reduces to
classical gravity in 9+1 dimensions (`supergravity'). This limit is
generally not a good limit for studying scattering, since the respective
amplitudes are suppressed as $1/N_c^2$ \citep{Polchinski:2002jw,HIM1}.
Yet, this is meaningful for processes taking place in a deconfined
plasma, which involves $N_c^2$ degrees of freedom per unit volume, thus
yielding finite amplitudes when $N_c\to\infty$. The gravity dual of the
${\mathcal N}=4$ SYM plasma with temperature $T$ is obtained
\citep{Witten:1998zw} by introducing a black--hole (BH) in the radial
dimension of AdS$_5$ --- something that may look natural, given that a BH
has entropy and thermal (Hawking) radiation. The BH horizon is located at
$\chi=1/T$ and is parallel to the Minkowski boundary --- that is, the BH
is homogeneous in the physical 4 dimensions. One can see here a
manifestation of the {\em ultraviolet/infrared correspondence} (or
`holographic principle'), which is very useful for the physical
interpretation of the supergravity calculations: the presence of a
gravitational source at a distance $\chi_0$ in the bulk of AdS$_5$ (here
the BH horizon at $\chi_0=1/T$) corresponds to adding a energy/momentum
scale $1/\chi_0$ in the boundary gauge theory (here, the plasma with
temperature $T$).
\begin{figure*}[t]
\centerline{\includegraphics[width=0.5\textwidth]{WAVE_UVIR0.eps}
\includegraphics[width=0.48\textwidth]{WAVE_FALL_SL.eps}}
\caption{\small\sl Space--like current in the plasma: the trajectory of
the wave packet in $AdS_5$ and its `shadow' on the boundary. Left: low
energy
--- the Maxwell wave gets stuck near the
boundary. Right: high energy --- the wave falls into the BH.
\label{fig:WAVE}}
\end{figure*}
\section{Deep inelastic scattering at strong coupling}
\label{DIS}
To understand parton evolution at strong coupling, we need a
non--perturbative and gauge--invariant definition of the concept of
`parton', which can be extrapolated to AdS/CFT. This is provided by deep
inelastic scattering (DIS), which in its essence describes the absorption
of a space--like photon by the constituents of the target hadron which
carry electric charge. The respective cross--section --- the `structure
function' $F_2(x,Q^2)$ --- is a direct mesure of the distribution of
these charged partons (the quarks in the case of QCD) on the resolution
scales $x$ and $Q^2$ of the virtual photon. Here $Q^2$ is the photon
virtuality and fixes the typical transverse momentum (or inverse
transverse size) of the struck quark. Furthermore $x\equiv Q^2/(2q\cdot
P)\approx Q^2/s$, with $s$ the invariant energy squared of the
proton--photon system, is the Bjorken--$x$ variable and fixes the
longitudinal resolution of the photon: the struck quark carry a fraction
$x$ of the hadron longitudinal momentum $P$. At weak coupling at least,
the gluon distribution can be extracted too from the measured
$F_2(x,Q^2)$, by using the perturbative evolution equations for the
parton distributions.
Using the AdS/CFT correspondence we have computed DIS off the ${\mathcal
N}=4$ SYM plasma at temperature $T$ and in the strong coupling (or
large--$N_c$) limit \citep{HIM2,HIM3}. The virtual photon couples to the
constituents of the plasma which carry the ${\mathcal R}$--charge (the
analog of the electromagnetic charge in ${\mathcal N}=4$ SYM). The dual,
supergravity, picture of DIS is as follows (see Fig.~\ref{fig:WAVE}) :
the ${\mathcal R}$--current $J_\mu$ acts as a perturbation on the
Minkowski boundary of AdS$_5$ at $\chi=0$, thus inducing a massless,
vector, supergravity field $A_m$ (with $m=\mu$ or $\chi$) which
propagates towards the bulk of AdS$_5$ ($\chi>0$), according to Maxwell
equations in curved space--time\footnote{The 5--dimensional sphere
($S^5$) part of AdS$_5\times S^5$ plays no role for this particular
calculation.} :
\begin{eqnarray}\label{maxwell} \partial_m\big(\sqrt{-g}g^{mp}g^{nq}
F_{pq})\,=\,0\,,
\qquad\mbox{where}\quad F_{mn}=\partial_m A_n-\partial_n A_m\,.
\end{eqnarray}
These equations describe the gravitational interaction between the
Maxwell field $A_m$ and the BH (implicit in the 5--dimensional metric
tensor $g^{mn}$). Note that there is no explicit coupling constant in the
equations: the gravitational scattering is rather controlled by the
kinematics. Given the solution $A_m$, there is a well--identified
procedure to construct the current--current correlator $\langle J_\mu(x)
J_\nu(y)\rangle$ in the boundary gauge theory. Then, the DIS structure
function is finally obtained by taking the imaginary part of this
correlator in momentum space, like usual.
Eqs.~(\ref{maxwell}) remain non--trivial even at $T=0$, in which case
they describe the propagation of the virtual photon through the vacuum of
the strongly--coupled ${\mathcal N}=4$ SYM theory. The physical
interpretation of the results can be deduced using the UV/IR
correspondance alluded to above, which is more precisely formulated as
follows \citep{HIM2,HIM3}: the radial penetration $\chi$ of the Maxwell
field $A_m$ in $AdS_5$ is proportional to the transverse size $L$ of the
typical quantum fluctuations of the virtual photon in the boundary gauge
theory.
As an example, consider the $T=0$ case: as well known, a space--like
photon cannot decay into on--shell (massless) quanta in the vacuum,
because of energy--momentum conservation. Rather it fluctuates into a
virtual system of partons, whose complexity can be very high at strong
coupling, but whose space--time delocalization is fixed by the
uncertainty principle: in a frame where the photon has 4--momentum
$q^\mu=(\omega,0,0,q)$ and (space--like) virtuality $Q^2= q^2-\omega^2 >
0$, its virtual fluctuations have a typical transverse size $L\sim 1/Q$
and a longitudinal size, or lifetime, $\Delta t\sim \omega/Q^2$. And
indeed, the solutions to Eqs.~(\ref{maxwell}) at $T=0$ show that the
wave--packet representing $A_m$ penetrates into the bulk up to a maximal
distance $\chi\sim 1/Q$ and the propagation time from the boundary up to
that maximal distance is $\sim \omega/Q^2$ \citep{HIM3}. The requirement
of energy--momentum conservation enters the AdS calculation in the form
of a repulsive barrier around $\chi\sim 1/Q$ which prevents the wave
packet to penetrate further down into AdS$_5$.
The same repulsive barrier, with a height $\propto Q^2$, shows up also at
finite temperature\footnote{The photon has 4--momentum
$q^\mu=(\omega,0,0,q)$ in the plasma rest frame and we assume that
$Q^2\equiv q^2-\omega^2\gg T^2$, as appropriate for hard probes.}, but
in that case there is also an attractive interaction $\propto
\omega^2T^4$, namely the gravitational attraction by the BH. The physical
interpretation of the latter in the dual gauge theory is quite subtle
\citep{HIM2} and will emerge from the arguments below. The competition
between these two interactions depends upon the kinematics. For
sufficiently low energy $\omega$, such that $\omega T^2\ll Q^3$, the
repulsive barrier wins and then the Maxwell field is stuck within a
distance $\chi\lesssim 1/Q\ll 1/T$ from the Minkowski boundary, so like
in the vacuum (cf. Fig.~\ref{fig:WAVE} left). In this regime there is
essentially no interaction with the black hole, meaning no absorption of
the virtual photon by the plasma, and hence no DIS. But for higher
energies and/or temperatures, such that $\omega T^2\gtrsim Q^3$, the
attraction wins and then the wave--packet falls into the BH horizon, from
which it cannot escape back anymore (cf. Fig.~\ref{fig:WAVE} right): the
space--like photon is completely absorbed into the plasma.
To understand the existence of these two regimes, it is useful to note
that the `criticality' condition for having strong interactions, that is
$\omega T^2\sim Q^3$, can be rewritten as
\begin{eqnarray}\label{Qs}
Q\,\sim\,\frac{\omega}{Q^2}\ T^2\,,
\end{eqnarray}
with the following interpretation \citep{HIM2} : the scattering becomes
strong when the lifetime $\Delta t\sim {\omega}/{Q^2}$ of the partonic
fluctuation is large enough for the mechanical work $W=\Delta t\times
F_T$ done by the {\em plasma force} $F_T\sim T^2$ acting on these partons
to compensate for their energy deficit $\sim Q$. This mechanical work
allows the partons to become (nearly) on--shell --- or more precisely to
reduce their virtuality from the original value $Q\gg T$ down to a value
of order $T$. When this happens, the fluctuation {\em thermalizes} ---
the partons become a part of the thermal bath --- and the photon
disappears into the plasma.
This plasma force $F_T\sim T^2$ represents (in some average way) the
effect of the strongly--coupled plasma on partonic fluctuations and can
be viewed as a prediction of the AdS/CFT calculation. Note that one
cannot interpret this force in terms of individual collisions between the
virtual photon and some `plasma constituents' : the BH dual to the plasma
is homogeneous in the four physical dimensions, hence it cannot transfer
any 4--momentum to the photon, so like a genuine scattering would do
(recall, e.g., Fig.~\ref{fig:quench} left). Rather, this is a kind of
{\em tidal force} which pulls the partons apart until they disappear in
the plasma. The emergence of a tidal force, which is a hallmark of
gravitational interactions, in the context of a gauge theory may look
surprising, but in fact this is not more mysterious than the basic
paradigm of the gauge/gravity duality --- the fact that a gauge theory at
strong coupling can be effectively described as gravity. Further insight
in that sense comes from an argument \citep{Polchinski:2002jw} based on
the operator product expansion (OPE): among the infinitely many
leading--twist operators which {\em a priori} contribute to OPE for DIS,
there is only one which survives in the strong coupling limit --- the
energy--momentum tensor $T_{\mu\nu}$. All the other operators acquire
large, negative, anomalous dimensions $\propto\lambda^{1/4}$ and thus are
strongly suppressed when $\lambda\to\infty$. (See also the discussion in
Sect.~\ref{SAT}.) Accordingly, one expects the theory of scattering in a
gauge theory at strong coupling to be an effective theory for
$T_{\mu\nu}$. By covariance, this must be a gravity theory.
\section{Parton saturation at strong coupling}
\label{SAT}
The OPE argument alluded to above also helps clarifying the {\em partonic
picture} of the AdS/CFT results for DIS at strong coupling
\citep{HIM2,HIM3}, to which I now turn. To formulate this picture, one
needs to use the DIS variables $Q^2=q^2-\omega^2$ and $x= Q^2/(2\omega
T)$. As previously mentioned, the AdS calculation allows one to deduce
the DIS structure function $F_2(x,Q^2)$ from the imaginary part of the
current--current correlator. The discussion in the previous section
suggests that there should be a dramatic change in $F_2(x,Q^2)$ at the
critical kinematics defined by the condition in \eqnum{Qs}. The latter
can be rewritten as $Q_s(\omega)\simeq (\omega T^2)^{1/3}$ or, in terms
of the DIS variables,
\beq\label{xs}
Q_s(x)\,\simeq\,\frac{T}{x}\,,\qquad\mbox{or}\qquad x_s(Q)
\,\simeq\,\frac{T}{Q}\,.\eeq
Any of these equations defines a line in the kinematical plane
$(x,\,Q^2)$, which for reasons to shortly become clear is dubbed the {\em
saturation line}. The AdS results for DIS off the strongly coupled plasma
can then be summarized as follows \citep{HIM2} :
\texttt{(i)} For relatively low energy, or high $Q^2$, such that $x >
x_s(Q)$, the scattering is negligible and $F_2(x,Q^2)\approx 0$. (More
precisely, there is a small contribution to $F_2$ produced via tunneling
across the repulsive barrier, but this is exponentially suppressed.)
\texttt{(ii)} For higher energies, or lower $Q^2$, such that $x \lesssim
x_s(Q)$, the scattering is strong and the structure function is non--zero
and parametrically large: $F_{2}(x,Q^2)\sim x{N^2_cQ^2}$.
These results are consistent with the energy--momentum sum--rule, which
requires the integral $\int_0^1{\rm d} x\,F_2(x,Q^2)$ to have a finite
limit as $Q^2\to\infty$. (This is simply the statement that the total
energy per unit length in the plasma is the same whatever is the
resolution scale $Q^2$ on which one measures this energy: by varying
$Q^2$ one merely changes the nature and size of the partons which carry
that energy, cf. Fig.~\ref{fig:RG}, but their cumulated energy remains
the same.) Using the above results, one finds indeed
\begin{eqnarray} \int_0^1{\rm d}
x\,F_2(x,Q^2)\,\simeq\,\int_0^{x_s}{\rm d}
x\,F_2(x,Q^2)\,\simeq\,
\,x_sF_2(x_s,Q^2)\,\sim\, N_c^2 T^2
\,, \label{SRT} \end{eqnarray}
where the integral is dominated by values $x\simeq x_s(Q)$.
The physical interpretation of these results becomes transparent after
recalling that the variable $x$ represents the longitudinal momentum
fraction of the plasma constituent which absorbs the virtual photon. Then
the above statement \texttt{(i)} implies that there are no partons at
large $x$, or high $Q^2$ : the strongly coupled plasma has no point--like
constituents. Also, statement \texttt{(ii)} together with \eqnum{SRT}
show that the total energy of the plasma as measured on a hard resolution
scale $Q^2\gg T^2$ is carried by very soft constituents with small values
of $x\simeq x_s(Q)\ll 1$.
This picture at strong coupling is very different from that of an
energetic hadron in QCD, as predicted by perturbative QCD and confirmed
by many experimental data \citep{CGCreviews}. In that case, the hadronic
wavefunction at high energy is dominated by small--$x$ partons (mostly
gluons), as produced via bremsstrahlung from partons with larger $x$.
Yet, the hadron energy is concentrated in the few partons with larger
values of $x$ (the `valence partons'); that is, the energy--momentum sum
rule is saturated by $x\sim 0.3$. Moreover these valence partons are seen
on all scales of $Q^2$, that is, there are {\em point--like}.
This rises the following questions: why and how did partons disappear at
strong coupling ? And what is the nature of the small--$x$ constituents
which carry the plasma energy on the hard resolution scale $Q^2$ ? A
first hint in that sense comes again from the OPE for DIS
\citep{Polchinski:2002jw}. The twist--two operators which enters OPE
probe the distribution of energy among the partons inside the hadron: the
hadron expectation value of the spin--$n$ twist--2 operator
$\mathcal{O}^{(n)}$ is proportional to the $(n-1)$--th moment of the
longitudinal momentum fraction $x$ carried by the quark and gluon
constituents of that hadron:
\beq\label{On}
\langle x^{n-1} \rangle_{Q^2}\,\equiv\,\int_0^1\rmd x\,x^{n-2}\,
F_2(x,Q^2)\,\propto\,
\langle\mathcal{O}^{(n)}\rangle_{Q^2}\,.
\eeq
As indicated in this equation, the operators depend upon the resolution
scale $Q^2$, because of the quantum evolution illustrated in
Fig.~\ref{fig:RG}. In a conformal theory at strong coupling, all such
operators except for $T_{\mu\nu}$ are strongly suppressed at large
$Q^2$~: $\mathcal{O}^{(n)}(Q^2)\propto (T^2/{Q^2})^{\lambda^{1/4}}\,\to
0$. Via \eqnum{On} this implies $\langle x^{n-1} \rangle_{Q^2}\to 0$ for
any $n>2$, which suggest a very rapid evolution towards $x=0$. This is in
fact natural at strong coupling, where one expects a {\em very efficient
parton branching}. Unlike at weak coupling, where parton radiation is
suppressed by powers of the coupling and thus favors the emission of soft
(small $x$) and collinear (small $k_\perp$) gluons --- for which the
bremsstrahlung probability is kinematically large ---, at strong coupling
there is no such a suppression anymore, and phase--space considerations
alone favor a `quasi--democratic' branching \citep{HIM2,HIM3} : the
energy and momentum of the parent parton are almost equally divided among
the daughter partons. (Soft and collinear emissions play no special role
at strong coupling, since they happen very slowly.) Via successive
branchings, all the parton will rapidly fall to small values of $x$ ---
actually the smallest one which are still consistent with
energy--momentum conservation (in the sense of the sum rule \eqref{SRT}).
That is, at strong coupling partons should still exist, but their
distributions should be concentrated at very small values of $x$. This
picture is indeed consistent with our previous findings for the strongly
coupled plasma.
One may furthermore wonder what is the mechanism which is responsible for
stopping the parton branching at sufficiently small values of $x$ and
which determines the specific $Q$--dependence of the critical value
$x_s(Q)$ --- or, vice--versa, the specific $x$--dependence of the
`saturation momentum' $Q_s(x)$. In Ref.~\citep{HIM2} we have proposed
that this mechanism is {\em parton saturation} : the partons keep
branching until their phase--space occupation numbers --- the number of
partons of a given color per unit transverse phase space $(b_\perp,
k_\perp)$ and unit rapidity $Y\equiv \ln (1/x)$ --- become of
$\mathcal{O}(1)$ :
\beq
\frac{1}{N^2_c}\, \frac{\rmd N}{\rmd Y \rmd^2b_\perp \rmd^2k_\perp}
\,\simeq\,1\qquad\mbox{for}\qquad k_\perp\lesssim Q_s(x)\,=\,
\frac{T}{x}\,. \label{phisat} \eeq
This interpretation follows from the AdS/CFT result for the
$F_{2}(x,Q^2)$ at $Q\lesssim Q_s(x)$, as shown before, together with the
observation that the quantity $(1/x)F_{2}(x,Q^2)$ is essentially the
number of partons in the plasma per unit area per unit rapidity as `seen'
by a virtual photon with resolution $Q$ :
\beq
\frac{1}{x}\,F_2(x,Q^2)\,\simeq\,\int^{Q}\!\! \rmd^2k_\perp\
\frac{\rmd N}{\rmd Y \rmd^2b_\perp \rmd^2k_\perp}
\,\sim \,N_c^2Q^2\qquad\mbox{for}\qquad
Q\lesssim Q_s(x)\,.\label{nsat}
\eeq
This interpretation is appealing in that it suggests some continuity in
the physics of saturation and unitarization from weak to strong coupling:
saturation occurs when the occupation numbers are high enough for the
repulsive interactions among the partons to prevent further radiation
\citep{CGCreviews}. At weak coupling, this requires large gluon
occupation numbers, of order $1/\alpha_s$, in order to compensate for the
weakness of the repulsive interactions. But at strong coupling, one can
think of the individual partons as `hard disks' with transverse area
$1/k_\perp^2$; then saturation occurs when these disks start to touch
with each other, {\em i.e.} for occupation numbers of order one.
But, clearly, there are important differences between the parton picture
at weak and respectively strong coupling. These differences are most
striking outside the saturation region, at $k_\perp\gg Q_s(x)$, where at
strong coupling there are no partons at all, whereas at weak coupling the
parton distributions show large `leading--twist' tails, which in fact
dominate the phenomenology at both HERA and LHC. Another important
difference refers to the energy dependence of the saturation momentum
$Q_s$. At weak coupling, this is determined by the rate for gluon
emission via bremsstrahlung, and more precisely by the BFKL evolution.
This predicts $Q_s^2 \sim 1/x^{\omega}$ where the `BFKL intercept'
$\omega$ is parametrically of $\mathcal{O}(\alpha_s N_c)$ and numerically
$\omega\simeq 0.2\div 0.3$ --- in agreement with the HERA data
\citep{CGCreviews}. At strong coupling we have found a much faster
increase with $1/x$, namely $Q_s^2(x) \propto 1/x^2$, but one factor
$1/x$ out of this result is simply a kinematical effect, related to our
study of an {\em infinite} plasma. That is, one should understand the
above result as $Q_s^2(x) \propto \Delta t/x$, where $\Delta t\sim
{\omega}/{Q^2}\sim 1/xT$ is the lifetime of the partonic fluctuation:
since the medium is infinite, the effects of the interactions accumulate
all the way along the parton lifetime. As for the other factor $1/x$,
this exhibits the `graviton intercept' $j-1=1$ ($j=2$ is the spin of the
graviton) and reflects the fact that the interactions responsible for DIS
at strong coupling involve exchanges of the energy--momentum tensor ---
the only operator which survives in OPE at strong coupling.
The above discussion also suggests how our result for $Q_s^2(x)$ should
change when, instead of an infinite plasma, we consider a slice of the
plasma with longitudinal size $L_z \ll {\omega}/{Q^2}$. In that case one
expects
\beq\label{QsatTL}
Q_s^2(x, T,L_z)\,\sim\, \frac{T^3 L_z}{x}\qquad
\mbox{(slice of the plasma with $L_z\ll 1/xT$)}\,,\eeq
and this is indeed confirmed by the respective AdS/CFT calculation
\citep{Mueller:2008bt,Avsar:2009xf}. Such a `slice of the plasma' may be
viewed as a rough model for a `nucleus' in ${\mathcal N}=4$ SYM (which
however involves $N_c^2$ degrees of freedom per unit volume)
\citep{Albacete:2008vs,Albacete:2009ji,Dominguez:2008vd,Gubser:2008pc}.
\section{High--energy scattering at strong coupling}
The previously described parton picture implies that high--energy
processes taking place in the {\em vacuum} of an hypothetical world which
is conformal and strongly coupled would look quite different from the
corresponding processes in QCD. For instance, the absence of large--$x$
partons means that, in the collision between two strongly coupled
hadrons, there is no particle production at forward and backward
rapidities. This is in sharp contrast to the situation at RHIC, where the
large--$x$ partons from the incoming nuclei are seen to emerge from the
collision, as hadronic jets, along their original trajectories.
\begin{figure*}[htb]\centerline
\includegraphics[width=.8\textwidth]{EPEM_ADS.eps}}
\caption{\sl\small $e^+e^-$ annihilation. Left: weak coupling. Right:
strong coupling. }\label{Fig:ISOTROPY}
\end{figure*}
A related prediction of AdS/CFT is the absence of jets in
electron--positron annihilation at strong coupling
\citep{HIM2,Hofman:2008ar}. Fig.~\ref{Fig:ISOTROPY} exhibits the typical,
2--jet, final state in $e^+e^-$ annihilation at weak coupling (left)
together with what should be the corresponding state at strong coupling
(right). In both cases, the final state is produced via the decay of a
time--like photon into a pair of partons and the subsequent evolution of
this pair. At weak coupling this evolution typically involves the
emission of soft and collinear gluons, with the result that the leading
partons get dressed into a pair of well--collimated jets of hadrons (cf.
Fig.~\ref{Fig:ISOTROPY} left). At strong coupling, parton branching is
much more efficient, as previously explained, and rapidly leads to a
system of numerous and relatively soft quanta, with energies and momenta
of the order of the confinement scale, which are isotropically
distributed in space (cf. Fig.~\ref{Fig:ISOTROPY} right)
\citep{Hofman:2008ar}.
But such discrepancies should not come as a surprise: after all, we know
that hard processes in high energy QCD are governed by weak coupling and
perturbative QCD does a good deal in predicting the respective
cross--sections. As for the softer processes in the vacuum, so like
hadronization, for which $\alpha_s\sim 1$, these are largely controlled
by confinement and hence they remain out of the reach of the AdS/CFT
techniques. Remember however that our original motivation for studying
high--energy processes at strong coupling is rather related to the {\em
quark--gluon plasma}, where there is no confinement and the coupling
might indeed be quite strong.
So let me finally return to this topic and consider the propagation of a
`hard probe' --- say, a heavy quark --- through a strongly--coupled
plasma. The string theory dual of a heavy quark is a Nambu--Goto string
hanging down from the boundary of AdS$_5$ and propagating through the
AdS$_5$ black hole space--time. By solving the corresponding equations of
motion, it has been possible to compute both the quark energy loss
\citep{Herzog:2006gh,Gubser:2006bz} and its momentum broadening
\citep{CasalderreySolana:2006rq,CasalderreySolana:2007qw,Giecold:2009cg}
at strong coupling. The physical interpretation of the results
\citep{Dominguez:2008vd,Giecold:2009cg} turns out to be quite
interesting: unlike in perturbative QCD, the dominant mechanism at work
is not thermal rescattering (cf. Fig.~\ref{fig:quench} left), but rather
{\em medium--induced parton branching} (cf. Fig.~\ref{fig:quench} right).
The corresponding physical picture is in fact quite similar to that of
DIS, as discussed in Sect.~4. The heavy quark can emit space--like quanta
of ${\mathcal N}=4$ SYM, which in the vacuum would have only a finite
lifetime $\Delta t\simeq \omega/Q^2$; after that time, they would be
reabsorbed by the heavy quark. (As usual, $\omega$ and $Q$ refers to the
energy and the virtuality of the emitted quanta.) However, in the
presence of the plasma, those quanta having a virtuality $Q$ lower than
$Q_s(\omega)$ can escape into the medium, and thus provide energy loss
and momentum broadening. Here, $Q_s(\omega)\simeq (\omega
T^2)^{1/3}$ is the plasma saturation
momentum on the energy scale of the fluctuation, cf. \eqnum{Qs}.
Based on this physical picture, it is possible to estimate the rate for
energy loss: each emission brings in an energy loss $\Delta E\simeq
\omega$ over a time $\Delta t$, so the corresponding rate ${{\rm d}
E}/{{\rm d} t}$ is proportional to $\omega/\Delta t\simeq Q^2 \lesssim
Q_s^2(\omega)$. Hence, the energy loss is dominated by those fluctuations
having the maximal possible energy $\omega_{\rm max}$ and a virtuality
equal to the corresponding saturation momentum: $Q\simeq Q_s(\omega_{\rm
max})$. More precisely, the quantity which is limited is not the energy
$\omega$ of a quanta, but its `rapidity' $\gamma_p\equiv\omega/Q$ (the
Lorentz boost factor): this cannot exceed the rapidity
$\gamma=1/\sqrt{1-v^2}$ of the heavy quark (which is here assumed to
propagate at constant speed $v$). It is therefore appropriate to
reexpress $Q_s$ as a function of $\gamma_p$, by successively writing
$Q_s\simeq (\omega T^2)^{1/3}= (\gamma_p Q_s T^2)^{1/3} =
\sqrt{\gamma_p}\, T$. This quantity takes a maximal value $(Q_s)_{\rm
max}= \sqrt{\gamma}\, T$, thus yielding the following estimate for the
rate for energy loss (below, $Q_s\equiv (Q_s)_{\rm max}$) :
\begin{eqnarray}\label{dEdt}
-\,\frac{{\rm d} E}{{\rm d} t}\,\simeq\,\sqrt{\lambda}\,
\frac{ \omega}{(\omega/Q_s^2)}
\,\simeq\,\sqrt{\lambda}\,Q_s^2 \,\sim\,\sqrt{\lambda}\,\gamma\,T^2
\,.\end{eqnarray}
The additional factor $\sqrt{\lambda}$ comes from the fact that, at
strong coupling, the heavy quark does not radiate just a single quanta
per time $\Delta t$, but rather a large number $\sim {\sqrt{\lambda}}$.
Eq.~\eqref{dEdt} is parametrically consistent with the respective AdS/CFT
result \citep{Herzog:2006gh,Gubser:2006bz}.
One can similarly estimate the momentum broadening: the $\sqrt{\lambda}$
quanta emitted during $\Delta t$ are uncorrelated with each other, so
they randomly modify the transverse momentum of the heavy quark, by a
typical amount $\Delta k_\perp\sim Q_s$ per emission. Such random
changes add in quadrature, thus yielding
\begin{eqnarray}\label{dpTdt}
\frac{{\rm d} \langle k_\perp^2\rangle}{{\rm d} t}\,\sim\,
\frac{\sqrt{\lambda}\,Q_s^2}{(\omega/Q_s^2)} \,\sim\,
\sqrt{\lambda}\,\frac{Q_s^4}{\gamma Q_s}\,\sim\,
\sqrt{\lambda}\,\sqrt{\gamma}\,T^3\,,\end{eqnarray}
in agreement with the explicit calculations in
Refs.~\citep{CasalderreySolana:2006rq,CasalderreySolana:2007qw,Giecold:2009cg}.
Note the strong enhancement of the medium effects at high energy, as
expressed by the Lorentz $\gamma$ factor in Eqs.~\eqref{dEdt} and
\eqref{dpTdt}: this might qualitatively explain the strong suppression of
particle production seen in Au+Au collisions at RHIC (cf. Sect.~2).
\section{Conclusions}
The AdS/CFT calculations summarized here demonstrate that, at least in a
conformal world, parton saturation is a universal phenomenon, appearing
at both weak and strong coupling, and presumably for any intermediate
value of the coupling. More generally, the concept of `parton' in
relation with high energy scattering appears to be relevant at strong
coupling as well. There are significant differences with respect to the
respective picture at weak coupling, so like the absence of point--like
constituents. These differences can be intuitively understood as
consequences of parton evolution at strong coupling. These differences
and their consequences for high--energy scattering --- which look very
different from the known phenomenology in QCD --- rule out this conformal
strong--coupling scenario as a candidate theory for high--energy
processes in the vacuum. On the other hand, such methods can give us a
hint towards understanding the deconfined QCD matter to be copiously
produced in the intermediate stages of heavy ion collisions at LHC. In
particular, they could shed light on fundamental open questions, such as
the rapid thermalization, the small entropy--to--density ratio, or the
strong jet quenching observed in relation with this matter at RHIC. Last
but not least, by combining in a unified theoretical framework concepts
and methods coming from fields as different as gravity, string theory,
quantum field theory, statistical physics, and hydrodynamics, the
gauge/string duality teaches us the unity of physics.
\providecommand{\href}[2]{#2}\begingroup\raggedright
| -24,940.050478 |
[
-3.13671875,
2.912109375
] | 15.031056 |
[
-2.2578125,
0.8369140625,
-2.025390625,
-5.25390625,
-0.740234375,
7.296875
] |
[
2.2890625,
8,
1.4892578125,
5.05078125
] | 298 | 6,700 |
[
-1.6884765625,
1.798828125
] | 24.641174 |
[
-6.046875,
-4.5,
-5.0078125,
-2.900390625,
1.888671875,
13.8515625
] | 1.255935 | 11.377291 | 23.373134 | 1.700186 |
[
1.8712512254714966
] | -17,371.14741 | 5.407164 | -24,742.240896 | 1.413272 | 6.035648 |
[
-2.833984375,
-3.814453125,
-3.33203125,
-4.5546875,
2.23828125,
12.0859375
] |
[
-5.40234375,
-2.53515625,
-2.34765625,
-1.244140625,
3.732421875,
4.984375
] | |
BkiUeyY25V5jJ1pbbnv0
|
\section{Introduction}
\IEEEPARstart{R}{eflectivity} inversion is an important deconvolution problem in reflection seismology, helpful in characterizing the layered subsurface structure. The subsurface is modeled as having sparse reflectivity localized at the interfaces between two layers, assuming largely horizontal and parallel layers, each with constant impedance \cite{oldenburg1983recovery, yilmaz2001seismic}, or in other words, a piecewise-constant impedance structure. Figure \ref{fig:model} shows a $3$-layer model of the subsurface, with a wet sandstone between two shale layers \cite{russell2019machine}.
\begin{figure*}[!t]
\centering
\resizebox{0.9\linewidth}{!}{
\includegraphics{figures/model_durin_arxiv.pdf}}
\caption{\protect\includegraphics[height=1.5ex]{figures/rgb.png}~An ideal three-layer subsurface model. The operator $*$ denotes convolution.}\label{fig:model}
\end{figure*}
The reflection coefficients ($r_{i}$) (Fig.~\ref{fig:model}(c)) at the interface between two adjoining layers $i$ and $i+1$ (Fig.~\ref{fig:model}(a)) are related to the subsurface geology \cite{oldenburg1983recovery} through the relation:
\begin{equation}\label{rhov}
r_{i} = \frac{\rho_{i+1}V_{i+1} - \rho_{i}V_{i}}{\rho_{i+1}V_{i+1} + \rho_{i}V_{i}},
\end{equation}%
where $\rho_i$ and $V_i$ are the density and P-wave velocity, respectively, of the $i^{th}$ layer. The product of density ($\rho$) and P-wave velocity ($V$) is known as the acoustic impedance (Fig.~\ref{fig:model}(b)).
Such a layered subsurface is often recovered through the widely used convolutional model \eqref{convmodel}, wherein the observed seismic data is modeled as the convolution between the source pulse ${\boldsymbol{h}}$ (Fig.~\ref{fig:model}(d)) and the Earth response or reflectivity ${\boldsymbol{x}} \in \mathbb{R}^{n}$ (Fig.~\ref{fig:model}(c)) \cite{taylor1979deconvolution, shearer2009introduction}. The observation at the surface is a noisy seismic trace ${\boldsymbol{y}} \in \mathbb{R}^{n}$ (Fig.~\ref{fig:model}(e)) given by:
\begin{equation}\label{convmodel}
{\boldsymbol{y}} = {\boldsymbol{h}}*{\boldsymbol{x}} + {\boldsymbol{n}},
\end{equation}%
where ${\boldsymbol{n}}$ is the measurement noise, $*$ denotes convolution, and ${\boldsymbol{h}}$ is assumed to be a Ricker wavelet. When the wavelet is known, the linear inverse problem \eqref{convmodel} can be solved within the sparsity framework by modeling the reflectivity ${\boldsymbol{x}}$ as sparse, as the significant interfaces are important in the inversion. Further, the location of the interfaces (in other words, the support recovery) is prioritized over amplitude recovery \cite{shearer2009introduction}.
\subsection{Prior Art}
We broadly classify the prior work related to reflectivity inversion into two categories, namely, optimization-based techniques and data-driven techniques.
\subsubsection{Optimization Techniques}
The solution $\boldsymbol{x}$ in \eqref{convmodel} can be recovered by minimizing the classical least-squares (LS) objective function:%
\begin{equation}\label{leastsquares}
\arg\min_{{\boldsymbol{x}}}\frac{1}{2}{\left\| {\boldsymbol{h}}*{\boldsymbol{x}}-{\boldsymbol{y}} \right\|}_{2}^{2}.
\end{equation}%
However, the finite length and measurement noise, and the loss of low and high-frequency information due to convolution of the reflectivity with a bandlimited wavelet \cite{oldenburg1983recovery, berkhout1977least, debeye1990lp}, lead to non-unique solutions to the above problem \cite{yuan2019seismic}. This ill-posed inverse problem is tackled by employing a sparsity prior on the solution \cite{tarantola2005inverse}. The $\ell_1$-norm regularization is widely preferred by virtue of its convexity \cite{oldenburg1983recovery, taylor1979deconvolution, liu2014fast, zhang2017inversion}. The $\ell_1$-regularized reflectivity inversion problem is posed as follows:
\begin{equation}\label{l1}
\arg\min_{{\boldsymbol{x}}} \, \frac{1}{2}{\left\| {\boldsymbol{h}}*{\boldsymbol{x}}-{\boldsymbol{y}} \right\|}_{2}^{2} + \lambda {\left\| {\boldsymbol{x}} \right\|}_{1},
\end{equation}%
where $\lambda$ is the regularization parameter.
Zhang and Castagna \cite{zhang2011seismic} adopted basis-pursuit inversion (BPI) to solve the problem in \eqref{l1}, based on the basis-pursuit de-noising algorithm proposed by \cite{chen2001atomic}. Algorithms such as the Iterative Shrinkage-Thresholding Algorithm (ISTA) \cite{daubechies2004iterative} and its faster variant, FISTA \cite{beck2009fast}, have been proposed to solve the $\ell_1$-norm regularized deconvolution problem. Relevant to the problem under consideration, \cite{perez2012inversion} adopted FISTA for reflectivity inversion. Further studies by \cite{perez2013high} and \cite{li2020debiasing} adopted FISTA along with debiasing steps of LS inversion and ``adding back the residual'', respectively, to improve amplitude recovery after support estimation using FISTA.
The sparsity of seismic data is difficult to estimate in practice and the adequacy of the $\ell_1$ norm for the reflectivity inversion problem has not been established \cite{yuan2019seismic, li2019optimal}. Also, $\ell_1$-norm regularization results in a biased estimate of ${\boldsymbol{x}}$ \cite{candes2008enhancing, zhang2010analysis, selesnick2017sparse}. In their application of FISTA to the seismic reflectivity inversion problem, \cite{perez2013high} and \cite{li2020debiasing} observed an attenuation of reflection coefficient magnitudes. They adopted post-processing debiasing steps \cite{wright2009sparse} to tackle the bias introduced due to the $\ell_1$-norm regularization. Nonconvex regularizers such as the minimax concave penalty (MCP) \cite{zhang2010nearly} have been shown to overcome the shortcomings of $\ell_1$ regularization in inverse problems. The advantages of adopting nonconvex regularizers over $\ell_1$ have been demonstrated in sparse recovery problems \cite{selesnick2017sparse, zhang2010nearly, woodworth2016compressed}. Particularly pertinent to this discussion is the data-driven $\ell_{q}$-norm regularization $(0<q<1)$ for seismic reflectivity inversion proposed by \cite{li2019optimal}, wherein the optimal $q$ was chosen based on the input data.
\subsubsection{Data-driven Methods}
The application of sparse Bayesian learning (SBL) for reflectivity inversion has also been explored \cite{yuan2013spectral, yuan2019seismic}, where the sparse reflection coefficients are recovered by maximizing the marginal likelihood. This is done through a sequential algorithm-based approach (SBL-SA) to update the sparsity-controlling hyperparameters \cite{yuan2013spectral}, or through the expectation-maximization algorithm (SBL-EM) \cite{yuan2019seismic, wipf2004sparse}. The application of neural networks to inversion problems in geophysics \cite{adler2021deep}, particularly reflectivity inversion, is a recent development \cite{russell2019machine, kim2018geophysical}. Kim and Nakata \cite{kim2018geophysical} used an elementary feedforward neural network to recover the sparse reflectivity. They obtained superior support recovery but low amplitude recovery using the neural network compared to a least-squares approach. Further, \cite{russell2019machine} discussed the suitability of machine learning approaches such as feedforward neural networks \cite{kim2018geophysical}. He observed that data-driven techniques outperform conventional deconvolution approaches when knowledge about the underlying geology is limited while also providing a computational advantage. Although, in applications pertaining to geoscience, and specifically, geophysics, the level of model interpretability is critical as one aims to gain physical insights into the system under consideration \cite{bergen2019machine}. Insights into an inverse problem, in addition to savings in terms of computational times, can be gained through deep neural network architectures that are informed by the sparse linear inverse problem itself \cite{bergen2019machine}.
To that end, \cite{gregor2010learning} proposed a new class of data-driven framework architectures based on unfolding iterative algorithms into neural network architectures \cite{monga2021algorithm}. They proposed learned ISTA (LISTA), based on unrolling the update steps of ISTA \cite{daubechies2004iterative} into layers of a feedforward neural network. Subsequent studies have demonstrated the efficacy of this class of model-based deep learning architectures \cite{shlezinger2020modelbased} in compressed sensing and sparse-recovery applications \cite{monga2021algorithm, zhang2017ista, borgerding2017amp, sreter2018learned, liu2019deep, li2020efficient, pokala2020confirmnet, wang2020dnu, tolooshams2020deep, jawali2020cornet, tolooshams2021unfolding}. Deep-unfolding combines the advantages of both data-driven and iterative techniques.
\subsection{Motivation and Contribution}
The limitations of $\ell_1$ regularization \cite{candes2008enhancing, zhang2010analysis, selesnick2017sparse}, in addition to the challenge of estimating the sparsity of seismic reflection data \cite{yuan2019seismic, li2019optimal}, can be overcome through nonconvex regularization \cite{li2019optimal}. However, the corresponding nonconvex cost suffers from local minima, and optimization becomes quite challenging. The hyper-parameters of most iterative techniques are set heuristically, which is likely to yield suboptimal solutions. Machine learning approaches outperform conventional techniques in reflectivity inversion \cite{russell2019machine}, especially in support recovery \cite{kim2018geophysical}, which is given higher priority \cite{shearer2009introduction}. Deep-unrolled architectures \cite{monga2021algorithm}, which belong to a class of model-based neural networks, are not entirely dissociated from the underlying physics of the problem, which is a potential risk in elementary data-driven approaches such as feedforward neural networks \cite{russell2019machine, kim2018geophysical, bergen2019machine}. These factors motivate us to employ a data-driven nonconvex regularization strategy and solve the reflectivity inversion problem through deep-unrolled architectures.
The contributions of this paper are summarized as follows.
\begin{enumerate}
\item We construct an optimization cost for sparse seismic reflectivity inversion based on the weighted counterpart of the {\it minimax-concave} penalty (MCP) \cite{zhang2010nearly}, where each component of the sparse reflectivity ${\boldsymbol{x}}$ is associated with a different weight.
\item The resulting optimization algorithm, the iterative firm-thresholding algorithm (IFTA) \cite{pokala2019firmnet} is unfolded into the deep-unfolded reflectivity inversion network (DuRIN). To the best of our knowledge, model-based architectures have not been explored for solving the seismic reflectivity inversion problem. In fact, deep-unrolling has not been employed for solving seismic inverse problems \cite{adler2021deep}.
\item The efficacy of the proposed formulation with respect to the state of the art is demonstrated over synthetic $1$-D and $2$-D data and on simulated as well as real datasets.
\end{enumerate}
\section{Weighted-MCP-Regularized Reflectivity Inversion}
In this study, we use the weighted counterpart of the nonconvex minimax-concave penalty (MCP) \cite{zhang2010nearly} defined as:
\begin{equation}
g_{ {\boldsymbol{\gamma}} }( {\boldsymbol{x}}; {\boldsymbol{\mu}}) = \sum\limits_{i=1}^n g_{\gamma_i}(x_i; \mu_i),
\end{equation}
where ${\boldsymbol{x}} = \left[ x_1, x_2, x_3, \ldots, x_n \right]$, ${\boldsymbol{\mu}} = \left[ \mu_1, \mu_2, \mu_3, \ldots, \mu_n \right]$, ${\boldsymbol{\gamma}} = \left[ \gamma_1, \gamma_2, \gamma_3, \ldots, \gamma_n \right]$, and
\begin{equation}\label{MCP}
g_{{\gamma}_i}( {x}_i; {\mu}_i) = \begin{cases} {{{\mu}}_i \left( \left| {x}_i \right| - \cfrac{{\left| {x}_i \right|}^2}{2 {{\mu}}_i {\gamma}_i } \right),
\quad {\text{for} \left| {x}_i \right| \leq {\gamma}_i ,} \\
{\cfrac{{{{\mu}}_i }^{2}{\gamma}_i }{2},
\quad\quad\qquad\qquad {\text{for} \left| {x}_i \right| \geq {\gamma}_i {{\mu}}_i ,} \end{cases}
\end{equation}
$\forall~i \in [\![1,n]\!]$. The trainable parameters of $g_{ {\boldsymbol{\gamma}} }( {\boldsymbol{x}}; {\boldsymbol{\mu}})$, ${\boldsymbol{\mu}} \in \mathbb{R}^n_{>0}$ and ${\boldsymbol{\gamma}} \in \mathbb{R}^n_{>1}$, are learned in a data-driven setting. $\mathbb{R}^n_{>t}$ denotes the set of vectors in $\mathbb{R}^n$ with entries greater than $t$.
\begin{figure}
\centering
\includegraphics[width=0.8\linewidth]{figures/regularizers.pdf}
\caption{\protect\includegraphics[height=1.5ex]{figures/rgb.png}~The minimax-concave penalty corresponding to $\gamma=2$, $\gamma=3$ and $\gamma=100$.}
\label{fig:penalty_functions}
\end{figure}
\subsection{Problem Formulation}
Formulating the optimization problem using the weighted MCP regularizer defined above leads to the objective function:%
\begin{equation}\label{regularization_problem}
\arg\min_{{\boldsymbol{x}}} \, \Big\{ {J({\boldsymbol{x}})} = \underbrace{\frac{1}{2}{\left\| {\boldsymbol{h}}*{\boldsymbol{x}}-{\boldsymbol{y}} \right\|}_{2}^{2}}_{F({\boldsymbol{x}})} + g_{{\boldsymbol{\gamma}}}( {\boldsymbol{x}}; {\boldsymbol{\mu}}) \Big\}.
\end{equation}%
The objective function ${J}$, data-fidelity term $F$, and the regularizer $g$ satisfy the following properties, which are essential for minimizing ${J}$:
\begin{enumerate}
\item[P1.] $F : \mathbb{R}^{n} \rightarrow \left (- \infty,\infty \right] $ is proper, closed, and $L$-smooth, i.e., $\Vert \nabla F({\boldsymbol{x}}) - \nabla F({\boldsymbol{y}})\Vert_{2} \leq L \Vert {\boldsymbol{x}} - {\boldsymbol{y}} \Vert_{2}$.
\item[P2.] $g$ is lower semi-continuous.
\item[P3.] ${J}$ is bounded from below, i.e., inf ${J} > -\infty$.
\end{enumerate}
Next, we optimize the problem stated in \eqref{regularization_problem} through the Majorization-Minimization (MM) approach \cite{figueiredo2007majorization}, and the resulting algorithm, the iterative firm-thresholding algorithm (IFTA) \cite{pokala2019firmnet}. Unfolding the iterations of IFTA results in a learnable network called deep-unfolded reflectivity inversion network (DuRIN), which has a similar architecture to FirmNet \cite{pokala2019firmnet}.
\subsection{IFTA: Iterative firm-thresholding algorithm}
Due to P1, there exists $\eta < 1/L$ such that $F({\boldsymbol{x}})$ is upper-bounded locally by a quadratic expansion about ${\boldsymbol{x}} = {\boldsymbol{x}}^{(k)}$ as:
\begin{equation}
F({\boldsymbol{x}}) \leq {Q^{(k)} \left( \boldsymbol{x} \right)},
\end{equation}
where ${Q^{(k)} \left( \boldsymbol{x} \right)} = F({\boldsymbol{x}}^{(k)}) + ({\boldsymbol{x}} - {\boldsymbol{x}}^{(k)})^{\intercal}\nabla F({\boldsymbol{x}}^{(k)}) + \frac{1}{2 \eta} {\left\| {\boldsymbol{x}} - {\boldsymbol{x}}^{(k)} \right\|}_{2}^{2}$. The majorizer to the objective function $J$ at ${\boldsymbol{x}}^{(k)}$ is given by:
\begin{equation}\label{majorizer}
H^{(k)}({\boldsymbol{x}}) = Q^{(k)} \left( \boldsymbol{x} \right) + g_{{\boldsymbol{\gamma}}}( {\boldsymbol{x}}; {\boldsymbol{\mu}}),
\end{equation}
such that $H^{(k)}({\boldsymbol{x}})\geq J(\boldsymbol{x})$. The update equation for minimizing $H^{(k)}({\boldsymbol{x}})$ to obtain ${\boldsymbol{x}}^{(k+1)}$ is given by,%
\begin{align}
{\boldsymbol{x}}^{(k+1)} = \arg\min_{{\boldsymbol{x}}} \frac{1}{2} {\left\| {\boldsymbol{x}}^{(k)} + \eta{\boldsymbol{h}}'* ({\boldsymbol{y}}-{\boldsymbol{h}}*{\boldsymbol{x}}^{(k)}) - {\boldsymbol{x}} \right\|}_{2}^{2} \nonumber \\
+ \eta g_{{\boldsymbol{\gamma}}}( {\boldsymbol{x}}; {\boldsymbol{\mu}}).
\end{align}%
The proximal operator of the penalty function $g_{{\gamma}_i}( {x}_i; {\mu}_i)$ is the firm-thresholding operator \cite{voronin2013new} given by
\begin{align}
\mathcal{G}_{{\gamma}_i}^{g}({x}_i; {\mu}_i) = \begin{cases} {0,
\qquad\qquad\quad\, {\text{for} \; \left| {x}_i \right| \leq {{\mu}}_i,} \\[5pt]
{\text{sgn} \left({x}_i\right) \odot \cfrac{{\gamma}_i}{{\gamma}_i - 1} \left( \left| {x}_i \right| - {{\mu}}_i \right),}& \\[8pt]
\qquad\qquad\qquad
{\text{for} \; {{\mu}}_i < \left| {x}_i \right| \leq {\gamma}_i {{\mu}}_i,} \\%[5pt]
{{x}_i,
\qquad\qquad\quad{\text{for} \; \left| {x}_i \right| > {\gamma}_i {{\mu}}_i}. \end{cases}
\label{firm_thresh}
\end{align}
\begin{figure}
\centering
\includegraphics[width=0.8\linewidth]{figures/operators.pdf}
\caption{\protect\includegraphics[height=1.5ex]{figures/rgb.png}~The proximal operators corresponding to the MCP for $\gamma=2$, $\gamma=3$ and $\gamma=100$.}
\label{fig:prox_operators}
\end{figure}
The update step at the $(k+1)^{th}$ iteration is given by
\begin{equation}
\begin{split}
{\boldsymbol{x}}^{(k+1)} = \mathcal{G}_{{\boldsymbol{\gamma}}}^{g} \left( {\boldsymbol{x}}^{(k)} + \frac{\eta}{2} {\boldsymbol{h}}' * \left( {\boldsymbol{y}} - {\boldsymbol{h}} * {\boldsymbol{x}}^{(k)} \right) \right),
\end{split}
\label{ifta}
\end{equation}%
where ${\boldsymbol{h}}'$ is the flipped version of ${\boldsymbol{h}}$. Algorithm \eqref{alg:ifta} lists the steps involved in solving the optimization problem \eqref{regularization_problem}. The update in \eqref{ifta} involves convolutions followed by nonlinear activation (the firm threshold in this case), and can therefore be represented as a layer in a neural network.
\begin{algorithm}[t]
\caption{IFTA: Iterative Firm-Thresholding Algorithm}\label{alg:ifta}
\KwIn{$\eta = 1/{\left\| {\boldsymbol{h}} \right\|}_2^2,~ k_{max}$, data ${\boldsymbol{y}}$, wavelet ${\boldsymbol{h}}$, learnable parameters ${\boldsymbol{\mu}}$, ${\boldsymbol{\gamma}}$}
Initialization: $ {\boldsymbol{x}}^{(0)} = 0 $\;
\While{$k < k_{max}$}{
$\boldsymbol{z}^{(k+1)} = {\boldsymbol{x}}^{(k)} - \frac{\eta}{2}\left( {\boldsymbol{h}}'* ({\boldsymbol{y}} - {\boldsymbol{h}}*{{\boldsymbol{x}}^{(k)}}) \right)$\;
${\boldsymbol{x}}^{(k+1)} = \mathcal{G}_{{\boldsymbol{\gamma}}}^{g} \left( \boldsymbol{z}^{(k+1)}; {\boldsymbol{\mu}} \right)$\;
$k = k + 1$\;
}
\KwOut{$\hat{{\boldsymbol{x}}} \leftarrow {\boldsymbol{x}}^{(k)}$}
\end{algorithm}%
\subsection{DuRIN: Deep-unfolded Reflectivity Inversion Network}
\label{subsec:durin}%
As mentioned in the previous section, the update in IFTA \eqref{ifta} can be interpreted as a layer in a neural network, and hence, we unroll the iterations of the algorithm into layers of a neural network to solve the problem given in \eqref{regularization_problem}. The proposed deep-unrolled architecture for sparse seismic reflectivity inversion is called Deep-unfolded Reflectivity Inversion Network (DuRIN), which resembles FirmNet \cite{pokala2019firmnet}. The input-output for each layer in DuRIN is given by
\begin{equation}
{\boldsymbol{x}}^{(k+1)} = \mathcal{G}_{{\boldsymbol{\gamma}}}^{g} ( \mathbf{W} {\boldsymbol{y}} + \mathbf{S} {\boldsymbol{x}}^{(k)}),
\label{eq:durin}
\end{equation}
where \( \mathbf{W}=\eta\mathop{{\boldsymbol{h}}'} \) and \( \mathbf{S}=\mathbf{I}-\eta\mathop{{\boldsymbol{h}}' {\boldsymbol{h}}} \) \cite{gregor2010learning} are initialized as Toeplitz matrices, and are dense and unstructured in the learning stage. The parameters that need to be learned are the matrices $\mathbf{W}$ and $\mathbf{S}$ and the thresholds (${\boldsymbol{\mu}}, {\boldsymbol{\gamma}}$), subject to ${\boldsymbol{\mu}}>{\boldsymbol{0}}, {\boldsymbol{\gamma}}>{\boldsymbol{1}}$, where ${\boldsymbol{0}}$ is the null vector and ${\boldsymbol{1}}$ is the vector of all ones. For training, we employ the smooth $\ell_1$ loss $\left( \text{Smooth}~\ell_1 = \beta \left\|{{\boldsymbol{x}} - \hat{{\boldsymbol{x}}}} \right\|_{1} + (1-\beta) \left\|{{\boldsymbol{x}} - \hat{{\boldsymbol{x}}}} \right\|_{2}^{2},~0<\beta \leq 1 \right)$ as the training cost, computed between the true reflectivity ${\boldsymbol{x}}$ and prediction $\hat{{\boldsymbol{x}}}$. In our experimental setup, $\beta=1$, and it is not a trainable parameter.
To improve amplitude recovery of DuRIN during inference, we re-estimate the amplitudes of the estimated sparse vector $\hat{{\boldsymbol{x}}}$ over the supports given by DuRIN as: $\hat{{\boldsymbol{x}}}_{\hat{\mathcal{S}}_{\hat{\boldsymbol{x}}}} = \mathbf{H}_{\hat{\mathcal{S}}_{\hat{\boldsymbol{x}}}}^{\dagger}{\boldsymbol{y}}$, where $\hat{\mathcal{S}}_{\hat{\boldsymbol{x}}}$ is the support, or the non-zero locations, of $\hat{\boldsymbol{x}}$, and $\mathbf{H}_{\hat{\mathcal{S}}_{\hat{\boldsymbol{x}}}}^{\dagger}$ is the pseudo-inverse of the Toeplitz matrix $\mathbf{H}$ of the kernel ${\boldsymbol{h}}$ over the support $\hat{\mathcal{S}}_{\hat{\boldsymbol{x}}}$. The steps of DuRIN for $k_{max}$ layers are listed in Algorithm~\ref{alg:durin}. Figure \ref{fig:schematic} illustrates a 3-layer architecture for DuRIN
\begin{figure*}[t]
\centering
\includegraphics[width=0.8\linewidth]{figures/schematic2.pdf}
\caption{\protect\includegraphics[height=1.5ex]{figures/rgb.png}~Schematic for (a) one iteration of the iterative firm-thresholding algorithm (IFTA) \cite{pokala2019firmnet}, and (b) 2 layers of the deep-unfolded reflectivity inversion network (DuRIN), which corresponds to FirmNet \cite{pokala2019firmnet}. A layer in the network (b) corresponds to an iteration in (a). The operator $\oplus$ represents element-wise sum.} \label{fig:schematic}
\end{figure*}%
We consider two variants of DuRIN, depending on the value of $\gamma_i,~\forall~i \in [\![1,n]\!]$. The generalized variant of DuRIN based on IFTA \cite{pokala2019firmnet}, where ${\gamma}_i$ are different across components and layers, is referred to as DuRIN-$1$, and corresponds to FirmNet \cite{pokala2019firmnet}. As $\gamma_i \to \infty$, the firm-thresholding function approaches the soft thresholding function \cite{selesnick2017sparse}. We call this variant DuRIN-$2$, which resembles LISTA \cite{gregor2010learning}.
\begin{algorithm}[t]
\caption{DuRIN: Deep-unfolded Reflectivity Inversion Network}\label{alg:durin}
\KwIn{$ {\boldsymbol{y}},~ \eta = 1/{\left\| \mathbf{H} \right\|}_2^2,~ k_{max},~ {\boldsymbol{\mu}},~ {\boldsymbol{\gamma}}$}
Initialization: $\mathbf{W},~ \mathbf{S},~ {\boldsymbol{x}}^{(0)} = \mathcal{G}_{{\boldsymbol{\gamma}}}^{g} (\mathbf{W}{{\boldsymbol{y}}}) $\;
\While{$k < k_{max}$}{
$\mathbf{c}^{(k+1)} = \mathbf{W}{{\boldsymbol{y}}} + \mathbf{S}{{\boldsymbol{x}}^{(k)}} $\;
${\boldsymbol{x}}^{(k+1)} = \mathcal{G}_{{\boldsymbol{\gamma}}}^{g} (\mathbf{c}^{(k+1)}) $\;
$k = k + 1$\;
}
$\hat{{\boldsymbol{x}}}_{\hat{\mathcal{S}}_{\hat{\boldsymbol{x}}}} = \mathbf{H}_{\hat{\mathcal{S}}_{\hat{\boldsymbol{x}}}}^{\dagger}{\boldsymbol{y}}$\;
\KwOut{$\hat{{\boldsymbol{x}}} \leftarrow {\boldsymbol{x}}^{(k)}$}
\end{algorithm}%
\section{Experimental Results}
\label{sec:experiments}
We present experimental results for the proposed networks, namely, DuRIN-$1$ and DuRIN-$2$, on synthetic $1$-D traces and $2$-D wedge models \cite{hamlyn2014thin}, the simulated $2$-D Marmousi$2$ model \cite{martin2006marmousi2}, and real $3$-D field data from the Penobscot $3$D survey off the coast of Nova Scotia, Canada \cite{penobscot3d}. We evaluate the performance of DuRIN-$1$ and DuRIN-$2$ in comparison with the benchmark techniques, namely, basis-pursuit inversion (BPI) \cite{chen2001atomic, zhang2011seismic}, fast iterative shrinkage-thresholding algorithm (FISTA) \cite{beck2009fast, perez2012inversion}, and expectation-maximization-based sparse Bayesian learning (SBL-EM) \cite{wipf2004sparse, yuan2019seismic}. To quantify the performance, we employ the objective metrics listed in the following section.
\subsection{Objective Metrics}
\label{subsec:metrics}
We evaluate the results using statistical parameters that measure the amplitude and support recovery between the ground-truth sparse vector ${\boldsymbol{x}}$ and the predicted sparse vector $\hat{{\boldsymbol{x}}}$.
\begin{enumerate}[wide, labelwidth=0pt, labelindent=0pt, noitemsep, nolistsep]
\item {Correlation Coefficient (CC) \cite{freedman2007statistics}:%
\begin{align*}
\text{CC} = \frac{\langle {\boldsymbol{x}}, \hat{{\boldsymbol{x}}} \rangle - \langle {\boldsymbol{x}}, {{\boldsymbol{1}}} \rangle \langle \hat{{\boldsymbol{x}}}, {{\boldsymbol{1}}} \rangle}{\sqrt{\left( \left\| {{\boldsymbol{x}}} \right\|_2^2 - {\langle {\boldsymbol{x}}, {{\boldsymbol{1}}} \rangle}^2 \right) \left( \left\| \hat{{\boldsymbol{x}}} \right\|_2^2 - {\langle \hat{{\boldsymbol{x}}}, {{\boldsymbol{1}}} \rangle}^2 \right) }},
\end{align*}%
where $\langle \cdot, \cdot \rangle$ denotes inner product, and ${\boldsymbol{1}}$ denotes a vector of all ones.
}%
\item Relative Reconstruction Error (RRE) and Signal-to-Reconstruction Error Ratio (SRER), defined as follows:%
\begin{align*
\text{RRE} = \frac{{\left\| \hat{{\boldsymbol{x}}} - {\boldsymbol{x}} \right\|}_2^2}{{\left\| {\boldsymbol{x}} \right\|}_2^2}, \quad \text{SRER} = 10\mathop {\log\limits_{10} \left(\frac{{\left\| {{\boldsymbol{x}}} \right\|_2^2}}{{\left\| {\hat{{\boldsymbol{x}}} - {{\boldsymbol{x}}}} \right\|_2^2}}\right){\text{dB}}}.
\end{align*}
\item {Probability of Error in Support (PES):%
\begin{align*}
\text{PES} = \left( \frac{{{\text{max}}(\left| {\hat{\mathcal{S}}_{\hat{\boldsymbol{x}}}} \right|,\left| {{\mathcal{S}}_{\boldsymbol{x}}} \right|) - \left| {\hat{\mathcal{S}}_{\hat{\boldsymbol{x}}}\mathop {\cap} {\mathcal{S}_{\boldsymbol{x}}}} \right|}}{{{\text{max}}(\left| {\hat{\mathcal{S}}_{\hat{\boldsymbol{x}}}} \right|,\left| {\mathcal{S}_{\boldsymbol{x}}} \right|)}}\right),
\end{align*}%
where $|\cdot|$ denotes the cardinality of the argument, and ${\mathcal{S}}_{\boldsymbol{x}}$ and $\hat{\mathcal{S}}_{\hat{\boldsymbol{x}}}$ denote the supports of ${\boldsymbol{x}}$ and $\hat{{\boldsymbol{x}}}$, respectively.}
\end{enumerate}%
\subsection{Training Phase}\label{subsec:training_phase}
The generation or acquisition of training data appropriately representing observed seismic data is crucial for the application of a data-driven approach to the reflectivity inversion problem \cite{kim2018geophysical}. We generate synthetic training data as $5\times10^{5}$ seismic traces, each of length $300$ samples, obtained by the convolution between $1$-D reflectivity profiles and a Ricker wavelet. The reflectivity profiles consist of $200$ samples each, and are padded with zeros before convolution such that their length is equal to that of the seismic traces \cite{russell2019machine}. Amplitudes of the reflection coefficients range from $-1.0$ to $1.0$, and the sparsity factor ($k$), i.e., the ratio of the number of non-zero elements to the total number of elements in a trace, is set to $0.05$. The locations of the reflection coefficients are picked uniformly at random, with no constraint on the minimum spacing between two reflectors/spikes, or in other words, the reflectors can be away only by one sampling interval. The reflectivity values, chosen randomly from the defined range of $-1.0$ to $1.0$, are assigned to the selected locations. The sampling interval, the increment in the amplitudes of the reflection coefficients, and the wavelet frequency vary based on the dataset.
The initial hyperparameters and the optimum number of layers in the network also vary depending on the parameters of the dataset, such as the sampling interval and the wavelet frequency. We initially consider six layers for both DuRIN-$1$ and DuRIN-$2$ for training and keep appending five layers at a time to arrive at the optimum number of layers. For the training phase, we set the batch size to $200$, use the ADAM optimizer \cite{kingma2014adam} and set the learning rate to $1\times10^{-4}$, with an input measurement signal-to-noise ratio (SNR) of $20$ dB to ensure the robustness of the proposed models against noise in the testing data \cite{kim2018geophysical}. The proposed networks, DuRIN-$1$ and DuRIN-$2$, are trained on $1$-D data, and operate on $1$-D, $2$-D, and $3$-D data trace-by-trace. In the following sections, we provide experimental results on synthetic, simulated, and real data.
\subsection{Testing Phase: Synthetic Data}\label{subsec:testing_phase}
\subsubsection{Synthetic 1-D Traces}\label{subsubsec:1d_trace}
We validated the performance of DuRIN-$1$ and DuRIN-$2$ on $1000$ realizations of synthetic $1$-D traces, with a $30$-Hz Ricker wavelet, $1$-ms sampling interval, and amplitude increment $0.2$. Table \ref{table:1d_trace} shows the comparison of objective metrics for the proposed networks and the benchmark techniques. DuRIN-$1$ and DuRIN-$2$ recover the amplitudes with higher accuracy than the compared methods, quantified by the objective metrics CC, RRE, and SRER, with considerably superior support recovery indicated by PES. Figure \ref{fig:1d_trace} shows the corresponding illustration for a sample $1$-D seismic trace of the $1000$ test realizations. The figure illustrates two main advantages of the proposed networks. Both DuRIN-$1$ and DuRIN-$2$ resolve closely-spaced reflectors right after $200$ ms on the time axis in Figure \ref{fig:1d_trace} (e) and (f) and do not introduce spurious supports as the benchmark techniques.
\begin{table}[t]
\centering
\caption{Objective metrics averaged over $1000$ test realizations of synthetic $1$-D seismic traces. The proposed DuRIN-$1$ and DuRIN-$2$ are superior in both amplitude and support recovery. The best performance is highlighted in \textbf{boldface}, while the second best is \underline{underlined}.
\label{table:1d_trace}
\resizebox{0.85\columnwidth}{!}{
\begin{tabular}{l||c|c|c|c}
\toprule
\bfseries Method & \bfseries CC & \bfseries RRE & \bfseries SRER & \bfseries PES\\
\midrule
BPI & ${0.6611}$ & ${0.5723}$ & ${3.4148}$ & ${0.9697}$\\
FISTA & ${0.6734}$ & ${0.5489}$ & ${3.5554}$ & ${0.8309}$\\
SBL-EM & ${0.6561}$ & ${0.6611}$ & $\mathbf{4.2666}$ & ${0.9697}$\\
DuRIN-$1$ & $\mathbf{0.7259}$ & $\mathbf{0.4638}$ & ${4.0212}$ & $\underline{0.5309}$\\
DuRIN-$2$ & $\underline{0.7237}$ & $\underline{0.4660}$ & $\underline{4.0230}$ & $\mathbf{0.5308}$\\
\bottomrule
\end{tabular}}
\end{table}
\begin{figure*}[t]
\centering
\includegraphics[width=0.55\linewidth]{figures/fig_trace_57.pdf}
\caption{\protect\includegraphics[height=1.5ex]{figures/rgb.png}~Sample results for recovered reflectivity from a synthetic $1$-D seismic trace out of $1000$ test realizations reported in Table \ref{table:1d_trace}. (a) The true seismic traces (TRUE$_{SEIS}$) and reflectivity (TRUE$_{REF}$); (b)-(d) Recovered $1$-D reflectivity profiles for the benchmark techniques show that they fail to distinguish between closely-spaced spikes right after $200$ ms on the time axis, and introduce spurious supports; (e)-(f) DuRIN-$1$ and DuRIN-$2$, on the other hand, distinguish between the closely-spaced spikes and exhibit superior support recovery, not introducing any spurious, low-amplitude supports.}\label{fig:1d_trace}
\end{figure*}
Table \ref{table:time} gives a comparison of the computational testing time over $100$, $200$, $500$, and $1000$ test realizations of synthetic $1$-D traces. DuRIN-$1$ and DuRIN-$2$ require lower computational time than the benchmark techniques, over two orders of magnitude lower than FISTA, the next best technique to our proposed networks. Lower computation times are significant for reflection seismic processing, where the size of datasets analyzed is large. We note that the lower computational testing times for DuRIN-$1$ and DuRIN-$2$ come at the expense of longer training times, where the models are trained on a large number of synthetic seismic traces.
\begin{table}[t]
\centering
\caption{Computational testing time (s) for $100$, $200$, $500$, and $1000$ test realizations of synthetic $1$-D seismic traces. DuRIN-$1$ and DuRIN-$2$ require significantly lower compute time during the inference phase as compared with the benchmark techniques. \\ R = number realizations.}
\label{table:time}
\resizebox{0.9\columnwidth}{!}{
\begin{tabular}{l||c|c|c|c}
\toprule
\multirow{2}{*}{\bfseries Method} & \multicolumn{4}{c}{\bfseries Testing time ($\mathrm{s}$)}\\
\cmidrule{2-5}
& {\bfseries R} $=\mathbf{100}$ & {\bfseries R} $=\mathbf{200}$ & {\bfseries R} $=\mathbf{500}$ & {\bfseries R} $=\mathbf{1000}$ \\
\midrule
BPI & ${34.9307}$ & ${70.5120}$ & ${171.6517}$ & ${375.6346}$\\
SBL-EM & ${202.5606}$ & ${400.1070}$ & ${947.3621}$ & ${1959.4553}$\\
FISTA & ${7.1995}$ & ${13.1076}$ & ${34.1289}$ & ${64.9389}$\\
DuRIN-$1$ & $\mathbf{0.0220}$ & $\underline{0.0467}$ & $\underline{0.0804}$ & $\underline{0.1434}$\\
DuRIN-$2$ & $\underline{0.0276}$ & $\mathbf{0.0365}$ & $\mathbf{0.0732}$ & $\mathbf{0.1095}$\\
\bottomrule
\end{tabular}}
\end{table}
Figure \ref{fig:iters_1d_trace} illustrates the computational advantage of the proposed DuRIN-$1$ and DuRIN-$2$ over the benchmark techniques in terms of layers/iterations. The $6$-layer DuRIN-$1$ and DuRIN-$2$ models outperform the benchmark techniques in terms of the four objective metrics considered in this study, with their performance further enhanced by adding $5$ layers, up to $26$ layers. We observe that the performance starts deteriorating after $31$ layers (Figure \ref{fig:iters_1d_trace}). We hypothesize that this deterioration could be attributed to the increase in the number of network parameters while keeping the size of the training dataset fixed, leading to overfitting. Future work could explore if increasing the training examples allows us to append more layers to enhance the networks' performance further rather than diminish it.
\begin{figure*}[t]
\centering
\includegraphics[width=0.55\linewidth]{figures/fig_iter_trace_slogx.pdf}
\caption{\protect\includegraphics[height=1.5ex]{figures/rgb.png}~Objective metrics vs. number of iterations/layers for the five methods considered in the study. The stems represent layers ($6$, $11$, $16$, $21$, $26$, and $31$) for DuRIN-$1$ and DuRIN-$2$, and the lines represent iterations for BPI, FISTA, and SBL-EM. For (a) CC and (c) SRER, higher values indicate superior performance, while for (b) RRE and (d) PES, lower values indicate better performance.}\label{fig:iters_1d_trace}
\end{figure*}
\subsubsection{Synthetic 2-D Wedge Models}\label{subsubsec:wedge}
We use synthetic wedge models to test the resolving capability of the proposed networks on thin beds \cite{hamlyn2014thin}. Wedge models usually consist of two interfaces, one horizontal and another inclined, with the wedge thinning to $0$ ms. Depending on the polarity of the reflection coefficients of the two interfaces being either the same or opposite, we have even or odd wedge models, respectively. Further, based on the polarity being negative or positive, we obtain four possible combinations of polarities of the reflection coefficients of the wedge models (NP, PN, NN, PP). In our experimental setup, we consider wedge models with $26$ seismic traces, with the separation between the two interfaces reducing from $50$ ms to $0$ ms in $2$ ms increments, and amplitudes of the reflection coefficients set to $\pm~0.5$.
Table \ref{table:2D_np} and Figure \ref{fig:wedge_np} show the performance of the proposed networks in comparison with the benchmark techniques for an odd wedge model (NP), with negative reflection coefficients on the upper horizontal interface (N) and positive on the lower inclined interface (P). DuRIN-$1$ and DuRIN-$2$ outperform the other methods in terms of both amplitude and support recovery metrics (Table \ref{table:2D_np}). As highlighted in Figure \ref{fig:wedge_np}, BPI, FISTA, and SBL-EM fail to resolve the reflectors below $5$ m wedge thickness, which is below the tuning thickness of $13$ ms, corresponding to $6.5$ m in our experimental setup, for a $30$ Hz wavelet frequency \cite{chung1995frequency}. This low resolving capability also contributes to the poorer (higher) PES scores for these techniques (Table \ref{table:2D_np}). On the other hand, the proposed DuRIN-$1$ and DuRIN-$2$ maintain the lateral continuity of both interfaces even below the tuning thickness.
\begin{table}[t]
\caption{Metrics for a synthetic $2$-D odd (NP) wedge model. DuRIN-$1$ and DuRIN-$2$ outperform the benchmark techniques in terms of the amplitude and support recovery metrics considered. The best performance is highlighted in \textbf{boldface}. The second best scores are \underline{underlined}.}
\label{table:2D_np}
\centering
\resizebox{0.85\columnwidth}{!}{
\begin{tabular}{l||c|c|c|c}
\toprule
\bfseries Method & \bfseries CC & \bfseries RRE & \bfseries SRER & \bfseries PES\\
\midrule
BPI & ${0.8110}$ & ${0.2693}$ & ${12.4474}$ & ${0.9933}$\\
FISTA & ${0.8108}$ & ${0.2583}$ & ${20.5075}$ & ${0.8846}$\\
SBL-EM & ${0.8848}$ & ${0.1446}$ & ${34.1678}$ & ${0.9933}$\\
DuRIN-$1$ & $\mathbf{0.9875}$ & $\mathbf{0.0622}$ & $\underline{35.7206}$ & $\underline{0.1795}$\\
DuRIN-$2$ & $\underline{0.9774}$ & $\underline{0.0734}$ & $\mathbf{37.6269}$ & $\mathbf{0.1026}$\\
\bottomrule
\end{tabular}}
\end{table}
\begin{figure*}[t]
\centering
\includegraphics[width=0.6\linewidth]{figures/fig_wedge_np.pdf}
\caption{\protect\includegraphics[height=1.5ex]{figures/rgb.png}~Results for a synthetic $2$-D odd wedge model. (a) True reflectivity; (b)-(d) Recovered reflectivity profiles from the benchmark techniques show a divergence from the ground-truth at wedge thickness $<5$ m, highlighted by the rectangle in black; (e)-(f) Reflectivity profiles recovered by DuRIN-$1$ and DuRIN-$2$ show a better resolution of reflectors below a wedge thickness of $5$ m.}\label{fig:wedge_np}
\end{figure*}
Table \ref{table:2D_pn} and Figure \ref{fig:wedge_pn} show the results for an odd wedge model (PN) with positive polarity of reflection coefficients on the upper, horizontal interface (P) and negative polarity on the lower, inclined interface (N). Table \ref{table:2D_pn} shows the proposed DuRIN-$1$ and DuRIN-$2$ outperforming the benchmark techniques, namely, BPI, FISTA, and SBL-EM, in terms of the amplitude and support recovery metrics considered in this study. Similar to the NP wedge model, and as highlighted in Figure \ref{fig:wedge_pn}, the benchmark techniques do not accurately resolve the reflectors below a wedge thickness of $5$ m, whereas DuRIN-$1$ and DuRIN-$2$ preserve lateral continuity of both the interfaces of the wedge model below tuning thickness.
\begin{table}[t]
\caption{Results for a synthetic $2$-D odd (PN) wedge model. The proposed networks, namely, DuRIN-$1$ and DuRIN-$2$ are superior in both amplitude and support recovery accuracy.
\label{table:2D_pn}
\centering
\resizebox{0.85\columnwidth}{!}{
\begin{tabular}{l||c|c|c|c}
\toprule
\bfseries Method & \bfseries CC & \bfseries RRE & \bfseries SRER & \bfseries PES\\
\midrule
BPI & ${0.8369}$ & ${0.2265}$ & ${12.5753}$ & ${0.9933}$\\
FISTA & ${0.8387}$ & ${0.2123}$ & ${21.5266}$ & ${0.8832}$\\
SBL-EM & ${0.8469}$ & ${0.2106}$ & ${32.4996}$ & ${0.9933}$\\
DuRIN-$1$ & $\mathbf{0.9774}$ & $\mathbf{0.0795}$ & $\underline{34.4034}$ & $\underline{0.1923}$\\
DuRIN-$2$ & $\underline{0.9577}$ & $\underline{0.1074}$ & $\mathbf{36.0748}$ & $\mathbf{0.1256}$\\
\bottomrule
\end{tabular}}
\end{table}
\begin{figure*}[ht]
\centering
\includegraphics[width=0.6\linewidth]{figures/fig_wedge_pn.pdf}
\caption{\protect\includegraphics[height=1.5ex]{figures/rgb.png}~Results for recovered reflectivity from a synthetic $2$-D odd wedge model (PN). (a) True reflectivity; (b)-(d) Recovered reflectivity signatures from the benchmark methods show a distinct divergence from the ground-truth reflectivity (a) at wedge thickness $<~5$ m, highlighted by the rectangle in black; (e)-(f) Recovered reflectivity profiles from the proposed networks, namely, DuRIN-$1$ and DuRIN-$2$ resolve the reflectors below a wedge thickness of $5$ m.}\label{fig:wedge_pn}
\end{figure*}
For the even wedge models, i.e., NN (Table \ref{table:2D_nn} and Figure \ref{fig:wedge_nn}) and PP (Table \ref{table:2D_pp} and Figure \ref{fig:wedge_pp}), all techniques exhibit comparable performance in terms of the amplitude metrics of CC and RRE, and resolve reflectors below tuning thickness without any divergence from the ground-truth reflector locations. The proposed networks, namely, DuRIN-$1$ and DuRIN-$2$ outperform the benchmark techniques in SRER and PES scores.
\begin{table}[ht]
\centering
\caption{Results for an even wedge model (NN).
DuRIN-$1$ and DuRIN-$2$ are superior in terms of SRER and PES.}
\label{table:2D_nn}
\resizebox{0.85\linewidth}{!}{
\begin{tabular}{l||c|c|c|c}
\toprule
\bfseries Method & \bfseries CC & \bfseries RRE & \bfseries SRER & \bfseries PES\\
\midrule
BPI & ${0.9215}$ & ${0.1733}$ & ${15.1795}$ & ${0.9933}$\\
FISTA & ${0.9572}$ & $\mathbf{0.0916}$ & ${22.1490}$ & ${0.8643}$\\
SBL-EM & $\mathbf{0.9616}$ & $\underline{0.1120}$ & ${38.1103}$ & ${0.9933}$\\
DuRIN-$1$ & ${0.9235}$ & ${0.1568}$ & $\mathbf{38.8751}$ & $\underline{0.1603}$\\
DuRIN-$2$ & $\underline{0.9599}$ & ${0.1163}$ & $\underline{38.2828}$ & $\mathbf{0.1346}$\\
\bottomrule
\end{tabular}}
\end{table}
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{figures/fig_wedge_nn.pdf}
\caption{\protect\includegraphics[height=1.5ex]{figures/rgb.png}~Results for recovered reflectivity from a synthetic $2$-D even wedge model (NN). (a) True reflectivity; (b)-(f) Recovered reflectivity profiles. The proposed networks and the benchmark techniques resolve the even wedge model well for wedge thickness $>~3$ m. DuRIN-$1$ (e), along with SBL-EM (d), also resolves the two reflectors of trace $3$ (corresponding to wedge thickness $2$ m).}\label{fig:wedge_nn}
\end{figure}
\begin{table}[ht]
\caption{Results for an even wedge model (PP).
While FISTA has a better CC and RRE score, DuRIN-$1$ and DuRIN-$2$ outperform the benchmark techniques in terms of SRER and PES.}
\label{table:2D_pp}
\centering
\resizebox{0.85\linewidth}{!}{
\begin{tabular}{l||c|c|c|c}
\toprule
\bfseries Method & \bfseries CC & \bfseries RRE & \bfseries SRER & \bfseries PES\\
\midrule
BPI & ${0.9303}$ & ${0.1524}$ & ${15.5630}$ & ${0.9933}$\\
FISTA & $\mathbf{0.9548}$ & $\mathbf{0.0905}$ & ${23.1071}$ & ${0.8572}$\\
SBL-EM & ${0.9232}$ & ${0.1882}$ & ${36.2027}$ & ${0.9933}$\\
DuRIN-$1$ & ${0.9238}$ & ${0.1635}$ & $\mathbf{36.9497}$ & $\underline{0.1923}$\\
DuRIN-$2$ & $\underline{0.9295}$ & $\underline{0.1447}$ & $\underline{36.8566}$ & $\mathbf{0.1410}$\\
\bottomrule
\end{tabular}}
\end{table}
\begin{figure}[ht]
\centering
\includegraphics[width=\linewidth]{figures/fig_wedge_pp.pdf}
\caption{\protect\includegraphics[height=1.5ex]{figures/rgb.png}~Results for an even wedge model (PP). (a) True; and (b)-(f) Recovered reflectivity. BPI, FISTA, and DuRIN-$1$ resolve reflectors for wedge thickness $>~3$ m, while SBL-EM and DuRIN-$2$ also resolve reflectors at $2$ m.}\label{fig:wedge_pp}
\end{figure}
\subsection{Testing Phase: Marmousi2 Model}\label{subsec:marmousi2}
The Marmousi$2$ model \cite{martin2006marmousi2} (width $\times$ depth: $17$ km $\times~3.5$ km) is an expansion of the original Marmousi model \cite{versteeg1994marmousi} (width $\times$ depth: $9.2$ km $\times~3$ km depth). The model provides a simulated $2$-D collection of seismic traces, and is widely used in reflection seismic processing for the calibration of techniques in structurally complex settings. The model has a $2$ ms sampling interval, with traces at an interval of $12.5$ m. We obtained the ground-truth reflectivity profiles $({\boldsymbol{x}})$ from the P-wave velocity and density models \eqref{rhov}, and convolved them with a $30$ Hz Ricker wavelet to generate the input data $({\boldsymbol{y}})$, with measurement SNR of $20$ dB to test the robustness of the proposed networks to noisy conditions.
\begin{table}
\caption{Results for a portion of the Marmousi$2$ model, corresponding to Table \ref{table:mm_mute}, without muting low-amplitude spikes. DuRIN-$1$ and DuRIN-$2$ have higher CC and RRE. The lower PES scores for BPI and SBL-EM are due to the spurious supports introduced by these techniques coinciding with the low-amplitude spikes. The best performance is highlighted in \textbf{boldface}; the second best is \underline{underlined}.}
\label{table:mm_unmute}
\centering
\resizebox{0.85\linewidth}{!}{
\begin{tabular}{l||c|c|c|c}
\toprule
\bfseries Method & \bfseries CC & \bfseries RRE & \bfseries SRER & \bfseries PES\\
\midrule
BPI & ${0.9714}$ & ${0.0474}$ & ${22.2817}$ & $\mathbf{0.1751}$\\
FISTA & ${0.9716}$ & ${0.0466}$ & $\underline{23.1565}$ & $\underline{0.8879}$\\
SBL-EM & ${0.9808}$ & ${0.0305}$ & $\mathbf{26.3986}$ & $\mathbf{0.1751}$\\
DuRIN-$1$ & $\mathbf{0.9857}$ & $\mathbf{0.0243}$ & ${22.5103}$ & ${0.9716}$\\
DuRIN-$2$ & $\underline{0.9820}$ & $\underline{0.0302}$ & ${22.1734}$ & ${0.9702}$\\
\bottomrule
\end{tabular}}
\end{table}
\begin{table}[t]
\caption{Objective metrics for a portion of the Marmousi$2$ model \cite{martin2006marmousi2}. DuRIN-$1$ and DuRIN-$2$ exhibit superior amplitude recovery than the benchmark techniques in terms of CC and RRE, and outperform in support recovery in terms of PES.}
\label{table:mm_mute}
\centering
\resizebox{0.85\columnwidth}{!}{
\begin{tabular}{l||c|c|c|c}
\toprule
\bfseries Method & \bfseries CC & \bfseries RRE & \bfseries SRER & \bfseries PES\\
\midrule
BPI & ${0.9717}$ & ${0.0465}$ & ${23.4547}$ & ${0.9778}$\\
FISTA & ${0.9720}$ & ${0.0456}$ & $\underline{25.1384}$ & ${0.7762}$\\
SBL-EM & ${0.9811}$ & ${0.0300}$ & $\mathbf{30.5194}$ & ${0.9778}$\\
DuRIN-$1$ & $\mathbf{0.9860}$ & $\mathbf{0.0238}$ & ${24.5692}$ & $\mathbf{0.2175}$\\
DuRIN-$2$ & $\underline{0.9822}$ & $\underline{0.0298}$ & ${23.8769}$ & $\underline{0.2438}$\\
\bottomrule
\end{tabular}}
\end{table}%
Here, we present results from a region of the Marmousi$2$ model that contains a gas-charged sand channel \cite{martin2006marmousi2}. Table \ref{table:mm_unmute} shows a comparison of the performance of the proposed DuRIN-$1$ and DuRIN-$2$ with that of the benchmark techniques for the portion containing the gas-charged sand channel. The Marmousi$2$ model \cite{martin2006marmousi2} has a large number of low-amplitude reflections. Additionally, in Figure \ref{fig:1d_trace}, we observe that BPI and SBL-EM introduce spurious reflections. We attribute the low PES scores observed for BPI and SBL-EM in Table \ref{table:mm_unmute} to such spurious supports complementing the low-amplitude reflections in the Marmousi$2$ model. We re-evaluate the objective metrics after muting reflections with amplitudes $<~1\%$ of the absolute of the maximum amplitude, the results for which are reported in Table \ref{table:mm_mute} and Figure \ref{fig:marmousi}. The re-evaluation did not affect the CC, RRE, and SRER scores adversely, but, as the PES is computed over significant reflections, it is higher (worse) for BPI and SBL-EM, and lower (better) for DuRIN-$1$ and DuRIN-$2$. The objective metrics reported in Table \ref{table:mm_mute} show superior amplitude and support recovery accuracy of the proposed networks over the benchmark methods. Figure \ref{fig:marmousi} shows that DuRIN-$1$ and DuRIN-$2$, along with SBL-EM, preserve the lateral continuity of interfaces, highlighted in the insets at the edge of the sand channel. The insets also show that BPI and FISTA introduce spurious false interfaces due to the interference of multiple events. DuRIN-$1$ and DuRIN-$2$ show limited recovery of the low-amplitude reflection coefficients right below the gas-charged sand channel (Figure \ref{fig:marmousi} (f) and (g)), which could be investigated in future work.
\begin{figure*}[!ht]
\centering
\includegraphics[width=0.65\linewidth]{figures/fig_mm_final.pdf}
\caption{\protect\includegraphics[height=1.5ex]{figures/rgb.png}~Results for a portion of the Marmousi$2$ synthetic model, corresponding to Table \ref{table:mm_mute} and marked by a red rectangle in (a) The P-wave velocity profile for the complete model; (b) The true reflectivity; (c)-(g) recovered $2$-D reflectivity profiles from the methods compared in this study, with insets plots showing the zoomed-in portions of the selected region marked by black rectangles. Insets show that (c) BPI and (d) FISTA introduce spurious reflection coefficients around the true interfaces, while (e) SBL-EM, (f) DuRIN-$1$, and (g) DuRIN-$2$ preserve the lateral continuity better.}\label{fig:marmousi}
\end{figure*}
\subsection{Testing Phase: Real Data}\label{subsec:real_data}
To validate the proposed networks on real data, we use a $3$-D seismic volume from the Penobscot $3$D survey off the coast of Nova Scotia, Canada \cite{penobscot3d}. We present results from a smaller region of the $3$-D\\ volume, with $201$ Inlines ($1150$-$1350$) and $121$ Xlines ($1040$-$1160$), a region that includes the two wells of this survey (wells L-$30$ and B-$41$, \cite{bianco2014geophysical}). Along the time axis, the region contains $800$ samples between the time interval of $0$ to $3196$ ms, with a $4$ ms sampling interval, and a $25$ Hz Ricker wavelet fits the data well \cite{bianco2014geophysical}.
Figure \ref{fig:real} shows the observed seismic data and recovered reflectivity profiles for Xline $1155$ of the survey, and overlaid in black, the seismic and reflectivity profiles, respectively, computed from the sonic logs of well L-$30$ \cite{bianco2014geophysical}. The inverted reflectivity profiles for BPI and FISTA are smooth and lack detail. On the other hand, the predicted reflectivity profiles generated using SBL-EM, DuRIN-$1$ and DuRIN-$2$ provide more details for characterizing the subsurface. Additionally, DuRIN-$1$ and DuRIN-$2$ also resolve closely-spaced interfaces better and preserve the lateral continuity, evident from the interfaces around $1.1$ s on the time axis (Figure \ref{fig:real} (e) and (f)). These results demonstrate the capability of the proposed networks, namely, DuRIN-$1$ and DuRIN-$2$, on real data from the field. We note that the models are trained on synthesized seismic traces before being tested on the real data.
\begin{figure*}[t]
\centering
\includegraphics[width=0.65\linewidth]{figures/fig_real.pdf}
\caption{\protect\includegraphics[height=1.5ex]{figures/rgb.png}~(a) Observed seismic data; (b)-(f) predicted $2$-D reflectivity profiles for the region marked by a black rectangle in (a), for Xline $1155$ of the Penobscot $3$D survey \cite{penobscot3d}. The overlaid waveforms in black are the seismic (in (a)) and reflectivity (in (b)-(f)), respectively, computed from the sonic logs of well L-$30$ \cite{bianco2014geophysical}. BPI (b) and FISTA (c) provide a smooth profile without many details, while SBL-EM (d), DuRIN-$1$ (e) and DuRIN-$2$ (f) generate a detailed reflectivity profile.}\label{fig:real}
\end{figure*}%
\section{Conclusions}\label{sec:conclusions}
We developed a nonconvex optimization cost for reflectivity inversion based on a weighted counterpart of the minimax concave penalty. We also developed the deep-unfolded reflectivity inversion network (DuRIN) for solving the problem under consideration. Experimental validation on synthetic, simulated, and real data demonstrated the superior resolving capability of DuRIN, especially in support recovery (represented by PES), which is crucial for characterizing the subsurface. DuRIN also preserves lateral continuity of interfaces despite being a single-trace approach. However, the trade-off between the number of layers of the network and training examples, and the suboptimal recovery of low-amplitude reflection coefficients require further attention. One could also consider data-driven prior learning \cite{bergen2019machine} based on a multi-penalty formulation \cite{jawali2020cornet} for solving the reflectivity inversion problem.
\section*{Acknowledgments}
This work is supported by Ministry of Earth Sciences, Government of India; Centre of Excellence in Advanced Mechanics of Materials, Indian Institute of Science (IISc), Bangalore; and Science and Engineering Research Board (SERB), India. The authors would like to thank Jishnu Sadasivan for fruitful technical discussions.
| -43,695.918509 |
[
-2.46875,
2.318359375
] | 26.888889 |
[
-4.015625,
0.110595703125,
-1.9765625,
-5.52734375,
-0.84521484375,
8.1875
] |
[
3.095703125,
7.8203125,
1.8896484375,
6.38671875
] | 467 | 5,615 |
[
-3.275390625,
3.7109375
] | 31.810911 |
[
-6.375,
-4.3515625,
-4.546875,
-2.375,
2.2421875,
12.6328125
] | 0.838684 | 14.088436 | 26.411398 | 7.423939 |
[
3.163388729095459
] | -28,290.926742 | 6.795191 | -42,609.738227 | 1.375966 | 6.148485 |
[
-2.751953125,
-3.85546875,
-4.23828125,
-5.13671875,
2.51953125,
12.734375
] |
[
-5.89453125,
-2.37109375,
-2.537109375,
-2.20703125,
3.810546875,
5.85546875
] | |
BkiUfovxK7kjXIK5t9Ch
|
\section{Introduction}
\label{intro}
Consider a system of $q$ partial differential equations (PDEs),
\begin{equation}\label{PDE}
\mathcal{A}(x,t,[\mathbf{u}])=\mathbf{0},
\end{equation}
where $\mathcal{A}$ is a row vector, $\mathbf{u}$ has components $u^\alpha,\ \alpha=1,\dots,q$, and square brackets around a differentiable expression denote the expression and finitely many of its derivatives\footnote{To simplify the presentation, we consider here only one spatial variable, $x$. The extension to PDEs with more spatial variables is outlined in Section~\ref{genericCL} and illustrated in Section~\ref{pKPsec}.}.
We assume that (\ref{PDE}) is totally nondegenerate (see \cite{olverbook}). A local conservation law is a divergence expression,
\begin{equation}\label{CLaw}
\mbox{Div}\, \mathbf{F}=D_x\{F(x,t,[\mathbf{u}])\}+D_t\{G(x,t,[\mathbf{u}])\},
\end{equation}
that vanishes on all solutions of \eqref{PDE}. The functions $F$ and $G$ are the flux and the density of the conservation law, respectively, and $D_x$ and $D_t$ denote the total derivatives with respect to $x$ and $t$, respectively. The conservation law \eqref{CLaw} is in characteristic form if there exists a column vector $\mathcal{Q}$ such that
\begin{equation}\label{CLchar}
\mbox{Div}\,\mathbf{F}=\mathcal{A}\mathcal{Q},
\end{equation}
in which case $\mathcal{Q}$ is called the characteristic. The space of total divergences forms the kernel of the Euler operator, $\mathcal{E}$, whose $\alpha$-th entry is
\begin{equation}\label{Eul}
\mathcal{E}_\alpha=\sum_{i,j}(-D_x)^i(-D_t)^j\frac{\partial}{\partial(D_x^iD_t^ju^\alpha)}.
\end{equation}
Hence
\begin{equation}\label{eulcond}
\mathcal{E}(\mathcal{A}\mathcal{Q})=\mathbf{0}
\end{equation}
if and only if there exists $\mathbf{F}$ such that \eqref{CLchar} holds. These results generalize immediately to PDE systems with more than two independent variables.
The literature on the numerical solution of PDEs is rich in numerical methods that preserve global invariants, but there are relatively few results on the preservation of local conservation laws. Arguably, local conservation laws are more necessary: they hold throughout the domain, apply to the set of all solutions, and provide much stronger constraints than are needed to preserve the corresponding global invariants. Moreover, when the domain and boundary conditions are suitable, conservation of such invariants is automatically achieved.
A new approach for developing finite difference schemes that preserve conservation laws of (\ref{PDE}) was introduced recently in \cite{FCHydon}. It exploits the fact that discrete conservation laws form the kernel of a discrete version of the Euler operator (\ref{Eul}). Discretizations of the PDE (\ref{PDE}) having discrete versions of the desired conservation laws are obtained by requiring that a discrete version of condition (\ref{eulcond}) is satisfied.
This requires the symbolic solution of a large system of nonlinear equations that is impractical in general.
The complexity of the symbolic calculations can be reduced by introducing compactness requirements on the schemes, and this direct approach has been applied to a range of PDEs with different structure in \cite{FCHydon,FCHmkdv,FCHnls}. However, the direct approach is greatly limited by the capacity of symbolic computation; it has been applied only to second-order approximations of PDEs with two independent variables.
In this paper we modify the approach in \cite{FCHydon} by finding semidiscretizations of (\ref{PDE}) that preserve semidiscrete local conservation laws. The reduction to one discrete space dimension significantly reduces the complexity of the computations, to the point that the determining system can be solved easily without introducing any restrictions on the form of the schemes. After this, a suitable integrator in time needs to be chosen to create a fully-discrete scheme; this depends on the form of the conservation laws that one aims to preserve.
If the PDE is equipped with conservative boundary conditions, it is known that the quadratic invariants of its space discretizations are preserved by symplectic Runge-Kutta methods \cite{Coop,SSC,CIZ}. In this paper we extend this result to prove that if $G$ is quadratic in $[u]$ then any symplectic Runge--Kutta method preserves the conservation law (\ref{CLaw}) locally, regardless of the boundary conditions.
There are various results on local conservation for Hamiltonian PDEs,
\begin{equation}\label{HamPDE}
D_t \mathbf{u}=\mathcal{D}\left([\mathbf{u}]_x\right)\mathcal{E}(H\left([\mathbf{u}]_x\right)),
\end{equation}
where $[\mathbf{u}]_x$ denotes $\mathbf{u}$ and its spatial derivatives only, $\mathcal{D}$ is a skew-adjoint operator that satisfies the Jacobi identity, and $H$ is the Hamiltonian function.
Multisymplectic schemes \cite{BR} and their generalizations \cite{Sun} can preserve local conservation laws with quadratic flux and density. Requiring the flux to be quadratic is however a strong constraint that is not satisfied by local momentum conservation laws of many important equations in physics such as the nonlinear Schr\"odinger (NLS) equation, the Korteweg-de Vries (KdV) equation, the Benjamin-Bona-Mahony (BBM) equation, the modified Korteweg-de Vries (mKdV) equation, and the Boussinesq equation. The strategy introduced in this paper does not suffer from this restriction, as no assumption is needed about the flux, so it can be applied to the preservation of these conservation laws as well.
Another popular approach is to use a discrete gradient method for the time integration of (\ref{HamPDE}).
These are obtained from a semidiscretization of $H$ and a skew-adjoint discretization of $\mathcal{D}$, and preserve a discrete conservation law of the energy \cite{QMcL}. One widely-used discrete gradient method is the Average Vector Field (AVF) method \cite{Cell,Cell2,McLaren}. We show that the AVF method yields the local conservation law of the Hamiltonian under constraints on the discretization of $\mathcal{D}$ that are milder than skew-adjointness. Consequently, conservation of the local Hamiltonian can be achieved for a larger class of discretizations.
Although the discussion so far has focused on PDEs with two independent variables, the approach of discretizing one dimension at a time works equally well for PDEs on higher-dimensional spaces. We discuss this and give an illustration.
The paper is organised as follows. Section~\ref{spacedis} introduces a method for obtaining conservative spatial semidiscretizations. Section~\ref{timeint} focuses on time integration. In particular, we show the following.
\begin{itemize}
\item[$\bullet$] Conservation laws with quadratic density (without any assumption on the flux) are preserved by any symplectic method in time locally and independently of the boundary conditions. Conservation laws for mass, charge and momentum are typically in this class.
\item[$\bullet$] For Hamiltonian PDEs, the AVF method preserves the local semidiscrete conservation law of the energy for a wide class of semidiscretizations that generalizes the result in \cite{QMcL}.
\item[$\bullet$] For other types of conservation law, fully-discrete methods can be found by introducing relatively few parameters and fixing them by requiring that the conservation law is in the kernel of a fully-discrete Euler operator. This approach can be iterated for dimensions, by using a sequence of semidiscrete and discrete Euler operators.
\end{itemize}
In Section~\ref{GBsec} we apply this new approach to the Boussinesq equation and introduce methods of order two and four that preserve three conservation laws. Section~\ref{Numsec} describes numerical benchmark tests, including evidence of stability and comparison with other methods from the literature. In Section~\ref{pKPsec}, we apply the new technique to a two-dimensional PDE, the potential Kadomtsev-Petviashvili (pKP) equation and introduce two families of schemes that preserve two conservation laws. Finally, we draw some conclusions in Section~\ref{concl}.
\section{Conservative space discretizations}\label{spacedis}
We begin with a regular spatial grid. The stencil consists of $M=B-A+1$ nodes, \begin{equation}\label{nodesx}
x_{m}=x_0+m\Delta x, \qquad m=A,\ldots,B,
\end{equation}
where $x_0$ is a generic grid point; let $\mathbf{x}$ denote the vector of the nodes.
The forward shift operator $S_m$ acts as follows on any semidiscrete function $f$:
\begin{equation*}
S_m:f(x_m,t)\mapsto f(x_{m+1},t);
\end{equation*}
the forward difference, forward average, and centered difference operators are
\begin{equation}\label{DmMm}
D_m=\tfrac{1}{\Delta x}(S_m-I),\qquad \mu_m=\tfrac{1}2(S_m+I),\qquad D^{(c)}_m=\tfrac{1}{2\Delta x}(S_m-S_m^{-1})
\end{equation}
respectively, where $I$ is the identity operator.
The semidiscretizations of $u^\alpha\!(x,\!t)$ are given by the column vector $\mathbf{U}\in\mathbb{R}^{Mq}$ with the {$(m+\alpha M-B)$-th} entry
$$U_{m}^\alpha(t)\approx u^\alpha(x_m,t),\qquad m=A,\ldots,B,\quad \alpha=1,\ldots,q.$$
The semidiscrete problem is
\begin{equation}\label{SDPDE}
\widetilde{\mathcal{A}}(\mathbf{x},t,[\mathbf{U}])=\mathbf{0},
\end{equation}
where here and henceforth tildes represent approximations to the corresponding continuous quantities, and square brackets around a semidiscrete expression denote the expression and a finite number of its time derivatives.
A semidiscrete conservation law of (\ref{SDPDE}) is a semidiscrete divergence,
\begin{equation}\label{SDCL}
\text{Div}\,\widetilde{\mathbf{F}}=D_m\{\widetilde{F}(\mathbf{x},t,[\mathbf{U}])\}+D_t\{\widetilde{G}(\mathbf{x},t,[\mathbf{U}])\},
\end{equation}
such that $$\text{Div}\,\widetilde{\mathbf{F}}=0,\,\,\,\text{when}\,\,\,[\widetilde{\mathcal{A}}=\mathbf{0}].$$
The functions $\widetilde{F}$ and $\widetilde{G}$ are the semidiscrete flux and density of the conservation law (\ref{SDCL}), respectively. Similarly, as in the continuous case, we say that (\ref{SDCL}) is in characteristic form if there exists $\widetilde{\mathcal{Q}}=\widetilde{\mathcal{Q}}(\mathbf{x},t,[\mathbf{U}])$, called the characteristic, such that $$\text{Div}\,\widetilde{\mathbf{F}}=\widetilde{\mathcal{A}}\widetilde{\mathcal{Q}}.$$
The following result is crucial for obtaining semidiscretizations that preserve conservation laws (see \cite{olverbook} and \cite{kuper} for analogous results in the continuous and totally discrete setting, respectively).
\begin{theo}\label{th1}
The kernel of the semidiscrete Euler operator $\mathsf{E}_\mathbf{U}$, whose $\alpha$-th entry is
\begin{equation*}
(\mathsf{E}_\mathbf{U})_\alpha=\sum_{i,j}S_m^{-i}(-D_t)^j\frac{\partial}{\partial (D_t^jU_{i}^\alpha)},
\end{equation*}
is the space of semidiscrete divergences (\ref{SDCL}).
\end{theo}
\begin{proof}
Let $L=L(\mathbf{x},t,[\mathbf{U}])$ such that $\mathsf{E}_\mathbf{U}(L)=\mathbf{0}$, and consider the derivative
\begin{equation}\label{dLdeps}
\frac{\mathrm{d}}{\mathrm{d} \varepsilon}L(\mathbf{x},t,\varepsilon [\mathbf{U}])=\sum_{\alpha,i,j}(D_t^jU_i^\alpha)\frac{\partial L(\mathbf{x},t,\varepsilon[\mathbf{U}])}{\partial(D_t^jU_i^\alpha)}.
\end{equation}
Integrating by parts yields
\begin{align*}
(D_t^jU_i^\alpha)\frac{\partial L}{\partial(D_t^jU_i^\alpha)}&\,=U_i^\alpha(-D_t)^j\frac{\partial L}{\partial(D_t^jU_i^\alpha)}+D_t \hat G\\
&\,=U_0^\alpha{S_m^{-i}(-D_t)^j\frac{\partial L}{\partial(D_t^jU_i^\alpha)}}+D_m \hat F+D_t \hat G=D_m \hat F+D_t \hat G,
\end{align*}
for some functions $\hat F=\hat F(\mathbf{x},t,\varepsilon,[\mathbf{U}])$ and $\hat G=\hat G(\mathbf{x},t,\varepsilon,[\mathbf{U}])$ whose precise expression is not of importance. Substituting this into (\ref{dLdeps}) and integrating over $\varepsilon\in [0,1]$ shows that $L$ is a semidiscrete divergence.
If $L$ is of the form (\ref{SDCL}), $\mathsf{E}_\mathbf{U}(L)=\mathbf{0}$ follows from the linearity of the Euler operator and from the fact that for any $k$ (see, e.g., \cite{hydonbook}),
\begin{align*}
\left(\sum_{i}S_m^{-i}\frac{\partial}{\partial (D_t^k\mathbf{U}_{i}^\alpha)}\right)(D_m\widetilde{F})=0,\,\,\, \left(\sum_{j}(-D_t)^j\frac{\partial}{\partial (D_t^j\mathbf{U}_{k}^\alpha)}\right)(D_t\widetilde{G})=0. \tag*{$\square$}
\end{align*}
\end{proof}
Based on the result in Theorem~\ref{th1}, the approach used in \cite{FCHydon, FCHmkdv,FCHnls} to preserve fully-discrete conservation laws, is adapted here to the preservation of semidiscrete conservation laws of (\ref{PDE}) with characteristics $\mathcal{Q}_\ell$, as follows:
\begin{enumerate}
\item Select a stencil that is large enough to support generic semidiscretizations for $\mathcal{A}$ and every $\mathcal{Q}_\ell$, having the desired order of accuracy. These approximations depend on a number of free parameters to be determined.
\item Find some of the parameters by imposing consistency, up to the desired order of accuracy, $p$.
\item Use symbolic algebra to determine the values of the free parameters that satisfy
\begin{equation}\label{sdeulcond}
\mathsf{E}_\mathbf{U}(\widetilde{\mathcal{A}}\widetilde{\mathcal{Q}}_\ell)=\mathbf{0},
\end{equation}
for $\ell=1$. This guarantees that the first conservation law is locally preserved. As both $\widetilde{\mathcal{A}}$ and $\widetilde{\mathcal{Q}}_\ell$ are accurate to order $p$, the discrete conservation law has the same order of accuracy.
\item Iterate the previous step, replacing $\widetilde{\mathcal{Q}}_1$ with $\widetilde{\mathcal{Q}}_\ell$, to obtain further constraints on the parameters. If (\ref{sdeulcond}) has no solution for some $\ell$, the corresponding conservation law cannot be preserved without violating one of the previous conservation laws. Typically, the more complicated a conservation law is, the more parameters need to be fixed to preserve it.
\end{enumerate}
\begin{rem}
It might seem appealing to identify a set of conservation laws that one wishes to preserve and use brute force symbolic computation to solve all constraints simultaneously. (This was our approach initially.) However, this takes far longer than the sequential approach and commonly comes up with a null result, with no indication as to which subsets of conservation laws can be preserved. The sequential algorithm above enables the user to decide which conservation laws should be prioritized. At each iteration, the computation is simplified by the fact that some parameters have already been fixed.
\end{rem}
\begin{rem}
If the algorithm above does not produce any schemes for a given stencil, one could try preserving the same conservation laws using a wider stencil. However, the wider the stencil is, the more the computational cost increases. Moreover, if one finds a conservative semidiscretization, a time integrator that preserves all the conservation laws is also needed. For example, in the next section we prove that some known time integrators preserve conservation laws whose density is either quadratic, or is a Hamiltonian, but to the best of our knowledge there are no methods that preserve both of these types.
\end{rem}
\section{Time integration}\label{timeint} We begin by considering one-step time integrators. For fully-discrete schemes the stencil is $$(x_{m},t_n),\quad m=A,\ldots,B,\quad n=0,1, \quad t_1=t_0+\Delta t,$$
and the forward shift operators in space and time are
$$S_m:f(x_m,t_n)\mapsto f(x_{m+1},t_n),\qquad S_n:f(x_m,t_0)\mapsto f(x_m,t_{1}),$$
respectively. The forward difference and forward average operators in space are defined by (\ref{DmMm}) and the corresponding operators in time are
$$D_n=\tfrac{1}{\Delta t}(S_n-I),\qquad \mu_n=\tfrac{1}2(S_n+I).$$
Let $\mathbf{u}_{n}\in\mathbb{R}^{Mq}$ be the column vector whose $(m+\alpha M-B)$-th entry is
$$u_{m,n}^\alpha\approx u^\alpha(x_m,t_n), \qquad m=A,\ldots,B,\quad \alpha=1,\ldots,q,$$
and let $\mathbf{u}_{m,n}\in\mathbb{R}^{q}$ be the column vector with entries
$$u_{m,n}^\alpha\approx u^\alpha(x_m,t_n), \qquad \alpha=1,\ldots,q.$$
\subsection{Conservation laws with quadratic density}\label{Clquadden}
Here attention is restricted to PDEs of the form
\begin{equation}\label{PDEtimesec}
D_t\left\{\mathbf{g}(x,[\mathbf{u}]_x)\right\}=\mathbf{h}(x,t,[\mathbf{u}]_x),
\end{equation}
where $\mathbf{g}$ is linear homogeneous in $[\mathbf{u}]_x$; these include Hamiltonian PDEs.
Consider a conservation law of (\ref{PDEtimesec}) of the form
\begin{equation}\label{momentum}
D_x\{F_2(x,t,[u]_x,[u_t]_x)\}+D_t\{G_2(x,[u]_x)\}=0,
\end{equation}
where the density, $G_2$, is a polynomial of degree two in $[u]_x$. (Without loss of generality, assume that no terms in $G_2$ depend on $x$ only.)
For many differential problems of importance in physics (such as KdV, NLS and BBM equations), the conservation laws of mass (or charge) and momentum are of the form (\ref{momentum}) with linear and quadratic density, respectively.
Let $P(\mathbf{x})$ be an invertible operator such that
\begin{equation*}
\widetilde{\mathbf{g}}(\mathbf{x},\mathbf{U})=P^{-1}(\mathbf{x})\mathbf{U}
\end{equation*}
and $\widetilde{\mathbf{h}}(\mathbf{x},t,\mathbf{U})$ are two column vectors whose $(m+\alpha M-B)$-th entry is a spatial discretization of the $\alpha$-th component of $\mathbf{g}(x,[\mathbf{u}]_x)$ and $\mathbf{h}(x,t,[\mathbf{u}]_x)$ at $x_m$, respectively. Let
\begin{equation}\label{SDcon}
D_t\{\widetilde{\mathbf{g}}(\mathbf{x},\mathbf{U})\}=\widetilde{\mathbf{h}}(\mathbf{x},t,P(\mathbf{x})\widetilde{\mathbf{g}}(\mathbf{x},\mathbf{U}))
\end{equation}
be a semidiscretization of the PDE (\ref{PDEtimesec}) with the following approximation to the conservation law (\ref{momentum}):
\begin{equation}\label{SDmom}
D_m\{\widetilde{F}_2(\mathbf{x},t,\mathbf{U},\mathbf{U}_t)\}+D_t\{\widetilde{G}_2(\mathbf{x},\mathbf{U})\}=0.
\end{equation}
Such a semidiscretization can be obtained using the technique in Section~\ref{spacedis}.
The flux and density of (\ref{SDmom}) have the general form
\begin{align*}
\widetilde{F}_2(\mathbf{x},t,\mathbf{U},\mathbf{U}_t)&=\widetilde{F}_2\left(\mathbf{x},t,\mathbf{U},P(\mathbf{x})\widetilde{\mathbf{h}}(\mathbf{x},t,\mathbf{U})\right),\\
\widetilde{G}_2(\mathbf{x},\mathbf{U})&=\tfrac{1}{2}\mathbf{U}^T S(\mathbf{x})\mathbf{U}+\mathbf{w}(\mathbf{x})^T\mathbf{U},
\end{align*}
where $S(\mathbf{x})=S(\mathbf{x})^T\in\mathbb{R}^{Mq\times Mq}$ and $\mathbf{w}(\mathbf{x})\in\mathbb{R}^{Mq}$ is a column vector.
The following theorem shows that symplectic Runge--Kutta methods preserve local conservation laws with quadratic density. The proof adapts Calvo, Iserles and Zanna's proof that such methods preserve quadratic invariants of systems of ODEs \cite{CIZ}, to take contributions from the flux into account.
\begin{theo}\label{th2}
The solution of any symplectic Runge--Kutta method applied to (\ref{SDcon}) satisfies a discrete version of (\ref{SDmom}).
\end{theo}
\begin{proof} The conservation law (\ref{SDmom}) amounts to
\begin{align}\label{cond1}\nonumber
D_m&\!\left\{\widetilde{F}_2\left(\mathbf{x},t,P(\mathbf{x})\widetilde \mathbf{g},P(\mathbf{x})\widetilde \mathbf{h}\left(\mathbf{x},t,P(\mathbf{x})\widetilde \mathbf{g}\right)\right)\right\}=-D_t\{\widetilde{G}_2(\mathbf{x},P(\mathbf{x})\widetilde \mathbf{g})\}\\
&=-\left((P(\mathbf{x})\widetilde\mathbf{g})^TS(\mathbf{x})+\mathbf{w}(\mathbf{x})^T\right)P(\mathbf{x})\widetilde\mathbf{h}(\mathbf{x},t,P(\mathbf{x})\widetilde\mathbf{g}).
\end{align}
Solving (\ref{SDcon}) using a $s$-stage symplectic Runge--Kutta method,
\begin{equation}\label{RK}
\widetilde\mathbf{g}_{n+1}=\widetilde\mathbf{g}_{n}+\Delta t\sum_{i=1}^sb_i \widetilde\mathbf{h}(\mathbf{x},t_n+c_i\Delta t,P(\mathbf{x})\mathbf{k}_i)\equiv\widetilde\mathbf{g}_{n}+\Delta t\sum_{i=1}^sb_i \widetilde\mathbf{h}_i,
\end{equation}
with internal stages
\begin{equation}\label{kstage}
\mathbf{k}_i=\widetilde\mathbf{g}_n+\Delta t\sum_{j=1}^sa_{i,j}\widetilde\mathbf{h}_j,\quad i=1,\ldots,s,
\end{equation}
we obtain $\mathbf{u}_n=P(x)\widetilde\mathbf{g}_n$. Moreover,
\begin{align*}
\widetilde{G}_2&(\mathbf{x},\mathbf{u}_{n+1})=\tfrac{1}2\mathbf{u}^T_{n+1} S(\mathbf{x})\mathbf{u}_{n+1}+\mathbf{w}(\mathbf{x})^T\mathbf{u}_{n+1}=\left(\tfrac{1}2 (P(\mathbf{x})\widetilde\mathbf{g}_{n+1})^T S(\mathbf{x})+\mathbf{w}(\mathbf{x})^T\right)P(\mathbf{x})\widetilde\mathbf{g}_{n+1}\\
\,=&\left(\tfrac{1}2 \mathbf{u}_n^T S(\mathbf{x})+\mathbf{w}(\mathbf{x})^T\right)\mathbf{u}_n+\Delta t\sum_{i=1}^sb_i\left((P(\mathbf{x})\widetilde\mathbf{g}_n)^T S(\mathbf{x})+\mathbf{w}(\mathbf{x})^T\right)P(\mathbf{x})\widetilde\mathbf{h}_i\\
&+\tfrac{\Delta t^2}2\sum_{i,j=1}^sb_ib_j \left(P(\mathbf{x})\widetilde\mathbf{h}_j\right)^TS(\mathbf{x})P(\mathbf{x})\widetilde\mathbf{h}_i.
\end{align*}
Using (\ref{kstage}) to eliminate $\widetilde\mathbf{g}_n$ from the first sum and rearranging, gives
\begin{align*}
\widetilde{G}_2(\mathbf{x},\mathbf{u}_{n+1})=&\,\widetilde{G}_2(\mathbf{x},\mathbf{u}_{n})+\Delta t\sum_{i=1}^sb_i\left( (P(\mathbf{x})\mathbf{k}_i)^T S(\mathbf{x})+\mathbf{w}(\mathbf{x})^T\right)P(\mathbf{x})\widetilde\mathbf{h}_i\\
&+\!\tfrac{\Delta t^2}2\sum_{i,j=1}^s\!(b_jb_i\!-\!b_ia_{i,j}\!-\!b_ja_{j,i})\left(P(\mathbf{x})\widetilde\mathbf{h}_j\right)^T\!\!\!S(\mathbf{x})P(\mathbf{x})\widetilde\mathbf{h}_i.
\end{align*}
The condition of symplecticity,
\begin{equation*}
b_ia_{i,j}+b_ja_{j,i}-b_ib_j=0,\quad \forall\, i,j=1,2,\ldots,s,
\end{equation*}
and (\ref{cond1}) give
\begin{equation*}
D_m\left\{\sum_{i=1}^s b_i\widetilde{F}_2\left(\mathbf{x},t_n+c_i\Delta t,P(\mathbf{x})\mathbf{k}_i,P(\mathbf{x})\widetilde\mathbf{h}_i\right)\right\}+D_n\{\widetilde{G}_2(\mathbf{x},\mathbf{u}_n)\}=0,
\end{equation*}
which is an approximation of (\ref{momentum}). $\hfill\ensuremath{\square}$
\end{proof}
\begin{rem}
Multisymplectic methods preserve conservation laws whose density and flux are both quadratic. By contrast, Theorem~\ref{th2} applies to all conservation laws that have quadratic density. As no assumption is needed on the flux, a larger class of conservation laws can be preserved.
\end{rem}
The following results follow directly from the proof of Theorem~\ref{th2}.
\begin{cor}\label{cor1}
Any Runge--Kutta method preserves semidiscrete local conservation laws whose density is linear in $[\mathbf{u}]_x$.
\end{cor}
\begin{cor}
The symplectic implicit midpoint method (defined by (\ref{RK})-(\ref{kstage}) with $s=1, b_1=1,$ and $a_{1,1}=c_1=1/2$) applied to (\ref{SDcon}) preserves the conservation law
\[D_m\left\{\widetilde{F}_2\left(\mathbf{x},t_n+\tfrac{1}2{\Delta t},\mu_n\mathbf{u}_n,D_n\mathbf{u}_n\right)\right\}+D_n\{\widetilde{G}_2(\mathbf{x},\mathbf{u}_n)\}=0.\]
\end{cor}
\subsection{Conservation law for the Hamiltonian}\label{CLHamsect}
We consider here the system of Hamiltonian PDEs (\ref{HamPDE}) defined by a Hamiltonian function $H$ on a domain with periodic boundary conditions. This assumption is introduced only for simplicity: the preservation of conservation laws is local and therefore independent of the specific boundary conditions assigned to the differential problem. The following local conservation law for the energy is satisfied by all solutions of (\ref{HamPDE}):
\begin{align}\nonumber
D_t(H)=&\,\sum_{\alpha,j}\frac{\partial H}{\partial(D_x^ju^\alpha)}(D_tD_x^ju^\alpha)=\sum_{\alpha,j}\frac{\partial H}{\partial(D_x^ju^\alpha)}D_x^j(D_tu^\alpha)\\\label{Hclaw}
=&\,D_x(\psi)+\mathcal{E}(H)^T\mathcal{DE}(H)\equiv D_x(F)
\end{align}
with $$\psi=\sum_{\alpha,i,j} (D_x^iD_tu^\alpha)(-D_x)^j\frac{\partial H}{\partial D_x^{i+j+1}u^\alpha}.$$
Among the best-known energy-conserving discrete gradient methods is the Average Vector Field (AVF) method \cite{McLaren}. We can use this in two different ways, depending on the number of points, $M$, in the stencil in (\ref{nodesx}). If $M$ is odd, let $A=-B$ so that the stencil is centred on $x_0$; we denote a semidiscretization of $H([\mathbf{u}]_x)$ on such a stencil by $\widetilde{H}(\mathbf{U})$.
The AVF method approximates (\ref{HamPDE}) by
\begin{equation}\label{AVF}
D_n\mathbf{u}_{0,0}=\widetilde{\mathcal{D}}(\mathbf{u}_0,\mathbf{u}_1)\widetilde{\delta}(\mathbf{u}_0,\mathbf{u}_1)\equiv\widetilde{\mathcal{D}}(\mathbf{u}_0,\mathbf{u}_1)\int_0^1\mathsf{E}_\mathbf{U} \left(\widetilde H(\mathbf U)\right)\Big\vert_{\mathbf{U}=(1-\xi)\mathbf{u}_0+\xi\mathbf{u}_1}\mathrm{d}\xi.
\end{equation}
If $M$ is even, let $A=1-B$ so that the stencil is centred at the midpoint of $x_0$ and $x_1$. Denote a semidiscretization of $H([\mathbf{u}]_x)$ as $\widetilde{H}(\mu_m\mathbf{U})$. We define the AVF method on such a stencil to be
\begin{align}\label{AVF2}
\!\!D_n\mu_m\mathbf{u}_{0,0}=\widetilde{\mathcal{D}}(\mathbf{u}_0,\mathbf{u}_1)\widetilde{\delta}(\mu_m\mathbf{u}_0,\mu_m\mathbf{u}_1):=\widetilde{\mathcal{D}}(\mathbf{u}_0,\mathbf{u}_1)\!\!\int_0^1\!\!\mathsf{E}_{\mu_m \mathbf{U}} \left(\widetilde H(\mu_m\mathbf U)\right)\Big\vert_{\mathbf{U}=(1-\xi)\mathbf{u}_0+\xi\mathbf{u}_1}\!\!\mathrm{d}\xi.
\end{align}
McLachlan and Quispel proved in \cite{QMcL} that discrete gradient methods preserve a discrete version of (\ref{Hclaw}) provided that $\widetilde{\mathcal{D}}$ is a skew-adjoint approximation of $\mathcal{D}$. The following theorem proves that the AVF method (\ref{AVF}) preserves the local conservation law for the energy (\ref{Hclaw}) under a milder assumption.
\begin{theo}\label{theoavf}
The AVF methods (\ref{AVF}) and (\ref{AVF2}) preserve a discrete energy conservation law if there exists a function $f$ defined on the stencil such that
\begin{equation}\label{oddmild}
\widetilde{\delta}(\mathbf{u}_0,\mathbf{u}_1)^T\widetilde{\mathcal{D}}(\mathbf{u}_0,\mathbf{u}_1)\widetilde{\delta}(\mathbf{u}_0,\mathbf{u}_1)=D_m(f),
\end{equation}
or
\begin{equation}\label{evenmild}
\widetilde{\delta}(\mu_m\mathbf{u}_0,\mu_m\mathbf{u}_1)^T\widetilde{\mathcal{D}}(\mathbf{u}_0,\mathbf{u}_1)\widetilde{\delta}(\mu_m\mathbf{u}_0,\mu_m\mathbf{u}_1)=D_m(f),
\end{equation}
respectively.
\end{theo}
\begin{proof}
Equation (\ref{oddmild}) yields a discrete energy conservation law for (\ref{AVF}), namely
\begin{align*}
\!\!D_n\widetilde{H}(\mathbf{u}_0)=&\sum_{i,\alpha}\!(D_n u^\alpha_{i,0})\int_0^1\!\frac{\partial}{\partial U_i^\alpha}\widetilde{H}(\mathbf{U})\Big\vert_{\mathbf{U}=(1-\xi)\mathbf{u}_0+\xi\mathbf{u}_{1}}\!\mathrm{d}\xi=\widetilde{\delta}(\mathbf{u}_0,\mathbf{u}_1)^T(D_n \mathbf{u}_{0,0})\!\\
&+\!D_m(\widetilde{\psi})=\,\widetilde{\delta}(\mathbf{u}_0,\mathbf{u}_1)^T\widetilde{\mathcal{D}}(\mathbf{u}_0,\mathbf{u}_1)\widetilde{\delta}(\mathbf{u}_0,\mathbf{u}_1)\!+\!D_m(\widetilde\psi)\!=\!D_m(\widetilde{F}),
\end{align*}
where
\begin{equation*}\widetilde{\psi}=\sum_{j\neq 0,\alpha}\frac{j\Delta x}{|j|}\sum_{i=\min\{j,0\}}^{\max\{j,0\}-1}(D_nu_{i,0}^\alpha)S_m^{i-j}\int_0^1\left(\frac{\partial}{\partial U_j^\alpha} \widetilde H(\mathbf U)\right)\Big\vert_{\mathbf{U}=(1-\xi)\mathbf{u_0}+\xi\mathbf{u_1}}\mathrm{d}\xi.
\end{equation*}
Similarly, from (\ref{evenmild}), the conservation law preserved by (\ref{AVF2}) is
\[D_n\widetilde{H}(\mu_m\mathbf{u}_0)=\widetilde{\delta}(\mu_m\mathbf{u}_0,\mu_m\mathbf{u}_1)^T\widetilde{\mathcal{D}}(\mathbf{u}_0,\mathbf{u}_1)\widetilde{\delta}(\mu_m\mathbf{u}_0,\mu_m\mathbf{u}_1)+D_m(\widetilde\phi)=D_m(\widetilde{F}),\]
where
\begin{equation*}\widetilde{\phi}\!=\!\sum_{j\neq 0,\alpha}\!\!\frac{j\Delta x}{|j|}\sum_{i=\min\{j,0\}}^{\max\{j,0\}-1}(D_n\mu_mu_{i,0}^\alpha)S_m^{i-j}\!\!\int_0^1\!\!\left(\!\frac{\partial}{\partial \mu_mU_j^\alpha} \widetilde H(\mu_m\mathbf U)\!\right)\!\!\Big\vert_{\mathbf{U}=(1-\xi)\mathbf{u_0}+\xi\mathbf{u_1}}\!\!\!\!\mathrm{d}\xi.\tag*{$\square$}
\end{equation*}
\end{proof}
\begin{rem}
Theorem~\ref{theoavf} holds true in particular when $\widetilde{\mathcal{D}}$ is skew-adjoint.
\end{rem}
\begin{rem}
Condition (\ref{oddmild}) is satisfied if and only if
\begin{equation}\label{testodd}
\mathsf{E}_{\mathbf{u}_n}\left(\widetilde{\delta}(\mathbf{u}_0,\mathbf{u}_1)^T\widetilde{\mathcal{D}}(\mathbf{u}_0,\mathbf{u}_1)\widetilde{\delta}(\mathbf{u}_0,\mathbf{u}_1)\right)=0,\qquad n=0,1.
\end{equation}
Similarly, condition (\ref{evenmild}) holds true if and only if
\begin{equation}\label{testeven}
\mathsf{E}_{\mathbf{u}_n}\left(\widetilde{\delta}(\mu_m\mathbf{u}_0,\mu_m\mathbf{u}_1)^T\widetilde{\mathcal{D}}(\mathbf{u}_0,\mathbf{u}_1)\widetilde{\delta}(\mu_m\mathbf{u}_0,\mu_m\mathbf{u}_1)\right)=0,\qquad n=0,1.
\end{equation}
Conditions (\ref{testodd}) and (\ref{testeven}) provide simple practical tests for analysing whether the assumptions of Theorem~\ref{theoavf} are satisfied by a given scheme.
\end{rem}
\begin{rem}
The following Hamiltonian-preserving schemes can be obtained from (\ref{AVF}) or (\ref{AVF2}), where the operator $\widetilde{\mathcal D}$ is not skew-adjoint but satisfies (\ref{oddmild}) or (\ref{evenmild}), respectively:
\begin{itemize}
\item[$\bullet$] EC$_8$ and the family of schemes MC$_8$ for KdV in \cite{FCHydon},
\item[$\bullet$] EC$_8$(0) (which preserves a quartic density) and MC$_8$(0) for mKdV in \cite{FCHmkdv},
\item[$\bullet$] EC$_6$ for BBM in \cite{FCHnls}.
\end{itemize}
\end{rem}
\subsection{Conservation laws of other types and multidimensional domains}\label{genericCL}
Fully-discrete methods that preserve other types of conservation law can be obtained again by using an Euler operator. For simplicity, we restrict the discussion to the case of second-order schemes for a PDE with polynomial nonlinearity. These can be obtained by following the steps below:
\begin{enumerate}
\item Let $P=\prod_{i=1}^r L_i$ be a polynomial of degree $r$ in the semidiscretization, where without loss of generality $L_i$ is a linear approximation of a single monomial factor in the continuous counterpart of $P$. Therefore, $L_i$ can depend on either $\mathbf{U}$ or $\mathbf{U}_t$. Assuming that the stencil has $N$ points in the time dimension, discretize $P$ using
\begin{equation}\label{fulld}
\sum_{j=0}^{N^r-1} \prod_{i=1}^r \alpha_j L_i(\mathbf{u}_{i_j}), \qquad \alpha_j\in\mathbb{R},
\end{equation}
where $i_j$ is the $i$-th digit of the representation of $N^r-1-j$ in base $N$, ordered from right to left, and setting $i_j=0$ if $N^r-1-j$ has less then $i$ digits. By varying $i_j$ one obtains all possible combinations of $N$ digits of length $r$. At this stage, we assume that the coefficients $\alpha_j$ only satisfy the requirements for consistency. For example, in a one-step method linear quantities are approximated by
$$L_1(\mathbf{U})=L_1(\mu_n\mathbf{u}_0),\qquad L_1(\mathbf{U}_t)=L_1(D_n\mathbf{u}_0),$$
and quadratic quantities by
\begin{align*}L_1L_2\approx&\, \alpha_0L_1(\mathbf{u}_1)L_2(\mathbf{u}_1)+\alpha_1L_1(\mathbf{u}_0)L_2(\mathbf{u}_1)+\alpha_2L_1(\mathbf{u}_1)L_2(\mathbf{u}_0)+\alpha_3L_1(\mathbf{u}_0)L_2(\mathbf{u}_0).
\end{align*}
\item The values of the parameters $\alpha_j$ are obtained by solving
\begin{equation}\label{fulleu}
\mathsf{E}_{\mathbf{u}}(\widetilde{\mathcal{A}}\widetilde{\mathcal{Q}}_\ell)=\mathbf{0}, \qquad \ell=1,2,\ldots,
\end{equation}
where $\widetilde{\mathcal{A}}$ and $\widetilde{\mathcal{Q}}_\ell$ are the approximations of the PDE and the characteristic obtained after steps 1 and 2, and $\mathsf{E}_{\mathbf{u}}$ is the difference Euler operator,
\begin{equation}\label{totdisc}
\mathsf{E}_{\mathbf{u}}=\sum_{i,j}S_m^{-i}S_n^{-j}\frac{\partial}{\partial u_{i,j}},
\end{equation}
whose kernel consists of all fully-discrete conservation laws \cite{kuper}.
\end{enumerate}
\begin{rem}
In total, net of consistency requirements, for each monomial of degree $r$ one needs:
\begin{itemize}
\item[$\bullet$] $\left(\begin{array}{c} M+r-1\\ r\end{array}\right)$ parameters for the semidiscretization. After solving (\ref{sdeulcond}), only a few of these will still be undetermined.
\item[$\bullet$] $N^r$ new parameters for the full discretization (\ref{fulld}), to be determined by solving (\ref{fulleu}).
\end{itemize}
By contrast, if one searches directly for all full discretizations without semidscretizing first, the strategy in \cite{FCHydon} introduces $\left(\begin{array}{c} NM+r-1\\ r\end{array}\right)$ variables for each monomial of degree $r$. This in general yields a huge nonlinear system whose solution is impractical.
\end{rem}
The algorithm above can be iterated, to preserve conservation laws for PDEs with more than two independent variables, by discretizing a single variable at each iteration. At the $k$-th iteration the Euler operator in (\ref{totdisc}) is replaced by
$$\mathsf{E}_{\mathbf{U}}^k=\sum S_{m_1}^{-i_1}S_{m_2}^{-i_2}\ldots S_{m_k}^{-i_k}(-D_{x_{k+1}})^{j_1}(-D_{x_{k+2}})^{j_2}\ldots(-D_{x_d})^{j_k},$$
whose kernel consists in the space of conservation laws of the form
$$D_{m_1}\widetilde{F}_1+D_{m_2}\widetilde{F}_2+\ldots+D_{m_k}\widetilde{F}_k+D_{x_{k+1}}F_{k+1}+\ldots+D_{x_{d}}F_{d}.$$
The proof is similar to that of Theorem~\ref{th1}.
\section{The Boussinesq equation}\label{GBsec}
We consider here the (Good) Boussinesq equation
\begin{equation}\label{Bousseq}
{u_{tt}-u_{xx} -(u^2)_{xx}+u_{xxxx}}=0,\quad (x,t)\in [a,b]\times[0,\infty),
\end{equation}
cast as a system of two PDEs
\begin{equation}\label{Bouss}
\mathcal{A}=(u_t-v_x, v_t-u_x-(u^2)_{x}+u_{xxx})=\mathbf{0}.
\end{equation}
This system can be written in Hamiltonian form,
$$D_t\,\mathbf{u}\equiv D_t\left(\begin{array}{c}u\\v\end{array}\right)=\mathcal{D}(\mathcal{E}(H)),$$
with
\begin{equation*}
\mathcal{D}=\left[\begin{array}{cc}
0&D_x\\D_x&0
\end{array}\right],\qquad H=\tfrac{1}3u^3+\tfrac{1}2(v^2+u^2+u_{x}^2).
\end{equation*}
System (\ref{Bouss}) has infinitely many independent conservation laws \cite{PC}. The first four are $D_xF_i+D_tG_i=0$, with
\begin{align}\label{CL1}
F_1=& -v,\qquad G_1=u,\\\label{CL2}
F_2=& u_{xx}-u-u^2,\qquad G_2=v,\\\label{CL3}
F_3=& uu_{xx}-\tfrac{1}2(v^2+u^2+u_x^2)-\tfrac{2}3u^3,\qquad G_3=uv,\\\label{CL4}
F_4=& vu_{xx}-u_xu_t-uv-u^2v,\qquad G_4=H.
\end{align}
with characteristics
\begin{equation}\label{dischar}
\mathcal{Q}_1=(1,0)^T, \quad\mathcal{Q}_2=(0,1)^T, \quad \mathcal{Q}_3=(v,u)^T,\quad \mathcal{Q}_4=(u+u^2-u_{xx},v)^T,
\end{equation}
respectively. When the boundary conditions are conservative (e.g. periodic), integrating in space (\ref{CL1})--(\ref{CL4}) gives the preservation of the following invariants,
\begin{equation}\label{invar}
I_1=\int u\,\mathrm{d}x,\qquad I_2=\int v\,\mathrm{d}x, \qquad I_3=\int uv\,\mathrm{d}x, \qquad I_4=\int H\,\mathrm{d}x;
\end{equation}
here $I_3$ and $I_4$ are the global momentum and the global energy, respectively.
\subsection{Conservative methods for the Boussinesq equation}
We look for semidiscretizations of (\ref{Bouss}) of the form
\begin{equation}\label{sdgen}
\widetilde{\mathcal{A}}:=(D_m \widetilde{F}_1+D_t \widetilde{G}_1, D_m \widetilde{F}_2+D_t \widetilde{G}_2)=\mathbf{0}.
\end{equation}
Solutions of (\ref{sdgen}) satisfy semidiscrete versions of the conservation laws (\ref{CL1}) and (\ref{CL2}) with $\widetilde{\mathcal{Q}}_1=\mathcal{Q}_1$ and $\widetilde{\mathcal{Q}}_2=\mathcal{Q}_2$.
The linear and quadratic terms in (\ref{Bouss}) and (\ref{dischar}) are approximated as
\begin{align*}
\sum_{i=A}^B \alpha_i Z_i, \qquad \sum_{i=A}^B\sum_{j=i}^B \beta_{i,j}Z_iZ_j,
\end{align*}
respectively, where $Z_i\in \{U_i, V_i, D_t U_i, D_t V_i\}$, and the coefficients $\alpha_i$ and $\beta_{i,j}$ are chosen by requiring the desired order of accuracy and the preservation of the conservation law of either the momentum (\ref{CL3}) or the energy (\ref{CL4}), according to the strategy outlined in Section~\ref{spacedis}. We have not found any semidiscrete scheme that preserves both of these conservation laws, as the constraints on the approximations of the nonlinear terms that we have obtained from (\ref{sdeulcond}) are not compatible with each other.
In the following, we present the components of \eqref{sdgen} and the characteristic $\widetilde{\mathcal{Q}}_{3/4}$ and density $\widetilde{G}_{3/4}$ of the remaining preserved conservation law. The corresponding flux $\widetilde{F}_{3/4}$ does not contribute to our global error estimates; in most cases, it has many terms and gives little insight, so we omit this.
Fully-discrete schemes that preserve the local momentum or the local energy are then obtained by applying Gauss-Legendre method or AVF method, respectively. As these are Runge--Kutta methods, the conservation laws (\ref{CL1}) and (\ref{CL2}) are preserved as a consequence of Corollary~\ref{cor1}.
\subsubsection*{Second-order schemes}\label{IIord}
Here we introduce new families of second-order schemes that preserve three conservation laws. The stencil in space consists of four points and we set $A=-1$ and $B=2$ in (\ref{nodesx}). All the free parameters in the formulae below $(\alpha,\beta,\gamma,\xi)$ are $O(\Delta x^2).$ Free parameters corresponding to higher degree perturbations are set equal to zero as their contribution is negligible.\\[1\baselineskip]
\textit{Momentum-conserving schemes}\\[1\baselineskip]
We have obtained six families of semidiscretizations that preserve the conservation law (\ref{CL3}), split by two different forms of the characteristic and the three parameter values $s\in\{0,1/3,1/2\}$.
The first three families are
\begin{align}\label{MC2}
\widetilde{F}_1=&-(V_0+\alpha D_m^2V_{-1}),\qquad \widetilde{G}_1=\mu_m(U_0+s{\Delta x^2}D_m^2U_{-1}), \\\nonumber
\widetilde{F}_2=&\,(1+\beta)D_m^2 U_{-1}-U_0-(U_0+\tfrac{(1-s)\Delta x^2}{3-5s}D_m^2U_{-1})(U_0+\tfrac{s\Delta x^2}{5s-1}D_m^2U_{-1})\\\nonumber
&+\{\tfrac{s\Delta x^2}{s+1}+\xi(3s-1)(1-2s)\}\{D_m^2(U_{-1}^2)-(D_mU_{-1})(D_mU_{0})\},\\\nonumber
\widetilde{G}_2\!=&\mu_m\!\left(V_0\!+\!\gamma D_m^2V_{-1}\right)\!,\quad\!
\widetilde{G}_3\!=\!\widetilde G_1\widetilde G_2,\quad \! \widetilde{\mathcal{Q}}_3\!=\!(\widetilde G_2,\widetilde G_1)^T\!.
\end{align}
Three remaining are given by $\widetilde{F}_1, \widetilde{F}_2$ and $\widetilde{G}_2$ as in (\ref{MC2}) together with
\begin{align*}
\widetilde{G}_1=&\,\mu_m\left(U_0+\gamma D_m^2U_{-1}\right),\\ \widetilde{\mathcal{Q}}_3=&\,\left(\mu_m(V_0+s{\Delta x^2}D_m^2V_{-1}),\,\mu_m(U_0+s{\Delta x^2}D_m^2U_{-1})\right)^T\!\!,\\
\widetilde{G}_3=&\,(\mu_mU_0)(\mu_mV_0)\!-\!(\gamma+s{\Delta x^2})\mu_m\{(D^{(c)}_mU_{0})(D^{(c)}_mV_{0})\}+{s\gamma\Delta x^2}(D_mD^{(c)}_mU_{0})(D_mD^{(c)}_mV_{0}).
\end{align*}
In the numerical tests section we limit our investigations to the semidiscretions obtained from (\ref{sdgen}) with (\ref{MC2}) and $s=\alpha=\beta=\xi=0$, $\gamma=\lambda_1\Delta x^2$; we use MC$_2(\lambda_1)$ to denote the family of finite difference schemes obtained by using the symplectic implicit midpoint method (Gauss-Legendre method of order two) to discretize in time. The iterative technique described in Section~\ref{genericCL} also finds these schemes, but no others.
\bigskip
\noindent\textit{Energy conserving schemes}
\medskip
There is only one family of semidiscretizations of the form (\ref{sdgen}) that preserves the local conservation law of the energy. For this family, $\widetilde{F}_1$ and $\widetilde{G}_2$ are defined as in (\ref{MC2}) and
\begin{align}\nonumber
\widetilde{G}_1=&\,\mu_mU_0,\,\,\, \widetilde{F}_2=(1\!+\!\beta)D_m^2 U_{-1}\!-\!U_0\!-\!(\mu_m^2U_{-1})^2,\,\,\, \widetilde{\mathcal{Q}}_4=(-\mu_m \widetilde F_2,-\mu_m \widetilde F_1)^T\\
\widetilde{G}_4=&\,\tfrac{1}2\left\{(\mu_m U_0)^2+(1+\beta)\mu_m\big((D^{(c)}_mU_{0})^2\big)\right\}\\\nonumber
&+\!\tfrac{1}2\left\{\mu_m(V_{0}+\alpha D_m^2V_{-1})\mu_m(V_{0}+\gamma D_m^2V_{-1})\right\}\!+\!\tfrac{1}3(\mu_mU_0)\mu_m\big((\mu_m^2 U_{-1})^2\big).
\end{align}
The resulting system of ODEs can be written in the form
\begin{equation}\label{Hamode}
D_t\left(\begin{array}{c} \mu_m U_0\\ \mu_m V_0\end{array}\right)=
\widetilde{\mathcal D}\left(\begin{array}{c} \mathsf{E}_{\mu_m \mathbf{U}} (\widetilde{H})\\ \mathsf{E}_{\mu_m \mathbf V} (\widetilde{H})\end{array}\right),
\end{equation}
with
\[ \widetilde{H}=\widetilde{G}_4,\qquad \widetilde{\mathcal D}=\left(\begin{array}{cc} 0 & \widetilde{D}_x\\ \widetilde{D}_x & 0\end{array}\right),\qquad \widetilde D_x=D_m(\mu_m+\gamma D_mD^{(c)}_mS_m^{-1})^{-1}.\]
The operator $\widetilde{\mathcal D}$ is not skew-adjoint, but satisfies (\ref{evenmild}). Therefore, by applying the AVF method (\ref{AVF2}) to (\ref{Hamode}) we obtain a family of fully-discrete schemes that preserve the local conservation law of the energy. We use EC$_2(\lambda_2)$ to denote the schemes with $\alpha=\gamma=0$ and $\beta =\lambda_2\Delta x^2$. Again, the iterative technique from Section~\ref{genericCL} yields only these schemes.
\subsubsection*{Fourth-order schemes}\label{IVord}
The families of fourth-order schemes introduced here preserve three conservation laws and depend on free parameters ($\alpha,\beta,\gamma,\xi$) that are all $O(\Delta x^4)$. Parameters introducing only perturbations of higher degree are set equal to~zero.
For each of the semidiscretizations introduced in this section, the discrete fluxes $\widetilde{F}_j$ are second-order accurate, but $D_m \widetilde F_j$ and the three preserved conservation laws,
$$\widetilde\mathcal{A}\widetilde\mathcal{Q}_j=D_m \widetilde F_j + D_t \widetilde G_j,$$
are approximated with fourth-order accuracy.
\bigskip
\noindent\textit{Momentum-conserving schemes}
\medskip
On a spatial stencil with six points ($A=-2$ and $B=3$ in (\ref{nodesx})) there are two families of semidiscretizations that preserve the local momentum conservation law. Let
\begin{align}\nonumber
\Phi(Z;p):=&\,Z_{-1}+p\Delta x^2D_m^2Z_{-2},\\\nonumber
\!\Theta(Z;p_1,p_2,p_3,p_4,p_5,p_6):=&D_m^2\Phi(Z;p_1)(D_m^2\Phi(Z;p_2))\!+\!p_3(S_m\Phi(Z;p_4))D_m^4Z_{-2}\\\nonumber
&\,+p_5D_m\mu_m(\Phi(Z;p_6))D_m^3\mu_mZ_{-2}.
\end{align}
The first family of semidiscretizations and their preserved conservation laws is given by
\begin{align}\label{MC4}
\widetilde{F}_1=&-\!(V_0\!-\!\tfrac{\Delta x^2}{24}D_m^2V_{-1}\!+\!\alpha D_m^4V_{-2}),\,\,\, \widetilde{G}_1=\!\mu_m(U_0\!-\!\tfrac{\Delta x^2}8\!D_m^2U_{-1}),\\\nonumber
\widetilde{F}_2=&\,D_m^2U_{-1}-U_0-U_0^2+\tfrac{\Delta x^2}{24}D_m^2(U_{-1}+U_{-1}^2-3D_m^2U_{-2})+\beta D_m^4U_{-2}\\\nonumber
&-\tfrac{\gamma}2\Theta(U;-\tfrac{1}{4},-\tfrac{3}{8},-2,-\tfrac{3}{16},-2,-\tfrac{1}{16})+\tfrac{7\Delta x^4}{192}\Theta(U;0,0,\tfrac{8}7,-\tfrac{3}{8},0,0),\\\nonumber
\widetilde{G}_2=&\,\mu_m(V_0-\tfrac{\Delta x^2}8D_m^2 V_{-1}+\xi D_m^4V_{-2}),\quad \widetilde{\mathcal{Q}}_3=(\widetilde{G}_2,\widetilde{G}_1)^T,\quad \widetilde{G}_3=\widetilde G_1\widetilde G_2.
\end{align}
The second family has $\widetilde{F}_1,\widetilde{F}_2$ and $\widetilde{G}_2$ as in (\ref{MC4}), together with
\begin{align*}
\widetilde{G}_1=&\,\mu_m(U_0\!-\!\tfrac{\Delta x^2}8D_m^2U_{-1}\!+\!\xi D_m^4U_{-2}),\\
\widetilde{\mathcal{Q}}_3=&\,\big(\mu_m(V_0\!-\!\tfrac{\Delta x^2}8D_m^2V_{-1}),\mu_m(U_0\!-\!\tfrac{\Delta x^2}8D_m^2U_{-1})\big)^T\!,\\
\widetilde{G}_3=&\,\{\mu_m(U_0\!-\!\tfrac{\Delta x^2}8D_m^2U_{-1})\}\{\mu_m(V_0\!-\!\tfrac{\Delta x^2}8D_m^2V_{-1})\}\\
&+\xi(D_mD^{(c)}_mU_{0})(D_mD^{(c)}_mV_{0})+\tfrac{\xi\Delta x^2}8\mu_m\{(D_m^2D^{(c)}_mU_{-1})(D_m^2D^{(c)}_mV_{-1})\}.
\end{align*}
We use MC$_4(\lambda_3)$ to denote the schemes obtained by applying the Gauss-Legendre method of order four to (\ref{MC4}), with $\alpha=\beta=\gamma=0$ and $\xi=\lambda_3\Delta x^4$.
\bigskip
\noindent\textit{Energy conserving schemes}
\medskip
On the most compact six-point stencil, there are no semidiscretizations of the form (\ref{sdgen}) that preserve the local conservation law for energy. However, a seven-point stencil ($B=-A=3$ in (\ref{nodesx})) is more fruitful. For $n\in\mathbb{N}$, let
$$ \varphi_n(k)=\begin{cases}\lceil \tfrac{n}2\rceil, & \text {if}\quad k\geq \tfrac{n}2,\\
k+1, & \text {if}\quad k< \tfrac{n}2, \end{cases}$$
and define the operators
$$\nu_m^\pm=I\pm\tfrac{\Delta x^2}6D_m^2S_m^{-1}$$
and the functions
\begin{align*}
\widehat{F}_1=&\, -(\nu_m^- V_{-1}+\alpha D_m^4V_{-3}),\\
\widehat{F}_2=&\, D_m^2 U_{-2}-\nu_m^-U_{-1}-(\nu_m^+ U_{-1})^2+(\beta-\tfrac{\Delta x^2}4)D_m^4U_{-3}\\
&+\tfrac{\Delta x^2}6\left\{2(\nu_m^+ U_{-1})(D_m^2\nu_m^+ U_{-2})+D_m^2\big((\nu_m^+ U_{-2})^2\big)\right\}.
\end{align*}
The family of semidiscretizations and conservation laws is as follows:
\begin{align}\label{EC4}
\widetilde{F}_1=&\, \mu_m\widehat{F}_1,\qquad \widetilde{G}_1=U_0,\\\nonumber
\widetilde{F}_2=&\, \mu_m\widehat{F}_2,\qquad \widetilde{G}_2=\,V_0+\gamma D_m^4V_{-2},\\\nonumber
\widetilde \mathcal{Q}_4=&\,(-\nu_m^+S_m\widehat{F}_2,-\nu_m^+S_m\widehat{F}_1)^T,\\\nonumber
\widetilde G_4=&\,\tfrac{1}2\{(V_0+\gamma D_m^4V_{-2})\nu_m^+(\nu_m^-V_0+\alpha D_m^4V_{-2})\\\nonumber
&+\mu_m\{(D_mU_{-1})D_m\nu_m^+(U_{-1}-\tfrac{\Delta x^2}4D_m^2U_{-2})\}+U_0\nu_m^+(\nu_m^-U_0-\beta D_m^4U_{-2})\}\\\nonumber
&+\tfrac{1}3U_0\nu_m^+\left((\nu_m^+U_0)^2-\tfrac{\Delta x^2}6\left(2(\nu_m^+U_0)(D_m^2\nu_m^+U_{-1})+D_m^2((\nu_m^+U_{-1})^2)\right)\right).
\end{align}
The systems of ODEs defined by (\ref{sdgen}) with (\ref{EC4}) amounts to
\begin{equation}\label{Hamode4}
D_t\left(\begin{array}{c} U_0\\ V_0\end{array}\right)=
\widetilde{\mathcal D}\left(\begin{array}{c} \mathsf{E}_{\mathbf{U}} (\widetilde{H})\\ \mathsf{E}_{ \mathbf V} (\widetilde{H})\end{array}\right),
\end{equation}
with
\[ \widetilde{H}=\widetilde{G}_4,\qquad \widetilde{\mathcal D}=\left(\begin{array}{cc} 0 & \widetilde{D}_x\\ \widetilde{D}_x & 0\end{array}\right),\qquad \widetilde D_x=D_m\mu_m(S_m\nu_m^++\gamma D_m^4S_m^{-1}\nu_m^+)^{-1}.\]
The operator $\widetilde{\mathcal D}$, although not skew-adjoint, satisfies (\ref{oddmild}). We use EC$_4(\lambda_4)$ to denote the family of schemes obtained by applying the AVF method of order four (see \cite{McLaren}) to (\ref{Hamode4}) with $\alpha =\lambda_4\Delta x^4$ and $\beta=\gamma=0$.
\section{Numerical Tests}\label{Numsec}
In this section we solve a couple of benchmark problems to show the effectiveness and conservation properties of the numerical methods in Section~\ref{GBsec}.
The results are compared with the following second-order structure-preserving methods:
\begin{itemize}
\item[$\bullet$] The multisymplectic scheme for (\ref{Bousseq}),
\begin{align*}
\text{PS}\equiv &\,D_n^2\mu_m^4u_{-2,-1}-D_m^2\mu_m^2\mu_n^2u_{-2,-1}-D_m^2\mu_m\mu_n(\mu_m\mu_n u_{-2,-1})^2+D_m^4\mu_n^2u_{-2,-1}=0,
\end{align*}
developed in \cite{Zeng} and equivalent to the well-known Preissmann scheme.
\item[$\bullet$] The symplectic scheme for (\ref{Bouss}) in \cite{Chen},
\begin{align*}
\text{MP}:= \big(&\,D_n u_{0,0}-D^{(c)}_m\mu_n v_{0,0},D_n v_{0,0}+D^{(c)}_m(D_m^2\mu_n u_{-1,0}- (\mu_n u_{0,0})^2-\mu_n u_{0,0} \,\big)=\mathbf{0},
\end{align*}
obtained by applying the midpoint rule to a suitable spatial discretization.
\item[$\bullet$] The energy-conserving scheme for (\ref{Bouss}) in \cite{Matsuo},
\begin{align*}
\text{DVD}:= \big(&\,D_n u_{0,0}-D_m\mu_n v_{0,0},\, D_n v_{0,0}\\&\!+\!D_m(D_m^2\mu_n u_{-2,0}\!-\!\tfrac{1}3 (u_{-1,0}^2\!+\!u_{-1,0}u_{-1,1}\!+\!u_{-1,1}^2) \!-\!\mu_n u_{-1,0} )\big)\!=\!\mathbf{0},
\end{align*}
obtained using a discrete variational derivative method. This scheme can be obtained also by applying the AVF method to the Hamiltonian system of ODEs defined by
$$ \widetilde{\mathcal D}=\left(\begin{array}{cc} 0 & D_mS_m^{-1}\\ D_m & 0\end{array}\right), \qquad H=\tfrac{1}2(U_{0}^2+V_{0}^2+\mu_m((D_mU_{-1})^2)+\tfrac{1}3U_{0}^3.$$
\end{itemize}
To the best of our knowledge, there are no schemes in the literature
for the Boussinesq equation that are fourth-order accurate in both space and time.
Therefore, we compare the performance of the fourth-order schemes in Section~\ref{GBsec} with the following finite difference scheme for (\ref{Bousseq}) introduced in \cite{Ismail}:
\begin{align*}
\text{FD}_4:=\!(1\!+\!\tfrac{\Delta x^2}{12}D_m^2S_m^{-1})^2D_n^2 u_{0,0}\!-\!D_m^2\mu_n \left\{ (1\!+\!\tfrac{\Delta x^2}{12}D_m^2S_m^{-1})(\mu_nu_{-1,0}\!+\!(\mu_nu_{-1,0})^2)\!-\!D_m^2\mu_nu_{-2,0}\right\}\!.
\end{align*}
The scheme FD$_4$ is fourth-order accurate in space and second-order accurate in time, so to have a fair comparison we will use this scheme with a time step equal to $\Delta t^2$.
We consider $(x,t)\in \Omega\equiv [a,b]\times [0,T]$ and periodic boundary conditions. We introduce on $\Omega$ a grid with $I+1$ nodes, $x_i$, in space and $J+1$ nodes, $t_j$, in time. Henceforth subscripts denote shifts from the point $(x_0,t_0)=(a,0)$ (e.g., $u_{i,j}\simeq u(a+i\Delta x,j\Delta t)$).
As the computational time is similar for all the schemes of the same order of accuracy, our comparisons are based on the error in the solution at the final time $t = T$, evaluated as
\begin{equation}\label{relerr}
\left.\frac{\|u-u_\mathrm{exact}\|}{\|u_\mathrm{exact}\|}\right|_{t=T},
\end{equation}
where $\|\cdot\|$ denotes the Euclidean norm.
We also compare the errors in the global invariants (\ref{invar}) defined as
$$\text{Err}_\alpha\!=\!\Delta x\! \max_{j=1,\ldots,J}\left|\sum_{i=0}^I\left(\widetilde{G}_\alpha\Big\vert_{U_m=u_{m+i,j},V_m=v_{m+i,j}}\!-\widetilde{G}_\alpha\Big\vert_{U_m=u_{m+i,0},V_m=v_{m+i,0}}\right)\right|,$$
where $\alpha=1,2,3,4.$ For the methods introduced in this paper, $\widetilde{G}_\alpha$ is given in Section~\ref{GBsec}. For all the other schemes, we set
$$\widetilde{G}_1\!=\!U_0,\quad\! \widetilde{G}_2\!=\!V_0,\quad\! \widetilde{G}_3\!=\! U_{0}V_{0},\quad\! \widetilde{G}_4\!=\!\tfrac{1}2(U_{0}^2\!+\!V_{0}^2\!+\!\mu_m((D_mU_{-1})^2)\!+\!\tfrac{1}3U_{0}^3\!.$$
\subsubsection*{Single soliton}
For the first problem we set $\Omega=[-60,60]\times[0,25]$ and the initial conditions given by the single soliton solution over $\mathbb{R}$,
\begin{equation*}
u_{\mathrm{exact}}(x,t)\!=\!-\tfrac{3p^2}2\mathrm{sech}^2\!\left(\tfrac{p}2(x\!-\!ct\!+\!d)\right),\quad\! v_{\mathrm{exact}}(x,t)\!=\!\tfrac{3cp^2}2\mathrm{sech}^2\!\left(\tfrac{p}2(x\!-\!ct\!+\!d)\right)\!,
\end{equation*}
where $c=\sqrt{1-p^2}.$ We choose $$p=\frac{1}{\sqrt{3}}, \qquad d=10.$$ \begin{table}[htb]
\caption{Single soliton problem: $\pi(\Delta x)$ for each scheme}\label{tabord}
\small
\centerline{\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline
$\Delta x=\Delta t$ & 0.2 & 0.3 & 0.4 & 0.5 & 0.6 & 0.7 & 0.8 & 0.9 \\
\hline
MC$_2$(0) & 1.99 & 2.00 & 1.97 & 1.92 & 1.96 & 1.94 & 1.70 & 2.00 \\
EC$_2$(0) & 1.99 & 1.97 & 1.94 & 1.94 & 1.91 & 1.88 & 1.66 & 1.88 \\
MC$_4$(0) & 4.00 & 3.99 & 3.98 & 3.96 & 3.95 & 3.99 & 3.80 & 3.94 \\
EC$_4$(0) & 4.01 & 4.00 & 4.05 & 4.13 & 4.06 & 4.21 & 4.32 & 4.26 \\
\hline
\end{tabular}}
\end{table}
\begin{figure}[htbp]
\begin{center}
\input{order1sol.tex}
\end{center}
\caption{Single soliton problem: plot of $\pi(\Delta x)$ showing convergence of the schemes.}
\label{ord1}
\end{figure}
We first examine the convergence and stability of the schemes found in Section~\ref{GBsec}, setting all parameters to zero. The order of convergence at various step sizes is measured by
$$\pi(\Delta x)=\frac{(k-1)\log(\mathrm{error}_{k}/\mathrm{error}_{k-1})}{k}, \qquad \text{where}\ k=10\Delta x,$$
and error$_k$ denotes the error obtained from (\ref{relerr}) with $\Delta x=\Delta t=k/10$, for $k=1,2,\ldots,9$.
The results in Table~\ref{tabord} and Figure~\ref{ord1} show that all the methods tend to the exact solution with the expected maximum order of convergence as the grid is refined, and are stable also for the largest stepsizes.
\begin{table}
\caption{Single soliton problem with $\Delta x=0.5$ and $\Delta t=0.5$ (except $\text{FD}_4$).}
\label{tab1}
\centerline{\begin{tabular}{cccccc}
\hline\noalign{\smallskip}
Method & $\text{Err}_1$ & $\text{Err}_2$ & $\text{Err}_3$ & $\text{Err}_4$ & Sol. Err. \\
\noalign{\smallskip}\hline\noalign{\smallskip}
$\mbox{MC}_2(0)$ & 6.22e-15 & 6.22e-15 & 1.22e-15 & 7.90e-04 & 0.0293\\
$\mbox{MC}_2(-0.21)$ & 5.33e-15 & 1.33e-15 & 1.22e-15 & 3.77e-04 & 0.0059\\
$\mbox{EC}_2(0)$ & 4.44e-15 & 4.00e-15 & 1.08e-05 & 6.66e-16 & 0.0415 \\
$\mbox{EC}_2(-0.20)$ & 3.55e-15 & 3.55e-15 & 2.63e-06 & 1.22e-15 & 0.0062 \\
PS & 5.88e-16 & 0.0194 & 5.79e-04 & 0.0013 & 0.0238\\
MP & 5.77e-15 & 4.89e-15 & 2.77e-04 & 0.0030 & 0.0706 \\
DVD & 5.77e-15 & 4.00e-15 & 0.0053 & 3.44e-15 & 0.0740\\
$\mbox{MC}_4(0)$ & 5.77e-15 & 5.77e-15 & 2.00e-15 & 3.18e-05 & 7.84e-04\\
$\mbox{MC}_4(0.06)$ & 5.77e-15 & 9.33e-15 & 2.89e-15 & 8.50e-06 & 1.23e-04 \\
$\mbox{EC}_4(0)$ & 6.66e-15 & 1.36e-11 & 4.02e-08 & 5.03e-14 & 4.12e-04\\
$\mbox{EC}_4(0.03)$ & 6.21e-15 & 1.57e-12 & 1.47e-08 & 5.57e-14 & 1.94e-04\\
$\mbox{FD}_4$\,\,($\Delta t=0.5^2$) & 9.97e-12 & 0.0036 & 1.07e-04 & 2.84e-04 & 0.0038\\
\noalign{\smallskip}\hline
\end{tabular}}
\end{table}
Table~\ref{tab1} shows the error in the conservation laws and the solution for the different methods with $\Delta x=\Delta t=0.5$. For this problem, the values of the free parameters that minimize the error in the solution of MC$_2(\lambda_1)$, EC$_2(\lambda_2)$, MC$_4(\lambda_3)$, EC$_4(\lambda_4)$ are $\lambda_1=-0.21,$ $\lambda_2=-0.20$, $\lambda_3=0.06$, $\lambda_4=0.03$. Such optimization is easy given that the solution is known, but is not currently feasible more generally. Therefore, the results obtained by setting the above parameters to zero are shown for comparison. This benchmark test illustrates that:
\begin{itemize}
\item[$\bullet$] All schemes preserve the first conservation law, but only those based on the formulation (\ref{Bouss}) preserve the second conservation law;
\item[$\bullet$] The schemes introduced in this paper preserve three conservation laws up to machine accuracy;
\item[$\bullet$] The new second-order schemes compare favourably with the methods from the literature in terms of accuracy in both the solution and the invariants that are not exactly conserved, even without optimising the parameters;
\item[$\bullet$] Choosing the optimal value of the free parameter, the error in the solution is roughly four times smaller (or more) than any other second-order method;
\item[$\bullet$] The fourth-order methods are all more much accurate than FD$_4$ with timestep $\Delta t=0.5^2$, even for non-optimal parameters.
\end{itemize}
\begin{figure}
\begin{center}
\input{1sol.tex}
\end{center}
\caption{Top: initial condition (dashed line) and solution of MC$_2(-0.21)$ at $T=25$ (solid line). Bottom: comparison of different schemes around the top of the soliton.}
\label{1solfig}
\end{figure}
\begin{figure}[htbp]
\begin{center}
\input{1sol2.tex}
\end{center}
\caption{Single soliton: error of fourth-order schemes.}
\label{1solerr}
\end{figure}
In Figure~\ref{1solfig}, the upper plot shows the solution obtained by the most accurate scheme, MC$_2(-0.21)$. The motion of the soliton does not produce any spurious oscillations. These can be seen (with amplitude of about $10^{-2}$) in the solutions of MP and DVD.
The lower plot shows the exact solution and the solution of MC$_2(-0.21)$ compared to the solutions of MP and PS around the top of the soliton (we omit the solution of DVD as it is the least accurate). The approximate solutions have been reconstructed using cubic spline interpolation of the values at the grid points denoted with markers. This figure shows how well the solution of MC$_2(-0.21)$ matches both the phase and the amplitude of the soliton.
Figure~\ref{1solerr} shows the difference between the exact solution and the solutions of MC$_4$(0.06) and FD$_4$ (with $\Delta t=0.5^2$). The error in MC$_4$ is roughly 30 times smaller; is located mainly at the peak of the soliton, and can be ascribed to a small phase error. The error of FD$_4$ is more widespread.
\subsubsection*{Interaction of two solitons}
We now study the interaction of two solitons over $\Omega=[-150,150]\times[0,50]$. The exact solution over $\mathbb{R}$ is \cite{MMM},
\begin{equation}\label{ex2sol}
u_{\text{exact}}=-6D_x^2\log \omega(x,t),\quad \omega(x,t)=1+\exp(\eta_1)+\exp(\eta_2)+A\exp(\eta_1+\eta_2),
\end{equation}
where
$$\eta_j=p_j(x-c_jt+d_j),\quad c_j=(-1)^j\sqrt{1-p_j^2},\quad A=\frac{(c_1-c_2)^2-3(p_1-p_2)^2}{(c_1-c_2)^2-3(p_1+p_2)^2}.$$
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=4.2in,height=5.7cm]{2sol.eps}\vspace{0.4cm}
\input{2sol.tex}
\end{center}
\caption{Solution of EC$_2(-0.18)$ on $\Omega$ and at the time $T=50$. }\label{2sol}
\end{figure}
\begin{figure}[htbp]
\begin{center}
\input{2sol2.tex}
\end{center}
\caption{Comparison of different schemes around the top of the two solitons.}
\label{2solz}
\end{figure}
\begin{figure}[htbp]
\begin{center}
\input{2solerr.tex}
\end{center}
\caption{Interaction of two solitons: error of fourth-order methods}
\label{2solerr}
\end{figure}
\begin{table}[tb]
\caption{Interaction of two solitons with $\Delta x=0.5$ and $\Delta t=0.5$ (except $\text{FD}_4$).}\label{tab2}
\centerline{\begin{tabular}{cccccc}
\hline\noalign{\smallskip}
Method & $\text{Err}_1$ & $\text{Err}_2$ & $\text{Err}_3$ & $\text{Err}_4$ & Sol. Err. \\
\noalign{\smallskip}\hline\noalign{\smallskip}
$\mbox{MC}_2(0)$ & 1.69e-14 & 7.83e-15 & 1.62e-15 & 0.0461 & 0.0257\\
$\mbox{MC}_2(-0.19)$ & 1.60e-14 & 9.44e-16 & 9.58e-16 & 0.0479 & 0.0061\\
$\mbox{EC}_2(0)$ & 1.33e-14 & 3.80e-15 & 7.32e-05 & 1.44e-15 & 0.0362 \\
$\mbox{EC}_2(-0.18)$ & 1.60e-14 & 4.86e-15 & 7.88e-05 & 1.55e-15 & 0.0057 \\
PS & 3.11e-16 & 3.44e-05 & 2.84e-04 & 0.0519 & 0.0176\\
MP & 1.07e-14 & 3.66e-15 & 1.55e-04 & 0.0532 & 0.0676 \\
DVD & 1.07e-14 & 3.72e-15 & 0.0300 & 1.44e-15 & 0.0422\\
$\mbox{MC}_4(0)$ & 1.33e-14 & 4.55e-15 & 1.71e-15 & 0.0494 & 5.08e-04 \\
$\mbox{MC}_4(0.06)$ & 1.42e-14 & 4.02e-15 & 1.86e-15 & 0.0495 & 1.12e-04 \\
$\mbox{EC}_4(0)$ & 1.15e-14 & 3.16e-15 & 3.91e-07 & 1.21e-14 & 3.20e-04\\
$\mbox{EC}_4(0.02)$ & 1.15e-14 & 6.16e-15 & 3.89e-07 & 1.25e-14 & 2.56e-04\\
$\mbox{FD}_4$\,\,($\Delta t=0.5^2$) & 2.98e-12 & 4.10e-04 & 4.23e-05 & 0.0522 & 0.0040\\
\noalign{\smallskip}\hline
\end{tabular}}
\end{table}
We obtain the initial conditions from (\ref{ex2sol}) setting
$$p_1=\frac{1}{\sqrt{6}},\qquad p_2=\frac{1}{\sqrt{5}},\qquad d_2=-d_1=20,$$
and solve this problem with $\Delta x=\Delta t=0.5$. The optimal values of the free parameters for each of the families MC$_2(\lambda_1)$, EC$_2(\lambda_2)$, MC$_4(\lambda_3)$, EC$_4(\lambda_4)$ are $\lambda_1=-0.19,$ $\lambda_2=-0.18$, $\lambda_3=0.06$, $\lambda_4=0.02$. The results in Table~\ref{tab2} are consistent with those in Table~\ref{tab1}, and analogous remarks apply.
Figure~\ref{2sol} shows the solution of the most accurate second-order scheme, EC$_2(-0.18)$, on the whole domain $\Omega$ (upper plot) and at the final time (lower plot). The schemes MP and DVD produce oscillations (amplitude $\simeq 0.005$), where the exact solution is flat. These do not occur in the solution of EC$_2(-0.18)$.
Figure~\ref{2solz} shows how the different schemes approximate the peak of the two solitons (omitting the least accurate solution of MP). The solution of EC$_2(-0.18)$ best reproduces the speed and the amplitude of the two waves.
Finally, Figure~\ref{2solerr} compares the difference between the exact solution and the approximations given by MC$_4$(0.06) and FD$_4$ (with $\Delta t=0.5^2$). Just as for the single soliton, the error of FD$_4$ has a higher amplitude and spreads far from the final location of the two solitons.
\section{The potential Kadomtsev-–Petviashvili (pKP) equation}\label{pKPsec}
This section briefly demonstrates that the novel strategy described in Section~\ref{genericCL} is practicable for PDEs with more than two independent variables. We seek to preserve two conservation laws,
\[
D_xF_i+D_yG_i+D_tH_i=0,
\]
of the pKP equation,
\begin{equation}\label{pKP}
u_{xt}+\tfrac{3}2u_{x}u_{xx}+\tfrac{1}4u_{xxxx}+\tfrac{3}4 u_{yy}=0.
\end{equation}
\noindent\textbf{Note}:\ Throughout this section, $H_i$ and $\widetilde{H}_i$ are components of conservation laws, not Hamiltonians.
The characteristics
$\mathcal{Q}_1=1, \mathcal{Q}_2=u_x,$
correspond respectively to the conservation laws with components
\begin{align}\label{pKPcl1}
F_1=&\tfrac{3}4 u_x^2+\tfrac{1}4 u_{xxx},\quad G_1=\tfrac{3}{4}u_y,\quad H_1=u_x,\\\label{pKPcl2}
F_2=&\tfrac{1}{2}u_x^3+\tfrac{1}{4}u_xu_{xxx}-\tfrac{1}{8}u_{xx}^2+\tfrac{3}{8}uu_{yy},\quad G_2=\tfrac{3}{8}(u_yu_x-uu_{xy}),\quad H_2=\tfrac{1}{2}u_x^2.
\end{align}
We introduce a uniform grid in space with nodes $(x_m,y_n)$ and use $U_{m,n}(t)$ to denote a semidiscrete approximation of $u(x_m,y_n,t)$. In this section, $D_n$ and $\mu_n$ are the forward difference and forward average operators acting on the second index, respectively.
The approach in Section~\ref{genericCL} can be applied to a full 15-point rectangular stencil; this yields a wide range of families of methods. For brevity, we present here only those schemes for which all spatial derivatives are approximated on a one-dimensional spatial stencil consisting of three and five points respectively for the $y$- and $x$-derivatives. There are just two one-parameter families, both of the form
\begin{equation}\label{methpKP}
D_m \widetilde{F}_1+D_n \widetilde{G}_1 + D_t\widetilde{H}_1=0
\end{equation}
(so $\widetilde{\mathcal{Q}}_1=1$), that preserve semidiscrete versions of (\ref{pKPcl1}) and (\ref{pKPcl2}).
The first family is defined by
\begin{equation}\label{MC1pKP}
\widetilde F_1=\tfrac{3}{4}(D_m^{(c)}U_{-1,0})(D_m^{(c)}U_{0,0}) + \tfrac{1}{4}D_m^3U_{-2,0},
\quad\widetilde G_1=\tfrac{3}{4}D_nU_{0,-1},
\quad\widetilde H_1=(I+\alpha D_m^2S_m^{-1})D_m^{(c)}U_{0,0};
\end{equation}
the semidiscrete version of (\ref{pKPcl2}) is
\begin{align*}
\widetilde{\mathcal{Q}}_2=&\,\tfrac{1}{3}D^{(c)}_m(U_{-1,0}+ U_{0,0}+ U_{1,0}),\\
\widetilde F_2=&\, (\tfrac{\alpha}{2}-\tfrac{\Delta x^2}{6})\left\{ (D^{(c)}_m\mu_mU_{-1,0})(D_tD_mD^{(c)}_mU_{-1,0})-(D_mD^{(c)}_mU_{-1,0})(D_tD^{(c)}_m\mu_mU_{-1,0})\right\}\\
& + \tfrac{1}{2}(D^{(c)}_m\mu_mU_{-1,0})(D^{(c)}_mU_{-1,0})(D^{(c)}_mU_{0,0})+\tfrac{1}{12} (D_m^3U_{-2,0})D_m(U_{-2,0}+ U_{-1,0}+ U_{0,0})\\
& - \tfrac{1}{24}\{ (D_m^2U_{-2,0})^2+(D_m^2U_{-2,0})(D_m^2U_{-1,0})+(D_m^2U_{-1,0})^2 \}\\
&+\tfrac{1}{8}\mu_m\{ U_{-2,0}D_n^2U_{0,-1}+U_{0,0}D_n^2U_{-2,-1} \}+\tfrac{1}{16}\{ U_{-1,0}D_n^2U_{0,-1}+U_{0,0}D_n^2U_{-1,-1} \},\\
\widetilde G_2=&\,\tfrac{1}{8}(D_nU_{0,-1})D^{(c)}_m\mu_n(U_{-1,-1}+ U_{0,-1}+ U_{1,-1})-\tfrac{1}{8}(\mu_nU_{0,-1})D^{(c)}_mD_n(U_{-1,-1}+ U_{0,-1}+ U_{1,-1}),\\
\widetilde H_2=&\,\tfrac{1}{2}(D^{(c)}_mU_{0,0})^2+\{\tfrac{\alpha}{6}D^{(c)}_m(U_{-1,0}+ U_{1,0})+\tfrac{\alpha+\Delta x^2}{6}D^{(c)}_mU_{0,0}\} D_m^2D^{(c)}_mU_{-1,0}.
\end{align*}
The second family has $\widetilde{G}_1$ and $\widetilde{H}_1$ as above, together with
\begin{align}\label{MC2pKP}
\widetilde F_1= (D^{(c)}_mU_{-1,0})(D^{(c)}_mU_{0,0})-\tfrac{1}{4}(D_mU_{-1,0})^2 + \tfrac{1}{4}D_m^3U_{-2,0}.
\end{align}
The semidiscrete version of (\ref{pKPcl2}) is
\begin{align*}
\widetilde{\mathcal{Q}}_2=&\,\tfrac{1}{2}D^{(c)}_m(U_{-1,0}+ U_{1,0}),\\
\widetilde F_2=&\, (\tfrac{\alpha}{2}-\tfrac{\Delta x^2}{4})\left\{ (D^{(c)}_m\mu_mU_{-1,0})(D_tD_mD^{(c)}_mU_{-1,0})-(D_mD^{(c)}_mU_{-1,0})(D_tD^{(c)}_m\mu_mU_{-1,0})\right\}\\
& + \tfrac{1}{4}(D^{(c)}_mU_{-1,0})(D^{(c)}_mU_{0,0})D_m(U_{-2,0}+U_{0,0})+\tfrac{1}{8} (D_m^3U_{-2,0})D_m(U_{-2,0}+ U_{0,0})\\
& - \tfrac{1}{16}\{ (D_m^2U_{-2,0})^2+(D_m^2U_{-1,0})^2 \}+\tfrac{3}{16}\mu_m\{ U_{-2,0}D_n^2U_{0,-1}+U_{0,0}D_n^2U_{-2,-1} \},\\
\widetilde G_2=&\,\tfrac{3}{16}(D_nU_{0,-1})D^{(c)}_m\mu_n(U_{-1,-1}+ U_{1,-1})-\tfrac{3}{16}(\mu_nU_{0,-1})D^{(c)}_mD_n(U_{-1,-1}+ U_{1,-1}),\\
\widetilde H_2=&\,\tfrac{1}{2}(D^{(c)}_mU_{0,0})^2+\{\tfrac{\alpha}{4}D^{(c)}_m(U_{-1,0}+ U_{1,0})+\tfrac{\Delta x^2}{4}D^{(c)}_mU_{0,0}\} D_m^2D^{(c)}_mU_{-1,0}.
\end{align*}
For both families, $\alpha=O(\Delta x^2,\Delta y^2).$ Let MC$_1(\alpha)$ and MC$_2(\alpha)$ denote the two families of fully-discrete schemes obtained by applying implicit midpoint in time to (\ref{methpKP}) with (\ref{MC1pKP}) and (\ref{MC2pKP}), respectively.
\subsubsection*{Numerical test}
As a brief test of the above schemes for the pKP equation, we use the following travelling wave solution of (\ref{pKP}) \cite{BK}:
\begin{equation}\label{wave}
u(x,y,t)=2\tanh(x + y - \tfrac{7}4t+5) + 2.
\end{equation}
We apply methods MC$_1(0)$ and MC$_2(0)$ to the pKP equation on the domain $\Omega=[-0.5,0.5]\times [-10,10]\times [0,5]$ with initial and Dirichlet boundary conditions given by (\ref{wave}), using step lengths $\Delta x=0.01, \Delta y=0.2$ and $\Delta t=0.05$.
\begin{figure}
\begin{center}
\includegraphics[scale=0.47]{mpn101_f5_init.eps}
\includegraphics[scale=0.47]{mpn101_f5.eps}
\end{center}
\caption{Initial condition (left) and solution of MC$_1(0)$ at time $t=5$.}\label{pKPtest}
\end{figure}
Figure~\ref{pKPtest} shows the profile of the wave at the initial time and the numerical solution of MC$_1(0)$ at the final time $t=5$. Both MC$_1(0)$ and MC$_2(0)$ simulate the motion of the wave to the required accuracy, with a maximum absolute error in the solution of $3.40\times 10^{-4}$ and $3.41\times 10^{-4}$, respectively.
\section{Conclusions}\label{concl}
In this paper we have introduced a new approach to constructing finite difference approximations to a system of PDEs that preserve multiple conservation laws. This is based on discretising one dimension at a time, using semidiscrete Euler operators to find constraints that simplify the remaining symbolic computations. This is much cheaper than the approach introduced in \cite{FCHydon}, and can be iterated to apply to PDEs with more than two independent variables. We have proved that any symplectic Runge--Kutta method preserves local conservation laws with quadratic density and that the AVF method preserves the local conservation law of the energy under milder conditions than the skew-adjointness of the discrete operator $\widetilde{\mathcal{D}}$.
The new strategy has been applied to obtain methods that preserve either the momentum or the energy of the Boussinesq equation. These are obtained as families that depend on a number of free parameters. Numerical tests have shown that the new schemes are competitive with respect to other methods in the literature and confirmed their conservation properties.
Very accurate solutions can be obtained by selecting optimal parameter values. However, these values depend strongly on the choice of initial condition.
Finally, we have given an example that the new approach is practicable for PDEs with three independent variables, by finding two new families of schemes that preserve two conservation laws for the pKP equation.
\subsection*{Acknowledgements} The authors would like to thank the Isaac Newton Institute for Mathematical Sciences for support and hospitality during the programme {\em Geometry, Compatibility and Structure Preservation in Computational Differential Equations}, when work on this paper was undertaken. This work was supported by EPSRC grant number EP/R014604/1.
The authors are grateful to Dr. Pranav Singh (University of Bath) and the referees for their constructive suggestions which have helped to improve this paper.
| -67,085.824163 |
[
-2.76953125,
2.39453125
] | 24.867021 |
[
-2.625,
0.35546875,
-2.5703125,
-6.640625,
-1.3330078125,
9.5390625
] |
[
1.3916015625,
8.1484375,
-1.18359375,
4.18359375
] | 596 | 6,457 |
[
-3.501953125,
4.01171875
] | 34.093289 |
[
-5.69140625,
-4.53515625,
-5.20703125,
-2.5546875,
2.041015625,
13.203125
] | 0.787436 | 11.016154 | 28.662124 | 6.00109 |
[
1.7450129985809326
] | -40,114.622774 | 7.178721 | -65,804.633589 | 0.427157 | 6.237727 |
[
-1.9560546875,
-3.669921875,
-4,
-5.375,
2.2109375,
12.78125
] |
[
-5.3046875,
-2.5703125,
-2.544921875,
-1.951171875,
3.9140625,
5.51953125
] | |
BkiUaHLxK0-nUGYe3P7a
|
\section{Introduction}
We begin with a few definitions.
Suppose that $v,b,r$ and $k$ are positive integers such that
$2 \leq k < v$.
A \emph{$(v,b,r,k)$-incomplete block design} (or \emph{$(v,b,r,k)$-IBD})
is a pair $(X, \ensuremath{\mathcal{B}})$, where $X$ is a set of
$v$ \emph{points} and $\ensuremath{\mathcal{B}}$ is a collection (i.e., a multiset) of $b$ subsets of $X$ (called \emph{blocks}),
such that
\begin{enumerate}
\item every block contains exactly $k$ points, and
\item every point is contained in exactly $r$ blocks.
\end{enumerate}
It follows that $bk=vr$ in a $(v,b,r,k)$-IBD.
Here are a few definitions relating to additional properties that IBDs might satisfy.
\begin{itemize}
\item Suppose $k \mid v$.
A \emph{parallel class} in a $(v,b,r,k)$-IBD $(X, \ensuremath{\mathcal{B}})$ is a set of $v/k$ disjoint blocks.
$(X, \ensuremath{\mathcal{B}})$ is \emph{resolvable} if $\ensuremath{\mathcal{B}}$ can be partitioned into $r$ parallel classes.
\item
A $(v,b,r,k)$-IBD $(X, \ensuremath{\mathcal{B}})$ is \emph{simple} if $\ensuremath{\mathcal{B}}$ does not contain any
repeated blocks, i.e., $\ensuremath{\mathcal{B}}$ is a set.
\item
A $(v,b,r,k)$-IBD $(X, \ensuremath{\mathcal{B}})$ is \emph{trivial} if $\ensuremath{\mathcal{B}}$
consists of all $\binom{v}{k}$ possible $k$-subsets of points.
\end{itemize}
A \emph{$(v, b, r, k, \ensuremath{\lambda})$-balanced incomplete block
design} (or a \emph{$(v, b, r, k, \ensuremath{\lambda})$-BIBD}) is a $(v, b, r, k)$-IBD
in which every pair of points occurs in exactly $\ensuremath{\lambda}$ blocks.
It is necessarily the case that
$\ensuremath{\lambda} = r(k-1)/(v-1)$.
Sometimes the notation $(v, k, \ensuremath{\lambda})$-BIBD is used, since the
parameters $r$ and $b$ can be computed from $v,k$ and $\ensuremath{\lambda}$
(specifically, $r = \lambda (v-1)/(k-1)$ and $b = vr/k$).
Suppose that $\ensuremath{\lambda}, t,k,$ and $v$ are positive integers such that
$t \leq k < v$.
A \emph{$t$-$(v,k,\ensuremath{\lambda})$-design} is a pair $(X, \ensuremath{\mathcal{B}})$, where $X$ is a set of
$v$ \emph{points} and $\ensuremath{\mathcal{B}}$ is a collection (i.e., a multiset) of $k$-subsets of $X$ (called \emph{blocks}),
such that every subset of $t$ points occurs in exactly $\lambda$ blocks.
It is well-known that a subset of $j\leq t-1$ points occurs in $\lambda_j$ blocks in $\ensuremath{\mathcal{B}}$, where
\begin{equation}
\label{lambda-j.eq} \lambda_j = \frac{\lambda \binom{v-j}{t-j}}{\binom{k-j}{t-j}}.
\end{equation}
If we define $b = \lambda_0$ and $r = \lambda_1$, then a $t$-$(v,k,\ensuremath{\lambda})$-design
is a $(v,b,r,k)$-IBD. A $2$-$(v,k,\ensuremath{\lambda})$-design is the same thing as a $(v, k, \ensuremath{\lambda})$-BIBD.
Also, a $1$-$(v,k,r)$-design is equivalent to a $(v,vr/k,r,k)$-IBD. A resolvable $2$-$(v,2,1)$-design is equivalent to a \emph{one-factorization} of the complete graph
$K_v$. The parallel classes in this design are called \emph{one-factors}.
\subsection{Related Work}
There are some examples of related work, which we mention now.
There is an unpublished construction due to Lonz and Vanstone of
$3$-$(v,4,3)$-designs from resolvable $2$-$(v,2,1)$-designs (see \cite{JV86}).
They take all possible
unions of two blocks from a parallel class of a $2$-$(v,2,1)$-design; this
immediately yields a $3$-$(v,4,3)$-design. In \cite{PSV89},
this method was extended to show the existence of simple $3$-$(v,4,3)$-designs.
Recently, this approach was generalized by Jimbo, Kunihara, Laue and Sawa in \cite{JKLS11}.
The master design is
a resolvable $3$-$(v,k,\ensuremath{\lambda})$-design and the indexing design is a $3$-design on
$v/k$ points. The construction \cite{JKLS11} can also be applied when the master design is a
one-factorization (this is a trivial $2$-design that can be viewed as a
degenerate $3$-design with $\ensuremath{\lambda} = 0$).
We employ the same construction as in \cite{JKLS11}, except the master design is a resolvable
$2$-design instead of a $3$-design. We still end up with a
$3$-design provided that the other ingredient in the construction is
a $3$-design on $v/k$ points with block size $v/(2k)$. All of our constructions,
as well as the other constructions mentioned above, can be considered to be
applications of the Shrikhande-Raghavarao construction.
\subsection{Organization of the Paper}
In Section \ref{construction.sec}, we present the main construction and discuss when this
construction will yield (resolvable) $3$-designs.
Section \ref{nontrivial.sec} presents conditions that guarantee that the main construction will yield a nontrivial design.
In Section \ref{simple.sec},
we investigate the simplicity of the constructed designs. We give some conditions that will guarantee
that the constructed designs are simple. We also look at several specific constructed
3-designs and show by computer that they
are simple even though we cannot prove theoretically that they are.
Simple 3-designs for these parameters were not previously known to exist.
Finally, Section \ref{conclusion.sec} is a brief conclusion.
\section{The Construction}
\label{construction.sec}
We begin by describing a construction of Shrikhande and Raghavarao published in 1963 \cite{SR63}.
\begin{construction}
\label{main}
We need two ingredients for the construction:
\begin{enumerate}
\item
Let $(X, \ensuremath{\mathcal{B}})$ be a resolvable $(v, b, r, k)$-IBD and let $\Pi_1, \ldots, \Pi_r$
denote the parallel classes in the resolution of $(X, \ensuremath{\mathcal{B}})$.
There are $w = v/k$ blocks in each parallel class.
Let the blocks in $\Pi_i$ be named $B_i^j$, $1 \leq j \leq w$.
We will call $(X,\ensuremath{\mathcal{B}})$ the \textbf{master design}.
\item
Suppose $(Y,\ensuremath{\mathcal{C}})$ is a $(w, b', r', k')$-IBD,
where $Y = \{ 1, \dots , w\}$.
We will call $(Y,\ensuremath{\mathcal{C}})$ the \textbf{indexing design}.
\end{enumerate}
Now, for each $i$, $1 \leq i \leq r$, and for each $C \in \ensuremath{\mathcal{C}}$,
define
\[ D_{i,C} = \bigcup _{j \in C} B_i^j .\]
That is, for every block $C$ of the indexing design and for every parallel class $\Pi_i$ of
the master design, we construct a block $D_{i,C}$ by taking the union of the blocks in $\Pi_i$ indexed
by $C$. Finally, let \[ \ensuremath{\mathcal{D}} = \{ D_{i,C} : 1 \leq i \leq r, C \in \ensuremath{\mathcal{C}} \}.\]
$(X, \ensuremath{\mathcal{D}})$ is the \textbf{constructed design}.
\end{construction}
We will study the design $(X,\ensuremath{\mathcal{D}})$ obtained from
Construction \ref{main}.
The following easily proven result was first shown in \cite{SR63}.
\begin{theorem}
\label{ibd}
Suppose that $(X, \ensuremath{\mathcal{B}})$ is a resolvable $(v,b,r,k)$-IBD,
and suppose $(Y,\ensuremath{\mathcal{C}})$
is a $(w, b', r', k')$-IBD, where $w = v/k$.
Let $(X, \ensuremath{\mathcal{D}})$ be defined as in Construction \ref{main}.
Then the constructed design $(X, \ensuremath{\mathcal{D}})$ is
a $(v, b'', r'', k'')$-IBD, where
\begin{eqnarray*}
b'' &=& r b',\\
r'' &=& r r', \mbox{and} \\
k'' &=& k k'.
\end{eqnarray*}
\end{theorem}
If the two input designs are BIBDs, then
so is the constructed design. The following result is also from \cite{SR63}.
\begin{theorem}
\label{bibd}
Suppose that $(X, \ensuremath{\mathcal{B}})$ is a resolvable $(v,k,\ensuremath{\lambda})$-BIBD,
and suppose $(Y,\ensuremath{\mathcal{C}})$
is a $(w,k',\ensuremath{\lambda}'_2)$-BIBD, where $w = v/k$.
Let $(X, \ensuremath{\mathcal{D}})$ be defined as in Construction \ref{main}.
Then the constructed design $(X, \ensuremath{\mathcal{D}})$ is
a $(v, b'', r'', k'', \lambda'')$-BIBD, where $b'', r''$ and $k''$ are
the same as in Theorem \ref{ibd}, and
\[ \lambda'' = \lambda r' + (r- \lambda)\lambda_2'.\]
\end{theorem}
We are interested in determining conditions
which guarantee that the constructed design is a $3$-design. For this
analysis, we will assume that the indexing design $(Y,\ensuremath{\mathcal{C}})$
is a $3$-$(w,k',\ensuremath{\lambda}')$-design, which requires $k' \geq 3$.
Let $x,y,z \in X$ be three distinct points and suppose these points occur in $\alpha$ blocks of
the master design $(X, \ensuremath{\mathcal{B}})$.
There are $3(\lambda - \alpha)$ parallel classes in which exactly
two points of $\{x, y, z\}$ occur in a block, and $r - \alpha - 3(\lambda - \alpha)$
parallel classes in which $x$, $y$, and $z$ occur in different blocks.
For a set of three points $x,y,z \in X$, let $\lambda_{x,y,z}$ denote the number of
blocks in the constructed design $\ensuremath{\mathcal{D}}$ that contain $x,y$ and $z$.
The following lemma is obtained by simple counting.
\begin{lemma}
\label{lem1}
Suppose that $(X, \ensuremath{\mathcal{B}})$ is a resolvable $(v,k,\ensuremath{\lambda})$-BIBD and $(Y,\ensuremath{\mathcal{C}})$
is a $3$-$(w,k',\ensuremath{\lambda}')$-design where $w=v/k$.
Let $(X, \ensuremath{\mathcal{D}})$ be defined as in Construction \ref{main}.
Suppose that three points
$x,y,z$ occur in exactly $\alpha$ blocks in $\ensuremath{\mathcal{B}}$. Then,
\begin{align}\label{lambda.3.2}
\lambda_{x,y,z} = \alpha r' + 3 \left (\lambda-\alpha \right )\lambda_2' + \left (r-\alpha -3(\lambda-\alpha) \right)\lambda'.
\end{align}
\end{lemma}
\begin{theorem}
\label{thm1}
Suppose that $(X, \ensuremath{\mathcal{B}})$ is a resolvable $(v,k,\ensuremath{\lambda})$-BIBD and $(Y,\ensuremath{\mathcal{C}})$
is a $3$-$(w,k',\ensuremath{\lambda}')$-design where $w=v/k$.
Let $(X, \ensuremath{\mathcal{D}})$ be defined as in Construction \ref{main}.
Then $(X, \ensuremath{\mathcal{D}})$ is a $3$-design if and only if one of the following conditions is
satisfied:
\begin{enumerate}
\item $(X, \ensuremath{\mathcal{B}})$ is a $3$-design,
\item $k=2$, or
\item $k' = v/(2k)$.
\end{enumerate}
\end{theorem}
\begin{proof}
Since $k' > 2$, if we substitute (\ref{lambda-j.eq}) into (\ref{lambda.3.2}), we see that
\begin{equation}
\lambda_{x,y,z} = \lambda' \left( 3\lambda \frac{w-2}{k'-2} + r-3\lambda \right)
+ \lambda' \alpha \left(\frac{(w-1)(w-2)}{(k'-1)(k'-2)} - 3 \frac{w-2}{k'-2} +2 \right).
\label{lambda.3.2.1}
\end{equation}
Therefore $\lambda_{x,y,z}$
is of the form $c_1 + \alpha c_2$, for constants $c_1$ and $c_2$.
Clearly, the design $(X, \ensuremath{\mathcal{D}})$ is a $3$-design if and only if $\alpha$ is a constant or $c_2 = 0$.
There are three possible cases that can occur:
\begin{enumerate}
\item If $\alpha>0$ is a constant, this implies that the master design $(X, \ensuremath{\mathcal{B}})$ is a $3$-design.
In this situation $(X, \ensuremath{\mathcal{D}})$
will be a $3$-design for any choice of $k' > 2$.
\item If $\alpha=0$, then it must be the case that $k=2$
(here, the master design can be thought of as a ``degenerate'' $3$-design). Again,
$(X, \ensuremath{\mathcal{D}})$
will be a $3$-design for any choice of $k' > 2$.
\item
We now use (\ref{lambda.3.2.1}) to derive the conditions that yield $c_2 = 0$.
Because $k' > 2$,
we have
\begin{align*}
& \frac{(w-1)(w-2)}{(k'-1)(k'-2)} - 3 \frac{w-2}{k'-2} +2 = 0 \\
\iff & (w-1)(w-2) - 3(w-2)(k'-1) + 2(k'-1)(k'-2) = 0 \\
\iff & w^2 - 3wk' + 2(k')^2 = 0 \\
\iff & (w-k')(w-2k') = 0.
\end{align*}
That is, $c_2=0$ when $k' = v/k$ or $k' = v/(2k)$. Note that if $k' = v/k$,
then $(X, \ensuremath{\mathcal{D}})$ is not a $3$-design since each block in $\ensuremath{\mathcal{B}}$ would consist of all the points of $X$. So
$(X, \ensuremath{\mathcal{D}})$ is a $3$-design if and only if $k' = v/(2k)$,
in which case $2k \mid v$.
\end{enumerate}
\end{proof}
The first case in Theorem \ref{thm1}, where the master design is a $3$-design, is the case that was studied
in \cite{JKLS11}. The second case, where $k=2$, was also studied
in \cite{JKLS11}.
We will study the third case in more detail. First, we record the designs
that are constructed in this case.
\begin{corollary}
\label{cor}
Suppose the following designs exist:
\begin{enumerate}
\item a resolvable $(v, b, r, k, \lambda)$-BIBD, and
\item a $3$-$(w,w/2,\lambda')$ design, where $w = v/k >4$ is even.
\end{enumerate}
Then there exists a $3$-$(v,v/2, \mu)$ design, where
\begin{equation}
\label{lambda3} \mu =
\lambda' \left( \frac{3\lambda w}{w-4} + r \right)\end{equation}
\end{corollary}
\begin{proof}
The value of $\mu$ can be obtained from (\ref{lambda.3.2.1}), where the coefficient of
$\alpha$ is equal to $0$. We have
\[ \mu =
\lambda' \left( 3\lambda \frac{w-2}{\frac{w}{2}-2} + r-3\lambda \right) =
\lambda' \left( \frac{3\lambda w}{w-4} + r \right).\]
\end{proof}
Theorem \ref{thm1} requires that $k' > 2$.
It is also useful to consider the variant
where $k' = 2$; in this case, we can take $(Y,\ensuremath{\mathcal{C}})$ to be a (trivial) $2$-$(w,2,1)$-design.
As in Lemma \ref{lem1}, suppose that
three points
$x,y,z$ occur in exactly $\alpha$ blocks in $\ensuremath{\mathcal{B}}$.
From (\ref{lambda.3.2}) with $\lambda'=0$, we have that
\begin{align}\label{lambda.3.1}
\lambda_{x,y,z} = \alpha (w-1) + 3 \left (\lambda-\alpha \right ) = \alpha (w-4) + 3 \lambda.
\end{align}
If $\alpha$ is not constant, it is clear that $\lambda_{x,y,z}$ will be constant if and only if
$w=4$. In this case, the indexing design is a $2$-$(4,2,1)$-design, which is
a one-factorization of $K_4$.
Therefore, we have the following variant of Corollary \ref{cor}.
\begin{corollary}
\label{corw4}
Suppose there is a resolvable $(v, b, r, k, \lambda)$-BIBD, where
$v/k = 4$. Then there is a $3$-$(v,v/2,3\lambda)$-design.
\end{corollary}
\subsection{Resolvability of the Constructed Designs}
In this section, we consider when the constructed design will be resolvable.
This question is easy to answer; we state the following simple result without proof.
\begin{lemma}
The constructed design $(X, \ensuremath{\mathcal{D}})$ in Construction \ref{main} is resolvable whenever the
indexing design $(Y,\ensuremath{\mathcal{C}})$ is resolvable.
\end{lemma}
However, if we look at Corollaries \ref{cor} and \ref{corw4}, we can proceed in
a slightly different manner. Both of these corollaries yield $3$-designs with block
size $v/2$. A result of Tran \cite[Theorem 2.7]{Tr01} shows that a
resolvable $3$-$(v,v/2, \mu)$-design exists whenever a $3$-$(v,v/2, \mu)$-design exists.
Thus we have the following extensions of Corollaries \ref{cor} and \ref{corw4}.
\begin{corollary}
\label{existence}
Suppose the following designs exist:
\begin{enumerate}
\item a resolvable $(v, b, r, k, \lambda)$-BIBD, and
\item a $3$-$(w,w/2,\lambda')$ design, where $w = v/k >4$ is even.
\end{enumerate}
Then there exists a resolvable $3$-$(v,v/2, \mu)$ design, where
\[\mu = \lambda' \left( \frac{3\lambda w}{w-4} + r \right).\]
\end{corollary}
\begin{corollary}
\label{existence2}
Suppose there is a resolvable $(v, b, r, k, \lambda)$-BIBD where
$v/k = 4$. Then there is a resolvable $3$-$(v,v/2,3\lambda)$-design.
\end{corollary}
\section{Nontriviality of the Constructed Designs}
\label{nontrivial.sec}
It is easy to see that Construction \ref{main} will often
produce nontrivial designs. Here is a small example to illustrate.
\begin{example}
{\rm Suppose we start with the trivial $(18,816,136,3)$-IBD consisting of all the
$3$-subsets of an $18$-set. This design
is resolvable into $r = 136$ parallel classes by Baranyi's Theorem. Apply Construction \ref{main} where the indexing
design is a $(6,20,10,3)$-IBD consisting of all the
$3$-subsets of a $6$-set.
The constructed design contains $2720$ blocks.
However, a trivial IBD with $18$ points and block size $9$
contains $\binom{18}{9} = 48620$ blocks. Therefore the constructed design is
nontrivial.
\qed
}
\end{example}
More generally, we will show that the constructed design
will be nontrivial whenever the master design and indexing design
are both \emph{simple} designs.
This is because a trivial design with block size $v/2$ contains $\binom{v}{v/2}$ blocks,
and the number of blocks in the constructed design will turn out to be smaller than that.
Suppose $2k$ divides $v$ and $v/k > 4$.
We apply Construction \ref{main} with the master design being a simple
design. The number of blocks in the master design is at
most $\binom{v}{k}$ and therefore $r \leq \binom{v-1}{k-1}$.
Since the indexing design is simple, it follows that the number of blocks in the constructed design
is at most
\[ r \binom{\frac{v}{k}}{\frac{v}{2k}} \leq \binom{v-1}{k-1} \binom{\frac{v}{k}}{\frac{v}{2k}}.\]
If we can prove the inequality
\begin{equation}
\label{NT.eq}
\binom{v-1}{k-1} \binom{\frac{v}{k}}{\frac{v}{2k}} < \binom{v}{\frac{v}{2}},
\end{equation}
then we will have shown that the constructed design is nontrivial. To do this, we will make use of the
well-known inequalities
\begin{equation}
\label{bin1.eq}
\binom{v-1}{k-1} < \left( \frac{v}{k} \right)^{k-1}
\end{equation}
and
\begin{equation}
\label{bin2.eq}
\frac{4^n}{2 \sqrt{n}} < \binom{2n}{n} < \frac{4^n}{\sqrt{\pi n}}.
\end{equation}
From (\ref{bin1.eq}) and (\ref{bin2.eq}), it is clear that
(\ref{NT.eq}) will hold provided that we can prove that the following inequality holds:
\begin{equation}
\label{NT2.eq}
\left( \frac{v}{k} \right)^{k-1} \frac{4^{v/2k}}{\sqrt{\pi (v/2k)}} < \frac{4^{v/2}}{2 \sqrt{v/2}}.
\end{equation}
After some simplification, it is easy to see that (\ref{NT2.eq}) is equivalent to
\begin{equation}
\label{NT3.eq}
\left( \frac{v}{k} \right)^{k-1} < 2^{v(k-1)/k} \sqrt{\frac{\pi}{4k}}.
\end{equation}
The inequality (\ref{NT3.eq}) can be rewritten as
\begin{equation}
\label{NT4.eq}
\left( \frac{2^{v/k}}{ \frac{v}{k}} \right)^{k-1} > \sqrt{\frac{4k}{\pi}}.
\end{equation}
Since $v/k \geq 4$, the left side of (\ref{NT4.eq}) is at least $4^{k-1}$.
Since $k \geq 2$, it is easy to see that $4^{k-1}\geq 2k$.
Finally it is obvious that $2k > \sqrt{\frac{4k}{\pi}}$.
Therefore (\ref{NT4.eq}) holds, and we have proven the following.
\begin{theorem}
\label{nontrivial}
Suppose that $2k$ divides $v$ and $v/k \geq 4$.
Suppose we apply Construction \ref{main} where the master and indexing designs
are both simple designs.
Then the constructed design is nontrivial.
\end{theorem}
\section{Simple Designs}
\label{simple.sec}
In certain situations, Corollaries \ref{cor} and \ref{corw4} yield simple designs.
This is the theme we pursue in this section.
Suppose $(X, \ensuremath{\mathcal{B}})$ is a resolvable $(v, b, r, k)$-IBD and let $w = v/k$.
Suppose the assumed resolution of $(X, \ensuremath{\mathcal{B}})$ consists of
$r$ parallel classes $\Pi_1, \ldots, \Pi_r$,
where $\Pi_i = \{ B_i^1, \dots , B_i^w\}$ for $1 \leq i \leq r$.
Let $1 \leq \alpha \leq w-1$ be an integer.
We say that two parallel classes $\Pi_i$ and $\Pi_j$ satisfy the
\emph{$\alpha$-partial replacement property} (or \emph{PRP}) if
there exist two parallel classes $\Pi_i^*$ and $\Pi_j^*$ of blocks in $\ensuremath{\mathcal{B}}$
such that
\begin{equation}
\label{PRP-eq}
\Pi_i^* \cup \Pi_j^* = \Pi_i \cup \Pi_j \quad \text{and} \quad
|\Pi_i^* \cap \Pi_i| = \alpha.
\end{equation} We say that the resolution of $(X, \ensuremath{\mathcal{B}})$
is \emph{$\alpha$-PRP-free} if there do not exist two parallel classes
in $\{ \Pi_1, \ldots, \Pi_r \}$
that satisfy $\alpha$-PRP. Further, we say that the resolution of $(X, \ensuremath{\mathcal{B}})$
is \emph{PRP-free} if there do not exist two parallel classes
in $\{ \Pi_1, \ldots, \Pi_r \}$
that satisfy $\alpha$-PRP for any positive integer $\alpha < w$.
As an example of a resolvable BIBD that is \emph{not} PRP-free,
consider a one-factorization of $K_{4n}$ that contains a
sub-one-factorization of $K_{2n}$. Each one-factor of a $K_{4n}$ has $2n$ edges
and each one-factor of a $K_{2n}$ contains $n$ edges.
There are $2n-1$ one-factors of the $K_{4n}$
that contain a one-factor of the $K_{2n}$. The $n$ edges in any one-factor of
$K_{2n}$ may be swapped with the $n$ edges in any other one-factor of $K_{2n}$.
Therefore, the two corresponding one-factors in the one-factorization of $K_{4n}$ satisfy $n$-PRP.
\begin{lemma}
\label{unique}
If a $(v,b,r,k)$-IBD has a unique resolution, then it is PRP-free.
\end{lemma}
\begin{proof} Let $w = v/k$. Suppose there are two parallel classes $\Pi_i$ and $\Pi_j$
that satisfy $\alpha$-PRP for some $\alpha$, where $1 \leq \alpha \leq w-1$.
Let $\Pi_i^*$ and $\Pi_j^*$ be the two parallel classes that
satisfy (\ref{PRP-eq}). If we replace $\Pi_i$ and $\Pi_j$ by $\Pi_i^*$ and $\Pi_j^*$,
then we obtain a second resolution of the given design.
\end{proof}
\begin{theorem}
\label{PRP}
Suppose we apply Construction \ref{main} with the master design having a resolution
that is $k'$-PRP-free, where the indexing design has block size $k'$.
Then the constructed design is simple.
Furthermore, in the case where the indexing design is trivial, the constructed design is
simple if and only if the resolution of the master design is $k'$-PRP-free.
\end{theorem}
\begin{proof}
Suppose that $(X,\ensuremath{\mathcal{D}})$ is not simple; then $D_{i,C} = D_{i',C'}$
where $(i,C) \neq (i',C')$. If $i = i'$ and $D_{i,C} = D_{i',C'}$,
then $C = C'$. Therefore, we can assume $i \neq i'$.
If we define
\[ \Pi_i^* = \{ B_i^j : j \in C\} \cup \{ B_{i'}^j : j \not\in C'\} \] and
\[ \Pi_{i'}^* = \{ B_{i'}^j : j \in C'\} \cup \{ B_{i}^j : j \not\in C\} ,\]
then we see that
the parallel classes $\Pi_i$ and $\Pi_{i'}$ in $(X,\ensuremath{\mathcal{B}})$ satisfy $k'$-PRP.
Now, assume that the indexing design is trivial and there are two parallel classes
$\Pi_i$ and $\Pi_{i'}$ in the resolution of $(X,\ensuremath{\mathcal{B}})$ that satisfy $k'$-PRP. So there exist
$\Pi_i^*$ and $\Pi_{i'}^*$ such that
\[\Pi_i^* \cup \Pi_{i'}^* = \Pi_i \cup \Pi_{i'} \quad \text{and} \quad
|\Pi_i^* \cap \Pi_i| = k'.\]
Let $C = \{j: B_i^j \in \Pi_i^* \cap \Pi_i\}$ and $C' = \{j: B_{i'}^j \in \Pi_{i'}^* \cap \Pi_{i'}\}$.
Note that $C$ and $C'$ are both blocks in the indexing design, because the indexing design is trivial.
Then it is easy to see that $D_{i,C} = D_{i',C'}$, so the constructed design is not simple.
\end{proof}
We give a well-known class of BIBDs having unique resolutions.
\begin{lemma}
\label{ARBIBD}
For all prime powers $q$ and for all integers $m \geq 2$, there exists
a PRP-free resolvable $\left( q^m, q^{m-1}, (q^{m-1}-1) / (q-1) \right)$-BIBD.
\end{lemma}
\begin{proof}
The hyperplanes of the
$m$-dimensional affine geometry $\mathsf{AG}(m,q)$ %
over $\mathbb{F}_{q}$ yield a
resolvable $\left( q^m, q^{m-1}, (q^{m-1}-1) / (q-1) \right)$-BIBD
where each parallel class consists of $q$ blocks.
Furthermore, any two blocks from different parallel classes intersect in
exactly $q^{m-2}$ points. From this fact, it is easily seen that this design has a unique resolution.
Hence, from Lemma \ref{unique}, the resolution is PRP-free.
\end{proof}
\begin{remark}
Any affine resolvable BIBD has a unique resolution and therefore is PRP-free.
\end{remark}
\begin{corollary}
Suppose $q = 2^n > 4$ and there exists
a $3$-$(q,q/2,\lambda')$ design.
Then there exists a simple $3$-$(q^m,q^{m-1}, \mu)$ design, where
\[ \mu =
\lambda' \left(\frac{q^m-4}{q-4}\right) .\]
\end{corollary}
\begin{proof}
We apply Theorem \ref{PRP},
starting with the resolvable $\left( q^m, q^{m-1}, (q^{m-1}-1) / (q-1) \right)$-BIBD
from Lemma \ref{ARBIBD}. This design has
$r = (q^{m}-1) / (q-1)$ and $w = q$. Corollary \ref{cor} establishes that the constructed design will
be a $3$-$(q^m,q^{m-1}, \mu)$-design, where
\begin{eqnarray*}
\mu &=&\lambda' \left( \frac{3\lambda w}{w-4} + r \right)\\
& = & \lambda' \left( \frac{3(q^{m-1}-1)}{q-1} \times \frac{q}{q-4} + \frac{q^{m}-1}{q-1} \right)\\
& = & \lambda' \left( \frac{3(q^{m}-q) + (q^{m}-1)(q-4)}{(q-1)(q-4)} \right)\\
& = & \lambda' \left( \frac{q^{m+1} - q^m - 4q + 4}{(q-1)(q-4)} \right)\\
& = & \lambda' \left( \frac{q^m - 4}{q-4} \right).
\end{eqnarray*}
Finally, Theorem \ref{PRP} shows that the constructed design is simple.
\end{proof}
As an example, suppose we take $q = 8$ and $m=2$. Here, the master design is the affine plane of order
$8$, i.e., a resolvable $(64,72,9,8,1)$-BIBD. If the indexing design is
a $3$-$(8,4,1)$-design, then the constructed
$3$-$(64,32,15)$-design is simple. If the indexing design is the trivial
$3$-$(8,4,5)$-design, then the constructed
$3$-$(64,32,75)$-design is again simple.
It also turns out that many $3$-designs constructed from Corollary
\ref{existence} are simple, even if we
cannot prove theoretically that they are. In fact, we have checked several
designs by computer and verified computationally that they are simple $3$-designs.
These designs all arise from trivial
indexing designs. They are listed in Table \ref{tab2} and details are provided in the rest of this section.
\begin{table}
\caption{Simple $3$-designs constructed from Corollaries
\ref{cor} and \ref{corw4}}
\label{tab2}
\begin{center}
\begin{tabular}{c|c|c}
master design & $w$ & constructed design\\ \hline
$2$-$(24,4,3)$ & 6 & $3$-$(24,12,50)$\\
$2$-$(24,6,5)$ & 4 & $3$-$(24,12,15)$\\
$2$-$(24,3,2)$ & 8 & $3$-$(24,12,175)$\\
$2$-$(28,7,6)$ & 4 & $3$-$(28,14,18)$\\
$2$-$(30,5,4)$ & 6 & $3$-$(30,15,65)$\\
$2$-$(30,3,2)$ & 10 & $3$-$(30,15,819)$\\
$2$-$(32,8,7)$ & 4 & $3$-$(32,16,21)$\\
$2$-$(36,9,8)$ & 4 & $3$-$(36,18,24)$
\end{tabular}
\end{center}
\end{table}
\subsection{A Simple $3$-$(24,12,15)$-design}
We construct a simple $3$-$(24,12,15)$-design on
point set $X=\{0,1, \dots, 22, \infty\}$. The cyclic group $\mathbb{Z}_{23}$
acts on the design. $\mathbb{Z}_{23}$ permutes
the points $\{0,1, \dots ,22\}$ (cyclically) and fixes the point $\infty$.
We start with a resolvable $(24,6,5)$-BIBD having a cyclic
automorphism of order 23 (see \cite[p.\ 408]{CD06}).
The four base blocks of the BIBD form a parallel class:
\[ \{\infty,0,1,7,15,20\}, \{2,3,4,5,10,14\},
\{6,11,13,17,19,22\}, \{8,9,12,16,18,21\} .\]
Here are the six base blocks of the constructed $3$-$(24,12,15)$ design:
\[
\begin{array}{l} B_1=\{\infty,0,1,7,15,20, 2,3,4,5,10,14\}\\
B_2=\{\infty,0,1,7,15,20, 6,11,13,17,19,22\}\\
B_3=\{\infty,0,1,7,15,20, 8,9,12,16,18,21\}\\
B_4=\{2,3,4,5,10,14, 6,11,13,17,19,22\}\\
B_5=\{2,3,4,5,10,14, 8,9,12,16,18,21\}\\
B_6=\{6,11,13,17,19,22, 8,9,12,16,18,21\}.
\end{array}
\]
The other blocks are obtained by developing the base blocks through $\mathbb{Z}_{23}$.
The next line gives data about intersection numbers of the design:
\[69 , 0 , 46 , 0, 506 ,2208, 3864 ,2208 ,506 , 0 ,46 , 0 , 0.\]
There are 13 numbers corresponding to 13 possible intersection numbers,
namely $0,1,2, \dots, 12$.
For example, 69 pairs of blocks have zero intersection; they form
the parallel classes of the design.
The second value equals 0, which says that there are no pairs of blocks
intersecting in 1 point.
The third value equals 46 pairs, which says that there are 46 pairs of blocks intersecting
in 2 points, etc.
The data show that any two blocks of the design intersect in at most
10 points, so the design is simple.
\subsection{A Simple $3$-$(28,14,18)$-design}
We construct a simple $3$-$(28,14,18)$-design on
point set $X=\{0,1, \dots, 26, \infty\}$. The cyclic group $\mathbb{Z}_{27}$
acts on the design. $\mathbb{Z}_{27}$ permutes
the points $\{0,1, \dots ,26\}$ (cyclically) and fixes the point $\infty$.
We start with a resolvable $(28,7,6)$-BIBD having a cyclic
automorphism of order 27 (see \cite[p.\ 408]{CD06}).
The four base blocks of the BIBD form a parallel class:
\[ \{\infty,0,5,14,15,24,25\}, \{1,10,17,20,22,23,26\},
\{2,6,8,9,13,19,21\}, \{3,4,7,11,12,16,18\} .\]
Here are the six base blocks of the constructed $3$-$(28,14,18)$ design:
\[
\begin{array}{l} B_1=\{\infty,0,5,14,15,24,25, 1,10,17,20,22,23,26\}\\
B_2=\{\infty,0,5,14,15,24,25, 2,6,8,9,13,19,21\}\\
B_3=\{\infty,0,5,14,15,24,25, 3,4,7,11,12,16,18\}\\
B_4=\{1,10,17,20,22,23,26, 2,6,8,9,13,19,21\}\\
B_5=\{1,10,17,20,22,23,26, 3,4,7,11,12,16,18\}\\
B_6=\{2,6,8,9,13,19,21, 3,4,7,11,12,16,18\}.
\end{array}
\]
The other blocks are obtained by developing the base blocks through $\mathbb{Z}_{27}$.
The next line gives the intersection numbers of the design:
\[81 , 0 , 0 , 0 , 54, 1080 ,3132, 4428 ,3132 ,1080, 54 , 0 , 0 , 0 , 0 .\]
This data shows that the design is simple.
\subsection{A Simple $3$-$(30,15,65)$-design}
We construct a simple $3$-$(30,15,65)$-design on
point set $X=\{0,1, \dots, 28, \infty\}$. The cyclic group $\mathbb{Z}_{29}$
acts on the design. $\mathbb{Z}_{29}$ permutes
the points $\{0,1, \dots ,28\}$ (cyclically) and fixes the point $\infty$.
We start with a resolvable $(30,5,4)$-BIBD having a cyclic
automorphism of order 29 (see \cite[p.\ 408]{CD06}).
The six base blocks of the BIBD form a parallel class:
\[ \begin{array}{l} \{\infty,4,8,17,18\}, \{0,1,7,24,27\},
\{2,12,15,16,23\},\\ \{9,11,13,21,26\},
\{3,14,20,22,25\}, \{5,6,10,19,28\} .\end{array}\]
Here are the 20 base blocks of the constructed $3$-$(30,15,65)$ design:
\[
\begin{array}{l}
B_1=\{\infty,4,8,17,18, 0,1,7,24,27, 2,12,15,16,23\} \\
B_2=\{\infty,4,8,17,18, 0,1,7,24,27, 9,11,13,21,26\} \\
B_3=\{\infty,4,8,17,18, 0,1,7,24,27, 3,14,20,22,25\} \\
B_4=\{\infty,4,8,17,18, 0,1,7,24,27, 5,6,10,19,28\} \\
B_5=\{\infty,4,8,17,18, 2,12,15,16,23, 9,11,13,21,26\} \\
B_6=\{\infty,4,8,17,18, 2,12,15,16,23, 3,14,20,22,25\} \\
B_7=\{\infty,4,8,17,18, 2,12,15,16,23, 5,6,10,19,28\} \\
B_8=\{\infty,4,8,17,18, 9,11,13,21,26, 3,14,20,22,25\} \\
B_9=\{\infty,4,8,17,18, 9,11,13,21,26, 5,6,10,19,28\} \\
B_{10}=\{\infty,4,8,17,18, 3,14,20,22,25, 5,6,10,19,28\} \\
B_{11}=\{0,1,7,24,27, 2,12,15,16,23, 9,11,13,21,26\} \\
B_{12}=\{0,1,7,24,27, 2,12,15,16,23, 3,14,20,22,25\} \\
B_{13}=\{0,1,7,24,27, 2,12,15,16,23, 5,6,10,19,28\} \\
B_{14}=\{0,1,7,24,27, 9,11,13,21,26, 3,14,20,22,25\}
\end{array}\]
\[ \begin{array}{l}
B_{15}=\{0,1,7,24,27, 9,11,13,21,26, 5,6,10,19,28\} \\
B_{16}=\{0,1,7,24,27, 3,14,20,22,25, 5,6,10,19,28\} \\
B_{17}=\{2,12,15,16,23, 9,11,13,21,26, 3,14,20,22,25\} \\
B_{18}=\{2,12,15,16,23, 9,11,13,21,26, 5,6,10,19,28\} \\
B_{19}=\{2,12,15,16,23, 3,14,20,22,25, 5,6,10,19,28\} \\
B_{20}=\{9,11,13,21,26, 3,14,20,22,25, 5,6,10,19,28\}.
\end{array}
\]
The other blocks are obtained by developing the base blocks through $\mathbb{Z}_{29}$.
The next line gives the intersection numbers of the design:
\[290 , 0 , 0 , 58 , 1044 ,10382 ,24940 ,47386 ,47386 ,24940 ,10382 ,1044, 58 , 0 , 0 , 0.\]
This data shows that the design is simple.
\subsection{A Simple $3$-$(24,12,50)$-design}
A $3$-$(24,12,50)$-design
constructed from a resolvable $(24,4,3)$-BIBD having a cyclic
automorphism of order 23 (see \cite[p.\ 407]{CD06}).
The six base blocks of the BIBD form a parallel class:
\[ \{\infty, 0 ,7 ,10\}, \{ 1, 8 ,12, 22\}, \{2 ,5, 6 ,11\},
\{3 ,9 ,14, 18\}, \{4 ,16 ,17 ,19\}, \{13, 15, 20 ,21\} .\]
The intersection numbers of the $3$-$(24,12,50)$-design (from 0 to 12 respectively) are
\[230, 0 , 0 , 1242 , 9982 , 23414 , 36064 , 23414 , 9982 , 1242,
0 , 0 , 0.\]
\subsection{A Simple $3$-$(24,12,175)$-design}
A $3$-$(24,12,175)$-design
constructed from a resolvable $(24,3,2)$-BIBD having a cyclic
automorphism of order 23 (see \cite[p.\ 407]{CD06}).
The eight base blocks of the BIBD form a parallel class:
\[ \{ \infty ,16, 20\}, \{0, 7 ,21\}, \{1 ,3 ,11\}, \{4, 5, 18\}, \{6 ,12, 17\}, \{2 ,10, 13\},
\{8 ,9, 14\}, \{15, 19 ,22 \} .\]
The intersection numbers of the $3$-$(24,12,175)$-design (from 0 to 12 respectively) are:
\[ 805 , 46 , 1380 , 30314 , 99958 , 289432 ,452180 , 289432 , 99958 , 30314,
1380 , 46 , 0 .\]
\subsection{A Simple $3$-$(30,15,819)$-design}
A $3$-$(30,15,819)$-design
constructed from a resolvable $(30,3,2)$-BIBD having a cyclic
automorphism of order 29 (see \cite[p.\ 407]{CD06}).
The ten base blocks of the BIBD form a parallel class:
\[
\begin{array}{c}
\{ \infty, 2 ,22\}, \{ 0 ,1 ,19\}, \{ 6 ,8 ,27\}, \{ 9 ,15 ,16\}, \{ 7, 10 ,14\},\\
\{ 11 ,25 ,28\}, \{ 12 ,18 ,23\}, \{ 13, 21 ,26\}, \{ 4, 20 ,24\}, \{ 3, 5 ,17\}.
\end{array}
\]
The intersection numbers of the $3$-$(30,15,819)$ are
\[
\begin{array}{c}
3654 , 0 , 928 , 115594 , 242730 ,1356272, 4482530 ,7150008, \\
7150008, 4482530 , 1356272 , 242730 ,115594 , 928 , 0 , 0 .
\end{array}\]
\subsection{A Simple $3$-$(32,16,21)$-design}
A $3$-$(32,16,21)$-design
constructed from a resolvable $(32,8,7)$-BIBD having a cyclic
automorphism of order 31 (see \cite[p.\ 408]{CD06}).
The four base blocks of the BIBD form a parallel class:
\[ \begin{array}{c} \{\infty , 0, 1 , 5 ,16 ,18 ,25 ,28\}, \{3 , 9 ,17 ,19 ,20, 21 ,24 ,26\},\\
\{2 ,6, 7 ,12 ,14 ,15, 23 ,27\}, \{4, 8 ,10 ,11 ,13, 22 ,29, 30) \} .\end{array}\]
The intersection numbers of the $3$-$(32,16,21)$-design are
\[
93 , 0 , 0 , 0 , 0 , 372 , 930, 4836, 4836, 4836, 930 , 372 , 0 , 0 , 0, 0 , 0.\]
\subsection{A Simple $3$-$(36,18,24)$-design}
A $3$-$(36,18,24)$-design
constructed from a resolvable $(36,9,8)$-BIBD having a cyclic
automorphism of order 35 (see \cite[p.\ 408]{CD06}).
The four base blocks of the BIBD form a parallel class:
\[ \begin{array}{c} \{\infty ,1 ,10, 11, 13 ,15, 25, 26 ,29\}, \{0, 9, 16 ,17, 18 ,19, 22, 24 ,30\},\\
\{ 3 ,4 ,6 ,12 ,21, 23, 27, 31 ,34\}, \{2 ,5 ,7 ,8, 14 ,20 ,28, 32 ,33 \} .\end{array}\]
The intersection numbers of the $3$-$(36,18,24)$-design are
\[105 , 0 , 0 , 0 , 0 , 70 , 280, 2100, 4970, 7000, 4970 ,2100 , 280 ,70 , 0 , 0, 0 , 0, 0 .\]
\section{Conclusion}
\label{conclusion.sec}
There are probably many other possible applications of the constructions in this paper
to obtain (new) simple $3$-designs. It would be very nice to find additional easily checked
conditions that would guarantee that a constructed $3$-design is simple.
| -41,228.762705 |
[
-2.29296875,
2.021484375
] | 36.399474 |
[
-2.619140625,
1.6787109375,
-2.271484375,
-5.53515625,
-1.759765625,
7.625
] |
[
0.75537109375,
7.1953125,
0.41943359375,
6.48046875
] | 293 | 4,272 |
[
-2.96875,
3.4453125
] | 42.651508 |
[
-5.09375,
-3.427734375,
-4.05078125,
-2.234375,
1.447265625,
10.8671875
] | 0.657985 | 16.799005 | 20.786517 | 11.483203 |
[
2.0716464519500732
] | -27,637.593459 | 5.122893 | -40,456.431904 | 1.736349 | 5.725145 |
[
-1.7216796875,
-2.94921875,
-3.76953125,
-5.3515625,
2.0859375,
11.6328125
] |
[
-5.7734375,
-1.8037109375,
-2.044921875,
-1.478515625,
3.23046875,
3.927734375
] | |
BkiUc7w4eIXhwh1BwyuE
|
\section{Entanglement measures}
\emph{Linear entropy}.
The linear entropy of a pure state $\psi$ is given by $E_{\rm lin}(\psi)=2[({\rm tr}\,\rho_A)^ 2-{\rm tr}\,(\rho_A^2)]$, where $\rho_A = {\rm tr}\,_A \ketbrad{\psi}$. As for any
pure-state entanglement measure, this definition can be extended to mixed states via the convex roof~\cite{Uhlmann2010}
\begin{equation}\label{Elin_gen}
E_{\rm lin}(\rho) = \min_{\{p_k,\psi_k\}} \sum_k p_k E_{\rm lin}(\psi_k)\,,
\end{equation}
where the minimization is taken over all decompositions $\rho=\sum_k p_k \ketbrad{\psi_k}$ into pure states. While
finding the optimal decomposition
for an arbitrary $\rho$ is generally a daunting challenge, the problem becomes tractable for families of states that exhibit certain symmetries~\cite{Vollbrecht2001}. This is the case of states $\rho^\diamond$. Albeit the techniques that we use are applicable to the whole family $\rho^\diamond$, in this paper we focus our attention on the subfamily of states
\begin{equation}
\sigma\ =\ z \ketbrad{\Phi}\ +\
(1-z)\left[\frac{1+\bar{r}}{2} \rho^\diamond_+ + \frac{1-\bar{r}}{2}\rho^\diamond_-
\right] \,,
\label{eq:sigma}
\end{equation}
where $0\leqslant z\leqslant 1$ and $-1\leqslant \bar{r}\leqslant 1$. These are the states
that lie on the lower facet of the tetrahedron, that is, the
triangle enclosed by states $\Phi$, $\rho^\diamond_{+}$, and
$\rho^\diamond_{-}$
(cf.~Fig.~\ref{fig:tetrahedron}); we will extend the analysis to all states $\rho^\diamond$ elsewhere.
Recall that, due to the $\bar{r}\leftrightarrow -\bar{r}$
symmetry, it suffices to study the parameter range $\bar{r}\in [0,1]$.
The boundary between PPT and NPT states in the $(\bar{r},z)$ parameter space is given by
\begin{equation}
z_{\mathrm{boundary}}(\bar{r})\ =\
\frac{-1+\bar{r}^2+2\sqrt{1-\bar{r}^2}}
{3+\bar{r}^2}
\ \ .
\label{eq:sigma_PPTborder}
\end{equation}
We first calculate the linear entropy $E_{\rm lin}(\sigma)$ using the method of the convex characteristic curve \cite{Osterloh2008}, with $\sigma$
given in Eq.~\eqref{eq:sigma}.
The method is built upon the fact that the pure states in an optimal decomposition of $\rho$, i.e., one achieving the minimum in Eq.~\eqref{Elin_gen}, can be expressed as linear combinations of the elements of any other pure-state decomposition. Note that $\sigma$ is a rank-7 state, and that the (pure) elements $\psi_\sigma$ of its decompositions are restricted to live
within its span.
Thus an arbitrary state $\psi_\sigma$ can always be written as a linear combination of the vectors that span the facet, i.e.,
\begin{align}
\!\!\ket{\psi_\sigma} = \sqrt{z} \ket{\Phi} + \sqrt{1-z}
\left[ \sqrt{(1+\bar{r})/2} \left(a\ket{01}+b\ket{20} \right.\right. \nonumber\\
\left.+ c\ket{12}\right) + \left.\sqrt{(1-\bar{r})/2} \left(d\ket{10}+e\ket{02}+f\ket{21}\right) \right] \,, \label{psi_sigma}
\end{align}
where
$\bar{r},z$ are the parameters that specify the corresponding symmetrized state $\sigma$ [cf. Eq.~\eqref{eq:sigma}],
and $\{a,b,c,d,e,f\}\equiv\xi$ are
complex free parameters subject to the normalization constraints $|a|^2+|b|^2+|c|^2=|d|^2+|e|^2+|f|^2=1$ [hence the number of real free parameters in Eq.~\eqref{psi_sigma} is 12, as it corresponds to an arbitrary state of dimension 7].
The first part of the method consists in computing the characteristic curve for the linear entropy $\tilde{E}_{\rm lin}(\bar{r},z)=\min_\xi E_{\rm lin}(\psi_\sigma)$ at each point in the facet.
We start with the expression of the linear entropy for an arbitrary bipartite pure state $\psi$ in terms of the state coefficients $E_{\rm lin}(\psi) =\sum_{jklm} \left|\psi_{jk}\psi_{lm}-\psi_{jm}\psi_{lk}\right|^2$, where $\psi_{jk}=\braket{jk}{\psi}$ for some orthonormal basis $\{\ket{jk}\}$. For the state $\psi_\sigma$ defined in Eq.~\eqref{psi_sigma},
the linear entropy is a function of the family coordinates $\bar{r}$ and $z$, and it depends on the free complex parameters $\xi$, i.e., $E_{\rm lin}(\psi_\sigma)=E_{\rm lin}(\bar{r},z,\xi)$.
The explicit form of this function immediately reveals that the complex phases of $\xi$ only play a role inside cosine multiplicative factors. Fixing all of them to the value $\pi$ minimizes each cosine factor independently, thus it suffices to consider real parameters. Taking this into account, the characteristic curve reads
\begin{align}\label{supp:cc}
\tilde{E}_{\rm lin}(\bar{r},z)=& \min_{\xi\in\mathbb{R}} \; \frac{4}{3} \Bigl\{ z -z(1-z)\sqrt{1-\bar{r}^2}(ad+be+cf) \nonumber\\
&-\sqrt{3z/2}(1-z)^{3/2}\left[ (1+\bar{r})\sqrt{1-\bar{r}}\right. \nonumber\\
&\times (bcd+ace+abf) \nonumber\\
&\left. +\sqrt{1+\bar{r}}(1-\bar{r})(cde+bdf+aef)\right]\nonumber\\
&+\frac{3}{4}(1-z)^2\left[(1+\bar{r})^2(a^2b^2+b^2c^2+a^2c^2)\right. \nonumber\\
&+(1-\bar{r}^2)(a^2d^2+b^2e^2+c^2f^2)\nonumber\\
&\left.+(1-\bar{r})^2(d^2e^2+e^2f^2+d^2f^2)\right]\Bigr\} \,.
\end{align}
Note that there are four free parameters left to minimize over (recall the normalization conditions $a^2+b^2+c^2=d^2+e^2+f^2=1$). Furthermore, a numerical evaluation of Eq.~\eqref{supp:cc} shows that, for all values of $\bar{r}$ and $z$, the minimum is still achieved if one sets $c=a$ and $f=d$,
thus we can reduce the free parameters to only $\{b,e\}$.
Although this is a great simplification of the original problem, the minimization turns out to be highly nontrivial, with at least three different regimes of solutions. An analytical formula for $\tilde{E}_{\rm lin}(\bar{r},z)$ hence becomes out of reach, so we resort to a numerical nonlinear multiparameter constrained minimization. As we will discuss, in principle this approach presents a problem of reliability that we nonetheless are able to circumvent.
The computed $\tilde{E}_{\rm lin}$ describes a nonconvex surface over the $(\bar{r},z)$ domain. The second part of the method is to compute its convexification $\tilde{E}^c_{\rm lin}$, which is, by construction, a lower bound of the true linear entropy of a generic (non-symmetric) state $\rho\in {\rm span}\{\ket{\psi_\sigma}\}\equiv\mathcal{S}$ \cite{Osterloh2008}. More precisely, at each point $(\bar{r},z)$ in the facet, the equation $\tilde{E}^c_{\rm lin}(\bar{r},z) \leqslant E_{\rm lin}(\rho)$ holds, where ${\rm tr}\,\rho\Phi = z$ and ${\rm tr}\,\rho\rho^\diamond_{+}=(1+\bar{r})(1-z)/6$.
Here we go one step further and prove that, for a subset $\mathcal{K}\subset\mathcal{S}$ of states that are invariant under some entanglement-preserving symmetry, as the states $\sigma$ are, $\tilde{E}^c_{\rm lin}$ also represents an upper bound to their true linear entropy, hence we have the result
\begin{equation}\label{eq:ccc_is_exact}
\tilde{E}^c_{\rm lin}(\bar{r},z) = E_{\rm lin}(\sigma(\bar{r},z)) \, .
\end{equation}
That is, at this point, up to the reliability of our numerical calculation \footnote{Note that we have produced $\tilde{E}^c_{\rm lin}$ by means of a nonlinear constrained multiparameter minimization, for which no algorithm guarantees to find the global minimum. Were the minimization to fail at some of the explored coordinates $(\bar{r},z)$, the corresponding values $\tilde{E}^c_{\rm lin}(\bar{r},z)$ would upper bound the exact linear entropy.}, $\tilde{E}^c_{\rm lin}$ is the true linear entropy for the states on the facet.
\begin{proof}[\indent Proof of Eq.~\eqref{eq:ccc_is_exact}]
We now proceed to prove that Eq.~\eqref{eq:ccc_is_exact} holds, independently of the chosen entanglement measure. Let $E$ be an entanglement measure and $\bar{r}, z, \ldots$ parametrize
the fidelities ${\rm tr}\,(\rho \rho_j)=f_j(\bar{r},z,\ldots )$ of some arbitrary state $\rho$ with some fixed given set of states
$\{\rho_j\}$.
While in general the value of the convex characteristic curve
\begin{equation}
\tilde{E}^c(\bar{r},z,\ldots)\leqslant E(\rho)
\label{eq:lower}
\end{equation}
represents a lower bound to $E(\rho)$~\cite{Osterloh2008}, it
{\em coincides} with the true value $E(\sigma)$
\begin{equation}
\tilde{E}^c(\bar{r},z,\ldots)\ =\ E(\sigma(\bar{r},z,\ldots))
\label{eq:equal}
\end{equation}
if the states $\sigma$ and $\{\rho_j\}$
are invariant under a group $G$ of entanglement-preserving symmetries, that
is, if $g \sigma g^{-1}=\sigma$, $ g \rho_j g^{-1}=\rho_j$,
and also $E(g \rho g^{-1})=E(\rho)$ for $g\in G$ and any $\rho$.
We first consider the case that $\tilde{E}^c(\bar{r},z,\ldots)=
E(\psi^c_{\sigma}(\bar{r},z,\ldots))$, i.e., the minimum value $\tilde{E}^c(\bar{r},z,\ldots)$
is realized by a single (not necessarily symmetric under $G$) pure state
$\psi^c_{\sigma}(\bar{r},z,\ldots)$ from the span of $\sigma$
with the same fidelity parameters. Now $\sigma(\bar{r},z,\ldots)$
can be obtained from
$\psi^c_{\sigma}(\bar{r},z,\ldots)$ simply by symmetrization with
respect to $G$
\begin{equation}\label{supp:symmetrized}
\sigma(\bar{r},z,\ldots) \ =\ \int dg\ g \psi^c_\sigma(\bar{r},z,\ldots) g^{-1}\ .
\end{equation}
Note that the fidelity parameters remain unchanged under symmetrization
if the states $\rho_j$ are $G$-invariant:
${\rm tr}\,\left[\left( g \psi_{\sigma}^c g^{-1}\right)\rho_j\right]
={\rm tr}\,\left[\psi_{\sigma}^c \left(g^{-1}\rho_j g\right)\right]
={\rm tr}\,\left[\psi_{\sigma}^c \rho_j \right]$.
The symmetrization~\eqref{supp:symmetrized} corresponds
to an example pure-state decomposition of $\sigma(\bar{r},z,\ldots)$,
so that
\begin{align}
E(\sigma(\bar{r},z,\ldots)) & \leqslant
\int dg\ E\left(g \psi^c_\sigma(\bar{r},z,\ldots) g^{-1}\right)
\nonumber\\
& =
\int dg\ E\left(\psi^c_\sigma(\bar{r},z,\ldots) \right)
\nonumber\\
& =
E\left(\psi^c_\sigma(\bar{r},z,\ldots) \right)
\nonumber\\
& =
\tilde{E}^c(\bar{r},z,\ldots)\ \ .
\label{supp:Esymmetrized}
\end{align}
But since, according to our assumption,
$\tilde{E}^c(\bar{r},z,\ldots)$ {\em is} the minimum achievable
value for {\em any} decomposition of $\sigma(\bar{r},z,\ldots)$ [cf. Eq.~\eqref{eq:lower}], we must have
$E(\sigma(\bar{r},z,\ldots))=\tilde{E}^c(\bar{r},z,\ldots)$.
Alternatively, we may have the case that
$\tilde{E}^c(\bar{r},z,\ldots)= \sum_k q_k E(\psi^c_{\sigma,k})$
with $\sum q_k=1$, i.e.,
the achievable minimum of $E$ at $\bar{r},z,\ldots$
results from a convex combination of pure states
$\psi^c_{\sigma,k}(\bar{r}_k,z_k,\ldots)$ within the span of $\sigma$.
While the
fidelity parameters $\bar{r}_k,z_k,\ldots$ are different from $\bar{r},z,
\ldots$, those of their convex combination are the same~\cite{Osterloh2008}:
${\rm tr}\,\left[\left(\sum_k q_k\psi^c_{\sigma,k}\right)\rho_j\right]
={\rm tr}\,\left[\sigma\rho_j\right]$.
Now, in analogy with the approach above, we may construct an
example decomposition of $\sigma(\bar{r},z,\ldots)$ by symmetrization,
because $\sigma$ is uniquely defined by the fidelity parameters $\bar{r},z,
\ldots$:
\begin{equation}\label{supp:Ksymmetrized}
\sigma(\bar{r},z,\ldots) \ =\ \sum_k q_k \int dg\ g \psi^c_{\sigma,k}(\bar{r},z,\ldots) g^{-1}\ .
\end{equation}
Correspondingly, for the entanglement we have, by virtue of
the convexity of $E$,
\begin{align}
\label{supp:KEsymmetrized}
E(\sigma(\bar{r},z,\ldots)) & \leqslant
\sum_k q_k \int dg\ E\left(g \psi^c_{\sigma,k}(\bar{r},z,\ldots) g^{-1}\right)
\nonumber\\
& =
\sum_k q_k \int dg\ E\left(\psi^c_{\sigma,k}(\bar{r},z,\ldots) \right)
\nonumber\\
& =
\sum_k q_k E\left(\psi^c_{\sigma,k}(\bar{r},z,\ldots) \right)
\nonumber\\
& =
\tilde{E}^c(\bar{r},z,\ldots)\ \ .
\end{align}
Again, since $\tilde{E}^c(\bar{r},z,\ldots)$ already represents the
smallest possible value for those fidelity parameters the
equal sign must hold. Thus, we have shown that, for states that are invariant under the action of an entanglement-preserving group of symmetry operations
$G$, the convex characteristic curve not only gives a lower
bound to an entanglement measure, but there is always
a decomposition that realizes this lower bound, confirming
that the convex characteristic curve coincides with
the exact entanglement measure. We remark that, in case such invariance is lacking, the convex characteristic curve holds only as a lower bound to the entanglement measure.
\end{proof}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{triangle.pdf}
\caption{(Color online). The convex set of states $\sigma$,
i.e., those that belong to the lower facet of the tetrahedron (cf.
Fig.~\ref{fig:tetrahedron}), for $\bar{r}\geqslant 0$.
The border of the set of separable states is given
by the straight line $z=1/3[1-\bar{r}(1-z)]$ (dotted gray line),
while the boundary between PPT and
NPT states, Eq.~\eqref{eq:sigma_PPTborder},
is represented by the dashed red line.
The filled region corresponds to all states $\sigma$ that are PPT entangled.
The border of the triangle (solid blue line) delimits the set of physical states.
The solid horizontal line (green)
at $z=2/7$ represents the one-parameter
family of $3\times 3$ bound entangled states of Horodecki
{\em et al.}~\cite{Horodecki1999a}. }
\label{fig:triangle}
\end{figure}
In addition to the numerical treatment of $\tilde{E}^c_{\rm lin}$ described above, we note that it is actually possible to characterize analytically the set of separable states. For $\tilde{E}_{\rm lin}(\bar{r},z)=0$ one can find
an analytical condition in terms of a higher-order polynomial.
The resulting zero line contains the separable points
at $(\bar{r},z)=(0,1/3)$ and $(1,0)$ and
lies below the straight line $z=1/3[1-\bar{r}(1-z)]$. Since
the set of separable states has to be convex, its border must
be given by that straight line
(cf.~Fig.~\ref{fig:triangle}). Alternatively, one could use the
computable cross norm to obtain the same result~\cite{Bertlmann2008,Bertlmann2009}.
\begin{figure}[h]
\centering
\includegraphics[width=.49\textwidth]{Elin.pdf}
\caption{(Color online). Exact convex roof for the
linear entropy $\tilde{E}^c_{\rm lin}$ of the family
$\sigma$, Eq.~\eqref{eq:sigma}. Notably, the boundary between
PPT and NPT states [red solid line in the $(\bar{r},z)$ plane]
is not at all special for the dependence of the linear
entropy on the parameters.}\label{fig:Elin}
\end{figure}
To finish this section, we now compute a lower bound to the linear entropy using a recent method proposed in~\cite{Moroder2014} and show that it coincides with $\tilde{E}^c_{\rm lin}$, thereby certifying our numerical minimization. The method is based on semidefinite programming (SDP)~\cite{Vandenberghe1996}, and is able to produce a lower bound for the convex roof extension of any measure that, for pure states, can be written as a low-order polynomial of operator expectation values. Given a test state $\rho\in\mathcal{H}$, the method requires that the measure is cast as a suitable operator acting on a symmetric extension $\omega_{12\ldots N}\in\mathcal{H}^ {\otimes N}$, with $N$ large enough for the output to be polynomial. Then, an SDP minimization of the measure is carried out over the variable $\omega_{12\ldots N}$ subject to a (necessary) separability constraint, i.e., the PPT criterion \cite{Peres1996,Horodecki1996a}. This is in the same spirit as the PPT-symmetric extensions hierarchy introduced
by Doherty {\em et al.}~\cite{Doherty2004,*Doherty2005}.
Fortunately, the linear entropy becomes a polynomial with only a two-copy extension \cite{Horodecki2003}, hence the dimension of $\omega_{12}$ is limited to 17 (that is the dimension of the symmetric subspace of two copies of $3\times 3$ states) and the SDPs are solved efficiently.
The result is shown in Fig.~\ref{fig:Elin}.
At this point it is worth stressing that the surface $\tilde{E}^c_{\rm lin}$ is obtained numerically from the convex hull of a finite set of three dimensional points, i.e., those where we have evaluated $\tilde{E}_{\rm lin}(\bar{r},z)$. Thus $\tilde{E}^c_{\rm lin}$ is not a smooth surface, for it depends on the chosen resolution of the grid of points. This is the reason behind the observed discrepancy of order $10^{-5}$ at worst between the SDP-computed lower bound and $\tilde{E}^c_{\rm lin}$, for a grid of $\sim10^5$ points. Remarkably, the discrepancy for $\sim 90\%$ of the grid is below $10^{-9}$.
\emph{Concurrence}.
Next, we show how the calculations we have done so far provide us also with the exact concurrence of states $\sigma$.
The concurrence of a pure state $\psi$ is directly related to its linear entropy by $C(\psi)=\sqrt{E_{\rm lin}(\psi)}$. Hence the characteristic curve of the concurrence for the facet is simply $\tilde{C}(\bar{r},z)=\min_\xi \sqrt{E_{\rm lin}(\psi_\sigma)} = \sqrt{\tilde{E}_{\rm lin}(\bar{r},z)}$. Since the SDP bound endorses $\tilde{E}^c_{\rm lin}$ as the exact linear entropy, in turn derived from $\tilde{E}_{\rm lin}$, we conclude
that $\tilde{C}$ is as good. In the same vein as before, its convexified version $\tilde{C}^c$ coincides with the exact concurrence \cite{Osterloh2008}.
It turns out that $\tilde{C}^ c$ is much simpler than $\tilde{E}^c_{\rm lin}$.
The concurrence of the states at points (0,1/3) and (1,0) (diagonal, thus separable) as well of the state at (0,1) (the maximally entangled state) are known, and define a plane that gives an upper bound to $\tilde{C}^c$. As it turns out, $\tilde{C}$ nowhere is below that plane, therefore said plane is indeed the convexified function in that region. This is summarized by the simple formula
$
\tilde{C}^c(\bar{r},z) = [(3-\bar{r})z-(1-\bar{r})]\sqrt{3}
$ for $z \geqslant (1-\bar{r})/(3-\bar{r})$, and $\tilde{C}^c(\bar{r},z) = 0$ otherwise.
{\em Discussion and outlook}.
We have presented a family of highly symmetric $d \times d$
bipartite mixed states for which we have specifically
studied the case of two qutrits ($d=3$). We have exploited the symmetry of the
states to calculate the exact linear entropy for the states $\sigma$
in a two-parameter
subfamily, Eq.~\eqref{eq:sigma}, which includes also PPT-entangled
states. To this end, we have employed a combination of the
convex characteristic curve~\cite{Osterloh2008} and an SDP
method~\cite{Moroder2014}. Moreover, based on the {\em a posteriori}
certification of the straightforward numerical minimization (yet without
analytical proof) we obtain also
the concurrence for the same subset of states as a simple affine function.
These results display several remarkable features. First, we note that
neither the linear entropy nor the concurrence exhibit any peculiarity
in their behavior along the boundary, Eq.~\eqref{eq:sigma_PPTborder},
between PPT and NPT-entangled states.
This may be viewed as an indication that the resources quantified
by those entanglement measures do not bear a direct relation to the
concept of distillability. Apart from that, the functional dependence of
the concurrence---as a polynomial entanglement measure of homogeneous
degree 1 in the density matrix---is the simplest one possible: an affine
function. An interesting technical point is that the SDP method for the
linear entropy, in principle, uses a hierarchy in the number of symmetric
extensions~\cite{Moroder2014}. Notably, our work shows that for the states
$\sigma$ in Eq.~\eqref{eq:sigma}
the exact solution is obtained already at the first level of the hierarchy.
Last but not least it is worth mentioning that for the
one-parameter family of $3\times 3$
bound-entangled states described by Horodecki {\em et al.}~\cite{Horodecki1999a}
there are numerous studies in the literature.
Our exact result now provides a firm
ground from quantitative entanglement theory for those investigations.
Apart from the obvious extensions of this work,
such as an investigation of the interior of the $3\times 3$ tetrahedron,
the most interesting aspect for PPT entanglement, and bipartite entanglement
in general, results from the following observation. The first exact
characteristics of $d\times d$ bipartite entanglement were obtained for
isotropic states~\cite{Horodecki1999}.
Relaxing that symmetry leads to the axisymmetric states
which are still quite similar to the isotropic ones, and allow for
exact treatment~\cite{Eltschka2015a}. Relaxing, in turn, the axisymmetry
as in the present study includes also PPT entanglement in the properties
amenable to an analysis without approximations.
That is, the relaxed axisymmetry provides
the grounds for a systematic investigation of PPT entanglement
and thus for a deeper insight into the structure of the state space
of bipartite systems.
\emph{Acknowledgments}.
This work was funded by the German Research Foundation within
SPP 1386 (C.E.), by Basque Government Grant No. IT-472-10,
MINECO Grants No. FIS2012-36673-C03-01 and No. FIS2015-67161-P, by EU ERC Starting Grant No. 258647/GEDENTQOPT,
and UPV/EHU program UFI 11/55 (G.S.\ and J.S.).
The authors would like to thank O.\ G\"uhne, M.\ Huber, R.\ Mu\~noz Tapia,
A.\ Monr\`as, G.\ T\'oth, and T.\ V\'ertesi for stimulating discussions
and comments.
| -20,952.070946 |
[
-1.58984375,
1.3408203125
] | 29.936306 |
[
-2.63671875,
0.3759765625,
-2.091796875,
-6.203125,
-1.7607421875,
8.71875
] |
[
2.341796875,
7.78125,
0.2493896484375,
5.625
] | 171 | 2,570 |
[
-3.51953125,
4.08203125
] | 30.841916 |
[
-5.7890625,
-4.45703125,
-4.62890625,
-2.482421875,
1.8720703125,
12.546875
] | 1.171039 | 14.378849 | 32.996109 | 1.953305 |
[
1.7806265354156494
] | -14,727.59129 | 5.814786 | -20,644.748547 | 1.17773 | 5.734958 |
[
-2.24609375,
-3.400390625,
-4.02734375,
-5.578125,
2.189453125,
12.8515625
] |
[
-5.08984375,
-1.6845703125,
-1.9482421875,
-0.98828125,
2.9609375,
3.3359375
] | |
BkiUa1LxK6-gDz87Nvcz
|
\section{Introduction}
Recently we have investigated the quantitative effect of embedding on gravitational lensing observations by resorting to a mixture of analytic work with a few numerical applications \cite{Kantowski10,Chen10,Chen11}.
The analytic results for quantities like the bending angle $\alpha$ produced by a point mass were given as functions of two impact variables $r_0$ and $\tilde{\phi}_1$ (see Fig.1).
These two parameters are not independent if the source and deflector redshifts are fixed.
Because of the non-linearity of the expressions we were only able to give an iterative procedure that allowed us to numerically evaluate the conventional minimum impact Schwarzschild coordinate $r_0$ as a function of $\tilde{\phi}_1$ \cite{Chen11}.
We have since been able to analytically carry out this iterative procedure (see Eq.\,(\ref{r0}) in the appendix) and hence obtain all lensing properties such as position, shear, etc., as functions of the single impact angle $\tilde{\phi}_1$.
The solution of the embedded lens equation and comparison with classical lensing theory is therefore greatly simplified.
Because the dependence of observable quantities on this angle is highly nonlinear, we are not able to eliminate $\tilde{\phi}_1$ in favor of $r_0$.
Derivations of our current results follow the steps given in \cite{Kantowski10,Chen10,Chen11} which we will not repeat but we will instead simply present the new results and use them on two examples.
The point mass lens is the simplest lens to use to demonstrate the effects of embedding; however, all lenses will require corrections. An embedded point mass lens is constructed by condensing a comoving sphere of pressureless dust of a standard homogeneous cosmology to a singular point mass m at the sphere's center, a construction first made by Einstein himself \cite{Einstein45,Schucking54,Kantowski69,Kantowski95}. When the cosmology contains a cosmological constant $\Lambda$ the gravity field inside the evacuated sphere is described by the Kottler metric \cite{Kottler18,Dyer74} rather than the Schwarzschild metric. In this paper we restrict ourselves to a flat background cosmology whose Friedman-Lema\^itre-Robertson-Walker (FLRW) metric is
\begin{equation}
ds^2=-c^2dT^2+R(T)^2\left[{d\chi^2}+\chi^2(d\theta^2+\sin^2\theta d\phi^2)\right].
\label{FLRW}
\end{equation}
The embedded condensation is described by the Kottler or Schwarzschild-de Sitter metric
\begin{equation}
\label{Kottler}
ds^2=-\gamma(r)^{-2}c^2dt^2+
\gamma(r)^2dr^2+
r^2(d\theta^2+\sin^2\theta\, d\phi^2),
\end{equation}
where $\gamma^{-1}(r)\equiv\sqrt{1-\beta^2(r)}$ and $\beta^2(r)\equiv r_s/r+\Lambda r^2/3$.
The constants $r_s$ and $\Lambda$ are the Schwarzschild radius ($2G{\rm m}/c^2$) of the condensed mass and the cosmological constant respectively.
By matching the first fundamental forms at the Kottler-FLRW boundary, angles $(\theta,\phi)$ of equations (\ref{FLRW}) and (\ref{Kottler}) are identified and the expanding Kottler radius $r_b$ of the void is related to its comoving FLRW boundary $\chi_b$ by
\begin{equation}\label{rb-T}
r_b=R(T)\chi_b.
\end{equation}
By matching the second fundamental forms the comoving FLRW radius $\chi_b$ is related to the Schwarzschild radius $r_s$ of the Kottler condensation by
\begin{equation}\label{second-form}
r_s=\Omega_{\rm m}\frac{H_0^2}{c^2}(R_0\chi_b)^3.
\end{equation}
Here $H_0$ is the familiar Hubble constant and the cosmological constant $\Lambda$ is constrained to be the same inside and outside of the Kottler hole.
\begin{figure*}
\includegraphics[width=0.8\textwidth,height=0.26\textheight]{fig1.eps}
\caption{A photon travels left to right entering a Kottler hole at $r=r_1,\, \phi=\pi-\tilde{\phi}_1$
and returns to the FLRW dust at $r=r_1+\Delta r,\, \phi=\tilde{\phi}_1 +\Delta\phi$.
The photon's orbit has been chosen symmetric in Kottler about the point of closest approach $r=r_0$, $\phi=\pi/2$. Due to the cosmological expansion, $\Delta r>0$.
The slope of the photon's co-moving trajectory in the x-y plane is $\xi_1$
when incoming and $\xi_1+\alpha$ after exiting.
The resulting deflection angle as seen by a comoving observer in the FLRW background is $\alpha$, which is negative by convention.
Expressions for $r_1,$ $\Delta r,$ $\xi_1,$ $\Delta\phi,$ and $\alpha$ as functions of the two impact parameters, $r_0$ and $\tilde{\phi}_1$, can be found in \cite{Kantowski10,Chen10,Chen11}.}
\label{fig:fig1}
\end{figure*}
In the following sections we will give image locations and image properties of small sources seen through Kottler voids in an otherwise flat FLRW universe (an embedded lens). We assume that the source and deflector are located at fixed FLRW comoving distances $\chi_s$ and $\chi_d$ from the observer which correspond to angular diameter distances $D_s$ and $D_d$, and redshifts $1+z_s=R_0/R_s$ and $1+z_d=R_0/R_d$, see Fig.\,2. These quantities are computed just as if the void didn't exist. Any quantity with a subscript `$d$' means that it is evaluated at redshift $z_d$ when the radius of the universe was $R_d=R(T_d)=R_0/(1+z_d)$. We give lensing properties such as the bending angle $\alpha$ of Eq.\,(\ref{alpha}) that are a series of smaller and smaller terms, sufficient to see both the shielding effect of embedding and the effect of the expansion rate $\beta_d=v_d/c$ of the void's Kottler radius $r_d=R_d\chi_b$ that existed at FLRW time $T_d$. The expansion rate $v_d$ is the speed of the expanding void boundary as measured by a stationary Kottler observer at $r_d$. It is given by evaluating $\beta(r)$ defined below Eq.\,(\ref{Kottler}) at $r=r_d$
\begin{equation}
\beta_d=\sqrt{\frac{r_s}{r_d}+\frac{\Lambda r_d^2}{3}}.
\label{beta_d}
\end{equation}
When expanding quantities such as $\alpha$ in a series we have taken parameters $\beta_d$ and $\chi_b/\chi_d=r_d/D_d$ (the angular radius of the Kottler hole, see Fig.~\ref{fig:fig2}) to be first order and $r_s/r_d$ and $\Lambda r_d^2/3$ to be second order. In our results, e.g., Eq.\,(\ref{alpha}), we have used a parameter $\mbox{\boldmath$\delta$}$ to keep track of each order. In Table 1 we give values for these and other parameters for two lens masses, ${\rm m}=10^{12}M_\odot$ (a large galaxy) and ${\rm m}=10^{15}M_\odot$ (a rich cluster) both at redshift $z_d=0.5$ in a flat $\Omega_{\rm m}=0.3$, $\Omega_\Lambda=0.7$ universe with $H_0=70\, \rm km\, s^{-1}\,Mpc^{-1}$. We refer to these as the galaxy lens and the cluster lens throughout the paper.
\begin{figure*}
\includegraphics[width=1.0\textwidth,height=0.17\textheight]{fig2.eps}
\caption{A photon travels from a source at comoving distance $\chi_s$ from the observer and then enters a Kottler hole of comoving radius $\chi_b$ centered at comoving distance $\chi_d$ from the observer. The photon is deflected by an angle $\alpha <0$
and returns to the FLRW dust on its way to the observer. Because this is a comoving picture the orbit inside the void is only representative and because the true orbit inside the void is symmetric about $\phi=\pi/2$ (see Fig. 1) the optical axis is rotated clockwise by the angle $-\rho$ (see Eq.\,(14) of \cite{Chen11}).}
\label{fig:fig2}
\end{figure*}
\begin{table}
\caption{\label{tab:parameters}Embedded lens parameters for two deflectors (a galaxy and a cluster) at $z_d=0.5$ when viewing a source at $z_s=1.0$ in a $\Omega_{\rm m}=0.3$, $\Omega_\Lambda=0.7$ universe with $H_0=70\, \rm km\, s^{-1}\,Mpc^{-1}$. Choosing $R_0=1$, gives $\chi_d=1.89\times 10^{3}$ Mpc and $\chi_s=3.31\times 10^{3}$ Mpc.}
\begin{ruledtabular}
\begin{tabular}{cccccccc}
Lens & m & $\beta_d$ & $r_d/D_d=\chi_b/\chi_d$ & $r_s/r_d$ & $\Lambda r_d^2/3$ & $\theta_E$(rad) & $\tilde{\phi}_E$(rad) \\
galaxy& $10^{12}M_\odot$ & $3.67\times 10^{-4}$&$9.54\times 10^{-4}$&$7.98\times 10^{-8}$&$5.51\times 10^{-8}$&$8.07\times 10^{-6}$&$8.45\times 10^{-3}$\\
cluster& $10^{15}M_\odot$& $3.67\times 10^{-3}$&$9.54\times 10^{-3}$&$7.98\times 10^{-6}$&$5.51\times 10^{-6}$&$2.55\times 10^{-4}$&$2.65\times 10^{-2}$\\
\end{tabular}
\end{ruledtabular}
\end{table}
Shielding typically causes the most significant embedding effect on images (i.e., the lowest order effect) and analytically appears as combinations of trig functions of the impact angle $\tilde{\phi}_1$ in quantities like the bending angle $\alpha$ (see the first $\cos^3\tilde{\phi}_1 $ term in Eq.\,(\ref{alpha}) below). This decrease in $\alpha$ is caused by the shortened period a passing photon is influenced by the mass condensation (recall that in conventional lensing the deflecting force has $``\infty$'' range). Corrections caused by the presence of $\Lambda$ first appear in the void's expansion rate $\beta_d$ and are typically smaller than shielding corrections (i.e., are higher order). It was the search for $\Lambda$'s effect on light deflections \cite{Rindler07,Sereno08,Ishak08a,Ishak08b,Sereno09,Ishak10a,Ishak10b} that prompted investigations of embedded lensing \cite{Schucker09a,Schucker09b,Kantowski10,Boudjemaa}.
\section{The Embedded Lens Equation}
In our results we have introduced an order parameter \mbox{\boldmath$\delta$} whose value is equal to 1 but whose purpose is to keep track of terms of similar orders as defined in the previous section. For weak lensing impact angles $\tilde{\phi}_1$, the higher the power of \mbox{\boldmath$\delta$} the smaller the respective terms. For strong lensing, when $\tilde{\phi}_1$ is sufficiently small, not all terms of a given order are of the same magnitude.
By using steps developed in \cite{Kantowski10,Chen10,Chen11} we find that to order $\mbox{\boldmath$\delta$}^4$ the source and image positions $\theta_S$ and $\theta_I$, as functions of the single impact parameter $\tilde{\phi}_1$, can be written as
\begin{eqnarray}
\theta_S&=&\theta_I+\frac{D_{ds}}{D_s}\,\alpha\Biggl\{1+\mbox{\boldmath$\delta$}^2\frac{\chi_b}{2(\chi_s-\chi_d)}\times\cr
&&\left[\frac{\chi_b}{\chi_d}\sin^2\tilde{\phi}_1+\frac{2}{3}\beta_d\left(4-\sin^2\tilde{\phi}_1+3\sec\tilde{\phi}_1\log\left[\tan\frac{\tilde{\phi}_1}{2}\right]\right)\tan^2\tilde{\phi}_1 \right]+{\cal O} \bigl(\mbox{\boldmath$\delta$}^3\bigr)\Biggr\},
\label{theta_S}
\end{eqnarray}
(see Fig. 2), where the bending angle is
\begin{eqnarray}
\alpha &=&-2\,\mbox{\boldmath$\delta$}^2\frac{r_s}{r_d}\csc\tilde{\phi}_1\Biggl\{\cos^3\tilde{\phi}_1+\mbox{\boldmath$\delta$}\Biggl[\beta_d\cos^2\tilde{\phi}_1(1+2\sin^2\tilde{\phi}_1)\Biggr]
+\mbox{\boldmath$\delta$}^2\biggl[-\beta_d\frac{\chi_b}{\chi_d}\frac{1}{2}\cos^3\tilde{\phi}_1\sin^2\tilde{\phi}_1\cr
&+&\frac{\Lambda r_d^2}{3}\cos\tilde{\phi}_1\sin^2\tilde{\phi}_1(-1+4\sin^2\tilde{\phi}_1)+\frac{r_s}{r_d}\Biggl(\frac{15}{16}\left(\frac{\pi}{2}-\tilde{\phi}_1\right)\csc\tilde{\phi}_1-\cot\tilde{\phi}_1\csc\tilde{\phi}_1\cr
&-&\frac{3}{2}\log\left[\cot\frac{\tilde{\phi}_1}{2}\right]\sin^2\tilde{\phi}_1
+\cos\tilde{\phi}_1\left(\frac{3}{16}+\frac{9}{8}\sin^2\tilde{\phi}_1+\frac{13}{4}\sin^4\tilde{\phi}_1 \right)\Biggr)\Biggr]+{\cal O} \bigl(\mbox{\boldmath$\delta$}^3\bigr)\Biggr\},
\label{alpha}
\end{eqnarray}
and
\begin{eqnarray}\label{theta_I}
\theta_I&=& \mbox{\boldmath$\delta$}\frac{\chi_b}{\chi_d}\sin\tilde{\phi}_1\Biggl\{1- \mbox{\boldmath$\delta$}\beta_d\cos\tilde{\phi}_1+
\mbox{\boldmath$\delta$}^2\Biggl[ \frac{1}{6}\left(\frac{\chi_b}{\chi_d}\right)^2\sin^2\tilde{\phi}_1+\frac{r_s}{r_d}\left(\cot^2\tilde{\phi}_1 -\frac{1}{2}\sin^2\tilde{\phi}_1\right) +\frac{\Lambda r_d^2}{3}\left(1-\frac{3}{2}\sin^2\tilde{\phi}_1\right)\Biggr]\cr
&+&\mbox{\boldmath$\delta$}^3\Biggl[\frac{\chi_b}{\chi_d}\left(\frac{1}{4}\frac{r_s}{r_d}-\frac{1}{2}\frac{\Lambda r_d^2}{3}\right)\cos\tilde{\phi}_1\sin^2\tilde{\phi}_1
-\beta_d\Biggl(\frac{1}{2}\left(\frac{\chi_b}{\chi_d}\right)^2\cos\tilde{\phi}_1\sin^2\tilde{\phi}_1 + \frac{\Lambda r_d^2}{3}\cos\tilde{\phi}_1\left(1-3\sin^2\tilde{\phi}_1\right)\cr
&-&\frac{r_s}{r_d}\left(\cos\tilde{\phi}_1\left(\frac{41}{12}-\frac{11}{12}\sin^2\tilde{\phi}_1\right)+2\log\left[\tan\frac{\tilde{\phi}_1}{2}\right]\right)\Biggr)\Biggr]+{\cal O} \bigl(\mbox{\boldmath$\delta$}^4\bigr)\Biggr\}.
\end{eqnarray}
We identify the pair of equations (\ref{theta_S}) and (\ref{theta_I}) above as the
embedded lens equation in parametric form. They can be used to obtain image positions and properties just as in conventional lensing theory. The conventional non-embedded lens equation is recovered by keeping only the lowest order terms in each expression and assuming $\cos\tilde{\phi}_1 \rightarrow 1$ and $r_d\sin\tilde{\phi}_1\rightarrow r_0$.
\begin{figure*}
\includegraphics[width=0.6\textwidth,height=0.4\textheight]{fig3.eps}
\caption{ Primary and secondary image positions as functions of source position $\theta_S$ for the cluster lens of Table 1. Beyond $\theta_S\sim 3.4 \theta_E$ the smallness of the secondary image's impact begins to violate the orbit approximation condition $\sin\tilde{\phi}_1 \gg r_s/r_0$.}
\label{fig:fig3}
\end{figure*}
Because of the dependence of each $\mbox{\boldmath$\delta$}$ order on the impact angle $\tilde{\phi}_1$, higher order terms that contain trig functions like $\csc\tilde{\phi}_1$ or $\cot\tilde{\phi}_1$ can become comparable in magnitude with the next lower order terms for sufficiently small values of $\tilde{\phi}_1$.
This happens in strong lensing.
For example the embedded Einstein ring size $\theta_E^\prime$ is found by first finding the value of
$\tilde{\phi}_1=\tilde{\phi}_E$ that makes $\theta_S$ of Eq.(\ref{theta_S}) vanish (see $\tilde{\phi}_E$ values in Table 1) and then evaluating $\theta_E^\prime=\theta_I(\tilde{\phi}_E)$ using Eq.\,(\ref{theta_I}). For $\theta_S$ to vanish, $\mbox{\boldmath$\delta$}$ and $\mbox{\boldmath$\delta$}^2$ terms must cancel. The result is
\begin{equation}
\left(\theta^\prime_E\right)^2=\theta_E^2 \cos\tilde{\phi}_E^3\left(1+3 \mbox{\boldmath$\delta$}\beta_d\tan\tilde{\phi}_E\sin\tilde{\phi}_E+{\cal O} (\mbox{\boldmath$\delta$}^2)\right),
\label{theta_E^prime}
\end{equation}
where $\theta_E$ is the conventional Einstein ring radius defined by
\begin{equation}
\theta_E^2=\frac{2\,r_s D_{ds}}{D_d\,D_s}.
\end{equation}
As we have found with most strongly lensed image properties, this value differs only slightly from the conventional value. For the Einstein ring radius the embedded value differs somewhat more than 0.05\% for the cluster lens and 0.005\% for the galaxy lens.
In Figure 3 we have used the embedded lens equation to locate primary and secondary images for the cluster lens.
Primary and secondary images positions are given by Eq.\,(\ref{theta_I}) and correspond respectively to impact angles $\tilde{\phi}_{+}$ and $\tilde{\phi}_-$ (the Einstein impact angle $\phi_E$ separates the two image domains, i.e.,\, $\tilde{\phi}_-<\phi_E<\tilde{\phi}_+$).
For a given source position $\Theta_S$, primary and secondary image impact angels $\tilde{\phi}_\pm$ are found by solving $\theta_S(\tilde{\phi}_\pm)=\pm \Theta_S$ (i.e., by inverting Eq.\,(\ref{theta_S})).
The two images are then located at $\pm\theta_I(\tilde{\phi}_\pm)$ (i.e., by using Eq.\,(\ref{theta_I})).
These two values of $\tilde{\phi}_1$ can then be used to determine primary and secondary image properties.
\section{Image Properties of The Embedded Lens}
To evaluate standard image properties the reader only has to compute the azimuthal and radial eigenvalues $(a_\phi,a_r)$ of the image matrix $\partial\boldsymbol\theta_S/\partial\boldsymbol\theta_I$ using equations (\ref{theta_S}) and (\ref{theta_I}).
We give them in Equations (\ref{aphi}) and (\ref{ar}) of the appendix. The primary and secondary values for $a_\phi$ and $a_r$ can then be used to obtain image amplification, effective surface density, shear, and eccentricity, respectively $\mu, \kappa, \gamma_s$, and $\epsilon$ (see \cite{Bourassa75,Ehlers}) by evaluating
\begin{equation}
\mu^{-1}=a_\phi a_r,
\label{mu}
\end{equation}
\begin{equation}
\kappa=1-\frac{1}{2}(a_\phi+a_r),
\label{kap}
\end{equation}
\begin{equation}
\gamma_s=\frac{1}{2}(a_r-a_\phi),
\label{gam}
\end{equation}
\begin{equation}
\epsilon=\sqrt{1-\left(\frac{a_\phi}{a_r}\right)^2}.
\label{eps}
\end{equation}
The above expressions give image properties for all values of impact angle $\tilde{\phi}_1$ such that the photon's orbit approximation is valid ($\sin^2\tilde{\phi}_1\gg r_s/r_d$), but because of the lengths of the resulting expressions, we find it appropriate to make two approximations in the next section, one for weak lensing and one for strong. The effective surface mass density $\kappa$ for the embedded lens is the one property that does not vanish as it does for the conventional Schwarzschild lens and is plotted in Figure\,4 for both weak and strong lensing of the primary cluster image. By a conventional Schwarzschild lens we mean conventional linear lensing theory applied to a point mass superimposed on a FLRW background. Even for strong lensing by the cluster the magnitude of $\kappa$ is only $\sim $ 0.1\% of the critical value. For the galaxy lens a $\kappa$ plot is similar to the cluster plot but is approximately a factor of 10 smaller.
\begin{figure*}
\includegraphics[width=0.6\textwidth,height=0.35\textheight]{fig4.eps}
\caption{The effective surface mass density $\kappa$ for the primary image of the embedded cluster lens of Table 1.}
\label{fig:fig4}
\end{figure*}
\section{Weak and Strong Approximations}
We found it necessary to keep terms to order $\mbox{\boldmath$\delta$}^4$ in expressions such as $\alpha$ and $\theta_I$ to obtain sufficiently accurate results for most strong lensing quantities.
Most weak observable quantities do not require such accuracy.
By dividing the domain for $\tilde{\phi}_1$ into strong and weak parts we are able to give shorter expressions for the two eigenvalues $(a_\phi,a_r)$ of equations (\ref{aphi}) and (\ref{ar}) and hence simpler expressions for $\mu,$ etc. For the strong domain we take $0.4\phi_E<\tilde{\phi}_1<5\phi_E$ and for the weak $5\phi_E<\tilde{\phi}_1<\pi/2$.
The maximum value for $\theta_I$ is approximately the ratio $r_d/D_d$ which from Table 1 is $\sim$$117$ times the Einstein ring radius $\theta_E$ for the cluster lens and $\sim$$37$ times for the galaxy.
Strong lensing consequently occurs for $\theta_I$ values up to $\sim$$5\theta_E$ and weak lensing begins to occur when $\theta_I$ exceeds that value.
To obtain shortened expressions for weak lensing we need only keep terms of order $\mbox{\boldmath$\delta$}^2$. This allows us to determine the lowest order effects of lens shielding and void expansion (the $\beta_d$ term) on image properties in the weak domain $5\phi_E<\tilde{\phi}_1<\pi/2$. We find that the approximate expressions are accurate to at least 0.1\% down to $ \tilde{\phi}_1=5\,\phi_E$ for the cluster and to at least 0.03\% for the galaxy.
For weak lensing Eqs.~(\ref{aphi}) and (\ref{ar}) simplify to
\begin{eqnarray}
a_\phi^{weak}&=&1-\mbox{\boldmath$\delta$}\left(\frac{\theta_E D_d}{r_d}\right)^2\csc^2\tilde{\phi}_1\cos^3\tilde{\phi}_1\Biggl\{1+\mbox{\boldmath$\delta$}\beta_d\sec\tilde{\phi}_1(2+\sin^2\tilde{\phi}_1)+{\cal O} \bigl(\mbox{\boldmath$\delta$}^2\bigr)\Biggr\},\\
a_r^{weak}&=&1+\mbox{\boldmath$\delta$}\left(\frac{\theta_E D_d}{r_d}\right)^2\csc^2\tilde{\phi}_1\Biggl\{\cos\tilde{\phi}_1(1+2\sin^2\tilde{\phi}_1)+\mbox{\boldmath$\delta$}\beta_d(2-\sin^2\tilde{\phi}_1+2\sin^4\tilde{\phi}_1)
+{\cal O} \bigl(\mbox{\boldmath$\delta$}^2\bigr)\Biggr\}.\nonumber
\label{arw}
\end{eqnarray}
From these the following image properties result:
\begin{eqnarray}
(\mu^{weak})^{-1}&=&1+3\,\mbox{\boldmath$\delta$} \left(\frac{\theta_E D_d}{r_d}\right)^2\Biggl\{\cos\tilde{\phi}_1+\, \mbox{\boldmath$\delta$} \beta_d\sin^2\tilde{\phi}_1+{\cal O} \bigl(\mbox{\boldmath$\delta$}^2\bigr) \Biggr\}\cr
&&-\mbox{\boldmath$\delta$}^2 \left(\frac{\theta_E D_d}{r_d}\right)^4\cot^4\tilde{\phi}_1\Biggl\{(1+2\sin^2\tilde{\phi}_1)
+{\cal O} \bigl(\mbox{\boldmath$\delta$}\bigr)\Biggr\},\label{muw}\\
\kappa^{weak} &=&-\frac{3}{2}\, \mbox{\boldmath$\delta$} \left(\frac{\theta_E D_d}{r_d}\right)^2\Biggl\{\cos\tilde{\phi}_1+\, \mbox{\boldmath$\delta$}\, \beta_d\sin^2\tilde{\phi}_1+{\cal O} \bigl(\mbox{\boldmath$\delta$}^2\bigr) \Biggr\},\\
\gamma^{weak}&=&\mbox{\boldmath$\delta$} \left(\frac{\theta_E D_d}{r_d}\right)^2\csc^2\tilde{\phi}_1\Biggl\{\cos\tilde{\phi}_1\left(1+\frac{1}{2}\sin^2\tilde{\phi}_1\right)+\,\mbox{\boldmath$\delta$} \beta_d\left(2-\sin^2\tilde{\phi}_1+\frac{1}{2}\sin^4\tilde{\phi}_1\right)+{\cal O} \bigl(\mbox{\boldmath$\delta$}^2\bigr) \Biggr\},\hspace{19pt}\\
\epsilon^{weak}&=&
\Biggl\{\sqrt{\mbox{\boldmath$\delta$}}\left(\frac{\theta_E D_d}{r_d}\right)\csc\tilde{\phi}_1\sqrt{2\cos\tilde{\phi}_1(2+\sin^2\tilde{\phi}_1)+\, \mbox{\boldmath$\delta$}\, \beta_d(4-2\sin^2\tilde{\phi}_1+\sin^4\tilde{\phi}_1)+{\cal O} \bigl(\mbox{\boldmath$\delta$}^2\bigr)}\Biggr\}\cr
&\times& 1/\Biggl\{1+\mbox{\boldmath$\delta$}\left(\frac{\theta_E D_d}{r_d}\right)^2\csc^2\tilde{\phi}_1\Bigl[\cos\tilde{\phi}_1(1+2\sin^2\tilde{\phi}_1)+{\cal O} \bigl(\mbox{\boldmath$\delta$}\bigr)\Bigr]\Biggr\}.
\label{epsilonw}
\end{eqnarray}
In Figure 5 we have compared image shear and ellipticity of the embedded lens images with conventional (non-embedded) Schwarzschild values.
The reader can see that beyond $\tilde{\phi}_1\sim 45^\circ$ the embedded lens differs from Schwarzschild by over 10\%.
This is caused primarily by the shielding of the embedded mass and increases as the transiting light ray's minimum impact $r_0$ approaches the void boundary.
The embedded amplification $\mu$ differs from conventional Schwarzschild by less than 0.2\% for the cluster and 0.02\% for the galaxy for the weak lensing domain and only increases to 2.5\% for the secondary cluster image in the strong lensing limit $\tilde{\phi}_1\rightarrow 0.4\phi_E$ where $\mu\rightarrow -0.03$.
The galaxy lens' numbers are significantly less and neither are plotted.
\begin{figure*}
\begin{center}$
\begin{array}{cc}
\includegraphics[width=0.52\textwidth,height=0.3\textheight]{fig5a.eps}
\hspace{10pt}
\includegraphics[width=0.5\textwidth,height=0.3\textheight]{fig5b.eps}
\end{array}$
\end{center}
\caption{ Corrections to image shear (left panel) and ellipticity (right panel) caused by embedding of the cluster lens of Table 1.
The solid blue curves in the left and right panels are respectively the $\gamma_s$ and $\epsilon$ for the conventional (non-embedded) Schwarzschild lens.
The fractional difference in the image shear, $\Delta\gamma_s/\gamma_s$, and ellipticity, $\Delta\epsilon/\epsilon,$ caused by embedding, are the dashed red curves in the left and right panels, plotted as a functions of $\tilde{\phi}_1$.
Differences are computed at the same primary image positions $\theta_I(\tilde{\phi}_1).$
}
\label{fig:fig5}
\end{figure*}
Using the order parameter $\mbox{\boldmath$\delta$}$ to track terms of equal importance is problematic for strong lensing.
As discussed above for strongly lensed images, small values of $\tilde{\phi}_1$ in trig functions like $\csc^2\tilde{\phi}_1$ increase the numerical magnitudes of some of the terms in Eqs.~(\ref{aphi}) and (\ref{ar}).
In the following strong lensing approximation we have kept terms based on their numerical size at the Einstein ring value $\tilde{\phi}_1=\phi_E$, and ordered them using another parameter $\Delta$ whose value is also 1.
For the cluster lens the $\Delta^1$ terms are of numerical order 0.1, $\Delta^2$ terms are of numerical order 0.01 and so on. For the galaxy lens all terms are $\sim 1/10$ those of the cluster.
The principal eigenvalues $a_\phi$ and $a_r$ of Eqs.~(\ref{aphi}) and (\ref{ar}) are approximated by
\begin{eqnarray}
a_\phi^{strong}&=&\Delta^2\Biggl[1-\left(\frac{\theta_E D_d}{r_d}\right)^2\csc^2\tilde{\phi}_1\Biggl\{\cos^3\tilde{\phi}_1-2\,\frac{r_s}{r_d}\csc^2\tilde{\phi}_1\cos\tilde{\phi}_1+2\,\Delta\beta_d\cos^2\tilde{\phi}_1+{\cal O}\bigl(\Delta^2\bigr)\Biggr\}\Biggr],\\
a_r^{strong}&=&1+\left(\frac{\theta_E D_d}{r_d}\right)^2\Biggl\{\csc^2\tilde{\phi}_1\cos\tilde{\phi}_1-2\,\Delta^2\frac{r_s}{r_d} \csc^4\tilde{\phi}_1\cos\tilde{\phi}_1+2\,\Delta^3(\cos\tilde{\phi}_1+\beta_d \csc^2\tilde{\phi}_1)+{\cal O}\bigl(\Delta^4\bigr)\Biggr\}.\nonumber
\label{ars}
\end{eqnarray}
These approximate expressions are accurate to at least 0.2\% for the strong domain $0.4\,\phi_E<\tilde{\phi}_1<5\,\phi_E$ for the cluster lens and accurate to 0.01\% for the galaxy.
Strong lensing image properties given in equations (\ref{mu})-(\ref{eps}) differ from conventional Schwarzschild values by only a fraction of a percent and are not separately approximated.
The effective surface mass density $\kappa$ of Eq.~(\ref{kap}), which no longer vanishes as it does for the non-embedded Schwarzschild lens, can be approximated to an accuracy of more than 0.01\% for the strong domain as
\begin{equation}
\kappa^{strong}=-\frac{3}{2}\left(\frac{\theta_E D_d}{r_d}\right)^2\Biggl(\cos\tilde{\phi}_1+\frac{5\pi}{32}\frac{r_s}{r_d}\csc^3\tilde{\phi}_1\Biggr).
\end{equation}
An additional strong lensing property of importance is the time delay.
It contains a geometric part and a potential part, i.e.,\, $\Delta T=\Delta T|_g+\Delta T|_p$, see \cite{Ehlers,Cooke75,Schucker10a}.
The arrival time differences for the two images caused by the difference in geometrical path lengths for our embedded Swiss cheese (SC) lens is $\Delta T_{\rm SC}|_g$ and when computed to maximum accuracy as described in \cite{Chen10}, proves to be almost indistinguishable from the conventional Schwarzschild value $\Delta T_{\rm Sch}|_g$
\begin{equation}
\frac{c\Delta T_{\rm SC}|_g}{r_s}\approx\frac{c\Delta T_{\rm Sch}|_g}{r_s}=(1+z_d)\left(\frac{\theta_S}{\theta_E}\right)\sqrt{ \left(\frac{\theta_S}{\theta_E}\right)^2+4}.
\end{equation}
The potential part of the embedded lens delay, $\Delta T_{\rm SC}|_p$, as defined in \cite{Chen10}, is given by taking the difference in the following for the primary and secondary images
\begin{equation}
\frac{c\Delta T_p}{r_s}=2\,(1+z_d)\Biggl\{ \log\left[\cot\frac{{\tilde{\phi}_1}}{2}\right]-\cos\tilde{\phi}_1\left(1+\frac{1}{3}\cos^2\tilde{\phi}_1\right)+\, \mbox{\boldmath$\delta$}\, \beta_d\cos^4\tilde{\phi}_1 +{\cal O} \bigl(\mbox{\boldmath$\delta$}^2\bigr)\Biggl\}.
\label{T_p}
\end{equation}
\begin{figure*}
\includegraphics[width=0.6\textwidth,height=0.36\textheight]{fig6.eps}
\caption{A comparison of the potential parts of the time delay ($\Delta T_{\rm SC}|_p$) of the embedded cluster lens with the conventional theory.
The comparison is for sources at same positions even though the image positions for the two theories are different.
The solid blue line is the conventional potential part of the time delay and the dashed red line is the fractional difference of the embedded and the conventional theories. }
\label{fig:fig6}
\end{figure*}
In Figure 6 we have compared $\Delta T_{\rm SC}|_p=\Delta T_p^{\rm secondary}-\Delta T_p^{\rm primary}$ (i.e.,\, the potential part of the time delay) of the cluster lens with the corresponding conventional (non-embedded) Schwarzschild value. The reader can see that there is a 2--4\% difference in arrival times between the theories.
\section{Conclusions}
This paper is one of a series of investigations of the differences in image properties caused by including the gravitational lens's mass in the cosmic mean density.
We call such a lens an embedded lens.
In this paper we have eliminated one of the two impact parameters previously required to give embedded point mass lensing quantities such as the bending angle $\alpha$ and the lens equation itself.
The theory remains more complicated than the conventional lensing theory, but is now much easier to use.
The new analytical expressions for image properties agree with the lowest order results given in \cite{Chen11}.
They can also be compared with the higher order results in \cite{Kantowski10,Chen10} that were given as functions of the two impact parameters $r_0$ and $\tilde{\phi}_1$.
To eliminate $r_0$ in our prior results for quantities such as $\alpha(\tilde{\phi}_1,r_0)$ in Eq.~(32) of \cite{Kantowski10} and obtain results such as Eq.\,(\ref{alpha}) given in this paper we had to analytically iterate Eq.~(17) of \cite{Chen11} to determine $r_1(\tilde{\phi}_1)$ and then use the orbit equation (11) of \cite{Kantowski10} to determine $r_0(\tilde{\phi}_1)$.
The result is given in Eq.\,(\ref{r0}) of the appendix for completeness and to allow the reader to eliminate $r_0$ in other quantities of interest.
We have found that with the exception of the potential part of the time delay and the effective surface mass density $\kappa$, strong lensing quantities are only minimally altered by making the lens mass a contributor to the mean mass density of the universe. Even there the effect is less than 5\% on the time-delay for a huge cluster lens, see Fig.\,6. For weak lensing most effects are also small; however, shear and image ellipticity begin to differ significantly ($>10$\%, see Fig.\,5) for large impact angles $\tilde{\phi}_1>45^\circ$. The one quantity that doesn't vanish in embedded point mass lensing is $\kappa$. It turns out to be negative, presumably accounting for the missing FLRW mass density in the Kottler void.
All results given here depend on having a flat ($\Omega=1$) background. Extending them to $\Omega\ne1$ is clearly possible. We expect that many results will differ trivially from what we have given here. The applicability of all results given here also depends on the lens being sufficiently condensed so as to be approximated by a point mass. The effects of embedding on extended lenses remains to be investigated \cite{Schucker10b}.
To correct for embedding we have used the Swiss cheese cosmologies which are commonly criticized for their unrealistic mass distributions, i.e., holes with masses at their centers that abruptly appear in otherwise uniform backgrounds.
The abrupt discontinuity that appears in the cheese is certainly an unrealistic representation of the true matter distribution; however, this is primarily an aesthetic complaint.
Fortunately for Swiss cheese, its purpose is not to represent the mass distribution but instead to account for the effects of mass inhomogeneities on the local/global dynamics of the geometry and on the optics of transiting light rays.
In those two aspects Swiss cheese does quite well.
The real shortcoming of a simple Swiss cheese type embedded lens (a single condensation moving with the Hubble flow) is the absence of any shear at the site of the embedded lens.
For such a simple embedded lens, neighboring inhomogeneities can only be distributed so as to produce a homogenized gravity field at the lens site.
Consequently the accuracy of our predictions can be questioned. Stated simply, the shortcoming of our lens model, and with standard Swiss cheese itself, is that neighboring and distant inhomogeneities produce an homogenized background at the point where the lens inhomogeneity is inserted.
We suspect this ``average'' lens is not representative because it does not account for effects of local shear.
We currently do not have a good estimate of how much de-homogenization alters the shielding radius (which is the major source of embedding effects) because there are no simple Einstein solutions which accurately model local distortions.
Such distortions can easily be accommodated in conventional lensing theory, but how they would alter the embedding radius is completely unknown.
Exact Einstein solutions containing a local shear can be constructed by using hierarchical models built from Swiss cheese itself.
Such a construction will probably be necessary to dependably estimate how the spherical shielding radius $r_d$ is distorted and possibly extended by a local shear and hence how it modifies predictions made here.
\begin{acknowledgments}
NSF AST-0707704, and US DOE Grant DE-FG02-07ER41517 and Support for Program number HST-GO-12298.05-A was provided by NASA through a grant from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Incorporated, under NASA contract NAS5-26555.
\end{acknowledgments}
| -32,332.773417 |
[
-3.447265625,
3.0859375
] | 43.661972 |
[
-3.220703125,
0.5703125,
-1.802734375,
-5.8359375,
-0.6552734375,
7.484375
] |
[
3.26171875,
8.0703125,
1.6796875,
5.140625
] | 250 | 3,673 |
[
-3.521484375,
4.14453125
] | 29.476037 |
[
-5.6015625,
-3.578125,
-3.4609375,
-2.384765625,
1.451171875,
10.3046875
] | 1.083723 | 14.188471 | 28.178601 | 3.106238 |
[
2.4456300735473633
] | -21,737.716952 | 6.406207 | -31,724.298473 | 0.497238 | 5.82671 |
[
-2.7109375,
-3.5625,
-3.56640625,
-4.81640625,
2.17578125,
11.8125
] |
[
-5.4765625,
-2.12890625,
-2.310546875,
-1.466796875,
3.1015625,
4.390625
] | |
BkiUdZDxK4tBVicoC6ij
|
\section{Introduction}
In this paper, we are concerned with the following continuous-time AWGN channel:
\begin{equation} \label{ct}
Y(t) = \int_0^t X(s) ds + B(t),~~~t \geq 0,
\end{equation}
where $\{B(t)\}$ denotes the standard Brownian motion. We will examine the channel (\ref{ct}) for both the non-feedback and feedback cases. More specifically, the non-feedback case refers to the scenario when the feedback is not allowed in the channel, and thereby the channel input $\{X(s)\}$ takes the following form:
\begin{equation} \label{non-feedback-case}
X(s) = g(s, M),
\end{equation}
where $g$ is a real-valued deterministic function, $M$ is a random variable, independent of $\{B(t)\}$ and often interpreted as the {\it message} to be transmitted through the channel. By contrast, for the feedback case, $X(s)$ may also depend on the previous output with the following form:
\begin{equation} \label{feedback-case}
X(s) = g(s, M, Y_0^s),
\end{equation}
where $Y_0^s \triangleq \{Y(r): 0 \leq r \leq s\}$ is the channel output up to time $s$ that is fed back to the sender, which will be referred to as the {\em channel feedback} up to time $s$. Here, we remark that for the non-feedback case as in (\ref{non-feedback-case}), it can be verified that for any $T > 0$,
$$
I(X_0^T; Y_0^T) = I(M; Y_0^T),
$$
which however does not hold true for the feedback case as in (\ref{feedback-case}) when the channel is intricately characterized by the following stochastic differential equation:
\begin{equation} \label{SDE}
Y(t) = \int_0^t g(s, M, Y_0^s) ds + B(t),~~~t \geq 0,
\end{equation}
rather than a simple input-output equation.
For any $T > 0$, we say that $t_{0}, t_{1}, \dots, t_{n} \in \mathbb{R}$~\footnote{Here and hereafter, all $t_i$ depend on $T$ and $n$, however we suppress this notational dependence for brevity.} are {\em evenly spaced} over $[0, T]$ if $t_{0} = 0$, $t_{n} = T$ and $\delta_{T, n} \triangleq t_{i} - t_{i-1} = T/n$ for all feasible $i$. Sampling the continuous-time channel (\ref{ct}) over the time window $[0, T]$ with respect to evenly spaced $t_{0}, t_{1}, \dots, t_{n}$, we obtain the following discrete-time Gaussian channel~\footnote{The sampler associated with (\ref{dt}) has been examined from a communication system design perspective and termed as the {\em integrate-and-dump} filter; see, e.g.,~\cite{CarlsonCrilly2009} and references therein.}
\begin{equation} \label{dt}
Y(t_{i}) = \int_{0}^{t_{i}} X(s) ds + B(t_{i}), \quad i = 1, 2, \dots, n.
\end{equation}
It turns out that if the sampling is ``fine'' enough, the mutual information of the continuous-time Gaussian channel (\ref{ct}) over $[0, T]$ can be ``well-approximated'' by that of the discrete-time Gaussian channel (\ref{dt}). More precisely, it has been established in~\cite{LiuHan2019} that under some mild assumptions,
\begin{equation} \label{limit-version}
\lim_{n \to \infty} I(M; Y(\Delta_{T, n})) = I(M; Y_0^T),
\end{equation}
where
$$
\Delta_{T, n} \triangleq \{t_{0}, t_{1}, \dots, t_{n}\}, \quad Y(\Delta_{T, n}) \triangleq \{Y(t_{0}), Y(t_{1}), \dots, Y(t_{n})\}.
$$
This result connects a continuous-time Gaussian channels with its associative discrete-time versions, which has been used to recover a classical information-theoretic formula~\cite{FourPersons}.
Strictly speaking, the above result is not new; as a matter of fact, it has been ``known'' for decades. Indeed, though not explicitly stated, a more general result, which implies (\ref{limit-version}) as a special corollary, follows from an appropriately modified argument in~\cite{GelfandYaglom1957} (see more details in Section~\ref{already-by-GY}). Moreover, the functional analysis approach as in~\cite{GelfandYaglom1957} tackles more general channels and thus reveals the essence beneath the connection as in (\ref{limit-version}). By comparison, when it comes to merely establishing the result (\ref{limit-version}), our stochastic approach in~\cite{LiuHan2019} is indirect and cumbersome, only scratching the surface of the connection.
That being said, the stochastic calculus approach does allow us to capitalize on the peculiar characteristics of the continuous-time AWGN channel formulated as in (\ref{ct}), which is already evidenced by the approximation theorems established therein that do not seem to follow from the aforementioned functional analysis approach. The present paper will continue to employ the stochastic calculus approach to conduct a closer examination of (\ref{ct}) and quantitatively strengthen (\ref{limit-version}), particularly by zooming in on its convergence behavior. Our results encompass both the non-feedback case (Theorem~\ref{non-feedback-theorem}) and the feedback case (Theorem~\ref{feedback-theorem}), which may be used for a finer analysis of the channel (\ref{ct}) with possible feedback from an information-theoretic perspective. In particular, among other possible implications, our results characterize how over-sampling approaches the true mutual information of a continuous-time band-limited AWGN channel over a finite time window (Corollary~\ref{non-feedback-corollary}).
We would like to add that the assumptions imposed in our results are rather mild. Indeed, the celebrated Shannon-Nyquist sampling theorem~\cite{sh49, ny24}, a widely used tool to connect a continuous-time non-feedback AWGN channel to its discrete-time versions, requires the band-limitedness of the channel input. By comparison, with considerably stronger conclusions, Theorem~\ref{non-feedback-theorem} only require some integrability conditions on the power spectral density function of the channel input and Theorem~\ref{feedback-theorem} only requires some mild regularity conditions that are more or less standard in the theory of stochastic differential equations.
\section{Notation and Preliminaries} \label{notations}
We use $(\Omega,\mathcal{F},\PX)$ to denote the underlying probability space, and $\EX$ to denote the expectation with respect to the probability measure $\PX$. As is typical in the theory of SDEs, we assume the probability space is equipped with a filtration $\{\mathcal{F}_t: 0 \leq t < \infty\}$, which satisfies the {\em usual conditions}~\cite{ka91} and is rich enough to accommodate the standard Brownian motion $\{B(t): 0 \leq t < \infty\}$. Throughout the paper, we will use uppercase letters (e.g., $X$, $Y$, $Y^{(n)}$) to denote random variables or processes, and their lowercase counterparts (e.g., $x$, $y$, $y^{(n)}$) to denote their realizations.
Let $C[0, \infty)$ denote the space of all continuous functions over $[0, \infty)$, and for any $t > 0$, let $C[0, t]$ denote the space of all continuous functions over $[0, t]$. As usual, we will equip the space $C[0, \infty)$ with the filtration $\{\mathcal{B}_t\}_{0 \leq t < \infty}$, where $\mathcal{B}_{\infty}$ denotes the standard Borel $\sigma$-algebra on the space $C[0, \infty)$ and $\mathcal{B}_t = \pi_t^{-1}(\mathcal{B}_{\infty})$, where $\pi_t : C[0, \infty) \to C[0, t]$ is defined as $(\pi_tx)(s) = x(t\wedge s)$.
For any $\varphi \in C[0, \infty)$, we use $\varphi(\{t_1, t_2, \dots, t_m\})$ to denote $\{\varphi(t_1), \varphi(t_2), \dots, \varphi(t_n)\}$ and $\varphi_0^t$ to denote $\{\varphi(s): 0 \leq s \leq t\}$. The sup-norm of $\varphi_0^t$, denoted by $\|\varphi_0^t\|$, is defined as $\|\varphi_0^t\| \triangleq \sup_{0 \leq s \leq t} |\varphi(s)|$; and similarly, we define $\|\varphi_0^t-\psi_0^t\| \triangleq \sup_{0 \leq s \leq t} |\varphi(s)-\psi(s)|$. For any $\varphi, \psi \in C[0, \infty)$, slightly abusing the notation, we define $\|\varphi_0^s-\psi_0^t\| \triangleq \|\hat{\varphi}_0^{\infty}-\hat{\psi}_0^{\infty}\|$, where $\hat{\varphi}, \hat{\psi} \in C[0, \infty)$ are ``stopped'' versions of $\varphi, \psi$ at time $s, t$, respectively, with $\hat{\varphi}(r) \triangleq \varphi(r \wedge s)$ and $\hat{\psi}(r) \triangleq \psi(r \wedge t)$ for any $r \geq 0$.
Let $X, Y, Z$ be random variables defined on the probability space $(\Omega,\mathcal{F},\PX)$, which will be used to illustrate most of the notions and facts in this section (note that the same notations may have different connotations in other sections). Note that in this paper, a random variable can be discrete-valued with a probability mass function, real-valued with a probability density function or path-valued (more precisely, $C[0, \infty)$ or $C[0, t]$-valued).
For any two probability measures $\mu$ and $\nu$, we write $\mu\sim\nu$ to mean they are equivalent, namely, $\mu$ is absolutely continuous with respect to $\nu$ and vice versa. For any two path-valued random variables $X_0^t = \{X(s); 0 \leq s \leq t\}$ and $Y_0^t =\{Y(s); 0 \leq s \leq t\}$, we use $\mu_{X_0^t}$ and $\mu_{Y_0^t}$ to denote the probability distributions on $\mathcal{B}_t$ induced by $X_0^t$ and $Y_0^t$, respectively; and if $\mu_{Y_0^t}$ is absolutely continuous with respect to $\mu_{X_0^t}$, we write the Radon-Nikodym derivative of $\mu_{Y_0^t}$ with respect to $\mu_{X_0^t}$ as $d\mu_{Y_0^t}/d\mu_{X_0^t}$. We use $\mu_{Y_0^t|Z=z}$ denote the probability distribution on $\mathcal{B}_t$ induced by $Y_0^t$ given $Z=z$, and $d\mu_{Y_0^t|Z=z}/d\mu_{X_0^t|Z=z}$ to denote the Radon-Nikodym derivative of $Y_0^t$ with respect to $X_0^t$ given $Z=z$. Obviously, when $Z$ is independent of $X$, $d\mu_{Y_0^t|Z=z}/d\mu_{X_0^t|Z=z}= d\mu_{Y_0^t|Z=z}/d\mu_{X_0^t}$.
By definition, for $\EX[X|\sigma(Y, Z)]$, the conditional expectation of $X$ with respect to the $\sigma$-algebra generated by $Y$ and $Z$, there exists a $\sigma(Y) \otimes \sigma(Z)$-measurable function $\Psi(\cdot, \cdot)$ such that $\Psi(Y, Z)=\EX[X|\sigma(Y, Z)]$.
For notational convenience, we will in this paper simply write $\EX[X|\sigma(Y, Z)]$ as $\EX[X|Y, Z]$, and $\Psi(y, z)$ as $\EX[X|y, z]$ and furthermore, $\Psi(Y, z)$ as $\EX[X|Y, z]$.
A {\em partition} of the probability space $(\Omega,\mathcal{F},\PX)$ is a disjoint collection of elements of $\mathcal{F}$ whose union is $\Omega$. It is well known there is a one-to-one correspondence between finite partitions and finite sub-$\sigma$-algebras of $\mathcal{F}$. For a finite sub-$\sigma$-algebra $\mathcal{H} \subset \mathcal{F}$, let $\eta(\mathcal{H})$ denote the corresponding finite partition. The entropy of a finite partition $\xi=\{A_1, A_2, \cdots, A_m\}$, denoted by $H(\xi)$, is defined as $H(\xi) \triangleq \sum_{i=1}^m -\PX(A_i)\log \PX(A_i)$, whereas the conditional entropy of $\xi$ given another finite partition $\zeta=\{B_1, B_2, \dots, B_n\}$, denoted by $H(\xi|\zeta)$, is defined as $H(\xi|\zeta) \triangleq \sum_{j=1}^n \sum_{i=1}^m -\PX(A_i \cap B_j) \log \PX(A_i|B_j)$. The mutual information between the above-mentioned two partitions $\xi$ and $\zeta$, denoted by $I(\xi; \zeta)$, is defined as $I(\xi; \zeta) \triangleq \sum_{j=1}^n \sum_{i=1}^m -\PX(A_i \cap B_j) \log \PX(A_i \cap B_j)/\PX(A_i) \PX(B_j)$.
For the random variable $X$, we define
$$
\eta(X) \triangleq \{\eta(\mathcal{H}): \mathcal{H} \mbox{ is a finite sub-}\sigma\mbox{-algebra of } \sigma(X)\}.
$$
The {\em entropy} of the random variable $X$, denoted by $H(X)$, is defined as
$$
H(X) \triangleq \sup_{\xi \in \eta(X)} H(\xi).
$$
The {\em conditional entropy} of $Y$ given $X$, denoted by $H(Y|X)$, is defined as
$$
H(Y|X) \triangleq \inf_{\xi \in \eta(X)} \sup_{\zeta \in \eta(Y)} H(\zeta|\xi).
$$
Here, we note that if $X$ and $Y$ are independent, then obviously it holds that
\begin{equation} \label{Y-X-Y}
H(Y|X)=H(Y).
\end{equation}
The {\em mutual information} between $X$ and $Y$, denoted by $I(X; Y)$, is defined as
$$
I(X; Y) \triangleq \sup_{\xi \in \eta(X), \; \zeta \in \eta(Y)} I(\xi; \zeta).
$$
A couple of properties of mutual information are in order. First, it can be shown, via a concavity argument, that the mutual information is always non-negative. Second, the mutual information is determined by the $\sigma$-algebras generated by the corresponding random variables. For example, for any random variables $X', Y', X'', Y''$,
\begin{equation} \label{I=I}
I(X'; Y')=I(X'; Y'') \mbox{ if } \sigma(X') = \sigma(X'') \mbox{ and } \sigma(Y')=\sigma(Y'')
\end{equation}
and
\begin{equation} \label{I<=I}
I(X'; Y') \leq I(X'; Y'') \mbox{ if } \sigma(X') \subset \sigma(X'') \mbox{ and } \sigma(Y') \subset \sigma(Y'').
\end{equation}
For another example, we have
$$
I(X; Y)=I(X, X; Y, Y+X),~~~I(X; Y) \leq I(X; Y, Z).
$$
It turns out that for the case that $X, Y, Z$ are all discrete random variables, all the above-mentioned notions are well-defined and can be computed rather explicitly: $H(X)$ can be computed as $H(X) = \EX[-\log p_X(X)]$, where $p_X(\cdot)$ denotes the probability mass function of $X$; $H(Y|X)$ can be computed as $H(Y|X) = \EX[-\log p_{Y|X}(Y|X)]$, where $p_{Y|X}(\cdot|\cdot)$ denotes the conditional probability mass function of $Y$ given $X$; $I(X; Y)$ can be computed as
\begin{equation} \label{mutual-information}
I(X; Y) = \EX\left[\log \frac{p_{Y|X}(Y|X)}{p_Y(Y)}\right].
\end{equation}
The mutual information is intimately related to entropy. As an example, one verifies that
\begin{equation} \label{I-H}
I(X; Y)=H(Y)-H(Y|X).
\end{equation}
Note that the quality (\ref{I-H}) may fail if non-discrete random variables are involved, since the corresponding entropies $H(Y)$ and $H(Y|X)$ can be infinity. For the case of real-valued random variables with density, this issue can be circumvented using the notion of differential entropy, as elaborated below.
Now, assume that $Y$ is real-valued with probability density function $f_Y(\cdot)$. The {\em differential entropy} of $Y$, denoted by $h(Y)$, is defined as $h(Y) \triangleq \EX[-\log f_Y(Y)]$. And the {\em differential conditional entropy} of $Y$ given a finite partition $\xi$, denoted by $h(Y|\zeta)$, is defined as $h(Y|\zeta) \triangleq \sum_{j=1}^n \PX(A_i) \int f_{Y|A_i}(x) \log f_{Y|A_i}(x) dx$. The {\em differential conditional entropy} of $Y$ given $X$ (which can be discrete-, real- or path-valued), denoted by $h(Y|X)$, is defined as $h(Y|X) \triangleq \inf_{\xi \in \eta(X)} h(Y|\xi)$ (which, similarly as in (\ref{Y-X-Y}), reduces to $h(Y)$ if $Y$ is independent of $X$); in particular, if the conditional probability density function $f_{Y|X}(\cdot|\cdot)$ exists, then $h(Y|X)$ can be explicitly computed as $\EX[-\log f_{Y|X}(Y|X)]$. As mentioned before, the aforementioned failure of (\ref{I-H}) can be salvaged with the notion of differential entropy:
\begin{equation} \label{I-h}
I(X; Y)=h(Y)-h(Y|X).
\end{equation}
Here we emphasize that all the above-mentioned definitions naturally carry over to the setting when some/all of the invovled random variables are replaced by vectors of random variables. For a quick example, let $Y = \{Y_1, Y_2, \dots, Y_n\}$, where each $Y_i$ is a real-valued random variable with density. Then, the differential entropy $h(Y)$ of $Y$ is defined as
$$
h(Y)=h(Y_1, Y_2, \dots, Y_n) \triangleq \EX[- \log f_{Y_1, Y_2, \dots, Y_n}(Y_1, Y_2, \dots, Y_n)],
$$
where $f_{Y_1, Y_2, \dots, Y_n}$ is the joint probability density function of $Y_1, Y_2, \dots, Y_n$.
The notion of mutual information can be further extended to generalized random processes, which we will only briefly describe and we refer the reader to~\cite{GelfandYaglom1957} for a more comprehensive exposition.
The mutual information between two generalized random processes $X=\{X(t)\}$ and $Y=\{Y(t)\}$ is defined as
\begin{equation} \label{general-definition}
I(X; Y) = \sup I(X(\phi_1), X(\phi_2), \dots, X(\phi_m); Y(\psi_1), Y(\psi_2), \dots, Y(\psi_n)),
\end{equation}
where the supremum is over all possible $n, m \in \mathbb{N}$ and all possible testing functions $\phi_1, \phi_2, \dots, \phi_m$ and $\psi_1, \psi_2, \dots, \psi_n$, and we have defined
$$
X(\phi_i) = \int X(t) \phi_i(t) dt, \quad i=1, 2, \dots, m, \mbox{ and } Y(\psi_j) = \int Y(t) \psi_j(t) dt, \quad j=1, 2, \dots, n.
$$
It can be verified that the general definition of mutual information as in (\ref{general-definition}) includes all previous definitions as special cases; moreover, when one of $X$ and $Y$, say, $Y$, is a random variable, the general definition boils down to
$$
I(X; Y) = \sup I(X(\phi_1), X(\phi_2), \dots, X(\phi_m); Y),
$$
where the supremum is over all possible $n \in \mathbb{N}$ and all possible testing functions $\phi_1, \phi_2, \dots, \phi_m$.
For the channel (\ref{ct}) with the input as in (\ref{non-feedback-case}) or (\ref{feedback-case}) , it is known that its mutual information over $[0, T]$ can be computed as (see, e.g., ~\cite{pi64,ih93}):
\begin{equation} \label{definition-mutual-information}
I(M; Y_0^T)=\begin{cases}
\EX\left[\log \frac{d \mu_{M, Y_0^T}}{d \mu_{M} \times \mu_{Y_0^T}}(M, Y_0^T)\right], &\mbox{ if } \frac{d \mu_{M, {Y_0^T}}}{d \mu_{M} \times \mu_{Y_0^T}} \mbox{ exists },\\
\infty, &\mbox{ otherwise },
\end{cases}
\end{equation}
where $d \mu_{M, Y_0^T}/d \mu_M \times \mu_{Y_0^T}$ denotes the Radon-Nikodym derivative of $\mu_{M, Y_0^T}$ with respect to $\mu_M \times \mu_{Y_0^T}$.
\section{The Non-Feedback Case}
In this section, we examine the AWGN channel (\ref{ct}) for the non-feedback case and give quantitative strengthenings of (\ref{limit-version}), detailed in the following theorem.
\begin{thm} \label{non-feedback-theorem}
For the continuous-time AWGN channel (\ref{ct}), suppose that the channel input $\{X(t)\}$ is a stationary stochastic process with power spectral density function $f(\cdot)$~\footnote{More precisely, the channel input $\{X(t): t \geq 0\}$ can be extended to a bi-infinite stationary stochastic process $\{X(t): -\infty < t < \infty\}$ with power spectral density function $f(\cdot)$.}. Then, the following two statements hold:
\begin{enumerate}
\item[(a)] Suppose $\int f(\lambda) |\lambda| d\lambda < \infty$. Then, for any $T$ and $n$,
$$
\sqrt{I(X_0^T; Y_0^T)} \leq \frac{\sqrt{2 T \delta_{T, n} \int f(\lambda) |\lambda| d \lambda}+\sqrt{2 T \delta_{T, n} \int f(\lambda) |\lambda| d\lambda + 4 I(X_0^T; Y(\Delta_{T, n}))}}{2}.
$$
\item[(b)]
Suppose $\int f(\lambda) |\lambda| d\lambda < \infty$ and $\int f(\lambda) d\lambda < \infty$. Then, for any $T$ and $n$,
$$
I(X_0^T; Y_0^T) - I(X_0^T; Y(\Delta_{T, n})) \leq T \sqrt{\delta_{T, n}} \left(\int f(\lambda) |\lambda| d\lambda \right)^{1/2} \left(\int f(\lambda) d\lambda \right)^{1/2}.
$$
Consequently, for any $T$, choosing $n = n(T)$ such that $\lim_{T \to \infty} \delta_{T, n(T)}= 0$, we have
$$
\frac{I(X_0^T; Y_0^T)}{T} - \frac{I(X_0^T; Y(\Delta_{T, n(T)}))}{T} = O\left(\sqrt{\delta_{T, n(T)}}\right),
$$
as $T$ tends to infinity.
\end{enumerate}
\end{thm}
\begin{proof}
Consider the following parameterized version of the channel (\ref{ct}):
\begin{equation} \label{snr-ct}
Z(t) = \sqrt{snr} \int_0^t X(s) ds + B(t), \quad t \in [0, T],
\end{equation}
where the parameter $snr > 0$ can be regarded as the signal-to-noise ratio of the channel. Obviously, when $snr$ is fixed to be $1$, $Z(t) = Y(t)$ for any $t \in [0, T]$, and moreover, the channel (\ref{snr-ct}) is exactly the same as the channel (\ref{ct}).
Sampling the channel (\ref{snr-ct}) with respect to sampling times $t_{0}, t_{1}, \dots, t_{n}$ that are evenly spaced over $[0, T]$, we obtain the following discrete-time Gaussian channel:
\begin{equation} \label{snr-dt}
Z(t_{i})-Z(t_{i-1})=\sqrt{snr} \int_{t_{i-1}}^{t_{i}} X(s) ds + B(t_{i}) - B(t_{i-1}), \quad i = 1, 2, \dots, n,
\end{equation}
which can be ``normalized'' as follows:
\begin{equation} \label{normalized-snr-dt}
\frac{Z(t_{i})-Z(t_{i-1})}{\sqrt{\delta_{T, n}}} = \sqrt{snr} \frac{\int_{t_{i-1}}^{t_{i}} X(s) ds}{\sqrt{\delta_{T, n}}} + \frac{B(t_{i}) - B(t_{ i-1})}{\sqrt{\delta_{T, n}}}, \quad i = 1, 2, \dots, n,
\end{equation}
where, at each time $i$, the channel noise $\frac{B(t_{i}) - B(t_{i-1})}{\sqrt{\delta_{T, n}}}$ is a standard Gaussian random variable, $\frac{\int_{t_{ i-1}}^{t_{i}} X(s) ds}{\sqrt{\delta_{T, n}}}$ and $\frac{Z(t_{i})-Z(t_{i-1})}{\sqrt{\delta_{T, n}}}$ should be regarded as the channel input and output, respectively.
By the continuous-time I-MMSE relationship~\cite{gu05} applied to the channel (\ref{snr-ct}), the mutual information of the channel (\ref{ct}) can be computed as
$$
I(X_0^T; Y_0^T) =\frac{1}{2} \int_0^1 \int_0^T \E[\left(X(s)-\E[X(s)|Z_0^T]\right)^2] ds d snr.
$$
And by the discrete-time I-MMSE relationship~\cite{gu05} (or more precisely, its extension~\cite{HanSong2016} to Gaussian memory channels) applied to the channel (\ref{normalized-snr-dt}), we have
$$
I(X_0^T; Y(\Delta_{T, n})) = \frac{1}{2} \int_0^1 \sum_{i=1}^n \E\left[\left(\frac{\int_{t_{i-1}}^{t_{i}} X(s) ds}{\sqrt{\delta_{T, n}}}-\E\left[\left.\frac{\int_{t_{i-1}}^{t_{i}} X(s) ds}{\sqrt{\delta_{T, n}}}\right|Z(\Delta_{T, n})\right]\right)^2\right] d snr.
$$
Obviously, by (\ref{I<=I}), it holds true that $I(X_0^T; Y_0^T) \geq I(X_0^T; Y(\Delta_{T, n}))$. In the following, we will give an upper bound on their difference $I(X_0^T; Y_0^T) - I(X_0^T; Y(\Delta_{T, n}))$, thereby characterizing the closeness between the two quantities. Towards this goal, using the following easily verified that for each $i$,
$$
\E\left[\left(\int_{t_{i-1}}^{t_i} X(s) - \E[X(s)|Z_0^T] \; ds \right)^2\right] \leq \E\left[\left(\int_{t_{i-1}}^{t_i} X(s) - \E[X(s)|Z(\Delta_{T, n})] \; ds \right)^2\right],
$$
we first note that
\begin{align*}
&I(X_0^T; Y_0^T) - I(X_0^T; Y(\Delta_{T, n}))\\
&= \frac{1}{2} \int_0^1 \int_0^T \E[\left(X(s)-\E[X(s)|Z_0^T]\right)^2] ds - \sum_{i=1}^n \E\left[\left(\int_{t_{i-1}}^{t_i} \frac{X(s)-\E[X(s)|Z(\Delta_{T, n})]}{\sqrt{\delta_{T, n}}} ds \right)^2 \right] d snr\\
&\leq \frac{1}{2} \int_0^1 \int_0^T \E[\left(X(s)-\E[X(s)|Z_0^T]\right)^2] ds - \sum_{i=1}^n \E\left[\left(\int_{t_{i-1}}^{t_i} \frac{X(s)-\E[X(s)|Z_0^T]}{\sqrt{\delta_{T, n}}} ds \right)^2 \right] d snr\\
&= \frac{1}{2} \int_0^1 \int_0^T \E[R^2[X(s); Z_0^T]] ds - \sum_{i=1}^n \E\left[\left(\int_{t_{i-1}}^{t_i} \frac{R[X(s); Z_0^T]}{\sqrt{\delta_{T, n}}} ds \right)^2 \right] d snr\\
& = \frac{1}{2} (S_1 + S_2)
\end{align*}
where we have used the shorthand notation $R[X(s); Z_0^T]$ for $X(s)-\E[X(s)|Z_0^T]$ and
{\small $$
S_1 \triangleq \sum_{i=1}^n \int_0^1 \int_{t_{i-1}}^{t_i} \E[R^2[X(s); Z_0^T]] ds - \sum_{i=1}^n \int_{t_{i-1}}^{t_i} \E\left[R[X(s); Z_0^T] R[X(t_{i-1}); Z_0^T] \right] ds d snr,
$$}
{\small $$
S_2 \triangleq \sum_{i=1}^n \int_0^1 \E\left[\left(\int_{t_{i-1}}^{t_i} R[X(s); Z_0^T] ds \right) R[X(t_{i-1}); Z_0^T] \right] - \sum_{i=1}^n \E\left[\left(\int_{t_{i-1}}^{t_i} \frac{R[X(s); Z_0^T]}{\sqrt{\delta_{T, n}}} ds \right)^2 \right] d snr.
$$}
For the first term, we have
\begin{align}
S_1^2 &= \left(\sum_{i=1}^n \int_{t_{i-1}}^{t_i} \int_0^1 \E[R^2[X(s); Z_0^T]] dsnr ds - \sum_{i=1}^n \int_{t_{i-1}}^{t_i} \int_0^1 \E\left[R[X(s); Z_0^T] R[X(t_{i-1}); Z_0^T] \right] dsnr ds \right)^2 \nonumber \\
&= \left(\sum_{i=1}^n \int_{t_{i-1}}^{t_i} \int_0^1 \E[R[X(s); Z_0^T] R[X(s)-X(t_{i-1}); Z_0^T]] dsnr ds \right)^2 \nonumber \\
&\leq n \sum_{i=1}^n \left(\int_{t_{i-1}}^{t_i} \int_0^1 \E[R[X(s); Z_0^T] R[X(s)-X(t_{i-1}); Z_0^T]] dsnr ds\right)^2 \nonumber \\
&\leq n \sum_{i=1}^n \int_{t_{i-1}}^{t_i} \int_0^1 \E[R^2[X(s); Z_0^T]] dsnr ds \int_{t_{i-1}}^{t_i} \int_0^1 \E[R^2[X(s)-X(t_{i-1}); Z_0^T]] dsnr ds, \label{r1}
\end{align}
where we have used the Cauchy-Scharz inequality for the last inequality. Now, noticing the fact that
$$
\E[R^2[X(s)-X(t_{i-1}); Z_0^T]] \leq \E[(X(s)-X(t_{i-1}))^2],
$$
we continue as follows:
\begin{align}
S_1^2&\leq n \sum_{i=1}^n \int_{t_{i-1}}^{t_i} \int_0^1 \E[R^2[X(s); Z_0^T]] dsnr ds \int_{t_{i-1}}^{t_i} \int_0^1 \E[(X(s)-X(t_{i-1}))^2] dsnr ds \label{r2}\\
&= n \sum_{i=1}^n \int_{t_{i-1}}^{t_i} \int_0^1 \E[R^2[X(s); Z_0^T]] dsnr ds \int_{t_{i-1}}^{t_i} \int_0^1 \E[X^2(s)+X^2(t_{i-1})-2 X(s) X(t_{i-1})] dsnr ds. \label{r3}
\end{align}
Now, using the fact that, for any $u, v \in \mathbb{R}$,
$$
\E[X(u) X(u+v)] = \int f(\lambda) e^{i v \lambda} d\lambda,
$$
we have
\begin{align}
S_1^2 &\leq n \sum_{i=1}^n \int_{t_{i-1}}^{t_i} \int_0^1 \E[R^2[X(s); Z_0^T]] dsnr ds \int_{t_{i-1}}^{t_i} \int_0^1 \int 2 f(\lambda) |1 - e^{i \lambda (s-t_{i-1})}| d\lambda dsnr ds \label{r4}\\
&\leq n \sum_{i=1}^n \int_{t_{i-1}}^{t_i} \int_0^1 \E[R^2[X(s); Z_0^T]] dsnr ds \int_{t_{i-1}}^{t_i} \int_0^1 \int 2 f(\lambda) |\lambda| (s-t_{i-1}) d\lambda dsnr ds \label{r5} \\
&\leq n \delta_{T, n}^2 \int f(\lambda) |\lambda| d\lambda \int_0^1 \sum_{i=1}^n \int_{t_{i-1}}^{t_i} \E[R^2[X(s); Z_0^T]] ds dsnr \label{r6}\\
&= T \delta_{T, n} \int f(\lambda) |\lambda| d\lambda \int_0^1 \int_0^T \E[(X(s)-\E[X(s)|Z_0^T])^2] ds dsnr \label{r7}\\
&= 2 T \delta_{T, n} \; I(X_0^T; Y_0^T) \int f(\lambda) |\lambda| d\lambda. \label{r8}
\end{align}
For the second term, we have
$$
S_2^2 = \left(\sum_{i=1}^n \int_{t_{i-1}}^{t_i} \int_0^1 \E \left[R[X(s); Z_0^T] \int_{t_{i-1}}^{t_i} \frac{R[X(t_{i-1})-X(u); Z_0^T]}{\delta_{T, n}} du \right] dsnr ds \right)^2.
$$
Starting from this and proceeding in a similar fashion as in (\ref{r1})-(\ref{r8}), we derive
$$
S_2^2 \leq 2 T \delta_{T, n} I(X_0^T; Y_0^T) \int f(\lambda) |\lambda| d\lambda.
$$
Combining the bounds on $S_1$ and $S_2$, we have
\begin{equation} \label{before-two-results}
I(X_0^T; Y_0^T) - I(X_0^T; Y(\Delta_{T, n})) \leq \sqrt{2 T \delta_{T, n}} \left(I(X_0^T; Y_0^T) \int f(\lambda) |\lambda| d\lambda \right)^{1/2}.
\end{equation}
It then immediately follows that
$$
\sqrt{I(X_0^T; Y_0^T)} \leq \frac{\sqrt{2 T \delta_{T, n} \int f(\lambda) |\lambda| d \lambda}+\sqrt{2 T \delta_{T, n} \int f(\lambda) |\lambda| d\lambda + 4 I(X_0^T; Y_{t_0}^{t_n})}}{2},
$$
establishing ($a$). Moreover, together with the fact that
$$
I(X_0^T; Y_0^T) \leq \frac{1}{2} \int_0^T \E[X^2(s)] ds = \frac{T}{2} \int f(\lambda) d\lambda,
$$
the inequality (\ref{before-two-results}) implies that
$$
\left|I(X_0^T; Y_0^T) - I(X_0^T; Y(\Delta_{T, n})) \right| \leq T \sqrt{\delta_{T, n}} \left(\int f(\lambda) |\lambda| d\lambda \right)^{1/2} \left(\int f(\lambda) d\lambda \right)^{1/2},
$$
establishing ($b$).
\end{proof}
The following corollary, which immediately follows from Theorem~\ref{non-feedback-theorem}, characterizes, among others, how over-sampling approaches the true mutual information of the channel (\ref{ct}) with bandwidth limit.
\begin{co} \label{non-feedback-corollary}
For the continuous-time AWGN channel (\ref{ct}), suppose that the channel input has bandwidth limit $W$ and average power $P$, more precisely, $f(\lambda) = 0$ for any $\lambda \in (-\infty, -W] \cup [W, \infty)$ and $\E[X^2(s)] \leq P$ for any $s \geq 0$. Then, the following two statements hold:
\begin{enumerate}
\item[(a)] For any $T$ and $n$,
$$
\sqrt{I(X_0^T; Y_0^T)} \leq \frac{\sqrt{2 T P W \delta_{T, n}}+\sqrt{2 T P W \delta_{T, n} + 4 I(X_0^T; Y(\Delta_{T, n}))}}{2}.
$$
\item[(b)]
For any $T$ and $n$,
$$
I(X_0^T; Y_0^T) - I(X_0^T; Y(\Delta_{T, n})) \leq T P \sqrt{W \delta_{T, n}}.
$$
Consequently, for each $T$, choosing $n = n(T)$ such that $\lim_{T \to \infty} \delta_{T, n(T)}= 0$, we have
$$
\frac{I(X_0^T; Y_0^T)}{T} - \frac{I(X_0^T; Y(\Delta_{T, n}))}{T} = O\left(\sqrt{\delta_{T, n(T)}}\right),
$$
as $T$ tends to infinity.
\end{enumerate}
\end{co}
\begin{rem} \label{extension-feedback}
The I-MMSE relationship has played an important role in deriving our results for non-feedback AWGN channels. This powerful tool has been extended~\cite{HanSong2016} to feedback AWGN channels in both discrete and continuous time. A natural question is whether the extended relationship can help us to derive counterpart results to Theorem~\ref{non-feedback-theorem} for the feedback case. As tempting and promising as this idea may look, we failed to find a way to effectively apply the extended I-MMSE relationship, and our treatment for the feedback case, to be detailed in the next section, will use other tools and techniques from the theory of stochastic calculus. Here, we remark that the formulas derived in~\cite{HanSong2016} are valid only with some extra assumption imposed and yet fail to hold true in general. For more detailed explanations and corrected formulas, see Arxiv:1401.3527.
\end{rem}
\section{The Feedback Case} \label{extensions}
In this section, we give quantitative strengthenings of (\ref{limit-version}) for the AWGN channel (\ref{ct}) in the feedback case, which we remind the reader is characterized by the stochastic differential equation in (\ref{SDE}).
The following regularity conditions may be imposed:
\begin{itemize}
\item[(a)] The solution $\{Y(t)\}$ to the stochastic differential equation (\ref{SDE}) uniquely exists.
\item[(b)]
$$
\PX\left(\int_0^T g^2(t, M, Y_0^t) dt < \infty \right)=\PX\left(\int_0^T g^2(t, M, B_0^t) dt < \infty \right)=1.
$$
\item[(c)]
$$
\int_0^T \EX[|g(t, M, Y_0^t)|] dt < \infty.
$$
\item[(d)] \textbf{The uniform Lipschitz condition:} There exists a constant $L > 0$ such that for any $0 \leq s_1, s_2, t_1, t_2 \leq T$, any $Y_0^T, Z_0^T$,
$$
|g(s_1, M, Y_0^{s_2})-g(t_1, M, Z_0^{t_2})| \leq L (|s_1-t_1|+ \|Y_{0}^{s_2}- Z_0^{t_2}\|).
$$
\item[(e)] \textbf{The uniform linear growth condition:} There exists a constant $L > 0$ such that for any $M$ and any $Y_0^T$,
$$
|g(t, M, Y_0^t)| \leq L (1+\|Y_0^t\|).
$$
\end{itemize}
We will need the following lemma, which has already been established~\cite{LiuHan2019} in a slightly more general setting.
\begin{lem} \label{improved-liptser-1}
Assume Conditions (d)-(e). Then, there exists a unique strong solution of (\ref{SDE}) with initial value $Y(0)=0$. Moreover, there exists $\varepsilon > 0$ such that
\begin{equation} \label{exponential-finiteness-1}
\EX [e^{\varepsilon \|Y_0^T\|^2}] < \infty,
\end{equation}
which immediately implies Conditions (b) and (c).
\end{lem}
We will also need the following lemma, whose proof is deferred to Section~\ref{proof-square-exponential}.
\begin{lem} \label{square-exponential}
Let $Z_{max} = \max \{Z_1, Z_2, \dots, Z_n\}$, where the i.i.d. random variables $Z_i \sim N(0, 1/n)$. Then, we have:
\begin{itemize}
\item[a)] For any $0 < \varepsilon < 1$, $\EX[Z_{max}^2] = O(n^{-(1-\varepsilon)})$.
\item[b)] For any $0 < \varepsilon < 1$, $\EX[Z_{max}^4] = O(n^{-(2-\varepsilon)})$.
\item[b)] For any $0 < \varepsilon < 1$, $\EX[e^{Z_{max}^2}] = 1 + O(n^{-(1-\varepsilon)})$.
\end{itemize}
\end{lem}
As detailed below, the following theorem gives the aforementioned quantitative strengthening of (\ref{limit-version}). Here, we mention that with the channel input as in (\ref{SDE}), the discrete-time channel (\ref{dt}) obtained by sampling the channel (\ref{ct}) over $[0, T]$ with respect to $\Delta_{T, n}$ takes the following form:
\begin{equation} \label{after-sampling-with-feedback}
Y(t_i)=\int_0^{t_i} g(s, M, Y_0^s) ds + B(t_i), \quad i=0, 1, \ldots, n.
\end{equation}
\begin{thm} \label{feedback-theorem}
Fix $T > 0$ and assume Conditions (d)-(e). Then, for any $0 < \varepsilon < 1/2$, we have
$$
I(M; Y_0^T) = I(M; Y(\Delta_{T, n})) + O(\delta_{\Delta_{T, n}}^{1/2-\varepsilon}),
$$
where we recall from Section~\ref{notations} that $Y(\Delta_{T, n}) = \{Y(t_0), Y(t_1), \ldots, Y(t_n)\}$.
\end{thm}
\begin{proof}
First of all, recall from Section~\ref{notations} that for a stochastic process $\{X(s)\}$ and any $t \in \mathbb{R}_+$, we use $\mu_{X_0^t}$ to denote the distribution on $C[0, t]$ induced by $X_0^t$. Throughout the proof, we only have to deal with the case $t=T$, and so we will simply write $\mu_{X_0^T}$ as $\mu_X$. Moreover, we will rewrite $\Delta_{T, n}$ as $\Delta_n$ and $\delta_{T, n}$ as $\delta_n$ for notational simplicity.
Note that an application of Theorem $7.14$ of~\cite{li01} with Conditions ($b$) and ($c$) yields that
\begin{equation} \label{crazy}
\PX\left(\int_0^T \EX^2[g(t, M, Y_0^t)|Y_0^t] dt < \infty \right)=1.
\end{equation}
Then one verifies that the assumptions of Lemma $7.7$ of~\cite{li01} are all satisfied (this lemma is stated under very general assumptions, which are exactly Conditions (b), (c) and (\ref{crazy}) when restricted to our settings), which implies that for any $m$,
\begin{equation} \label{two-tilde}
\mu_Y \sim \mu_{Y|M=m} \sim \mu_B,
\end{equation}
and moreover, with probability $1$,
{\small \begin{equation} \label{RN-1}
\frac{d\mu_{Y|M}}{d\mu_B}(Y_0^T|M)=\frac{1}{\EX[e^{\rho_1(M, Y_0^T)}|Y_0^T, M]}, \quad \frac{d\mu_{Y}}{d\mu_B}(Y_0^T)=\frac{1}{\EX[e^{\rho_1(M, Y_0^T)}|Y_0^T]},
\end{equation}}
where
$$
\rho_1(m, Y_0^T) \triangleq -\int_0^T g(s, m, Y_0^s) dY(s)+\frac{1}{2} \int_0^T g(s, m, Y_0^s)^2 ds.
$$
Here we note that $\EX[e^{\rho_1(M, Y_0^T)}|Y_0^T, M]$ is in fact equal to $e^{\rho_1(M, Y_0^T)}$, but we keep it the way it is as above for an easy comparison.
Note that it follows from $\EX[d\mu_B/d\mu_Y(Y_0^T)]=1$ that $\EX[e^{\rho_1(M, Y_0^T)}]=1$, which is equivalent to
\begin{equation} \label{weak-Novikov}
\EX[e^{-\int_0^T g(s, M, Y_0^s) dB(s)-\frac{1}{2} \int_0^T g(s, M, Y_0^s)^2 ds}]=1.
\end{equation}
Then, a parallel argument as in the proof of Theorem $7.1$ of~\cite{li01} (which requires the condition (\ref{weak-Novikov})) further implies that, for any $n$,
{\small \begin{equation} \label{RN-2}
\hspace{-1.0cm} \frac{d\mu_{Y(\Delta_n)|M}}{d\mu_{B(\Delta_n)}}(Y(\Delta_n)|M) = \frac{1}{\EX[e^{\rho_1(M, Y_0^T)}|Y(\Delta_n), M]}, \quad \frac{d\mu_{Y(\Delta_n)}}{d\mu_{B(\Delta_n)}}(Y(\Delta_n))=\frac{1}{\EX[e^{\rho_1(M, Y_0^T)}|Y(\Delta_n)]}, ~~\mbox{a.s.}.
\end{equation}}
Next, by the definition of mutual information, we have
\begin{align}
I(M; Y(\Delta_n)) &= \EX\left[\log f_{Y(\Delta_n)|M}(Y(\Delta_n)|M)\right]-\EX\left[\log f_{Y(\Delta_n)}(Y(\Delta_n))\right] \nonumber\\
& = \EX\left[\log \frac{d\mu_{Y(\Delta_n)|M}}{d\mu_{B(\Delta_n)}}(Y(\Delta_n)|M)\right]-\EX\left[\log \frac{d\mu_{Y(\Delta_n)}}{d\mu_{B(\Delta_n)}}(Y(\Delta_n))\right], \label{l-3}
\end{align}
and
\begin{align}
I(M; Y_0^T) & = \EX\left[\log \frac{d \mu_{M, Y_0^T}}{d \mu_{M} \times \mu_{Y_0^T}}(M, Y_0^T)\right] \nonumber\\
& =\EX\left[\log \frac{d\mu_{Y|M}}{d\mu_{B}}(Y_0^T|M)\right]-\EX\left[\log \frac{d\mu_{Y}}{d\mu_B}(Y_0^T)\right], \label{l-4}
\end{align}
where the well-definedness of the Radon-Nikodym derivatives are guaranteed by (\ref{two-tilde}).
By (\ref{l-3}), (\ref{RN-1}) and (\ref{RN-2}), we have
{\small \begin{align*}
I(M; Y(\Delta_n)) &= \EX\left[\log \frac{d\mu_{Y(\Delta_n)|M}}{d\mu_{B(\Delta_n)}}(Y(\Delta_n)|M)\right]-\EX\left[\log \frac{d\mu_{Y(\Delta_n)}}{d\mu_{B(\Delta_n)}}(Y(\Delta_n))\right]\\
&= -\EX[\log \EX[e^{\rho_1(M, Y_0^T)}|Y(\Delta_n), M]] + \EX[\log \EX[e^{\rho_1(M, Y_0^T)}|Y(\Delta_n)]],
\end{align*}}
and
{\small \begin{align*}
I(M; Y_0^T) &= \EX\left[\log \frac{d\mu_{Y|M}}{d\mu_{B}}(Y_0^T|M)\right]-\EX\left[\log \frac{d\mu_{Y}}{d\mu_{B}}(Y_0^T)\right]\\
&= -\EX[\log \EX[e^{\rho_1(M, Y_0^T)}|Y_0^T, M]] + \EX[\log \EX[e^{\rho_1(M, Y_0^T)}|Y_0^T]]\\
&= -\rho_1(M, Y_0^T) + \EX[\log \EX[e^{\rho_1(M, Y_0^T)}|Y_0^T]].
\end{align*}}
We next proceed in the following two steps.
{\bf Step $\bf 1$.} In this step, we show that for any $0 < \varepsilon < 1/2$,
$$
\EX\left[\log \frac{d\mu_{Y|M}}{d\mu_{B}}(Y_0^T|M)\right] - \EX\left[\log \frac{d\mu_{Y(\Delta_n)|M}}{d\mu_{B(\Delta_n)}}(Y(\Delta_n)|M)\right] = O(\delta^{1/2-\varepsilon}_{n}).
$$
Apparently, it suffices to show that for any $0 < \varepsilon < 1/2$,
\begin{equation} \label{Fn-Conv}
\EX[|\log \EX[e^{\rho_1(M, Y_0^T)}|Y(\Delta_n), M] - \rho_1(M, Y_0^T)|] = O(\delta^{1/2-\varepsilon}_{n}).
\end{equation}
Let $\bar{Y}_{\Delta_n, 0}^T$ denote the piecewise linear version of $Y_0^T$ with respect to $\Delta_n$; more precisely, for any $i=0, 1, \dots, n$, let $\bar{Y}_{\Delta_n}(t_{i})=Y(t_{i})$, and for any $t_{i-1} < s < t_{i}$ with $s=\lambda t_{i-1}+(1-\lambda) t_{i}$ for some $0 < \lambda < 1$, let $\bar{Y}_{\Delta_n}(s)=\lambda Y(t_{i-1})+(1-\lambda) Y(t_{i})$. Let $\bar{g}_{\Delta_n}(s, M, \bar{Y}_{\Delta_n, 0}^s)$ denote the piecewise ``flat'' version of $g(s, M, \bar{Y}_{\Delta_n, 0}^s)$ with respect to $\Delta_n$; more precisely, for any $t_{i-1} \leq s < t_{i}$, $\bar{g}_{\Delta_n}(s, M, \bar{Y}_{\Delta_n, 0}^s)=g(t_{i-1}, M, \bar{Y}_{\Delta_n, 0}^{t_{i-1}})$.
Letting
$$
\rho_2(\Delta_n, m, Y_0^T) \triangleq -\int_0^T \bar{g}_{\Delta_n}(s, m, \bar{Y}_{\Delta_n, 0}^s) dY(s)+\frac{1}{2} \int_0^T \bar{g}^2_{\Delta_n}(s, m, \bar{Y}_{\Delta_n, 0}^s) ds,
$$
we have
\begin{align*}
\log \EX[e^{\rho_1(M, Y_0^T)}|Y(\Delta_n), M] & = \log \EX[e^{\rho_2(\Delta_n, M, Y_0^T) +\rho_1(M, Y_0^T)-\rho_2(\Delta_n, M, Y_0^T)}|Y(\Delta_n), M]\\
& = \log e^{\rho_2(\Delta_n, M, Y_0^T)} \EX[e^{\rho_1(M, Y_0^T)-\rho_2(\Delta_n, M, Y_0^T)}|Y(\Delta_n), M]\\
& = \rho_2(\Delta_n, M, Y_0^T) +\log \EX[e^{\rho_1(M, Y_0^T)-\rho_2(\Delta_n, M, Y_0^T)}|Y(\Delta_n), M],
\end{align*}
where we have used the fact that
\begin{equation} \label{rho-2-adapted}
\EX[e^{\rho_2(\Delta_n, M, Y_0^T)}|Y(\Delta_n), M]=e^{\rho_2(\Delta_n, M, Y_0^T)},
\end{equation}
since $\rho_2(\Delta_n, M, Y_0^T)$ only depends on $M$ and $Y(\Delta_n)$.
We now prove the following rate of convergence:
\begin{equation} \label{conv-1}
\EX\left[(\rho_1(M, Y_0^T)-\rho_2(\Delta_n, M, Y_0^T))^2\right] = O(\delta_n^{1-\varepsilon}),
\end{equation}
where $0 < \varepsilon < 1$. To this end, we note that
\begin{equation} \label{rho1-rho2}
\rho_1(M, Y_0^T)-\rho_2(\Delta_n, M, Y_0^T) = -\int_0^T (g(s)-\bar{g}_{\Delta_n}(s)) dB(s) - \frac{1}{2} \int_0^T (g(s)-\bar{g}_{\Delta_n}(s))^2 ds,
\end{equation}
where we have rewritten $g(s, M, Y_0^s)$ as $g(s)$, and $\bar{g}_{\Delta_n}(s, M, \bar{Y}_{\Delta_n, 0}^s)$ as $\bar{g}_{\Delta_n}(s)$. It then follows that (\ref{conv-1}) boils down to
\begin{equation} \label{conv-1a}
\EX\left[\left(-\int_0^T (g(s)-\bar{g}_{\Delta_n}(s)) dB(s) - \frac{1}{2} \int_0^T (g(s)-\bar{g}_{\Delta_n}(s))^2 ds \right)^2 \right] = O(\delta_n^{1-\varepsilon}).
\end{equation}
To establish (\ref{conv-1a}), notice that, by the It\^{o} isometry~\cite{ok95}, we have
$$
\EX\left[\left(\int_0^T (g(s)-\bar{g}_{\Delta_n}(s)) dB(s) \right)^2\right] = \EX\left[\int_0^T (g(s)-\bar{g}_{\Delta_n}(s))^2 ds \right].
$$
We next prove that for any $0 < \varepsilon < 1$,
\begin{equation} \label{conv-1b}
\EX\left[\int_0^T (g(s)-\bar{g}_{\Delta_n}(s))^2 ds\right] = O(\delta_n^{1-\varepsilon}).
\end{equation}
To see this, we note that, by Conditions ($d$) and ($e$), there exists $L_1 > 0$ such that for any $s \in [0, T]$ with $t_{i-1} \leq s < t_{i}$,
\begin{align}
\hspace{-1cm} |g(s, M, \bar{Y}_{\Delta_n, 0}^s)-\bar{g}_{\Delta_n}(s, M, \bar{Y}_{\Delta_n, 0}^s)| & = |g(s, M, \bar{Y}_{\Delta_n, 0}^s)-g(t_{i-1}, M, \bar{Y}_{\Delta_n, 0}^{t_{i-1}})| \nonumber\\
& \leq L_1 (|s-t_{i-1}| +\|\bar{Y}_{\Delta_n, 0}^s-\bar{Y}_{\Delta_n, 0}^{t_{i-1}}\|) \nonumber\\
& \leq L_1 (|s-t_{i-1}| + |Y(t_{i})-Y(t_{i-1})|) \nonumber\\
& \leq L_1 \delta_n+ L_1 \delta_n+L_1 \delta_n \|Y_0^T\|+|B(t_{i})-B(t_{i-1})|. \label{diff-2}
\end{align}
Similarly, there exists $L_2 > 0$ such that for any $s \in [0, T]$,
\begin{align*}
& \hspace{-1cm} |g(s, M, Y_{0}^s)-g(s, M, \bar{Y}_{\Delta_n, 0}^s)| \\
& \leq L_2 \delta_n+L_2 \delta_n \|Y_0^T\|+\max_i \max_{s \in [t_{i-1}, t_i]} \max\{|B(s)-B(t_{i-1})|, |B(s)-B(t_{i})|\}\\
& \leq L_2 \delta_n+L_2 \delta_n \|Y_0^T\| + \max_i \left(\max\{\max_{s \in [t_{i-1}, t_i]} (B(s)-B(t_{i-1})), - \min_{s \in [t_{i-1}, t_i]} (B(s)-B(t_{i-1})), \right.\\
& \hspace{6cm} \left. \max_{s \in [t_{i-1}, t_i]} (B(s)-B(t_{i})), - \min_{s \in [t_{i-1}, t_i]} (B(s)-B(t_{i})) \} \right)\\
& \leq L_2 \delta_n+L_2 \delta_n \|Y_0^T\| + \max_i \left\{\max_{s \in [t_{i-1}, t_i]} (B(s)-B(t_{i-1})) \right\} + \max_i \left\{- \min_{s \in [t_{i-1}, t_i]} (B(s)-B(t_{i-1})) \right\}\\
& \hspace{5cm} + \max_i \left\{\max_{s \in [t_{i-1}, t_i]} (B(s)-B(t_{i})) \right\} + \max_i \left\{- \min_{s \in [t_{i-1}, t_i]} (B(s)-B(t_{i})) \right\}.
\end{align*}
It is well-known (see, e.g., Theorem $2.21$ in~\cite{MortersPeres2010}) that $\max_{s \in [t_{i-1}, t_i]} (B(s)-B(t_{i-1}))$ is distributed as $|B(t_{i})-B(t_{i-1})|$, which, together with the fact that $\{B(t)\}$ has independent increments, leads to
\begin{align*}
& \hspace{-2cm} \E\left[\left(\max_i \left\{\max_{s \in [t_{i-1}, t_i]} (B(s)-B(t_{i-1})) \right\}\right)^2\right] = \E\left[\left(\max_i \left\{|B(t_{i})-B(t_{i-1})| \right\} \right)^2 \right] \\
& \leq \E\left[\left(\max_i \left\{B(t_{i})-B(t_{i-1}) \right\} \right)^2 \right] + \E\left[\left(\min_i \left\{B(t_{i})-B(t_{i-1}) \right\} \right)^2 \right].
\end{align*}
Now, applying Lemma~\ref{square-exponential}~$a)$, we conclude that for any $0 < \varepsilon < 1$,
$$
\E\left[\left(\max_i \left\{\max_{s \in [t_{i-1}, t_i]} (B(s)-B(t_{i-1})) \right\}\right)^2\right] = O(\delta_n^{1-\varepsilon}).
$$
And a completely parallel argument yields that for any $0 < \varepsilon < 1$,
$$
\E\left[\left(\max_i \left\{-\min_{s \in [t_{i-1}, t_i]} (B(s)-B(t_{i-1})) \right\}\right)^2\right] = O(\delta_n^{1-\varepsilon}),
$$
and moreover,
$$
\E\left[\left(\max_i \left\{-\min_{s \in [t_{i-1}, t_i]} (B(s)-B(t_{i})) \right\}\right)^2\right] = O(\delta_n^{1-\varepsilon}),
$$
$$
\E\left[\left(\max_i \left\{-\min_{s \in [t_{i-1}, t_i]} (B(s)-B(t_{i})) \right\}\right)^2\right] = O(\delta_n^{1-\varepsilon}),
$$
where for the latter two, we have used, in addition, the well-known fact that the time reversed Brownian motion is still a Brownian motion. Noticing that, by Lemma~\ref{improved-liptser-1}, $\|Y_0^T\|^2$ is integrable, we arrive at (\ref{conv-1b}), up to an arbitrarily small $\varepsilon > 0$.
A similar argument as above, coupled with Lemma~\ref{square-exponential}~$b)$ (rather than Lemma~\ref{square-exponential}~$a)$), will yield that for any $0 < \varepsilon < 1$,
\begin{equation} \label{conv-1f}
\EX\left[\left(\int_0^T (g(s)-\bar{g}_{\Delta_n}(s))^2 ds \right)^2\right] = O(\delta_n^{2-\varepsilon}),
\end{equation}
which, together with (\ref{conv-1a}) and (\ref{conv-1b}), implies (\ref{conv-1}), as desired.
We now prove the following rate of convergence:
\begin{equation} \label{conv-2}
\EX[\log \EX[e^{\rho_1(M, Y_0^T)-\rho_2(\Delta_n, M, Y_0^T)}|Y(\Delta_n), M]] = O(\delta_n^{1-\varepsilon}),
\end{equation}
where $0 < \varepsilon < 1$. To this end, we first note that
\begin{align*}
\EX[\log \EX[e^{\rho_1(M, Y_0^T)-\rho_2(\Delta_n, M, Y_0^T)}|Y(\Delta_n), M]] & \leq \log \EX[\EX[e^{\rho_1(M, Y_0^T)-\rho_2(\Delta_n, M, Y_0^T)}|Y(\Delta_n), M]] \\
&= \log \EX[e^{\rho_1(M, Y_0^T)-\rho_2(\Delta_n, M, Y_0^T)}]\\
&\stackrel{(a)}{=} \log \EX[e^{-\int_0^T (g(s)-\bar{g}_{\Delta_n}(s)) dB(s) - \frac{1}{2} \int_0^T (g(s)-\bar{g}_{\Delta_n}(s))^2 ds}]\\
&\leq 0,
\end{align*}
where for ($a$), we have used (\ref{rho1-rho2}), and for the last inequality, we have used the fact that
$$
\EX[e^{-\int_0^T (g(s)-\bar{g}_{\Delta_n}(s)) dB(s) - \frac{1}{2} \int_0^T (g(s)-\bar{g}_{\Delta_n}(s))^2 ds}] \leq 1,
$$
which is a well-known fact that follows from Fatou's lemma. For another direction, we have
\begin{align*}
\EX[\log \EX[e^{\rho_1(M, Y_0^T)-\rho_2(\Delta_n, M, Y_0^T)}|Y(\Delta_n), M]] & \geq \EX[\EX[\log e^{\rho_1(M, Y_0^T)-\rho_2(\Delta_n, M, Y_0^T)}|Y(\Delta_n), M]] \\
&= \EX[\EX[\rho_1(M, Y_0^T)-\rho_2(\Delta_n, M, Y_0^T)|Y(\Delta_n), M]] \\
&= \EX[\rho_1(M, Y_0^T)-\rho_2(\Delta_n, M, Y_0^T)]\\
&= - \frac{1}{2} \int_0^T \EX[(g(s)-\bar{g}_{\Delta_n}(s))^2] ds,
\end{align*}
which, together with (\ref{conv-1b}), leads to (\ref{conv-2}).
{\bf Step $\bf 2$.} In this step, we will prove that for any $0 < \varepsilon < 1/2$, there exists a constant $C > 0$ such that for all $n$,
\begin{equation} \label{Gn-Conv}
-\EX\left[\log \frac{d\mu_{Y}}{d\mu_B}(Y_0^T)\right] + \EX\left[\log \frac{d\mu_{Y(\Delta_n)}}{d\mu_{B(\Delta_n)}}(Y(\Delta_n))\right] \leq C \delta_n^{1/2-\varepsilon}.
\end{equation}
First of all, note that by Theorem $6.2.2$ in~\cite{ih93}, we have,
$$
\frac{d \mu_{Y}}{d\mu_B}(Y_0^T) = e^{\int_0^T \hat{g}(Y_0^s) dY(s)-\frac{1}{2} \int_0^T \hat{g}^2(Y_0^s) ds},
$$
where $\hat{g}(Y_0^s)=\EX[g(s, M, Y_0^s)|Y_0^s]$. Moreover, by Theorem $7.23$ of~\cite{li01}, we have,
$$
\frac{d \mu_{Y}}{d\mu_B}(Y_0^T) = \int \frac{d \mu_{Y|M}}{d\mu_B}(Y_0^T|m) d\mu_M(m),
$$
where
$$
\frac{d \mu_{Y|M}}{d\mu_B}(Y_0^T|m) = e^{\int_0^T g(s, m, Y_0^s) dY(s)-\frac{1}{2} \int_0^T g^2(s, m, Y_0^s) ds}.
$$
Similarly, we have
\begin{align*}
\frac{d\mu_{Y(\Delta_n)}}{d\mu_{B(\Delta_n)}}(Y(\Delta_n)) &= \int \frac{d\mu_{Y(\Delta_n)|M}}{d\mu_{B(\Delta_n)}}(Y(\Delta_n)|m) d\mu_M(m)\\
&= \int \frac{1}{\EX[e^{\rho_1(M, Y_0^T)}|Y(\Delta_n), m]} d\mu_M(m).
\end{align*}
It then follows that
\begin{align*}
\hspace{-1cm} -\EX\left[\log \frac{d\mu_{Y}}{d\mu_B}(Y_0^T)\right] + \EX\left[\log \frac{d\mu_{Y(\Delta_n)}}{d\mu_{B(\Delta_n)}}(Y(\Delta_n))\right] & = -\EX\left[\log e^{-\hat{\rho}_1(Y_0^T)}\right] + \EX \left[ \log \int \frac{1}{\EX[e^{\rho_1(M, Y_0^T)}|Y(\Delta_n), m]} d\mu_M(m) \right]\\
& = \EX \left[ \log \int \frac{e^{\hat{\rho}_1(Y_0^T)-\rho_2(\Delta_n, m, Y_0^T)}}{\EX[e^{\rho_1(M, Y_0^T)-\rho_2(\Delta_n, M, Y_0^T)}|Y(\Delta_n), m]} d\mu_M(m) \right],
\end{align*}
where we have used the shorthand notation $\hat{\rho}_1(Y_0^T)$ for $-\int_0^T \hat{g}(Y_0^s) dY(s)+\frac{1}{2} \int_0^T \hat{g}^2(Y_0^s) ds$, and we have used (\ref{rho-2-adapted}) in deriving the last equality. It then follows that
\begin{align*}
& \hspace{-1cm} \EX \left[ \log \int \frac{e^{\hat{\rho}_1(Y_0^T)-\rho_2(\Delta_n, m, Y_0^T)}}{\EX[e^{\rho_1(M, Y_0^T)-\rho_2(\Delta_n, M, Y_0^T)}|Y(\Delta_n), m]} d\mu_M(m) \right] \\
& \leq \EX \left[ \log \int e^{\hat{\rho}_1(Y_0^T)-\rho_2(\Delta_n, m, Y_0^T)} \EX[e^{-\rho_1(M, Y_0^T)+\rho_2(\Delta_n, M, Y_0^T)}|Y(\Delta_n), m] d\mu_M(m) \right]\\
& \leq \log \EX \left[ e^{\hat{\rho}_1(Y_0^T)-\rho_2(\Delta_n, M, Y_0^T)} \EX[e^{-\rho_1(M, Y_0^T)+\rho_2(\Delta_n, M, Y_0^T)}|Y(\Delta_n), M] \left(\frac{d\mu_Y}{d\mu_B}(Y_0^T)\right)/\left(\frac{d\mu_{Y|M}}{d\mu_B}(Y_0^T|M)\right) \right]\\
& =\log \EX \left[ e^{\rho_1(M, Y_0^T)-\rho_2(\Delta_n, M, Y_0^T)} \EX[e^{-\rho_1(M, Y_0^T)+\rho_2(\Delta_n, M, Y_0^T)}|Y(\Delta_n), M] \right].
\end{align*}
Now, applying the Cauchy-Schwarz inequality, we have
\begin{align*}
& \EX^2 \left[ e^{\rho_1(M, Y_0^T)-\rho_2(\Delta_n, M, Y_0^T)} \EX[e^{-\rho_1(M, Y_0^T)+\rho_2(\Delta_n, M, Y_0^T)}|Y(\Delta_n), M] \right] \\
& \hspace{2cm} \leq \EX \left[ e^{2 \rho_1(M, Y_0^T)- 2 \rho_2(\Delta_n, M, Y_0^T)} \right] \EX\left[\EX^2[e^{-\rho_1(M, Y_0^T)+\rho_2(\Delta_n, M, Y_0^T)}|Y(\Delta_n), M] \right]\\
& \hspace{2cm} \leq \EX \left[ e^{2 \rho_1(M, Y_0^T)- 2 \rho_2(\Delta_n, M, Y_0^T)} \right] \EX\left[\EX [e^{-2\rho_1(M, Y_0^T)+2\rho_2(\Delta_n, M, Y_0^T)}|Y(\Delta_n), M] \right]\\
& \hspace{2cm} = \EX \left[ e^{2 \rho_1(M, Y_0^T)- 2 \rho_2(\Delta_n, M, Y_0^T)} \right] \EX\left[e^{-2\rho_1(M, Y_0^T)+2\rho_2(\Delta_n, M, Y_0^T)} \right].
\end{align*}
Again, applying the Cauchy-Schwarz inequality, we have
\begin{align*}
\EX^2[e^{2 \rho_1(M, Y_0^T)-2 \rho_2(\Delta_n, M, Y_0^T)}] & = \EX^2[e^{-2 \int_0^T (g(s)-\bar{g}_{\Delta_n}(s)) dB(s) - \int_0^T (g(s)-\bar{g}_{\Delta_n}(s))^2 ds}]\\
& = \EX^2[e^{-2 \int_0^T (g(s)-\bar{g}_{\Delta_n}(s)) dB(s) - 4 \int_0^T (g(s)-\bar{g}_{\Delta_n}(s))^2 ds + 3 \int_0^T (g(s)-\bar{g}_{\Delta_n}(s))^2 ds}]\\
& \leq \EX[e^{-4 \int_0^T (g(s)-\bar{g}_{\Delta_n}(s)) dB(s) - 8 \int_0^T (g(s)-\bar{g}_{\Delta_n}(s))^2 ds}] \EX[e^{6 \int_0^T (g(s)-\bar{g}_{\Delta_n}(s))^2 ds}]\\
& \leq \EX[e^{6 \int_0^T (g(s)-\bar{g}_{\Delta_n}(s))^2 ds}],
\end{align*}
where for the last inequality, we have used Fatou's lemma. Now, using a largely parallel argument as in Step $1$ coupled with Lemma~\ref{square-exponential}~$c)$, we conclude that for any $0 < \varepsilon < 1$,
$$
\EX^2[e^{2 \rho_1(M, Y_0^T)-2 \rho_2(\Delta_n, M, Y_0^T)}] \leq \EX[e^{6 \int_0^T (g(s)-\bar{g}_{\Delta_n}(s))^2 ds}] = 1 + O(\delta_n^{1-\varepsilon}),
$$
which further leads to (\ref{Gn-Conv}).
The theorem then immediately follows from Steps $1$, $2$ and the fact that
$$
0 \leq I(M; Y_0^T) - I(M; Y(\Delta_{T, n})).
$$
\end{proof}
\section{Concluding Remarks}
As opposed to the Brownian motion formulation in (\ref{ct}), a continuous-time AWGN channel can be alternatively characterized by the following white noise formulation:
$$
Y(t) = X(t) + Z(t),~~~t \in \mathbb{R}
$$
where $\{Z(t)\}$ is a white Gaussian noise with flat spectral density $1$, and slightly abusing the notation, we have still used $X$ and $Y$, parameterized by $t \in \mathbb{R}$, to represent the channel input and output, respectively. While there is a comprehensive comparison between these two models in~\cite{LiuHan2019}, we emphasize here that the Browninan motion formulation enables a more rigorous information-theoretic examination of AWGN channels and empowers the tools in the theory of stochastic calculus that seem to be essential for more quantitative results for such channels.
Indeed, we believe the framework and techniques developed in this work can be applied to a more quantitative investigation of sampling a wider range of Gaussian channels possibly with different input constraints and related issues, detailed below. First, it is possible that our approach can be applied to obtain a faster rate of convergence for sampling of a peak-power constrained AWGN channel since, as observed in~\cite{Kenneth2018}, the peak-power constraint can provide some much needed uniformity property.
Second, a result by Elias~\cite{Elias1961} has been used to re-derive~\cite{GallagerNakiboglu2010} the somewhat surprising result that for feedback AWGN channels, the Schalkwijk-Kailath scheme yields the decoding error probability that decreases as a second-order exponent in block length. Noticing that Elias' argument in fact used discrete-time MMSE and our treatment essentially quantifies the difference between discrete-time and continuous-time MMSEs, it is worthwhile to investigate whether a rate of convergence result on the decoding error probability can be established for continuous-time feedback AWGN channels. Third, we can also consider sampling of continuous-time additive non-white Gaussian channels. To the best of our knowledge, results in this direction are scarce, but there are a great deal of efforts devoted to discrete-time feedback non-white Gaussian channels (see, e.g.,~\cite{kim10} and references therein). Given the recently obtained counterpart results in discrete time~\cite{LiuTao2019}, it is promising that our treatment can be adapted to such channels, in particular, additive channels with continuous-time auto-regressive and moving average Gaussian noises.
\bigskip \bigskip
{\bf Acknowledgement.} This work is supported by the Research Grants Council of the Hong Kong Special Administrative Region, China, under Project 17301017 and by the National Natural Science Foundation of China, under Project 61871343.
\section*{Appendices}
| -71,777.110259 |
[
-2.744140625,
2.40625
] | 26.666667 |
[
-3.453125,
0.57177734375,
-1.9091796875,
-7.27734375,
-0.916015625,
9.71875
] |
[
0.6337890625,
7.8203125,
-0.81689453125,
4.19921875
] | 207 | 5,861 |
[
-3.5859375,
4.27734375
] | 39.454359 |
[
-5.8515625,
-4.1796875,
-4.88671875,
-2.509765625,
1.9462890625,
12.9765625
] | 0.733864 | 14.230299 | 22.197577 | 3.449929 |
[
1.6530693769454956
] | -46,315.80423 | 5.440369 | -72,197.671908 | 0.639779 | 6.100064 |
[
-2.056640625,
-3.435546875,
-4.16015625,
-5.71875,
2.013671875,
13.09375
] |
[
-5.9140625,
-2.5390625,
-2.13671875,
-1.767578125,
3.884765625,
4.8828125
] | |
BkiUdqw25V5jEEL8tP0l
|
\section{Introduction}
A simple random walk on a finite connected graph $G$ with at least two vertices is a reversible Markov chain that starts at some vertex $v\in G$, and at each step moves with equal probability to any vertex adjacent to its present position. The mixing and the cover time of the random walk are among the graph parameters which have been extensively studied. To these parameters, Winkler and Zuckerman \cite{zuckermanmultiple} added the $\varepsilon$-blanket time variable (an exact definition will be given later in \eqref{blank1}) as the least time such that the walk has spent at every vertex at least an $\varepsilon$ fraction of time as much as expected at stationarity. Then, the $\varepsilon$-blanket time of $G$ is defined as the expected $\varepsilon$-blanket time variable maximized over the starting vertex.
The necessity of introducing and studying the blanket time arises mainly from applications in computer science. For example, suppose that a limited access to a source of information is randomly transferred from (authorized) user to user in a network. How long does it take for each user to own the information for as long as it is supposed to? To answer this question under the assumption that each user has to be active processing the information equally often involves the consideration of the blanket time. To a broader extent, viewing the internet as a (directed) graph, where every edge represents a link, a web surfer can be regarded as a walker who visits and records the sites at random. In a procedure that resembles Google's PageRank (PR), one wishes to rank a website according to the amount of time such walkers spend on it. A way to produce such an estimate is to rank the website according to the number of visits. The blanket time is the first time at which we expect this estimate to become relatively accurate.
Obviously, for every $\varepsilon\in (0,1)$, the $\varepsilon$-blanket time is larger than the cover time since one has to wait for all the vertices to have been visited at least once. Winkler and Zuckerman \cite{zuckermanmultiple} made the conjecture that, for every $\varepsilon\in (0,1)$, the $\varepsilon$-blanket time and the cover time are equivalent up to universal constants that depend only on $\varepsilon$ and not on the particular underlying graph $G$. This conjecture was resolved by Ding, Lee and Peres \cite{ding2011cover} who provided a highly non-trivial connection between those graph parameters and the discrete Gaussian free field (GFF) on $G$ using Talagrand's theory of majorizing measures. Recall that the GFF on $G$ with vertex set $V$ is a centered Gaussian process $(\eta_v)_{v\in V}$ with $\eta_{v_0}=0$, for some $v_0\in V$, and covariance structure given by the Green kernel of the random walk killed at $v_0$.
Recent years have witnessed a growing interest in studying the geometric and analytic properties of random graphs partly motivated by applications in research areas ranging from sociology and systems biology to interacting particle systems as well as by the need to present convincing models to gain insight into real-world networks. One aspect of this develpoment consists of examining the metric structure and connectivity of random graphs at criticality, that is precicely when we witness the emergence of a giant component that has size proportional to the number of vertices of the graph. Several examples of trees, including critical Galton-Watson trees, possess the Brownian Continuum Random Tree (CRT) as their scaling limit. A program \cite{bhamidi2014scaling} has been launched in the last few years having as its general aim to prove that the maximal components in the critical regime of a number of fundamental random graph models, with their distances scaling like $n^{1/3}$, fall into the basin of attraction of the Erd\H{o}s-R\'enyi random graph. Their scaling limit is a multiple of the scaling limit of the Erd\H{o}s-R\'enyi random graph in the critical window, that is a tilted version of the Brownian CRT where a finite number of vertices have been identified. Two of the examples that belong to the Erd\H{o}s-R\'enyi universality class are the configuration model in the critical scaling window and critical inhomogeneous random graphs, where different vertices have different proclivity to form edges, as it was shown in the recent work of \cite{bhamidi2016geometry} and \cite{bhamidi2017inhomo} respectively.
In \cite{croydon2012convergence}, Croydon, Hambly and Kumagai established criteria for the convergence of mixing times for random walks on general sequences of finite graphs. Furthermore, they applied their mixing time results in a number of examples of random graphs, such as self-similar fractal graphs with random weights, critical Galton-Watson trees, the critical Erd\H{o}s-R\'enyi random graph and the range of high-dimensional random walk. Motivated by their approach, starting with the strong assumption that the sequences of graphs, associated measures, walks and local times converge appropriately, we provide asymptotic bounds on the distribution of the blanket times of the random walks in the sequence.
To state the aforementioned assumption, we continue by introducing the graph theoretic framework in which we work. Firstly, Let $G=(V(G),E(G))$ be a finite connected graph with at least two vertices, where $V(G)$ denotes the vertex set of $G$ and $E(G)$ denotes the edge set of $G$. We endow the edge set $E(G)$ with a symmetric weight function $\mu^G: V(G)^2\rightarrow \mathbb{R_{+}}$ that satisfies $\mu_{xy}^G>0$ if and only if $\{x,y\}\in E(G)$. Now, the weighted random walk associated with $(G,\mu^G)$ is the Markov chain $((X^G_t)_{t\ge 0},\mathbf{P}_{x}^G,x\in V(G))$ with transition probabilities $(P_G(x,y))_{x,y\in V(G)}$ given by
\[
P_{G}(x,y):=\frac{\mu^G_{xy}}{\mu_x^G},
\]
where $\mu^G_x=\sum_{y\in V(G)} \mu_{xy}^G$. One can easily check that this Markov chain is reversible and has stationary distribution given by
\[
\pi^G(A):=\frac{\sum_{x\in A} \mu_x^G}{\sum_{x\in V(G)} \mu_x^G},
\]
for every $A\subseteq V(G)$. The process $X^G$ has corresponding local times $(L_t^G(x))_{x\in V(G),t\ge 0}$ given by $L_0^G(x)=0$, for every $x\in V(G)$, and, for $t\ge 1$
\[
L_t^G(x):=\frac{1}{\mu_x^G} \sum_{i=0}^{t-1} \mathbf{1}_{\{X_i^G=x\}}.
\]
The simple random walk on this graph is a Markov chain with transition probabilities given by $P(x,y):=1/\text{deg}(x)$ for all $x\in V(G)$, such that $\{x,y\}\in E(G)$. Observe that the simple random walk on $G$ is a weighted random walk on $G$ that assigns a constant unit weight to each edge.
To endow $G$ with a metric, we can choose $d_G$ to be the shortest path distance, which counts the number of edges in the shortest path between a pair of vertices in $G$. But this is not the most convenient choice in many examples. Another typical graph distance that arises from the view of $G$ as an electrical network equipped with conductances $(\mu^G_{xy})_{\{x,y\}\in E(G)}$ is the so-called resistance metric. For $f, g: V(G)\rightarrow \mathbb{R}$ let
\begin{equation} \label{resist1}
\mathcal{E}_G(f,g):=\frac{1}{2} \sum_{\substack{x,y\in V(G): \\ \{x,y\}\in E(G)}} (f(x)-f(y))(g(x)-g(y)) \mu_{xy}^G
\end{equation}
denote the Dirichlet form associated with the process $X^G$. Note that the sum in the expression above counts each edge twice. One can give the following interpretation of $\mathcal{E}_G(f,f)$ in terms of electrical networks. Given a voltage $f$ on the network, the current flow $I$ associated with $f$ is defined as $I_{xy}:=\mu_{xy}^G (f(x)-f(y))$, for every $\{x,y\}\in E(G)$. Then, the energy dissipation of a wire connecting $x$ and $y$ is $\mu^G_{xy} (f(x)-f(y))^2$. So, $\mathcal{E}_G(f,f)$ is the total energy dissipation of $G$. We define the resistance operator on disjoint sets $A, B\in V(G)$ through the formula
\begin{equation} \label{resist2}
R_{G}(A,B)^{-1}:=\inf \{\mathcal{E}_G(f,f): f: V(G)\rightarrow \mathbb{R}, f|_{A}=0, f|_{B}=1\}.
\end{equation}
Now, the distance on the vertices of $G$ defined by $R_G(x,y):=R_G(\{x\},\{y\})$, for $x\neq y$, and $R_G(x,x):=0$ is indeed a metric on the vertices of $G$. For a proof and a treatise on random walks on electrical networks see \cite[Chapter 9]{levin2017markov}.
For some $\varepsilon\in (0,1)$, define the $\varepsilon$-blanket time variable by
\begin{equation} \label{blank1}
\tau_{\text{bl}}^G(\varepsilon):=\inf\{t\ge 0: m^G L_t^G(x)\ge \varepsilon t, \ \forall x\in V(G)\},
\end{equation}
where $m^G$ is the total mass of the graph with respect to the measure $\mu^G$, i.e. $m^G:=\sum_{x\in V(G)} \mu_x^G$.
Taking the mean over the random walk started from the worst possible vertex defines the $\varepsilon$-blanket time, i.e.
\[
t_{\text{bl}}^G(\varepsilon):=\max_{x\in V(G)} \mathbf{E}_x \tau_{\text{bl}}^G(\varepsilon).
\]
Secondly, let $(K,d_K)$ be a compact metric space and let $\pi$ be a Borel measure of full support on $(K,d_K)$. Take $((X_t)_{t\ge 0},\mathbf{P}_x,x\in K)$ to be a $\pi$-symmetric Hunt process that admits local times $(L_t(x))_{x\in K,t\ge 0}$ continuous at $x$, uniformly over compact time intervals in $t$, $\mathbf{P}_x$-a.s. for every $x\in K$. A Hunt process is a strong Markov process that possesses useful properties such as the right-continuity and the existense of the left limits of sample paths (for definitions and other properties see \cite[Appendix A.2]{fukushima2010dirichlet}). Analogously, it is possible to define the $\varepsilon$-blanket time variable of $K$ as
\begin{equation} \label{blacken1}
\tau_{\text{bl}}(\varepsilon):=\inf \{t\ge 0: L_t(x)\ge \varepsilon t, \ \forall x\in K\}
\end{equation}
and check that is a non-trivial quantity (see Proposition \ref{nontrivexp}).
The following assumption encodes the information that, properly rescaled, the discrete state spaces, invariant measures, random walks, and local times, converge to $(K,d_k)$, $\pi$, $X$, and $(L_t(x))_{x\in K,t\in [0,T]}$ respectively, for some fixed $T>0$. This formulation will be described in terms of the extended Gromov-Hausdorff topology constructed in Section \ref{extendedsec}.
\begin{assum} \label{Assum1}
Fix $T>0$. Let $(G^n)_{n\ge 1}$ be a sequence of finite connected graphs that have at least two vertices, for which there exist sequences of real numbers $(\alpha(n))_{n\ge 1}$ and $(\beta(n))_{n\ge 1}$, such that
\[
\left(\left(V(G^n),\alpha(n) d_{G^n},\rho^n\right),\pi^n,\left(X^n_{\beta(n) t}\right)_{t\in [0,T]},\left(L^n_{\beta(n) t}(x)\right)_{\substack{x\in V(G^n), \\ t\in [0,T]}}\right)\longrightarrow \left(\left(K,d_K,\rho\right),\pi,X,\left(L_t(x)\right)_{\substack{x\in K, \\ t\in [0,T]}}\right)
\]
in the sense of the extended pointed Gromov-Hausdorff topology, where $\rho^n\in V(G^n)$ and $\rho\in K$ are distinguished points. In the above expression the definition of the discrete local times is extended to all positive times by linear interpolation.
\end{assum}
In most of the examples that will be discussed later, we will consider random graphs. In this context, we want to verify that the previous convergence holds in distribution. Our first conclusion is the following.
\begin{theorem} \label{Mth}
Suppose that Assumption \ref{Assum1} holds in such a way that the time and space scaling factors satisfy $\alpha(n) \beta(n)=m^{G^n}$, for every $n\ge 1$. Then, for every $\varepsilon\in (0,1)$, $\delta\in (0,1)$ and $t\in [0,T]$,
\begin{equation} \label{blacken2}
\limsup_{n\rightarrow \infty} \mathbf{P}_{\rho_n}^n \left(\beta(n)^{-1} \tau_{\textnormal{bl}}^n(\varepsilon)\le t\right)\le \mathbf{P}_{\rho} \left(\tau_{\textnormal{bl}}(\varepsilon(1-\delta))\le t\right),
\end{equation}
\begin{equation} \label{blacken3}
\liminf_{n\rightarrow \infty} \mathbf{P}_{\rho_n}^n \left(\beta(n)^{-1} \tau_{\textnormal{bl}}^n(\varepsilon)\le t\right)\ge \mathbf{P}_{\rho} \left({\tau}_{\textnormal{bl}}(\varepsilon)<t\right),
\end{equation}
where $\mathbf{P}_{\rho}$ is the law of $X$ on $K$, started from $\rho$.
\end{theorem}
The mapping $\varepsilon\mapsto \tau_{\textnormal{bl}}(\varepsilon)$ is increasing in $(0,1)$, so it posseses left and right limits at each point. It also becomes clear that if $\tau_{\textnormal{bl}}(\varepsilon)$ is continuous with probability one at $\varepsilon$, then
$\lim_{\delta\to 0}\tau_{\textnormal{bl}}(\varepsilon(1-\delta))=\tau_{\textnormal{bl}}(\varepsilon)$,
holds with probability one. Letting $\delta\to 0$ on both \eqref{blacken2} and \eqref{blacken3} demonstrates that $\beta(n)^{-1} \tau^n_{\textnormal{bl}}(\varepsilon)\to \tau_{\textnormal{bl}}(\varepsilon)$ in distribution.
\begin{corollary} \label{verification}
Suppose that Assumption \ref{Assum1} holds in such a way that the time and space scaling factors satisfy $\alpha(n) \beta(n)=m^{G^n}$, for every $n\ge 1$. If $\tau_{\textnormal{bl}}(\varepsilon)$ is continuous with probability one at $\varepsilon$, then
$
\beta(n)^{-1} \tau^n_{\textnormal{bl}}(\varepsilon)\to \tau_{\textnormal{bl}}(\varepsilon)
$
in distribution.
\end{corollary}
To demonstrate our main results consider first $T$, a critical Galton-Watson tree (with finite variance $\sigma^2$). The following result on the cover time of the simple random walk was obtained by Aldous (see \cite[Proposition 15]{aldous1991random}), which we apply to the blanket time in place of the cover time. The two parameters are equivalent up to universal constants as was conjectured in \cite{zuckermanmultiple} and proved in \cite{ding2011cover}.
\begin{theorem} [\textbf{Aldous \cite{aldous1991random}}]
Let $T$ be a critical Galton-Watson tree (with finite variance $\sigma^2$). For any $\delta>0$ there exists $A=A(\delta,\varepsilon,\sigma^2)>0$ such that
\[
P(A^{-1} k^{3/2}\le t_{\textnormal{bl}}^{T}(\varepsilon)\le A k^{3/2}||T|\in [k,2k])\ge 1-\delta,
\]
for every $\varepsilon\in (0,1)$.
\end{theorem}
Now let $G(n,p)$ be the resulting subgraph of the complete graph on $n$ vertices obtained by $p$-bond percolation. If $p=n^{-1}+\lambda n^{-4/3}$ for some $\lambda\in \mathbb{R}$, that is when we are in the so-called critical window, the largest connected component $\mathcal{C}_1^n$, as a graph, converges to a random compact metric space $\mathcal{M}$ that can be constructed directly from the Brownian CRT $\mathcal{T}_e$ (see the work of \cite{addario2012continuum}). The following result on the blanket time of the simple random walk on $\mathcal{C}_1^n$ is due to Barlow, Ding, Nachmias and Peres \cite{barlow2011evolution}.
\begin{theorem} [\textbf{Barlow, Ding, Nachmias, Peres \cite{barlow2011evolution}}]
Let $\mathcal{C}_1^n$ be the largest connected component of $G(n,p)$, $p=n^{-1}+\lambda n^{-4/3}$, $\lambda\in \mathbb{R}$ fixed. For any $\delta>0$ there exists $B=B(\delta,\varepsilon)>0$ such that
\[
P(B^{-1} n\le t_{\textnormal{bl}}^{\mathcal{C}_1^n}(\varepsilon)\le B n)\ge 1-\delta,
\]
for every $\varepsilon\in (0,1)$.
\end{theorem}
Our contribution refines the previous existing tightness results on the order of the blanket time. In what follows $\mathbb{P}_{\rho^n}$, $n\ge 1$ as well as $\mathbb{P}_{\rho}$ are the annealed measures, that is the probability measures obtained by integrating out the randomness of the state spaces involved. We remark also here that we can apply our main result due to the recent work of \cite{croydon2016scaling} that generalises previous work done in \cite{croydon2008convergence} and \cite{croydon2012scaling}.
\begin{theorem} \label{finish1}
Let $\mathcal{T}_n$ be a critical Galton-Watson tree (with finite variance) conditioned to have total progeny $n+1$. Fix $\varepsilon\in (0,1)$. If $\tau_{\textnormal{bl}}^n(\varepsilon)$ is the $\varepsilon$-blanket time variable of the simple random walk on $\mathcal{T}_n$, started from its root $\rho^n$, then
\[
\mathbb{P}_{\rho^n}\left(n^{-3/2} \tau_{\textnormal{bl}}^n(\varepsilon)\le t\right)\to \mathbb{P}_{\rho}\left(\tau_{\textnormal{bl}}^e(\varepsilon)\le t\right),
\]
for every $t\ge 0$, where $\tau_{\textnormal{bl}}^e(\varepsilon)\in (0,\infty)$ is the $\varepsilon$-blanket time variable of the Brownian motion on $\mathcal{T}_e$, started from a distinguished point $\rho\in \mathcal{T}_e$. Equivalently, for every $\varepsilon\in (0,1)$, $n^{-3/2} \tau_{\textnormal{bl}}^n(\varepsilon)$ under $\mathbb{P}_{\rho^n}$ converges weakly to $\tau_{\textnormal{bl}}^e(\varepsilon)$ under $\mathbb{P}_{\rho}$.
\end{theorem}
\begin{theorem}
Fix $\varepsilon\in (0,1)$. If $\tau_{\textnormal{bl}}^{n}(\varepsilon)$ is the $\varepsilon$-blanket time variable
of the simple random walk on $\mathcal{C}_1^n$, started from its root $\rho^n$, then
\[
\mathbb{P}_{\rho^n}\left(n^{-1} \tau_{\textnormal{bl}}^n(\varepsilon)\le t\right)\to \mathbb{P}_{\rho}\left(\tau_{\textnormal{bl}}^{\mathcal{M}}(\varepsilon)\le t\right),
\]
for every $t\ge 0$, where $\tau_{\textnormal{bl}}^{\mathcal{M}}(\varepsilon)\in (0,\infty)$ is the $\varepsilon$-blanket time variable of the Brownian motion on $\mathcal{M}$, started from $\rho$.
\end{theorem}
Moreover, to present our last result we consider the configuration model. Let $M^n(d)$ be the random multigraph labelled by $[n]$ such that the $i$-th vertex has degree $d_i$, $i\ge 1$, for every $1\le i\le n$, which is constructed as follows. Assign $d_i$ half-edges to each vertex $i$, labelling them in an arbitrary way. Then, the configuration model is produced by a uniform pairing of the half-edges to create full edges. If the degree sequence satisfies certain conditions that would be precicely given later in Assumption \ref{gracias}, it was shown in the work of \cite{dhara2017critical} that the largest connected component $M_1^n(d)$, is of order $n^{2/3}$. Recently in \cite{bhamidi2016geometry} its scaling limit, $\mathcal{M}_D$, was proven to exist and to belong to the Erd\H{o}s-R\'enyi universality class.
\begin{theorem} \label{finish3}
Fix $\varepsilon\in (0,1)$. If $\tau_{\textnormal{bl}}^{n}(\varepsilon)$ is the $\varepsilon$-blanket time variable of the simple random walk on $M_1^n(d)$, started from its root $\rho^n$, then
\[
\mathbb{P}_{\rho^n}\left(n^{-1} \tau_{\textnormal{bl}}^n(\varepsilon)\le t\right)\to \mathbb{P}_{\rho}\left(\tau_{\textnormal{bl}}^{\mathcal{M}_D}(\varepsilon)\le t\right),
\]
for every $t\ge 0$, where $\tau_{\textnormal{bl}}^{\mathcal{M}_D}(\varepsilon)\in (0,\infty)$ is the $\varepsilon$-blanket time variable of the Brownian motion on $\mathcal{M}_D$, started from $\rho$.
\end{theorem}
The paper is organized as follows. In Section \ref{extendedsec}, we introduce the extended Gromov-Hausdorff topology and derive some useful properties. In Section \ref{blankbounds}, we prove Theorem \ref{Mth} under Assumption \ref{Assum1} and present Assumption \ref{Assum3}, a weaker sufficient assumption when the sequence of spaces is equipped with resistance metrics. In Section \ref{maintheorems}, we verify the assumptions of Corollary \ref{verification}, and therefore prove convergence of blanket times for the series of critical random graphs mentioned above, thus effectively proving Theorem \ref{finish1}-Theorem \ref{finish3}. The main tool we employ to prove continuity of the $\varepsilon$-blanket time of the diffusion on the limiting spaces that appear in the statements of the preceding theorems is to exploit scale invariance properties of the finite measures, such as It\^{o}'s excursion measure (see Section \ref{needed} for key facts of its theory), that give rise to realizations of those spaces. For that reason we believe our results to easily transfer when considering Galton-Watson trees with critical offspring distribution in the domain of attraction of a stable law with index $\alpha\in (1,2)$ (see \cite[Theorem 4.3]{le2006random}) and random stable looptrees (see \cite[Theorem 4.1]{curien2014loop}). Also, we hope our work to be seen as a stepping stone to deal with the more delicate problem of establishing convergence in distribution of the rescaled cover times of the discrete-time walks in each application of our main result. See \cite[Remark 7.4]{croydon2015moduli} for a thorough discussion on the demanding nature of this project.
\section{Extended Gromov-Hausdorff topologies} \label{extendedsec}
In this section we define an extended Gromov-Hausdorff distance between quadruples consisting of a compact metric space , a Borel measure, a time-indexed right-continuous path with left-hand limits and a local time-type function. This allows us to make precise the assumption under which we are able to prove convergence of blanket times for the random walks on various models of critical random graphs. In Lemma \ref{2.2}, we give an equivalent characterization of the assumption that will be used in Section \ref{blankbounds} when proving distributional limits for the rescaled blanket times. Also, Lemma \ref{2.3} will be useful when it comes checking that the examples we treat satisfy the assumption.
Let $(K,d_K)$ be a non-empty compact metric space. For a fixed $T>0$, let $X^K$ be a path in $D([0,T],K)$, the space of c\`adl\`ag functions, i.e. right-continuous functions with left-hand limits, from $[0,T]$ to $K$. We say that a function $\lambda$ from $[0,T]$ onto itself is a time-change if it is strictly increasing and continuous. Let $\Lambda$ denote the set of all time-changes. If $\lambda\in \Lambda$, then $\lambda(0)=0$ and $\lambda(T)=T$. We equip $D([0,T],K)$ with the Skorohod metric $d_{J_1}$ defined as follows:
\[
d_{J_1}(x,y):=\inf_{\lambda\in \Lambda} \bigg\{\sup_{t\in [0,T]} |\lambda(t)-t|+\sup_{t\in [0,T]} d_{K} (x(\lambda(t)),y(t))\bigg\},
\]
for $x,y\in D([0,T],K)$. The idea behind going from the uniform metric to the Skorohod metric $d_{J_1}$ is to say that two paths are close if they are uniformly close in $[0,T]$, after allowing small perturbations of time.
Moreover, $D([0,T],K)$ endowed with $d_{J_1}$ becomes a separable metric space (see \cite[Theorem 12.2]{billingsley2013convergence}). Let $\mathcal{P}(K)$ denote the space of Borel measures on $K$. If $\mu,\nu\in \mathcal{P}(K)$ we set
\[
d_{P}(\mu,\nu)=\inf\{\varepsilon>0: \mu(A)\le \nu(A^{\varepsilon})+\varepsilon\text{ and }\nu(A)\le \mu(A^{\varepsilon})+\varepsilon,\text{ for any }A\in \mathcal{M}(K)\},
\]
where $\mathcal{M}(K)$ is the set of all closed subsets of $K$. This expression gives the standard Prokhorov metric between $\mu$ and $\nu$. Moreover, it is known , see \cite{daley2007introduction} Appendix A.2.5, that $(\mathcal{P}(K),d_{P})$ is a Polish metric space, i.e. a complete and separable metric space, and the topology generated by $d_P$ is exactly the topology of weak convergence, the convergence against bounded and continuous functionals.
Let $\pi^K$ be a Borel measure on $K$ and $L^K=(L_t^K(x))_{x\in K,t\in [0,T]}$ be a jointly continuous function of $(t,x)$ taking positive real values. Let $\mathbb{K}$ be the collection of quadruples $(K,\pi^K,X^K,L^K)$. We say that two elements $(K,\pi^K,X^K,L^K)$ and $(K',\pi^{K'},X^{K'},L^{K'})$ of $\mathbb{K}$ are equivalent if there exists an isometry $f:K\rightarrow K'$ such that
\begin{itemize}
\item $\pi^{K}\circ f^{-1}=\pi^{K'}$,
\item $f\circ X^K=X^{K'},$ which is a shorthand of $f(X_t^K)=X_t^{K'}$, for every $t\in [0,T]$.
\item $L_t^{K'}\circ f=L_t^K$, for every $t\in [0,T]$, which is a shorthand of $L_t^{K'}(f(x))=L_t^K(x)$, for every $t\in [0,T]$, $x\in K$.
\end{itemize}
Not to overcomplicate our notation, we will often identify an equivalence class of $\mathbb{K}$ with a particular element of it. We now introduce a distance $d_{\mathbb{K}}$ on $\mathbb{K}$ by setting
\begin{align*}
d_{\mathbb{K}}&((K,\pi^K,X^K,L^K),(K',\pi^{K'},X^{K'},L^{K'}))
\\
&:=\inf_{Z,\phi,\phi',\mathcal{C}}\bigg\{d_P^Z(\pi^K\circ \phi^{-1},\pi^{K'}\circ \phi'^{-1})+d_{J_1}^Z(\phi(X_t^K),\phi'(X_t^{K'}))
\\
&+\sup_{(x,x')\in \mathcal{C}}\bigg(d_{Z}(\phi(x),\phi'(x'))+\sup_{t\in [0,T]} |L_t^K(x)-L_t^{K'}(x')|\bigg)\bigg\},
\end{align*}
where the infimum is taken over all metric spaces $(Z,d_Z)$, isometric embeddings $\phi:K\rightarrow Z$, $\phi':K'\rightarrow Z$ and correspondences $\mathcal{C}$ between $K$ and $K'$. A correspondence between $K$ and $K'$ is a subset of $K\times K'$, such that for every $x\in K$ there exists at least one $x'$ in $K'$ such that $(x,x')\in \mathcal{C}$ and conversely for every $x'\in K'$ there exists at least one $x\in K$ such that $(x,x')\in \mathcal{C}$. In the above expression $d^Z_P$ is the standard Prokhorov distance between Borel measures on $Z$, and $d_{J_1}^Z$ is the Skorohod metric $d_{J_1}$ between c\`adl\`ag paths on $Z$.
In the following proposition we check that the definition of $d_{\mathbb{K}}$ induces a metric and that the resulting metric space is separable. The latter fact will be used repeatedly later when it comes to applying Skorohod's represantation theorem on sequences of random graphs to prove statements regarding their blanket times or the cover times. Before proceeding to the proof of Proposition \ref{Prop1.1}, let us first make a few remarks about the ideas behind the definition of $d_{\mathbb{K}}$. The first term along with the Hausdorff distance on $Z$ between $\phi(K)$ and $\phi'(K')$ is that used in the Gromov-Hausdorff-Prokhorov distance for compact metric spaces (see \cite[Section 2.2, (6)]{abraham2013note}). Though, in our definition of $d_{\mathbb{K}}$ we did not consider the Hausdorff distance between the embedded compact metric spaces $K$ and $K'$, since it is absorbed by the first part of the third term in the expression for $d_{\mathbb{K}}$. Recall here the equivalent definition of the standard Gromov-Hausdorff distance via correspondences as a way to relate two compact metric spaces (see \cite[Theorem 7.3.25]{burago2001course}). The motivation for the second term comes from \cite{croydon2012scaling}, where the author defined a distance between pairs of compact length spaces (for a definition of a length space see \cite[Definition 2.1.6]{burago2001course}) and continuous paths on those spaces. The restriction on length spaces is not necessary, as we will see later, on proving that $d_{\mathbb{K}}$ provides a metric. Considering c\`adl\`ag paths instead of continuous paths and replacing the uniform metric with the Skorohod metric $d_{J_1}$ allows us to prove separability without assuming that $(K,d_K)$ is a non-empty compact length space. The final term was first introduced in \cite[Section 6]{duquesne2005probabilistic} to define a distance between spatial trees equipped with a continuous function.
\begin{proposition} \label{Prop1.1}
$(\mathbb{K},d_{\mathbb{K}})$ is a separable metric space.
\end{proposition}
\begin{proof}
That $d_{\mathbb{K}}$ is non-negative and symmetric is obvious. To prove that is also finite, for any choice of $(K,\pi^K,X^K,L^K)$, $(K',\pi^{K'},X^{K'},L^{K'})$ consider the disjoint union $Z=K\sqcup K'$ of $K$ and $K'$. Then, set $d_{Z}(x,x'):=\text{diam}_{K}(K)+\text{diam}_{K'}(K')$, for any $x\in K$, $x'\in K'$, where
\[
\text{diam}_{K}(K)=\sup_{y,z\in K} d_K(y,z)
\]
denotes the diameter of $K$ with respect to the metric $d_K$. Since $K$ and $K'$ are compact their diameters are finite. Therefore, $d_Z$ is finite for any $x\in K$, $x'\in K'$. To conclude that $d_{\mathbb{K}}$ is finite, simply suppose that $\mathcal{C}=K\times K'$.
Next, we show that $d_{\mathbb{K}}$ is positive-definite. Let $(K,\pi^K,X^K,L^K), (K',\pi^{K'},X^{K'},L^{K'})$ be in $\mathbb{K}$ such that $d_{\mathbb{K}}((K,\pi^K,X^K,L^K),(K',\pi^{K'},X^{K'},L^{K'}))=0$. Then, for every $\varepsilon>0$ there exist $Z,\phi,\phi',\mathcal{C}$ such that the sum of the quantities inside the infimum in the definition of $d_{\mathbb{K}}$ is bounded above by $\varepsilon$. Furthermore, there exists $\lambda_{\varepsilon}\in \Lambda$ such that the sum of the quantities inside the infimum in the definition of $d_{J_1}^Z$ is bounded above by $2 \varepsilon$. Recall that for every $t\in [0,T]$, $L^K_t:K\to \mathbb{R}_{+}$ is a continuous function and since $K$ is a compact metric space, then it is also uniformly continuous. Therefore, there exists a $\delta\in (0,\varepsilon]$ such that
\begin{equation} \label{2.1}
\sup_{\substack{x_1,x_2\in K:\\ d_{K}(x_1,x_2)<\delta}} \sup_{t\in [0,T]} |L_t^K(x_1)-L_t^K(x_2)|\le \varepsilon.
\end{equation}
Now, let $(x_i)_{i\ge 1}$ be a dense sequence of disjoint elements in $K$. Since $K$ is compact, there exists an integer $N_{\varepsilon}$ such that the collection of open balls $(B_{K}(x_i,\delta))_{i=1}^{N_{\varepsilon}}$ covers $K$. Defining $A_1=B_{K}(x_1,\delta)$ and $A_i=B_{K}(x_i,\delta)\setminus \cup_{j=1}^{i-1} B_{K}(x_j,\delta)$, for $i=2,...,N_{\varepsilon}$, we have that $(A_i)_{i=1}^{N_{\varepsilon}}$ is a disjoint cover of $K$. Consider a function $f_{\varepsilon}:K\rightarrow K'$ by setting
\[
f_{\varepsilon}(x):=x_i'
\]
on $A_i$, where $x_i'$ is chosen such that $(x_i,x_i')\in \mathcal{C}$, for $i=1,...,N_{\varepsilon}$. Note that by definition $f_{\varepsilon}$ is a measurable function defined on $K$. For any $x\in K$, such that $x\in A_i$ for some $i=1,...,N_{\varepsilon}$, we have that
\begin{align} \label{mine1}
d_{Z}(\phi(x),\phi'(f_{\varepsilon}(x)))&=d_{Z}(\phi(x),\phi'(x_i')) \nonumber
\\
&\le d_{Z}(\phi(x),\phi(x_i))+d_{Z}(\phi(x_i),\phi'(x_i'))\le \delta+\varepsilon\le 2 \varepsilon.
\end{align}
From \eqref{mine1}, it follows that for any $x\in K$ and $y\in K$
\begin{align*}
|d_{Z}(\phi(x),\phi(y))-d_{Z}(\phi'(f_{\varepsilon}(x)),\phi'(f_{\varepsilon}(y))|&\le d_{Z}(\phi(y),\phi'(f_{\varepsilon}(y)))+d_{Z}(\phi(x),\phi'(f_{\varepsilon}(x)))
\\
&\le 2 \varepsilon+2 \varepsilon=4 \varepsilon.
\end{align*}
This immediately yields
\begin{equation} \label{2.2}
\sup_{x,y\in K} |d_{K}(x,y)-d_{K'}(f_{\varepsilon}(x),f_{\varepsilon}(y))|\le 4 \varepsilon.
\end{equation}
From \eqref{2.2}, we deduce the bound
\begin{equation} \label{2.3}
d_{P}^{K'}(\pi^{K}\circ f_{\varepsilon}^{-1},\pi^{K'})\le 5 \varepsilon
\end{equation}
for the Prokhorov distance between $\pi^K\circ f_{\varepsilon}^{-1}$ and $\pi^{K'}$ in $K'$. Using \eqref{2.1} and the fact that the last quantity inside the infimum in the definition of $d_{\mathbb{K}}$ is bounded above by $\varepsilon$, we deduce
\begin{equation} \label{2.4}
\sup_{x\in K,t\in [0,T]} |L_t^K(x)-L_t^{K'}(f_{\varepsilon}(x))|\le 2 \varepsilon.
\end{equation}
Using \eqref{mine1} and the fact that the second quantity in the infimum is bounded above by $\varepsilon$, we deduce that for any $t\in [0,T]$
\begin{align*}
d_{Z}(\phi'(f_{\varepsilon}(X^K_{\lambda_{\varepsilon}(t)})),\phi'(X_t^{K'}))&\le d_{Z}(\phi'(f_{\varepsilon}(X^K_{\lambda_{\varepsilon}(t)})),\phi(X^K_{\lambda_{\varepsilon}(t)}))+d_{Z}(\phi(X^K_{\lambda_{\varepsilon}(t)}),\phi'(X_t^{K'}))
\\
&\le 2 \varepsilon+2 \varepsilon=4 \varepsilon.
\end{align*}
Therefore,
\begin{equation} \label{Sk1}
\sup_{t\in [0,T]} d_{K'}(f_{\varepsilon}(X_{\lambda_{\varepsilon}(t)}^K),X_t^{K'})\le 4 \varepsilon.
\end{equation}
Using a diagonalization argument we can find a sequence $(\varepsilon_n)_{n\ge 1}$ such that $f_{\varepsilon_n}(x_i)$ converges to some limit $f(x_i)\in K'$, for every $i\ge 1$. From \eqref{2.2} we immediately get that $d_{K}(x_i,x_j)=d_{K'}(f(x_i),f(x_j))$, for every $i,j\ge 1$. By \cite[Proposition 1.5.9]{burago2001course}, this map can be extended continuously to the whole $K$. This shows that $f$ is distance-preserving. Reversing the roles of $K$ and $K'$, we are able to find also a distance-preserving map from $K'$ to $K$. Hence $f$ is an isometry. We are now able to check that $\pi^K\circ f^{-1}=\pi^{K'}$, $L_t^{K'}\circ f=L_t^K$, for all $t\in [0,T]$, and $f\circ X^K=X^{K'}$. Since $f_{\varepsilon_n}(x_i)$ converges to $f(x_i)$ in $K'$, we can find $\varepsilon'\in (0,\varepsilon]$ such that $d_{K'}(f_{\varepsilon'}(x_i),f(x_i))\le \varepsilon$, for $i=1,...,N_{\varepsilon}$. Recall that $(x_i)_{i=1}^{N_{\varepsilon}}$ is an $\varepsilon$-net in $K$. Then, for $i=1,...,N_{\varepsilon}$, such that $x\in A_i$, using \eqref{2.2} and the fact that $f$ is an isometry, we deduce
\begin{equation} \label{2.5}
d_{K'}(f_{\varepsilon'}(x),f(x))\le d_{K'}(f_{\varepsilon'}(x),f_{\varepsilon'}(x_i))+d_{K'}(f_{\varepsilon'}(x_i),f(x_i))+d_{K'}(f(x_i),f(x))\le 7 \varepsilon.
\end{equation}
This, combined with \eqref{2.3} implies
\[
d_{P}^{K'}(\pi^K\circ f^{-1},\pi^{K'})\le d_{P}^{K'}(\pi^K\circ f^{-1},\pi^{K}\circ f_{\varepsilon'}^{-1})+d_{P}^{K'}(\pi^K\circ f_{\varepsilon'}^{-1},\pi^{K'})\le 12 \varepsilon.
\]
Since $\varepsilon>0$ was arbitrary, $\pi^{K}\circ f^{-1}=\pi^{K'}$. Moreover, from \eqref{2.4} and \eqref{2.5} we have that
\begin{align*}
&\sup_{x\in K,t\in [0,T]} |L_t^K(x)-L_t^{K'}(f(x))|
\\
&\le \sup_{x\in K,t\in [0,T]} |L_t^{K}(x)-L_t^{K'}(f_{\varepsilon'}(x))|+\sup_{x\in K,t\in [0,T]} |L_t^{K'}(f_{\varepsilon'}(x))-L_t^{K'}(f(x))|
\\
&\le 2 \varepsilon+\sup_{\substack{x_1',x_2'\in K':\\ d_{K'}(x_1',x_2')\le 7\varepsilon}} \sup_{t\in [0,T]} |L_t^{K'}(x_1')-L_t^{K'}(x_2')|.
\end{align*}
Now, this and the uniform continuity of $L^{K'}$ (replace $L^K$ by $L^{K'}$ in \eqref{2.1}) gives $L_t^{K'}\circ f=L_t^K$, for all $t\in [0,T]$. Finally, we verify that $f\circ X^K=X^{K'}$. For any $t\in [0,T]$
\[
d_{K'}(f(X_{\lambda_{\varepsilon}(t)}^K),X_t^{K'})\le d_{K'}(f(X_{\lambda_{\varepsilon}(t)}^K),f_{\varepsilon'}(X_{\lambda_{\varepsilon}(t)}^K))+d_{K'}(f_{\varepsilon'}(X_{\lambda_{\varepsilon}(t)}^K),X_t^{K'})\le 7 \varepsilon+4 \varepsilon=11 \varepsilon,
\]
where we used \eqref{Sk1} and \eqref{2.5}. Therefore,
\begin{equation} \label{Sk2}
\sup_{t\in [0,T]} d_{K'}(f(X_{\lambda_{\varepsilon}(t)}^K),X_t^{K'})\le 11 \varepsilon.
\end{equation}
Recall that $\sup_{t\in [0,T]} |\lambda_{\varepsilon}(t)-t|\le 2 \varepsilon$. From this and \eqref{Sk2}, it follows that for every $t\in [0,T]$, there exists a sequence $(z_n)_{n\ge 1}$, such that $z_n\rightarrow t$ and $d_{K'}(f(X_{z_n}^{K}),X_t^{K'})\rightarrow 0$, as $n\rightarrow \infty$. If $t$ is a continuity point of $f\circ X^K$, then $d_{K'}(f(X_{z_n}^{K}),f(X_t^K))\rightarrow 0$, as $n\rightarrow \infty$. Thus, $f(X_t^K)=X_t^{K'}$. If $f\circ X^K$ has a jump at $t$ and $(z_n)_{n\ge 1}$ has a subsequence $(z_{n_k})_{k\ge 1}$, such that $z_{n_k}\ge t$, for any $k\ge 1$, then $d_{K'}(f(X_{z_{n_k}}^{K}),X_t^{K'})\rightarrow 0$, as $n\rightarrow \infty$, and $d_{K'}(f(X_{z_{n_k}}^{K}),f(X_t^K))\rightarrow 0$, as $n\rightarrow \infty$. Therefore, $f(X_t^K)=X_t^{K'}$. Otherwise, $z_n<t$, for $n$ large enough and $d_{K'}(f(X_{z_n}^{K}),f(X_{t-}^{K}))\rightarrow 0$, as $n\rightarrow \infty$, which implies $f(X_{t-}^{K})=X_t^{K'}$. Essentially, what we have proved is that if $f\circ X^K$ has a jump, then either $f(X^K_t)=X_t^{K'}$ or $f(X^K_{t-})=X_t^{K'}$. But, since $X^{K'}$ is c\`adl\`ag, $f\circ X^K=X^{K'}$. This completes the proof that the quadruples $(K,\pi^K,X^K,L^K)$ and $(K',\pi^{K'},X^{K'},L^{K'})$ are equivalent in $(\mathbb{K},d_{\mathbb{K}})$, and consequently that $d_{\mathbb{K}}$ is positive-definite.
For the triangle inequality we follow the proof of \cite[Proposition 7.3.16]{burago2001course}, which proves the triangle inequality for the standard Gromov-Hausdorff distance. Let $\mathcal{K}^i=(K^i,\pi^i,X^i,L^i)$ be an element of $(\mathbb{K},d_{\mathbb{K}})$ for $i=1,2,3$. Suppose that
\[
d_{\mathbb{K}}(\mathcal{K}^1,\mathcal{K}^2)<\delta_1.
\]
Thus, there exists a metric space $Z_1$, isometric embeddings $\phi_{1,1}: K^1\rightarrow Z_1$, $\phi_{2,1}:K^2\rightarrow Z_1$ and a correspondence $\mathcal{C}_1$ between $K^1$ and $K^2$ such that the sum of the quantities inside the infimum that defines $d_{\mathbb{K}}$ is bounded above by $\delta_1$. Similarly, if
\[
d_{\mathbb{K}}(\mathcal{K}^2,\mathcal{K}^3)<\delta_2,
\]
there exists a metric space $Z_2$, isometric embeddings $\phi_{2,2}: K^2\rightarrow Z_2$, $\phi_{2,3}: K^3\rightarrow Z_2$ and a correspondence $\mathcal{C}_2$ between $K^2$ and $K^3$ such that the sum of the quantities inside the infimum that defines $d_{\mathbb{K}}$ is bounded above by $\delta_2$. Next, we set $Z=Z_1\sqcup Z_2$ to be the disjoint union of $Z_1$ and $Z_2$ and we define a distance on $Z$ in the following way. Let $d_{Z|Z_i\times Z_i}=d_{Z_i}$, for $i=1,2$, and for $x\in Z_1$, $y\in Z_2$ set
\[
d_{Z}(x,y):=\inf_{z\in K^2} \{d_{Z_1}(x,\phi_{2,1}(z))+d_{Z_2}(\phi_{2,2}(z),y)\}.
\]
It is obvious that $d_Z$ is symmetric and non-negative. It is also easy to check that $d_Z$ satisfies the triangle inequality. Identifying points that are separated by zero distance and slightly abusing notation, we turn $(Z,d_{Z})$ into a metric space, which comes with isometric embeddings $\phi_i$ of $Z_i$ for $i=1,2$. Using the triangle inequality of the Prokhorov metric on $Z$, gives us that
\[
d_P^Z(\pi^1\circ (\phi_1\circ \phi_{1,1})^{-1},\pi^3\circ (\phi_2\circ \phi_{3,2})^{-1})
\]
\[
\le d_P^Z(\pi^1\circ (\phi_1\circ \phi_{1,1})^{-1},\pi^2\circ (\phi_1\circ \phi_{2,1})^{-1})+d_P^Z(\pi^2\circ (\phi_1\circ \phi_{2,1})^{-1},\pi^3\circ (\phi_2\circ \phi_{3,2})^{-1}).
\]
Now, since $\phi_1(\phi_{2,1}(y))=\phi_2(\phi_{2,2}(y))$, for all $y\in K^2$, we deduce
\begin{equation} \label{tria1}
d_P^Z(\pi^1\circ (\phi_1\circ \phi_{1,1})^{-1},\pi^3\circ (\phi_2\circ \phi_{3,2})^{-1}) \le d_P^{Z_1}(\pi^1\circ \phi_{1,1}^{-1},\pi^2\circ \phi_{2,1}^{-1})+d_P^{Z_2}(\pi^2\circ \phi_{2,2}^{-1},\pi^3\circ \phi_{3,2}^{-1}).
\end{equation}
A similar bound also applies for the embedded c\`adl\`ag paths. Namely, using the same methods as above, we deduce
\begin{equation} \label{tria2}
d_{J_1}^Z((\phi_1\circ \phi_{1,1})(X^1),(\phi_2\circ \phi_{3,2})(X^3))\le d_{J_1}^Z(\phi_{1,1}(X^1),\phi_{2,1}(X^2))+d_{J_1}^Z(\phi_{2,2}(X^2),\phi_{3,2}(X^3)).
\end{equation}
Now, let
\[
\mathcal{C}:=\{(x,z)\in K^1\times K^3: (x,y)\in \mathcal{C}_1,(y,z)\in \mathcal{C}_2,\text{ for some }y\in K^2\}.
\]
Observe that $\mathcal{C}$ is a correspondence between $K^1$ and $K^3$. Then, if $(x,z)\in \mathcal{C}$, there exists $y\in K^2$ such that $(x,y)\in \mathcal{C}_1$ and $(y,z)\in \mathcal{C}_2$, and noting again that $\phi_1(\phi_{2,1}(y))=\phi_2(\phi_{2,2}(y))$, for all $y\in K^2$, we deduce
\begin{equation} \label{tria3}
d_Z(\phi_1(\phi_{1,1}(x)),\phi_2(\phi_{3,2}(z)))\le d_{Z_1}(\phi_{1,1}(x),\phi_{2,1}(y))+d_{Z_2}(\phi_{2,2}(y),\phi_{3,2}(z)).
\end{equation}
Using the same arguments one can prove a corresponding bound involving $L^i$, $i=1,2,3$. Namely, if $(x,z)\in \mathcal{C}$, there exists $y\in K^2$ such that $(x,y)\in \mathcal{C}_1$ and $(y,z)\in \mathcal{C}_2$, and moreover
\begin{equation} \label{tria4}
\sup_{t\in [0,T]}|L_t^1(x)-L_t^3(z)|\le \sup_{t\in [0,T]} |L_t^1(x)-L_t^2(y)|+\sup_{t\in [0,T]} |L_t^2(y)-L_t^3(z)|.
\end{equation}
Putting \eqref{tria1}, \eqref{tria2}, \eqref{tria3} and \eqref{tria4} together gives
\[
d_{\mathbb{K}}(\mathcal{K}^1,\mathcal{K}^3)\le \delta_1+\delta_2,
\]
and the triangle inequality follows. Thus, $(\mathbb{K},d_{\mathbb{K}})$ forms a metric space.
To finish the proof, we need to show that $(\mathbb{K},d_{\mathbb{K}})$ is separable. Let $(K,\pi,X,L)$ be an element of $\mathbb{K}$. First, let $K^n$ be a finite $n^{-1}$-net of $K$, which exists since $K$ is compact. Furthermore, we can endow $K^n$ with a metric $d_{K^n}$, such that $d_{K^n}(x,y)\in \mathbb{Q}$, and moreover $|d_{K^n}(x,y)-d_K(x,y)|\le n^{-1}$, for every $x,y \in K^n$. Since, $K^n$ is a finite $n^{-1}$-net of $K$ we can choose a partition for $K$, $(A_x)_{x\in K^n}$, such that $x\in A_x$, and $\text{diam}_{K}(A_x)\le 2 n^{-1}$. We can even choose the partition in such a way that $A_x$ is measurable for all $x\in K^n$ (see for example the definition of $(A_i)_{i=1}^{N_{\varepsilon}}$ after \eqref{2.1}). Next, we construct a Borel measure $\pi^n$ in $K^n$ that takes rational mass at each point, i.e. $\pi^n(\{x\})\in \mathbb{Q}$, and $|\pi^n(\{x\})-\pi(A_x)|\le n^{-1}$. Define $\varepsilon_n$ by
\[
\varepsilon_n:=\sup_{\substack{s,t\in [0,T]:\\ |s-t|\le n^{-1}}} \sup_{\substack{x,x'\in K:\\ d_{K}(x,x')\le n^{-1}}} |L_s(x)-L_t(x')|.
\]
By the joint continuity of $L$, $\varepsilon_n\rightarrow 0$, as $n\rightarrow \infty$. Let $0=s_0<s_1<\cdot \cdot \cdot <s_r=T$ be a set of rational times such that $|s_{i+1}-s_i|\le n^{-1}$, for $i=0,...,r-1$. Choose $L_{s_i}^n(x)\in \mathbb{Q}$ with $|L_{s_i}^n(x)-L_{s_i}(x)|\le n^{-1}$, for every $x\in K^n$. We interpolate linearly between the finite collection of rational time points in order to define $L^n$ to the whole domain $K^n\times [0,T]$. Let $\mathcal{C}^n:=\{(x,x')\in K\times K^n: d_K(x,x')\le n^{-1}\}$. Clearly $\mathcal{C}^n$ defines a correspondence between $K$ and $K^n$. Let $(x,x')\in \mathcal{C}^n$ and $s\in [s_i,s_{i+1}]$, for some $i=0,...,r-1$. Then, using the triangle inequality we observe that
\begin{equation} \label{correction1}
|L_s^n(x)-L_s(x')|\le |L_s^n(x)-L_s(x)|+|L_s(x)-L_s(x')|\le |L_s^n(x)-L_s(x)|+\varepsilon_n.
\end{equation}
Since we interpolated linearly to define $L^n$ beyond rational time points on the whole space $K^n\times [0,T]$ we have that
\begin{equation} \label{correction2}
|L_s^n(x)-L_s(x)|\le |L_{s_{i+1}}^n(x)-L_s(x)|+|L_{s_i}^n(x)-L_s(x)|.
\end{equation}
Applying the triangle inequality again yields
\begin{align*}
|L_{s_i}^n(x)-L_s(x)|&\le |L_{s_i}^n(x)-L_{s_i}(x)|+|L_{s_i}(x)-L_s(x)|
\\
&\le n^{-1}+\varepsilon_n.
\end{align*}
The same upper bound applies for $|L_{s_{i+1}}^n(x)-L_s(x)|$, and from \eqref{correction1} and \eqref{correction2} we conclude that for $(x,x')\in \mathcal{C}^n$ and $s\in [s_i,s_{i+1}]$, for some $i=0,...,r-1$,
\[
|L_s^n(x)-L_s(x')|\le 2 n^{-1}+3 \varepsilon_n.
\]
For $X\in D([0,T],K)$ and $A\subseteq [0,T]$ put
\[
w(X;A):=\sup_{s,t\in A} d_K(X_t,X_s).
\]
Now, for $\delta\in (0,1)$, define the c\`adl\`ag modulus to be
\[
w'(X;\delta):=\inf_{\Sigma} \max_{1\le i\le k} w(X;[t_{i-1},t_i)),
\]
where the infimum is taken over all partitions $\Sigma=\{0=t_0<t_1<\cdot \cdot \cdot <t_k=T\}$, $k\in \mathbb{N}$, with $\min_{1\le i\le k}(t_i-t_{i-1})>\delta$. For a function to lie in $D([0,T],K)$, it is necessary and sufficient to satisfy $w'(X;\delta)\to 0$, as $\delta\to 0$. Let $B_n$ be the set of functions having a constant value in $K^n$ over each interval $[(u-1)T/n,uT/n)$, for some $n\in \mathbb{N}$ and also a value in $K^n$ at time $T$. Take $B=\cup_{n\ge 1} B_n$, and observe that is countable. Clearly, putting $z=(z_u)_{u=0}^{n}$, with $z_u=uT/n$, for every $u=0,...,n$ satisfies $0=z_0<z_1<\cdot \cdot \cdot <z_n=T$. Let $T_z: D([0,T],K)\to D([0,T],K)$ be the map that is defined in the following way. For $X\in D([0,T],K)$ take $T_zX$ to have a constant value $X(z_{u-1})$ over the interval $[z_{u-1},z_u)$ for $1\le u\le n$ and the value $X(T)$ at $t=T$. From an adaptation of \cite[Lemma 3, p.127]{billingsley2013convergence}, considering c\`adl\`ag paths that take values on metric spaces, we have that
\begin{equation} \label{separ1}
d_{J_1}(T_zX,X)\le T n^{-1}+w'(X;T n^{-1}).
\end{equation}
Also, there exists $X^n\in B_n$, for which
\begin{equation} \label{separ2}
d_{J_1}(T_zX,X^n)\le T n^{-1}.
\end{equation}
Combining \eqref{separ1} and \eqref{separ2}, we have that
\[
d_{J_1}(X^n,X)\le d_{J_1}(X^n,T_zX)+d_{J_1}(T_zX,X)\le 2 T n^{-1}+w'(X;T n^{-1}).
\]
With the choice of the sequence $(K^n,\pi^n,X^n,L^n)$, we find that
\[
d_{\mathbb{K}}((K^n,\pi^n,X^n,L^n),(K,\pi,X,L))\le (4 +2 T) n^{-1}+3 \varepsilon_n+w'(X;T n^{-1}).
\]
Recalling that $w'(X;T n^{-1})\to 0$, as $n\to \infty$, and noting that our sequence was drawn from a countable subset of $\mathbb{K}$ completes the proof of the proposition.
\end{proof}
Fix $T>0$. Let $\tilde{\mathbb{K}}$ be the space of quadruples of the form $(K,\pi^K,X^K,L^K)$, where $K$ is a non-empty compact pointed metric space with distinguished vertex $\rho$, $\pi^K$ is a Borel measure on $K$, $X^K=(X^K_t)_{t\in [0,K]}$ is a c\`adl\`ag path on $K$ and $L^K=(L_t(x))_{x\in K,t\in [0,T]}$ is a jointly continuous positive real-valued function of $(t,x)$. We say that two elements of $\tilde{\mathbb{K}}$, say $(K,\pi^K,X^K,L^K)$ and $(K',\pi^{K'},X^{K'},L^{K'})$, are equivalent if and only there is a root-preserving isometry $f:K\to K'$, such that $f(\rho)=\rho'$, $\pi^K\circ f^{-1}=\pi^{K'}$, $f\circ X^K=X^{K'}$ and $L_t^{K'}\circ f=L_t^K$, for every $t\in [0,T]$. It is possible to define a metric on the equivalence classes of $\tilde{\mathbb{K}}$ by imposing in the definition of $d_{\mathbb{K}}$ that the infimum is taken over all correspondences that contain $(\rho,\rho')$. The incorporation of distinguished points to the extended Gromov-Hausdorff topology leaves the proof of Proposition \ref{Prop1.1} unchanged and it is possible to show that $(\tilde{\mathbb{K}},d_{\tilde{\mathbb{K}}})$ is a separable metric space.
The aim of the following lemmas is to establish a sufficient condition for Assumption \ref{Assum1} to hold, as well as to show that if Assumption \ref{Assum1} holds then we can isometrically embed the rescaled graphs, measures, random walks and local times into a common metric space such that they all converge to the relevant objects. To be more precise we formulate this last statement in the next lemma.
\begin{lemma} \label{lem2.2}
If Assumption \ref{Assum1} is satisfied, then we can find isometric embeddings of $(V(G^n),d_{G^n})_{n\ge 1}$ and $(K,d_K)$ into a common metric space $(F,d_F)$ such that
\begin{equation} \label{extra}
\lim_{n\rightarrow \infty} d_H^F(V(G^n),K)=0, \qquad \lim_{n\to \infty} d_F(\rho^n,\rho)=0,
\end{equation}
where $d_H^F$ is the standard Hausdorff distance between $V(G^n)$ and $K$, regarded as subsets of $(F,d_F)$,
\begin{equation} \label{2.6}
\lim_{n\rightarrow \infty} d_P^F(\pi^n,\pi)=0,
\end{equation}
where $d_P^F$ is the standard Prokhorov distance between $V(G^n)$ and $K$, regarded as subsets of $(F,d_F)$,
\begin{equation} \label{2.7}
\lim_{n\rightarrow \infty} d_{J_1}^F(X^n,X)=0,
\end{equation}
where $d_{J_1}^F$ is the Skorohod $d_{J_1}$ metric between $V(G^n)$ and $K$, regarded as subsets of $(F,d_F)$. Also,
\begin{equation} \label{2.8}
\lim_{\delta\rightarrow 0} \limsup_{n\rightarrow \infty} \sup_{\substack{x^n\in V(G^n),x\in K: \\ d_F(x^n,x)<\delta}} \sup_{t\in [0,T]} |L^n_{\beta(n) t}(x^n)-L_t(x)|=0.
\end{equation}
For simplicity we have identified the measures and the random walks in $V(G^n)$ with their isometric embeddings in $(F,d_F)$.
\end{lemma}
\begin{proof}
Since Assumption \ref{Assum1} holds, for each $n\ge 1$ we can find metric spaces $(F_n,d_n)$, isometric embeddings $\phi_n:V(G^n)\rightarrow F_n$, $\phi_n':K\rightarrow F_n$ and correspondences $\mathcal{C}^n$ (that contain $(\rho^n,\rho)$) between $V(G^n)$ and $K$ such that (identifying the relevant objects with their embeddings)
\begin{align} \label{2.9}
d_{P}^{F_n}(\pi^n,\pi)+d_{J_1}^{F_n}(X^n,X)+\sup_{(x,x')\in \mathcal{C}^n}\bigg (d_n(x,x')+\sup_{t\in [0,T]}|L^n_{\beta(n) t}(x)-L_t(x')|\bigg )\le \varepsilon_n,
\end{align}
where $\varepsilon_n\rightarrow 0$, as $n\rightarrow \infty$. Now, let $F=\sqcup_{n\ge 1} F_n$, be the disjoint union of $F_n$, and define the distance $d_{F|F_n\times F_n}=d_{n}$, for $n\ge 1$, and for $x\in F_n$, $x'\in F_{n'}$, $n\neq n'$
\[
d_F(x,x'):=\inf_{y\in K}\{d_n(x,y)+d_{n'}(y,x')\}.
\]
This distance, as the distance that was defined in order to prove the triangle inequality in Proposition \ref{Prop1.1}, is symmetric and non-negative, so identifying points that are separated by a zero distance, we turn $(F,d_F)$ into a metric space, which comes with natural isometric embeddings of $(V(G^n),d_{G^n})_{n\ge 1}$ and $(K,d_K)$. In this setting, under the appropriate isometric embeddings \eqref{extra}, \eqref{2.6} and \eqref{2.7} readily hold from \eqref{2.9}. Thus, it only remains to prove \eqref{2.8}. For every $x\in V(G^n)$, since $\mathcal{C}^n$ is a correspondence in $V(G^n)\times K$, there exists an $x'\in K$ such that $(x,x')\in \mathcal{C}^n$. Then, \eqref{2.9} implies that $d_F(x,x')\le \varepsilon_n$. Now, let $(y,y')\in \mathcal{C}^n$, $(z,z')\in \mathcal{C}^n$ and note that
\begin{align*}
&\sup_{t\in [0,T]} |L_{\beta(n) t}^n(y)-L_{\beta(n) t}^n(z)|\\
&\le \sup_{t\in [0,T]} |L_{\beta(n) t}^n(y)-L_t(y')|+\sup_{t\in [0,T]} |L_{\beta(n) t}^n(z)-L_t(z')|+\sup_{t\in [0,T]} |L_t(y')-L_t(z')|\\
&\le 2 \varepsilon_n+\sup_{t\in [0,T]} |L_t(y')-L_t(z')|.
\end{align*}
For any $\delta>0$ and $y, z\in V(G^n)$, such that $d_{G^n}(y,z)<\delta$, we have that
\[
d_K(y',z')\le d_F(y,y')+d_F(z,z')+d_{G^n}(y,z)<2 \varepsilon_n+\delta.
\]
Therefore,
\begin{align} \label{2.10}
&\sup_{\substack{y,z\in V(G^n): \\ d_{G^n}(y,z)<\delta}} \sup_{t\in [0,T]} |L_{\beta(n) t}^n(y)-L_{\beta(n) t}^n(z)|\nonumber \\
&\le 2 \varepsilon_n+\sup_{\substack{y,z\in K: \\ d_K(y,z)< 2 \varepsilon_n+\delta}} \sup_{t\in [0,T]} |L_t(y)-L_t(z)|.
\end{align}
Also, for every $x\in K$ there exists an $x'\in V(G^n)$ such that $(x',x)\in \mathcal{C}^n$, and furthermore $d_F(x',x)\le \varepsilon_n$. Let $x^n\in V(G^n)$ such that $d_F(x^n,x)<\delta$. Then,
\[
d_F(x^n,x')\le d_F(x^n,x)+d_F(x',x)<2 \epsilon_n+\delta.
\]
More generally, if we denote by $B_F(x,r)$, the open balls of radius $r$, centered in $x$, we have the following inclusion
\[
B_F(x,\delta)\cap V(G^n)\subseteq B_F(x',2 \varepsilon_n+\delta)\cap V(G^n).
\]
For $x\in K$, and $x'\in V(G^n)$ with $d_F(x',x)\le \varepsilon_n$, using \eqref{2.9}, we deduce
\begin{align*}
&\sup_{t\in [0,T]} |L_{\beta(n) t}^n(x^n)-L_t(x)|\\
&\le \sup_{t\in [0,T]} |L_{\beta(n) t}^n(x^n)-L_{\beta(n) t}^n(x')|+\sup_{t\in [0,T]} |L_{\beta(n) t}^n(x')-L_t(x)|\\
&\le \varepsilon_n+\sup_{t\in [0,T]} |L_{\beta(n) t}^n(x^n)-L_{\beta(n) t}^n(x')|.
\end{align*}
Since $x^n \in B_F(x',2 \varepsilon_n+\delta)\cap V(G^n)$, taking the supremum over all $x^n\in V(G^n)$ and $x\in K$, for which $d_F(x^n,x)<\delta$ and using \eqref{2.10}, we deduce
\begin{align*}
&\sup_{\substack{x^n\in V(G^n),x\in K: \\ d_F(x^n,x)<\delta}} \sup_{t\in [0,T]} |L_{\beta(n) t}^n(x^n)-L_t(x)|\\
&\le \varepsilon_n+\sup_{\substack{y,z\in V(G^n): \\ d_{G^n}(y,z)<2 \varepsilon_n+\delta}} \sup_{t\in [0,T]} |L_{\beta(n) t}^n(y)-L_{\beta(n) t}^n(z)|\\
&\le 3 \varepsilon_n+\sup_{\substack{y,z\in K: \\ d_K(y,z)< 4 \varepsilon_n+\delta}} \sup_{t\in [0,T]} |L_t(y)-L_t(z)|.
\end{align*}
Using the continuity of $L$, as $n\rightarrow \infty$
\begin{equation} \label{explicit}
\limsup_{n\rightarrow \infty} \sup_{\substack{x^n\in V(G^n),x\in K: \\ d_F(x^n,x)<\delta}} \sup_{t\in [0,T]} |L_{\beta(n) t}^n(x^n)-L_t(x)|\le \sup_{\substack{y,z\in K: \\ d_K(y,z)\le \delta}} \sup_{t\in [0,T]} |L_t(y)-L_t(z)|.
\end{equation}
Again appealing to the continuity of $L$, the right-hand side converges to 0, as $\delta\rightarrow 0$. Thus, we showed that \eqref{2.8} holds, and this finishes the proof of Lemma \ref{lem2.2}.
\end{proof}
In the process of proving \eqref{2.8} we established a useful equicontinuity property. We state and prove this property in the next corollary.
\begin{corollary}
Fix $T>0$ and suppose that Assumption \ref{Assum1} holds. Then,
\begin{equation} \label{2.11}
\lim_{\delta\rightarrow 0} \limsup_{n\rightarrow \infty} \sup_{\substack{y,z\in V(G^n): \\ d_{G^n}(y,z)<\delta}} \sup_{t\in [0,T]} |L_{\beta(n) t}^n(y)-L_{\beta(n) t}^n(z)|=0.
\end{equation}
\end{corollary}
\begin{proof}
As we hinted upon when deriving \eqref{explicit}, using the continuity of $L$,
\[
\limsup_{n\rightarrow \infty} \sup_{\substack{y,z\in V(G^n): \\ d_{G^n}(y,z)<\delta}} \sup_{t\in [0,T]} |L_{\beta(n) t}^n(y)-L_{\beta(n) t}^n(z)|\le \sup_{\substack{y,z\in K: \\ d_K(y,z)\le \delta}} \sup_{t\in [0,T]} |L_t(y)-L_t(z)|.
\]
Sending $\delta\rightarrow 0$ gives the desired result.
\end{proof}
Next, we prove that if we reverse the conclusions of Lemma \ref{lem2.2}, more specifically if \eqref{extra}-\eqref{2.8} hold, then also Assumption \ref{Assum1} holds.
\begin{lemma} \label{lem2.3}
Suppose that \eqref{extra}-\eqref{2.8} hold. Then so does Assumption \ref{Assum1}.
\end{lemma}
\begin{proof}
There exist isometric embeddings of $(V(G^n),d_{G^n})_{n\ge 1}$ and $(K,d_K)$ into a common metric space $(F,d_F)$, under which the assumptions \eqref{extra}-\eqref{2.8} hold. Since \eqref{extra} gives the convergence of spaces under the Hausdorff metric, \eqref{2.6} gives the convergence of measures under the Prokhorov metric and \eqref{2.7} gives the convergence of paths under $d_{J_1}$, it only remains to check the uniform convergence of local times. Let $\mathcal{C}^n$ be the set of all pairs $(x,x')\in K\times V(G^n)$, for which $d_F(x,x')\le n^{-1}$. Since \eqref{extra} holds, $\mathcal{C}^n$ are correspondences for $n\ge 1$. Then, for $(x,x')\in \mathcal{C}^n$
\[
\sup_{t\in [0,T]} |L_{\beta(n) t}^n(x')-L_t(x)|\le \sup_{\substack{x^n\in V(G^n),x\in K: \\ d_F(x^n,x)<n^{-1}}} \sup_{t\in [0,T]} |L^n_{\beta(n) t}(x^n)-L_t(x)|,
\]
and using \eqref{2.8} completes the proof.
\end{proof}
\section{Blanket time-scaling and distributional bounds} \label{blankbounds}
In this section, we show that under Assumption \ref{Assum1}, and as a consequence of the local time convergence in Lemma \ref{emblem}, we are able to establish asymptotic bounds on the distribution of the blanket times of the graphs in the sequence. The argument for the cover time-scaling was provided first in \cite[Corollary 7.3]{croydon2015moduli} by restricting to the unweighted Sierpi\'nski gasket graphs. The argument is applicable to any other model as long as the relevant assumptions are satisfied.
\subsection{Proof of Theorem \ref{Mth}} \label{Sec2.2}
First, let us check that the $\varepsilon$-blanket time variable of $K$ as written in \eqref{blacken1} is well-defined.
\begin{proposition} \label{nontrivexp}
Fix $\varepsilon\in (0,1)$. For every $x\in K$, $\mathbf{P}_x$-a.s. we have that $\tau_{\textnormal{bl}}(\varepsilon)\in (0,\infty)$.
\end{proposition}
\begin{proof}
Fix $x,y\in K$. There is a strictly positive $\mathbf{P}_x$-probability that $L_t(x)>0$ for $t$ large enough, which is a consequence of \cite[Lemma 3.6]{marcus1992sample}. From the joint continuity of $(L_t(x))_{x\in K,t\ge 0}$, there exist $r\equiv r(x)$, $\delta\equiv \delta(x)>0$ and $t_{*}\equiv t_{*}(x)<\infty$ such that
\begin{equation} \label{stem1}
\mathbf{P}_x\left(\inf_{z\in B(x,r)} L_{t_{*}}(z)>\delta\right)>0.
\end{equation}
Now, let $\tau_{x,y}(t_{*}):=\inf\{t>t_{*}+\tau_x: X_t=y\}$, where $\tau_x:=\inf \{t>0: X_t=x\}$ is the hitting time of $x$. In other words, $\tau_{x,y}(t_{*})$ is the first hitting of $y$ by $X$ after time $t_{*}+\tau_x$. The commute time identity for a resistance derived in the proof of \cite[Lemma 2.9]{croydon2016time} yields that $\mathbf{E}_x \tau_y+\mathbf{E}_y \tau_x=R(x,y) \pi(K)$, for every $x,y\in K$, which in turn deduces that $\mathbf{E}_x \tau_y<\infty$. Applying this observation about the finite first moments of hitting times, one can check that $\tau_{x,y}(t_{*})<\infty$, $\mathbf{P}_y$-a.s., and also that
\[
\mathbf{P}_y\left(\inf_{z\in B(x,r)} L_{\tau_{x,y}(t_{*})}(z)>\delta\right)>0.
\]
This simply follows from an application of \eqref{stem1} and the Strong Markov property. The additivity of local times and the Strong Markov property implies that
\begin{equation} \label{insecure}
\liminf_{t\to \infty} \inf_{z\in B(x,r)} \frac{L_t(z)}{t}\ge \left(\sum_{i=1}^{\infty}\xi_i^1\right) \left(\sum_{i=1}^{\infty} \xi_i^2\right)^{-1},
\end{equation}
where $(\xi_i^1)_{i\ge 1}$ are independent random variables distributed as $\inf_{z\in B(x,r)} L_{\tau_{x,y}(t_{*})}(z)$
and $(\xi_i^2)_{i\ge 1}$ are independent random variables distributed as $\tau_{x,y}(t_{*})$. The strong law of large numbers deduces that the right-hand side of \eqref{insecure} converges to $\mathbf{E}_y \left(\inf_{z\in B(x,r)} L_{\tau_{x,y}(t_{*})}(z)\right) (\mathbf{E}_y \tau_{x,y}(t_{*}))^{-1}$, $\mathbf{P}_x$-a.s. Using the occupation density formula and the commute time identity for a resistance, observe that
\begin{align*}
\mathbf{E}_y L_{\tau_{x,y}(t_{*})}(x) \cdot (\mathbf{E}_y \tau_{x,y}(t_{*}))^{-1}=\mathbf{E}_x L_{\tau_y}(x) \cdot (\mathbf{E}_y \tau_{x,y}(t_{*}))^{-1}
=R(x,y)\cdot (R(x,y) \pi(K))^{-1}=\pi(K)^{-1},
\end{align*}
and therefore, fixing some $\varepsilon_{*}\in (\varepsilon,1)$, the joint continuity of the local time process lets it to be deduced that
\begin{equation} \label{stem2}
\liminf_{t\to \infty} \inf_{z\in B(x,r)} \frac{L_t(z)}{t}\ge \varepsilon_{*},
\end{equation}
$\mathbf{P}_x$-a.s. To extend this statement that holds uniformly over $B(x,r)$, $\mathbf{P}_x$-a.s., we use the compactness of $K$. Consider the open cover $(B(x,r))_{x_i\in K}$ the open cover for $K$, which say admits a finite subcover $(B(x_i,r_i))_{i=1}^{N}$. Since the right-hand of \eqref{stem2} is greater than $\varepsilon_{*}$, $\mathbf{P}_x$-a.s., the result clearly follows as
\[
\lim_{t\to \infty} \frac{L_t(x)}{t}=\min_{1\le i\le N} \lim_{t\to \infty} \inf_{x\in B(x_i,r_i)} \frac{L_t(x)}{t}.
\]
\end{proof}
We are now ready to prove one of our main results.
\begin{proof}[Proof of Theorem \ref{Mth}]
Let $\varepsilon\in (0,1)$, $\delta\in (0,1)$ and $t\in [0,T]$. Suppose that $t<{\tau}_{\textnormal{bl}}(\varepsilon(1-\delta))$. Then, there exists a $y\in K$ for which $L_t(y)<\varepsilon (1-\delta) t$. Using the Skorohod representation theorem, we can suppose that the conclusions of Lemma \ref{lem2.2} holds in an almost-surely sense. From \eqref{extra}, there exists $y^n\in V(G^n)$ such that, for $n$ large enough, $d_F(y^n,y)<2 \varepsilon$. Then, the local convergence at \eqref{2.8} implies that, for $n$ large enough
\[
\alpha(n) L_{\beta(n) t}^n(y^n)\le L_t(y)+\varepsilon \delta t.
\]
Thus, for $n$ large enough, it follows that $\alpha(n) L_{\beta(n) t}^n(y^n)\le L_t(y)+\varepsilon \delta t< \varepsilon t$. Using the time and space scaling identity, we deduce $m^{G^n} L^n_{\beta(n) t}(y^n)<\varepsilon \beta(n) t$, for $n$ large enough, which in turn implies that $\beta(n) t\le \tau_{\text{bl}}^n(\varepsilon)$, for $n$ large enough. As a consequence, we get that $\tau_{\text{bl}}(\varepsilon (1-\delta))\le \liminf_{n\rightarrow \infty} \beta(n)^{-1} \tau_{\text{bl}}^n(\varepsilon)$, which proves \eqref{blacken2}.
Assume now that $\tau_{\text{bl}}(\varepsilon(1+\delta))<t$. Then, for some $\tau_{\text{bl}}(\varepsilon(1+\delta))\le t_0<t$, it is the case that $L_{t_0}(x)\ge \varepsilon(1+\delta) t_0$, for every $y\in K$. As in the previous paragraph, using the Skorohod represantation theorem, we suppose that the conclusions of Lemma \ref{lem2.2} holds almost-surely. From \eqref{extra}, for every $y^n\in V(G^n)$, there exists a $y\in K$ such that, for $n$ large enough, $d_F(y^n,y)<2 \varepsilon$. From the local convergence statement at \eqref{2.8}, we have that, for $n$ large enough
\[
\alpha(n) L_{\beta(n) t_0}^n(y^n)\ge L_{t_0}(y)-\varepsilon \delta t_0.
\]
Therefore, for $n$ large enough, it follows that $\alpha(n) L^n_{\beta(n) t_0}(y^n)\ge L_{t_0}(y)-\varepsilon \delta t_0\ge \varepsilon t_0$, for every $y\in K$. As before, using the time and space scaling identity yields $m^{G^n} L^n_{\beta(n) t_0}(y^n)\ge \varepsilon \beta(n) t_0$, for every $y^n\in V(G^n)$ and large enough $n$, which in turn implies that $\beta(n) t_0\ge \tau^n_{\text{bl}}(\varepsilon)$, for $n$ large enough. As consequence we get that $\limsup_{n\rightarrow \infty} \beta(n)^{-1} \tau^n_{\text{bl}}(\varepsilon)\le \tau_{\text{bl}}(\varepsilon(1+\delta))$, from which \eqref{blacken3} follows by observing that $\tau_{\text{bl}}$ is right-continuous at $\varepsilon$.
\end{proof}
\subsection{Local time convergence}
To check that Assumption \ref{Assum1} holds we need to verify that the convergence of the rescaled local times in \eqref{2.8}, as suggested by Lemma \ref{lem2.2}. Due to work done in a more general framework in \cite{croydon2016time}, we can weaken the local convergence statement of \eqref{2.8} and replace it by the equicontinuity condition of \eqref{2.11}. In \eqref{resist2} we defined a resistance metric on a graph viewed as an electrical network. Next, we give the definition of a regular resistance form and its associated resistance metric for arbitrary non-empty sets, which is a combination of \cite[Definition 2.1]{croydon2016time} and \cite[Definition 2.2]{croydon2016time}.
\begin{definition} \label{resistf}
Let $K$ be a non-empty set. A pair $(\mathcal{E},\mathcal{K})$ is called a regular resistance form on $K$ if the following six conditions are satisfied.
\begin{enumerate}[label=\roman{*})]
\item $\mathcal{K}$ is a linear subspace of the collection of functions $\{f:K\to \mathbb{R}\}$ containing constants, and $\mathcal{E}$ is a non-negative symmetric quadratic form on $\mathcal{K}$ such that $\mathcal{E}(f,f)=0$ if and only if $f$ is constant on $K$.
\item Let $\sim$ be an equivalence relation on $\mathcal{K}$ defined by saying $f\sim g$ if and only if the difference $f-g$ is constant on $K$. Then, $(\mathcal{K}/\sim,\mathcal{E})$ is a Hilbert space.
\item If $x\neq y$, there exists $f\in \mathcal{K}$ such that $f(x)\neq f(y)$.
\item For any $x,y\in K$,
\[
R(x,y):=\sup \left\{\frac{|f(x)-f(y)|^2}{\mathcal{E}(f,f)}: f\in \mathcal{K}, \ \mathcal{E}(f,f)>0\right\}<\infty.
\]
\item If $\bar{f}:=(f\wedge 1)\vee 0$, then $f\in \mathcal{K}$ and $\mathcal{E}(\bar{f},\bar{f})\le \mathcal{E}(f,f)$ for any $f\in \mathcal{K}$.
\item If $\mathcal{K}\cap C_0(K)$ is dense in $C_0(K)$ with respect to the supremum norm on $K$, where $C_0(K)$ denotes the space of compactly supported, continuous (with respect to $R$) functions on $K$.
\end{enumerate}
\end{definition}
It is the first five conditions that have to be satisfied in order for the pair $(\mathcal{E},\mathcal{K})$ to define a resistance form. If in addition the sixth condition is satisfied then $(\mathcal{E},\mathcal{K})$ defines a regular resistance form. Note that the fourth condition can be rewritten as $R(x,y)^{-1}=\inf\{\mathcal{E}(f,f): f:K\to \mathbb{R},f(x)=0,f(y)=1\}$, and it can be proven that it is actually a metric on $K$ (see \cite[Proposition 3.3]{kigami2012resistance}). It also clearly resembles the effective resistance on $V(G)$ as defined in \eqref{resist2}. More specifically, taking $\mathcal{K}:=\{f:V(G)\to \mathbb{R}\}$ and $\mathcal{E}_G$ as defined in \eqref{resist1} one can prove that the pair $(\mathcal{E}_G,\mathcal{K})$ satisfies the six conditions of Definition \ref{resistf}, and therefore is a regular resistance form on $V(G)$ with associated resistance metric given by \eqref{resist2}. For a detailed proof of this fact see \cite[Example 1.2.5]{fukushima2010dirichlet}. Finally, in this setting given a regular Dirichlet form, standard theory gives us the existence of an associated Hunt process $X=((X_t)_{t\ge 0},\mathbf{P}_x,x\in K)$ that is defined uniquely everywhere (see \cite[Theorem 7.2.1]{fukushima2010dirichlet} and \cite[Theorem 9.9]{kigami2012resistance}).
Suppose that the discrete state spaces $(V(G^n))_{n\ge 1}$ are equipped with resistances $(R_{G^n})_{n\ge 1}$ as defined in \eqref{resist2} and that the limiting non-empty metric space $K$, that appears in Assumption \ref{Assum1}, is equipped with a resistance metric $R$ as in Definition \ref{resistf}, such that
\begin{itemize}
\item $(K,R)$ is compact,
\item $\pi$ is a Borel measure of full support on $(K,R)$,
\item $X=((X_t)_{t\ge 0},\mathbf{P}_x,x\in K)$ admits local times $L=(L_t(x))_{x\in K,t\ge 0}$ continuous at $x$, uniformly over compact intervals in $t$, $\mathbf{P}_x$ -a.s. for every $x\in K$.
\end{itemize}
In the following extra assumption we input the information encoded in the first three conclusions of Lemma \ref{lem2.2}, given that we work in a probabilistic setting instead. For simplicity as before we identify the various objects with their embeddings.
\begin{assum} \label{Assum3}
Fix $T>0$. Let $(G^n)_{n\ge 1}$ be a sequence of finite connected graphs that have at least two vertices, for which there exist sequences of real numbers $(\alpha(n))_{n\ge 1}$ and $(\beta(n))_{n\ge 1}$, such that
\[
\left(\left(V(G^n),\alpha(n) R_{G^n},\rho^n\right),\pi^n,\left(X^n_{\beta(n) t}\right)_{t\in [0,T]}\right)\longrightarrow \left(\left(K,R,\rho\right),\pi,X\right)
\]
in the sense of the pointed extended pointed Gromov-Hausdorff topology, where $\rho^n\in V(G^n)$ and $\rho\in K$ are distinguished points. Furthermore, suppose that for every $\varepsilon>0$ and $T>0$,
\begin{equation} \label{emb3}
\lim_{\delta\to 0} \limsup_{n\to \infty} \sup_{x\in V(G^n)} \mathbf{P}_x^n\left(\sup_{\substack{y,z\in V(G^n): \\ R_{G^n}(y,z)<\delta}} \sup_{t\in [0,T]} \alpha(n) |L_{\beta(n) t}^n(y)-L_{\beta(n) t}^n(z)|\ge \varepsilon\right)=0.
\end{equation}
\end{assum}
It is Assumption \ref{Assum3} we have to verify in the examples of random graphs we will consider later. As we prove below in the last lemma of this subsection, if Assumption \ref{Assum3} holds then the finite dimensional local times converge in distribution (see \eqref{2.8}). Given that $(V(G^n),R_{G^n})_{n\ge 1}$ and $(K,R)$ can be isometrically embedded into a common metric space $(F,d_F)$ such that $X^n$ under $\mathbf{P}_{\rho^n}^n$ converges weakly to the law of $X$ under $\mathbf{P}_{\rho}$ on $D([0,T],F)$ (see Lemma \ref{lem2.2}),
we can couple $X^n$ started from $\rho^n$ and $X$ started from $\rho$ into a common probability space such that $(X^n_{\beta(n) t})_{t\in [0,T]}\to (X_t)_{t\in [0,T]}$ in $D([0,T],F)$, almost-surely.
Denote by $\mathbf{P}$ the joint probability measure under which the convergence above holds. Proving the convegence of finite dimensional distributions of local times is then an application of three lemmas that appear in \cite{croydon2016time}, which we summarize below.
\begin{lemma} [\textbf{Croydon, Hambly, Kumagai \cite{croydon2016time}}] \label{three}
For every $x\in F$, $\delta>0$, introduce the function $f_{\delta,x}(y):=\max\{0,\delta-d_F(x,y)\}$. Then, under Assumption \ref{Assum3},
\begin{enumerate}[label=\roman{*})]
\item $\mathbf{P}$-a.s., for each $x\in K$ and $T>0$, as $\delta\to 0$,
\[
\sup_{t\in [0,T]} \left|\frac{\int_{0}^{t} f_{\delta,x}(X_s) ds}{\int_K f_{\delta,x}(y) \pi(dy)}
-L_t(x)
\right|\to 0.
\]
\item $\mathbf{P}$-a.s., for each $x\in K$, $T>0$ and $\delta>0$, as $n\to \infty$,
\[
\sup_{t\in [0,T]} \left|\frac{\int_{0}^{t} f_{\delta,x}(X_s) ds}{\int_K f_{\delta,x}(y) \pi(dy)}
-\frac{\int_{0}^{t} f_{\delta,x}(X_{\beta(n) s}^n) ds}{\int_{V(G^n)} f_{\delta,x}(y) \pi^n(dy)}
\right|\to 0.
\]
\item For each $x\in K$ and $T>0$, if $x^n\in V(G^n)$ is such that $d_F(x^n,x)\to 0$, as $n\to \infty$, then
\[
\lim_{\delta\to 0} \limsup_{n\to \infty} \mathbf{P}\left(\sup_{t\in [0,T]} \left|\frac{\int_{0}^{t} f_{\delta,x}(X_{\beta(n) s}^n) ds}{\int_{V(G^n)} f_{\delta,x}(y) \pi^n(dy)}-\alpha(n) L_{\beta(n) t}^n(x^n)\right|>\varepsilon \right)=0.
\]
\end{enumerate}
\end{lemma}
By applying the conclusions of Lemma \ref{three}, one deduces that for any $x\in K$ and $T>0$, if $x^n\in V(G^n)$ such that $d_F(x^n,x)\to 0$, as $n\to \infty$, then $(\alpha(n) L^n_{\beta(n) t}(x^n))_{t\in [0,T]}\to (L_t(x))_{t\in [0,T]}$ in $\mathbf{P}$-probability in $C([0,T],\mathbb{R})$. This result extends to finite collections of points, and this is enough to establish the convergence of finite dimensional distributions of local times.
\begin{lemma} \label{emblem}
Suppose that Assumption \ref{Assum3} holds. Then, if the finite collections $(x_i^n)_{i=1}^{k}$ in $V(G^n)$, for $n\ge 1$, are such that $d_F(x_i^n,x_i)\to 0$, as $n\to \infty$, for some $(x_i)_{i=1}^{k}$ in $K$, then it holds that
\begin{equation} \label{emb4}
(\alpha(n) L_{\beta(n) t}^n(x_i^n))_{i=1,...,k,t\in [0,T]}\to (L_t(x_i))_{i=1,...,k,t\in [0,T]},
\end{equation}
in distribution in $C([0,T],\mathbb{R}^k)$.
\end{lemma}
\section{Examples} \label{maintheorems}
In this section we demonstrate that it is possible to apply our main results in a number of examples where the graphs, and the limiting spaces are random. These examples include critical Galton-Watson trees, the critical Erd\H{o}s-R\'enyi random graph and the critical regime of the configuration model. The aforementioned models of sequences of random graphs exhibit a mean-field behavior at criticality in the sense that the scaling exponents for the walks, and consequently for the local times, are a multiple of the volume and the diameter of the graphs. In the first few pages of each subsection we quickly survey some of the key features of each example that will be helpful when verifying Assumption \ref{Assum3}. Our method used in proving continuity of the blanket time of the limiting diffusion is generic in the sense that it applies on each random metric measure space and a corresponding $\sigma$-finite measure that generates realizations of the random metric measure space in such a way that rescaling the $\sigma$-finite measure by a constant factor results in generating the same space with its metric and measure perturbed by a multiple of this constant factor.
\subsection{Critical Galton-Watson trees} \label{CGWT}
We start by briefly describing the connection between critical Galton-Watson trees and the Brownian continuum random tree (CRT). Let $\xi$ be a mean 1 random variable with variance $0<\sigma_{\xi}^2<+\infty$, whose distribution is aperiodic (its support generates the lattice $\mathbb{Z}$, not just a strict subgroup of $\mathbb{Z}$). Let $\mathcal{T}_n$ be a Galton-Watson tree with offspring distribution $\xi$ conditioned to have total number of vertices $n+1$, which is well-defined from the aperiodicity of the distribution of $\xi$. Then, it is the case that
\begin{equation} \label{con1}
\left(V(\mathcal{T}_n),n^{-1/2} d_{\mathcal{T}_n}\right)\rightarrow \left(\mathcal{T}_e,d_{\mathcal{T}_e}\right),
\end{equation}
in distribution with respect to the Gromov-Hausdorff distance between compact metric spaces, where $d_{\mathcal{T}_n}$ is the shortest path distance on the vertex set $V(\mathcal{T}_n)$ (see \cite{aldous1993continuum} and \cite{le1993uniform}). To describe the limiting object in \eqref{con1}, let $(e_t)_{0\le t\le 1}$ denote the normalised Brownian excursion, which is informally a linear Brownian motion, started from zero, conditioned to remain positive in $(0,1)$ and come back to zero at time 1. We extend the definition of $(e_t)_{0\le t\le 1}$ by setting $e_t=0$, if $t>1$. We define a distance $d_{\mathcal{T}_e}$ in $[0,1]$ by setting
\begin{equation} \label{dist}
d_{\mathcal{T}_e}(s,t)=e(s)+e(t)-2 \min_{r\in [s\wedge t,s\vee t]} e(r).
\end{equation}
Introducing the equivalence relation $s\sim t$ if and only if $e(s)=e(t)=\min_{r\in [s\wedge t,s\vee t]} e(r)$ and defining $\mathcal{T}_e:=[0,1]/\sim $ it is possible to check that $(\mathcal{T}_e,d_{\mathcal{T}_e})$ is almost-surely a compact metric space. Moreover, $(\mathcal{T}_e,d_{\mathcal{T}_e})$ is a random real tree , called the CRT. For the notion of compact real trees coded by functions and a proof of the previous result see \cite[Section 2]{le2006random}. There is a natural Borel measure upon $\mathcal{T}_e$, $\pi^e$ say, which is the image measure on $\mathcal{T}_e$ of the Lebesgue measure on $[0,1]$ by the canonical projection of $[0,1]$ onto $\mathcal{T}_e$.
Upon almost-every realization of the metric measure space $(\mathcal{T}_e,d_{\mathcal{T}_e},\pi^e)$, it is possible to define a corresponding Brownian motion $X^e$. The way this can be done is described in \cite[Section 2.2]{croydon2008convergence}. Now if we denote by $\mathbf{P}_{\rho^n}^{\mathcal{T}_n}$ the law of the simple random walk in $\mathcal{T}_n$, started from a distinguished point $\rho^n$, and by $\pi^n$ the stationary probability measure, then as it was shown in \cite{croydon2010scaling} the scaling limit in \eqref{con1} can be extended to the distributional convergence of
\[
\bigg(\big(V(\mathcal{T}_n),n^{-1/2} d_{\mathcal{T}_n},\rho^n\big),\pi^n(n^{1/2} \cdot ),\mathbf{P}_{\rho^n}^{\mathcal{T}_n}\big((n^{-1/2} X^n_{\lfloor n^{3/2} t \rfloor})_{t\in [0,1]}\in \cdot \big)\bigg)
\]
to $\bigg(\big(\mathcal{T}_e,d_{\mathcal{T}_e},\rho\big),\pi^e,\mathbf{P}_{\rho}^e\bigg)$, where $\mathbf{P}_{\rho}^e$ is the law of $X^e$, started from a distinguished point $\rho$. This convergence described in \cite{croydon2010scaling} holds after embedding all the relevant objects nicely into a Banach space. We can reformulate this result in terms of the pointed extended Gromov-Hausdorff topology that incorporates distinguished points. Namely
\begin{equation}\label{con2}
\left(\left(V(\mathcal{T}_n),n^{-1/2} d_{\mathcal{T}_n},\rho^n\right),\pi^n,\left(n^{-1/2} X^n_{\lfloor n^{3/2} t \rfloor}\right)_{t\in [0,1]}\right)\longrightarrow \left(\left(\mathcal{T}_e,d_{\mathcal{T}_e},\rho\right),\pi^e,\left(X^e_{(\sigma_{\xi}/2) t}\right)_{t\in [0,1]}\right),
\end{equation}
in distribution in an extended pointed Gromov-Hausdorff sense.
Next, we introduce the contour function of $\mathcal{T}_n$. Informally, it encodes the trace of the motion of a particle that starts from the root at time $t=0$ and then explores the tree from left to right, moving continuously at unit speed along its edges. Formally, we define a function first for integer arguments as follows:
\[
f(0)=\rho^n.
\]
Given $f(i)=v$, we define $f(i+1)$ to be, if possible, the leftmost child that has not been visited yet, let's say $w$. If no child is left unvisited, we let $f(i+1)$ be the parent of $v$. Then, the contour function of $\mathcal{T}_n$, is defined as the distance of $f(i)$ from the root $\rho_n$, i.e.
\[
V_n(i):=d_{\mathcal{T}_n}(\rho^n,f(i)), \qquad 0\le i\le 2 n.
\]
We extend $V_n$ continuously to $[0,2 n]$ by interpolating linearly between integral points. The following theorem is due to Aldous.
\begin{theorem} [\textbf{Aldous \cite{aldous1993continuum}}]
Let $v_n$ denote the normalized contour function of $\mathcal{T}_n$, defined by
\[
v_n(s):=\frac{V_n(2 n s)}{\sqrt{n}}, \qquad 0\le s\le 1.
\]
Then, the following convergence holds in distribution in $C([0,1])$:
\[
v_n\xrightarrow{(d)} v:=\frac{2}{\sigma_{\xi}} e,
\]
where $e$ is a normalized Brownian excursion.
\end{theorem}
An essential tool in what follows will be a universal concentration estimate of the fluctuations of local times that holds uniformly over compact time intervals. For the statement of this result let
\[
r(\mathcal{T}_n):=\sup_{x,y\in V(\mathcal{T}_n)} d_{\mathcal{T}_n}(x,y)
\]
denote the diameter of $\mathcal{T}_n$ in the resistance metric and $m(\mathcal{T}_n)$ denote the total mass of $\mathcal{T}_n$. Also, we introduce the rescaled resistance metric $\tilde{d}_{\mathcal{T}_n}(x,y):=r(\mathcal{T}_n)^{-1} d_{\mathcal{T}_n}(x,y)$.
\begin{theorem} [\textbf{Croydon \cite{croydon2015moduli}}] \label{prepar1}
For every $T>0$, there exist constants $c_1$ and $c_2$ not depending on $\mathcal{T}_n$ such that
\begin{equation} \label{proof1}
\sup_{y,z\in V(\mathcal{T}_n)} \mathbf{P}_{\rho^n}^{\mathcal{T}_n} \left(r(\mathcal{T}_n)^{-1} \sup_{t\in [0,T]} \left|L_{r(\mathcal{T}_n) m(\mathcal{T}_n) t}^n(y)-L_{r(\mathcal{T}_n) m(\mathcal{T}_n) t}^n(z) \right|\ge \lambda \sqrt{\tilde{d}_{\mathcal{T}_n}(y,z)}\right)\le c_1 e^{-c_2 \lambda}
\end{equation}
for every $\lambda\ge 0$. Moreover, the constants can be chosen in such a way that only $c_1$ depends on $T$.
\end{theorem}
We remark here that the product $m(\mathcal{T}_n) r(\mathcal{T}_n)$, that is the product of the volume and the diameter of the graph, which is also the maximal commute time of the random walk, gives the natural time-scaling for the various models of sequences of critical random graphs we are going to consider. The last ingredient we are going to make considerable use of is the tightness of the sequence $||v_n||_{H_{\alpha}}$ of H\"older norms, for some $\alpha>0$. The proof of \eqref{proof2} is based on Kolmogorov's continuity criterion (and its proof to get uniformity in $n$).
\begin{theorem} [\textbf{Janson and Marckert \cite{janson2005convergence}}] \label{prepar2}
There exists $\alpha\in (0,1/2)$ such that for every $\varepsilon>0$ there exists a finite real number $K_{\varepsilon}$ such that
\begin{equation} \label{proof2}
P\left(\sup_{s,t\in [0,1]} \frac{|v_n(s)-v_n(t)|}{|t-s|^{\alpha}}\le K_{\varepsilon}\right)\ge 1-\varepsilon,
\end{equation}
uniformly on $n$.
\end{theorem}
\begin{remark}
Building upon \cite{gitten2003state}, Janson and Marckert proved this precise estimate on the geometry of the trees when the offspring distribution has finite exponential moments. Relaxing this strong condition to only a finite variance assumption, the recent work of Marzouk and more specifically \cite[Lemma 1]{marzouk2018snake} implies that Theorem \ref{prepar2} holds for the normalized height function of $\mathcal{T}_n$, which constitutes an alternative encoding of the trees. That Theorem \ref{prepar2} can be stated as well in the terms of the normalized contour function of $\mathcal{T}_n$, with only a finite variance assumption, is briefly achieved using that the normalized contour process is arbitrarily close to a time-changed normalized height process. See the equation that appears as (15) in \cite[Theorem 1.7]{legall2005applications} and refer to \cite[Section 1.6]{legall2005applications} for a detailed discussion.
\end{remark}
Since $(\mathcal{T}_n)_{n\ge 1}$ is a collection of graph trees it follows that the shortest path distance $d_{\mathcal{T}_n}$, $n\ge 1$ is identical to the resistance metric on the vertex set $V(\mathcal{T}_n)$, $n\ge 1$. In this context, we make use of the full machinery provided by the theorems above in order to prove that the local times are equicontinuous with respect to the annealed law, which is formally defined for suitable events as
\begin{equation} \label{exemplary}
\mathbb{P}_{\rho^n}(\cdot):=\int \mathbf{P}_{\rho^n}^{\mathcal{T}_n}(\cdot) P(d \mathcal{T}_n).
\end{equation}
\begin{proposition} \label{own}
For every $\varepsilon>0$ and $T>0$,
\[
\lim_{\delta\rightarrow 0} \limsup_{n\to \infty} \mathbb{P}_{\rho^n} \left(\sup_{\substack{y,z\in V(\mathcal{T}_n): \\ n^{-1/2} d_{\mathcal{T}_n}(y,z)<\delta}} \sup_{t\in [0,T]} n^{-1/2} |L_{n^{3/2} t}^n(y)-L_{n^{3/2} t}^n(z)|\ge \varepsilon\right)=0.
\]
\end{proposition}
\begin{proof}
Let us define, similarly to $d_{\mathcal{T}_e}$, the distance $d_{v_n}$ in $[0,1]$ by setting
\[
d_{v_n}(s,t):=v_n(s)+v_n(t)-2 \min_{r\in [s\wedge t,s\vee t]} v_n(r).
\]
Using the terminology introduced to describe the CRT, it is a fact that $\mathcal{T}_n$ coincides with the tree coded by the function $n^{1/2} v_n$, equipped with the metric $n^{1/2} d_{v_n}$ and the equivalence relation $s\sim t$ if and only if $v_n(s)=v_n(t)=\min_{r\in [s\wedge t,s\vee t]} v_n(r)$. We denote by $p_{v_n}: [0,1]\rightarrow \mathcal{T}_n$ the canonical projection that maps every time point in $[0,1]$ to its equivalence class on $\mathcal{T}_n$. From Theorem \ref{prepar2}, there exist $C>0$ and $\alpha\in (0,1/2)$ such that
\begin{equation} \label{proof3}
d_{v_n}(s,t)\le C |t-s|^{\alpha}, \qquad \forall s,t\in [0,1]
\end{equation}
with probability arbitrarily close to 1. We condition on $v_n$, assuming that it satisfies \eqref{proof3}. Given $t_1,t_2\in [0,1]$, with $2 n t_1$ and $2 n t_2$ integers, such that $p_{v_n}(t_1)=y$ and $p_{v_n}(t_2)=z$, the total length of the path between $y$ and $z$ is, using \eqref{proof3}
\begin{align*}
V_n(2 n t_1)+V_n(2 n t_2)-2 \min_{r\in [t_1\wedge t_2,t_1\vee t_2]} V_n(2 n r)&=n^{1/2} (v_n(t_1)+v_n(t_2)-2 \min_{r\in [t_1\wedge t_2,t_1\vee t_2]} v_n(r))\\
&=n^{1/2} d_{v_n}(t_1,t_2)\le C n^{1/2} |t_1-t_2|^{\alpha}.
\end{align*}
By Theorem \ref{prepar1}, for any fixed $p>0$,
\begin{align*}
&\mathbf{E}_{\rho^n}^{\mathcal{T}_n} \left[\sup_{t\in [0,T]} r(\mathcal{T}_n)^{-1} \left|L_{r(\mathcal{T}_n) m(\mathcal{T}_n) t}^n(y)-L_{r(\mathcal{T}_n) m(\mathcal{T}_n) t}^n(z) \right|^p\right]\\
&=\int_{0}^{\infty} \mathbf{P}_{\rho^n}^{\mathcal{T}_n} \left(\sup_{t\in [0,T]} r(\mathcal{T}_n)^{-1} \left|L_{r(\mathcal{T}_n) m(\mathcal{T}_n) t}^n(y)-L_{r(\mathcal{T}_n) m(\mathcal{T}_n) t}^n(z) \right|\ge \varepsilon^{1/p}\right) d \varepsilon\\
&\le c_1 \int_{0}^{\infty} e^{-c_2 \frac{\varepsilon^{1/p}}{\sqrt{r(\mathcal{T}_n)^{-1} d_{\mathcal{T}_n}(y,z)}}} d \varepsilon.
\end{align*}
Changing variables, $\lambda^{1/p}=\frac{\varepsilon^{1/p}}{\sqrt{r(\mathcal{T}_n)^{-1} d_{\mathcal{T}_n}(y,z)}}$ yields
\[
\int_{0}^{\infty} e^{-c_2 \frac{\varepsilon^{1/p}}{\sqrt{r(\mathcal{T}_n)^{-1} d_{\mathcal{T}_n}(y,z)}}} d \varepsilon=(r(\mathcal{T}_n)^{-1} d_{\mathcal{T}_n}(y,z))^{p/2} \cdot \int_{0}^{\infty}e^{-c_2 \lambda^{1/p}} d \lambda\le c_3 (r(\mathcal{T}_n)^{-1} d_{\mathcal{T}_n}(y,z))^{p/2},
\]
where $c_3$ is a constant depending only on $p$. Therefore,
\begin{equation} \label{proof4}
\mathbf{E}_{\rho^n}^{\mathcal{T}_n} \left[\sup_{t\in [0,T]} r(\mathcal{T}_n)^{-1} \left|L_{r(\mathcal{T}_n) m(\mathcal{T}_n) t}^n(y)-L_{r(\mathcal{T}_n) m(\mathcal{T}_n) t}^n(z) \right|^p\right]\le c_3 (r(\mathcal{T}_n)^{-1} d_{\mathcal{T}_n}(y,z))^{p/2}.
\end{equation}
Conditioned on the event that $v_n$ satisfies \eqref{proof3}, observe that the total length of the path between $y$ and $z$ is bounded above by $C n^{1/2} |t_1-t_2|^{\alpha}$, and consequently the diameter of $\mathcal{T}_n$ in the resistance metric is bounded above by a multiple of $n^{1/2}$. More specifically, $r(\mathcal{T}_n)\le C n^{1/2} 2^{\alpha}$. Moreover, $m(\mathcal{T}_n)=2n$. Hence, by \eqref{proof4}, we have shown that, conditioned on $v_n$ satisfying \eqref{proof3}, for any fixed $p>0$,
\begin{align*}
\mathbf{E}_{\rho^n}^{\mathcal{T}_n} \left[\sup_{t\in [0,T]} n^{-1/2} \left|L_{n^{3/2} t}^n(y)-L_{n^{3/2} t}^n(z) \right|^p\right]&\le c_4 (n^{-1/2} d_{\mathcal{T}_n}(y,z))^{p/2} 2^{-\alpha p}\\
&\le c_5 \left|(t_1/4)-(t_2/4)\right|^{\alpha p/2}.
\end{align*}
This holds for all $t_1$ and $t_2$, with $2 n t_1$ and $2 n t_2$ integers, such that $p_{v_n}(t_1)=y$ and $p_{v_n}(t_2)=z$. Choosing $p$ such that $\alpha p>2$, and using the moment condition (13.14) of \cite[Theorem 13.5]{billingsley2013convergence} yields that, under the event that $v_n$ satisfies \eqref{proof3}, the sequence of the rescaled local times $\{(n^{-1/2} L^n_{n^{3/2} t}(2 n \cdot))_{t\in [0,T]}\}_{n\ge 1}$ is tight in $C[0,1]$. Therefore, for every $\varepsilon>0$ and $T>0$,
\[
\lim_{\delta\rightarrow 0} \limsup_{n\to \infty} \mathbb{P}_{\rho^n} \left(\sup_{\substack{y,z\in V(\mathcal{T}_n): \\ n^{-1/2} d_{\mathcal{T}_n}(y,z)<\delta}} \sup_{t\in [0,T]} n^{-1/2} |L_{n^{3/2} t}^n(y)-L_{n^{3/2} t}^n(z)|\ge \varepsilon\Bigg |\sup_{s,t\in [0,1]} \frac{d_{v_n}(s,t)}{|t-s|^{\alpha}}\le C \right)=0.
\]
To complete the proof, note that
\[
\mathbb{P}_{\rho^n}(A)\le \mathbb{P}_{\rho^n}\left(A\Bigg|\sup_{s,t\in [0,1]} \frac{d_{v_n}(s,t)}{|t-s|^{\alpha}}\le C\right)+P(d_{v_n}(s,t)>C |t-s|^{\alpha}, \ \forall s,t\in [0,1]),
\]
measurable $A\subseteq \mathbb{K}$. The desired result now follows using the tightness of the rescaled local times, conditioned on $v_n$ satisfying \eqref{proof3}, which was shown before, and using the fact that the second probability on the right-hand side above, by \eqref{proof3}, is arbitrarily small.
\end{proof}
\subsubsection{It\^o's excursion theory of Brownian motion} \label{needed}
We recall some key facts of It\^{o}'s excursion theory of reflected Brownian motion. Our main interest here lies on the scaling property of the It\^{o} excursion measure. Let $(L_t^0)_{t\ge 0}$ denote the local time process at level 0 of the reflected Brownian motion $(|B_t|)_{t\ge 0}$. It can be shown that
\[
L_t^0=\lim_{\varepsilon\rightarrow 0}\frac{1}{2 \varepsilon} \int_{0}^{t} \mbox{1\hspace{-4.25pt}\fontsize{12}{14.4}\selectfont\textrm{1}}_{[0,\varepsilon]}(|B_s|) ds,
\]
for every $t\ge 0$, a.s. The local time process at level 0 is increasing, and its set of points of increase coincides with the set of time points for which the reflected Brownian is identical to zero. Now, introducing the right-continuous inverse of the local time process at level 0, i.e.
\[
\tau_k:=\inf\{t\ge 0: L_t^0>k\},
\]
for every $k\ge 0$, we have that the set of points of increase of $(L_t^0)_{t\ge 0}$ alternatively belong to the set
\[
\{\tau_k:k\ge 0\}\cup \{\tau_{k-}: k\in D\},
\]
where $D$ is the set of countable discontinuities of the mapping $k\mapsto \tau_k$. Then, for every $k\in D$ we define the excursion $(e_k(t))_{t\ge 0}$ with excursion interval $(\tau_{k-},\tau_k)$ away from 0 as
\[
e_k(t):=
\begin{cases}
|B_{t+\tau_{k-}}| & \text{if } 0\le t\le \tau_k-\tau_{k-}, \\
0 & \text{if } t>\tau_k-\tau_{k-}.
\end{cases}
\]
Let $E$ denote the space of excursions, namely the space of functions $e\in C(\mathbb{R}_+,\mathbb{R}_+)$, satisfying $e(0)=0$ and $\zeta(e):=\sup\{s>0: e(s)>0\}\in (0,\infty)$. By convention $\sup \emptyset=0$. Observe here that for every $k\in D$, $e_k\in E$, and moreover $\zeta(e_k)=\tau_k-\tau_{k-}$. The main theorem of It\^o's excursion theory adapted in our setting is the existence of a $\sigma$-finite measure $\mathbb{N} (d e)$ on the space of positive excursions of linear Brownian motion, such that the point measure
\[
\sum_{k\in D}\delta_{(k,e_k)}(ds \ de)
\]
is a Poisson measure on $\mathbb{R}_{+}\times E$, with intensity $ds \otimes \mathbb{N}(d e)$. The It\^o excursion measure has the following scaling property. For every $a>0$ consider the mapping $\Theta_a: E\rightarrow E$ defined by setting $\Theta_a(e)(t):=\sqrt{a}e(t/a)$, for every $e\in E$, and $t\ge 0$. Then, we have that
\begin{equation} \label{sc1}
\mathbb{N}\circ \Theta^{-1}_a=\sqrt{a} \mathbb{N}
\end{equation}
Versions of the It\^o excursion measure $\mathbb{N}(d e)$ under different conditionings are possible. For example one can define conditionings with respect to the height or the length of the excursion. For our purposes we focus on the fact that there exists a unique collection of probability measures $(\mathbb{N}_s: s>0)$ on $E$, such that $\mathbb{N}_s(\zeta=s)=1$, for every $s>0$, and moreover for every measurable event $A\in E$,
\begin{equation} \label{sc2}
\mathbb{N}(A)=\int_{0}^{\infty} \mathbb{N}_s(A) \frac{ds}{2 \sqrt{2 \pi s^3}}.
\end{equation}
In other words $\mathbb{N}_s(d e)$ is the It\^o excursion measure $\mathbb{N}(d e)$, conditioned on the event $\{\zeta=s\}$. We might write $\mathbb{N}_1=\mathbb{N}(\cdot |\zeta=1)$ to denote law of the normalized Brownian excursion. It is straightforward from \eqref{sc1} and \eqref{sc2} to check that $\mathbb{N}_s$ satisfies the scaling property
\begin{equation} \label{sc3}
\mathbb{N}_s\circ \Theta_a^{-1}=\mathbb{N}_{as},
\end{equation}
for every $s>0$ and $a>0$. To conclude our recap on It\^o's excursion theory we highlight the fact that for every $t>0$ the process $(e(t+r))_{r\ge 0}$ is Markov under the conditional probability measure $\mathbb{N}(\cdot |\zeta>t)$. The transition kernel of the process is the same with the one of a Brownian motion killed upon the first time it hits zero.
\subsubsection{Continuity of blanket times of Brownian motion on the CRT} \label{unes}
We are primarily interested in proving continuity of the $\varepsilon$-blanket time variable of the Brownian motion on the CRT. For every $\varepsilon \in (0,1)$ we let
\[
\mathcal{A}_{\varepsilon}:=\left\{(\mathcal{T}_e)_{e\in E}: \mathbf{P}_{\rho}^{e}\left(\tau_{\text{bl}}^e(\varepsilon-)=\tau_{\text{bl}}^e(\varepsilon)\right)=1\right\}
\]
denote the the collection of random trees coded by positive excursions that have continuous $\varepsilon$-blanket times $\mathbf{P}_{\rho}^{e}$-a.s. Recall that the mapping $\varepsilon\mapsto \tau_{\text{bl}}^e(\varepsilon)$ is increasing in $(0,1)$, and therefore it posseses left and right limits, so $\mathcal{A}_{\varepsilon}$ is well-defined for every $\varepsilon\in (0,1)$. Moreover, $\varepsilon\mapsto \tau_{\text{bl}}^e(\varepsilon)$ has at most a countably infinite number of discontinuities $\mathbf{P}_{\rho}^e$-a.s. Using Fubini, we immediately get
\[
\int_{0}^{1} \int \mathbf{P}_{\rho}^{e}\left(\tau_{\text{bl}}^e(\varepsilon-)\neq \tau_{\text{bl}}^e(\varepsilon)\right) \mathbb{N}(d e) d \varepsilon=\mathbf{E}_{\mathbb{P}} \left[\int_{0}^{1} \mbox{1\hspace{-4.25pt}\fontsize{12}{14.4}\selectfont\textrm{1}}_{\left\{\tau_{\text{bl}}^e(\varepsilon-)\neq \tau_{\text{bl}}^e(\varepsilon)\right\}} d \varepsilon \right]=0,
\]
where by $\mathbf{E}_{\mathbb{P}_{\rho}}$, we denote the expectation with respect to the measure
\begin{equation} \label{sc5}
\mathbb{P}_{\rho}(\cdot):=\int \mathbf{P}_{\rho}^{e}(\cdot) \mathbb{N}(d e).
\end{equation}
Therefore, if we denote the Lebesgue measure by $\lambda$, we deduce that for $\lambda$-a.e. $\varepsilon$
\[
\int \left(1-\mathbf{P}_{\rho}^{e}\left(\tau_{\text{bl}}^e(\varepsilon-)=\tau_{\text{bl}}^e(\varepsilon)\right)\right) \mathbb{N}(d e)=0.
\]
The fact that $\mathbb{N}(d e)$ is a sigma-finite measure on $E$ yields that for $\lambda$-a.e. $\varepsilon$, $\mathbb{N}$-a.e. $e$
\begin{equation} \label{sc6}
\mathbf{P}_{\rho}^{e}\left(\tau_{\text{bl}}^e(\varepsilon-)=\tau_{\text{bl}}^e(\varepsilon)\right)=1.
\end{equation}
Thus, we inferred that for $\lambda$-a.e. $\varepsilon$, $\mathbb{N}$-a.e. $e$, $\mathcal{T}_e\in \mathcal{A}_{\varepsilon}$.
To be satisfactory for our purposes, we need to improve this statement to hold for every $\varepsilon\in (0,1)$. Let $e\in E$ and consider the random real tree $(\mathcal{T}_e,d_{\mathcal{T}_e},\pi^e)$, where $d_{\mathcal{T}_e}$ is defined as in \eqref{dist} and $\pi^e$ is the image measure on $\mathcal{T}_e$ of the Lebesgue measure on $[0,\zeta]$ by the canonical projection $p_e$ of $[0,\zeta]$ onto $\mathcal{T}_e$. First, we observe that by applying the mapping $\Theta_a$ to $e$ for some $a>0$, results in perturbing $d_{\mathcal{T}_e}$ by a factor of $\sqrt{a}$ and $\pi^e$ by a factor of $a$. Recalling that $\Theta_a(e)(t)=\sqrt{a} e(t/a)$, for every $t\ge 0$, it is immediate that $d_{\mathcal{T}_{\Theta_a(e)}}(s,t)=\sqrt{a} d_{\mathcal{T}_e}(s,t)$, for every $s,t\in [0,\zeta]$. To see how $\pi^e$ rescales consider the set
\[
I=\{t\in [0,\zeta]: p_e(t)\in A\},
\]
where $A$ is a Borel set of $\mathcal{T}_e$. Then, by definition $\pi^e(A)=\lambda(I)$. Moreover, $\pi^{\Theta_a(e)}(A)=\lambda(I')$, where
\[
I=\{t\in [0,a \zeta]: p_e(t)\in A\}.
\]
By the scaling property of the Lebesgue measure, we have that $\pi^{\Theta_a(e)}(A)=a \lambda(I)$, and therefore $\pi^{\Theta_a(e)}(A)=a \pi^e(A)$. For simplicity, for the random real tree $\mathcal{T}= (\mathcal{T}_e,d_{\mathcal{T}_e},\pi^e)$, we write $\Theta_a \mathcal{T}$ to denote the resulting random real tree $(\mathcal{T}_e,\sqrt{a} d_{\mathcal{T}_e},a \pi^e)$ after rescaling.
Next, if the Brownian motion on $\mathcal{T}$ admits local times $(L_t(x))_{x\in \mathcal{T}_e,t\ge 0}$ that are jointly continuous $\mathbf{P}_{\rho}^e$-a.s., then it is the case that the Brownian motion on $\Theta_a \mathcal{T}$ admits local times distributed as $(\sqrt{a} L_{a^{-3/2} t}(x))_{x\in \mathcal{T}_e,t\ge 0}$ that are jointly continuous $\mathbf{P}_{\rho}^e$-a.s. To justify this check that, for every $t>0$
\[
\int \sqrt{a} L_{a^{-3/2 } t}(x) \pi^{\Theta_a(e)}(d x)=a^{3/2} \int L_{a^{-3/2} t}(x) \pi^ e(d x)=t.
\]
Now, for every $\varepsilon\in (0,1)$ fraction of time and every scalar parameter $a>1$, for the $\varepsilon$-blanket time variable of the Brownian motion on $\Theta_a \mathcal{T}$ as defined in \eqref{blacken1}, we have that
\begin{align*}
\tau^{\Theta_a(e)}_{\text{bl}}(a^{-1} \varepsilon) &\,{\buildrel (d) \over =}\, \inf \{t\ge 0: \sqrt{a} L_{a^{-3/2} t}(x)\ge \varepsilon a^{-1} t,\ \forall x\in \mathcal{T}_e\} \\
&\,{\buildrel (d) \over =}\, \inf \{t\ge 0: L_{a^{-3/2} t}(x)\ge \varepsilon a^{-3/2} t,\ \forall x\in \mathcal{T}_e\}\ \,{\buildrel (d) \over =}\, a^{-3/2} \tau_{\text{bl}}^e(\varepsilon).
\end{align*}
This fact asserts that $\mathcal{T}\in \mathcal{A}_{\varepsilon}$ if and only if $\Theta_a \mathcal{T}\in \mathcal{A}_{a^{-1} \varepsilon}$. In other words, $\tau_{\text{bl}}^e(\varepsilon)$ is continuous at $\varepsilon$, $\mathbf{P}_{\rho}^{e}$-a.s. if and only if $\tau_{\text{bl}}^{\Theta_a(e)}(a^{-1} \varepsilon)$ is continuous at $\varepsilon$, $\mathbf{P}_{\rho}^e$-a.s. Using the precise way in which the blanket times above relate as well as the scaling properties of the usual and the normalized It\^o excursion we prove the following proposition.
\begin{proposition} \label{pr2}
For every $\varepsilon\in (0,1)$, $\mathbb{N}$-a.e. $e$, $\tau_{\textnormal{bl}}^e(\varepsilon)$ is continuous at $\varepsilon$, $\mathbf{P}_{\rho}^e$-a.s. Moreover, $\mathbb{N}_1$-a.e. $e$, $\tau_{\textnormal{bl}}^e(\varepsilon)$ is continuous at $\varepsilon$, $\mathbf{P}_{\rho}^e$-a.s.
\end{proposition}
\begin{proof}
Fix $\varepsilon\in (0,1)$. We choose $a>1$ in such a way that $a^{-1} \varepsilon\in \Omega_0$, where $\Omega_0$ is the set for which the assertion in \eqref{sc6} holds $\lambda$-a.e. $\varepsilon$. Namely, $\mathbb{N}$-a.e. $e$, $\mathcal{T}\in \mathcal{A}_{a^{-1} \varepsilon}$. Using the scaling property of It\^o's excursion measure as quoted in \eqref{sc1} yields $\sqrt{a} \mathbb{N}$-a.e. $e$, $\Theta_a \mathcal{T}\in \mathcal{A}_{a^{-1} \varepsilon}$, and consequently $\mathbb{N}$-a.e. $e$, $\mathcal{T}\in \mathcal{A}_{\varepsilon}$, where we exploited the fact that $\Theta_a \mathcal{T}\in \mathcal{A}_{a^{-1} \varepsilon}$ if and only if $\mathcal{T}\in \mathcal{A}_{\varepsilon}$. Since $\varepsilon\in (0,1)$ was arbitrary, this establishes the first conclusion that $\mathbb{N}$-a.e. $e$, $\tau_{\text{bl}}^e(\varepsilon)$ is continuous at $\varepsilon$, $\mathbf{P}_{\rho}^e$-a.s.
What remains now is to prove a similar result but with $\mathbb{N}(d e)$ replaced with its version conditioned on the length of the excursion. Following the same steps we used in order to prove \eqref{sc6}, we infer that for $\lambda$-a.e. $\varepsilon$, $\mathbb{N}(\cdot |\zeta\in [1,2])$-a.e. $e$, $\mathcal{T}\in \mathcal{A}_{\zeta^{-1} \varepsilon}$, and consequently $\lambda$-a.e. $\varepsilon$, $\mathbb{N}(\cdot |\zeta\in [1,2])$-a.e. $e$, $\Theta_{\zeta} \mathcal{T}\in \mathcal{A}_{\varepsilon}$. Using the scaling property of the normalised It\^o excursion measure (see \eqref{sc3}), we deduce that $\lambda$-a.e. $\varepsilon$, $\mathbb{N}_1$-a.e. $e$, $\mathcal{T}\in \mathcal{A}_{\varepsilon}$, where $\mathbb{N}_1$ is the law of the normalized Brownian excursion. To conclude, we proceed using the same argument as in the first part of the proof.
Fix an $\varepsilon\in (0,1)$ and choose $a>1$ such that $a^{-1} \varepsilon\in \Phi_0$, where $\Phi_0$ is the set for which the assertion $\mathbb{N}_1$-a.e. $e$, $\mathcal{T}\in \mathcal{A}_{\varepsilon}$ holds $\lambda$-a.e $\varepsilon$. Namely, $\mathbb{N}_1$-a.e. $e$, $\mathcal{T}\in \mathcal{A}_{a^{-1} \varepsilon}$, which from the scaling property of the normalised It\^o excursion measure yields $a \mathbb{N}_1$-a.e. $e$, $\Theta_a \mathcal{T}\in \mathcal{A}_{a^{-1} \varepsilon}$. As before this gives us that $\mathbb{N}_1$-a.e. $e$, $\mathcal{T}\in \mathcal{A}_{\varepsilon}$, or in other words that $\mathbb{N}_1$-a.e. $e$, $\tau_{\text{bl}}^e(\varepsilon)$ is continuous at $\varepsilon$, $\mathbf{P}_{\rho}^e$-a.s.
\end{proof}
Since the space in which the convergence in \eqref{con2} takes place is separable we can use Skorohod's coupling to deduce that there exists a common metric space $(F,d_F)$ and a joint probability measure $\tilde{\mathbf{P}}$ such that, as $n\to \infty$,
\[
d_H^F(V(\tilde{\mathcal{T}}_n),\tilde{\mathcal{T}}_e)\to 0, \qquad d_P^F(\tilde{\pi}^n,\tilde{\pi}^e)\to 0, \qquad d_F(\tilde{\rho}^n,\tilde{\rho})\to 0, \qquad \tilde{\mathbf{P}}\text{ -a.s.},
\]
where $(V(\mathcal{T}_n),\pi^n,\rho^n)\,{\buildrel (d) \over =}\, (V(\tilde{\mathcal{T}_n}),\tilde{\pi}^n,\tilde{\rho}^n)$ and $(\mathcal{T}_e,\pi^e,\rho)\,{\buildrel (d) \over =}\, (\tilde{\mathcal{T}_e},\tilde{\pi}^e,\tilde{\rho})$. Moreover, $X^n$ under $\mathbf{P}_{\tilde{\rho}^n}^{\tilde{\mathcal{T}_n}}$ converges weakly to the law of $X^e$ under $\mathbf{P}_{\tilde{\rho}}^{\tilde{e}}$ on $D([0,1],F)$.
In Proposition \ref{own} we proved equicontinuity of the local times with respect to the annealed law. Then, reexamining the proof of Lemma \ref{emblem}, one can see that in this case $L^n$ under $\mathbb{P}_{\tilde{\rho}^n}(\cdot):= \int \mathbf{P}_{\tilde{\rho}^n}^{\tilde{\mathcal{T}_n}}(\cdot ) d \mathbf{\tilde{P}}$ will converge weakly to $L$ under $\mathbb{P}_{\tilde{\rho}}(\cdot):=\int \mathbf{P}_{\tilde{\rho}}^{\tilde{e}}(\cdot ) d \mathbf{\tilde{P}}$ in the sense of the local convergence as stated in \eqref{emb4}. It was this precise statement that was used extensively in the derivation of asymptotic distributional bounds for the rescaled blanket times in Section \ref{Sec2.2}. Then, the statement of Theorem \ref{Mth} translates as follows. For every $\varepsilon\in (0,1)$, $\delta\in (0,1)$ and $t\in [0,1]$,
\begin{equation*}
\limsup_{n\rightarrow \infty} \int \mathbf{P}_{\tilde{\rho}^n}^{\tilde{\mathcal{T}_n}} \left(n^{-3/2} \tau_{\textnormal{bl}}^n(\varepsilon)\le t\right) d \mathbf{\tilde{P}}\le \int \mathbf{P}_{\tilde{\rho}}^{\tilde{e}} \left(\tau_{\textnormal{bl}}^e(\varepsilon(1-\delta))\le t\right) d \mathbf{\tilde{P}},
\end{equation*}
\begin{equation*}
\liminf_{n\rightarrow \infty} \int \mathbf{P}_{\tilde{\rho}^n}^{\tilde{\mathcal{T}_n}} \left(n^{-3/2} \tau_{\textnormal{bl}}^n(\varepsilon)\le t\right) d \mathbf{\tilde{P}}\ge \int \mathbf{P}_{\tilde{\rho}}^{\tilde{e}} \left({\tau}_{\textnormal{bl}}^e(\varepsilon)<t\right) d \mathbf{\tilde{P}}.
\end{equation*}
From Proposition \ref{pr2} and the dominated convergence theorem we have that for every $\varepsilon\in (0,1)$ and $t\in [0,1]$,
\begin{align*}
\lim_{\delta\to 0} \int \mathbf{P}_{\tilde{\rho}}^{\tilde{e}} \left(\tau_{\textnormal{bl}}^e(\varepsilon(1-\delta))\le t\right) d \mathbf{\tilde{P}}&=\lim_{\delta\to 0} \int \mathbf{P}_{\tilde{\rho}}^{\tilde{e}} \left({\tau}_{\textnormal{bl}}^e(\varepsilon)<t\right) d \mathbf{\tilde{P}}=\int \mathbf{P}_{\rho}^e\left(\tau_{\text{bl}}^e(\varepsilon)\le t\right) \mathbb{N}(d e)
\\
&=\mathbb{P}_{\rho}\left(\tau_{\text{bl}}^e(\varepsilon)\le t\right).
\end{align*}
Therefore, we deduce that for every $\varepsilon\in (0,1)$ and $t\in [0,1]$,
\begin{align*}
\lim_{n\to \infty} \mathbb{P}_{\rho^n} \left(n^{-3/2} \tau_{\textnormal{bl}}^n(\varepsilon)\le t\right)&=\lim_{n\to \infty} \int \mathbf{P}_{\rho^n}^{\mathcal{T}_n} \left(n^{-3/2} \tau_{\textnormal{bl}}^n(\varepsilon)\le t\right) P(d \mathcal{T}_n)=\mathbb{P}_{\rho}\left(\tau_{\text{bl}}^e(\varepsilon)\le t\right).
\end{align*}
In the theorem below we state the $\varepsilon$-blanket time variable convergence result we have just proved.
\begin{theorem}
Fix $\varepsilon\in (0,1)$. If $\tau_{\textnormal{bl}}^n(\varepsilon)$ is the $\varepsilon$-blanket time variable of the random walk on $\mathcal{T}_n$, started from its root $\rho^n$, then
\[
\mathbb{P}_{\rho^n}\left(n^{-3/2} \tau_{\textnormal{bl}}^n(\varepsilon)\le t\right)\to \mathbb{P}_{\rho}\left(\tau_{\textnormal{bl}}^e(\varepsilon)\le t\right),
\]
for every $t\ge 0$, where $\tau_{\textnormal{bl}}^e(\varepsilon)\in (0,\infty)$ is the $\varepsilon$-blanket time variable of the Brownian motion on $\mathcal{T}_e$, started from $\rho$. Equivalently, for every $\varepsilon\in (0,1)$, $n^{-3/2} \tau_{\textnormal{bl}}^n(\varepsilon)$ under $\mathbb{P}_{\rho^n}$ converges weakly to $\tau_{\textnormal{bl}}^e(\varepsilon)$ under $\mathbb{P}_{\rho}$.
\end{theorem}
\subsection{The critical Erd\texorpdfstring{\H{o}}{H}s-R\'enyi random graph} \label{lalala}
Our interest in this section shifts to the Erd\H{o}s-R\'enyi random graph at criticality. Take $n$ vertices labelled by $[n]:=\{1,...,n\}$ and put edges between any pair independently with fixed probability $p\in [0,1]$. Denote the resulting random graph by $G(n,p)$. Let $p=c/n$ for some $c>0$. This model exhibits a phase transition in its structure for large $n$. With probability tending to $1$, when $c<1$, the largest connected component has size $O(\log n)$. On the other hand, when $c>1$, we see the emergence of a giant component that contains a positive proportion of the vertices. We will focus here on the critical case $c=1$, and more specifically, on the critical window $p=n^{-1}+\lambda n^{-4/3}$, $\lambda\in \mathbb{R}$. The most significant result in this regime was proven by Aldous \cite{aldous1997brownian}. Fix $\lambda\in \mathbb{R}$ and let $(C_i^n)_{i\ge 1}$ denote the sequence of the component sizes of $G(n,n^{-1}+\lambda n^{-4/3})$. For reasons that are inherent in understanding the structure of the components, we track the surplus of each one, that is the number of vertices that have to be removed in order to obtain a tree. Let $(S_i^n)_{n\ge 1}$ be the sequence of the corresponding surpluses.
\begin{theorem} [\textbf{Aldous \cite{aldous1997brownian}}] \label{PhT}
As $n\to \infty$,
\begin{equation} \label{PhT1}
\left(n^{-2/3} (C_i^n)_{i\ge 1},(S_i^n)_{i\ge 1}\right)\longrightarrow \left((C_i)_{i\ge 1},(S_i)_{i\ge 1}\right)
\end{equation}
in distribution, where the convergence of the first sequence takes place in $\ell^2_{\downarrow}$, the set of positive, decreasing sequences $(x_i)_{i\ge 1}$ with $\sum_{i=1}^{\infty} x_i^2<\infty$. For the second sequence it takes place in the product topology.
\end{theorem}
The limit is described by stochastic processes that encode various aspects of the structure of the random graph. Consider a Brownian motion with parabolic drift, $(B^{\lambda}_t)_{t\ge 0}$, where
\begin{equation} \label{parabola}
B^{\lambda}_t:=B_t+\lambda t-\frac{t^2}{2}
\end{equation}
and $(B_t)_{t\ge 0}$ is a standard Brownian motion. Then, the limiting sequence $(C_i)_{i\ge 1}$ has the distribution of the ordered sequence of lengths of excursions of the process $B^{\lambda}_t-\inf_{0\le s\le t} B^{\lambda}_s$, that is the parabolic Brownian motion reflected upon its minimum. Finally, $(S_i)_{i\ge 1}$ is recoved as follows. Draw the graph of the reflected process and scatter points on the place according to a rate 1 Poisson process and keep those that fall between the $x$-axis and the function. Then, $S_i$, $i\ge 1$ are the Poisson number of points that fell in the corresponding excursion with length $C_i$. Observe that the distribution of the limit $(C_i)_{i\ge 1}$ depends on the particular value of $\lambda$ chosen.
The scaling limit of the largest connected component of the Erd\H{o}s-R\'enyi random graph on the critical window arises as a tilted version of the CRT. Given a pointset $\mathcal{P}$, that is a subset of the upper half plane that contains only a finite number of points in any compact subset, and a positive Brownian excursion $e$, we define $\mathcal{P}\cap e$ as the number of points from $\mathcal{P}$ that fall under the area of $e$. We construct a glued metric space $\mathcal{M}_{e,\mathcal{P}}$ as follows. For each point $(t,x)\in \mathcal{P}\cap e$, let $u_{(t,x)}$ be the unique vertex $p_e(t)\in \mathcal{T}_e$ and $v_{(t,x)}$ be the unique vertex on the path from the root to $u_{(t,x)}$ at a distance $x$ from the root. Let $E_{\mathcal{P}}=\{(u_{(t,x)},v_{(t,x)}):(t,x)\in \mathcal{P}\cap e\}$ be the finite set that consists of the pairs of vertex identifications. Let $\{v_i,u_i\}_{i=1,...,k}$ be $k$ pairs of points that belong to $E_{\mathcal{P}}$. We define a quasi-metric on $\mathcal{T}_e$ by setting
\[
d_{\mathcal{M}_{e,\mathcal{P}}}(x,y):=\min\left\{d_{\mathcal{T}_e}(x,y),\inf_{i_1,...,i_r} \left\{d_{\mathcal{T}_e}(x,u_{i_1})+\sum_{j=1}^{r-1} d_{\mathcal{T}_e}(v_{i_j},u_{i_{j+1}})+d_{\mathcal{T}_e}(v_r,y)\right\}\right\},
\]
where the infimum is taken over $r$ positive integers, and all subsets $\{i_1,...,i_r\}\subseteq \{1,...,k\}$. Moreover, note that the vertices $i_1,...,i_k$ can be chosen to be distinct. The metric defined above gives the shortest distance between $x,y\in \mathcal{T}_e$ when we glue the vertices $v_i$ and $u_i$ for $i=1,...,k$. It is clear that $d_{\mathcal{M}_{e,\mathcal{P}}}$ defines only a quasi-metric since $d_{\mathcal{M}_{e,\mathcal{P}}}(u_i,v_i)=0$, for every $i=1,...,k$, but $u_i\neq v_i$, for every $i=1,...,k$. We define an equivalence relation on $\mathcal{T}_e$ by setting $x\sim_{E_{\mathcal{P}}} y$ if and only if $d_{\mathcal{M}_{e,\mathcal{P}}}(x,y)=0$. This makes the vertex identification explicit and $\mathcal{M}_{e,\mathcal{P}}$ is defined as
\[
\mathcal{M}_{e,\mathcal{P}}:=(\mathcal{T}_e/\sim_{E_{\mathcal{P}}},d_{\mathcal{M}_{e,\mathcal{P}}}).
\]
To endow $\mathcal{M}_{e,\mathcal{P}}$ with a canonical measure let $p_{e,\mathcal{P}}$ denote the canonical projection from $\mathcal{T}_e$ to the quotient space $\mathcal{T}_e/\sim_{E_{\mathcal{P}}}$. We define $\pi_{e,\mathcal{P}}:=\pi^e\circ p_{e,\mathcal{P}}^{-1}$, where $\pi^e$ is the image measure on $\mathcal{T}_e$ of the Lebesgue measure $\lambda$ on $[0,\zeta]$ by the canonical projection $p_e$ of $[0,\zeta]$ onto $\mathcal{T}_e$. So, $\pi_{e,\mathcal{P}}=\lambda\circ p_e^{-1} \circ p_{e,\mathcal{P}}^{-1}$. We note that the restriction of $p_{e,\mathcal{P}}$ to $\mathcal{T}_e$ is $p_e$.
For every $\zeta>0$, as in \cite{addario2012continuum}, we define a tilted excursion of length $\zeta$ to be a random variable that takes values in $E$ whose distribution is characterized by
\[
\mathbf{P}(\tilde{e}\in \mathcal{E})=\frac{\mathbf{E}\left(\mbox{1\hspace{-4.25pt}\fontsize{12}{14.4}\selectfont\textrm{1}}_{\{e\in \mathcal{E}\}}\exp\left(\int_{0}^{\zeta} e(t) dt\right)\right)}{\mathbf{E}\left(\exp\left(\int_{0}^{\zeta} e(t) dt\right)\right)},
\]
for every measurable $\mathcal{E}\subseteq E$. We note here that the $\sigma$-algebra on $E$ is the one generated by the open sets with respect to the supremum norm on $C(\mathbb{R}_{+},\mathbb{R}_{+})$. Write $\mathcal{M}^{(\zeta)}$ for the random compact metric space distributed as $(\mathcal{M}_{\tilde{e},\mathcal{P}},2 d_{\mathcal{M}_{\tilde{e},\mathcal{P}}})$, where $\tilde{e}$ is a tilted Brownian excursion of length $\zeta$ and the random pointset of interest $\mathcal{P}$ is a Poisson point process on $\mathbb{R}_{+}^2$ of unit intensity with respect to the Lebesgue measure independent of $\tilde{e}$. We now give an alternative description of $\mathcal{M}_{\tilde{e},\mathcal{P}}$. It is easy to prove that the number $|\mathcal{P}\cap \tilde{e}|$ of vertex identifications is a Poisson random variable with mean $\int_{0}^{\zeta} \tilde{e}(u) du$. Given that $|\mathcal{P}\cap \tilde{e}|=k$, the co-ordinate $u_{(t,x)}$ has density
\[
\frac{\tilde{e}(u)}{\int_{0}^{\zeta} \tilde{e}(t) dt}
\]
on $[0,\zeta]$, and given $u_{(t,x)}$, its pair $v_{(t,x)}$ is uniformly distributed on $[0,\tilde{e}(u_{(t,x)})]$. The other $k-1$ vertex identifications are distributed accordingly and independently of the pair $(u_{(t,x)},v_{(t,x)})$. After introducing notation, we are in the position to write the limit of the largest connected component, say $\mathcal{C}_1^n$, as $\mathcal{M}^{(C_1)}$, where $C_1$ has the distribution of the length of the longest excursion of the reflected upon its minimum parabolic Brownian motion as defined in \eqref{parabola}. Moreover, the longest excursion, when conditioned to have length $C_1$, is distributed as a tilted excursion $\tilde{e}$ with length $C_1$. The following convergence is a simplified version of \cite[Theorem 2]{addario2012continuum}. As $n\to \infty$,
\begin{equation}
\left(n^{-2/3} C_1^n,\left(V(\mathcal{C}_1^n),n^{-1/3} d_{\mathcal{C}_1^n}\right)\right)\longrightarrow \left(C_1,\left(\mathcal{M},d_{\mathcal{M}}\right)\right),
\end{equation}
in distribution, where conditional on $C_1$, $\mathcal{M}\,{\buildrel (d) \over =}\,\mathcal{M}^{(C_1)}$. Moreover, it was shown in \cite{croydon2012scaling}, that the discrete-time simple random walks $X^{\mathcal{C}_1^n}$ on $\mathcal{C}_1^n$, started from a distinguished vertex $\rho^n$ satisfy a distributional convergence of the form
\begin{equation}
\left(n^{-1/3} X_{\lfloor n t \rfloor}^{\mathcal{C}_1^n}\right)_{t\ge 0}\to \left(X_t^{\mathcal{M}}\right)_{t\ge 0},
\end{equation}
where $X^{\mathcal{M}}$ is a diffusion on $\mathcal{M}$, started from a distinguished point $\rho\in \mathcal{M}$. The convergence of the associated stationary probability measures, say $\pi^n$, was not directly proven in \cite{croydon2012scaling}, although the hard work to this direction has been done. More specifically, see \cite[Lemma 6.3]{croydon2012scaling}. The results above can be reformulated in the following distributional convergence in terms of the pointed extended Gromov-Hausdorff topology.
\begin{equation}
\left(\left(V(\mathcal{C}_1^n),n^{-1/3} d_{\mathcal{C}_1^n},\rho^n\right),\pi^n,\left(n^{-1/3} X^{\mathcal{C}_1^n}_{\lfloor n t \rfloor}\right)_{t\ge 0}\right)\longrightarrow \left(\left(\mathcal{M},d_{\mathcal{M}},\rho\right),\pi^{\mathcal{M}},X^{\mathcal{M}}\right).
\end{equation}
Now, we describe how to generate a connected component on a fixed number of vertices. To any such component we can associate a spanning subtree, the depth-first tree by considering the following algorithm. The initial step places the vertex with label 1 in a stack and declares it open. In the next step vertex 1 is declared as explored and is removed from the start of the stack, where we place in increasing order the neighbors of 1 that have not been seen (open or explored) yet. We proceed inductively. When the set of open vertices becomes empty the procedure finishes. It is obvious that the resulting graph that consists of edges between a vertex that was explored at a given step and a vertex that has not been seen yet at the same step, is a tree. For a connected graph $G$ with $m$ vertices, we refer to this tree as the depth-first tree and write $T(G)$. For $i=0,...,m-1$, let $X(i):=|O(i)|-1$ be the number of vertices seen but not yet fully explored at step $i$. The process $(X(i): 0\le i<m)$ is called the depth-first walk of the graph $G$.
Let $\mathbb{T}_m$ be the set of (unordered) tree labelled by $[m]$. For $T\in \mathbb{T}_m$, its associated depth-first tree is $T$ itself. We call an edge permitted by the first-depth procedure run on $T$ if its addition produces the same depth-first tree. Exactly $X(i)$ edges are permitted at step $i$, and therefore the total number of permitted edges is given by
\[
a(T):=\sum_{i=0}^{m-1} X(i),
\]
which is called the area of $T$. Given a tree $T$ and a connected graph $G$, $T(G)=T$ if and only if $G$ can be obtained from $T$ by adding a subset of permitted edges by the depth-first procedure. Therefore, writing $\mathbb{G}_T$ for the set of connected graphs $G$ that satisfy $T(G)=T$, we have that $\{\mathbb{G}_T: T\in \mathbb{T}_m\}$ is a partition of the connected graphs on $[m]$, and that the cardinality of $\mathbb{G}_T$ is $2^{a(T)}$, since every permitted edge is included or not.
Back to the question on how to generate a connected component, write $G_m^p$ for the graph with the same distribution as $G(m,p)$ conditioned to be connected. Thus, we focus on generating $G_m^p$ instead.
\begin{lemma} [\textbf{Addario-Berry, Broutin, Goldschmidt \cite{addario2012continuum}}] \label{gener}
Fix $p\in (0,1)$. Pick a random tree $\tilde{T}_m^p$ that has a ``tilted'' distribution which is biased in favor of trees with large area. Namely, pick $\tilde{T}_m^p$ in such a way that
\[
P(\tilde{T}_m^p=T)\propto (1-p)^{-a(T)}, \qquad T\in \mathbb{T}_m.
\]
Add to $\tilde{T}_m^p$ each of the $a(\tilde{T}_m^p)$ permitted edges independently with probability $p$. Call the graph generated $\tilde{G}_m^p$. Then, $\tilde{G}_m^p$ has the same distribution as $G_m^p$.
\end{lemma}
We use $\rho^m$ to denote the root of $\tilde{T}_m^p$. In what follows we give a detailed description on how we can transfer the results proved in Section \ref{CGWT}. We denote by $\tilde{V}_m=(\tilde{V}_m(i): 0\le i\le 2 m)$ the contour process of $\tilde{T}_m^p$, and by $\tilde{v}_m=(((m/\zeta)^{-1/2} \tilde{V}_m(2 (m/\zeta) s): 0\le s\le \zeta)$ the rescaled contour process of positive length $\zeta$ as well. We start by showing that, for some $\alpha>0$, the sequence $||\tilde{v}_m||_{H_{\alpha}}$ of H\"older norms is tight.
\begin{lemma} \label{tiltcont1}
Suppose that p=p(m) in such a way that $m p^{2/3}\to \zeta$, as $m\to \infty$. There exists $\alpha\in (0,1/2)$ such that for every $\varepsilon>0$ there exists a finite real number $M_{\varepsilon}$ such that
\begin{equation}
P\left(\sup_{s,t\in [0,1]}\frac{|\tilde{v}_m(s)-\tilde{v}_m(t)|}{|t-s|^{\alpha}}\le M_{\varepsilon}\right)\ge 1-\varepsilon.
\end{equation}
\end{lemma}
\begin{proof}
For notational simplity we assume that $\zeta=1$. The general result follows by Brownian scaling. Let $T_m$ be a tree chosen uniformly from $[m]$. Write $V_m$ and $v_m$ for its associated contour process and normalized contour process respectively. We note here that Theorem \ref{prepar2} is stated in the more general situation of Galton-Watson trees with critical offspring distribution that has finite variance and exponential moments, conditioned to have $m$ vertices. If the offspring distribution to be Poisson with mean $1$, then the conditioned tree is a uniformly distributed labelled tree. Then, by Lemma \ref{gener}, if $K_{\varepsilon}$ is the finite real number for which \eqref{proof2} holds,
\[
P\left(\sup_{s,t\in [0,1]}\frac{|\tilde{v}_m(s)-\tilde{v}_m(t)|}{|t-s|^{\alpha}}\ge K_{\varepsilon}\right)=\frac{E\left[\mbox{1\hspace{-4.25pt}\fontsize{12}{14.4}\selectfont\textrm{1}}_{\left\{\sup_{s,t\in [0,1]}\frac{|v_m(s)-v_m(t)|}{|t-s|^{\alpha}}\ge K_{\varepsilon}\right\}} (1-p)^{-a(T_m)}\right]}{E\left[(1-p)^{-a(T_m)}\right]}.
\]
Since $m p^{2/3}\to 1$, as $m\to \infty$, there exists $c>0$ such that $p\le c m^{-3/2}$, for every $m\ge 1$. Moreover, from \cite[Lemma 14]{addario2012continuum} we can find universal constants $K_1$, $K_2>0$ such that $E\left[(1-p)^{-\xi a(T_m)}\right]<K_1 e^{K_2 c^2 \xi^2}$, for every $\xi>0$. Using this along with the Cauchy-Schwarz inequality yields
\begin{align} \label{taram}
&P\left(\sup_{s,t\in [0,1]}\frac{|\tilde{v}_m(s)-\tilde{v}_m(t)|}{|t-s|^{\alpha}}\ge K_{\varepsilon}\right) \nonumber \\
&\le \frac{P\left(\sup_{s,t\in [0,1]}\frac{|v_m(s)-v_m(t)|}{|t-s|^{\alpha}}\ge K_{\varepsilon}\right)^{1/2} \left(E\left[(1-p)^{-2 a(T_m)}\right]\right)^{1/2}}{E\left[(1-p)^{-a(T_m)}\right]}\le \frac{(\varepsilon K_1e^{4 K_2 c^2})^{1/2}}{E\left[(1-p)^{-a(T_m)}\right]}.
\end{align}
Finally, recall that $a(T_m)=\sum_{i=0}^{m-1} X_m(i)$, where $(X_m(i): 0\le i\le m)$ is the depth-first walk associated with $T_m$ (for convenience we have put $X_m(m)=0$). From \cite[Theorem 3]{marckert2003depth} we know that, as $m\to \infty$
\[
(m^{-1/2} X_m(\lfloor m t \rfloor))_{t\in [0,1]}\to (e(t))_{t\in [0,1]},
\]
in distribution in $D([0,1],\mathbb{R}_{+})$, where $(e(t))_{t\in [0,1]}$ is a normalized Brownian excursion. Writing
\[
(1-p)^{-a(T_m)}=(1-p)^{-\sum_{i=0}^{m-1} X_m(i)}=(1-p)^{-m^{3/2} \int_{0}^{1} m^{-1/2} X_m(\lfloor m t \rfloor) dt}
\]
and using that the sequence $(1-p)^{-a(T_m)}$ is uniformly integrable, we deduce that
\[
E\left[(1-p)^{-a(T_m)}\right]\to E\left[\exp \left(\int_{0}^{1} e(u) du\right)\right]>0,
\]
as $m\to \infty$. The desired result follows from \eqref{taram} and the fact that the sequence of random variables $\{||\tilde{v}_m||_{H_{\alpha}}\}_{m\in I}$ is bounded in probability (tight) whenever $I$ is finite.
\end{proof}
It is now immediate to check that the local times $(L^m_t(x))_{x\in V(G_m^p),t\ge 0}$ of the corresponding simple random walk on $G_m^p$ are equicontinuous under the annealed law (which is defined similar to \eqref{exemplary}) after rescaling. The proof of the next lemma relies heavily on the same methods used to establish Proposition \ref{own}, and therefore we will make use of the parts that remain unchanged.
\begin{lemma} \label{tiltcomp}
Suppose that p=p(m) in such a way that $m p^{2/3}\to \zeta$, as $m\to \infty$. For every $\varepsilon>0$ and $T>0$,
\[
\lim_{\delta\rightarrow 0} \limsup_{m\to \infty} \mathbb{P}_{\rho^m} \left(\sup_{\substack{y,z\in V(G_m^p): \\ m^{-1/2} R_{G_m^p}(y,z)<\delta}} \sup_{t\in [0,T]} m^{-1/2} |L_{m^{3/2} t}^m(y)-L_{m^{3/2} t}^m(z)|\ge \varepsilon \Bigg | s(G_m^p)=s\right)=0.
\]
\end{lemma}
\begin{proof}
Again, for simplicity let $\zeta=1$. The general result follows by Brownian scaling. Call $\tilde{G}_m^p$ the graph generated by the process of adding $\text{Bin}(a(\tilde{T}_m^p),p)$ number of surplus edges to $\tilde{T}_m^p$. We view $\tilde{G}_m^p$ as the metric space $\tilde{T}_m^p$ that includes the edges (of length 1) that have been added. Recall that $\tilde{G}_m^p$ has the same distribution as $G_m^p$. From Lemma \ref{tiltcont1}, there exists $C_1>0$ and $\alpha\in (0,1/2)$ such that
\begin{equation} \label{tiltcont2}
\tilde{v}_m(s)+\tilde{v}_m(t)-2 \min_{r\in [s\wedge t,s\vee t]} \tilde{v}_m(r)\le C_1 |t-s|^{\alpha}, \qquad \forall s,t\in [0,1]
\end{equation}
with probability arbitrarily close to 1. Conditioned on $\tilde{v}_m$ satisfying \eqref{tiltcont2}, and given $t_1,t_2\in [0,1]$, with $2 m t_1$ and $2 m t_2$ integers, such that $p_{\tilde{v}_m}(t_1)=y$ and $p_{\tilde{v}_m}(t_2)=z$, the resistance on $\tilde{G}_m^p$ between $y$ and $z$ is smaller than the total length of the path between $y$ and $z$ on $\tilde{T}_m^p$. Therefore, using \eqref{tiltcont2}
\begin{align*}
R_{\tilde{G}_m^p}(y,z)&\le \tilde{V}_m(2 m t_1)+\tilde{V}_m(2 m t_2)-2 \min_{r\in [t_1\wedge t_2,t_1\vee t_2]} \tilde{V}_m(2 m r)\\
&=m^{1/2} (\tilde{v}_m(t_1)+\tilde{v}_m(t_2)-2 \min_{r\in [t_1\wedge t_2,t_1\vee t_2]} \tilde{v}_m(r))=m^{1/2} d_{\tilde{v}_m}(t_1,t_2)\le C_1 m^{1/2} |t_1-t_2|^{\alpha},
\end{align*}
which also indicates that, on the event that \eqref{tiltcont2} holds, the maximum resistance $r(\tilde{G}_m^p)$ is bounded above by $C m^{1/2} 2^{\alpha}$. Moreover, conditioning on $s(\tilde{G}_m^p)=s$, $m(\tilde{G}_m^p)=2 (m+s)$. Hence, by \eqref{proof4}, which still holds by replacing $\mathcal{T}_n$ by $\tilde{G}_m^p$, we can show that, conditioned on $\tilde{v}_m$ satisfying \eqref{tiltcont2}, for any fixed $q>0$,
\begin{align*}
\mathbf{E}_{\rho^m}^{\tilde{G}_m^p} \left[\sup_{t\in [0,T]} m^{-1/2} \left|L_{m^{3/2} t}^m(y)-L_{m^{3/2} t}^m(z) \right|^q\bigg| s(\tilde{G}_m^p)=s\right]&\le C_2 (m^{-1/2} R_{\tilde{G}_m^p}(y,z))^{q/2} 2^{-\alpha q}\\
&\le C_3 \left|(t_1/4)-(t_2/4)\right|^{\alpha q/2},
\end{align*}
where $t_1$ and $t_2$ are such that $2 m t_1$ and $2 m t_2$ are integers, such that $p_{\tilde{v}_m}(t_1)=y$ and $p_{\tilde{v}_m}(t_2)=z$. The rest of proof is finished in the manner of Proposition \ref{own}, and therefore we omit it.
\end{proof}
For notational simplicity, the next result is stated for the largest connected component on the critical window. In fact, it holds for the family of the $i$-th largest connected components, $i\ge 1$. In this case, let us denote by $\mathcal{C}_1^n$ the largest connected component of $G(n,n^{-1}+\lambda n^{-4/3})$, and by $(L_t^n(x))_{x\in V(\mathcal{C}_1^n),t\ge 0}$ the local times of the simple random walk on $\mathcal{C}_1^n$.
\begin{proposition} \label{ownown}
For every $\varepsilon>0$ and $T>0$,
\[
\lim_{\delta\rightarrow 0} \limsup_{n\to \infty} \mathbb{P}_{\rho^n} \left(\sup_{\substack{y,z\in V(\mathcal{C}_1^n): \\ n^{-1/3} R_{\mathcal{C}_1^n}(y,z)<\delta}} \sup_{t\in [0,T]} n^{-1/3} |L_{n t}^n(y)-L_{n t}^n(z)|\ge \varepsilon\right)=0.
\]
\end{proposition}
\begin{proof}
Conditioning on the size and the surplus of $\mathcal{C}_1^n$, we observe that for every $\varepsilon_1>0$ and $T_1>0$,
\begin{align} \label{boring}
&\mathbb{P}_{\rho^n} \left(\sup_{\substack{y,z\in V(\mathcal{C}_1^n): \\ n^{-1/3} R_{\mathcal{C}_1^n}(y,z)<\delta_1}} \sup_{t\in [0,T_1]} n^{-1/3} |L_{n t}^n(y)-L_{n t}^n(z)|\ge \varepsilon_1\right)
\\ \nonumber \le
&\sum_{m=1}^{\lfloor A n^{2/3} \rfloor} \sum_{s=0}^{S} \mathbb{P}_{\rho^n} \left(\sup_{\substack{y,z\in V(\mathcal{C}_1^n): \\ n^{-1/3} R_{\mathcal{C}_1^n}(y,z)<\delta_1}} \sup_{t\in [0,T_1]} n^{-1/3} |L_{n t}^{n}(y)-L_{n t}^{n}(z)|\ge \varepsilon_1\bigg ||V(\mathcal{C}_1^n)|=m,s(\mathcal{C}_1^n)=s\right)
\\ \nonumber \cdot
&P(|V(\mathcal{C}_1^n)|=m,s(\mathcal{C}_1^n)=s)+P(|V(\mathcal{C}_1^n)|>\lfloor A n^{2/3} \rfloor)+P(s(\mathcal{C}_1^n)>S),
\end{align}
for large enough constants $A$ and $S$. Since the largest components on the critical window $p=n^{-1}+\lambda n^{-4/3}$, $\lambda\in \mathbb{R}$ have sizes of order $n^{2/3}$, we have that $\mathcal{C}_1^n$ conditioned to have size $m$ and surplus $s$ has the same distribution with $G_m^p$, where $m$ and $p=p(m)$ are in such a way that $m p^{2/3}\to \zeta$, as $m\to \infty$, for some $\zeta>0$. Therefore, we acan bound \eqref{boring} by
\begin{align*}
&\sup_{m\ge 1} \mathbb{P}_{\rho^n} \left(\sup_{\substack{y,z\in V(G_m^p): \\ m^{-1/2} R_{G_m^p}(y,z)<\delta_2}} \sup_{t\in [0,T_2]} m^{-1/2} |L_{m^{3/2} t}^{m}(y)-L_{m^{3/2} t}^{m}(z)|\ge \varepsilon_2\bigg |s(G_m^p)=s\right)
\\ \\+
&P(|V(\mathcal{C}_1^n)|>\lfloor A n^{2/3} \rfloor)+P(s(\mathcal{C}_1^n)>S).
\end{align*}
From Theorem \ref{PhT1},
\begin{equation} \label{boring2}
\lim_{A\to \infty} \limsup_{n\to \infty} P(n^{-2/3} |V(\mathcal{C}_1^n)|>A)=0.
\end{equation}
Furthermore, as $n\to \infty$,
\[
s(\mathcal{C}_1^n)\to \text{Poi}\left(\int_{0}^{\zeta} \tilde{e}(u) du\right),
\]
where $\text{Poi}\left(\int_{0}^{\zeta} \tilde{e}(t) dt\right)$ denotes a Poisson random variable with mean the area under a tilted excursion of length $\sigma$ (see \cite[Corollary 23]{addario2012continuum}), and as a consequence tightness of process that encodes the surplus of $\mathcal{C}_1^n$ follows, i.e.
\begin{equation} \label{boring3}
\lim_{S\to \infty} \limsup_{n\to \infty} P(s(\mathcal{C}_1^n)>S).
\end{equation}
Now, the proof is finished by combining \eqref{boring2} and \eqref{boring3} with the equicontinuity result of Lemma \ref{tiltcomp}.
\end{proof}
\subsubsection{Continuity of blanket times of Brownian motion on \texorpdfstring{$\mathcal{M}$}{M}} \label{excm}
To prove continuity of the $\varepsilon$-blanket time of the Brownian motion on $\mathcal{M}$, we first define a $\sigma$-finite measure on the product space of positive excursions and random pointsets of $\mathbb{R}_{+}^2$. Throughout this section we denote the Lebesgue measure on $\mathbb{R}_{+}^2$ by $\ell$. We define the aforementioned measure $\mathbf{N}(d(e,\mathcal{P}))$ by setting
\begin{equation} \label{meas1}
\mathbf{N}(d e,|\mathcal{P}|=k, (d x_1,...,d x_k)\in A_1\times...\times A_k):=\int_{0}^{\infty} f_L(l) \mathbb{N}_l (d e) \frac{e^{-1}}{k!} \prod_{i=1}^{k} \frac{\ell(A_i\cap A_e)}{\ell(A_e)},
\end{equation}
where $f_L(l):=d l/\sqrt{2 \pi l^3}$, $l\ge 0$ gives the density of the length of the excursion $e$ and $A_e:=\{(t,x): 0\le x\le e(t)\}$ denotes the area under its graph. In other words, the measure picks first an excursion length according to $f_L(l)$ and, given $L=l$, it picks a Brownian excursion of that length. Then, independently of $e$ it chooses $k$ points according to a Poisson with unit mean, which distributes uniformly on the area under the graph of $e$.
It turns out that this is an easier measure to work with when applying our scaling argument to prove continuity of the blanket times. Also, as will see later, $\mathbf{N}$ is absolutely continuous with respect to the canonical measure $\mathbf{N}^{t,\lambda}(d(e,\mathcal{P}))$ that first at time $t$ picks a tilted Brownian excursion $e$ of a randomly chosen length $l$, and then independently of $e$ chooses $k$ points distributed as a Poisson random variable with mean $\int_{0}^{l} e(t) dt$, which as before are distributed uniformly on the area under the graph of $e$. To fully describe this measure let $\mathbb{N}^{t,\lambda}$ denote the measure (for excursions starting at time $t$) associated to $B^{\lambda}_t-\inf_{0\le s\le t} B^{\lambda}_s$, first stated by Aldous in \cite{aldous1997brownian}. We note that $\mathbb{N}^{t,\lambda}=\mathbb{N}^{0,\lambda-t}$ and thus it suffices to write $\mathbb{N}^{0,\lambda}$ for every $\lambda\in \mathbb{R}$. For every measurable $A\in E$,
\[
\mathbb{N}^{0,\lambda}(A)=\int_{0}^{\infty} \mathbb{N}^{0,\lambda}_l(A) f_{L}(l) F_{\lambda}(l) \mathbb{N}_l\left(\exp\left(\int_{0}^{l} e(u) du\right)\right),
\]
where $\mathbb{N}_{l}^{0,\lambda}$ is a shorthand for the excursion measure $\mathbb{N}^{0,\lambda}$, conditioned on the event $\{\tilde{L}=l\}$ and $F_{\lambda}(l):=\exp\left(-1/6\left(\lambda^3+(l-\lambda)^3\right)\right)$. For simplicity, let $g_{\tilde{L}}(l,\lambda):=f_{L}(\lambda) F_{\lambda}(l) \mathbb{N}_l\left(\exp\left(\int_{0}^{l} e(u) du\right)\right)$. In analogy with \eqref{meas1} we characterize $\mathbf{N}^{t,\lambda}(d(e,\mathcal{P}))$ by setting
\begin{align} \label{meas3}
&\mathbf{N}^{t,\lambda}(d e,|\mathcal{P}|=k, (d x_1,...,d x_k)\in A_1\times...\times A_k)
\nonumber \\ :=
&\int_{0}^{\infty} g_{\tilde{L}}(l,\lambda-t) \mathbb{N}^{t,\lambda}_l (d e) \exp\left(-\int_{0}^{l} e(u) du\right)\frac{\left(\int_{0}^{l} e(u) du\right)^k}{k!} \prod_{i=1}^{k} \frac{\ell(A_i\cap A_e)}{\ell(A_e)}.
\end{align}
After calculations that involve the use of the Cameron-Martin-Girsanov formula \cite[Chapter IX, (1.10) Theorem]{revuz1999continuous} (for the entirety of those calculations one can consult \cite[Section 5]{addario2012continuum}), one deduces that
\[
\mathbb{N}_l^{t,\lambda}(d e)=\exp\left(\int_{0}^{l} e(u) du\right) \frac{\mathbb{N}_l(d e)}{\mathbb{N}_l\left(\exp\left(\int_{0}^{l} e(u) du\right)\right)},
\]
and as a consequence the following expression for the Radon-Nikodym derivative is valid.
\begin{equation} \label{meas4}
\frac{d \mathbf{N}^{t,\lambda}}{d \mathbf{N}}=\frac{F_{\lambda-t}(l) \left(\int_{0}^{l} e(u) du\right)^k/k!}{e^{-1}/k!}=\exp\left(1-\frac{1}{6}\left(\lambda^3+(l-\lambda+t)^3\right)\right) \left(\int_{0}^{l} e(u) du\right)^k.
\end{equation}
The main reason to consider $\mathbf{N}$ instead of $\mathbf{N}^{t,\lambda}$ is that it can easily be seen to enjoy the same scaling property with the normalized It\^o excursion measure. Recall that for every $b>0$, the mapping $\Theta_b:E\to E$ defined by setting $\Theta_b(e)(t):=\sqrt{b} e(t/b)$, for every $e\in E$, and $t\ge 0$. As we saw in subsection \ref{unes} acts on the real tree coded by $e$ scaling its distance and invariant probability measure appropriately. Let $\mathcal{M}_{e,\mathcal{P}}$, where $e$ is a Brownian excursion of length $\zeta$ and pointset $\mathcal{P}$, that is a Poisson process on $\mathbb{R}_{+}^2$ of intensity one with respect to the Lebesgue measure independent of $e$. If $|\mathcal{P}\cap e|$ is distributed as a Poisson random variable with mean $\int_{0}^{\zeta} e(u) du$, then $|\mathcal{P}\cap \Theta_b(e)|$ has law given by a Poisson distribution with mean $b^{3/2} \int_{0}^{\zeta} e(u) du$. Moreover, conditioned on $|\mathcal{P}\cap e|$, if the coordinates of a point $(u_{(t,x)},v_{(t,x)})$ in $\mathcal{P}\cap e$ have densities proportional to $e(u)$ for $u_{(t,x)}$ and, conditioned on $u_{(t,x)}$, uniformly on $[0,e(u_{(t,x)})]$ for $v_{(t,x)}$ respectively, then conditioned on $|\mathcal{P}\cap \Theta_b(e)|$, the coordinates of a point $(u^b_{(t,x)},v^b_{(t,x)})$ in $\mathcal{P}\cap \Theta_b(e)$ has coordinates distributed as $b u_{(t,x)}$ in the case of $u^b_{(t,x)}$, and conditioned on $u_{(t,x)}$, uniform on $[0,\sqrt{b} u_{(t,x)}]$ in the case of $v^b_{(t,x)}$ respectively. From now on we use $\Theta_b(e,\mathcal{P})$ to denote the mapping from the product space of positive excursions and pointsets of the upper half plance onto itself that rescales $e$ as $\Theta_b(e)$ and repositions the collection of points in $\mathcal{P}$ as described above.
Starting with how the distance rescales under the application of $\Theta_b$, $b>0$, it is immediate from the definition of the quasi-metric $d_{\mathcal{M}_{e,\mathcal{P}}}$ that $d_{\mathcal{M}_{\Theta_b(e,\mathcal{P})}}=b^{1/2} d_{\mathcal{M}_{e,\mathcal{P}}}$. Let $\mathcal{L}(\mathcal{T}_e)$ denote the set of leaves of $\mathcal{T}_e$. Then, $\pi^e$ has full support on $\mathcal{L}(\mathcal{T}_e)$ and $\pi^e(\mathcal{L}(\mathcal{T}_e))=\zeta(e)$. Consider the set
\[
I=\{\sigma\in \mathcal{L}(\mathcal{T}_e): p_{e,\mathcal{P}}(\sigma)\in A\},
\]
for a measurable subset $A$ of $\mathcal{M}_{e,\mathcal{P}}$. Then, from the definition of $\pi_{e,\mathcal{P}}$, we have that $\pi_{e,\mathcal{P}}(A)=\pi^e(I)$, and consequently $\pi_{\Theta_b(e,\mathcal{P})}(A)=\pi^{\Theta_b(e)}(I)$. As we examined before $\pi^{\Theta_b(e)}(I)=b \pi^e(I)$, and from this it follows that $\pi_{\Theta_b(e,\mathcal{P})}(A)=b \pi_{e,\mathcal{P}}(A)$, for every measurable subset $A$ of $\mathcal{M}_{e,\mathcal{P}}$. Finally, since $\mathbb{N}\circ \Theta_b^{-1}=\sqrt{b} \mathbb{N}$ and using the fact that $\ell(A_i\cap A_e)/\ell(A_e)$ in \eqref{meas1} is scale invariant under $\Theta_b$, it follows that
$\mathbf{N}\circ \Theta_b^{-1}=\sqrt{b} \mathbf{N}$, i.e. $\mathbf{N}(d(e,\mathcal{P}))$ rescales exactly like $\mathbb{N}(d e)$.
We now have all the ingredients to prove continuity of the blanket times of the Brownian motion on $\mathcal{M}$. We briefly describe the arguments that have been already used in establishing Proposition \ref{pr2}. Let $\tau_{\text{bl}}^{e,\mathcal{P}}(\varepsilon)$ denote the $\varepsilon$-blanket time of the Brownian motion $X^{e,\mathcal{P}}$ on $\mathcal{M}_{e,\mathcal{P}}$ started from a distinguished vertex $\bar{\rho}$, for some $\varepsilon\in (0,1)$. Taking the expectation of the law of $\tau_{\text{bl}}^{e,\mathcal{P}}(\varepsilon)$, $\varepsilon\in (0,1)$ against the $\sigma$-finite measure $\mathbf{N}$ (see \eqref{sc5}), using Fubini and the monotonicity of the blanket times, yields
\[
\mathbf{P}_{\bar{\rho}}^{e,\mathcal{P}}\left(\tau_{\text{bl}}^{e,\mathcal{P}}(\varepsilon-)=\tau_{\text{bl}}^{e,\mathcal{P}}(\varepsilon)\right)=1,
\]
$\lambda$-a.e. $\varepsilon$, $\mathbf{N}$-a.e. $(e,\mathcal{P})$, where $\mathbf{P}_{\bar{\rho}}^{e,\mathcal{P}}$ denotes the law of $X^{e,\mathcal{P}}$. The rest of the argument relies on impoving such a statement to hold for every $\varepsilon\in (0,1)$ by using scaling.
In the transformed glued metric space $\mathcal{M}_{\Theta_b(e,\mathcal{P})}$ the Brownian motion admits $\mathbf{P}_{\bar{\rho}}^{e,\mathcal{P}}$-a.s. jointly continuous local times $(\sqrt{b} L_{b^{-3/2} t}(x))_{x\in \mathcal{M}_{e,\mathcal{P}},t\ge 0}$. This enough to infer that, for every $\varepsilon\in (0,1)$ and $b>1$, the continuity of the $\varepsilon$-blanket time variable of $\mathcal{M}_{e,\mathcal{P}}$ is equivalent to the continuity of the $b^{-1} \varepsilon$-blanket time variable of $\mathcal{M}_{\Theta_b(e,\mathcal{P})}$, and consequently as in the proof of Proposition \ref{pr2} applying our scaling argument implies
\[
\mathbf{P}_{\bar{\rho}}^{e,\mathcal{P}}\left(\tau_{\text{bl}}^{e,\mathcal{P}}(\varepsilon-)=\tau_{\text{bl}}^{e,\mathcal{P}}(\varepsilon)\right)=1,
\]
$\mathbf{N}$-a.e. $(e,\mathcal{P})$. Recall that, conditional on $C_1$, $\mathcal{M}\,{\buildrel (d) \over =}\,\mathcal{M}^{(C_1)}$, where $C_1$ is the length of the longest excursion of the process defined in \eqref{parabola}, which is distributed as a tilted excursion of that length. Then, applying again our scaling argument as in the end of the proof of Proposition \ref{pr2}, conditional on $C_1$, we deduce
\[
\mathbf{P}_{\rho}^{\mathcal{M}}\left(\tau_{\text{bl}}^{\mathcal{M}}(\varepsilon-)=\tau_{\text{bl}}^{\mathcal{M}}(\varepsilon)\right)=1,
\]
$\mathbf{N}_{C_1}$-a.e. $(e,\mathcal{P})$, where $\mathbf{N}_l$ is the version of $\mathbf{N}$ defined in \eqref{meas1} conditioned on the event $\{L=l\}$. Since the canonical measure $\mathbf{N}^{0,\lambda}_{C_1}$ is absolutely continuous with respect to $\mathbf{N}_{C_1}$ as it was shown in \eqref{meas4}, the above also yields that conditional on $C_1$, $\mathbf{N}^{0,\lambda}_{C_1}$-a.e. $(e,\mathcal{P})$, $\varepsilon\mapsto \tau_{\text{bl}}^{\mathcal{M}}(\varepsilon)$ is continuous $\mathbf{P}_{\rho}^{\mathcal{M}}$-a.s.
Given this result and Proposition \eqref{ownown} the following convergence theorem follows from Theorem \ref{Mth}. Here, for a particular real value of $\lambda$ and conditional on $C_1$,
\begin{equation} \label{lastbits}
\mathbb{P}_{\rho}(\cdot):=\int \mathbf{P}_{\rho}^{\mathcal{M}}(\cdot ) \mathbf{N}_{C_1}^{0,\lambda}(d(e,\mathcal{P})),
\end{equation}
formally defines the annealed measure for suitable events.
\begin{theorem}
Fix $\varepsilon\in (0,1)$. If $\tau_{\textnormal{bl}}^{n}(\varepsilon)$ is the $\varepsilon$-blanket time variable of the random walk on $\mathcal{C}_1^n$, started from its root $\rho^n$, then
\[
\mathbb{P}_{\rho^n}\left(n^{-1} \tau_{\textnormal{bl}}^n(\varepsilon)\le t\right)\to \mathbb{P}_{\rho}\left(\tau_{\textnormal{bl}}^{\mathcal{M}}(\varepsilon)\le t\right),
\]
for every $t\ge 0$, where $\tau_{\textnormal{bl}}^{\mathcal{M}}(\varepsilon)\in (0,\infty)$ is the $\varepsilon$-blanket time variable of the Brownian motion on $\mathcal{M}$, started from $\rho$.
\end{theorem}
\subsection{Critical random graph with prescribed degrees}
Let $\text{G}_{n,d}$ denote the space of all simple graphs labelled by $[n]$ such that the $i$-th vertex has degree $d_i$, $i\ge 1$, for $1\le i\le n$. We denote the vector $(d_i: i\in [n])$ of the prescribed degree sequence by $d$, where $\ell_n:=\sum_{i\in [n]} d_i$ is assumed even. Write $\bar{\text{G}}_{n,d}$ for $\text{G}_{n,d}$ with the difference that we allow self-loops as well as the occurence of multiple edges between the same pair of vertices. Then, the configuration model is the random multigraph in $\text{G}_{n,d}$ constructed as follows. Assign each vertex $i$ with $d_i$ half-edges, labelling them arbitrarily by $1,...,\ell_n$. The multigraph $\text{M}^n(d)$ produced by a uniform random pairing of the half-edges to create full edges is called the configuration model. In particular, we look at prescribed degree sequences that satisfy the following assumptions.
\begin{assum} \label{gracias}
Let $D_n$ is a random variable with distribution given by
\[
P(D_n=i)=\frac{\# \{j: d_j=i\}}{n}.
\]
In other words, $D_n$ has the law of the degree of a vertex chosen uniformly at random from $[n]$. Suppose that $D_n\xrightarrow{(d)} D$, for a limiting random variable $D$ such that $P(D=1)>0$. Moreover, assume the following as $n\to \infty$,
\begin{align*}
&\text{(i) Convergence of the third moments: }E(D_n^3)\to E(D^3)<\infty,
\\
&\text{(ii) Scaling critical window: }\frac{E(D_n (D_n-1))}{E(D_n)}=1+\lambda n^{-1/3}+o(n^{-1/3}), \text{ for some } \lambda\in \mathbb{R}. \text{ In particular, }\\
&E(D^2)=2 E(D).
\end{align*}
\end{assum}
\begin{remark}
We remark here that the configuration model with random i.i.d. degrees sampled from a distribution with $E(D^3)<\infty$ treated in \cite{joseph2014component} meets the assumptions introduced above almost surely. Especially, (ii) is satisfied for $\lambda=0$ and corresponds to the critical case, i.e. if $E(D^2)<2 E(D)$ there is no giant component with probability tending to 1, as $n\to \infty$. In addition, $E(D^2)>2 E(D)$ sees the emergence of a unique giant component with probability tending to 1, as $n\to \infty$.
\end{remark}
Write $c=(c_1,c_2,c_3)\in \mathbb{R}_{+}^3$ and define $(B^{c,\lambda}_t)_{t\ge 0}$, a Brownian motion with parabolic drift by
\begin{equation} \label{pard}
B^{c,\lambda}_t:=\frac{\sqrt{c_2}}{c_1} B_t+\lambda t-\frac{c_2 t^2}{2 c_1^3},
\end{equation}
where $(B_t)_{t\ge 0}$ is a standard Brownian motion. The most general result under minimum assumptions for the joint convergence of the component sizes and the corresponding surpluses was proven in \cite{dhara2017critical}. Fix $\lambda\in \mathbb{R}$ and let $(M_i^n)_{i\ge 1}$ and $(R_i^n)_{i\ge 1}$ denote the sequence of the sizes and surpluses of the components of $\text{M}^n(d)$ respectively.
\begin{theorem}[\textbf{Dhara, Hofstad, Leeuwaarden, Sen \cite{dhara2017critical}}] \label{Van}
As $n\to \infty$,
\begin{equation}
\left(n^{-2/3} (M_i^n)_{i\ge 1},(R_i^n)_{i\ge 1}\right)\longrightarrow \left((M_i^{c_D})_{i\ge 1},(R_i^{c_D})_{i\ge 1}\right)
\end{equation}
in distribution, where the convergence of the first sequence takes place in $\ell^2_{\downarrow}$ and for the second in the product topology.
\end{theorem}
The limiting sequence $(M_i^{c_D})_{i\ge 1}$ is distributed as the ordered sequence of lengths of excursions of the process $(B_t^{c_D,\lambda})_{t\ge 0}$ reflected upon its minimum, where $c_D$ has coordinates depending only on the first three moments of $D$ and are given exactly by $c_1^D=E(D)$, $c_2^D=E(D^3) E(D)-(E(D^2))^2$ and $c_3^D=1/E(D)$. Drawing the graph of the reflected process, scattering points on the plane according to a Poisson with rate $c_3^D$ and keeping only those that fell between the $x$-axis and the function, describes $R_i^{c_D}$, $i\ge 1$ as the number of those points that fell in the excursion with length $M_i^{c_D}$. Note that Theorem \ref{PhT} follows as a corollary of Theorem \ref{Van} for the special choice of $c_{\text{ER}}=(1,1,1)$.
In Section \ref{lalala} we introduced $\mathcal{M}^{(\zeta)}$ as the random real tree coded by a tilted Brownian excursion of length $\zeta$ to which a number of point identifications to create cycles is added. The number of point identifications is a Poisson random variable with mean given by the area under the tilted excursion. Given that number, say $k\ge 0$, $x_i$ is picked with a density proportional to the height of the tilted excursion in an i.i.d. fashion for every $1\le i\le k$ and $y_i$ is picked uniformly from the path that connects the root to $x_i$. Then, $x_i$ and $y_i$ are identified, $1\le i\le k$. Let $\mathcal{M}^{(\zeta,c_3)}$ denote $\mathcal{M}^{(\zeta)}$ if the number of point identifications is instead Poisson with mean given by the area under the tilted excursion multiplied by $c_3$, i.e. $c_3 \int_{0}^{\zeta} \tilde{e}(u) du$.
Then, the limit of the largest connected component of the configuration on the scaling critical window, say $\text{M}_1^n(d)$, can be written as a scalar of $\mathcal{M}^{(M_1^{c_D},c_3^D)}$, where $M_1^{c_D}$ is distributed according to the length of the longest excursion of the reflected upon its minimum parabolic Brownian motion with coefficients dependent on $c_D$ as defined in \eqref{pard}. This statement is made precise as a simplified version of \cite[Theorem 2.4]{bhamidi2016geometry}, which we quote. As $n\to \infty$,
\begin{equation} \label{uni1}
\left(n^{-2/3} M_1^n,\left(V(\text{M}^n_1(d)),n^{-1/3} d_{\text{M}_1^n(d)},\rho^n\right)\right)\longrightarrow \left(M_1^{c_D},\left(\mathcal{M}_D,\frac{c_1^D}{\sqrt{c_2^D}}d_{\mathcal{M}_D},\rho\right)\right),
\end{equation}
in distribution, where conditional on $M_1^{c_D}$, $\mathcal{M}_D\,{\buildrel (d) \over =}\,\mathcal{M}^{(M_1^{c_D},c_3^D)}$. Actually \cite[Theorem 2.4]{bhamidi2016geometry} holds also by considering the largest connected component of a uniform element of $G_{n,d}$ with $d$ satisfying the minimum assumptions in (i) and (ii). Again, the limit of the maximal components of $G(n,n^{-1}+\lambda n^{-4/3})$ can be recover by considering $D_{\text{ER}}$ to be a mean 1 Poisson random variable ($c_1^{D_{\text{ER}}}=c_2^{D_{\text{ER}}}=c_3^{D_{\text{ER}}}=1$) and the fact that $G(n,n^{-1}+\lambda n^{-4/3})$ conditioned on the degree sequence being $d$ is uniformly distributed over $G_{n,d}$. The result in \eqref{uni1} is to be regarded as a stepping stone to a general program that seeks to prove universality for the metric structure of the maximal components at criticality for a number of random graphs models with their distances scaling like $n^{1/3}$. For more details see \cite{bhamidi2014scaling}.
We turn now our interest on how to sample uniformly a connected component with a given degree sequence $(\tilde{d}_i: i\in [\tilde{m}])$ that satisfies the following assumption.
\begin{assum} \label{assum4}
(i) Let $\tilde{d}_1=1$, and $\tilde{d}_i\ge 1$, for every $1\le i\le \tilde{m}$.
\\
(ii) There exists a probability mass function $(\tilde{p}_i)_{i\ge 1}$ with the properties
\[
\tilde{p}_1>0, \qquad \sum_{i\ge 1} i \tilde{p}_i=2, \qquad \sum_{i\ge 1} i^2 \tilde{p}_i<\infty
\]
such that
\[
\frac{1}{\tilde{m}} \# \{j: \tilde{d}_j=i\}\to \tilde{p}_i, \text{ for all } i\ge 1, \text{ and } \qquad \frac{1}{\tilde{m}} \sum_{i\ge 1} \tilde{d}_i^2\to \sum_{i\ge 1} i^2 \tilde{p}_i.
\]
In particular, $\max_{1\le 1\le \tilde{m}} \tilde{d}_i=o(\sqrt{m})$.
\end{assum}
For a given rooted plane tree $\theta$ with root $\rho$, let $\text{c}(\theta)=(\text{c}_v(\theta))_{v\in \theta}$, where $\text{c}_v(\theta)$ gives the number of children of $v$ in $\theta$ and let $\text{s}(\theta)=(\text{s}_i(\theta))_{i\ge 0}$ be the empirical children distribution (ECD) of $\theta$, i.e. $\text{s}_i(\theta):=\# \{v\in \theta: c_v(\theta)=i\}$, for every $i\ge 0$. Note that $\text{s}_0(\theta)=\# \mathcal{L}(\theta)$ gives exactly the number of leaves of $\theta$. Now, given a sequence of integers $\text{s}=(\text{s}_i)_{i\ge 0}$, it is easy to check that it is tenable for a tree $\theta$ if and only if $\text{s}_0\ge 1$, $\text{s}_i\ge 0$ for every $i\ge 1$, and
\[
\sum_{i\ge 0} \text{s}_i=1+\sum_{i\ge 1} i \text{s}_i<\infty.
\]
Given $\text{s}$, let $\mathbf{T}_{\text{s}}$ denote the collection of all plane trees having ECD $\text{s}$.
Let $x,y\in \mathcal{L}({\theta})$. We say that the ordered pair of leaves $(x,y)$ is admissible if $\text{par}(x)<_{\text{DF}} \text{par}(y)$, i.e. if the parent of $x$ is explored before the parent of $y$ during a depth-first search of $\theta$, and if $\text{gpar}(y)\in [[\rho,\text{gpar}(x)]]$, i.e. if the grandparent of $y$ belongs to ancestral line that connects the root to the grandparent of $x$. Let $(\mathbf{A}(\theta),<<)$ denote the collection of pairs of admissible leaves of $\theta$ endowed with the linear order << that declares $(x_1,y_1)<<(x_2,y_2)$ if and only if $x_1=x_2$ or $x_1<_{\text{DF}} x_2$ and $y_1<_{\text{DF}} y_2$. For $k\ge 1$, we denote by $\mathbf{A}_k(\theta)$ the collection of admissible $k$-tuples of $2 k$ distinct leaves and by $\mathbf{T}_{\text{s}}^k$ the pairs $(\theta,\text{z})$ for which $\theta\in \mathbf{T}_{\text{s}}$ and $\text{z}\in \mathbf{A}_k(\theta)$. Finally, for a rooted plane tree $\theta$ and $\text{z}=\{(x_1,y_1),...,(x_k,y_k)\}\in \mathbf{A}_k(\theta)$, we denote by $L(\theta,\text{z})$ the rooted plane tree obtained from $\theta$ performing the following operation. Delete $x_i$, $y_i$ together with the edges adjacent to them for each $i=1,...,k$ and add an edge between $\text{par}(x_i)$ and $\text{par}(y_i)$. We equip $L(\theta,\text{z})$ with the shortest path distance and the uniform probability measure on its set of vertices. Finally, let
We will work with connected graphs with a fixed surplus, so suppose that
\[
\sum_{i\in [\tilde{m}]} \tilde{d}_i=2(\tilde{m}-1)+2 k,
\]
for some fixed $k\ge 0$. Under Assumption \ref{assum4}, the lowest labelled vertex has one descendant and for the remaining vertices $2,...,\tilde{m}$ we form the children sequence $\text{c}:=(\text{c}_i)_{i=2}^{\tilde{m}+2 k}$ via $\text{c}_i:=\tilde{d}_i-1$, for every $2\le i\le \tilde{m}$, and $\text{c}_{\tilde{m}+j}:=0$, for every $1\le j\le 2 k$. Observe that
\begin{equation} \label{childish}
\sum_{i=2}^{\tilde{m}+2 k} c_i=\sum_{i=2}^{\tilde{m}} \tilde{d}_i-(\tilde{m}-1)=(\tilde{m}-1+2 k)-1,
\end{equation}
and thus $\text{c}$ can be seen as representing a children sequence for a plane tree with $m:=\tilde{m}-1+2 k$ vertices. Write $\text{s}:=(\text{s}_i)_{i\ge 0}$ for its ECD. The following lemma is due to Bhamidi and Sen.
\begin{lemma} [\textbf{Bhamidi, Sen \cite{bhamidi2016geometry}}] \label{assf}
To generate uniformly a connected graph with prescibed degree sequence $\tilde{d}$ satisfying Assumption \ref{assum4}
\begin{enumerate}
\item Generate first $(\tilde{T}_{\textnormal{s}},\tilde{Z})$ uniformly from $\mathbf{T}_{\textnormal{s}}^k$. If $\tilde{Z}=\{(x_1,y_1),...,(x_k,y_k)\}$ with $(x_1,y_1)<<...<<(x_k,y_k)$, label $x_i$ as $\tilde{m}+2 i-1$ and $y_i$ as $\tilde{m}+2 i$, $1\le i\le k$. Label the rest of the $\tilde{m}-1$ vertices uniformly using the remaining labels $2,...,\tilde{m}$ so that in the resulting labelled tree the vertex $j$ has exactly $\tilde{d}_j-1$ children. Call this labelled tree $\tilde{T}_{\textnormal{s}}^{\textnormal{lb}}$.
\item Construct $L(\tilde{T}_{\textnormal{s}}^{\textnormal{lb}},\tilde{Z})$, attach a vertex labelled 1 to the root and forget about the planar order and the root. Call $\mathcal{G}$ the resulting graph.
\end{enumerate}
Then, $\mathcal{G}$ is distributed uniformly over the set of connected graph with prescribed degree sequence $\tilde{d}$.
\end{lemma}
We use $\rho^m$ to denote the root of $\tilde{T}_{\text{s}}$. We denote by $\tilde{V}_m^{\text{s}}=(\tilde{V}_m^{\text{s}}(i): 0\le i\le 2 m)$ the contour process of $\tilde{T}_{\text{s}}$, the random tree generated according to 1 in the statement of Lemma \ref{assf}, and by $\tilde{v}_m^{\text{s}}=(m^{-1/2} \tilde{V}_m^{\text{s}}(2 m s): 0\le s\le 1)$ the rescaled contour process as well. In the next lemma we show that, for some $\alpha>0$, the sequence $||\tilde{v}_m^{\text{s}}||_{H_{\alpha}}$ of H\"older norms is tight.
\begin{lemma} \label{highon}
There exists $\alpha\in (0,1/2)$ such that for every $\varepsilon>0$ there exists a finite real number $M_{\varepsilon}$ such that
\begin{equation}
P\left(\sup_{t\in [0,1]}\frac{|\tilde{v}_m^{\textnormal{s}}(s)-\tilde{v}_m^{\textnormal{s}}(t)|}{|t-s|^{\alpha}}\le M_{\varepsilon}\right)\ge 1-\varepsilon.
\end{equation}
\end{lemma}
\begin{proof}
From the definition of $(\tilde{T}_{\text{s}},\tilde{Z})$, it is clear that
\[
P(\tilde{T}_{\text{s}}=\theta)=\frac{|\mathbf{A}_k(\theta)|}{|\mathbf{T}_{\text{s}}^k|},
\]
for any $\theta\in \mathbf{T}_{\text{s}}$, i.e. $\tilde{T}_{\text{s}}$ is a random tree that has ``tilted'' distribution which is biased in favor of trees with large collection of admissible $k$-tuples between $2 k$ distinct leaves. Hence for any $f: C([0,1],\mathbb{R}_{+})\to \mathbb{R}_{+}$ bounded and continuous function,
\begin{equation} \label{tiltme}
E\left(f\left(\tilde{v}_{m}^{\text{s}}\right)\right)=\frac{E\left(f\left(v_m^{\text{s}}\right)|\mathbf{A}_k(T_{\text{s}})|\right)}{E\left(|\mathbf{A}_k(T_{\text{s}})|\right)},
\end{equation}
where $T_{\text{s}}$ a uniform plane tree having ECD $\text{s}$, which is specified by the children sequence described in \ref{childish}. Here, $v_m^{\text{s}}$ is the normalized contour function that encodes $T_{\text{s}}$. Note that when $\tilde{d}$ satisfies Assumption \ref{assum4}, $\text{s}$ satisfies the following.
\[
\sum_{i\ge 0} \text{s}_i=m, \qquad \frac{\text{s}_i}{m}\to p_i, \text{ for all } i\ge 0,\text{ and } \qquad \frac{1}{m} \sum_{i\ge 0}i^2 \text{s}_i\to \sum_{i\ge 0} i^2 p_i.
\]
In particular, $\max \{i: \text{s}_i\neq 0\}=o(\sqrt{m})$. Also, $p:=(p_i)_{i\ge 0}$ is a probability mass function with $p_i:=\tilde{p}_{i+1}$, for every $i\ge 0$, and therefore it satisfies the properties
\begin{equation} \label{sleeping}
p_0>0, \qquad \sum_{i\ge 0} i p_i=1, \qquad \sum_{i\ge 0} i^2 p_i<\infty.
\end{equation}
Now, by \eqref{tiltme} along with the Cauchy-Schwarz inequality, if $K_{\varepsilon}$ is the finite real number for which \eqref{proof2} holds, we deduce
\begin{align} \label{sleep}
P\left(\sup_{s,t\in [0,1]}\frac{|\tilde{v}_m^{\text{s}}(s)-\tilde{v}_m^{\text{s}}(t)|}{|t-s|^{\alpha}}\ge K_{\varepsilon}\right)&=\frac{E\left[\mbox{1\hspace{-4.25pt}\fontsize{12}{14.4}\selectfont\textrm{1}}_{\left\{\sup_{t\in [0,1]}\frac{|v_m^{\text{s}}(s)-v_m^{\text{s}}(t)|}{|t-s|^{\alpha}}\ge K_{\varepsilon}\right\}} |\mathbf{A}_k(T_{\text{s}})|\right]}{E\left(|\mathbf{A}_k(T_{\text{s}})|\right)}\nonumber \\
&\le \frac{P\left(\sup_{s,t\in [0,1]}\frac{|v_m^{\text{s}}(s)-v_m^{\text{s}}(t)|}{|t-s|^{\alpha}}\ge K_{\varepsilon}\right)^{1/2} \left(E\left[\left(\frac{|\mathbf{A}_k(T_{\text{s}})|}{\text{s}_0^{k} m^{k/2}}\right)^{2}\right]\right)^{1/2} }{E\left[\frac{|\mathbf{A}_k(T_{\text{s}})|}{\text{s}_0^{k} m^{k/2}}\right]}.
\end{align}
Using \cite[Lemma 6.3(ii)]{bhamidi2016geometry}, we have that
\[
\sup_{m} E\left[\left(\frac{|\mathbf{A}_k(T_{\text{s}})|}{\text{s}_0^k m^{k/2}}\right)^{2}\right]\le \frac{1}{k!} \sup_{m} E\left[\left(\frac{|\mathbf{A}(T_{\text{s}})|}{\text{s}_0 \sqrt{m}}\right)^{2 k}\right]<\infty,
\]
for every $k\ge 1$. Furthermore, using \cite[Lemma 6.3(iii)]{bhamidi2016geometry}, and \cite[Lemma 6.3(v)]{bhamidi2016geometry} together with the uniform integrability from above, we conclude that
\[
E\left[\frac{|\mathbf{A}_k(T_{\text{s}})|}{\text{s}_0^{k} m^{k/2}}\right]\to \left(\frac{p_0 \sigma}{2}\right)^k E\left[\left(\int_{0}^{1} 2 e(u) du\right)^k\right]>0,
\]
as $m\to \infty$, where $(e(t))_{t\in [0,1]}$ is a normalized Brownian excursion and $\sigma^2=\sum_{i\ge 0} i^2 p_i-1$. It only remains to deal with the quantity $P\left(||v_m^{\text{s}}||_{H_{\alpha}}\ge K_{\varepsilon}\right)$ that appears on the right-hand side of \eqref{sleep}. It turns out that plane trees chosen uniformly from $\mathbf{T}_{\text{s}}$ are related to G-W trees by a simple conditioning. The uniform distribution on $\mathbf{T}_{\text{s}}$ coincides with the distribution of a G-W tree $\theta$ with offspring distribution $\mu:=(\mu_i)_{i\ge 0}$, which must satisfy $\mu_i>0$ if $\text{s}_i>0$, conditioned on the event $\cap_{i\ge 0} \{\text{s}_i(\theta)=\text{s}_i\}$. Take $\mu=p$ as in \eqref{sleeping} to be the critical offspring distribution with finite variance of a G-W tree $\theta$. Then, if $P_p$ is the probability distribution of $\theta$,
\[
P\left(||v_m^{\text{s}}||_{H_{\alpha}}\ge K_{\varepsilon}\right)=P_p\left(||v_m||_{H_{\alpha}}\ge K_{\varepsilon}|\text{s}_i(\theta)=\text{s}_i, i\ge 0\right),
\]
where $||v_m||_{H_{\alpha}}$ denotes the $\alpha$-H\"older norm of the normalized contour function $v_m$ that encodes $\theta$. The proof is completed as a result of Theorem \ref{prepar2}.
\end{proof}
In Lemma \ref{assf} we saw that $\mathcal{G}$, a uniformly chosen connected graph with prescribed degree sequence $\tilde{d}$ that satisfies Assumption \ref{assum4} is distributed as $L(\tilde{T}_{\text{s}}^{\text{lb}},\tilde{Z})$, where $(\tilde{T}_{\text{s}}^{\text{lb}},\tilde{Z})$ is a uniform labelled element of $\mathbf{T}_{\text{s}}^k$. Recall that to obtain $L(\tilde{T}_{\text{s}}^{\text{lb}},\tilde{Z})$ from $(\tilde{T}_{\text{s}}^{\text{lb}},\tilde{Z})$, where $\tilde{Z}=\{(x_i,y_i),...,(x_k,y_k)\}$ with $(x_1,y_1)<<...<<(x_k,y_k)$, for every pair of admissible leaves we add an edge between $\text{par}(x_i)$ and $\text{par}(y_i)$, and delete $x_i$ and $y_i$ and the two edges incident to them for $1\le i\le k$. The resistance on $L(\tilde{T}_{\text{s}}^{\text{lb}},\tilde{Z})$ between two vertices is smaller than the total length of the path between them on $\tilde{T}_{\text{s}}^{\text{lb}}$. This observation together with Lemma \ref{highon} is enough to establish equicontinuity of the rescaled local times $(L^{\mathcal{G}}_t(x))_{x\in V(\mathcal{G}),t\ge 0}$ of the simple random walk on $\mathcal{G}$ under the annealed law (which is defined similar to \eqref{exemplary}). Since the proof relies heavily on arguments that are present in the proof of Proposition \ref{own} we omit it.
\begin{lemma} \label{sizeb3}
For every $\varepsilon>0$ and $T>0$,
\[
\lim_{\delta\rightarrow 0} \limsup_{m\to \infty} \mathbb{P}_{\rho^m} \left(\sup_{\substack{y,z\in V(\mathcal{G}): \\ m^{-1/2} R_{\mathcal{G}}(y,z)<\delta}} \sup_{t\in [0,T]} m^{-1/2} |L_{m^{3/2} t}^{\mathcal{G}}(y)-L_{m^{3/2} t}^{\mathcal{G}}(z)|\ge \varepsilon\right)=0.
\]
\end{lemma}
Assume that $d$ satisfies Assumption \ref{gracias} with limiting random variable $D$, and let $D^*$ denote its size-biased distribution given by
\[
p^{*}_{i}:=P(D^*=i)=\frac{i P(D=i)}{E(D)}, \qquad i\ge 1.
\]
Then, for $M_1^n(d)$, the largest connected component of the configuration model $M^n(d)$,
\begin{equation} \label{sizeb1}
\frac{\# \{v\in M_1^n(d): d_v=i\}}{|V(M_1^n(d))|}\xrightarrow{\text{P}} p_i^{*}, \text{ for all } i\ge 1, \qquad \frac{1}{|V(M_1^n(d))|} \sum_{v\in M_1^n(d)} d_v^2\xrightarrow{\text{P}} \sum_{i\ge 1} i^2 p_i^{*}<\infty,
\end{equation}
\begin{equation} \label{sizeb2}
P(M_1^n(d) \text{ is simple})\to 1.
\end{equation}
For a justification of \eqref{sizeb1} and \eqref{sizeb2} see \cite[Proposition 8.2]{bhamidi2016geometry}. Note that $P(D=1)>0$ under Assumption \ref{gracias}, and hence $p_1^{*}>0$. Furthermore, under Assumption \ref{gracias},
\[
\sum_{i\ge 1} i p_i^{*}=\frac{E(D^2)}{E(D)}=2,
\]
and this shows along with \eqref{sizeb1} that $(d_v: v\in M_1^n(d))$ satisfies Assumption \ref{assum4} (after a possible) with limiting probability mass function $p^{*}:=(p_i^{*})_{i\ge 1}$. Let $\mathcal{P}$ denote the partition of $M^n(d)$ into different components. Conditional on the event $\{M_1^n(d) \text{ is simple}\}\cap \{\mathcal{P}=P\}$, $M_1^n$ is uniformly distributed over the set of simple, connected graphs with degree sequence decided by the partition $P$ \cite[Proposition 7.7]{van2016random}. Since $P(M_1^n(d) \text{ is a multigraph})\to 0$ by \eqref{sizeb2}, the following Proposition simply follows as a combination of Theorem \ref{Van} and Lemma \ref{sizeb3}.
\begin{proposition}
Under Assumption \ref{gracias}, for every $\varepsilon>0$ and $T>0$, the rescaled local times of the simple random walk on $M_1^n(d)$ are equicontinuous under the annealed law, i.e.
\[
\lim_{\delta\rightarrow 0} \limsup_{n\to \infty} \mathbb{P}_{\rho^n} \left(\sup_{\substack{y,z\in V(M_1^n(d)): \\ n^{-1/3} R_{M_1^n(d)}(y,z)<\delta}} \sup_{t\in [0,T]} n^{-1/3} |L_{n t}^n(y)-L_{n t}^n(z)|\ge \varepsilon\right)=0.
\]
\end{proposition}
\subsubsection{Convergence of the walks} \label{newresult}
Croydon \cite{croydon2016scaling} used regular resistance forms to describe the scaling limit of the associated random walks on scaling limits of sequences of spaces equipped with resistance metrics and measures provided that they converge with respect to a suitable Gromov-Hausdorff topology, and under the assumption that a non-explosion condition is satisfied. For families of random graphs that are nearly trees and their scaling limit can be described as a tree `glued' at a finite number of pairs of points, a useful corollary of \cite[Theorem 1.2]{croydon2016scaling} combined with \cite[Proposition 8.4]{croydon2016scaling} yields the convergence of the processes associated with the fused spaces.
To see that the conclusion of \cite[Proposition 8.4]{croydon2016scaling} holds, recall that under Assumption \ref{gracias}, jointly with Theorem \ref{Van},
\[
\left(V(\text{M}^n_1(d)),n^{-1/3} d_{\text{M}_1^n(d)},\rho^n\right)\longrightarrow \left(\mathcal{M}_D,\frac{c_1^D}{\sqrt{c_2^D}}d_{\mathcal{M}_D},\rho\right),
\]
as $n\to \infty$ in the Gromov-Hausdorff-Prokhorov sense. Let $\mathcal{P}$ denote the partition of $M^n(d)$ into different components. Conditional on the event $\{M_1^n(d) \text{ is simple}\}\cap \{\mathcal{P}=P\}$, $M_1^n$ is uniformly distributed over the set of simple, connected graphs with degree sequence decided by the partition $P$, and therefore the convergence above is valid with $M_1^n(d)$ replaced by $L(\tilde{T}_{\textnormal{s}},\tilde{Z})$ (see Lemma \ref{assf} for its construction). If $\tilde{Z}=\{(x_1,y_1),...,(x_{R_1^n},y_{R_1^n})\}$ with $(x_1,y_1)<<...<<(x_{R_1^n},y_{R_1^n})$, let $D(\tilde{T}_{\textnormal{s}},\tilde{Z})$ be the space obtained by fusing $x_i$ and $\text{gpar}(y_i)$, $1\le j\le R_1^n$, endowed with the graph distance and the push-forward of the uniform probability measure on $\tilde{T}_{\textnormal{s}}$, and observe that
\[
d_{\text{GHP}}(L(\tilde{T}_{\textnormal{s}},\tilde{Z}),D(\tilde{T}_{\textnormal{s}},\tilde{Z}))\le 5 R_1^n.
\]
Thus, jointly with Theorem \ref{Van}, the convergence above is valid with $M_1^n(d)$ replaced with the `glued' tree $D(\tilde{T}_{\textnormal{s}},\tilde{Z})$, and since $\mathcal{M}_D$ is also a `glued' tree, this shows that the conclusion of \cite[Proposition 8.4]{croydon2016scaling} is valid.
Fix $r\ge 0$. It remains to show that
\[
\lim_{r\to \infty} \liminf_{n\to \infty} P\left(n^{-1/3} d_{M_1^n(d)}(\rho_n,B_n(\rho_n,r)^c)\ge \lambda\right)=1,
\]
for every $\lambda\ge 0$. Indeed,
\begin{align*}
P\left(n^{-1/3} d_{M_1^n(d)}(\rho_n,B_n(\rho_n,r)^c)\ge \lambda\right)&\ge P\left(n^{-1/3} d_{M_1^n(d)}(\rho_n,B_n(\rho_n,r)^c)\ge \lambda,n^{-1/3} D_1^n(d)\le r\right)
\\
&=P(n^{-1/3} D_1^n(d)\le r),
\end{align*}
where $D_1^n(d):=\text{diam}(M_1^n(d))$. Letting $D_1(d):=\text{diam}(\mathcal{M}_D)$, since
\[
|n^{-1/3} D_1^n(d)-D_1(d)|\le 2 d_{\textnormal{GH}}(n^{-1/3} M_1^n(d),\mathcal{M}_D),
\]
we deduce
\[
\liminf_{n\to \infty} P\left(n^{-1/3} d_{M_1^n(d)}(\rho_n,B_n(\rho_n,r)^c)\ge \lambda\right)\ge \liminf_{n\to \infty} P(n^{-1/3} D_1^n(d)\le r)=P(D_1(d)\le r).
\]
As $r\to \infty$ the right-hand-side tends to 1, and this shows that the non-explosion condition is fulfilled. As a consequence we have the convergence of the processes associated with the fused spaces.
\begin{theorem}
It is possible to isometrically embed $(M_1^n(d),d_{M_1^n(d)})$, $n\ge 1$ and $(\mathcal{M}_D,d_{\mathcal{M}_D})$ into a common metric space $(F,d_F)$ such that
\begin{equation}
\mathbf{P}^{M_1^n(d)}_{\rho^n}\left(n^{-1/3} (X_{\lfloor n t\rfloor}^{M_1^n(d)})_{t\ge 0}\in \cdot \right)\to \mathbf{P}^{\mathcal{M}_D}_{\rho}\left((X_t)^{\mathcal{M}_D}_{t\ge 0}\in \cdot \right)
\end{equation}
weakly as probability measures in $D(\mathbb{R}_{+},F)$.
\end{theorem}
\subsubsection{Continuity of blanket times of Brownian motion on \texorpdfstring{$\mathcal{M}_{D}$}{MD}}
In Section \ref{excm} we presented $\mathbb{N}^{t,\lambda}$, the inhomogeneous excursion (for excursions starting at time $t$) measure associated with a Brownian motion with parabolic drift as defined in \eqref{parabola}. Denote by $\mathbb{N}^{c,\lambda}_{t}$ the excursion measure associated with $B^{c,\lambda}$ as defined in \eqref{pard}. Write $(e(u): 0\le u\le t)$ for the canonical process under $\mathbb{N}$. By the Cameron-Martin-Girsanov formula \cite[Chapter IX, (1.10) Theorem]{revuz1999continuous}, applied under $\mathbb{N}$,
\[
\frac{d \mathbb{N}^{c,\lambda}_{0}}{d \mathbb{N}}=\exp\left(\frac{\sqrt{c_2}}{c_1} \int_{0}^{t}\gamma(u) d e(u)-\frac{1}{2} \int_{0}^{t} \gamma^2(u) du\right),
\]
where $\gamma(u):=\lambda-\frac{c_2}{c_1^3} u$ is the drift. On the sets of excursions of length $t$, using integration by parts we have that
\[
\frac{\sqrt{c_2}}{c_1} \int_{0}^{t} \left(\lambda-\frac{c_2}{c_1^3}u\right)d e(u)=\frac{c_2^{3/2}}{c_1^4} \int_{0}^{t} e(u) du,
\]
a multiplicative of the area under the excursion of length $t$. So, the density becomes
\[
\frac{d \mathbb{N}^{c,\lambda}_{0}}{d \mathbb{N}}=\exp\left(\frac{c_2^{3/2}}{c_1^4} \int_{0}^{t} e(u) du-\frac{1}{6}\left(\left(\frac{c_2}{c_1^3} t-\lambda\right)^3+\lambda^3\right)\right).
\]
There is a corresponding probability measure $\mathbb{N}_{0,l}^{c,\lambda}:=\mathbb{N}_0^{c,\lambda}(\cdot |\tilde{L}=l)$, which for a Borel set $\mathcal{B}$ on the space of positive excursions of finite length, is determined by
\[
\mathbb{N}^{c,\lambda}_{0,l}(\mbox{1\hspace{-4.25pt}\fontsize{12}{14.4}\selectfont\textrm{1}}_{\mathcal{B}})=\frac{\mathbb{N}_l\left(\exp\left(\frac{c_2^{3/2}}{c_1^4} \int_{0}^{l} e(u) du\right) \mbox{1\hspace{-4.25pt}\fontsize{12}{14.4}\selectfont\textrm{1}}_{\mathcal{B}} \right)}{\mathbb{N}_l\left(\exp\left(\frac{c_2^{3/2}}{c_1^4} \int_{0}^{l} e(u) du\right)\right)}.
\]
To determine $\mathbb{N}^{c,\lambda}_0(\tilde{L}\in dl)$ recall that $\mathbb{N}(L\in dl)=f_L(\lambda)=dl/\sqrt{2 \pi l^3}$, $l\ge 0$, and therefore
\[
\mathbb{N}^{c,\lambda}_0(\tilde{L}\in dl)=f_L(l) \exp\left(-\frac{1}{6}\left(\left(\frac{c_2}{c_1^3}l-\lambda\right)^3+\lambda^3\right)\right) \mathbb{N}_l\left(\exp\left(\frac{c_2^{3/2}}{c_1^4} \int_{0}^{l} e(u) du\right)\right).
\]
Let $\mathbf{N}^{c,\lambda}_t$ denote the canonical measure that first at time $t$ picks a tilted Brownian excursion of a randomly chosen length $l$, and then independently of $e$ chooses a number of points according to a Poisson random variable with mean $c_3 \int_{0}^{l} e(t) dt$, which subsequently are distributed uniformly on the area under the graph of $e$. In comparison with \eqref{meas3} we characterize $\mathbf{N}^{c,\lambda}_t(d(e,\mathcal{P}))$ by setting
\begin{align*}
&\mathbf{N}^{c,\lambda}_t(d e,|\mathcal{P}|=k, (d x_1,...,d x_k)\in A_1\times...\times A_k)
\nonumber \\ :=
&\int_{0}^{\infty} \mathbb{N}^{c,\lambda-t}_0(\tilde{L}\in dl) \mathbb{N}^{c,\lambda}_{t,l} (d e) \exp\left(-c_3 \int_{0}^{l} e(u) du\right)\frac{\left(c_3 \int_{0}^{l} e(u) du\right)^k}{k!} \prod_{i=1}^{k} \frac{\ell(A_i\cap A_e)}{\ell(A_e)}.
\end{align*}
It is easy to see that $\mathbf{N}^{c,\lambda}_t$ is absolutely continuous with respect to $\mathbf{N}$ as defined in \eqref{meas1}. More specifically,
\[
\frac{d \mathbf{N}_t^{c,\lambda}}{d \mathbf{N}}=\exp \left(1-\frac{1}{6}\left(\lambda^3+\left(\frac{c_2}{c_1^3 }l-\lambda+t\right)^3\right)\right) \left(c_3 \int_{0}^{l} e(u) du\right)^k.
\]
Using the same methods as in Section \ref{excm}, we can prove that
\begin{theorem}
Fix $\varepsilon\in (0,1)$. If $\tau_{\textnormal{bl}}^{n}(\varepsilon)$ is the $\varepsilon$-blanket time variable of the random walk on $M_1^n(d)$, started from its root $\rho^n$, then
\[
\mathbb{P}_{\rho^n}\left(n^{-1} \tau_{\textnormal{bl}}^n(\varepsilon)\le t\right)\to \mathbb{P}_{\rho}\left(\tau_{\textnormal{bl}}^{\mathcal{M}_D}(\varepsilon)\le t\right),
\]
for every $t\ge 0$, where $\tau_{\textnormal{bl}}^{\mathcal{M}_D}(\varepsilon)\in (0,\infty)$ is the $\varepsilon$-blanket time variable of the Brownian motion on $\mathcal{M}_D$, started from $\rho$ and $\mathbb{P}_{\rho}$ is the annealed law defined similarly as in \eqref{lastbits}.
\end{theorem}
\section*{Acknowledgements}
I would like to thank my supervisor Dr David Croydon for suggesting the problem, his support and many useful discussions.
\bibliographystyle{abbrv}
| -164,846.059544 |
[
-3.30859375,
2.984375
] | 32.340116 |
[
-3.017578125,
0.5341796875,
-2.263671875,
-6.73046875,
-0.892578125,
9.265625
] |
[
3.330078125,
7.609375,
2.599609375,
7.28515625
] | 1,082 | 18,496 |
[
-3.41796875,
4.01171875
] | 34.938659 |
[
-5.93359375,
-4.36328125,
-5.50390625,
-2.7578125,
2.091796875,
14.2421875
] | 0.890509 | 15.322244 | 18.041739 | 2.820896 |
[
2.245209217071533
] | -88,531.654868 | 5.858834 | -165,299.145859 | 0.597979 | 6.469414 |
[
-2.130859375,
-3.369140625,
-3.79296875,
-5.28125,
2.134765625,
12.46875
] |
[
-5.5703125,
-2.29296875,
-2.41015625,
-1.6484375,
3.8125,
5.13671875
] | |
BkiUdvbxaKgQL0dMaDiQ
|
\section{Introduction and main results}
It is known \cite{6} that the Euler gamma-function
$$\Gamma(z)=\int_{0}^{+\infty}t^{z-1}e^{-t}dt, ~~~ \Re z> 0$$
(analytically continued as a meromorphic function in the complex plane $\mathbb{C}$) does not satisfy any non-trivial algebraic differential equation whose coefficients are polynomials in $\mathbb{C}$. That is, if $P(v_{0}, v_{n}, ..., v_{l})$ is a polynomial of $n+1$ variables with polynomial coefficients in $z\in\mathbb{C}$ such that
$P(\Gamma, \Gamma', ..., \Gamma^{(n)})(z)\equiv 0$ for $z\in\mathbb{C}$, then $P\equiv 0$. Hilbert\cite{5}, in Problem 18 of his famous list of 23 problems, stated that Riemann zeta function
$$\zeta(z)=\sum_{n=1}^{+\infty}\frac{1}{n^{z}}$$
does not satisfy any non-trivial algebraic differential equation whose coefficients are polynomials in $\mathbb{C}$, the problem was solved in great generality by Ostrowski\cite{11}. Since then, these results have been extended in many different directions(see e.g. \cite{2}, \cite{3}, \cite{7}, \cite{12}), to list a few. It is well known that $\zeta$ is associated with $\Gamma$ by the Riemann functional equation
\begin{equation}\label{eq0.1}
\begin{aligned}
\zeta(1-z)=2^{1-z}\pi^{-z}\cos \frac{\pi z}{2}\Gamma(z)\zeta(z).
\end{aligned}
\end{equation}
By virtue of (\ref{eq0.1}), one knows from Bank-Kaufman \cite{1} and Liao-Yang \cite{9} that neither $\zeta$ nor $\Gamma$ satisfy any non-trivial algebraic differential equation whose coefficients are meromorphic functions, growing strictly slower than $e^{z}$ as $|z|\rightarrow +\infty$ or having period $1$, detailed description of this matter can be found in Li and Ye \cite{8}. The result \cite{1} can not applied to algebraic differential equations involving both $\zeta$ and $\Gamma$ simultaneously, since $\zeta$ and $\Gamma$ grow strictly faster than $e^{z}$ as $|z|\rightarrow +\infty$; see Ye \cite{16}.
Recently, Markus \cite{10} proved that $\zeta(\sin(2\pi z))$ cannot satisfy any non-trivial algebraic differential equations whose coefficients are polynomials in $\Gamma$ and its derivatives, and he conjectured that $\zeta$ itself cannot satisfy any non-trivial algebraic differential equations whose coefficients are polynomials in $\Gamma$ and its derivatives. Thus, we are interested in knowing whether there is a non-trivial polynomial $P(u_{0}, u_{1}, ..., u_{m}; v_{0}, v_{1}, ..., v_{n})$
such that
$$P(\zeta, \zeta', ..., \zeta^{(m)}; \Gamma, \Gamma', ..., \Gamma^{(n)})(z)\equiv 0, ~~~ z\in\mathbb{C}.$$
In other words, one wants to know whether $\zeta$ satisfies any non-trivial algebraic differential equation whose coefficients are differential polynomials of $\Gamma$.
In this paper, we prove the main result as follows.
\begin{theorem}\label{M-thm}
Let $m, n, l$ be positive integers, $n<l$. $P(u_{0}, u_{1}, ..., u_{m}; v_{0}, v_{n}, v_{l})$ is a polynomial of $m+4$ variables with polynomial coefficients in $z\in\mathbb{C}$ such that
\begin{equation}\label{eq1.1}
\begin{aligned}
P(\zeta, \zeta', ..., \zeta^{(m)}; \Gamma, \Gamma^{(n)}, \Gamma^{(l)})(z)\equiv 0
\end{aligned}
\end{equation}
for $z\in\mathbb{C}$. Then the polynomial $P$ must be identically equal to zero.
\end{theorem}
Theorem~\ref{M-thm} shows that $\zeta$ does not satisfy any non-trivial algebraic differential equation whose coefficients could be polynomials in $\Gamma, \Gamma^{(n)}, \Gamma^{(l)}$. Our theorem extended the result that $\zeta$ does not satisfy any non-trivial algebraic differential equation whose coefficients could be polynomials in $\Gamma, \Gamma', \Gamma''$, which is proved by Li and Ye \cite{8}. As of \cite{8B},using its terminology, only $\zeta$ and the family of distinguished polynomials of $\Gamma, \Gamma', ..., \Gamma^{(n)}$ were considered.
\section{Preliminary}
To prove our theorem, we need the following celebrated theorem from Voronin \cite{14}.
\begin{lemma}\label{lem2.1}
Fix $x\in\left(\frac{1}{2}, 1\right)$ for $z=x+i y\in\mathbb{C}$. Define
$$\gamma(y):=(\zeta(x+i y), \zeta'(x+i y), ..., \zeta^{(m)}(x+i y))$$
to be a curve in $y$. Then, $\gamma(\mathbb{R})$ is everywhere dense in $\mathbb{C}^{m+1}$.
\end{lemma}
\section{Proof of Theorem~\ref{M-thm}}
Let
$P(u_{0}, u_{1}, ..., u_{m}; v_{0}, v_{n}, v_{l})$
be a polynomial in its arguments with coefficients in the field $\mathbb{C}$ such that (\ref{eq1.1}) is satisfied. In view of the discussions in Section $2$ of \cite{10}, one can simply assume that the coefficients of $P$ are constants. Denote by
\begin{equation*}
\begin{aligned}
\Lambda:=\{\lambda:=(\lambda_{0}, \lambda_{n}, \lambda_{l}): \lambda_{0}, \lambda_{n}, \lambda_{l} ~\text{are non-negative integers}~\},
\end{aligned}
\end{equation*}
a triple-index set having a finite cardinality, and define
\begin{equation*}
\begin{aligned}
\Lambda_p:=\{\lambda\in\Lambda:|\lambda|=p ~\text{with}~ |\lambda|:=\lambda_{0}+\lambda_{n}+\lambda_{l}\},
\end{aligned}
\end{equation*}
\begin{equation*}
\begin{aligned}
\Lambda_{q}^\ast:=\{\lambda\in\Lambda:|\lambda|^\ast=q ~\text{with}~ |\lambda|^\ast:=n\lambda_{n}+l\lambda_{l}\}.
\end{aligned}
\end{equation*}
Then, there is a non-negative integer $L$ such that
$$P(u_{0}, u_{1}, ..., u_{m}; v_{0}, v_{n}, v_{l})=\sum_{p=0}^{L}\sum_{\lambda\in\Lambda_{p}}a_{\lambda}(u_{0}, u_{1}, ..., u_{m})v_{0}^{\lambda_{0}}v_{n}^{\lambda_{n}}v_{l}^{\lambda_{l}},$$
where $a_{\lambda}(u_{0}, u_{1}, ..., u_{m})$ is a polynomial of $m+1$ variables with coefficients in $\mathbb{C}$. Set, for each $p=0, 1, ..., L$,
$$P_{p}(u_{0}, u_{1}, ..., u_{m}; v_{0}, v_{n}, v_{l})=\sum_{\lambda\in\Lambda_{p}}a_{\lambda}(u_{0}, u_{1}, ..., u_{m})v_{0}^{\lambda_{0}}v_{n}^{\lambda_{n}}v_{l}^{\lambda_{l}}.$$
Arranging $P_{p}(u_{0}, u_{1}, ..., u_{m}; v_{0}, v_{n}, v_{l})$ in $v:=(v_{0}, v_{n}, v_{l})$ in the ascending order of $q=|\lambda|^{\ast}$, we get that there is a non-negative integer $M_{p}$ such that
\begin{equation}\label{eq1.2}
\begin{aligned}
P_{p}(u_{0}, u_{1}, ..., u_{m}; v_{0}, v_{n}, v_{l})=\sum_{q=0}^{M_{p}}\sum_{\lambda\in\Lambda_{p}\cap\Lambda_{q}^{\ast}}a_{\lambda}(u_{0}, u_{1}, ..., u_{m})v_{0}^{\lambda_{0}}v_{n}^{\lambda_{n}}v_{l}^{\lambda_{l}}.
\end{aligned}
\end{equation}
Consequently,
\begin{equation}\label{eq1.3}
\begin{aligned}
P(u_{0}, u_{1}, ..., u_{m}; v_{0}, v_{n}, v_{l})=\sum_{p=0}^{L}\sum_{q=0}^{M_{p}}\sum_{\lambda\in\Lambda_{p}\cap\Lambda_{q}^{\ast}}a_{\lambda}(u_{0}, u_{1}, ..., u_{m})v_{0}^{\lambda_{0}}v_{n}^{\lambda_{n}}v_{l}^{\lambda_{l}}.
\end{aligned}
\end{equation}
\begin{claim}\label{cla 3.1}
Assume (\ref{eq1.1}) holds, then, for each $0\leq p\leq L$ and $z\in \mathbb{C}$, one has
\begin{equation*}
P_{p}(\zeta, \zeta', ..., \zeta^{(m)}; \Gamma, \Gamma^{(n)}, \Gamma^{(l)})(z)\equiv 0.
\end{equation*}
\end{claim}
\emph{Proof.}
Suppose $p_0$ is the smallest index among $\{0, 1, ..., L\}$ such that
\begin{equation*}
P_{p_0}(\zeta, \zeta', ..., \zeta^{(m)}; \Gamma, \Gamma^{(n)}, \Gamma^{(l)})(z)\not\equiv 0.
\end{equation*}
Then, we have
\begin{equation}\label{eq1.4}
\begin{aligned}
P_{p_0}\left(\zeta, \zeta', ..., \zeta^{(m)}; 1, \frac{\Gamma^{(n)}}{\Gamma}, \frac{\Gamma^{(l)}}{\Gamma}\right)(z)=\frac{P_{p_0}(\zeta, \zeta', ..., \zeta^{(m)}; \Gamma, \Gamma^{(n)}, \Gamma^{(l)})(z)}{\Gamma^{p_0}(z)}\not\equiv 0.
\end{aligned}
\end{equation}
Define the digamma function $f:=\frac{\Gamma'}{\Gamma}$, and introduce inductively
\begin{equation*}
\begin{aligned}
\frac{\Gamma'}{\Gamma}=&f\\
=&f\big[1+\frac{f'}{f^2}(c_1+\varepsilon_1)\big] ~ \text{for}~ c_1=0 ~ \text{and} ~ \varepsilon_1=0,\\
\frac{\Gamma''}{\Gamma}=&\left(\frac{\Gamma'}{\Gamma}\right)'+\left(\frac{\Gamma'}{\Gamma}\right)^{2}=f'+f^{2}\\
=&f^{2}\big[1+\frac{f'}{f^2}(c_2+\varepsilon_2)\big] ~ \text{for}~ c_2=1 ~ \text{and} ~ \varepsilon_2=0,\\
\frac{\Gamma'''}{\Gamma}=&\left(\frac{\Gamma''}{\Gamma}\right)'+\frac{\Gamma''}{\Gamma}\cdot\frac{\Gamma'}{\Gamma}=f''+3ff'+f^{3}\\
=&f^{3}\big[1+\frac{f'}{f^2}(c_3+\varepsilon_3)\big] ~ \text{for}~ c_3=3 ~ \text{and} ~ \varepsilon_3=\frac{f''}{ff'},\\
\frac{\Gamma^{(4)}}{\Gamma}=&\left(\frac{\Gamma'''}{\Gamma}\right)'+\frac{\Gamma'''}{\Gamma}\cdot\frac{\Gamma'}{\Gamma}=f'''+4ff''+3(f')^{2}+6f^{2}f'+f^{4}\\
=&f^{4}\big[1+\frac{f'}{f^2}(c_4+\varepsilon_4)\big] ~ \text{for}~ c_4=6 ~ \text{and} ~ \varepsilon_4=\frac{f'''}{f^{2}f'}+4\frac{f''}{ff'}+3\frac{f'}{f^{2}},\\
\frac{\Gamma^{(5)}}{\Gamma}=&\left(\frac{\Gamma^{(4)}}{\Gamma}\right)'+\frac{\Gamma^{(4)}}{\Gamma}\cdot\frac{\Gamma'}{\Gamma}\\
=&f^{4}+5ff'''+10f'f''+10f^{2}f''+15f(f')^{2}+10f^{3}f'+f^{5}\\
=&f^{5}\big[1+\frac{f'}{f^2}(c_5+\varepsilon_5)\big] ~ \text{for}~ c_5=10 ~ \text{and} \\
&\varepsilon_5=\frac{f^{(4)}}{f^{3}f'}+5\frac{f'''}{f^{2}f'}+10\frac{f''}{ff'}+10\frac{f''}{f^{3}}+15\frac{f'}{f^{2}},\\
\vdots\\
\frac{\Gamma^{(n)}}{\Gamma}=&f^{n}\big[1+\frac{f'}{f^2}(c_n+\varepsilon_n)\big]~\text{upon assumption, and then we can deduce that}\\
\frac{\Gamma^{(n+1)}}{\Gamma}=&\left(\frac{\Gamma^{(n)}}{\Gamma}\right)'+\frac{\Gamma^{(n)}}{\Gamma}\cdot\frac{\Gamma'}{\Gamma}
=f^{n+1}+f^{n-1}f'(c_n+n+\varepsilon_n)\\
&+f^{n-2}f'\varepsilon'_n +[(n-2)f^{n-3}(f')^{2}+f^{n-2}f''](c_n+\varepsilon_n)\\
=&f^{n+1}\big[1+\frac{f'}{f^2}(c_{n+1}+\varepsilon_{n+1})\big] ~ \text{for}~ c_{n+1}=c_n+n=\frac{n(n+1)}{2} ~ \text{and}\\
& \varepsilon_{n+1}=\varepsilon_n+\frac{\varepsilon'_n}{f}+\big[(n-2)\frac{f'}{f^{2}}+\frac{f''}{ff'}\big](c_n+\varepsilon_n).
\end{aligned}
\end{equation*}
Next, a classical result of Stirling (\cite{13}, p. 151), says
$$\log \Gamma(z)=\left(z-\frac{1}{2}\right)\log z-z+\frac{1}{2}\log (2\pi)+\int_{0}^{+\infty}\frac{[u]-u+\frac{1}{2}}{u+z}du,$$
we have, by Lebesgue's convergence theorem, that
$$f(z)=\frac{\Gamma'}{\Gamma}(z)=\log z-\frac{1}{2z}-\int_{0}^{+\infty}\frac{[u]-u+\frac{1}{2}}{(u+z)^{2}}du,$$
so for $n=1, 2, ...$, one deduces inductively
\begin{equation*}
\begin{aligned}
f^{(n)}(z)=(-1)^{n-1}\left\{\frac{(n-1)!}{z^{n}}+\frac{n!}{2z^{n+1}}+(n+1)!\int_{0}^{+\infty}\frac{[u]-u+\frac{1}{2}}{(u+z)^{n+2}}du \right\}.
\end{aligned}
\end{equation*}
It is thus quite straightforward to verify
\begin{equation}\label{eq1.5}
\begin{aligned}
f(z)=\log z+o(1)=\log z(1+o(1))
\end{aligned}
\end{equation}
and
\begin{equation*}
\begin{aligned}
f^{(n)}(z)=\frac{(-1)^{n-1}(n-1)!}{z^{n}}(1+o(1))
\end{aligned}
\end{equation*}
for $n=1, 2, ...$, and hence
\begin{equation}\label{eq1.6*}
\begin{aligned}
\frac{f'}{f^{2}}(z)=\frac{1}{z(\log z)^{2}}(1+o(1)) ~~\text{and}~~\frac{f''}{ff'}(z)=-\frac{1}{z\log z}(1+o(1)),
\end{aligned}
\end{equation}
uniformly for all $z\in \mathbb{C}\setminus\{z:|\arg z-\pi|\leq\delta\}$ for some $\delta>0$, where $o(1)$ stands for a quantity that goes to $0$ as $|z|\rightarrow +\infty$.
Now, recall $c_n=\frac{n(n-1)}{2}$ and $\varepsilon_1=\varepsilon_2=0$. It can be deduced from (\ref{eq1.6*}) that, as $|z|\rightarrow +\infty$,
\begin{equation*}
\begin{aligned}
\varepsilon_{3}&=\frac{f''}{ff'}=-\frac{1}{z\log z}(1+o(1)),\\
\varepsilon_{4}&=\varepsilon_3+\frac{\varepsilon'_3}{f}+\big(\frac{f'}{f^{2}}+\frac{f''}{ff'}\big)(c_3+\varepsilon_3)=-\frac{4}{z\log z}(1+o(1)),\\
\varepsilon_{5}&=\varepsilon_4+\frac{\varepsilon'_4}{f}+\big(2\frac{f'}{f^{2}}+\frac{f''}{ff'}\big)(c_4+\varepsilon_4)=-\frac{10}{z\log z}(1+o(1)),\\
\vdots\\
\varepsilon_{n}&=-\frac{n(n-1)(n-2)}{6}\frac{1}{z\log z}(1+o(1))~\text{upon assumption, and thus}\\
\varepsilon_{n+1}&=\varepsilon_n+\frac{\varepsilon'_n}{f}+\big[(n-2)\frac{f'}{f^{2}}+\frac{f''}{ff'}\big](c_n+\varepsilon_n)\\
&=-\big[\frac{n(n-1)(n-2)}{6}+c_n\big]\frac{1}{z\log z}(1+o(1))\\
&=-\big[\frac{n(n-1)(n-2)}{6}+\frac{n(n-1)}{2}\big]\frac{1}{z\log z}(1+o(1))\\
&=-\frac{n(n^{2}-1)}{6}\frac{1}{z\log z}(1+o(1)).
\end{aligned}
\end{equation*}
The above formulas illustrate that for $n=1, 2, ...$,
\begin{equation}\label{eq1.6**}
\begin{aligned}
\varepsilon_{n}(z)=o(1) ~~ \text{as} ~~ |z|\rightarrow +\infty.
\end{aligned}
\end{equation}
Noting $\frac{\Gamma^{(n)}}{\Gamma}=f^{n}\big[1+\frac{f'}{f^2}(c_n+\varepsilon_n)\big]$,$\frac{\Gamma^{(l)}}{\Gamma}=f^{l}\big[1+\frac{f'}{f^2}(c_l+\varepsilon_l)\big]$ by using (\ref{eq1.2}) and (\ref{eq1.4}), we get
\begin{equation}\label{eq1.7}
\begin{aligned}
&P_{p_0}\left(\zeta, \zeta', ..., \zeta^{(m)}; 1, \frac{\Gamma^{(n)}}{\Gamma}, \frac{\Gamma^{(l)}}{\Gamma}\right)\\
=&\sum_{q=0}^{M_{p_0}}\sum_{\lambda\in\Lambda_{p_0}\cap\Lambda_{q}^{\ast}}a_{\lambda}(\zeta, \zeta', ..., \zeta^{(m)})\left(\frac{\Gamma^{(n)}}{\Gamma}\right)^{\lambda_{n}}\left(\frac{\Gamma^{(l)}}{\Gamma}\right)^{\lambda_{l}}\\
=&\sum_{q=0}^{M_{p_0}}\sum_{\lambda\in\Lambda_{p_0}\cap\Lambda_{q}^{\ast}}a_{\lambda}(\zeta, \zeta', ..., \zeta^{(m)})f^{n\lambda_{n}}\big[1+\frac{f'}{f^2}(c_n+\varepsilon_n)\big]^{\lambda_{n}}f^{l\lambda_{l}}\big[1+\frac{f'}{f^2}(c_l+\varepsilon_l)\big]^{\lambda_{l}}\\
=&\sum_{q=0}^{M_{p_0}}f^{q}\sum_{\lambda\in\Lambda_{p_0}\cap\Lambda_{q}^{\ast}}a_{\lambda}(\zeta, \zeta', ..., \zeta^{(m)})\big[1+\frac{f'}{f^{2}}(c_{n}+\varepsilon_{n})\big]^{\lambda_{n}}\big[1+\frac{f'}{f^{2}}(c_{l}+\varepsilon_{l})\big]^{\lambda_{l}}.
\end{aligned}
\end{equation}
Then, we define
$$\Lambda_{j}^{\ast\ast}:=\{\lambda\in\Lambda: |\lambda|^{\ast\ast}=j ~\text{with}~|\lambda|^{\ast\ast}:=\lambda_{n}+\lambda_{l}\},$$
and assume that the largest possible $\lambda_{n}+\lambda_{l}$ is $N_{p_0}$. For any $\lambda=(\lambda_{0},\lambda_{n},\lambda_{l})\in\Lambda_{p_0}\cap\Lambda_{q}^{\ast}\cap\Lambda_{j}^{\ast\ast}$, we have
\begin{equation}\label{eq1.7.1}
\left\{
\begin{aligned}
\lambda_{0}+\lambda_{n}+\lambda_{l}=p_0, \\
n\lambda_{n}+l\lambda_{l}=q, \\
\lambda_{n}+\lambda_{l}=j,
\end{aligned}
\right.
\end{equation}
and denote the matrix of coefficients of system \eqref{eq1.7.1} by
\begin{equation*}
\begin{aligned}
B=\left(\begin{array}{cccc}
1& 1& 1 \\
0& n& l\\
0& 1& 1\\
\end{array}\right).
\end{aligned}
\end{equation*}
Thus, we have $\det(B)=n-l\neq 0$. By Cramer's Rule, we know $\lambda=(\lambda_{0}, \lambda_{n},\lambda_{l})\in\Lambda_{p_0}\cap\Lambda_{q}^{\ast}\cap\Lambda_{j}^{\ast\ast}$ is uniquely determined by any fixed $p_0, q, j$, and vice versa. So, we can rewrite (\ref{eq1.7}) as
\begin{equation}\label{eq1.8}
\begin{aligned}
&P_{p_0}\bigg(\zeta, \zeta', ..., \zeta^{(m)}; 1, \frac{\Gamma^{(n)}}{\Gamma}, \frac{\Gamma^{(l)}}{\Gamma}\bigg)(z)\\
=&\sum_{q=0}^{M_{p_0}}f^{q}(z)\sum_{j=0}^{N_{p_0}}a_{\lambda\in\Lambda_{p_0}\cap\Lambda_{q}^{\ast}\cap\Lambda_{j}^{\ast\ast}}(\vec{\zeta}) \big[1+H(c_{n}+\varepsilon_{n})\big]^{\lambda_{n}}\big[1+H(c_{l}+\varepsilon_{l})\big]^{\lambda_{l}}(z)\\
=&\sum_{q=0}^{M_{p_0}}f^{q}(z)\sum_{t=0}^{N_{p_0}}\big[b_{q,t}(\vec{\zeta})H^{t}\big](z)\\
=&f^{M_{p_0}}(z)\big[b_{M_{p_0},0}(\vec{\zeta})+b_{M_{p_0},1}(\vec{\zeta})H+\cdots+b_{M_{p_0},N_{p_0}}(\vec{\zeta})H^{N_{p_0}}\big](z)\\
&+f^{M_{p_0}-1}(z)\big[b_{M_{p_0}-1,0}(\vec{\zeta})+b_{M_{p_0}-1,1}(\vec{\zeta})H+\cdots+b_{M_{p_0}-1,N_{p_0}}(\vec{\zeta})H^{N_{p_0}}\big](z)\\
&+\cdots+f(z)\big[b_{1,0}(\vec{\zeta})+b_{1,1}(\vec{\zeta})H+\cdots+b_{1,N_{p_0}}(\vec{\zeta})H^{N_{p_0}}\big](z)\\
&+\big[b_{0,0}(\vec{\zeta})+b_{0,1}(\vec{\zeta})H+\cdots+b_{0,N_{p_0}}(\vec{\zeta})H^{N_{p_0}}\big](z),
\end{aligned}
\end{equation}
with $H=\frac{f'}{f^{2}}$, $(\vec{\zeta})$ being the abbreviation for the vector function $(\zeta, \zeta', ..., \zeta^{(m)})$, and for some $p_0, q, j$, a term with $|\lambda|=p_0, |\lambda|^{\ast}=q, |\lambda|^{\ast\ast}=j$ may not appear in (\ref{eq1.7}), if so, we simply regard the coefficient $a_{\lambda\in\Lambda_{p_0}\cap\Lambda_{q}^{\ast}\cap\Lambda_{j}^{\ast\ast}}(u_{0}, u_{1}, ..., u_{m})$ to be identically zero.
Here, for fixed $p_0, q$, set $u:=(u_0, u_1,..., u_m)$, the polynomials $a_{\lambda\in\Lambda_{p_0}\cap\Lambda_{q}^{\ast}\cap\Lambda_{j}^{\ast\ast}}(u)$, $b_{q, t}(u)$ satisfy the relations as following:
\begin{equation}\label{eq1.8.1}
\begin{aligned}
b_{q,N_{p_0}}(u)=&a_{\lambda\in\Lambda_{p_0}\cap\Lambda_{q}^{\ast}\cap\Lambda_{N_{p_0}}^{\ast\ast}}(u)(c_{n}+\varepsilon_{n})^{\lambda_{n}} (c_{l}+\varepsilon_{l})^{\lambda_{l}}, \\
b_{q,N_{p_0}-1}(u)=&a_{\lambda\in\Lambda_{p_0}\cap\Lambda_{q}^{\ast}\cap\Lambda_{N_{p_0}-1}^{\ast\ast}}(u)(c_{n}+\varepsilon_{n})^{\lambda_{n}}(c_{l}+\varepsilon_{l})^{\lambda_{l}}+a_{\lambda\in\Lambda_{p_0}\cap\Lambda_{q}^{\ast}\cap\Lambda_{N_{p_0}}^{\ast\ast}}(u)\\
&[\lambda_{l}(c_{n}+\varepsilon_{n})^{\lambda_{n}-1}(c_{l}+\varepsilon_{l})^{\lambda_{l}}+(c_{n}+\varepsilon_{n})^{\lambda_{n}} \lambda_{n}(c_{n}+\varepsilon_{n})^{\lambda_{n}-1}],\\
\vdots\\
b_{q,1}(u)=&\sum_{j=1}^{N_{p_0}}a_{\lambda\in\Lambda_{p_0}\cap\Lambda_{q}^{\ast}\cap\Lambda_{j}^{\ast\ast}}(u)[\lambda_{n}(c_{n}+\varepsilon_{n})+\lambda_{l}(c_{l}+\varepsilon_{l})],\\
b_{q,0}(u)=&\sum_{j=0}^{N_{p_0}}a_{\lambda\in\Lambda_{p_0}\cap\Lambda_{q}^{\ast}\cap\Lambda_{j}^{\ast\ast}}(u).
\end{aligned}
\end{equation}
Recall (\ref{eq1.4}), suppose $b_{q_{0},t_{0}}(u_{0}, u_{1}, ..., u_{m})$ is the first non-zero term in the ordered sequence as following:
$$b_{M_{p_0},0}(u), b_{M_{p_0}-1,0}(u), ..., b_{1,0}(u), b_{0,0}(u),$$
$$b_{M_{p_0},1}(u), b_{M_{p_0}-1,1}(u), ..., b_{1,1}(u), b_{0,1}(u),$$
$$\ddots$$
$$b_{M_{p_0},N_{p_0}-1}(u), b_{M_{p_0}-1,N_{p_0}-1}(u), ..., b_{1,N_{p_0}-1}(u), b_{0,N_{p_0}-1}(u).$$
$$b_{M_{p_0},N_{p_0}}(u), b_{M_{p_0}-1,N_{p_0}}(u), ..., b_{0,N_{p_0}}(u), b_{0,N_{p_0}}(u).$$
Since $b_{q_{0},t_{0}}(u_0, u_1,..., u_m)\not\equiv 0$, we can find an $\varepsilon_0>0$ and a (sufficiently small) subset $\mathbf{\Omega}$ of $\mathbb{C}^{m+1}$ such that,
\begin{equation*}
\begin{aligned}
|b_{q_{0},t_{0}}(u_0, u_1,..., u_m)|\geq\varepsilon_0,
\end{aligned}
\end{equation*}
uniformly for all $u:=(u_0, u_1,..., u_m)\in \mathbf{\Omega}\subsetneq \mathbb{C}^{m+1}$.
Furthermore, since $a_{\lambda\in\Lambda_{p_0}\cap\Lambda_{q}^{\ast}\cap\Lambda_{j}^{\ast\ast}}$ is a polynomial of $u_{0}, u_{1}, ..., u_{m}$, in view of the finiteness of indices, we can find a constant $C_0> 1$ such that
\begin{equation*}
\begin{aligned} |a_{\lambda\in\Lambda_{p_0}\cap\Lambda_{q}^{\ast}\cap\Lambda_{j}^{\ast\ast}}(u_0, u_1,..., u_m)|\leq C_0
\end{aligned}
\end{equation*}
uniformly for all $u:=(u_0, u_1,..., u_m)\in \mathbf{\Omega}\subsetneq \mathbb{C}^{m+1}$
and for all $0\leq p_0 \leq L$, $0\leq q \leq M_{p_0}$, $0\leq j \leq N_{p_0}$.
Then, using Lemma \ref{lem2.1}, there exists a sequence of real numbers $\{y_{k}\}_{k=1}^{+\infty}$ with $|y_{k}|\rightarrow +\infty$ such that $\gamma(y_k)\in \mathbf{\Omega}\subsetneq \mathbb{C}^{m+1}$ when $x=\frac{3}{4}$. It follows from (\ref{eq1.6**}) and (\ref{eq1.8.1}) that there exists a constant $C_1>1$ such that,
for $z_{k}:=\frac{3}{4}+i y_{k}\in \mathbb{C}$, one has
\begin{equation}\label{eq2.1}
\begin{aligned}
|b_{q_{0},t_{0}}(\zeta, \zeta',..., \zeta^{(m)})(z_{k})|\geq\varepsilon_0 \ \ and \ \ |b_{q,t}(\zeta, \zeta',..., \zeta^{(m)})(z_{k})|\leq C_1,
\end{aligned}
\end{equation}
uniformly for all large $k$ and for all $q, t$ with $0\leq q\leq M_{p_0}$, $0\leq t\leq N_{p_0}$. In view of (\ref{eq1.5}) and (\ref{eq1.6*}), we have
$$f^{q}(z_{k})b_{q,t}(\zeta, \zeta',..., \zeta^{(m)})(z_{k})\left(\frac{f'(z_{k})}{f^2(z_{k})}\right)^{t}$$
is equal to
$$b_{q,t}(\zeta, \zeta',..., \zeta^{(m)})(z_{k})\frac{(\log z_{k})^{q-2t}}{(z_{k})^{t}}(1+o(1))$$
when $k\rightarrow +\infty$, where the indices here either satisfy $t=t_{0}$ with $0\leq q\leq q_{0}$ or satisfy $t_{0}<t\leq N_{p_0}$ with $0\leq q\leq M_{p_0}$. As a result, the term
$$f^{q_{0}}(z_{k})b_{q_{0},t_{0}}(\zeta(z_{k}), \zeta'(z_{k})..., \zeta^{(m)}(z_{k}))\left(\frac{f'(z_{k})}{f^2(z_{k})}\right)^{t_{0}},$$
among all possible terms in (\ref{eq1.8}), dominates in growth when $k\rightarrow +\infty$. In fact, for sufficiently large $k$, if $t=t_0$ with $0\leq q< q_{0}$, one derives,
$$\frac{|\log z_{k}|^{q}}{|z_{k}|^{t_0}|\log z_{k}|^{2t_0}}\ll \frac{|\log z_{k}|^{q_0}}{|z_{k}|^{t_0}|\log z_{k}|^{2t_0}},$$
while if $t_{0}<t\leq N_{p_0}$ with $0\leq q\leq M_{p_0}$, one has
$$|\log z_{k}|^{q}\leq |\log z_{k}|^{M_{p_0}+q_0},$$
$$|z_{k}|^{t}|\log z_{k}|^{2t}\gg |z_{k}|^{t_0}|\log z_{k}|^{M_{p_0}+2t_0},$$
and then one also derives,
$$\frac{|\log z_{k}|^{q}}{|z_{k}|^{t}|\log z_{k}|^{2t}}\ll \frac{|\log z_{k}|^{q_0}}{|z_{k}|^{t_0}|\log z_{k}|^{2t_0}}.$$
We then derive from (\ref{eq1.8}) and (\ref{eq2.1}), as $k\rightarrow \infty$, that
\begin{equation}\label{eq2.2}
\begin{aligned}
\left|P_{p_0}\left(\zeta, \zeta', ..., \zeta^{(m)}; 1, \frac{\Gamma^{(n)}}{\Gamma}, \frac{\Gamma^{(l)}}{\Gamma}\right)(z_{k})\right|\geq \frac{\varepsilon_0}{2} \left|\frac{(\log z_{k})^{q_{0}-2t_{0}}}{(z_{k})^{t_{0}}}\right|.
\end{aligned}
\end{equation}
If $p_0=L$, then, by the definition of $p_0$ and ($\ref{eq2.2}$), it yields that
\begin{equation*}
\begin{aligned}
&P(\zeta, \zeta', ..., \zeta^{(m)}; \Gamma, \Gamma^{(n)}, \Gamma^{(l)})(z_{k})\\
=&P_{L}(\zeta, \zeta', ..., \zeta^{(m)}; \Gamma, \Gamma^{(n)}, \Gamma^{(l)})(z_{k})\\
=&\Gamma^{L}(z_{k})P_{L}\left(\zeta, \zeta', ..., \zeta^{(m)}; 1, \frac{\Gamma^{(n)}}{\Gamma}, \frac{\Gamma^{(l)}}{\Gamma}\right)(z_{k})\neq0,
\end{aligned}
\end{equation*}
for sufficiently large $k$. This contradicts the hypothesis (\ref{eq1.1}). Claim \ref{cla 3.1} is proved in this case.
If $p_0<L$, by virtue of another result of Stirling (\cite{13}, p.151) which says
\begin{equation}\label{eq2.2.1}
\begin{aligned}
\left|\Gamma\left(\frac{3}{4}+i y \right)\right|=e^{\frac{-\pi|y|}{2}} |y|^{\frac{1}{4}} \sqrt{2\pi}(1+o(1)), ~ \text{as} ~ y\rightarrow +\infty,
\end{aligned}
\end{equation}
and for all $t>0$,
$f^{q}(z_{k})\left(\frac{f'(z_{k})}{f^{2}(z_{k})}\right)^{t}\rightarrow 0$ as $k\rightarrow \infty$,
one easily observes, in view of (\ref{eq1.3}), (\ref{eq1.4}), (\ref{eq2.2}),(\ref{eq2.2.1}) and (\ref{eq2.1}), that there is a constant $C>0$ such that, for all large $q$,
\begin{equation*}
\begin{aligned}
&\left|\frac{P(\zeta, \zeta', ..., \zeta^{(m)}; \Gamma, \Gamma^{(n)}, \Gamma^{(l)})(z_{k})}{\Gamma^{L}(z_{k})}\right|\\
=&\left|\sum_{s=p_0}^{L}\frac{1}{\Gamma^{L-s}(z_{k})}P_{s}\left(\zeta, \zeta', ..., \zeta^{(m)}; 1, \frac{\Gamma^{(n)}}{\Gamma}, \frac{\Gamma^{(l)}}{\Gamma}\right)(z_{k}) \right|\\
\geq& e^{\frac{(L-p_0)\pi|y_{k}|}{2}}|y_{k}|^{\frac{1}{4}(p_0-L)}(\sqrt{2\pi})^{p_0-L}\eta \frac{|\log z_{k}|^{q_{0}-2t_{0}}}{|z_{k}|^{t_{0}}} \\
&-C_2 e^{\frac{(L-p_0-1)\pi|y_{k}|}{2}}|y_{k}|^{\frac{1}{4}(p_0+1-L)}(\sqrt{2\pi})^{(p_0+1-L)}|\log z_{k}|^{C_3}\rightarrow +\infty,
\end{aligned}
\end{equation*}
as $k\rightarrow \infty$ for some constants $C_2, C_3> 0$ depending on $C_1, L$. Hence,
$$P(\zeta, \zeta', ..., \zeta^{(m)}; \Gamma, \Gamma^{(n)}, \Gamma^{(l)})(z_{k})\neq 0$$
for sufficiently large $z_{k}$, which again contradicts the hypothesis (\ref{eq1.1}). Thus, the proof of Claim \ref{cla 3.1} is completed.
\begin{claim}\label{cla 3.2}
For each $0\leq p\leq L$, when
\begin{equation}\label{eq2.3}
\begin{aligned}
P_{p}(\zeta, \zeta', ..., \zeta^{(m)}; \Gamma, \Gamma^{(n)}, \Gamma^{(l)})(z)\equiv 0
\end{aligned}
\end{equation}
for $z\in \mathbb{C}$, then the polynomial $P_{p}(u_{0}, u_{1}, ..., u_{m}; v_{0}, v_{n}, v_{l})$ vanishes identically.
\end{claim}
\emph{Proof.}
When $p=0$, by definition, $P_{0}(u_{0}, u_{1}, ..., u_{m}; v_{0}, v_{n}, v_{l})$ is a polynomial in $u_{0}, u_{1}, ..., u_{m}$ only; so, from the solution \cite{11} to the question posed by Hilbert, $P_0(\zeta, \zeta', ..., \zeta^{(m)})(z)\equiv 0$ leads to $P_0(u_{0}, u_{1}, ..., u_{m})\equiv 0.$
Henceforth, assume $p>0$. For simplicity, we write $p=p_0$ and use the expression (\ref{eq1.8}) and its associated notations. Next, we prove that $b_{q,t}(u_{0}, u_{1}, ..., u_{m})\equiv 0$ for all $0\leq q\leq M_{p_0}$, $0\leq t\leq N_{p_0}$. To this end, we first prove that each of
$$b_{M_{p_0},0}(u_{0}, u_{1}, ..., u_{m}), b_{M_{p_0}-1,0}(u_{0}, u_{1}, ..., u_{m}), ..., b_{0,0}(u_{0}, u_{1}, ..., u_{m}) $$
must be identically equal to zero. Let's start from $b_{M_{p_0},0}(u_{0}, u_{1}, ..., u_{m})$,
\begin{equation*}
\begin{aligned}
b_{M_{p_0},0}(u_{0}, u_{1}, ..., u_{m})=\sum_{j=0}^{N_{p_0}}a_{\lambda\in\Lambda_{p_0}\cap\Lambda_{M_{p_0}}^{\ast}\cap\Lambda_{j}^{\ast\ast}}(u_{0}, u_{1}, ..., u_{m}),
\end{aligned}
\end{equation*}
a polynomial of $u_{0}, u_{1}, ..., u_{m}$, and assume that it does not vanish identically. Then, following what we have done in Claim \ref{cla 3.1}, one has $(\ref{eq2.1})$ (or its analogue for this newly chosen $P_0$).
Among all the terms of $P_{p_0}\big(\zeta, \zeta', ..., \zeta^{(m)}; 1, \frac{\Gamma^{(n)}}{\Gamma}, \frac{\Gamma^{(l)}}{\Gamma}\big)(z_{k})$ described as in (\ref{eq1.8}), the term
$$f^{M_{p_{0}}}(z_{k})b_{M_{p_{0}},0}(\zeta, \zeta', ..., \zeta^{(m)})(z_{k})\sim b_{M_{p_{0}},0}(\zeta, \zeta', ..., \zeta^{(m)})(z_{k})(\log z_{k})^{M_{p_{0}}},$$
dominates in growth for large $k$, since for all $t>0$, as $k\rightarrow \infty$,
$f^{q}(z_{k})\left(\frac{f'(z_{k})}{f^{2}(z_{k})}\right)^{t}\rightarrow 0$. Thus, analogous to (\ref{eq2.2}), we deduce from (\ref{eq1.8}) and (\ref{eq2.1}) that
\begin{equation*}
\begin{aligned}
&P_{p_0}(\zeta, \zeta', ..., \zeta^{(m)}; \Gamma, \Gamma^{(n)}, \Gamma^{(l)})(z_k)\\
=&\Gamma^{p_0}(z_{k})P_{p_0}\left(\zeta, \zeta', ..., \zeta^{(m)}; 1, \frac{\Gamma^{(n)}}{\Gamma}, \frac{\Gamma^{(l)}}{\Gamma}\right)(z_{k})\neq0
\end{aligned}
\end{equation*}
for sufficiently large $k$, which contradicts with the hypothesis (\ref{eq2.3}). Therefore,
$b_{M_{p_{0}},0}(u_{0}, u_{1}, ..., u_{m})\equiv 0.$ The next term is $b_{M_{p_{0}}-1,0}(u_{0}, u_{1}, ..., u_{m})$ with $$f^{M_{p_{0}}-1}(z_{k})b_{M_{p_{0}}-1,0}(\zeta, \zeta', ..., \zeta^{(m)})(z_{k})\sim b_{M_{p_{0}}-1,0}(\zeta, \zeta', ..., \zeta^{(m)})(z_{k})(\log z_{k})^{M_{p_{0}}-1},$$
so that one can derive $b_{M_{p_{0}}-1,0}(u_{0}, u_{1}, ..., u_{m})\equiv 0$ in exactly the same manner; repeating this process, we have $b_{q,0}(u_{0}, u_{1}, ..., u_{m})\equiv 0$ for each $0\leq q\leq M_{p_{0}}$. Next, after the elimination of $\frac{f'}{f^2}$, one can perform the preceding procedure again for
$$b_{M_{p_{0}},1}(u_{0}, u_{1}, ..., u_{m}), b_{M_{p_{0}}-1,1}(u_{0}, u_{1}, ..., u_{m}), ..., b_{0,1}(u_{0}, u_{1}, ..., u_{m})$$
and obtain that $b_{q,1}(u_{0}, u_{1}, ..., u_{m})\equiv 0$ for each $0\leq q \leq M_{p_{0}}$. Continuing like this, we get $b_{q,t}(u_{0}, u_{1}, ..., u_{m})\equiv 0$ for all $0\leq q \leq M_{p_{0}}$, $0\leq t \leq N_{p_{0}}$. Thus, it can deduce from $(\ref{eq1.8.1})$ that all the coefficients $a_{\lambda\in\Lambda_{p_0}\cap\Lambda_{q}^{\ast}\cap\Lambda_{j}^{\ast\ast}}(u_{0}, u_{1}, ..., u_{m})$ in $P_{p_0}(u_{0}, u_{1}, ..., u_{m}; v_{0}, v_{n}, v_{l})$ are identically zeros. Therefore,
$$P_{p_0}(u_{0}, u_{1}, ..., u_{m}; v_{0}, v_{n}, v_{l})\equiv 0,$$
and Claim \ref{cla 3.2} is proved completely.
It follows from $(\ref{eq1.2})$ and $(\ref{eq1.3})$ that the proof of Theorem~\ref{M-thm} is a straightforward consequence of Claim \ref{cla 3.1} and Claim \ref{cla 3.2}, so that $(\ref{eq1.1})$ indeed leads to $P(u_{0}, u_{1}, ..., u_{m}; v_{0}, v_{n}, v_{l})\equiv 0$.
| -50,255.068094 |
[
-1.9677734375,
1.638671875
] | 14.697406 |
[
-4.07421875,
0.515625,
-2.619140625,
-6.0390625,
-0.6240234375,
8.6640625
] |
[
2.041015625,
9.3125,
1.3154296875,
3.73828125
] | 254 | 2,105 |
[
-3.337890625,
3.87109375
] | 46.974306 |
[
-5.421875,
-3.408203125,
-3.869140625,
-2.501953125,
1.583984375,
10.9375
] | 0.779373 | 6.526693 | 33.586698 | 5.201162 |
[
1.4662532806396484
] | -34,257.815969 | 7.314489 | -49,751.228208 | 0.506592 | 5.7594 |
[
-2.564453125,
-3.130859375,
-4.0390625,
-5.7421875,
2.228515625,
12.609375
] |
[
-5.40234375,
-1.3349609375,
-1.716796875,
-1.126953125,
3.197265625,
3.265625
] | |
BkiUdZM4dbghZzegCCP7
|
\section{Introduction}
The conventional theory of superconductivity consists of two distinct pieces. The first describes the formalism of pairing, premised on the idea that two electrons effectively attract one another; this pairing is well described by the Bardeen-Cooper-Schrieffer (BCS) wave function,\cite{bardeen57} and continues to be used extensively to describe almost all known superconductors. The second piece concerns the origin of the attractive interaction, suggested originally by
Fr\"ohlich \cite{frohlich50} and Bardeen\cite{bardeen50} to be the electron-phonon interaction. In the BCS calculation the additional piece concerning the origin (i.e. {\em mechanism}) of pairing enters only in that a simple (though confusing) cutoff of order the Debye (phonon) frequency is included in the attractive interaction. It is confusing because it is added in momentum space, whereas the physical cutoff is in frequency space, to reflect the retardation effects expected from relatively sluggish phonons interacting with nimble electrons. The cutoff occurs naturally in frequency space in the later formulation of Eliashberg\cite{eliashberg60} and others.\cite{nambu60,morel62,schrieffer63}
A very solid and coherent case has been made for this description of phonon-mediated mechanism of superconductivity
for simple metals like Lead.\cite{parks69,allen82} However, even though Eliashberg theory is often referred to as a microscopic theory of superconductivity (and, indeed, it is {\em more} microscopic than BCS theory), it retains some elements of a phenomenology. This occurs in two distinct areas in the theory; the first concerns the treatment of direct Coulomb interactions, which in both BCS and Eliashberg treatments tends to be modelled through the use of a single (renormalized) parameter, $\mu^\ast$, and the second concerns the treatment of the phonons. Traditionally, the phonons are taken from experiment, notably tunneling\cite{mcmillan69} or neutron scattering.\cite{brockhouse62} In either case all the ``calculations'' concerning the phonons have been done by nature, and handed over to the theorist to incorporate into Eliashberg theory. Notable exceptions are, for example, treatments in Ref. (\onlinecite{marsiglio90}) and more recently in Ref. (\onlinecite{bauer11}), where an attempt is made to compute modification to the properties of the phonons alongside those of the electrons, due to strong electron-phonon interactions.
As an example we show in Fig.~1 a plot of the pairing susceptibility (a measure of the superconducting critical temperature) vs. electron filling, using a Migdal-Eliashberg calculation with and without phonon renormalization.\cite{marsiglio91} When phonon renormalization is included, the susceptibility (here plotted for an inverse temperature,
$\beta = 6$), decreases, particularly as one approaches half filling. In fact our calculations stop because a charge density wave instability occurs for a density less than half filling. This makes apparent that phonon renormalization has an important but deleterious effect on superconductivity, at least for the model considered in Ref. (\onlinecite{marsiglio91}). This is an important point which will have to be reconsidered in future work. For the present, however, we wish to focus on a simpler model, that of a {\em single} electron interacting with many phonons. This object is usually referred to as a polaron (though not in the literature pertaining to superconductivity\cite{remark1}), and we regard the polaron as the basic building block, from which a theory of interacting electrons and phonons must be built. This is {\em not} how the theory of superconductivity progressed historically. A weak coupling approach was assumed at the onset; this means that a (large) Fermi sea is taken as `given', and then Migdal-like\cite{migdal58} arguments are adopted to essentially use weak coupling perturbation theory.\cite{remark2}
\begin{figure}
\begin{center}
\includegraphics[height=3.3in,width=2.5in, angle = -90]{fig1.eps}
\caption{(Color online)
Pairing susceptibility vs. electron filling for a fixed bare coupling strength, $\lambda_0 = 2$, and a fixed temperature, given by $\beta \equiv 1/(k_B T) = 6$. Results for two different frequencies are shown; usual Migdal calculations of the electronic properties with no changes to the phonons are given by the curves, while results that allow the phonons to renormalize due to the electron phonon coupling are given by the symbols. Clearly allowing for phonon renormalization leads to a suppression of the pairing susceptibility, and therefore superconductivity.}
\end{center}
\end{figure}
In this work we wish to illustrate the polaronic nature of the electron when coupled to phonons, using the Holstein model.\cite{holstein59} We will summarize results in one, two, and three dimensions, adopting various strategies: weak and strong coupling perturbation theory,\cite{marsiglio95,li10} the adiabatic approximation,\cite{kabanov93} and an exact diagonalization method which incorporates Bloch's Theorem which we refer to as the Trugman method.\cite{trugman90,bonca99,ku02} Very little work has been done beyond the Holstein model. When a non-local interaction is used, the coupling in momentum space acquires momentum dependence, making all techniques more difficult. Very recently we have succeeded in obtaining analytical results for the polaron in the weak coupling limit\cite{li11} of the
Bari\u si\'c, Labb\' e, and Friedel\cite{barisic70}-Su-Schrieffer-Heeger\cite{su79} (BLF-SSH) model, which models acoustic phonons coupled to electrons through their itineracy. Other, more complicated models have also been proposed. For example in Ref. (\onlinecite{hague06}) a longer range electron phonon interaction is proposed, and Jahn-Teller-type distortions in the presence of repulsive Coulomb interactions also lead to polaron formation.\cite{mertelj07} Very recently, Innocenti {\it et al.}\cite{innocenti10} have proposed a two electron band scenario, where tuning of the chemical potential can lead to polaron formation; a companion proposal\cite{poccia11} also suggests that so-called `dopant self-organization' can lead to trapping of polarons, resulting in some kind of inhomogeneous phase.
These are all interesting possibilities, but here we focus on rigorous results for the simplest model describing polarons.
In the next section we describe the various techniques used to attack the Holstein polaron, followed by results for two specific properties --- the ground state energy and the effective mass. It is seen, for example, that the electron is always polaronic in one dimension. More precisely, this statement means the following: as the phonon frequency is taken to be smaller and smaller (this is necessarily a process taken as a limit) more and more phonons are required to properly describe the ground state, irrespective of the coupling strength. In the past researchers have referred to this as a self-localized (or self-trapped) state; this is a misnomer, as no such violation of Bloch's Theorem occurs. Partly this nomenclature has arisen from variational studies (and the adiabatic limit, where the phonon frequency $\omega_E$ is taken to be zero at the onset), where then a localized state does arise. The result is that the entity of interest emerging out of the ground state is hardly recognizable as an electron; rather it contains a considerable admixture of phonons, and hence is renamed a polaron. After presenting results in two and three dimensions, where a regime of weak coupling exists in which the electron definitely retains free-electron-like ( and therefore `non-polaron-like') character for weak coupling strengths, we will conclude with a summary.
\section{Methods}
Perhaps the simplest starting point for all studies such as this one is the weak coupling approximation. This is well-defined with straightforward perturbative methods, or one can adopt the language of Green functions, where the electron propagator, $G(k,\omega)$, can be written in momentum ($k$) and frequency ($\omega$) space in terms of a self energy, $\Sigma(k,\omega)$:
\begin{equation}
\Sigma(k,\omega+ i\delta) = -\sum_{k^\prime} |g(k,k^\prime)|^2 G_0(k^\prime,\omega + i\delta - \omega(k-k^\prime)),
\label{selfenergy}
\end{equation}
where $G_0(k,\omega+i\delta) \equiv \bigl[ \omega + i\delta - \epsilon_k \bigr]^{-1}$ is the non-interacting electron retarded propagator, $\epsilon_k$ ($\omega_q$) gives the dispersion relation for the non-interacting electrons (phonons), respectively, and $g(k,k^\prime)$ describes any possible momentum dependence in the electron-phonon coupling. Note that the infinitesimal imaginary part to the frequency, $i\delta$, is necessary for the analyticity of the Green function; here we use the retarded functions (positive imaginary part). This is written quite generally, and for any dimension. Here, we wish to discuss specifically the Holstein model, with Hamiltonian written in momentum space as
\begin{eqnarray}
H&=&\sum_k \epsilon_k c_k^\dagger c_k + \omega_E \sum_q a_q^\dagger a_q \nonumber \\
&-& {g\omega_E \over \sqrt{N}} \sum_{k,k^\prime} c_k^\dagger c_{k^\prime}(a_{k-k^\prime} + a_{-(k - k^\prime)}^\dagger),
\label{hol_ham}
\end{eqnarray}
where $\omega_E$ is the Einstein phonon frequency and $g$ is the (dimensionless) electron-phonon coupling, and in this case is clearly a constant. We will tend to use the (also dimensionless) parameter $\lambda$, which will have a different definition depending on dimension. In one dimension, we use $\lambda \equiv g^2\omega_E/(W/2) = g^2\omega_E/(2t) $, where $W \equiv 4t$ is the electron bandwidth written in terms of the nearest neighbour hopping parameter, $t$. Hereafter $t$ will be set to unity.
\begin{figure*}[tp]
\centering
\includegraphics[height=2.8in,width=3.0in]{1D_ENERGY-1.EPS}
\includegraphics[height=2.8in,width=3.0in]{1D_mass-1.EPS}
\includegraphics[height=2.8in,width=3.0in]{2D_energy-1.EPS}
\includegraphics[height=2.8in,width=3.0in]{2D_mass-1.EPS}
\includegraphics[height=2.8in,width=3.0in]{3D_energy-1.EPS}
\includegraphics[height=2.8in,width=3.0in]{3D_mass-1.EPS}
\caption{The ground state energy and effective mass vs electron phonon coupling strength for the Holstein model. The energy ((a) (c) (e)) and effective mass ((b) (d) (f)) are plotted in one, two, and three dimensions, respectively. Note the very abrupt crossover to polaron-like behaviour in 2D and 3D, both at relatively low coupling strengths. The strong coupling limit agrees very well with the adiabatic limit in all three dimensions. Also note that in (a), (c) and (e) there are three curves that are indistinguishable for $\lambda {{ \atop {> \atop \sim}} \atop } 1.5$, showing that the strong coupling and adiabatic regime is readily achieved for moderate $\lambda$ and non-zero $\omega_E$.}
\label{fig2}
\end{figure*}
In one and two dimensions, the momentum sums can be evaluated analytically. The results are
\begin{equation}
\Sigma_{\mathrm{1D}}(\omega+i\delta) = {\frac{ \lambda \omega_E \mathrm{sgn}%
(\omega - \omega_E) }{\sqrt{\biggl({\frac{\omega - \omega_E }{2t}}\biggr)^2
- 1}}},
\label{sig_weak_1D}
\end{equation}
so that the ground state energy is
\begin{equation}
E_0 = -2t\biggl( 1 + \lambda \sqrt{\frac{\omega_E }{4t + \omega_E}} \biggr),
\label{e0_weak_1D}
\end{equation}
and the effective mass $m^\ast$ is given by
\begin{equation}
m/m^\ast = 1 - {\frac{\lambda }{2}} \sqrt{\frac{t }{\omega_E}} {\frac{1 + {\frac{%
\omega_E }{2t}} }{(1 + {\frac{\omega_E }{4t}})^{3/2}}}.
\label{eff_mass_1D}
\end{equation}
In two dimensions we obtain\cite{li10}
\begin{equation}
\Sigma _{\mathrm{2D}}(\omega+i\delta) = {\frac{ \lambda }{2}}{\frac{%
8t\omega_E }{\omega - \omega_E}} K\Biggl[ \biggl({\frac{4t }{\omega -
\omega_E}}\biggr)^2\Biggr], \label{sig_weak_2D}
\end{equation}
where $K(x) \equiv \int_0^{\pi/2} d\theta {\frac{1 }{\sqrt{1 - x \sin^2{%
\theta}}}}$ is the complete Elliptic integral of the first kind. This leads
to a ground state energy, which, in weak coupling, is:
\begin{equation}
E_0 = -4t \Biggl( 1 + {\frac{\lambda }{4}}{\frac{\omega_E }{t}}{\frac{1 }{1
+ \omega_E/(4t)}} K\Biggl[ {\frac{1 }{\bigl(1 + \omega_E/(4t)\bigr)^2}} %
\Biggr] \Biggr). \label{e0_weak_2D}
\end{equation}
We can take the derivative of Eq. (\ref{sig_weak_2D}) to obtain:
\begin{equation}
m^\ast/m = 1 + {\frac{\lambda }{2}} {\frac{1 }{1 + \omega_E/(8t)}} E\biggl[{1 \over
(1+\omega_E/(4t))^2}\biggr], \label{mass_weak_2D}
\end{equation}
where $E(x) \equiv \int_0^{\pi/2} d\theta \sqrt{1 - x \sin^2{\theta}}$ is
the complete Elliptic integral of the second kind.
Three dimensions is straightforward, but requires numerical integration.
The calculation in the strong coupling limit is outlined in Ref. (\onlinecite{marsiglio95}).
The unperturbed state is:
\begin{equation}
|\psi \rangle = e^{-g^2/2} \sum_\ell e^{ikR_\ell} e^{-g\hat{a}_\ell^\dagger} \hat{c}%
_\ell^\dagger |0 \rangle, \label{sc_zero}
\end{equation}
where the sum is over all lattice sites.\cite{remark3} Straightforward application of perturbation theory yields
a dispersion, albeit with exponentially suppressed bandwidth. This constitutes an estimate for the effective mass, which is far too high, and will be omitted from the plots. However, second order perturbation theory gives a substantive correction to the ground state energies; these are given, for large coupling values, as
\begin{equation}
E_0 \approx -\omega_E g^2 - {dt^2 \over \omega_E g^2},
\label{strong_energy}
\end{equation}
where $d$ is the dimension of the space. In the parts of Fig.~2 showing the ground state energy, the curve displaying Eq. (\ref{strong_energy}) is indistinguishable from the adiabatic limit for $\lambda {{ \atop {> \atop \sim}} \atop } 1$.
Adiabatic limit results are determined through a decoupling procedure, whereby the ion displacements are treated as
c-numbers.\cite{kabanov93,marsiglio95}
Finally, exact results are obtained using the Trugman method,\cite{bonca99,ku02} We are able to obtain converged results for very small values of the dimensionless parameter $\omega_E/t$ by using Eq. (\ref{sc_zero}) as a `seed' state for the state construction. Iteration is considerably improved as well by starting in the strong coupling limit, converging results there, and using these results as subsequent `seeds' as we span the coupling by lowering the coupling strength a little at a time.
\section{Results and Discussion}
Results are shown in Fig.~2 for the one dimensional (a) and (b), two dimensional (c) and (d), and three dimensional (e) and (f) ground state energy and effective mass, respectively, vs. electron phonon coupling strength, for a variety of phonon frequencies. Note that we use $\omega_E/t$ and $\lambda$ as the two variables to characterize the adiabaticity and coupling strength, respectively, of the electron phonon system. Here
$\lambda \equiv \omega_E g^2/(W/2)$ in one and three dimensions, where $W$ is the bandwidth (here, for a tight binding model with nearest neighbour hopping only, $W = 4t$ in 1D and $W = 12 t$ in 3D. In two dimensions we defined $\lambda$ slightly differently,\cite{li10} as $\lambda \equiv \omega_E g^2/(2 \pi t)$; this is because the non-interacting electron density of states (EDOS) is a constant at the bottom of the band, and so we adopted this value to multiply the bare value, $\lambda_0$, rather than the average EDOS, which is what we did in 1D and 3D. With this definition, for example, the effective mass becomes $1 + \lambda/2$ in 2D in the weak coupling adiabatic limit.\cite{li10}
The main results can be summarized as follows: in 1D the electron is {\em always} polaron-like. Note in particular that as the phonon frequency decreases the effective mass increases dramatically (Fig. 2(b)), and essentially diverges in the adiabatic limit, even for infinitesimal coupling strength. The results for the ground state go smoothly over to the adiabatic limit, as is apparent from the disappearance of two of the finite frequency curves into the adiabatic curve in Fig. 2(a).
In 2D and 3D there are clear delineations of free electron-like behaviour at weak coupling and polaron-like behaviour at intermediate and strong coupling. However all the results are `smooth' for a non-zero phonon frequency --- only when an adiabatic calculation is performed (with the phonon frequency set equal to zero at the onset) does the curve display a kink (and therefore a transition). So no phase transition occurs as long as $\omega_E \ne 0$. The remarkable result is that the crossover to polaron-like behaviour occurs at such an intermediate coupling strength,\cite{alexandrov_proc} well below the value of coupling strength normally assigned to conventional superconductors (even in 2D, if the average EDOS is used in the definition of $\lambda$, the crossover coupling strength only moves up to about $0.86$, just a little lower than the value in 3D). Ongoing work is investigating this crossover behaviour when acoustic modes are involved.\cite{li11}
\label{Acknowledgments}
This work was supported in part by the Natural Sciences and Engineering
Research Council of Canada (NSERC), by ICORE (Alberta), and by the Canadian
Institute for Advanced Research (CIFAR).
| -11,768.914881 |
[
-3.296875,
2.97265625
] | 25.984252 |
[
-2.873046875,
0.62255859375,
-2.28515625,
-5.9140625,
-0.8408203125,
8.7421875
] |
[
2.88671875,
8.234375,
2.72265625,
6.8359375
] | 124 | 2,309 |
[
-3.580078125,
4.30078125
] | 24.307123 |
[
-5.79296875,
-4.125,
-4.6171875,
-2.552734375,
1.8603515625,
12.7421875
] | 0.825764 | 17.808862 | 34.430489 | 3.998975 |
[
2.244561195373535
] | -8,574.79167 | 5.769164 | -11,525.355338 | 0.735681 | 5.725901 |
[
-2.416015625,
-3.830078125,
-3.642578125,
-4.90625,
2.146484375,
12.25
] |
[
-5.48046875,
-2.1015625,
-2.341796875,
-1.4990234375,
3.78515625,
5.05078125
] | |
BkiUdTvxK6wB9mpb2KKI
|
\section{Preface}
This note follows closely
the talk I gave at the ONEPAS and then repeated in a modified form at the MCQM\footnote{ONEPAS and MCQM stand for Online Northeast PDE and Analysis Seminar and Mathematical Challenges in Quantum Mechanics.} seminar series. I inserted a few details and explanations in the text (often as footnotes), expanded the comments on the literature at the end of the main text along with three appendices giving some precise definitions omitted in the talk (and the main text).
My goal was to describe the main mathematical structures and results of an important area of Quantum Mechanics involving PDEs and to formulate open problems.
Some of the open problems are closely related to the mainstream PDEs; others would draw blank for most of the PDE practitioners.
The problems of the first type
deal with quantum fluids such as superconductors, superfluids and Bose-Einstein condensates which are natural to describe in terms of
operators (density operators, etc), rather than functions, though passing to integral kernels one could produce the standard PDE description.
The problems of the second type concern particle systems interacting with quantized radiation, i.e. photons. The key goal here is to describe the processes
of emission and absorption of radiation, say light, by systems of matter such as atoms and molecules. These are, obviously, not very fanciful problems and they
deserve close attention.
Naturally, the material selected in this article adheres closely to my own research and is not comprehensive whatever interpretation of this word is used.
All references to the literature are collected in Section \ref{sec:liter}. I tried to be fair and acknowledge all recent contributions to the subjects covered.
However, I am sure I missed many worthy works and I will appreciate any information about those.
\subsection*{Acknowledgments}
I am grateful to S\'ebastien Breteaux, Thomas Chen, Ilias Chenn, J\'er\'emy Faupin, Zhou Gang, Marcel Griesemer, Stephen Gustafson Lars Jonsson, Tim Tzaneteas and, especially, Volker Bach, J\"urg Fr\"ohlich and Avy Soffer for enjoyable collaboration and to Rupert Frank, Gian Michele Graf, Christian Hainzl, for stimulating discussions on the topics touched upon in this review.
It is a pleasure to thank Javier Gomez-Serrano, Benoit Pausader, Fabio Pusateri, Ian Tice and Claudio Cacciapuoti, Raffaele Carlone, Michele Correggi for the invitations to speak in their seminar series and to Javier, Benoit, Fabio and Ian for suggesting to write a review and encouragement, and Benoit, for many constructive remarks.
The author's research
is supported in part by NSERC Grant No. NA7901.
\section{Schr\"odinger equation}
Quantum systems of $n$ particles in the space $\mathbb{R}^3$ are described by the {\it Schr\"odinger equation}\footnote{The relation of solutions of the Schr\"odinger equation
to quantum observable effects is
a subject matter of Quantum Mechanics.}
\begin{equation*}i \partial_t \Psi_t=H_n \Psi_t, \tag{SE}\end{equation*}
where $\Psi_t=\Psi_t(x_1, \dots, x_n)$, a family of Sobolev functions of the particle coordinates $x_1, \dots, x_n$ in $\mathbb{R}^3$ and $H_n $ is an $n-$particle Schr\"odinger operator. For
$n$ particles of masses $m_1, \dots, m_n$ interacting via $2$-body potentials $v_{ij}$ and with external potentials $V_i$, $H_n $ has the form
\begin{equation}\label{Hn}H_n:=\sum_1^n h_{x_i} +\sum_{i<j} v_{ij}(x_i-x_j),\end{equation}
where $h_{x_i}=\frac{-1}{2m_{i}}\Delta_{x_i}+V_i(x_i)$ is a one-particle Schr\"odinger oprerator with an external potential $V_i(x)$ acting on the $i$-th particle variable $x_i$.
The function $\Psi_t$, called the wave function, gives probability distributions at time $t$ for various physical observables.
\medskip
One of the first results going back to J. von Neumann and elaborated
by B. Simon can be formulated as
\[\text{ global existence $\Longleftrightarrow$ self-adjointness of }H_n.\]
\medskip
By the Kato-Rellich theorem, the operator $H_n$ is self-adjoint for a fairly large class of potentials. Hence, the mathematical task,
as suggested by the underlying physics,
is to \[\text{ describe the space-time behaviour of solutions of (SE). }\]
\smallskip
\noindent The {\it main dichotomy} here can be formulated as
\[\text{ stability vs. collapse or disintegration}.\]
\medskip
By stability we mean the {\it localization in space} and periodicity in time. This can be further enlarged on
and reduced to spectral properties of $H_n$:
\medskip
\begin{itemize} \item stability w. r. to collapse ($\inf H_n
>-\infty$)
\item stability w. r. to break-up
($\inf H$ is an eigenvalue).\footnote{The condition $H_n\ge -C>-\infty$, which follows from a refined uncertainty relation, says that the particles are not sucked into attractive Coulomb singularities, contrary to the prediction of classical physics. The condition gap($\inf H$, rest)$>0$ implies that, for initial conditions close to the ground state (eigenfunctions corresponding to the lowest energy $\inf H$),
the solution to (SE)
remains concentrated in a bounded set in the configuration space $\mathbb{R}^{3n}$ (with the motion of the centre of mass factored out)
and that this property is stable w.r. to perturbations of the Hamiltonian.}
\end{itemize}
Thus, the stability explains the existence of structures of matter such as atoms, molecules,
..., stars.
One can refine the first type of stability as the extensivity of the energy, i.e. $H_n\ge -c n.$
The latter is known as the stability of matter.
\medskip
We say a system undergoes decay, or, more precisely, local decay, if the probability ($\int_Q|\Psi_t|^2$) that it occupies any bounded domain ($Q\subset \mathbb{R}^3$) in the physical space vanishes with time. The description of all possible scenarios of such an evolution is provided by the scattering theory.
\bigskip
\paragraph{\bf Scattering.} The main mathematical problem of the scattering theory -- the {\it asymptotic completeness} -- states that
\bigskip
As time progresses, a quantum system settles in a superposition of states
consisting of collections of stable freely moving fragments.
\begin{thm}[Asymptotic completeness]\label{thm:AC} Suppose that the pair potentials $v_{ij}(x_i-x_j)$ entering $H_n$ satisfy $v_{ij}(y)=O(|y|^{-\mu})$, as $|x|\rightarrow \infty$, with $\mu>\sqrt3-1$. Then the asymptotic completeness holds.
\end{thm}
\bigskip
\noindent {\it Open problem}: Prove the asymptotic completeness for $v_{ij}(y)=O(|y|^{-\mu})$, with $\mu\le \sqrt3-1$.
\section{Including photons (Nonrelativistic QED)}\label{sec:QED}
To describe the real (or at least visible) world, we have to couple the particles to {\it photons} (i.e to {\it quantized electromagnetic field}).
The dynamics of the resulting system is described again by
the Schr\"odinger equation,\footnote{This Schr\"odinger equation was used by P.A. Dirac and E. Fermi already in the early days of Quantum Mechanics, see Fermi's review \cite{Fermi}.}
\begin{equation}\label{SEqed}
i \partial_t\Psi_t =H_\kappa\Psi_t,
\end{equation}
but with the more complex Hamiltonian $H_\kappa$ acting on the state space
${\mathcal{H}}:=\mathcal{H}_p\otimes \mathcal{H}_f$, which is the tensor product of the spaces of particles and photons. This Hamiltonian is given by
\begin{equation}\label{Hqed}
H_\kappa = \sum_{j=1}^n \frac{1}{2 m_j} \big( - i \nabla_{x_j} - \kappa A_\xi ( x_j ) \big)^2 + U (x) + H_f .
\end{equation}
Here,
$\kappa$ is the
particle charge, $U(x)$, $x = ( x_1 , \ldots , x_n )$, is the
total potential effecting the particles,\\ \medskip
\qquad $A_\xi =\xi * A$ is the
UV-regularized, {\it quantized vector
potential} $ A$ and\\ \medskip
\qquad $H_{f}$ is the
photon Hamiltonian.\footnote{For the definitions of $A$ and $H_f$, see Appendix \ref{sec:QED-ham}.}
\medskip
The key phenomena one would like to describe are
emission and absorption of the electromagnetic radiation. Physical description of these processes translates into the following mathematical problems:
- the existence of the ground state
- the instability of excited states
- the emergence of resonances.
The ground and excited states are eigenstates of $H_\kappa$ with the smallest and remaining
eigenvalues (energies), respectively. The resonances are thought of physically as `metastable' states, or `bound states with finite life-times'. Mathematically, they correspond to complex poles of an analytic continuation of the resolvent of $H_{\kappa}$ across the continuous spectrum to the second Riemann sheet.
\footnote{The instability of excited states means that $H_\kappa$ has no eigenvalues in a small neighbourhood of the excited eigenvalues of $H_{\kappa=0}$. As $\kappa$ is `turned on', the real poles of the resolvent of $H_{\kappa=0}$ corresponding to the excited eigenvalues migrate to the second Riemann sheet.
To connect this instability to, say, the spontaneous emission of photons, one shows that the solution starting with the initial condition which is an excited eigenvector of $H_{\kappa=0}$, i.e. the tensor product of an excited eigenvector of the particle system and the photon vacuum,
describes the particle system descending
to the ground state, with the difference in energy carried out by a departing photon(s).}
Establishing the properties above and giving estimates of the renormalized
energies and life-times are the main tasks of mathematical theory of radiation.
\begin{thm}[Problem of radiation]\label{thm:rad} Assume $U(x)$ is a Kato-type potential and the Fermi Golden rule holds. Then for $\kappa>0$ sufficiently small,
(a) $H_\kappa$ has a (unique) ground state exponentially localized in the particle coordinates,
(b) $H_\kappa$ has no excited states outside of a sufficiently small neighbourhood of the continuous spectrum, and
(c) $H_\kappa$ has resonances of the multiplicities equal to those of the vanished eigenvalues and which converge to the latter as $\kappa\rightarrow 0$.
\end{thm}
The Fermi Golden rule expresses the effectiveness of the coupling of the particles to the electromagnetic field (that there is no accidental decoupling). The uniqueness of the ground state depends on symmetries present and could fail. Statements (b) and (c) express the instability of excited states and their turning to the resonances.
We discuss briefly difficulties arising in proving the theorem above. Since the photons are massless, they can be born out of the vacuum and absorbed back to the latter, as well as emitted and absorbed by the particle systems, in arbitrary large numbers and form dense fluctuating clouds around the particles. This leads to divergences
in the formal perturbation series for various physical quantities (e.g. the energies of the ground states and resonances), the phenomenon known as the {\it infrared problem}.\footnote{The photon clouds lead to renormalizations of various physical quantities (such as the mass of electron and energies of the ground state and resonances), which would diverge if the ultra-violet cut-off (UV) is removed.}
If we think of $H_\kappa$ as a perturbation of the Hamiltonian $H_{\kappa=0}$ of the decoupled system in which the particles and photons do not interact,
then we run into another, related, manifestation of the infrared problem:
the bound state energies of the Hamiltonian $H_{\kappa=0}$
are not isolated.\footnote{Indeed, the bound state energies of $H_{\kappa=0}$ are exactly the bound state energies, $E^{(k)}_{\rm part}$, of the particle Hamiltonian \[H_{\rm part} = \sum_{j=1}^n \frac{1}{2 m_j} \big( - i \Delta_{x_j} \big) + U (x).\] However, the values $E^{(k)}_{\rm part}+\lam, $ where $\lam\in \s(H_f)$ is an arbitrary photon energy,
also belong to the spectrum of $H_{\kappa=0}$. Since the photon energies fill in the semi-axis $[0, \infty)$, for each bound state energy $E^{(k)}_{\rm part}$,
there is a continuous spectrum branch $[E^{(k)}_{\rm part}, \infty)$ of $H_{\kappa=0}$.
}
The standard perturbation theory fails in treating such eigenvalues and one needs a {\it new} theory. The {\it spectral renormalization-group theory} developed to prove Theorem \ref{thm:rad}
is exactly the theory which deals with this problem.
The next problem is to show that, as we experience in everyday life, most of the photons born out by a particle system escape it at some characteristic time and travel freely unless they
encounter with another particle system. The mathematical formulation of this property leads to the problem of asymptotic completeness which we have already met in the previous section.
To formulate the result here, let $N_{\rm ph}$ be the quantum observable (self-adjoint operator) of the number of photons, $\Sigma:=\inf\s_{\rm ess}(H_{\rm part})$, the ionization threshold, and $\Psi_t$, a solution to \eqref{SEqed}. Then, we have
\begin{thm}[Rayleigh scattering]\label{thm:Rayl}
Assume that initially the energy is localized below the ionization threshold, $\Sigma,$ of the particle system and
that $\sup_t\lan \Psi_t, N_{\rm ph} \Psi_t \ran <\infty$
(satisfied in special cases). Then the asymptotic completeness holds.
\end{thm}
\bigskip
\noindent {\it Open problem}: Prove that $\sup_t\lan \Psi_t, N_{\rm ph} \Psi_t \ran <\infty$ for general particle systems (like atoms).
\section{Effective Equations} \label{sec:eff-eqs}
To describe systems of a large number of particles, say from $10$ to $10^{20}$, it is necessary
to design effective approaches giving some aggregate, or `collective', information.
Often, such approaches provide information which is practically impossible to extract from solving the original equations even if this was feasible. In the quantum many-body problem, the key effective approach is the Hartree-Fock approximation, trading the number of particles for the nonlinearity. This approximation, its natural extension and the effective equations it leads to are described
below.
\bigskip
\paragraph{\bf Hartree and Hartree-Fock equations and their extensions.} Consider a system of $n$ identical {\it bosons} or {\it fermions} whose evolution is described by
the Schr\"odinger equation \begin{equation*}i \partial_t \Psi_t=H_n \Psi_t, \tag{SE}\end{equation*}
where $H_n$ is the Schr\"odinger operator given in \eqref{Hn}, with $m_{i}=m$, $v_{ij}=v$ and $V_{j}=V$.
To obtain an effective, one-particle approximation for large $n$, we restrict (SE) to the {\it Hartree} and {\it Hartree-Fock} states given by symmetric and anti-symmetric products of one-particle wave functions:
\begin{equation}\otimes_1^n \psi\ \quad \text{ and }\ \quad \wedge_1^n \psi_i,\end{equation}
This leads to equations for $\psi$ and $\psi_1, \dots, \psi_n$ - the {\it Hartree} (H) and {\it Hartree-Fock} (HF) equations,
widely used in physics and chemistry.
\bigskip
However, these equations fail to account for quantum fluids i.e. quantum gases exhibiting some quantum behaviour at the macroscopic scale such as superconductivity, superfluidity and Bose-Einstein (BE) condensation.
For this, one needs {\it another conceptual step}.
\bigskip
\paragraph{\bf Non-Abelian random Gaussian fields and Wick states.}
In rigorous quantum statistical mechanics, the states are described by linear, positive (normalized) functionals, $\om$, on a $C^*$-algebra of observables, $\mathcal{A}$ and their evolution is given by the
von~Neumann-Landau equation
\begin{align}\label{vN-eq}
\partial_t\om_t(A) =- \om_t(i[H, A]) \,,\ \forall A\in \mathcal{A},
\end{align}
where $\om_t$ is the state at time $t$.
The simplest and in a sense a single most important class of (fixed-time) states, $\chi$, consists of quantum (non-Abelian) generalization of random Gaussian fields, i.e. states
uniquely determined by the {\it two-point correlations}\footnote{For a moment, we ignore the expectation $\chi (\psi(x))$.}:
\begin{equation} \label{psi*psi-correl}\
\chi (\psi^{*}(y ) \, \psi(x)), \end{equation}
where $\psi(x)$ and $\psi^{*}(x )$ are quantum fields adjoint to each other (the annihilation and creation operators).
These are exactly the {\it HF states}: If
$\chi_t$ is an evolving HF state, then the {\it operator $\g$ with the integral kernel} $\chi_t(\psi^{*}(y ) \, \psi(x))$
satisfies the {\it HF equation}
\begin{align}\label{HF-eq}
& i\partial_{t}\gamma =[h_{\gamma},\gamma],
\end{align} where $h_{\gamma}$ is the $\g$-dependent, one-particle Schr\"odinger operator,
\begin{align}\label{hgam}
& h_{\g} =-\Delta+V
+ v * \rho_\gamma - v^\sharp \g, \end{align}
with $\rho_\gamma(x, t):= \gamma(x, x, t)$ (=\ --charge density) and $v^\sharp \g $ the operator with the integral kernel $(v^\sharp \g)(x, y) = v(x - y)\g (x;y)$. The terms $v * \rho_\gamma$ and $v^\sharp\, \gamma$ are the direct (`electrostatic') and exchange (HF) self-interaction energies.
However, the above states are not the most general `quadratic' (or quantum Gaussian) states. The most general ones are defined by
{\it all} one- and two-point correlations\footnote{For more details see \cite{BBCFS} and Appendix E of \cite{CS1}.}
\begin{align}
\begin{split} &
\chi_t (\psi(x)), \quad \chi_t( \psi^{*}(y) \, \psi(x)) \quad \text{and}\ \quad
\chi_t(\psi(x) \, \psi(y)). \end{split} \end{align}
This type of states were introduced by Bardeen-Cooper-Schrieffer
for fermions and by Bogolubov, for bosons.
For such states all correlations are either $0$ or are f
sums of products of one- and two-point
ones. This is the {\it Wick} property from quantum field theory (QFT)
and consequently, we call such states the {\it Wick} states. (In mathematical literature, they are called the quasifree states.) They give the most general effective one-body description
to the $n-$body dynamics.
\medskip
\noindent {\bf Remark.} If we replace the exchange energy operator, $ex(\g):=v^\sharp \g$ in the operator \eqref{hgam} entering the Hartree-Fock equation \eqref{HF-eq}
by an multiplication operator by a local function $xc(\rho_\g)$ of $\rho_\g$
which models {\it exchange} and {\it correlation} energies, we arrive at the time-dependent Kohn-Sham equation, a key equation of the {\it density functional theory} (DFT).
\medskip
\paragraph{\bf Dynamics.}
Restricting the von~Neumann-Landau evolution \eqref{vN-eq} to Wick states
yields a system of coupled nonlinear PDE's for the functions
\begin{align} \label{phi-def}
&\phi (x, t) := \chi_t( \psi(x)),\\
\label{gam-def} &\gamma (x,y, t) := \chi_t( \psi^{*}(y) \, \psi(x)),\\
\label{al-def} &\al (x,y, t):= \chi_t( \psi(x) \, \psi(y)). \end{align}
These are the (time-dependent) \textit{Bogolubov-\-de Gennes} (fermions) and \textit{Hartree-\-Fock-\-Bogo\-lubov} (bosons) equations\footnote{For more details see Appendix \ref{sec:HFBsyst}.}.
\medskip
For the {\it Bose-Einstein condensation},
$\phi$ is the wave function of the {\it BE condensate} and $\gamma (x,y, t)$ and $\al (x,y, t)$
yield the {\it density matrix}
of the {\it non-condensed atoms} and the {\it `pair wave function'}
for the superfluid component.
For {\it fermions}, $\phi (x, t) := \chi} %{\varphi_t( \psi(x))=0$ and $\g (x,y, t)$ describes the normal electrons, while $\al (x,y, t)$ superconducting ones (more precisely, Cooper pairs of electrons).
In what follows, we associate with the functions $\gamma (x,y, t)$ and $\al (x,y, t)$, the operators $\gamma$ and $ \al$, whose integral kernels these functions are.
As usual for evolution equations, the first mathematical problem here is
\smallskip
{\it The well-posedness of the initial value problem.}
\smallskip
Physically, the first problem one would like to address is existence of a ground/Gibbs state and its symmetry.
\medskip
\paragraph{\bf Ground/Gibbs state and symmetry breaking.} It is easy to show that the energy, $E(\chi} %{\varphi_t):=\chi} %{\varphi_t(H)$, if it exists, is conserved. For systems with non-compact symmetries, like translations, the energy is infinite. In this case, one considers `local' or `renormalized' energy.
The notion of energy allows us to define the key state - the ground
state: a {\it static solution minimizing the (local/renormalized)
energy}.
\bigskip
\noindent {\it Key problems}: Existence and symmetry
of the ground state, the excitation spectrum and the dynamics nearby.
\medskip
The most important symmetry is the translational one which holds in the absence of external interactions (potentials).
Nothing is known about breaking of this symmetry in the ground states for both equations.
Whenever systems are considered for positive temperatures, the energy and the ground state are replaced by the free energy and the Gibbs equilibrium state.
\section{Hartree-\-Fock-\-Bogo\-lubov system}
We describe the full {\it Hartree-\-Fock-\-Bogo\-lubov (HFB) system} in Appendix \ref{sec:HFBsyst}. Here, we consider the reduced HFB system ($2-$gas model) resulting from neglecting the $\al$-component:
\begin{align}
i\partial_{t}\phi & = h\phi +v*(|\phi|^{2} +2\rho_{\gamma })\phi, \tag{GP}\\
i\partial_{t}\gamma & =[h_{\gamma , {\phi} },\gamma ], \tag{HF} \end{align}
where
$h=-\Delta+V$ is a one-particle Schr\"odinger operator, $\rho_\gamma(x, t):= \gamma (x;x, t)$, the one-particle density
and
\begin{align} \notag & h_{\gamma , \phi }:=h+v* (\rho_{\gamma }+ |\phi |^{2}).
\end{align}
These are coupled {\it Gross-Pitaevskii} and {\it Hartree} equations. The term $v* (\rho_{\gamma }+ |\phi |^{2})$ is the direct (`electrostatic') self-interaction produced by the combined charge density $\rho_{\gamma }+ |\phi |^{2}$ of non-condensed and condensed particles (atoms).
\medskip
In addition to the general problems formulated in the previous section, the following {\it problems} are of a special interest for the HFB system:
\medskip
- Bose-Einstein condensation,
- Collapse oscillations for $\lam<0$ ({\it correction to the Papanicolaou-Sulem-Sulem collapse law?}).
\medskip
Physically there are two important {\it set-ups} here:
\medskip
External (attractive or confined) potentials $V$ vs translational invariance ($V=0$).
\section{Bogolubov-de Gennes system}\label{sec:BdGeqs}
For {\it fermions}, $\phi (x, t) := \chi} %{\varphi_t( \psi(x))=0$ and, since the Wick states describe superconductors,
$(\g, \al)$ are coupled to the {\it electromagnetic field}. With the latter described by the {\it magnetic} potential $a$
and the {\it electrostatic potential},
$\varphi$, and, with the former taken in the Coulomb gauge ($\operatorname{div} a=0$) and the latter absorbed in the inter-particle (pair) interaction potential,
the equations for $\g, \al$ and
$a$ read\footnote{For a discussion of gauges, the origin
of the BdG equations (in particular, an elimination of the electric potential) and properties of $\g$ and $\al$ see Appendix \ref{sec:BdGsyst}.}
\begin{align} \label{BdG-gam}
& i\partial_{t}\gamma =[h_{a, \g},\gamma ]_- + [ v^\sharp \al, \al]_-,\\
\label{BdG-al} & i\partial_{t}\alpha = [h_{a, \g},\alpha ]_{+} - [ v^\sharp \al, \g ]_{+} + v^\sharp \al,\\
\label{Amp-Maxw-eq}
- &\partial_t^2 a = \operatorname{curl}^* \operatorname{curl} a + j(\g, a),
\end{align}
where $[A,B]_-=A B^* - B A^*$, $[A,B]_+=A\bar B^* + B\bar A^*$, with $\bar A:=\mathcal{C} A\mathcal{C}$, with $\mathcal{C}$ being the complex conjugation,
$v(x - y)$ is a pair potential,
$v^\sharp \al $ is, recall, the operator with the integral kernel $(v^\sharp \al)(x, y) = v(x - y)\al (x;y)$, $j(\g, a)(x) := [-i \nabla_a,\g]_+(x, x)$ is the
current density, and, finally (cf. Eq. \eqref{hgam}),
\begin{align}\label{hgam-a}
& h_{a, \g} =-\Delta_a+
v * \rho_\gamma- v^\sharp \g,
\end{align}
with $\Delta_a:=(\nabla + ia)^2$.
Here we have assumed, for simplicity, that the external potential is zero, $V=0$, and have chosen the unit electric charge to be $e=-1$.
\medskip
These are the celebrated {\it Bogolubov-de Gennes (BdG) system}.
They give the `mean-field' (BCS) theory of superconductivity.
Eq. \eqref{BdG-gam} is essentially the HF equation coupled to the other two equations and Eq. \eqref{Amp-Maxw-eq} comes from two Maxwell equations (Amp\`{e}re's and Faraday's laws).
\bigskip
As was pointed out in Section \ref{sec:eff-eqs}, the key problem here is the {\it existence and symmetry of the ground state and the description of the nearby dynamics}.
Experiments show that at the lowest energy (locally), states of quantum matter typically enjoy the maximal available symmetry. With this in mind, we begin with describing the symmetry group of the BdG system.
\bigskip
\paragraph{\bf Gauge (magnetic) translational invariance.}
Arguably, the simplest and most important symmetry is the translational one. In the magnetic fields this symmetry is broken. However, for constant magnetic fields, there is a non-abelian symmetry replacing it.
The BdG equations
are invariant under the $t$-independent
{\it gauge} transform
\begin{equation}\label{gauge-transf'}
T^{\rm gauge}_\chi : (\g, \al, a)\rightarrow (e^{i\chi }\g e^{-i\chi } , e^{i\chi } \al e^{i\chi }, a + \nabla\chi),
\end{equation}
where $\chi\in H^2(\mathbb{R}^d, \mathbb{R})$. (This defines a natural equivalence relation one should keep in mind.)
To preserve the Coulomb gauge, we could take $\chi$ {\it linear} in $x$.
Thus, we define the {\it gauge (magnetically) translationally (MT) invariant} states as states whose translations stay in the same gauge-equivalence class, i.e. which are invariant under the transformations
\begin{equation*}
T_{ s}: ( \g, \al, a) \rightarrow (T^{\rm gauge}_{\chi_s})^{-1}T^{\rm transl}_{ s} ( \g, \al, a),\end{equation*}
for any $s\in \mathbb{R}^d$, with $d=2, 3$, and for some (linear) functions $\chi_s(x)$. Here $T^{\rm transl}_{ s},\ s\in \mathbb{R}^d,$ is the group of translations.
We require $T_{ s}$ to be a projective group representation of $\mathbb{R}^d$. Then, the function $\chi_s(x)$ (of $x$ and $s$) satisfies the {\it co-cycle relation}
\begin{equation}\label{co-cycle'}
\chi_{s+t}(x) -\chi_s (x+t) - \chi_t(x) =\frac 12 b (s, t),
\end{equation}
$\forall s, t\in \mathbb{R}^d$, where $b(s, t)$ is a constant
two-form on $\mathbb{R}^d$.\footnote{In this case, $T_{ s}$ is, in general, a non-abelian group with the generators which are the components of the operator-vector $i(-i\nabla -\frac 12 b\wedge x)$.} Two-forms on $\mathbb{R}^d$, $d=2, 3$, are gauge equivalent to the 2-form $b\cdot (s\wedge t)$, where $b$ is a constant vector if $d=3$ and a scalar if $d=2$.\footnote{$b\cdot (t\wedge s)$ is the flux of the constant vector field $b$ through the area $t\wedge s$ spanned by the vectors $s$ and $t$.}
The function $\chi_s(x):= \frac 12 b\cdot (x\wedge s)$
clearly solves the equation \eqref{co-cycle'}. (This fixes a special - symmetric - gauge.)
The two-form $b(s, t)$, or vector/scalar $b$, is identified with a {\it constant external magnetic field}.
\bigskip
This extends the
translational symmetry to the $\operatorname{curl} a\neq 0$ regime.
For $\operatorname{curl} a\neq 0$, gauge-translationally invariant states yield the simplest solutions and candidates for the ground state for the BdG system.
\bigskip
\paragraph{\bf Ground state.}
As was alluded to above, typically, the ground state (GS) has the maximal symmetry. Hence,
depending on the magnetic field $b$, one expects that:
\medskip
GS is {\it translationally invariant} for $b= 0$,
\medskip
GS is {\it magnetically translationally invariant} for $b\ne 0$.
\medskip
\noindent Candidates for the ground (or equilibrium Gibbs)
state are:
\smallskip
\begin{enumerate}
\item {\it Normal states}: $(\g, \al, a)$, with $\al=0$ ($\Rightarrow \g$ is `Gibbs state').
\item {\it Superconducting states}: $(\g, \al, a)$, with $\al \ne 0$ and $a = 0$ (`Meissner states').
\end{enumerate}
\begin{thm}\label{HHSS}
For $b=0$,
there is a
superconducting, normal, {\rm translationally invariant} solution. \end{thm}
\begin{thm} \label{CS}
For $b\neq 0$,
the MT-invariance implies the normality $(\al=0)$.
\end{thm}
\begin{cor} For $b\neq 0$, the superconductivity ($\al \ne 0$) implies the symmetry breaking.\end{cor}
An addendum to the maximal symmetry paradigm formulated above could be stated as:
in the
ground state, whenever the maximal symmetry is broken, it is broken in the most minimal
way. For the simplest and most important
case of the translationally invariant systems, breaking of the translational invariance leads to formation of crystalline structures with lattice symmetries.
We are far from being able to prove
that this happens in reasonable models. However, the natural problem at one level below - the existence and stability of such structures - is possibly approachable.
Note that the stability would imply that a crystalline solution
is a local minimizer (but not necessarily minimizing locally) of a `renormalized' energy. Though it does not give the global minimizing property, one might say that it is the next best thing.
\medskip
\paragraph{\bf Vortex lattices.}
Physical experiments show (see below) that most of superconductors in magnetic fields have, in their lowest (locally) energy states, lattice symmetry, yielding minimal symmetry breaking as discussed above.
Here we define and discuss such states.
For $b\neq 0$, we define states, which we call
the vortex lattice states (or just {\it vortex lattices}), as
\medskip
\begin{itemize}
\item
{\it Vortex lattice}:
$(\g, \al, a)$ s.t. $T^{\rm transl}_{ s} (\g, \al, a) = T^{\rm gauge}_{\chi_s} (\g, \al, a)$,
$\forall s \in \mathcal{L}} %{\lambda}$ (some lattice in $\mathbb{R}^2$), with $\chi_s : \mathcal{L}} %{\lambda}\times\mathbb{R}^2 \to \mathbb{R}$, and
{$\alpha \not= 0$}.
\end{itemize}
\medskip
The fact that $T^{\rm transl}_{ s}$ is a group representation implies that
the map $\chi_s$ satisfies the {\it co-cycle relation} (see \eqref{co-cycle'}):
\begin{equation}\label{co-cycle}
\chi_{s+t}(x) -\chi_s (x+t) - \chi_t(x) \in 2\pi \mathbb{Z},\ \forall s, t\in \mathcal{L}} %{\lambda}.
\end{equation}
\noindent Co-cycle relation \eqref{co-cycle} implies that the {\it magnetic flux is quantized}:
$$\frac{1}{2\pi} \int_{\Om^{\mathcal{L}} %{\lambda}}} \operatorname{curl} a \in \mathbb{Z}.$$
Here $\Om^\mathcal{L}} %{\lambda}$ is a fundamental cell of $\mathcal{L}} %{\lambda}$ and the left-hand side is the 1st Chern number. The latter can be expressed directly in terms of $\chi$.
\bigskip
The existence result for such solutions is recorded in the following
\begin{thm} \label{thm:BdGExistence}For the BdG system without the self-interaction term:
\medskip
(i)
$\forall n, T>0$ and $\mathcal{L}} %{\lambda}$\
there is a static
solution $u_{T n \mathcal{L}} %{\lambda}}:=(\g, \al, a)$ satisfying
\begin{align}
& u_{T n \mathcal{L}} %{\lambda}}\ \text{ is $\mathcal{L}} %{\lambda}$-equivariant: }\ T^{\rm transl}_{ s} u_{T n \mathcal{L}} %{\lambda}} = T^{\rm gauge}_{\chi_s} u_{T n \mathcal{L}} %{\lambda} }, \forall s\in \mathcal{L}} %{\lambda},\\
&\text{the 1st Chern number is $n$: }\ \int_{\Om^{\mathcal{L}} %{\lambda}}}\operatorname{curl} a =2\pi n,\end{align}
\qquad $u_{n T \mathcal{L}} %{\lambda}}$ minimizes the {\it free energy}\footnote{For the definition of the free energy see Appendix \ref{sec:BdGsyst}.} $F_T=E-TS-\mu N$ on $\Om^\mathcal{L}} %{\lambda}$ for
$c_1=n$;
\bigskip
(ii) For the pair potential $v\le 0, v\not\equiv 0$ and $T$ and $b$ sufficiently small,
$u_{T n \mathcal{L}} %{\lambda}}$ is a {\it vortex lattice} (i.e. {$\alpha \not= 0$});
\bigskip
(iii) For $n>1$, there is a {\it finer lattice}, $\mathcal{L}} %{\lambda}' \supset \mathcal{L}} %{\lambda}$
for which $u_{T n \mathcal{L}} %{\lambda}}=u_{T 1 \mathcal{L}} %{\lambda}'}$, i.e.
$u_{T n \mathcal{L}} %{\lambda}}$ is $\mathcal{L}} %{\lambda}'$-equivariant with $c_1=1$.
\end{thm}
\paragraph{\it Open problem:} Show that for superconductors of Type II and for $n=1$, the vortex lattice solution is stable. (For discussion of Type I and II superconductors and the notion of stability in the context of the BdG, see \cite{CS1}. These notions as well as the notion of self-duality still need to be elucidated in the BdG theory.)
On the first step, one could address stability w.r. to symmetry preserving (periodic) perturbations is usually accessible. Showing that a crystalline solution is stable under general perturbations (deforming the lattice in various ways) is a rather subtle, highly non-trivial matter.
\bigskip
\section{Ginzburg-Landau equations}
In the leading approximation (close to the critical temperature and after {\it `integrating out'} $\g$ and the relative coordinate $x-y$ of $\al(x, y)$),
the {\it BdG system} leads to the {\it time-dependent Ginzburg-Landau equations}
\begin{equation*}
\bs
\g \partial_{t} \psi &= \Delta_a \psi + \kappa^2 (1-|\psi|^2) \psi, \\
\nu \partial_t a &= -\operatorname{curl}^2 a + \Im ( \bar{\psi} \nabla_a \psi ).
\end{split}
\end{equation*}
Here $\Re \g, \Re \nu \geq 0$ are constants arising in the approximation. In the application to superconductivity,\footnote{Besides describing equilibrium states of {\it superconductors} (mesoscopically), the static GLE describe also the (static) {\it $U(1)$ Yang-Mills-Higgs model} of particle physics (a part of {\it Weinberg-Salam model of electro-weak interactions}/a standard model). In particle physics, $\psi$ and $a$ are the Higgs and $U(1)$ gauge
(electro-magnetic) fields, respectively.}
\medskip
\qquad {$|\psi |^2$} is the density of
superconducting electrons; \\
\qquad $a : \mathbb{R}^{d+1} \to \mathbb{R}^d$ is the magnetic potential;\\
\qquad $\Im ( \bar{\psi} \nabla_a \psi )$ is the superconducting current.
\bigskip
\noindent The second GLE equation comes from two Maxwell equations (Amp\`{e}re's and Faraday's laws).
\bigskip
\paragraph{\bf Vortex lattices.}As with the BdG and HFB systems, the major open problem is the symmetry of the ground state. Experiments show that the ground state of a superconductor of Type II in a constant magnetic field is in the form of
the hexagonal vortex lattice (see Figure 1).
\begin{figure}[h]\label{fig:AL}
\begin{center}
\includegraphics[height=5.0cm]{magvortpicture.pdf}
\caption{Experimental picture of the Abrikosov lattice obtained using a tunnelling microscope. Different colours
signify different densities of super-conducting electrons.
Theoretical description of this experiment is given solving the BdG system based on the coarse-scale approximation given by the
Ginzburg-Landau system.}
\end{center}
\end{figure}
For the GLE, the vortex lattice is defined as a {\it static} solution equivariant under the lattice translations in the sense that
\[T^{\rm transl}_{ s} (\psi, a) = T^{\rm gauge}_{\chi_s} (\psi, a),\] for every
$ s \in \mathcal{L}} %{\lambda}$ (some lattice in $\mathbb{R}^2$),
with $\chi_s : \mathcal{L}} %{\lambda}\times\mathbb{R}^2 \to \mathbb{R}$ (satisfying \eqref{co-cycle}).
A vortex lattice solution is formed by {\it magnetic vortices} ({\it localized finite energy} solutions of a fixed degree, see Fig. 2), arranged in a (mesoscopic) lattice $\mathcal{L}} %{\lambda}$.
\begin{figure}[h]
\begin{center}
\qquad \qquad
\includegraphics[height=3.0cm]{linevort.pdf} \qquad \qquad \text{ }
\includegraphics[height=3.0cm]{vortex.jpg}
\caption{The left figure is a 2-dimensional
section of a picture of a line vortex center. The figure on the right shows density of the
super-conducting electrons $|\psi|^2=n_s$, the magnitude of the magnetic field and the circulation of the super-conducting current.}
\end{center}
\end{figure}
As for the BdG system,
we are far from being able to prove that the ground state of the GLE is given by a vortex lattice. One step below it is the problem of existence and
{\it stability} of vortex lattice solutions.
The {\it existence} of vortex lattice solutions is well known by now.
Thus, we are left with the problem:
\medskip
\begin{itemize}
\item Stability of vortex lattice solutions. \end{itemize}
\medskip
Since the
VL {\it are not localized}, the {\it stability is a delicate matter}.
\smallskip
\begin{thm} \label{thm:GLvortlattstab} The vortex lattices are stable under lattice-periodic and local
perturbations of the same parity as the vortex lattice.\end{thm}
\bigskip
\noindent {\it Open problem}: Show stability/instability under more general lattice deformations.
\section{Summary}
\begin{itemize}
\item We reviewed some basic properties of the {\it Schr\"odinger equation} which encodes {\it all the information} about quantum systems\footnote{Extraction of the physical information from the solutions is done according to a fixed quantum mechanical procedure.
The theoretical results agree to remarkable precision with experimental ones, while the physical theory connecting the two, initiated by J. von Neumann soon after advent of Quantum Mechanics, remains largely under construction.}.
\bigskip
\item While we learned much about
the general structure\footnote{Such as the structure of the spectrum, scattering theory, theory of resonances.} of this equation, our understanding
of {\it specific quantum systems} with the number of particles $\ge 3$ is spotty and the progress, with some notable exceptions like the stability of matter and the derivation of mean-field type equations, is very slow.
\bigskip
\item However, there is one important direction where
robust progress is possible -- {\it `effective'
equations} for quantum systems of large number of identical particles (quantum gases),
especially, those describing quantum fluids (i.e. quantum gases exhibiting some quantum behaviour at the macroscopic scale) such as
{\it superconductors, superfluids and Bose-Einstein condensates -- the HFB and BdG equations}.
\medskip
\bigskip
\item We (a) formulated
some key mathematical problems related to the HFB and BdG equations,
(b) described the key stationary solutions of BdG equations, the competitors for the ground/Gibbs state: {\it normal, superconducting and mixed (or intermediate) states}, and
(c) presented an important class of the mixed states -- the {\it vortex lattices} -- demonstrating the symmetry breaking.
\end{itemize}
\medskip
\section{Remarks on literature}\label{sec:liter}
{\it $n-$particle scattering}. Theorem \ref{thm:AC} is due to {A.~Soffer and I.M.~Sigal ($\mu>1$), and J.~Derezi\'nski ($\sqrt3-1< \mu < 1$)} (\cite{Der}, see also \cite{SigSof1}). The proofs use important earlier results and ideas
of P.~Deift and B.~Simon, V.~Enss, C.~G\'erard, G.M.~Graf,
E.~Mourre, D.~Yafaev (\cite{DS, Enss, FrHe, GGM1, Ger1, Gra, SigSof3, SigSof4, Skib1, Skib2, Tam, Va1, Va2, Yaf}), see \cite{DerGer1, GerLa, HunSig2} for books and a review.
\bigskip
For some of the extensions, see \cite{APSS, HeMoSkib, HeSk, Skib1, Skib2, Skib4, SigSof}).
Many ideas and techniques from this field
successfully entered
Quantum Electrodynamics (QED), see below,
nonlinear evolution equations (\cite{BlSof1, BlSof2, Cucc, CuccMa, DonSchSof1, DonSchSof2, LindSof2, LindSof3, LindSof3, RodSchSof, SchSofSt, Sof}), wave propagation (\cite{DeBievHisSig, DeBievPra1, DeBievPra2}) and
the energy transfer in the non-autonomous Schr\"odinger equations
(\cite{Bamb, BambGrMa, BambGrMaRob, BambLanMon, BerMas, Mas1, Mas2, MasRob, Mon}).
For an important parallel development see \cite{Va1, Va2}.
\bigskip
{\it NR QED,
Radiation}.
Theorem \ref{thm:rad} was proven by V.~Bach, J.~Fr\"ohlich and I.M.~Sigal (\cite{BFS1, BFS2, BFS3}). Many important extensions and improvements were obtained in
\cite{AlFau, AlFauGui,
ArHirHiros, BFP, BarFauGui, BalFauFrSchu, CaHai, CorrFalOlRoug, DFP1, DFP3, FGSig1, GrHas, GGM2, GuMaMo, Ger1, GrHas, Hai, HaiHirSp, HaiSei1, HaiVouVu, HaHe1, HaHe2, HaHe3, HaHe5, HaSie, Hiro1, Hiro2, LiLo1, LiLo2, Mo3, TeWa}, to mention some of more recent results, see e.g \cite{Ar2, Gr2, Sig3, Sig5, GS, Sp4} for reviews and book presentations.
The main ingredient in the proof of Theorem \ref{thm:rad}
is the spectral renormalization group introduced in \cite{BFS1, BFS2}.
A different approach was developed by V.~Bach, M.~Balestros, J.~Fr\"ohlich, A.~Pizzo, which the authors call the multiscale or Pizzo method (\cite{BachBalInPi, BachBalKoMen, BachBalMen, BachBalPi}).
The infrared (IR) problem was addressed by Th.~Chen, J.~Fr\"ohlich
and A.~Pizzo (\cite{ChF, ChFP1, ChFP2}).
For the renormalization of the electron mass see \cite{Ch, BCFS}.
Ideas and techniques from the NR QED were extended to the Nelson and spin-boson models of the condensed matter physics\footnote{These models have a form similar to the NR QED, with photons replaced by (acoustic) phonons (quantized (longitudinal) oscillations of the underlying medium) and the interaction somewhat modified. The main problems for these models are exactly the same as those formulated in Section \ref{sec:QED}. The main difference is that these models do not have the gauge invariance which allows to lower the IR singularity in the QED Hamiltonian.} in
\cite{BaDecPiz, BalDeckHan, BalDeckHan2, BalDeckFauHan, Mo2, GeoRas, DamMo, HaHe4, HaHinSie, LorMinSp, Skib3, Sp1, Sp3}.
\bigskip
{\it NR QED, Asymptotic completeness}. Theorem \ref{thm:Rayl} is due to J.~Faupin and I.M.~Sigal and W.~De Roeck, M.~Griesemer and A.~Kupiainen (\cite{FaupSig, DeRoGriKu}).
These works used ideas and results of M.~H\"ubner-H.~Spohn, J.~Derezi\'nski-C.~G\'erard, J.~Fr\"ohlich-M.~Griesemer-B.~Schlein (\cite{DerGer2, FrGrSchl1, FrGrSchl2, FrGrSchl3, FrGrSchl4, HuSp, Sp2}) as well as those from the $n$-particle scattering theory discussed above. For further developments
\cite{DybMo, DybPi1, DybPi2, DybPi3, BFP, GrZ} and review \cite{Sig5}.
The finiteness of mean number of photons for the spin-boson model was proven by W.~De~Roeck and A.~Kupiainen (\cite{DeRoKu})
\bigskip
{\it The HFB and BdG systems.} Stationary versions of these systems (written in eigenfunction expansion representations) are used extensively in the physics literature.
For the HFB system, one inserts the delta-function potentials and in the BdG case one sets $a=0$.
The full, time-dependent systems in the general form as they appear in this paper were written out and formally derived in \cite{BBCFS} and \cite{CS1, BenSokSolov}, respectively.
Clearly, the HFB and BdG systems generalize the Hartree and Hartree-Fock equations. For the relation between the Hartree and Hartree-Fock approximations, on one hand, and quasi-free (Wick) states, on the other, see \cite{BenPorSchl} and Appendix E of \cite{CS1}.
For a rigorous derivation of effective equations similar to the HFB and BdG systems, see the books \cite{LSSY, BenPorSchl}, reviews \cite{Lew2, Nap, Roug} and some recent
articles \cite{BenNamPorSchlSeir2, CorrLundRoug1, ChrHaiNam, DecFrPicPiz1, DecFrPicPiz2, DeuSeir, GrillMach2, GrillMach3, NamNap}.
The rigorous theory started (from very different perspectives) with works of K. Hepp, E. H. Lieb and B. Simon and P.-L. Lions \cite{He, LS1, LS2, Lio} (see also \cite{GinVelo1, GinVelo2} for early follow up work).
For the {\it HFB system, the static, homogeneous case} (i.e. $V=0$ and $\g$ and $\s$ are translation-invariant)
was treated rigorously by
M.~Napi{\'o}rkowski, R.~Reuvers and J.P.~Solovej, see \cite{NapReuvSolov1, NapReuvSolov2}.
In the {\it general case}, general properties and the well-posedness were established in \cite{BBCFS}.
\medskip
The {\it BdG system without the (dynamic) electromagnetic field, i.e. with} $a=0$. See \cite{HaiSei2} for an excellent review.
Theorem \ref{HHSS} was proven by Ch.~Hainzl, E. Hamza, R.~Seiringer and J.P.~Solovej (\cite{HHSS}). The Cauchy problem was investigated by N.~Benedikter, J.~Sok and J.P.~Solovej (\cite{BenSokSolov}).
For the {\it full BdG system}, general properties of the system and classification and properties of the ground states (and more generally, static solutions) were established by I. Chenn and I.M. Sigal (\cite{CS1}). In particular, Theorem \ref{CS} was proven in \cite{CS1}. An asymptotic behaviour of critical temperature in weak magnetic fields was established in \cite{FrHaiLang, DeuHaiScha}.
For $\al=0$, the BdG system leads to the Hartree-Fock equation coupled to the Maxwell equations. Closely related to the Hartree-Fock equation is the important and widely used {\it Kohn-Sham equation}, the main tool in the {\it density functional theory (DFT)}, see \cite{AnCan, CanM, CLS, CS0, ELu2, ELu3, ELu, ELY, PrNord, PusS}
and references therein for the former and \cite{Lew3, LewinLiebSeir, LewLiebSeir2} for the latter.
In a remarkable work, R.~Frank, Ch.~Hainzl, R.~Seiringer, J.P.~Solovej (\cite{FHSS}) have shown that, for non-dynamical magnetic fields, the nanoscopic approximation of the BdG system is given by the Ginzburg-Landau one (see also \cite{FHSS2, FHSS3, CFHS}).
\medskip
{\it Vortex lattices}. {Existence of vortex lattices for the BdG equations} was proven by I.~Chenn and I.M.~Sigal (\cite{CS1}).
\smallskip
For the existence results for the {\it Ginzburg-Landau system},
see review \cite{Sig2}.
Stability of vortex lattices for the Ginzburg-Landau system under lattice-periodic and local perturbations was proven by
I. M.~Sigal and T.~Tzaneteas (\cite{ST1, ST2}).
\medskip
Important
results on asymptotic behaviour of solutions, for $\kappa \to\infty$ and applied magnetic fields, $h$, satisfying $h\le \frac{1}{2}\ln \kappa$+const (the London limit), were obtained in \cite{ASand}.
Further extensions to the Ginzburg-Landau equations for anisotropic and high temperature superconductors in the $\kappa \to\infty$ regime can be found in \cite{ABS1, ABS2}.
For
the $\kappa\rightarrow \infty$ (the semi-classical)
regime in the Ginzburg-Landau equations
and the linear eigenvalue problem related
to the second critical magnetic field, see the books
\cite{SS}\footnote{For the precursor of the development described in this book, see \cite{BBH}.} and \cite{FournHelffer}, respectively, and see \cite{AftSe, AlHe, AlHePa, CorrRoug1, CorrRoug2, CorrRoug3, FouHe1, FouHe2, FouHePe, FournKach1, FournKach2}, for some additional and more recent results.
\medskip
There are many similarities between the Ginzburg-Landau and {\it Gross-Pitaevskii} equations and more specifically between the phenomena of superconductivity and the Bose-Einstein condensations (BEC) described by these equations, respectively. The key results for the BEC
vortices in the Gross-Petaevski equation are due to A.~Aftalion, R.~Jerrard, M.~Correggi, N. Rougerie, J.~Yngvasson et al,
see the book \cite{Aft} and the more recent papers \cite{AftBlJe, AftMaWe, AftNoSou, AftSa, AftSou, CorrRougYng, CorrDubLundRoug}.
For recent work on vortices in the somewhat similar {\it Landau-Lifshitz}-type equations and the Weinberg-Salam model of electro-weak interactions (the $U(2)$ Yang-Mills-Higgs system), see \cite{GuWa, LiMe, GarS}.
The HFB, BdG, GL and GP equations are obtained from the original many-body Schr\"odinger equation by `integrating out' some degrees of freedom corresponding to finer length scales (the GL and GP equations) or faster dynamics (the HFB and BdG equations).
Such equations are called the effective equations and the dynamics described by them, the effective dynamics. One can continue further integrating out degrees of freedom
to obtain even coarser (say, nanoscopic) effective equations, see e.g. \cite{BrJer, CollJer, CLaLe, CCoSc, CSc, CorrRougYng, CorrDubLundRoug, DFP2, ELu2, ELu3, ELu, ELY, FeffWein3, FeffLThWein2, FeffLThWein3, FrGJS, GangS1, GangS2, Jer, JerSm, JFrGS, Holm, HMZw, HPZw, HZw, HZw2, WaLuWein}.
For effective equations approach to other quantum-mechanical problems, see \cite{Lew1, Lew4, Li2, Li4, SimK, Te} for reviews and books and \cite{AnaLew, AnaSig, BrMa, FrKnSchSo, FrLem, FrNamVan1, FrNamVan2, LiSei, MaSo, Nier, Pan, PST1, PST2},
for some recent work.
Of important aspects of the quantum many-body problem lying further a field
and not discussed above,
we mention the Coulomb systems, positive temperatures, topological techniques,
random Schr\"odinger operators
and perturbation theory. For some general reviews and books in these areas, see \cite{AtJoPil, Avr, AFG, CFKS, DiKiKleKlo, DSS, Feff1, Feff2, Fr3, Fr4, GJ, His, JP, Kir, LeBL, Li1, Li3, LiSei, SimH, SimK}.
| -28,546.871818 |
[
-0.8271484375,
1.095703125
] | 60.814743 |
[
-2.826171875,
-0.2193603515625,
-2.298828125,
-5.265625,
0.0416259765625,
7.7421875
] |
[
3.958984375,
6.1953125,
1.8330078125,
6.79296875
] | 401 | 6,459 |
[
-3.501953125,
4.06640625
] | 28.822197 |
[
-5.6328125,
-3.931640625,
-4.70703125,
-2.4765625,
1.8037109375,
12.5703125
] | 1.057112 | 38.94407 | 28.456417 | 0.645257 |
[
1.564988613128662
] | -18,347.53207 | 5.492181 | -28,054.061369 | 0.591983 | 6.212667 |
[
-2.4765625,
-3.51953125,
-3.294921875,
-4.609375,
2.107421875,
11.7265625
] |
[
-5.6171875,
-1.9794921875,
-1.8203125,
-0.72900390625,
3.17578125,
3.677734375
] | |
BkiUd0c5qWTBLNsCq9jW
|
\section{KMOS Overview}
\label{sec:overview}
KMOS\cite{sha10a,sha10b} is a fully cryogenic seeing limited multi-object integral field spectrometer,
which is being built by a consortium of German and British institutes
as one of the second generation VLT instruments.
It is equipped with 24 integral field units, each with a
2.8\arcsec$\times$2.8\arcsec\ field of view, that can be deployed
on-the-fly using robotic arms anywhere within a 7\arcmin\ patrol
field.
The number and size of the IFUs, and the spectral resolution, have
been chosen according to the key science driver, to study the morphology and kinematics of emission lines in high redshift galaxies.
Each of the 3 identical segments in the instrument comprises 8
pick-off arms (including K-mirrors to keep the fields oriented
correctly on sky), filter wheels, the integral field unit (that
uses mirrors to slice the image and re-arrange it along a
pseudo-slit), and a spectrograph with a 2k$\times$2k HAWAII-2RG detector.
A number of gratings are available, allowing observations across
various bands from 0.8--2.5\micron\ at a resolution of $R\sim3500$ in
order to enable science observations to probe between the bright
atmospheric OH lines.
An additional grating enables observation of the H and K bands
simultaneously at a lower resolution.
Since the pixel scale is nominally 0.2\arcsec, a single exposure with
KMOS generates 336 slit spectra, each about 2000 pixels in the
spectral direction and 14 pixels wide spatially.
Each slit spectrum is separated from its neighbours with a gap of a
few blank pixels.
Although there are a large number of spectra, the absolute data volume
is not excessive: in a typical night the instrument will produce only
$\sim$5\,Gbyte of raw data (although it could be up to $\sim$50\,Gbyte in
the extreme case of short exposures).
The pipeline modules (recipes) are written in C using the ESO
Common Pipeline Library.
They can be executed using the standard ESO pipeline tools, either in the
command line with {\em EsoRex} or with the {\em Gasgano} GUI.
KMOS is currently in its manufacturing phase, and has already had its
technical first light\cite{sha10a,sha10b}.
The preliminary acceptance in Europe and first light on sky at the
VLT in Chile are both expected during 2011.
Further details on specific aspects of the instrument, including
hardware, software, control, and testing, can be found elsewhere in
these proceedings.
\section{Instrument Configuration and Observing Modes}
KMOS itself is opto-mechanically a very complex instrument that slices
images from 24 deployable pick-off arms in order to perform integral
field spectroscopy (also known as hyperspectral imaging).
However, from an observer's perspective, configuring and using the
instrument is rather straightforward.
Unlike many other integral field spectrometers, there is no choice of
pixel scale.
Neither is it necessary to adjust the centre of the spectral bandpass, or
optimise the slit width against resolution as can be the case for
longslit spectrometers.
And in contrast to most `workhorse' instruments, there are no
additional options such as imaging or polarimetry.
The only complex operation, allocating the 24 arms to specific science
targets in the patrol field, is greatly assisted by an automated tool
KARMA\cite{weg08} that optimizes the assignment taking into account
optical and mechanical constraints, as well as target priorities.
This simplicity in terms of configuration has a direct impact on the
data processing, since each set of frames can be processed in
essentially the same way.
However, there are more subtle features, related to the observing
modes\cite{weg08}, that need to be considered.
One of the observing modes that is expected to be used frequently is
`nod to sky'.
Here, 24 targets are observed in every pair of pointings, with most of
the arms on targets at the first position, and the remainder on
targets during the second exposure after a nod.
In this case, the sky background is removed simply by subtracting
alternate exposures for each arm.
But this means it is not always possible to classify an entire
exposure as `sky' or `object' since some arms may be on sky and others
on targets.
Consequently, classifications are made separately for each arm and
the data for each IFU are processed independently.
With the `stare' mode, the arms are on the same science target for
multiple sequential exposures.
For this mode, the observer has the option of occasionally observing a
sky pointing -- in which case that can be used to remove the background.
Alternatively, if the target is compact enough that it
can be dithered within the IFU's field of view, the background can be
recovered from sky pixels around the science target.
Another option, potentially one of the most efficient and effective
uses of the `stare' mode, would be to have a single dedicated sky arm, so
that the other 23 could always be on targets.
Ideally, one would subtract this sky exposure from all the other arms
-- with the advantage that it has monitored the background
(particularly the highly variable atmospheric OH lines) {\em during} the
science exposure and so provides the best background measurement.
However, there are two issues that mean this method can only be applied
in a rather restricted sense:
(i) the effect of spatial and spectral curvature means that one must match
corresponding arms in each segment (e.g. arms 1, 9, and 17), since only
then do the detailed spectral profiles of the OH lines match exactly;
(ii) the IFUs have differing orientations on sky so that, for example,
the data for IFU\,9 would need to be rotated by 180$^\circ$, and that
for IFU\,17 flipped, before they can subtracted from IFU\,1.
The third mode is `mapping' in which the arms are positioned as closely
as possible and, with a sequence of 9 or 16 pointings, are used to map a
large (arcmin scale) contiguous field.
This imposes a requirement that the data processing software be able to
handle a large number of cubes: 24 IFUs and several exposures at each
of 16 pointings leads to in excess of 1000 data cubes that must
eventually be combined.
\section{Calibration}
Various calibrations are required to process KMOS data.
The way in which they are associated, the input data
required, and the output produced, are summarised in
Fig.~\ref{fig:cascade}.
The specific recipes or modules in this calibration cascade are
described below.
In order to make sure that the various calculations and fits performed
are robust, we employ a routine that flags and rejects deviant
values very efficiently, even when there are only 4--5 values in a
sample.
The key aspect is to begin with an estimate of the standard
deviation based on a percentile clipping, and then improve on this
with a few iterations.
\begin{figure}
\begin{center}
\includegraphics[height=5.5cm]{KMOS_calibcascade.eps}
\end{center}
\caption{ \label{fig:cascade}
Calibration Cascade showing how the various calibration recipes are
associated, what input data they require (both from the instrument and
externally), and what calibrations they produce.
}
\end{figure}
\subsection{Calibration Recipes}
\begin{description}
\item{\em kmo\_dark} combines several raw dark frames, and in the
process makes a first pass at identifying bad pixels. If the
exposure time is very short, it also yields the read noise. And for
very long exposures it provides a measure of the dark current.\vspace{-1mm}
\item{\em kmo\_flat} has two functions.
The first is to combine several `lamp on' and `lamp off' frames to make a
flatfield, in the process identifying additional bad pixels.
For KMOS, a bad pixel is defined as one that should be
ignored (for any reason) during the cube reconstruction.
As such, it also includes pixels that are simply not illuminated.
The second function is to trace the edges of each
slitlet in each IFU, and hence map the spatial curvature.
This is a key process, and is implemented in a robust way with no
prior assumptions about the location of the slitlet edges.
As such, it works even if several IFUs are out of action.
The routine creates a pair of calibration frames called XCAL (see
Fig.~\ref{fig:xcal}) and YCAL in which the value of pixel denotes its
spatial location with respect to the centre of its IFU along the
right ascension and declination axes respectively.
Each pixel is also tagged with the identification of the IFU to which it
belongs.\vspace{-1mm}
\item{\em kmo\_wave\_cal} calculates the wavelength of each pixel,
treating each slitlet completely independently.
The initial identification of arc lines within a slitlet is done by
comparing every possible pair of line separations with those from a
linelist. With only the approximate scaling as a prior assumption,
this is a robust method that yields an unambiguous match between the
data and line list.
From this point, the lines are traced across each slitlet and hence
the wavelength at every pixel can be interpolated.
These are written into a frame called LCAL.\vspace{-1mm}
\item{\em kmo\_illuminate} is a spatial sky flat.
It measures variations in the illumination across the field of
each IFU in order to correct for vignetting, either at the edge of the
patrol field or if an arm is in the shadow of another.\vspace{-1mm}
\item{\em kmo\_std\_star} is a standard recipe that both derives the
atmospheric and instrument transmission as a function of wavelength, and also
calculates the scaling factor for flux calibration.
The crucial issue here is to remove the spectral features.
This is assisted by temporarily dividing out a model atmosphere.
One then divides out a solar spectrum that has been
convolved to the same resolution (or, for AB stars observed in the
K-band, interpolate across Br$\gamma$).
Afterwards, the model atmosphere is multiplied back in.
A look-up table enables the code to estimate the stellar temperature
and remove the intrinsic spectral slope.
\end{description}
\begin{figure}
\begin{center}
\includegraphics[width=15cm]{KMOS_xcal.eps}
\end{center}
\caption{ \label{fig:xcal}
The calibration frames for XCAL in which the value of each
pixel indicates the spatial distance along the X-axis (right
ascension) of the IFU. The numbers along the top indicate to which IFU
each section of data belongs.
Each IFU is arranged in 14 separate 2D slit spectra.
For the first two detectors the IFUs are oriented vertically on sky;
so each slitlet has a single value and values increase/decrease
between slitlets (left inset).
The 3rd detector appears different because the orientation of the IFUs
is 90$^\circ$ offset and so values increase across each slitlet
(right inset).
}
\end{figure}
\subsection{Quality Control}
Monitoring the consistency and trends in the quality control (QC)
parameters provides
information about the status and health of the instrument.
These parameters show that the instrument is functioning
correctly and that data can be calibrated properly (but do not
necessarily reflect the quality of the data itself).
Rather than list them all explicitly, a summary of the main
classes, the recipe that generates them, and their purpose is given:
\begin{tabular}{p{2.2cm}p{4.4cm}p{8.9cm}}
{\em kmo\_dark} &
Bias, readnoise, dark current, number of bad pixels &
Indicates general status of detector and readout electronics \\
{\em kmo\_flat} &
Lamp efficiency, saturation, signal-to-noise &
Determines whether brightness of lamp is changing, if it is too
bright, and if there were enough exposures \\
&
Mean \& rms for coefficients of fits to slitlet edges &
Reflects stability of spectra on detector (e.g. 0$^{th}$ \& 1$^{st}$
order coefficients indicate location \& tilt) \\
{\em kmo\_wave\_cal} &
Ar \& Ne lamp efficiencies, saturation, signal-to-noise &
Determines whether brightness of lamps is changing, if they are too
bright, and if there were enough exposures \\
&
Mean \& rms for coefficients of dispersion solution &
Reflects stability of spectra on detector (e.g. 0$^{th}$ order
coefficients indicate location) \\
&
Spectral resolution &
Useful for science; a change may indicate intrument is no longer
focussed \\
{\em kmo\_illuminate} &
Inter- and intra-IFU uniformity &
Indicates impact of, and changes in, vignetting \\
{\em kmo\_std\_star} &
Zeropoint, spatial resolution &
Indicates whether instrument throughput has changed \\
\end{tabular}
\section{Data Processing}
\label{sec:dataproc}
Generating fully reduced cubes from the raw KMOS data is an intricate
processes that has to fulfil a number of roles.
For KMOS, the software has been designed using the experience gained
at MPE over more than a decade from both 3D\cite{wei96} and
SINFONI\cite{eis03,bon04,abu06}. It incorporates the perspective of
astronomers as well as programmers.
\subsection{General Requirements}
The detailed processing steps may vary considerably between different
observations, depending on the targets and the science that is being
addressed.
The software must be flexible enough to provide a high quality
reduction for all possible science cases without overwhelming the
astronomer with endless parameters.
For KMOS, this is achieved by making the pipeline modular.
In practice, the pipeline can be run either in a monolithic way (so
that the appropriate modules are automatically executed sequentially)
or controlled on the command line, perhaps using a script.
In the latter case, users can stop at any point, execute their own
code, and then continue.
A high fidelity `science grade' reduction may take some time to
process.
However, the pipeline must also provide rapid
processing for the real-time display.
This is achieved in several ways.
Unncessary steps can be omitted -- for example flatfielding or
telluric correction may not be needed to reconstruct an image of the
target that is good enough for acquisition purposes.
The speed of the reconstruction depends on the algorithm used.
The fastest and simplest, `nearest neighbour', is sufficient for
acquisition.
The most time consuming step is generating the neighbourhood map,
which is the list of pixels on the detector that are closest to each
pixel in the reconstructed cube (see Section~\ref{sec:recon}).
In the context here, re-using the
neighbourhood map can make the reconstruction much faster, providing
the 24 images for the real-time display within a few seconds.
Error propagation, to yield an estimate of the uncertainty in the
final reduced cube, is performed in a fairly simple manner.
An initial estimate of the error is made for each raw frame using the
counts in each pixel, together with the known read noise and gain.
This is then propagated in a standard way (but ignoring cross terms when
adding or multiplying frames).
The same method can also be applied for most of the interpolation
schemes outlined in Section~\ref{sec:recon}, in which the output value
is just a weighted sum of input values.
It is in the combining stage that the situation becomes more complex.
Here, systematics mean it is likely that combining the input noise
estimates will underestimate the true noise.
Instead, if there are enough values, one can derive a better estimate
of the noise directly from their variance.
This requires that each input value has the same weight, which is a
reasonable assumption for near-infrared data taken with the same
exposure times.
Estimating the noise for integrated spectra is also non-trivial, because
of the correlations between pixels.
This has been analysed in some detail and it has been
shown\cite{for09} that the noise does not decrease as one might
naively expect; and that a scaling factor, which depends on the
aperture size, must be applied.
The data format itself is an additional complexity for the user,
because of the need to handle, and associate in a single file, either
data from 3 detectors or datacubes for 24 arms.
For this reason, each KMOS file can have multiple extensions, as
illustrated in Fig.~\ref{fig:dataformat}.
For the detector based format (e.g. raw data, dark frames,
flatfields), there will be either three or six extensions depending on
whether noise frames are included;
and each extension will have 2D data.
For data products (e.g. cubes, collapsed images, extracted spectra),
there can be up to 48 extensions; and the data in each extension can
have 1, 2, or 3 dimensions.
For this reason, there is a keyword in each extension that indicates
what data is stored there.
It also means that it is no longer straightforward for astronomers to
work on the data without modifying their tools to deal with these data
formats.
\begin{figure}
\begin{center}
\includegraphics[height=5cm]{KMOS_dataformat.eps}
\end{center}
\caption{ \label{fig:dataformat}
Overview of the KMOS data format. A keyword EXTNAME in the extension indicates
what data is there. Example string values for this keyword are shown
in the figure.
Far left: initially there are 3
raw frames based on the detector format (i.e. 2k$\times$2k) stored in 3
extensions.
Centre left: once the associated noise is calculated, the additional
frames are interleaved to make 6 extensions.
Centre right: The reconstructed cubes (or collapsed images or extracted
spectra) are each stored in a separate extension.
Far right: interleaving noise cubes yields up to 48 extensions.
}
\end{figure}
Simulations are a key aspect of testing the data processing.
In order to make the recipes as robust as possible, we have tested
them using two different sets of simulations.
The first set aims to test the functionality using data that is
representative of KMOS.
These data are generated directly in the 2D detector based format.
As such, it is quick and easy to change anything, for example the
curvature or dispersion.
These simulations are useful for testing different scenarios, and for
developing the recipes since the processing procedures cannot rely on
specific {\em a priori} knowledge about the data format.
The second set of simulations aim to be much more realistic and
quantitatively resemble KMOS.
Generating them takes time since a 3D source is mapped onto the 2D
detector using the known instrumental properties.
These are important for testing the validity of the reconstruction.
Requiring that the pipeline recipes can process both sets of
simulations leads to robustness and is a good indication that it will
be able to deal with the real KMOS data.
\subsection{Flexure}
Flexure is a potential issue for KMOS since the
instrument itself rotates to maintain the field orientation, and so
the gravity vector is constantly changing.
The impact of flexure is simply to shift the location on the
detector (along both spectral and spatial axes) at which data is
recorded.
Preliminary estimates, based on models and end-to-end tests, indicate that
within an individual frame, it should not be a problem.
But across a long sequence of frames, in the worst case the shift may
both be as much as about 0.5\,pixel.
At this level, it must be corrected.
Fortunately, along both spectral and spatial axes flexure can be
tracked using the OH emission lines.
As we show below, both spatial and spectral flexure can be fully compensated using
2-pass processing without compromising the data quality.
This is reflected in the workflow given in Fig.~\ref{fig:workflow2}.
Uncorrected spectral flexure will result in poor background
subtraction and also broadening of emission lines in the science target.
It can be measured most easily in a reconstructed cube
by comparing the wavelength of several spatially integrated OH lines to a
reference.
A method to do exactly this\cite{dav07} is frequently used on SINFONI data.
With KMOS, we have the advantage that the offset can be fed back into
the calibration data and the cube reconstructed again, thus avoiding
additional interpolations.
Uncorrected spatial flexure will result in a mismatch between the
flatfield and the data (i.e. some data will be lost), and also a shift
of the target in the field of view.
The former effect can be dealt with by taking flatfields when KMOS is
rotated to several different angles (the number depending on the
severity of the flexure).
The software then needs to select the flatfield taken at the angle
that best matches those of the science observations.
This can therefore be fully compensated, although at the expense of
longer daytime calibration requirements.
The latter effect can be corrected in a similar way to the spectral
flexure, by feeding the measured offset (again most easily derived
from a reconstructed cube) into the calibration data and reconstructing the
cube again.
\subsection{Pipeline Modules and Workflows}
\begin{figure}
\begin{center}
\includegraphics[height=8cm]{KMOS_workflow1.eps}
\end{center}
\caption{ \label{fig:workflow1}
One possible workflow for the KMOS pipeline.
This is a standard workflow
in which subtracting the sky is the first step, and in which each cube
is reconstructed before they are shifted and aligned.}
\end{figure}
The data processing pipeline is constructed from various modules
(recipes) as illustrated in Fig.~\ref{fig:workflow1}.
A user can either execute each recipe in turn, or use the pipeline
recipe to execute them all automatically in sequence.
The function of the main recipes used here are outlined below.
Each of these recipes works on all the extensions in the given data file.\vspace{-3mm}
\begin{description}
\item{\em kmo\_noise\_map} makes an initial estimate of the noise for
raw frames from the counts, using the known read noise and gain.\vspace{-1mm}
\item{\em kmo\_arithmetic} performs simple mathematical operations on
cubes using scalars, spectra, images, or other cubes; and on
images using either scalars or other images. The associated
noise frames are propagated appropriately. \vspace{-1mm}
\item{\em kmo\_reconstruct} creates a 3D cube from its 2D data using
the calibration files. The scheme and interpolation methods are
described in Section~\ref{sec:recon}.\vspace{-1mm}
\item{\em kmo\_shift} performs sub-pixel shifts to align a set of
cubes on the same WCS. Noise is also propagated.\vspace{-1mm}
\item{\em kmo\_combine} combines cubes that are on the same WCS
(i.e. have only integer pixel shifts between them). As default,
only corresponding IFUs are combined; but for the `mapping' mode, all
IFUs are combined into a single product.\vspace{-1mm}
\end{description}
\begin{figure}
\begin{center}
\includegraphics[height=8cm]{KMOS_workflow2.eps}
\end{center}
\caption{ \label{fig:workflow2}
One possible workflow for the KMOS pipeline.
This is an advanced workflow in which all the data are combined and
reconstructed simultaneously in a single step, and the sky is
subtracted afterwards
in an optimal fashion (i.e. minimising OH residuals). In the scheme
shown here, the data are also corrected for spatial and spectral
offsets/flexure, which are derived for each dataset after a
temporary reconstruction step.}
\end{figure}
In the second workflow shown in Fig.~\ref{fig:workflow2} there are some
additional recipes which are based on those described above, which
allow one to reconstruct a cube from multiple data sets, or to perform
an optimal sky subtraction.
In addition, there are a number of recipes and tools that are used
only indirectly in the pipeline, but perform tasks that
are frequently required.\vspace{-3mm}
\begin{description}
\item{\em kmo\_stats} calculates a number of useful statistical measures.\vspace{-1mm}
\item{\em kmo\_make\_image} collapses a cube spectrally to create an
image. One can specify a global wavelength range, and also omit
specific regions around bright OH lines.\vspace{-1mm}
\item{\em kmo\_fit\_profile} fits various profile forms to a spectral or
spatial (1D and 2D) profile. This is useful for measuring the
seeing, the spectral resolution, or aligning compact bright objects.\vspace{-1mm}
\item{\em kmo\_extract\_spec} extracts an integrated spectrum, and can use
different methods based on apertures or (non-binary) masks.\vspace{-1mm}
\item{\em kmo\_rotate} rotates a cube, which can be useful if KMOS was
set at a non-zero rotator offset during observations.\vspace{-1mm}
\end{description}
\subsection{Additional Tools}
Handling datacubes can be complex so we provide some extra more
advanced tools that may be useful for
an astronomer but are not used in the pipeline.
These may require significant user interaction in order to optimise
the parameters.\vspace{-3mm}
\begin{description}
\item{\em kmo\_cosmic} is a 3D version of the very effective gradient method
implemented in LACosmic\cite{dok01} to detect deviant pixels.\vspace{-1mm}
\item{\em kmo\_voronoi} is a spatial binning scheme based on Voronoi
tessellations that aims to achieve uniform
signal-to-noise\cite{cap03}. This routine does not affect the
spectral resolution.\vspace{-1mm}
\item{\em kmo\_extract\_pv} creates a position-velocity diagram from
the cube, as one would obtain from a 2D longslit spectrum.\vspace{-1mm}
\item{\em kmo\_extract\_moments} derives the kinematics for an
emission line using both moment calculations, and also by convolving
the instrumental line profile with a Gaussian (so as to derive the
intrinsic line properties).\vspace{-1mm}
\item{\em kom\_fit\_continuum} allows the user to fit and subtract the
continuum from each spectrum in a cube, in order to leave just the
emission (and absorption) lines.
\end{description}
\section{Data Cube Reconstruction}
\label{sec:recon}
\subsection{Philosophy}
Interpolation is a crucial issue for integral field spectroscopy,
since reconstructing 3D datacubes requires a significant amount of
interpolation\cite{dav08}.
In the classical approach, one needs to correct for bad pixels,
straighten the spectral traces, linearise the dispersion, and finally
align the slitlets (or pixels).
Tuning the wavelength scale to correct for flexures between frames (as
described in Section~\ref{sec:dataproc}) can introduce an additional
interpolation step.
Poor management of the interpolation strategy or poor choice of the
interpolation scheme can degrade the quality of the final data.
For this reason, KMOS makes use of an alternative perspective on the purpose
of calibrations which allows one to view the data reconstruction in a
different way.
In principle this enables one to perform all the interpolation in a
single step while at the same time permitting a far greater
flexibility.
This can be summarised as follows:\vspace{-3mm}
\begin{description}
\item{\em Standard View}:
Calibrations allow one to create the mathematical functions
(e.g. polynomials) needed to correct the spectral and spatial curvature on
the detector. With this perspective
(i) bad pixels must be first corrected,
(ii) the raw data itself sets the pixel scale of the calibrated
product, and
(iii) frames must be individually corrected first before they can be
aligned and combined.\vspace{-1mm}
\item{\em Alternative View}:
Calibrations allow one to create look-up tables which associate each
measured value on the detector with its spectral and spatial location
in the final reconstructed data. This perspective means that
(i) bad pixels are simply ignored because they do not appear in the look-up
tables,
(ii) one has the freedom to choose any spectral and spatial sampling
in the processed product, and
(iii) by including multiple frames in the loop-up table, one can
combine and interpolate all the data simultaneously in one step.\vspace{-1mm}
\end{description}
\begin{figure}
\begin{center}
\includegraphics[height=6cm]{fig_interpolation.eps}
\end{center}
\caption{ \label{fig:interp}
Interpolation scheme illustrated in 2-dimensions.
(a) observed data are sampled regularly in the reference frame of the detector.
(b) this sampling is irregular in the reference frame of the
reconstructed cube; bad pixels can simply be omitted from the set
of sampled points.
(c) one can freely specify the required gridding
(i.e. spatial/spectral pixel scale)
for the reconstructed data; it is independent of the actual sampling.
(d) each required grid point is interpolated from sampled points which
lie in its local neighbourhood. Any suitable algorithm can be used
for the interpolation.
}
\end{figure}
This alternative perspective, which is used for KMOS, is outlined
graphically in Fig.~\ref{fig:interp}.
The most important realisation is that in `detector space' there can
be no concept of a wavelength or spatial axis.
These concepts apply only to the final reconstructed cube.
The detector is nothing more than the medium on which raw data values are
recorded.
The calibrations allow one to assign each measured value on the
detector to a spatial/spectral location in the
reconstructed cube.
Together, these locations provide an irregularly spaced sampling of
that cube.
The aim is thus to reduce the raw data and the calibrations to
a list of values with their associated locations:
\[
\begin{array}{cccc}
value_0, & x_0, & y_0, & \lambda_0 \\
value_1, & x_1, & y_1, & \lambda_1 \\
\vdots & \vdots & \vdots & \vdots \\
value_n, & x_n, & y_n, & \lambda_n \\
\end{array}
\]
Data associated with bad pixels is simply excluded from the list and
so does not contribute to the set of sampled locations.
Creation of this list is the first step.
The second step is to specify the regular sampling -- i.e. the spatial
and spectral pixel size -- that is required for the reconstructed
cube.
The third step is to interpolate each of these regularly gridded
positions from sampled locations in the local neighbourhood.
In a fourth step, one can determine any spectral (or spatial) offsets
in the reconstructed cube and feed these back to create a
new list with updated locations for each measured value.
One can then re-interpolate the regular grid of points, leading to a
final cube which has been reconstructed in a single interpolation and
which has no offsets.
It is possible to take this scheme further, by adding multiple frames
to the look-up table that lists the data values and their associated
locations.
In doing so, one needs to include the spatial offsets between the
frames, but this can be determined either from keywords in the frame
headers or by measuring offsets directly from the temporarily
reconstructed cubes of each individual frame.
But ultimately, this allows one to perform a single interpolation that
simultaneously reconstructs and combines (and rotates) multiple frames.
This is the essence of the workflow illustrated Fig.~\ref{fig:workflow2}.
\subsection{Interpolation Methods}
There are several specific intepolation schemes that have been
considered for KMOS. Not all of these will necessarily be implemented.\vspace{-3mm}
\begin{description}
\item{\em Nearest Neighbour} is a fast but approximate method. It may
be a good choice for noisy data (since it does not degrade the
signal-to-noise at all) when the highest spatial or spectral
resolution is not required.\vspace{-1mm}
\item{\em Modified Shepard's Method}\cite{ren88} is a localised
quadratic inverse distance weighted method. It is a relatively fast
and accurate method\cite{yan04}. We have implemented a simplified
version which interpolates from the data itself, rather than from a smooth
quadratic function that is fit to the data.\vspace{-1mm}
\item{\em Kriging}\cite{cla00} is a distance weighted scheme that is
optimal in the sense that interpolated values have the smallest
variance. Its strength is that it takes into account
correlations between neighbouring pixels based on the seeing and
spectral resolution, and provides an estimate of the uncertainty.\vspace{-1mm}
\item{\em Drizzling}\cite{fru02} is a linear reconstruction method
introduced to cope with undersampled
data. It is an ideal method to optimise the spatial resolution of
the combined data when the observations were performed in excellent
seeing, or the data sets taken at sub-pixel offsets.\vspace{-1mm}
\item{\em Cubic Spline} interpolation is a standard technique. For
irregularly sampled data (i.e. when reconstructing a cube) it has to
be applied in 3 successive 1D interpolations. But for shifting or
rotating data that is already on a regular grid, a single bicubic
spline interpolation is possible.
\end{description}
\subsection{The Neighbourhood Map}
One step that is implicit in all of the above methods is the creation
of a neighbourhood map.
This is the list of pixels in the detector based format that lie close
to each pixel (voxel) in the reconstructed cube.
For the nearest neighbour, there will be only one detector pixel
for each cube voxel.
But the other methods require lists of 20 or 30 detector pixels for
each voxel in order to perform an appropriate interpolation.
It is the creation of this neighbourhood map that takes most of the
time.
However, the mapping is fixed for any given set calibration frames
(XCAL, YCAL, and LCAL) and so can be re-used.
For acquisition, when speed is of the essence, the cubes will be
reconstructed using a set of master neighbourhood maps that are
updated only occasionally.
By doing so, it will be possible to see the reconstructed data in
quasi-real time.
Similarly, if astronomers are processing their data at home, the
first time the interpolation is performed it will take some time.
However, thereafter the neighbourhood map can be re-used, and
reconstructing additional data sets will be very much faster.
\section{Visualisation}
There are now many tools available for visualising 3D datacubes.
However, the format of the data and the fact that there are 24 cubes
for each exposure mean that most of these tools are not compatible with, or ideal for, KMOS.
QFitsView is a popular and verstile tool, that has been developed at
MPE and has been included in ESO's SciSoft releases for several years.
It is available from
\verb+http://www.mpe.mpg.de/~ott/QFitsView+,
and is now being upgraded to handle KMOS data.
QFitsView already enables simple versions of many detailed analyses to be
performed.
However, it is not yet clear how best these should be applied when
there are multiple datasets within a buffer.
Such issues will be considered on an `as-needs' basis, and the initial
upgrade will address only more essential aspects as follows:\vspace{-3mm}
\begin{itemize}
\item
recognise KMOS files and open all extensions as subsets of a single buffer.\vspace{-1mm}
\item
show a list of the extensions, with their IFU identification as well
as whether the contents refer to are data or noise; allow the user to
select all or some of the datasets and display them.\vspace{-1mm}
\item
simultaneously display a representation of as many spectra, images, or
cubes as are selected, with an appropriate scaling.\vspace{-1mm}
\item
for multiple cubes, enable the user to change the
representation of all datasets simultaneously to be a single
wavelength slice, a combination of slices within a given wavelength
range, or a continuum-subtracted line map.\vspace{-1mm}
\item
save datasets as individual `standard' fitsfiles, i.e. having no
extensions, and compatible with the tools with which most astronomers
are more familiar).
\end{itemize}
| -14,988.135848 |
[
-2.890625,
2.677734375
] | 36.8758 |
[
-2.572265625,
0.86865234375,
-1.6123046875,
-5.640625,
-1.0751953125,
7.40234375
] |
[
2.525390625,
5.8125,
3.59765625,
5.6484375
] | 297 | 5,276 |
[
-2.640625,
2.76953125
] | 21.241152 |
[
-5.3984375,
-2.345703125,
-2.21875,
-1.013671875,
0.9951171875,
7.53515625
] | 0.948869 | 15.043285 | 24.829416 | 0.697572 |
[
2.381986141204834
] | -12,445.960991 | 5.293404 | -14,895.574662 | 0.257806 | 5.941671 |
[
-3.140625,
-3.51171875,
-2.23046875,
-3.431640625,
2.619140625,
9.203125
] |
[
-6.14453125,
-2.93359375,
-2.68359375,
-1.759765625,
4.0390625,
5.6953125
] | |
BkiUfCLxK0iCl4WD5TiB
|
\section{Further discussion in two dimension}
\section{Introduction}
Let $\Omega\subset\mathbb{R}^n$ be a domain with Lipschitz boundary. In this paper, we study the $H(\od)\cap H(\boldsymbol{\delta})$ elliptic problem: given $\boldsymbol{f}\in L^2\Lambda^k(\Omega)$, find $\boldsymbol{\omega}\in H\Lambda^k(\Omega)\cap H^*_0\Lambda^k(\Omega)$, such that
\begin{equation}\label{eq:primalhodge}
\langle\od^k\boldsymbol{\omega},\od^k\boldsymbol{\mu}\rangle_{L^2\Lambda^{k+1}}+\langle\boldsymbol{\delta}_k\boldsymbol{\omega},\boldsymbol{\delta}_k\boldsymbol{\mu}\rangle_{L^2\Lambda^{k-1}}+\langle\boldsymbol{\omega},\boldsymbol{\mu}\rangle_{L^2\Lambda^k}=\langle \boldsymbol{f},\boldsymbol{\mu}\rangle_{L^2\Lambda^k},\ \ \forall\,\boldsymbol{\mu}\in H\Lambda^k(\Omega)\cap H^*_0\Lambda^k(\Omega).
\end{equation}
Here, following \cite{Arnold.D2018feec}, we denote by $\Lambda^k(\Xi)$ the space of differential $k$-forms on an $n$-dimensional domain $\Xi$, and $L^2\Lambda^k(\Xi)$ consists of differential $k$-forms with coefficients in $L^2(\Xi)$ component by component, and $\langle\cdot,\cdot\rangle_{L^2\Lambda^k(\Xi)}$ is the inner product of the Hilbert space $L^2\Lambda^k(\Xi)$. In this paper, we will occasionally drop $\Omega$ for differential forms on $\Omega$. The exterior differential operator $\od^k:\Lambda^k(\Xi)\to \Lambda^{k+1}(\Xi)$ is unbounded from $L^2\Lambda^k(\Xi)$ to $\Lambda^{k+1}(\Xi)$. Denote,
$$
H\Lambda^k(\Xi)=H(\od^k,\Xi):=\left\{\boldsymbol{\omega}\in L^2\Lambda^k(\Xi):\od^k\boldsymbol{\omega}\in L^2\Lambda^{k+1}(\Xi)\right\},
$$
then $H\Lambda^k(\Xi)$ is a Hilbert space with the norm $\|\boldsymbol{\omega}\|_{L^2\Lambda^k(\Xi)}+\|\od^k\boldsymbol{\omega}\|_{L^2\Lambda^{k+1}(\Xi)}$. Denote by $H_0\Lambda^k(\Xi)$ the closure of $P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{C}_0^\infty\Lambda^k(\Xi)$ in $H\Lambda^k(\Xi)$. The Hodge star operator $\star$ maps $L^2\Lambda^k(\Xi)$ isomorphically to $L^2\Lambda^{n-k}(\Xi)$ for each $0\leqslant k\leqslant n$. The \emph{codifferential operator} $\boldsymbol{\delta}_k$ defined by $\boldsymbol{\delta}_k\boldsymbol{\mu}=(-1)^{kn}\star\od^{n-k}\star\boldsymbol{\mu}$ is unbounded from $L^2\Lambda^k(\Xi)$ to $L^2\Lambda^{k-1}(\Xi)$. Denote
$$
H^*\Lambda^k(\Xi)=H(\boldsymbol{\delta}_k,\Xi):=\left\{\boldsymbol{\mu}\in L^2\Lambda^k(\Xi):\boldsymbol{\delta}_k\boldsymbol{\mu}\in L^2\Lambda^{k-1}(\Xi)\right\},
$$
and $H^*_0\Lambda^k(\Xi)$ the closure of $P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{C}_0^\infty\Lambda^k(\Xi)$ in $H^*\Lambda^k(\Xi)$. Then $H^*\Lambda^k(\Xi)=\star H\Lambda^{n-k}(\Xi)$, and $H^*_0\Lambda^k(\Xi)=\star H_0\Lambda^{n-k}(\Xi)$. The model problem \eqref{eq:primalhodge} corresponds to a strong form that
$$
\od^k\boldsymbol{\omega}\in H^*_0\Lambda^{k+1}(\Omega),\ \ \ \boldsymbol{\delta}_k\boldsymbol{\omega}\in H\Lambda^{k-1}(\Omega),
$$
and
\begin{equation}
\boldsymbol{\delta}_{k+1}\od^k\boldsymbol{\omega}+\od^{k-1}\boldsymbol{\delta}_k\boldsymbol{\omega}+\boldsymbol{\omega}=\boldsymbol{f}.
\end{equation}
The model problem arises in many applied sciences, including electromagnetics\cite{Monk.P2003mono,Hiptmair.R2002acta}, fluid-structure interaction \cite{Bathe.K;Nitikitpaiboon.C;Wang.X1995cs,Bermudez.A;Rodriguez.R1994cmame,Hamdi.M;Ousset.Y;Verchery.G1978ijnme}, and others. Particularly, known as Hodge-Laplacian operator, the finite element methods associated with $\boldsymbol{\delta}_{k+1}\od^k+\od^{k-1}\boldsymbol{\delta}_k$ have been a central topic of the finite element exterior calculus (FEEC) discussed in many aspects, and we refer to \cite{Arnold.D;Falk.R;Winther.R2006acta,Arnold.D;Falk.R;Winther.R2010bams,Arnold.D2018feec} for a thorough introduction to FEEC.
~\\
It is well recognized that, the conforming finite element scheme for \eqref{eq:primalhodge} may lead to a spurious solution that converges to a wrong limit when the exact solution $\boldsymbol{\omega}$ is not smooth enough. Indeed, taking the $H\Lambda^1\cap H^*\Lambda^1$ in two dimension for example, a piecewise polynomial subspace of $H\Lambda^1\cap H^*_0\Lambda^1$ is contained in $H^1\Lambda^1\cap H^*_0\Lambda^1$, while $H^1\Lambda^1\cap H^*_0\Lambda^1$ is a closed subspace of $H\Lambda^1\cap H^*_0\Lambda^1$, and thus the singular part of $\boldsymbol{\omega}$ can not be captured by the conforming finite element space. To cope with this situation, a well-developed approach is to use mixed finite element method. Again, the main approach can be found in detail in \cite{Arnold.D;Falk.R;Winther.R2006acta,Arnold.D;Falk.R;Winther.R2010bams,Arnold.D2018feec}; some recent progress can be found in \cite{Li.Y2019sinum,Demlow.A;Hirani.A2014} for {\it a posteriori} error estimation and adaptive methods, and in \cite{Hong.Q;Li.Y;Xu.J2022mc} for a detailed analysis of Discontinuous Galerkin (DG) methods in FEEC in the newly-presented eXtended Galerkin (XG) framework.
On the other hand, to discretize directly the primal formulation \eqref{eq:primalhodge} has been still attracting research interests. Virtual element methods are designed for the three dimensional vector potential formulation of magnetostatic problems \cite{Beiroa.L;Brezzi.F;Marini.D;Alessandro.R2018}, with the major interests restricted to cases in which the solution is reasonably smooth and hence where higher order methods could be more profitable. Nonconforming element methods and discontinuous Galerkin methods are also designed which can lead to a correct approximation of the nonsmooth solution for the $H({\rm curl})\cap H({\rm div})$ problem in two dimension; readers are referred to \cite{Brenner.S;Sung.L;Cui.J2008} for an interior penalty method, to \cite{brenner2009quadratic} for a nonconforming finite element method, and to \cite{Brenner.S;Cui.J;Li.F;Sung.L2008nm} for a nonconforming finite element used with inter-element penalties. Recent works also include \cite{Barker.M2022thesis,Barker.M;Cao.S;Stern.A2022arxiv,Mirebeau.J2012aml}.
~\\
In this paper, we present a unified family of nonconforming finite element schemes for the primal formulation \eqref{eq:primalhodge} for any $n\geqslant 2$ and $1\leqslant k\leqslant n-1$, and on the subdivision of the domain by simplexes. The main feature of the finite element schemes is, all the finite element functions are defined by local shape function spaces and the continuity conditions, and no penalty term or stabilization is used in the schemes. The local shape function spaces are a slightly enrichment by $P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{H}^2_{\od}(T)$ (see \eqref{eq:defhd}) based on \ $P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}_0\Lambda^k(T)+\boldsymbol{\kappa}(P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}_0\Lambda^{k+1}(T))+\star\boldsymbol{\kappa}\star(P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}_0\Lambda^{k-1}(T))$, namely the minimal local space for $\boldsymbol{\delta}_{k+1}\od^k+\od^{k-1}\boldsymbol{\delta}_k$, but they do not contain the complete linear polynomial spaces which are used in \cite{Brenner.S;Sung.L;Cui.J2008,Brenner.S;Cui.J;Li.F;Sung.L2008nm,brenner2009quadratic,Barker.M2022thesis,Barker.M;Cao.S;Stern.A2022arxiv,Mirebeau.J2012aml}. Another difference from them, particularly \cite{brenner2009quadratic,Mirebeau.J2012aml}, is that the finite element functions in this present paper possess a different kind of inter-element continuity. As precisely described in \eqref{eq:fems}, the continuity is imposed in a dual way, and the consistency error can this way be controlled. This dual way makes the finite element functions not correspond to a ``finite element" in the sense of Ciarlet's triple \cite{Ciarlet.P1978book}, and the analysis and implementation would rely on non-standard techniques. For one thing, a stable interpolator is given, which works for functions in $H\Lambda^k(\Omega)\cap H^*_0\Lambda^k(\Omega)$ with minimal regularity, and an optimal approximation error estimation can be proved. Different from classical interpolators discussed in \cite{Arnold.D;Falk.R;Winther.R2006acta,Licht.M2019mc,Christiansen.S;Winther.R2008smoothed,Falk.R;Winther.R2014mc}, the interpolator is cell-wise defined, namely for $\boldsymbol{\omega},\boldsymbol{\mu}\in H\Lambda^k\cap H^*_0\Lambda^k$, if $\boldsymbol{\omega}$ and $\boldsymbol{\mu}$ are equal on a cell, then their interpolations are equal on this cell. Besides, a precise set of basis functions can be presented, the supports of which are each contained in a vertex patch, and the programming of the scheme can be done in a standard routine as for the standard ``finite element" method. Indeed, locally supported basis functions have been also found and implemented for many specific non-Ciarlet type finite element spaces \cite{Fortin.M;Soulie.M1983,Park.C;Sheen.D2003,Zhang.S2020IMA,Zhang.S2021SCM,Liu.W;Zhang.S2022arxiv,Xi.Y;Ji.X;Zhang.S2020jsc,Xi.Y;Ji.X;Zhang.S2021cicp}.
Finally we remark that, the construction of the finite element scheme is fit for quite general situations. For example, a quite the same scheme can be constructed where the local shape function space is a slight enrichment by $P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{H}^2_{\boldsymbol{\delta}}(T)$ (see \eqref{eq:defhdelta}) based on \ $P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}_0\Lambda^k(T)+\boldsymbol{\kappa}(P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}_0\Lambda^{k+1}(T))+\star\boldsymbol{\kappa}\star(P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}_0\Lambda^{k-1}(T))$ $\mathbf{P}_{\od\cap\odelta}^{\rm m+\odelta}\Lambda^k(T)$. The construction is also fit for other boundary conditions, such as the elliptic problems on $H_0\Lambda^k(\Omega)\cap H^*\Lambda^k(\Omega)$.
~\\
The remaining of the paper is organized as follows. In Section \ref{sec:fes}, we define the finite element spaces and prove its optimal approximation to functions in $H\Lambda^k\cap H^*_0\Lambda^k$ by constructing a cell-wise defined interpolator. In Section \ref{sec:fescheme}, we define the finite element schemes, present the error estimation and illustrate that the scheme can practically implemented by presenting a set of locally supported basis functions.
\section{Finite element space for $H\Lambda^k(\Omega)\cap H^*_0\Lambda^k(\Omega)$}
\label{sec:fes}
\subsection{Polynomial spaces on a simplex}
Denote the set of $k$-indices as
$$
\mathbb{IX}_{k,n}:=\{\boldsymbol{\alpha}=(\boldsymbol{\alpha}_1,\dots,\boldsymbol{\alpha}_k)\in\mathbb{N}^k:1\leqslant \boldsymbol{\alpha}_1<\boldsymbol{\alpha}_2<\dots<\boldsymbol{\alpha}_k\leqslant n\},\ \ \mathbb{N}\ \mbox{the\ set\ of\ integers}.
$$
Then
$$
\od^k(\boldsymbol{\kappa}(\,{\rm dx}^{\boldsymbol{\alpha}_1}\wedge\dots\wedge\,{\rm dx}^{\boldsymbol{\alpha}_{k+1}}))=(k+1)\,{\rm dx}^{\boldsymbol{\alpha}_1}\wedge\dots\wedge\,{\rm dx}^{\boldsymbol{\alpha}_{k+1}},\ \ \mbox{for}\ \ \boldsymbol{\alpha}\in\mathbb{IX}_{k+1,n},
$$
and
$$
\boldsymbol{\delta}_k(\star\boldsymbol{\kappa}\star(\,{\rm dx}^{\boldsymbol{\alpha}_1}\wedge\dots\wedge \,{\rm dx}^{\boldsymbol{\alpha}_{k-1}}))=(-1)^{kn-n-1} (n-k+1)(\,{\rm dx}^{\boldsymbol{\alpha}_1}\wedge\dots\wedge\,{\rm dx}^{\boldsymbol{\alpha}_{k-1}}),\ \ \mbox{for}\ \ \boldsymbol{\alpha}\in\mathbb{IX}_{k-1,n},
$$
where $\boldsymbol{\kappa}$ is the Koszul operator
$$
\boldsymbol{\kappa}(\,{\rm dx}^{\boldsymbol{\alpha}_1}\wedge\dots\wedge\,{\rm dx}^{\boldsymbol{\alpha}_k})=\sum_{j=1}^k(-1)^{j+1}x^{\boldsymbol{\alpha}_j}\,{\rm dx}^{\boldsymbol{\alpha}_1}\wedge\dots\wedge\,{\rm dx}^{\boldsymbol{\alpha}_{j-1}}\wedge\,{\rm dx}^{\boldsymbol{\alpha}_{j+1}}\wedge\dots\wedge\,{\rm dx}^{\boldsymbol{\alpha}_k},\ \ \mbox{for}\ \ \boldsymbol{\alpha}\in\mathbb{IX}_{k,n}.
$$
\subsubsection{Structures of polynomials on a simplex}
Given $T$ a simplex, denote on $T$ $\tilde{x}^j=x^j-c_j$ where $c_j$ is a constant such that $\int_T\tilde{x}^j=0$. Denote a simplex dependent Koszul operator
$$
\boldsymbol{\kappa}_T(\,{\rm dx}^{\boldsymbol{\alpha}_1}\wedge\dots\wedge\,{\rm dx}^{\boldsymbol{\alpha}_k}):=\sum_{j=1}^k(-1)^{(j+1)}\tilde{x}^{\boldsymbol{\alpha}_j}\,{\rm dx}^{\boldsymbol{\alpha}_1}\wedge\dots\,{\rm dx}^{\boldsymbol{\alpha}_{j-1}}\wedge\,{\rm dx}^{\boldsymbol{\alpha}_{j+1}}\wedge\dots\wedge\,{\rm dx}^{\boldsymbol{\alpha}_k},\ \ \mbox{for}\ \ \boldsymbol{\alpha}\in\mathbb{IX}_{k,n}.
$$
Then
$$
\od^{k-1}\boldsymbol{\kappa}_T(\,{\rm dx}^{\boldsymbol{\alpha}_1}\wedge\dots\,{\rm dx}^{\boldsymbol{\alpha}_k})=k\,{\rm dx}^{\boldsymbol{\alpha}_1}\wedge\dots\,{\rm dx}^{\boldsymbol{\alpha}_k}.
$$
\begin{lemma}\label{lem:pikappa}
There exists a constant $C_{k,n}$, depending on the regularity of $T$, such that
\begin{equation}\label{eq:pimud}
\|\boldsymbol{\mu}\|_{L^2\Lambda^k(T)}\leqslant C_{k,n}h_T\|\od^k\boldsymbol{\mu}\|_{L^2\Lambda^{k+1}(T)},\ \ \mbox{for}\ \ \boldsymbol{\mu}\in\boldsymbol{\kappa}_T(P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}_0\Lambda^{k+1}(T)),
\end{equation}
and
\begin{equation}\label{eq:pimude}
\|\boldsymbol{\mu}\|_{L^2\Lambda^k(T)}\leqslant C_{k,n}h_T\|\boldsymbol{\delta}_k\boldsymbol{\mu}\|_{L^2\Lambda^{k+1}(T)}\ \ \mbox{for}\ \boldsymbol{\mu}\in\star\boldsymbol{\kappa}_T\star(P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}_0\Lambda^{k-1}(T)).
\end{equation}
\end{lemma}
\begin{proof}
Given $
\displaystyle \boldsymbol{\mu}=\sum_{\boldsymbol{\alpha}\in\mathbb{IX}_{k+1,n}}C_{\boldsymbol{\alpha}} \left(\sum_{j=1}^{k+1}(-1)^{j+1}\tilde x^{\boldsymbol{\alpha}_j}\,{\rm dx}^{\boldsymbol{\alpha}_1}\wedge\,{\rm dx}^{\boldsymbol{\alpha}_2}\wedge\dots\wedge\,{\rm dx}^{\boldsymbol{\alpha}_{j-1}}\wedge\,{\rm dx}^{\boldsymbol{\alpha}_{j+1}}\wedge\dots\wedge \,{\rm dx}^{\boldsymbol{\alpha}_{k+1}}\right)$,
\begin{multline*}
|\boldsymbol{\mu}|^2_{H^1\Lambda^k(T)}=\left\|\sum_{\boldsymbol{\alpha}\in\mathbb{IX}_{k,n}}C_{\boldsymbol{\alpha}}\sum_{j=1}^{k+1}(-1)^{j+1}\nabla \tilde{x}^{\boldsymbol{\alpha}_j}\,{\rm dx}^{\boldsymbol{\alpha}_1}\wedge\dots\wedge\,{\rm dx}^{\boldsymbol{\alpha}_{j-1}}\wedge\,{\rm dx}^{\boldsymbol{\alpha}_{j+1}}\wedge\dots\wedge \,{\rm dx}^{\boldsymbol{\alpha}_{k+1}}\right\|_{L^2\Lambda^k(T)}^2
\\
=\left\langle \sum_{\boldsymbol{\alpha}\in\mathbb{IX}_{k,n}}C_{\boldsymbol{\alpha}}\sum_{j=1}^{k+1}(-1)^{j+1}\nabla \tilde{x}^{\boldsymbol{\alpha}_j}\,{\rm dx}^{\boldsymbol{\alpha}_1}\wedge\dots\wedge\,{\rm dx}^{\boldsymbol{\alpha}_{j-1}}\wedge\,{\rm dx}^{\boldsymbol{\alpha}_{j+1}}\wedge\dots\wedge \,{\rm dx}^{\boldsymbol{\alpha}_{k+1}}, \right.\qquad
\\
\qquad\qquad\left.\sum_{\boldsymbol{\alpha}'\in\mathbb{IX}_{k,n}}C_{\boldsymbol{\alpha}'}\sum_{i=1}^{k+1}(-1)^{i+1}\nabla \tilde{x}^{\boldsymbol{\alpha}'_i}\,{\rm dx}^{\boldsymbol{\alpha}'_1}\wedge\dots\wedge\,{\rm dx}^{\boldsymbol{\alpha}'_{i-1}}\wedge\,{\rm dx}^{\boldsymbol{\alpha}'_{i+1}}\wedge\dots\wedge \,{\rm dx}^{\boldsymbol{\alpha}'_{k+1}}\right\rangle_{L^2\Lambda^k(T)}
\\
=\sum_{\boldsymbol{\alpha}\in\mathbb{IX}_{k,n}}\sum_{\boldsymbol{\alpha}'\in\mathbb{IX}_{k,n}} C_{\boldsymbol{\alpha}}C_{\boldsymbol{\alpha}'}\sum_{j=1}^{k+1}\sum_{i=1}^{k+1} (-1)^{j+i}e^{\boldsymbol{\alpha}_j}\cdot e^{\boldsymbol{\alpha}_i} \Bigg\langle \,{\rm dx}^{\boldsymbol{\alpha}_1}\wedge\dots\wedge\,{\rm dx}^{\boldsymbol{\alpha}_{j-1}}\wedge\,{\rm dx}^{\boldsymbol{\alpha}_{j+1}}\wedge\dots\wedge \,{\rm dx}^{\boldsymbol{\alpha}_{k+1}},
\\
\,{\rm dx}^{\boldsymbol{\alpha}'_1}\wedge\dots\wedge\,{\rm dx}^{\boldsymbol{\alpha}'_{i-1}}\wedge\,{\rm dx}^{\boldsymbol{\alpha}'_{i+1}}\wedge\dots\wedge \,{\rm dx}^{\boldsymbol{\alpha}'_{k+1}}\Bigg\rangle_{L^2\Lambda^k(T)}=(k+1)|T|\sum_\alpha C_\alpha^2
\end{multline*}
and
$$
\left\|\od^k\mu\right\|_{L^2\Lambda^{k+1}(T)}^2=(k+1)^2\left\|\sum_\alpha C_\alpha \,{\rm dx}^{\boldsymbol{\alpha}_1}\wedge\,{\rm dx}^{\boldsymbol{\alpha}_2}\wedge\dots\wedge\,{\rm dx}^{\boldsymbol{\alpha}_{k+1}}\right\|_{L^2\Lambda^{k+1}(T)}^2=(k+1)^2|T|\sum_{\boldsymbol{\alpha}}C_{\boldsymbol{\alpha}}^2.
$$
Namely
$$
\|\od^k\mu\|_{L^2\Lambda^{k+1}(T)}=\sqrt{k+1}|\boldsymbol{\mu}|_{H^1\Lambda^k(T)}.
$$
Therefore, by noting that $\int_T\tilde{x}^j=0$, with a constant $C_n$ depending on the regularity of $T$, we obtain
$$
\|\boldsymbol{\mu}\|_{L^2\Lambda^k(T)}\leqslant C_nh_T|\boldsymbol{\mu}|_{H^1\Lambda^k(T)}=C_n(k+1)^{-1/2}h_T\|\od^k\mu\|_{L^2\Lambda^{k+1}(T)}.
$$
This proves \eqref{eq:pimud}. Similarly can \eqref{eq:pimude} be proved.
\end{proof}
In this part and in the sequel, we make the convention that, for $\boldsymbol{\alpha}\in\mathbb{IX}_{k,n}$, we use $\boldsymbol{\beta}$ for one in $\mathbb{IX}_{n-k,n}$, such that $\boldsymbol{\alpha}$ and $\boldsymbol{\beta}$ partition $\{1,2,\dots,n\}$. For $\boldsymbol{\alpha}\in\mathbb{IX}_{k,n}$, denote
$$
\tilde\boldsymbol{\mu}_{\boldsymbol{\delta},T}^{\boldsymbol{\alpha}}=\sum_{j=1}^k[(\tilde{x}^{\boldsymbol{\alpha}_j})^2-c^{\boldsymbol{\alpha}_j}]\,{\rm dx}^{\boldsymbol{\alpha}_1}\wedge\dots\wedge\,{\rm dx}^{\boldsymbol{\alpha}_k},
$$
and
$$
\tilde\boldsymbol{\mu}_{\od,T}^{\boldsymbol{\alpha}}=\sum_{j=1}^{n-k}[(\tilde{x}^{\boldsymbol{\beta}_j})^2-c^{\boldsymbol{\beta}_j}]\,{\rm dx}^{\boldsymbol{\alpha}_1}\wedge\,{\rm dx}^{\boldsymbol{\alpha}_2}\wedge\dots\wedge\,{\rm dx}^{\boldsymbol{\alpha}_k},
$$
where $c^{\boldsymbol{\alpha}_j}$ and $c^{\boldsymbol{\beta}_j}$ are constants such that $\int_T [(\tilde x^{\boldsymbol{\alpha}_j})^2-c^{\boldsymbol{\alpha}_j}]=0$ and $\int_T[(\tilde x^{\boldsymbol{\beta}_j})^2-c^{\boldsymbol{\beta}_j}]=0$. Then
\begin{equation}
\od^k\tilde\boldsymbol{\mu}_{\boldsymbol{\delta},T}^{\boldsymbol{\alpha}}=0,\quad \boldsymbol{\delta}_k\tilde\boldsymbol{\mu}_{\boldsymbol{\delta},T}^{\boldsymbol{\alpha}}=(-1)^n\cdot 2\boldsymbol{\kappa}_T(\,{\rm dx}^{\boldsymbol{\alpha}_1}\wedge\,{\rm dx}^{\boldsymbol{\alpha}_2}\wedge\dots\wedge \,{\rm dx}^{\boldsymbol{\alpha}_k}),
\end{equation}
\begin{equation}
\boldsymbol{\delta}_k\tilde\boldsymbol{\mu}_{\od,T}^{\boldsymbol{\alpha}}=0,\quad \mbox{and},\quad \od^k\tilde\boldsymbol{\mu}_{\od,T}^{\boldsymbol{\alpha}}=2(-1)^{n(1+k)+1}\star(\boldsymbol{\kappa}_T(\star (\,{\rm dx}^{\boldsymbol{\alpha}_1}\wedge\dots\,{\rm dx}^{\boldsymbol{\alpha}_k}))).
\end{equation}
We refer to the appendix, particularly \eqref{eq:Aodelta} and \eqref{eq:Aod}, for some detailed calculations.
Denote
\begin{equation}\label{eq:defhd}
P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{H}^2_{\od}\Lambda^k(T):={\rm span}\{\tilde\boldsymbol{\mu}_{\od,T}^{\boldsymbol{\alpha}}:\boldsymbol{\alpha}\in\mathbb{IX}_{k,n}\},
\end{equation}
and
\begin{equation}\label{eq:defhdelta}
P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{H}^2_{\boldsymbol{\delta}}\Lambda^k(T):={\rm span}\{\tilde\boldsymbol{\mu}_{\boldsymbol{\delta},T}^{\boldsymbol{\alpha}}:\boldsymbol{\alpha}\in\mathbb{IX}_{k,n}\}.
\end{equation}
\begin{lemma}\label{lem:surjecdde}
\begin{enumerate}
\item $\od^k$ is bijective from $\boldsymbol{\kappa}_T(P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}_0\Lambda^{k+1})$ onto $P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}_0\Lambda^{k+1}$, and bijective from $P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{H}^2_{\od}\Lambda^k(T)$ onto $\star\boldsymbol{\kappa}_T\star(P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}_0\Lambda^k(T))$.
\item $\boldsymbol{\delta}_k$ is bijective from $\star\boldsymbol{\kappa}_T\star(P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}_0\Lambda^{k-1})$ onto $P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}_0\Lambda^{k-1}$, and bijective from $P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{H}^2_{\boldsymbol{\delta}}\Lambda^k(T)$ onto $\boldsymbol{\kappa}_T(P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}_0\Lambda^k(T))$.
\end{enumerate}
\end{lemma}
\begin{lemma}\label{lem:pih}
There exists a constant $C_{k,n}$, depending on the regularity of $T$, such that
\begin{equation}\label{eq:hfhde}
\|\boldsymbol{\mu}\|_{L^2\Lambda^k(T)}\leqslant C_{k,n}h_T\|\boldsymbol{\delta}_k\boldsymbol{\mu}\|_{L^2\Lambda^{k-1}(T)},\ \ \mbox{for}\ \boldsymbol{\mu}\in P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{H}^2_{\boldsymbol{\delta}}\Lambda^k(T),
\end{equation}
and
\begin{equation}\label{eq:hfhd}
\|\boldsymbol{\mu}\|_{L^2\Lambda^k(T)}\leqslant C_{k,n}h_T\|\od^k\boldsymbol{\mu}\|_{L^2\Lambda^{k+1}(T)},\ \ \mbox{for}\ \boldsymbol{\mu}\in P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{H}^2_{\od}\Lambda^k(T).
\end{equation}
\end{lemma}
\begin{proof}
For $\displaystyle\boldsymbol{\mu}=\sum_{\boldsymbol{\varepsilon}\in\mathbb{IX}_{k,n}}C_{\boldsymbol{\varepsilon}}\tilde\boldsymbol{\mu}_{\boldsymbol{\delta},T}^{\boldsymbol{\varepsilon}}$,
$$
\|\boldsymbol{\mu}\|_{L^2\Lambda^k(T)}^2\leqslant C_{k,n}h_T^2\|\sum_{\boldsymbol{\varepsilon}}C_{\boldsymbol{\varepsilon}}\sum_{1\leqslant j\leqslant k}\nabla(\tilde{x}^{\boldsymbol{\varepsilon}_j})^2\,{\rm dx}^{\boldsymbol{\varepsilon}_1}\wedge\,{\rm dx}^{\boldsymbol{\varepsilon}_2}\wedge\dots\wedge\,{\rm dx}^{\boldsymbol{\varepsilon}_k}\|_{L^2\Lambda^k(T)}^2\leqslant C_{k,n}h_T^4|T|\sum_{\boldsymbol{\varepsilon}}C_{\boldsymbol{\varepsilon}}^2.
$$
Note that
$$
\boldsymbol{\delta}_k\boldsymbol{\mu}=2(-1)^n\sum_{\boldsymbol{\varepsilon}}C_{\boldsymbol{\varepsilon}}\boldsymbol{\kappa}_T(\,{\rm dx}^{\boldsymbol{\varepsilon}_1}\wedge\dots\wedge\,{\rm dx}^{\boldsymbol{\varepsilon}_k})
$$
and
$$
\od^{k-1}\boldsymbol{\delta}_k\boldsymbol{\mu}=2k(-1)^n\sum_{\boldsymbol{\varepsilon}\in\mathbb{IX}_{k,n}}C_{\boldsymbol{\varepsilon}}\,{\rm dx}^{\boldsymbol{\varepsilon}_1}\wedge\dots\wedge\,{\rm dx}^{\boldsymbol{\varepsilon}_k}.
$$
Therefore, by the inverse inequality,
$$
4k^2h_T^2|T|\sum_{\boldsymbol{\varepsilon}\in\mathbb{IX}_{k,n}}C_{\boldsymbol{\varepsilon}}^2=h_T^2\|\od^{k-1}\boldsymbol{\delta}_k\boldsymbol{\mu}\|_{L^2\Lambda^k(T)}^2\leqslant C_{k,n}\|\boldsymbol{\delta}_k\boldsymbol{\mu}\|_{L^2\Lambda^{k-1}(T)}^2.
$$
The proof of \eqref{eq:hfhde} is thus completed. Similarly can \eqref{eq:hfhd} be proved.
\end{proof}
\begin{lemma}
There exists a constant $C_{k,n}$, depending on the regularity of $T$, such that
\begin{equation
\|\boldsymbol{\mu}\|_{L^2\Lambda^k(T)}\leqslant C_{k,n}h_T\|\boldsymbol{\delta}_k\boldsymbol{\mu}\|_{L^2\Lambda^{k-1}(T)},\ \ \mbox{for}\ \boldsymbol{\mu}\in \star\boldsymbol{\kappa}_T\star(P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}_0\Lambda^{k-1}(T))+P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{H}^2_{\boldsymbol{\delta}}\Lambda^k(T),
\end{equation}
and
\begin{equation
\|\boldsymbol{\mu}\|_{L^2\Lambda^k(T)}\leqslant C_{k,n}h_T\|\od^k\boldsymbol{\mu}\|_{L^2\Lambda^{k+1}(T)},\ \ \mbox{for}\ \boldsymbol{\mu}\in \boldsymbol{\kappa}_T(P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}_0\Lambda^{k+1}(T))+P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{H}^2_{\od}\Lambda^k(T).
\end{equation}
\end{lemma}
\begin{proof}
Note that $\boldsymbol{\delta}_k(\star\boldsymbol{\kappa}_T\star(P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}_0\Lambda^{k-1}(T)))$ and $\boldsymbol{\delta}_k(P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{H}^2_{\boldsymbol{\delta}}\Lambda^k(T))$ are orthogonal, and $\od^k(\boldsymbol{\kappa}_T(P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}_0\Lambda^{k+1}(T)))$ and $\od^k(P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{H}^2_{\od}\Lambda^k(T))$ are orthogonal. The lemma follows by Lemmas \ref{lem:pikappa} and \ref{lem:pih}.
\end{proof}
It is well known the lowest-degree trimmed space is
$$
P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}^-_1\Lambda^k(T)=P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}_0\Lambda^k(T)+\boldsymbol{\kappa}(P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}_0\Lambda^{k+1}(T)),
$$
and denote
$$
P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}^{*,-}_1\Lambda^k(T)=\star(P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}^-_1\Lambda^k(T))=P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}_0\Lambda^k(T)+\star\boldsymbol{\kappa}\star(P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}_0\Lambda^{k-1}(T)).
$$
Here we introduce, for the $H\Lambda^k\cap H^*_0\Lambda^k$ problem, an enriched trimmed space, defined by
\begin{equation}
\mathbf{P}_{\od\cap\odelta}^{\rm m+\od}\Lambda^k(T):=P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}_0\Lambda^k(T)\oplus\boldsymbol{\kappa}_T(P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}_0\Lambda^{k+1}(T))\oplus\star\boldsymbol{\kappa}_T\star(P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}_0\Lambda^{k-1}(T))\oplusP}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{H}^2_{\od}\Lambda^k(T).
\end{equation}
Then
$$
\od^k(\mathbf{P}_{\od\cap\odelta}^{\rm m+\od}\Lambda^k(T))=P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}^{*,-}_1\Lambda^{k+1}(T),\ \ \mbox{and}\ \ \boldsymbol{\delta}_k(\mathbf{P}_{\od\cap\odelta}^{\rm m+\od}\Lambda^k(T))=P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}_0\Lambda^{k-1}(T).
$$
\subsubsection{A projective interpolator}
Define the interpolator
$$
\mathbb{I}^{\rm m+\od,k}_{\od\cap\boldsymbol{\delta},T}:H\Lambda^k(T)\cap H^*\Lambda^k(T)\to \mathbf{P}_{\od\cap\odelta}^{\rm m+\od}\Lambda^k(T),
$$
such that, for $\boldsymbol{\mu}\in H\Lambda^k(T)\cap H^*\Lambda^k(T)$,
\begin{multline}\label{eq:int-1}
\langle\od^k \mathbb{I}^{\rm m+\od,k}_{\od\cap\boldsymbol{\delta},T}\boldsymbol{\mu},\boldsymbol{\eta}\rangle_{L^2\Lambda^{k+1}(T)}-\langle \mathbb{I}^{\rm m+\od,k}_{\od\cap\boldsymbol{\delta},T}\boldsymbol{\mu}, \boldsymbol{\delta}_{k+1} \boldsymbol{\eta}\rangle_{L^2\Lambda^k(T)}
\\
=
\langle\od^k \boldsymbol{\mu},\boldsymbol{\eta}\rangle_{L^2\Lambda^{k+1}(T)}-\langle \boldsymbol{\mu}, \boldsymbol{\delta}_{k+1} \boldsymbol{\eta}\rangle_{L^2\Lambda^k(T)},\ \forall\,\boldsymbol{\eta}\inP}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}^{*,-}_1\Lambda^{k+1}(T),
\end{multline}
and
\begin{multline}\label{eq:int-2}
\langle\boldsymbol{\delta}_k \mathbb{I}^{\rm m+\od,k}_{\od\cap\boldsymbol{\delta},T}\boldsymbol{\mu},\boldsymbol{\tau}\rangle_{L^2\Lambda^{k-1}(T)}-\langle \mathbb{I}^{\rm m+\od,k}_{\od\cap\boldsymbol{\delta},T}\boldsymbol{\mu}, \od^{k-1} \boldsymbol{\tau}\rangle_{L^2\Lambda^k(T)}
\\
=
\langle\boldsymbol{\delta}_k \boldsymbol{\mu},\boldsymbol{\tau}\rangle_{L^2\Lambda^{k-1}(T)}-\langle \boldsymbol{\mu}, \od^{k-1} \boldsymbol{\tau}\rangle_{L^2\Lambda^k(T)},\ \forall\,\boldsymbol{\tau}\inP}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}^-_1\Lambda^{k-1}(T).
\end{multline}
\begin{lemma}\label{lem:welldintpprojec}
The interpolator $\mathbb{I}^{\rm m+\od,k}_{\od\cap\boldsymbol{\delta},T}$ is well defined, and $\mathbb{I}^{\rm m+\od,k}_{\od\cap\boldsymbol{\delta},T}\boldsymbol{\mu}=\boldsymbol{\mu}$ for $\boldsymbol{\mu}\in \mathbf{P}_{\od\cap\odelta}^{\rm m+\od}\Lambda^k(T)$.
\end{lemma}
\begin{proof}
Elementarily,
$$
P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}^{*,-}_1\Lambda^{k+1}(T)=P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}_0\Lambda^{k+1}(T)\oplus^\perp\star\boldsymbol{\kappa}_T\star(P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}_0\Lambda^{k}(T)),
$$
and
$$
P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}^-_1\Lambda^{k-1}(T)=P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}_0\Lambda^{k-1}(T)\oplus^\perp\boldsymbol{\kappa}_T(P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}_0\Lambda^k(T)).
$$
Given $\boldsymbol{\eta}\in P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}^{*,-}_1\Lambda^{k+1}(T)$, decompose it to $\boldsymbol{\eta}=\boldsymbol{\eta}_0+\boldsymbol{\eta}_{\boldsymbol{\delta}}$, with $\boldsymbol{\eta}_0\in P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}_0\Lambda^{k+1}(T)$ and $\boldsymbol{\eta}_{\boldsymbol{\delta}}\in \star\boldsymbol{\kappa}_T\star(P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}_0\Lambda^{k}(T))$, and decompose $\boldsymbol{\tau}\in P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}^-_1\Lambda^{k-1}(T)$ to $\boldsymbol{\tau}=\boldsymbol{\tau}_0+\boldsymbol{\tau}_{\od}$ with $\boldsymbol{\tau}_0\inP}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}_0\Lambda^{k-1}(T)$ and $\boldsymbol{\tau}_{\od}\in \boldsymbol{\kappa}_T(P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}_0\Lambda^k(T))$.
Now, given $\boldsymbol{\mu}\in H\Lambda^k(T)\cap H^*\Lambda^k(T)$, there exists a unique $\boldsymbol{\mu}_{\boldsymbol{\delta}}\in \star\boldsymbol{\kappa}_T\star(P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}_0\Lambda^{k-1}(T))$, such that
$$
\langle\boldsymbol{\delta}_k\boldsymbol{\mu}_{\boldsymbol{\delta}},\boldsymbol{\tau}_0\rangle_{L^2\Lambda^{k-1}(T)}=\langle\boldsymbol{\delta}_k\boldsymbol{\mu},\boldsymbol{\tau}_0\rangle_{L^2\Lambda^{k-1}(T)},\ \forall\,\boldsymbol{\tau}_0\inP}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}_0\Lambda^{k-1}(T),
$$
and there exists a unique $\boldsymbol{\mu}_0\inP}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}_0\Lambda^k(T)$, such that
$$
-\langle\boldsymbol{\mu}_0,\od^{k-1}\boldsymbol{\tau}_{\od}\rangle_{L^2\Lambda^k(T)}=\langle\boldsymbol{\delta}_k\boldsymbol{\mu},\boldsymbol{\tau}_{\od}\rangle_{L^2\Lambda^{k-1}(T)}-\langle\boldsymbol{\mu},\od^{k-1}\boldsymbol{\tau}_{\od}\rangle_{L^2\Lambda^k(T)},\ \forall\,\boldsymbol{\tau}_{\od}\in \boldsymbol{\kappa}_T(P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}_0\Lambda^k(T)).
$$
Then, there exists a unique $\boldsymbol{\mu}_{\od}\in \boldsymbol{\kappa}_T(P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}_0\Lambda^{k+1}(T))$, such that
\begin{equation}\label{eq:dkmueta0}
\langle \od^k\boldsymbol{\mu}_{\od},\boldsymbol{\eta}_0\rangle_{L^2\Lambda^{k+1}(T)}=\langle\od^k\boldsymbol{\mu},\boldsymbol{\eta}_0\rangle_{L^2\Lambda^{k+1}(T)},\ \forall\,\boldsymbol{\eta}_0\inP}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}_0\Lambda^{k+1}(T),
\end{equation}
and there exists a unique $\boldsymbol{\mu}_{\od+}\inP}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{H}^2_{\od}\Lambda^k(T)$, such that
\begin{multline}\label{eq:dkmuetaod}
\langle\od^k\boldsymbol{\mu}_{\od+},\boldsymbol{\eta}_{\boldsymbol{\delta}}\rangle_{L^2\Lambda^{k+1}(T)}-\langle\boldsymbol{\mu}_0,\boldsymbol{\delta}_{k+1}\boldsymbol{\eta}_{\boldsymbol{\delta}}\rangle_{L^2\Lambda^k(T)}
\\
=\langle\od^k\boldsymbol{\mu},\boldsymbol{\eta}_{\boldsymbol{\delta}}\rangle_{L^2\Lambda^{k-1}(T)}-\langle\boldsymbol{\mu},\boldsymbol{\delta}_{k+1}\boldsymbol{\eta}_{\boldsymbol{\delta}}\rangle_{L^2\Lambda^k(T)},\ \forall\,\boldsymbol{\eta}_{\boldsymbol{\delta}}\in\star\boldsymbol{\kappa}_T\star(P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}_0\Lambda^k(T)).
\end{multline}
Now set $\boldsymbol{\mu}_K:=\boldsymbol{\mu}_0+\boldsymbol{\mu}_{\od}+\boldsymbol{\mu}_{\boldsymbol{\delta}}+\boldsymbol{\mu}_{\od+}$, and $\boldsymbol{\mu}_K$ satisfies all requirements \eqref{eq:int-1} and \eqref{eq:int-2} of $\mathbb{I}^{\rm m+\od,k}_{\od\cap\boldsymbol{\delta},T}\boldsymbol{\mu}$; namely, $\mathbb{I}^{\rm m+\od,k}_{\od\cap\boldsymbol{\delta},T}\boldsymbol{\mu}=\boldsymbol{\mu}_0+\boldsymbol{\mu}_{\od}+\boldsymbol{\mu}_{\boldsymbol{\delta}}+\boldsymbol{\mu}_{\od+}$, uniquely determined. Evidently, if $\boldsymbol{\mu}\in \mathbf{P}_{\od\cap\odelta}^{\rm m+\od}\Lambda^k(T)$, $\boldsymbol{\mu}=\mathbb{I}^{\rm m+\od,k}_{\od\cap\boldsymbol{\delta},T}\boldsymbol{\mu}$. The proof is completed.
\end{proof}
\begin{remark}\label{rem:nodalf}
Denote a set of quantities
\begin{equation}\label{eq:detcond}
\left\{\langle\od^k \boldsymbol{\mu},\boldsymbol{\eta}\rangle_{L^2\Lambda^{k+1}(T)}-\langle \boldsymbol{\mu}, \boldsymbol{\delta}_{k+1} \boldsymbol{\eta}\rangle_{L^2\Lambda^k(T)},\ \ \langle\boldsymbol{\delta}_k \boldsymbol{\mu},\boldsymbol{\tau}\rangle_{L^2\Lambda^{k-1}(T)}-\langle \boldsymbol{\mu}, \od^{k-1} \boldsymbol{\tau}\rangle_{L^2\Lambda^k(T)}\right\}.
\end{equation}
Then, according to the proof of Lemma \ref{lem:welldintpprojec},
\begin{itemize}
\item given $\boldsymbol{\mu}\in \mathbf{P}_{\od\cap\odelta}^{\rm m+\od}\Lambda^k(T)$, $\boldsymbol{\mu}=0$ if and only if all quantities in \eqref{eq:detcond} vanish for any $(\boldsymbol{\eta},\boldsymbol{\tau})\in P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}^{*,-}_1\Lambda^{k+1}(T)\timesP}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}^-_1\Lambda^{k-1}(T)$;
\item given $(\boldsymbol{\eta},\boldsymbol{\tau})\in P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}^{*,-}_1\Lambda^{k+1}(T)\timesP}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}^-_1\Lambda^{k-1}(T)$, $(\boldsymbol{\eta},\boldsymbol{\tau})=(0,0)$, if and only if all quantities in \eqref{eq:detcond} vanish for any $\boldsymbol{\mu}\in \mathbf{P}_{\od\cap\odelta}^{\rm m+\od}\Lambda^k(T)$.
\end{itemize}
This way, $\boldsymbol{\mu}\in\mathbf{P}_{\od\cap\odelta}^{\rm m+\od}\Lambda^k(T)$ is unisolvent by \eqref{eq:detcond} with $(\boldsymbol{\eta},\boldsymbol{\tau})\in P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}^{*,-}_1\Lambda^{k+1}(T)\timesP}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}^-_1\Lambda^{k-1}(T)$, and $(\boldsymbol{\eta},\boldsymbol{\tau})\in P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}^{*,-}_1\Lambda^{k+1}(T)\timesP}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}^-_1\Lambda^{k-1}(T)$ is unisolvent by \eqref{eq:detcond} with $\boldsymbol{\mu}\in \mathbf{P}_{\od\cap\odelta}^{\rm m+\od}\Lambda^k(T)$.
\end{remark}
\begin{lemma}
There exists a constant $C_{k,n}$, depending on the regularity of the simplex $T$, such that,
\begin{multline*}
\|\mathbb{I}^{\rm m+\od,k}_{\od\cap\boldsymbol{\delta},T}\boldsymbol{\mu}\|_{L^2\Lambda^k(T)} + \|\od^k\mathbb{I}^{\rm m+\od,k}_{\od\cap\boldsymbol{\delta},T}\boldsymbol{\mu}\|_{L^2\Lambda^{k+1}(T)} +
\|\boldsymbol{\delta}_k\mathbb{I}^{\rm m+\od,k}_{\od\cap\boldsymbol{\delta},T}\boldsymbol{\mu}\|_{L^2\Lambda^{k-1}(T)}
\\
\leqslant C_{k,n} (\|\boldsymbol{\mu}\|_{L^2\Lambda^k(T)} + \|\od^k\boldsymbol{\mu}\|_{L^2\Lambda^{k+1}(T)} + \|\boldsymbol{\delta}_k\boldsymbol{\mu}\|_{L^2\Lambda^{k-1}(T)}).
\end{multline*}
\end{lemma}
\begin{proof}
Given $\boldsymbol{\mu}\in H\Lambda^k(T)\cap H^*\Lambda^k(T)$, for the interpolation, we use $\boldsymbol{\mu}_0$, $\boldsymbol{\mu}_{\od}$, $\boldsymbol{\mu}_{\boldsymbol{\delta}}$ and $\boldsymbol{\mu}_{\od+}$ as in the proof of Lemma \ref{lem:welldintpprojec}, and $\mathbb{I}^{\rm m+\od,k}_{\od\cap\boldsymbol{\delta},T}\boldsymbol{\mu}=\boldsymbol{\mu}_0+\boldsymbol{\mu}_{\od}+\boldsymbol{\mu}_{\boldsymbol{\delta}}+\boldsymbol{\mu}_{\od+}$. Then
$$
\boldsymbol{\delta}_k \mathbb{I}^{\rm m+\od,k}_{\od\cap\boldsymbol{\delta},T}\boldsymbol{\mu}=\boldsymbol{\delta}_k\boldsymbol{\mu}_{\boldsymbol{\delta}}=\mathbf{P}^k_0 \boldsymbol{\delta}_k\boldsymbol{\mu}.
$$
Further,
$$
\langle \boldsymbol{\mu}-\boldsymbol{\mu}_0,\od^{k-1}\boldsymbol{\tau}_{\od}\rangle_{L^2\Lambda^k(T)}=\langle\boldsymbol{\delta}_k\boldsymbol{\mu},\boldsymbol{\tau}_{\od}\rangle_{L^2\Lambda^{k-1}(T)},\ \forall\,\boldsymbol{\tau}_{\od}\in\boldsymbol{\kappa}_T(P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}_0\Lambda^k).
$$
Then, by Lemmas \ref{lem:surjecdde} and \ref{lem:pikappa}, we have
$$
\|\mathbf{P}^k_0\boldsymbol{\mu}-\boldsymbol{\mu}_0\|_{L^2\Lambda^k(T)}\leqslant C_{k,n}h_T\|\boldsymbol{\delta}_k\boldsymbol{\mu}\|_{L^2\Lambda^{k-1}(T)}.
$$
By \eqref{eq:dkmueta0} and \eqref{eq:dkmuetaod}, we have
$$
\langle \od^k\mathbb{I}^{\rm m+\od,k}_{\od\cap\boldsymbol{\delta},T}\boldsymbol{\mu}-\od^k\boldsymbol{\mu},\boldsymbol{\eta}_0\rangle_{L^2\Lambda^{k+1}(T)}=0,\ \forall\,\boldsymbol{\eta}_0\inP}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}_0\Lambda^{k+1}(T),
$$
and
$$
\langle\od^k\mathbb{I}^{\rm m+\od,k}_{\od\cap\boldsymbol{\delta},T}\boldsymbol{\mu}-\od^k\boldsymbol{\mu},\boldsymbol{\eta}_{\boldsymbol{\delta}}\rangle_{L^2\Lambda^{k+1}(T)}=\langle\boldsymbol{\mu}_0-\boldsymbol{\mu},\boldsymbol{\delta}_{k+1}\boldsymbol{\eta}_{\boldsymbol{\delta}}\rangle_{L^2\Lambda^k(T)},\ \forall\,\boldsymbol{\eta}_{\boldsymbol{\delta}}\in\star\boldsymbol{\kappa}_T\star(P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}_0\Lambda^k(T)).
$$
Therefore, by Lemma \ref{lem:pikappa},
$$
\|\mathbf{P}^{k+1}_{P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}^{*,-}_1\Lambda^{k+1}(T)}\od^k\boldsymbol{\mu}-\od^k\mathbb{I}^{\rm m+\od,k}_{\od\cap\boldsymbol{\delta},T}\boldsymbol{\mu}\|_{L^2\Lambda^{k+1}(T)}\leqslant C_{k,n}h_T^{-1}\|\mathbf{P}^k_0\boldsymbol{\mu}-\boldsymbol{\mu}_0\|_{L^2\Lambda^k(T)}\leqslant C_{k,n}\|\boldsymbol{\delta}_k\boldsymbol{\mu}\|_{L^2\Lambda^{k-1}(T)}.
$$
Summing all above leads to the assertion and completed the proof.
\end{proof}
\begin{lemma}\label{lem:approxcell}
There exists a constant $C_{k,n}$, depending on the regularity of the simplex $T$, such that,
\begin{multline}
\|\boldsymbol{\omega}-\mathbb{I}^{\rm m+\od,k}_{\od\cap\boldsymbol{\delta},T}\boldsymbol{\omega}\|_{L^2\Lambda^k(T)} + \|\od^k\boldsymbol{\omega}-\od^k\mathbb{I}^{\rm m+\od,k}_{\od\cap\boldsymbol{\delta},T}\boldsymbol{\omega}\|_{L^2\Lambda^{k+1}(T)} +
\|\boldsymbol{\delta}_k\boldsymbol{\omega}-\boldsymbol{\delta}_k\mathbb{I}^{\rm m+\od,k}_{\od\cap\boldsymbol{\delta},T}\boldsymbol{\omega}\|_{L^2\Lambda^{k-1}(T)}
\\
\leqslant C_{k,n} \inf_{\boldsymbol{\mu}\in\mathbf{P}_{\od\cap\odelta}^{\rm m+\od}\Lambda^k(T)}\left[\|\boldsymbol{\omega}-\boldsymbol{\mu}\|_{L^2\Lambda^k(T)} + \|\od^k(\boldsymbol{\omega}-\boldsymbol{\mu})\|_{L^2\Lambda^{k+1}(T)} + \|\boldsymbol{\delta}_k(\boldsymbol{\omega}-\boldsymbol{\mu})\|_{L^2\Lambda^{k-1}(T)}\right].
\end{multline}
\end{lemma}
\begin{proof}
The proof follows immediately by the projection and stability.
\end{proof}
\begin{remark}
It can be proved that
\begin{multline*}
\inf_{\boldsymbol{\mu}\in\mathbf{P}_{\od\cap\odelta}^{\rm m+\od}\Lambda^k(T)}\|\boldsymbol{\omega}-\boldsymbol{\mu}\|_{L^2\Lambda^k(T)} + \|\od^k(\boldsymbol{\omega}-\boldsymbol{\mu})\|_{L^2\Lambda^{k+1}(T)} + \|\boldsymbol{\delta}_k(\boldsymbol{\omega}-\boldsymbol{\mu})\|_{L^2\Lambda^{k-1}(T)}
\\
\leqslant \inf_{\boldsymbol{\mu}\inP}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}_0\Lambda^k(T)}\|\boldsymbol{\omega}-\boldsymbol{\mu}\|_{L^2\Lambda^k(T)} + \inf_{\boldsymbol{\eta}\inP}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}^{*,-}_1\Lambda^{k+1}(T)}\|\od^k\boldsymbol{\omega}-\boldsymbol{\eta}\|_{L^2\Lambda^{k+1}(T)} + \inf_{\boldsymbol{\tau}\inP}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}_0\Lambda^{k-1}(T)}\|\boldsymbol{\delta}_k\boldsymbol{\omega}-\boldsymbol{\tau}\|_{L^2\Lambda^{k-1}(T)}
\\
+Ch_T(\|\od^k\boldsymbol{\omega}\|_{L^2\Lambda^{k+1}(T)}+\|\boldsymbol{\delta}_k\boldsymbol{\omega}\|_{L^2\Lambda^{k-1}(T)}).
\end{multline*}
\end{remark}
\subsection{Finite element space and global approximation}
For $\Xi$ a subdomain of $\Omega$, we denote by $E_\Xi^\Omega$ the extension from $L^1_{\rm loc}(\Xi)$ to $L^1_{\rm loc}(\Omega)$, the spaces of locally integrable functions, respectively. Namely,
$$
E_\Xi^\Omega: L^1_{\rm loc}(\Xi)\to L^1_{\rm loc}(\Omega),\ \
E_\Xi^\Omega v=\left\{\begin{array}{ll}
v,&\mbox{on}\,\Xi,
\\
0,&\mbox{else},
\end{array}\right.
\ \mbox{for}\,v\in L^1_{\rm loc}(\Xi).
$$
We use the same notation $L^1_{\rm loc}$ for both scalar and non-scalar locally integrable functions, and, here and in the sequel, use the same notation $E_\Xi^\Omega$ for both scalar and non-scalar functions.
Let $P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{G}_\Omega=\left\{P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{G}_h\right\}$ be a set of shape regular subdivisions of $\Omega$ by simplexes. For a subdivision $P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{G}_h$, define formally the product of a set of function spaces $\left\{\Upsilon(T)\right\}_{T\inP}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{G}_h}$ defined cell by cell such that $E_T^\Omega\Upsilon(T)$ for all $T\inP}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{G}_h$ are compatible,
$$
\prod_{K\inP}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{G}}\Upsilon(T):=\sum_{K\inP}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{G}} E_T^\Omega\Upsilon(T),
$$
and the summation is direct. $\displaystyle\prod_{K\inP}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{G}}\Upsilon(T)$ defined this way is actually the tensor product of all $\Upsilon(T)$.
Denote
\begin{itemize}
\item $\displaystyleP}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}^-_1\Lambda^k(P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{G}_h):=\prod_{T\inP}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{G}_h}P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}^-_1\Lambda^k(T)$;
\item $\displaystyleP}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}^{*,-}_1\Lambda^k(P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{G}_h):=\prod_{T\inP}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{G}_h}P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}^{*,-}_1\Lambda^k(T)$;
\item $\displaystyle\mathbf{P}_{\od\cap\odelta}^{\rm m+\od}\Lambda^k(P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{G}_h):=\prod_{T\inP}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{G}_h}\mathbf{P}_{\od\cap\odelta}^{\rm m+\od}\Lambda^k(T)$.
\end{itemize}
Denote the conforming space by Whitney forms by
$$
\boldsymbol{W}^*_{h0}\Lambda^{k+1}=P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}^{*,-}_1\Lambda^{k+1}(P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{G}_h)\cap H^*_0\Lambda^{k+1},\ \mbox{and}\ \
\boldsymbol{W}_h\Lambda^{k-1}=P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}^-_1\Lambda^{k-1}(P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{G}_h)\cap H\Lambda^{k-1}.
$$
On $P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{G}_h$, define the finite element space:
\begin{multline}\label{eq:fems}
\boldsymbol{V}_{\od\cap\boldsymbol{\delta}}^{\rm m, +\od}\Lambda^k:=\Big\{\boldsymbol{\mu}_h\in\mathbf{P}_{\od\cap\odelta}^{\rm m+\od}\Lambda^k(P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{G}_h):
\langle\od^k_h\boldsymbol{\mu}_h,\boldsymbol{\eta}_h\rangle_{L^2\Lambda^{k+1}}-\langle\boldsymbol{\mu}_h,\boldsymbol{\delta}_{k+1}\boldsymbol{\eta}_h\rangle_{L^2\Lambda^k}=0,\ \forall\,\boldsymbol{\eta}_h\in\boldsymbol{W}^*_{h0}\Lambda^{k+1},
\\
\mbox{and}\ \ \langle\boldsymbol{\delta}_{k,h}\boldsymbol{\mu}_h,\boldsymbol{\tau}_h\rangle_{L^2\Lambda^{k-1}}-\langle\boldsymbol{\mu}_h,\od^{k-1}\boldsymbol{\tau}_h\rangle_{L^2\Lambda^k}=0,\ \forall\,\boldsymbol{\tau}_h\in\boldsymbol{W}_h\Lambda^{k-1}
\Big\}.
\end{multline}
Here and in the sequel we use the subscript $\cdot_h$ to denote the piecewise operation on $P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{G}_h$.
Define the interpolator
$$
\mathbb{I}^{\rm m+\od,k}_{\od\cap\boldsymbol{\delta},h}:\prod_{T\inP}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{G}_h}H\Lambda^k(T)\cap H^*\Lambda^k(T)\to \prod_{T\inP}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{G}_h} \mathbf{P}_{\od\cap\odelta}^{\rm m+\od}\Lambda^k(T),
$$
such that
$$
(\mathbb{I}^{\rm m+\od,k}_{\od\cap\boldsymbol{\delta},h}\boldsymbol{\mu})|_T=\mathbb{I}^{\rm m+\od,k}_{\od\cap\boldsymbol{\delta},T}(\boldsymbol{\mu}|_T),\ \forall\,T\inP}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{G}_h.
$$
\begin{lemma}
If $\boldsymbol{\mu}\in H\Lambda^k\cap H^*_0\Lambda^k$, then \ \ $\mathbb{I}^{\rm m+\od,k}_{\od\cap\boldsymbol{\delta},h}\boldsymbol{\mu}\in \boldsymbol{V}_{\od\cap\boldsymbol{\delta}}^{\rm m, +\od}\Lambda^k$.
\end{lemma}
\begin{proof}
Provided $\boldsymbol{\mu}\in H\Lambda^k\cap H^*_0\Lambda^k$, it holds that
$$
\left\{
\begin{array}{ll}
\langle\od^k\boldsymbol{\mu},\boldsymbol{\eta}_h\rangle_{L^2\Lambda^{k+1}}-\langle\boldsymbol{\mu},\boldsymbol{\delta}_{k+1}\boldsymbol{\eta}_h\rangle_{L^2\Lambda^k}=0,&\forall\,\boldsymbol{\eta}_h\in\boldsymbol{W}^*_{h0}\Lambda^{k+1}
\\
\langle\boldsymbol{\delta}_k\boldsymbol{\mu},\boldsymbol{\tau}_h\rangle_{L^2\Lambda^{k-1}}-\langle\boldsymbol{\mu},\od^{k-1}\boldsymbol{\tau}_h\rangle_{L^2\Lambda^k}=0,&\forall\,\boldsymbol{\tau}_h\in\boldsymbol{W}_h\Lambda^{k-1}
\end{array}
\right. .
$$
By the definitions of $\mathbb{I}^{\rm m+\od,k}_{\od\cap\boldsymbol{\delta},h}$ and $\mathbb{I}^{\rm m+\od,k}_{\od\cap\boldsymbol{\delta},T}$, it holds that
\begin{multline*}
\langle\od^k_h\mathbb{I}^{\rm m+\od,k}_{\od\cap\boldsymbol{\delta},h}\boldsymbol{\mu},\boldsymbol{\eta}_h\rangle_{L^2\Lambda^{k+1}}-\langle \mathbb{I}^{\rm m+\od,k}_{\od\cap\boldsymbol{\delta},h}\boldsymbol{\mu},\boldsymbol{\delta}_{k+1}\boldsymbol{\eta}_h\rangle_{L^2\Lambda^k}
=\sum_{T\inP}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{G}_h}\langle\od^k\boldsymbol{\mu},\boldsymbol{\eta}_h\rangle_{L^2\Lambda^{k+1}(T)}-\langle\boldsymbol{\mu},\boldsymbol{\delta}_{k+1}\boldsymbol{\eta}_h\rangle_{L^2\Lambda^k(T)}
\\
=\langle\od^k\boldsymbol{\mu},\boldsymbol{\eta}_h\rangle_{L^2\Lambda^{k+1}(\Omega)}-\langle\boldsymbol{\mu},\boldsymbol{\delta}_{k+1}\boldsymbol{\eta}_h\rangle_{L^2\Lambda^k(\Omega)}=0,\forall\,\boldsymbol{\eta}_h\in\boldsymbol{W}^*_{h0}\Lambda^{k+1},
\end{multline*}
and similarly
$$
\langle\boldsymbol{\delta}_{k,h}\mathbb{I}^{\rm m+\od,k}_{\od\cap\boldsymbol{\delta},h}\boldsymbol{\mu},\boldsymbol{\tau}_h\rangle_{L^2\Lambda^{k-1}}-\langle \mathbb{I}^{\rm m+\od,k}_{\od\cap\boldsymbol{\delta},h}\boldsymbol{\mu},\od^{k-1}\boldsymbol{\tau}_h\rangle_{L^2\Lambda^k}=0,\ \forall\,\boldsymbol{\tau}_h\in\boldsymbol{W}_h\Lambda^{k-1}.
$$
Namely, $\mathbb{I}^{\rm m+\od,k}_{\od\cap\boldsymbol{\delta},h}\boldsymbol{\mu}\in \boldsymbol{V}_{\od\cap\boldsymbol{\delta}}^{\rm m, +\od}\Lambda^k$. The proof is completed.
\end{proof}
By Lemma \ref{lem:approxcell}, we immediately obtain the lemma below.
\begin{lemma}\label{lem:globalinterror}
For $\boldsymbol{\omega}\in H\Lambda^k\cap H^*_0\Lambda^k$, with a constant $C_{k,n}$ depending on the shape regularity of $P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{G}_h$,
\begin{multline}
\|\boldsymbol{\omega}-\mathbb{I}^{\rm m+\od,k}_{\od\cap\boldsymbol{\delta},h}\boldsymbol{\omega}\|_{L^2\Lambda^k} + \|\od^k\boldsymbol{\omega}-\od^k_h\mathbb{I}^{\rm m+\od,k}_{\od\cap\boldsymbol{\delta},h}\boldsymbol{\omega}\|_{L^2\Lambda^{k+1}} +
\|\boldsymbol{\delta}_k\boldsymbol{\omega}-\boldsymbol{\delta}_{k,h}\mathbb{I}^{\rm m+\od,k}_{\od\cap\boldsymbol{\delta},h}\boldsymbol{\omega}\|_{L^2\Lambda^{k-1}}
\\
\leqslant C_{k,n} \inf_{\boldsymbol{\mu}_h\in \mathbf{P}_{\od\cap\odelta}^{\rm m+\od}\Lambda^k(P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{G}_h)}\left[\|\boldsymbol{\omega}-\boldsymbol{\mu}_h\|_{L^2\Lambda^k} + \|\od^k_h(\boldsymbol{\omega}-\boldsymbol{\mu}_h)\|_{L^2\Lambda^{k+1}} + \|\boldsymbol{\delta}_{k,h}(\boldsymbol{\omega}-\boldsymbol{\mu}_h)\|_{L^2\Lambda^{k-1}}\right].
\end{multline}
\end{lemma}
\section{A primal finite element scheme of the $\mathbf{H}(\od)\cap \mathbf{H}(\boldsymbol{\delta})$ elliptic problem}
\label{sec:fescheme}
\subsection{Finite element scheme and error estimate}
Given a simplicial subdivision $P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{G}_h$ of $\Omega$, we consider the finite element problem: find $\boldsymbol{\omega}_h\in \boldsymbol{V}_{\od\cap\boldsymbol{\delta}}^{\rm m, +\od}\Lambda^k$, such that
\begin{equation}\label{eq:primalhodgedis}
\langle\od^k_h\boldsymbol{\omega}_h,\od^k_h\boldsymbol{\mu}_h\rangle_{L^2\Lambda^{k+1}}+\langle\boldsymbol{\delta}_{k,}\boldsymbol{\omega}_h,\boldsymbol{\delta}_{k,h}\boldsymbol{\mu}_h\rangle_{L^2\Lambda^{k-1}}+\langle\boldsymbol{\omega}_h,\boldsymbol{\mu}_h\rangle_{L^2\Lambda^k}=\langle \boldsymbol{f},\boldsymbol{\mu}_h\rangle_{L^2\Lambda^k},\ \forall\,\boldsymbol{\mu}_h\in \boldsymbol{V}_{\od\cap\boldsymbol{\delta}}^{\rm m, +\od}\Lambda^k.
\end{equation}
The well-posed-ness of \eqref{eq:primalhodgedis} is immediate. The main theoretical result is the theorem below.
\begin{theorem}
Let $\boldsymbol{\omega}$ and $\boldsymbol{\omega}_h$ be the solutions of \eqref{eq:primalhodge} and \eqref{eq:primalhodgedis}, respectively. Then
\begin{multline}
\|\boldsymbol{\omega}-\boldsymbol{\omega}_h\|_{L^2\Lambda^k}+\|\od^k_h(\boldsymbol{\omega}-\boldsymbol{\omega}_h)\|_{L^2\Lambda^{k+1}}+\|\boldsymbol{\delta}_{k,h}(\boldsymbol{\omega}-\boldsymbol{\omega}_h)\|_{L^2\Lambda^{k-1}}
\\
\leqslant C\inf_{\boldsymbol{\mu}_h\in \mathbf{P}_{\od\cap\odelta}^{\rm m+\od}\Lambda^k(P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{G}_h)}\left[\|\boldsymbol{\omega}-\boldsymbol{\mu}_h\|_{L^2\Lambda^k}+\|\od^k_h(\boldsymbol{\omega}-\boldsymbol{\mu}_h)\|_{L^2\Lambda^{k+1}}+\|\boldsymbol{\delta}_{k,h}(\boldsymbol{\omega}-\boldsymbol{\mu}_h)\|_{L^2\Lambda^{k-1}}\right]
\\
+\inf_{\boldsymbol{\eta}_h\in \boldsymbol{W}^*_{h0}\Lambda^{k+1}}\left[\|\od^k\boldsymbol{\omega}-\boldsymbol{\eta}_h\|_{L^2\Lambda^{k+1}}+\|\boldsymbol{\delta}_{k+1}(\od^k\boldsymbol{\omega}-\boldsymbol{\eta}_h)\|_{L^2\Lambda^k}\right]
\\
+\inf_{\boldsymbol{\tau}_h\in \boldsymbol{W}_h\Lambda^{k-1}}\left[\|\boldsymbol{\delta}_k\boldsymbol{\omega}-\boldsymbol{\tau}_h\|_{L^2\Lambda^{k-1}}+\|\od^{k-1}(\boldsymbol{\delta}_k\boldsymbol{\omega}-\boldsymbol{\tau}_h)\|_{L^2\Lambda^k}\right].
\end{multline}
\end{theorem}
\begin{proof}
According to Strang's lemma, the consistency error part is
\begin{multline*}
CE(\boldsymbol{\mu}_h)=\langle\od^k\boldsymbol{\omega},\od^k_h\boldsymbol{\mu}_h\rangle_{L^2\Lambda^{k+1}}+\langle\boldsymbol{\delta}_k\boldsymbol{\omega},\boldsymbol{\delta}_{k,h}\boldsymbol{\mu}_h\rangle_{L^2\Lambda^{k-1}}+\langle\boldsymbol{\omega},\boldsymbol{\mu}_h\rangle_{L^2\Lambda^k}-\langle \boldsymbol{f},\boldsymbol{\mu}_h\rangle_{L^2\Lambda^k}
\\
=\langle\od^k\boldsymbol{\omega},\od^k_h\boldsymbol{\mu}_h\rangle_{L^2\Lambda^{k+1}}-\langle\boldsymbol{\delta}_{k+1}\od^k\boldsymbol{\omega},\boldsymbol{\mu}_h\rangle_{L^2\Lambda^k}+\langle\boldsymbol{\delta}_k\boldsymbol{\omega},\boldsymbol{\delta}_{k,h}\boldsymbol{\mu}_h\rangle_{L^2\Lambda^{k-1}}-\langle\od^{k-1}\boldsymbol{\delta}_k\boldsymbol{\omega},\boldsymbol{\mu}_h\rangle_{L^2\Lambda^k}.
\end{multline*}
By the definition of $\boldsymbol{V}_{\od\cap\boldsymbol{\delta}}^{\rm m, +\od}\Lambda^k$, we have for any $\boldsymbol{\eta}_h\in \boldsymbol{W}^*_{h0}\Lambda^{k+1}$ and $\boldsymbol{\tau}_h\in \boldsymbol{W}_h\Lambda^{k-1}$,
\begin{multline}
CE(\boldsymbol{\mu}_h)=\langle\od^k\boldsymbol{\omega}-\boldsymbol{\eta}_h,\od^k_h\boldsymbol{\mu}_h\rangle_{L^2\Lambda^{k+1}}-\langle\boldsymbol{\delta}_{k+1}(\od^k\boldsymbol{\omega}-\boldsymbol{\eta}_h),\boldsymbol{\mu}_h\rangle_{L^2\Lambda^k}
\\
+\langle\boldsymbol{\delta}_k\boldsymbol{\omega}-\boldsymbol{\tau}_h,\boldsymbol{\delta}_{k,h}\boldsymbol{\mu}_h\rangle_{L^2\Lambda^{k-1}}-\langle\od^{k-1}(\boldsymbol{\delta}_k\boldsymbol{\omega}-\boldsymbol{\tau}_h),\boldsymbol{\mu}_h\rangle_{L^2\Lambda^k}
\\
\leqslant (\|\od^k\boldsymbol{\omega}-\boldsymbol{\eta}_h\|_{L^2\Lambda^{k+1}}+\|\boldsymbol{\delta}_{k+1}(\od^k\boldsymbol{\omega}-\boldsymbol{\eta}_h)\|_{L^2\Lambda^k})(\|\boldsymbol{\mu}_h\|_{L^2\Lambda^k}+\|\boldsymbol{\delta}_{k,h}\boldsymbol{\mu}_h\|_{L^2\Lambda^{k-1}})
\\
+(\|\boldsymbol{\delta}_k\boldsymbol{\omega}-\boldsymbol{\tau}_h\|_{L^2\Lambda^{k-1}}+\|\od^{k-1}(\boldsymbol{\delta}_k\boldsymbol{\omega}-\boldsymbol{\tau}_h))(\|\boldsymbol{\mu}_h\|_{L^2\Lambda^k}+\|\od^k_h\boldsymbol{\mu}_h\|_{L^2\Lambda^{k+1}}).
\end{multline}
This estimate, together with the approximation estimate Lemma \ref{lem:globalinterror}, leads to the total error estimation, and completes the proof.
\end{proof}
\subsection{Implementation of the scheme: a set of locally supported basis functions}
The finite element space $\boldsymbol{V}_{\od\cap\boldsymbol{\delta}}^{\rm m, +\od}\Lambda^k$ does not correspond to a ``finite element" defined as Ciarlet's triple \cite{Ciarlet.P1978book}. Though, in this section, we present a set of basis functions of $\boldsymbol{V}_{\od\cap\boldsymbol{\delta}}^{\rm m, +\od}\Lambda^k$ which are tightly supported. Therefore, the finite element scheme can still be implemented by the standard routine.
To illustrate the general procedure in Section \ref{sec:generalproce}, a two-dimensional example is given in Section \ref{sec:examples}, where we particularly refer to Figures \ref{fig:basisinteriorvertex} and \ref{fig:boundaryvertex} for the illustration of the local supports of the basis functions.
\subsubsection{A general procedure}\label{sec:generalproce}
Denote
$$
\mathbf{P}_{\od\cap\odelta,\od}^{\rm m+\od}\Lambda^k(T):=\left\{\langle\boldsymbol{\delta}_k \boldsymbol{\mu},\boldsymbol{\tau}\rangle_{L^2\Lambda^{k-1}(T)}-\langle \boldsymbol{\mu}, \od^{k-1} \boldsymbol{\tau}\rangle_{L^2\Lambda^k(T)}=0,\ \forall\,\boldsymbol{\tau}\in P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}^-_1\Lambda^{k-1}(T)\right\},
$$
and
$$
\mathbf{P}_{\od\cap\odelta,\odelta}^{\rm m+\od}\Lambda^k(T):=\left\{\langle\od^k \boldsymbol{\mu},\boldsymbol{\eta}\rangle_{L^2\Lambda^{k+1}(T)}-\langle \boldsymbol{\mu}, \boldsymbol{\delta}_{k+1} \boldsymbol{\eta}\rangle_{L^2\Lambda^k(T)}=0, \forall\,\boldsymbol{\eta}\inP}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}^{*,-}_1\Lambda^{k+1}(T)\right\}.
$$
Then $\mathbf{P}_{\od\cap\odelta,\od}^{\rm m+\od}\Lambda^k(T)$ is unisolvent by $\left\{\langle\od^k \boldsymbol{\mu},\boldsymbol{\eta}\rangle_{L^2\Lambda^{k+1}(T)}-\langle \boldsymbol{\mu}, \boldsymbol{\delta}_{k+1} \boldsymbol{\eta}\rangle_{L^2\Lambda^k(T)},\ \ \boldsymbol{\eta}\inP}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}^{*,-}_1\Lambda^{k+1}(T)\right\}$, $\mathbf{P}_{\od\cap\odelta,\odelta}^{\rm m+\od}\Lambda^k(T)$ is unisolvent by $\left\{\langle\boldsymbol{\delta}_k \boldsymbol{\mu},\boldsymbol{\tau}\rangle_{L^2\Lambda^{k-1}(T)}-\langle \boldsymbol{\mu}, \od^{k-1} \boldsymbol{\tau}\rangle_{L^2\Lambda^k(T)},\ \boldsymbol{\tau}\in P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}^-_1\Lambda^{k-1}(T)\right\}$. Denote
$$
\mathbf{P}_{\od\cap\odelta,\od}^{\rm m+\od}\Lambda^k(P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{G}_h)=\prod_{T\inP}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{G}_h}\mathbf{P}_{\od\cap\odelta,\od}^{\rm m+\od}\Lambda^k(T),\ \ \mathbf{P}_{\od\cap\odelta,\odelta}^{\rm m+\od}\Lambda^k(P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{G}_h)=\prod_{T\inP}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{G}_h}\mathbf{P}_{\od\cap\odelta,\odelta}^{\rm m+\od}\Lambda^k(T).
$$
Then
$$
\mathbf{P}_{\od\cap\odelta}^{\rm m+\od}\Lambda^k(T)=\mathbf{P}_{\od\cap\odelta,\od}^{\rm m+\od}\Lambda^k(T)\oplus \mathbf{P}_{\od\cap\odelta,\odelta}^{\rm m+\od}\Lambda^k(T)\ \ \mbox{and}\ \ \mathbf{P}_{\od\cap\odelta}^{\rm m+\od}\Lambda^k(P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{G}_h)=\mathbf{P}_{\od\cap\odelta,\od}^{\rm m+\od}\Lambda^k(P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{G}_h)\oplus \mathbf{P}_{\od\cap\odelta,\odelta}^{\rm m+\od}\Lambda^k(P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{G}_h).
$$
It follows that
\begin{multline*}
\boldsymbol{V}_{\od\cap\boldsymbol{\delta}}^{\rm m, +\od}\Lambda^k=\Big\{\boldsymbol{\mu}_h\in\mathbf{P}_{\od\cap\odelta}^{\rm m+\od}\Lambda^k(P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{G}_h):
\langle\od^k_h\boldsymbol{\mu}_h,\boldsymbol{\eta}_h\rangle_{L^2\Lambda^{k+1}}-\langle\boldsymbol{\mu}_h,\boldsymbol{\delta}_{k+1}\boldsymbol{\eta}_h\rangle_{L^2\Lambda^k}=0,\ \forall\,\boldsymbol{\eta}_h\in\boldsymbol{W}^*_{h0}\Lambda^{k+1},\quad
\\
\mbox{and}\ \ \langle\boldsymbol{\delta}_{k,h}\boldsymbol{\mu}_h,\boldsymbol{\tau}_h\rangle_{L^2\Lambda^{k-1}}-\langle\boldsymbol{\mu}_h,\od^{k-1}\boldsymbol{\tau}_h\rangle_{L^2\Lambda^k}=0,\ \forall\,\boldsymbol{\tau}_h\in\boldsymbol{W}_h\Lambda^{k-1}
\Big\}
\\
=\Big\{\boldsymbol{\mu}_h\in\mathbf{P}_{\od\cap\odelta,\od}^{\rm m+\od}\Lambda^k(P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{G}_h)\oplus \mathbf{P}_{\od\cap\odelta,\odelta}^{\rm m+\od}\Lambda^k(P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{G}_h):
\langle\od^k_h\boldsymbol{\mu}_h,\boldsymbol{\eta}_h\rangle_{L^2\Lambda^{k+1}}-\langle\boldsymbol{\mu}_h,\boldsymbol{\delta}_{k+1}\boldsymbol{\eta}_h\rangle_{L^2\Lambda^k}=0,\ \forall\,\boldsymbol{\eta}_h\in\boldsymbol{W}^*_{h0}\Lambda^{k+1},
\\
\mbox{and}\ \ \langle\boldsymbol{\delta}_{k,h}\boldsymbol{\mu}_h,\boldsymbol{\tau}_h\rangle_{L^2\Lambda^{k-1}}-\langle\boldsymbol{\mu}_h,\od^{k-1}\boldsymbol{\tau}_h\rangle_{L^2\Lambda^k}=0,\ \forall\,\boldsymbol{\tau}_h\in\boldsymbol{W}_h\Lambda^{k-1}
\Big\}
\\
=\Big\{\boldsymbol{\mu}_h\in \mathbf{P}_{\od\cap\odelta,\odelta}^{\rm m+\od}\Lambda^k(P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{G}_h):\langle\boldsymbol{\delta}_{k,h}\boldsymbol{\mu}_h,\boldsymbol{\tau}_h\rangle_{L^2\Lambda^{k-1}}-\langle\boldsymbol{\mu}_h,\od^{k-1}\boldsymbol{\tau}_h\rangle_{L^2\Lambda^k}=0,\ \forall\,\boldsymbol{\tau}_h\in\boldsymbol{W}_h\Lambda^{k-1}
\Big\}\qquad\qquad\qquad
\\
\oplus \Big\{\boldsymbol{\mu}_h\in\mathbf{P}_{\od\cap\odelta,\od}^{\rm m+\od}\Lambda^k(P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{G}_h):\ \langle\od^k_h\boldsymbol{\mu}_h,\boldsymbol{\eta}_h\rangle_{L^2\Lambda^{k+1}}-\langle\boldsymbol{\mu}_h,\boldsymbol{\delta}_{k+1}\boldsymbol{\eta}_h\rangle_{L^2\Lambda^k}=0,\ \forall\,\boldsymbol{\eta}_h\in\boldsymbol{W}^*_{h0}\Lambda^{k+1}
\Big\}:=\boldsymbol{V}_{\boldsymbol{\delta}}\oplus\boldsymbol{V}_{\od}.
\end{multline*}
Now we figure out the basis functions of $\boldsymbol{V}_{\boldsymbol{\delta}}$ and $\boldsymbol{V}_{\od}$, respectively. They two form the whole set of basis functions of $\boldsymbol{V}_{\od\cap\boldsymbol{\delta}}^{\rm m, +\od}\Lambda^k$.
\paragraph{\bf Basis functions of $\boldsymbol{V}_{\boldsymbol{\delta}}$} The basis functions of $\boldsymbol{V}_{\boldsymbol{\delta}}$ are determined by a 3-step procedure.
\begin{description}
\item[Step 1]
Let $\mathbf{B}_{\bf W}^{k-1}=\{\boldsymbol{\psi}_j\}_{j=1}^{\dim(\boldsymbol{W}_h\Lambda^{k-1})}$ be the set of nodal basis functions of $\boldsymbol{W}_h\Lambda^{k-1}$; then on every simplex $T$, the restrictions $\boldsymbol{\psi}_j|_T$ of those $\boldsymbol{\psi}_j$ that are nonzero on $T$ are linearly independent, and such $\boldsymbol{\psi}_j|_T$ expand $P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}^-_1\Lambda^{k-1}(T)$;
\item[Step 2] for any $T\inP}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{G}_h$, set $I^T:=\left\{1\leqslant i\leqslant \dim(\boldsymbol{W}_h\Lambda^{k-1}):\mathring{T}\cap {\rm supp}(\boldsymbol{\psi}_i)\neq\emptyset\right\}$, and there exist a set of functions $\left\{\boldsymbol{\mu}^T_i:i\in I^T\right\}\subset \mathbf{P}_{\od\cap\odelta,\odelta}^{\rm m+\od}\Lambda^k(T)$, such that $\langle\boldsymbol{\delta}_k\boldsymbol{\mu}^T_i,\boldsymbol{\psi}_j|_T\rangle_{L^2\Lambda^{k-1}}-\langle\boldsymbol{\mu}^T_i,\od^{k-1}\boldsymbol{\psi}_j|_T\rangle_{L^2\Lambda^k}=\delta_{ij}$. Then $\mathbf{P}_{\od\cap\odelta,\odelta}^{\rm m+\od}\Lambda^k(T)={\rm span}\{\boldsymbol{\mu}^T_i:\ i\in I^T\}$, and all $\boldsymbol{\mu}_i^T$ are linearly independent.
\item[Step 3] Now, by definition,
\begin{multline*}
\boldsymbol{V}_{\boldsymbol{\delta}}=\left\{\boldsymbol{\mu}_h\in \mathbf{P}_{\od\cap\odelta,\odelta}^{\rm m+\od}\Lambda^k(P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{G}_h):\langle\boldsymbol{\delta}_{k,h}\boldsymbol{\mu}_h,\boldsymbol{\tau}_h\rangle_{L^2\Lambda^{k-1}}-\langle\boldsymbol{\mu}_h,\od^{k-1}\boldsymbol{\tau}_h\rangle_{L^2\Lambda^k}=0,\ \forall\,\boldsymbol{\tau}_h\in\boldsymbol{W}_h\Lambda^{k-1}\right\}
\\
=\left\{\boldsymbol{\mu}_h\in \sum_{T\inP}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{G}_h}\sum_{i\in I^T}{\rm span}\{E_T^\Omega\boldsymbol{\mu}_i^T\}:\langle\boldsymbol{\delta}_{k,h}\boldsymbol{\mu}_h,\boldsymbol{\psi}_j\rangle_{L^2\Lambda^{k-1}}-\langle\boldsymbol{\mu}_h,\od^{k-1}\boldsymbol{\psi}_j\rangle_{L^2\Lambda^k}=0,\ \forall\,\boldsymbol{\psi}_j\in \mathbf{B}_{\bf W}^{k-1}\right\}
\\
=\sum_{1\leqslant j\leqslant \dim(\boldsymbol{W}_h\Lambda^{k-1})}\left\{\boldsymbol{\mu}_h\in\sum_{\mathring{T}\cap{\rm supp}(\boldsymbol{\psi}_j)\neq\emptyset}{\rm span}\{E_T^\Omega\boldsymbol{\mu}_j^T\}:\langle\boldsymbol{\delta}_{k,h}\boldsymbol{\mu}_h,\boldsymbol{\psi}_j\rangle_{L^2\Lambda^{k-1}}-\langle\boldsymbol{\mu}_h,\od^{k-1}\boldsymbol{\psi}_j\rangle_{L^2\Lambda^k}=0\right\}.
\end{multline*}
\end{description}
Namely, a set of basis functions of $\boldsymbol{V}_{\boldsymbol{\delta}}$ consists of, for $1\leqslant j\leqslant \dim(\boldsymbol{W}_h\Lambda^{k-1})$, functions
\begin{equation}
\boldsymbol{\mu}_h\in\sum_{\mathring{T}\cap{\rm supp}(\boldsymbol{\psi}_j)\neq\emptyset}{\rm span}\{E_T^\Omega\boldsymbol{\mu}_j^T\},\ \ \mbox{such\ that}\ \langle\boldsymbol{\delta}_{k,h}\boldsymbol{\mu}_h,\boldsymbol{\psi}_j\rangle_{L^2\Lambda^{k-1}}-\langle\boldsymbol{\mu}_h,\od^{k-1}\boldsymbol{\psi}_j\rangle_{L^2\Lambda^k}=0.
\end{equation}
Note that the supports of these functions are contained in the support of $\boldsymbol{\psi}_j$.
\paragraph{\bf Basis functions of $\boldsymbol{V}_{\od}$} The basis functions of $\boldsymbol{V}_{\od}$ are determined by a 3-step procedure, slightly different from the procedure for $\boldsymbol{V}_{\boldsymbol{\delta}}$. They basis functions have basically two categories.
\begin{description}
\item[Step 1]
Let $\mathbf{B}_{\bf W,0}^{*,k+1}=\{\boldsymbol{\psi}^*_j\}_{j=1}^{\dim(\boldsymbol{W}^*_{h0}\Lambda^{k+1})}$ be the set of nodal basis functions of $\boldsymbol{W}^*_{h0}\Lambda^{k+1}$; then on every simplex $T$, the restrictions $\boldsymbol{\psi}^*_j|_T$ of those $\boldsymbol{\psi}^*_j$ that are nonzero on $T$ are linearly independent, and for interior $T$, such $\boldsymbol{\psi}^*_j|_T$ expand $P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{P}^{*,-}_1\Lambda^{k+1}(T)$;
\item[Step 2] For any $T\inP}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{G}_h$, set $I^T:=\left\{1\leqslant i\leqslant \dim(\boldsymbol{W}^*_h\Lambda^{k+1}):\mathring{T}\cap {\rm supp}(\boldsymbol{\psi}^*_i)\neq\emptyset\right\}$, and there exist, not necessarily unique, a set of functions $\left\{\boldsymbol{\mu}^T_i:i\in I^T\right\}\subset \mathbf{P}_{\od\cap\odelta,\od}^{\rm m+\od}\Lambda^k(T)$, such that $\langle\od^k\boldsymbol{\mu}^T_i,\boldsymbol{\psi}^*_j|_T\rangle_{L^2\Lambda^{k+1}}-\langle\boldsymbol{\mu}^T_i,\boldsymbol{\delta}_{k+1}\boldsymbol{\psi}^*_j|_T\rangle_{L^2\Lambda^k}=\delta_{ij}$. Denote $({\rm span}\{\boldsymbol{\mu}^T_i:\ i\in I^T\})^{\rm c}=\{\boldsymbol{\mu}\in \mathbf{P}_{\od\cap\odelta,\od}^{\rm m+\od}\Lambda^k(T): \langle\od^k\boldsymbol{\mu},\boldsymbol{\psi}^*_j|_T\rangle_{L^2\Lambda^{k+1}}-\langle\boldsymbol{\mu},\boldsymbol{\delta}_{k+1}\boldsymbol{\psi}^*_j|_T\rangle_{L^2\Lambda^k}=0,\ j\in I^T\}$, not necessarily non-empty. Then $$
\mathbf{P}_{\od\cap\odelta,\od}^{\rm m+\od}\Lambda^k(T)={\rm span}\{\boldsymbol{\mu}^T_i:\ i\in I^T\}+({\rm span}\{\boldsymbol{\mu}^T_i:\ i\in I^T\})^{\rm c}.
$$
\item[Step 3] By definition,
$$
\begin{array}{rl}
\boldsymbol{V}_{\od}=&\displaystyle\Big\{\boldsymbol{\mu}_h\in\mathbf{P}_{\od\cap\odelta,\od}^{\rm m+\od}\Lambda^k(P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{G}_h):\ \langle\od^k_h\boldsymbol{\mu}_h,\boldsymbol{\eta}_h\rangle_{L^2\Lambda^{k+1}}-\langle\boldsymbol{\mu}_h,\boldsymbol{\delta}_{k+1}\boldsymbol{\eta}_h\rangle_{L^2\Lambda^k}=0,\ \forall\,\boldsymbol{\eta}_h\in\boldsymbol{W}^*_{h0}\Lambda^{k+1}
\Big\}
\\
=&\displaystyle\prod_{T\inP}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{G}_h} ({\rm span}\{\boldsymbol{\mu}^T_i:\ i\in I^T\})^{\rm c}
\\
&\ \displaystyle\oplus\left\{\boldsymbol{\mu}_h\in \sum_{T\inP}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{G}_h}\sum_{i\in I^T}{\rm span}\{E_T^\Omega\boldsymbol{\mu}_i^T\}:\langle\od^k_h\boldsymbol{\mu}_h,\boldsymbol{\psi}^*_h\rangle_{L^2\Lambda^{k+1}}-\langle\boldsymbol{\mu}_h,\boldsymbol{\delta}_{k+1}\boldsymbol{\psi}^*_h\rangle_{L^2\Lambda^k}=0,\ \forall\,\boldsymbol{\psi}^*_j\in \mathbf{B}_{\bf W,0}^{k-1}\right\}
\\
=&\displaystyle\prod_{T\inP}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{G}_h} ({\rm span}\{\boldsymbol{\mu}^T_i:\ i\in I^T\})^{\rm c}
\\
&\ \displaystyle\oplus \sum_{1\leqslant j\leqslant \dim(\boldsymbol{W}_{h0}\Lambda^{k+1})}\left\{\boldsymbol{\mu}_h\in\sum_{\mathring{T}\cap{\rm supp}(\boldsymbol{\psi}_j)\neq\emptyset}{\rm span}\{E_T^\Omega\boldsymbol{\mu}_j^T\}:\langle\od^k_h\boldsymbol{\mu}_h,\boldsymbol{\psi}^*_h\rangle_{L^2\Lambda^{k+1}}-\langle\boldsymbol{\mu}_h,\boldsymbol{\delta}_{k+1}\boldsymbol{\psi}^*_h\rangle_{L^2\Lambda^k}=0\right\}.
\end{array}
$
\end{description}
Namely, a set of basis functions of $\boldsymbol{V}_{\od}$ consists of two parts:
\begin{enumerate}
\item Part I: $\displaystyle\prod_{T\inP}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{G}_h} ({\rm span}\{\boldsymbol{\mu}^T_i:\ i\in I^T\})^{\rm c}$. The supports of these functions are each contained in a single simplex.
\item Part II: for $1\leqslant j\leqslant \dim(\boldsymbol{W}^*_{h0}\Lambda^{k+1})$, functions
$$
\displaystyle\boldsymbol{\mu}_h\in\sum_{\mathring{T}\cap{\rm supp}(\boldsymbol{\psi}^*_j)\neq\emptyset}{\rm span}\{E_T^\Omega\boldsymbol{\mu}_j^T\}\ \ \mbox{such\ that}\
\langle\od^k_h\boldsymbol{\mu}_h,\boldsymbol{\psi}^*_j\rangle_{L^2\Lambda^{k+1}}-\langle\boldsymbol{\mu}_h,\boldsymbol{\delta}_{k+1}\boldsymbol{\psi}^*_j\rangle_{L^2\Lambda^k}=0.
$$
Note that the supports of these functions are contained in the support of $\boldsymbol{\psi}^*_j$.
\end{enumerate}
\subsubsection{Examples}\label{sec:examples}
We take the two-dimensional $H\Lambda^1\cap H^*_0\Lambda^1$ problem for example. Let $\Omega$ be a polygon. The problem reads: find $\boldsymbol{\omega}\in H({\rm rot},\Omega)\cap H_0({\rm div},\Omega)$, such that
\begin{equation}
({\rm rot}\boldsymbol{\omega},{\rm rot}\boldsymbol{\mu})+({\rm div}\boldsymbol{\omega},{\rm div}\boldsymbol{\mu})+(\boldsymbol{\omega},\boldsymbol{\mu})=(\boldsymbol{f},\boldsymbol{\mu}),\ \forall\,\boldsymbol{\mu}\in H({\rm rot},\Omega)\cap H_0({\rm div},\Omega).
\end{equation}
The corresponding spaces are $H^1(\Omega)=H({\rm grad},\Omega)$ for $H\Lambda^0$ and $H^1_0(\Omega)=H_0({\rm curl},\Omega)$ for $H^*_0\Lambda^2$, respectively. We use for the conforming spaces of Whitney forms the linear element space $V^1_h$ for $H({\rm grad},\Omega)$ and $V^1_{h0}=V^1_h\cap H^1_0(\Omega)$ for $H_0({\rm curl},\Omega)$.
Let $P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{T}_h$ be a shape-regular triangular subdivision of $\Omega$ with mesh size $h$, such that $\overline\Omega=\cup_{T\inP}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{T}_h}\overline T$, and every boundary vertex is connected to at least one interior vertex. Denote by $P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{E}_h$, $P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{E}_h^i$, $P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{E}_h^b$, $P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{X}_h$, $P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{X}_h^i$, $P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{X}_h^b$ and $P}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{X}_h^c$ the set of edges, interior edges, boundary edges, vertices, interior vertices, boundary vertices and corners, respectively.
In the setting, the shape function space is
$$
\mathbf{P}_{\od\cap\odelta}^{\rm m+\od}\Lambda^1(T)=\mathbf{P}^{\rm m+rot}_{{\rm rot}\cap{\rm div}}(T)={\rm span}\left\{\left(\begin{array}{c}1\\ 0\end{array}\right),\ \left(\begin{array}{c}0\\ 1\end{array}\right),\ \left(\begin{array}{c}\tilde{x}\\ \tilde{y}\end{array}\right),\ \left(\begin{array}{c}\tilde{y}\\ -\tilde{x}\end{array}\right),\ \left(\begin{array}{c}\tilde{y}^2\\ 0\end{array}\right),\ \left(\begin{array}{c}0\\ \tilde{x}^2\end{array}\right)\right\}.
$$
\begin{figure}[htbp]
\begin{tikzpicture}[scale=1.1]
\path coordinate (a0b0) at (0,0)
coordinate (a0b1) at (0,1)
coordinate (a0b2) at (0,2)
coordinate (a0b3) at (0,3)
coordinate (a0b4) at (0,4)
coordinate (a1b0) at (1,0)
coordinate (a1b1) at (1,1)
coordinate (a1b2) at (1,2)
coordinate (a1b3) at (1,3)
coordinate (a1b4) at (1,4)
coordinate (a2b0) at (2,0)
coordinate (a2b1) at (2,1)
coordinate (a2b2) at (2,2)
coordinate (a2b3) at (2,3)
coordinate (a2b4) at (2,4)
coordinate (a3b0) at (3,0)
coordinate (a3b1) at (3,1)
coordinate (a3b2) at (3,2)
coordinate (a3b3) at (3,3)
coordinate (a3b4) at (3,4)
coordinate (a4b0) at (4,0)
coordinate (a4b1) at (4,1)
coordinate (a4b2) at (4,2)
coordinate (a4b3) at (4,3)
coordinate (a4b4) at (4,4);
\draw[line width=.4pt] (a0b0) -- (a0b4) ;
\draw[line width=.4pt] (a1b0) -- (a1b4) ;
\draw[line width=.4pt] (a2b0) -- (a2b4) ;
\draw[line width=.4pt] (a3b0) -- (a3b4) ;
\draw[line width=.4pt] (a4b0) -- (a4b4) ;
\draw[line width=.4pt] (a0b0) -- (a4b0) ;
\draw[line width=.4pt] (a0b1) -- (a4b1) ;
\draw[line width=.4pt] (a0b2) -- (a4b2) ;
\draw[line width=.4pt] (a0b3) -- (a4b3) ;
\draw[line width=.4pt] (a0b4) -- (a4b4) ;
\draw[line width=.4pt] (a0b0) -- (a4b0) ;
\draw[line width=.4pt] (a0b3) -- (a1b4) ;
\draw[line width=.4pt] (a0b2) -- (a2b4) ;
\draw[line width=.4pt] (a0b1) -- (a3b4) ;
\draw[line width=.4pt] (a1b0) -- (a4b3) ;
\draw[line width=.4pt] (a2b0) -- (a4b2) ;
\draw[line width=.4pt] (a3b0) -- (a4b1) ;
\draw[line width=.4pt] (a0b0) -- (a4b4) ;
\draw[fill] (a3b2) circle [radius=0.05];
\node[below right] at (a3b2) {$A$};
\draw[fill] (a2b1) circle [radius=0.05];
\node[below right] at (a2b1) {$B$};
\draw[fill] (a1b0) circle [radius=0.05];
\node[below] at (a1b0) {$C$};
\begin{scope}
\fill[pattern=horizontal lines] (a2b1)--(a3b1)--(a4b2)--(a4b3)--(a3b3)--(a2b2)--(a2b1);
\fill[pattern=vertical lines] (a1b0)--(a2b0)--(a3b1)--(a3b2)--(a2b2)--(a1b1)--(a1b0);
\fill[pattern=north west lines] (a0b0)--(a2b0)--(a2b1)--(a1b1)--(a0b0);
\end{scope}
\path coordinate (ca0b0) at (5,0)
coordinate (ca1b0) at (6,0)
coordinate (ca2b0) at (7,0)
coordinate (ca2b1) at (7,1)
coordinate (ca1b1) at (6,1);
\draw[line width=.4pt] (ca2b1) -- (ca2b0) -- (ca1b0) -- (ca0b0) -- (ca1b1) -- (ca2b1) -- (ca1b0) -- (ca1b1)
\path coordinate (ba1b0) at (8,0)
coordinate (ba2b0) at (9,0)
coordinate (ba3b1) at (10,1)
coordinate (ba3b2) at (10,2)
coordinate (ba2b2) at (9,2)
coordinate (ba1b1) at (8,1)
coordinate (ba2b1) at (9,1);
\draw[line width=.4pt] (ba1b1) -- (ba2b2);
\draw[line width=.4pt] (ba1b0) -- (ba3b2);
\draw[line width=.4pt] (ba2b0) -- (ba3b1);
\draw[line width=.4pt] (ba2b2) -- (ba3b2);
\draw[line width=.4pt] (ba1b1) -- (ba3b1);
\draw[line width=.4pt] (ba1b0) -- (ba2b0);
\draw[line width=.4pt] (ba1b0) -- (ba1b1);
\draw[line width=.4pt] (ba2b0) -- (ba2b2);
\draw[line width=.4pt] (ba3b1) -- (ba3b2);
\path coordinate (aa2b1) at (11,1)
coordinate (aa3b1) at (12,1)
coordinate (aa4b2) at (13,2)
coordinate (aa4b3) at (13,3)
coordinate (aa3b3) at (12,3)
coordinate (aa2b2) at (11,2)
coordinate (aa3b2) at (12,2);
\draw[line width=.4pt] (aa2b2) -- (aa3b3);
\draw[line width=.4pt] (aa2b1) -- (aa4b3);
\draw[line width=.4pt] (aa3b1) -- (aa4b2);
\draw[line width=.4pt] (aa3b3) -- (aa4b3);
\draw[line width=.4pt] (aa2b2) -- (aa4b2);
\draw[line width=.4pt] (aa2b1) -- (aa3b1);
\draw[line width=.4pt] (aa2b1) -- (aa2b2);
\draw[line width=.4pt] (aa3b1) -- (aa3b3);
\draw[line width=.4pt] (aa4b2) -- (aa4b3);
\begin{scope}
\fill[pattern=horizontal lines] (aa2b1)--(aa3b1)--(aa4b2)--(aa4b3)--(aa3b3)--(aa2b2)--(aa2b1);
\fill[pattern=vertical lines] (ba1b0)--(ba2b0)--(ba3b1)--(ba3b2)--(ba2b2)--(ba1b1)--(ba1b0);
\fill[pattern=north west lines] (ca0b0)--(ca2b0)--(ca2b1)--(ca1b1)--(ca0b0);
\end{scope}
\draw[fill] (aa3b2) circle [radius=0.05];
\node[below right] at (aa3b2) {$A$};
\draw[fill] (ba2b1) circle [radius=0.05];
\node[below right] at (ba2b1) {$B$};
\draw[fill] (ca1b0) circle [radius=0.05];
\node[below] at (ca1b0) {$C$};
\end{tikzpicture}
\caption{Illustration of supports of $\psi_A$, $\psi_{\rm B}$ and $\psi_C$, as well as $\psi^*_A$ and $\psi^*_{\rm B}$. $A$ (as well as $B$) denotes an interior vertex, and $C$ denotes a boundary vertex.}\label{fig:vertices}
\end{figure}
\begin{multline}\label{eq:defspaceintwod}
\boldsymbol{V}_{\od\cap\boldsymbol{\delta}}^{\rm m, +\od}\Lambda^k=\boldsymbol{V}_{{\rm rot}\cap{\rm div}}^{\rm m, +{\rm rot}}:=\Bigg\{\boldsymbol{\mu}_h\in\prod_{T\inP}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{G}_h}\mathbf{P}^{\rm m+rot}_{{\rm rot}\cap{\rm div}}(T):
({\rm rot}_h\boldsymbol{\mu}_h,\boldsymbol{\eta}_h)-(\boldsymbol{\mu}_h,{\rm curl}\boldsymbol{\eta}_h)=0,\ \forall\,\boldsymbol{\eta}_h\in V^1_{h0},
\\
\mbox{and}\ \ ({\rm div}_h\boldsymbol{\mu}_h,\boldsymbol{\tau}_h)-(\boldsymbol{\mu}_h,{\rm grad}\boldsymbol{\tau}_h)=0,\forall\,\boldsymbol{\tau}_h\in V^1_h. \Bigg\}
\end{multline}
Now we present the basis functions of $\boldsymbol{V}_{\od\cap\boldsymbol{\delta}}^{\rm m, +\od}\Lambda^k$ by a 3-step procedure.
\paragraph{\bf Step 1} Both $V^1_h$ and $V^1_{h0}$ admit locally supported basis functions, denoted by $\phi_a$ and $\psi_a$ associated with vertices $a\inP}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{X}_h$ and $a\inP}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{X}_h^i$, respectively. The restrictions of $\phi_a$ and $\psi_a$ on a triangle are each one of the barycentric coordinates on the triangle. We refer to Figure \ref{fig:vertices} for an illustration of interior and boundary vertices, and also the supports of $\phi_a$($\psi_a$).
\paragraph{\bf Step 2} On a cell $T$, with $a_i\inP}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{X}_h$ being its vertices, let $\lambda^{a_i}_T$ be the barycentric coordinates of $T$, and find $\boldsymbol{\mu}^{{\rm div}}_{a_i,T},\ \boldsymbol{\mu}^{{\rm rot}}_{a_i,T}\in \mathbf{P}_{\od\cap\odelta}^{\rm m+\od}\Lambda^1(T)$, $1\leqslant i\leqslant 3$, such that
$$
({\rm rot} \boldsymbol{\mu}^{{\rm div}}_{a_i,T},\lambda_T^{a_j})_T-(\boldsymbol{\mu}^{{\rm div}}_{a_i,T},{\rm curl}\lambda_T^{a_j})=0,\ \ ({\rm div}\boldsymbol{\mu}^{{\rm div}}_{a_i,T},\lambda_T^{a_j})_T-(\boldsymbol{\mu}^{{\rm div}}_{a_i,T},{\rm grad}\lambda_T^{a_j})=\delta_{ij}, \ \ 1\leqslant i,j\leqslant 3,
$$
and
$$
({\rm rot} \boldsymbol{\mu}^{{\rm rot}}_{a_i,T},\lambda_T^{a_j})_T-(\boldsymbol{\mu}^{{\rm div}}_{a_i,T},{\rm curl}\lambda_T^{a_j})=\delta_{ij},\ \ ({\rm div}\boldsymbol{\mu}^{{\rm rot}}_{a_i,T},\lambda_T^{a_j})_T-(\boldsymbol{\mu}^{{\rm div}}_{a_i,T},{\rm grad}\lambda_T^{a_j})=0,\ \ 1\leqslant i,j\leqslant 3.
$$
\paragraph{\bf Step 3} The global basis functions of $\boldsymbol{V}_{\od\cap\boldsymbol{\delta}}^{\rm m, +\od}\Lambda^k$ fall into two categories, corresponding to $\boldsymbol{V}_{\boldsymbol{\delta}}$ ($\boldsymbol{V}_{{\rm div}}$ here) and $\boldsymbol{V}_{\od}$ ($\boldsymbol{V}_{{\rm rot}}$ here), respectively.
\begin{description}
\item[Category I] All these functions in ${\rm span}\{\boldsymbol{\mu}^{{\rm div}}_{a,T}:a\inP}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{X}_h,\ T\inP}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{G}_h\}$ such that conditions in \eqref{eq:defspaceintwod} are satisfied with respect to every $\phi_a\in V^1_h$ a basis function. Particularly, with respect to any vertex $a$, the associated basis functions of $\boldsymbol{V}_{\od\cap\boldsymbol{\delta}}^{\rm m, +\od}\Lambda^k$ are all these functions in ${\rm span}\{\boldsymbol{\mu}^{{\rm div}}_{a,T}:\mathring{T}\cap {\rm supp}(\phi_a)\neq\emptyset\}$, namely, functions
$$
\boldsymbol{\omega}_h=\sum_{\partial T\ni a}c_TE_T^\Omega\boldsymbol{\mu}^{{\rm div}}_{a,T},\ \ \mbox{such\ that}\
\sum_{\partial T\ni a}({\rm div}\boldsymbol{\omega}_h,\phi_a|_T)_T+(\boldsymbol{\omega}_h,\nabla\phi_a|_T)_T=0.
$$
Those $\boldsymbol{\omega}_h$ in this category are each supported in a two-successive-cell patch. We refer to Figure \ref{fig:basisinteriorvertex} for the case $a\inP}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{X}_h^i$, and to Figure \ref{fig:boundaryvertex} (right) for an illustration that $a\inP}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{X}_h^b$.
\item[Category II] All these functions in ${\rm span}\{\boldsymbol{\mu}^{{\rm rot}}_{a,T}:a\inP}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{X}_h,\ T\inP}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{G}_h\}$ such that conditions in \eqref{eq:defspaceintwod} are satisfied with respect to every $\psi_a\in V^1_{h0}$ a basis function. Particularly, with respect to an interior vertex $a$, the associated basis functions of $\boldsymbol{V}_{\od\cap\boldsymbol{\delta}}^{\rm m, +\od}\Lambda^k$ are all these functions in ${\rm span}\{\boldsymbol{\mu}^{{\rm rot}}_{a,T}:\mathring{T}\cap {\rm supp}(\phi_a)\neq\emptyset\}$, namely, functions
$$
\boldsymbol{\omega}_h=\sum_{\partial T\ni a}c_TE_T^\Omega\boldsymbol{\mu}^{{\rm rot}}_{a,T},\ \mbox{such\ that}\ \sum_{\partial T\ni a}({\rm rot}\boldsymbol{\omega}_h,\phi_a|_T)_T-(\boldsymbol{\omega}_h,{\rm curl}\phi_a|_T)_T=0.
$$
Those $\boldsymbol{\omega}_h$ in this category are each supported in a two-successive-cell patch. We refer to Figure \ref{fig:basisinteriorvertex} for the case $a\inP}^{*,-}_1\Lambda{\mathbf{P}_{\odelta}^{\rm m}\Lambda{X}_h^i$.
With respect to a boundary vertex $a$, the associated basis functions of $\boldsymbol{V}_{\od\cap\boldsymbol{\delta}}^{\rm m, +\od}\Lambda^k$ are all these functions in ${\rm span}\{\boldsymbol{\mu}^{{\rm rot}}_{a,T}:\mathring{T}\cap {\rm supp}(\phi_a)\neq\emptyset\}$. The supports of the functions are each a single triangle. We refer to Figure \ref{fig:boundaryvertex}, left, for an illustration.
\end{description}
\begin{figure}
\begin{tikzpicture}[scale=1.3]
\path
coordinate (a1mb0) at (0.9,0)
coordinate (a1pb0m) at (1.1,-0.1)
coordinate (a1pb0p) at (1.1,0.1)
coordinate (a1mb0m) at (0.9,-0.1)
coordinate (a1mb0p) at (0.9,0.1)
coordinate (a2pb0p) at (2.1,0.1)
coordinate (a2pb0m) at (2.1,-0.1)
coordinate (a2mb0p) at (1.9,0.1)
coordinate (a2mb0m) at (1.9,-0.1)
coordinate (a1b1m) at (1,0.9)
coordinate (a1mb1) at (0.9,1)
coordinate (a1pb1p) at (1.1,1.1)
coordinate (a1pb1m) at (1.1,0.9)
coordinate (a1mb1p) at (0.9,1.1)
coordinate (a1mb1m) at (0.9,0.9)
coordinate (a2b1m) at (2,0.9)
coordinate (a2pb1p) at (2.1,1.1)
coordinate (a2pb1m) at (2.1,0.9)
coordinate (a2mb1p) at (1.9,1.1)
coordinate (a2mb1m) at (1.9,0.9)
coordinate (a3pb1p) at (3.1,1.1)
coordinate (a3pb1m) at (3.1,0.9)
coordinate (a3mb1p) at (2.9,1.1)
coordinate (a3mb1m) at (2.9,0.9)
coordinate (a1mb2) at (0.9,2)
coordinate (a1pb2p) at (1.1,2.1)
coordinate (a1pb2m) at (1.1,1.9)
coordinate (a1mb2p) at (0.9,2.1)
coordinate (a1mb2m) at (0.9,1.9)
coordinate (a2pb2) at (2.1,2)
coordinate (a2b2p) at (2,2.1)
coordinate (a2b2m) at (2,1.9)
coordinate (a2mb2) at (1.9,2)
coordinate (a2pb2p) at (2.1,2.1)
coordinate (a2pb2m) at (2.1,1.9)
coordinate (a2mb2p) at (1.9,2.1)
coordinate (a2mb2m) at (1.9,1.9)
coordinate (a3pb2) at (3.1,2)
coordinate (a3pb2p) at (3.1,2.1)
coordinate (a3pb2m) at (3.1,1.9)
coordinate (a3mb2p) at (2.9,2.1)
coordinate (a3mb2m) at (2.9,1.9)
coordinate (a2b3p) at (2,3.1)
coordinate (a2pb3p) at (2.1,3.1)
coordinate (a2pb3m) at (2.1,2.9)
coordinate (a2mb3p) at (1.9,3.1)
coordinate (a2mb3m) at (1.9,2.9)
coordinate (a3pb3) at (3.1,3)
coordinate (a3b3p) at (3,3.1)
coordinate (a3pb3p) at (3.1,3.1)
coordinate (a3pb3m) at (3.1,2.9)
coordinate (a3mb3p) at (2.9,3.1)
coordinate (a3mb3m) at (2.9,2.9);
\draw[line width=.4pt] (a1mb1) -- (a1mb2) -- (a2mb2) -- cycle;
\draw[line width=.4pt] (a1b1m) -- (a2b2m) -- (a2b1m) -- cycle;
\draw[line width=.4pt] (a1mb2p) -- (a2mb2p) -- (a2mb3p) -- cycle;
\draw[line width=.4pt] (a2b2p) -- (a2b3p) -- (a3b3p) -- cycle;
\draw[line width=.4pt] (a2pb2) -- (a3pb2) -- (a3pb3) -- cycle;
\draw[line width=.4pt] (a2pb2m) -- (a3pb2m) -- (a2pb1m) -- cycle;
\node[left] at (2,2.6) {$+$};
\node[above] at (1.5,2) {$+$};
\node[left] at (1.4,2.6) {$-$};
\node at(1.5,2.4){$T_1$};
\node[right] at (1.9,2.7) {$+$};
\node[above] at (2.45,2.5) {$+$};
\node[above] at (2.5,3) {$-$};
\node at(2.3,2.9) {$T_6$};
\node[right] at (3,2.4) {$-$};
\node[above] at (2.7,1.9) {$+$};
\node[right] at (2.4,2.4) {$+$};
\node at(2.9,2.3) {$T_5$};
\node[right] at (2,1.35) {$+$};
\node[right] at (2.5,1.3) {$-$};
\node[below] at (2.6,2) {$+$};
\node at(2.4,1.5){$T_4$};
\node[right] at (1.3,1.3) {$+$};
\node[left] at (2.1,1.3) {$+$};
\node[below] at (1.5,1) {$-$};
\node at(1.7,1.1){$T_3$};
\node[left] at (1,1.5) {$-$};
\node[left] at (1.55,1.5) {$+$};
\node[below] at (1.3,2.1) {$+$};
\node at(1.1,1.7){$T_2$};
\draw[fill] (2,2) circle [radius=0.05];
\node[below right] at (2,2) {$A$};
\path coordinate (aa2b1) at (5,1)
coordinate (aa3b1) at (6,1)
coordinate (aa4b2) at (7,2)
coordinate (aa4b3) at (7,3)
coordinate (aa3b3) at (6,3)
coordinate (aa2b2) at (5,2)
coordinate (aa3b2) at (6,2);
\draw[line width=.4pt] (aa2b2) -- (aa3b3);
\draw[line width=.4pt] (aa2b1) -- (aa4b3);
\draw[line width=.4pt] (aa3b1) -- (aa4b2);
\draw[line width=.4pt] (aa3b3) -- (aa4b3);
\draw[line width=.4pt] (aa2b2) -- (aa4b2);
\draw[line width=.4pt] (aa2b1) -- (aa3b1);
\draw[line width=.4pt] (aa2b1) -- (aa2b2);
\draw[line width=.4pt] (aa3b1) -- (aa3b3);
\draw[line width=.4pt] (aa4b2) -- (aa4b3);
\path coordinate (ba2b1) at (9,1)
coordinate (ba3b1) at (10,1)
coordinate (ba4b2) at (11,2)
coordinate (ba4b3) at (11,3)
coordinate (ba3b3) at (10,3)
coordinate (ba2b2) at (9,2)
coordinate (ba3b2) at (10,2);
\draw[line width=.4pt] (ba2b2) -- (ba3b3);
\draw[line width=.4pt] (ba2b1) -- (ba4b3);
\draw[line width=.4pt] (ba3b1) -- (ba4b2);
\draw[line width=.4pt] (ba3b3) -- (ba4b3);
\draw[line width=.4pt] (ba2b2) -- (ba4b2);
\draw[line width=.4pt] (ba2b1) -- (ba3b1);
\draw[line width=.4pt] (ba2b1) -- (ba2b2);
\draw[line width=.4pt] (ba3b1) -- (ba3b3);
\draw[line width=.4pt] (ba4b2) -- (ba4b3);
\path coordinate (ca2b1) at (1,-2)
coordinate (ca3b1) at (2,-2)
coordinate (ca4b2) at (3,-1)
coordinate (ca4b3) at (3,0)
coordinate (ca3b3) at (2,0)
coordinate (ca2b2) at (1,-1)
coordinate (ca3b2) at (2,-1);
\draw[line width=.4pt] (ca2b2) -- (ca3b3);
\draw[line width=.4pt] (ca2b1) -- (ca4b3);
\draw[line width=.4pt] (ca3b1) -- (ca4b2);
\draw[line width=.4pt] (ca3b3) -- (ca4b3);
\draw[line width=.4pt] (ca2b2) -- (ca4b2);
\draw[line width=.4pt] (ca2b1) -- (ca3b1);
\draw[line width=.4pt] (ca2b1) -- (ca2b2);
\draw[line width=.4pt] (ca3b1) -- (ca3b3);
\draw[line width=.4pt] (ca4b2) -- (ca4b3);
\path coordinate (da2b1) at (5,-2)
coordinate (da3b1) at (6,-2)
coordinate (da4b2) at (7,-1)
coordinate (da4b3) at (7,0)
coordinate (da3b3) at (6,0)
coordinate (da2b2) at (5,-1)
coordinate (da3b2) at (6,-1);
\draw[line width=.4pt] (da2b2) -- (da3b3);
\draw[line width=.4pt] (da2b1) -- (da4b3);
\draw[line width=.4pt] (da3b1) -- (da4b2);
\draw[line width=.4pt] (da3b3) -- (da4b3);
\draw[line width=.4pt] (da2b2) -- (da4b2);
\draw[line width=.4pt] (da2b1) -- (da3b1);
\draw[line width=.4pt] (da2b1) -- (da2b2);
\draw[line width=.4pt] (da3b1) -- (da3b3);
\draw[line width=.4pt] (da4b2) -- (da4b3);
\path coordinate (ea2b1) at (9,-2)
coordinate (ea3b1) at (10,-2)
coordinate (ea4b2) at (11,-1)
coordinate (ea4b3) at (11,0)
coordinate (ea3b3) at (10,0)
coordinate (ea2b2) at (9,-1)
coordinate (ea3b2) at (10,-1);
\draw[line width=.4pt] (ea2b2) -- (ea3b3);
\draw[line width=.4pt] (ea2b1) -- (ea4b3);
\draw[line width=.4pt] (ea3b1) -- (ea4b2);
\draw[line width=.4pt] (ea3b3) -- (ea4b3);
\draw[line width=.4pt] (ea2b2) -- (ea4b2);
\draw[line width=.4pt] (ea2b1) -- (ea3b1);
\draw[line width=.4pt] (ea2b1) -- (ea2b2);
\draw[line width=.4pt] (ea3b1) -- (ea3b3);
\draw[line width=.4pt] (ea4b2) -- (ea4b3);
\draw[fill] (aa3b2) circle [radius=0.05];
\node[below right] at (aa3b2) {$A$};
\draw[fill] (ba3b2) circle [radius=0.05];
\node[below right] at (ba3b2) {$A$};
\draw[fill] (ca3b2) circle [radius=0.05];
\node[above left] at (ca3b2) {$A$};
\draw[fill] (da3b2) circle [radius=0.05];
\node[above left] at (da3b2) {$A$};
\draw[fill] (ea3b2) circle [radius=0.05];
\node[below right] at (ea3b2) {$A$};
\begin{scope}
\fill[pattern=horizontal lines] (aa2b1)--(aa3b2)--(aa3b3)--(aa2b2)--(aa2b1);
\fill[pattern=horizontal lines] (ba2b1)--(ba3b1)--(ba3b2)--(ba2b2)--(ba2b1);
\fill[pattern=horizontal lines] (ca2b1)--(ca3b1)--(ca4b2)--(ca3b2)--(ca2b1);
\fill[pattern=horizontal lines] (da3b1)--(da4b2)--(da4b3)--(da3b2)--(da3b1);
\fill[pattern=horizontal lines] (ea3b2)--(ea4b2)--(ea4b3)--(ea3b3)--(ea3b2);
\end{scope}
\end{tikzpicture}
\caption{$A$ is an interior vertex; cf. Figure \ref{fig:vertices}. Five basis functions associated with the interior vertex $A$. The shadowed parts are respectively the supports of the basis functions.}\label{fig:basisinteriorvertex}
\end{figure}
\begin{figure}[htbp]
\begin{tikzpicture}[scale=1.5]
\path coordinate (ca0mb0) at (0.9,0)
coordinate (ca1mb0) at (1.9,0)
coordinate (ca1mb1) at (1.9,1)
coordinate (ca1b0p) at (2,0.1)
coordinate (ca1b1p) at (2,1.1)
coordinate (ca2b1p) at (3,1.1)
coordinate (ca1pb0) at (2.1,0)
coordinate (ca2pb0) at (3.1,0)
coordinate (ca2pb1) at (3.1,1);
\draw[line width=.4pt] (ca0mb0) -- (ca1mb0) -- (ca1mb1) --cycle;
\draw[line width=.4pt] (ca1b0p) -- (ca1b1p) -- (ca2b1p) --cycle;
\draw[line width=.4pt] (ca1pb0) -- (ca2pb0) -- (ca2pb1) --cycle;
\node[left] at (2,0.5) {$+$};
\node[above] at (1.5,-0.1) {$+$};
\node[left] at (1.5,.55) {$-$};
\node[right] at (1.9,.7) {$+$};
\node[left] at (2.65,.7) {$+$};
\node[above] at (2.5,1) {$-$};
\node[right] at (2.5,.5) {$+$};
\node[above] at (2.65,-0.1) {$+$};
\node[right] at (3,.5) {$-$};
\draw[fill] (2,0) circle [radius=0.05];
\node[below] at (2,0) {$C$};
\node[below] at (1.8,-0.3) {(a) no essential boundary condition};
\path coordinate (ca0b0) at (5,0)
coordinate (ca1b0) at (6,0)
coordinate (ca2b0) at (7,0)
coordinate (ca2b1) at (7,1)
coordinate (ca1b1) at (6,1);
\draw[line width=.4pt] (ca2b1) -- (ca2b0) -- (ca1b0) -- (ca0b0) -- (ca1b1) -- (ca2b1) -- (ca1b0) -- (ca1b1)
\path coordinate (cra0b0) at (8,0)
coordinate (cra1b0) at (9,0)
coordinate (cra2b0) at (10,0)
coordinate (cra2b1) at (10,1)
coordinate (cra1b1) at (9,1);
\draw[line width=.4pt] (cra2b1) -- (cra2b0) -- (cra1b0) -- (cra0b0) -- (cra1b1) -- (cra2b1) -- (cra1b0) -- (cra1b1)
\begin{scope}
\fill[pattern=north west lines] (ca0b0)--(ca1b0)--(ca2b1)--(ca1b1)--(ca0b0);
\fill[pattern=north west lines] (cra1b0)--(cra2b0)--(cra2b1)--(cra1b1)--(cra1b0);
\end{scope}
\draw[fill] (ca1b0) circle [radius=0.05];
\node[below] at (ca1b0) {$C$};
\draw[fill] (cra1b0) circle [radius=0.05];
\node[below] at (cra1b0) {$C$};
\node[below] at (7.5,-0.3) {(b) with essential boundary condition };
\end{tikzpicture}
\caption{ C is boundary vertex with a three-cell patch; cf. Figure \ref{fig:vertices}. The basis functions associated with $C$ are: (a) with no essential boundary condition, supported on a single cell; the case applies for $\boldsymbol{V}_{{\rm rot}}$; (b) with essential boundary condition, supported on the shadowed parts; the case applies for $\boldsymbol{V}_{{\rm div}}$.}\label{fig:boundaryvertex}
\end{figure}
| -161,439.61228 |
[
-2.470703125,
2.142578125
] | 26.818182 |
[
-3.40625,
0.67041015625,
-2.66015625,
-6.46875,
-1.16015625,
9.484375
] |
[
1.568359375,
8.8984375,
-1.40625,
3.673828125
] | 539 | 5,123 |
[
-3.5078125,
3.986328125
] | 38.65974 |
[
-5.63671875,
-4.3046875,
-5.41015625,
-2.671875,
1.9541015625,
13.46875
] | 2.646274 | 18.506393 | 28.791724 | 8.902634 |
[
0.6757093071937561
] | -115,544.879862 | 10.975991 | -159,939.385453 | 0.455273 | 6.24504 |
[
-1.9306640625,
-3.5859375,
-4.62890625,
-6.06640625,
1.9560546875,
13.8515625
] |
[
-5.7890625,
-1.6298828125,
-2.38671875,
-1.208984375,
3.203125,
3.859375
] | |
BkiUdvU5qsBDHTHZ3jMg
|
\section{Introduction}
Complex oxides have long been viewed as possible
candidates for the next generation of electronic
devices, which require reduced feature sizes, enhanced operating speeds
and low consumption.
Amongst oxides, ferroelectrics offer
the ability to store information in a non-volatile manner
via fast reversible polarization switching in ferroelectric random-access
memory (FeRAM).
The observation of giant tunneling electroresistance (TER)~\cite{Garcia2009}
in ultrathin (3 unit cells) ferroelectric films has recently opened a novel
paradigm for device design based on these materials~\cite{Zubko2009,Segal2009}.
Although the experiments~\cite{Garcia2009} ascribed TER
to ferroelectricity, which appeared robust and switchable, how
the polar state is stabilized in such thin films is by no
means established.
In principle, a ferroelectric film with an exposed surface cannot sustain a
monodomain polarization perpendicular to the surface, because of
the strong depolarizing field that would inevitably arise~\footnote{
The polarization is clearly observed to be perpendicular to the interface, consistent
with the expected behavior of compressively strained films~\cite{Schlom2007}.
}.
Charged particles from the environment could in principle
cancel the depolarizing field~\cite{Fridkin1980} (Fig. 1 left).
So far, however, the only chemical control of switching in air relates to neutral
species, O$_2$~\cite{Wang2009,Fong2006,Spanier2006} (Fig. 1 center).
It is then not clear how neutral gas-phase molecules could interact with a
biased atomic force microscopy (AFM) tip to produce the polar state.
Here we argue that the voltage applied with the AFM tip
induces electrochemical switching (Fig. 1 right), i.e. redox processes that are essential to
liberate free charge and therefore screen the depolarizing field.
This process would act as a nanobattery, rather than a nanocapacitor.
Note that the same mechanism could explain other effects at oxide interfaces,
such as the switchable two-dimensional electron gas (2DEG) at the LaAlO$_3$/SrTiO$_3$ interface~\cite{Cen2008,Bristowe2011a},
where the switching appears to be mediated by surface charge~\cite{Xie2010}.
\begin{figure}[b]
\includegraphics[width=0.45\textwidth]{1.pdf}
\caption{\label{BANDS}{
Schematic illustration of the conventional (left) and redox
(center) mechanisms for ferroelectric screening in the absence of a top
electrode. The presence of a biased tip can promote an alternative
redox mechanism that provides an
external circuit for the screening electrons (right).
}}
\end{figure}
To explore this mechanism we consider the system
studied experimentally in Ref.~\onlinecite{Garcia2009}, consisting of
a compressively strained nanometer-thick BaTiO$_3$ (BTO) film
on a La$_{0.7}$Sr$_{0.3}$MnO$_3$ (LSMO) bottom electrode.
Here we show, using first principles calculations,
that
(i) the pristine system (clean BTO surface with an ideal TiO$_2$ termination)
does not allow for a ferroelectric polarization, $P$, normal to the surface despite the
large compressive strain; (ii) a
non-zero $P$
is crucially dependent on the presence of a surface
external ionic charge,
in the form of defects
or adsorbates; and (iii)
the energetics for the formation of
oxidized or reduced surface defects support the
electrochemical switching model.
We also find (iv) a systematic change in band offset with screening
charge density, which we identify as the microscopic mechanism behind
the experimentally observed TER~\cite{Garcia2009},
and (v) a large magnetoelectric coupling,
due to the accumulation or depletion of spin-polarized carriers at
the interface with ferromagnetic LSMO.
The connection between these effects can be summarized as follows:
under open-circuit boundary conditions
the electric displacement field $D$ within the film,
the change in magnetization at the interface $\Delta M$
and the interface dipole,
are all proportional (or equal) to the external ionic charge density,
$Q$ per unit surface $S$, produced by the redox processes.
\section{Methods}
The density-functional theory (DFT) calculations are performed using the
spin-polarized Wu-Cohen (WC) exchange-correlation functional~\cite{Wu2006a},
as implemented in the {\sc Siesta} code~\cite{Ordejon1996,Soler2002}~\footnote{
Details of the pseudopotentials, numerical atomic orbitals and LSMO doping are given in Ref.~\onlinecite{Ferrari2006} and~\onlinecite{Junquera2003}.}.
We find GGA-WC to reproduce bulk~\cite{Ferrari2006} and
surface~\cite{Ferrari2006,Pruneda2007} properties of LSMO that were calculated
using the Perdew-Burke-Ernzerhof (PBE) scheme~\cite{Perdew1996}; at the same
time, GGA-WC is more appropriate for ferroelectric oxides.
The LSMO/BTO system consists of 5.5 unit cells of LSMO
(MnO$_2$-terminated) stacked with 3 unit cells of BTO along the
$c$ direction in a slab geometry.
The supercell contains a 15 \AA{} thick
vacuum layer and has either 2$\times$2 or $\sqrt{2}$$\times$$\sqrt{2}$
in-plane periodicity (see Fig. 2).
The 5.5 unit cells of LSMO are thick enough to show bulk-like features in the center,
and 3 unit cells of BTO was experimentally shown
to be thick enough for ferroelectricity~\cite{Garcia2009}.
We use a dipole correction to simulate open-circuit
boundary conditions, enforcing zero macroscopic electric field
in the vacuum layer.
We constrain the in-plane lattice parameter to experimental bulk NdGaO$_3$ (NGO) to
reproduce the experimental conditions of Ref.~\cite{Garcia2009}; this
imposes a large (3\%) compressive strain on BTO.
Based on this slab geometry, we perform a number of calculations where
we vary the surface composition by introducing defects or adsorbates.
In particular, we simulate the clean TiO$_2$-terminated surface (we
shall refer to this structure as ``pristine'' henceforth); one O vacancy
(``O-vac'') or adatom (``O-ads'') per 2$\times$2 surface cell; one H
adatom (``H'') or OH group (``OH'') per $\sqrt{2} \times \sqrt{2}$ cell
~\footnote{Atomic forces were relaxed to less than 40 meV/\AA.}.
Hereafter we shall discuss
the results with special regard for the presence or absence of
ferroelectric polarization in each case.
\begin{figure}[t]
\includegraphics[width=0.45\textwidth]{2.pdf}
\caption{\label{BAND}{(Color online)
Cation-anion splittings, $\delta z=z_{\mathrm{cation}}-z_{\mathrm{anion}}$
through the LSMO/BTO slab (the bottom half of LSMO is not shown).
The dotted lines correspond to the average of the AO and BO$_2$ layer
anion-cation splitting for the inwards and outwards $P$ in bulk BTO, strained to NGO.
}}
\end{figure}
\section{Discussion}
\subsection{The pristine system}
Fig. 2 shows the relaxed out-of-plane structural distortions as
a function of the surface chemical environment.
The pristine system is characterized by negligible distortions
in the interior of the BTO film, suggesting the absence of
macroscopic $P$ in this system. Only a surface rumpling is
present, resulting in a small net inwards dipole (non-switchable)
that decays rapidly towards the bulk (a surface rumpling is a known
general feature of oxide surfaces, in particular the TiO$_2$
termination of BTO~\cite{Padilla1997,Heifets1997}).
A vanishing $P$ is consistent with the open-circuit boundary
conditions, despite the large compressive strain.
In absence of a top electrode the macroscopic electric
displacement field $D$ in BTO is equal and opposite to the
density of external surface charge. As this charge is zero at the
clean TiO$_2$ surface, the film is constrained to a paraelectric
state.
\subsection{Chemical switching}
To illustrate possible screening scenarios, we now include representative
surface defects~\footnote{
Sampling the entire phase space (redox species and density, temperature,
partial pressure, polarization) is beyond the scope of this work, but
can be done within thermodynamic theory (see Ref.~\onlinecite{Stephenson2011}).}.
The O-vac and O-ads systems are both characterized by
large ferroelectric distortions (Fig. 2). These
are comparable to the strained bulk, where we calculate a
spontaneous polarization $P_0$=0.369 C/m$^2$ (0.35 $e/S$).
This result is again consistent with the constraint that $D=-Q/S$.
In fact, one oxygen defect for every 2$\times$2 unit cells
(0.5 $e/S$) yields a larger surface charge than what would be
sufficient to screen $P_0$. This justifies the larger
cation-anion rumplings that we obtain in the film compared with the
bulk (Fig. 2).
OH and H adatoms (with $\sqrt2$$\times$$\sqrt2$ coverage
to maintain $Q/S$) produce distortions of similar magnitude (Fig. 2).
This confirms the generality of the ferroelectric switching mechanism:
the ferroelectric state really depends on the net surface charge, and not on the chemical
identity of the adsorbed species.
In order to study the electrochemical switching (Fig. 1 right), we commence
by analyzing $chemical$ switching (Fig. 1 center).
Both are controlled by redox processes that transform bound charge into free charge, allowing for an electronic transfer
between the surface defect and the metal substrate,
but have different associated chemical sources/drains and energetics.
Chemical switching was recently shown in a system consisting of PbTiO$_3$ on SrRuO$_3$~\cite{Fong2006,Wang2009}
and BTO films on Au or vacuum~\cite{Spanier2006}.
To assess whether these redox reactions are thermodynamically accessible
in typical experimental conditions, we estimate the formation energy of
the defective systems
taking the reactions: 1) Slab(pristine) $\rightarrow$ Slab(O-vac)+1/2O$_2$
and equivalent for O-ads, 2) 1/2H$_2$O+1/4O$_2$+Slab(pristine)
$\rightarrow$ Slab(OH) and 3) 1/2H$_2$O+Slab(pristine) $\rightarrow$ Slab(H)+1/4O$_2$.
The chemical potential of the relevant molecular species is set to the calculated total energy of the
spin-polarized molecule in a large cubic box.
The results are summarized in Table 1.
They suggests that, whilst the oxygen adatom is likely
to form under oxygen-rich conditions, the formation energy
for the oxygen vacancy is possibly too high to form even in oxygen-poor conditions.
The calculated OH and H formation energies
suggest that water is a very likely redox
intermediate.
Note that H$_2$O is ubiquitous in most experiments performed in air,
and was recently found to play a crucial role in AFM experiments performed
on LaAlO$_3$/SrTiO$_3$~\cite{Bi2010}.
Since both sets of reactions involve oxygen, we therefore expect
that altering the surrounding oxygen partial pressure would affect the stability
of reduction or oxidation processes, consistent with the recently observed chemical
switching~\cite{Wang2009}.
\begin{table}[b
\caption{\label{multilayers}{The formation energy, $E_f$, of the defective systems
for O$_2$ and H$_2$O rich conditions (see text for definitions).
}}
\begin{ruledtabular}
\begin{tabular}{ c | c c c c}
& O-vac & O-ads & OH & H \\ [3pt]
\hline
$E_f$ (eV) & +3.6 & -0.4 & -1.5 & +0.9 \\
\end{tabular}
\end{ruledtabular}
\end{table}
\begin{figure}[t]
\includegraphics[width=0.42\textwidth]{3.pdf}
\caption{\label{BAND}{(Color online)
Change in Mn magnetic moment (left), $\Delta M$,
near the interface (the interface MnO$_2$ layer is on the right).
Open symbols represent the change in 3$d$ $e_g$ occupation,
and closed symbols the total magnetic moment.
A schematic illustration (right) of the effect of O-vac and O-ads on the BTO polarization (arrows)
and the Mn 3$d$ $e_g$ occupation (blue lobes) and total Mn magnetic moment (numbers).
}}
\end{figure}
\subsection{Electrochemical switching}
Now we discuss how the
electrochemical processes could proceed in practice during the AFM switching
experiments of Ref.~\onlinecite{Garcia2009} (general electrochemical processes
on oxide surfaces are reviewed in Ref.~\onlinecite{Kalinin,Waser2009}).
As schematically shown in Fig. 1 (right), a biased tip close to contact can
remove surface ions. These would then undergo a redox reaction at the tip surface.
This process is favored by the energy associated with the biased external
circuit, $QV_{ext}$, but costs an energy equal to the change in binding
energy of the ion to the ferroelectric surface and to the tip surface, $\Delta E_f$
(this effectively redefines the relevant chemical potential).
By minimizing the Gibbs free energy of the system
(see e.g. Ref.~\onlinecite{Stephenson2011} or~\onlinecite{Morozovska2010})
it can be shown that poling can stabilize redox
defects if $V_{ext}>\Delta E_f/Q$, after which the equilibrium redox charge
density, $Q/S$, and polarization both grow with $V_{ext}$.
This electrochemical process would then act as a nanobattery,
rather than a nanocapacitor.
By controlling the environment (species and chemical potential)
and $V_{ext}$, one may be also able to selectively control the
active redox reaction, potentially opening new routes to surface
redox catalysis.
After removal of the tip, the surface redox density from poling can remain,
since the reverse reaction is now blocked by
key reactants being removed with the tip.
This would explain the observation of Ref.~\onlinecite{Garcia2009} that the
domains are stable for a very long time after ``writing''.
Of course, lateral charge diffusion across domain boundaries~\cite{Kalinin2004} may
still occur in principle, but kinetic barriers are likely to hinder such processes.
Therefore the bulk polarization, $P_0$, is expected to be an estimate of the equilibrium
polarization after poling.
We note that unlike in the LaAlO$_3$/SrTiO$_3$ system where the polarization is driving the
surface chemistry~\cite{Bristowe2011a}, in ferroelectric films we expect it is the surface chemistry (and poling)
that is driving the polarization.
This is because the energy scale for changing the polarization is much larger
in LaAlO$_3$ than in the ferroelectric.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.4\textwidth]{4a.pdf}
\includegraphics[
width=0.4\textwidth,viewport= 50 5 800 620,clip]{4b}
\caption{\label{TRAP}{
Top: Schematic illustration of the change in
band offset, $E_{\mathrm{VO}}$, with polarization reversal.
Bottom: Tunnel electroresistance (TER)
vs BTO thickness.
Experimental points taken from Garcia $et$ $al$.~\cite{Garcia2009}
(squares with dashed line fit)
are compared with a theoretical expression~\cite{Gruverman2009}
which uses the tunneling barrier height expected
from the BTO bulk polarization,
$P_0$ (solid line).
Inset: Calculated band offset against
electric displacement field for the three BTO states.
The straight line fit is used to determine the band offset (and hence barrier height)
at $\pm P_0$ for the TER plot.
}}
\end{center}
\end{figure}
\subsection{Magnetoelectric coupling}
The electronic transfer mechanism can be quantitively
estimated through the change in magnetization of LSMO.
LSMO is a half-metal with only Mn 3$d$ $e_g$ majority spin levels around
the Fermi level.
As the screening carriers are fully spin-polarized, an electronic transfer between LMSO
and the BTO surface results in a systematic change of the magnetization near the interface.
We calculate the change in magnetization from the pristine to the O-vac and O-ads
systems and to the 2OH and 2H systems, $\Delta M$, as $\pm$1.7 $\mu_B$
and $\pm$1.5 $\mu_B$ in the supercell, equivalent to $\pm$0.42 $e/S$
and $\pm$0.37 $e/S$ respectively
(the remaining 0.1 electrons/holes stay in BTO, see Appendix).
This extra electron density (which
corresponds to the electric displacement, $D$, because of the
half-metallic nature of LSMO)
resides in the interface region, decaying into the electrode
with an associated Thomas-Fermi screening length (see Fig. 3).
This situation is similar to the carrier-mediated magnetoelectricity already
predicted at SrTiO$_3$/SrRuO$_3$ interfaces~\cite{Rondinelli2007}
and in LSMO/BTO superlattices~\cite{Burton2009}.
In agreement with Ref.~\onlinecite{Burton2009} a competing interface antiferromagnetic type phase
(called A$_1$ in Table 1 of Ref.~\onlinecite{Burton2009}) was found for the outwards BTO polarization.
A similar magnetoelectric effect has recently been experimentally
realized~\cite{Vaz2010,Molegraaf2009}.
\subsection{Tunneling Electroresistance}
We now discuss how the electrochemical switching process may lead to the
giant TER observed in the LSMO/BTO system~\cite{Garcia2009}.
In the simplest semiclassical approximation,
TER has an exponential dependence on the tunneling barrier shape~\cite{Gruverman2009}.
The interface dipole, and hence band offset ($E_{\mathrm{VO}}=E_{\mathrm{VBM}}-E_F$),
at a metal/ferroelectric interface depends linearly on the electric displacement field, $D$,
in a way that can be expressed with an effective screening length~\cite{Junquera2003a,Zhuravlev2005,Kohlstedt2005}, $\lambda_{eff}$.
For LSMO-BTO we calculate $\lambda_{eff}=0.11$ \AA.
Using the calculated values of the band offset (Fig. 4 inset) and the
experimental band gap of BTO, we obtain the change in
barrier height upon complete polarization reversal (for $D=\pm P_0$ the
potential in BTO is flat, i.e. the tunneling barrier shape is rectangular),
$\Delta \varphi$, and the average barrier height,
$\bar\varphi=(\varphi_{\mathrm{out}}+\varphi_{\mathrm{in}})/2$.
These values then yield an estimate of the TER using the exponential
dependence~\cite{Gruverman2009} on the barrier thickness, $d$, for large TER,
\begin{equation}
\mathrm{TER} \approx \mathrm{exp}\left[\frac{\sqrt{2m}}{\hbar}\frac{\Delta\varphi}{\sqrt{\bar{\varphi}}}d\right].
\end{equation}
Fig. 4 compares this estimate with the experimental data~\cite{Garcia2009}
showing that this simple model captures remarkably well the essential
physics of TER in this system.
We note a recent study reported comparable shifts in $E_{\mathrm{VO}}$ (measured using
photoelectron spectroscopy) on a similar ferroelectric/LSMO system upon polarization
reversal~\cite{Wu2011}.
The origin of electroresistance effects in oxide nanotubes has also recently
been suggested as redox reactions~\cite{Nonnenmann2010}.
However the redox arguments there are fundamentally different - it is proposed that
the electrons yielded by oxygen vacancies are directly available for conduction.
\section{Conclusions}
In conclusion we have studied an electrochemical mechanism
for ferroelectric switching in thin films and proposed it as the origin of
switchable ferroelectricity, TER and magnetoelectricity
in a prototypical system.
This work opens several avenues for future research.
From the experimental point of view, it would be interesting to investigate
the composition of a ferroelectric surface before and after switching
(e.g. via the AFM tip), to verify whether reduced or oxidized gas-phase
species are present (as suggested by our results).
Also, this point could be indirectly checked by performing the AFM-mediated switching
experiments in a controlled atmosphere, in analogy to the experiments
of Bi et al. [19] on LAO/STO.
From the theoretical point of view, a natural next step would be to
perform a more detailed thermodynamic analysis of the stability of a
ferroelectric surface (either pristine or decorated with adsorbates).
This would involve exploring different coverages, possible inhomogeneous
polarization states, and the effect of temperature and other external
perturbations.
We hope that our results will stimulate further
investigations along these (and possibly other) directions.
\begin{acknowledgments}
We acknowledge G Catalan, J \'{I}\~{n}iguez, M Bibes, V Garcia,
N Mathur, X Moya, J Junquera, C Ocal and S Streiffer for valuable discussions,
the support of EPSRC, NANOSELECT and MCINN FIS2009-12721-C04-01
and computing resources of CamGRID at Cambridge, the
Spanish Supercomputer Network and HPC Europa.
PBL acknowledges DOE support under FWP 70069.
\end{acknowledgments}
\begin{figure*
\centering
\begin{minipage}{.32\textwidth}
\includegraphics[width=\textwidth]{5a.pdf}
\end{minipage}
\begin{minipage}{.32\textwidth}
\includegraphics[width=\textwidth]{5b.pdf}
\end{minipage}
\begin{minipage}{.32\textwidth}
\includegraphics[width=\textwidth]{5c.pdf}
\end{minipage}
\begin{minipage}{.32\textwidth}
\includegraphics[width=\textwidth]{5d.pdf}
\end{minipage}
\begin{minipage}{.32\textwidth}
\includegraphics[width=\textwidth]{5e.pdf}
\end{minipage}
\begin{minipage}{.32\textwidth}
\includegraphics[width=\textwidth]{5f.pdf}
\end{minipage}
\caption{(Color online) Spin-resolved layer-by-layer density of states centered around $E_\mathrm{f}$ for the LSMO/BTO systems.
Positive DOS represents majority spin, negative DOS minority spin. Only the BTO layers (and 1 LSMO layer)
are shown for clarity.
Panels (a)-(e) correspond to pristine, O-ads, O-vac, 2OH and 2H systems respectively.
Panel (f) shows a profile of $\rho_{free}$ through the various LSMO/BTO systems.}
\end{figure*}
| -12,261.832255 |
[
-3.37890625,
3.11328125
] | 39.519651 |
[
-2.619140625,
0.339111328125,
-2.154296875,
-5.1875,
-0.55810546875,
7.72265625
] |
[
3.71484375,
7.55859375,
3.22265625,
6.23046875
] | 180 | 2,720 |
[
-2.892578125,
3.376953125
] | 24.44455 |
[
-6.03125,
-4.01953125,
-3.916015625,
-2.51171875,
1.83984375,
12.078125
] | 0.702033 | 14.652643 | 35.882353 | 2.409251 |
[
2.55610728263855
] | -9,480.457195 | 6.022426 | -11,876.884373 | 0.470057 | 5.938731 |
[
-3.12109375,
-3.791015625,
-3.236328125,
-4.0859375,
2.52734375,
11.3046875
] |
[
-5.23046875,
-1.5546875,
-1.591796875,
-0.8779296875,
3,
3.443359375
] | |
BkiUbAU5qsJBjjX4UzLd
|
\section{Introduction}
\label{sec:intro}
Generative adversarial networks (GANs)~\cite{GAN} are a representative type of generative models, which push the fake data distribution towards the real one via adversarial learning, making the sampled data (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, images) more similar to the real one.
In general, the GAN models are comprised of two parts, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, a generator $\mathit{G}$ and a discriminator $\mathit{D}$.
Specifically, the generator aims at generating fake samples as realistic as possible, while the discriminator, in turn, aims at distinguishing between real and fake samples to avoid being fooled by the generator.
During the adversary process, the ability of $\mathit{G}$ and $\mathit{D}$ can be constantly improved.
Finally, they may reach a balanced state (\emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, Nash equilibrium), where $\mathit{G}$ is capable of generating images that $\mathit{D}$ cannot tell between real and fake.
The past few years have witnessed the rapid development of GANs~\cite{GAN}, and a series of works make tremendous efforts to improve the quality of GAN-generated images in terms of network structures~\cite{LapGAN,DCGAN,StackGAN,SAGAN}, loss functions~\cite{LSGAN,BEGAN,RaGAN,WGAN,WGAN-GP,SNGAN}, and so on.
Recently, several GAN models (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, PGGAN~\cite{PGGAN}, BigGAN~\cite{BigGAN}, and StyleGAN series~\cite{StyleGAN,StyleGAN2,StyleGAN2-Ada,StyleGAN3}) have shown superior image generation ability to produce photo-realistic high-resolution (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, $1024\times1024$) images.
Therefore, the exploration on leveraging GANs for enhancing image quality has drawn upsurging attention.
Prior works often simply take the adversarial loss as an extra term in their learning objectives.
On the one hand, the discriminator and adversarial loss can serve as out-of-the-box components, which could be readily incorporated into many other models, and bring no extra complexity during inference.
On the other hand, applying adversarial training generally can avoid the models being optimized to a blurry average solution, which makes the output image contain more sharp and clear details.
As such, the effectiveness has been verified in many vision tasks such as image editing~\cite{Pix2pix,CycleGAN,StarGAN,AttGAN,STGAN,StarGANv2}, image super-resolution~\cite{SRGAN,ESRGAN,Real-ESRGAN,BSRGAN}, and image deblurring~\cite{DeblurGAN,DeblurGANv2,DBGAN}.
However, with the development of GANs, it becomes much harder for such a utilization scheme to fully explore the potential of GAN models.
Therefore, many recent works turn to explore and leverage pre-trained GAN models, whose advantages are analyzed as follows.
\begin{figure}[!t]
\centering
\footnotesize
\begin{tabular}{cccccc}
\includegraphics[width=0.16\textwidth]{./figures/Figure1/A.jpg} \hspace{-2.75mm} & \hspace{-2.75mm}
\includegraphics[width=0.16\textwidth]{./figures/Figure1/inter_0.2.jpg} \hspace{-2.75mm} & \hspace{-2.75mm}
\includegraphics[width=0.16\textwidth]{./figures/Figure1/inter_0.4.jpg} \hspace{-2.75mm} & \hspace{-2.75mm}
\includegraphics[width=0.16\textwidth]{./figures/Figure1/inter_0.6.jpg} \hspace{-2.75mm} & \hspace{-2.75mm}
\includegraphics[width=0.16\textwidth]{./figures/Figure1/inter_0.8.jpg} \hspace{-2.75mm} & \hspace{-2.75mm}
\includegraphics[width=0.16\textwidth]{./figures/Figure1/B.jpg} \\[-2mm]
Image $\textbf{x}_1$ \hspace{-2.75mm} & $\alpha=0.2$ \hspace{-2.75mm} & $\alpha=0.4$ \hspace{-2.75mm} & $\alpha=0.6$ \hspace{-2.75mm} & $\alpha=0.8$ \hspace{-2.75mm} & Image $\textbf{x}_2$ \\
\includegraphics[width=0.16\textwidth]{./figures/Figure1/A_inv.jpg} \hspace{-2.75mm} & \hspace{-2.75mm}
\includegraphics[width=0.16\textwidth]{./figures/Figure1/0.2.jpg} \hspace{-2.75mm} & \hspace{-2.75mm}
\includegraphics[width=0.16\textwidth]{./figures/Figure1/0.4.jpg} \hspace{-2.75mm} & \hspace{-2.75mm}
\includegraphics[width=0.16\textwidth]{./figures/Figure1/0.6.jpg} \hspace{-2.75mm} & \hspace{-2.75mm}
\includegraphics[width=0.16\textwidth]{./figures/Figure1/0.8.jpg} \hspace{-2.75mm} & \hspace{-2.75mm}
\includegraphics[width=0.16\textwidth]{./figures/Figure1/B_inv.jpg} \\[-2mm]
$G(E(\textbf{x}_1))$ \hspace{-2.75mm} & $\beta=0.2$ \hspace{-2.75mm} & $\beta=0.4$ \hspace{-2.75mm} & $\beta=0.6$ \hspace{-2.75mm} & $\beta=0.8$ \hspace{-2.75mm} & $G(E(\textbf{x}_2))$
\end{tabular}
\caption{Image interpolation in the pixel space and the latent space. $\textbf{x}_1$ and $\textbf{x}_2$ are two images of the same object, while $G(E(\mathbf{x}))$ means the inversion result of $\mathbf{x}$. The first row shows the results of interpolating directly in the pixel space via $\mathbf{x}=\alpha\cdot\mathbf{x}_1+(1-\alpha)\cdot\mathbf{x}_2$. In contrast, the second row shows the results of interpolating in the latent space via $\mathbf{x}=G(\beta\cdot\mathit{E}(\mathbf{x}_1)+(1-\beta)\cdot\mathit{E}(\mathbf{x}_2))$. It can be seen that the interpolation in the latent space is semantically more plausible. The sample images are from the tower subset of LSUN dataset~\cite{LSUN}, and the results in the second row are generated via the officially released model of IDInvert~\cite{IDInvert}.}
\label{fig:intro_interpolation}
\end{figure}
\vspace{0.5em}
\noindent\textbf{Disentanglement of GAN Latent Space.}
For many vision tasks like image editing, it is a basic problem to identify image contents and the manipulation mode of a concept to be edited.
Since the concepts are entangled with each other and form an interlaced group of editing spaces (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, changing a facial image to \textit{smile} or \textit{old} can both cause wrinkles).
Even with the well-labeled datasets (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, CelebA~\cite{CelebA}), it is still a very challenging task to describe the manipulation mode directly on the pixels~\cite{StarGAN,AttGAN,STGAN,StarGANv2}.
As a remedy, pre-trained GAN models provide yet another possible solution, where the disentanglement of latent spaces has been explored recently~\cite{StyleGAN,voynov2020unsupervised}.
As shown in \cref{fig:intro_interpolation}, interpolation between images will cause obvious artifacts, while the same operation in the latent space will be semantically more plausible.
Such a phenomenon clearly illustrates the latent space disentanglement ability of recent GAN models.
\vspace{0.5em}
\noindent\textbf{Leveraging pre-trained GAN Priors.}
For tasks with image outputs, regularization and natural image priors are often applied to enhance the output image quality, \emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, local smoothness~\cite{rudin1992nonlinear}, non-local self-similarity~\cite{buades2005non}, sparsity~\cite{elad2006image}, \emph{etc}\onedot} \def\vs{\emph{vs}\onedot.
Nevertheless, they are either hand-crafted with manually tuned parameters, or based on strong assumptions that are usually not applicable in real applications, making the performance and flexibility greatly limited.
On the contrary, the learning objective of GAN models is to generate high-quality diverse images from the latent code (which is generally a random noise).
In other words, given a vector located in the latent space, the well-trained GAN models are very likely to generate a high-quality output.
Therefore, the pre-trained GAN models are naturally able to provide stronger and more reliable high-quality natural image priors.
Furthermore, the GAN priors can be helpful to a wide range of tasks, making it with much more practical significance.
\vspace{0.5em}
\noindent\textbf{Training Scheme \& Development Potentials.}
Collecting pairwise training data is extremely time-consuming and expensive, and it is impossible to collect abundant training data for all tasks.
Fortunately, GAN models are trained in an unsupervised manner, where the training images are much easier to collect without requiring ground-truth labels or reference images.
Indeed, there are already a huge amount of unlabeled images, and the unsupervised training scheme of GAN models makes it easier to apply the extremely large-scale training sets and to train models with greater capacity for better performance.
As a result, many recent works show rising interest on pre-trained GAN models and have developed a series of methods.
In this paper, we review the recent progress on leveraging pre-trained GAN models, from understanding GAN models~\cite{GANDissection,GANSeeing,GANalyze,GANSpace} to leveraging them for subsequent tasks such as image editing~\cite{suzuki2018spatially,GANPaint,StyleRig,StyleFlow} and restoration~\cite{PULSE,pSp,GLEAN,GFPGAN,GPEN}, mainly involving,
\begin{itemize}
\item We survey the characteristics of pre-trained GAN models, and the efforts for exploring and acquiring these characteristics are also introduced.
\item Focusing on image restoration and editing tasks, we summarize and compare relevant methods to utilize pre-trained GAN models.
\item We discuss the open problems and potential research directions in this field.
\end{itemize}
The rest of this paper is organized as follows.
To begin with, \cref{sec:GANs} introduces recent progress of GAN models, including network architecture, training mechanism, and so on.
Then, by exploring the characteristics of pre-trained GAN models (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, latent space exploration), the relevant methods and experiments for understanding pre-trained GAN models are reviewed in \cref{sec:understanding}.
Based on the impressions of latent space, we introduce and summarize the methods in \cref{sec:applications}, which leverage pre-trained GANs for subsequent applications, especially the image editing and restoration tasks.
Finally, we discuss the open problems in this field, and point out some potential directions for future research in \cref{sec:discussion}.
\section{Pre-trained Generative Adversarial Networks}
\label{sec:GANs}
\subsection{Preliminary}
\label{sec:GANs_preliminary}
In the deep learning~\cite{deep_learning} era, when researchers are referring to a task, generally there accompany two fundamental problems, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, \textit{data sets} and \textit{evaluation metrics}.
For GAN models, these two problems are even more important.
On the one hand, numerous training samples are typically required to avoid over-fitting and to tap the potential of their superior capacity.
On the other hand, the evaluation of generative models is more difficult since there are no ground-truth or labels for assessment, and generating thousands or even millions of images for user study is a mission impossible due to the time and labor costs.
Therefore, before introducing the large-scale GAN models, we brief the relevant datasets, as well as the metrics for evaluating the pre-trained GAN models.
Besides, we summarize the symbols used in this survey in \cref{tab:symbols}.
\begin{table}[t]
\centering
\caption{A summary on the symbols used in this survey.}\label{tab:symbols}
\scalebox{0.54}{
\begin{tabular}{ccc}
\toprule
Format & Definition & Examples \\
\midrule
Upper-case italic letters (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, $\mathit{G}$) & Models or networks & Generator $\mathit{G}$; discriminator $\mathit{D}$; encoder $\mathit{E}$; MLP $\mathit{F}$; segmentation model $\mathit{S}$. \\
Bold letters (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, $\mathbf{z}$) & Images or tensors & Image $\mathbf{x}$, $\mathbf{I}$; latent code (or random noise) $\mathbf{z}$, $\mathbf{w}$, $\mathbf{p}$, $\mathbf{n}$, \emph{etc}\onedot} \def\vs{\emph{vs}\onedot; learned input $\mathbf{m}$; Fourier feature input $\mathbf{F}$; condition $\mathbf{c}$. \\
Handwritten letter (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, $\mathcal{Z}$) & Latent spaces or distributions & Latent spaces $\mathcal{Z}$, $\mathcal{C}$, $\mathcal{W}$, $\mathcal{S}$, $\mathcal{P}$, $\mathcal{N}$, $\mathcal{F}$; loss functions $\mathcal{L}$ \\
Greek letters (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, $\alpha$) & Parameters or specified & Interpolation parameters or hyperparameters $\alpha$, $\beta$, $\lambda$; $\bm{\theta}$ model parameters; any possible network inputs $\bm{\delta}$. \\
Double struck letter (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, $\mathbb{R}$) & Following mathematical definitions & Mathematical expectation $\mathbb{E}$; Spatial dimension $\mathbb{R}$. \\
\bottomrule
\end{tabular}}
\end{table}
\subsubsection{Image Datasets for Training Large-scale GAN Models}
\vspace{0.5em}
\noindent\textbf{ImageNet~\cite{ImageNet}} is the very first large-scale natural image dataset, which contains over 14M images from more than 21K classes (referred to as ImageNet-21K), and the average image size is $469\times387$.
The commonly used configuration is a 1,000-class subset with $\sim$1K images per class (denoted by ImageNet-1K), and the images are usually resized to $224\times224$ or $256\times256$ in practice.
\vspace{0.5em}
\noindent\textbf{CelebA~\cite{CelebA}} (Large-scale CelebFaces Attributes Dataset) contains more than 200K facial images from 10k celebrities, each of the images is annotated with 40 attributes, and the aligned images are resized and cropped to $178\times218$.
A subset of CelebA is post-processed to form the CelebA-HQ~\cite{PGGAN} dataset, containing 30K facial images with the resolution of $1024\times1024$.
The dataset is also extended with annotations like CelebAMask-HQ~\cite{MaskGAN} via pixel-wise facial component labeling (face parsing).
\vspace{0.5em}
\noindent\textbf{LSUN~\cite{LSUN} \& AFHQ~\cite{StarGANv2}} are another two datasets of scene and object categories.
The LSUN dataset is composed of 10 scene categories and 20 object categories, each of which contains around 1M labeled images, and the images are provided by resizing the shorter edge to 256 pixels and compressing to JPEG image quality of 75.
On the contrary, AFHQ (Animal Faces-HQ Dataset) is composed of only three categories (\emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, cat, dog, and wildlife), each of which contains around 5K high-quality animal face images with a resolution of $512\times512$.
Recent methods often take a specific category of these two datasets for training their GAN models, \emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, \textit{bedroom}, \textit{church} from LSUN and \textit{cat} from AFHQ.
\vspace{0.5em}
\noindent\textbf{FFHQ~\cite{StyleGAN}} (Flickr-Faces-HQ Dataset) is a high-quality image dataset collected from Flickr\footnote{\url{https://www.flickr.com/}}, containing 70K automatically aligned high-resolution ($1024\times1024$) facial images with diverse data distribution.
To date, FFHQ is the most popular dataset for training GAN models for facial image generation. Image editing and restoration methods based on pre-trained GANs are often trained on FFHQ dataset and evaluated on CelebA-HQ~\cite{PGGAN} to show the generalization ability.
\vspace{0.5em}
\noindent\textbf{Other Datasets}
Apart from the aforementioned ones, some other datasets~\cite{MNIST,SVHN,CIFAR,StyleGAN2-Ada,DeepFashion,Cityscapes} are also employed for prototyping and/or training large-scale GAN models.
To sum up, large scale and favorable image quality are two key ingredients of datasets for training high-quality large-scale GAN models.
Therefore, some other datasets like Objects 365~\cite{Objects365}, Places 365~\cite{Places365}, and Open Images~\cite{OpenImages} may also be used for training GAN models with greater capacity.
\subsubsection{Evaluation Metrics for Assessing GAN models}
\vspace{0.5em}
\noindent\textbf{Mean Opinion Score} (MOS) is the most intuitive index to assess the perceptual quality of images, since the image quality assessment is conducted by human raters.
However, assessing via MOS is very expensive, and the results may be biased due to factors like subjective perceiving differences.
Furthermore, the time-consuming procedure makes it suitable for small-scale assessment, \emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, user study, but hard to be employed for the evaluation during training and broader comparison.
\vspace{0.5em}
\noindent\textbf{Inception Score~\cite{InceptionScore}} (IS) is a commonly used metric for evaluating the image quality and class diversity.
Considering a well-trained classifier (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, Inception-v3~\cite{Inception-v3} trained on ImageNet~\cite{ImageNet} for calculating IS), the high confidence of classifying a generated image $\mathbf{\hat{x}}$ into a class $\mathit{c}$ (\emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, smaller entropy of $\mathit{p}(\mathit{c}|\mathbf{\hat{x}})$) generally implies decent image generation quality.
On the other hand, better class diversity requires that the marginal probability distribution of class label approximates uniform distribution, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, higher entropy of $\mathit{p}(\mathit{c})$.
Therefore, the Inception Score is defined by
\begin{equation}
\mathit{IS} = \exp(\mathbb{E}_{\mathbf{\hat{x}}\sim\mathit{p}_\mathit{g}}\mathit{D}_\mathit{KL}(\mathit{p}(\mathit{c}|\mathbf{\hat{x}})\|\mathit{p}(c))),
\end{equation}
where $\mathbf{\hat{x}}$ is sampled from the generated image distribution $\mathit{p}_\mathit{g}$, $\mathit{D}_\mathit{KL}(\cdot\|\cdot)$ denotes Kullback-Leibler (K-L) divergence.
However, IS is unable to detect mode collapse, \emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, a high Inception Score will be derived when the generator produces one and only one high-quality image for each class.
\vspace{0.5em}
\noindent\textbf{Fr\'{e}chet Inception Distance~\cite{FID}}
Considering the drawbacks of IS, Fr\'{e}chet Inception Distance (FID) takes the real images into consideration and calculates the distance between generated images and real ones.
Specifically, FID is defined by
\begin{equation}
\mathit{FID}=\|\mu_\mathit{r}-\mu_\mathit{g}\|_2^2+\mathit{Tr}(\Sigma_\mathit{r}+\Sigma_\mathit{g}-2(\Sigma_\mathit{r}\Sigma_\mathit{g})^{\frac{1}{2}}),
\end{equation}
where $\mathit{Tr}(\cdot)$ denotes trace of a matrix.
The real and generated image features extracted by Inception-v3~\cite{Inception-v3} are both assumed to follow the multivariate Gaussian distribution, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, $\phi(\mathbf{x})\sim\mathcal{N}(\mu_\mathit{r},\Sigma_\mathit{r})$, $\phi(\mathbf{\hat{x}})\sim\mathcal{N}(\mu_\mathit{g},\Sigma_\mathit{g})$.
By utilizing the real samples, FID is not restricted by the training dataset (\emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, ImageNet~\cite{ImageNet} for Inception-v3~\cite{Inception-v3}), and is much more sensitive to mode collapse.
\vspace{0.5em}
\noindent\textbf{Sliced Wasserstein Distance~\cite{SWD}}
It is worth noting that the Gaussian assumption of FID does not always hold true.
To avoid such assumption, Bonneel \emph{et al}\onedot~\cite{SWD} proposed to calculate the Sliced Wasserstein distance (SWD) to approximate the Wasserstein distance, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot,
\begin{equation}
\mathit{SWD}(\mathbf{I}_\mathit{x},\mathbf{I}_\mathit{y})=(\int_{\mathbb{S}^{\mathit{d}-1}}\mathit{W}^\mathit{p}_\mathit{p}(\mathcal{R}\mathbf{I}_\mathit{x}(.,\theta),\mathcal{R}\mathbf{I}_\mathit{y}(.,\theta))\mathrm{d}\theta)^{\frac{1}{\mathit{p}}},
\label{eqn:SWD}
\end{equation}
where $\mathbb{S}^{\mathit{d}-1}$ is the unit sphere in $\mathbb{R}^\mathit{d}$, $p$ is set to 2 in practice, $\mathcal{R}$ denotes the Radon transform.
Please refer to \cite{GSWD} for more details.
\vspace{0.5em}
\noindent\textbf{GAN-train \& GAN-test~\cite{GAN-train}}
Another problem in GAN training is over-fitting, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, the model memorizes the training samples for generation.
The GAN-train and GAN-test indices provide a more comprehensive evaluation on GAN model.
In particular, GAN-train is the accuracy of a classifier trained on a generated dataset $\mathcal{D}_\mathit{g}$ and evaluated on a real-image validation dataset $\mathcal{D}_\mathit{r}^\mathit{val}$, while GAN-test is the accuracy of a classifier trained on real-image training set $\mathcal{D}_\mathit{r}^\mathit{train}$ and tested on $\mathcal{D}_\mathit{g}$.
These two metrics can identify problems such as mode dropping, poor image quality, over-fitting, \emph{etc}\onedot} \def\vs{\emph{vs}\onedot.
\vspace{0.5em}
\noindent\textbf{Fr\'{e}chet Segmentation Distance~\cite{GANSeeing}}
As shown in \cite{GANSeeing}, the mode collapse occurs not only at the distribution level, but also at the instance level.
In particular, certain objects may not be generated by a GAN model, and this phenomenon is termed by mode dropping.
For assessing the mode dropping in a GAN model, the Fr\'{e}chet Segmentation Distance (FSD) is defined by modifying FID, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot,
\begin{equation}
\mathit{FSD}=\|\mu_\mathit{g}-\mu_\mathit{t}\|_2^2+\mathit{Tr}(\Sigma_\mathit{g}+\Sigma_\mathit{t}-2(\Sigma_\mathit{g}\Sigma_\mathit{t})^{\frac{1}{2}}),
\end{equation}
where $\mu_\mathit{t}$ ($\mu_\mathit{g}$) is the mean pixel count for each object class over a sample of training images (generated images), and $\Sigma_\mathit{t}$ ($\Sigma_\mathit{g}$) is the covariance of these pixels.
Besides, many other metrics like Perceptual Path Length~\cite{StyleGAN}, Linear Separability~\cite{StyleGAN}, Precision and Recall~\cite{PrecRecall}, and Geometry Score~\cite{GeometryScore} are also explored for better evaluating GANs. We recommend \cite{GAN_Metrics} for a more comprehensive survey on GAN metrics.
\begin{figure
\centering
\begin{overpic}[width=.95\linewidth]{./figures/GAN_arch.pdf}
\tiny
\put(3.5, 98.5){\color{Blue}$\mathbf{z}\!\in\!\mathbb{R}^{512}$}
\put(42.2, 98.5){\color{Blue}Score}
\put(22.5, 83.5){\color{Blue}$\alpha$}
\put(21.5, 85.5){\color{Blue}$1\!-\!\alpha$}
\put(43, 85.5){\color{Blue}$\alpha$}
\put(38.5, 88){\color{Blue}$1\!-\!\alpha$}
\put(60, 98.5){\color{red}$\mathbf{c}\!\in\!\mathbb{R}^{128}$}
\put(66, 98.5){$\mathbf{z}\!\in\!\mathbb{R}^{160}$}
\put(80, 98.5){Score}
\put(56.5, 95.5){$\mathbb{R}^{140}$}
\put(56.5, 93.8){\color{red}$\mathbb{R}^{128}$}
\put(68.5, 92.5){$\mathbb{R}^{128+20}$}
\put(56.5, 84.3){\color{red}$\mathbb{R}^{128}$}
\put(56.8, 82.6){$\mathbb{R}^{20}$}
\put(56.5, 77.4){\color{red}$\mathbb{R}^{128}$}
\put(56.8, 75.7){$\mathbb{R}^{20}$}
\put(56.5, 68.8){\color{red}$\mathbb{R}^{128}$}
\put(56.8, 67.1){$\mathbb{R}^{20}$}
\put(2, 58.5){$\mathbf{z}\!\in\!\mathbb{R}^{512}$}
\put(12.5, 58.5){$\mathbf{m}\!\in\!\mathbb{R}^{4\!\times\!4\!\times\!512}$}
\put(22.2, 58.5){$\mathbf{n}\!\in\!\mathbb{R}^{\mathit{H}\!\times\!\mathit{W}}$}
\put(2, 26){$\mathbf{w}\!\in\!\mathbb{R}^{512}$}
\put(54.3, 58.5){Score}
\put(67.5, 58.5){$\mathbf{z}\!\in\!\mathbb{R}^{512}$}
\put(67.5, 26){$\mathbf{w}\!\in\!\mathbb{R}^{512}$}
\put(0, 4){$\mathcal{Z}$ Space}
\put(27.5, 4){$\mathcal{P}$ Space}
\put(36, 4){$\mathcal{W}$ Space}
\put(50.5, 4){$\mathcal{W}+$ Space}
\put(57.9, 4){$\mathcal{F}$ Space}
\put(63.7, 4){$\mathcal{N}$ Space}
\end{overpic}
\caption{Illustration of recent GAN models (see (a)$\sim$(d)) and the latent spaces of StyleGAN series~\cite{StyleGAN} (see (e)). (a)~For PGGAN~\cite{PGGAN}, the {\color{Blue}blue part} denotes the progressive growing procedure from $4\times4$ to $8\times8$. The components with dash lines are employed for the fade-in strategy, where $\alpha$ is gradually growing to 1. They are discarded when the model grows to a higher-resolution. (b)~For BigGAN~\cite{BigGAN}, a specific noise is delivered to each layer together with the class embedding, and the model is end-to-end trained without the progressive growing procedure. (c)~For StyleGAN~\cite{StyleGAN}, a series of FC layers are deployed to map $\mathbf{z}$ into $\mathbf{w}$. The {\color{Green}green part} only belongs to StyleGAN2. (d)~For StyleGAN3~\cite{StyleGAN3}, the generator is largely modulated to improve the translational and rotation equivariance. The discriminator is omitted since it is identical with that used in StyleGAN2~\cite{StyleGAN2}. (e)~For simplicity, here we take the StyleGAN series as an example to show the latent spaces based on GAN inversion task.}
\label{fig:GAN_arch}
\end{figure}
\subsection{Large-scale GAN models}
\label{sec:GANs_development}
GANs are originally proposed for image generation.
As such, we take image generation as an example to introduce the basic concepts and training schemes of GANs, which could be naturally generalized to other domains.
A typical GAN model~\cite{GAN,DCGAN} is composed of a generator and a discriminator, which are denoted by $\mathit{G}$ and $\mathit{D}$, respectively.
Given a random vector $\mathbf{z}\in\mathcal{Z}$, the generator maps it to an image $\mathbf{\hat{x}}$,
\begin{equation}
\mathbf{\hat{x}}=\mathit{G}(\mathbf{z}, \bm{\delta};\bm{\theta}_\mathit{G}),
\label{eqn:GAN}
\end{equation}
where $\bm{\delta}$ denotes other possible inputs (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, class condition, layer-wise noise, \emph{etc}\onedot} \def\vs{\emph{vs}\onedot), $\bm{\theta}_\mathit{G}$ represents the parameters of $\mathit{G}$, $\mathbf{\hat{x}}\in\mathbb{R}^{\mathit{H}\times\mathit{W}\times\mathit{C}}$ denotes the generated $\mathit{C}$-channel image with the size of $\mathit{H}\times\mathit{W}$.
Then, the discriminator $\mathit{D}$ takes $\mathbf{\hat{x}}$ (or a real image $\mathbf{x}$) as input, and predicts the probability that the input is from the distribution of real images or the distance to the real image distribution\footnote{Precisely speaking, the metric is \textit{diversity} rather than \textit{distance} for some adversarial loss functions such as vanilla GAN~\cite{GAN}.}, and the learning objective is defined by,
\begin{equation}
\min_\mathit{G}\max_\mathit{D}\mathcal{L}_\mathit{GAN}(\mathit{G},\mathit{D})=\mathbb{E}_{\mathbf{x}\sim\mathit{p}_\mathit{data}(\mathbf{x})}[\log\mathit{D}(\mathbf{x})]+\mathbb{E}_{\mathbf{z}\sim\mathit{p}_\mathit{\mathbf{z}}(\mathbf{z})}[\log(1-\mathit{D}(\mathit{G}(\mathbf{z})))].
\label{eqn:loss_GAN}
\end{equation}
In the following, we mainly introduce a handful of recent GAN models that are deployed in the methods reviewed, and we recommend \cite{GANSurvey,StudioGAN} for a comprehensive review of GANs.
\vspace{0.5em}
\noindent\textbf{PGGAN~\cite{PGGAN}}
Leveraging the progressive growing strategy, PGGAN (or ProGAN) achieves the $1024\times1024$ image generation resolution.
Specifically, at the beginning of the training procedure, a $4\times4$ ``image'' is generated by the initial generator $\mathit{G}^{(4)}$, and the discriminator $\mathit{D}^{(4)}$ accordingly takes $4\times4$ input as well.
Once the model converges at a resolution, another layer is deployed at the end of the generator, which generates images with $2\times$ resolution.
Finally we can obtain $\mathit{G}^{(1024)}$, which could generate high-quality $1024\times1024$ images.
The overall architecture is shown in \cref{fig:GAN_arch}(a).
\vspace{0.5em}
\noindent\textbf{BigGAN~\cite{BigGAN}}
As the name implies, BigGAN analyzes the influence of GAN model size on generation quality and proposes an architecture with more parameters and end-to-end trained with large batch-size.
For more stable training, the top singular value of parameters are used to monitor the mode collapse during training, and a novel regularization term on the generator $\mathit{G}$ is designed according to the analysis.
The $\mathit{R}_1$ zero-centered gradient penalty~\cite{GAN_convergence} is also leveraged in the discriminator $\mathit{D}$.
Besides, to achieve class-conditioned generation, BigGAN delivers the class embedding and noise vector in to each residual block of the generator.
Finally, with the large capacity, BigGAN could generate appealing images with the resolution of $512\times512$.
The overall architecture is shown in \cref{fig:GAN_arch}(b).
\vspace{0.5em}
\noindent\textbf{StyleGAN~\cite{StyleGAN}}
The StyleGAN series~\cite{StyleGAN,StyleGAN2,StyleGAN3} are the most popular GAN models in recent years.
Instead of taking noise as input, Karras~\emph{et al}\onedot proposed to learn a constant input $\mathbf{m}$ (as shown in \cref{fig:GAN_arch}(c)).
And they used a mapping network composed of several fully-connected layers to map the noise $\mathbf{z}$ into a latent representation $\mathbf{w}$, which is delivered to the AdaIN~\cite{AdaIN} layer in each scale.
In addition, layer-wise per-pixel noise was introduced in each scale for further performance improvement and achieved stochastic variation on generated details.
StyleGAN follows the progressive growing training scheme of PGGAN~\cite{PGGAN}, the final-stage structure of StyleGAN is shown as the black part in \cref{fig:GAN_arch}(c).
\vspace{0.5em}
\noindent\textbf{StyleGAN2~\cite{StyleGAN2}}
The overall structure of StyleGAN2 is similar to StyleGAN, and the block structure and regularization terms are modulated for better generalization quality.
Besides, StyleGAN2 deploys a ``toRGB'' module in each scale for substituting the progressive growing strategy.
The lazy regularization and perceptual path length regularization are also deployed.
Furthermore, Karras~\emph{et al}\onedot~\cite{StyleGAN2-Ada} proposed StyleGAN2-Ada based on a delicately designed discriminator augmentation mechanism, which largely improves the generated image quality and significantly stabilizes training in limited data regimes.
\vspace{0.5em}
\noindent\textbf{StyleGAN3~\cite{StyleGAN3}}
Since the StyleGAN architecture was proposed, utilizing pre-trained GAN models for downstream tasks has drawn much attention, making the equivariance a necessity for many applications.
However, the synthesis process of previous GAN models depends on the coordinates of pixels rather than the surface of the generated objects.
Karras~\emph{et al}\onedot analyzed the cause of such conditions, and designed a model with translational equivariance and rotation equivariance, which achieves comparable performance against StyleGAN2 but with even less parameters.
The comparison between these GAN models are shown in \cref{fig:GAN_arch} and \cref{tab:GAN_comparison}.
It can be seen that, in order to achieve realistic generation, these methods keep exploring smoother and more stable \textit{information flow pipelines and architectures}, more disentangled and controllable \textit{input and condition forms}, as well as better \textit{training schemes}.
In the next section, we will delve more deeply into the GAN models, trying to analyze and understand the characteristics and properties of pre-trained GANs.
\begin{table
\centering
\caption{Comparison of recent GAN models. The two numbers of parameters for StyleGAN3~\cite{StyleGAN3} are respectively for the translational equivariant configuration and the rotation equivariant configuration. $\mathbf{z}$, $\mathbf{m}$, and $\mathbf{F}$ denote random noise, learned constant, and Fourier feature input~\cite{FourierFeature}, respectively.}
\label{tab:GAN_comparison}
\scalebox{.75}{
\begin{tabular}{cccccccc}
\toprule
GAN Model & Publication & Resolution & \# Params & Training Scheme & Input & Condition & Regularization \\
\midrule
PGGAN~\cite{PGGAN} & ICLR 2018 & 1024 & 23.1M & Growing & $\mathbf{z}$ & - & Grad. Penalty (GP) \\
BigGAN~\cite{BigGAN} & ICLR 2019 & 512 & 82.5M & End-to-end & $\mathbf{z}$ & $\mathbf{c}+\mathbf{z}$ & Ortho. Reg. \& $\mathit{R}_1$ \\
StyleGAN~\cite{StyleGAN} & CVPR 2019 & 1024 & 26.2M & Growing & $\mathbf{m}$ & $\mathbf{w}+\mathbf{n}$ & Mixing Reg., GP \& $\mathit{R}_1$ \\
StyleGAN2~\cite{StyleGAN2} & CVPR 2020 & 1024 & 30.0M & Multi-scale & $\mathbf{m}$ & $\mathbf{w}+\mathbf{n}$ & Mixing Reg., Lazy $\mathit{R}_1$ \& PPL \\
StyleGAN3~\cite{StyleGAN3} & NeurIPS 2021 & 1024 & 22.3M/15.8M & End-to-end & $\mathbf{F}$ & $\mathbf{w}$ & $\mathit{R}_1$ \\
\bottomrule
\end{tabular}}
\end{table}
\section{Analyzing and Understanding Large-scale GAN Models}
\label{sec:understanding}
In this section, we first analyze the GAN models from the perspective of image generation, and review the recent progress on understanding the neurons of pre-trained GAN models.
Then, to leverage the priority of the recent GAN architectures (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, StyleGAN series~\cite{StyleGAN,StyleGAN2,StyleGAN3}), the frequently used latent spaces are introduced, together with a series of methods for mapping images back to the corresponding latent space (\emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, GAN inversion).
Finally, we explore the current progress on latent space disentanglement, one of the key problems of image manipulation using pre-trained GANs.
\subsection{Neuron Understanding}
\label{subsec:Understanding_neuron}
\subsubsection{Preliminary}
Analogous to other neural networks, GAN~\cite{GAN} models are also trained in a data-driven manner, making them a black box which is hard to understand and interpret.
To open the black box and better understand the image generation process in GAN generators, a series of works~\cite{GANDissection,GANInverting,GANSeeing,GANCorrection} propose to disassemble the pre-trained GAN models and see how objects in the output images are generated.
In particular, as introduced in \cref{sec:GANs_development}, the generation process in the generator $\mathit{G}$ could be written as $\mathbf{\hat{x}}=\mathit{G}(\mathbf{z})$.
Following the main model paradigm in the deep learning era, a typical GAN model is composed of a stack of neural layers (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, a $\mathit{N}$-layer GAN model), thus we can decompose the generator $\mathit{G}$ into two parts at the $\mathit{i}$-th layer, and \cref{eqn:GAN} can be rewritten as,
\begin{equation}
\mathbf{\hat{x}}=\mathit{G}(\mathbf{z})=(\mathit{G}_{\mathit{i}+1}^\mathit{N}\circ\mathit{G}_{1}^\mathit{i})(\mathbf{z})=\mathit{G}_{\mathit{i}+1}^\mathit{N}(\mathit{G}_{1}^\mathit{i}(\mathbf{z})),
\end{equation}
where $\mathit{G}_\mathit{i}$ is the $\mathit{i}$-th layer of $\mathit{G}$, and $\mathit{G}_\mathit{a}^\mathit{b}=\mathit{G}_\mathit{b}\circ\mathit{G}_{\mathit{b}-1}\circ\cdots\circ\mathit{G}_{\mathit{a}+1}\circ\mathit{G}_\mathit{a}$.
Then the features map $\mathbf{f}_\mathit{i}$ at the $\mathit{i}$-th layer (\emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, $\mathbf{f}_\mathit{i}=\mathit{G}_{1}^\mathit{i}(\mathbf{z})$) contains all information for generating the objects in the output images.
\subsubsection{GAN Dissection}
\label{subsubsec:GANDissection}
For a certain convolution layer, the activation at each position is obtained by the sum-of-product operation between the previous features and the convolution kernels.
Since each output channel has a particular kernel, we can see it as a neuron, which potentially controls the generation of some objects, and the discussions in this subsection are based on this observation.
Bau~\emph{et al}\onedot~\cite{GANDissection} first built the relationship between the generated objects and the feature maps from $\mathbf{f}_\mathit{i}$ via network dissection techniques~\cite{NetworkDissection}.
Specifically, for each object class $\mathit{c}$, a semantic segmentation model $\mathit{S}_\mathit{c}$ is deployed to get the binary segmentation result $\mathit{S}_\mathit{c}(\mathbf{\hat{x}})$.
Then, the relationship between this class and a feature map $\mathbf{f}_\mathit{i,u}$ (aka, a neuron) is evaluated by the \textit{spatial agreement} between the thresholded feature map and the segmentation result, which is evaluated by the intersection-over-union (IoU) measure, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot,
\begin{equation}
\mathit{IoU}_\mathit{u,c}=\frac{\mathbb{E}_\mathbf{z}\left|(\mathbf{f}^{\uparrow}_\mathit{i,u} > \mathit{t}_\mathit{u,c})\land\mathit{S}_\mathit{c}(\mathbf{\hat{x}})\right|}{\mathbb{E}_\mathbf{z}\left|(\mathbf{f}^{\uparrow}_\mathit{i,u} > \mathit{t}_\mathit{u,c})\lor\mathit{S}_\mathit{c}(\mathbf{\hat{x}})\right|},\ \mathrm{where}\ \mathit{t}_\mathit{u,c}={\arg\max}_\mathit{t}\frac{\mathbf{I}(\mathbf{f}^{\uparrow}_\mathit{i,u} > \mathit{t};\mathit{S}_\mathit{c}(\mathbf{\hat{x}}))}{\mathbf{H}(\mathbf{f}^{\uparrow}_\mathit{i,u} > \mathit{t};\mathit{S}_\mathit{c}(\mathbf{\hat{x}}))},
\end{equation}
where $\land$ and $\lor$ represent intersection and union operations, $\uparrow$ denotes the upsampling operation, and $\mathit{t}_\mathit{u,c}$ is determined by maximizing the portion of the mutual information $\mathbf{I}$ in the joint entropy $\mathbf{H}$.
\begin{figure}[t]
\centering
\footnotesize
\begin{minipage}{0.475\linewidth}
\centering
\begin{tabular}{cccc}
\hspace{-2.75mm}
\includegraphics[width=0.24\textwidth]{./figures/GANDissection/input.png} \hspace{-2.75mm} & \hspace{-2.75mm}
\includegraphics[width=0.24\textwidth]{./figures/GANDissection/output.png} \hspace{-2.75mm} & \hspace{-2.75mm}
\includegraphics[width=0.24\textwidth]{./figures/GANDissection/input2.png} \hspace{-2.75mm} & \hspace{-2.75mm}
\includegraphics[width=0.24\textwidth]{./figures/GANDissection/output2.png} \\[-1.5mm]
\hspace{-2.75mm}
Input \hspace{-2.75mm} & \hspace{-2.75mm}
Remove Trees \hspace{-2.75mm} & \hspace{-2.75mm}
Input \hspace{-2.75mm} & \hspace{-2.75mm}
Add Grasses
\end{tabular}
\captionof{figure}{Removing or adding particular classes via relevant unit discovery in GANDissection~\cite{GANDissection}.}
\label{fig:GANDissection}
\end{minipage}
\hspace{2mm}
\begin{minipage}{0.475\linewidth}
\centering
\begin{tabular}{cccc}
\hspace{-2.75mm}
\includegraphics[width=0.24\textwidth]{./figures/GANRewritting/input.png} \hspace{-2.75mm} & \hspace{-2.75mm}
\includegraphics[width=0.24\textwidth]{./figures/GANRewritting/input2.png} \hspace{-2.75mm} & \hspace{-2.75mm} \includegraphics[width=0.24\textwidth]{./figures/GANRewritting/output.png} \hspace{-2.75mm} & \hspace{-2.75mm}
\includegraphics[width=0.24\textwidth]{./figures/GANRewritting/output2.png} \\[-1.5mm]
\multicolumn{2}{c}{\hspace{-2.75mm}Two inputs\hspace{-2.75mm}} & \multicolumn{2}{c}{\hspace{-2.75mm} \texttt{church}$\to$\texttt{dome}$\Rightarrow$\texttt{church}$\to$\texttt{tree}}
\end{tabular}
\captionof{figure}{Manipulating the geneartion rules via \cite{GANRewriting}, the two input images are modified via the same model.}
\label{fig:GANRewritting}
\end{minipage}
\end{figure}
In order to further verify the causal effects between the neurons and the generation results, after identifying the units which are relevant to a specific class of objects, Bau~\emph{et al}\onedot~\cite{GANDissection} further ablated (or inserted) the units by setting the activation of the relevant neurons to zero (or a class-specific constant).
\cref{fig:GANDissection} shows the results of such modifications.
It can be seen that the whole church appears after ablating the trees, and the grasses are added to the other image.
The result indicates that
1)~the generation is under the control of the relevant neurons, whose activation values determine the absence of the objects in the final generated image, and
2)~the previously unseen objects may still be generated, but sheltered by other objects.
Tousi~\emph{et al}\onedot~\cite{GANCorrection} further explored this phenomenon in multiple layers, and achieved a more accurate manipulation.
This procedure could be regarded as an inverse procedure of the classification task to some extent, where the final prediction also shelters other classes, and we recommend \cite{ActivationAtlas} for an intuitive and interactive understanding.
\subsubsection{Generation Rule Manipulation}
\label{subsubsec:GANrewriting}
When adding some objects in an image as stated in \cref{subsubsec:GANDissection}, it will fail to generate the desired objects somewhere (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, adding a door in the sky).
Since the objects could be successfully inserted in other regions (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, adding a door on the wall), Bau~\emph{et al}\onedot~\cite{Rewriting} asserted that there are some rules in the generation process learned from the data, and they seek to manipulate the rules for verification and a general modification method at the model level.
Specifically, as shown in \cref{fig:GANRewritting}, suppose the rule in a generator is like $\mathbf{I}_\mathtt{dome}=\mathit{G}_{\mathtt{church}\rightarrow\mathtt{dome}}(\mathit{G}_{\mathbf{z}\rightarrow\mathtt{church}}(\mathbf{z}))$, where the generator could be regarded as an associative memory that stores a set of key-value pairs $\{(\mathbf{k}_\mathit{i}, \mathbf{v}_\mathit{i})\}$, and is formulated by
\begin{equation}
\mathbf{v}_\mathit{i}\approx\mathbf{W}\mathbf{k}_\mathit{i},
\end{equation}
we can modulate the parameter $\mathbf{W}$ (aka the associative memory) to reorganize the key-value pairs.
For example, we could change the rule $\mathtt{church}\rightarrow\mathtt{dome}$ to $\mathtt{church}\rightarrow\mathtt{tree}$, and the results are shown in \cref{fig:GANRewritting}.
The results clearly show the generation process, and help to open the black box of GAN models.
\begin{table}
\newcolumntype{L}[1]{>{\raggedright\arraybackslash}p{#1}}
\caption{A summary of GAN inversion and methods leveraging pre-trained GANs for image editing and restoration. For the inversion method, ``O'', ``L'', ``T'' represent optimization-based, learning-based, and training-based (or fine-tuning) methods, while ``/'' means no inversion is performed in this method, and the numbers (without square brackets) are the indices of methods used for inversion in this table. Note that the methods are ordered (roughly) according to publicly accessible time (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, the appear time on \href{https://arxiv.org}{ArXiv}, \href{https://openreview.net}{openreview.net}, \href{https://openaccess.thecvf.com/}{CVF Open Access}, \emph{etc}\onedot} \def\vs{\emph{vs}\onedot.). We have also highlighted several works that are frequently used for GAN inversion or image editing/restoration.}
\label{tab:methods}
\centering
\scalebox{0.52}{
\begin{tabular}{cccccccc}
\toprule
\textbf{No.} & \textbf{Method} & \textbf{Publication} & \textbf{Backbone} & \textbf{\begin{tabular}[c]{@{}c@{}}Latent\\ Space\end{tabular}} & \textbf{\begin{tabular}[c]{@{}c@{}}Inversion\\ Method\end{tabular}} & \textbf{Dataset$^\ast$} & \textbf{Application$^\dagger$} \\
\midrule
1 & BiGAN~\cite{BiGAN} & ICLR 2017 & / & $\mathcal{Z}$ & T & MN, IN & Inv \\
2 & ALI~\cite{ALI} & ICLR 2017 & / & $\mathcal{Z}$ & T & CF, SV, CA, IN & Inv, Int \\
3 & Zhu~\emph{et al}\onedot~\cite{zhu2016generative} & ECCV 2016 & DCGAN & $\mathcal{Z}$ & L, O & SH, LS, PL$^\ddagger$ & Inv, Int, AE \\
\rowcolor{green!20}4 & IcGAN~\cite{IcGAN} & NeurIPSw 2016 & cGAN & $\mathcal{Z}$, $\mathcal{C}$ & L & MN, CA & Inv, AT, AE \\
5 & Creswell~\emph{et al}\onedot~\cite{creswell2018inverting} & T-NNLS 2018 & DCGAN, WGAN-GP & $\mathcal{Z}$ & O & OM, UT, CA & Inv \\
6 & Lipton~\emph{et al}\onedot~\cite{lipton2017precise} & ICLRw 2017 & DCGAN & $\mathcal{Z}$ & O & CA & Inv \\
7 & PGD-GAN~\cite{PGD-GAN} & ICASSP 2018 & DCGAN & $\mathcal{Z}$ & O & MN, CA & Inv \\
8 & Ma~\emph{et al}\onedot~\cite{ma2018invertibility} & NeurIPS 2018 & DCGAN & $\mathcal{Z}$ & O & MN, CA & Inv, IP \\
9 & Suzuki~\emph{et al}\onedot~\cite{suzuki2018spatially} & ArXiv 2018 & SNGAN, BigGAN, StyleGAN & $\mathcal{F}$ & 3 & IN, FL, FF, DA & CO \\
\rowcolor{green!20}10 & GANDissection~\cite{GANDissection} & ICLR 2019 & PGGAN & $\mathcal{F}$ & / & LS, AD & AE, AR \\
11 & NPGD~\cite{NPGD} & ICCV 2019 & DCGAN, SAGAN & $\mathcal{Z}$ & L, O & MN, CA, LS & Inv, SR, IP \\
\rowcolor{green!20}12 & Image2StyleGAN~\cite{Image2StyleGAN} & ICCV 2019 & StyleGAN & $\mathcal{W}+$ & O & FF$^\ddagger$ & Inv, Int, AE, ST \\
13 & Bau~\emph{et al}\onedot~\cite{bau2019inverting} & ICLRw 2019 & PGGAN, WGAN-GP, StyleGAN & $\mathcal{Z}$, $\mathcal{W}$ & L, O & LS & Inv \\
14 & GANPaint~\cite{GANPaint} & ToG 2019 & PGGAN & $\mathcal{Z}$, $\bm{\Theta}$ & L, O, T & LS & Inv, AE \\
\rowcolor{green!20}15 & InterFaceGAN~\cite{InterFaceGAN} & CVPR 2020 & PGGAN, StyleGAN & $\mathcal{Z}$, $\mathcal{W}$ & 3, 8 & CH & AE, AR \\
\rowcolor{green!20}16 & GANSeeing~\cite{GANSeeing} & ICCV 2019 & PGGAN, WGAN-GP, StyleGAN & $\mathcal{Z}$, $\mathcal{W}$ & 13 & LS & Inv \\
17 & YLG~\cite{YLG} & CVPR 2020 & SAGAN & $\mathcal{Z}$ & O & IN & Inv \\
\rowcolor{green!20}18 & Image2StyleGAN++~\cite{Image2StyleGAN++} & CVPR 2020 & StyleGAN & $\mathcal{W}+$, $\mathcal{N}$ & O & LS, FF & Inv, CO, IP, AE, ST \\
19 & mGANPrior~\cite{mGANPrior} & CVPR 2020 & PGGAN, StyleGAN & $\mathcal{Z}$ & O & FF, CH, LS & Inv, IC, SR, IP, DN, AE \\
20 & MimicGAN~\cite{MimicGAN} & IJCV 2020 & DCGAN & $\mathcal{Z}$ & O & CA, FF, LF & Inv, UDA, AD, AN \\
\rowcolor{green!20}21 & PULSE~\cite{PULSE} & CVPR 2020 & StyleGAN & $\mathcal{Z}$ & O & FF, CH & Inv, SR \\
22 & DGP~\cite{DGP} & ECCV 2020 & BigGAN & $\mathcal{Z}$ & O, T & IN, P3 & Inv, Int, IC, IP, SR, AD, TR, AE \\
23 & StyleGAN2Distillation~\cite{StyleGAN2Distillation} & ECCV 2020 & StyleGAN2, pix2pixHD & $\mathcal{W}+$ & / & FF & AT, AE \\
24 & EditingInStyle~\cite{EditingInStyle} & CVPR 2020 & PGGAN, StyleGAN, StyleGAN2 & $\mathcal{F}$ & / & FF, LS & AT \\
25 & StyleRig~\cite{StyleRig} & CVPR 2020 & StyleGAN & $\mathcal{W}+$ & / & FF & AT \\
26 & ALAE~\cite{ALAE} & CVPR 2020 & StyleGAN & $\mathcal{W}$ & T & MN, FF, LS, CH & Inv, AT \\
\rowcolor{green!20}27 & IDInvert~\cite{IDInvert} & ECCV 2020 & StyleGAN & $\mathcal{W}+$ & L, O & FF, LS & Inv, Int, AE, CO \\
28 & pix2latent~\cite{pix2latent} & ECCV 2020 & BigGAN, StyleGAN2 & $\mathcal{Z}$ & O & IN, CO, CF, LS & Inv, TR, AE \\
29 & IDDistanglement~\cite{IDDisentanglement} & ToG 2020 & StyleGAN & $\mathcal{W}$ & L & FF & Inv, AT \\
30 & WhenAndHow~\cite{WhenAndHow} & ArXiv 2020 & MLP & $\mathcal{Z}$ & O & MN & Inv, IP \\
31 & Guan~\emph{et al}\onedot~\cite{guan2020collaborative} & ArXiv 2020 & StyleGAN & $\mathcal{W}+$ & L, O & CH, CD & Inv, Int, AT, IC \\
\rowcolor{green!20}32 & SeFa~\cite{SeFa} & CVPR 2021 & PGGAN, BigGAN, StyleGAN & $\mathcal{Z}$ & 19, 27 & FF, CH, LS, IN, SS, DA & AE \\
33 & GH-Feat~\cite{GH-Feat} & CVPR 2021 & StyleGAN & $\mathcal{S}$ & L & MN, FF, LS, IN & Inv, AT, AE \\
\rowcolor{green!20}34 & pSp~\cite{pSp} & CVPR 2021 & StyleGAN2 & $\mathcal{W}+$ & L & FF, AF, CH, CM & Inv, FF, SI, SR \\
35 & StyleFlow~\cite{StyleFlow} & ToG 2021 & StyleGAN, StyleGAN2 & $\mathcal{W}+$ & 12 & FF, LS & AT, AE \\
36 & PIE~\cite{PIE} & ToG 2020 & StyleGAN & $\mathcal{W}+$ & O & FF & AT, AE \\
37 & Bartz~\emph{et al}\onedot~\cite{bartz2020one} & BMVC 2020 & StyleGAN, StyleGAN2 & $\mathcal{Z}$, $\mathcal{W}+$ & L & FF, LS & Inv, DN \\
38 & StyleIntervention~\cite{StyleIntervention} & ArXiv 2020 & StyleGAN2 & $\mathcal{S}$ & O & FF & Inv, AE \\
\rowcolor{green!20}39 & StyleSpace~\cite{StyleSpace} & CVPR 2021 & StyleGAN2 & $\mathcal{S}$ & O & FF, LS & Inv, AE \\
40 & Hijack-GAN~\cite{Hijack-GAN} & CVPR 2021 & PGGAN, StyleGAN & $\mathcal{Z}$ & / & CH & AE \\
41 & NaviGAN~\cite{NaviGAN} & CVPR 2021 & pix2pixHD, BigGAN, StyleGAN2 & $\bm{\Theta}$ & \cite{StyleGAN2} & FF, LS, CS, IN & AE \\
42 & GLEAN~\cite{GLEAN} & CVPR 2021 & StyleGAN & $\mathcal{W}+$ & L & FF, LS & Inv, SR \\
\rowcolor{green!20}43 & ImprovedGANEmbedding~\cite{ImprovedGANEmbedding} & ArXiv 2020 & StyleGAN, StyleGAN2 & $\mathcal{P}$ & O & FF, MF$^\ddagger$ & Inv, IC, IP, SR \\
44 & GFPGAN~\cite{GFPGAN} & CVPR 2021 & StyleGAN2 & $\mathcal{W}$ & L & FF & Inv, SR \\
45 & EnjoyEditing~\cite{EnjoyEditing} & ICLR 2021 & PGGAN, StyleGAN2 & $\mathcal{Z}$ & 12 & FF, CA, CH, P3, TR & Inv, AE \\
46 & SAM~\cite{SAM} & ToG 2021 & StyleGAN & $\mathcal{W}+$ & L & CA, CH & AE \\
47 & e4e~\cite{E4E} & ToG 2021 & StyleGAN2 & $\mathcal{W}+$ & L & FF, CH, LS, SC & Inv, AE \\
\rowcolor{green!20}48 & StyleCLIP~\cite{StyleClip} & ICCV 2021 & StyleGAN2 & $\mathcal{W}+$, $\mathcal{S}$ & 47, O & FF, CH, LS, AF & AE \\
49 & LatentComposition~\cite{LatentComposition} & ICLR 2021 & PGGAN, StyleGAN2 & $\mathcal{Z}$ & L & FF, CH, LS & Inv, IP, AT \\
50 & GANEnsembling~\cite{GANEnsembling} & CVPR 2021 & StyleGAN2 & $\mathcal{W}+$ & L, O & CH, SC, PT & Inv, AT \\
51 & ReStyle~\cite{ReStyle} & ICCV 2021 & StyleGAN2 & $\mathcal{W}+$ & L & FF, CH, SC, LS, AF & Inv, AE \\
52 & E2Style~\cite{baseline} & T-IP 2022 & StyleGAN2 & $\mathcal{W}+$ & L & FF, CH & Inv, SI, PI, AT, IP, SR, AE, IH \\
\rowcolor{green!20}53 & GPEN~\cite{GPEN} & CVPR 2021 & StyleGAN2 & $\mathcal{W}+$, $\mathcal{N}$ & L & FF, CH & Inv, SR \\
54 & Consecutive~\cite{Consecutive} & ICCV 2021 & StyleGAN & $\mathcal{W}+$ & O & FF, RA & Inv, Int, AE \\
\rowcolor{green!20}55 & BDInvert~\cite{BDInvert} & ICCV 2021 & StyleGAN, StyleGAN2 & $\mathcal{F}$/$\mathcal{W}+$ & O & FF, CH, LS & Inv, AE \\
56 & HFGI~\cite{HFGI} & CVPR 2022 & StyleGAN2 & $\mathcal{W}+$, $\mathcal{F}$ & L & FF, CH, SC & Inv, AE \\
57 & VisualVocab~\cite{VisualVocab} & ICCV 2021 & BigGAN & $\mathcal{Z}$ & / & P3, IN & AE \\
58 & HyperStyle~\cite{HyperStyle} & CVPR 2022 & StyleGAN2 & $\mathcal{W}+$ & L & FF, CH, AF & Inv, AE, ST \\
\rowcolor{green!20}59 & GANGealing~\cite{GANGealing} & CVPR 2022 & StyleGAN2 & $\mathcal{W}$ & / & LS, FF, AF, CH, CU & TR \\
60 & HyperInverter~\cite{HyperInverter} & CVPR 2022 & StyleGAN2 & $\mathcal{W}$, $\bm{\Theta}$ & L & FF, CH, LS & Inv, Int, AE \\
\rowcolor{green!20}61 & InsetGAN~\cite{InsetGAN} & CVPR 2022 & StyleGAN2 & $\mathcal{W}+$ & O & FF, DF$^\ddagger$ & CO, IG \\
62 & HairMapper~\cite{HairMapper} & CVPR 2022 & StyleGAN2 & $\mathcal{W}+$ & 47 & FF, CM$^\ddagger$ & AE \\
63 & SAMInv~\cite{SAMInv} & CVPR 2022 & BigGAN-deep, StyleGAN2 & $\mathcal{W}+$, $\mathcal{F}$ & L & FF, LS, IN & Inv, AE
\\
\bottomrule
\multicolumn{8}{L{1.85\linewidth}}{$^\ast$Abbreviations: AD (\href{https://groups.csail.mit.edu/vision/datasets/ADE20K/}{ADE20K}~\cite{ADE20K}), AF (\href{https://github.com/clovaai/stargan-v2}{AFHQ}~\cite{StarGANv2}), CA (\href{http://mmlab.ie.cuhk.edu.hk/projects/CelebA.html}{CelebA}~\cite{CelebA}), CD (\href{https://bcsiriuschen.github.io/CARC/}{CACD}~\cite{CACD}), CF (\href{http://www.cs.toronto.edu/~kriz/cifar.html}{CIFAR}~\cite{CIFAR}), CH (\href{https://github.com/tkarras/progressive\_growing\_of\_gans}{CelebA-HQ}~\cite{PGGAN}), CM (\href{https://github.com/switchablenorms/CelebAMask-HQ}{CelebAMask-HQ}~\cite{MaskGAN}), CO (\href{https://cocodataset.org/\#home}{MS COCO}~\cite{COCO}), CS (\href{https://www.cityscapes-dataset.com/}{CityScapes}~\cite{Cityscapes}), CU (\href{https://authors.library.caltech.edu/27452/}{Caltech-UCSD Birds}~\cite{CUB}), DA (\href{https://www.gwern.net/Danbooru}{Danbooru}~\cite{Danbooru}, aka Anime Faces), DF (\href{http://mmlab.ie.cuhk.edu.hk/projects/DeepFashion.html}{DeepFashion}~\cite{DeepFashion}), FF (\href{https://github.com/nvlabs/stylegan}{FFHQ}~\cite{StyleGAN}), FL (\href{https://www.robots.ox.ac.uk/~vgg/data/flowers/}{Flowers}~\cite{Flowers}), IN (\href{https://image-net.org/}{ImageNet}~\cite{ImageNet}), LF (\href{http://vis-www.cs.umass.edu/lfw/}{LFW}~\cite{LFW}), LS (\href{https://www.yf.io/p/lsun}{LSUN}~\cite{LSUN}), MF (\href{https://github.com/NVlabs/metfaces-dataset}{MetFaces}~\cite{StyleGAN2-Ada}), MN (\href{http://yann.lecun.com/exdb/mnist/}{MNIST}~\cite{MNIST}), OM (\href{https://github.com/brendenlake/omniglot}{Omniglot}~\cite{Omniglot}), P3 (\href{http://places2.csail.mit.edu/}{Places365}~\cite{Places365}), PL (\href{http://places.csail.mit.edu/}{Places}~\cite{Places}), PT (\href{https://www.robots.ox.ac.uk/~vgg/data/pets/}{Oxford-IIIT Pet}~\cite{CatsAndDogs}, aka Cats and Dogs), RA (\href{https://zenodo.org/record/1188976}{RAVDESS}~\cite{RAVDESS}), SC (\href{http://ai.stanford.edu/~jkrause/cars/car\_dataset.html}{Stanford Cars}~\cite{StanfordCars}), SS (\href{http://streetscore.media.mit.edu/static/files/streetscore\_data.zip}{Streetscape}~\cite{Streetscape}), SV (\href{http://ufldl.stanford.edu/housenumbers/}{SVHN}~\cite{SVHN}), TR (\href{http://transattr.cs.brown.edu/}{Transient}~\cite{Transient}), UT (\href{https://vision.cs.utexas.edu/projects/finegrained/utzap50k/}{UT Zappos50K}~\cite{Shoes})}\\
\multicolumn{8}{L{1.85\linewidth}}{$^\dagger$Abbreviations: AD (Adversarial Defense), AE (Attribute Editing, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, w/o reference), AN (Anomaly Detection), AR (Artifacts Removal), AT (Attribute Transfer, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, w/ reference), CO (Image Crossover), [U]DA ([Unsupervised] Domain Adaptation), DN (Image Denoising), FF (Face Frontalization), IC (Image Colorization), IG (Image Generation), IH (Information Hiding), Int (Interpolation), Inv (Inversion), IP (Inpainting), PI (Parsing or Segmentation to Image), SI (Sketch to Image), SR (Image Super-resolution), ST (Style Transfer), TR (Transform and Random Jittering).}\\
\multicolumn{8}{L{1.85\linewidth}}{$^\ddagger$Some custom datasets collected or regenerated by the authors are omitted since they are not publicly available or can be generated automatically based on current public datasets.}
\end{tabular}}
\end{table}
\subsection{GAN Inversion}
\label{subsec:GAN_Inversion}
As introduced in \cref{sec:intro}, a natural choice for achieving better output image quality in recent years is to apply adversarial losses, especially the image editing and restoration methods~\cite{Pix2pix,CycleGAN,SRGAN,DeblurGAN}.
However, such a method is insufficient to fully explore the image generation ability of GAN models.
Since the working scheme of GAN models is to map the random vectors (latent representations) into images (\cref{eqn:GAN}), the very first thing to utilize pre-trained GAN models is to invert the input images back into a meaningful latent space (\emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, GAN inversion), then the latent representations could be manipulated or optimized for achieving tasks like image editing or restoration.
Thus we first introduce the commonly used latent spaces in the literature, and then introduce the typical GAN inversion methods.
\subsubsection{Latent Spaces}
\label{subsubsec:latent_space}
In order to invert an image back to the latent representation, the first thing is to determine the inversion target, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, the latent space.
For this purpose, the reconstruction accuracy, interpretability, as well as editability should be considered.
In the following, we introduce the commonly used latent spaces in the literature, and an illustration can be found in \cref{fig:GAN_arch}(e).
\vspace{0.5em}
\noindent\textbf{$\mathcal{Z}$ Space \& $\mathcal{C}$ Space}
Following the vanilla GAN~\cite{GAN}, early GAN models~\cite{CGAN,LapGAN,DCGAN,StackGAN,SAGAN,BigGAN,PGGAN} take random noise vectors as the input, and generate fake images via a stack of convolution layers, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot,
\begin{equation}
\mathbf{\hat{x}}=\mathit{G}(\mathbf{z},\mathbf{c};\bm{\theta}_\mathit{G}),
\label{eqn:cGAN}
\end{equation}
where $\mathbf{c}$ is the condition information (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, class label, attribute annotations) for conditional GANs~\cite{CGAN,BigGAN}.
Therefore, a natural choice is to directly invert the generation process, and map the image back to a random noise $\mathbf{z}$, which spans the $\mathcal{Z}$ space.
Since $\mathbf{z}$ is sampled from a very simple distribution (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, the standard Gaussian distribution), the semantic features are largely entangled in the $\mathcal{Z}$ space, and the simple $\mathcal{Z}$ space is too limited to simultaneously represent both content and semantic information.
An alternative way is to jointly use the $\mathcal{Z}$ space and the $\mathcal{C}$ space, where $\mathbf{c}\in\mathcal{C}$.
Thus some methods (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, IcGAN~\cite{IcGAN}) could invert the image into both $\mathcal{Z}$ and $\mathcal{C}$ spaces, showing extraordinary performance in disentangling the content and attribute representations.
However, these methods require massive human efforts in labeling the attributes to model the $\mathcal{Z}$ space.
\vspace{0.5em}
\noindent\textbf{$\mathcal{W}$ Space \& $\mathcal{W}+$ Space}
To separate the semantic conditions (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, class, style, \emph{etc}\onedot} \def\vs{\emph{vs}\onedot) with image contents (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, identity), Karras~\emph{et al}\onedot proposed the StyleGAN series~\cite{StyleGAN,StyleGAN2,StyleGAN3}, which could be formulated by
\begin{equation}
\mathbf{\hat{x}}=\mathit{G}(\mathbf{m}, \mathbf{n}, \mathit{F}(\mathbf{z}); \bm{\theta}_\mathit{G})=\mathit{G}(\mathbf{m},\mathbf{n},\mathbf{w};\bm{\theta}_\mathit{G}),
\label{eqn:StyleGAN}
\end{equation}
where $\mathbf{m}$ and $\mathbf{n}$ are learnable constant features and layer-wise noises\footnote{For StyleGAN3~\cite{StyleGAN3}, $\mathbf{m}$ denotes the Fourier feature~\cite{FourierFeature}, while $\mathbf{n}$ is deprecated.}, $\mathit{F}$ is the multi-layer perceptron (MLP) for mapping $\mathbf{z}$ to $\mathbf{w}$.
Due to the superior generation quality of StyleGAN models, they become the most popular GAN architectures.
Instead of mapping back to the $\mathcal{Z}$ space, the latent representation $\mathbf{w}\in\mathcal{W}$ is generally utilized for StyleGAN inversion, which is proven semantically better disentangled with the affine transmissions by the mapping network $\mathit{F}$~\cite{StyleGAN}.
Based on the $\mathcal{W}$ space, some researchers~\cite{Image2StyleGAN} proposed to predict an individual latent representation for each generator layer (see \cref{fig:GAN_arch}(c)), resulting in the $\mathcal{W}+$ space, which shows a better inversion accuracy comparing to the $\mathcal{W}$ space.
Note that the latent codes in the earlier layers (\emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, the ones near the input end) control coarser-grained attributes, while the finer-grained attributes are controlled by the latent codes in the latter layers (\emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, the ones near the output end), and we refer to \cite{StyleGAN} for detailed illustrations.
\vspace{0.5em}
\noindent\textbf{$\mathcal{S}$ Space \& $\mathcal{P}$ Space}
Although $\mathcal{W}$ and $\mathcal{W}+$ spaces have better properties, the changes in $\mathbf{w}$ tend to influence the whole image, and different dimensions in $\mathbf{w}$ follow various distributions.
As introduced in \cref{subsec:Understanding_neuron}, Bau~\emph{et al}\onedot have explored the relationship between neurons and objects in the generated images.
Inspired by the discovery that each neuron (channel) may be related to a specific class or semantic component, several works choose to predict channel-wise style parameters~\cite{StyleIntervention,StyleSpace,xu2020hierarchical} denoted by $\mathcal{S}$ space, with which one can control the local content of the images.
Besides, the $\mathcal{S}$ space is also extended by considering both channel-wise and spatial-wise diversity~\cite{SalS-GAN}, which is denoted by the $\mathcal{SA}$ space, but we still regard it as a kind of $\mathcal{S}$ space.
Zhu~\emph{et al}\onedot~\cite{ImprovedGANEmbedding} focused on the distribution of the $\mathcal{W}$ space, and proposed to use the latent representation (denoted by $\mathbf{p}\in\mathcal{P}$) before the last leaky ReLU layer of the mapping network $\mathit{F}$.
Compared to $\mathcal{W}$, $\mathcal{P}$ has a simpler structure (\emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, similar to a multivariate Gaussian distribution).
And they further propose a $\mathcal{P}_\mathit{N}$ space by mapping the distribution of each latent representation to be of zero mean and unit variance.
The $\mathcal{P}_\mathit{N}$ space is better filled by the latent codes, and the image generation quality could be evaluated by the distance between the latent representation and the average latent code.
\vspace{0.5em}
\noindent\textbf{$\mathcal{N}$ Space \& $\mathcal{F}$ Space}
For current GAN models, the largest dimension of the latent representation is $18\times512$ (the 512-dim $\mathbf{w}$ for 18 layers, and the dimension of $\mathbf{z}$ is much smaller).
Even though the most prominent features of the image have been reconstructed, the high-frequency features cannot be faithfully reproduced by such latent spaces.
Therefore, some works~\cite{Image2StyleGAN++,GPEN} proposed to leverage the layer-wise noise maps in StyleGAN and StyleGAN2 (\emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, $\mathbf{n}\in\mathcal{N}$), and achieved much more accurate reconstructions.
Yet the $\mathcal{N}$ space weakens the editability to some extent, thus another category considers the features directly.
For example, Bau~\emph{et al}\onedot~\cite{GANSeeing} inverted the images into the features at a particular layer to investigate the mode dropping problem.
Kang~\emph{et al}\onedot~\cite{BDInvert} proposed to map the image back to the feature map $\mathbf{f}$ at a certain scale (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, $8\times8$), and jointly use the $\mathcal{F}$ and $\mathcal{W}+$ spaces (denoted by $\mathcal{F}/\mathcal{W}+$ space).
Some methods also consider fine-tuning part of~\cite{GANPaint,NaviGAN} or the whole~\cite{GPEN,NearPerfect} GAN model, and we denote the model parameters as $\bm{\Theta}$ space.
As analyzed in \cref{subsec:Understanding_neuron}, the $\bm{\Theta}$ space is more like the generation rules rather than a latent space, and we mention it for completeness.
\subsubsection{GAN Inversion Methods}
In the literature, there are mainly three kinds of GAN inversion methods, including optimization-based, learning-based, and hybrid methods.
\cref{tab:methods} also summarizes the characteristic of these methods, which shows the used backbone, latent space, inversion method, dataset, and their applications.
In the following, the relevant methods are introduced in detail.
\vspace{0.5em}
\noindent\textbf{Optimization-based Methods}
Given a pre-trained GAN model, the objective of GAN inversion is to find an image $\mathbf{\hat{x}}$ generated by the GAN model which approximates a given image $\mathbf{x}$, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot,
\begin{equation}
\mathbf{\hat{x}}^\ast=\arg\min_{\mathbf{\hat{x}}\in\mathcal{X}}\ell(\mathbf{\hat{x}}, \mathbf{x}),
\label{eqn:inversion_image}
\end{equation}
where $\ell$ is a distance like $\ell_1$ or $\ell_2$, and $\mathcal{X}$ is the image space that are spanned by the images generated by the GAN model.
We require the image is in the GAN space in order to leverage the properties of GANs.
Thus with \cref{eqn:GAN}, it can be rewritten as,
\begin{equation}
\mathbf{\color{red}z}^\ast=\arg\min_{\color{red}{\mathbf{z}\in\mathcal{Z}}}\ell({\color{red}\mathit{G}(\mathbf{z}, \bm{\delta};\bm{\theta}_\mathit{G})}, \mathbf{x}),
\label{eqn:inversion_opt}
\end{equation}
where the difference against \cref{eqn:inversion_image} has been highlighted with {\color{red}red}.
Here we use $\mathcal{Z}$ space to illustrate the problem, and the formulation can be generalized to other latent spaces.
Due to that the generator $\mathit{G}$ is differentiable, \cref{eqn:inversion_opt} can be directly optimized via gradient descent algorithms~\cite{creswell2018inverting,lipton2017precise,PGD-GAN,ma2018invertibility,Image2StyleGAN,YLG,Image2StyleGAN++,mGANPrior,MimicGAN,PULSE,DGP,pix2latent,WhenAndHow,PIE,StyleIntervention,StyleSpace,ImprovedGANEmbedding,Consecutive,BDInvert}, and the invertibility of GAN models is discussed by Aberdam~\emph{et al}\onedot~\cite{WhenAndHow} and Ma~\emph{et al}\onedot~\cite{ma2018invertibility} with MLP-based and conv-based generators, respectively.
To promote the inversion accuracy and training stability, and boost the subsequent applications such as image editing and restoration, there are many improvements based on the basic formula in \cref{eqn:inversion_opt}.
Zhu~\emph{et al}\onedot~\cite{zhu2016generative} adopted L-BFGS~\cite{L-BFGS} algorithm instead of Adam~\cite{Adam}.
Lipton~\emph{et al}\onedot~\cite{lipton2017precise} and Shah~\emph{et al}\onedot~\cite{PGD-GAN} proposed to use projected gradient descent (PGD) for better optimization in the latent space.
Later, the inner loop of PGD was replaced by a neural network by Raj~\emph{et al}\onedot~\cite{NPGD}.
Daras~\emph{et al}\onedot~\cite{YLG} leveraged the attention map of the discriminator to boost the inversion.
Huh~\emph{et al}\onedot~\cite{pix2latent} and Kang~\emph{et al}\onedot~\cite{BDInvert} further considered the geometric transformations of the subjects.
An obvious problem of optimization-based methods is efficiency.
Since the optimization procedure usually requires thousands of update iterations, it takes a very long time for inversion.
To eliminate this problem, Abdal~\emph{et al}\onedot~\cite{Image2StyleGAN} proposed to initialize with the average latent code (\emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, $\mathbf{\bar{w}}$ obtained when training StyleGAN~\cite{StyleGAN}).
However, it still takes even minutes to invert a $1024\times1024$ image with a high-end GPU (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, NVIDIA Tesla V100).
An alternative way is to leverage the learning-based methods, which predict the latent representation in a single forward pass.
\vspace{0.5em}
\noindent\textbf{Learning-based Methods}
Learning-based GAN inversion methods~\cite{IcGAN,IDDisentanglement,GH-Feat,pSp,StyleFlow,bartz2020one,GLEAN,GFPGAN,SAM,E4E,LatentComposition,ReStyle,baseline,GPEN} are generally equipped with an additional encoder $\mathit{E}$, and the latent code is optimized in an indirect way.
Specifically, they optimize the parameters of the encoder, which is used to map the image back to the latent code, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, $\mathit{E}:\mathbf{x}\to\mathbf{z}$.
Therefore, \cref{eqn:inversion_opt} can be re-written as,
\begin{equation}
{\color{red}\bm{\theta}}_\mathit{\color{red}E}^\ast=\arg\min_{\color{red}\bm{\theta}_\mathit{E}}{\color{red}{\sum}_\mathit{i}}\ell(\mathit{G}({\color{red}\mathit{E}(\mathbf{x}_\mathit{i};\bm{\theta}_\mathit{E})}, \bm{\delta};\bm{\theta}_\mathit{G}), \mathbf{x}_{\color{red}\mathit{i}}),
\label{eqn:inversion_learning}
\end{equation}
where $\bm{\theta}_\mathit{E}$ denotes the parameters of $\mathit{E}$, and the difference against \cref{eqn:inversion_opt} has been highlighted in {\color{red}red}.
It can be seen that the learning-based methods train the encoder with a whole dataset to learn the mapping from image to latent code, rather than optimize on a single image.
Such a scheme yields several advantages.
First, the optimization becomes smoother, which avoids the latent code from being trapped into a local optimum.
Then, since the encoder has access to a large batch of images, it is easier for the encoder to learn some patterns from the data, \emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, the semantic directions for image manipulation in the latent space~\cite{IcGAN,IDDisentanglement,GH-Feat,StyleFlow,E4E,LatentComposition}.
On the contrary, the optimization-based methods generally rely on other methods (will be introduced in \cref{sec:disentanglement}).
Moreover, while the optimization-based methods can only be optimized with real images (the latent code of the synthetic images are already available), the learning-based methods can be trained with synthetic images.
In this way, they directly leverage the ground-truth latent code~\cite{IcGAN}, and \cref{eqn:inversion_learning} can be written as,
\begin{equation}
\bm{\theta}_\mathit{E}^\ast=\arg\min_{\bm{\theta}_\mathit{E}}{\sum}_\mathit{i}\ell({\color{red}\mathit{E}}({\color{red}\mathit{G}}({\color{red}\mathbf{z}}_\mathit{i}, \bm{\delta};\bm{\theta}_\mathit{G}); \bm{\theta}_\mathit{E}), {\color{red}\mathbf{z}}_\mathit{i}).
\label{eqn:inversion_learning_latent_code}
\end{equation}
Some methods also introduce extra models as auxiliary information or guidance.
For example,
Nitzan~\emph{et al}\onedot~\cite{IDDisentanglement} incorporated a face landmark detection model to control the pose of the generated face images.
Many works~\cite{IDDisentanglement,pSp,GPEN} introduced face recognition~\cite{ArcFace,CurricularFace} or contrastive learning~\cite{MoCo} models as ID/content similarity metrics, while some of them~\cite{baseline,GANDissection} also introduced semantic segmentation or face parsing models.
Furthermore, vision-language models are also used for inversion and editing~\cite{StyleClip}.
With the encoder, the model can get the latent code in a single forward pass, but the inversion is generally less accurate than optimization-based methods.
Therefore, Wei~\emph{et al}\onedot~\cite{baseline} proposed to use a multi-stage refinement scheme to keep both accuracy and efficiency, Alaluf~\emph{et al}\onedot~\cite{ReStyle} introduced an iterative procedure which is initialized with the average face latent $\mathbf{\bar{w}}$.
\begin{figure}
\centering
\scriptsize
\begin{overpic}[width=.99\linewidth]{./figures/GAN_Inversion.pdf}
\put(5, 17){$\mathbf{x}$}
\put(43, 17){$\mathbf{\hat{x}}$}
\put(56, 17){$\mathbf{x}$}
\put(94, 17){$\mathbf{\hat{x}}$}
\put(5, 2){$\mathbf{x}$}
\put(43, 2){$\mathbf{\hat{x}}$}
\put(56, 2){$\mathbf{x}$}
\put(94, 2){$\mathbf{\hat{x}}$}
\end{overpic}
\caption{Illustration of GAN inversion methods. $\mathbf{x}$ and $\mathbf{\hat{x}}$ are given real image and generated image, respectively. The {\color{red}red} dotted line means supervision. It can be seen that the in-domain constraint~\cite{IDInvert} requires the generated image $\mathbf{\hat{x}}$ can be inverted back into the latent space. Here, $\mathbf{z}$ is not restricted in $\mathcal{Z}$ space, and may refer to more generic latent code (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, $\mathbf{w}$, $\mathbf{f}$, \emph{etc}\onedot} \def\vs{\emph{vs}\onedot).}
\label{fig:inversion_scheme}
\end{figure}
\vspace{0.5em}
\noindent\textbf{Hybrid Methods}
For both efficiency and accuracy, Zhu~\emph{et al}\onedot~\cite{zhu2016generative} proposed the hybrid method that initializes the latent code for optimization via learning-based methods, which was followed by a series of works~\cite{NPGD,GANPaint,IDInvert,guan2020collaborative,GANEnsembling}.
Among these methods, we would like to highlight the method named IDInvert~\cite{IDInvert}, which leverages the encoder for in-domain inversion.
Specifically, an encoder is trained like the learning-based methods following \cref{eqn:inversion_learning}, and the discriminator is also leveraged for training.
Then, following the optimization-based methods, the latent code $\mathbf{z}_0$ obtained by the encoder is used to initialize the optimization procedure, and then the learning objective can be formulated by,
\begin{equation}
\mathbf{z}^\ast=\argmin_{\mathbf{z}\in\mathcal{Z}}\ell(\mathit{G}(\mathbf{z}, \bm{\delta};\bm{\theta}_\mathit{G}), \mathbf{x}) {\color{red}+ \lambda\|\mathbf{z}-\mathit{E}(\mathit{G}(\mathbf{z}, \bm{\delta};\bm{\theta}_\mathit{G}); \bm{\theta}_\mathit{E})\|_2},
\label{eqn:inversion_idinvert}
\end{equation}
where $\lambda$ is the balancing hyper-parameter, and the difference between \cref{eqn:inversion_opt} has been highlighted with {\color{red}red}.
It can be seen from \cref{eqn:inversion_idinvert}, that the second term requires that the image generated with the optimized latent code can be mapped back via the encoder.
In other words, the optimization is guaranteed to be performed within the manifold supported by the encoder (as well as the generator), and it is called \textit{in-domain inversion}.
Please refer to \cref{fig:inversion_scheme}(d) for an illustration.
Guan~\emph{et al}\onedot~\cite{guan2020collaborative} leveraged the encoder during the optimization process in a collaborative way.
In particular, the encoder (\emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, the embedding network in their paper) and the iterator participate in each other's procedure.
In each iteration, the encoder predicts a latent code as a reasonable initialization for the iterator, which obviously speeds up the optimization.
Then, the iterator obtains a better latent code via optimization, which conversely serves as supervision for training the encoder.
Both the encoder and the iterator achieve better performance/efficiency, and the method naturally supports online learning.
Apart from the above methods, some works choose to learn an encoder together with the generator~\cite{BiGAN,ALI,ALAE}, which combine autoencoders with GANs and are out of the scope of this review.
Some other generative models are invertible by design~\cite{GLOW,DDPM}, however, the latent representation is typically a tensor of the same size as the image, which is sampled from a very simple distribution (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, Gaussian), making the latent representation less meaningful, and is beyond the scope of this paper.
\subsection{Semantic Disentanglement and Discovery}
\label{sec:disentanglement}
Understanding visual concepts is vital for many vision tasks like image editing and restoration, which helps the models to generate semantically consistent results.
In this section, we review the relevant methods from two perspectives, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, semantic disentanglement and discovery~\cite{ramesh2018spectral}.
The former cares more about encouraging the independence of latent variables, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, a perturbation to any component of a latent variable should result in a change to the output that is distinct, while the latter focuses on discovering disentangled latent representations of highly interpretability, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, a perturbation to one dimension should result in a change along exactly one meaningful attribute.
\subsubsection{Semantic Disentanglement}
\label{subsubsec:semantic_disentanglement}
In early GAN models (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, DCGAN~\cite{DCGAN}), the concepts or attributes are generally entangled to a large extent in the rather simple latent space (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, standard Gaussian).
Although some methods~\cite{IcGAN} achieve disentangled attributes by leveraging the conditional GANs~\cite{CGAN,BigGAN} trained on datasets with class labels (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, ImageNet~\cite{ImageNet}, LSUN~\cite{LSUN}, \emph{etc}\onedot} \def\vs{\emph{vs}\onedot) or attribute annotations (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, CelebA~\cite{CelebA}, RaFD~\cite{RaFD}, \emph{etc}\onedot} \def\vs{\emph{vs}\onedot), massive human efforts are required, which is unacceptable for all tasks.
Thus a series of methods~\cite{InfoGAN,SpectralRegularizer,HessianPenalty,VP-GAN,WhereAndWhat,OroJaR,EigenGAN} are proposed for better disentanglement in GANs.
A route of methods achieves disentanglement by maximizing the mutual information.
InfoGAN~\cite{InfoGAN} is an early exploration on this task, which introduces an extra latent code $\mathbf{c}$ alongside the original random variable $\mathbf{z}$.
The model architecture is the same as \cref{eqn:cGAN}, while the meaning of $\mathbf{c}$ is automatically learned via mutual information maximization when optimizing the GAN models, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot,
\begin{equation}
\min_\mathit{G}\max_\mathit{D}\mathcal{L}_\mathit{InfoGAN}(\mathit{G},\mathit{D})=\mathcal{L}_\mathit{GAN}(\mathit{G},\mathit{D})-\lambda\mathit{I}(\mathit{G}(\mathbf{z},\mathbf{c};\bm{\theta}_\mathit{G}); \mathbf{c}),
\label{eqn:loss_infogan}
\end{equation}
where $\mathit{I}(\mathit{x};\mathit{y})$ denotes the mutual information between $\mathit{x}$ and $\mathit{y}$, and $\mathcal{L}_\mathit{GAN}$ is defined in \cref{eqn:loss_GAN}.
In particular, a regressor is trained to predict the input $\mathbf{c}$.
As such, the changes in $\mathbf{c}$ are required to influence the generation in a distinct way, which forces $\mathbf{c}$ to be traceable and disentangled.
Zhu~\emph{et al}\onedot~\cite{VP-GAN} argued that the latent variation predictability can represent the disentanglement, and improved InfoGAN by predicting the relative changes between two generated images with only one dimension of the latent vector changed, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot,
\begin{equation}
\min_\mathit{G}\max_\mathit{D}\mathcal{L}_\mathit{VP-GAN}(\mathit{G},\mathit{D})=\mathcal{L}_\mathit{GAN}(\mathit{G},\mathit{D})-\lambda\mathit{I}(\mathit{G}(\mathbf{z};\bm{\theta}_\mathit{G}), \mathit{G}(\mathbf{z}+ \mathbf{\epsilon}\mathbf{d};\bm{\theta}_\mathit{G}); \mathbf{d}),
\label{eqn:loss_vp_gan}
\end{equation}
where $\mathbf{d}$ is a one-hot vector with the same dimension as $\mathbf{z}$, and $\epsilon\!\in\!\mathcal{N}(0, 1)$.
Note that the first metric of GAN disentanglement is proposed based on the observation.
They further assumed that the interpretable representation should be spatially consistent, and improved the variation predictability in \cite{WhereAndWhat}.
Another category focuses on the gradient of the GAN models with respect to the inputs.
Ramesh~\emph{et al}\onedot~\cite{SpectralRegularizer} focused on the coordinate directions in the latent space, and proposed a spectral regularizer to align the leading right-singular vectors of the Jacobian of GAN models w.r.t\onedot} \def\dof{d.o.f\onedot the inputs with the coordinate axes, so that the trajectories are semantically meaningful.
Peebles~\emph{et al}\onedot~\cite{HessianPenalty} achieved disentanglement by encouraging the Hessian matrix of the GAN models w.r.t\onedot} \def\dof{d.o.f\onedot the inputs to be diagonal, and the Hutchinson's estimator is leveraged to approximate the regularization.
It is worth noting that the authors also showed the ability of the Hessian Penalty to ``deactivate'' redundant components.
Wei~\emph{et al}\onedot~\cite{OroJaR} required that when perturbing a single dimension of the network input, the change in the output should be independent (and uncorrelated) with those caused by the other input dimensions.
Based on this assumption, they changed the assumption in \cite{HessianPenalty} from $\frac{\partial}{\partial\mathit{z}_\mathit{j}}(\frac{\partial\mathit{G}}{\partial\mathit{z}_\mathit{i}})=0$ to $[\frac{\partial\mathit{G}}{\partial\mathit{z}_\mathit{j}}]^\mathsf{T}\frac{\partial\mathit{G}}{\partial\mathit{z}_\mathit{i}}=0$.
He~\emph{et al}\onedot~\cite{EigenGAN} embedded a linear subspace in each layer of the GAN model by directly adding a group of basis $\mathbf{U}_\mathit{i}$, where the $\mathit{i}$-th noise $\mathbf{z}_\mathit{i}$ is rewritten as $\mathbf{U}_\mathit{i}\mathbf{L}_\mathit{i}\mathbf{z}_\mathit{i}+\bm{\mu}_\mathit{i}$, and $\mathbf{L}_\mathit{i}$ is a diagonal matrix indicating the importance of basis vectors, $\bm{\mu}_\mathit{i}$ denotes the origin of the subspace.
They required that the basis is orthogonal, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, minimizing $\|\mathbf{U}_\mathit{i}^\mathsf{T}\mathbf{U}_\mathit{i}-\mathbf{I}\|^2_\mathit{F}$.
As introduced in \cref{sec:GANs_development}, recent large-scale GAN models (especially the StyleGAN~\cite{StyleGAN} series) achieve disentangled latent representations by sending the conditional information to the generator layers (see \cref{fig:GAN_arch}), and the StyleGAN model maps the latent code into an intermediate latent space, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, $\mathcal{W}$ space via an MLP.
A series of regularization terms are deployed for more stable training and better disentanglement, \emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, the mixing regularization and perceptual path length regularization.
Please refer to \cref{tab:GAN_comparison} for more information.
However, since the disentanglement is implicitly achieved in the $\mathcal{W}$ space, researchers also focused on finding meaningful semantic directions in the GAN models, which are introduced as follows.
\subsubsection{Semantic Discovery}
To find meaningful semantic directions in the latent spaces, there are two main streams in the literature, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, the supervised and unsupervised methods, which are introduced respectively as follows.
\vspace{0.5em}
\noindent\textbf{Supervised Methods}
The most intuitive way to leverage the manually annotated supervision is to model the semantic space explicitly via conditional GANs, where the attribute vector is delivered into the GAN model.
Perarnau~\emph{et al}\onedot~\cite{IcGAN} followed this intuition in IcGAN and adopted two encoders to map images into both the random noise space and the attribute space.
However, this scheme is unavailable for unconditional GANs, which only synthesize images from random noise.
Besides, the methods require massive human efforts for data annotation, which makes it inflexible and expensive.
Therefore, researchers sought to find indirect ways to leverage the annotations.
For example, Shen~\emph{et al}\onedot~\cite{InterFaceGAN} assumed that there is a hyperplane in the latent space for binary attributes (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, male and female), which serves as the separation boundary between the opposite attributes.
Thus they proposed to leverage the \textit{normal vector of the hyper-plane} as the editing direction, and the support vector machine (SVM) is deployed for obtaining the hyper-plane.
\cite{Hijack-GAN,StyleGAN2Distillation, EnjoyEditing} followed a similar assumption.
Wang~\emph{et al}\onedot~\cite{Hijack-GAN} trained a proxy model which can map the input noise to the attribute space, and the \textit{gradient} of the proxy model w.r.t\onedot} \def\dof{d.o.f\onedot the input noise is used as the non-linear editing direction.
Viazovetskyi~\emph{et al}\onedot~\cite{StyleGAN2Distillation} leveraged the pre-trained image attribute classifier to find the \textit{class centers} of attributes in the latent space, and used the direction between different centers to represent the editing direction.
Zhuang~\emph{et al}\onedot~\cite{EnjoyEditing} introduced a group of learnable \textit{editing directions} $\mathbf{d}$, and the pre-trained image attribute classifier is used to align each of the directions with an attribute.
Instead of learning editing directions, Abdal~\emph{et al}\onedot~\cite{StyleFlow} introduced a conditional normalizing flow model in $\mathcal{W}$-space of StyleGAN, which takes the attributes as conditions to map the latent $\mathbf{w}$ back into the noise space.
To edit the attributes, one can obtain the edited latent $\mathbf{w}'$ by inverting the noise with new attributes.
Besides, several methods focused on editing specific attributes in similar ways, for example, Alaluf~\emph{et al}\onedot~\cite{SAM} sought to edit the \textit{age} via an age regressor, and Goetschalckx~\emph{et al}\onedot~\cite{GANalyze} tried to make the images \textit{more memorable} with a memorability predictor.
The methods mentioned above still find the attribute directions directly related to the supervision (\emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, the manually annotated attributes), and some researchers proposed to edit some attributes with indirect supervision.
As introduced in \cref{subsec:Understanding_neuron}, Bau~\emph{et al}\onedot~\cite{GANDissection} have revealed the relationship between the GAN model neurons and the appearance of objects, which inspires a series of disentanglement methods to achieve \textit{spatially precise control}~\cite{GH-Feat,StyleSpace,StyleIntervention}.
Nitzan~\emph{et al}\onedot~\cite{IDDisentanglement} leveraged a pre-trained \textit{identification} model and a \textit{landmark detection} model, with which the attributes that can be described via landmarks (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, expression) are decomposed with the identity, thus the attributes can be transferred to other faces.
Tewari~\emph{et al}\onedot~\cite{StyleRig} shared a similar idea but employed a pre-trained three-dimensional morphable face model (\textit{3DMM}) to learn synthetic face image editing, which is later extended to real faces in \cite{PIE}.
Huh~\emph{et al}\onedot~\cite{pix2latent} learned transformation of input shift, rescaling, \emph{etc}\onedot} \def\vs{\emph{vs}\onedot, and estimate spatial transformation together with the latent variable.
In particular, the spatial positions of the objects are obtained via a pre-trained \textit{object detection} model.
Xu~\emph{et al}\onedot~\cite{Consecutive} bypassed the requirement of the pre-trained classifier by leveraging video data and optical flow.
Jahanian~\emph{et al}\onedot~\cite{Steerability} applied data augmentation methods to the original data and train the model in a self-supervised manner, but they are limited to these simple semantic features.
\vspace{0.5em}
\noindent\textbf{Unsupervised Methods}
Compared to supervised methods, the unsupervised scheme has two main advantages.
First, it avoids the need for manual labeling, thus is more applicable to generalize to other categories, which greatly promotes the practical value of the methods.
Second, the lack of supervision information also implies the lifting of restrictions, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, the models are encouraged to find new semantic features, which are not necessarily labeled by human annotators.
Some methods for semantic disentanglement introduced in \cref{subsubsec:semantic_disentanglement} also support semantic discovery, which mainly considers the gradient w.r.t\onedot} \def\dof{d.o.f\onedot inputs or mutual information.
For example, Ramesh~\emph{et al}\onedot~\cite{SpectralRegularizer} found trajectories corresponding to the \textit{principal eigenvectors}, which is extended by Wang~\emph{et al}\onedot~\cite{wang2021geometric} via further introducing the \textit{Riemannian geometry metric}.
Peebles~\emph{et al}\onedot~\cite{HessianPenalty} leveraged their \textit{Hessian-based regularization} term to identify interpretable directions in BigGAN's latent space.
Voynov~\emph{et al}\onedot~\cite{voynov2020unsupervised} adopted similar strategies as \cite{VP-GAN}, which learns a group of editing basis.
By adding a perturbation to the latent code, the model is required to \textit{predict the perturbation} given the original and modified image pairs.
Tzelepis~\emph{et al}\onedot~\cite{WarpedGANSpace} further argued that existing methods assume the latent directions are linear, which limits the disentanglement, thus they introduce RBF kernel to map the latent representations into a \textit{non-linear space}, and learn a non-linear RBF path.
As analyzed in \cref{subsubsec:GANDissection}, the final output is determined by the features in the previous layers, thus GAN features can be considered for easier semantic discovery than the image domain.
Following this idea, H{\"a}rk{\"o}nen~\emph{et al}\onedot~\cite{GANSpace} proposed that the \textit{principal components of early layer features} represent important factors of variations, so that they find semantic directions by PCA operation over the early feature space.
Collins~\emph{et al}\onedot~\cite{EditingInStyle} sought to find attributes relevant to a region of interest (ROI), and proposed to cluster the features in a layer and sort the channels by the \textit{relevance with the ROI regions} (following the idea of \cite{GANDissection} but in an unsupervised way).
From another perspective, the parameters also play an important role in the generation process, which has also been introduced in \cref{subsubsec:GANrewriting}.
Therefore, there are also some methods leveraging the parameter space for semantic discovery.
Shen and Zhou~\cite{SeFa} formulated the perturbation in a layer and the corresponding changes in the output image, and finally the relationship between the semantic directions and the parameter matrix is derived, leading to a \textit{closed-form solution}, which achieves an efficient and effective semantic factorization method (termed by SeFa).
Cherepkov~\emph{et al}\onedot~\cite{NaviGAN} followed similar strategies to \cite{voynov2020unsupervised} and \cite{wang2021geometric}, and proposed two methods in the parameter domain by considering parameter perturbation and the Hessian of LPIPS w.r.t\onedot} \def\dof{d.o.f\onedot the GAN parameters.
It is worth noting that the large-scale vision-language models are introduced by Patashnik~\emph{et al}\onedot~\cite{StyleClip}, which leverages the common latent space for visual-linguist features, and combines the image-domain semantic discovery with the intrinsic semantic information in languages.
\section{Applications to Image Editing and Restoration}
\label{sec:applications}
Since most GAN models are trained in an unsupervised manner, they are not limited to a specific task and can serve as a generic prior for many other tasks with visual outputs (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, images).
In this section, we will go through these applications, where a brief summary can be found in \cref{tab:methods}.
\subsection{Image Editing}
\label{subsec:app_image_editing}
As introduced in \cref{subsec:GAN_Inversion,sec:disentanglement}, the latent spaces of recent GAN models have been widely explored in the literature.
With the exhilarating attribute disentanglement ability of the latest GAN models, we are able to discover and traverse the latent spaces in a more semantic-aware way, making image editing applicable with pre-trained GAN models.
In this section, we briefly review the image editing applications relying on semantic disentanglement and discovery methods, including image interpolation, style transfer, image crossover, attribute editing, image transform, \emph{etc}\onedot} \def\vs{\emph{vs}\onedot.
\subsubsection{Image Interpolation (Image Morphing)}
Image interpolation, also known as image morphing, aims at interpolating two images semantically.
By observing the generated images, we can have an intuitive sense of the space used for interpolation, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, a well-characterized space should be reflected in smooth and semantically appropriate transitions in the interpolated images.
Therefore, apart from generating new images, image interpolation is often utilized to evaluate the quality of the discovered latent space.
For example, as shown in \cref{fig:intro_interpolation}, the interpolation operation in the pixel space (\emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, $\mathbf{I}=\alpha\cdot\mathbf{x}_1+(1-\alpha)\cdot\mathbf{x}_2$) leads to obvious artifacts, and the results cannot meet the semantic expectations.
On the contrary, the interpolation in the latent space (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, \cite{zhu2016generative,Image2StyleGAN,Consecutive,IDInvert}) is able to well describe the change of view, which can be formulated by,
\begin{equation}
\mathbf{I}=\mathit{G}(\beta\cdot\mathbf{z}_1+(1-\beta)\cdot\mathbf{z}_2); \bm{\theta}_\mathit{G}),
\label{eqn:interpolation}
\end{equation}
where $\mathbf{z}_1$ and $\mathbf{z}_2$ can be obtained by the inversion methods (\cref{subsec:GAN_Inversion}) from two images $\mathbf{x}_1$ and $\mathbf{x}_2$.
There are two special cases that are worth mentioning.
As revealed by Abdal~\emph{et al}\onedot~\cite{Image2StyleGAN}, when performing GAN inversion via optimization-based methods, a GAN model trained with face datasets can well reconstruct an image from other classes (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, cats).
However, since the optimization-based methods are less constrained and can have a much larger accessible latent space, interpolating between these reconstructed images will not produce meaningful results (see \cref{fig:interpolation}), which further verifies the propositions of IDInvert~\cite{IDInvert} about domain-regularized optimization (see \cref{eqn:inversion_idinvert}).
Another thing is about the training/fine-tuning based inversion methods (see \cref{tab:methods}).
For more accurate reconstruction, some of these methods (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, \cite{DGP}) will derive a specific group of model parameters for each sample.
In this way, there will be two sets of $\bm{\theta}_\mathit{G}$ when performing image interpolation operations, which is inconsistent with \cref{eqn:interpolation}.
Fortunately, Wang~\emph{et al}\onedot~\cite{wang2019deep} showed that when fine-tuned from the same checkpoint, the parameters of two models can be interpolated to achieve an intermediate function.
Thus, the image interpolation can be performed as,
\begin{equation}
\mathbf{I}=\mathit{G}(\beta\cdot\mathbf{z}_1+(1-\beta)\cdot\mathbf{z}_2); {\color{red}\beta\cdot\bm{\theta}_{\mathit{G}_1}+(1-\beta)\cdot\bm{\theta}_{\mathit{G}_2}}),
\end{equation}
where $\bm{\theta}_{\mathit{G}_1}$ and $\bm{\theta}_{\mathit{G}_2}$ are the specialized parameters for reconstructing $\mathbf{x}_1$ and $\mathbf{x}_2$, respectively, and the difference from \cref{eqn:interpolation} is highlighted with {\color{red}red}.
\begin{figure}[t]
\centering
\scriptsize
\begin{tabular}{cccccc}
\hspace{-2.75mm} \includegraphics[width=.15\linewidth]{figures/interpolation/0.png} \hspace{-2.75mm} & \hspace{-2.75mm} \includegraphics[width=.15\linewidth]{figures/interpolation/1.png} \hspace{-2.75mm} & \hspace{-2.75mm} \includegraphics[width=.15\linewidth]{figures/interpolation/2.png} \hspace{-2.75mm} & \hspace{-2.75mm} \includegraphics[width=.15\linewidth]{figures/interpolation/3.png} \hspace{-2.75mm} & \hspace{-2.75mm} \includegraphics[width=.15\linewidth]{figures/interpolation/4.png} \hspace{-2.75mm} & \hspace{-2.75mm} \includegraphics[width=.15\linewidth]{figures/interpolation/5.png}\hspace{-2.75mm} \\[-1mm]
\hspace{-2.75mm} $\mathit{G}(\mathbf{z}_1)$ \hspace{-2.75mm} & \hspace{-2.75mm} $\lambda=$0.2 \hspace{-2.75mm} & \hspace{-2.75mm} $\lambda=$0.4 \hspace{-2.75mm} & \hspace{-2.75mm} $\lambda=$0.6 \hspace{-2.75mm} & \hspace{-2.75mm} $\lambda=$0.8 \hspace{-2.75mm} & \hspace{-2.75mm} $\mathit{G}(\mathbf{z}_2)$ \hspace{-2.75mm}
\end{tabular}
\caption{Inversion and interpolation~\cite{Image2StyleGAN} between two cat images A and B from the AFHQ dataset~\cite{StarGANv2} with a GAN model trained on the FFHQ dataset~\cite{StyleGAN}. It can be seen that even the image can be well reconstructed, the interpolation results ($\mathit{G}(\lambda\cdot\mathbf{z}_1+(1-\lambda)\cdot\mathbf{z}_2)$) are looming the shape of human faces. $\mathbf{z}_1$ and $\mathbf{z}_2$ are latent codes obtained from $\mathbf{x}_1$ and $\mathbf{x}_2$ via \cite{Image2StyleGAN}, respectively.}
\label{fig:interpolation}
\end{figure}
\subsubsection{Style and Attribute Transfer, Image Crossover}
\label{subsubsec:app_style_transfer}
As shown in \cite{StyleGAN}, the features in earlier layers control coarser-grained attributes like structure and pose, while the ones in latter layers control finer-grained attributes like appearance and textures.
For the StyleGAN series, the latent codes are corresponding to these layers in the $\mathcal{W}$ space, and with the style mixing regularization during training, the latent codes in different layers can be distinct~\cite{StyleGAN}.
Therefore, the pre-trained StyleGAN models are ready-to-use for style transfer tasks by replacing the latent codes in latter layers with the ones from the style image~\cite{Image2StyleGAN,StyleGAN2Distillation,guan2020collaborative,GH-Feat,baseline}.
In this way, the characters regarded as style (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, color, stroke, texture, \emph{etc}\onedot} \def\vs{\emph{vs}\onedot) are transferred to the content image.
Instead of simply mixing style codes from different images, some works also perform a more precise control by specifying the attributes to be transferred~\cite{StyleRig,IDDisentanglement}, which can be combined with the methods introduced in \cref{sec:disentanglement} for disentangling the attributes.
Some methods~\cite{pSp,baseline} also considered image-to-image translation tasks such as transferring sketch or parsing (segmentation) images into natural ones, or generating toonified versions of a human face.
Apart from the global tasks of style and attribute transfer, some methods focus on local modifications from two perspectives, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, local style transfer and image crossover.
The former is the style transfer task in a local region, while the latter is to combine multiple images and regenerate the stitched image to make it visually plausible.
For example, \cite{LatentComposition} introduced random masks (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, $\mathbf{m}_1$ and $\mathbf{m}_2$) during training, such that the encoder will be compatible with masked input images, making it possible to combine multiple incomplete images by corresponding mask directly,
\begin{equation}
\mathbf{I}=\mathit{G}(\mathit{E}(\mathbf{m}_1\circ\mathbf{x}_1+\mathbf{m}_2\circ\mathbf{x}_2)),
\end{equation}
which generates more realistic and natural images than blending in pixel or latent spaces like,
\begin{align}
\mathbf{I}&=\mathit{G}(\mathit{E}(\alpha\cdot\mathbf{x}_1+(1-\alpha)\cdot\mathbf{x}_2)),&(\mathtt{pixel\ space\ blend})\label{eqn:pixel_blend}\\ \mathbf{I}&=\mathit{G}(\alpha\cdot\mathit{E}(\mathbf{x}_1)+(1-\alpha)\cdot\mathit{E}(\mathbf{x}_2)).&(\mathtt{latent\ space\ blend})\label{eqn:latent_blend}
\end{align}
Note that Image2StyleGAN++~\cite{Image2StyleGAN++} performs local style transfer in both $\mathcal{W}+$ and $\mathcal{N}$ spaces. Some methods directly transfer the features rather than latent codes~\cite{suzuki2018spatially,EditingInStyle}.
Visual results can be found in \cref{fig:app_style_transfer}.
\begin{figure}[t]
\centering
\scriptsize
\begin{tabular}{cc}
\begin{tabular}{ccccc}
\hspace{-3mm} & \hspace{-3mm} \includegraphics[width=.1\linewidth]{figures/StyleTransfer/input1.jpg} \hspace{-3mm} & \hspace{-3mm} \includegraphics[width=.1\linewidth]{figures/StyleTransfer/input2.jpg} \hspace{-3mm} & \hspace{-3mm} \includegraphics[width=.1\linewidth]{figures/StyleTransfer/input3.jpg} \hspace{-3mm} & \hspace{-3mm} \includegraphics[width=.1\linewidth]{figures/StyleTransfer/input4.jpg} \hspace{-3mm} \\
\hspace{-3mm} \includegraphics[width=.1\linewidth]{figures/StyleTransfer/style1.jpg} \hspace{-3mm} & \hspace{-3mm} \includegraphics[width=.1\linewidth]{figures/StyleTransfer/output1-1.jpg} \hspace{-3mm} & \hspace{-3mm} \includegraphics[width=.1\linewidth]{figures/StyleTransfer/output2-1.jpg} \hspace{-3mm} & \hspace{-3mm} \includegraphics[width=.1\linewidth]{figures/StyleTransfer/output3-1.jpg} \hspace{-3mm} & \hspace{-3mm} \includegraphics[width=.1\linewidth]{figures/StyleTransfer/output4-1.jpg} \hspace{-3mm} \\
\hspace{-3mm} \includegraphics[width=.1\linewidth]{figures/StyleTransfer/style2.jpg} \hspace{-3mm} & \hspace{-3mm} \includegraphics[width=.1\linewidth]{figures/StyleTransfer/output1-2.jpg} \hspace{-3mm} & \hspace{-3mm} \includegraphics[width=.1\linewidth]{figures/StyleTransfer/output2-2.jpg} \hspace{-3mm} & \hspace{-3mm} \includegraphics[width=.1\linewidth]{figures/StyleTransfer/output3-2.jpg} \hspace{-3mm} & \hspace{-3mm} \includegraphics[width=.1\linewidth]{figures/StyleTransfer/output4-2.jpg} \hspace{-3mm} \\
\hspace{-3mm} \includegraphics[width=.1\linewidth]{figures/StyleTransfer/style3.jpg} \hspace{-3mm} & \hspace{-3mm} \includegraphics[width=.1\linewidth]{figures/StyleTransfer/output1-3.jpg} \hspace{-3mm} & \hspace{-3mm} \includegraphics[width=.1\linewidth]{figures/StyleTransfer/output2-3.jpg} \hspace{-3mm} & \hspace{-3mm} \includegraphics[width=.1\linewidth]{figures/StyleTransfer/output3-3.jpg} \hspace{-3mm} & \hspace{-3mm} \includegraphics[width=.1\linewidth]{figures/StyleTransfer/output4-3.jpg} \hspace{-3mm} \\
\end{tabular} &
\begin{tabular}{ccc}
\hspace{-3mm} \includegraphics[width=.1\linewidth]{figures/CrossOver/target5.jpg} \hspace{-3mm} & \hspace{-3mm} \includegraphics[width=.1\linewidth]{figures/CrossOver/target6.jpg} \hspace{-3mm} & \hspace{-3mm} \includegraphics[width=.1\linewidth]{figures/CrossOver/target7.jpg} \hspace{-3mm} \\
\hspace{-3mm} \includegraphics[width=.1\linewidth]{figures/CrossOver/context5.jpg} \hspace{-3mm} & \hspace{-3mm} \includegraphics[width=.1\linewidth]{figures/CrossOver/context6.jpg} \hspace{-3mm} & \hspace{-3mm} \includegraphics[width=.1\linewidth]{figures/CrossOver/context7.jpg} \hspace{-3mm} \\
\hspace{-3mm} \includegraphics[width=.1\linewidth]{figures/CrossOver/mix5.jpg} \hspace{-3mm} & \hspace{-3mm} \includegraphics[width=.1\linewidth]{figures/CrossOver/mix6.jpg} \hspace{-3mm} & \hspace{-3mm} \includegraphics[width=.1\linewidth]{figures/CrossOver/mix7.jpg} \hspace{-3mm} \\
\hspace{-3mm} \includegraphics[width=.1\linewidth]{figures/CrossOver/output5.jpg} \hspace{-3mm} & \hspace{-3mm} \includegraphics[width=.1\linewidth]{figures/CrossOver/output6.jpg} \hspace{-3mm} & \hspace{-3mm} \includegraphics[width=.1\linewidth]{figures/CrossOver/output7.jpg} \hspace{-3mm} \\
\end{tabular} \\[-1mm]
\hspace{-3mm}Style Transfer & Image Crossover\hspace{-3mm}
\end{tabular}
\caption{Style transfer and image crossover. The images are processed by IDInvert~\cite{IDInvert}.}
\label{fig:app_style_transfer}
\end{figure}
\subsubsection{Attribute Editing}
Attribute editing aims to modify certain attributes of an image (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, age, gender, \emph{etc}\onedot} \def\vs{\emph{vs}\onedot), which is a popular application of GAN models in recent years.
The difference between attribute editing and attribute transfer in \cref{subsubsec:app_style_transfer} is that the reference image is not necessary for attribute editing tasks.
Some methods focus on \textit{physical attributes}.
For example, Zhu~\emph{et al}\onedot~\cite{zhu2016generative} and Abdal~\emph{et al}\onedot~\cite{Image2StyleGAN++} took color strokes provided by users as input and optimize the image towards a reasonable appearance where the color or shape of regions covered by the strokes are modified, while the background regions are kept unchanged.
Jahanian~\emph{et al}\onedot~\cite{Steerability} focused on self-supervised training for zooming, shifting, rotating, brightness adjustment, and \emph{etc}\onedot} \def\vs{\emph{vs}\onedot.
To manipulate the attributes more closely related to \textit{semantic concepts}, some methods~\cite{IcGAN,DGP,GANalyze,StyleGAN2Distillation,StyleFlow,SAM} leveraged pre-trained attribute classifiers or attribute labels of the datasets to identify the manipulation direction in the latent space.
Some methods~\cite{InterFaceGAN,mGANPrior,IDInvert,StyleSpace} chose to first define an attribute separation hyperplane and take the normal vector as editing directions. Considering the non-linearity of the latent space, especially when manipulating the attributes to a large extent, Wang~\emph{et al}\onedot~\cite{Hijack-GAN} proposed to learn a proxy model to estimate the gradient of the pre-trained GAN models and modify the attributes step by step.
The primary drawback of the aforementioned methods is that they need annotated labels (or classification models trained with these labels) for training.
To eliminate the dependence on these labels, some works~\cite{SeFa,NaviGAN} proposed unsupervised methods to disentangle different attributes, which have been introduced in detail in \cref{sec:disentanglement}.
Besides, some methods~\cite{GANDissection,GANPaint,StyleIntervention} achieved local attribute editing based on the observations introduced in \cref{subsubsec:GANDissection}.
There are also some other interesting methods to achieve attribute editing.
Image2StyleGAN~\cite{Image2StyleGAN} uses a one-shot method that obtains the manipulation direction vector via the difference between two images.
\cite{GH-Feat} sampled global or local features in the latent space which results in random new attributes.
E4E~\cite{E4E} proposes to leverage an encoder for generating target latent codes directly instead of the previous inversion+editing methods.
The visual results are given in \cref{fig:app_attribute_editing}.
\begin{figure}[t]
\centering
\scriptsize
\begin{tabular}{cc}
\includegraphics[width=.45\linewidth]{figures/Attribute Editing/input.jpg} & \includegraphics[width=.45\linewidth]{figures/Attribute Editing/output1.jpg} \\[-1mm]
Input & To \textit{Old} \\
\includegraphics[width=.45\linewidth]{figures/Attribute Editing/output2.jpg} & \includegraphics[width=.45\linewidth]{figures/Attribute Editing/output3.jpg} \\[-1mm]
To \textit{Not Smile} & To \textit{Profile}
\end{tabular}
\caption{Attribute editing with manipulation directions found by InterFaceGAN~\cite{InterFaceGAN}.}
\label{fig:app_attribute_editing}
\end{figure}
\subsubsection{Image Transform and Jittering, Visual Alignment}
\label{subsubsec:app_transform}
Since StyleGAN~\cite{StyleGAN} introduces layer-wise noises (\emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, the $\mathcal{N}$ space), it naturally supports random jittering on different layers.
Pan~\emph{et al}\onedot~\cite{DGP} further added Gaussian noise to the $\mathcal{Z}$ space of BigGAN model~\cite{BigGAN} and achieved image jittering with larger variance.
Jahanian~\emph{et al}\onedot~\cite{Steerability} also explored the image transformation with pre-trained GANs.
As introduced in \cref{sec:GANs_development}, StyleGAN3~\cite{StyleGAN3} adopts equivariant layers, which greatly facilitates the output image quality when performing transform operations like shifting.
Based on the image transform operations, some methods leverage pre-trained GANs for visual alignment.
For example, pSp~\cite{pSp} constrains that face images are inverted to the same latent code with its horizontally flipped counterpart, which leads to a frontal face for profile ones.
Peebles~\emph{et al}\onedot~\cite{GANGealing} further extended the method to more general cases by clustering one or several center points (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, frontal for faces), and finally they can obtain a spatial transformer network that can perform visual alignment as well as the reverse process.
In other words, we can edit the transformed (aligned) images for dealing with a certain object in all video frames.
In \cref{fig:app_image_transform}, we show several examples of these applications.
\begin{figure}[t]
\centering
\scriptsize
\begin{tabular}{ccc}
\hspace{-3mm} \includegraphics[width=.33\linewidth]{figures/TR/average1.png} \hspace{-3mm} & \hspace{-3mm} \includegraphics[width=.33\linewidth]{figures/TR/average2.png} \hspace{-3mm} & \hspace{-3mm} \includegraphics[width=.33\linewidth]{figures/TR/average3.png} \hspace{-3mm} \\[-1mm]
\hspace{-3mm} Averaging Original Images \hspace{-3mm} & \hspace{-3mm} Affine Transform \hspace{-3mm} & \hspace{-3mm} Elastic Transform \hspace{-3mm} \\
\multicolumn{3}{c}{\includegraphics[width=.95\linewidth]{figures/TR/TR.png}}\\[-1mm]
\multicolumn{3}{c}{Editing on Aligned Images}
\end{tabular}
\caption{Dense visual alignment method~\cite{GANGealing} trained with GAN supervision. Please zoom in for better observation.}
\label{fig:app_image_transform}
\end{figure}
\subsection{Image Restoration}
Unlike the applications introduced in \cref{subsec:app_image_editing}, which mainly leverage the exhilarating latent space disentanglement ability of recent GAN models, image restoration tasks rely more on the high-quality natural image priors provided by the pre-trained large-scale GAN models.
Note that the GAN model is trained to learn the mapping $\mathit{g}:\mathbf{z}\mapsto\mathbf{x}$, where $\mathbf{z}$ is generally sampled from a simple distribution (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, the standard Gaussian distribution $\mathbf{z}\sim\mathcal{N}(\mathbf{0}, \mathbf{I})$), while $\mathbf{x}\in\mathcal{X}$ represents the high-quality training datasets.
With the millions or billions of training iterations, we assume that $\{\mathbf{z}_\mathit{i}\}$ covers at least a truncated Gaussian space, or more precisely, a manifold, which is denoted by $\mathcal{M}$.
In other words, given a latent code sampled in $\mathcal{M}$, for a well-trained GAN model helps to constrain the output to the $\mathcal{X}$ space, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, the output tends to be a high-quality image following the training image distribution.
In this way, given a corrupted image, we can find a latent code in the latent space, which has the smallest ``distance'' from this corrupted image.
Then the image generated from the latent code can be regarded as the one with the highest likelihood among all possible high-quality counterparts.
The relevant image restoration applications include image super-resolution, image denoising, image inpainting, image colorization, artifacts removal, \emph{etc}\onedot} \def\vs{\emph{vs}\onedot., and we will go through these applications in this subsection.
\subsubsection{Image Super-resolution, Image Denoising, Image Inpainting, and Image Colorization}
Considering the common methods used for image super-resolution, image denoising, image inpainting, and image colorization, we introduce these applications together in this part.
Overall, these methods can be divided into two classes, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, unsupervised and supervised.
Specifically, for unsupervised methods~\cite{mGANPrior,PULSE,DGP}, the optimization objective is typically defined by,
\begin{equation}
\mathcal{L}_\mathit{unsup}=\argmin_\mathbf{z}\|\zeta(\mathit{G}(\mathbf{z};\bm{\theta}_\mathit{G})) - \mathbf{x}_\mathit{LQ}\|_\mathit{p}^\mathit{p}+\rho(\mathbf{z}),
\end{equation}
where $\zeta$ denotes the degradation function (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, down-sampling for image super-resolution, color-to-gray for image colorization, masking for image inpainting), and $\rho$ denotes the regularization on $\mathbf{z}$ derived from the distribution of the latent space (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, Gaussian).
Here we note that, as shown in \cref{fig:app_restoration}, the denoising task here is actually inpainting tasks with random defective pixels rather than additive noise suppressing, and for super-resolution tasks the degradation process is approximated by simple down-sampling algorithms like bicubic.
These drawbacks make the optimization-based unsupervised methods limited to a certain range of tasks, where the ground-truth images are unavailable during inference.
To solve this problem, with the development of learning-based methods, the ground-truth (reference) images can be utilized for training the encoder to obtain the restoration results, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, the supervised methods.
With the reference, the encoder can directly learn the mapping $\mathit{f}: \mathbf{x}_\mathit{LQ}\mapsto\mathbf{z}_\mathit{HQ}$, which is then delivered into the pre-trained GAN model to obtain the high-quality output image $\mathbf{x}_\mathit{HQ}=\mathit{G}(\mathbf{z}_\mathit{HQ}; \bm{\theta}_\mathit{G})$~\cite{pSp,bartz2020one,GLEAN,ImprovedGANEmbedding,GFPGAN,baseline,GPEN}.
In particular, the loss function is defined by,
\begin{equation}
\mathcal{L}_\mathit{sup}=\sum_\mathit{i}\|\xi_\mathit{i}(\mathit{G}(\mathit{E}(\mathbf{x}_\mathit{LQ};\bm{\theta}_\mathit{E});\bm{\theta}_\mathit{G}) - \xi_\mathit{i}(\mathbf{x}_\mathit{HQ})\|_\mathit{p}^\mathit{p}+\rho(\mathbf{z}),
\end{equation}
where $\xi_\mathit{i}$ denotes the identity operation or models for loss functions like identity loss~\cite{ArcFace}, LPIPS loss~\cite{LPIPS}, and face parsing loss~\cite{MaskGAN}, which are utilized for better identity preserving or perceptual image quality.
It is worth noting that, unlike other methods which produce the output images directly by the pre-trained GAN model, GLEAN~\cite{GLEAN} stacks an extra decoder on the top of the GAN model, where the features from the pre-trained GAN model as taken as intermediate features to improve the quality of the output generated by the extra decoder.
Visual results can be found in \cref{fig:app_restoration}.
\begin{figure}[t]
\centering
\scriptsize
\begin{tabular}{ccc}
\hspace{-2.75mm} \includegraphics[width=.32\linewidth]{figures/Restoration/Restoration1.png} \hspace{-2.75mm} & \hspace{-2.75mm} \includegraphics[width=.32\linewidth]{figures/Restoration/Restoration2.jpg} \hspace{-2.75mm} & \hspace{-2.75mm} \includegraphics[width=.32\linewidth]{figures/Restoration/Restoration3.jpg} \hspace{-2.75mm}\\[-1mm]
\hspace{-2.75mm} Real Image Super-resolution \hspace{-2.75mm} & \hspace{-2.75mm} Image Colorization \hspace{-2.75mm} & \hspace{-2.75mm} Image Inpainting \hspace{-2.75mm}
\end{tabular}
\caption{Image restoration with GPEN~\cite{GPEN} pre-trained with FFHQ~\cite{StyleGAN} dataset.}
\label{fig:app_restoration}
\end{figure}
\subsubsection{Artifact Removal}
As shown by Bau~\emph{et al}\onedot~\cite{GANDissection}, similar to other objects, the generation of some artifacts is also controlled by the neurons of the GAN model.
Therefore, the network dissection technique can also be used to identify the neurons that cause the artifacts, which can bring an obvious image quality improvement (the FID~\cite{FID} improves from $\sim$43 to $\sim$27).
Shen~\emph{et al}\onedot~\cite{InterFaceGAN} further explored artifact removal in the latent space, where a linear-SVM is trained with 4K bad results to obtain the separation hyperplane, and the artifacts are successfully suppressed by using the normal vector.
\cite{GANCorrection} trained an artifact classifier with a proposed dataset, and identified the flawed regions via Grad-CAM~\cite{Grad-CAM}. Then the artifacts are suppressed in a sequential process.
The visual results are shown in \cref{fig:app_artifacts_removal}.
\begin{figure}[t]
\centering
\scriptsize
\begin{tabular}{cccccc}
\hspace{-2.75mm} \includegraphics[width=.16\linewidth]{figures/ArtifactsRemoval/input1.jpg} \hspace{-2.75mm} & \hspace{-2.75mm} \includegraphics[width=.16\linewidth]{figures/ArtifactsRemoval/output1.jpg} \hspace{-2.75mm} & \hspace{-2.75mm} \includegraphics[width=.16\linewidth]{figures/ArtifactsRemoval/input2.jpg} \hspace{-2.75mm} & \hspace{-2.75mm} \includegraphics[width=.16\linewidth]{figures/ArtifactsRemoval/output2.jpg} \hspace{-2.75mm} & \hspace{-2.75mm} \includegraphics[width=.16\linewidth]{figures/ArtifactsRemoval/input3.jpg} \hspace{-2.75mm} & \hspace{-2.75mm} \includegraphics[width=.16\linewidth]{figures/ArtifactsRemoval/output3.jpg} \hspace{-2.75mm} \\[-1mm]
\hspace{-2.75mm} Input \hspace{-2.75mm} & \hspace{-2.75mm} Output \hspace{-2.75mm} & \hspace{-2.75mm} Input \hspace{-2.75mm} & \hspace{-2.75mm} Output \hspace{-2.75mm} & \hspace{-2.75mm} Input \hspace{-2.75mm} & \hspace{-2.75mm} Output \hspace{-2.75mm}
\end{tabular}
\caption{Artifacts removal via GANDissection~\cite{GANDissection}, where the neurons controlling the generation of artifacts are set to zero.}
\label{fig:app_artifacts_removal}
\end{figure}
\subsection{Other Applications}
Apart from the aforementioned image editing and restoration applications, the pre-trained GAN models can also be applied to other tasks.
For example, information hiding~\cite{baseline}, unsupervised domain adaptation~\cite{MimicGAN}, adversarial defense~\cite{MimicGAN,DGP}, anomaly detection~\cite{MimicGAN}, 3D reconstruction~\cite{pan20202d,zhang2021unsupervised}, and so on.
Besides, a new trend in leveraging pre-trained GAN models is to combine multiple GAN models for generating different parts (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, the face region and the whole body)~\cite{InsetGAN}, which has great potential for generating images with complex scenes and rich contents.
\section{Discussion}
\label{sec:discussion}
\subsection{Challenges and Open Problems}
\subsubsection{GAN Inversion}
For leveraging pre-trained GAN models on real images, it is a fundamental task to invert them into the latent space, with which the images should be well reconstructed via the GAN models.
Although existing GAN inversion methods have demonstrated decent performance, they are still far from perfect, and there are many challenging problems that remain unsolved.
\vspace{0.5em}
\noindent\textbf{Fidelity and Editability}
When inverting a real image into latent space, there exists a trade-off between the reconstruction fidelity and latent code editability.
On the one hand, for methods that only use the $\mathcal{W}$ space (or the derived spaces like $\mathcal{W}+$ or $\mathcal{S}$ spaces), their reconstruction accuracy is greatly limited due to the narrow latent space in comparison to the image space.
Besides, mode dropping~\cite{GANSeeing} is a severe problem influencing the inversion quality.
On the other hand, for methods that leverage spatial dimensions (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, the $\mathcal{F}$~\cite{BDInvert}, $\bm{\Theta}$~\cite{NaviGAN,HyperStyle}, and $\mathcal{N}$~\cite{Image2StyleGAN++,GPEN} spaces), the editability becomes a main concern as these spaces are less disentangled.
Wang~\emph{et al}\onedot~\cite{HFGI} took a step forward by controlling the passage of spatial-dimension features with the latent code in $\mathcal{W}$ space, which achieves a better trade-off between fidelity and editability.
However, it still remains to be better solved for real-world applications, especially when there are geometry changes (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, zoom, rotation) or camera movements.
\vspace{0.5em}
\noindent\textbf{Out-of-distribution Generalization}
Since mode dropping~\cite{GANSeeing} occurs even when a subject has shown in the training images, the out-of-distribution generalization ability (on unseen classes or domains) is a main concern of pre-trained GANs.
Indeed, with less constrained optimization, the GAN models are able to generate images from classes that have no intersection with the training set~\cite{Image2StyleGAN}.
Nevertheless, the resultant latent code does not fall into the meaningful latent space, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, the semantic directions in the latent space are inconsistent with the out-of-distribution images.
So that we are unable to edit these images following the prior learned in the training set.
Some methods have tried to adapt GAN models to new domains via few-shot learning via cross-domain correspondence~\cite{ojha2021few}, but the properties like editability require further exploration.
\subsubsection{Image Generation, Editing, and Restoration}
Current methods have achieved appealing performance on image generation, editing, and restoration tasks, where the results are photo-realistic and the semantic information is well represented.
Here we conclude some potential directions for better leveraging pre-trained GANs.
\vspace{0.5em}
\noindent\textbf{Multi-GAN Composition}
Due to the limited model capacity and training mechanism, it is rather hard for a single GAN model to generate a complex scene with every object photo-realistic and natural.
Therefore, Fr{\"u}hst{\"u}ck~\emph{et al}\onedot~\cite{InsetGAN} tried to combine two pre-trained GAN models together for a compositional GAN model, one of which is for generating the whole body of a person, while the other is responsible for providing a better face region.
The two GAN models are combined by joint-optimizing the latent codes.
It is of great potential to generate complex scenes via the combination of multiple GAN models.
\vspace{0.5em}
\noindent\textbf{Physical Rules and 3D Awareness}
For the purpose of image editing or video generation, the GAN models should be aware of the physical rules for realistic results, and the 3D awareness also benefits to produce reasonable structures.
As introduced in \cref{subsubsec:app_transform}, the pre-trained GAN models are able to manipulate the geography properties of the images, in other words, they have implicitly learned some physical rules from the training set.
Besides, the 3D priors (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, 3DMM) have been implied for learning face editing models~\cite{StyleRig,PIE}, which shows appealing results.
Furthermore, the embedded physical rules and 3D awareness will also help to combine multiple GAN models for generating complex scenes.
\vspace{0.5em}
\noindent\textbf{Diverse and Reliable Results}
A major problem of existing image editing and restoration methods based on pre-trained GANs is that only one result is obtained for a given input.
Yet these two tasks are both ill-posed, which prefer diverse results for more flexible choices.
More importantly, for image restoration tasks, since the input image is usually greatly degraded, the results are sometimes less reliable.
In this condition, we argue that providing multiple results with confidence scores and/or precise editing directions recommended based on the degraded input will be more applicable, where timely feedback and adjustment can be obtained through user participation (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, eyewitness).
\vspace{0.5em}
\noindent\textbf{Multi-modality Combination}
It is common sense that the generated images should be semantically consistent, and we hope that we can directly provide the semantic descriptions to modify the generated (edited, restored) results.
To this end, many works~\cite{StyleClip,DALLE2,imagen} have sought to combine language models with generative models, and exhibit extraordinary performance in text-guided image editing.
It is challenging yet exciting for the pre-trained GAN models to be well-aligned with semantic concepts in the languages, which will make the real-world application based on pre-trained GANs more convenient.
\subsubsection{Others}
\vspace{0.5em}
\noindent\textbf{Combining with Other Generative Models}
Comparing to other generative methods (\emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot, VAE~\cite{VAE}, Normalizing Flow~\cite{GLOW}, and Diffusion Models~\cite{DDPM}), the advantages of GAN models lie in the superior image quality (\textit{vs} VAEs), flexible structure (\textit{vs} normalizing flows) and efficient sampling (\textit{vs} diffusion models).
However, GAN models approximate the data distribution in an implicit way, and they are unstable to train and can easily lead to mode collapse.
A potential way is to combine the GAN models with other generative methods, which can give full play to the advantages of both methods.
For example, Lyu~\emph{et al}\onedot~\cite{lyu2022accelerating} proposed to use GANs to generate an intermediate result for diffusion models, Grover~\emph{et al}\onedot~\cite{grover2018flow} combined the maximum likelihood of normalizing flow with GANs.
\vspace{0.5em}
\noindent\textbf{Evaluation Metrics}
For boosting the development of GANs and techniques of leveraging pre-trained GAN models, the evaluation metrics play an important role.
For intuitive evaluations such as reconstruction quality and mode dropping, there have been some methods like FID~\cite{FID} and FSD~\cite{GANSeeing}.
However, indirect or more complex evaluations like the detail and semantic consistency after editing and restoration are less explored.
Besides, for the aforementioned challenging problems, corresponding evaluation metrics are also desired to evaluate the effectiveness.
\subsection{Conclusion}
Behind the superior generation ability, GAN models have shown the concept abstraction ability and the learned generative image priors, which have been extensively explored in recent methods.
In this paper, we give a comprehensive survey of recent progress on leveraging pre-trained GAN models for image editing and restoration tasks, where the two main features of recent GAN models are utilized, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, the disentanglement ability of the latent space and the generative image priors inherently learned by the GAN models.
By introducing the neuron understanding methods, we try to construct an intuitive and in-depth impression of pre-trained GAN models.
Subsequently, we review the GAN inversion methods as well as the semantic disentanglement and discovery techniques, and the relevant applications of image editing and restoration tasks are introduced.
Finally, we discuss some challenges and open problems in leveraging pre-trained GAN models for image editing and restoration.
\Acknowledgements{This work was supported by National Natural Science Foundation of China (Grant Nos. U19A2073 and 62006064) and Hong Kong RGC RIF (Grant No. R5001-18).}
\bibliographystyle{SCIS2022}
| -98,950.945316 |
[
-2.896484375,
2.5859375
] | 53.014184 |
[
-2.771484375,
2.283203125,
-0.91845703125,
-3.970703125,
-1.197265625,
4.8671875
] |
[
2.5,
5.14453125,
-0.0689697265625,
4.2734375
] | 1,230 | 12,980 |
[
-2.2890625,
2.400390625
] | 34.303104 |
[
-6.2265625,
-3.75,
-3.361328125,
-1.3203125,
2.19921875,
10.1796875
] | 0.819811 | 19.000772 | 21.417565 | 3.716086 |
[
2.2810416221618652
] | -67,187.754623 | 7.273652 | -97,840.375318 | 0.523239 | 6.405584 |
[
-3.1875,
-3.33203125,
-3.259765625,
-3.9140625,
2.638671875,
10.5078125
] |
[
-5.62890625,
-1.3583984375,
-1.859375,
-1.369140625,
3.326171875,
4.23046875
] | |
BkiUdqE4eIOjRwMTjGno
|
\section{Introduction}
\IEEEPARstart{W}{ith} \ankit{the introduction of Bitcoin~\cite{nakamoto2008bitcoin} in 2008, blockchain technology has been adopted by almost every scenario in today's digital world with an aim to keep temper-proof records~\cite{conti2018economic}. Originally, the concept of distributed consensus - upon which the blockchain technology is founded - dates back to the~'90s~\cite{haber1990time, szabo2005bitgold, Wei1998money, finney2004rpow}. But, after the success of Bitcoin, a multitude of other areas have been} transformed by the idea of a transparent and distributed public ledger, such as elections, contract management, insurance, charity, and ownership transfer through inheritance and will~\cite{blockchain_app}.
\par
The Bitcoin system combines together different well-known concepts, such as digital signatures, hashing functions, PoW, and Merkle trees. However, \ankit{with an exponential growth in the number of users and Bitcoin-miners,} the Bitcoin network observed collateral effects of its consensus protocol. \ankit{In particular, it suffers from} scalability~\cite{croman2016scaling}, hashing power centralization, mining strategies exposures~\cite{rosenfeld2011analysis, eyal2015miner, luu2015power}, and energy- and computational-deficiency~\cite{malone2014energy}. \ankit{To this end, a} variety of solutions have been proposed over time. Some of these solutions \ankit{aim to improve the existing version of PoW ({e.g.,~} Litecoin\footnote{Litecoin adopted its \textit{scrypt} PoW function's design to increase the decentralization of the system and further optimize the mining process~\cite{percival2009stronger}.}, Primecoin\footnote{Primecoin exploits the computational power used for the mining activity to contribute to mathematical research topics.}~\cite{king2013primecoin}}) while other proposals used completely different concepts, such as PoS, PoB~\cite{wiki_pob, slimcoin}, and PoET~\cite{intel}. \ankit{As a representative example, PoS systems (both stake modification-based~\cite{king2012ppcoin, cloakcoin2014, ren2014proof, pikepost} and actual balance-based~\cite{nxt, vasin2014blackcoin}) have serious concerns from the security point-of-view~\cite{li2017securing}; such issues have been partially solved by providing formally-proved security guarantees~\cite{kiayias2017ouroboros, bentov2016snow} or by discouraging users to act maliciously through punitive procedures~\cite{buterin2017casper, buterin2014slasher}. For the sake of completeness, some other works~\cite{sompolinsky2016spectre, popov2016iota} tried to achieve better scalability and throughput by implementing a block graph structure based on Directed Acyclic Graph (DAG), albeit, these solutions have their own demerits.}
\par
\ankit{On the other side, Algorand~\cite{micali2016algorand_tec, micali2017algorand_pap} - an innovative protocol for the blockchain technology -} \rankit{aims to solve the ``blockchain trilemma'' of decentralization, scalability, and security.} It relies on PoS integrated in a BA layout to achieve optimal resilience against faulty nodes. It guarantees an overwhelming probability of the blockchain's linearity and a block rate of nearly a minute. It is comparable to the approaches presented in~\cite{kiayias2017ouroboros, bentov2016snow} \ankit{and has the potential to shape the future of the blockchain technology.}
\par
\textit{Motivation:}
Given the promising properties of the Algorand protocol in terms of decentralization and scalability, the security aspects of \ankit{this consensus algorithm} are crucial. To the best of our knowledge, our work is the first analysis of Algorand's security. We aim to at least start a discussion about a possible security flaw on the message validation process, which could possibly expose honest nodes' bandwidth and memory resources to Distributed Denial-of-Service \ankit{(DDoS)} attacks.
\par
\textit{Contribution:}
The major contributions of our work are as follows:
\begin{enumerate}
\item \rankit{We demonstrate a practically feasible attack on the Algorand protocol. The attack has the potential to target an arbitrarily sized group of honest users and slow down the consensus process; leaving the targeted nodes behind in the chain as compared to the non-attacked nodes.}
\item We implemented the protocol in our simulator created in Java and evaluated the feasibility of our attack. The simulator is available on request.
\end{enumerate}
\textit{Organization:}
The rest of the paper is organized as follows: Section~\ref{section:algorand} gives a thorough description of the Algorand protocol. We elucidate our attack scenario in Section~\ref{section:attack}. Section~\ref{section:evaluation} presents the implementation details of our simulator, evaluation settings, and our results. Section~\ref{section:analysis} discusses the feasibility of our attack and the possible countermeasures. Finally, Section~\ref{section:conclusion} concludes the paper by listing potential future works.
\section{Algorand}
\label{section:algorand}
\michele{Since the main focus of our work is on Algorand, we first elucidate the Algorand protocol. We introduce the main limitations of current blockchain technologies in Section~\ref{section:limits}. We present the distinct features of Algorand in Section~\ref{section:properties}. Section~\ref{section:assumptions} discusses the assumptions specified by the Algorand protocol with respect to the adversary model. Section~\ref{section:network} gives an overview of the network topology, communication protocol, and the definition of messages. Then, consensus mechanism of Algorand is described in Section~\ref{section:consensus}. Finally, the technical details of ``cryptographic sortition'' - a critical component of the consensus process - are discussed in Section~\ref{section:sortition}.}
\subsection{The limitations of current Blockchain}
\label{section:limits}
\ankit{The main motivation for the development of Algorand lies in the underlying assumption and technical problems that
are essentially shared by most of the PoW-based blockchains~\cite{micali2016algorand_tec}. Those limitations include:
\begin{enumerate}
\item $\lbrack$Assumption$\rbrack$ Majority of nodes contributing to the computational power of the network will be honest.
\item $\lbrack$Technical Problem$\rbrack$ Wastage of computational power as well as electrical energy.
\item $\lbrack$Technical Problem$\rbrack$ Concentration of power, {i.e.,~} the total computing power for block generation lies within just few mining pools.
\item $\lbrack$Technical Problem$\rbrack$ Ambiguity in the consensus reached by different network nodes on the confirmed transactions, which leads to fork of blockchain.
\end{enumerate}
}
\subsection{Salient features of Algorand}
\label{section:properties}
The protocol has distinct and promising features:
\begin{enumerate}
\item It is designed to be fully decentralized and democratic. \michele{Moreover, there is no distinction between the role of different groups of users, {e.g.,~} miners {vs.~} ``normal'' users.}
\item Each user runs the same functions with negligible hardware requirements ({i.e.,~} with negligible computational effort) \ankit{as the concept of miner/mining does not exist in Algorand.}
\item \ankit{It is scalable with the number of nodes as well as in terms of confirmed transactions throughput. The authors~\cite{micali2017algorand_pap} have shown that the throughput of Algorand is fifty times that of Bitcoin.}
\item The blockchain does not fork with overwhelming probability ($P_{fork}=10^{-12}$).
\end{enumerate}
\subsection{Protocol assumptions}
\label{section:assumptions}
\ankit{To guarantee security of the blockchain, Algorand relies on the conventional assumption of PoS systems, {i.e.,~} Honest Majority of Money~(HMM). In particular, Algorand assumes a continuum in the ownership of currency belonging to honest users, }\michele{{i.e.,~} at any round $r$ at least a fraction $h$ (greater than two-third) of the money held by honest accounts on round $r-k$ \ankit{must still belong to the} honest users, for some integers $h$ and $k$ that are \ankit{defined by} the protocol.}
Together with HMM, Algorand requires a partial synchronization assumption on the honest nodes' clocks. It states that the honest nodes' clocks are not required to be strictly synchronized, but they are assumed to have the same speed with a bounded tolerance.
\par
\ankit{Furthermore}, Algorand was built to face a very strong adversary. The adversary is capable to perfectly synchronize all malicious nodes \ankit{that} he\footnote{In this paper, we have mentioned different entities as a masculine entity without any gender-bias.} controls and can corrupt honest users indiscriminately and instantly. \ankit{However, the attacker's capabilities are bounded by the following three main facts}: (1)~he can corrupt honest users until the HMM property is maintained; (2)~he cannot forge secret keys of the honest users that he have not corrupted; and (3)~\ankit{he cannot prevent the honest recipient to receive messages.}
\subsection{Network communication}
\label{section:network}
The Algorand network is a Bitcoin-like peer-to-peer network. Each user is identified by a public/private key pair, and each key pair corresponds to a node in the topology. \ankit{For simplicity, we will use the terms ``user'' and ``node'' interchangeably}. Each node establishes and maintains a different TCP connection with each of his peers. \ankit{The current specifications of Algorand do not specify if a node has to choose his peers based on their stake, which indeed exposes the network to a Sybil attack\footnote{\ankit{In a sybil attack, the attacker creates a large number of nodes at zero cost ({i.e.,~} with no stake) to increase the probability of connecting with his targets.}}.}
\par
Nodes broadcast messages through the network using a standard gossip protocol - each node gossips his message to his peers, who in turn relays it to their neighbors. A message has to be validated before it can be relayed, and it is never sent twice to the same user.
The protocol defines four types of messages: (1)~transactions; (2)~voting messages; (3)~block proposals; and \rmichele{(4)~credential messages}. A transaction (t) from user $i$ to user $j$ is defined as:
\begin{equation}
t=sig_i(pk_i, pk_j, a, I, H(SI)),
\end{equation}
where $pk_u$ is the public key of the user $u$, $a$ stands for the amount of Algorand units to transfer from $pk_i$ to $pk_j$, $I$ is an optional string to describe the transfer, $H(SI)$ is a 256-bit hash produced by hashing
a secret string $SI$ (not specified in the body of the message), and $sig_u(x)$ \ankit{denotes digitally signed data~$x$ using the private key of user $u$.}
\par
A voting message \ankit{(vm)} from a user $i$ in round $r$ and step $s$ is defined as:
\begin{equation}
\label{definition:voting_message}
vm_i^{r,s}=(sig(v_i^{r,s}), \sigma_i^{r,s}),
\end{equation}
where $\sigma_i^{r,s}$ represents the sortition proof of the \ankit{user $i$ in $s^{th}$~step of $r^{th}$~round} (further details in Section~\ref{section:sortition}), $v_i^{r,s}$ defines the vote of user $i$ in round $r$ and step $s$.
\par
\michele{A block proposal \ankit{(bp)} for round $r$ from user $i$ is defined as:
\begin{equation}
\label{definition:block_proposal}
bp^r_i=(B^r_i, sig_i(H(B^r_i)), \sigma^{r,1}_i),
\end{equation}
where $B^r_i$ is defined as
\begin{equation}
B^r_i=
\begin{cases}
(r, PAY^r, sig_i(Q^{r-1}), H(B^{r-1})), & \quad \text{if } B^r_i\neq E^r;\\
(r,Q^{r-1},H(B^{r-1})), & \quad \text{otherwise;}
\end{cases}
\end{equation}
where $PAY^r$ is the set of transactions $t$, while $Q^r$ is called \textit{seed} and defined as
\begin{equation}
\label{definition:seed}
Q^r =
\begin{cases}
H(sig_{l^r}(Q^{r-1},r)), & \quad \text{if } B^r\neq E^r;\\
H(Q^{r-1},r), & \quad \text{otherwise;}
\end{cases}
\end{equation}
where $l^r$ is the user who created the consensus block $B^r$ for round $r$. The seed is a parameter needed in the sortition process (more details in Section~\ref{section:sortition}). $E^r$ is the default block for round $r$, and it is defined as
\begin{equation}
E^r=(r,Q^{r-1},H(B^{r-1})).
\end{equation}
}
\par
\rmichele{Finally, the credential message (cm) \rankit{from a user $i$ for a round $r$} is defined as:
\begin{equation}
\label{definition:credentials}
cm^r_i=(\sigma^{r,1}_i).
\end{equation}
It consists of the sortition proof of the block proposer (\rankit{in this case, user $i$}). Each sortition proof is associated with a priority value in a deterministic way \michele{(further details in Section~\ref{section:sortition})}. Credential messages are far smaller in size than the block proposals. Thus, they propagate faster through the network. Credential messages are gossiped by the block proposers at the beginning of a round along with their block proposals. Since the block proposals and the credential messages from the same block proposer contain the same sortition proof, the peer nodes can leverage the priority values from the credential messages to not relay block proposals with low priorities - which helps in preventing congestion in the network.
}
\subsection{Consensus algorithm}
\label{section:consensus}
Algorand's consensus algorithm consists of a synchronous protocol that combines the concepts of PoS systems with a Byzantine fault tolerance agreement. The protocol virtually schedules the time into rounds. At each round, all the network nodes attempt to reach consensus on a new block of transactions. Each round is composed of the following actions:
\begin{itemize}
\item Each node in the network must first check its role to determine whether he has to propose a block for a given round. To do so, the nodes use cryptographic sortition, which requires them to run a trivial (a single hash) computation challenge. In case a node has to propose a block, he collects all the pending transactions inside a block proposal and gossips it together with the sortition proof. The block's size is limited by a fixed protocol parameter.
\item \michele{Each node waits for incoming block proposals from the previous step for a predefined duration of time. Among all the valid blocks he receives, he selects the one with the highest priority sortition proof. Then, he computes an hash using this block, sets this hash as the input for the BA of the next steps.}
\item All nodes in the network try to reach consensus on one block through a BA protocol. The BA has two key phases. Both the phases are composed by subsequent steps. At each step, a distinct group of users (called committee members) is elected in an unpredictable way through cryptographic sortition. These committee members spread their voting messages based on votes received from the previous step. The committee members attach a sortition proof that proves their legitimacy while spreading their voting messages. The two phases of BA are as follows:
\begin{enumerate}
\item The first phase is called the \textit{reduction} phase that comprises two steps. \ankit{In the first step, each committee member votes for the hash of the blocks proposed for consideration. In the second step, committee members vote for the hash that received votes over a certain threshold (defined by protocol). In case none of the hash/block receives enough votes, committee members vote for the hash of the default empty block.
}
\item The next phase consists of another binary BA to ensure that each node agrees on the same consensus; the consensus can conclude on the output of the previous phase \rankit{or} on the default value in the absence of a majority.
\end{enumerate}
\end{itemize}
\subsection{Cryptographic sortition}
\label{section:sortition}
\michele{Cryptographic sortition is used by each network node at the beginning of every step of each phase to determine whether he has a role in that particular step. \ankit{It is a simple and lightweight process that involves computing a single hash and a digital signature}. The sortition algorithm is implemented using a Verifiable Random Function (VRF\footnote{\ankit{VRF functions allow users to produce verifiable computations. The users produce a proof using their private keys, which can be verified by other users via the producer's public key.}
)~\cite{micali1999verifiable}}. In particular, the hash for a certain step $s$ and round $r$ is computed by the user~$i$ as
\begin{equation}
y=H(sig_i(r,s,Q^{r-1})),
\end{equation}
\michele{where $H$ is a hashing function, $y$ a 256-bit long hash, and $Q^{r-1}$ is the seed quantity defined by Eq.~\ref{definition:seed}. $sig_i(r,s,Q^{r-1})$ is called the sortition proof and is attached to voting messages or block proposals to prove the legitimacy of the corresponding voters or block proposers. The hash is used by the nodes to verify a certain sortition condition. The possibility that the condition is verified is directly proportional to the stake of the node and depends on a constant parameter \michele{$\pi_{role}$} fixed in the protocol, which guarantees that on average a fixed number of nodes are elected at each step for a certain role. For this reason, the protocol allows users to be elected more than once in the same step. A user $i$ can possibly be elected at each step $w_i$ times, where $w_i$ is the number of user's units of currency. Formally, the interval $[0,1)$ is partitioned into $w_i$ intervals \michele{$I$, where the $j$-th interval $I_j$ is defined as:}}
\begin{equation}
I_j=\Big[\displaystyle\sum_{x=0}^j B(x; w_i; p), \displaystyle\sum_{x=0}^{j + 1} B(x; w_i; p)\Big), j \in (0,1,...w_i)
\end{equation}
\michele{where $B(x; w_i; p)$ is the binomial probability that user~$i$ is elected exactly $x$ times on $w_i$ chances, each with probability $p=\pi_{role}/{W}$. Here, $\pi_{role}$ estimates the expected number of sorted users for that role and $W$ is the total amount of currency in the network. The number of votes are found by applying the binary point operator to the hash ({i.e.,~} $y/2^{y.length}$) and searching the interval $I_j$ with which the value is associated. If the value lies on the $j$-th interval, then $i$ can vote $j$ times.
There are at least three interesting observations about this procedure:
\begin{enumerate}
\item Since on average a fixed number of nodes are elected at each step, the traffic of voting messages in the network does not increase with the number of participants.
\item The procedure to decide whether a user~$i$ is sorted for a certain step in a round can be executed only by the user~$i$ as it involves a signature process.
\item Since $Q^{r-1}$ can be computed by a node only when that node reaches consensus on round $r-1$, the result of the sortition of user~$i$ in a certain step of round $r$ is unpredictable and unknown even to the user~$i$. This avoids the possibility of collusion between voters of one or more steps to manipulate the consensus process on a certain round.
\end{enumerate}}
\section{Our attack}
\label{section:attack}
\michele{We explain our attack in this section, starting from attack preliminaries in Section~\ref{section:attack_assumption} followed by a brief introduction to flooding attacks in Section~\ref{section:attack_scenario}, and the description of our attack in Section~\ref{section:attack_undecidable}.}
\subsection{Attack preliminaries}
\label{section:attack_assumption}
In our adversary model, we assume that all the malicious nodes are coordinated by a single/unique entity. \michele{This translates into the following two properties of the malicious nodes:
\begin{enumerate}
\item Each malicious node can sign messages using any other malicious node's private key.
\item The malicious nodes participate in the protocol either passively or actively. These malicious nodes coordinate and use the messages they receive from their honest peers to be aware of the protocol's execution status.
\end{enumerate}
We assume that all the malicious nodes and their public/private key pairs were created by the adversary sufficiently in advance, {i.e.,~} before the attack takes place. This enables all of them to become the voters or block proposers in the protocol}\footnote{This assumption is necessary because Algorand increases the unpredictability of the outcome of sortition process by limiting the set of keys that can participate in the process. The eligible keys must be created at least $k$ rounds before.}. As mentioned in Section~\ref{section:network}, we can also safely assume that the network is vulnerable to the Sybil attack, meaning that the honest nodes choose their peers randomly without relying on their stake. In this way, the malicious nodes have the same probability as of the other nodes to connect to honest peers. \michele{ On the other side, honest peers can control/limit the number of incoming connections, denoted by \textit{max-connections}.}
\subsection{A typical flooding attack}
\label{section:attack_scenario}
\ankit{In a typical network scenario, a flooding attack targets honest nodes' bandwidth and memory resources by sending a huge number of message to him.} Typically, the attack is performed at the application layer and messages are sent over previously established connections. Generally, a flooding attack sends a huge number of fake or invalid messages towards the target nodes to slow down their message reception and message processing. Crafting such attacks from a single nodes is not very effective because nodes in Bitcoin-like decentralized networks - where no level of trust is assumed between participants - often rely on ``misbehaviour'' scores to label and identify possible malicious users. As soon as an honest node decides that one of his peers is malicious, he can stop processing messages from that peer, drop the connection, and blacklist him to prevent future connection requests from that peer for a certain time.
\ankit{Hence, the attacker needs a large number of malicious nodes connected to the target. So he can send some messages from each connection in a coordinated manner and create the same impact without being detected (or at least fully detected).}
\subsection{Magnifying attack's impact using undecidable messages}
\label{section:attack_undecidable}
\ankit{Our attack aims to increase the impact of the flooding attacks by exploiting the message validation process of Algorand. Due its design, our attack also enables an adversary to have a longer connection time before getting blacklisted by the target node.}
\michele{In Algorand, each message that is a part of the consensus algorithm ({i.e.,~} voting messages (Eq.~\ref{definition:voting_message}) and block proposals (Eq.~\ref{definition:block_proposal})) is subject to two distinct checks during the validation process:
\begin{enumerate}
\item \textit{Stateless checks}: A set of checks on a message/packet that a validator node can perform to know its current status ({e.g.,~} packet's structure, packet's authentication, check for duplicated messages).
\item \textit{Sortition proof check}: The sortition proof must be checked to verify that the sender/author of the message had the right to gossip it.
\end{enumerate}
As described Section~\ref{section:sortition}, the sortition proof is a byte-string that represents the signature of a user $i$ in the form $sig_i(r,s,Q^{r-1})$}, the validator node should hash this string to verify whether the given output satisfies the sortition property. However, before blindly hashing the signature, the protocol requires him to verify it first. Thus, he needs all the parameters - that were originally used to create it - including the seed quantity $Q^{r-1}$. This is a stateful check because the validator should have already reached consensus on the round $r-1$ to independently compute $Q^{r-1}$. If that is not the case, then the message cannot be fully validated at the current status of the validator node. In such a situation, the only option that the validator node has is to store the message and wait until it can be fully validated because messages that are not yet fully validated, cannot be discarded. We refer to these messages as \textit{undecidable messages}
\par
\michele{At this point, the aim of the attacker is to flood honest targets with as many undecidable messages as possible to saturate their bandwidth and possibly their memory. These two objectives decide the type and number of messages used to construct our attack vector. With respect to the type, we chose the largest possible message, {i.e.,~} block proposals containing max-sized blocks. Since messages need to pass the stateless checks, each public key can gossip only a single block proposal per round. However, keys are created at zero cost and each malicious node can sign messages with any of the adversary keys. Moreover, each node can include an arbitrary number of block proposals in his payload - one for each key\footnote{As long as the messages are authenticated and sent only once, a single public key is able to create a different undecidable message for every step in each round. It means that it can create a block proposal on the first step and a voting message for every step of BA (until the maximum $m$-th step allowed by the protocol). However, sending one message for every step implies that the given public key is sorted at every step of that round, which is very unlikely. A similar argument can be used by the target node to statistically label malicious users without waiting for their messages to be fully verifiable. Moreover, given a reasonable max-block size (order of MB) and $m$ set to 150 steps~\cite{micali2017algorand_pap}, a single block proposal containing a max-size block would make up the large majority of the total payload weight. Hence, choosing a single block proposal for each key would imply that each user is elected only once in a given round, and cheaters are more difficult to identify a-priori.}.}
\par
\michele{Now, we describe the attack's execution in practice. The attacker connects to his target as one of his peers. Since the target node is assumed to honestly execute the Algorand protocol, as soon as he receives and validates a message for a certain round $r$, he has to relay it to all of his peers including the malicious node. Hence, the attacker just needs to (1)~prepare its payload in advance\footnote{The attacker does not need any information to pre-compute a payload for a certain round $r$. He just needs to sign some random messages declaring them as if they belong to round $r+1$. All additional information will be checked only once the message becomes fully verifiable in round $r+1$, by that time the attacker would have already achieved his goal.} for the attack in round $r$; (2)~wait to receive a block proposal \rmichele{or a credential message~(Eq.~\ref{definition:credentials})} from his target\footnote{Block proposals together with the credential messages are gossiped first through the network at each round.} in round $r$ to be aware of the fact that the target has started the execution of round~$r$; and (3)~send the payload to the target \rankit{from each connected malicious node}.}
\section{Evaluation}
\label{section:evaluation}
\michele{In this section we present the evaluation of our proposed attack. Section~\ref{section:implementation} provides the technical details of our simulator while the evaluation settings and the results are presented in Section~\ref{section:evaluation_settings} and Section~\ref{section:results}, respectively.}
\subsection{Simulator}
\label{section:implementation}
\ankit{Since, the source code or an official simulator for Algorand was not available at the time of our experiments, we built our own simulator that is completely written in Java. Both the works~\cite{micali2016algorand_tec, micali2017algorand_pap} on Algorand propose slightly different, but overlapping versions of the protocol. The differences are in the terminating conditions to reach the final majority consensus within BA. However, our threat model is independent of the particulars of the chosen version of BA. We referred to the protocol described in the technical paper~\cite{micali2017algorand_pap} for the implementation in our simulator.}
\par
We model an honest node as an entity composed of three main components: \michele{(1)~a network interface to \ankit{send/}receive messages from peers; (2)~a validator process to verify the received messages; and (3)~the process that runs the protocol}. Since our attack focuses on flooding block proposals or voting messages, transactions are never sent/received directly in our simulation. Instead, the throughput (transactions confirmed per second) is simulated by creating block proposals containing a fixed number of \ankit{arbitrary} transactions\michele{\footnote{We assume that in a real implementation of the protocol, messages and transactions are processed by independent processes.}}.
\par
To reduce the impact of simultaneous/parallel signature verification during simulation, we simulate block validation by using a sleep procedure. Furthermore, nodes are provided with a cache memory not to verify the same transaction twice\footnote{Without a dedicated cache, transactions are validated at least twice. The time when the transaction is gossiped by its original creator and the time when it is received in a block.}.
\rankit{While block signatures are simulated other checks, {e.g.,~} packet authentication and sortition proof validation, are actually computed legitimately.} \ankit{We do not assume any latency in the network communication for the sake of presenting the minimum impact of our proposed attack, which would further enhance in the presence of latency.}
\subsection{Evaluation settings}
\label{section:evaluation_settings}
All simulations were done on a virtual machine on OpenStack~\cite{openstack} running Ubuntu 16.04 with 16 Intel Core Processors (Hashwell, no TSX, IBRS) and 64 GB of RAM. In our experiments, we simulated a network of \rrmichele{500} nodes, where each node had a 30 Mbps upload and 30 Mbps download bandwidth. \ankit{The target nodes were connected to all the malicious nodes involved in the attack. \rankit{However, the number of malicious nodes does not represent the \textit{max-connections} parameter controlled by the honest nodes as honest nodes were connected to malicious as well as other honest peers.} For the sake of simplicity, we assumed that once an undecidable message fails in the verification process, then all messages from the sender of that message are discarded without being processed.} \rrankit{Each simulation was run for $\sim$45 minutes, and the attack payload was prepared using different blocks sizes during different experiments.}
\par
To evaluate the effectiveness of our attack, we primarily focus on the number of \ankit{``legitimate''} messages received and validated in due time by the honest nodes \ankit{as well as on the average time taken to complete a round}. \michele{The two evaluation metrics are correlated in a way that if too many messages are not received or \ankit{processed} in due time for a given step, the execution time of the corresponding step increases until a timeout is reached.} We thoroughly evaluated our attack scenario by varying the number of malicious nodes involved in the attack as well as the number of block proposals sent from each malicious node. In particular, the maximum number of tested block proposals (sent from one malicious node at the time of attack) was 70, following the analysis of the cryptographic sortition for block proposers by the authors of the protocol~\cite{micali2017algorand_pap}. \ankit{We set the block proposal receiving time and the step timeout (explained in the next section) to 150 and 60~seconds, respectively.} \rrmichele{Finally, we also evaluated the impact of our attack by varying the size of the attack payload.}
\subsection{Results}
\label{section:results}
\michele
}
\par
\ankit{The execution time for a round consists of two parts: block proposal receiving time and step timeout. The block proposal receiving time is constant and is defined as the maximum window of time for the nodes to receive \rankit{the block proposals for a given round}. The step timeout is the maximum amount of time by which each step must complete. Here, each step finishes as soon as enough voting messages are processed or the step timeout occurs.} In ``no attack'' scenario, the largest contribution to round time should come from the block proposal receiving time and the steps should execute before the step timeout occurs. The same is reflected in our results shown in \figurename~\ref{chart:time}.
Here, in the absence of malicious nodes,
nearly \rrankit{83}\% of the round time was taken by the fixed block \rankit{proposal} receiving window. \ankit{On the other side, the round time remains nearly the same as ``no attack'' scenario with a lower number of malicious nodes/keys per malicious node. However, the protocol's performance starts to deteriorate with increasing number of malicious nodes/keys per malicious node, starting from the attack payload size of just 500 blocks (10 malicious nodes with 50 malicious keys). \rankit{In facts, the average round time with all the remaining higher attack configurations was over \rrmichele{390}~seconds; which means that these configurations caused at least \rrmichele{four} step timeouts in addition to the block proposal receiving time.} As mentioned earlier, \rankit{our proposed attack does not prevent nodes from reaching consensus. Instead, it aims to significantly increase the execution time of each round, which eventually leads the attacked nodes to reach their consensus on the default value, leaving the targeted nodes behind the non-attacked nodes in the chain.}}
\begin{figure}[H]
\centering
\includegraphics[trim = 2mm 2mm 2mm 2mm, clip, width=.95\columnwidth]{./graphs/out/attack_time_comparison}
\caption{\ankit{Effect of the number of malicious nodes and keys per malicious nodes upon average round time
}}
\label{chart:time}
\end{figure}
\par
\ankit{The next important criteria to evaluate the impact of an attack on Algorand is the percentage of ``legitimate'' messages received and ``legitimate'' messages validated in due time. \figurename~\ref{chart:percentage} shows the effect of our attack upon ``legitimate'' messages received and validated with respect to the number of malicious nodes and keys per malicious nodes. Also here, with an attack payload of just 500 blocks (10 malicious nodes with 50 malicious keys), the percentage of ``legitimate'' messages received/validated starts to degrade considerably, which eventually lead to step failures.}
\begin{figure}[H]
\centering
\includegraphics[trim = 2mm 2mm 2mm 2mm, clip, width=.95\columnwidth]{./graphs/out/attack_perc_comparison}
\caption{\ankit{Effect of the number of malicious nodes and keys per malicious nodes upon percentage of ``legitimate'' messages received and validated}}
\label{chart:percentage}
\end{figure}
\par
It is important to mention that an adversary can tune our attack on the number of malicious nodes/keys to prevent the detection while still creating significant disturbance in the network. Moreover, the size of the blocks is another important parameter to tune the attack. \rrmichele{
The results presented in the Figures~\ref{chart:time}~and~\ref{chart:percentage} were obtained using 1~MB of attack payload. Tables~\ref{table:block_size_timing}~and~\ref{table:block_size_messages} show the impact of attack payload's size upon the average round time and the percentage of ``legitimate'' messages received as well as validated in due time, respectively.
Our results show that the size of the block does not significantly affect the performance of honest nodes in the absence of the attack. However, increasing size of the attack payload severely affect the performance of honest nodes in term of both the average round time and messages processed. It happens because the block's size directly affects the download time of a block, which is further worsened by the number of malicious nodes and the keys per malicious nodes involved in the attack.
}
\begin{table}[!htbp]
\centering
\caption{\rrankit{Effect of the attack payload's size upon average round time}}
\label{table:block_size_timing}
\resizebox{.95\columnwidth}{!}
{
\begin{tabular}{cc|c|c|c|c|}
\hline
\multicolumn{1}{|c|}{\textbf{\begin{tabular}[c]{@{}c@{}}Block size\\ (in MB)\end{tabular}}} & \textbf{\begin{tabular}[c]{@{}c@{}}Keys/malicious\\ node\end{tabular}} & \textbf{\begin{tabular}[c]{@{}c@{}}5 malicious\\ nodes\end{tabular}} & \textbf{\begin{tabular}[c]{@{}c@{}}10 malicious\\ nodes\end{tabular}} & \textbf{\begin{tabular}[c]{@{}c@{}}15 malicious\\ nodes\end{tabular}} & \textbf{\begin{tabular}[c]{@{}c@{}}No\\ attack\end{tabular}} \\ \hline
\multicolumn{1}{|c|}{\multirow{3}{*}{0.5}} & 25 & 178.65 & 180.55 & 182.36 & \multirow{3}{*}{181.42} \\ \cline{2-5}
\multicolumn{1}{|c|}{} & 50 & 180.73 & 181.36 & 181.72 & \\ \cline{2-5}
\multicolumn{1}{|c|}{} & 70 & 178.98 & 180.45 & 237.27 & \\ \hline
\multicolumn{1}{|c|}{\multirow{3}{*}{1.0}} & 25 & 182.21 & 182.87 & 183.24 & \multirow{3}{*}{181.16} \\ \cline{2-5}
\multicolumn{1}{|c|}{} & 50 & 182.49 & 232.15 & 391.16 & \\ \cline{2-5}
\multicolumn{1}{|c|}{} & 70 & 181.69 & 391.12 & 391.45 & \\ \hline
\multicolumn{1}{|c|}{\multirow{3}{*}{1.5}} & 25 & 180.16 & 180.97 & 241.52 & \multirow{3}{*}{181.54} \\ \cline{2-5}
\multicolumn{1}{|c|}{} & 50 & 181.22 & 391.58 & 391.66 & \\ \cline{2-5}
\multicolumn{1}{|c|}{} & 70 & 235.76 & 391.28 & 391.54 & \\ \hline
\multicolumn{1}{|c|}{\multirow{3}{*}{2.0}} & 25 & 178.97 & 226.63 & 391.31 & \multirow{3}{*}{182.31} \\ \cline{2-5}
\multicolumn{1}{|c|}{} & 50 & 226.59 & 390.88 & 391.57 & \\ \cline{2-5}
\multicolumn{1}{|c|}{} & 70 & 391.39 & 391.83 & 391.93 & \\ \hline
\multicolumn{1}{l}{} & \multicolumn{1}{l|}{\textbf{}} & \textbf{Time (s)} & \textbf{Time (s)} & \textbf{Time (s)} & \textbf{Time (s)} \\ \cline{3-6}
\end{tabular}
}
\end{table}
\begin{table*}[!htbp]
\centering
\caption{\rrankit{Effect of the attack payload's size upon the percentage of ``legitimate'' messages received and validated}}
\label{table:block_size_messages}
\resizebox{1.95\columnwidth}{!}
{
\begin{tabular}{cc|c|c|c|c|c|c|c|c|}
\hline
\multicolumn{1}{|c|}{\textbf{Block size (in MB)}} & \textbf{Keys/malicious node} & \multicolumn{2}{c|}{\textbf{5 malicious nodes}} & \multicolumn{2}{c|}{\textbf{10 malicious nodes}} & \multicolumn{2}{c|}{\textbf{15 malicious nodes}} & \multicolumn{2}{c|}{\textbf{No attack}} \\ \hline
\multicolumn{1}{|c|}{\multirow{3}{*}{0.5}} & 25 & 96.98 & 91.61 & 97.26 & 92.49 & 97.08 & 90.16 & \multirow{3}{*}{97.51} & \multirow{3}{*}{90.53} \\ \cline{2-8}
\multicolumn{1}{|c|}{} & 50 & 97.37 & 92.87 & 96.07 & 90.90 & 94.96 & 90.43 & & \\ \cline{2-8}
\multicolumn{1}{|c|}{} & 70 & 97.85 & 91.25 & 96.04 & 90.21 & 75.03 & 64.99 & & \\ \hline
\multicolumn{1}{|c|}{\multirow{3}{*}{1.0}} & 25 & 97.70 & 91.02 & 97.98 & 92.41 & 92.46 & 87.77 & \multirow{3}{*}{96.32} & \multirow{3}{*}{89.7} \\ \cline{2-8}
\multicolumn{1}{|c|}{} & 50 & 96.81 & 89.75 & 80.52 & 72.72 & 45.47 & 44.06 & & \\ \cline{2-8}
\multicolumn{1}{|c|}{} & 70 & 94.24 & 88.41 & 42.31 & 41.46 & 10.66 & 10.55 & & \\ \hline
\multicolumn{1}{|c|}{\multirow{3}{*}{1.5}} & 25 & 99.32 & 94.85 & 93.62 & 89.97 & 65.62 & 63.32 & \multirow{3}{*}{97.43} & \multirow{3}{*}{90.77} \\ \cline{2-8}
\multicolumn{1}{|c|}{} & 50 & 93.25 & 86.65 & 44.63 & 43.10 & 8.75 & 8.02 & & \\ \cline{2-8}
\multicolumn{1}{|c|}{} & 70 & 73.57 & 67.28 & 25.93 & 24.32 & 6.79 & 5.82 & & \\ \hline
\multicolumn{1}{|c|}{\multirow{3}{*}{2.0}} & 25 & 97.83 & 92.01 & 80.03 & 73.21 & 42.96 & 41.59 & \multirow{3}{*}{96.80} & \multirow{3}{*}{90.09} \\ \cline{2-8}
\multicolumn{1}{|c|}{} & 50 & 79.99 & 75.12 & 26.66 & 25.21 & 1.05 & 1.00 & & \\ \cline{2-8}
\multicolumn{1}{|c|}{} & 70 & 55.37 & 49.71 & 4.96 & 4.83 & 0.95 & 0.91 & & \\ \hline
& & \textbf{\begin{tabular}[c]{@{}c@{}}Messages\\ received (\%)\end{tabular}} & \textbf{\begin{tabular}[c]{@{}c@{}}Messages\\ validated (\%)\end{tabular}} & \textbf{\begin{tabular}[c]{@{}c@{}}Messages\\ received (\%)\end{tabular}} & \textbf{\begin{tabular}[c]{@{}c@{}}Messages\\ validated (\%)\end{tabular}} & \textbf{\begin{tabular}[c]{@{}c@{}}Messages\\ received (\%)\end{tabular}} & \textbf{\begin{tabular}[c]{@{}c@{}}Messages\\ validated (\%)\end{tabular}} & \textbf{\begin{tabular}[c]{@{}c@{}}Messages\\ received (\%)\end{tabular}} & \textbf{\begin{tabular}[c]{@{}c@{}}Messages\\ validated (\%)\end{tabular}} \\ \cline{3-10}
\end{tabular}
}
\end{table*}
\section{Feasibility of the attack}
\label{section:analysis}
\ankit{The attack costs practically nothing to an adversary. However, the adversary has to overcome one minor challenge, {i.e.,~} to connect at least one malicious node with the honest target. This malicious node can then flood the target with an arbitrarily large number of block proposals - one for each public key controlled by the adversary. However, the recipient can monitor and label connections from which he receives a suspiciously large number of messages for a round. Then, it is likely that the honest nodes would simply drop (or, at least temporarily stop listening) from such a connection. Hence, the success of our attack depends on how many malicious connections an attacker is able to establish with the target node.
}
\par
The other important aspects of our attack are: (1)~since TCP connections are required to be established, IP spoofing is not an option\footnote{Establishing TCP connections via IP spoofing is problematic as one cannot complete the TCP three-way handshake protocol; because the responses ({i.e.,~} the SYN-ACK packets) from the target nodes are not addressed to the real location of a malicious node. If we assume such possibility, then the adversary has control over at least one router on the path between the fake address and the target, or the target node's network has allowed source-routing~\cite{sunshine1977source} of the messages, or the Initial Sequence Number (ISN) in the handshake protocol is vulnerable to the prediction attack. All of these assumptions are quite strong hypothesis.} and real addresses should be used;
(2)~the target's \textit{max-connections} parameter defines the maximum number of connections allowed towards it. We believe that the former issue can be solved by using botnet or relying on hidden services, such as Tor, to protect the address identity inside the network. However, the \textit{max-connections} parameter is out of the adversary's control. Hence, \michele{it is considered as an attack variable in our attack scenario. Consequently, establishing and maintaining a high number of connections with the target nodes is somehow a bit more challenging. But, the goal is achievable due to the current design of the protocol, where peers are not weighted on their stake.}
\section{Conclusion and future works}
\label{section:conclusion}
\ankit{The Algorand protocol was proposed to overcome the limitations of conventional blockchain technologies. The protocol has great potential and is in under active development.} In this paper, we present a particular security flaw of Algorand protocol that exploits its process of validating messages. In particular, we evaluated a possible DDoS-like flooding scenario \ankit{under practical assumptions}, where honest nodes suffer significant delays in the execution of protocol. Furthermore, we also discuss the major factors that make this attack scenario more challenging for the honest nodes. Finally, we also present possible solutions to prevent such an attack. The rigorous formalization of these countermeasures is kept as the future work.
\bibliographystyle{IEEEtran}
| -25,673.618519 |
[
-0.317626953125,
0.8740234375
] | 22.483221 |
[
-3.123046875,
0.70458984375,
-1.720703125,
-4.95703125,
0.0567626953125,
6.8359375
] |
[
2.767578125,
6.47265625,
2.814453125,
5.25
] | 409 | 5,857 |
[
-1.6328125,
1.6328125
] | 25.351729 |
[
-5.40625,
-2.791015625,
-3.5234375,
-1.8828125,
1.623046875,
10.171875
] | 0.887915 | 11.557649 | 26.566502 | 5.103584 |
[
2.133434295654297
] | -19,158.029385 | 5.768653 | -24,991.111539 | 0.488354 | 6.034496 |
[
-3.017578125,
-3.01953125,
-3.01171875,
-4.15234375,
2.666015625,
10.1484375
] |
[
-5.51171875,
-1.9326171875,
-1.58984375,
-1.0771484375,
3.69921875,
4.3046875
] | |
BkiUdVw4eIZijW5EQLXO
|
\section{Introduction}
\label{sec:introduction}
A modification is proposed herein to passively alter the re-radiation angle of a Van Atta retro-directive array \cite{Atta1959}. Here, the re-radiated beam is steered to a desired angle by imparting an additional phase gradient into the received signal. While the idea of changing a Van Atta array's output angle is not new, previous proposals have focused on active techniques like phase-conjugating mixing \cite{Miyamoto2002} or require non-reciprocal elements \cite{Davies1963}. In contrast, the proposed technique can be performed passively and without the use of non-reciprocal elements, external power, or control signals.
Passive beam re-direction has also become a popular focus in recent metasurface research with applications in communications and radar cross-section (RCS) reduction \cite{Modi2017}, \cite{Li2014}. These can be fabricated to achieve highly efficient wavefront retro-reflection \cite{Wong17}, \cite{Estakhri2017} and re-direction \cite{Diaz-Rubioe2017}, \cite{Epstein2016}. However, current re-direction metasurface designs only operate effectively at a single set of incidence and re-radiation angles. In contrast, the proposed method consistently applies the same angular offset over a wide range of incident angles at a fixed frequency.
As a demonstration, a 10 GHz patch antenna based reflector was designed to re-radiate a beam offset $-15^{\circ}$ with respect to incidence. The reflector's performance was then assessed through simulation and experimental measurements.
\section{Re-Directive Array Theory}
\label{sec:theory}
\begin{figure}[!tb]
\centering
\includegraphics[width=3.25in]{ang1.PNG}
\caption{Polarized-duplexed re-directing reflector}
\label{fig:VRPol}
\end{figure}
A uniformly spaced N-element phased array is capable of controlling its direction of radiation by maintaining a phase gradient $\alpha=-k_{0}d\sin{\theta}$ between each element \cite{Balanis2005}. Here, $k_{0}=2\pi/\lambda_{0}$ is the free space wavenumber, $d$ is the inter-element spacing, and $\theta$ is the desired radiation angle referenced to the array's normal axis. The input phase for each element ($\phi_{Tx}$) is thus an increasing multiple of this gradient.
A Van Atta retro-reflector can be considered an array that attains its power and phase gradient from a received incident wave and interconnecting lines. Due to differences in propagation distances, a wave with an off-boresight incident angle of $\psi$ is captured with a phase gradient of $\tau=-k_{0}d\sin{\psi}$. Usually, both elements in each pair receive and transmit simultaneously; capturing and feeding electromagnetic energy to its partner via interconnecting lines to be re-radiated. If the lines are of equal lengths, or differ by multiples of the guided wavelength, the phase gradient is inverted when the signals reach their partner antennas and the array radiates towards the source. As the phase gradient is induced externally, only one connecting line is required per pair \cite{Larsen1964}.
Consequently, the direction of re-radiation can be altered by changing this naturally induced phase gradient. Fig. \ref{fig:VRPol} details such a concept where an additional phase gradient $\delta$ can be passively imparted with phase shifters or by adding or removing segments of interconnecting lines. The total transmitted phase gradient then becomes:
\begin{equation}
\alpha=\tau+\delta=-k_{0}d\sin{\psi}+\delta
\end{equation}
resulting in a re-radiated beam angle of:
\begin{equation}
\theta=\frac{-\arcsin(\alpha)}{k_{0}d}
\end{equation}
Thus, the additional phase gradient for a re-radiation angle of $\theta$ when the array is illuminated at incident angle $\psi$ is:
\begin{equation}
\delta=k_{0}d\left(\sin{\psi}-\sin{\theta}\right)
\label{eqn:SteeringAngle}
\end{equation}
To achieve re-direction, the added phase must be an increasing multiple of $\delta$ at each element. This requires \textit{isolated connecting networks so a separate phase offset can be imparted to each counter-propagating wave}. While the use of non-reciprocal circuits like circulators has been suggested \cite{Davies1963}, a simpler solution is proposed in Fig. \ref{fig:VRPol}. In this design, pairs of dual polarized antennas are connected with two lines and polarization duplexing is exploited to achieve cross-propagating isolation. A multiplicative phase shift can then be added to each line to augment the phase gradient. The advantage of this design is that no non-reciprocal or active components are required. However, each element pair now requires two connecting lines, increasing routing complexity.
\section{Design and Simulation}
\label{sec:simulation}
\begin{figure}[!tb]
\centering
\includegraphics[width=3.25in]{ang2.PNG}
\caption{$\gamma=-15^{\circ}$ re-directing Van Atta array}
\label{fig:VRPol4El}
\end{figure}
To test the proposed principle, a polarized-duplexed, single 4-element row reflector with a $\gamma=\theta-\psi=-15^{\circ}$ offset angle was modeled at 10 GHz with ANSYS HFSS (Fig. \ref{fig:VRPol4El}). The square patch antennas measure 8.28 mm with a center-to-center spacing of 16.8 mm and matched to 1.21 mm wide microstrip lines with 0.15 mm wide quarter-wave transformers. These were modeled as 35 $\mu$m thick copper traces on a 35 x 82 mm, 0.5 mm thick Rogers RO3003 ($\epsilon_{r}=3$) substrate. Inter-connecting line lengths (including transformers) were initially sized to achieve retro-reflection: $a=48.08$ mm, $b=86.54$ mm. A planar design limited the range of line extension and $\gamma=-15^{\circ}$ was the widest offset angle achievable without overlap. The required phase gradient was calculated to be $\delta=52.5^{\circ}$, using (\ref{eqn:SteeringAngle}) with $\psi=5^{\circ}$ and $\theta=-10^{\circ}$. This was added by extending the connecting lines by multiples of $\delta/\beta_{g}=2.81$ mm, where $\beta_{g}$ is the guided propagation constant.
\section{Fabrication \& Experiment Setup}
\label{sec:prototype}
Real-world functionality was verified by fabricating and measuring an 8-row re-directing reflector. Each row was separated by a patch center-to-center spacing of 30 mm and printed on a 242.5 x 77 mm substrate. The fabricated reflector along with an angular reference is shown in Fig. \ref{fig:MeasurementRfl} where trace damage was noted. Measurements were performed using the bistatic apparatus in Fig. \ref{fig:MeasurementLayout}. A vertically polarized source horn was fixed to the right arm and kept stationary, while a probe horn was attached to a movable left arm. The orientation of the probe antenna could also be rotated to switch between measuring vertical (co-polarized) and horizontal (cross-polarized) polarizations. The following measurements were conducted over a frequency range of 8-12 GHz (Fig. \ref{fig:MeasurementTypes}):
\begin{enumerate}
\item Fixed $\gamma=-15^{\circ}$ between source and probe, cross-polarized output measured across incidence angles
\item Probe: bistatic sweep, Source: $\psi=30^{\circ}$ incidence
\item Probe: bistatic sweep, Source: $\psi=15^{\circ}$ incidence
\item Probe: bistatic sweep, Source: $\psi=-5^{\circ}$ incidence
\item Probe: bistatic sweep, Source: $\psi=-15^{\circ}$ incidence
\end{enumerate}
Measurement 1 fixed the probe at a constant $-15^{\circ}$ offset from the source. The reflector was rotated by $5^{\circ}$ to position the source at a specific incidence angle where the probe would measure the re-radiated cross-polarized power at the designed $-15^{\circ}$ offset. This allowed reflector performance to be evaluated over a range of frequencies and incidence angles.
Measurements 2-5 fixed the source at a specific incident angle and then azimuthally swept the probe antenna at $5^{\circ}$ increments to obtain a scattered radiation pattern. Sweep resolution was reduced to $1^{\circ}$ within $\pm5^{\circ}$ of the expected specular and re-radiation angles. Three cuts were taken per measurement: Co and cross-polarized sweeps with the reflector to obtain specular and re-radiated patterns respectively and a co-polarized cut with an equally sized copper plate for a baseline. Sweep ranges and number of feasible incidence angles were limited by apparatus physical constraints.
\begin{figure*}[!htb]
\centering
\begin{minipage}{.2\textwidth}
\vspace*{\fill}
\centering
\includegraphics[width=\textwidth]{ang3.png}
\subcaption{Fabricated re-director}
\label{fig:MeasurementRfl}
\end{minipage}\hspace{0.01\textwidth}
\begin{minipage}{.66\textwidth}
\vspace*{\fill}
\centering
\includegraphics[width=\textwidth]{ang4.PNG}
\subcaption{Experimental apparatus}
\label{fig:MeasurementLayout}\par\vfill
\includegraphics[width=\textwidth]{ang5.PNG}
\subcaption{Measurement types}
\label{fig:MeasurementTypes}
\end{minipage}
\caption{Fabrication \& experimental measurement}
\label{fig:Measurement}
\end{figure*}
\section{Results \& Discussion}
\label{sec:results}
Fig. \ref{fig:FixedBistatic} plots the re-radiated power from Measurement 1 over a range of incident angles. The array's power output fluctuates slightly within $\pm(50-60)^{\circ}$ and drops off outside this envelope. These edge drop-offs are expected as conventional patch antennas cannot operate at extreme angles while the $\pm10^{\circ}$ fluctuation is attributed to resonance effects. Optimal performance is achieved at $30^{\circ}$ incidence as both the incident and re-radiated lobes are within the $\pm(50-60)^{\circ}$ envelope. In contrast, performance at negative incidence angles is slightly reduced since the re-radiated lobe is pushed to the $-50^{\circ}$ range. Furthermore, the fabricated reflector's optimal frequency shifted to 9.88 GHz; likely attributed to etching tolerances and naturally occurring variations in the substrate's material properties. Power outputs for both 9.88 and 10 GHz are plotted to highlight this difference in performance.
\begin{figure}[!b]
\centering
\includegraphics[width=3in]{ang6.png}
\caption{M1: Measured cross-polarized re-radiated power}
\label{fig:FixedBistatic}
\end{figure}
Fig. \ref{fig:BistaticRCS} plots the radiation patterns from Measurements 2-5 as bistatic RCS. Both simulated and measured RCS values are normalized to account for differences in reflector size, number of rows, and fabrication trace damage. These normalization patterns are obtained by simulating and measuring copper plates of the same dimensions as their re-directive counterparts. To reduce clutter, only the measured copper RCS is shown. For easier comparison, the radiation patterns are plotted at optimal frequency; 10 GHz for simulated and 9.88 GHz for the measured. Both the simulated and measured patterns show a re-radiated, cross-polarized main lobe approximately $-15^{\circ}$ from incidence, demonstrating functionality over multiple incidence angles. Re-direction accuracy falls within $\pm5^{\circ}$ and can be attributed to the non-linear nature of (\ref{eqn:SteeringAngle}). In particular, the required gradient is calculated from a specific incident and re-radiation angle pair rather than a specified offset. Consequently, minor drifting may occur at other incident angles. Due to antenna and transmission line properties, some power is specularly reflected by the array. The polarization-duplexing makes it easy to separate these specular components as the re-radiated main lobe is cross-polarized. Absorption of incident power is evident as the (co-polarized) specular reflection lobes of the arrays are lower than those of the copper plate. Furthermore, measured array co-polarized power is inversely proportional to cross-polarized power, suggesting that any power not received is specularly reflected. As shown in Fig. \ref{fig:FixedBistatic}, optimal performance occurs at $30^{\circ}$ incidence where measured re-radiated power is approximately equal to the specularly reflected power. The worst performance occurs at $-5^{\circ}$, where the measured peak specular reflection is 7.4 dBsm higher than the re-radiated peak. At all other incidence angles, the specular/re-radiation difference is approximately 3-5 dBsm.
Array performance can be improved by increasing re-radiated power or reducing specular reflection. The former is dependent on element quantity and matching while minimizing exposed non-radiating surfaces reduces the latter. Trimming the upper left and lower right corners of the model in Fig. \ref{fig:VRPol4El} reduced the simulated specular/re-radiation difference by up to 3 dB over all incidence angles.
\begin{figure*}[!htb]
\centering
\begin{subfigure}{0.4\linewidth}
\includegraphics[width=\linewidth, height=0.9\linewidth]{ang7.png}
\caption{M2: $\psi=30^{\circ}$ incidence}
\label{fig:Bistatic30}
\end{subfigure}
\begin{subfigure}{0.4\linewidth}
\includegraphics[width=\linewidth, height=0.9\linewidth]{ang8.png}
\caption{M3: $\psi=15^{\circ}$ incidence}
\label{fig:Bistatic15}
\end{subfigure}
\begin{subfigure}{0.4\linewidth}
\includegraphics[width=\linewidth, height=0.9\linewidth]{ang9.png}
\caption{M4: $\psi=-5^{\circ}$ incidence}
\label{fig:Bistatic_5}
\end{subfigure}
\begin{subfigure}{0.4\linewidth}
\includegraphics[width=\linewidth, height=0.9\linewidth]{ang10.png}
\caption{M5: $\psi=-15^{\circ}$ incidence}
\label{fig:Bistatic_15}
\end{subfigure}
\caption{Normalized scattering patterns of $\gamma=-15^{\circ}$ re-directive reflector}
\label{fig:BistaticRCS}
\end{figure*}
\section{Conclusions}
\label{sec:conclusion}
In this Letter, array theory was used to passively modify the re-radiation angle of a Van Atta retro-reflector. Unlike other passive re-direction techniques, this method effectively applies the same angular offset over a wide range of incident angles at a fixed frequency. Further verification was provided through simulation and experimental measurements on a $-15^{\circ}$ offset re-directive reflector. Polarization-duplexing was used to passively and reciprocally achieve the required isolation in the interconnecting networks. While the principles behind the modified Van Atta array are universal, reflector performance was limited by the properties of the planar antennas and microstrip lines used. In particular, the amount of power received and re-radiated decreased significantly at steeper incident angles. Moreover, peak specular levels were 0.8-7.4 dBsm higher than their re-radiated counterparts over the tested incidence angles.
Future work is expected to focus on improving reflector performance and angular range. These can be accomplished by strategies such as using wider accepting-angle antennas, increasing elements, and decreasing exposed non-radiating surfaces. In addition, variable phase shifters can be added to the interconnecting lines, in place of line length adjustments, to enable variable beam steering and tuning.
\IEEEtriggeratref{3}
\printbibliography[title={References}]
\end{document}
| -11,856.516001 |
[
-3.220703125,
2.845703125
] | 28.873239 |
[
-4.59765625,
-2.16796875,
-2.38671875,
-6.3125,
0.458251953125,
10.2890625
] |
[
2.93359375,
6.7890625,
1.927734375,
5.12109375
] | 138 | 1,820 |
[
-3.0859375,
3.666015625
] | 22.798931 |
[
-6.09765625,
-4.953125,
-3.287109375,
-1.2919921875,
2.833984375,
10.3671875
] | 1.320914 | 20.881875 | 36.373626 | 2.337341 |
[
2.1389317512512207
] | -9,346.361969 | 6.489011 | -11,615.316559 | 0.270957 | 5.745012 |
[
-3.017578125,
-3.564453125,
-3.9375,
-4.22265625,
2.6875,
10.5625
] |
[
-5.984375,
-4.08203125,
-3.31640625,
-2.87109375,
4.4609375,
8.515625
] | |
BkiUdJc5qsBC_MvyfL35
|
\section{Introduction \label{sec:intro}}
\par
The large samples of $\ensuremath{\mathrm{Z^0}}$
events collected at $\ensuremath{\mathrm{e^+}}\ensuremath{\mathrm{e^-}}$ colliders over the past ten years have
made it possible to study resonance dynamics and test low energy QCD using
the decays of $\tau$ leptons to kaons.
In this paper, measurements of the branching ratios of the
{\small $\taupkz$}, {\small $\taupkzp$} and {\small $\taukkz$} decay
modes are presented.\footnote{Charge conjugation is implied throughout
this paper.} These measurements are based on two samples that identify
$\tau$ decays with \ensuremath{\mathrm{K^0_L}}\ and \ensuremath{\mathrm{K^0_S}}\ mesons. The $\ensuremath{\mathrm{K^0_L}}$ mesons are
identified by their one-prong
nature accompanied by a large deposition of energy in the hadron calorimeter
while the $\ensuremath{\mathrm{K^0_S}}$ mesons are identified through their decay into two charged
pions. The selected number of {\small $\taupkzkz$} decays is very small
and is treated as background in this analysis.
The results presented here are extracted from the
data collected between 1991 and 1995 at energies close to
the \ensuremath{\mathrm{Z^0}}\ resonance, corresponding to an integrated luminosity of
163 pb$^{-1}$, with the OPAL detector at LEP.
A description of the OPAL detector can be found in~\cite{opaldetector}.
The performance and particle identification capabilities of the
OPAL jet chamber are described in~\cite{opaltracker}.
The $\tau$ pair Monte Carlo samples used in this analysis are
generated using the KORALZ 4.0 package \cite{koralz}.
The dynamics of the $\tau$ decays are simulated with the
Tauola 2.4 decay library \cite{tauola}.
The Monte Carlo events are then passed through the
OPAL detector simulation \cite{gopal}.
\section{Selection of {\boldmath $\tp\tm$} events \label{sec:tausel}}
The procedure used to select $\ensuremath{\mathrm{Z^0}} \rightarrow \tp\tm$ events is identical
to that described in previous OPAL publications~\cite{opal:e,opal:tp}.
The decay of the $\mathrm{Z}^0$ produces two back-to-back taus.
The taus are highly relativistic so that the decay products
are strongly collimated. As a result it is convenient to
treat each $\tau$ decay as a jet, where
charged tracks and clusters are assigned to cones of half-angle
$35^\circ$~\cite{opal:e,opal:tp}. In order to avoid regions of
non-uniform calorimeter response, the two $\tau$ jets are
restricted to the barrel region of the OPAL detector by requiring that the
average polar angle\footnote{In the
OPAL coordinate system the e$^-$ beam direction
defines the $+z$ axis, and the centre of the LEP ring defines
the $+x$ axis.
The polar angle $\theta$ is measured from the $+z$ axis,
and the azimuthal angle $\phi$ is measured from the $+x$ axis.}
of the two jets satisfies
$\overline{|cos\theta|} < 0.68$.
The level of contamination from
multihadronic events ($\ee \!\! \rightarrow \! \qq$) is significantly reduced by requiring not more
than six tracks and ten electromagnetic clusters per event.
Bhabha $(\ee \!\! \rightarrow \! \ee)$ and muon pair $(\ee \!\! \rightarrow \! \mm)$ events are removed by
rejecting events where the total electromagnetic energy and the
scalar sum of the track momenta are close to the centre-of-mass energy.
Two-photon events, $\ee \!\! \rightarrow \! (\ee)\ee$ or $\ee \!\! \rightarrow \! (\ee)\mm$, are removed by
rejecting events which have little visible energy in the electromagnetic
calorimeter and a large acollinearity angle\footnote{
The acollinearity angle is the supplement of the angle between the
decay products of each jet.}
between the two jets.
A total of $100925$ events are selected for the $\kk{}^0_{\mathrm L}$ sample
and $ 85789$ events are selected for the $\kk{}^0_{\mathrm S}$ sample from the
1991-1995 data set.
The different number of $\tp\tm$ events in the two samples is due to
different detector status requirements used in each selection.
The fraction of background from non-$\tau$ sources is
$(1.6 \pm 0.1)$\%~\cite{opal:e,opal:tp}.
\section{Selection of $\taukl$ decays \label{sec:klong}}
The selection of $\tau$ decays into to a final state containing at
least one $\ensuremath{\mathrm{K^0_L}}$ meson follows a simple cut-based procedure.
Some \ensuremath{\mathrm{K^0_S}}\ mesons will be
selected in this \ensuremath{\mathrm{K^0_L}}\ selection.
In particular, \ensuremath{\mathrm{K^0_S}}\ mesons that decay
late in the jet chamber, the solenoid or the electromagnetic calorimeter,
will be indistinguishable from \ensuremath{\mathrm{K^0_L}}\ mesons and are
considered to be part of the \ensuremath{\mathrm{K^0_L}}\ signal.
The Monte Carlo simulation predicts that the $\ensuremath{\mathrm{K^0}}$ component
identified by the $\ensuremath{\mathrm{K^0_L}}$ selection is composed of 86\%
$\ensuremath{\mathrm{K^0_L}}$ and 14\% $\ensuremath{\mathrm{K^0_S}}$ mesons.
First, each jet must contain exactly one track pointing to the
primary vertex and its momentum divided by the beam energy ($p/\mbox{$E_{\mathrm{beam}}$}$)
must be less than 0.5, see fig.~\ref{plot:precuts}(a).
This requirement removes high-momentum pion decays from the one-prong sample.
In order to exclude leptonic background, there
must be at least one cluster in the hadron calorimeter and the total
amount of energy measured by the hadronic calorimeter within the
jet must be greater than 7.5 GeV,
see fig.~\ref{plot:precuts}(b).
The $\taukl$ decays will deposit on average more energy in the hadron
calorimeter than most other $\tau$ decays.
This property is exploited using
the variable, $S_{\mathrm{H}} = (E_{\mathrm{H}} - p)/\sigma_{\mathrm{H}}$,
where $E_{\mathrm{H}}$ is the total energy deposited in the
hadron calorimeter for the jet, $p$ is the momentum of the track
and $\sigma_{\mathrm{H}}/E_{\mathrm{H}} = 0.165 + 0.847/\sqrt{E_{\mathrm{H}}}$
is the hadron calorimeter resolution. The energy is calibrated and
the resolution is measured using pions from $\tau$ decays
that do not interact in the electromagnetic calorimeter.
Events with $S_{\mathrm{H}} \ge 2.0$ are classified
as $\mathrm{X}^- \; \overline{\kk}{}^0_{\mathrm L}$ decays, see fig.~\ref{plot:precuts}(c).
A total of $305$ candidates are selected using the above requirements.
The background is estimated to be
24\% from other $\tau$ decays and 6\% from $\ee \!\! \rightarrow \! \qq$ events.
The primary $\tau$ background consists of $\tau^- \rightarrow \pi^- \nu_\tau$,
$\tau^- \rightarrow \rho(770)^- \nu_\tau$ and
$\tau^- \rightarrow \mbox{a}_1(1260)^-\nu_\tau$ decays.
The sample of $\taukl$ decays is subdivided into two sets:
one in which the track is identified as a pion and another in
which the track is identified as a kaon.
The sample with charged pions is then passed through an additional
selection which identifies those decays that include a $\pi^0$ meson.
The identification of the charged hadron uses the normalized
specific energy loss defined as
$(({\mathrm{d}} E/{\mathrm{d}}x)_{\mathrm{measured}} - ({\mathrm{d}} E/{\mathrm{d}}x)_{\mathrm{expected}}) / \sigma_{{\mathrm{d}} E/{\mathrm{d}}x}$,
where $\sigma_{{\mathrm{d}} E/{\mathrm{d}}x}$ is the ${\mathrm{d}} E/{\mathrm{d}}x$ resolution. Using this quantity,
it is possible to separate charged pions and kaons
at a level of $2\sigma$ in the momentum range of 2-30 GeV.
The expected ${\mathrm{d}} E/{\mathrm{d}}x$ is calculated using the Bethe-Bloch equation
parameterised for the OPAL jet chamber~\cite{opaltracker}.
The parameterisation is checked using one-prong
hadronic $\tau$ decays by comparing the mean values and widths of the
normalised ${\mathrm{d}} E/{\mathrm{d}}x$ distributions in bins of $\beta = p/E$, with $E^2 = p^2 + m_\pi^2$.
It is found that a small $\beta$-dependent correction is to
be applied to the Monte Carlo. The correction shifts the mean value
of the expected ${\mathrm{d}} E/{\mathrm{d}}x$ by up to 10\% and the widths by approximately
5\%. Charged pions and kaons are separated using a ${\mathrm{d}} E/{\mathrm{d}}x$ probability
variable $W$, which is calculated from the normalized ${\mathrm{d}} E/{\mathrm{d}}x$ for each
particle species. These are combined into pion and kaon probabilities,
\begin{eqnarray*}
\mathrm{P}_\pi & = & W_\pi/(W_\pi + W_\ensuremath{\mathrm{K}}) \\
\mathrm{P}_{\mathrm{K}} & = & W_\ensuremath{\mathrm{K}}/(W_\pi + W_\ensuremath{\mathrm{K}}).
\end{eqnarray*}
The distributions of the difference $\mathrm{P}_\pi-\mathrm{P}_{\mathrm{K}}$ is shown in
fig.~\ref{fig:dedx:kl}(a). A track
is considered to be a pion if $\mathrm{P}_\pi>\mathrm{P}_{\mathrm{K}}$.
A neural network algorithm is used to identify the $\taupkz$ and $\taupkzp$
decay modes.
The neural network algorithm uses 7 variables to identify the $\tau$ jets:
\begin{itemize}
\item The total energy of the jet in the electromagnetic calorimeter divided
by the beam energy, $E/\mbox{$E_{\mathrm{beam}}$}$.
\item The total energy of the jet in the electromagnetic calorimeter divided
by the momentum of the track, $E/p$.
\item The number of electromagnetic clusters in the jet with an energy
greater than 1 GeV.
\item The minimum fraction of active lead glass blocks which together
contains more than 90\% of the total electromagnetic energy of the jet,
$F_{90}$.
\item The difference in the azimuthal angle between the track and the
presampler signal farthest away from the track but still within the jet,
$\phi_{\mathrm{PS}}$.
\item The difference in theta ($\Delta \theta$)
and phi ($\Delta \phi$) between the track and the vector obtained
by adding together all the electromagnetic calorimeter clusters in the jet.
\end{itemize}
The variables used in the neural network and the output are
shown in fig.~\ref{plot:klnn}. If the neural network output is larger
than 0.2 then the decay is considered to contain a $\pi^0$ meson.
\section{Selection of $\tauks$ decays \label{sec:kshort}}
The algorithm for identifying $\kk{}^0_{\mathrm S}$ candidates is similar to those
used in other OPAL analyses (for example, see \cite{opal:bose}).
The algorithm begins by pairing tracks with opposite charge.
Each track must have a minimum transverse momentum of 150 MeV and
more than 40 out of a possible 159 hits in the jet chamber.
Intersection points of track pairs in the plane perpendicular to the
beam axis are considered to be secondary vertex candidates.
Each secondary vertex is then
required to satisfy the following criteria:
\begin{itemize}
\item
The radial distance $R_V$ from the secondary vertex to the
primary vertex must be greater than 10 cm and less than 150 cm.
\item
The reconstructed momentum vector of the $\kk{}^0_{\mathrm S}$
candidate in the plane perpendicular to the beam axis must point to
the beam axis within 1$^\circ$.
\item
If $R_V$ is between 30 and 150 cm
(i.e. the secondary vertex is inside the jet chamber volume), then
the radius of the first jet chamber hit associated with either of the two
tracks ($R_1$) must satisfy $R_V - R_1 < 5$ cm.
\item
If $R_V$ is between 10 and 30 cm
(not inside the jet chamber volume), then
the impact parameter of the track is required to exceed 1 mm.
\item
The invariant mass of the pair of tracks, assuming both
tracks to be electrons from a photon conversion, is required to be
greater than 100 MeV.
\end{itemize}
The $\tau$ jet is required to have at least one \ensuremath{\mathrm{K^0_S}}\ candidate. If there
is more than one candidate then only the secondary vertex with an invariant
mass closest to the true \ensuremath{\mathrm{K^0_S}}\ mass is retained.
In addition, each jet is required to have only one additional
track, called the primary track.
A number of additional criteria are applied to reduce the
background from other $\tau$ decays.
Each track associated with the \ensuremath{\mathrm{K^0_S}}\
must have $p>1$ GeV and must not have any hits in the axial regions of
the vertex drift chamber, which extends radially from 10.3 to 16.2 cm. If
the radial distance to the secondary vertex is between 30 and 150 cm,
tracks with hits in the stereo region of the vertex drift chamber, which
extends radially from 18.8 to 21.3 cm, are rejected.
Candidate decays containing photon conversions identified with
an algorithm described in \cite{idncon}, are rejected.
Finally, the mass of the jet (assuming that the primary track is a pion)
must be less than 2 GeV and the invariant mass of the $\kk{}^0_{\mathrm S}$ candidate
is required to be between 0.4 and 0.6 GeV, see fig.~\ref{plot:precuts}(d).
A total of $349$ candidates are obtained with a
background of approximately 10\%, consisting primarily of
$\tau^- \rightarrow \rho(770)^- \nu_\tau$ and
$\tau^- \rightarrow \mbox{a}_1(1260)^-\nu_\tau$ decays.
The sample of $\tauks$ decays is subdivided into two sets:
one in which the primary track is identified as a pion and another in
which the primary track is identified as a kaon,
as described in section~\ref{sec:klong}.
In fig.~\ref{fig:dedx:kl}(b), the difference of the pion ($\mathrm{P}_\pi$)
and kaon ($\mathrm{P}_{\mathrm{K}}$)
probability weight ratios is shown for tracks identified as charged
pions and kaons. A decay is considered to contain a $\pi^-$ if
$\mathrm{P}_\pi > \mathrm{P}_{\mathrm{K}}$.
A neural network algorithm is used to identify the $\taupkz$ and $\taupkzp$
decay modes.
The neural network algorithm is similar to the one used in the $\ensuremath{\mathrm{K^0_L}}$
analysis;
the differences are due to the different topologies of the two
selections.
The neural network algorithm for this selection uses 6 variables to identify
the $\tau$ jets:
\begin{itemize}
\item The total energy of the jet in the electromagnetic calorimeter divided by
the scalar sum of the momenta of the tracks, $E/p$.
\item The number of clusters in the electromagnetic calorimeter
with energy greater than 1 GeV in the jet.
\item The minimum fraction of active lead glass blocks which together
contains more than 90\% of the total electromagnetic energy of the jet,
$F_{90}$.
\item The total presampler multiplicity in the jet.
\item The difference in theta ($\Delta \theta$)
and phi ($\Delta \phi$) between the vector obtained by adding
together all the tracks and the vector obtained by adding together all
the clusters in the electromagnetic calorimeter associated to the jet.
\end{itemize}
The variables used in the neural network algorithm are shown in
figs.~\ref{plot:nn}(a-f).
A decay is considered to contain a $\pi^0$ if the neural
network output, shown in fig.~\ref{plot:nn}(g), is greater than 0.5.
\section{Branching ratios \label{sec:br}}
The branching ratios of the {\small $\taupkz$}, {\small $\taupkzp$}
and {\small $\taukkz$} decay modes are calculated
independently for the $\tau$ jets containing \ensuremath{\mathrm{K^0_L}}\ and \ensuremath{\mathrm{K^0_S}}\ decays.
However, for a given data sample, the three branching ratios are
calculated simultaneously.
Each selection can be characterised in terms of the efficiency for
detecting each decay mode $i$, the branching ratio of each mode and the number
of events selected in the data:
\[
\label{br_1}
\epsilon_{i1} B_1 + \epsilon_{i2} B_2 + \epsilon_{i3} B_3
+ \sum_{k=4}^{M} \epsilon_{ik} B_k =
\frac{N_i-N_i^{{\mbox{\scriptsize non}}-\tau}}{N_{\tau}
(1-f^{{\mbox{\scriptsize non}}-\tau})}
\]
where $N_i$ is the number of data events that pass the selection $i$,
$\epsilon_{ij}$ ($j=1,3$) are the efficiencies for selecting signal
$j$ using selection $i$,
$\epsilon_{ik}$ ($k=4,\ldots,M$) are the efficiencies for selecting
the $\tau$ background modes using selection $i$ and $M$ is the number
of the $\tau$ decay modes.
The branching ratios of the signal channels and backgrounds are
$B_j$ ($j=1,3$) and $B_k$ ($k=4,\dots,M$), respectively.
The fraction of non-$\tau$ events in the $\tau$ pair sample is
$f^{{\mbox{\scriptsize non}}-\tau}$,
$N_\tau$ is the total number of taus in the data that pass the
$\tau$ pair selection and $N_i^{{\mbox{\scriptsize non}}-\tau}$ is the
non-$\tau$ background present in each selection $i$.
The selection efficiencies ($\epsilon_{ij}$) for both signal and background
are determined from Monte Carlo simulation.
The $\tau$ background branching ratios are taken from the
Particle Data Group compilation \cite{pdg}.
Solving the three simultaneous equations yields the
branching ratios in each sample of selected events.
A small correction is applied to the branching ratios to account for
any biases introduced into the $\tau$ pair sample by the $\tau$ pair selection.
The bias factor is defined as the ratio of the fraction of the selected decays in a
sample of $\tau$ decays after the $\tau$ selection is applied to the fraction
before the selection. The bias factors are calculated using approximately
2.2 million simulated $\tau^+\tau^-$ events.
The bias factors for the {\small $\taupkz$}, {\small $\taupkzp$} and
{\small $\taukkz$} decays are found to be $\mbox{$0.99\pm0.01$}$, $\mbox{$1.00\pm0.01$}$ and $\mbox{$1.00\pm0.01$}$
for the branching ratios obtained from the $\kk{}^0_{\mathrm L}$ sample,
and $0.99 \pm 0.01$, $1.01 \pm 0.03$ and $0.99 \pm 0.02$
for the branching ratios obtained from the $\kk{}^0_{\mathrm S}$ sample. The
uncertainties on the measurements are statistical only.
The $\ensuremath{\mathrm{K^0_L}}$ ($\ensuremath{\mathrm{K^0_S}}$) selection identifies
$178$ ($199$) {\small $\taupkz$} decays, $ 81$ ($ 67$)
{\small $\taupkzp$} decays
and $ 41$ ($ 83$) {\small $\taukkz$} decays.
The efficiency matrix for each sample is given in table~\ref{table:eff}.
The branching ratios obtained from the $\kk{}^0_{\mathrm L}$ sample are
\begin{eqnarray*}
B(\taupkz) & = & \ensuremath{(9.1\pm 0.9\pm 0.6})\times 10^{-3} , \\
B(\taupkzp) & = & \ensuremath{(3.6\pm 1.3\pm 1.0})\times 10^{-3} , \\
B(\taukkz) & = & \ensuremath{(3.3\pm 0.9\pm 0.7})\times 10^{-3} ,\\[-3mm]
\end{eqnarray*}
while the branching ratios obtained from the $\kk{}^0_{\mathrm S}$ sample are
\begin{eqnarray*}
B(\taupkz) & = & ( 9.6\pm 1.0\pm 0.7)\times 10^{-3} , \\
B(\taupkzp) & = & ( 3.0\pm 0.9\pm 0.9)\times 10^{-3} , \\
B(\taukkz) & = & ( 3.3\pm 0.7\pm 0.5)\times 10^{-3} ,\\[-3mm]
\end{eqnarray*}
where the first error is statistical and the second is systematic.
\section{Systematic errors \label{sec:errors}}
The systematic uncertainties on the branching ratios are
presented in table~\ref{table:syst}. The dominant contributions
to the systematic uncertainty arises from the efficiency of
the two selections, the uncertainty of the backgrounds, the modelling
of the ${\mathrm{d}} E/{\mathrm{d}}x$, the identification of the $\pi^0$ and the modelling
of Monte Carlo. These uncertainties are discussed in more detail
below. In addition,
there are straightforward contributions from the limited statistics
of the Monte Carlo samples used to estimate the selection efficiencies
and from the uncertainties on the bias factors.
The systematic error on the branching ratios due to the Monte Carlo
statistics is calculated directly from the statistical uncertainties
on the elements of the inverse efficiency matrix~\cite{matrix:inversion}.
The systematic error on each branching ratio due to the
bias factor is calculated directly from the bias factor error.
\noindent{\bfseries \ensuremath{\mathrm{K^0_L}}\ and \ensuremath{\mathrm{K^0_S}}\ selection efficiencies:}
\noindent
The \ensuremath{\mathrm{K^0_L}}\ selection efficiency is sensitive to the calibration of the
momentum, the energy measured by the hadron calorimeter and
the resolution of the hadron calorimeter.
The uncertainty on the momentum scale is typically better than
1\%~\cite{opal:tp}.
The uncertainty in the energy scale of the hadron calorimeter
is obtained by studying a sample of single charged hadrons from
$\tau$ decays, the level of agreement between the data and Monte Carlo
is 1.5\%. The uncertainty due to the measurement of the resolution
of the hadron calorimeter is estimated by varying the resolution
within its uncertainties. Also, the shower containment is examined by looking at the
leakage of energy out of the back of the hadron calorimeter.
It is found that about 8\% of $\ensuremath{\mathrm{K^0_L}}$ decays may not be
fully contained, these decays are well modelled by the Monte Carlo
and does not result in a systematic uncertainty.
The \ensuremath{\mathrm{K^0_S}}\ selection efficiency is sensitive to the requirements on
the impact parameter, the momentum and the number of hits in the stereo
and axial regions of the vertex chamber on the tracks associated to the \ensuremath{\mathrm{K^0_S}}\ .
The systematic error on the \ensuremath{\mathrm{K^0_S}}\ selection efficiency is determined
by dropping each relevant criterion except for the impact parameter resolution.
The impact parameter resolution has been shown to have an uncertainty that
is typically better than $\pm 20\%$~\cite{impact}. Variations of the
impact parameter resolution are found to have almost no contribution
to the systematic error on the \ensuremath{\mathrm{K^0_S}}\ selection efficiency.
\noindent{\bfseries Background estimation:}
\noindent
The systematic error due to the background in the \ensuremath{\mathrm{K^0_L}}\ sample
includes the uncertainty in the branching ratios of the background
decays, including the {\small $\taupkzkz$}
and {\small $\taupkzkzp$} decays, as well as the uncertainty from the
Monte Carlo statistics~\cite{pdg,aleph-k0}.
The non-$\ensuremath{\mathrm{K^0}}$ background consists primarily of $\pi^-$, $\rho(770)^-$
and $\mathrm{a}_1(1260)^-$ decays in which the decays have a low momentum
track with at least one of the final $\pi$ mesons leaving some energy
in the hadron calorimeter. To investigate this background, the
$\ensuremath{\mathrm{K^0_L}}$ selection cut $S_H$ is reversed and the invariant mass spectra
are studied for each decay mode. The ratios of the data to the Monte Carlo
simulation are consistent with unity:
$0.97 \pm 0.02$, $1.04 \pm 0.02$ and $0.94\pm0.06$ for the
$\pi^-\overline{\kk}{}^0$, $\pi^-\overline{\kk}{}^0 \ge 1\pi^0$ and
$\ensuremath{\mathrm{K^-}}\ensuremath{\mathrm{K^0}}\ge0\pi^0$ selections, respectively. The various contributions
to the systematic error from the background are added in quadrature.
The background in the \ensuremath{\mathrm{K^0_S}}\ sample includes {\small $\taupkzkz$}
and {\small $\taupkzkzp$} decays,
which contain \ensuremath{\mathrm{K^0_S}}\ mesons, and other $\tau$ decays, the uncertainty
is composed of the Monte Carlo statistical uncertainty plus a
component due to the uncertainty in the branching ratios of these
decays~\cite{pdg,aleph-k0}.
A study of the sidebands of the $m_{\pi\pi}$
distribution (see fig.~\ref{plot:precuts}(d)) showed that the
background prediction from other $\tau$ decays is observed to be about 20\%
smaller in the Monte Carlo simulation than in the data.
As a result, the background is scaled upward by a factor of 1.2
and a 20\% uncertainty is assigned to the background estimate.
The background estimate is cross-checked using the invariant mass
distributions of the tracks associated with the \ensuremath{\mathrm{K^0_S}}\ candidate for each
of the exclusive channels. The ratios of the data to the Monte Carlo
simulation are consistent: $1.07 \pm 0.12$,
$1.09 \pm 0.06$ and $0.93\pm0.11$ for the $\pi^-\overline{\kk}{}^0_{\mathrm S}$,
$\pi^-\overline{\kk}{}^0_{\mathrm S} \ge 1\pi^0$ and $\ensuremath{\mathrm{K^-}}\ensuremath{\mathrm{K^0_S}}\ge0\pi^0$ selections,
respectively. The various contributions
to the systematic error from the background are added in quadrature.
\noindent{\bfseries Modelling of ${\mathrm{d}} E/{\mathrm{d}}x$:}
\noindent
For both samples, the normalized ${\mathrm{d}} E/{\mathrm{d}}x$ distributions are studied using the
sample of single charged hadrons from $\tau$ decays.
The uncertainty in the branching ratios is estimated by varying
the means of the normalised
${\mathrm{d}} E/{\mathrm{d}}x$ distributions by $\pm1$ standard deviation of their
central values. In addition, to account for possible differences
in the ${\mathrm{d}} E/{\mathrm{d}}x$ resolution,
the widths of the normalized ${\mathrm{d}} E/{\mathrm{d}}x$ distributions are varied by $\pm30$\%.
Due to the three tracks present in the \ensuremath{\mathrm{K^0_S}}\ sample, an additional
contribution to the systematic error is obtained by measuring the difference
in the branching ratios when
two different corrections are applied to the Monte Carlo. The first
correction is
estimated from the one-prong hadronic tau decays while the second
correction is estimated using the sample of pions from the decay of the
$\ensuremath{\mathrm{K^0_S}}$. The various contributions
to the systematic error from the ${\mathrm{d}} E/{\mathrm{d}}x$ modelling are added in quadrature.
\noindent{\bfseries Identification of $\pi^0$:}
\noindent
Both the \ensuremath{\mathrm{K^0_L}}\ and \ensuremath{\mathrm{K^0_S}}\ samples use a neural network algorithm
to separate the {\small $\taupkz$} and {\small $\taupkzp$} decay modes.
The most powerful variable for distinguishing between these two decays
is the energy deposited in the electromagnetic calorimeter.
The systematic error in the branching ratio is evaluated by shifting
the electromagnetic energy scale by $\pm 1\%$; this variation is assigned
after studying the differences between data and Monte Carlo in $E/p$
distributions for 3-prong $\tau$ decays.
The uncertainty affecting the $\pi^0$ identification also includes the maximum
uncertainty when each variable (except in those which
include the electromagnetic energy) is individually dropped
from the neural network
algorithm. These uncertainties are added in quadrature with those obtained
from the energy scale uncertainty. The stability of the neural network
algorithm is studied by removing all but the two most significant
variables from the neural network, the results are within the systematic
uncertainties for both samples. As a cross check, the cut on the neural
network output for both
the \ensuremath{\mathrm{K^0_L}}\ and \ensuremath{\mathrm{K^0_S}}\ samples is varied between 0.1 and 0.8,
with the result being consistent within the total
systematic uncertainties.
\noindent{\bfseries Monte Carlo modelling:}
\noindent
The models used in the Monte Carlo generator can effect both the pion and
kaon momentum spectra. This effect can produce biases when
determining the $\ensuremath{\mathrm{K^0}}$ identification efficiency, the
$\ensuremath{\mathrm{K}}/\pi$ separation and the $\pi^0$ identification.
The dynamics of the $\pi^-\overline{\kk}{}^0$ decay mode are
well understood. The $\pi^-\overline{\kk}{}^0$ decay mode is generated by
Tauola via the $\ensuremath{\mathrm{K}}^*(892)^-$
resonance. The $\ensuremath{\mathrm{K^-}}\ensuremath{\mathrm{K^0}}$ final state is generated by Tauola using phase
space only.
The $\taupkzp$ decay mode is composed of
$\tau^- \! \rightarrow \pi^- \overline{\kk}{}^0 \pi^0 \nu_\tau$ and
$\tau^- \! \rightarrow \! \pi^- \overline{\kk}{}^0 \pi^0 \pi^0 \nu_\tau$ decays.
The $\taupkp$ channel is modelled by Tauola assuming
that the decay proceeds via the $\ensuremath{\mathrm{K}}_1(1400)$ resonance. Recent results
from ALEPH~\cite{aleph-oneprong} on one-prong $\tau$ decays with kaons,
and OPAL~\cite{sherry-towers} using
$\tau^- \rightarrow \ensuremath{\mathrm{K^-}} \pi^-\pi^+\nu_\tau$ decays, suggest that
the $\taupkp$ decay will also proceed via the $\ensuremath{\mathrm{K}}_1(1270)$ resonance.
A special Monte Carlo simulation is generated in which the final state
is created using the $\ensuremath{\mathrm{K}}_1(1270)$ and $\ensuremath{\mathrm{K}}_1(1400)$ resonances, using
the algorithm developed for the analysis described in~\cite{sherry-towers}.
The selection efficiency of the $\taupkp$ final state is estimated
from the special Monte Carlo for both resonances. For the $\ensuremath{\mathrm{K^0_L}}$ analysis,
the efficiencies agree at a level of 10\%. For the $\ensuremath{\mathrm{K^0_S}}$ analysis, the
selection efficiencies agree at a level of 5\%.
The $\taupkpp$ decay mode is not modelled by Tauola.
The branching ratio of this mode was recently measured to be
$(0.26\pm0.24)\times10^{-3}$~\cite{aleph-k0}. A special Monte Carlo
sample of the $\taupkpp$ decay mode is generated using flat phase space
and it is found that the efficiency of the $\taupkpp$ decay mode agrees
within 30\% of the efficiency of the $\taupkp$ decay mode. For the
systematic uncertainty associated with this decay mode, 30\% of the
$\taupkpp$ branching ratio is used.
The $\tau^- \rightarrow \, \kk^- \kk{}^0 {\mbox{\small $[\geq 1\pi^0]$}} \nu_\tau$
decay mode is composed of $\tau^- \rightarrow \, \kk^- \kk{}^0 \pi^0 \nu_\tau$ and
$\tau^- \rightarrow \, \kk^- \kk{}^0 \pi^0\pi^0 \nu_\tau$ decays.
The $\taukkc$ decay mode is generated by Tauola through a combination
of the $\rho(1700)$ and $\mathrm{a}_1(1260)$ resonances.
Monte Carlo simulations of these two modes are generated
separately, again using the algorithm developed for the analysis described
in~\cite{sherry-towers}. The selection efficiencies of the $\taukkc$ decay mode
are calculated for these two samples and are equivalent
within statistical errors. No systematic uncertainty is included for this
channel. The $\taukkd$ decay mode is not
modelled by Tauola. The Particle Data Group~\cite{pdg}
give an upper bound of $0.18\times10^{-3}$ for this channel. A special Monte Carlo
sample of the $\taukkd$ decay mode is generated using flat phase space
and the efficiency of the $\taukkd$ decay mode is observed to be within 30\% of
the efficiency of the $\taukkc$ decay mode. For the
systematic uncertainty associated with this decay mode, 30\% of the $\taukkd$
branching ratio is used.
Finally, the $\taukkb$ selection efficiency may depend on the relative
$\taukka$ and $\taukkc$ branching ratios. Using the current world averages
from~\cite{pdg}, the relative contribution of each channel is varied
by $\pm 25\%$. For the $\ensuremath{\mathrm{K^0_S}}$ analysis, no effect is observed
on the branching ratio, as the efficiency for selecting the two channels is
very similar; hence, no systematic error is included.
\section{Summary}
The branching ratios of the decays of the $\tau$ leptons to neutral kaons
are measured using the OPAL data recorded at centre-of-mass energies near
the $\mathrm{Z}^0$ resonance from a recorded luminosity of 163 pb$^{-1}$.
The measurement is based on two samples which identify $\tau$ decays
with $\ensuremath{\mathrm{K^0_L}}$ and $\ensuremath{\mathrm{K^0_S}}$ mesons.
The branching ratios obtained from the $\kk{}^0_{\mathrm L}$ sample are
\begin{eqnarray*}
B(\taupkz) & = & \ensuremath{(9.1\pm 0.9\pm 0.6})\times 10^{-3}, \\
B(\taupkzp) & = & \ensuremath{(3.6\pm 1.3\pm 1.0})\times 10^{-3}, \\
B(\taukkz) & = & \ensuremath{(3.3\pm 0.9\pm 0.7})\times 10^{-3},
\end{eqnarray*}
while the branching ratios obtained from the $\kk{}^0_{\mathrm S}$ sample are
\begin{eqnarray*}
B(\taupkz) & = & ( 9.6\pm 1.0\pm 0.7)\times 10^{-3}, \\
B(\taupkzp) & = & ( 3.0\pm 0.9\pm 0.9)\times 10^{-3}, \\
B(\taukkz) & = & ( 3.3\pm 0.7\pm 0.5)\times 10^{-3}.
\end{eqnarray*}
In each case the first error is statistical and the second systematic.
The combined results are
\begin{eqnarray*}
B(\taupkz) & = & {(9.33\pm0.68\pm0.49})\times 10^{-3}, \\
B(\taupkzp) & = & {(3.24\pm0.74\pm0.66})\times 10^{-3}, \\
B(\taukkz) & = & {(3.30\pm0.55\pm0.39})\times 10^{-3}.
\end{eqnarray*}
The branching ratios are compared with existing measurements and
theoretical predictions in
fig.~\ref{plot:bratios} for the $\taupkz$ and $\taupkzp$ decay modes
\cite{aleph-oneprong,k0-measurents}.
The solid band is the new average branching ratio
of the OPAL, ALEPH, CLEO and L3 measurements.
The results of this work are
in good agreement with previous measurements.
The branching ratios of the decay modes
are predicted from various theoretical models. The measurement
of the decay fraction of the $\taupkz$ decay agrees well with the
range $(8.9-10.3) \times 10^{-3}$ estimated by Braaten {\it et~al.}\
in~\cite{langrangian} and falls in the range of
$(6.6-9.6) \times 10^{-3}$ predicted by Finkemeier and Mirkes in~\cite{fink}.
The decay $\taupkzp$, assuming that the decay contains only
one $\pi^0$, is predicted to be in the range of $(0.9-3.7) \times 10^{-3}$ from
\cite{langrangian} and in the range of $(8.1-9.6)\times 10^{-3}$
from~\cite{fink}. The $\taupkp$ branching ratio prediction by
Finkemeier and Mirkes is significantly higher than the
experimental results, however they argue that the widths of the
$\ensuremath{\mathrm{K}}_1$ resonance~\cite{pdg} used in their calculation are unusually
narrow and that increasing the $\ensuremath{\mathrm{K}}_1$ width would give a prediction that
agrees with the experimental measurements~\cite{fink2}.
The branching ratio of the $\taukkz$ decay mode is
the sum of the $\taukka$ and $\taukkc$ decay modes. The decay fraction
agrees well with the estimated range $(2.4-4.0)\times10^{-3}$ predicted
by~\cite{langrangian} and $(2.3-2.7)\times10^{-3}$ predicted by~\cite{fink}.
The $\taupkz$ decay mode is assumed to be dominated by the $\ensuremath{\mathrm{K}}^*(892)^-$
resonance. This can be observed from the $\pi^-\ensuremath{\mathrm{\overline{K}^0}}$ invariant
mass distributions shown in fig.~\ref{plot:kstmass}, for the decay modes
{\small $\taupks$} and {\small $\taupkl$}, respectively.
Assuming that the $\taupkz$ decay mode proceeds entirely through the
$\ensuremath{\mathrm{K}}^*(892)^-$ resonance, then using isospin invariance
the branching ratio of the $\taukstar$ decay mode
is calculated to be $0.0140 \pm 0.0013$.
This value is consistent with the current world average
$0.0128 \pm 0.0008$~\cite{pdg}.
Finally, the ratio of the decay constants $f_\rho$ and $f_{\ensuremath{\mathrm{K}}^*}$
can be estimated using the $\taukstar$ branching ratio and the OPAL
$\ensuremath{\tau^{-}\rightarrow \mathrm{h}^{-} \pi^0\nu_\tau}$ branching ratio of $0.2589 \pm 0.0034$~\cite{opal:tp}.
The $\ensuremath{\tau^{-}\rightarrow \mathrm{h}^{-} \pi^0\nu_\tau}$ decay mode is the sum of the decay
modes $\tau^- \rightarrow \pi^-\pi^0\nu_\tau$ and $\tau^- \rightarrow \ensuremath{\mathrm{K^-}}\pi^0\nu_\tau$.
The branching ratio of the $\tau$ to the final state $\ensuremath{\mathrm{K^-}}\pi^0\nu_\tau$
is calculated to be $(4.67 \pm 0.42)\times10^{-3}$ using isospin invariance
and the $\taupkz$ branching ratio. Consequently,
$B(\tau^- \rightarrow \pi^-\pi^0\nu_\tau)$ is derived to be $0.2543 \pm 0.0034$.
Using these results, $\tan\theta_c = 0.227$ for the Cabibbo angle and the
particle masses from~\cite{pdg}, the decay constant ratio
\begin{displaymath}
\frac{f_\rho}{f_{\ensuremath{\mathrm{K}}^*}} = \tan\theta_c\sqrt{\frac{B(\ensuremath{\tau^{-}\rightarrow\rho^{-} \nu_\tau})}{B(\taukstar)}}
\left(\frac{m^2_\tau - m^2_{\ensuremath{\mathrm{K}}^*}}{m^2_\tau - m^2_\rho}\right)
\sqrt{\frac{m^2_\tau + 2 m^2_{\ensuremath{\mathrm{K}}^*}}{m^2_\tau + 2m^2_\rho}}
= 0.93 \pm 0.05
\end{displaymath}
is obtained.
The error is dominated by the uncertainties on the branching
ratios. The recent result from ALEPH~\cite{aleph-k0},
$0.94 \pm 0.03$, agrees well with the new OPAL
result. Finally, this ratio has been predicted by Oneda~\cite{oneda} using
the Das-Mathur-Okubo sum rule relations~\cite{dmo} between the spectral
functions based on assumptions of $SU(3)_f$ symmetry.
At the $SU(3)_f$ symmetry limit
($m_\mathrm{u} =m_\mathrm{d} =m_\mathrm{s}$), the decay constant ratio
is expected to be unity, $f_\rho = f_{\ensuremath{\mathrm{K}}^*}$. In the asymptotic $SU(3)_f$
symmetry limit at high energies, Oneda predicts that $f_\rho/f_{\ensuremath{\mathrm{K}}^*}
=m_\rho/m_{\ensuremath{\mathrm{K}}^*} = 0.86$.
\bigskip
\section*{Acknowledgements}
We particularly wish to thank the SL Division for the efficient operation
of the LEP accelerator at all energies
and for their continuing close cooperation with
our experimental group. We thank our colleagues from CEA, DAPNIA/SPP,
CE-Saclay for their efforts over the years on the time-of-flight and trigger
systems which we continue to use. In addition to the support staff at our own
institutions we are pleased to acknowledge the \\
Department of Energy, USA, \\
National Science Foundation, USA, \\
Particle Physics and Astronomy Research Council, UK, \\
Natural Sciences and Engineering Research Council, Canada, \\
Israel Science Foundation, administered by the Israel
Academy of Science and Humanities, \\
Minerva Gesellschaft, \\
Benoziyo Center for High Energy Physics,\\
Japanese Ministry of Education, Science and Culture (the
Monbusho) and a grant under the Monbusho International
Science Research Program,\\
Japanese Society for the Promotion of Science (JSPS),\\
German Israeli Bi-national Science Foundation (GIF), \\
Bundesministerium f\"ur Bildung, Wissenschaft,
Forschung und Technologie, Germany, \\
National Research Council of Canada, \\
Research Corporation, USA,\\
Hungarian Foundation for Scientific Research, OTKA T-029328,
T023793 and OTKA F-023259.\\
\clearpage
| -29,875.212031 |
[
-1.5927734375,
1.5986328125
] | 32.034632 |
[
-2.349609375,
0.94677734375,
-1.494140625,
-5.55078125,
-1.501953125,
7.5859375
] |
[
1.4951171875,
6.234375,
2.625,
4.75
] | 360 | 4,894 |
[
-3.2265625,
3.87890625
] | 28.622497 |
[
-5.3671875,
-2.220703125,
-2.4296875,
-1.8564453125,
0.6083984375,
8.890625
] | 1.829291 | 13.471075 | 21.597875 | 3.225201 |
[
1.8065820932388306
] | -21,532.380933 | 5.585002 | -28,920.716813 | 1.382944 | 5.600508 |
[
-3.1015625,
-3.66015625,
-3.23046875,
-4.2890625,
2.2734375,
11.09375
] |
[
-5.35546875,
-1.873046875,
-2.07421875,
-1.52734375,
3.099609375,
4.34765625
] | |
BkiUervxK1Thg9qFezh8
|
\section{Introduction}
The existence of the large--scale clustering of galaxies had already
been well established by the early 1970's mainly due to the pioneering
work of {\scite{totsuji:correlation}} and {\scite{peebles:nature}} who
showed that the two--point correlation function for galaxies in the
Lick and Zwicky catalogues was positive and had the power-law form
$\xi(r)\propto{r^{-1.8}}$ on scales $\lesssim10\ifmmode{h^{-1}{\rm Mpc}}\else{$h^{-1}{\rm Mpc}$}\fi$. Their result
was later extended to three--dimensional galaxy catalogues as well.
Although the clustering of galaxies is now a well--known fact, a
complete description of clustering which includes its geometrical
features has so far eluded researchers. This is perhaps due to the
fact that the galaxy density field which we observe appears to be
strongly non--Gaussian. A Gaussian random field is uniquely described
by its power spectrum $P(k)$, or its two--point correlation function
$\xi(r)$ since $\xi(r)$ and $P(k)$ form a Fourier transform pair.
This is no longer true for a non--Gaussian field for which $\xi$ must
be complemented by other statistical descriptors which are sensitive
to the structure of matter on large scales. In the so-called
``standard model'' of structure formation, an initially Gaussian
density distribution becomes non--Gaussian due to mode-coupling and
the resultant build up of phase correlations during the non--linear
regime. These phase correlations give rise to the amazing diversity
of form, which is characteristic of a highly evolved distribution of
matter, and is often referred to as being cellular, filamentary,
sheet--like, network--like, sponge--like, a cosmic web etc. Most of
these descriptions are based either on a visual appearance of
large--scale structure or on the presence of features that are absent
in the reference Gaussian distribution which, by definition, is
assumed to be featureless. Gravitational instability, for example,
may cause CDM--like Gaussian initial perturbations to evolve towards a
density field that percolates at a higher density threshold, i.e.\ at
a lower filling factor, than a Gaussian field {\cite{melott:cluster}}.
Such distributions display greater connectivity and are sometimes
referred to as being ``network--like'' {\cite{yess:universality}}.
In order to come to grips with the rich textural possibilities
inherent in large--scale structure, a number of geometrical indicators
of clustering have been proposed in the past including minimal
spanning trees {\cite{barrow:minimalspanning}}, the genus curve
{\cite{gott:sponge}}, percolation theory
{\cite{zeldovich:maximum,shandarin:percolation}} and shape analysis
(\pcite{sathyaprakash:morphology,sahni:approximation} and references
therein). A major recent advance in our understanding of
gravitational clustering has been associated with the application of
Minkowski functionals (MFs) to cosmology {\cite{mecke:robust}}. The
four MFs $V_0,\ldots,V_3$ provide an excellent description of the
geometrical properties of a collection of point objects (galaxies) or,
alternatively, of continuous distributions such as density fields in
large--scale structure or brightness contours in the cosmic microwave
background. The scope and descriptive power of the MFs is enhanced by
the fact that both percolation analysis and the genus curve are
members of the family. Additionally, as demonstrated by
{\scite{sahni:shapefinders}}, ratios of MFs provide us with an
excellent ``shape statistic'' with which one may attempt to quantify
the morphology of large scale structure, including the shapes of
individual superclusters and voids. Spurred by the success of MFs in
quantifying the geometrical properties of large--scale structure we
apply the MFs to scale--invariant N--body simulations of gravitational
clustering, with an attempt to probe both the global properties and
the individual ``bits and pieces'' that might make up the ``cosmic
web'' {\cite{bond:filament}}.
\section{Method}
\subsection{Minkowski functionals}
Minkowski functionals (named after {\pcite{minkowski:volumen}}) were
introduced into cosmology by {\scite{mecke:robust}} employing the
generalized Boolean grain model. This model associates a body with a
point process (in our case given by the location of galaxies or
clusters) by decorating each point with a ball of radius $r$. The
union set of the covering balls is then studied morphologically,
whereby the radius of the ball serves as a diagnostic parameter
probing the spatial scale of the body.
In this paper we shall use the excursion set approach which is
applicable to continuous fields (that may be constructed from point
processes). {\scite{schmalzing:beyond}} pointed out how to apply
Minkowski functionals to isocontours of continuous fields, where the
contour level (the threshold) is employed as diagnostic parameter.
The excursion set approach inherits two diagnostic parameters, because
we may also vary the smoothing scale used in constructing the
continuous field.
In three dimensions there exist four Minkowski functionals $V_\mu$,
$\mu=0,1,2,3$. They provide a complete and unique description of a
pattern's global morphology in the sense of Hadwiger's theorem
{\cite{hadwiger:vorlesung}}. While reducing the information contained
in the full hierarchy of correlation functions, this small set of
numbers incorporates correlations of arbitrary order and therefore
provides a complementary look on large--scale structure. The
geometric interpretations of all Minkowski functionals in three
dimensions are summarized in Table~\ref{tab:minkowski}.
We calculate global Minkowski functionals of the isodensity contours
of the density field as described in Appendix~\ref{app:minkowski}.
Furthermore, we separately calculate the partial Minkowski functionals
of each isolated part of the isodensity contour. Since the total
isodensity contour is the union of all its parts, the global
functionals are given as sums of the partial functionals at the same
threshold. This follows the spirit of {\scite{mecke:robust}}, where
partial Minkowski functionals are introduced to measure local
morphology for the generalized Boolean grain model.
Partial Minkowski functionals (PMF) offer the possibility of probing
the morphology of individual objects, or the object's environmental
morphological properties, respectively. We expect that this concept
will be more powerful when applied to continuous fields at high
spatial resolution so that the details of structures are not smoothed
out. Their application to point processes also delivers more direct
information. PMF provide a wide range of possibilities for
morphological studies which we shall explore in a forthcoming paper.
\subsection{Shapefinders}
One important task of morphological statistics consists in quantifying
strongly non--Gaussian features such as filaments and pancakes. Given
the four Minkowski functionals, we aim at reducing their morphological
information content to two measures of planarity and filamentarity,
respectively, as has been done for example with various geometrical
quantities {\cite{mo:statistical}}, moments of inertia
{\cite{babul:filament}}, and cumulants of counts--in--cells
{\cite{luo:shape}}.
Recently, {\scite{sahni:shapefinders}} proposed a set of
shapefinders derived from Minkowski functionals. One starts from
the three independent ratios of Minkowski functionals that have
dimension of length. Requiring that they yield the radius $R$ if
applied to a ball, we define
\begin{equation}
\text{Thickness }T:=\frac{V_0}{2V_1},\qquad
\text{Width }W:=\frac{2V_1}{{\pi}V_2},\qquad
\text{Length }L:=\frac{3V_2}{4V_3}.
\end{equation}
By the isoperimetric inequalities~(\ref{eq:isoperimetric}), we have
$L{\ge}W{\ge}T$ for any convex body.
Going one step further, {\scite{sahni:shapefinders}} also define
dimensionless shapefinders by
\begin{equation}
\label{eq:shapefinders}
\text{Planarity }{\mathcal{P}}:=\frac{W-T}{W+T},\qquad
\text{Filamentarity }{\mathcal{F}}:=\frac{L-W}{L+W}.
\end{equation}
Some examples are in order.
\subsection{Simple examples}
Let us consider some simple families of convex bodies in
three--dimensional space that can take both filamentary and planar
shape.
A spheroid with two axes of length $r$ and one axis of length
$\lambda{r}$ has Minkowski functionals
\begin{equation}
V_0= \frac{4\pi}{3}r^3\lambda ,\qquad
V_1= \frac{\pi}{3}r^2\left(1+f(1/\lambda)\right) ,\qquad
V_2= \frac{2}{3}r\left(\lambda+f(\lambda)\right) ,\qquad
V_3= 1,
\label{eq:spheroid}
\end{equation}
where\footnote{Note that for arguments $x>1$, one can use the relation
$i\arccos{x}=\ln\left(x+\sqrt{x^2-1}\right)$ to recover an explicitly
real--valued expression.}
\begin{equation}
f(x)=\frac{\arccos{x}}{\sqrt{1-x^2}}.
\end{equation}
By varying the parameter $\lambda$ from zero to infinity, we can
change the morphology of the spheroid from a filament to a pancake via
a spherical cluster.
A different way of deforming a filament into a pancake goes via
generic triaxial ellipsoids; thereby one of the smaller axes of a
strongly prolate spheroid is increased in size until it matches the
larger axis and an oblate spheroid has been reached. An integral
expression can be found in {\cite{sahni:shapefinders}}.
Yet another transition from prolate to oblate shape is provided by
cylinders of radius $r$ and height $\lambda{r}$. Here, the Minkowski
functionals are given by
\begin{equation}
V_0=\pi r^3\lambda ,\qquad
V_1=\frac{\pi}{3}r^2(1+\lambda) ,\qquad
V_2=\frac{1}{3}r(\pi+\lambda) ,\qquad
V_3=1.
\label{eq:cylinder}
\end{equation}
A Blaschke diagram, that is a plot of planarity ${\mathcal{P}}$ and
filamentarity ${\mathcal{F}}$, summarizing the morphological properties of these
simple convex bodies is shown in Figure~\ref{fig:blaschke.simple}.
\section{A set of $N$--body simulations}
\subsection{Description}
We start from a family of initial power law spectra $P(k)\propto{k}^n$
with $n\in\{-2,-1,0,+1\}$ set before an Einstein--de~Sitter background
($\Omega$=1, $\Lambda$=0). We conduct numerical experiments using a
PM code (consult {\pcite{melott:controlled}} for details). Four sets
of phases were used for each model, making a total of 16 simulation
runs. Each run consists of $128^3$ particles sampled at an epoch well
in the non--linear regime. This epoch is chosen such that the scale
of non--linearity $k_{\text{nl}}$, defined in terms of the evolved
spectrum
\begin{equation}
\sigma^2_{k_{\text{nl}}} = \int_0^{k_{\text{nl}}}{\mathrm{d}}^3k P(k) = 1,
\end{equation}
is equal to eight in units of the fundamental mode of the simulation
box. By using the stage $k_{\text{nl}}=8$, we make sure that
structure is already sufficiently developed on scales much larger than
the simulation's resolution, while it is not yet influenced by
boundary effects.
Using a cloud--in--cell kernel, these particles were put onto a
$256^3$ grid, which is the maximum value an ordinary workstation can
tackle with acceptable time and memory consumption. Subsequently, the
density field was smoothed with a Gaussian kernel
$\propto\exp\left(-x^2/2\lambda^2\right)$, where $x$ is the distance
in mesh units and the width $\lambda$ is set to 3. Tests have shown
that this value both leads to a reasonably smooth field, and preserves
at least some detail on smaller scales. Throughout the article, we
re--scale the density to the density contrast $\delta$, ranging from
$-1$ to infinity with zero mean.
\subsection{The global field}
The global Minkowski functionals calculated from the density fields
described in the previous section are shown in
Figure~\ref{fig:epoch.f}. The four different line styles correspond
to the different spectral indices. Figure~\ref{fig:epoch.f.rescaled}
shows the Minkowski functionals for the same set of models, but
instead of the density threshold $\delta$, the rescaled threshold
$\nu$ is used as the $x$--axis. $\nu$ is calculated from the volume
Minkowski functional $v_0$, that is the filling factor $f$, as
described by {\scite{gott:quantitative}}. Essentially, its use forces
exact Gaussian behavior of the volume $v_0$, by the implicit
connection
\begin{equation}
v_0(\delta)=\frac{1}{2}-\frac{1}{2}\Phi\left(\frac{\nu}{\sqrt{2}}\right).
\end{equation}
Thus the deviations from Gaussianity that are due to changes in the
one--point probability distribution function are removed, and
deviations due to higher--order correlations are emphasized.
Obviously, the global functionals clearly discriminate between the
various models. However, in order to make this statement more
quantitative, let us take a closer look at the individual coherent
objects composing the isodensity contours.
\subsection{The largest objects}
At intermediate thresholds, the excursion sets consist of numerous
isolated objects. We identify them by grouping adjacent occupied grid
cells into one object, where adjacent means that the cells have a
common face. Since the Minkowski functionals of the global field are
calculated by integrating over quantities that can be assigned to
individual grid cells, the partial Minkowski functionals of each
object can be obtained at no extra cost once the cells belonging to
each object have been identified.
Several plots in Figures~\ref{fig:clusters.abcd},
{\ref{fig:clusters.ijkl}}, {\ref{fig:clusters.uvwx}}, and
{\ref{fig:clusters.qrst}} illustrate the behavior of the Minkowski
functionals of these objects.
Obviously, the contribution of smaller objects to the volume is almost
negligible compared to the largest one. Note that in all figures, the
mean and standard deviation over all four realizations are shown
instead of the individual curves. It is worth noting that the
variance is largest in the $n=-2$ and $-1$ models, which are dominated
by structures on large scales and hence show the strongest sample
variance.
The models with various initial spectral indices $n$ show
qualitatively a similar behavior. At small filling factors (high
density thresholds) the two largest clusters give negligible
contribution to each of the global characteristics. Then, at
percolation transition the largest cluster quickly becomes the
dominant structure in terms of volume, area, and integrated mean
curvature. The second largest cluster also grows at the percolation
threshold but just a little and then quickly diminishes. The
percolation transition is clearly marked in all three characteristics
of the largest cluster by their sudden growth. However, this
transition does not happen at a well--defined threshold. Instead,
clusters gradually merge into the largest objects as the threshold is
decreased (the filling factor grows). This continuous transition has
also been observed using percolation analysis, i.e.\ the zeroth
Minkowski functional alone
{\cite{shandarin:topology,shandarin:detection,klypin:percolation,sahni:probing}}
and is explained by the finite size of the sample.
Nevertheless, the percolation transitions happen within fairly narrow
ranges of the filling factor that are clearly distinct for different
models in question: The filling factors are approximately
$0.03\pm0.01,0.07\pm0.015,0.11\pm 0.015,0.14\pm0.015$ in the
$n=-2,-1,0,+1$ models, respectively. It is remarkable that all
percolation transitions occur at smaller filling factors than in
Gaussian fields (about 0.16) indicating that even in the most
hierarchical model ($n=+1$) the structures tend to be more connected
than in the ``structureless'' Gaussian field.
{\scite{pauls:hierarchical}} showed positive correlation with networks
based on the same phases all the way to $n=+3$. This confirms the
conclusion of {\scite{yess:universality}}: The universality of the
network structures results from the evolution of Gaussian initial
conditions through gravitational instability. The Euler
characteristic of the largest cluster also marks the percolation
threshold but in a different manner: before percolation it is zero and
after percolation it becomes negative in every model, however, in the
$n=0$ and $n=+1$ models it grows to a small positive peak before
becoming negative. All global functionals have no particular features
at the percolation threshold.
\subsection{Small objects}
As an example, the Blaschke diagram for the model $n=-2$ is shown in
Figure~\ref{fig:blaschke.abcd}. The distributions for the other models
look qualitatively very similar and the average quantities for other
models are presented in Figures~\ref{fig:mass.x} and
{\ref{fig:mass.y}}. Figure~\ref{fig:mass.x} shows that most of the
small objects are either spherical or slightly planar (two largest
dots in Figure~\ref{fig:blaschke.abcd}). There is also a considerable
number of elongated clusters with filamentarities from 0.1 to 0.5. In
some cases filamentarity reaches large values $\sim1$. On the
contrary planarity is much weaker, it hardly reaches the value of 0.2
(which is partly a consequence of the smoothing). There is a hint of
a small correlation between filamentarity and planarity: the objects
with the largest filamentarity also tend to have larger planarity.
Figures~\ref{fig:mass.x} and {\ref{fig:mass.y}} display the
shapefinders as functions of the cluster mass. The curves give
averages over the realizations of each model. Apparently, the signal
for filamentarity is much stronger than for planarity, regardless of
the model which is in full agreement with
Figure~\ref{fig:blaschke.abcd}. The planarity and filamentarity
distributions qualitatively look very similar except the amplitude.
Small objects ($5\times10^{-6}\lesssim{m}\lesssim5\times10^{-4}$)
display stronger planarity and filamentarity for models with more
power on large scales. However, for greater masses
($m\gtrsim5\times10^{-4}$) the situation is reversed: the less
large-scale power the greater the filamentarity and planarity. If the
former seems to be natural and was expected, the latter has been
unexpected. Both the planarity and filamentarity monotonically grow
and reach their maxima at largest clusters: ${\mathcal{P}}_m\approx0.1$ and
${\mathcal{P}}_m\approx0.5$ in all models. As expected the largest objects
possess the largest planarities and filamentarities, but the
independence of the maxima from the model again was unexpected.
Figures~\ref{fig:blaschke.x} and {\ref{fig:blaschke.y}} show the
histograms for the shapefinders ${\mathcal{P}}$ and ${\mathcal{F}}$, respectively. They
clearly show the large difference in the total number of structures:
the more power on small scales the greater the abundance of clusters.
These figures are in general agreement with the
Figures~\ref{fig:mass.x} and {\ref{fig:mass.y}}.
\section{Conclusion and Outlook}
Global Minkowski functionals do discriminate
(Figures~\ref{fig:epoch.f} and {\ref{fig:epoch.f.rescaled}}). Only
the $n=-2$ model shows slight drawbacks as far as robustness is
concerned, but that is due to the method's sensitivity to large--scale
features of the smoothed density field. The total area, mean
curvature, and Euler characteristic are sensitive to abundances of the
structures and are easy to interpret: the more power on small scales
(greater $n$) the more abundant structures and therefore the greater
the amplitude of the curve.
Even more valuable information is obtained from looking at the
Minkowski functionals of the largest coherent object at each threshold
(Figures~\ref{fig:clusters.abcd}, {\ref{fig:clusters.ijkl}},
{\ref{fig:clusters.uvwx}}, and {\ref{fig:clusters.qrst}}). All four
Minkowski functionals of the largest cluster clearly consistently
detect the percolation transition. Two points are worth stressing: 1)
in all models the percolation transition happens at smaller filling
factors than in the ``structureless'' Gaussian fields and 2) the more
power on the large scales (i.e. the smaller $n$) the smaller the
filling factor at percolation. Both conclusions confirm the results of
{\scite{yess:universality}} about the universality of the network
structures in the power law models with $n\le1$. The results of
{\scite{pauls:hierarchical}} present evidence that this should be
expected all the way up to $n=+3$; at $n=+4$ mode coupling effects
from smaller scales should begin to fully disrupt the network
structure.
Small objects, on the other hand, give different results. Their
abundance discriminates well, but is already determined by the
difference in the Euler characteristic as well as by the total area
and mean curvature of the whole contour (Figures~\ref{fig:epoch.f} and
{\ref{fig:epoch.f.rescaled}}) that are also sensitive to the abundance
of structures. The morphology of small objects as measured by
shapefinders shows little differences between models so far
(Figures~\ref{fig:mass.x}, {\ref{fig:mass.y}}, {\ref{fig:blaschke.x}}
and {\ref{fig:blaschke.y}}). Both the maximum average planarity
(${\mathcal{P}}\approx0.1$) and the maximum average filamentarity
(${\mathcal{F}}\approx0.5$) are reached in the most massive non-percolating
objects. None of the model showed ribbon--like objects characterized
by both large planarity and large filamentarity. We may speculate that
the smaller objects are ones which formed earlier, are more nonlinear,
and therefore more decoupled from initial conditions.
However, all models used Gaussian initial conditions, and evolve under
the influence of gravity. Hence the similar morphology of the clumps
may point towards universal behavior. Note that things such as string
wakes might produce totally different results.
The grouping and measurement techniques used in this study may be less
accurate for small objects than for large clusters. It is worth
trying to study the morphology of small objects by applying more
accurate methods of measuring partial Minkowski functionals such as a
Boolean grain model (\pcite{schmalzing:diplom}, and a follow--up to
this article) or the interpolation method of
{\scite{novikov:minkowski}} generalized to three dimensions.
\section*{Acknowledgments}
JS wishes to thank Martin Kerscher for interesting discussions and
valuable comments. This work is part of a project in the
``Sonderforschungsbereich 375--95 f\"ur Astroteilchenphysik'' of the
Deutsche Forschungsgemeinschaft. ALM and SFS acknowledge support of
the NSF--EPSCoR program and the GRF program at the University of
Kansas. SFS acknowledges support from TAC in Copenhagen.
| -12,182.212073 |
[
-3.1875,
2.93359375
] | 23.798627 |
[
-2.69140625,
0.41015625,
-2.255859375,
-5.96484375,
-0.5546875,
7.90625
] |
[
3.111328125,
6.8828125,
1.8349609375,
4.96875
] | 190 | 2,974 |
[
-3.42578125,
3.923828125
] | 26.678386 |
[
-5.9140625,
-4.140625,
-4.1328125,
-2.3203125,
1.7919921875,
11.7578125
] | 1.215582 | 14.772848 | 32.985878 | 1.312528 |
[
2.8551292419433594
] | -9,318.588133 | 5.947209 | -11,908.358228 | 0.870696 | 5.848413 |
[
-2.400390625,
-3.568359375,
-3.646484375,
-4.38671875,
2.521484375,
11.484375
] |
[
-5.2578125,
-1.771484375,
-2.533203125,
-0.83740234375,
3.55859375,
4.16796875
] | |
BkiUfxXxK1ThhCdy-5v3
|
\section{Introduction and results}
In this paper we consider the following question which was presented by D. McDuff and L. Polterovich (question 4 in ~\cite{McD:Prblm}).
Given a closed symplectic manifold $(M, \omega)$, is there an upper bound for Hofer's norm of a commutator
of two Hamiltonian diffeomorphisms of $M$? Clearly, a similar question for $\RR^{2n}$ and the standard symplectic
form has negative answer: it is sufficient to pick two Hamiltonians $f, g$ with nontrivial commutator $[f, g]$.
Then the norm of the rescaled commutator $\left\|[s f(\frac{\cdot}{s}), s g(\frac{\cdot}{s})] \right\| \to \infty$ as $s$ goes to infinity. However, this rescaling trick cannot be applied
to closed manifolds and the question becomes not obvious. In this article we construct commutators with arbitrarily large norm in a class of
closed manifolds which contains surfaces of positive genus and direct products of such surfaces:
\begin{thm} \label{T:main}
Let $(\Sigma, \omega)$ be a closed surface of positive genus and $(N, \omega')$ be a torus or a closed symplectic manifold which admits a pair of transverse
closed Lagrangians whose union is weakly exact.
Then the product manifold $(\Sigma \times N, \omega \oplus \omega')$ admits Hamiltonian commutators with arbitrarily large Hofer norm.
\end{thm}
\medskip
Main tool used in the argument is the theorem below which appeared in ~\cite{Usher:subm-Hnorm}.
Given two Lagrangians $L, L'$ we define the \emph{separation energy}
\[
E_{sep} (L, L') = \inf_{g L \cap L' = \emptyset} \| g \|
\]
to be the least Hofer energy needed to move $L$ away from $L'$ by a compactly supported Hamiltonian.
In the case no such deformation exists we set $E_{sep} (L, L') = \infty$. This notion is a natural generalization
of the displacement energy of a Lagrangian.
\begin{thm} \label{T:bound}\emph{(M. Usher)}
Let $(M, \omega)$ be a geometrically bounded symplectic manifold, $L, L' \subset M$ be two compact Lagrangians that
intersect transversely. Pick an intersection point $p \in L \cap L'$ and
a tame almost complex structure $J$. Then the separation energy satisfies
\[
E_{sep} (L, L') \geq \min (\sigma_M, \sigma_L, \sigma_{L'}, \sigma_p)
\]
where $\sigma_M$ denotes the minimal energy of a nonconstant $J$-holomorphic sphere in $M$, $\sigma_L, \sigma_{L'}$ stand for the minimal
energy of nonconstant $J$-holomorphic discs with boundary on $L, L'$, respectively. $\sigma_p$ denotes the minimal energy
of a nonconstant $J$-holomorphic strip
\[
u: (\RR \times [0, 1], i) \to (M, J),
\]
with boundary conditions $u (s, 0) \in L$, $u (s, 1) \in L'$ for all $s \in \RR$ and which converges to $p$ either as $s \to +\infty$ or as $s \to -\infty$.
Moreover, in the case $J$ is regular we may consider for $\sigma_p$ only those strips that are isolated in the corresponding nonparametrized moduli space.
\end{thm}
Theorem 4.9 in ~\cite{Usher:subm-Hnorm} claims $E_{sep} (L, L') > 0$ which is weaker than the statement above.
Nevertheless, the argument taken verbatim implies Theorem~\ref{T:bound} except for the last sentence regarding regular
almost complex structures and isolated holomorphic strips. This additional property can be established by restricting attention in the proof
to holomorphic strips of index zero.
This theorem can be seen as an adaptation of Chekanov's bound ~\cite{Chekanov:Lag-energy-1} for the displacement energy:
$E_{disp} (L) \geq \min (\sigma_M, \sigma_L)$. As in Chekanov's argument, here is no requirement of existence of
Floer homology.
Theorem~\ref{T:bound} gives a tool to compute lower bounds for Hofer's norm. Given a Hamiltonian $g$, one may consider
two disjoint Lagrangians $L, L'$ such that $g L$ intersects $L'$ transversely. Then
$\| g \| \geq E_{sep} (g L, L') \geq \min (\sigma_M, \sigma_{g L}, \sigma_{L'}, \sigma_p)$ for all $p \in gL \cap L'$.
In certain cases (like the case when $M$ is a surface) the righthandside expression can be easily computed.
\medskip
The rest of this paper is organized as follows. Section~\ref{S:def} provides basic definitions for Hofer's geometry and Floer theory
and in Section~\ref{S:con} we construct commutators for Theorem~\ref{T:main}.
\medskip
\emph{Acknowledgements:}
The author is grateful to L. Polterovich and M. Usher for useful discussions and comments.
He also thanks the referee for his/her careful work.
\section{Definitions}\label{S:def}
A symplectic manifold $(M, \omega)$ is called \emph{geometrically bounded} (see ~\cite{ALP}) if there exist a complete Riemannian metric $g$ and
an almost complex structure $J$ such that $(M, g)$ has bounded sectional curvature and injectivity radius bounded away from zero.
In addition we ask that $\omega (v, J v) > c_1 g(v, v)$ and $|\omega (v, w)|^2 < c_2 g(v,v)g(w,w)$ for some constants $c_1, c_2 > 0$.
$J$ is called a \emph{tame} almost complex structure. Note that for closed $M$ it is enough to assume that $\omega (v, Jv) > 0$.
Let $\Sigma$ be a Riemannian surface (possibly with boundary), $u : \Sigma \to M$ a $J$-holomorphic map.
The \emph{energy} or \emph{symplectic area} of $u$
is defined by $E(u) = \int_{\Sigma} u^* \omega$.
Given two Lagrangian submanifolds $L_1, L_2 \subset M$ that intersect transversely and a tame almost complex structure $J$ we consider the
set of nonconstant $J$-holomorphic strips
$u:\RR \times [0, 1] \to M$ that satisfy $u(s, 0) \in L_1$, $u (s, 1) \in L_2$ for all $s \in \RR$. Every strip $u$ with $E(u) < \infty$ converges to a pair of
intersection points $p, q \in L_1 \cap L_2$ as $s \to \pm \infty$. We denote by $\mathcal{M}^* (p, J)$ the set of all
nonparametrized finite-energy $J$-holomorphic strips as above which converge to a given point $p \in L_1 \cap L_2$ either as $t \to +\infty$ or as $t \to -\infty$. The general theory
(see ~\cite{Fl:Morse-theory}, ~\cite{McD-Sa:Jhol-2}) states that for \emph{regular} almost complex structures $J$, $\mathcal{M}^* (p, J)$ admits a manifold structure.
The set of regular structures $J$ is generic in the space of all tame almost complex structures on $M$.
Assume that $(\Sigma, \omega)$ is a closed symplectic surface. Then every tame complex structure $J$ on $\Sigma$ is regular (see Theorem 12.2 in ~\cite{Si-Ro-Sa:CombiFloer}). Let $L_1, L_2$ be two
simple closed connected curves in $\Sigma$ intersecting transversely. Denote $D_+ = \{ z \in \CC \, \big| \, |z| \leq 1 \, , \, \text{Im} (z) \geq 0\}$.
Following ~\cite{Si-Ro-Sa:CombiFloer} we define a \emph{smooth lune} to be an orientation preserving immersion $u : D_+ \to \Sigma$ such that
$u (D_+ \cap \RR) \subset L_1$, $u (D_+ \cap S^1) \subset L_2$. $u(1), u(-1) \in L_1 \cap L_2$ are the endpoints of $u$.
We pick $p \in L_1 \cap L_2$. By Theorem 12.1, ~\cite{Si-Ro-Sa:CombiFloer} there is a bijection between isolated nonparametrized holomorphic strips in $\mathcal{M}^* (p, J)$
and equivalence classes (under reparametrization) of smooth lunes with endpoint at $p$. Moreover, holomorphic strips and corresponding lunes share the same image in $\Sigma$.
In particular, they have the same energy which is equal to the area covered by the image (counted with multiplicity).
In this situation we may replace the definition of $\sigma_p$ in Theorem~\ref{T:bound} by the minimal area of a smooth lune with endpoint at $p$.
\medskip
We call a closed symplectic manifold $(N, \omega_N)$ \emph{admissible} if it satisfies the following.
\begin{itemize}
\item
There exist two Lagrangian submanifolds
$K_1, K_2$ that intersect transversely, an intersection point $p \in K_1 \cap K_2$ and a regular tame
almost complex structure $J_N$ such that $\mathcal{M}^* (p, J_N)$ has no isolated strips.
\item
All $J_N$-holomorphic spheres and disks with boundary either in $K_1$ or in $K_2$ are constant. (This condition holds automatically
when $\omega_N$ vanishes on $\pi_2 (N, K_1)$ and on $\pi_2 (N, K_2)$.)
\end{itemize}
Suppose that a closed symplectic manifold $(N, \omega)$ admits a pair of transverse Lagrangians whose union is weakly exact
(namely, $\omega$ vanishes on $\pi_2 (N, K_1 \cup K_2)$).
It is admissible by the definition above. A closed surface of positive genus is also an example of such manifold:
we pick $K_1, K_2$ to be a pair of simple closed curves that intersect transversely at single point $p$. Let $J_N$ be an arbitrary complex structure.
Let $u$ be a $J_N$-holomorphic strip with bounded energy and boundary on $K_1 \cup K_2$. Its lift $\tilde{u}$ to the universal cover
maps boundary of the strip $\RR \times [0, 1]$ to lifts $\widetilde{K}_1 \cup \widetilde{K}_2$.
The maximum principle implies that boundary of the image of $\tilde{u}$ is contained in $\widetilde{K}_1 \cup \widetilde{K}_2$, hence
$\im(\tilde{u}) \subset \widetilde{K}_1 \cup \widetilde{K}_2$. By the open mapping theorem $\tilde{u}$ is
a constant map, hence $u \equiv p$. All holomorphic discs and spheres are constant since $\pi_2 (N, K_1) = \pi_2 (N, K_2) = \pi_2 (N) = \{0\}$.
\begin{lem} \label{lem:prod}
Admissible manifolds are closed under direct product.
\end{lem}
\begin{proof}
Let $N_1, N_2$ be admissible manifolds, $K_{1, N_i}, K_{2, N_i}$ the corresponding Lagrangians.
Consider the products $K_{j, N_1} \times K_{j, N_2}$, $j=1,2$ of the respective Lagrangians and the product almost complex structure.
The regularity of the almost complex structure is achieved by surjectivity of the linearized $\bar{\partial}$ operator (see ~\cite{McD-Sa:Jhol-2}).
A simple computation shows that surjectivity for the product structure follows from that for $J_{N_1}$ and $J_{N_2}$.
The product almost complex structure is tame with respect to the product symplectic form.
The projections $N_1 \times N_2 \to N_i$, $i = 1, 2$ are $J$-holomorphic,
therefore every $(J_{N_1}, J_{N_2})$-holomorphic strip in $N_1 \times N_2$ projects to a pair of $J$-holomorphic strips in $N_1$, $N_2$. Conversely,
given a pair of such strips one may lift them to a strip in $N_1 \times N_2$.
This implies that isolated strips in $\mathcal{M}^* \left((p_1, p_2), (J_{N_1}, J_{N_2}); N_1 \times N_2 \right)$ must
project to a pair of isolated strips in $\mathcal{M}^* (p_i, J_{N_i}; N_i) \cup \{ \text{const} \}$ which are necessarily constant ones.
Similarly, $J$-holomorphic spheres and disks project to those in $N_1$ and $N_2$ hence are constant ones.
The lemma follows.
\end{proof}
By this lemma products of positive genus surfaces are admissible.
In the next section we construct commutators in direct products $M \times N$ where $M$ is a closed
symplectic surface of positive genus and $N$ is admissible. This proves Theorem~\ref{T:main}.
\medskip
Let $(M, \omega)$ be a symplectic manifold, $g$ be a Hamiltonian diffeomorphism with compact support in $M$.
The Hofer norm $\|g\|$ (cf. ~\cite{Hof:TopProp}) is defined by
\[
\|g\| = \inf \left\{ \int_0^1 \max G(\cdot, t) - \min G(\cdot, t) \dd t \right\}
\]
where the infimum goes over all compactly supported Hamiltonian functions $G : M \times [0,1] \to \RR$
such that $g$ is the time-1 map of the corresponding flow.
\section{Construction}\label{S:con}
First we construct commutators with large Hofer norm on a torus $T^2$.
We use the following convention: $S^1 = \RR / \ZZ$, $T^2 = S^1 \times S^1 = \RR^2 / \ZZ^2$ equipped with $x, y$
coordinates. Sometimes we will consider $x+iy$ as a single complex coordinate.
$T^2$ is equipped with the standard symplectic form $\omega = \dd x \wedge \dd y$ so it has area $1$.
Let $\eta': S^1 \to \RR$ be the piecewise linear function given by
\[
\eta'(s) = \left\{ \begin{array}{ll}
4s & \textrm{if $s \in [0, \frac{1}{4}]$}\\
1 & \textrm{if $s \in [\frac{1}{4}, \frac{1}{2}]$}\\
3-4s & \textrm{if $s \in [\frac{1}{2}, \frac{3}{4}]$}\\
0 & \textrm{if $s \in [\frac{3}{4}, 1]$} \end{array}\right.
\]
and $\eta : S^1 \to \RR$ be a $C^0$-close smooth approximation of $\eta'$ given by rounding the four corners in
their $\varepsilon$-small neighborhoods (see Figure~\ref{F:setup}).
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=0.8\textwidth]{setup}
\caption{}
\label{F:setup}
\end{center}
\end{figure}
We define two Hamiltonian functions $F, G : T^2 \to \RR$ by $F(x, y) = \eta (x)$ and $G (x, y) = \eta (y)$ and denote
by $f_t, g_t$ the time-$t$ maps of the corresponding autonomous flows. Geometrically, $f_t$ rotates upwards all points
in the annulus $-\varepsilon<x<1/4+\varepsilon$ (with the maximal rotation length equal to $4t$), rotates downwards
the points of the annulus $1/2-\varepsilon<x<3/4+\varepsilon$ and leaves the rest of the torus in place. $g_t$ performs
the same deformation in the horizontal direction.
\begin{prop}\label{p:torus}
The Hofer norm of the commutator $[f_t, g_s] = f_{-t} g_{-s} f_t g_s$ satisfies
\[
\min(t, s)-1 \leq \left\| [f_t, g_s] \right\| \leq 2 \min (t, s).
\]
\end{prop}
Clearly, the proposition implies the desired result for $T^2$ by letting $s, t \to \infty$.
\begin{proof}
The upper bound follows from a standard computation:
\[
\|f_{-t}\| = \|f_t\| \leq \int_0^t \max (F) - \min(F) \dd t = t.
\]
By conjugation invariance of Hofer's norm we have $\|g_{-s} f_t g_s\| = \|f_t\| \leq t$ as well. Therefore
$\|[f_t, g_s]\| \leq \|f_{-t}\| + \|g_{-s} f_t g_s\| \leq 2 t$ by the triangle inequality.
A similar computation shows $\|[f_t, g_s]\| \leq 2 s$.
\medskip
We prove the lower bound.
Note that $\|[f_t, g_s]\|$ depends continuously on $s$ and $t$, hence it is enough to show the
statement for a dense subset of values.
\medskip
\underline{Step I:}
We introduce two Lagrangians meridians
\[
L = \{1-2\varepsilon\} \times S^1, \quad L' = \{1-\varepsilon\} \times S^1.
\]
($\varepsilon$ is the same as in construction of $\eta$.)
$L \cap L' = \emptyset$, hence
\[
\|[f_t, g_s]\| \geq E_{sep} (L, [f_t, g_s] L')
\]
and it is enough to show
that
\[
E_{sep} (L, [f_t, g_s] L') \geq \min(t, s)-1.
\]
Bi-invariance of Hofer's norm implies
\begin{eqnarray} \label{Eq:esep}
E_{sep} (L, [f_t, g_s] L') &=& E_{sep} (L, f_{-t} g_{-s} f_t g_s (L')) = E_{sep} (g_s f_t (L), f_t g_s (L')) = \\
&=& E_{sep} (g_s (L), f_t g_s (L')). \nonumber
\end{eqnarray}
The last equality follows from the fact that $f_t$ leaves $L$ invariant.
We use Theorem~\ref{T:bound} to get a lower bound for the righthandside expression in (\ref{Eq:esep}).
Equip $T^2$ with multiplication by $i$. As
\[
\pi_2 (T^2) = \pi_2 (T^2, g_s (L)) = \pi_2 (T^2, f_t g_s (L')) = \{0\},
\]
all holomorphic spheres and holomorphic disks with boundary on Lagrangians have zero energy hence are constant ones. Therefore
\[
\sigma_{T^2} = \sigma_{g_s (L)} = \sigma_{f_t g_s (L')} = \infty.
\]
It is enough to pick an intersection point $p$ so that $\sigma_p \geq \min(t, s)-1$.
\medskip
\underline{Step II:}
Put $L_s := g_s (L)$, $L_{ts}' := f_t g_s (L')$.
Consider the universal cover $\pi : \RR^2 \to T^2$. Let
\[
\widetilde{L} = \{1-2\varepsilon\} \times \RR, \quad \widetilde{L}' = \{1-\varepsilon\} \times \RR
\]
be lifts of $L$ and $L'$, denote by $\tilde{f}_t$, $\tilde{g}_s$ the lifts of $f_t$, $g_s$ to $\RR^2$.
$\widetilde{L}_s := \tilde{g}_s (\widetilde{L})$ depicted on Figure~\ref{F:lift} is a lift of $L_s$. It looks like an infinite two-sided comb whose `teeth'
have area $s$ each. $\widetilde{L}'_{ts} := \tilde{f}_t \tilde{g}_s (\widetilde{L}')$ is obtained from a similar `comb'
$\tilde{g}_s (\widetilde{L}')$ whose
`teeth' are vertically deformed by $\tilde{f}_t$ in a periodic way. The area bounded between each oscillation of $\widetilde{L}'_{ts}$ and a `tooth'
of $\widetilde{L}_s$ is in the interval $[t-1, t+1]$.
The righthand side of Figure~\ref{F:lift} gives a rough description of the half-plane to the left of $\widetilde{L}_{ts}'$. That is,
$\widetilde{L}_{ts}'$ is the boundary of the shaded region. Figure~\ref{F:def} gives a
more detailed description of the intersection pattern of two arcs of $\widetilde{L}_s$ and
$\widetilde{L}_{st}'$ (fat lines). In order to simplify the picture some of the remaining parts of
curves are hidden, the rest are sketched either by thin or by dotted lines.
\begin{figure}[!tbp]
\begin{center}
\includegraphics[width=\textwidth, trim=0 150 100 250]{lift}
\caption{}
\label{F:lift}
\end{center}
\end{figure}
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=0.7\textwidth]{deformed_meridians}
\caption{}
\label{F:def}
\end{center}
\end{figure}
We may assume that $g_s L \pitchfork f_t g_s (L')$ as this property holds for generic $t$. Pick an intersection point
$\tilde{p} \in \widetilde{L}_s \cap \widetilde{L}_{ts}'$ in the neighborhood
of $\{0\}\times \RR$ as described in Figure~\ref{F:def}. Put $p := \pi (\tilde{p}) \in L_s \cap L_{ts}'$.
\medskip
\underline{Step III:}
We show that $\sigma_p \geq \min(t, s)-1$. Following the discussion in Section~\ref{S:def}, we may compute $\sigma_p$
by considering the minimal energy of a smooth lune with endpoint at $p$ instead of the energy of holomorphic strips.
Note that every such lune lifts to a smooth lune in $\RR^2$ with endpoint at $\tilde{p}$ and appropriate boundary conditions in $\widetilde{L}_s$ and
$\widetilde{L}_{ts}'$. The lift preserves the energy (area) of a lune, so we proceed with computations on the universal cover instead of $T^2$.
In Remark 6.11, ~\cite{Si-Ro-Sa:CombiFloer}
the authors propose the following algorithm to locate the lunes.
Given an intersection point $\tilde{q} \in \widetilde{L}_s \cap \widetilde{L}_{ts}'$, consider the
two arcs $\gamma \subset \widetilde{L}_s$, $\gamma' \subset \widetilde{L}_{ts}'$ connecting $\tilde{p}$ with $\tilde{q}$. $\gamma \cup \gamma'$ bounds
a smooth lune $u : D_+ \to \RR^2$, $u(\RR \cap D_+) \subseteq \gamma$, $u(S^1 \cap D_+) \subseteq \gamma'$ with $u(-1) = \tilde{p}, u(1) = \tilde{q}$ if and only if the following conditions hold:
\begin{enumerate}
\item
Orient $\gamma, \gamma'$ from $\tilde{p}$ to $\tilde{q}$. $\tilde{p}, \tilde{q}$ must have opposite intersection indices.
\item
$\gamma$ must be homotopic to $\gamma'$ relative endpoints.
\item
The winding number $w:\RR^2 \setminus (\gamma \cup \gamma') \to \ZZ$ of the closed curve $\gamma * -\gamma'$ must be non-negative
and satisfy $w(z) \in \{0, 1\}$ near $\tilde{p}$ and $\tilde{q}$.
\end{enumerate}
Moreover, if $\gamma, \gamma'$ satisfy the conditions above, such a lune is unique up to reparametrization.
We may find lunes with $u(-1) = \tilde{q}, u(1) = \tilde{p}$ in a similar way by interchanging the roles of $\tilde{p}$ and $\tilde{q}$.
Figure~\ref{F:lunes} describes six lunes with endpoint at $\tilde{p}$ (the proportions are not precise). The lunes (c-f) cover roughly
either a straight `tooth' or a vertically deformed one and have energy
$E(u) \geq s-1$ while the lunes (a) and (b) satisfy $E(u) \geq t-1$. Therefore $E(u) \geq \min(t, s)-1$ holds for all six.
\begin{figure}[!htbp]
\begin{center}
\includegraphics[width=0.6\textwidth]{lunes}
\caption{}
\label{F:lunes}
\end{center}
\end{figure}
\medskip
\underline{Step IV:}
We show that there are no other lunes. This implies $\sigma_p \geq \min(t, s)-1$ and finishes the proof of the proposition.
Note that in our setup the second condition of the algorithm is always true and the first one holds for many candidate points $\tilde{q}$.
The only significant constraint is imposed by the third condition.
Let $\tilde{q} \in \widetilde{L}_s \cap \widetilde{L}_{ts}'$ be a candidate for the second endpoint of a lune,
$\gamma \subset \widetilde{L}_s$, $\gamma' \subset \widetilde{L}_{ts}'$ be the arcs connecting $\tilde{p}$ with $\tilde{q}$.
Observation I: let $j$ be one of (a-f), denote by $u_j$ the corresponding lune. Put $\tilde{q}_j$, $\gamma_j \subset \widetilde{L}_s$,
$\gamma'_j \subset \widetilde{L}_{ts}'$ to be the second endpoint and boundary arcs of $u_j$.
If $\gamma \supsetneq \gamma_j$ and $\gamma' \supsetneq \gamma'_j$ then $\tilde{q}_j \in \gamma \cap \gamma'$ and the winding number
$w: \RR^2 \setminus (\gamma \cup \gamma') \to \ZZ$ of $\gamma * -\gamma'$ near $\tilde{q}_j$ will attain both positive and negative values. This
is a contradiction to the third condition for existence of a lune in the list above.
Observation II: note that $\gamma$ traverses $\widetilde{L}_s$ which goes from $\tilde{p}$ either to the left or to the right to the
approximate distance $4s$ and then turns back. Similarly, $\gamma' \subset \widetilde{L}_{ts}'$ and $\widetilde{L}_{ts}'$ goes either up or down to the distance $4t$ and back.
We call $\gamma$ `short' if $\gamma \subsetneq \gamma_c$ or $\gamma \subsetneq \gamma_d$, otherwise it is `long'.
Similarly, $\gamma'$ is `short' if $\gamma' \subsetneq \gamma'_a$ or $\gamma' \subsetneq \gamma'_b$ and `long' otherwise.
If the curve $\gamma$ is short then $\gamma'$ must be long to arrive to the same endpoint $\tilde{q}$.
And vice versa: a short $\gamma'$ implies that $\gamma$ is long. Therefore at least one of
$\gamma, \gamma'$ is long. That is, either $\gamma$ traverses a full `tooth' or $\gamma'$ goes along a full oscillation.
We assume by contradiction that $\tilde{q}$ is an endpoint which is different from the six endpoints in (a-f). There are eight possible cases:
\begin{itemize}
\item
$\gamma$ is long and goes to the right from $\tilde{p}$ while $\gamma'$ goes down from $\tilde{p}$.
Then $\gamma \supsetneq \gamma_c$ hence by observation I, $\gamma' \subset \gamma'_c$.
But $\gamma'_c$ contains no intersection points with $\widetilde{L}_s$ other than $\tilde{p}$ and $\tilde{q}_c$, a contradiction.
\item
$\gamma$ is long and goes to the right, $\gamma'$ goes up.
Note that all points on $\gamma$ have their $y$ coordinate below that of $\tilde{p}$.
Hence $\gamma'$ must traverse at least one full oscillation to arrive to a point $\tilde{q}$ which is below $\tilde{p}$.
Moreover, $\tilde{q}$ is different from $\tilde{q}_a$. But then $\gamma' \supsetneq \gamma'_a$, $\gamma \supsetneq \gamma_a$, a contradiction to observation I.
\item
$\gamma$ is long and goes to the left, $\gamma'$ goes up.
Then $\gamma \supsetneq \gamma_d$ hence by observation I, $\gamma' \subset \gamma'_d$.
But $\gamma'_d$ contains no intersection points with $\widetilde{L}_s$ other than $\tilde{p}$ and $\tilde{q}_d$, a contradiction.
\item
$\gamma$ is long and goes to the left, $\gamma'$ goes down.
Then all points on $\gamma$ have their $y$ coordinate above that of $\tilde{p}$.
Hence $\gamma'$ must traverse at least one full oscillation to arrive to $\tilde{q}$ which is above $\tilde{p}$. Moreover,
$\tilde{q}$ is different from $\tilde{q}_b$.
Then $\gamma \supsetneq \gamma_b$, $\gamma' \supsetneq \gamma'_b$, a contradiction.
\item
$\gamma'$ is long and goes up from $\tilde{p}$, $\gamma$ goes right.
Then $\gamma' \supsetneq \gamma_a$ hence by observation I, $\gamma \subset \gamma_a$.
Therefore $\tilde{q} \in \gamma_a$. However, on the way up from $\tilde{p}$, $\widetilde{L}_{ts}'$ intersects $\gamma_a$ only at
the point $\tilde{q}_a$.
\item
$\gamma'$ is long and goes up, $\gamma$ goes left.
Then $\gamma \supsetneq \gamma_f$ as $\gamma_f$ has no interior intersection points with $\widetilde{L}_{ts}'$.
We may assume that $\gamma$ is short (the case of a long $\gamma$ was already considered above), namely, $\tilde{q} \in \gamma_d$.
$\gamma'_f$ intersects $\gamma_d$ at three points other than $\tilde{p}$, two of them are $\tilde{q}_d, \tilde{q}_f$ and the third has the same intersection index as
$\tilde{p}$, hence cannot be an endpoint of a lune (contradiction to condition (1) of the algorithm).
Therefore $\gamma'$ must traverse $\gamma'_f$, in contradiction to observation I.
\item
$\gamma'$ is long and goes down, $\gamma$ goes right.
Then $\gamma \supsetneq \gamma_e$ as $\gamma_e$ has no interior intersection points with $\widetilde{L}_{ts}'$.
We may assume that $\gamma$ is short, that is, $\tilde{q} \in \gamma_c$. $\gamma'_e$ intersects $\gamma_c$ at four points:
$\tilde{p}, \tilde{q}_c, \tilde{q}_e$ and the forth has inappropriate intersection index. Therefore $\gamma'$ must traverse
$\gamma'_e$, in contradiction to observation I.
\item
$\gamma'$ is long and goes down, $\gamma$ goes left.
Then $\gamma' \supsetneq \gamma_b'$. By observation I, $\gamma \subset \gamma_b$. However, on the way down from $\tilde{p}$,
$\widetilde{L}_{ts}'$ intersects $\gamma_b$ only at the point $\tilde{q}_b$.
\end{itemize}
\end{proof}
\begin{rem}
$f_1, g_1$ constructed above generate a free group $F_2$ in $\Ham(T^2)$:
$\supp(F) \cup \supp (G)$ is homotopic to the number eight figure. Pick a common periodic point $x \in T^2$ of period one for both
$f_t, g_t$. Then the trajectory of $x$ under the action of $<f_1, g_1>$ gives an isomorphism
$<f_1, g_1> \simeq \pi_1 (\supp(F) \cup \supp (G), x)$.
Denote by $S$ the generating set consisting of
$f_1, g_1, f_{-1}, g_{-1}$ and their conjugates in $F_2$. D. Calegari observed
that the word metric with respect to $S$ satisfies all estimates known to the author of the restriction of Hofer's metric to $F_2$.
It is easy to show that $\|\cdot\|_S \geq \| \cdot \|_H$. It would be interesting to know if these two metrics are comparable.
\end{rem}
\medskip
We adapt Proposition~\ref{p:torus} to handle closed surfaces $(M, \omega)$ of genus $g > 1$. Without loss of generality
we assume that $Area (M) > 1$. Let $F, G : T^2 \to \RR$ be the Hamiltonian functions from Proposition~\ref{p:torus}.
We present $M$ as a connected sum $T^2 \# \Sigma_{g-1}$ of the torus with a surface of genus $g-1$ where
$\Sigma_{g-1}$ is glued to $T^2$ along a small circle which does not intersect the supports of $F, G$ and the curves $L$, $L'$. This allows us to push
$F, G, f_t, g_s$ forward to $M$. We continue to denote objects on $M$ with the same notation. We claim that the same bounds on the Hofer norm hold in $M$.
\begin{prop}\label{p:surf}
\[
\min(t, s)-1 \leq \|[f_t, g_s]\| \leq 2 \min (t, s).
\]
\end{prop}
\begin{proof}
The proof follows the same lines as Proposition~\ref{p:torus}. We indicate only the necessary changes.
Computation of the upper bound is the same.
\underline{Step I:} we push $L$, $L'$, $p$ from $T^2$ forward to $M$ and equip $M$ with a complex structure which restricts to $i$ on $T^2$.
As before, we would like to show $E_{sep} (g_s (L), f_t g_s (L')) \geq \min(t, s) - 1$ using Theorem~\ref{T:bound}.
We note that $\sigma_{M} = \sigma_{g_s (L)} = \sigma_{f_t g_s (L')} = \infty$.
\underline{Step II:} We work with a cover $\pi: \widehat{M} \to M$ which is obtained by gluing to $\RR^2$ infinitely many copies
of $\Sigma_{g-1}$ along $\ZZ^2$-periodic lattice. The lifts $\widetilde{L}_s, \widetilde{L}_{ts}', \tilde{p}$ are pushed
from $\RR^2$ forward to $\widehat{L}_s, \widehat{L}_{ts}', \hat{p}$ in $\widehat{M}$. We get the same picture as in Figure~\ref{F:def} up to copies of $\Sigma_{g-1}$ attached in appropriate places.
\underline{Steps III-IV:}
We show that $\sigma_p \geq \min(t, s)-1$. We consider smooth lunes with endpoint at $p$ and claim that they arise as pushforward
of [some of] the lunes (a-f) described in Figure~\ref{F:lunes}. We observe that pushforward is possible only for those lunes that do not cover the attaching circle
of $\Sigma_{g-1}$. When it is defined, the pushforward preserves the energy, therefore the desired bound for $\sigma_p$ in $M$ follows from that in $T^2$.
Let $u_M : D_+ \to M$ be a lune with endpoint at $p$, $\hat{u}_M$ be its lift to a lune in $\widehat{M}$ with endpoint at $\hat{p}$.
As $\hat{u}_M (\partial D_+) \subset \widehat{L}_s \cup \widehat{L}_{ts}'$, the degree of $\hat{u}_M$ is a locally constant function
in $\widehat{M} \setminus (\widehat{L}_s \cup \widehat{L}_{ts}')$.
We claim that the degree vanishes in all connected components which contain a copy of $\Sigma_{g-1}$.
Indeed, let $\alpha$ be a noncontractible loop in $\Sigma_{g-1}$. If $\hat{u}_M$ has nonzero degree at points of $\alpha$, we lift the picture to the universal cover
and get a lift of $\alpha$ contained inside the lift of $\hat{u}_M$. However, the lift of $\hat{u}_M$ is a lune hence it is bounded while the lift of $\alpha$ is not bounded, a contradiction.
Therefore the degree of $\hat{u}_M$ is zero in connected components containing copies of $\Sigma_{g-1}$.
$\hat{u}_M$ is an orientation preserving immersion by definition of a smooth lune, hence these connected components are outside of the image of
$\hat{u}_M$. This implies that $\hat{u}_M$ can be obtained by pushing forward a lune in $\RR^2$.
\end{proof}
Let $M = \Sigma \times N$ where $\Sigma$ is a closed surface of positive genus and $N$ is an admissible manifold as defined in Section~\ref{S:def}.
We equip $M$ with a product symplectic form $\omega$. Denote by $\pi_\Sigma : M \to \Sigma$, $\pi_N : M \to N$ the natural projections.
The Hamiltonians $f_t, g_s$ defined in Proposition~\ref{p:surf} lift to $\hat{f}_t = \pi_\Sigma^* f_t$, $\hat{g}_s = \pi_\Sigma^* g_s$. They
satisfy the same inequality in Hofer norm:
\begin{prop}\label{p:prod}
\[
\min(t, s)-1 \leq \|[\hat{f}_t, \hat{g}_s]\| \leq 2 \min (t, s).
\]
\end{prop}
\begin{proof}
Computation of the upper bound is the same.
To prove the lower bound we pick Lagrangians $\widehat{L} := L \times K_1$, $\widehat{L}' := L' \times K_2$
where $L, L'$ are the Lagrangians in $\Sigma$ defined in Proposition~\ref{p:surf} and $K_1, K_2 \subset N$ are given by the definition of
an admissible manifold. We pick a product almost complex structure $\widehat{J} := (i, J_N)$ where $i$ is a
complex structure in $\Sigma$ and $J_N$ is as in the definition of an admissible manifold. $\widehat{J}$ is regular by the same argument as
used in Lemma~\ref{lem:prod}.
Note that $\widehat{L}_s := \hat{g}_s \widehat{L} = L_s \times K_1$, ($L_s \subset \Sigma$ is the same as in the proof of Proposition~\ref{p:surf}),
$\widehat{L}_{ts}' := \hat{f}_t \hat{g}_s (\widehat{L}') = L_{ts}' \times K_2$.
Let $\hat{p} := (p_\Sigma, p_N)$ where $p_\Sigma \in L_s \cap L_{ts}'$ is as in Proposition~\ref{p:surf} and
$p_N \in K_1 \cap K_2$ is taken from the definition of $N$. As before, $\|[\hat{f}_t, \hat{g}_s]\| \geq E_{sep} (\widehat{L}_s, \widehat{L}_{ts}')$
and we apply Theorem~\ref{T:bound} to prove that $E_{sep} (\widehat{L}_s, \widehat{L}_{ts}') \geq \min(t, s) - 1$.
$\pi_2 (\Sigma) = \{0\}$ together with the second property of admissible manifolds imply that $\widehat{J}$-holomorphic spheres in $M$
are constant, hence $\sigma_M = \infty$. A similar argument for disks implies $\sigma_{\widehat{L}} = \sigma_{\widehat{L}_{ts}'} = \infty$.
We note that all $\widehat{J}$-holomorphic strips in $M$ are projected by $\pi_\Sigma$ and $\pi_N$ to pseudo-holomorphic strips in $\Sigma$ and in $N$.
Vice versa, given two pseudo-holomorphic strips $u_1$ in $\Sigma$ and $u_2$ in $N$, they lift to $\hat{u} = (u_1, u_2)$ in $M = \Sigma \times N$.
Isolated $\widehat{J}$-holomorphic strips $\hat{u}$ in $M$ with endpoint at $\hat{p}$ are presented by a pair $(u_1, u_2)$ where $u_1$ is an isolated holomorphic strip
in $\Sigma$ with endpoint at $p_\Sigma$ and $u_2$ is an isolated $J_N$ holomorphic strip with endpoint at $p_N$. We assume that $\hat{u}$ is not constant.
$u_2$ is constant by definition of $N$ while $E(u_1) \geq \min(t, s) - 1$ by computation in Proposition~\ref{p:surf}. We note that
$E(u) = E(u_1) + E (u_2) \geq \min(t, s) - 1$ hence $\sigma_{p} \geq \min(t, s) - 1$ and the proposition follows.
\end{proof}
| -32,209.924024 |
[
-2.94921875,
2.70703125
] | 46.92623 |
[
-2.583984375,
0.70263671875,
-2.00390625,
-5.98046875,
-1.0283203125,
8.7265625
] |
[
3.67578125,
8.9296875,
1.78515625,
6.296875
] | 264 | 4,529 |
[
-3.6015625,
4.265625
] | 33.209144 |
[
-4.93359375,
-3.6953125,
-5.02734375,
-2.35546875,
1.5556640625,
12.71875
] | 0.629429 | 14.237528 | 19.849856 | 1.168995 |
[
1.0670257806777954
] | -21,124.136946 | 4.735703 | -32,007.059642 | 0.718016 | 5.794755 |
[
-1.576171875,
-3.271484375,
-3.884765625,
-5.51953125,
1.80078125,
12.609375
] |
[
-5.65234375,
-1.619140625,
-2.16015625,
-1.029296875,
3.53515625,
3.66796875
] | |
BkiUfTHxK0fkXPSOrRn8
|
\section{Introduction}
\label{sec:intro}
The historical detection of gravitational waves (GWs) by the LIGO/Virgo Collaboration has marked the beginning of a new era in astrophysics
and the birth of GW astronomy~\cite{Abbott:2016blz}. The next generation of detectors will routinely observe the coalescence
of compact objects, such as black holes (BHs) and neutron stars. These observations will probe, for the first time, the highly dynamical
regime of strong-field gravity and may provide the answer to long-standing
issues~\cite{Cardoso:2016rao,Nakano:2016sgf,Yunes:2016jcc}. Is cosmic censorship
preserved in violent gravitational interactions? Do GW observations carry incontrovertible evidence for the event horizon of BHs? Can we pinpoint, in gravitational waveforms, the signature of the light ring or of ergosurfaces?
Simultaneously, the entire coalescence process can be used to constrain gravity theories in novel ways~\cite{Berti:2015itd,TheLIGOScientific:2016src,Yunes:2016jcc}. That general relativity (GR) is not the ultimate theory of gravity is a possibility that should be entertained
in the light of several observations (such as those related to the dark-matter and the dark-energy problems),
and of the difficulty to reconcile GR with quantum field theory~\cite{Berti:2015itd}.
Although such an extension of GR is unknown---and a robust spacetime parametrization in strong-field gravity is lacking---GW observations will help us to exclude or to strongly constrain
wide classes of alternative theories. The inspiral stage, for example, when the two objects are far apart, can teach us about possible extra radiation channels~\cite{TheLIGOScientific:2016src,Barausse:2016eii,Cardoso:2016olt,Yunes:2016jcc}, while the final ringdown stage---when the end-product is relaxing to its final state---provides for remarkable tests of GR, through the measurement of the characteristic quasinormal modes (QNMs)~\cite{Berti:2009kk}.
In GR, as well as in essentially any relativistic theory of gravity, BHs are extremely simple objects described by only a handful of parameters. Accordingly, their QNMs are completely characterized
by only a few parameters as well. For example, Kerr BHs in GR are characterized by their mass and angular momentum, and so are their QNMs.
In a nutshell, measurement of one single QNM (i.e, a ringing frequency and a decay time scale~\cite{Berti:2005ys,Berti:2009kk}) allows for a determination of the BH mass and angular momentum. The measurement of a second QNM tests GR~\cite{Berti:2005ys,Berti:2007zu,Gossan:2011ha,Meidam:2014jpa}. In the context of modified theories of gravity, a second QNM can be used to measure possible extra coupling parameters, as was shown recently
for a theory with an extra vector degree of freedom~\cite{Cardoso:2016olt}.
Some of the most viable and appealing modifications of gravity are those obtained via the inclusion of extra scalar fields---such as scalar-tensor theories of gravity---or of higher-curvature terms in the action, or both. Higher-order gravity is generically motivated by UV corrections, which also arise naturally in some low-energy truncations of string theories. The paradigmatic case, and the one we focus on here, is Einstein-dilaton-Gauss-Bonnet (EDGB) gravity, described by the action~\cite{Kanti:1995vq,Berti:2015itd}
\begin{equation}
S=\int d^4 x\frac{\sqrt{-g}}{16\pi}\left(R-\frac{1}{2}\partial_a\phi\partial^a\phi+\frac{\alpha}{4}e^{\phi}{\cal R}_{\rm GB}^2\right)+S_{m}\,,
\label{eq:action}
\end{equation}
where
\begin{equation}
{\cal R}_{\rm GB}^2={R_{abcd}} R^{abcd}-4 {R_{ab}} R^{ab}+R^2\,,
\end{equation}
is the Gauss-Bonnet topological term, $S_m$ represents the matter sector and we use (throughout this work) units for which the Newton's constant and the speed of light are unity, $G=c=1$.
Current best constraints on the coupling constant $\alpha$ are $\sqrt{\alpha}<10\,{\rm km}$~\cite{Yagi:2012gp,Berti:2015itd}\footnote{We note a typo in the review~\cite{Berti:2015itd}. In the notation used in the review, Eq.~(2.26) should read $\sqrt{|\alpha_{\rm GB}|}\lesssim 5\times10^5\,{\rm cm}$.}.
We will close two important gaps in the literature concerning BHs in this theory:
we first find strong evidence that static EDGB BHs are linearly stable,
and then compute gravitational waveforms from plunging particles,
which can be argued to be an indicator of how BH collisions proceed in this theory.
Finally, we discuss the constraints on the coupling parameter $\alpha$ from current and future ringdown observations.
\section{Framework}
\label{sec:review}
The equations of motion obtained by extremizing \eqref{eq:action} with respect to the metric and dilaton field are given by~\cite{Kanti:1995vq}
\begin{align}
\Box \phi&=\frac{\alpha}{4}e^{\phi}{\cal R}_{\rm GB}^2\label{eq:scalarphi}\,,\\
G_{ab}&=\frac{1}{2}\partial_a\phi\partial_b\phi-\frac{1}{4}g_{ab}
(\partial_c\phi) (\partial^c\phi)
-\alpha {\cal K}_{ab}+8\pi T_{ab} \label{eq:einsteineq}\,,
\end{align}
where $G_{ab}=R_{ab}-\frac{1}{2}g_{ab}R$ is the Einstein tensor, $T_{ab}$ is the matter stress-energy tensor,
and
\begin{equation}
{\cal K}_{ab}=(g_{ac}g_{bd}+g_{ad}g_{bc})\epsilon^{idjk}\nabla_{l}\left(\tilde{R}^{cl}_{~~jk}\partial_i e^{\phi}\right)\,,
\end{equation}
where $\epsilon^{abcd}$ is the contravariant Levi-Civita tensor, $\tilde{R}^{ab}_{~~cd}=\epsilon^{abij}R_{ijcd}$.
\subsection{BH solutions in EDGB gravity}
BHs in EDGB gravity are scalar-vacuum solutions of the above equations, which were first constructed analytically in spherical symmetry, in the small-coupling regime~\cite{Mignemi:1992nt},
\begin{equation}
\zeta:=\frac{\alpha}{M^2}\ll1 \,,
\end{equation}
where $M$ is the BH mass. In spherical symmetry, the line element reads
\begin{equation}
ds^2=-A(r)dt^2+B(r)^{-1}dr^2+r^2d\Omega^2\,,
\label{eq:linel}
\end{equation}
where $d\Omega^2$ is the standard unit 2-sphere line element.
Both functions $A(r)$ and $B(r)$ and the scalar field $\phi$ can be expanded in powers of the coupling parameter $\zeta$, and the corresponding solutions can be found by solving the field equations~\eqref{eq:scalarphi}--\eqref{eq:einsteineq} perturbatively (see also Ref.~\cite{Yunes:2011we}).
Further details are given in Appendix \ref{app:motion}.
Nonperturbative solutions were investigated numerically in Ref.~\cite{Kanti:1995vq} for static geometries and in Ref.~\cite{Pani:2009wy} for slowly rotating BHs to first order in the spin.
It was shown that static BH solutions exist only up to a maximum value of $\zeta$, namely~\cite{Pani:2009wy}
\begin{equation}
0\leq \zeta \lesssim 0.691\,.
\end{equation}
Because $\zeta$ is strictly less than unity, higher-order perturbative expansions~\cite{Maselli:2015tta} are
accurate almost in the entire parameter space.
Slowly rotating solutions were described numerically in Ref.~\cite{Pani:2009wy}, and analytically in the small-coupling regime in Refs.~\cite{Pani:2011gy,Ayzenberg:2014aka}, and were recently extended to higher order in the coupling and in the spin parameter~\cite{Maselli:2015tta}. In the latter case, the line element
(as well as the scalar field) can be expanded in a complete basis of orthogonal functions according to their symmetry properties~\cite{Maselli:2015tta}.
Numerical solutions describing rotating BHs for arbitrary coupling and spin were found in Ref.~\cite{Kleihaus:2011tg} and have been recently thoroughly discussed in Ref.~\cite{Kleihaus:2015aje}.
The linearized mode stability of spherically symmetric BHs against radial fluctuations was studied in Ref.~\cite{Kanti:1997br}, whereas axial gravitational perturbations were studied in Ref.~\cite{Pani:2009wy}. In the polar sector, the linear stability of EDGB BHs was analyzed in Ref.~\cite{Ayzenberg:2013wua}, focusing in the particular regimes where perturbations are dominantly gravitational or scalar, as well using a high-frequency analysis of the perturbations.
In addition, axial quasinormal modes of neutron stars in EDGB theory were studied in Ref.~\cite{Blazquez-Salcedo:2015ets}.
Due to the cumbersome field equations, there is currently no result concerning the stability of nonrotating EDGB BHs for the most relevant gravitational polar sector, and there is no stability analysis for rotating solutions.
In this work we partly fill this gap by performing a full linear (mode) stability analysis of static EDGB BHs.
\subsection{Perturbed BHs in EDGB}\label{subsec:pert}
We are interested in understanding how BHs in EDGB theory respond to small perturbations, as those induced by a
fluctuation of the metric or of the dilaton or by a small external perturbing object. We focus our attention in both
dilaton fluctuations in vacuum and those induced by a small pointlike particle plunging into the BH. The first case will describe the late-time behavior of perturbed EDGB
BHs, which also dominates the ringdown signal from a distorted BH formed in a coalescence. Pointlike particles, on the
other hand, are a good proxy for small BHs or neutron stars falling into massive BHs, but are also known to provide
reasonably accurate estimates even for equal-mass BH
collisions~\cite{Berti:2010ce,Sperhake:2011ik,Tiec:2014lba}.
Pointlike particles of mass $\mu\ll M$ are modeled by~\cite{Zerilli:1971wd,Poisson:2011nh,Breuer:1974uc}
\begin{align}
S_m=\mu \int d\tau\sqrt{-g}\,,
\end{align}
with $d\tau=d\lambda \sqrt{-g_{ab}\dot x^a\dot x^b}$. We will consider pointlike objects with a scalar charge which is entirely due to the Gauss-Bonnet coupling. In other words, we will investigate BH spacetimes of which the scalar charge arises purely from the Gauss-Bonnet term. The theory we consider contemplates no other couplings to matter, and therefore
pointlike particles follow geodesics. This need not be the case generically, and nontrivial couplings to matter can be envisioned~\cite{Quinn:2000wa,Poisson:2011nh}. These couplings will certainly influence the motion of particles and the radiation in collisions but will not affect the intrinsic ringdown properties of the spacetime.
We consider a spherically symmetric EDGB BH distorted by either the pointlike particle or through some fluctuation in the metric or scalar field. At the linearized level, the full geometry is described by
\begin{align}
g_{ab}&=g_{ab}^{(0)}+\varepsilon\, h_{ab}\,,\\
\phi&=\phi_0(r)+\varepsilon\,\delta\phi\,,
\end{align}
where $\varepsilon \ll 1$ is a bookkeeping parameter, $g_{ab}^{(0)}$ is described by \eqref{eq:linel} and $\phi_0(r)$ is the corresponding background scalar. As background solution, we consider both a perturbative solution~\cite{Maselli:2015tta} up to ${\cal O}(\zeta^6)$, and a numerical solution for arbitrary values of $\zeta$.
The fluctuations $h_{ab}$ and $\delta \phi$ are functions of $(t,r,\theta,\varphi)$. Einstein's equations can be further simplified by Fourier transforming these quantities
and by expanding them in (tensor and scalar) spherical harmonics, e.g.,
\begin{equation}
\delta\phi(t,r)=\frac{1}{\sqrt{2\pi}}\int d\omega \frac{\phi_1(\omega,r)}{r}Y^{lm} e^{-i\omega t}\,,\label{Fourier}
\end{equation}
where $Y^{lm}$ are the standard spherical harmonics and $\omega$ is a Fourier frequency.
The metric perturbations can be decomposed in terms of tensorial spherical harmonics~\cite{Regge:1957td,Zerilli:1971wd}. By using this decomposition, perturbations naturally split into two sectors according to their parity (either axial or polar). Axial perturbations of EDGB BHs are simpler, because they are decoupled from the scalar-field perturbations~\cite{Pani:2009wy}. Here, we shall consider the two sectors of the perturbations
In the Regge-Wheeler gauge~\cite{Regge:1957td}, the polar sector of the metric perturbations is given by
\begin{align}
&h_{ab}=\left[
\begin{array}{cccc}
A H_0 & H_1 & 0 & 0 \\
H_1 & H_2/B & 0 & 0\\
0 & 0 & r^2 K & 0\\
0 & 0 & 0 & r^2 \sin^2\theta K
\end{array}\right] Y^{lm} \,,
\label{hab13}
\end{align}
while the axial sector reads
\begin{align}
&h_{ab}=\left[
\begin{array}{cccc}
0 & 0 & 0 & \sin\theta~ h_0\partial_\theta\\
0 & 0 & 0 & \sin\theta~ h_1\partial_\theta\\
0 & 0 & 0 & 0\\
\sin\theta~ h_0\partial_\theta & \sin\theta~ h_1\partial_\theta & 0 & 0
\end{array}\right]\,Y^{lm} \,.
\end{align}
In the above definitions, the metric perturbations depend only on $t$ and $r$. Note that we have already specialized the spacetime to axial symmetry: for the cases handled here, one can always rotate the coordinate axis such
that the spacetime is axially symmetric.
We shall Fourier-decompose these perturbation functions
as, e.g., $X(t,r)=(2\pi)^{-1/2}\int d\omega X(\omega,r)e^{-i\omega t}$.
Likewise, the stress-energy tensor can be decomposed into spherical harmonics~\cite{Zerilli:1971wd,Sago:2002fe}. In the radial plunging case considered here, the particle only disturbs the spacetime in the polar sector, and its stress-energy tensor can be written as
\begin{equation}
T_{ab}=
\left[
\begin{array}{cccc}
A_{lm}^{(0)} & \frac{i}{\sqrt{2}}A_{lm}^{(1)} & 0 & 0 \\
\frac{i }{\sqrt{2}}A_{lm}^{(1)} & A_{lm}& 0 & 0 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
\end{array}
\right]Y^{lm}\,,
\end{equation}
where the functions $A_{lm}^{(0)}$, $A_{lm}^{(1)}$ and $A_{lm}$, like the dilaton and metric perturbations, can be Fourier decomposed, such that they depend only on $r$ and $\omega$. The explicit form of these functions is given in Appendix~\ref{app:line}.
The linearized dynamics is governed by a coupled system which can be obtained by expanding the field equations \eqref{eq:scalarphi}--\eqref{eq:einsteineq} up to first order in the perturbation functions $h_{ab}$ and $\delta\phi$. The equations take the schematic form (no sum on $j$)
\begin{equation}
\frac{d}{dr}{\bm{\Psi}}_j+\bm{V}_{j}\bm{\Psi}_j=\bm{S}_j\,,
\label{eq:perturbations}
\end{equation}
where $j=({\rm p},{\rm a})$ for the polar and axial sectors, respectively, $\bm{\Psi}_j$ is a column vector, the components of which are $\bm{\Psi}_{\rm p}\equiv (H_1,K,\phi_1,\phi_1')$ for the polar sector and $\bm{\Psi}_{\rm a}\equiv (h_1,h_0)$ for the axial sector, respectively. The matrices $\bm{V}_{j}$ describe the coupling among the perturbations and depend on the background fields. The vectors $\bm{S}_j$ represent the source terms associated to the point-particle stress-energy tensor. The explicit forms of $\bm{V}_{j}$ and $\bm{S}_j$ are derived in Appendix \ref{app:line}, where we also show that the functions $H_2$ and $H_0$ can be completely determined in terms of $\bm{\Psi}_{\rm p}$. For the axial sector, the source terms vanish identically and we can write the coupled first-order equations into a second-order Schr\"odinger-like equation with an effective potential.
\subsection{Computation of the QNMs}\label{sec:qnms}
The late-time gravitational signal from a perturbed BH is dominated by a sum of exponentially damped sinusoids, the QNMs, which correspond to the characteristic vibration modes of the spacetime~\cite{Nollert:1999ji,Berti:2009kk}.
The QNMs are solutions of the sourceless wave equations~\eqref{eq:perturbations}
(i.e., $\bm{S}_j=0$), along with proper boundary conditions---purely outgoing
waves at infinity and ingoing waves at the horizon.
The latter correspond to the following behavior of the wave function at the
boundaries
\begin{equation} \bm \Psi_j\propto \left\{ \begin{array}{ll} e^{-i\omega r_*}\,,&r\sim r_h \,,\\
e^{i\omega r_*}\,, &r\to\infty\,, \end{array}\right. \end{equation}
where $r_h$ is the horizon radius in these coordinates and the ``tortoise''
coordinate $r_*$ is defined by
\begin{equation} \frac{dr_*}{dr}=\frac{1}{\sqrt{A B}}\,. \end{equation}
To compute the QNMs, we use a direct-integration method. We construct a square matrix $\bm X$ (four and two dimensional in the polar and axial case, respectively) of which the columns are independent solutions of Eq.~\eqref{eq:perturbations}. This matrix can be constructed by a certain combination of the solution which is regular at the horizon and the one which is regular at infinity (cf. Refs.~\cite{Pani:2013pma,Macedo:2016wgh} for details). For the polar sector, we note that the boundary conditions defining the QNM eigenvalue problem depend on two parameters, related to the amplitudes of the scalar and gravitational perturbations.
Two of the columns of $\bm X$ can be constructed by integrating the equations from the horizon outward. Likewise, two other solutions can be constructed by integrating the equations from infinity inward. In general, the solutions integrated from the horizon are linearly independent from the ones integrated from infinity, unless $\omega$ is the QNM frequency. In other words, the QNM frequencies are obtained by imposing
\begin{equation}
{\rm det} (\bm X)|_{r=r_{\rm m}}=0\,,
\end{equation}
where $r_{\rm m}$ is an arbitrary matching point of the order of the horizon
radius. The same procedure can be done for the axial modes, with the difference
that the boundary conditions apply only to the gravitational amplitude and therefore the problem is technically
less involved.
We have computed the lowest QNMs of static BHs using both the full numerical background and the perturbative analytical solution given by Eqs.~\eqref{eq:expme}--\eqref{eq:expphi}, up to $N=4$, i.e. up to ${\cal O}(\zeta^6)$. We checked the numerical stability of the QNM frequencies against changes in the values of the numerical horizon $r_h$, numerical infinity and the matching radius $r_{\rm m}$ (typically, we use $r_{\rm m}\sim 4 r_h$).
One of the advantages of the above procedure is that it can be applied to both the perturbative solution and the full numerical one.
\subsection{Plunging particles}\label{sec:plunge}
To obtain the metric and dilaton perturbations due to a particle plunging into a BH, we need to solve the inhomogeneous system~\eqref{eq:perturbations}. Two common methods used in the literature to solve this problem are a direct integration and a Green's function approach (see, e.g., Ref.~\cite{Cardoso:2016olt}).
The Green's function method relies on the fundamental matrix $\bm X$, as constructed by using the homogeneous solutions as discussed in the previous section. The formal solution of \eqref{eq:perturbations} can be written as \cite{boyce2008elementary}
\begin{equation}
\bm \Psi_j=\bm X \bm \beta +\bm X\int dr \bm X^{-1}\bm S\,,
\label{eq:formal_sol}
\end{equation}
where $\bm \beta$ is a constant vector to be determined by imposing the proper boundary conditions. Thus, once the fundamental matrix $\bm X$ of the homogeneous problem is computed, its convolution with the source term $\bm S$ in \eqref{eq:formal_sol} yields the solution of the inhomogeneous problem. However, in many problems, the source term might converge slowly at the boundaries or even diverge, as in our case in which the source term diverges at the event horizon. These problems can be avoided by performing a suitable nontrivial transformations of the perturbation functions~\cite{Sasaki:1981kj,Cardoso:2002jr}, or with a careful choice of Green's function~\cite{Poisson:1996ya}.
A different scheme which avoids this problem consists in integrating directly the full inhomogeneous system, by imposing the proper boundary conditions for the full solution. First, we expand the perturbation functions near the horizon as
\begin{equation}
\bm \Psi_j(r\to r_h)\approx \sum_{k=0}^{N}(r-r_h)^{k+p}\bm \psi_{j,k}e^{-i\omega r_*} +\bm \Psi_{j,H}\,,
\end{equation}
where $p$ is a constant chosen such that the above expansion satisfies the inhomogeneous equations near the horizon and $\bm \Psi_{j,H}$ is the (ingoing) solution of the homogeneous equation. The above expansion is solved iteratively near the event horizon for the coefficients $\bm \psi_{j,k}$ up to $k=N$, and they generically depend on constants (say $\bm \psi_{j,0}$) which are related to the amplitude of the fields at the horizon. The amplitudes at the horizon are then used as shooting parameters; i.e. we chose them such that the numerical solution satisfies the proper boundary conditions also at infinity. For arbitrary amplitudes at the horizon, the numerical solution far from the BH is a combination of ingoing and outgoing waves, i.e.,
\begin{equation}
\bm \Psi_j=\bm\Psi_j^{out}+\bm\Psi_j^{in}\,,
\end{equation}
and the required solution is obtained by setting the amplitude of the ingoing waves to zero. In the EDGB case, the amplitudes are related to the gravitational and dilaton perturbations, and therefore the problem is a two-parameter shooting problem for the amplitudes at the horizon. Note that for radial plunging, since the source terms for the axial sector are zero, the particle only induces perturbations in the polar sector.
With the numerical solution at hand, one can compute the gravitational and scalar energy spectra at infinity. This is achieved through the effective stress-energy tensor for the gravitational perturbations (i.e., the Isaacson tensor) and the stress-energy tensor of the dilaton field \cite{Isaacson:1968zza}. As shown in Ref.~\cite{Stein:2010pn}, the Isaacson tensor is the same in GR and in a class of theories including EDGB gravity. The gravitational flux, for a given multipole $l$, can be written as \cite{Zerilli:1971wd}
\begin{equation}
\frac{dE_g}{d\omega}= \frac{1}{32\pi} \frac{(l+2)!}{(l-2)!}|K(r\to\infty)|^2\,,
\end{equation}
where $K(r)$ is the polar perturbation given by Eq. (\ref{hab13}).~\footnote{Note that, formally, $K$ is a gauge-dependent quantity whereas the use of gauge-independent quantities is obviously desired. Moreover, at large distances the Regge-Wheeler gauge is not well defined.
However, in this limit the function $K$ yields the so-called Zerilli function $Z(r)$~\cite{Zerilli:1971wd} (which is a gauge-invariant quantity~\cite{Moncrief:1974am}) since $K(r\rightarrow\infty)=i\omega Z(r\rightarrow\infty)$.}
The dilaton flux reads
\begin{equation}
\frac{dE_\phi}{d\omega}=\frac{\omega^2}{16\pi}|\phi_1(r\to\infty)|^2\,,
\end{equation}
where $\phi_1(r)$ is given by the dilaton perturbation in Eq. (\ref{Fourier}).
As shown in Appendix~\ref{app:line}, since the particle does not have a direct coupling to the dilaton field, dilaton radiation only exists due to the gravitational perturbations. In this sense, the gravitational perturbations work as a ``source'' for the dilaton radiation. Since in the dilaton equation they are proportional either to the parameter $\zeta$ or the background scalar field (and derivative), it is natural to expect that the dilaton radiation scales with $\zeta^2$. Additionally, for $l=0$ and $l=1$ the perturbations of EDGB BHs can be written as a single second-order ordinary differential equation, which represents the dilaton perturbation (see Appendix~\ref{app:line}).
\section{Stability and QNMs of EDGB black holes}
\begin{figure*}
\includegraphics[width=\columnwidth]{axial_real_finite.pdf}\includegraphics[width=\columnwidth]{axial_imaginary_finite.pdf}%
\caption{Real (left) and imaginary (right) parts of the axial $l=2$ fundamental mode, normalized by the Schwarzschild ($\zeta=0$) values. We compare three different approaches: the geodesic (black dotted line), the small-coupling limit (red dashed line), and the finite $\zeta$ regime (blue solid line). In the inset we show that for small values of $\zeta$ the small-coupling limit (red squared) agrees well with the finite-coupling result up to $\zeta\approx 0.3$.}%
\label{fig:axial_modes}
\end{figure*}
As previously discussed, the QNMs govern the late-time behavior of any small fluctuation away from axisymmetry of the BH. Since the time dependence is of the form $e^{-i\omega t}$, the {\it absence} of a QNM frequency with the {\it positive imaginary} part in the spectrum implies that all fluctuations decay exponentially with time. Thus, a criterion for linearized mode stability of the spacetime is that all its QNM frequencies have a negative imaginary part~\cite{Berti:2009kk}. In addition, the late-time dynamics is controlled by the fundamental QNM, i.e., the mode with the smallest imaginary component (equivalently, with the longest decay time).
The Schwarzschild spacetime is stable, and its fundamental QNM frequency, $\omega^{S}=M\omega_{R}^{S}+iM\omega_{I}^{S}$, for the $l=2$ mode reads~\cite{Chandrasekhar:1975zza,Berti:2009kk,ref:webpage}
\begin{eqnarray}
M\omega^{S}&\approx&0.3737-i\,0.08896 \quad {\rm gravitational}\label{eq:qnm_s}\\
M\omega^{S}&\approx&0.4836-i\,0.09676 \quad {\rm scalar}\label{eq:qnm_ss}
\end{eqnarray}
for the gravitational and scalar fundamental modes, respectively, where $M$ is the BH mass. In EDGB gravity, because the coupling $\zeta$ is smaller than unity, we expect that the fundamental QNM frequencies\footnote{In the $\zeta\to0$ limit the scalar sector is decoupled and we expect to recover the scalar QNMs of a Schwarzschild BH. This is confirmed by the computation presented in this section.} are only slightly different from Eqs.~\eqref{eq:qnm_s} and \eqref{eq:qnm_ss}. In other words, we expect EDGB BHs to be stable for sufficiently small coupling.
We will now study if these BHs are stable throughout all values of $\zeta$, and also quantify the deviation from the corresponding Schwarzschild value.
We have computed the QNMs of EDGB BHs with two independent codes, both in a small-$\zeta$ expansion and using the full numerical background.
In the small-coupling limit, where the expressions can be expanded in powers of $\zeta$ (see Appendix~\ref{app:motion}), the real and imaginary parts of the QNM frequencies can be written as
\begin{equation}
\frac{\omega_{R}}{\omega_{R}^{S}}=1+\sum_{j=1}^{N}R_{j}\zeta^j \,,\quad\frac{\omega_I}{\omega_{I}^{S}}=1+\sum_{j=1}^{N}I_{j}\zeta^j \,,
\label{eq:expan_modes}
\end{equation}
where $\omega_{R}^{S}$ and $\omega_{I}^{S}$ are, respectively, the real and imaginary parts of the modes of Schwarzschild BH with the same mass $M$.
The coefficients $R_j$ and $I_j$ are obtained by fitting the numerical data with the above expression and depend on $l$ and on the nature of the mode
\subsection{QNMs, light ring, and geodesic correspondence} \label{sec:geodesics}
Computing the QNMs of spacetimes known only numerically can be challenging. For
spherically symmetric spacetimes, a WKB-type analysis has shown that, in the
eikonal ($l\gg 1$) limit, the QNMs can be obtained using only properties of the
light ring (which defines the radius of the photon sphere of the BH). This ``null geodesic
correspondence''~\cite{Ferrari:1984zz,Cardoso:2008bp,Yang:2012he} is useful as
it requires only manipulation of background quantities which are easy to obtain,
and provides a clear physical insight into the QNMs of BHs: they correspond to
waves trapped near the peak of the potential barrier for null particles (i.e., within the photon sphere), slowly
leaking out on a time scale given by the geodesic instability time scale.
The geodesic correspondence only works, formally, in the $l\gg 1$ regime, but can be used even at low $l$ using an appropriate calibration. In fact, this approach has proven to provide reliable results also for low-$l$ modes for a variety of BH spacetimes~\cite{Cardoso:2008bp}, including Kerr-Newman BHs~\cite{Cardoso:2016olt}.
By extending the analysis of Refs.~\cite{Ferrari:1984zz,Cardoso:2008bp,Yang:2012he}, Ref.~\cite{Cardoso:2016olt} recently showed that the complex QNMs of a stationary and axisymmetric BH in the eikonal limit can be written as
\begin{equation}
\omega_R+i\omega_I \sim \Omega l-i(n+1/2)|\lambda|\,, \label{eikonal}
\end{equation}
where $n$ is the overtone number,
\begin{eqnarray}
\Omega &=&\frac{-g_{t\varphi}'+\sqrt{{g_{t\varphi}'}^2-g_{tt}' g_{\varphi\varphi}'}}{g_{\varphi\varphi}'}\,,
\end{eqnarray}
is the orbital frequency at the light ring on the orbital plane, and
\begin{eqnarray}
\lambda = -\frac{1}{\dot t}\sqrt{\frac{V''}{2}}\,, \qquad \dot t=-\frac{E^2 g_{\varphi\varphi}+L g_{t\varphi}}{g_{t\varphi}^2-g_{tt}g_{\varphi\varphi}}
\end{eqnarray}
is the Lyapunov coefficient evaluated at the light-ring location on the equatorial plane. In the above expression, a prime denotes radial derivative, whereas $E$ and $L$ are the (conserved) specific energy and angular momentum of the geodesic, and $V$ is the effective radial potential. The expression~\eqref{eikonal} is valid for $l=m\gg1$ modes. A more involved result for the QNMs of a Kerr BH with generic $l\gg1$ and $|m|\leq l$ is derived in Ref.~\cite{Yang:2012he}.
The geodesic correspondence has been formally proven for Kerr BHs and for a variety of spacetimes, but it has never been checked for BH solutions in modified gravity. This is particularly interesting in light of the breaking of the isospectrality of the axial and polar QNMs of BHs in EDGB theory, as we discuss in the next section. It is important at this stage to point out that the geodesic approximation must fail to capture some of the features of the full problem. It is well adapted, in principle, to describe
the effects of rotation, but cannot take into account (at least not blindly) the presence of extra degrees of freedom like scalars in EDGB. Thus, using this correspondence to extract the QNMs of BHs in modified gravity should be done carefully.
As previously discussed, the background solution and the metric of a spinning BH in EDGB theory is known analytically up to ${\cal O}(\chi^5,\zeta^7)$~\cite{Maselli:2015tta} and numerically for any value of $\chi$ and $\zeta$~\cite{Kleihaus:2011tg,Kleihaus:2015aje}, where $\chi$ is the dimensionless angular momentum parameter,
\begin{equation}
\chi=J/M^2\,.
\end{equation}
We will use this result in Sec. \ref{detectability}, to estimate the modes of spinning EDGB BHs.
\subsection{Axial modes}
\begin{table*}%
\caption{Numerical value of the coefficients $R_j$ and $I_j$ for the expansions in the small-coupling limit, cf.~\eqref{eq:expan_modes} for the axial QNMs. The geodesic coefficients are computed from the exact analytical solution for small $\zeta$ limit, while the QNM frequencies coefficients are obtained through a polynomial fit with the data.}
\label{tab:coeff_axial}
\begin{tabular}{ccccc}
\hline\hline
& $j$ & $l=2$ & $l=3$ & Geodesic \\
\hline
\multirow{4}{*}{$R_j$}
& 1 & 0 & 0 & 0 \\
& 2 & $1.002\times10^{-3}$ & $1.173\times 10^{-2}$ & $1.257\times 10^{-2}$ \\
& 3 & $1.906\times 10^{-3}$ & $5.035\times 10^{-3}$ & $6.872\times 10^{-3}$ \\
& 4 & $1.131\times 10^{-3}$ & $1.353\times 10^{-2}$ & $5.537\times 10^{-3}$ \\ \hline
\multirow{4}{*}{$I_j$}
& 1 & 0 & 0 & 0\\
& 2 & $-5.174\times 10^{-3}$ & $-4.774\times 10^{-3}$ & $-5.267\times 10^{-3}$ \\
& 3 & $5.766\times 10^{-3}$ & $7.590\times 10^{-4}$ & $-7.184\times 10^{-3}$ \\
& 4 & $-7.091\times 10^{-3}$ & $-3.282\times 10^{-3}$ & $-7.822\times 10^{-3}$ \\
\hline \hline
\end{tabular}
\end{table*}
The axial sector of gravitational perturbations is decoupled from the scalar-field perturbations, and hence is simpler to study. Using the direct-integration procedure described in Sec.~\ref{sec:qnms}, we have computed the axial QNMs both in a small-$\zeta$ expansion and in the full numerical background. Our results are summarized in Fig.~\ref{fig:axial_modes} and in Table~\ref{tab:coeff_axial}.
In Fig.~\ref{fig:axial_modes}, we show the behavior of the axial $l=2$
fundamental mode as a function of the coupling $\zeta$, normalized by the
corresponding Schwarzschild quantity. In most of the range of $\zeta$, the
behavior of the modes is smooth and given by a corresponding small deformation
of the Schwarzschild QNMs. The only exception occurs close to the critical value
of the coupling constant $\zeta \approx 0.691$, where the QNMs have a very
sensitive dependence on $\zeta$. For small values of $\zeta$, say
for $\zeta\lesssim 0.4$, analytically
expanded backgrounds [up to ${\cal O}(\zeta^6)$] yield QNMs which are in very
good agreement with the full numerical solution. This provides a nontrivial
check for both our (independent) codes.
Figure~\ref{fig:axial_modes} also shows the result of the geodesic algorithm
described in Sec.~\ref{sec:geodesics}.
For the axial modes, which are decoupled from the scalar perturbations,
the geodesic predictions are in good agreement with the results of
the full numerical solution.
Although not shown in Fig.~\ref{fig:axial_modes}, the agreement is better for higher multipoles, as
expected (cf. Table~\ref{tab:coeff_axial}).
Finally, Table~\ref{tab:coeff_axial} shows the results of the polynomial
fit~\eqref{eq:expan_modes} to the axial QNM of EDGB BHs, which is specially
accurate for small $\zeta$. By analyzing the perturbation equations, it is easy
to show that $R_1=I_1=0$ in the expansion~\eqref{eq:expan_modes}. The full
numerical results are available online~\cite{ref:webpage}.
In Table~\ref{tab:coeff_axial}, we also give the coefficients obtained by the geodesic algorithm, which help to quantify the accuracy of the geodesic approximation; for instance at $\zeta=0.5$ the difference in percentage between the real and the
imaginary parts of the (not-normalized) frequency, relative to the numerical result,
is, respectively, less than $3\%$ and less than $8\%$ for $l=2$, and improves for $l>2$. Similar deviations are obtained in the GR limit, $\zeta=0$.
As explained in Appendix~\ref{app:line}, similarly to what happens in GR, there are no axial QNMs for $l=0$ and $l=1$.
\begin{figure*}[ht]%
\includegraphics[width=\columnwidth]{polar_grav}\includegraphics[width=\columnwidth]{polar_scal}%
\caption{Real (left) and imaginary (right) parts of the polar quasinormal modes for $l=2$, for the gravitational- and scalar-led modes, as functions of the coupling $\zeta$, normalized by the Schwarzschild-limit quantities. The insets show a closeup in order to see the comparison for small values of $\zeta$.
}%
\label{fig:polar_modes}%
\end{figure*}
\subsection{Polar modes}
\begin{table}%
\caption{Numerical value of the coefficients $R_j$ and $I_j$ for the polar gravitational-led and scalar-led modes.
}
\label{tab:coeff}
\begin{tabular}{c c c c}
\hline\hline
& $j$ & {\footnotesize Polar, gravitational $l=2$} & {\footnotesize Polar, scalar $l=2$} \\
\hline
\multirow{4}{*}{$R_j$}
& 1 & 0 & $-1.408\times 10^{-2}$ \\
& 2 & $-3.135 \times 10^{-2}$ & $1.127\times 10^{-1}$ \\
& 3& $-9.674\times 10^{-2}$ & $-1.462\times 10^{-1}$ \\
& 4 & $2.375\times 10^{-1}$ & $5.334\times 10^{-1}$\\ \hline
\multirow{4}{*}{$I_j$}
& 1 & 0 &$5.580\times 10^{-2}$\\
& 2 & $4.371\times 10^{-2}$ &$-6.780\times 10^{-2}$ \\
& 3 & $1.794\times 10^{-1}$ &$1.042\times 10^{-1}$\\
& 4 & $-2.947\times 10^{-1}$ & $-2.868\times 10^{1}$\\
\end{tabular}
\begin{tabular}{c c c c}
\hline\hline
& $j$ & {\footnotesize Polar, gravitational $l=3$} & {\footnotesize Polar, scalar $l=3$} \\
\hline
\multirow{4}{*}{$R_j$}
& 1 & 0 & $-6.361 \times 10^{-3}$ \\
& 2 & $-9.911\times 10^{-2}$ & $1.442\times 10^{-1}$ \\
& 3& $-4.907 \times 10^{-2}$ & $1.168\times 10^{-1}$ \\
& 4 & $9.286\times 10^{-2}$ & $-1.803\times 10^{-1}$\\ \hline
\multirow{4}{*}{$I_j$}
& 1 & 0 &$2.906\times 10^{-3}$\\
& 2 & $7.710\times 10^{-2}$ &$-5.670\times 10^{-2}$ \\
& 3 & $1.399\times 10^{-1}$ &$-1.445\times 10^{-1}$\\
& 4 & $-3.450\times 10^{-1}$ & $2.105\times 10^{-1}$\\
\hline \hline
\end{tabular}
\end{table}
Unlike the axial sector, the polar gravitational sector of the metric perturbations couples to the scalar-field perturbations.
The system of ODEs is more complex and finding the QNM frequencies is therefore more challenging. Even for arbitrarily small values of $\zeta$ the QNMs contain two families: (i) \emph{gravitational-led} modes, which reduce to the gravitational QNMs of Schwarzschild BHs in the $\zeta\to0$ limit, and (ii) \emph{scalar-led} modes, which reduce to the QNMs of a test scalar field on a Schwarzschild metric when $\zeta\to0$ (see Ref.~\cite{Molina:2010fb} for a similar situation in another theory). The extent to which each of these modes is excited in actual physical setups
is discussed in the next section.
Our results for the quadrupole modes ($l=2$) are summarized in Fig.~\ref{fig:polar_modes} and
Table~\ref{tab:coeff}, where we show the fundamental gravitational-led and
scalar-led QNMs. The deviations from the GR case are larger than in the axial
case. This is probably due to the extra coupling between gravitational and
scalar degrees of freedom. From the fits given in Table~\ref{tab:coeff}, it is
interesting to note that the leading-order correction to the $l=2$
gravitational-led mode has an opposite sign compared to the axial mode. In
particular, since the geodesic correspondence predicts only one type of modes
and the latter are in good agreement with the axial modes, we find that the
behavior of the polar modes is not captured by the geodesic correspondence; therefore, we do not plot the geodesics results in the figure.
This qualitative difference is
expected, because the axial potential resembles the geodesic potential at large
$l$, whereas in the polar sector the coupling to scalar perturbations
drastically changes the dynamics of the perturbations. Likewise, there is no
reason to expect that the behavior of scalar-led perturbations is well captured
by the geodesic correspondence, at least for small values of $l$.
Due to the coupling between the dilaton and gravitational perturbations, there are also nontrivial $l=0,\,1$ scalar-led modes for EDGB BHs. These reduce to their respective scalar modes in the Schwarzschild spacetime when $\zeta\to 0$. The results at finite coupling follow the trend of higher multipoles.
\subsection{Mode stability}
From the above results, it is clear that the fundamental QNMs of an EDGB BH change at most by a few percent relative to the Schwarzschild case. As a consequence, these modes are stable for any value of $\zeta$ in the domain of existence of static EDGB BHs. We have investigated this issue also for higher multipoles ($l\geq2$) and our numerical search has found no unstable modes in the entire parameter space. This strongly indicates that static EDGB BHs are linearly mode stable, just like Schwarzschild BHs.
\section{Radial plunge}
In this section, by using the procedures depicted in Sec.~\ref{sec:plunge}, we discuss the gravitational and dilaton radiation emitted by a particle plunging radially into an EDGB BH. For practical reasons, this computation was done within the small-$\zeta$ approach for the background, and the equations are expanded up to order ${\cal O} (\zeta^6)$. As discussed in the previous section, this higher-order perturbative approach gives precise results even for relatively large values of $\zeta$.
\begin{figure}%
\includegraphics[width=\columnwidth]{flux_g.pdf}%
\caption{Gravitational quadrupolar flux for radial plunges into an EDGB BH with different boosts for $\zeta=0.1$. For small couplings the changes in the gravitational fluxes are very small. In the inset, we plot the ratio of the fluxes for $\zeta=0.1$ and for the Schwarzschild spacetime. The vertical dotted lines in the inset are the values of the gravitational- and scalar-led QNMs.}
\label{fig:flux_g}%
\end{figure}
\begin{figure}%
\includegraphics[width=\columnwidth]{flux_phi.pdf}
\caption{Dilaton quadrupolar flux for radial plunges into an EDGB BH with different boosts for $\zeta=0.1$. As expected, the scalar flux starts to present an exponential suppression for frequencies roughly larger than the dilaton mode. In the inset, we plot the ratio between the fluxes for $\zeta=0.1$ and $\zeta=0.05$, which confirms that the scalar flux scales dominantly with $\zeta^2$.
}%
\label{fig:flux_phi}%
\end{figure}
In Fig.~\ref{fig:flux_g} we show the gravitational flux, by considering different
initial boosts for the particle (see Appendix~\ref{app:motion}).
The deviations from the GR case are very small,
of order of $\sim 1\%$, at least for $\zeta\lesssim0.1$.
Moreover, we note that all additional terms appearing in the metric perturbation
equations (see Appendix~\ref{app:line})---sources included---are at least of
order $\sim\zeta^2$, and therefore the corrections to the flux are proportional
to $\sim\zeta^2$. Therefore, in the small-coupling limit, the gravitational flux
can be written as
\begin{equation} \frac{dE_g}{d\omega}\sim \frac{dE_g^{S}}{d\omega}\left[1+{\cal
O}(\zeta^2)\right]\,, \end{equation}
where $dE_g^{S}/d\omega$ is the corresponding Schwarzschild flux.
Although the overall gravitational corrections are small, the dilaton perturbations can also be radiated by the plunging particles, similarly to the case of a neutral particle plunging into a charged BH~\cite{Johnston:1973cd,Johnston:1974vf,Cardoso:2016olt}. This is due to the coupling between the gravitational and dilaton perturbations, cf. Eq.~\eqref{eq:dilaton_per}. We show the dilaton flux for $\zeta=0.1$ in Fig.~\ref{fig:flux_phi}. Note that the flux displays a cutoff roughly at $\omega\sim\omega_R^{\phi}$, where $\omega_R^{\phi}$ is the scalar-led QNM. Also in this case the dilaton flux scales dominantly as $\zeta^2$. Additionally, because the source terms in the dilaton field are only due to the gravitational perturbations, the dilaton radiation is also dominantly quadrupolar (see Ref.~\cite{Johnston:1973cd} for a similar setup with perturbations in Reissner-Nordstr\"om BHs).
Although the total radiated energy due to the dilaton radiation scales as
$\zeta^2$, it can still be considerably high, depending on the radiating source. For
instance, for the source GW150914 the luminosity due to the GWs was
${dE_g}/{dt}\approx 3.6 \times 10^{56}{\rm erg/s}$~\cite{Abbott:2016blz}.
Therefore, even if the scalar radiation is small compared to the gravitational
one, it can still have a considerable value. The implication of a burst of
dilaton radiation depends on how the environment (plasma, surrounding stars,
etc.) interacts with the dilaton field.
\section{Constraints on the EDGB coupling from ringdown observations\label{detectability}}
The results of the previous section show that plunges of point particles do not excite considerably the scalar-led QNMs. This suggests that such modes might be only mildly excited during the coalescence of two BHs of equal mass and that the main signature of EDGB theory would be a shift of the ringdown frequencies, the latter being governed by the fundamental gravitational-led modes.
Thus, the use of ringdown measurements in EDGB theory to estimate the magnitude of the coupling
$\zeta$ would rely only on the deviation of the gravitational-led modes from their GR counterpart. These deviations are parametrized in terms of the fit~\eqref{eq:expan_modes} of which the coefficients are given in Tables~\ref{tab:coeff_axial} and \ref{tab:coeff} for the $l=2,3$ fundamental axial and polar modes, respectively.
The results of the previous sections refer to nonspinning BHs, whereas the end product of the coalescence is a spinning compact object. Thus, ringdown tests require the knowledge of the first dominant modes of a spinning BH as a function of the spin $\chi$ and of the coupling constant of the modified theory of gravity. Computing the QNMs of generic spinning BHs in modified gravity is a very challenging task\footnote{The task of computing the QNMs of a Kerr BH is enormously simplified by the fact that the gravitational perturbation equations are separable, due to
special properties of the background Kerr geometry which does not necessarily hold for other spinning BH solutions.}, which has witnessed some developments only recently (cf. Ref.~\cite{Pani:2013pma} for an overview). Nonetheless, most of the results are obtained within a perturbative expansion valid for $\chi\ll1$ and quickly become intractable at the higher perturbative order. The latter is required to extrapolate the perturbative result up to $\chi\approx 0.7$, which is roughly the spin of the final BH measured in the two coalescence events detected by aLIGO to date~\cite{Abbott:2016blz,Abbott:2016nmj}.
To overcome this limitation and estimate how rotation affects the QNMs of an EDGB BH, we rely on the geodesic correspondence and on our knowledge of the metric of a spinning BH in EDGB theory. As previously mentioned, the latter is known analytically up to ${\cal O}(\chi^5,\zeta^7)$~\cite{Maselli:2015tta} and numerically for any value of $\chi$ and $\zeta$~\cite{Kleihaus:2011tg,Kleihaus:2015aje}.
In the previous section we have checked that the geodesic correspondence works reasonably well for axial modes, so in this section we will focus on the latter. However, we may argue that the order of magnitude of our estimates should be correct also for the more relevant polar modes.
By using the geodesic correspondence for $l=m$ modes described in Sec.~\ref{sec:geodesics}, we obtain the following result
\begin{widetext}
\begin{eqnarray}
\frac{\omega_R}{\omega_R(\chi=0)} &=& 1\, +\left(0.3849\, +0.0326 \zeta ^2\right) \chi +\left(0.2038\,
+0.0264 \zeta ^2\right) \chi ^2+\left(0.1283\, +0.0169 \zeta^2\right) \chi ^3\nonumber\\
&&+\left(0.0897\, +0.0105 \zeta ^2\right) \chi^4+\left(0.0671\, +0.0054 \zeta ^2\right) \chi ^5+{\cal O}(\chi^6,\zeta^3)\,,\\
\frac{\omega_I}{\omega_I(\chi=0)} &=& 1\, -0.0059 \zeta ^2 \chi -\left(0.0741+0.0066 \zeta ^2\right)
\chi ^2-\left(0.0713-0.0002 \zeta ^2\right) \chi^3\nonumber\\
&&-\left(0.0604-0.0050 \zeta ^2\right) \chi^4-\left(0.0504-0.0079 \zeta ^2\right) \chi^5 +{\cal O}(\chi^6,\zeta^3)\,,\label{geodesic_correspondence}
\end{eqnarray}
\end{widetext}
where $\omega_{R,I}(\chi=0)$ are the corresponding axial modes for a nonspinning EDGB BH shown in Table~\ref{tab:coeff_axial}.
The above expressions are valid up to ${\cal O}(\chi^5,\zeta^2)$ and for any $l=m\gg1$; the only difference enters in the normalization factors on the left-hand side.
The detection of two ringdown modes is necessary to estimate the mass, spin and
coupling $\zeta$. Here we adopt the same Fisher-matrix technique presented in
Ref.~\cite{Cardoso:2016olt}. Let us consider first the axial modes, which are, generically,
excited during the merger phase.
The measurement of the two most dominant modes (which we take to be $l=m=2,3$, excited with a relative amplitude of $3:1$~\cite{Berti:2007zu}) gives us sufficient information to extract the mass and spin of the BH, as well as the coupling parameter of the theory. To do this, we use the geodesic correspondence, Eq.~\eqref{geodesic_correspondence}. {\it Assuming} that both modes are excited to detectable amplitude (this can be quantified using the methods of Ref.~\cite{Berti:2007zu}), then a Fisher-matrix computation allows us to estimate the uncertainties in the parameters determining the ringdown~\cite{Cardoso:2016olt}: $M, \chi, \zeta$. This then allows us to constrain the magnitude of the coupling parameter $\zeta$,
\begin{equation} \zeta \lesssim \frac{3.98 -0.718 \chi+0.181\chi^2-0.045
\chi^3}{\sqrt{\rho}}\,, \label{boundzeta} \end{equation}
where $\rho$ is the signal-to-noise ratio in
the ringdown waveform and the ${\cal O}(\chi^4)$ and ${\cal O}(\chi^5)$ terms
are negligible. The numerator in the expression above is a (mildly) decreasing
function of the spin and ranges from $\approx 4$ to $\approx 3.5$ in the region
$0\leq\chi\approx0.8$.
This analysis can be extended to polar modes. In fact, one might even argue that corrections to polar modes are higher, since the polar QNMs of nonrotating BHs are more affected than the axial, cf. Tables~\ref{tab:coeff_axial} and \ref{tab:coeff}.
However, there is little evidence that they follow the geodesic correspondence,
but we expect that the order of magnitude change in the modes remains the same.
Therefore, it is reasonable to believe that, even when polar modes are excited, a mode analysis of the ringing signal can set constraints on $\zeta$ of the order of those given by Eq.~(\ref{boundzeta}).
As a reference, the final BH spin of GW150914 was $\chi\approx0.67$~\cite{Abbott:2016blz} and $\rho\approx7.7$ in the ringdown part~\cite{TheLIGOScientific:2016src}.
This [assuming, as a first approximation, that Eq.~(\ref{boundzeta}) also holds for polar modes] yields roughly $\zeta\lesssim 1.3$, which is weaker than the theoretical bound~\cite{Pani:2009wy} for the finite-$\zeta$ solution, $\zeta\lesssim0.691$, and therefore meaningless.
This constraint is nevertheless comparable with that derived from the orbital decay rate of low-mass x-ray binaries~\cite{Yagi:2012gp}
and it is slightly larger than the projected bound achievable in the near future from the measurements of quasiperiodic oscillations in the spectrum of accreting BHs~\cite{Maselli:2014fca}, although the latter might be affected by astrophysical systematics.
Our estimate suggests that $\rho\gtrsim25$ in the ringdown waveform is needed to obtain an upper limit which is more stringent than the theoretical bound using GW ringdown detections.
Of course, in order to set these upper limits, an accurate computation of both polar and axial modes of rotating EDGB BHs will be needed.
\section{Discussion and conclusions}
EDGB gravity is a simple and viable higher-curvature correction to GR which predicts BH solutions with scalar charge. Since strong-curvature corrections are suppressed at large distance, it is natural to expect that the most stringent constraints on this theory come from the strong-curvature, highly dynamical regime as the one involved in a BH coalescence. Furthermore, compact stars in this theory possess only a very small scalar charge~\cite{Yagi:2011xp,Kleihaus:2016dui} and therefore EDGB gravity evades the stringent constraints on the dipole radiation coming from current binary-pulsar systems~\cite{Berti:2015itd}.
The estimate~\eqref{boundzeta} translates to the following upper bound on the dimensionful EDGB coupling\footnote{Since one of the parameters of our Fisher-matrix analysis is $\zeta=\alpha/M^2$, propagation of errors implies a relative uncertainty $\delta\alpha/\alpha=\delta\zeta/\zeta+2\delta M/M$. However, in the large-$\rho$ limit the term $\delta M/M$ is negligible because it scales as $1/\rho$, compared to the $1/\sqrt{\rho}$ behavior of the error on $\zeta$ [cf. Eq.~\eqref{boundzeta}]. Therefore, in this limit $\delta \alpha\lesssim \delta \zeta M^2$.},
\begin{equation}
\alpha^{1/2}\lesssim 11 \left(\frac{50}{\rho}\right)^{1/4}\left(\frac{M}{10 M_\odot}\right)\,{\rm km}\,, \label{boundalpha}
\end{equation}
where the prefactor changes by less than $10\%$ depending on the final BH spin. This result is in agreement with the simple estimates derived in Ref.~\cite{Barausse:2014tra}. As a consequence, our analysis also confirms that in most cases modified-gravity effects can be distinguished from environmental effects~\cite{Barausse:2014tra}.
Future GW detectors will greatly increase the signal-to-noise ratio, a large value of which is necessary to perform ringdown tests of the Kerr metric~\cite{Berti:2016lat}. The signal-to-noise ratio of a ringdown waveform scales approximately (among its dependence on other quantities not shown here) as $\rho\sim M^{3/2}/S_n(f)^{1/2}$~\cite{Berti:2005ys}, where $M$ is the final BH mass and $S_n(f)$ is the detector noise power spectral density at a given frequency $f$. The best sensitivity of the future Voyager~\cite{Voyager} and Einstein Telescope~\cite{ET} detectors will be, respectively, roughly a factor of $10$ and a factor of $100$ better than in the first aLIGO observing run at the same optimal frequency $f\sim 10^2\,{\rm Hz}$~\cite{Berti:2016lat}.
Thus, the Einstein Telescope with an optimal design can achieve a signal-to-noise ratio of roughly $\rho\approx 100$ for the ringdown signal of a GW150914-like event.
From Eqs.~\eqref{boundalpha} and~\eqref{boundzeta}, this would translate into the bound $\alpha^{1/2}\lesssim 8 \left(\frac{M}{10 M_\odot}\right)\,{\rm km}$ and $\zeta\lesssim 0.4$. As expected, lighter BHs would provide a significantly more stringent constraint on $\alpha$, although their ringdown frequency might not fall into the optimal frequency range for ground-based detectors. Due to the small exponent of $\rho$ in Eq.~\eqref{boundalpha}, even an increase of $\rho$ of 1 order of magnitude will not provide a significantly more stringent constraint on the EDGB coupling. A stronger constraint may be set if future observations detect a light BH with a very large signal-to-noise ratio.
Given this scenario, electromagnetic observations of accreting BHs (like the one discussed in Ref.~\cite{Maselli:2014fca}) might provide more stringent constraints in the future, although the latter are affected by astrophysical systematics that are absent in the ringdown case.
Our estimates in the case of spinning BHs rely on the geodesic analogy for QNMs, which we verified only for axial modes in the static case and for Kerr BHs with any spin~\cite{Cardoso:2016olt}. It would be interesting to compute the modes of slowly rotating EDGB BHs (e.g. by adapting the methods discussed in Ref.~\cite{Pani:2013pma}) and to check the geodesic approximation in the spinning case. This computation will be required to place precise constraints on the EDGB coupling through future detections of BH ringing with high signal-to-noise ratio.
Another interesting extension of our work concerns the scalar waves emitted during the coalescence. Although the luminosity in scalar waves is significant, this radiation may be possibly detected only if the dilaton is coupled to matter. Such coupling is presumably small and would not give rise to any effects in the detectors. Nonetheless, if the dilaton-matter coupling is non-negligible, the scalar radiation might be investigated through the same techniques developed to study the scalar emission in scalar-tensor theories, e.g. by using a network of ground-based detectors~\cite{hayama}.
\begin{acknowledgments}
V.C. acknowledges financial support provided under the European Union's H2020 ERC Consolidator Grant ``Matter and strong-field gravity: New frontiers in Einstein's theory'' Grant No. MaGRaTh--646597.
C.M. acknowledges financial support from Conselho Nacional de Desenvolvimento Cient\'ifico e Tecnol\'ogico through Grant No.~232804/2014-1.
J.L.B.S., F.S.K. and J.K. gratefully acknowledge support by the DFG Research Training Group 1620 ``Models of Gravity''.
J.L.B.S. and J.K. gratefully acknowledge support by the grant FP7, Marie Curie Actions, People, International Research Staff Exchange Scheme (IRSES-606096).
Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Economic Development and Innovation.
This project has received funding from the European Union's Horizon 2020 research and innovation program under the Marie Sklodowska-Curie Grant No.~690904 and from FCT-Portugal through the Project No. IF/00293/2013.
This work was supported by the H2020-MSCA-RISE-2015 Grant No. StronGrHEP-690904.
\end{acknowledgments}
| -31,146.224273 |
[
-2.73046875,
2.6328125
] | 40.504202 |
[
-2.62109375,
0.6142578125,
-1.7998046875,
-6.57421875,
-1.0205078125,
8.75
] |
[
5.71484375,
8.625,
4.875,
7.7109375
] | 480 | 7,184 |
[
-2.078125,
2.130859375
] | 27.00087 |
[
-5.94140625,
-4.50390625,
-5.01953125,
-2.6875,
1.904296875,
13.15625
] | 1.15188 | 23.677795 | 24.109131 | 5.626969 |
[
2.3067147731781006
] | -20,160.60014 | 5.800529 | -30,355.299692 | 0.129587 | 6.032584 |
[
-2.447265625,
-3.9765625,
-4.05078125,
-5.28125,
2.322265625,
13.03125
] |
[
-5.5,
-2.451171875,
-2.33203125,
-1.8251953125,
3.369140625,
5.2734375
] | |
BkiUcKvxK6-gDz87O8I5
|
\section{INTRODUCTION}
In the past three decades, self-driving cars have witnessed considerable advancements in academic research and automotive development. Driving in highway traffic is a challenging task, even for a human, that requires intelligent decision-making for long-term goals and cautious short-term trajectory planning to execute decisions safely.
Advanced driving assistance system (ADAS) is a hierarchical architecture that incorporates object detection, sensor fusion, planning, and control modules. Automakers and researchers leverage ADAS to approach the autonomous driving problem in a modular manner \cite{ziegler2014making}. Several methods have been proposed for the decision-making of autonomous vehicles on a highway driving task. Most of the studies approached as a control problem \cite{taylor1999comparative, hatipoglu2003automated}. Recently, deep reinforcement learning (RL) approaches have presented a decent alternative to the optimal lane-changing problem \cite{alizadeh2019automated, hoel2018automated}. However, none of the solutions provide a reliable method to translate the generated decisions to safe trajectories.
Trajectory planning, on the other hand, has been addressed by multiple studies. Claussmann et al. \cite{claussmann2019review} distinguish the space configuration for the path planning into three main categories: i.e., sampling points \cite{li2014unified}, connected cells \cite{yu2016semantic}, and lattice representation \cite{werling2010optimal}. Sample-based decompositions normally provide a probabilistic method to randomly sample points from the feasible space and generate obstacle-free roadmaps. Although they are useful in local urban planning, the major drawback is that they do not guarantee that a solution will be found in a finite computation time, which would be disastrous in highway driving. Connected cells create an occupancy grid that is not memory efficient and introduce false indicative occupation with moving obstacles on the highway, making the approach a good option for decision-making but not for planning. On the contrary, lattice in motion planning provides a spatial structure of the path that includes the vehicle motion primitives. Lattice enables predictive planning based on the moving obstacles surrounding ego while considering the kinematic constraints, making this method a feasible representation for trajectory planning. In this work, we have utilized lattice representation to generate candidate trajectories and chose the optimal path among them.
\begin{figure}[!t]
\centering
\includegraphics[width=\linewidth]{figs/ADAS_Architecture.pdf}
\caption{The proposed hierarchical architecture for the long-term decision-making and short-term trajectory planning in the Frenet space.}
\label{fig:ADAS hierarchical arch.}
\end{figure}
The vast majority of the studies focused on generating collision-free trajectories using optimization techniques. For instance, in a recent distinguished work, Bertha-Benz \cite{ziegler2014trajectory} formalized the urban trajectory planning problem as a nonlinear optimization problem with constraint polygons for moving objects and used a Newton-type method to solve the optimal trajectory. Although Bertha's planning approach exhibited a promising outcome in urban driving, it may lack the intelligence and, as a result, lack safety on highway driving since the optimization attempts to find the short-term obstacle-free path, but it does not incorporate long-term goals.
In this work, we have provided a rigorous framework for planning in autonomous driving in the highway driving task. Fig. \ref{fig:ADAS hierarchical arch.} summarizes the overall architecture that has been developed. The architecture addresses long-term decision-making based on the traffic situation to maximize ego driving performance and improve the traffic flow by respecting other drivers' projected decisions. The framework also provides a simple and scalable motion planning algorithm on the Frenet frame \cite{werling2010optimal} to generate safe and feasible polynomial trajectories to address short-term goals. We have introduced a novel obstacle avoidance method for velocity obstacles on the Frenet space that enables the motion planner to explore the driving corridors \cite{bender2015combinatorial} and generate spatiotemporal trajectories. Transferring the calculations to Frenet space makes the driving behavior invariant to the road curvatures and road slopes in three dimensions, which improves the optimization computations significantly and simplifies the cost function manipulation. The framework also includes a scalable supervisor module that controls the safety of the generated decisions and trajectories. The supervisor sends a recalculation command to the modules if an unpredicted situation appears during the path following. This significantly improves the safety and reliability of the algorithm. We have also shown the simplicity of configuring various driving styles using intuitive parameters from the framework that resemble human behavior. We have employed CARLA \cite{dosovitskiy2017carla} as a high-fidelity simulation that correctly reproduces the real-world vehicle dynamics and the city highway design and environment stochasticity.
\section{Framework Hierarchical Architecture}
Fig. \ref{fig:ADAS hierarchical arch.} summarizes the implemented hierarchical architecture in CARLA that incorporates the proposed framework for trajectory planning in the Frenet space. The behavior planner utilizes the sensory measurements and commands high-level actions to the local planner to produce a feasible and optimal trajectory. A feedback controller stabilizes the vehicle dynamics while tracking the commanded trajectory. The framework also includes a supervisor where heuristic functions can be implemented to append multiple layers of safety and reliability, e.g. forward and side collision avoidance systems (CAS) and lane keeping assist (LKA). In the following, we will elaborate on each layer individually.
\subsection{Behavior Planning}
Behavior planner (BP) is a key component in the planning architecture of autonomous driving. It generates a set of high-level driving actions or maneuvers to safely achieve desired driving missions under various driving constraints such as lane keeping, lane change, etc. Behavior planning module generates safe and efficient actions or maneuvers subject to the various constraints such as rules of the road, and the surrounding static and dynamic objects. We use Intelligent Driving Model (IDM) \cite{kesting2010enhanced} as an adaptive cruise control (ACC) and Minimizing Overall Braking Induced by Lane changes (MOBIL) \cite{kesting2007general} algorithms to cover all standard behavior decisions such as track speed, follow leader, decelerate to stop, stop, and lane change as illustrated in Fig. \ref{f:FSM_IDM_MOBIL}.
\begin{figure} [!t]
\centering
\includegraphics[width=\linewidth]{figs/FSM_IDM_MOBIL.png}
\caption{State transitions in the behavioral planning context}
\label{f:FSM_IDM_MOBIL}
\end{figure}
As depicted in Fig. \ref{f:FSM_IDM_MOBIL}, the ego vehicle stays in the current lane with a desired speed computed by IDM module until MOBIL algorithm decides on a lane change. once lane change decision is made by MOBIL algorithm, the state of the ego transitions from cruising to lane change until lane change maneuver is done, then it continues to maintain its current lane in cruising mode.
\subsection{Local Planning}
\label{ss: local planning}
The decision-making layer in ADAS generates long-term decisions before sending them to the local planner (LP). The planner translates the commanded maneuver modes, such as LaneChangeRight and StayOnTheLane, along with the desired speed to optimal trajectories in a variable time-horizon window. The generated trajectories consist of two polynomials of time, concerning lateral and longitudinal movements. An optimization-based algorithm minimizes a user-defined cost function to generate an optimal trajectory. The heuristically manipulated cost function characterizes the commanded maneuver modes while capturing the optimal driving style, such as comfort and safety, and while constraining the trajectories to avoid dynamic obstacles surrounding the ego.
\newline
\subsubsection*{Frenet Frame}
The driving behavior is less variant to the road curvatures than the surrounding actors' dynamic and interactions. Thus, it is more efficient to transform the calculations from the Cartesian coordinate to the Frenet frame\cite{werling2010optimal}, which specifies the vehicle's position in terms of longitudinal displacement along the road's arc ($s$) and the lateral offset from the road shoulder ($d$). Figure \ref{f:frenet frame}. illustrates the vehicle 2D position on the Frenet frame.
\begin{figure}[!h]
\centering
\includegraphics[width=0.9\linewidth]{figs/frenet_frame.pdf}
\caption{Frenet Frame visualization}
\label{f:frenet frame}
\end{figure}
This representation is invariant to the road curvatures. i.e., a vehicle driving on the center of a lane with constant speed would have fixed $d$ and $\dot s$ values - dot indicates the time derivative throughout this paper. Following is the summarized procedure to transform from Cartesian to the Frenet frame,
\begin{itemize}
\item Manually create a global route using a set of $n$ points, $\{(x_1,y_1,z_1), ..., (x_n,y_n,z_n)\}$, on the inertial frame and define
\begin{equation}
\bar x = \{x_1,...,x_n\}, \bar y = \{y_1,...,y_n\}, \bar z = \{z_1,...,z_n\}
\end{equation}
\item Define vector $\bar s$, with length $n$, where each element indicates the distance traveled since the first waypoint, i.e., the $i^{th}$ element is
\begin{equation}
s_i = s_{i-1} + \sqrt{(x_i - x_{i-1})^2 + (y_i - y_{i-1})^2 + (z_i - z_{i-1})^2}
\end{equation}
\item Interpolate three cubic spline curves \cite{maekawa2010curvature} $s_x(s)$, $s_y(s)$, and $s_z(s)$ for $(\bar s, \bar x)$, $(\bar s, \bar y)$, and $(\bar s, \bar z)$ pairs of vectors, respectively.
\end{itemize}
The forward transformation, shown in Fig. \ref{f:frenet frame}, from Frenet ($\mathcal{F}$) to Cartesian ($\mathcal{C}$) can be performed using the calculated splines, i.e,
\begin{equation}
(s_0,d_0)^\mathcal{C} = (x_0,y_0,z_0)
\end{equation}
where,
\begin{align}
& x_0 = s_x(s_0) + d_0 \times \sin (s^{\prime}_x(s_0)) \nonumber \\
& y_0 = s_y(s_0) + d_0 \times \sin (s^{\prime}_y(s_0)) \nonumber \\
& z_0 = s_z(s_0) + d_0 \times \sin (s^{\prime}_z(s_0))
\label{e: frenet to cart}
\end{align}
in which, the prime indicates the derivative w.r.t. variable $s$. There is no analytical solution for the inverse transform. We have utilized the approach introduced in reference \cite{fickenscher2018path} to estimate the Frenet states from Cartesian values by dividing the global path into smaller segments and locally approximating the divergence from the route.
\newline
\subsubsection*{Motion Planning in Frenet Frame}
Now that we defined the forward and inverse transformations, the planner can generate the desired trajectory in the Frenet format. The low-level controller receives the transformed optimal trajectory in the inertial frame and drives the vehicle along the path. At each time-step, the planner generates a set of candidate trajectories $\mathcal{T}=\{\tau_1, ..., \tau_m\}$ known as lattices that are shown in Fig. \ref{f:lattices}.
\begin{figure}[!h]
\centering
\includegraphics[width=0.89\linewidth]{figs/lattices_carla.png}
\caption{Lattices in Frenet frame for one time-step that align with the road curvature. The optimal trajectory is shown in green which takes the ego to the center line \cite{werling2010optimal}.}
\label{f:lattices}
\end{figure}
The trajectories consist of $s(t)$ and $d(t)$, which are polynomials of time along Frenet axes
\begin{equation}
\begin{bmatrix}
s(t) & d(t)
\end{bmatrix} =
\begin{bmatrix}
\sum_{i=0}^{k} c_i t^i & \sum_{i=0}^{r} a_i t^i
\end{bmatrix}
\end{equation}
where, $a$ and $c$ are the polynomial coefficients. The following remark enables us to calculate the polynomial coefficients.
\begin{remark}
For a generated path,
\begin{equation}
\tau_i = \Big\{(s_i(t), d_i(t)) \in \mathbb{R}^2 \hspace{1mm}\vline \hspace{1mm} t \in [t_0, t_f]=[0, T_i]\Big\}
\end{equation}
to be continuous w.r.t. the previous trajectory and to be dynamically feasible, the following conditions must hold
\begin{align}
& \begin{bmatrix} d_i(t_0) & \dot d_i(t_0) & \ddot d_i(t_0) & d_i(t_f) & \dot d_i(t_f) & \ddot d_i(t_f)\end{bmatrix} = \nonumber \\
& \begin{bmatrix} d_{i-1}(T_{i-1}) & 0 & 0 & d_f & 0 & 0 \end{bmatrix}
\end{align}
\begin{align}
& \begin{bmatrix} s_i(t_0) & \dot s_i(t_0) & \ddot s_i(t_0) & \dot s_i(t_f) & \ddot s_i(t_f)\end{bmatrix} = \nonumber \\
& \begin{bmatrix} s_{i-1}(T_{i-1}) & 0 & 0 & v_f & 0 \end{bmatrix}
\end{align}
Here, we have defined $t_0 = 0$ as the initial time, $t_f = T_i$ as the time of arrival, $d_f$ as final lateral position, and $v_f$ as the vehicle velocity along $s$-axis at end of path. Note that we disregarded the lateral velocity at the beginning and the end of trajectories, so they align with the road's arc in both ends. Also, note that we defined six constraints for $d(t)$ and five for $s(t)$, which makes them form quintic ($r=5$) and quartic ($k=4$) polynomials, respectively. Since $t_0$ and $T_{i-1}$ are known values at each time-step, producing lattices boils down to identifying terminal manifolds: arrival time $t_f$, lateral position $d_f$, and speed $v_f$. The set $\mathcal{T}$ is produced by varying these three unknowns within the feasible ranges.
\newline
\end{remark}
\subsubsection*{Numerical Optimization}
Since we generated the lattices, we can select the optimal trajectory $\tau^*$ from the set $\mathcal{T}$. Two kinds of constraints have been leveraged in this section. Hard constraints are utilized to eliminate infeasible trajectories that violate the vehicle's dynamical capabilities or potentially can make a collision. Soft constraints penalize the objective function in terms of safety, reliability, and comfort.
To evaluate the trajectories in terms of the hard constraints and generate the tracking reference for the feedback controller, we should generate higher-order information from each $\tau$, that is trajectories in Cartesian coordinate $\tau^{\mathcal{C}} = \{x(t), y(t), z(t)\}$, curvature $k(t)$, heading angle $\psi(t)$, velocity $v(t)$, acceleration $a(t)$, and jerk $j(t)$. To this end, for each $\tau$ we sample points from $s(t)$ and $d(t)$ polynomials with a constant sampling rate (CARLA's $dt$) to calculate a vector of samples for each variable. We use the following equations for the curvature and heading angle \cite{ziegler2014trajectory}
\begin{equation}
k(t) = \frac{\dot{x}(t) \ddot{y}(t) - \dot{y}(t) \ddot{x}(t)}{\sqrt[3]{\dot{x}(t)^2 + \dot{y}(t)^2}}, \hspace{2mm}
\psi(t) = \arctan (\frac{\dot{y}(t)}{\dot{x}(t)})
\end{equation}
Processing information in these vectors to check for hard constraint violations eliminates infeasible trajectories. To check for collision with dynamic obstacles, we must be able to anticipate the objects' future positions. Since the obstacles are moving vehicles with known states and dynamics, we can propagate the surrounding actors' positions up to the maximum time of horizon, $T_{max}$, in $\mathcal{T}$ and eliminate unsafe lattices that potentially collide with other obstacles,
\begin{equation}
\mathcal{T} = \Big\{\tau \hspace{1mm} \vline \hspace{1mm} \tau \notin \mathcal{U} \Big\}
\end{equation}
where,
\begin{align}
\label{e: unsafe lattices}
\mathcal{U} = \bigg\{ & \tau (s(t), d(t)) \hspace{1mm} \vline \hspace{1mm}
\Big( \exists \hspace{0.5mm} t^\prime \in [0, T]\Big)
\Big( \exists \hspace{0.5mm} o(s_o(t), d_o(t)) \in \mathcal{O} \Big) \nonumber \\
& \sqrt{\Big(s(t^\prime) - s_o(t^\prime)\Big)^2 + \Big(d(t^\prime) - d_o(t^\prime)\Big)^2} < r_c^2 \bigg\}
\end{align}
is the set of unsafe lattices that foreseeably collide with at least one obstacle from the obstacle set $\mathcal{O}$, with $r_c$ being the collision radius. Discovering the existence of $t^\prime$ in eq. \ref{e: unsafe lattices} between two objects is not a trivial problem, since it requires geometrical calculations. Assume that $\tau (s(t), d(t))$ is an arbitrary lattice in $\mathcal{T}$, and $\tau_o (s_o(t), d_o(t))$ is the obstacle's predicted trajectory. The problem is to find the existence of a $t^\prime$ at which the euclidean distance between $\tau$ and $\tau_o$ is less than the collision radius, $r_c$. Here, each trajectory forms a curved cylinder shape with base radius, $r_c$, where, we are checking if two shapes intersect in three-dimensional world. Two trajectories $\tau$ and $\tau_o$ intersect if
\begin{equation}
\rho(t) = \sqrt{\Big(s(t) - s_o(t)\Big)^2 + \Big( d(t) - d_o(t)\Big)^2} - r_c
\end{equation}
has real roots. This can be discovered using Descarte's rule of signs, which indicates, a polynomial can have as many positive roots as it contains changes of sign, when the coefficients are ordered in terms of the exponents. Number of negative real roots can also be found by checking the $\rho(-t)$ polynomial. Repeating the same procedure for all pairs of lattices and obstacles eliminates the unsafe lattices.
This process is basically exploring driving corridors - Fig. \ref{f:driving corridor} - to discover feasible lattices. Driving corridors incorporate the actors $x-y$ positions w.r.t. time. This enables us to find safe spatiotemporal trajectories that pass through the corridors as illustrated in Fig. \ref{f:driving corridor}. The remaining trajectory candidates are examined in terms of velocity
\begin{equation}
\begin{matrix}
v_{min} \leq ||v(t)|| \leq v_{max} & \forall t \in [0, T]
\end{matrix}
\end{equation}
and acceleration
\begin{equation}
\begin{matrix}
0 \leq ||a(t)|| \leq a_{max} & \forall t \in [0, T]
\end{matrix}
\end{equation}
that are safe, dynamically feasible, and legally allowed. The violating candidates get eliminated, which results in an updated set of candidate trajectories, $\mathcal{T}$.
\begin{figure}[!h]
\centering
\includegraphics[width=\linewidth]{figs/Driving_Corridors.pdf}
\caption{Spatiotemporal trajectories for moving obstacles and candidate paths visualized in driving corridors}
\label{f:driving corridor}
\end{figure}
When it comes to the soft constraints, it is possible to design an objective function that incorporates all of the required characteristics for the optimal trajectory, and utilize a numerical optimization to find the best trajectory among the candidates. The cost function to characterize the optimal trajectory $\tau^*$ is defined as
\begin{align}
\mathbf{J}(\tau) = w_{o} J_{o} + w_{v} J_{v} + w_{a} J_a + w_j J_{j} + w_{\dot \psi} J_{\dot \psi}
\label{e: objective function}
\end{align}
where, $w_x$ weights identify the importance of each term. Tuning the coefficients is dependent to the vehicle dynamics and the trade-off between agility, safety, and comfort. In the following we discuss the individual terms in $\mathbf{J}$.
\begin{equation}
J_{o}(\tau) = (d(t) - d_{des})^2
\end{equation}
minimizes the vehicle's lateral offset to the target lane center. $d_{des}$ is a constant value, which, indicates the lateral position of the target lane on the Frenet coordinate. Although, MOBIL has already considered the safety before commanding $d_{des}$, here we append a second layer of safety for the Lane Change actions by incorporating the target lane safety in the cost function. Thus, it is possible for the LP to overlook the BP commands for the sake of safety and/or optimality.
\begin{equation}
J{v} = (v(t) - v_{des}(t))^2
\end{equation}
includes the vehicle speed error in the cost function to drive at the desired speed, which is generated by the BP layer. Similar to target lane, the speed commanded by the IDM can also be overwritten by the LP before being sent to low-level controller. Finally,
\begin{equation}
\begin{matrix}
J_a = a(t) ^ 2,
& J_j = j(t) ^ 2,
& J_{\dot \psi} = \dot{\psi} (t) ^2
\end{matrix}
\end{equation}
suppress the vehicle acceleration, jerk, and yawing rate to improve the safety and comfort.
Similar to Bertha \cite{ziegler2014trajectory}, it is possible to formulate the problem in convex optimization manner to find $\tau^*$ analytically. Although this approach is pragmatic and computationally efficient, the optimization is out of scope of this study. In addition, it is unclear how to incorporate the driving corridors in Bertha setup. Alternatively, we discretize $t_f, d_f,$ and $v_f$ within the feasible ranges and generate lattices $\mathcal{T}$ as shown in Fig.
\ref{f:lattices}. Checking for hard constraints shrinks $\mathcal{T}$ to few trajectories in a crowded traffic. Finally, we utilize a simple linear search within $\mathcal{T}$ to find the optimal trajectory $\tau^*$ that minimizes $\mathbf{J}$ in eq. \ref{e: objective function}.
\subsection{Supervisor}
The supervisor is the highest layer in the hierarchy that can overwrite the output of each submodule or request for recalculation. The supervisor uses mathematical and heuristic functions and restrictions to append another safety layer to the system. Once BP commands the high-level decision, LP generates the trajectory, and the remaining layers work together to drive the ego on the path. To this end, BP and LP execute at lower frequencies than other layers. LP generates the optimal path based on the current situation of the surrounding vehicles. However, the surrounding human/autonomous drivers may perform unexpected maneuvers that jeopardize the safety of the generated trajectory. We employed a copy of the IDM controller (IDM2) with more conservative parameters to avoid forward collisions in the supervisor layer. At each time-step, IDM2 tracks the time to collision (TTC) with the leading and following vehicles in the target lane. If TTC violates the safety threshold, IDM2 raises the safety violation flag, and the supervisor calls the LP to recalculate the trajectory based on the current situation. In addition to this, we also implemented a simple heuristic function that checks the traffic rules violation of the highway maximum speed. This function can be enhanced by supplementing more traffic rules into the function.
\subsection{Low-level Controller}
It is possible to sample from the optimal trajectory $\tau^* = \{s^*(t), d^*(t))\}$ to generate a list of waypoints to track. The queue consists of $m=\frac{T}{dt}$ waypoints, where $T$ is the time of horizon of the path, and $dt$ is the simulation sampling rate. Since vehicle dynamics and the controllers are defined in the Cartesian frame, a Frenet to Cartesian transformation (eq. \ref{e: frenet to cart}) enables the controllers to calculate the desired throttle ($T$) and the steering angle ($\theta$) values. A lateral PID model of the two-point visual controller \cite{salvucci2004two} is utilized to stabilize the steering angle. At each time-step, the controller pops the next two consecutive waypoints from the queue and calculates the desired steering angle. Since the waypoints are time labeled, the reference acceleration can be extracted from the list at each time-step. A longitudinal PID controller stabilizes the vehicle acceleration and speed by producing the desired throttle value.
\section{EXPERIMENTS}
This section provides a comprehensive evaluation and performance measurement for the proposed framework on CARLA v0.9.9.2 high-fidelity simulation environment. We selected TESLA model 3 as the ego dynamics and the highway loop available in TOWN04 to evaluate the proposed agents' performances. Trivially, the proposed agent's driving style is highly dependent on the parameter tuning of each layer of the hierarchy. This compromise introduces a trade-off between safety, comfort, and speed, i.e., fast agents tend to drive aggressively and vice versa. Fig. \ref{fig:agility trade-off} illustrates the trade-off and proposes a simple approach to achieve various driving styles by modifying only two parameters from IDM and MOBIL. Considering the measurements uncertainty, small safe-time-headway values can potentially cause accidents. The "Collision" region is provided to address this issue and prevent the user to chose driver parameter inside this region. The framework's simplicity and the proposed trade-off enabled us to introduce three different configurations: Agile, Moderate, and Conservative drivers. As the names suggest, each agent offers a different approach in the speed-maximization-lane-change-minimization trade-off. The remaining parameters are identical for all agents, however, the framework provides complete freedom in modifying parameters in all layers and designing various driving behaviors that match human characteristics. We utilized a common-sense method to tune the optimization soft constraint coefficients. Parameters that characterize hard constraints are selected according to the vehicle dynamics ($a_{max}$), highway regulations ($v_{max}$), and the safety criteria ($r_c$ and $v_{min}$).
\begin{figure}
\centering
\includegraphics[width=0.8\linewidth]{figs/agility_trade-off.pdf}
\caption{The Framework's trade-off between safety, agility, and the politeness among various driving styles}
\label{fig:agility trade-off}
\end{figure}
\subsection{Qualitative Analyses}
This section provides a qualitative analysis based on the drivers' performance in several case studies. The scenarios cover situations where a self-driving car may commonly face in highway driving task, in addition, they target the intelligence, maneuverability, and the safety of the agents.
\subsubsection*{Case Study 1 (Intelligence-Safety)}
The scenario starts with a situation where the traffic flows on each lane at a different speed. The ego is driving on the second lane at 20 m/s. The lanes speed increase from right to left, making lane four the fastest passage. As illustrated in Fig. \ref{fig:case studies}, Agile and Moderate drivers make two consecutive lane changes to reach the fastest lane. However, the Conservative driver stays in the current lane until the left lane traffic becomes less sever then makes the lane change to left. Since lane three traffic is dense and moves faster than ego lane, a safe merging maneuver would take ego to the adjacent lane, however, the traffic in lane three would become temporarily slower. Agile and Moderate agents prefer the speed gain to politeness, making them slow down the traffic flow temporarily but finish the track much earlier than the Conservative agent. The scenario shows how the agents perform strategic and safe merging maneuvers to navigate through the traffic and eventually gain speed. This scenario also demonstrates the drivers' different approaches toward the agility-politeness trade-off, which shows the framework's compatibility to implement various driving styles by tuning only two parameters from the framework (Fig. \ref{fig:agility trade-off}).
\begin{figure}[!t]
\centering
\includegraphics[width=\linewidth]{figs/Case_studies.pdf}
\caption{ Qualitative results for four proposed drivers in two case studies. Arrows show the agents tracked trajectories for two consecutive steps. Transparent regions show the convex shape of the generated lattices (recorded video of the agents' performances are submitted as supplementary files for review)}
\label{fig:case studies}
\end{figure}
\subsubsection*{Case Study 2 (Maneuverability)}
Safely navigating through the cars to escape a traffic cluster requires situational awareness to perform a complex sequence of escaping maneuvers. In this scenario, the ego is tailgating a vehicle in lane three with 22 m/s. The traffic after this vehicle in lane three is smooth. A vehicle in lane four is moving slightly faster with 23 m/s. The traffic in lanes one and two are dense and slow because of an upcoming exit. Fig. \ref{fig:case studies} shows that Moderate and Conservative drivers move to the slightly faster lane on the left and keep driving there. Agile driver, on the other hand, performs an overtaking maneuver that assists the driver to escape the traffic and gain speed eventually. The MP lattices for two consecutive steps of agile agent are highlighted in yellow regions. While driving in lane four, most of the lattices that end up in lane three potentially violate the hard constraints (collision with moving obstacles). The remaining lattices keep a short safe-time-headway in IDM and a small politeness factor in MOBIL. Following those trajectories would require an aggressive maneuver, which may disrupt the traffic in lane three. The agile agent favors these lattices to trajectories that stay in lane four because of the driver's nature. This scenario confirms how the framework performs a complex sequence of maneuvers to achieve the desired driving style.
\subsubsection*{Quantitative Analysis}
In the qualitative analyses, we showed that the framework's situational awareness helped the drivers to generate complex trajectories and the maneuverability of the architecture enabled the agents to follow the generated trajectories. We also showed the qualitative differences between the driver's behavior in two case studies. Here we compare the driver's performance on randomly generated scenarios. The following quantitative metrics (percentages) have been used to compare the agents' performance on all of the scenarios,
\begin{itemize}
\item Speed = $100(1 - \frac{\vert \text{average speed} - \text{target speed}\vert}{\text{max error}})$
\item Comfort = $100 (1-w_c\frac{\text{average jerk}}{\text{max jerk}} - (1-w_c)\frac{\text{average yaw rate}}{\text{max yaw rate}})$
\item Safety = $\frac{100}{n}\sum_{i=1}^{n} (1 - \frac{\text{min TTC}}{TTC_i})$
\end{itemize}
where $w_c$ is used to weigh the importance of factors and TTC stands for the frontal time-to-collision experiences. In the Safety equation we included $n$ steps of each scenario where ego tailgates a vehicle (TTC exists). For each scenario, the vehicles are spawned in random relative positions and target speeds to the ego. The scenarios start in an arbitrary position on the highway track, and the track length is 500 meters. Surrounding vehicles are randomly selected from a vehicle pool in CARLA, each one having different dynamics. We evaluated the agents' performance in 1000 scenarios and recorded the results in Table \ref{tab:random scenarios}. Overall, the agile driver showed a better approach toward gaining speed; however, it lacks safety and comfort. The safety issue becomes significant if the uncertainty level increases in the measurements. In contrast, the Conservative driver performed a more beneficial approach to safety and comfort but drove slow in most cases. The Moderate driver has displayed a satisfactory performance based on the provided metrics. This made the Moderate agent exhibit a better average percentage for all metrics in comparison with other drivers. Trivially, it is possible to migrate the Moderate driver point in Fig. \ref{fig:agility trade-off} to a desired sweet spot inside the region that matches the human driver's style. This demonstrates the flexibility of the proposed framework.
\begin{table}
\centering
\begin{tabular}{c | c c c}
& Conservative & Moderate & Agile\\
\hline
Speed
& 53 \DrawBlackPercentageBar{0.53} & 74 \DrawBlackPercentageBar{0.74} & 81 \DrawRedPercentageBar{0.81}\\
Safety
& 60 \DrawRedPercentageBar{0.60} & 52 \DrawBlackPercentageBar{0.49}& 28 \DrawBlackPercentageBar{0.28}\\
Comfort
& 83 \DrawRedPercentageBar{0.83} & 75 \DrawBlackPercentageBar{0.70} & 47 \DrawBlackPercentageBar{0.47}\\
\hline
Average & 65 \DrawBlackPercentageBar{0.65} & 67 \DrawRedPercentageBar{0.67} & 52 \DrawBlackPercentageBar{0.52}\\
\end{tabular}
\caption{Performance metrics as percentages for three driving styles in 1000 randomly generated test scenarios}
\label{tab:random scenarios}
\end{table}
\section{CONCLUSIONS}
In this paper, we introduced a hierarchical framework for decision-making and planning on highway driving tasks. IDM and MOBIL driving models have been utilized to maximize ego performance as well as the politeness wrt other drivers. A novel obstacle avoidance approach is introduced on the Frenet frame for the moving obstacles. The optimization explores the driving corridors to generate spatiotemporal polynomial trajectories to navigate through the traffic safely and obey the BP commands. The framework also introduced a heuristic supervisor that identifies unexpected situations and recalculates each module in case of a potential emergency. Experiments in CARLA simulation have shown the promising performance and the scalability of the framework in implementing various driving styles that match human behavior.
\bibliographystyle{IEEEtran}
| -18,391.669152 |
[
-3.263671875,
3.083984375
] | 24.06639 |
[
-3.748046875,
0.1727294921875,
-2.3359375,
-5.765625,
-0.34765625,
8.3359375
] |
[
3.44140625,
6.94921875,
2.9453125,
8.6875
] | 266 | 4,403 |
[
-2.859375,
3.5390625
] | 23.348602 |
[
-6.375,
-4.13671875,
-4.3359375,
-2.013671875,
2.671875,
12.015625
] | 0.544281 | 12.72477 | 28.707699 | 1.717483 |
[
2.645686149597168
] | -12,998.790052 | 5.841926 | -17,961.284291 | 0.544281 | 6.041848 |
[
-3.1015625,
-3.484375,
-3.439453125,
-4.45703125,
2.703125,
11.359375
] |
[
-5.4921875,
-1.884765625,
-2.4609375,
-1.6240234375,
3.634765625,
4.9921875
] | |
BkiUduE4eIXgu5sT-Z3g
|
\section{Introduction}
Time series with defined autocovariance functions are said to present long-memory or long-range dependency when their autocovariance function is decreasing very slowly, slower than an exponential decay. More precisely, let $g(\cdot)$ be the autocovariance function of a time series $X$. $X$ is said to be long-memory if there exists $\alpha$, $0<\alpha<1$, such that $g(t)$ is asymptotically equivalent to $|t|^{-\alpha}$ when $t \to +\infty$ (see \cite{beran.1994.1} and references therein). This definition implies that the covariance function is not summable. Equivalently, the spectral density $f(\cdot)$, if it exists, is such that $f(\lambda)$ is equivalent up to a constant to $|\lambda|^{1-\alpha}$ when $\lambda\to 0^+$. In this case, when trying to estimate the expectation using the empirical mean of long-memory time series, the variance of the estimator is not decreasing to 0 as $N^{-1}$ (where $N$ is the sample size). Hence it is crucial to take into account the presence of long-memory for defining good estimators \citep{beran.1994.1}. In the case of univariate time series, several very efficient approaches have been developed and validated. A web page entitled ``Time Series Analysis''\footnote{\url{https://cran.r-project.org/web/views/TimeSeries.html}} from the \textsf{R} software is providing a very exhaustive list of methods and softwares dealing with long-memory time series. Among others, we can cite the packages \pkg{fracdiff} \citep{haslett1989space}, \pkg{arfima} and \pkg{FGN} \citep{veenstra.2012.arfimaR}, \pkg{longmemo} \citep{beran.1994.1} and \pkg{forecast} \citep{hyndman.2008.JSS.forecastR}. For example, \pkg{fracdiff} is dedicated to simulation of fractional ARIMA time series and to estimation using regression of the periodogram. \pkg{longmemo} provides real data examples of time series with long-memory properties.
Approaches for multivariate long-memory time series are less developed. When dealing with multivariate time series, an important quantity to estimate is the covariance or correlation between pairs of time series. The effect of the presence of long-memory on this estimation is obvious, as stated by \cite{RobinsonDiscussion}. One \textsf{R} package, \pkg{waveslim}, is dedicated to the wavelet correlation analysis for pairs of random variables \citep{whitcher.2000.2} but long-range dependence properties are not considered. \cite{SelaHurvich2012} provide \textsf{R} code\footnote{freely available \url{http://pages.stern.nyu.edu/~rsela/VARFI/code.html}} for bivariate long-range dependent time series with parametric estimations. The objective of this paper is to provide an efficient \textsf{R} package, called \pkg{multiwave}, to estimate the long-memory parameters and covariance matrices for multivariate time series. The estimation procedures are based on a semi-parametric approach, which is robust to model misspecification.
The procedures are also suited for dealing with more than two dimensional data. Indeed they are based on Whittle approximation which provides a simple function to optimize. This function can be used for any dimension of the problem. In comparison, regression of the scalogram or periodogram \citep{Achard08} is based on a linear fit of pairs of time series, and thus there does not exist an easy way to extend to more than two dimensions.
\pkg{multiwave} package is based on \cite{AchardGannaz2014}, where we developed a wavelet-based approach using Whittle approximation for an efficient estimation of the long-memory parameters and the long-run covariance matrices. In addition, \pkg{multiwave} proposes an implementation of an alternative method using Fourier decomposition as described in \cite{Shimotsu07}\footnote{code available in \textsf{matlab} \url{http://shimotsu.web.fc2.com/Site/Matlab_Codes.html}}.
As it is described in this paper, \pkg{multiwave} is a very versatile package and opens the way to estimation of the long-memory parameters and the long-run covariance matrices using multivariate data sets. It is in particular not necessary to assume the stationarity of the time series as it is the case when using Fourier decomposition \citep{FayMoulinesRoueffTaqqu}. The Whittle approximation is computed using either the coefficients of wavelet decomposition or the coefficients of Fourier decomposition when the time series are stationary.
The package \pkg{multiwave} is divided in three parts. A first group of functions is dedicated to the simulation of multivariate long-memory time series; the main function is \code{fivarma}. A second group of functions is implementing the wavelet decomposition, through \code{DWTexact} and associated functions.
Finally the computation of the estimators are coded using the Fourier decomposition in \code{mfw} and its derivatives
and using the wavelet decomposition in \code{mww} and its derivatives.
The mathematical background is detailed in a separate Section \ref{sec:model}. The rest of the paper is dedicated to the description of the package \pkg{multiwave}. Simple examples of parametric models and real data are presented in Section \ref{sec:data} with corresponding functions of \pkg{multiwave} ready to apply. Core estimation functions using wavelets and Fourier transform are detailed in Section \ref{sec:estimation} along with pieces of code using simulated time series. Finally, practical considerations are discussed in the three last sections. Section \ref{sec:discussions} is discussing the practical choices of parameters. Comparaisons between wavelets and Fourier approaches are described in Section \ref{sec:simu}. And an application to real data in neuroscience is concluding the paper, Section \ref{sec:realdata}.
\section{Theoretical background}
\label{sec:model}
As in the univariate case, the definition of long-memory for a $p$-vector process is based on the asymptotic behaviour of the cross-spectral density in the neighbourhood of zero \citep{Moulines07SpectralDensity}. We consider $N$ observations of a long-memory $p$-vector process $\mbox{${\mathbf X}$}=\{X_{\ell}(k),k\in\mathbb{Z}, \ell=1,\dots,p\}$, namely $\mbox{${\mathbf X}$}(1),\dots \mbox{${\mathbf X}$}(N)$. $\mbox{${\mathbf X}$}$ is said to be a multivariate $M(\mbox{${\mathbf d}$})$ process when for each $\ell=1,\dots,p$ there exists $D_\ell\in\mathbb{N}$ such that the $D_\ell$-th order difference $\Delta^{D_\ell}X_\ell$ is covariance stationary. In addition, let us assume that for any $\ell,m=1,\dots,p$ the generalized cross-spectral density of $X_\ell$ and $X_m$ is
\begin{equation}
\label{eqn:density}
f_{\ell,m}(\lambda)=\frac{1}{2\pi}\Omega_{\ell,m}(1-e^{-i\lambda})^{-d_\ell}(1-e^{i\lambda})^{-d_m}f_{\ell,m}^S(\lambda),\qquad \lambda\in[-\pi,\pi],
\end{equation}
with $\mbox{\boldmath$\Omega$}=(\Omega_{\ell,m})_{\ell,m=1,\dots,p}$ an Hermitian matrix. The functions $f_{\ell,m}^S(\cdot)$ correspond to the short-memory dynamics of the process. The parameters $d_\ell$ satisfies $-1/2<d_\ell-D_\ell<1/2$.
More generally, the wavelet-based procedure is available for cross-spectral density satisfying an approximation
\begin{equation}\label{eqn:approx}\mbox{${\boldsymbol{f}}$}(\lambda)\sim \mbox{\boldmath$\Lambda$}(\mbox{${\mathbf d}$}) \mbox{\boldmath$\Omega$} \mbox{\boldmath$\Lambda$}(\mbox{${\mathbf d}$})^\ast,\quad \text{~~when~~}\lambda\to 0, \quad \text{~~with~~} \mbox{\boldmath$\Lambda$}(\mbox{${\mathbf d}$})=\mbox{diag}(|\lambda|^{-\mbox{${\mathbf d}$}}e^{-i \,\text{sign}(\lambda)\pi \mbox{${\mathbf d}$}/2}),
\end{equation}
where the exponent $\ast$ denotes the conjugate transpose operator. Here and subsequently $\sim$ means that the ratio of the left- and right-hand sides converges to one. Note that, the process $X_\ell$ is not necessarily stationary.
The long-range dependence parameter measures the power-like rate of decay of the autocovariance function. The long-run covariance matrix $\mbox{\boldmath$\Omega$}$ can be seen as the covariance at low frequencies between the time series. It gives a quantification of the link between the components of the multivariate time series. The long-run covariance parameter of the model is free from the difference in the autocorrelation behaviour of each component. It is linked with long-run correlations $\left(\Omega_{\ell,m}/\sqrt{\Omega_{\ell,\ell}\Omega_{m,m}}\right)_{\ell,m=1,\dots,p}$, which are also encountered in literature as power-law coherencies between two time series \citep{SelaHurvich2012} or as fractal connectivities \citep{Achard08}.
\subsection{A parametric example: FIVARMA}
\label{sec:fivarma}
Fractionally Integrated Vector Auto Regressive Moving Average (FIVARMA) processes are parametric models with a spectral density satisfying approximation~\eqref{eqn:approx}. They correspond to Model A of \cite{Lobato97}. We refer to this paper for a detailed mathematical description.
Let $\mbox{${\mathbf u}$}$ be a $p$-dimensional white noise with $\mathbb{E}[\mbox{${\mathbf u}$}(t)\mid \mathcal F_{t-1}]=0$ and $\mathbb{E}[\mbox{${\mathbf u}$}(t)\mbox{${\mathbf u}$}(t)^T\mid \mathcal F_{t-1}]=\mbox{\boldmath$\Sigma$}$, where $\mathcal F_{t-1}$ is the $\sigma$-field generated by $\{\mbox{${\mathbf u}$}(s),\,s<t\}$, and $\mbox{\boldmath$\Sigma$}$ is a positive definite matrix.
Let $(\mbox{${\mathbf A}$}_k)_{k\in\mathbb{N}}$ be a sequence of $\mathbb{R}^{p\times p}$-valued matrices with $\mbox{${\mathbf A}$}_0$ the identity matrix and $\sum_{k=0}^\infty \|\mbox{${\mathbf A}$}_k\|^2<\infty$. The discrete Fourier transform of the sequence is denoted $\mbox{${\mathbf A}$}(\cdot)$, that is $\mbox{${\mathbf A}$}(\lambda)=\sum_{k=0}^{\infty} \mbox{${\mathbf A}$}_k e^{i k\lambda}$. We assume that all the roots of $|\mbox{${\mathbf A}$}(\mathbb{L})|$ are outside the closed unit circle, where $\mathbb{L}$ denotes the lag operator. Let also $(\mbox{${\mathbf B}$}_k)_{k\in\mathbb{N}}$ be a sequence in $\mathbb{R}^{p\times p}$ with $\mbox{${\mathbf B}$}_0$ the identity matrix and $\sum_{k=0}^\infty \|\mbox{${\mathbf B}$}_k\|^2<\infty$. As defined for $\mbox{${\mathbf A}$}$, $\mbox{${\mathbf B}$}(\cdot)$ denotes the discrete Fourier transform of the sequence, $\mbox{${\mathbf B}$}(\lambda)=\sum_{k=0}^{\infty} B_k e^{i k\lambda}$.
Let $\mbox{${\mathbf X}$}$ be defined by
\begin{equation}
\label{formula:cointegration}
\mbox{${\mathbf A}$}(\mathbb{L})\,\mbox{diag}(\mbox{1\hspace{-.35em}1}-\mathbb{L})^{\mbox{${\mathbf d}$}}\,\mbox{${\mathbf X}$}(t)=\mbox{${\mathbf B}$}(\mathbb{L})\mbox{${\mathbf u}$}(t).
\end{equation}
The spectral density satisfies
\[
f_{\ell,m}(\lambda)\sim_{\lambda\to 0^+} \frac{1}{2\pi}\Omega_{\ell,m}e^{-i\pi/2(d_\ell-d_m)}\lambda^{-(d_\ell+d_m)}
\]
with \begin{equation}
\label{eqn:omega}
\mbox{\boldmath$\Omega$}=\mbox{${\mathbf A}$}(1)^{-1}\mbox{${\mathbf B}$}(1)\mbox{\boldmath$\Sigma$}\mbox{${\mathbf B}$}(1)^T{\mbox{${\mathbf A}$}(1)^T}^{-1}.
\end{equation} Then $\mbox{${\mathbf X}$}$ is called a $FIVARMA(d,q)$ process and satisfies approximation~\eqref{eqn:approx}.
\subsubsection*{Limits of the model}
Note that in definition \eqref{formula:cointegration} the operators are applied in a given order, where the lag operator is taken first. Changing the order of the lag operator and autoregression corresponds to model $B$ of \cite{Lobato97} and $VARFI$ models of \cite{SelaHurvich2012} where $\mbox{${\mathbf X}$}$ is obtained with equation $\mbox{diag}(\mbox{1\hspace{-.35em}1}-\mathbb{L})^{\mbox{${\mathbf d}$}}\,\mbox{${\mathbf A}$}(\mathbb{L})\,\mbox{${\mathbf X}$}(t)=\mbox{${\mathbf B}$}(\mathbb{L})\mbox{${\mathbf u}$}(t)$, with similar notations than above.
That is, $\mbox{${\mathbf X}$}$ is obtained by fractional integration after autoregression, which is also called \textit{cointegration}. The spectral density still satisfies the approximation \eqref{eqn:approx} however the matrix $\mbox{\boldmath$\Omega$}$ may no longer be Hermitian.
Clearly \pkg{multiwave} package is not built to deal with such cases. We refer to alternative methods in literature, among others \cite{Robinson08, SelaHurvich2012, Shimotsu2012}. Taking into account cointegration is a difficult problem that exceeds the scope of this paper. Future work is needed to handle this particular case.
\subsection{Fourier-based estimation (MFW)}
The discrete Fourier transform and
the periodogram of $\mbox{${\mathbf X}$}$ evaluated at frequency $\lambda$ are defined as in \cite{Shimotsu07}'s procedure
\begin{align*}
\mbox{${\mathbf W}$}^F(\lambda)&= \frac{1}{\sqrt{2\pi N}} \sum_{t=1}^N \mbox{${\mathbf X}$}(t)e^{i t\lambda},\\
\mbox{${\mathbf I}$}^F(\lambda) &= \mbox{${\mathbf W}$}^F(\lambda)\mbox{${\mathbf W}$}^F(\lambda)^\ast.
\end{align*}
Let $\lambda_j = 2\pi j/N$, $j=1, \dots, m$, be the Fourier frequencies used in estimation, $m\in\mathbb{N}$. Define $\mbox{\boldmath$\Lambda$}_j^F(\mbox{${\mathbf d}$})=\mbox{diag}\left(\lambda_j^{\mbox{${\mathbf d}$}}
e^{i(\pi-\lambda_j)\mbox{${\mathbf d}$}/2}\right)$.
The estimators $(\hat\mbox{${\mathbf d}$}^{MFW},\hat\mbox{\boldmath$\Omega$}^{MFW})$ are minimizers of the criterion
\begin{multline*}
\mathcal L^{MFW}(\mbox{${\mathbf d}$},\mbox{\boldmath$\Omega$})=\\
\frac{1}{m}\sum_{j=1}^{m} \left[ \log\det\left(\mbox{\boldmath$\Lambda$}_j^F(\mbox{${\mathbf d}$})\mbox{\boldmath$\Omega$}(\mbox{${\mathbf d}$})
\mbox{\boldmath$\Lambda$}_j^F(\mbox{${\mathbf d}$})^\ast\right)+\mbox{${\mathbf W}$}^{F}(\lambda_j)^{\ast}\left(\mbox{\boldmath$\Lambda$}_j^F(\mbox{${\mathbf d}$})\mbox{\boldmath$\Omega$}(\mbox{${\mathbf d}$})
\mbox{\boldmath$\Lambda$}_j^F(\mbox{${\mathbf d}$})^{\ast}
\right)^{-1}\mbox{${\mathbf W}$}^F(\lambda_j)\right].
\end{multline*}
The solution satisfies
\begin{align}
\label{eqn:dF}
\hat\mbox{${\mathbf d}$}^{MFW}&=\displaystyle \mathop{argmin}_{\mbox{${\mathbf d}$}} \log\det(\hat \mbox{\boldmath$\Omega$}^{MFW}(\mbox{${\mathbf d}$})) - 2\log(2)\left(\frac{1}{m}\sum_{j=1}^{m} \lambda_j\right),\\
\label{eqn:GF}
\hat \mbox{\boldmath$\Omega$}^{MFW}&=\hat \mbox{\boldmath$\Omega$}^{MFW}(\hat\mbox{${\mathbf d}$}^{MFW}),\\
\nonumber
\text{with~} \hat \mbox{\boldmath$\Omega$}^{MFW}(\mbox{${\mathbf d}$})& =\frac{1}{m} \sum_{j=1}^{m} Re\left(\mbox{\boldmath$\Lambda$}_j^F(\mbox{${\mathbf d}$})^{-1}
\mbox{${\mathbf I}$}^F(j)\mbox{\boldmath$\Lambda$}_j^F(\mbox{${\mathbf d}$})^{-1}\right).
\end{align}
The dynamics of the frequencies at the neighbourhood of the origin is given by the dynamics of the spectral density around the zero frequency. The form of the criterion is justified by a second-order approximation of the spectral density matrix \eqref{eqn:density}, rather than the approximation \eqref{eqn:approx}. For an estimation based on the first-order approximation \eqref{eqn:approx}, one should replace $\mbox{\boldmath$\Lambda$}_j^F(\mbox{${\mathbf d}$})$ by $\mbox{\boldmath$\Lambda$}_j^{F\,(1)}(\mbox{${\mathbf d}$})=\mbox{diag}\left(\lambda_j^{\mbox{${\mathbf d}$}}
e^{i\pi\mbox{${\mathbf d}$}/2}\right)$.
\cite{Shimotsu07} established the theoretical performance of this estimation procedure, for both the long-range dependence parameters and the long-run covariance matrix. It is shown that the variance for the estimation of the vector $\mbox{${\mathbf d}$}$ is decreased for the multivariate procedure with respect to a univariate one. It is worth mentioning that \cite{Lobato99} developed a similar estimation procedure, based on a rougher approximation of the cross-spectral density, $\mbox{\boldmath$\Lambda$}_j^{F}(\mbox{${\mathbf d}$})=\mbox{diag}\left(\lambda_j^{\mbox{${\mathbf d}$}}\right)$. Interestingly, the quality of estimation for the vector $\mbox{${\mathbf d}$}$ is similar. Nevertheless, the estimation of the long-run covariance matrix $\mbox{\boldmath$\Omega$}$ is biased since it does not take into account the phase-shift appearing in $\mbox{\boldmath$\Lambda$}_j^F(\mbox{${\mathbf d}$})$. We refer to \cite{Lobato99} and to \cite{Shimotsu07} for a more detailed study of these estimators and their consistency.
\subsection{Wavelet-based estimation (MWW)}
Wavelets are providing a very efficient tool because of their high flexibility to deal with nonstationary time series which is particularly useful for real data applications. Their good performances in comparison to Fourier have already been shown for example in univariate settings \citep{FayMoulinesRoueffTaqqu}.
Let $(\phi(\cdot),\psi(\cdot))$ be respectively a father and a mother wavelets, satisfying regularity conditions, as stated in \cite{AchardGannaz2014}.
At a given resolution $j\geq 0$, for $k\in\mathbb{Z}$, we define the dilated and translated functions $\phi_{j,k}(\cdot)=2^{-j/2}\phi(2^{-j}\cdot -k)$ and $\psi_{j,k}(\cdot)=2^{-j/2}\psi(2^{-j}\cdot -k)$. The wavelet coefficients of the process $\mbox{${\mathbf X}$}$ are defined by $$\mbox{${\mathbf W}$}_{j,k}=\int_\mathbb{R} \tilde{\mbox{${\mathbf X}$}}(t)\psi_{j,k}(t)dt\quad j\geq 0, k\in\mathbb{Z},$$ where $\tilde{\mbox{${\mathbf X}$}}(t)=\sum_{k\in\mathbb{Z}}\mbox{${\mathbf X}$}(k)\phi(t-k).$
For given $j\geq 0$ and $k\in\mathbb{Z}$, $\mbox{${\mathbf W}$}_{j,k}$ is a $p$-dimensional vector $\mbox{${\mathbf W}$}_{jk}=\begin{pmatrix}
W_{j,k}(1) & W_{j,k}(2) & \dots & W_{j,k}(p) \end{pmatrix}$ where $W_{j,k}(\ell)= \int_\mathbb{R} \tilde{\mbox{${\mathbf X}$}_\ell}(t)\psi_{j,k}(t)dt$.
For any $j\geq 0$, the process $(\mbox{${\mathbf W}$}_{j,k})_{k\in\mathbb{Z}}$ is covariance stationary \citep{AchardGannaz2014}. Let $\theta_{\ell,m}(j)$ denote the wavelet covariance at scale $j$ between processes $X_\ell$ and $X_m$, {\it i.e.} $\theta_{\ell,m}(j)=Cov(W_{j,k}(\ell), W_{j,k}(m))$ for any position $k$. Let us introduce the wavelet scalogram
\begin{equation}\label{eqn:scalogram}
\mbox{${\mathbf I}$}^W(j)=\sum_{k\in\mathbb{Z}} \mbox{${\mathbf W}$}_{j,k}\mbox{${\mathbf W}$}_{j,k}^T.
\end{equation}
The wavelet scalogram is the equivalent of the Fourier periodogram. Yet the scalogram is not normalized, on the contrary of the periodogram. We also introduce the function $K(\cdot)$, defined as
\begin{equation}
\label{eqn:K}
K(\delta)=\int_{-\infty}^{\infty}{|\lambda|^{-\delta}|\hat\psi(\lambda)|^2\,d\lambda},\quad\delta\in(-\alpha,M).
\end{equation}
The wavelet Whittle procedure is described in \cite{AchardGannaz2014}. Let $\mbox{\boldmath$\Lambda$}_j(\mbox{${\mathbf d}$})=\mbox{diag}\left(2^{j\mbox{${\mathbf d}$}},\, j_0\leq j\leq j_1\right)$. Let $\mbox{${\mathbf G}$}(\mbox{${\mathbf d}$})$ denote a $p\times p$-matrix with $(\ell,m)$-{th} element equal to
\begin{equation}
\label{eqn:Gdef}
G_{\ell,m}(\mbox{${\mathbf d}$})=f^S(0)\Omega_{\ell,m}K(d_\ell+d_m)cos({\pi(d_\ell-d_m)/2}).
\end{equation}
The estimators $(\hat\mbox{${\mathbf d}$}^{MWW},\hat\mbox{${\mathbf G}$}^{MWW})$ are defined by minimization of $\mathcal L^{MWM}(\mbox{${\mathbf d}$},\mbox{${\mathbf G}$})$, with
\begin{multline*}
\mathcal L^{MWM}(\mbox{${\mathbf d}$},\mbox{${\mathbf G}$})=\\ \frac{1}{n}\sum_{j=j_0}^{j_1} \left[ n_j \log\det\left(\mbox{\boldmath$\Lambda$}_j^W(\mbox{${\mathbf d}$})\mbox{${\mathbf G}$}(\mbox{${\mathbf d}$})\mbox{\boldmath$\Lambda$}_j^W(\mbox{${\mathbf d}$})\right)+\sum_k \mbox{${\mathbf W}$}_{j,k}^T\left(\mbox{\boldmath$\Lambda$}_j^W(\mbox{${\mathbf d}$})\mbox{${\mathbf G}$}(\mbox{${\mathbf d}$})\mbox{\boldmath$\Lambda$}_j^W(\mbox{${\mathbf d}$})
\right)^{-1}\mbox{${\mathbf W}$}_{j,k}\right].
\end{multline*}
The estimation is here based on a first-order approximation of the spectral density matrix around 0.
The solutions of this problem satisfy
\begin{align}
\label{eqn:dhat}
\hat\mbox{${\mathbf d}$}^{MWW}&=\displaystyle \mathop{argmin}_{\mbox{${\mathbf d}$}} \log\det(\hat \mbox{${\mathbf G}$}^{MWW}(\mbox{${\mathbf d}$})) + 2\log(2)\left(\frac{1}{n}\sum_{j=j_0}^{j_1} j n_j\right)\left(\sum_{\ell=1}^p d_\ell\right),\\
\label{eqn:G}
\hat \mbox{${\mathbf G}$}^{MWW}(\mbox{${\mathbf d}$}) &=\frac{1}{n} \sum_{j=j_0}^{j_1} \mbox{\boldmath$\Lambda$}_j^W(\mbox{${\mathbf d}$})^{-1}
\mbox{${\mathbf I}$}^W(j)\mbox{\boldmath$\Lambda$}_j^W(\mbox{${\mathbf d}$})^{-1},
\end{align}
where $\mbox{${\mathbf I}$}^W(j)$ is the wavelet scalogram at scale $j$ defined in~\eqref{eqn:scalogram}. The long-run covariance matrix can then be estimated by
\begin{equation}
\label{eqn:Ohat}
\hat\Omega_{\ell,m}^{MWW}=\hat G_{\ell,m}^{MWW}(\hat \mbox{${\mathbf d}$}^{MWW})/(\cos(\pi(\hat d_\ell^{MWW}-\hat d_m^{MWW})/2) K(\hat d_\ell^{MWW}+\hat d_m^{MWW})).
\end{equation}
This second step in the estimation of the long-run covariance matrix $\mbox{\boldmath$\Omega$}$ is needed because the wavelets used in this paper are real and cannot correct the phase-shift (given by \eqref{eqn:Ohat}). This is not the case for the Fourier Whittle estimation described in \cite{Shimotsu07}. Fortunately, the phase-shift can be expressed as a multiplicative cosine term in the covariance of the wavelet coefficients and a correction is still possible.
\cite{AchardGannaz2014} established that the MWW estimators \eqref{eqn:dhat} and \eqref{eqn:Ohat} are consistent under non-restrictive conditions. The rate of convergence for the estimation of the long-range parameters $\mbox{${\mathbf d}$}$ is similar to the MFW estimator and is minimax. We refer to \cite{AchardGannaz2014} for the detailed study of the asymptotic behaviour of MWW estimation.
\section{Examples of multivariate long-memory time series}
\label{sec:data}
This section is describing specific functions of \pkg{multiwave} for the user to be able to simulate multivariate long-memory processes.
Parametric models are defined and implemented. In addition a data sets of real data from neuroimaging is provided.
\subsection{Simulations of FIVARMA}
\label{sec:fivarma-code}
\pkg{multiwave} package proposes simulation functions for time series with a spectral density satisfying approximation \eqref{eqn:approx}. The main function is \code{fivarma} which computes a parametric FIVARMA process defined in Section~\ref{sec:fivarma}.
The input parameters of a $FIVARMA(q,d,r)$ process are the covariance matrix $\mbox{\boldmath$\Sigma$}$ of the innovation process $\mbox{${\mathbf u}$}$, the vector AR (AutoRegressive) $(\mbox{${\mathbf A}$}_k)_{k=0,\dots,q}$, $\mbox{${\mathbf A}$}_k \in \mathbb{R}^{p\times p}$, the vector MA (MovingAverage) $(\mbox{${\mathbf B}$}_k)_{k=0,\dots,r}$, $\mbox{${\mathbf B}$}_k \in \mathbb{R}^{p\times p}$, and the vector of long-range parameters $\mbox{${\mathbf d}$}\in\mathbb{R}^p$. FIVAR model of \cite{SelaHurvich2008} is a subcase, corresponding to MA coefficients equal to zero. The parameters of \code{fivarma} are thus, in order, \code{(N, d, cov\_matrix, VAR, VMA)} where \code{cov\_matrix}$=\mbox{\boldmath$\Sigma$}$ and \code{VAR} and \code{VMA} denote respectively the sequences of matrices $(\mbox{${\mathbf A}$}_k)_{k=0,\dots,q}$ and $(\mbox{${\mathbf B}$}_k)_{k=0,\dots,r}$.
The output of the function \code{fivarma} is a list with first the values $\mbox{${\mathbf X}$}(1), \mbox{${\mathbf X}$}(2), \ldots, \mbox{${\mathbf X}$}(N)$ obtained by equation~\eqref{formula:cointegration} with $\mbox{${\mathbf u}$}(t)$ white noise with centered Gaussian distribution and covariance $\mbox{\boldmath$\Sigma$}$. The second element of the list is the value of the matrix $\mbox{\boldmath$\Omega$}$ defined in \eqref{eqn:omega}.
\code{fivarma} is based on two other functions:
\begin{itemize}
\item \code{fracdiff} applies a vectorial fractional differencing procedure and corresponds to a FIVARMA(0,d,0).
\item \code{varma} computes a realisation of a multivariate ARMA process and corresponds to the case $\mbox{${\mathbf d}$}=0$.
\end{itemize}
Similar functions can be found in other packages ({\it e.g.} \pkg{fracdiff} and \pkg{MST} \citep{tsay2013multivariate}) but were re-implemented in \pkg{multiwave} package.
\vspace{\baselineskip}
\textit{Example.}
\begin{CODE}
\begin{CodeIn}
R> N <- 2^8
R> d0 <- c(0.2,0.4)
R> rho <- 0.8
R> cov <- matrix(c(1,rho,rho,1),2,2)
R> VMA <- diag(c(0.4,0.7))
R> VAR <- array(c(0.8,0.2,0,0.6),dim=c(2,2))
R> resp <- fivarma(N, d0, cov_matrix=cov, VAR=VAR, VMA=VMA)
R> x <- resp$x
R> long_run_cov <- resp$long_run_cov
R> long_run_cov
\end{CodeIn}
\begin{CodeOut}
[,1] [,2]
[1,] 0.6049383 0.5854938
[2,] 0.5854938 0.9730806
\end{CodeOut}
\end{CODE}
\begin{CodeIn}
R> par(mfrow=c(2,1),mai=c(0.5,1,0.5,0.5))
R> plot(x[,1],type='l',lty=1)
R> plot(x[,2],type='l',lty=1)
\end{CodeIn}
\includegraphics[width=10cm]{plotx}
\subsection{A real data set}
\label{sec:real_data_desc}
In order to describe how parameters can be chosen in a practical point of view, we provide a real data example (see Section~\ref{sec:realdata}).
Noninvasive data recorded from the brain are an example where the proposed methodology is efficient. The data consist of time series recording signals from the brain: electroencephalography (EEG) for the electrical signals, magnetoencephalography (MEG) for the magnetic signals or functional Magnetic Renonance Imaging (fMRI) for the Blood Oxygen Level Dependent (BOLD) signals. These data are intrinscally correlated because of the known interactions of the brain areas (also called regions of interest). Furthermore, it has already been shown that these time series present long-memory features \citep{maxim.2005.1}. Other data sets presenting similar features are coming from finance e.g. \cite{Songsiri}, where time series are correlated because of links between companies for example, and they also present long-memory characteristics. In this section, we observed time series extracted using fMRI facilities. The whole description of this data sets is detailed in \cite{Termenon2016}.
The data set called \code{brainHCP} contains the time series of 1200 points in time and 89 regions of the brain.
Figure \ref{fig:fmri.signal} displays 6 arbitrary signals from one subject in this data set.
\begin{CodeIn}
R> data(brainHCP)
R> dim(brainHCP)
\end{CodeIn}
\begin{CodeOut}
[1] 1200 89
\end{CodeOut}
\begin{figure}[!ht]
\caption{Plot of 6 arbitrary signals from a subject of fMRI data set.}
\label{fig:fmri.signal}
\begin{center}
{\includegraphics[scale=0.4]{signaux}}
\end{center}
\end{figure}
\section{Estimation of the long-memory parameters and covariance matrix}
\label{sec:estimation}
The objective of this package is to provide the implementation of two sets of methods based either on Fourier or wavelet decomposition. That is Multivariate Fourier Whittle (MFW) and Multivariate Wavelet Whittle (MWW) estimation procedures. The corresponding functions are respectively called \code{mfw} and \code{mww} in the package.
The output of the implemented methods consists in the estimation of two quantities, $\mbox{${\mathbf d}$}$ and $\mbox{\boldmath$\Omega$}$, where $\mbox{${\mathbf d}$}$ corresponds to the long-memory parameters of the time series and $\mbox{\boldmath$\Omega$}$ is reflecting the coupling between the pairs of time series.
\begin{figure}[!h]
\caption{Input and Output of \pkg{multiwave} package in a two-dimensional case.}
\begin{tabular}{m{7.5cm}m{2.2cm}m{0.4\textwidth}}
\begin{center}INPUT\end{center} & & \quad OUTPUT\\
\vspace{-0.5cm}\includegraphics[width=8cm]{plotx}&\begin{tabular}{c}
\pkg{multiwave}\\\includegraphics[width=2cm]{fleche}\end{tabular}
&\vline height 3cm depth 0.5cm \hspace{12pt}\begin{minipage}[b]{0.4\textwidth}
{Long-memory parameters
$\hat\mbox{${\mathbf d}$} = \begin{pmatrix}
0.17462\\ 0.24477
\end{pmatrix}$
\vspace{0.5cm}
Long-run covariance
$\hat\mbox{\boldmath$\Omega$} = \begin{pmatrix}
0.61064 & 0.55949 \\
0.55949 & 1.29167
\end{pmatrix}
$
}
\end{minipage}
\end{tabular}
\end{figure}
The two functions \code{mfw} and \code{mww} are implementing the semiparametric Whittle estimation using respectively Fourier decomposition and wavelet decomposition in order to estimate $\mbox{${\mathbf d}$}$ and $\mbox{\boldmath$\Omega$}$.
A fast execution of these two functions is
\begin{CODE}
\begin{CodeIn}
R> ## Fourier decomposition
R> m <- 57 ## default value of Shimotsu 2017
R> res_mfw <- mfw(x,m)
R> ## Wavelet decomposition
R> res_filter <- scaling_filter('Daubechies',8) ## choice of filter
R> filter <- res_filter$h
R> LU <- c(2,11) ## choice of wavelet scales
R> res_mww <- mww(x,filter,LU)
\end{CodeIn}
\end{CODE}
\subsection{Multivariate Fourier Whittle estimation}
\label{sec:mfw}
As a first implementation, it is natural to use Fourier decomposition to approximate the spectral density of time series.
Package \pkg{multiwave} proposes functions to compute MFW estimators:
\begin{itemize}
\item \code{mfw} computes the multivariate Fourier Whittle estimators of both the long-range dependence parameters and the long-run covariance matrix.
\item \code{mfw\_cov\_eval} computes the multivariate Fourier-based Whittle estimator for the long-run covariance matrix for a given value of the long-range dependence $\mbox{${\mathbf d}$}$.
\item \code{mfw\_eval} returns the value of the multivariate Fourier Whittle criterion with respect to $\mbox{${\mathbf d}$}$ at a given value of $\mbox{${\mathbf d}$}$.
\end{itemize}
The functions \code{mfw\_cov\_eval} and \code{mfw\_eval} are internal functions of \code{mfw}. In \code{mfw}, we apply first a minimum search of \code{mfw\_eval} with respect to $\mbox{${\mathbf d}$}$, and \code{mfw\_cov\_eval} is returning the estimation of $\mbox{\boldmath$\Omega$}$ for the estimated value $\mbox{${\mathbf d}$}$.
We only detail function \code{mfw} hereafter and refer to the package description for other functions.
Let $\mbox{${\mathbf X}$}$ be the $p \times N$-matrix of observations, with general term ${x}_{\ell,i}=X_\ell(i)$, $\ell=1,\dots, p$ and $i=1,\dots, N$. Let $m$ be the number of frequencies used in MFW procedure. Given $x$ and $m$, the function \code{mfw} computes the MFW estimators defined by \eqref{eqn:dF} and \eqref{eqn:GF}, with the frequencies $\lambda_j = 2\pi j/N$, $j=1, \dots, m$. The optimization in equation \eqref{eqn:dF} is done using \code{optimize} function of \textsf{R} in one-dimensional settings and a Newton-type algorithm through \code{nlm} function of \textsf{R} otherwise. The initialization of the algorithm is set equal to the vector of univariate Fourier-based Whittle estimations. Even if it increases the computational time, such an initialization is important in high-dimensional settings. For example, in the MEG data set studied in \cite{AchardGannaz2014}, the optimization is done in $\mathbb{R}^{274}$. An initialization at the origin may not be able to reach the minimum even with a high number of iterations, due to the high dimension.
Function \code{mfw} returns a list with first the $p$-dimensional vector $\hat \mbox{${\mathbf d}$}^{MFW}$ and second the $p\times p$-matrix $\hat \mbox{\boldmath$\Omega$}^{MFW}$. The quality of estimation is depending on the parameter $m$. Theoretical results show that $m$ must be small enough so the short-range properties of the time series do not bias the estimation. On the contrary a too small value will introduce variance in estimation since it decreases the number of frequency used in the procedure. \cite{Lobato99}, \cite{Shimotsu07} or \cite{Nielsen11} propose a default value $m=N^{0.65}$. This choice is discussed in the simulation study in Section \ref{sec:simu}.
\vspace{\baselineskip}
\textit{Example.}
\begin{CODE}
\begin{CodeIn}
R> # Simulation of the data
R> N <- 2^8
R> d0 <- c(0.2,0.4)
R> rho <- 0.8
R> cov <- matrix(c(1,rho,rho,1),2,2)
R> VMA <- diag(c(0.4,0.7))
R> VAR <- array(c(0.8,0.2,0,0.6),dim=c(2,2))
R> resp <- fivarma(N, d0, cov_matrix=cov, VAR=VAR, VMA=VMA)
R> x <- resp$x
R>
R> # Estimation
R> m <- N^(0.65) ## default value of Shimotsu
R> res_mfw <- mfw(x,m)
R> res_mfw
\end{CodeIn}
\begin{CodeOut}
$d
[1] 0.1239810 0.3497609
$cov
[,1] [,2]
[1,] 0.5004685 0.5840763
[2,] 0.5840763 1.2547056
\end{CodeOut}
\end{CODE}
\subsection{Multivariate Wavelet Whittle estimation}
\label{sec:mww}
The functions applying MWW estimation in package \pkg{multiwave} are the following:
\begin{itemize}
\item \code{mww} computes the multivariate wavelet Whittle estimators of the long-range dependence parameters and the long-run covariance matrix.
\item \code{mww\_cov\_eval} computes the multivariate wavelet-based Whittle estimator for the long-run covariance matrix for a given value of the long-range dependence $\mbox{${\mathbf d}$}$.
\item \code{mww\_eval} returns the value of the multivariate wavelet Whittle criterion with respect to $\mbox{${\mathbf d}$}$ at a given value of $\mbox{${\mathbf d}$}$.
\end{itemize}
\code{mww\_cov\_eval} and \code{mww\_eval} are internal functions of \code{mww}. In \code{mww}, we apply first a minimum search of \code{mww\_eval} with respect to $\mbox{${\mathbf d}$}$, and \code{mww\_cov\_eval} is returning the estimation of $\mbox{\boldmath$\Omega$}$ for the estimated value of $\mbox{${\mathbf d}$}$. MWW estimation is based on the wavelet transform of time series and \code{mww} needs the definition of a wavelet filter. The computation of a filter and of a wavelet transform are described below.
\subsubsection{Wavelet transform and scalogram}
The wavelet decomposition in package \pkg{multiwave} is implemented using an exact discrete wavelet transform.
\begin{itemize}
\item \code{scalingfilter} defines the wavelet filter (only Daubechies' wavelets are available).
\item \code{computenj} computes the number of wavelet coefficients for each individual scale.
\item \code{DWTexact} provides the wavelet transform of the data.
\item \code{psi\_hat\_exact} gives the Fourier transform of the wavelet function.
\item \code{K\_eval} evaluates the value of the integral \eqref{eqn:K}.
\end{itemize}
\textit{Example.}
To obtain the wavelet filter of a Daubechies' wavelet of order 4, that is with 2 vanishing moments, one should write:
\begin{CODE}
\begin{CodeIn}
R> res_filter <- scaling_filter('Daubechies',4);
R> filter <- res_filter$h
R> filter
\end{CodeIn}
\begin{CodeOut}
[1] 0.4829629 0.8365163 0.2241439 -0.1294095
\end{CodeOut}
\end{CODE}
Next, given an $N$ dimensional vector \code{x}, the wavelet coefficients of \code{x} are given by function \code{DWTexact}:
\begin{CODE}
\begin{CodeIn}
R> # Simulation of the data
R> N <- 2^8
R> d0 <- 0.2
R> resp <- fivarma(N, d0)
R> x <- resp$x
R>
R> # Wavelet decomposition
R> resw <- DWTexact(x,filter)
R> xwav <- resw$dwt # returns the vector of the wavelet coefficients
R> index <- resw$indmaxband
R> Jmax <- resw$Jmax
R> Jmax
\end{CodeIn}
\begin{CodeOut}
[1] 6
\end{CodeOut}
\begin{CodeIn}
R> index
\end{CodeIn}
\begin{CodeOut}
[1] 127 189 219 233 239 241
\end{CodeOut}
\begin{CodeIn}
R> length(x)
\end{CodeIn}
\begin{CodeOut}
[1] 256
\end{CodeOut}
\end{CODE}
\code{index} gives the index of the last coefficient at each scale and \code{Jmax} gives the maximal scale. The vector of coefficients \code{xwav} is $m$-dimensional, with $m$ maximum of \code{index}, equal to \code{index[Jmax]}. The coefficients of the third scale are for example given by:
\begin{CODE}
\begin{CodeIn}
R> xwav[seq(index[2]+1,index[3]),1]
\end{CodeIn}
\end{CODE}
Finally it is useful to compute the quantity $K(\delta)$ defined in equation \eqref{eqn:K}. Thus one needs to recover the Fourier transform of the wavelet, $\widehat{\psi}(\cdot)$. This is done using the function \code{psi\_hat\_exact}. Its inputs are the filter defined previously and an index of precision. It returns $(\widehat{\psi}(u_i))_{i=1,\dots,q*2^J}$ where $q$ is the length of the filter and $u_i$ are equally spaced points on the interval $[-\pi 2^{J-3} (q-1)/2; \,\pi 2^{J-3} (q-1)/2]$.
\begin{CODE}
\begin{CodeIn}
R> res_psi <- psi_hat_exact(filter,J=10)
R> psih <- res_psi$psih
R> gridh <- res_psi$grid
\end{CodeIn}
\end{CODE}
where \code{res\_psi\$grid} returns the values of the grid $(u_i)$ and \code{res\_psi\$psih} returns the corresponding values of $\widehat\psi(u_i)$. It is recommended to take $J\leq 15$ in practice and the default value is $J=10$. Indeed, a large value of \code{J} is increasing the computational time.\\
Given the function $\widehat{\psi}(\cdot)$, we are now able to evaluate $K(\mbox{${\mathbf d}$})$ for a given value of $\mbox{${\mathbf d}$}$:
\begin{CODE}
\begin{CodeIn}
R> K <- K_eval(psih,gridh,d0)
\end{CodeIn}
\end{CODE}
\subsubsection{Estimation}
\code{mww} is now described in detail.
Let $\mbox{${\mathbf X}$}$ be the $p\times N$-matrix of observations, with general entries ${x}_{\ell,i}=X_\ell(i)$, $\ell=1,\dots, p$ and $i=1,\dots, N$. Let $LU$ be the bivariate vector giving the lowest scale $j_0$ and the upper scale $j_1$ of the wavelet coefficients used in estimation. Given $\mbox{${\mathbf X}$}$ and $LU$, the function \code{mww} computes the MWW estimators defined by \eqref{eqn:dhat} and \eqref{eqn:Ohat}. As previously, the optimization in equation \eqref{eqn:dhat} is done using \code{optimize} function of \textsf{R} in one-dimensional settings and a Newton-type algorithm through \code{nlm} function of \textsf{R} otherwise. The initialization of the algorithm is set equal to the vector of univariate wavelet-based Whittle estimations. The reasons are identical to the ones given for the function \code{mfw}.
Function \code{mww} returns a list with first the $p$-dimensional vector $\hat \mbox{${\mathbf d}$}^{MWW}$ and second the $p\times p$-matrix $\hat \mbox{\boldmath$\Omega$}^{MWW}$. The quality of estimation is depending on the parameters $j_0$ and $j_1$ appearing in \eqref{eqn:dhat} and \eqref{eqn:G}. Default value for $j_0$ is set to $2$ and for $j_1$ to the highest integer lower than $\log_2(N)$. The critical value to choose for estimation is $j_0$, as it can be seen in the theoretical conditions for consistency \citep{AchardGannaz2014} and in simulations studies. Similarly to the choice of the parameter $m$ for MFW procedure, a compromise exists between choosing a small value of $j_0$, which would introduce a bias due to the short-range properties of the time series, and a high value that will reduce the number of frequencies and thus increase variance.
\vspace{\baselineskip}
\textit{Example.}
\begin{CODE}
\begin{CodeIn}
R> # Simulation of the data
R> N <- 2^8
R> d0 <- c(0.2,0.4)
R> rho <- 0.8
R> cov <- matrix(c(1,rho,rho,1),2,2)
R> VMA <- diag(c(0.4,0.7))
R> VAR <- array(c(0.8,0.2,0,0.6),dim=c(2,2))
R> resp <- fivarma(N, d0, cov_matrix=cov, VAR=VAR, VMA=VMA)
R> x <- resp$x
R>
R> # Parameter of estimation
R> res_filter <- scaling_filter('Daubechies',8);
R> filter <- res_filter$h
R> LU <- c(2,8)
R>
R> # Estimation
R> res_mww <- mww(x, filter, LU)
R> res_mww
\end{CodeIn}
\begin{CodeOut}
$d
[1] 0.1955683 0.4857156
$cov
[,1] [,2]
[1,] 0.5826133 0.5051330
[2,] 0.5051330 0.9584876
\end{CodeOut}
\end{CODE}
If one wants to apply several times the estimation on the same data set, or modifying the parameters of estimation, it is useful to separate the wavelet transform and the estimation scheme.
Wavelet-based estimation can be evaluated directly on the wavelet transform of the data using the following functions:
\begin{itemize}
\item \code{mww\_wav} computes the multivariate wavelet Whittle estimators of the long-range dependence parameters and the lon-run covariance matrix, given the wavelet transform of the data.
\item \code{mww\_wav\_cov\_eval} computes the MWW estimator for the long-run covariance matrix for a given value of the long-range dependence $\mbox{${\mathbf d}$}$, given the wavelet transform of the data.
\item \code{mww\_wav\_eval} returns the value of the multivariate wavelet-based Whittle criterion with respect to $\mbox{${\mathbf d}$}$ at a given value of $\mbox{${\mathbf d}$}$, for a specific wavelet transform of the data.
\end{itemize}
We refer to the description of the functions in the package for more details.
\vspace{\baselineskip}
\textit{Example.}
\begin{CODE}
\begin{CodeIn}
R> # Simulation of the data
R> N <- 2^8
R> d0 <- c(0.2,0.4)
R> rho <- 0.8
R> cov <- matrix(c(1,rho,rho,1),2,2)
R> VMA <- diag(c(0.4,0.7))
R> VAR <- array(c(0.8,0.2,0,0.6),dim=c(2,2))
R> resp <- fivarma(N, d0, cov_matrix=cov, VAR=VAR, VMA=VMA)
R> x <- resp$x
R>
R> N <- dim(x)[1]
R> k <- dim(x)[2]
R>
R> # Parameter of estimation
R> res_filter <- scaling_filter('Daubechies',8)
R> filter <- res_filter$h
R> LU <- c(2,8)
R>
R> ## Wavelet decomposition
R> xwav <- matrix(0,N,k)
R> for(j in 1:k){
R> xx <- x[,j]
R> resw <- DWTexact(xx,filter)
R> xwav_temp <- resw$dwt
R> index <- resw$indmaxband
R> Jmax <- resw$Jmax
R> xwav[1:index[Jmax],j] <- xwav_temp
R> }
R> ## we free some memory
R> new_xwav <- matrix(0,min(index[Jmax],N),k)
R> if(index[Jmax]<N){
R> new_xwav[(1:(index[Jmax])),] <- xwav[(1:(index[Jmax])),]
R> }
R> xwav <- new_xwav
R> index <- c(0,index)
R>
R> ##### Compute the wavelet functions
R> res_psi <- psi_hat_exact(filter,Jmax)
R> psih <- res_psi$psih
R> grid <- res_psi$grid
R>
R> # Estimation
R> res_mww_wav <- mww_wav(xwav,index,psih,grid,LU)
R> res_mww_wav
\end{CodeIn}
\begin{CodeOut}
$d
[1] 0.1684136 0.3693829
$cov
[,1] [,2]
[1,] 0.5177418 0.426430
[2,] 0.4264300 0.834756
\end{CodeOut}
\end{CODE}
\section{Practical choices of parameters for MWW estimation}
\label{sec:discussions}
MWW procedure implemented in \code{mww} depends on mainly two parameters: the choice of the wavelet bases \code{filter} and the choice of the wavelet scales \code{LU}. These parameters are not fixed in the package as the performances of the estimation may be improved by a careful choice. The possibility to choose the wavelet scales is particularly of interest when dealing with short-range dependence. We show that a simple graphical representation of the scalogram is able to guide the user in the choice of the wavelet scales.
\subsection{Choice of the wavelet bases}
\label{sec:choixbase}
Actually \pkg{multiwave} only proposes Daubechies' wavelets, which satisfy theoretical properties of \cite{AchardGannaz2014}. Other bases are possible but not implemented. The wavelet bases is imputed {\it via} the parameter \code{filter} in function \code{mww}, where \code{filter} is obtained by \code{filter <- scaling\_filter('Daubechies',2*M)\$h}.
The main parameter characterizing the Daubechies' bases is thus the number of vanishing moments, $M$. MWW estimation presents the advantage to be available even if the time series are nonstationary or with polynomial trends, as soon as $\max\mbox{${\mathbf d}$}\leq M$. For example, for stationary time series, a parameter $M=1$ is sufficient (which is equivalent to considering Haar bases). For real data application, when nonstationarity or trends are suspected, a higher value of $M$ is necessary.
As discussed in \cite{FayMoulinesRoueffTaqqu}, when $M$ increases, the quality of estimation (slightly) decreases. Depending on the data, a compromise is then needed between choosing a large enough number of vanishing moments $M$ to handle nonstationarity in the data and the quality of estimation.
On the contrary, MFW estimators are only suited to stationary time series. Some extensions of Fourier-based estimation were proposed in univariate setting such as tapered Fourier (see {\it e.g.} \cite{FayMoulinesRoueffTaqqu} and references therein). For multivariate estimation \cite{Nielsen11} proposes an extension of \cite{Shimotsu07} based on the transform defined in \cite{Abadir}. However, this approach gives satisfactory results only for $d < 1.5$ and we decided not to implement it in the package for simplicity.
\subsection{Choice of wavelet scales}
\label{sec:choixLU}
The second parameter we need to tune is \code{LU}, which corresponds to the range of scales used in estimation. \code{LU} is a two-dimensional vector, that is \code{LU<-c(j0, j1)}, with \code{j0} the lowest scale and \code{j1} the upper scale. Parameters \code{j0} and \code{j1} are respectively $j_0$ and $j_1$ defined in equations \eqref{eqn:dhat} and \eqref{eqn:G} in the estimation procedure.
One advantage of wavelets is to be able to qualitatively evaluate the choice of wavelet scales to estimate the long-memory parameters and correlation by inspection of wavelet scalogram. As mentioned in \cite{AbryVeitch98, FayMoulinesRoueffTaqqu} for univariate settings, the first and last scales may have to be discarded from the analysis. The first scale may be affected by the presence of short-memory phenomena. In the example of FIVARMA model, this is driven by the AR and MA coefficients. For the last scales, the impact is different and it comes from the finite length of the time series. Indeed, as derived in \cite{whitcher.2000.2}, the variance of the estimator is increasing with the wavelet scales.
The usual log-scalogram diagram used in univariate settings is showing the linear behaviour of the log variance with respect to the wavelet scales \citep{AbryVeitch98}. This is also true for the covariances as shown in Proposition~2 of \cite{AchardGannaz2014}. For all $k\in\mathbb{Z}$, $Cov(W_{j,k}(\ell),W_{j,k}(m))$ is equivalent to $2^{j(d_\ell+d_m)}G_{\ell,m}(\mbox{${\mathbf d}$})$ when $j$ goes to infinity, with $G_{\ell,m}(\mbox{${\mathbf d}$})$ defined in equation~\eqref{eqn:Gdef}. This property is illustrated in Figure~\ref{fig:scal} with a bivariate FIVARMA processes, as described in Section~\ref{sec:fivarma}. This figure represents the boxplots of the variance of the wavelet coefficients of each component of the time series at each scales (subfigures \ref{fig:scal}.(a) and \ref{fig:scal}.(b)). These plots correspond to the usual log-scalogram diagram \citep{AbryVeitch98}. Subfigure \ref{fig:scal}.(c) displays the analog representation of the covariance between the components. For both variance and covariance, the points satisfying the above approximation are aligned. Scales corresponding to non-aligned points should be removed from estimation as it can be seen on subfigure~\ref{fig:scal}.(a)~and~\ref{fig:scal}.(c) where the highest frequencies are modified by the presence of short-range dependence. Thus, one may discard the first scale from estimation to improve its quality.
\begin{figure}[!h]
\caption[Boxplots of the log-variance of the wavelet coefficients at different scales for multivariate long-memory processes]{Boxplots of the log2-variance of the wavelet coefficients at different scales for simulated bivariate $FIVARMA(1,(0.4,0.4),1)$; (a) for the first component and (b) for the second component. (c) Boxplots of the log2-absolute covariance for the same data. The red lines represent the theoretical linear prediction given by $j (d_\ell+d_m)+\log_2(G_{\ell,m}(\mbox{${\mathbf d}$}))$, with $\ell=m=1$ for subfigure~(a), $\ell=m=2$ for subfigure~(b) and $\ell=1$, $m=2$ for subfigure~(c). The horizontal axis corresponds to increasing scales, that is, decreasing frequencies. The indexes of the horizontal axis display the number of coefficients available.
Parameters in FIVARMA model were the following: the white noise is Gaussian with a covariance equal to $\mbox{\boldmath$\Sigma$}=\begin{pmatrix} 1 & 0.8 \\ 0.8 & 1\end{pmatrix}$, the AR coefficient is set equal to $A=\begin{pmatrix} 0.8 & 0 \\ 0.2 & 0.6\end{pmatrix}$ and the MA coefficient is set equal to $B=\begin{pmatrix} 0.4 & 0 \\0.2 & 0.7 \end{pmatrix}$.
Calculation was done on $N=512$ observations for $100$ replications.}
\label{fig:scal}
\begin{center}{\includegraphics[width=1\textwidth]{scal.pdf}}
\end{center}
\end{figure}
Using the results on the wavelet variance and covariance in terms of scales, we showed that the wavelet correlation is asymptotically constant with respect to the wavelet scales. Indeed, for all $\ell,m=1,\dots,p$, for all $k\in\mathbb{Z}$, $Cor(W_{j,k}(\ell),W_{j,k}(m))$ is equivalent to $G_{\ell,m}(\mbox{${\mathbf d}$})/\sqrt{G_{\ell,\ell}(\mbox{${\mathbf d}$})G_{m,m}(\mbox{${\mathbf d}$})}$ when $j$ goes to infinity \citep{AchardGannaz2014}. As for the log-scalogram diagram, the correlation between wavelet coefficients with respect to the scales can be plotted and scales where the observed correlation is not equal to the value obtained for the majority of scales should be removed from estimation. The wavelet correlation spectrum is a complementary way to qualitatively evaluate the range of scales where the analysis should be carried out. Figure \ref{fig:cov} illustrates this on four different data sets. Four different simulations of bivariate processes are applied using finite difference processes, FIVAR and FIVARMA processes, as described in Section~\ref{sec:fivarma}. Figure \ref{fig:cov} represents the boxplots of the correlation between wavelet coefficients of the two components of the time series at each scales. Again the presence of short-range dependence alters the highest frequencies (subfigures \ref{fig:cov}.(b) and \ref{fig:cov}.(c)). This is also observed with nonstationarity (subfigure \ref{fig:cov}.(d)). The first scales should then be removed from estimation. This free parameter of the package is particularly useful with the presence of short-range dependence or nonstationarity. Visual comparison to constant values may be easier for selection of the correct range of wavelet scale to use in the estimation.
\begin{figure}[!h]
\caption[Boxplots of the correlation of the wavelet coefficients at different scales for multivariate long-memory processes]{Boxplots of the correlation of the wavelet coefficients at different scales for four different simulated data: (a) bivariate $FIVARMA(0,(0.4,0.4),0)$ (b) bivariate $FIVARMA(1,(0.4,0.4),0)$; (c) bivariate $FIVARMA(1,(0.4,0.4),1)$; and finally, (d) an example with non stationary time series, $FIVARMA(0,(0.8,1.2),0)$. The horizontal red lines represent the true long-run correlation for each simulation. The horizontal axis corresponds to increasing scales, that is, decreasing frequencies. The indexes of the horizontal axis display the number of coefficients available.
Parameters in FIVARMA models were, if needed, the following: the white noise covariance is set equal to $\mbox{\boldmath$\Sigma$}=\begin{pmatrix} 1 & 0.8 \\ 0.8 & 1\end{pmatrix}$, the AR coefficient is set equal to $A=\begin{pmatrix} 0.8 & 0 \\ 0.2 & 0.6\end{pmatrix}$ and the MA coefficient is set equal to $B=\begin{pmatrix} 0.4 & 0 \\0.2 & 0.7 \end{pmatrix}$.
Calculation was done on $N=512$ observations for $100$ replications.}
\label{fig:cov}
\begin{center}{\includegraphics[width=\textwidth]{illust_choice.pdf}}
\end{center}
\end{figure}
A similar discussion is detailed in Section~\ref{sec:realdata} for a real neuroscience data set. With real data sets a bootstrap procedure is necessary to obtain boxplots, as it will be explained in Section~\ref{sec:realdata}. For the Fourier procedure, the equivalent parameter is the number of frequencies $m$. However, wavelets are providing a graphical way to choose the upper and lower scales. To our knowledge, no equivalent qualitative evaluation for Fourier procedure is available.
\subsection{Numerical examples}
In order to quantify the quality of the choice of the parameters, numerical examples are provided. The first tables illustrate the quality of the estimators for a long-memory process with no short-range dependence. Then, the simulations are complexified by adding short-range behaviour or nonstationarity.
In each example we simulated 1000 Monte-Carlo replications of $N=512$ observations of FIVARMA(q,d,r), with a dimension $p=2$, for a set of different parameter values. Simulations are done using the function \code{fivarma}. MWW estimators are computed using function \code{mww}. MWW procedure is done using a Daubechies' wavelet bases with $M=4$ vanishing moments. This choice is motivated by discussion of Section~\ref{sec:choixbase}, because it can handle different settings, including nonstationary ones.
The quality of estimation is measured {\it via} the bias, the standard deviation ({std}) and the RMSE which is equal to $\sqrt{bias^2+std^2}$. For clarity, all tables are displayed at the end of the paper.
\subsubsection{A reference example}
We first consider a simple example, with neither short-range dependence, nor nonstationarity. Time series were simulated using a bivariate $FIVARMA(0,\mbox{${\mathbf d}$},0)$ with a long-run correlation matrix $\mbox{\boldmath$\Omega$}=\begin{pmatrix}
1 & \rho \\ \rho & 1
\end{pmatrix}$ and $\rho =0.8$. The bivariate vector $\mbox{${\mathbf d}$}$ is chosen in $[0,0.5)^2$, such that the time series are stationary.
As shown in Figure \ref{fig:cov}(a), and discussed in Section~\ref{sec:choixLU}, all scales can be kept for estimation.
Table~\ref{tab:dwav} displays results for the MWW estimation of $\mbox{${\mathbf d}$}$. This illustrates that multivariate estimation improves the quality of estimation for $\mbox{${\mathbf d}$}$. Indeed, the last column gives the ratio between the RMSEs of the multivariate wavelet-based estimation and of the univariate wavelet-based estimation (ratio M/U). This ratio is always smaller than 1, that is, multivariate RMSE is always lower than univariate RMSE.
\subsubsection{Short-range dependence}
Consider a $FIVARMA(1,\mbox{${\mathbf d}$},0)$ obtained with the model described in section \ref{sec:fivarma} and given by the function \code{fivarma}. This case corresponds to a $FIVAR$ model of \cite{SelaHurvich2012}. The AR coefficient is taken equal to $\mbox{${\mathbf A}$}=\begin{pmatrix}
0.8 & 0 \\ 0.2 & 0.6
\end{pmatrix}$ and the correlation between the innovation processes equal to $\rho =0.8$. More precisely let $\mbox{\boldmath$\epsilon$}$ be a bivariate white noise process with covariance matrix $\mbox{\boldmath$\Sigma$}=\begin{pmatrix}
1 & \rho \\ \rho & 1
\end{pmatrix}$ and let $\mbox{${\mathbf u}$}$ be the AR(1) process defined by $\mbox{${\mathbf u}$}(t)+\mbox{${\mathbf A}$} \mbox{${\mathbf u}$}(t-1)=\mbox{\boldmath$\epsilon$}(t)$. The time series observation $\mbox{${\mathbf X}$}(t)$ at time $t$ satisfies $(1-\mathbb{L})^d\mbox{${\mathbf X}$}(t)= \mbox{${\mathbf u}$}(t).$ The matrix $\mbox{\boldmath$\Omega$}$ in equation \eqref{eqn:omega} is equal to
\[ \mbox{\boldmath$\Omega$} =(I+\mbox{${\mathbf A}$})^{-1}\mbox{\boldmath$\Sigma$} (I+\mbox{${\mathbf A}$})^{-1\,T} \simeq \begin{pmatrix}
0.3086 & 0.2392 \\
0.2392 & 0.3260
\end{pmatrix}.
\]
The corresponding long-run correlation is thus equal to $0.754$.
As explained above, the finest scales are influenced by the short-range dependence and they have to be discarded from the estimation. As it can be seen in Figure~\ref{fig:cov}(c), the first two scales should be removed. We obtained accordingly that the lowest RMSE in estimation is obtained taking $j_0=3$.
Numerical results for estimation are given in Table~\ref{tab:dwav2} and Table~\ref{tab:Owav2}. We can observe that estimation of $\hat\mbox{${\mathbf d}$}$ and $\hat\mbox{\boldmath$\Omega$}$ are still satisfactory. The RMSE is very similar to the previous case with no short-range behaviour.
\subsubsection{Nonstationarity}
Nonstationary examples are simulated using values of $\mbox{${\mathbf d}$}$ higher than $0.5$. We consider order 1 or 2 of nonstationarity, that is $\mbox{${\mathbf d}$}\in[0.5,2.5)^2$.
The behaviour of the wavelet correlations at each scale is illustrated in Figure \ref{fig:cov}(d). Contrary to stationary simulations where the optimal choice of $j_0$ was equal to $j_0=1$, the optimal choice of the parameter $j_0$ is $j_0=2$. Results are given in Table~\ref{tab:dwav3} and Table~\ref{tab:Owav3}.
Comparing Table~\ref{tab:dwav} and Table~\ref{tab:dwav3}, the quality of estimation of $\mbox{${\mathbf d}$}$ is still accurate in nonstationary settings, with similar values for the RMSE. As for the estimation of $\mbox{\boldmath$\Omega$}$, Table~\ref{tab:Owav} and Table~\ref{tab:Owav3} indicate that MWW still provides a good quality estimation of the long-run covariance matrix. The quality is slightly lower but still satisfactory.
\subsection{Discussion on identifiability}
\label{sec:identifiabilite}
In practical applications, it seems natural to assume that time series have the same order of stationarity. However, when two time series have long-memory parameters $d_\ell$ and $d_m$ satisfying $d_\ell-d_m=1$ the long-run covariance matrix $\mbox{\boldmath$\Omega$}$ is no longer identifiable with the wavelet-based procedure. Indeed, Proposition~2 in \cite{AchardGannaz2014} states that in this particular case the covariance $Cov(W_{j,k}(\ell),W_{j,k}(m))$ tends to 0 when the scale $j$ tends to infinity. Figure~\ref{fig:diff} illustrates this approximation for a bivariate $FIVARMA(0,(0.2,\, 1.2),0)$, a correlation matrix $\mbox{\boldmath$\Omega$}=\begin{pmatrix}
1 & \rho \\ \rho & 1
\end{pmatrix}$ and $\rho =0.8$.
When $\hat d_\ell- \hat d_m=1$, the estimator~\eqref{eqn:Ohat} is no longer defined. In practice, the quantity $\hat d_\ell- \hat d_m$ cannot be exactly equal to 1. Nevertheless, as dividing by a cosine function of this difference, a small error in the estimation of $(d_\ell, d_m)$ will lead to an important bias in the estimation of $\Omega_{\ell,m}$. As it can be seen in Figure~\ref{fig:bias.cov}, the resulting bias increases in the neighbourhood of the non-identifiable lines $d_\ell-d_m=\pm 1$.
{
\begin{figure
\caption[Boxplots of the covariance of the wavelet coefficients at different scales for a bivariate $FIVARMA(0,(0.2,1.2),0)$]{Boxplots of the covariance of the wavelet coefficients at different scales for a bivariate $ARFIMA(0,(0.2,1.2),0)$ with $\mbox{\boldmath$\Omega$}=\begin{pmatrix} 1 & 0.8 \\ 0.8 & 1\end{pmatrix}$. The index of the horizontal axis displays the number of coefficients available. Calculation was done on $N=512$ observations for $1000$ replications.}
\label{fig:diff}
\begin{center}{\includegraphics[width=12cm,height=7cm]{illust_identifiability.pdf}}
\end{center}
\end{figure}
\begin{figure}[!h]
\caption[RMSE in the estimation of the cross-covariance term $\Omega_{12}$ with respect to $(d_1,d_2)$.]{RMSE in the estimation of the cross-covariance term $\Omega_{12}$ with respect to $(d_1,d_2)$. Estimation was done using multivariate Wavelet Whittle estimator in a bivariate $FIVARMA(0,(d_1, d_2),0)$ with $\mathbf{\Omega}=\begin{pmatrix} 1 & 0.8\\ 0.8 & 1 \end{pmatrix}$. Subfigure~(b) represents an image plot of subfigure~(a) (with a different colour scale to improve visual quality). Blue lines on subfigure~(b) correspond to $d_2-d_1=\pm1$. Calculation was done on $N=512$ observations for $1000$ replications.}
\label{fig:bias.cov}
\begin{center}
\begin{tabular}{cc}
(a) & (b)\\
[-0cm]
{\includegraphics[width=8cm]{figure6a}}&
\begin{minipage}[b]{0.4\textwidth}{\vspace*{1cm}\includegraphics[width=7cm]{figure6b}\vspace{1.5cm}}\end{minipage}
\vspace{-1cm}
\end{tabular}
\end{center}
\end{figure}
}
When this situation occurs, say when the difference between $d_\ell-d_m$ is between 0.75 and 1.25, the estimation of $\mbox{${\mathbf d}$}$ is not affected. But the user must be careful for the estimation of $\mbox{\boldmath$\Omega$}$. One solution is to differentiate or integrate one of the two processes. For example, Table~\ref{tab:bias.cov} illustrates the non-identifiability of $\mbox{\boldmath$\Omega$}$ in a bivariate $FIVARMA(0,\begin{pmatrix}
0.2 & 1.2 \end{pmatrix},0)$. When differentiating the second component (with $d_2=1.2$) the estimator has again good performances.
\begin{table}
\caption{Multivariate Wavelet Whittle estimation of $\mbox{\boldmath$\Omega$}$ for a bivariate $FIVARMA(0,(0.2,\,1.2),0)$ with $\rho=0.8$, $N=512$ with 1000 repetitions. As the memory parameter of the second component is greater than one, one possibility is to differentiate the second component. $j_0$ is chosen to be equal to 1. }
\label{tab:bias.cov}
\begin{center}
\begin{tabular}{@{}lSSSSSSS@{}}
\toprule
& \multicolumn{3}{l}{\textbf{Without differentiation }} & & \multicolumn{3}{l}{\textbf{With differentiation} }\\
\cmidrule{2-4} \cmidrule{6-8}
$\mbox{\boldmath$\Omega$}$ & \multicolumn{1}{c}{\emph{bias}} & \multicolumn{1}{c}{\emph{std}} & \multicolumn{1}{c}{\emph{RMSE}} & & \multicolumn{1}{c}{\emph{bias}} & \multicolumn{1}{c}{\emph{std}} & \multicolumn{1}{c}{\emph{RMSE}} \\
\midrule
$\Omega_{1,1}$ & 0.0935 & 0.0762 & 0.1206 && 0.0349 & 0.0678 & 0.0762\\
$\Omega_{1,2}$ & 4.1863 & 7.2103 & 8.3375 && 0.0261 & 0.06 & 0.0654\\
$\Omega_{2,2}$ & 0.2215 & 0.0819 & 0.2362 && 0.0292 & 0.07 & 0.0758\\
correlation & 3.5255 & 6.2935 & 7.2137 && 0.0003 & 0.0155 & 0.0155\\
\bottomrule
\end{tabular}
\end{center}
\end{table}
\section{MFW estimation and comparison with MWW}
\label{sec:simu}
The comparison between Fourier-based and wavelet-based approach is presented now. Time series were simulated using a bivariate $FIVARMA(0,\mbox{${\mathbf d}$},0)$ with a long-run correlation matrix $\mbox{\boldmath$\Omega$}=\begin{pmatrix}
1 & \rho \\ \rho & 1
\end{pmatrix}$ and $\rho =0.8$. The bivariate vector $\mbox{${\mathbf d}$}$ is chosen in $[0,0.5)^2$, such that the time series are stationary. In such a setting, Fourier-based estimators are available.
For Fourier-based approach, the parameter to choose is \code{m} corresponding to the number of frequencies taken into account in the estimation. The default value in \cite{Shimotsu07} is $m=N^{0.65}$. We also make comparisons with an optimal value computed by minimizing the RMSE.
\subsection{Estimation of the long-memory parameters}
Table~\ref{tab:dfou} gives the results obtained for the estimation of $\mbox{${\mathbf d}$}$ using MFW procedure (to be compared to Table~\ref{tab:dwav} for wavelets). Comparison between Fourier and wavelet-based procedures is summarized by the ratio between the RMSE given by MWW estimation and the RMSE given by MFW estimation, denoted by ratio W/F. Taking the same number of frequencies as \cite{Shimotsu07}, that is, $m=N^{0.65}$, Table~\ref{tab:dfou} shows that the quality of MWW end MFW procedures are comparable, even if wavelet-based estimation slightly improves Fourier-based estimation with such a choice of $m$.
Next we also consider the number of frequencies leading to the minimal RMSE for MFW estimation. As it can be seen in Table~\ref{tab:dfou}, qualities of both procedures are very similar but MFW estimation then (slightly) surpasses MWW estimation.
Very precise comparisons of Fourier-based and wavelet-based approaches are described in \cite{FayMoulinesRoueffTaqqu} for a univariate setting. In particular, it is shown that since the time series are stationary, the use of Haar bases should improve MWW quality. The authors indeed obtained better results with the Haar-based procedure than with Fourier-based procedure. To highlight the versatility of wavelet-based procedures, we choose here wavelet bases with four vanishing moments. A fair comparison with a Fourier-based method should consider tapered Fourier of order 4 as it is detailed in \cite{FayMoulinesRoueffTaqqu}. They quantify the influence of the regularity of the wavelet bases and discuss the comparison with (tapered) Fourier bases. Similar results are expected to be obtained in the case of multivariate time series, however, this topic exceeds the scope of this paper.
\subsection{Estimation of the long-run covariance}
Finally, Table~\ref{tab:Ofou} and Table~\ref{tab:Ofou2} display results for the estimation of $\mbox{\boldmath$\Omega$}$ with MFW method (to be compared to Table~\ref{tab:Owav} for wavelets). When MFW estimation is applied with the usual number $m=N^{0.65}$ of frequencies, one can see that the wavelet-based procedure still estimates better the long-run covariance and the long-run correlation, with a ratio W/F always lower than $0.6$ for the estimation of $\mbox{\boldmath$\Omega$}$ terms. When the number of frequencies in Fourier-based estimation is chosen optimally, MFW and MWW procedures behave similarly and none appears significantly better than the other.
To conclude, MFW and MWW estimation procedures give very similar results. The slight improvement of Fourier-based procedure for the estimation of $\mbox{${\mathbf d}$}$ can be explained by the choice of the wavelet bases, however wavelets are efficient for a large set of applications, including time series with trends and nonstationarity features.
\section{Application on real neuroscience data}
\label{sec:realdata}
As already shown in Figure \ref{fig:cov} for simulated data, the advantage of representing the wavelet correlation in terms of scale is to qualitatively determine the scales necessary to estimate the long-memory parameters and long-range covariance matrix. When dealing with real data, bootstrap is providing a way to assess the variability of the estimators. Using the real data described in Section \ref{sec:real_data_desc}, sliding overlapping window of the time series were extracted containing 512 points and we repeated the estimation until reaching the final point of the time series. This is illustrated in Figure \ref{fig:cov_real_data}, where an example of four pairs of fMRI data from one subject is presented. Boxplots are constructed using the sliding window extractions. From these plots, and taking into account
neuroscientific hypothesis stating that the signal of interest for resting state is occurring for frequency below 0.1Hz, we chose to compute the long-memory parameters between scales 3 and 6.
\begin{figure}[!ht]
\caption[Boxplots of the correlation of the wavelet coefficients at different scales for fMRI data]{Boxplots of the correlation of the wavelet coefficients at different scales for real time series from fMRI data sets: (a) Time series 1 and 2; (b) Time series 13 and 14; (c) Time series 31 and 32; (d) Time series 47 and 48. boxplots were obtained using sliding windows with $N=512$ points, extracted from two fMRI time series with length equal to 1200 points, from a single subject. The estimated long parameters $d$ of the two time series are equal. fMRI data set is described in section~\ref{sec:realdata}. The index of the horizontal axis displays the number of coefficients available. The horizontal red lines represent the estimated long-run correlation. Calculation was done on $N=512$ observations for $100$ replications using sliding windows (with overlap).}
\label{fig:cov_real_data}
\begin{center}{\includegraphics[width=0.92\textwidth]{illust_choice_real_data.pdf}}
\end{center}
\end{figure}
Figures \ref{fig:hist} and \ref{fig:corr} display an example of long-memory parameter and long-run correlation estimated for one subject.
\begin{figure}[!ht]
\caption{Histogram of $\hat\mbox{${\mathbf d}$}$ from a subject of fMRI data set.}
\label{fig:hist}
\begin{center}
{\includegraphics[height=6cm]{histogram}}
\end{center}
\end{figure}
\begin{figure}[!h]
\caption{Estimation of $\mbox{\boldmath$\Omega$}$ from a subject of fMRI data set.}
\label{fig:corr}
\begin{center}
{\includegraphics[height=9cm]{correlation}}
\end{center}
\end{figure}
\newpage
\section*{Conclusion}
The \textsf{R} package \pkg{multiwave} provides a versatile wavelet-based approach, as well as a Fourier-based approach, for estimating long-memory parameters and long-run covariance matrices of multivariate time series. The two estimation procedures are based on semi-parametric approaches of \cite{Shimotsu07} and \cite{AchardGannaz2014}. The added value of the package is to provide estimations in long-range dependence multidimensional settings, which is not proposed presently by any \textsf{R} package to our knowledge. This paper describes the functions of the package \pkg{multiwave} and discusses some practical points for applications, including on a real data set. A simulation study shows first that multivariate estimation improves univariate estimation. The advantage of the wavelet-based procedure with respect to the Fourier-based estimation is its flexibility, allowing to take into account trends or nonstationarity.
\vspace{10pt}
{\bf Acknowledgements.} The authors are grateful to Shimotsu (\url{http://shimotsu.web.fc2.com/Site/Matlab\_Codes.html}) and to Fa\"y, Moulines, Roueff and Taqqu for kindly providing the codes of their respective papers. This work was partly supported by the project \textit{Graphsip} from Agence Nationale de la Recherche (ANR-14-CE27-0001). S.A. was partly funded by a grant from la Région Rhône-Alpes and a grant from
AGIR-PEPS, Université Grenoble Alpes–CNRS.
\bibliographystyle{plain}
| -51,567.856213 |
[
-2.712890625,
2.509765625
] | 42.162819 |
[
-3.40625,
0.2071533203125,
-1.8779296875,
-5.7890625,
-0.62548828125,
8.328125
] |
[
2.421875,
8.2265625,
0.76318359375,
6.2109375
] | 632 | 8,045 |
[
-2.63671875,
2.646484375
] | 28.651364 |
[
-6.14453125,
-3.8984375,
-4.390625,
-2.439453125,
1.9833984375,
12.234375
] | 1.458541 | 31.546973 | 21.085984 | 6.920075 |
[
2.369734287261963
] | -33,799.51628 | 6.221255 | -50,707.233148 | 0.503497 | 5.986225 |
[
-2.533203125,
-3.873046875,
-4.1484375,
-5.33984375,
2.25390625,
13.0234375
] |
[
-5.84375,
-2.6328125,
-2.568359375,
-1.974609375,
3.611328125,
5.375
] | |
BkiUdV84c3aisLf0PuG9
|
\section{0pt}{12pt plus 4pt minus 2pt}{0pt plus 2pt minus 2pt}
\begin{document}
\mainmatter
\title{Reconstructing degree distribution and triangle counts from edge-sampled graphs}
\titlerunning{Sampled graph reconstruction}
\author{Naomi A. Arnold \and Ra\'ul J. Mondrag\'on \and Richard G. Clegg}
\authorrunning{Naomi A. Arnold et al.}
\tocauthor{Naomi A. Arnold, Raul J. Mondrag\'on and Richard G. Clegg}
\institute{Queen Mary University of London, London, E1 4NS, UK\\
\email{[email protected]}}
\maketitle
\begin{abstract}
Often, due to prohibitively large size or to limits to data collecting APIs, it is not possible to work with a complete network dataset and sampling is required. A type of sampling which is consistent with Twitter API restrictions is uniform edge sampling. In this paper, we propose a methodology for the recovery of two fundamental network properties from an edge-sampled network: the degree distribution and the triangle count (we estimate the totals for the network and the counts associated with each edge). We use a Bayesian approach and show a range of methods for constructing a prior which does not require assumptions about the original network. Our approach is tested on two synthetic and two real datasets with diverse degree and triangle count distributions.
\keywords{network reconstruction, Bayesian statistics, sampling}
\end{abstract}
\section{Introduction}\label{sec:intro}\input{sections/intro}
\section{Related Work}\label{sec:relatedwork}\input{sections/relatedwork}
\section{Properties of edge sampled graphs}\label{sec:properties}\input{sections/properties}
\section{Estimators for the degree sequence and triangle count}\label{sec:estimators}\input{sections/estimators}
\section{Constructing a prior}\label{sec:priors}\input{sections/prior}
\section{Results}\label{sec:results}\input{sections/results}
\section{Conclusion}\label{sec:conclusion}\input{sections/conclusion}
\bibliographystyle{abbrv}
\subsection{Degree}~\label{sec:deriveddegree}
Let $k_i$, respectively $k_i'$ denote the degree of a node $v_i \in G$ and $G'$ respectively. Then $k_i'$ follows a binomial distribution $k_i' \sim B(k_i, p)$, with conditional probability given by
\begin{equation}\label{eqn:degreebinomial}
\mathbb{P}(k_i' = k' | k_i = k) = \binom{k}{k'} p^{k'} (1-p)^{k-k'},
\end{equation}
with expectation
$\mathbb{E}(k_i' | k_i =k) = kp$
and variance ${\textrm{Var}}\,(k_i'|k_i=k) = kp(1-p)$.
The probability that node $v_i \in V$ of degree $k_i$ has degree 0 in $G'$ is given by
$\mathbb{P}(k_i' = 0) = (1-p)^{k_i}$.
Nodes in $G$ that become isolated as part of the sampling process are invisible to observers of $G'$ and should be considered removed. In this way, let $\delta_i$ be the indicator random variable representing the removal of node $v_i$ from $G$, with probability $(1-p)^{k_i}$.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.5\linewidth]{plots/NodesRemoved.pdf}
\caption[Removed nodes in ER and BA networks by edge-sampling]{Proportion of nodes removed in Erd\H{o}s-R\'enyi and Barab\'asi-Albert graphs, of size 1000 nodes and 10000 (ER), 9900 (BA) edges (see~\cref{tab:datasets}), using the edge-sampling procedure.}
\label{fig:removednodes}
\end{figure}
Then, the expected number $N_0$ of removed nodes from $G$ is given by
${\mathbb{E}}\,(N_0) = \sum_{i=1}^N {\mathbb{E}}\,(\delta_i) = \sum_{k\geq 1} (1-p)^{k}N_k$,
where $N_k$ is the number of vertices in $G$ of degree $k$. This is dependent on the degree distribution; distributions with large numbers of low-degree nodes will experience higher numbers of nodes removed.
This brings to mind the friendship paradox~\cite{feld1991your}, where a node incident to a randomly chosen edge will on average have a higher degree than a randomly chosen node.
From~\cref{fig:removednodes} we see
Barab\'asi-Albert~\cite{barabasi1999emergence} networks experience more node removal than Erd\H{o}s-R\'enyi~\cite{erdHos1960evolution} networks.
The variance in the number of nodes removed is given by
\begin{align*}
{\textrm{Var}}\,(N_0) &= {\textrm{Var}}\,\left(\sum_{i=1}^N \delta_i \right) = \sum_{i=1}^N {\textrm{Var}}\,(\delta_i) + \sum_{1\leq i\neq j \leq N} {\textrm{Cov}}\,(\delta_i, \delta_j) \\
&= \sum_{k\geq 0} (1-p)^k\left[1 - (1-p)^k\right]N_k \notag \\
&\qquad + \sum_{ k, k' \geq 0} \left[(1-p^{k + k'-1} - (1-p)^{k + k'}\right] N_{k,k'}
\end{align*}
where $N_{k,k'}$ is the number of edges connecting vertices of degree $k$ and $k'$.
\subsection{Triangles}\label{sec:triangles}
Let $T_l$ be the number of triangles in $G$ which include edge $e_l \in E$. Then the number of triangles in $G$, denoted by $T$, is given by
\begin{equation}\label{eqn:totaltri}
T = \frac{1}{3} \sum_{e_l \in E} T_l
\end{equation}
where the factor of $\frac{1}{3}$ is present because each triangle in the sum is counted three times, once for each link.
Let $T_l'$ be the number of triangles which include edge $e_l$ in the sampled graph $G'$, defining $T_l' = 0$ if $e_l \notin E'$. In the case that edge $e_l$ remains in the sampled network, then each triangle that includes $e_l$ will remain in the sampled network if and only if the other two edges remain; this occurs with probability $p^2$. There are $T_l$ such triangles, so the number of these which remain in the sampled network is binomially distributed with $T_l$ trials and probability $p^2$. That is,
\begin{equation}\label{eqn:edgein}
\mathbb{P}(T_l' = t' | T_l = t, e_l \in E') = \binom{t}{t'} p^{2t'} (1-p^2)^{t-t'}.
\end{equation}
In the case that $e_l$ does not remain in the sampled network, the following holds:
\begin{equation}\label{eqn:edgeout}
\mathbb{P}(T_l' = t' | T_l = t, e_l \notin E') = \delta_{0,t'}
\end{equation}
where $\delta_{0,t'}$ is the Kronecker delta function, taking the value of 1 if $t'=0$ and 0 otherwise (since we defined $T_l'=0$ if $e_l \notin E'$).
We can use the law of total probability to remove the conditioning on $e_l$ from \cref{eqn:edgein,eqn:edgeout} and find $\mathbb{P}(T_l' = t')$ as follows,
\begin{align}
\mathbb{P}(T_l' = t' | T_l = t) &= \mathbb{P}(T_l' = t' | T_l = t, e_l \in E') \mathbb{P}(e_l \in E') \notag\\
&+ \mathbb{P}(T_l' = t' | T_l = t, e_l \notin E') \mathbb{P}(e_l \notin E') \notag \\
&= p\binom{t}{t'} p^{2t'} (1-p^2)^{t-t'} + \delta_{0,t'} (1-p).\label{eqn:T'givenT}
\end{align}
Therefore, the conditional probability mass function for $T_l'$ given $T_l$ is given by~\cref{eqn:T'givenT}.
The expected value of $T_l'$ given $T_l$ is given by
\begin{align}
\mathbb{E}(T_l'|T_l=t) &= \sum_{t'=0}^{t} t'\mathbb{P}(T_l'=t'|T_l = t)
\notag \\
&= \sum_{t'=0}^t t' \left[p\binom{t}{t'} p^{2t'} (1-p^2)^{t-t'} + \delta_{0,t'} (1-p)\right] \notag \\
&= p \sum_{t'=0}^t t' \binom{t}{t'} p^{2t'} (1-p^2)^{t-t'}
= p\times p^2t =p^3t \label{eqn:aha}
\end{align}
where \cref{eqn:aha} comes from noting that the sum in the lhs precisely evaluates the expected value of a binomial random variable with $t$ trials and probability $p^2$.
Let $T'$ be a random variable representing the triangle count of $G'$, then
$$
\mathbb{E}(T') = \mathbb{E}\left[\frac{1}{3} \sum_{e_l \in E} T_l'\right] = \frac{1}{3} \sum_{e_l \in E} p^3 T_l = p^3 T
$$
where $T'$ is the triangle count of the sampled network $G'$.
The variance of $T_l'$ given $T_l$ is then
\begin{equation}
{\textrm{Var}}\,(T_l'|T_l=t)
= p^3t(1 - p^2 + p^2t - p^3t) \label{eqn:TlVar}.
\end{equation}
An argument involving computation of the covariances ${\textrm{Cov}}\,(T_j, T_l)$ shows that the variance of the expected total triangle count of $G'$ given the individual triangle counts $T_1, \dots , T_M$ is given by
\begin{align}
{\textrm{Var}}\,(T'|T_1, \dots , T_M) &= \frac{1}{9}\bigg[ 3p^3(1-p^2)T + (p^3-p^2)\sum_{e_l \in E} T_l^2 \notag\\
&\qquad+ 6T(p^3-p^6) + 8k(p^5-p^6) \bigg].
\label{eqn:tdash_var}
\end{align}
where $k$ is the number of triangles which share a link. Full derivations of~\cref{eqn:TlVar,eqn:tdash_var} can be found in the first author's thesis~\cite{arnold2021studying} which also contains derivations for wedge counts and clustering coefficient.
An expression for this variance conditioned on total triangle count $T$ not edge triangle counts is given in~\cite{tsourakakis2009doulion}.
The number of triangles per node $T_i$ can be obtained from $T_{e_\ell}$ from
$2T_i=\sum_k T_{e_{\ell}=(i,k)\in E}$
meaning the estimators of the edge-sampled network can be extended to evaluate vertex statistics eg local transitivity of the nodes $c_i= \sum_k T_{e_{\ell}=(i,k)\in E}/(k_i(k_i-1))$.
\subsection{Method of moments estimators}~\label{sec:MME}
Let $X$ be a random variable associated with a statistic of $G$ and let $X'$ be that statistic on $G'$ with expected value $\mathbb{E}(X') = f(X,p)$. A naive `scale-up' estimator for $X$ given observed value $x'$ for $X'$ is the solution $\hat{x}$ to the equation $x' = f(\hat{x},p)$, provided a solution exists. Borrowing the terminology from~\cite{ganguly2017estimation}, we will refer to these estimators as \emph{method of moments estimators}(MME).
\subsubsection{Degree}
For a node of degree $k_i'$ in $G'$, the MME for $k_i$ is given by $k_i'/p$. This is an unbiased estimator with mean ${\mathbb{E}}\,(k_i'/p) = \frac{1}{p} k_i p = k_i$ and variance ${\textrm{Var}}\,(k_i'/p) = \frac{1}{p}k(1-p)$. Nodes with the lowest possible degree (one) in the sampled graph are estimated as having degree $1/p$ in the unsampled graph so as $p$ decreases, the estimation of low-degree nodes becomes poorer.
\subsubsection{Triangle count}
The expected triangle count $\mathbb{E}(T_l')$ of edge $e_l$ is $p^3T_l$. If in addition, $e_l$ remains in $G'$, its expected triangle count is given by $p^2 T_l$. Therefore the MME for $T_l$ is $p^{-3}T_l'$ or $p^{-2}T_l$, without and with the conditioning respectively. Similar to the MME for degree, it provides poor estimates for edges that have a low triangle count, as it disallows any estimates of $T_l$ in the range $(0, 1/p^2)$.
Similarly, an MME estimate proposed by Tsourakakis et al~\cite{tsourakakis2009doulion} for the total triangle count of a network is $p^{-3} T'$, which has expected value ${\mathbb{E}}\,(p^{-3} T') = T$. They found that this estimator has variance $\frac{1}{p^6}\left((p^3-p^6)T + 2k(p^5 - p^6)\right)$, where $k$ is the number of pairs of triangles which share a link.
\subsection{Bayes estimator}~\label{sec:bayesestimators}
\noindent This estimator relies on Bayes theorem, giving
\begin{equation}
\mathbb{P}(X=x|X'=x') = \frac{\mathbb{P}(X'=x'|X=x) \mathbb{P}(X=x)}{\mathbb{P}(X'=x')}.
\end{equation}
$\mathbb{P}(X'=x'|X=x)$ is the \emph{likelihood} which is determined by the edge sampling procedure and is known. $P(X=x)$ is the \emph{prior} function which will be denoted by $\pi(x)$; this is in general not known.
A posterior estimate for $X$ given $X'$ can then be given as the expected value
\begin{equation}
{\mathbb{E}}\,(X|X'=x') = \frac{\sum_x x\mathbb{P}(X'=x'|X=x)\pi(x)}{\mathbb{P}(X'=x')}.
\end{equation}
The immediate question arises of how to deal with the prior $\pi(x)$, as this may involve making assumptions about the structure of $G$. This will be discussed case by case for the degree and triangle count.
\subsubsection{Degree}\label{sec:degreebayes}
Using the likelihood function for the degree from \cref{eqn:degreebinomial}, a posterior estimate for the degree of node $v_i$ given it has degree $k_i'$ in $G'$ is
\begin{equation}\label{eqn:bayesdegree}
\mathbb{E}(k_i|k_i'=k') = \frac{\sum_{k=k'}^{\infty} k \binom{k}{k'}(1-p)^{k} \pi(k) }{\sum_{k=k'}^{\infty} \binom{k}{k'}(1-p)^{k} \pi(k)}
\end{equation}
where $\pi(k)$ is a prior for the degree distribution $P(k)$ of $G$.
\subsubsection{Triangle count}
Using the likelihood function from \cref{eqn:edgein}, a posterior estimate for the triangle count of edge $e_l$ in $G$ given it remains in $G'$ is
\begin{align}\label{eqn:posteriortriangles}
\mathbb{E}(T_l|T_l'=t',e_l \in G')
&= \cfrac{\sum_{t=t'}^{\infty} t \binom{t}{t'}(1-p^2)^{t} \pi(t) }{\sum_{t=0}^{\infty} \binom{t}{t'}(1-p^2)^{t} \pi(t) }
\end{align}
where $\pi(t)$ is a prior for the proportion of edges with triangle count $t$.~\footnote{In experimental runs, the native binomial functions introduced numerical inaccuracies for large powers of $(1-p)$. Therefore, an equivalent evaluation of binomial probabilities using the log-gamma function and laws of logs was used in practice.}
To establish the total triangle count, summing the value of this estimator over the remaining edges in $G'$ and dividing by 3, as in \cref{eqn:totaltri}, will provide an underestimate for the total triangle count of $G$, since there are potentially many missing edges in $G'$. To mitigate this, we scale this factor up to the estimated number of edges in $G$. That is, our estimate of the total triangle count becomes
$$
\hat{T} = \frac{1}{3p} \sum_{e_l \in G'} \hat{T}_l.
$$
\subsection{Degree distribution}
A prior could be obtained from chosen family of distributions such as the Zipf distribution or a power law distribution, but this baked-in assumption may not be desirable. Furthermore, it has been shown that the distribution of a sampled network may not even follow the distribution of the true network~\cite{stumpf2005subnets}. Therefore, we propose two different methods of constructing a prior which do not make assumptions about the degree distribution of the true network.
First, it is possible to estimate the prior using a Monte Carlo method to minimise the $\ell_2$ norm of the error. In this approach, we find a degree sequence $\{\kappa_i\}$ which minimises
$\min\left( ||p\kappa_i - k'_i||_2^2\right)$, with the restrictions that the degree is an integer number and the sum of the degrees is equal to twice the number of links. To do this, we start with $\kappa_i = \lfloor k_i'/p \rfloor$. If the sum $\sum_{v_i \in V'} \kappa_i$ of the estimated degrees is not equal to the estimated number of links $ \lfloor 2M'/p \rfloor$ then we increment or decrement the degree of nodes chosen uniformly at random until equality holds. Then we rewire links at random for a large number of iterations (15000 in our case), accepting each proposed rewire if it decreases the $\ell_2$ error. If $1/p$ is an integer, then the MME $\kappa_i = k_i'/p $ is a global minimum.
The MME (and hence sometimes the minimisation method) cannot estimate the degree $k_i\approx k'_i/p$ when $k'_i=0$; that is, the lowest possible estimated degree is $1/p$. If a good estimate for the original number of nodes is known then another prior we for capturing these low degree nodes is constructed by ``cascading" links from high degree to low degree nodes. More precisely, as with the minimisation method, we start with an estimated degree sequence $\kappa_i = \lfloor k_i'/p \rfloor$, redistributing links as before if the total estimated degree does not match twice the estimated number of links. Then, we place the nodes in descending order based on their estimated degree, with the knowledge of the original number of nodes in the network being used to append placeholder nodes which would have been removed by the sampling process. Finally, we pick the first occurring node in this list with degree zero, and increment this degree by simultaneously decrementing the degree of the node directly before it. This step is performed iteratively until there are no degree zero nodes. Finally, as a comparison point representing the best possible result achievable with the Bayes method, we use a true prior which is the degree frequencies of the original network as a probability distribution.
\subsection{Triangle per link distribution}
The two methods for prior construction of the degree distribution do not immediately translate to an analogue for triangles, and little is known about the triangle per edge distribution as a starting place for selecting a prior. As an initial approach therefore, we use a Poisson distribution $\text{Po}(\lambda)$ with $\lambda = 3\hat{T}/M'$, the average number of triangles per link in the MME estimator. As with the degree distribution, we include a result with a true prior as a comparison point.
\section*{The ensemble}
\section*{Computational considerations}
Evaluating the binomial term in~\cref{eqn:bayesdegree,eqn:posteriortriangles} using built-in functions would result in numerical inaccuracies since the $\binom{n}{k}$ term can become combinatorially large in large networks (for example, the maximum degree in the AS network is 2389), and both of $p^k$, $(1-p)^{(n-k)}$ can become very small. Instead, we rewrite the binomial density function using the natural logarithm as
\begin{equation*}
f(k,k';p) = \begin{cases}
\begin{aligned} & \exp\big[\ln(\Gamma(k+1)) - \ln(\Gamma(k-k'+1)) \\
& \qquad{} - \ln(\Gamma(k'+1)) + k'\log p \\
& \qquad{} + (k-k')\log(1-p) \big] \end{aligned} & 0 < p < 1, k' \leq k \\
0 & p=0, k\neq 0 \\
\delta_{kk'} & p=1
\end{cases}
\end{equation*}
where $\Gamma$ is the gamma function evaluating as $\Gamma(n) = (n-1)!$ for $n \in \mathbb{N}$.
| -17,772.520215 |
[
-1.69921875,
1.533203125
] | 36.363636 |
[
-3.337890625,
1.2470703125,
-0.08880615234375,
-3.501953125,
-1.1181640625,
4.0859375
] |
[
0.80126953125,
4.6171875,
0.62158203125,
5.47265625
] | 94 | 2,232 |
[
-3.0703125,
3.486328125
] | 30.631573 |
[
-5.7734375,
-3.30859375,
-3.34375,
-2.126953125,
1.6669921875,
10.421875
] | 1.185673 | 17.535079 | 31.675627 | 2.109386 |
[
1.6539788246154785
] | -11,798.299748 | 5.441308 | -17,588.54229 | 0.996295 | 5.583062 |
[
-2.697265625,
-3.357421875,
-3.49609375,
-4.8359375,
2.4453125,
11.59375
] |
[
-5.765625,
-1.583984375,
-1.94921875,
-1.6455078125,
3.154296875,
4.0859375
] | |
BkiUcY3xK1fBGsWmup8P
|
\section{Introduction}
\label{S-Intro}
Solar jets are small scale dynamic events observed as collimated
ejections of plasma material
from the lower solar atmosphere to coronal heights.
Many jets are observed in active regions
(for example see \opencite{Schmieder1983}, \opencite{Gu1994},
\opencite{Schmieder13},
\opencite{Guo13}, \opencite{Chandra17} and references cited therein).
They are observed in a wide range of electromagnetic spectra from the
optical to UV--EUV and X-rays.
Jets are particularly visible in coronal hole (CH)
regions \citep{Madjarska},
probably because these are regions of open magnetic fields with less EUV background emission.
In chromospheric spectral lines, solar
jets are called surges. Surges and jets are associated with base brightening, referred as ``jet bright points''
\citep{Sterling16}. For state-of-art observations, theory and
models of jets, we refer the review of \inlinecite{Raouafi16}.
The soft X-ray telescope onboard {\em Yohkoh} satellite opened a new view on understanding solar
jets with the pioneer paper of \inlinecite{Shibata92} followed by several studies
deriving the characteristics
of solar jets such as
velocity, height, temperature, width, and density \citep{Shimojo96,Shimojo98,Shimojo00,Schmieder13,Chandra15}.
Using the SECCHI/EUVI data \inlinecite{Nistico09}
performed a morphological study of 79 coronal jets and found that
31 have helical structures. Five jets were associated with micro--CMEs.
In some cases solar jets are associated with
eruption of the full
base region.
Such types of jets are called
``blow-out jets'' \citep{Moore10,Moore13,Hong11,Shen12,Chandra17}. When successive jets occur at the same
location and with similar morphological properties in some time interval, they are called
homologous recurrent jets. Usually they are ejected in the same direction and their footpoint structures are similar.
Several studies have been performed about recurrent homologous jets in different wavelengths such as
in H$\alpha$ \citep{Asai01,Uddin12,Chandra17}, in EUV \citep{Chae99,Jiang07,Schmieder13,Joshi17}, and in X-rays \citep{Kim01,Kamio07,Sterling16}.
Based on the X-ray jets and EUV/UV jets observed by {\em Hinode} and
SDO/AIA respectively, a new jet formation model has been proposed by
Sterling and co-workers \citep{Sterling17,Panesar17}. According to their model all the jets would be
originated by the eruption of a small--scale filament or ``minifilament", as already mentioned in
\inlinecite{Hong11} and \inlinecite{Shen12}.
\inlinecite{Sterling16} found that the minifilament eruption kinematics
was similar to the kinematics of most observed large filament eruptions. In large filament
eruptions it was commonly observed that the slow rise phase is followed by the fast rise
acceleration phase. The association of minifilament eruptions was also observed
in CH regions \citep{Sterling15} and in quiet regions \citep{Panesar17}.
However, \inlinecite{Sterling16} and \inlinecite{Sterling17} found that the existence of
minifilament eruptions in active region jets are sometimes difficult to observe.
The explosive dynamic nature, morphology and
magnetic configuration of the jets at their base
support the idea that they are the result of magnetic reconnection.
Magnetic reconnection at null--points is proposed in several jet models.
It is also confirmed by magnetic field extrapolations and simulations
\citep{Moreno08,Zhang12,Schmieder13}. In some of the studies circular ribbons have been
observed at the base of jets \citep{Wang12}. This topology also
supports the presence of magnetic null--points above jet locations.
However, using photospheric magnetic field extrapolations,
\inlinecite{Mandrini1996} and \inlinecite{Guo2013} explained the
jets by the presence of bald patch (BP) regions along
QSLs without the existence of null--points. QSLs
are thin layers with a finite gradient in the
connectivity of the magnetic field where magnetic reconnection can occur.
Very recently, \inlinecite{Chandra17} computed the magnetic topology of
the NOAA active region 10484 on 21--24 October 2003 and
found that the flares and jets occurring in this region were due to
magnetic reconnection at the BP separatrices.
BP separatrices are regions
where magnetic field lines are not anchored on the
photosphere but are tangent between two
different magnetic regions.
The global magnetic topology of active regions is commonly determined by
potential extrapolation which allows the computation of QSL locations, which are
relatively robustly determined \citep{Demoulin1996}.
QSLs are measured by the computation of the squashing degree {\em Q}, proposed by \inlinecite {Titov2002}.
The squashing degree increases with the growth of connectivity gradients, and
becomes infinite for separatrices.
QSLs footprints in the photosphere coincide with the flare
ribbons even if the shape is not always perfectly represented \citep{Dalmasse2015,Zuccarello17}.
Flux emergence in the photosphere could be one of the drivers of solar jets.
According to this scenario, it is the reconnection between the
newly emerged magnetic flux (EMF) and the pre--existing magnetic flux that leads to the formation of jets.
The reconnection can
be driven by the motions in the newly emerged flux. To explain this magnetic
reconnection the first dynamical model was proposed in two dimensions (2D)
\citep{Yokoyama95,Yokoyama96}.
Now EUV and X-ray jets are being modeled with 3D magnetohydrodynamic (MHD) simulations
\citep{Moreno08,Moreno13,Archontis13,Torok09}. The background corona
for both 2D and 3D models is an open magnetic field.
Another scenario proposed for the jet origin is the
loss--of--equilibrium mechanism. In this mechanism, the jet
occurs when the stressed closed flux under the
null--point reconnects with the surrounding quasi-potential flux exterior to the fan surface.
The reconnection starts when some
threshold altitude is reached. In this scenario, the jet displays an untwisting
motion \citep{Pariat09,Pariat10,Dalmasse12,Pariat15}.
Flux cancellation at the jet locations is also
frequently proposed as the driver of jets
\citep{Innes10,Liu11,Innes16,Adams14,Young14,Cheung15,Chen15}. \inlinecite{Zhang00} proposed such flux cancellation
between oppositely directed magnetic
field to explain macrospicules and microscopic jets.
\inlinecite{Adams14} also found magnetic flux convergence and cancellation along the polarity
inversion line (PIL), where the jets were initiated.
In \inlinecite{Guo13} the cancelling flux occurred at the edge of EMF
during its expansion. Therefore, it is still
discussed if jets are due to flux emerging or cancelling or both.
The NOAA AR 12035 had a very high productivity of activity
during 15--16 April 2014
with many confined and eruptive
flares in its northern part and jets in its southern part. The confined and eruptive flares from this active region have been studied in
\inlinecite{Zuccarello17}. They explained that the eruptive flares become more and more confined, when the overlying magnetic field becomes
less and less anti--parallel.
In this study, we investigate the morphology,
kinematics and magnetic causes of the solar jets
during 15--16 April 2014 from NOAA AR 12035.
The paper is organized as follows.
Section \ref{obs} presents the observational data, morphology,
kinematics and the magnetic configuration of the active region.
The magnetic topology of the active region is discussed in
Section \ref{mag_top}. Finally, in Section \ref{result} we discuss and conclude our results.
\section{Observations}
\label{obs}
The NOAA AR 12035 produced many solar jets during
15--16 April 2014 towards the south direction. We have selected
eleven clearly visible jets for our investigation.
Their description is given in Table 1.
These jets were observed by the {\em Atmospheric Imaging
Assembly} (AIA, \opencite{Lemen12}) onboard {\em Solar Dynamics Observatory} (SDO,
\opencite{Pesnell12}) in different
EUV and UV wavebands.
The spatial and temporal resolution of AIA
data are 1.2$^{^{\prime\prime}}$ and 12 s respectively.
To study the magnetic configuration around the location of the
jets we have used the line--of--sight (LOS) magnetic field data from the
{\em Heliospheric Magnetic Imager} (HMI,
\opencite{Schou11}) having a spatial resolution of 1$^{^{\prime\prime}}$
and a cadence of 45 s.
On 16 April 2014 two jets, that we will name jet J${_5}$
and J\ensuremath{'}${_5}$ in the next section,
were observed in H$\alpha$ by the 15 cm Coud\'e telescope
operating at Aryabhatta Research Institute of Observational Sciences (ARIES), Nainital, India (Figure~\ref{fig:hal}).
The pixel size and cadence are 0.58$^{^{\prime\prime}}$ and 1 min respectively.
For consistency, we have shifted all the images to 16 April 2014 10:30 UT.
For identifying the onset and peak time of the jets,
we look into the temporal evolution of intensity at the jet foot-point.
We create a box containing the bright jet base and calculate
the total intensity inside it. Then this total intensity is
normalized by the intensity of the quiet region.
The morphology, kinematics and magnetic configuration of the jet locations
are described in the following subsections.
\subsection{EUV Observations}
\label{obs1}
The eleven selected jets during 15--16 April 2014
are named as J${_1}$--J${_7}$ and J\ensuremath{'}${_3}$-- J\ensuremath{'}${_6}$
respectively.
Out of eleven jets, the first two jets J${_1}$ and J${_2}$ occur on 15 April and the remaining nine jets
are on 16 April 2014.
These jets originated from two locations in the south part of the NOAA AR 12035.
One location is at position [X,Y] = [-220$^{''}$, -215$^{''}$] and the other is at
[-205$^{''}$, -215$^{''}$] (Figure~\ref{fig:hal} left panel).
In this paper, we will refer them {\bf as}
site 1 jets (J) and site 2 jets (J\ensuremath{'}) respectively. The two sites are at a distance
of 15 arcsec from each other ($\approx$ 11 Mm).
Figure~\ref{fig:jet/morpho} displays images of the eleven jets observed
with the AIA filters at 304 {\AA} (top), 193 {\AA} (middle)
and 94 {\AA} (bottom) during their peak phase.
Almost all the jets are visible in all EUV channels, which
indicates the multi-thermal nature of the jets.
The onset and end time of each jet are summarized in Table \ref{recurrent}.
Jet J${_1}$ started on 15 April 2014 with a bright base at site 1.
The zoomed view of evolution of jet
J${_1}$ in AIA 211 \AA\ is shown in Figure~\ref{fig:jet1}.
Together with the ejection of bright material, we
have also observed the ejection of cool and
dark material in 211 {\AA} (see Figure~\ref{fig:jet1}c).
The dark jet is due to the presence of cool material
absorbing the coronal emission. The bright
and cool jet material were rotating clockwise (see the attached movie in 211 \AA).
After the onset of $\approx$ 2 min it started to rotate anti-clockwise.
This indicates the untwisting of the system to relax to a lower energy state by propagating
twist outwards.
The jet J${_2}$ started also with a bright base similar to
J${_1}$ from the same location.
This jet was also showing untwisting like as in the case of
J${_1}$. However, its base was less bright and less
broader than J${_1}$.
On 16 April 2014 J${_3}$ and J\ensuremath{'}${_3}$
started from site 1 and site 2 respectively.
The base of J\ensuremath{'}${_3}$ was like a circular ribbon and broader than J${_3}$.
Jet J\ensuremath{'}${_4}$ is bigger than J${_4}$.
Jet J${_4}$ peaks almost simultaneously in all EUV wavelengths around 07:17 UT.
The peak time for J\ensuremath{'}${_4}$ is five minutes later than J${_4}$, around
07:21 UT in all wavebands. One difference between
these jets and J${_1}$--J\ensuremath{'}${_3}$ was that they have no rotation.
The evolution of jets J${_5}$ and J\ensuremath{'}${_5}$ is presented in
Figure~\ref{fig:jet5} in AIA 211 \AA.
In contrast to the other jets, the peak time for jet J${_5}$ and J\ensuremath{'}${_5}$
is different for different wavebands. The peak
time for J${_5}$ at 304 \AA\ is 10:38:30 UT,
and at 94 \AA\ is 10:41 UT. For the case of J\ensuremath{'}${_5}$,
the peak time in 304 \AA\ is 10:37:30 and at 94 \AA\ is 10:38 UT.
For these jets, the peak for
cool plasma (longer wavelength 304 \AA) appears earlier than the hot plasma
material (shorter wavelength 94 \AA).
The temporal evolution of flux at the base of jet J$_5$ is shown in
Figure~\ref{fig:inten}.
This behavior is contrary to other jets reported in the literature
where hotter plasma appears before cooler plasma,
suggesting some mechanism of cooling versus time \citep{Alex99}.
These are the largest jets among the ones discussed in this study. As the event
progress, interestingly part of jet J\ensuremath{'}${_5}$ was detached from it and
moved towards site 1 and finally merged with jet
J${_5}$ (see the attached movie in 211 \AA).
We have estimated the speed of
J\ensuremath{'}${_5}$ towards J${_5}$ as $\approx$ 45 km s$^{-1}$.
Merging of the broken part of J\ensuremath{'}${_5}$
with J${_5}$ made it bigger as it was ejected in the south
as well as in the north direction at the same time.
We have also noticed the circular ribbon at the base of
J\ensuremath{'}${_5}$, shown in Figure~\ref{fig:jet5}e.
Looking at the evolution of the J\ensuremath{'}${_5}$ jet, we found
that the jet follows a sigmoidal-shape loop path visible in AIA 193 \AA\
(see Figure~\ref{fig:loop}).
These loops are originating from the sunspot and going towards the south with a
sigmoid-shape. The sigmoidal-shape of these loops could be due the the clockwise rotation of big positive
polarity spot (see Section 2.4).
We have observed the clockwise rotation in jet
J${_5}$ and the calculated rotation speed in four wavelengths
171 \AA\ , 193\AA\ , 211 \AA\ and in 304 \AA.
The speed varies from 90 km s$^{-1}$ to 130 km s$^{-1}$.
Jet J${_6}$ was small and of weak intensity whereas J\ensuremath{'}${_6}$ had
a strong intensity and was very bright, wide and had a circular base.
J\ensuremath{'}${_6}$ started to move towards site 1, but like in the case of
J\ensuremath{'}${_5}$, it could not reach up to J${_6}$ location. We did not find any rotation in this jet.
Jet J${_7}$ started from site 1 location and also showed clockwise rotation.
Let us summarize the different morphological features of these jets.
We have found that all jets from site 1 have similar morphology.
Jets from site 2 also have similar morphology, but this
is different from the morphology of jets ejected from site 1.
One common feature in all jets of site 2 on 16 April
was that after their trigger they all have a tendency to
move towards site 1 before or during the ejections.
It seems that for the case of J${_5}$ and J\ensuremath{'}${_5}$,
there is a connection between these two.
Another common feature of jets
originated from site 2 is
that they all follow the sigmoidal
loops visible in different AIA wavebands.
The jets from site 2 occur before or almost simultaneously
(in the case of J\ensuremath{'}${_4}$) with the jets from site 1.
This is true apart from the jet J$_6$. The main difference is
that J$_6$ is quite weak among all of the coupled jets.
We have noticed that jets observed in 193 \AA\
are thinner than in 304 \AA. All jets started
with a bright base, similar to common X-ray jet observations.
\subsection{H$\alpha$ Observations}
\label{obs2}
Two jets-- jet J${_5}$ and J\ensuremath{'}${_5}$ were also observed
in the H$\alpha$ line center (6563 \AA) by ARIES,
Nainital, India as a bright ejecta. The bandwidth of the H$\alpha$
filter was 0.5 \AA\ with a cadence of 1 min.
H$\alpha$ jets from both jets sites started at $\approx$ 10:35 UT,
one minute later than in the EUV wavebands and they faded
away at $\approx$ 10:50 UT. We also observed the dark
material ejection surge between the bright jets from the two
sites site 1 and site 2 respectively.
To compare the spatial location of H$\alpha$ with EUV images,
we have over-plotted the
contours of H$\alpha$ in AIA 304 \AA\ and 171 \AA\ .
The result is shown in Figure~\ref{fig:hal}.
We found that H$\alpha$ jets are coaligned with the EUV jets.
However, the position of the H$\alpha$ jet is
only at the origin of the EUV jets. It may be because the H$\alpha$ jets
are in the chromosphere and have
less height than the EUV jets. The bright
ejections in H$\alpha$ are followed by dark material ejections.
It could be the cooler part of the jet.
This cooler part is spatially shifted towards the west side from the bright jets.
\subsection{Height--Time Analysis}
\label{obs3}
To understand the kinematics and dynamics of the observed jets, we have made a time-distance analysis in
different AIA/EUV channels {\it i.e.} 94, 131, 171, 193, 211, and 304 \AA.
To perform this analysis, we adopted two methods, one is the time--slice technique and
the other is tracking of the leading edge of the jets. For the time--slice technique, we
fixed a slit at the center of the jets and observed the motion of plasma along the slit.
An example of this time--slice analysis of jet J${_1}$ on 15 April 2014 is presented in
Figure~\ref{fig:slice1}.
In the figure, the left image shows the location of the slit (drawn as a black dashed line) and the right
denotes the height-distance maps at different EUV wavelengths.
Using this time--slice technique, we have computed the heights and projected
speeds
for the different structures of all jets at different wavelengths,
shown in the right panel of the figures.
For the same jet of 15 April 2014 at 193 \AA, the height--time plot using the leading
edge procedure is presented in Figure~\ref{fig:leading_edge}.
The curve has an exponential behavior with two acceleration phases.
We decided to fit linearly the beginning of the expansion and then
the later phase in order to compared the values with those of the other
methods. Two speed values are derived. The first is 117 km s$^{-1}$ which is
nearly half of the speed derived after the acceleration phase
(252 km s$^{-1}$). The error is around 10 km s$^{-1}$ according to the points chosen
for the fits.
The time--slice technique indicates that every jet has
multi-speed structures
and the speeds of the jets are different in different EUV wavelengths.
The average speed by the time--slice technique varies in different wavelengths
for J${_1}$: 205 -- 295 km s$^{-1}$, J${_2}$:
177--235 km s$^{-1}$, J${_3}$: 132--163 km s$^{-1}$, J\ensuremath{'}${_3}$: 100 --121 km s$^{-1}$,
J${_4}$: 133 --164 km s$^{-1}$, J\ensuremath{'}${_4}$: 295 --325 km s$^{-1}$, J${_5}$: 174 --202 km s$^{-1}$,
J\ensuremath{'}${_5}$: 275 --364 km s$^{-1}$, J${_6}$: 156 --197 km s$^{-1}$
J\ensuremath{'}${_6}$: 304 --343 km s$^{-1}$, and for J${_7}$: 154 --217 km s$^{-1}$ respectively.
The dispersion of the values for one jet according to the different considered wavelengths is
between 10$\%$ to 50$\%$, with no rule.
It is difficult to understand
if it is just the range of the uncertainties of the measurements or if it really
corresponds to the existence of multi-components in the jet with multi-temperatures and speeds
or if the slit of the time distance diagram is crossing different components of the jets.
We have compared these results with the speed derived by the leading edge procedure.
We found that the fast speed fits with the time-distance derived
velocity. The time-distance technique with a straight slit ignores the
first phase of the jets. The values of speed derived by both methods
are presented in Table \ref{recurrent}.
We have also computed the lifetimes, widths, and the maximum heights
of each jet.
The lifetimes vary from 6 to 24 min.
J${_5}$ has the maximum lifetime (24 min), whereas J\ensuremath{'}${_4}$
has the minimum (6 min) lifetime.
In general, we found the narrow and long life--time jets achieved
more height than the wide and short lifetime jets.
We have also noticed that the lifetimes of jets are longer in 304 \AA\ and in 171 \AA\ than in 94 \AA.
The width ranges from 2.6--6 Mm.
The maximum height was attained by J${_5}$ and it was 217 Mm.
\subsection{Evolution of Photospheric Magnetic Field}
\label{obs3}
The NOAA AR 12035 appeared at the east limb on
11 April 2014 with a $\beta$ magnetic
configuration. It went over the west limb on
24 April 2014. Figure~\ref{fig:mag} presents the magnetic field evolution of
the active region observed by SDO/HMI during 15--16 April 2014.
The active region consists of two large positive polarity sunspots P1 and P2 followed by the negative
polarity spot N1 (see Figure~\ref{fig:top}a).
The positive polarity P1 behaves like a decaying sunspot with a ``moat region"
around it. When the
sunspot is close to its decay stage, it loses its polarity by dispersing
in all directions. This dispersed positive polarity
cancels a part of the negative magnetic polarity
left at the site of the jet location.
The origin of jet activity lies between the
two leading positive polarities. As seen in
Figure~\ref{fig:mag} the whole active region shows
clockwise rotation (see also the attached movie MOV3).
Together with the rotation of the whole active region, the
western big spot P1 also shows a rotation
in the same direction as the active region.
The sunspot rotation makes the active region
sheared and the loops connecting the preceding
positive polarity and the
following negative polarity become sigmoidal
(see Figure~\ref{fig:loop}). The jets are ejected in the
south-east direction instead of south direction probably because of the
upper part of the sigmoidal loops.
To investigate the magnetic field evolution at the jet's origin, we have
carefully analyzed the development of different polarities.
On 15 April, we observed the positive polarity P1 and in its south a bunch of
negative polarity N2 and a positive polarity p (Figure~\ref{fig:mag}).
Site 1 of the jet's origin is located between N2 and p.
Site 2 is located to the west of site 1.
The zoomed view of magnetic flux evolution at site 1 is shown in Figure
\ref{fig:mag1}. In the
figure, we have noticed several patches of emerging flux of positive and negative polarities.
In addition to this emerging flux, we have interestingly found that the
negative polarity N2 and the positive polarity p came closer and cancelled
each other. The jet's cancelling location is represented by the red arrow.
To examine the flux cancellation at the jet
site 1, we made a time-slice diagram along the slit
shown in Figure \ref{fig:mag1} as the yellow dashed line.
The result is presented in Figure \ref{fig:mag_slice}.
The positive and negative flux approached each other and cancelled afterwards.
We have drawn the start and end time of the jets from site 1 and this is
shown in the figure by vertical red dashed lines. We noticed that the jet
activity from site 1 was between this flux cancellation site.
Further, we did quantitative measurements in the box at jet site 1 drawn in Figure \ref{fig:mag} (top, middle panel).
The positive, negative and total signed flux variation as a function of time
calculated over the box is shown in Figure \ref{fig:mag_evo}a.
On 15 April the flux is constantly emerging even with some cancellation around 16:00 UT,
on 16 April the flux decreases due to cancellation.
For site 2, the enlarged version of magnetic field evolution is shown in
Figure \ref{fig:mag2}. The emergence of different positive
and negative patches is shown by green and cyan arrows respectively.
Around the site 2 location, we have observed positive
polarities surrounded by a kind of supergranule cell.
Inside this positive
polarity supergranule, small bipoles with mainly negative polarities are
continuously emerging.
For the quantitative evolution of the positive, negative and total magnetic flux at
the site 2 location, we have also calculated the magnetic flux as a function of time inside the
red box (top, middle) of Figure \ref{fig:mag}. The variation of magnetic flux with time
is shown in Figure \ref{fig:mag_evo}b. We have noticed that the emergence of magnetic field
is continuously followed by the cancellation. At site 2, all the four jets are present on
16 April and we have drawn the onset and end time of these jets. The jet duration over site 2 is
denoted by the green dashed lines in Figure \ref{fig:mag_evo}b. We found that the jets from site 2
were lying in between the emergence and the cancellation site.
In summary, we have noticed the continuous magnetic flux
emergence followed by cancellation at site 1 during the time of the jet on 15 April,
On 16 April flux emergence and cancellation are recurrent at both sites.
\begin{table}[t]\tiny
\caption{Different physical parameters derived from SDO/AIA data for
jets on 15--16 April 2014 at site 1 [-220$^{''}$, -215$^{''}$]
and site 2 [-205$^{''}$, -215$^{''}$] for different jets. The jets
J${_1}$ -- J${_7}$ are from site 1 and
J\ensuremath{'}${_3}$ --
J\ensuremath{'}${_6}$ are from site 2. The speeds without and within parentheses are
derived from the time--slice technique and the leading edge method respectively (see column 5).}
\label{recurrent}
\centering
\begin{tabular}{ccccccc}
\hline
Jet&~~Start/& Speed at different& Height& Width& Lifetime \\
number&end & wavelengths ($\lambda$) in km s$^{-1}$ & (Mm) & (Mm) & (min) \\
&(UT) &304\AA~~~211\AA~~~193\AA~~~171\AA~~~131\AA~~~~94\AA & &\\
\hline
J${_1}$ &~~14:55/ & 205~~~~ 295~~~~~ 257~~~~~ 249~~~~~ 268~~~~~ 206& 145 &4.4 &15 \\
&15:10 &~~~~~~~~~~~~~(252)~~~~~~~~~~~~~~~~~~~~~~~~~& & & \\
\hline
J${_2}$ &~~18:01/ & 234~~~~ 177~~~~~ 199~~~~~ 232~~~~~ 221~~~~~ 235& 124 &5.2 &08 \\
&18:09 &~~~~~~~~~~~~~(196)~~~~~~~~~~~~~~~~~~~~~~~~~& & & \\
\hline
J${_3}$ &~~06:33/ & 137~~~~ 140~~~~~ 132~~~~~ 153~~~~~ 148~~~~~ 163& 202 &4.3 &23 \\
&06:56 &~~~~~~~~~~~~~(126)~~~~~~~~~~~~~~~~~~~~~~~~~& & & \\
\hline
J\ensuremath{'}${_3}$ &~~06:10/ & 113~~~~ 121~~~~~ 109~~~~~ 110~~~~~ 105~~~~~ 100& 108 &3.5 &15 \\
&06:34 &~~~~~~~~~~~~~(100)~~~~~~~~~~~~~~~~~~~~~~~~~& & & \\
\hline
J${_4}$ &~~07:13/ &136~~~~ 139~~~~~ 147~~~~~ 133~~~~~ 164~~~~~ 138& 95 &2.6 &11 \\
&07:23 &~~~~~~~~~~~~~(147)~~~~~~~~~~~~~~~~~~~~~~~~~& & & \\
\hline
J\ensuremath{'}${_4}$ &~~07:12/ & 305~~~~ 295~~~~~ 325~~~~~ 300~~~~~ 296~~~~~ 303& 116 &4.5 &06\\
&07:18 &~~~~~~~~~~~~~(322)~~~~~~~~~~~~~~~~~~~~~~~~~& & & \\
\hline
J${_5}$ &~~10:36/ & 183~~~~ 202~~~~~ 174~~~~~ 185~~~~~ 184~~~~~ 182& 217 &6.0 &24\\
&10:50 &~~~~~~~~~~~~~(174)~~~~~~~~~~~~~~~~~~~~~~~~~& & & \\
\hline
J\ensuremath{'}${_5}$ &~~10:33/ & 275~~~~ 343~~~~~ 364~~~~~ 291~~~~~ 316~~~~~ 241&87 &3.6 &10 \\
&10:43 &~~~~~~~~~~~~~(357)~~~~~~~~~~~~~~~~~~~~~~~~~& & & \\
\hline
J${_6}$ &~~14:41/ & 197~~~~ 177~~~~~ 192~~~~~ 156~~~~~ 170~~~~~ 171&94 &3.0 &15 \\
&14:55 &~~~~~~~~~~~~~(187)~~~~~~~~~~~~~~~~~~~~~~~~~& & & \\
\hline
J\ensuremath{'}${_6}$ &~~14:47/ & 343~~~~ 326~~~~~ 340~~~~~ 323~~~~~ 307~~~~~ 304& 152 &4.0 &16 \\
&14:59 &~~~~~~~~~~~~~(335)~~~~~~~~~~~~~~~~~~~~~~~~~& & & \\
\hline
J${_7}$ &~~16:59/ & 183~~~~ 174~~~~~ 154~~~~~ 187~~~~~ 184~~~~~ 217& 145 & 5.1 &14\\
&17:13 &~~~~~~~~~~~~~(154)~~~~~~~~~~~~~~~~~~~~~~~~~& & & \\
\hline
\end{tabular}
\end{table}
\section{Magnetic Topology}
\label{mag_top}
The longitudinal magnetic field maps observed by HMI show a strong
complexity of the polarity pattern and a fast evolution
(Figures \ref{fig:mag}, \ref{fig:mag1}, \ref{fig:mag2}) in the two
sites where the two series of jets are initiated. Looking carefully at
the AIA movies we have detected a shift of the jets from site 2
to site 1 and never from site 1 to site 2. The shift is not always fully accomplished or fully observed. The best case is jet J\ensuremath{'}${_5}$ from site 2 which completely
merged with J${_5}$ from site 1.
It is important to understand why there is a
slippage of the magnetic field lines. Slippage reconnection has been
observed in many flares \citep {Priest1992, Berlicki2004,
Aulanier2005, Dudik2012}.
Commonly it is due to the slippage of magnetic field lines anchored along
QSL structures. The slippage occurs when and
where the squashing degree is high enough along the QSL to force the
reconnection. \inlinecite {Demoulin1996} and \inlinecite{Demoulin1998} has shown,
theoretically and
observationally, how it can be produced. \citet{Dalmasse2015} demonstrated that the QSLs
are robust structures and can be computed in potential configurations.
Qualitatively the results are very good with this approach
and there is a relatively good fit between the location of QSL footprints
with the observed flare ribbons \citep{Aulanier2005}. However when the
magnetic configuration is too complex, the QSL footprints do not fit
perfectly and a one to one comparison is increasingly difficult with smaller and smaller scale polarities
\citep {Dalmasse2015}.
The benefit of using linear force--free field (LFFF) extrapolation can be weak since QSLs are
robust to parameter changes. In the present case, LFFF extrapolation would perhaps help
to follow the path of the jets which shows some curvature at their bases
(mentioned as a sigmoidal shape in the text of Section 2.1 ) but at the expense of the geometry of loops at large heights.
However, the magnetic field strength is really fragmented in the jet regions.
Pre-processing the data would smear electric currents in their moving weak polarities, so it will
not help to derive better QSLs.
Therefore to investigate the magnetic topology of the jet producing regions,
we use the same
potential magnetic field extrapolation of AR 12035 calculated in \inlinecite {Zuccarello17}.
This method is based on the fast Fourier transform method of \inlinecite {Alissandrakis1981} and
the extrapolation is performed by using a large field-of-view that includes at its center the AR 12035.
This allows us to identify the key topological structures of the active region (see Figure~\ref{fig:top} left panels).
Maps of the squashing degree {\em Q} at the photospheric plane were calculated using the topology tracing code
topotr (see \inlinecite {Priest1995}, \inlinecite{Demoulin1996}, \inlinecite{Pariat2012} for more details).
The locations of the largest values of {\em Q} define the footprints of the QSLs
\citep{Demoulin1996,Aulanier2005} and they correspond to regions where electric current layers can easily develop.
In \inlinecite{Zuccarello17} we study the behavior of eruptive and failed eruptions
occurring in the north-west part of the AR. We have found two QSLs: the first one (Q1 in Figure~\ref{fig:top}b and c)
encircling the positive polarity P1 and separating the magnetic flux system from the external field and a
second one (Q2 in Figure~\ref{fig:top}b and \ref{fig:top}c) highlighting
the spine of a high-altitude coronal null-point similarly to what is seen in \inlinecite{Masson2009}.
The flares occurred mainly at the north-west edge of the large QSL (Q1).
Since the fan-like QSL (Q1) encircling the flare region was separated from the complex QSL system
around the jet producing region (at the south of P1), we have argued that the jet activity and the
flares were not really linked to each other, even if their timings seem to be related \citep{Zuccarello17}.
On 15 April the jets are initiated in site 1 and we find a well defined QSL surrounding the region of the jet.
On 16 April the configuration is much more complicated with
many QSLs which are in the site region of both jets. The zoom of the {\em Q} map of 16 April
around the jet producing region (red box of Figure~\ref{fig:top}c) is shown in Figure~\ref{fig:null}.
We find that the two sites of the jets, site 1 and site 2, are respectively inside the QSLs Q3 and Q4,
and that both were embedded in a larger QSL, labeled as Q0 in the figure.
We also identified several quasi photospheric null points, that are indicated by yellow circles.
The QSL map is very complex, and difficult to analyze and compare in detail with the observations due to
the small scale of the events and fast motion of the small polarities, both the moving p polarity in
site 1 and the emergence of small bipoles in site 2.
Since it looks quite possible that the 2 QSLs (Q3 and Q4) intersect or touch each other,
we conjecture that field line foot points could move from site 2 and site 1 by a sequence of
reconnections across QSLs as in \inlinecite{Dalmasse2015}.
This could produce the transfer/or tendency of movement of jets from site 2 to site 1, as we have
observed for jets J$_5$ and J\ensuremath{'}${_5}$ for example (Figure~\ref{fig:jet5}). In the case of the
other jets from site 2, they also show a tendency of slippage towards site 1.
On 15 April the topology of QSLs in the jet region is more simple.
That could explain perhaps why there are no jets from site 2.
\section{Discussion and Conclusions}
\label{result}
In this study, we have analyzed eleven solar jets originating from two
different sites in the same active region
(NOAA AR 12035) during 15--16 April 2014 using the
SDO/AIA, HMI, and ARIES H$\alpha$ data.
The magnetic topology of the active region was discussed using a potential
magnetic field extrapolation.
The extrapolation was done using HMI photospheric magnetic
field as the boundary condition.
The main conclusions of our study are as follows:
\begin{itemize}
\item
We found two sites for the different jet's activity.
On 15 April the jets originated from site 1 and we measure a large increase of
emerging flux and small cancellation. On 16 April site 1
and site 2 are associated with continuous emerging magnetic flux followed by cancellation at the jet time.
\item
The kinematics of jets at different EUV wavebands revealed that the speeds,
widths, heights and lifetimes of jets
are slightly different at different wavelengths. This can be interpreted as
the multi--temperature and multi-- velocity structure of
solar jets.
\item
Most of the jets showed clockwise rotation, which
indicates untwisting. As a result of this
untwisting, the twist/helicity was injected
in the upper solar atmosphere.
\item
We observed the slippage of jets at site 2 namely
J\ensuremath{'}${_3}$--J\ensuremath{'}${_6}$ towards the eastern site (site 1) and never the reverse movement.
Along with the movement of jets towards site 1, we found, in the case of
jet J\ensuremath{'}${_5}$ that a part
detached from it and moved towards the site 1 location and
finally merge into jet J${_5}$.
\item
On 16 April both jet sites are associated with
the QSLs. The possible intersection of the two
QSLs encircling each site could explain the slip
reconnection occurring along the
QSLs which favor the translation of jets from
site 2 to site 1.
\end{itemize}
The clockwise rotations (right-to-left) in some of the jets
indicate the untwisting of the jets.
The untwisting jets eject the helicity in the
higher solar atmosphere \citep{Pariat15}.
The injected helicity in the jets may
be part of the global emergence of twisted magnetic fields.
During the rotation like in the case of jet J${_1}$, we observed
the rotating jet material contains bright as
well as dark material.
This result is consistent with simulations done by \inlinecite{Fang14}.
In their simulation, they found the simulated jet consists
of untwisted field lines, with a mixture of cold and hot plasma.
The kinematics of the jets indicates that different jets have not only
different speeds but their speed also varies with different wavelengths.
This can be interpreted as multi-temperature and
multi-velocity structures in the solar jets.
Our calculated values of the speeds, widths
and lifetimes are consistent with earlier reported values
in the literature \citep{Shimojo96,Shimojo00,Schmieder13,Chandra15}.
We have also observed that the average lifetime is longer in 304 \AA\
than in shorter wavelength observations,
which suggests that the cooler component of jets have a longer lifetime
in comparison to the hotter component.
This supports the study of \inlinecite{Nistico09}.
\inlinecite{Nistico09} compared the lifetime between 171 \AA\ and
304 \AA\ and found that the lifetime is longer for the longer wavelength.
For all the studied jets, except for
J${_5}$ and J\ensuremath{'}${_5}$, the jet peak time is simultaneous at all wavelengths.
In the case of J${_5}$ and
J\ensuremath{'}${_5}$, the peaks at longer wavelengths are earlier than
at the shorter EUV wavelengths. The time delay between longer and shorter EUV wavelengths
in the case of jet J${_5}$ and J\ensuremath{'}${_5}$ can be interpreted as
during the reconnection, there could be a different heating time for
different threads.
Thanks to the SDO high spatial and temporal resolution,
we could examine the
dynamics of these jets in a more precise way.
As mentioned in Section \ref{obs1}, the detached part of J\ensuremath{'}${_5}$
moved towards J${_5}$ and finally merged with J${_5}$.
The motion of the broken part of J\ensuremath{'}${_5}$
towards the east can be interpreted
as the expansion of the reconnection region with time \citep{Raouafi16}.
As suggested by \inlinecite{Sterling15}, the coronal jets
may be due to the eruption of mini filaments and they predicted
that the spire of the jets moves away from the jet base bright point.
Our observation of the motion of the broken part of J\ensuremath{'}${_5}$
is away from the jet base. This supports the findings of \inlinecite{Savcheva09}
and the interpretation proposed by \inlinecite{Sterling15}.
We have observed the flux emergence followed by flux cancellation at site 1 during 15 April 2014.
Moreover, on 16 April 2014 flux emergence and cancellation are recurrent in both jet sites.
The observation of cool and hot material in our study
supports the hypothesis of small filament eruption and
a universal mechanism for eruptions at different scales \citep{Sterling15,Wyper2017}.
\newpage
\begin{figure}[t]
\vspace*{-0.4cm}
\hspace{-1.2cm}
\includegraphics[width=1.08\textwidth, clip=]{fig1.ps}
\vspace*{-0.4cm}
\caption{Jets J${_5}$ and J\ensuremath{'}${_5}$ from site 1 and site 2
respectively in the south of the main sunspot of AR 12035 on 16 April 2014 {\it i.e.} from left to right in H$\alpha$, observed in Nainital,
India (ARIES),
in 304 \AA\ and in 171 \AA\ observed with SDO/AIA.
The contours of H$\alpha$ are over-plotted on to 304 \AA\ and 171 \AA.
The jets look to be overlapping according to the scale resolution of the images. }
\label{fig:hal}
\end{figure}
\begin{figure}
\includegraphics[width=1.0\textwidth, clip=]{fig2.ps}
\caption{Recurrent jets in the active region AR 12035 on 15 and 16 April 2014.
Upper, middle and bottom panels are images of AIA
filters at 304 \AA, 193 \AA, and 94 \AA\ respectively.
The arrows indicate the bright emission of the jets in two different locations mentioned as site 1 and site 2 in Figure~\ref{fig:hal}.}
\label{fig:jet/morpho}
\end{figure}
\newpage
\begin{figure}[t]
\includegraphics[width=1.0\textwidth, clip=]{fig3.ps}
\caption{Evolution of jet J${_1}$ initiated in site 1
observed by AIA 211 \AA. The contours on the first image are HMI
longitudinal magnetic field with
yellow and cyan colors for positive and negative polarities respectively.
The contour levels are $\pm$100 Gauss.
The site 1 is indicated by a white arrow.
The cool and hot part of the ejected jet J${_1}$ are indicated
by black arrows.}
\label{fig:jet1}
\end{figure}
\newpage
\begin{figure}[t]
\includegraphics[width=1.0\textwidth, clip=]{fig4.ps}
\caption{Evolution of jets J${_5}$ and J\ensuremath{'}${_5}$
from site 1 and site 2 (indicated by white arrows)
observed by AIA 211 \AA. The contours on the first image are HMI
longitudinal magnetic field.
Yellow and cyan colors are for positive and negative polarities respectively.
The contour levels are $\pm$100 Gauss. The positions of jets J${_5}$, J\ensuremath{'}${_5}$ and the
circular ribbon are indicated by black arrows.}
\label{fig:jet5}
\end{figure}
\newpage
\begin{figure}[t]
\vspace*{-0.1cm}
\hspace{-0.5cm}
\includegraphics[width=1.0\textwidth, angle=0, clip=]{fig5.ps}
\vspace{-0.2cm}
\caption{Intensity profile of jet J${_5}$ at base location. Different colors
represent different SDO/AIA wavelengths. The peak of the cooler component is
earlier (304 \AA: longer wavelength) than the peak of the hotter component (94 \AA: shorter wavelength).}
\label{fig:inten}
\end{figure}
\newpage
\begin{figure}[t]
\includegraphics[width=0.70\textwidth, clip=]{fig6.ps}
\caption{Sigmoidal orientation of the loops in the active region
in the AIA 193 \AA\, filter (right panel) and in the difference image (left panel).
The jet J$_5$ follows the path of the loop.}
\label{fig:loop}
\end{figure}
\newpage
\begin{figure}[t]
\hspace{-0.8cm}
\includegraphics[width=1.06\textwidth, clip=]{fig7.ps}
\vspace{-0.6cm}
\caption{Time-slice analysis of jet J${_1}$ on 15 April 2014. Left: position of slit along the jet,
right: velocity comparison at different wavelengths.}
\label{fig:slice1}
\end{figure}
\begin{figure}[h]
\hspace{-0.8cm}
\includegraphics[width=1.00\textwidth,clip=]{fig8.ps}
\caption{An example of a height--time plot for Jet J${_1}$ derived using the
leading edge procedure.}
\label{fig:leading_edge}
\end{figure}
\newpage
\begin{figure}[t]
\includegraphics[width=1.00\textwidth,angle=0,clip=]{fig9.ps}
\caption{ Magnetic field evolution of NOAA active region 12035. The two jet
locations (site 1 between N2 and p, site 2 at the edge of the supergranule
indicated by the black contour) are indicated by white arrows.
The red boxes in the left of the bottom panel show the site 1 (left) and
site 2 (right) locations respectively and the field of view of Figures \ref{fig:mag1}
and \ref{fig:mag2}.}
\label{fig:mag}
\end{figure}
\newpage
\begin{figure}[t]
\includegraphics[width=0.70\textwidth,angle=90,clip=]{fig10.ps}
\caption{Evolution of magnetic field at the site 1 location.
The flux cancellation at site 1 between the negative polarity and the positive
polarity (p) is shown by a red arrow.
The horizontal yellow line indicates the location of the slit, where the time-distance evolution of flux is
calculated (see Figure \ref{fig:mag_slice}).}
\label{fig:mag1}
\end{figure}
\newpage
\begin{figure}[t]
\includegraphics[width=1.0\textwidth,angle=0,clip=]{fig11.ps}
\caption{Evolution of magnetic field at the site 2 location. The positive (negative) flux emergence is shown by green (cyan)
arrows respectively. The roundish black curve indicate kind of supergranule cell.}
\label{fig:mag2}
\end{figure}
\newpage
\begin{figure}[t]
\includegraphics[width=1.00\textwidth,clip=]{fig12.ps}
\vspace*{-4.5cm}
\caption{Magnetic field evolution along the slice shown in Figure 9 at site 1. Two vertical lines in the image
indicate the time of jet activity.}
\label{fig:mag_slice}
\end{figure}
\begin{figure}[t]
\vspace*{0.2cm}
\includegraphics[width=1.00\textwidth,clip=]{fig13.ps}
\vspace*{-0.1cm}
\caption{Magnetic flux as a function of time calculated over the red box of
Figure \ref{fig:mag} for the site 1 (a) and site 2 (b) location. Black, red and blue curves are for
total, positive and negative magnetic flux respectively.
Jets on 15 April and 16 April occur in between the yellow and green dashed
lines respectively.}
\label{fig:mag_evo}
\end{figure}
\begin{figure}[h]
\includegraphics[width=1.00\textwidth,clip=]{fig14.ps}
\vspace*{-0.1cm}
\caption{Magnetic field (line of sight component) of AR 12035 for
16 April 2014. Panel (a) shows the two leading positive polarities
P1 and P2 and the following negative polarity N1.
The magnetic field is overlaid by magnetic field lines obtained
from a potential extrapolation. Panels (b) and (c): Contours of the magnetic
field overlaid by the footprints of the QSL at {\em z}=0.4 Mm above the photosphere,
respectively for 15 April (b) and 16 April (c).
Q1 and Q2 indicate the QSLs related to the flares,
The magenta/cyan contours are drawn for
levels of the magnetic field equal to $\pm 100, \pm 300,
\pm 500, \pm 700, \pm 900$ Gauss. The red boxes indicate the region of the jets, site 1 on 15 April and both sites on 16 April.
(adapted from \inlinecite{Zuccarello17}).}
\label{fig:top}
\end{figure}
\begin{figure}[h]
\includegraphics[width=1.00\textwidth,clip=]{fig15.ps}
\vspace*{-0.1cm}
\caption{Zoom of the jet region
inside the red box of 16 April (see panel (c) in Figure \ref{fig:top}).
Q1 and Q2 indicate the QSLs related to the flares,
Q3 and Q4 to the site 1 and site 2 of the jets
respectively. The vertical color bar indicates the value of the
squashing degree {\em Q}. The yellow circles indicate the positions
of four null points. }
\label{fig:null}
\end{figure}
\begin{acks}
The authors are thankful to the referee for the
constructive comments and suggestions
which improved the manuscript significantly.
We acknowledge the SDO/AIA and HMI open data policy. RJ
acknowledges the Department of
Science and Technology (DST), Government of India
for an INSPIRE fellowship. This work was initiated during
the one month stay of RC at the Observatoire de Paris, LESIA, Meudon, France.
RC also acknowledges the support from SERB--DST project no. SERB/F/7455/ 2017-17.
\noindent {\bf Disclosure of Potential Conflicts of Interest} The authors declare that they have no conflicts of interest.
\end{acks}
| -26,160.713721 |
[
-3.06640625,
2.939453125
] | 38.427948 |
[
-2.67578125,
0.65966796875,
-1.4033203125,
-4.48046875,
-0.52978515625,
6.22265625
] |
[
2.37890625,
6.46875,
3.61328125,
4.9921875
] | 405 | 6,644 |
[
-3.21484375,
3.67578125
] | 29.577834 |
[
-5.640625,
-2.23828125,
-2.466796875,
-1.779296875,
1.02734375,
9.0625
] | 1.472147 | 15.67944 | 20.906081 | 5.626649 |
[
3.540297508239746
] | -21,363.786813 | 5.111981 | -25,244.285062 | 0.58297 | 5.819188 |
[
-3.322265625,
-3.353515625,
-2.677734375,
-3.486328125,
2.3671875,
9.9453125
] |
[
-5.78125,
-1.4775390625,
-2.01171875,
-0.65625,
3.2890625,
3.92578125
] | |
BkiUdMXxK3xgpZzQoFCv
|
\section{Introduction}
Binary transition metal compounds crystallizing in the $B20$
structure (space group $P2_13$) \cite{Boren33,Pauling48} show a
variety of interesting physical properties ranging from a
narrow-gapped semiconductor (FeSi) to diamagnetic (CoSi) or
paramagnetic (FeGe) metal. In the latter case recent experiments
have shown that the helical ordering temperature ($T_C=280$~K at
ambient pressure) can be suppressed by the application of external
pressure \cite{Pedrazzini07}. Furthermore, anomalies in internal
structural parameters in cubic FeGe ($\epsilon$-FeGe) for isothermal
compression seem to be related to the magnetic phase boundary line.
Here we present single crystal data at ambient conditions as well as
in detail the results of the powder diffraction data at high
pressure and temperatures as low as 82~K.
\section{Sample Growth and Characterization}
Single crystals of $\epsilon$-FeGe were grown by chemical vapor
transport using iodine as agent \cite{Richardson67} in a homemade
two zone furnace. $\epsilon$-FeGe crystallized very slowly by an
endothermal transport reaction from 850~K to 810~K although the
transport was made perpendicular to the tube axis \cite{Kraemer74}.
The resulting crystals had a volume of less than 1~mm$^3$. Pieces
used in the subsequent experiments were examined thoroughly by
various X-ray techniques and electron-beam microanalysis.
The wavelength-dispersive-X-ray-analysis (WDX) performed on a
microprobe (CAMECA SX100, W-cathode) confirms the ratio Fe:Ge = 1:1
of the crystals and X-ray spectra excited by electron beam (25 keV,
20 nA) showed only lines of both elements. The evaluation of the
background intensities at energies of Si-$K$ and I-$L$-lines allowed
us to exclude impurity concentrations larger then 0.08 wt.\% for
both elements involved in the preparation process. Microstructures
prepared from crystals by metallographic methods contained small,
disoriented areas which show unusual strong orientation in the
scanning electron microscope images using back-scattered electron
(BSE) contrast as well as in polarized light of the light microscope
(cf.~Fig.~\ref{fig:microprobe}).
\begin{figure}
\center{\includegraphics[width=0.35\textwidth]{wilhelm_fig1}}
\caption{Microstructure of well-facetted $\epsilon$-FeGe-sample
(scanning electron microscope image, BSE contrast) is dominated by
single crystalline area (homogenous grey) and cavity (black) at the
center. A small, disoriented area (darker grey) next to the central
hole shows weak contrast in the BSE image but no significant
differences in composition is found by WDX measurements.}
\label{fig:microprobe}
\end{figure}
\section{Single Crystal Structure Determination}
The crystal structure of $\epsilon$-FeGe was solved and refined on
single crystal intensity data (Rigaku R-Axis Spider, Ag-$K{\alpha}$
radiation, $\lambda=56.080$~pm, rotating anode-micro source, with
mirror optic, $\omega$-scan, $\Delta\omega=1.0^\circ$, 160 images;
$2\Theta_{max} = 122.6^\circ$, empirical absorption correction). For
the structure refinement a full-matrix least squares on $F^2$ on
eight parameter was performed using SHELXL-97 \cite{shelxs97}. A
summary of the refinement data is given in
Tab.~\ref{tab:crystallographicdata} and \ref{tab:atomiccoordinates}.
The lattice parameter were determined from powder data on 12
reflections with the program package CSD \cite{Akselrud93} (image
plate Guinier camera HUBER G670, Co-$K\alpha_1$, $\lambda$ =
178.8965 pm, LaB$_6$ ($a$ = 415.692 pm) as internal standard, $T$ =
295 K). All structural parameter are given in
Tab.~\ref{tab:structure}.
\begin{figure}[t]
\hspace*{-25mm}
\includegraphics[width=0.75\textwidth]{wilhelm_fig2}
\caption{Projection of the crystal structure of $\epsilon$-FeGe
along [100]. The symbols show the position of the $2_1$ screw axis.
The lines indicate the shortest Fe-Ge distance.}
\label{fig:projection}
\end{figure}
\begin{figure}[]
\hspace*{-10mm}
\includegraphics[width=0.575\textwidth]{wilhelm_fig3}
\caption{Coordination polyhedron of Fe (left) and Ge (right). The
shortest Fe-Ge distance is $d_1$.} \label{fig:coordination}
\end{figure}
\begin{table}[]
\caption{Details of the crystal structure analysis of
$\epsilon$-FeGe at ambient pressure and $T=295$~K.}
\label{tab:crystallographicdata}
\begin{tabular}{l|l}\hline
Density (g cm$^{-3}$) & 8.22 \\
$\mu$ (cm$^{-1}$) & 219 \\
measured and unique reflections & 3680; 1025\\
obs. reflections $> 4.0~\sigma(Fo)$ & 1001\\
$R_{int}$ & 0.018\\
$R1$; $wR2$ & 0.020; 0.026\\
Goof=S & 1.209\\
$\Delta\rho_{max}$;$\Delta\rho_{min}$ (e10$^6$~pm$^3$) & 2.05; -1.01
\\\hline
\end{tabular}
\end{table}
\begin{table}[]
\caption{Anisotropic displacement parameter $U_{ij}$ (in pm$^2$) for
$\epsilon$-FeGe. The $U_{eq}$ is defined as one third of the
orthogonalized $U_{ij}$ tensor.} \label{tab:atomiccoordinates}
\begin{tabular}{l|c|c|c}
& U$_{11}=U_{22}=U_{33}$ & $U_{23}=U_{13}=U_{12}$ & $U_{eq}$ \\
\hline
Fe & 57(2) & -6(1) & 57(2) \\
Ge & 55(2) & 0(1) & 55(2) \\ \hline
\end{tabular}
\end{table}
In the $B20$ structure (space group $P2_13$, No. 198, $Z=4$) both
atoms occupy $4a$ $(x,x,x)$ Wyckoff positions
(Fig.~\ref{fig:projection}). In a compound AB crystallizing in the
ideal $B20$ structure ($x_{\rm{A}}=0.1545$,
$x_{\rm{B}}=1-x_{\rm{A}}$) both atoms have a seven-fold coordination
which lies between that of the two simplest binary cubic structures
NaCl (CN = 6) and CsCl (CN = 8). $\epsilon$-FeGe however, attains a
distorted rock-salt structure where the atoms are slightly shifted
from their ideal position ($x_{\rm{Fe}}=0.13522(2)$ and
$x_{\rm{Ge}}=0.84186(1)$. This results in one short Fe-Ge distance,
$d_1=238.78$~pm, along the trigonal axis pertinent to the Fe atom as
well as two longer ones, $d_2=244.57$~pm, and $d_3=264.45$~pm,
between the three second and three third-nearest neighbors,
respectively (see Fig.~\ref{fig:coordination}). Thus, the
coordination sphere of Fe is build up of seven Ge atoms describing a
strong distorted mono capped trigonal prism.
\section{Powder data at High Pressure}
\begin{table}[]
\caption{Structural parameters of $\epsilon$-FeGe at ambient
conditions and for comparison data of $\epsilon$-FeSi taken from
Ref.~\cite{Wartchow97}. In the standard setting the fractional
coordinates are $x_{\rm{Fe}}=0.38522(2)$, $x_{\rm{Ge}}=0.09186(1)$
for $\epsilon$-FeGe and $x_{\rm{Fe}}=0.38650(2)$,
$x_{\rm{Si}}=0.09262(5)$ for $\epsilon$-FeSi. B denotes either Ge or
Si.} \label{tab:structure}
\begin{tabular}{l|c|c}
& $\epsilon$-FeGe & $\epsilon$-FeSi \\ \hline
$a$~(pm) & 469.95(2) & 449.5(2) \\
$V_0$~(10$^6$pm$^3)$ & 103.79(1) & 90.8 \\
$x_{\rm{Fe}}$ & 0.13522(2) & 0.13650(2) \\
$x_{\rm B}$ & 0.84186(1) & 0.84262(5) \\
$d_1$(pm) & 238.78(2) & 228.8 \\
$d_2$(pm) & 244.57(1) & 234.6 \\
$d_3$(pm) & 264.45(1) & 251.98 \\
$d_{\rm{Fe-Fe}}$(pm) & 288.11(1) & 275.6 \\
$d_{\rm{B-B}}$~(pm) & 291.13(1) & 278.3 \\ \hline
\end{tabular}
\end{table}
\begin{figure}[]
\begin{center}
\includegraphics[width=0.5\textwidth]{wilhelm_fig4a}
\includegraphics[width=0.5\textwidth]{wilhelm_fig4b}
\caption{(a) Diffraction pattern recorded at 82~K at different
pressures. The difference between raw and fitted data is also shown.
(b) Pressure dependence of the interatomic Fe-Ge distances $d_1$ and
$d_2$ (left scale) and $d_3$ (right scale). The lines represent a
Murnaghan EOS fit to the data. In $d_1(p)$ a discontinuity at about
15~GPa indicates a subtle change in the distances. Error bars
represent $3\sigma$. Inset: $V(p)$ data and EOS fit. Open symbols
represent data obtained on pressure release.}\label{fig:pattern}
\end{center}
\end{figure}
\begin{figure}[]
\begin{center}
\includegraphics[width=0.5\textwidth]{wilhelm_fig5a}
\includegraphics[width=0.5\textwidth]{wilhelm_fig5b}
\caption{Pressure dependence of the interatomic Fe-Ge distances
$d_1$ and $d_2$ (left scale) and $d_3$ (right scale). Error bars
represent $3\sigma$. The lines represent a Murnaghan EOS fit to the
data. (a) $T=230$~K data set, where $d_1(p)$ shows a discontinuity
at about 12~GPa. (b) $T=295$~K data set, where no anomaly in
$d_1(p)$ is observed. Insets: $V(p)$ data and EOS fit. Data obtained
on decreasing pressure are shown as open symbols.}
\label{fig:distanes230}
\end{center}
\end{figure}
The structural evolution with pressure at different temperatures was
determined by angle-dispersive X-ray diffraction experiments at the
beamline ID09 at the European Synchroton Radiation Facility. The
diffraction pattern were collected at a wavelength of
$\lambda=41.254$~pm during a 10~s exposure time with an image plate.
For these measurements well ground and annealed powder (at 670 K for
two days) was loaded in a diamond-anvil cell. Helium was used as
pressure medium and the pressure was determined via the fluorescence
peaks of SrB$_4$O$_7$:Sm$^{2+}$ \cite{Leger90}.
Figure~\ref{fig:pattern}(a) shows two examples of a full profile
structural refinement of the diffraction pattern recorded at 82~K.
The intensities can be very well refined though with little less
precision at the highest pressure. Thus, the fractional atomic
parameter $x_{\rm{Fe}}$ and $x_{\rm{Ge}}$, and hence the interatomic
distances could be deduced. The refinement of the diffraction
pattern measured at 82~K and 230~K revealed clear indications that
the shortest interatomic Fe-Ge distance, $d_1$, which is parallel to
$[111]$, changes its pressure dependence discontinuously
(Fig.~\ref{fig:pattern}(b) and Fig.~\ref{fig:distanes230}(a)). The
extrapolation of the low-pressure $d_1(p)$ behavior for $82\,$K and
$230\,$K described by an appropriate equation-of-state (EOS) clearly
fails to account for the pressure dependence above $15.8\,$GPa and
$12\,$GPa, respectively. These anomalies in $d_1(p)$ agree
remarkably well with the $T_C(p)$-phase boundary deduced from the
electrical resistivity measurements reported in
Ref.~\cite{Pedrazzini07}. The two remaining Fe-Ge distances as well
as all distances at room temperature (Fig.~\ref{fig:distanes230}(b))
decrease smoothly with pressure and no anomaly is seen within the
error bars.
\begin{table}
\caption{Bulk modulus $B_0$, and its pressure derivative $B'$ of
$\epsilon$-FeGe obtained from a fit of the Murnaghan EOS to the
$V(p)$ data. \label{tab:compressibility}}
\begin{tabular}{l|c|c}
T (K) & $B_0$~(GPa) & $B'$ \\ \hline
295 & 130(1) & 4.7(1) \\
230 & 135(1) & 4.7(1) \\
82 & 147(3) & 4.4(2) \\
\hline
\end{tabular}
\end{table}
The $V(p)$ dependencies of $\epsilon$-FeGe at all investigated
temperatures revealed no discontinuous pressure dependence and that
the compression is reversible (cf. insets to
Figs.~\ref{fig:pattern}(b) and \ref{fig:distanes230}). All $V(p)$
can be described by a Murnaghan EOS \cite{Murnaghan44}. The
resulting fit parameters are given in
Tab.~\ref{tab:compressibility}. The bulk modulus of $\epsilon$-FeGe
is considerably smaller than that of $\epsilon$-FeSi which is
ranging from $B_0=160(1)$~GPa ($B'=4.0$) \cite{Wood95} to 209(6)~GPa
($B'=5.3$) \cite{Knittle95}. A simple comparison of the unit-cell
volume and using the compressibility of $\epsilon$-FeGe shows that
$\epsilon$-FeGe attains the same unit-cell volume as $\epsilon$-FeSi
at about 25~GPa. Theoretical studies on $\epsilon$-FeSi predict a
phase transformation to the $B2$ structure (CsCl-type) at about
13~GPa \cite{Vocadlo99} or in the range of 30 to 40~GPa
\cite{Caracas04}. On the other hand, no phase transformation in
$\epsilon$-FeSi has been found up to 50~GPa \cite{Knittle95}. Thus,
more structural information at pressures well above 30~GPa is needed
to resolve the stability of $\epsilon$-FeGe.
We acknowledge W. Schnelle for characterizing the samples by
magnetic measurements and H. Borrmann for single crystal data
collection. We are grateful to R. Demchnya and A. Wosylus for their
commitment during the high pressure X-ray experiments.
| -11,407.866572 |
[
-2.625,
2.53515625
] | 16.27907 |
[
-5.63671875,
-2.978515625,
-2.943359375,
-8.203125,
1.1982421875,
12.703125
] |
[
2.96484375,
7.2734375,
4.125,
4.7109375
] | 148 | 1,444 |
[
-3.3671875,
3.97265625
] | 32.691523 |
[
-5.71484375,
-3.0859375,
-1.86328125,
-1.013671875,
1.6982421875,
7.80859375
] | 0.967338 | 12.750891 | 46.745152 | 8.401797 |
[
2.0600216388702393
] | -8,404.903092 | 6.08518 | -10,982.465994 | 0.364174 | 5.830409 |
[
-3.744140625,
-3.912109375,
-3.541015625,
-3.837890625,
2.837890625,
9.921875
] |
[
-6.65625,
-3.998046875,
-3.412109375,
-2.560546875,
4.90625,
8.171875
] | |
BkiUgBDxK7IADzpkVqhY
|
\section{Introduction}
\label{Introduction}
\textcolor{black}{Detonation structures in narrow channels are usually observed to exhibit a two-dimensional (2D) structure (\textcolor{black}{e.g., see Refs. \cite{austin2003, bhattacharjee2013, xiao2019}}).} Nevertheless, they \textcolor{black}{tend to} propagate with large velocity deficits due to losses \textcolor{black}{originating from} the side walls \cite{berets1950, manson1957}. The cell sizes are also reported to be much larger in \textcolor{black}{narrower} channels, due presumably to increased reaction zone lengths \cite{strehlow1967, monwar2007, ishii2011}, \textcolor{black}{and the propagation limits (i.e., the critical initial pressures, below which detonations fail to propagate) are impacted as well (e.g., see Refs. \cite{dove1974, radulescu2002failure,chao2009,zhang2016b}).}
Previous works focused on \textcolor{black}{modelling the wall losses of detonations in 1D \cite{zeldovich1950, fay1959, zhang1994, sow2014, faria2015}}. One approach, introduced by Zel'dovich \cite{zeldovich1950}, \textcolor{black}{was} to model the wall losses with volume-averaged friction and heat loss terms. Another, \textcolor{black}{due to Fay \cite{fay1959}}, \textcolor{black}{accounted} for the negative displacement thickness of the boundary layer, whose effect \textcolor{black}{appeared} as a source term of \textcolor{black}{mass} flow sink in the governing equations for the inviscid core \textcolor{black}{(i.e., the undisturbed free stream flow)}. Despite these efforts in modelling 1D detonations, in a realistic way, the effect of these wall losses on the dynamics of 2D cellular detonations in narrow channels remains unknown.
\textcolor{black}{On the other hand, the recent works of Tsuboi et al. \cite{tsuboi2013}, Chinnayya et al. \cite{chinnayya2013}, and Sow et al. \cite{sow2019} have showed that directly resolving the viscous boundary layers by the Navier-Stokes (NS) equations requires remarkably high resolution, which makes the computations considerably expensive. Moreover, for these NS calculations, the other difficulty lies in quantifying the wall loss effects on detonation cellular structures.}
In the present study, we adapt Mirels' technique \textcolor{black}{\cite{mirels1956}} by accounting for the wall-boundary-layer-induced loss in a 2D formulation of the problem. This permits \textcolor{black}{us} to \textcolor{black}{readily} compute the dynamics of unsteady 2D cellular detonations with a supplemental lateral loss.
The communication first reports experiments in a narrow channel, with variation of the channel width ($w$) and cell size ($\lambda$) ratios, i.e., $w/\lambda$. We then formulate the \textcolor{black}{governing} equations with a lateral loss, which is evaluated from Mirels' boundary layer theory \cite{mirels1956}. Comparisons between experiments and simulations follow. Finally, \textcolor{black}{effects of the boundary layer losses on dynamics of the unsteady 2D cellular detonations are explored}.
\section{Experiments in a Narrow Channel}
\subsection{Experimental details}
\label{Experimental details}
The experiments were performed in a 3.4-m-long thin rectangular aluminium channel with an internal height and width of 203 mm and 19 mm, respectively, as described in detail elsewhere \cite{bhattacharjee2013}. The shock tube comprises three parts, i.e., the detonation initiation section, the propagation section, and the test section (about 1.0 m in length). The mixture was ignited in the first section by a high voltage igniter, and mesh wires were inserted in this part for promoting the detonation formation. The detonation evolution process was visualized in the test section, and its mean propagation speed over the whole test part was obtained by \textcolor{black}{one 113B24 and five 113B27 piezoelectric PCB pressure sensors} using the time-of-arrival method. The presently investigated mixture is the very regular stoichiometric hydrogen-oxygen diluted with $70\%$ argon (i.e., 2H$_2$/O$_2$/7Ar). Since the mixture of 2H$_2$/O$_2$/7Ar has low reactive sensitivity, a more reactive driver gas of stoichiometric ethylene-oxygen (i.e., C$_2$H$_4$/3O$_2$) was used in the initiation section, which was separated from the propagation section with a diaphragm. For visualization, a Z-type schlieren setup \cite{bhattacharjee2013} was utilized with a light source of 360 W. The resolution of the high-speed camera was $384\times288$ px$^2$ with the framing rate of 77481 fps (about 12.9 $\mu s$ for each interval).
\subsection{Results}
\label{Results}
Figure \ref{ExpPhoto} shows the schlieren photos of the detonation reaction zone structures, at varied initial pressures ranging from 10.3 kPa to 3.1 kPa. By decreasing the initial pressure for reducing the kinetic sensitivity of the mixture, detonations can be clearly observed to propagate with considerably enlarged cellular structures, with the velocity deficits increased up to $20\% \sim 30\%$ of the ideal CJ detonation speed. At a relatively high initial pressure in Fig.\ \ref{ExpPhoto}a, where $w/\lambda$ is about 0.5, \textcolor{black}{one can observe possible 3D-like effects of the overall detonation structure by the presence of duplicate features that do not overlap in the schlieren image.} This adds to the difficulty in studying its dynamics under the present resolution. With the decrease of $w/\lambda$ to about 0.25 in Fig.\ \ref{ExpPhoto}b, the detonation structure is qualitatively similar. The appearance of double Mach stems sharing the same triple point indicates that detonation is still relatively unstable in this condition. When $w/\lambda$ is further reduced to less than 0.1, as shown in Fig.\ \ref{ExpPhoto}c and Fig.\ \ref{ExpPhoto}d, detonations become perfectly planar and \textcolor{black}{essentially} two-dimensional for the investigation. At these low initial pressures, detonations are organized with relatively large unburned induction zones, as can be observed behind both the leading shock and the transverse wave. The vortex structures characteristic of the Kelvin-Helmholtz instability can also be readily seen along the slip line from Fig.\ \ref{ExpPhoto}d. Since the single-head detonation in Fig.\ \ref{ExpPhoto}d experiences a much larger velocity deficit, being more expensive in CFD calculations, we will adopt the condition of Fig.\ \ref{ExpPhoto}c as the main benchmark for subsequent model development and validation of simulations.
\begin{figure}[]
\centering
{\includegraphics[width=1.0\textwidth]{Pictures/ExpPhoto.pdf}}
\caption{\textcolor{black}{The structure of cellular detonations with varied $w/\lambda$, IS: the incident shock, MS: Mach stem, TS: the transverse shock (video animations as Supplemental material illustrating the evolution process).} } \label{ExpPhoto}
\end{figure}
\section{Numerical Simulations}
\label{Numerical Simulations}
\subsection{Unsteady quasi-2D formulation}
\label{Unsteady quasi-2D formulation}
Although the viscous effects are responsible for the boundary layer losses of detonations in thin channels, the present work aims at modelling these effects into the inviscid core flow as a source term. For such transient inviscid reactive core flow behind detonations in a narrow channel, the area-averaged equations of motion across the $z$-direction (i.e., the channel width direction into/out of the page when viewing Fig.\ \ref{ExpPhoto}) can be expressed in the lab frame of reference $\left(x, y, t\right)$ as
\begin{subequations}
\begin{align}
&\pde{\rho}{t}+\pde{\left(\rho u\right)}{x} + \pde{\left(\rho v\right)}{y} = -\rho\dfrac{1}{A}\dfrac{DA}{Dt} \label{mass-2} \\
&\pde{\left(\rho u\right)}{t}+\pde{\left(\rho u^2 +p\right)}{x}+\pde{\left(\rho u v\right)}{y} = -\rho u\dfrac{1}{A}\dfrac{DA}{Dt} \label{momentum-x-2} \\
&\pde{\left(\rho v \right)}{t}+\pde{\left(\rho u v \right)}{x}+\pde{\left(\rho v^2 +p\right)}{y} = -\rho v\dfrac{1}{A}\dfrac{DA}{Dt} \label{momentum-y-2} \\
&\textcolor{black}{\pde{\left(\rho e\right)}{t} + \pde{\left(\rho e u +p u\right)}{x} + \pde{\left(\rho e v +p v\right)}{y} = -\left(\rho e+p\right) \dfrac{1}{A}\dfrac{DA}{Dt} -Q\dot{\omega}_R \label{energy-2}} \\
&\textcolor{black}{\pde{\left(\rho Y \right)}{t} + \pde{\left(\rho uY\right)}{x} +\pde{\left(\rho v Y\right)}{y} = -\rho Y \dfrac{1}{A}\dfrac{DA}{Dt} +\dot{\omega}_R} \label{species-2}
\end{align} \label{governing-equations-2}where $\rho, u, v, A, p,\textcolor{black}{Q, Y, \dot{\omega}_R}$ denote the mixture density, $x$-direction flow velocity, $y$-direction flow velocity, the cross section area, pressure, \textcolor{black}{heat release, mass fraction and rate of mass production of the single reactant $R$. Note that the present work adopts the reaction of single species with no intermediate radicals, i.e., in the form of $R \rightarrow P$, where $R$ is the only reactant and $P$ the product. The total sensible energy plus the kinetic energy is $e = \dfrac{p/\rho}{\gamma - 1}+\dfrac{1}{2}\left(u^2+v^2\right)$, where a calorically perfect gas with constant specific heats is assumed and $\gamma$ is the ratio of specific heats.} The material derivative $DA/Dt$ appearing in the right-hand source term is the cross-sectional area change. Of noteworthy is that one can find the formal derivation of the 1D version by Chesser \cite{chesser1992}.
\end{subequations}
For an observer travelling in the frame of reference attached to the leading shock at average speed $D_s$ and following the motion of the fluid, we have the source term in Eq.\ \ref{governing-equations-2} further expressed as
\begin{align}
\dfrac{D}{Dt}\left( \ln A\right) = \pde{}{t'}\left( \ln A\right) + u'\pde{}{x'}\left( \ln A\right)
\end{align}
where $A\left(x, t\right) = H\times W\left(x, t\right)$ is the effective cross section area. $H$ is the fixed channel height of 203 mm in this communication, while $ W\left(x, t\right)$ is the effective channel width in the $z$-direction. \textcolor{black}{Since the channel height $H$ is much larger than the width $w$ (i.e., $H/w \approx 10$) in the present work, the boundary layer effects in the channel height direction are negligible as compared to those from the channel width direction.} $x'$, $t'$, and $u'$ are, respectively, the space and time coordinates and the post-shock flow velocity in the shock-attached frame of reference. Since the motion is pseudo-steady (travelling wave with perturbation), we can neglect $\pde{}{t'}\left(lnA\right)$ to the leading order. We thus get
\begin{align}
\dfrac{D}{Dt}\left( \ln A\right) \approxeq u'\pde{}{x'}\left( \ln A \right) \label{DlnA/Dt}
\end{align}
which can be evaluated from Fay's boundary layer theory \cite{fay1959} by using Mirels' compressible laminar boundary layer solutions \cite{mirels1956}. The boundary layer displacement thickness $\delta^*\left(x'\right)$ behind a moving shock is \cite{mirels1956}
\begin{align}
\delta^*\left(x'\right) = K_M \sqrt{\dfrac{\mu_s x'}{\rho_0 D_s}}
\end{align}
where $x'$ is the distance from the shock, $\rho_0$ the density of the flow ahead of the shock, $\mu_s$ the post-shock dynamic viscosity, and $K_M$ the Mirels' constant. One can refer to the recent work of Xiao and Radulescu \cite{xiao2019} for details in evaluating $K_M$ for hydrogen-oxygen-argon detonations at varied initial pressures. For 2H$_2$/O$_2$/7Ar detonations, they found that $K_M \approx 4.0$ \cite{xiao2019}. Using this relation, the boundary layer displacement thickness $\delta^*$ in the experiment of Fig.\ \ref{ExpPhoto}c was calculated to be 1.9 mm, with $D_s = 0.8 D_{CJ}$ and $x'$ as the hydrodynamic thickness $x_H$ between the leading shock and the sonic surface. Note that $x_H$ was computed from the generalized ZND model with lateral losses \cite{klein1995, xiao2019}. Clearly, the boundary layer is much thinner than the channel width.
Since $ W\left(x'\right) = w + 2\delta^*\left(x'\right)$, where $w$ is the physical channel width of 19 mm, we can thus obtain
\begin{align}
\pde{}{x'}\left(lnA\right) = \dfrac{2}{w + 2\delta^*\left(x'\right)}\times\ode{\delta^*\left(x'\right)}{x'}
\end{align}
where $\delta^*\left(x'\right) \ll w$, Eq.\ \ref{DlnA/Dt} then changes to
\begin{align}
\dfrac{D}{Dt}\left( \ln A\right) = u' \dfrac{2}{w} \dfrac{K_M}{2}\sqrt{\dfrac{\mu_s }{\rho_0 D_s}}\left(x'\right)^{-0.5} \label{DlnA/Dt-2}
\end{align}
Since Mirels' model assumes that the post-shock state is uniform and steady, to the leading order, we can thus write $x'$ as $x' = u'\left(t-t_s\right)$, where $t_s$ is the time at which the particle crosses the shock. With the mass conservation across the shock $\rho_s u' = \rho_0 D_s$, where $\rho_s$ is the post-shock density, Eq.\ \ref{DlnA/Dt-2} can be greatly simplified as the following simple expression
\begin{align}
\dfrac{D}{Dt}\left( \ln A\right) =\dfrac{K_M}{w}\sqrt{\dfrac{\nu_s}{t-t_s}} \label{final-DADT}
\end{align}
where $\nu_s$ is the post-shock kinematic viscosity, and $t_{\textrm{elapse}} = t-t_s$ is the elapsed time since a particle has passed through the shock front. The shock time $t_s$ is recorded when the shock passes over, and convected with the motion of that particle:
\begin{align}
\pde{t_s}{t} + \vec{u}\boldsymbol{\cdot} \mathbb{\nabla}t_s = 0
\end{align}
\subsection{Two-step chemistry model}
\label{Two-step chemistry model}
In the experiment of Fig.\ \ref{ExpPhoto}c, we have calculated the post-shock temperatures using the experimentally measured shock speeds along the walls and cell axis. We found that the lowest post-shock temperature is about 900 K, which is still above its cross-over temperature of 800 K to 850 K. Thus, in this study, the two-step chain-branching reaction model \cite{short2003, leung2010} will be employed for describing the chemical kinetics. It consists of two components, i.e., a thermally neutral induction zone followed by an exothermic main reaction zone. The transport equations of the induction and reaction variables can be written as:
\begin{subequations}
\begin{align}
&\pde{\left(\rho \lambda_i \right)}{t} + \pde{\left(\rho u\lambda_i\right)}{x} +\pde{\left(\rho v \lambda_i\right)}{y} = -\rho \lambda_i \dfrac{1}{A}\dfrac{DA}{Dt} - \mathcal{H}\left(\lambda_i\right)k_i \rho^{\alpha+1} \textrm{exp}\left(-\dfrac{E_a}{R
T}\right) \label{induction}\\
&\pde{\left(\rho \lambda_r \right)}{t} + \pde{\left(\rho u\lambda_r\right)}{x} +\pde{\left(\rho v \lambda_r \right)}{y} = -\rho \lambda_r \dfrac{1}{A}\dfrac{DA}{Dt} - \left[1-\mathcal{H}\left(\lambda_i\right)\right]k_r \rho^{\beta+1}\lambda_r^{\nu} \label{reaction}
\end{align}\label{reaction-rate-equations}where $\lambda_i$ is the progress variable for the induction zone with a value of 1 in the reactants and 0 at the end of the induction zone, $\lambda_r$ \textcolor{black}{the reaction progress variable} with a value of 1 in the unburned zone and 0 in the burned products. $\mathcal{H}\left(\lambda_i\right)$ is the Heaviside function given as
\end{subequations}
\begin{align}
\textcolor{black}{\mathcal{H}\left(\lambda_i\right) =
\begin{cases}
0 & \text{if $\lambda_i=0$} \\
1 & \text{if $\lambda_i > 0$}
\end{cases}}
\end{align}which disables the progress of $\lambda_i$ at the end of the induction zone. $k_i$ and $k_r$ are rate constants, $E_a$ the activation energy controlling the temperature sensitivity of the induction zone duration, $\nu$ is the reaction order, while $\alpha$ and $\beta$ are further empirical reaction order parameters.
Table\ \ref{2-step-parameters} shows the non-dimensional parameters for the two-step model at three different initial pressures from experiments in Fig.\ \ref{ExpPhoto}. They were
\begin{table}[]
\footnotesize
\centering
\caption{The calibrated non-dimensional parameters for the two-step model from the detailed chemistry.}
\begin{tabular}{cccccc}
\toprule
$p_0$ (kPa) & $\gamma$ & $E_a/RT_0$ & $Q/RT_0$ & $k_i$ & $k_r$ \\
\midrule
4.1 & 1.5 & 31.2 & 11.5 & 45.6 & 0.078 \\
6.9 & 1.5 & 22.8 & 11.8 & 10.6 & 0.11 \\
10.3 & 1.5 & 24.2 & 12.0 & 12.7 & 0.14 \\
\bottomrule
\end{tabular}%
\label{2-step-parameters}%
\end{table}%
\begin{figure}[]
\centering
{\includegraphics[width=1.0\textwidth]{Pictures/ZND1.pdf}}
\caption{\textcolor{black}{ZND profiles of (a) density and (b) temperature obtained by using the 2-step chemistry model and the detailed chemistry at the initial pressure of 4.1 kPa.}} \label{ZND-structure}
\end{figure}calibrated from the detailed chemistry using the San Diego chemical reaction mechanism (Williams) \cite{Williams2014}, by using Shepherd's Shock and Detonation Toolbox (SDToolbox) \cite{SDtoolbox}. $\gamma$ was the post-shock isentropic exponent of the CJ detonation, while the heat release $Q$ was determined from the perfect gas relation recovering the correct Mach number \cite{fickett2000}. The effective activation energy $E_a$ was calculated from the \textcolor{black}{logarithmic} derivative of the ignition delay with respect to the inverse of post-shock temperature. Note that the present work adopts the initial state variables ($p_0, \rho_0, T_0$) and the ZND induction zone length ($\Delta_i$) as the normalization scales. As such, the rate constant $k_i$ can be directly obtained from Eq.\ (\ref{induction}) by scaling the dimensionless induction zone length to unity, while $k_r$ can be determined by recovering the correct induction/reaction time ratio from the detailed chemistry. Finally, $\alpha = 1.2, \beta = 1.0, \nu = 1.6$ were adopted for matching the detailed chemistry ZND structure, as can be easily seen from the \textcolor{black}{density and temperature profiles} in Fig.\ \ref{ZND-structure}. \textcolor{black}{For the minor discrepancies observed near the end of the reaction zone, it is due to the limitation of assuming the constant specific heats in the present work.}
\subsection {Computational details}
\label{Computational details}
The non-dimensional governing equations were solved employing the $MG$ code, developed by S. Falle of the University of Leeds, which uses a \textcolor{black}{second-order-accurate exact} Godunov solver \cite{falle1991} with adaptive mesh refinement. The computational domain height was held constant at the same height of experiments (203 mm in height), i.e., $72\Delta_i$ for $p_0 = 4.1$ kPa, $116\Delta_i$ for $p_0 = 6.9$ kPa, and $182\Delta_i$ for $p_0 = 10.3$ kPa. The domain length varied from $3000\Delta_i$ to $5000\Delta_i$. The detonation propagated from left to the right, with reflective boundary conditions imposed to the top and bottom sides, and zero-gradient boundary conditions applied to the left and right ends. The computations were started using a ZND profile placed $300\Delta_i$ in length from the left boundary. An initial density disturbance zone of $4\Delta_i$ was added ahead of the initial ZND solution for accelerating the evolution to cellular detonations. \textcolor{black}{This density perturbation method is the same as that of Maxwell et al. \cite{maxwell2017}, which is give by}
\begin{align}
\textcolor{black}{\rho\left(x, y, t = 0\right) =
\begin{cases}
\textnormal{ZND solution} & \text{if $x < 0$} \\
1.25 - 0.5n & \text{if $0 \le x \le 4$} \\
1 & \text{otherwise} \\
\end{cases}}
\end{align}\textcolor{black}{where $n$ is a random real number from 0 to 1.} As for the numerical resolution, 5 levels of mesh refinement were adopted with the coarsest and finest grid sizes of $1/2\Delta_i$ and $1/16\Delta_i$, respectively. Since the reaction zone length of the simulated cases in this work is in the order of $100\Delta_i$ to $1000\Delta_i$, as demonstrated in Fig.\ \ref{ZND-structure}, such resolution is adequate for obtaining reliable results. This has been verified by a resolution test using a higher level of mesh refinement for calculating the CJ detonation without loss, at the initial pressure of 4.1 kPa. We found that the final stable cell size does not change. In simulations, we ran all the cases for long enough time until we have obtained at least 10 repeated stable cycles of the detonation structures. Due to the large domain size, each case running with 100 to 200 cores in parallel in \textit{Cedar} of Compute Canada requires about one week to complete. More than 20 cases were involved in this study.
\subsection {Results and discussion}
\label{results}
\subsubsection{Comparison with experiments}
\label{Comparison with experiments}
Figure\ \ref{shutr-2} shows the numerically tracked maximum energy release rates for detonations with different losses, at the initial pressure of 4.1 kPa. This corresponds to the open shutter photograph in experiments. From the simulated results, it can be observed that the detonation cell size becomes larger as a result of increasing the Mirels' constant $K_M$, i.e., increasing the magnitude of boundary layer losses. As the ideal CJ detonation ($K_M = 0$) has four stable cells across the channel height, it can only accommodate a single-head detonation for $K_M = 2.0$. When $K_M $ is further increased to 2.5, the single-head detonation finally failed, as can be seen from Fig.\ \ref{shutr-2}e. According to the calculations performed by Xiao and Radulescu \cite{xiao2019}, the theoretical Mirels' constant for 2H$_2$/O$_2$/7Ar detonations is supposed to be $K_M \approx 4.0$. However, the present simulations show that $K_M = 1.75$ can very well recover the experiment (in Fig.\ \ref{ExpPhoto}c) in terms of the cell size and the velocity deficit. This also occurs for cases at other initial pressures of 6.9 kPa and 10.3 kPa, respectively, as shown in Fig. \ref{shutr-4}. While $K_M = 4.0$ results in a larger cell size and velocity deficit, $K_M \approx 2.5$ appears to be able to correctly recover them, when compared to the experiments from Fig.\ \ref{ExpPhoto}a and Fig.\ \ref{ExpPhoto}b. Such discrepancy from the theoretically computed $K_M$ presumably originates from Mirels' assumption of the uniform and steady state behind the shock. For detonations, significant gradients of pressure, temperature, \textcolor{black}{and velocity} exist. \textcolor{black}{Particularly, the flow acceleration (in the shock-attached reference) from being subsonic behind the detonation front to sonic in the reaction zone can contribute to thinned boundary layers, as already noted by Chinnayya et al. \cite{chinnayya2013}. In their 2D viscous simulations, they also found that the thickness of the computed boundary layers behind detonations is approximately half of that predicted by the uniform steady boundary layer theory.} Moreover, the theoretical calculations of $K_M$ assume the leading shock of the ideal CJ detonation speed, while the 2D simulations have detonations of significant velocity deficits. Thus, it results in a smaller $K_M$ than expected by theory. Future work should be devoted to refine the model to account for these non-idealities. Nevertheless, it is quite satisfying that the model works within a factor of 2, which can also be absorbed by uncertainties in chemical kinetics \cite{taylor2013}.
\begin{figure}[]
\centering
{\includegraphics[width=1.0\textwidth]{Pictures/shutr-2.pdf}}
\caption{The recorded maximum energy release rates of detonations at the initial pressure of 4.1 kPa with varied $K_M$. Their cell size and mean propagation speeds are: (a) $D/D_{CJ} = 1.0, \lambda = 51$ mm, (b) $D/D_{CJ} = 0.90, \lambda = 136$ mm, (c) $D/D_{CJ} = 0.85, \lambda = 203$ mm, (d) $D/D_{CJ} = 0.81, \lambda \approx 406$ mm , and (e) detonation failure. In the experiment, $D/D_{CJ} = 0.83, \lambda = 203$ mm . Note that the red symbol represents $50\Delta_i$ in length, and the length of the shown domain is $1500\Delta_i$.} \label{shutr-2}
\end{figure}
\begin{figure}[]
\centering
{\includegraphics[width=1.0\textwidth]{Pictures/shutr-4.pdf}}
\caption{The recorded maximum energy release rates of detonations at the initial pressure of 6.9 kPa and 10.3 kPa, respectively. Their cell size and mean propagation speeds are: (a) $D/D_{CJ} = 0.87, \lambda = 81$ mm, (b) $D/D_{CJ} = 0.78, \lambda = 203$ mm, (c) $D/D_{CJ} = 0.91, \lambda = 41$ mm, (d) $D/D_{CJ} = 0.87, \lambda \approx 58$ mm. Note that the red symbol represents $50\Delta_i$ in length, and the length of the shown domain is $1500\Delta_i$. } \label{shutr-4}
\end{figure}
Besides the velocity deficit and cell size, the experimentally visualized qualitative features of the detonation structure, as well as its cellular dynamics can be very well reproduced by the simulations, as shown in Fig.\ \ref{Exp-compare-0.6psi}. Figure\ \ref{Dspeed-compare-0.6psi} also shows the quantitative agreement in temporal velocity evolution at a single cell, when compared to experiments. This suggests the robustness of the proposed quasi-2D formulation as well as the two-step chemistry model in simulating the real detonations in experiments.
\begin{figure}[]
\centering
{\includegraphics[width=1.0\textwidth]{Pictures/Exp-compare-2.pdf}}
\caption{Comparisons of the gradient of the density (bottom column) from the quasi-2D simulation ( $p_0 = 4.1$ kPa, $K_M = 1.75$) with the schlieren photos (top column) from experiments at the initial pressure of 4.1 kPa. $H$ is the channel height of 203 mm. } \label{Exp-compare-0.6psi}
\end{figure}
\begin{figure}[]
\centering
\includegraphics[width=0.62\textwidth]{Pictures/Dspeed.pdf}
\caption{The evolution of detonation speeds in a cell at the initial pressure of 4.1 kPa. Note that the experimental speeds in a cell were reconstructed by Cheevers et al. \cite{cheevers2019}. The numerical data are the local time-average speed obtained with a running time average of 5 neighbouring points from more than 60 sampling points in a stable cell, for the case of $p_0 = 4.1$ kPa with $K_M = 1.75$. \textcolor{black}{$L$ is the cell length.}} \label{Dspeed-compare-0.6psi}
\end{figure}
\subsubsection{Dynamics of detonations with different losses}
The effect of boundary layer losses on detonation dynamics is shown in Fig.\ \ref{Dspeed-Km}, in terms of the normalized speed with respect to the position in a cell. More than 60 points were sampled in a cell for each case, and the local time-average speed was obtained with a running time average of 5 neighbouring points. Evidently, the presence of wall losses modifies the cellular dynamics. Compared to the CJ case, the cases with losses have a larger deviation from the larger maximum velocity at the beginning of the cell to the smaller minimum one before the collision of triple points. Whether it is the losses that directly modify the flow velocity inside the cellular structure, or the increased activation energy due to velocity deficits that results in such larger fluctuation, requires further confirmation.
\begin{figure}[]
\centering
{\includegraphics[width=0.62\textwidth]{Pictures/Dspeed-Km.pdf}}
\caption{Local time-average speed (along the cell axis) of detonations with different losses, at the initial pressure of 4.1 kPa. Note that the detonation speed is normalized by the mean propagation speed $D_{\textrm{avg}}$ in a cell, and $L$ is the cell length. } \label{Dspeed-Km}
\end{figure}
\subsubsection{The ${D}/{D_{CJ}}-K_M$ relationships}
The variation of the velocity deficit with respect to the Mirels' constant $K_M$ can be better appreciated from Fig.\ \ref{2step-ZND}. The theoretical predictions were obtained by solving the generalized ZND model with lateral flow divergence \cite{klein1995}, using the present two-step reaction model. Clearly, the ZND model underpredicts the velocity deficit obtained from the quasi-2D simulations. As the constant $K_M$ increases to the propagation limit, such discrepancy becomes more significant. These results are consistent with those reported previously by Sow et al. \cite{sow2014} and Reynaud et al. \cite{reynaud2017}, who also found the underprediction of the velocity deficit from the unsteady simulations by the analytical model for relatively regular detonations.
\begin{figure}[]
\centering
\includegraphics[width=0.6\textwidth]{Pictures/2step-ZND.pdf}
\caption{Comparisons of the quasi-2D simulation results with the steady 1D ZND model in terms of the ${D}/{D_{CJ}}-K_M$ relationships.} \label{2step-ZND}
\end{figure}
\subsubsection{The ${\lambda}/{\lambda_{CJ}} - {D}/{D_{CJ}}$ relationships}
We have further obtained the dimensionless relationships in terms of the detonation cell size ($\lambda$) and the characteristic induction zone length ($\Delta$) with respect to the velocity deficit, as shown in Fig.\ \ref{Cellsize-DDCJ}. $\lambda_{CJ}$ and $\Delta_{CJ}$ represent the corresponding length scales of the ideal CJ detonation. The induction length $\Delta$ was obtained through zero-dimensional constant-volume combustion calculations with the post-shock velocity (in the shock-attached frame of reference) multiplying by the time to the peak thermicity, as proposed by Shepherd \cite{shepherd1986}. The detailed chemistry mechanism was utilized in these calculations. The excellent agreement between the ${\lambda}/{\lambda_{CJ}}$ and ${\Delta}/{\Delta_{CJ}}$ correlations suggest that the increase in cell size due to wall losses is still controlled by the increase in the induction zone length as a result of the velocity deficits.
\begin{figure}[]
\centering
{\includegraphics[width=0.6\textwidth]{Pictures/Cellsize-DDCJ-2.pdf}}
\caption{The dimensionless cell size (${\lambda}/{\lambda_{CJ}}$) and characteristic induction zone length (${\Delta}/{\Delta_{CJ}}$) as a function of the velocity deficit. The symbols are from the quasi-2D simulations. The predictions (using Eq.\ (\ref{lamda-CJ-1})) with $E_a/RT_{s, CJ} = 5.0$ and $E_a/RT_{s, CJ} = 3.8$ correspond to initial pressures of 4.1 kPa and 10.3 kPa, respectively. \textcolor{black}{Note that the real-chemistry constant-volume combustion calculations ($\Delta / \Delta_{CJ}$) of 6.9 kPa (denoted by the purple dashed line) and 10.3 kPa (denoted by the green dashed line) follow almost the same curve.}} \label{Cellsize-DDCJ}
\end{figure}
Assuming that the cell size varies with the induction zone thickness \cite{shepherd1986} and that the ignition delay $t_{ig} \sim \textrm{exp}(E_a/RT_s)$, we can thus get
\begin{align}
\dfrac{\lambda}{\lambda_{CJ}} &\approxeq \dfrac{\Delta}{\Delta_{CJ}} = \dfrac{u'_s }{u'_{s, CJ}}\times \dfrac{t_{ig}}{t_{ig, CJ}} \notag \\
& \approxeq\left(\dfrac{u'_s}{u'_0}\right)\left(\dfrac{u'_0}{u'_{s, CJ}}\right)\textrm{exp}\left\lbrace \frac{E_a}{RT_{s, CJ}}\left[\left(\dfrac{T_{s, CJ}}{T_0}\right)\left(\dfrac{T_0}{T_{s}}\right) - 1\right] \right\rbrace \label{lamda-CJ-1}
\end{align}where $u'_s$ and $u'_0$ are the flow velocity behind and ahead of the shock in the shock-attached frame of reference, respectively, and $T_s$ the post-shock temperature. The terms in brackets can be readily evaluated from the shock-jump equations. In the limit of strong shock and high activation energy, we can further simplify Eq.\ (\ref{lamda-CJ-1}) to
${\lambda}/{\lambda_{CJ}} \approxeq \textrm{exp}\left\lbrace \left({2E_a}/{RT_{s, CJ}}\right)\left(1 - {D}/{D_{CJ}}\right) \right\rbrace \label{lamda-CJ-2}$. It thus highlights the exponential sensitivity of the cell size and induction zone length on velocity deficit, which is controlled by the global activation energy. This generalizes Desbordes' observations for overdriven detonations \cite{desbordes1988}. The results in Fig.\ \ref{Cellsize-DDCJ} show that the ${\lambda}/{\lambda_{CJ}} - {D}/{D_{CJ}}$ correlations can be well predicted by the simple expression \eqref{lamda-CJ-1}. The sensitivity of cell size on velocity deficits also highlights the importance of providing the detonation speed when reporting experimentally measured cell size, since the variation can be up to an order of magnitude, even for the weakly sensitive mixtures studied here.
\section{Conclusions}
\label{conclusion}
The present study has shown that the dynamics of 2D cellular detonations in narrow channels can be well captured using a quasi-2D approach modelling the lateral boundary layer losses using Mirels' theory. With an appropriate Mirels' constant, $K_M$, deviating by approximately a factor of 2 from the model proposed by Mirels for steady constant pressure boundary layers, the simulations are found in excellent agreement with experiment. \textcolor{black}{Compared to directly resolving the boundary layers of detonations in the present narrow channel experiments by calculating the 3D NS equations, this novel formulation can save the computational cost by up to five orders of magnitude.} We have also shown that the cellular cycle dynamics is also affected by the losses, which yield larger velocity fluctuations and more rapid decay rates of the lead shock. Finally, the increase in cell size with increasing velocity deficit follows the Arrhenius dependence of ignition delay on the temperature of an equivalent steady shock, in spite of the cellular dynamics, generalizing previous observations of Desbordes for overdriven detonations in generally regular mixtures.
\section*{Acknowledgments}
\label{Acknowledgments}
The authors wish to acknowledge the financial support from the Natural Sciences and Engineering
Research Council of Canada (NSERC) through the Discovery Grant ``Predictability of detonation wave dynamics in gases: experiment and model development", as well as the support of Compute Canada and \textcolor{black}{Core Facility for Advanced Research Computing at CWRU}.
\bibliographystyle{elsarticle-num-PROCI}
\section{Introduction}
\label{Introduction}
\textcolor{black}{Detonation structures in narrow channels are usually observed to exhibit a two-dimensional (2D) structure (\textcolor{black}{e.g., see Refs. \cite{austin2003, bhattacharjee2013, xiao2019}}).} Nevertheless, they \textcolor{black}{tend to} propagate with large velocity deficits due to losses \textcolor{black}{originating from} the side walls \cite{berets1950, manson1957}. The cell sizes are also reported to be much larger in \textcolor{black}{narrower} channels, due presumably to increased reaction zone lengths \cite{strehlow1967, monwar2007, ishii2011}, \textcolor{black}{and the propagation limits (i.e., the critical initial pressures, below which detonations fail to propagate) are impacted as well (e.g., see Refs. \cite{dove1974, radulescu2002failure,chao2009,zhang2016b}).}
Previous works focused on \textcolor{black}{modelling the wall losses of detonations in 1D \cite{zeldovich1950, fay1959, zhang1994, sow2014, faria2015}}. One approach, introduced by Zel'dovich \cite{zeldovich1950}, \textcolor{black}{was} to model the wall losses with volume-averaged friction and heat loss terms. Another, \textcolor{black}{due to Fay \cite{fay1959}}, \textcolor{black}{accounted} for the negative displacement thickness of the boundary layer, whose effect \textcolor{black}{appeared} as a source term of \textcolor{black}{mass} flow sink in the governing equations for the inviscid core \textcolor{black}{(i.e., the undisturbed free stream flow)}. Despite these efforts in modelling 1D detonations, in a realistic way, the effect of these wall losses on the dynamics of 2D cellular detonations in narrow channels remains unknown.
\textcolor{black}{On the other hand, the recent works of Tsuboi et al. \cite{tsuboi2013}, Chinnayya et al. \cite{chinnayya2013}, and Sow et al. \cite{sow2019} have showed that directly resolving the viscous boundary layers by the Navier-Stokes (NS) equations requires remarkably high resolution, which makes the computations considerably expensive. Moreover, for these NS calculations, the other difficulty lies in quantifying the wall loss effects on detonation cellular structures.}
In the present study, we adapt Mirels' technique \textcolor{black}{\cite{mirels1956}} by accounting for the wall-boundary-layer-induced loss in a 2D formulation of the problem. This permits \textcolor{black}{us} to \textcolor{black}{readily} compute the dynamics of unsteady 2D cellular detonations with a supplemental lateral loss.
The communication first reports experiments in a narrow channel, with variation of the channel width ($w$) and cell size ($\lambda$) ratios, i.e., $w/\lambda$. We then formulate the \textcolor{black}{governing} equations with a lateral loss, which is evaluated from Mirels' boundary layer theory \cite{mirels1956}. Comparisons between experiments and simulations follow. Finally, \textcolor{black}{effects of the boundary layer losses on dynamics of the unsteady 2D cellular detonations are explored}.
\section{Experiments in a Narrow Channel}
\subsection{Experimental details}
\label{Experimental details}
The experiments were performed in a 3.4-m-long thin rectangular aluminium channel with an internal height and width of 203 mm and 19 mm, respectively, as described in detail elsewhere \cite{bhattacharjee2013}. The shock tube comprises three parts, i.e., the detonation initiation section, the propagation section, and the test section (about 1.0 m in length). The mixture was ignited in the first section by a high voltage igniter, and mesh wires were inserted in this part for promoting the detonation formation. The detonation evolution process was visualized in the test section, and its mean propagation speed over the whole test part was obtained by \textcolor{black}{one 113B24 and five 113B27 piezoelectric PCB pressure sensors} using the time-of-arrival method. The presently investigated mixture is the very regular stoichiometric hydrogen-oxygen diluted with $70\%$ argon (i.e., 2H$_2$/O$_2$/7Ar). Since the mixture of 2H$_2$/O$_2$/7Ar has low reactive sensitivity, a more reactive driver gas of stoichiometric ethylene-oxygen (i.e., C$_2$H$_4$/3O$_2$) was used in the initiation section, which was separated from the propagation section with a diaphragm. For visualization, a Z-type schlieren setup \cite{bhattacharjee2013} was utilized with a light source of 360 W. The resolution of the high-speed camera was $384\times288$ px$^2$ with the framing rate of 77481 fps (about 12.9 $\mu s$ for each interval).
\subsection{Results}
\label{Results}
Figure \ref{ExpPhoto} shows the schlieren photos of the detonation reaction zone structures, at varied initial pressures ranging from 10.3 kPa to 3.1 kPa. By decreasing the initial pressure for reducing the kinetic sensitivity of the mixture, detonations can be clearly observed to propagate with considerably enlarged cellular structures, with the velocity deficits increased up to $20\% \sim 30\%$ of the ideal CJ detonation speed. At a relatively high initial pressure in Fig.\ \ref{ExpPhoto}a, where $w/\lambda$ is about 0.5, \textcolor{black}{one can observe possible 3D-like effects of the overall detonation structure by the presence of duplicate features that do not overlap in the schlieren image.} This adds to the difficulty in studying its dynamics under the present resolution. With the decrease of $w/\lambda$ to about 0.25 in Fig.\ \ref{ExpPhoto}b, the detonation structure is qualitatively similar. The appearance of double Mach stems sharing the same triple point indicates that detonation is still relatively unstable in this condition. When $w/\lambda$ is further reduced to less than 0.1, as shown in Fig.\ \ref{ExpPhoto}c and Fig.\ \ref{ExpPhoto}d, detonations become perfectly planar and \textcolor{black}{essentially} two-dimensional for the investigation. At these low initial pressures, detonations are organized with relatively large unburned induction zones, as can be observed behind both the leading shock and the transverse wave. The vortex structures characteristic of the Kelvin-Helmholtz instability can also be readily seen along the slip line from Fig.\ \ref{ExpPhoto}d. Since the single-head detonation in Fig.\ \ref{ExpPhoto}d experiences a much larger velocity deficit, being more expensive in CFD calculations, we will adopt the condition of Fig.\ \ref{ExpPhoto}c as the main benchmark for subsequent model development and validation of simulations.
\begin{figure}[]
\centering
{\includegraphics[width=1.0\textwidth]{Pictures/ExpPhoto.pdf}}
\caption{\textcolor{black}{The structure of cellular detonations with varied $w/\lambda$, IS: the incident shock, MS: Mach stem, TS: the transverse shock (video animations as Supplemental material illustrating the evolution process).} } \label{ExpPhoto}
\end{figure}
\section{Numerical Simulations}
\label{Numerical Simulations}
\subsection{Unsteady quasi-2D formulation}
\label{Unsteady quasi-2D formulation}
Although the viscous effects are responsible for the boundary layer losses of detonations in thin channels, the present work aims at modelling these effects into the inviscid core flow as a source term. For such transient inviscid reactive core flow behind detonations in a narrow channel, the area-averaged equations of motion across the $z$-direction (i.e., the channel width direction into/out of the page when viewing Fig.\ \ref{ExpPhoto}) can be expressed in the lab frame of reference $\left(x, y, t\right)$ as
\begin{subequations}
\begin{align}
&\pde{\rho}{t}+\pde{\left(\rho u\right)}{x} + \pde{\left(\rho v\right)}{y} = -\rho\dfrac{1}{A}\dfrac{DA}{Dt} \label{mass-2} \\
&\pde{\left(\rho u\right)}{t}+\pde{\left(\rho u^2 +p\right)}{x}+\pde{\left(\rho u v\right)}{y} = -\rho u\dfrac{1}{A}\dfrac{DA}{Dt} \label{momentum-x-2} \\
&\pde{\left(\rho v \right)}{t}+\pde{\left(\rho u v \right)}{x}+\pde{\left(\rho v^2 +p\right)}{y} = -\rho v\dfrac{1}{A}\dfrac{DA}{Dt} \label{momentum-y-2} \\
&\textcolor{black}{\pde{\left(\rho e\right)}{t} + \pde{\left(\rho e u +p u\right)}{x} + \pde{\left(\rho e v +p v\right)}{y} = -\left(\rho e+p\right) \dfrac{1}{A}\dfrac{DA}{Dt} -Q\dot{\omega}_R \label{energy-2}} \\
&\textcolor{black}{\pde{\left(\rho Y \right)}{t} + \pde{\left(\rho uY\right)}{x} +\pde{\left(\rho v Y\right)}{y} = -\rho Y \dfrac{1}{A}\dfrac{DA}{Dt} +\dot{\omega}_R} \label{species-2}
\end{align} \label{governing-equations-2}where $\rho, u, v, A, p,\textcolor{black}{Q, Y, \dot{\omega}_R}$ denote the mixture density, $x$-direction flow velocity, $y$-direction flow velocity, the cross section area, pressure, \textcolor{black}{heat release, mass fraction and rate of mass production of the single reactant $R$. Note that the present work adopts the reaction of single species with no intermediate radicals, i.e., in the form of $R \rightarrow P$, where $R$ is the only reactant and $P$ the product. The total sensible energy plus the kinetic energy is $e = \dfrac{p/\rho}{\gamma - 1}+\dfrac{1}{2}\left(u^2+v^2\right)$, where a calorically perfect gas with constant specific heats is assumed and $\gamma$ is the ratio of specific heats.} The material derivative $DA/Dt$ appearing in the right-hand source term is the cross-sectional area change. Of noteworthy is that one can find the formal derivation of the 1D version by Chesser \cite{chesser1992}.
\end{subequations}
For an observer travelling in the frame of reference attached to the leading shock at average speed $D_s$ and following the motion of the fluid, we have the source term in Eq.\ \ref{governing-equations-2} further expressed as
\begin{align}
\dfrac{D}{Dt}\left( \ln A\right) = \pde{}{t'}\left( \ln A\right) + u'\pde{}{x'}\left( \ln A\right)
\end{align}
where $A\left(x, t\right) = H\times W\left(x, t\right)$ is the effective cross section area. $H$ is the fixed channel height of 203 mm in this communication, while $ W\left(x, t\right)$ is the effective channel width in the $z$-direction. \textcolor{black}{Since the channel height $H$ is much larger than the width $w$ (i.e., $H/w \approx 10$) in the present work, the boundary layer effects in the channel height direction are negligible as compared to those from the channel width direction.} $x'$, $t'$, and $u'$ are, respectively, the space and time coordinates and the post-shock flow velocity in the shock-attached frame of reference. Since the motion is pseudo-steady (travelling wave with perturbation), we can neglect $\pde{}{t'}\left(lnA\right)$ to the leading order. We thus get
\begin{align}
\dfrac{D}{Dt}\left( \ln A\right) \approxeq u'\pde{}{x'}\left( \ln A \right) \label{DlnA/Dt}
\end{align}
which can be evaluated from Fay's boundary layer theory \cite{fay1959} by using Mirels' compressible laminar boundary layer solutions \cite{mirels1956}. The boundary layer displacement thickness $\delta^*\left(x'\right)$ behind a moving shock is \cite{mirels1956}
\begin{align}
\delta^*\left(x'\right) = K_M \sqrt{\dfrac{\mu_s x'}{\rho_0 D_s}}
\end{align}
where $x'$ is the distance from the shock, $\rho_0$ the density of the flow ahead of the shock, $\mu_s$ the post-shock dynamic viscosity, and $K_M$ the Mirels' constant. One can refer to the recent work of Xiao and Radulescu \cite{xiao2019} for details in evaluating $K_M$ for hydrogen-oxygen-argon detonations at varied initial pressures. For 2H$_2$/O$_2$/7Ar detonations, they found that $K_M \approx 4.0$ \cite{xiao2019}. Using this relation, the boundary layer displacement thickness $\delta^*$ in the experiment of Fig.\ \ref{ExpPhoto}c was calculated to be 1.9 mm, with $D_s = 0.8 D_{CJ}$ and $x'$ as the hydrodynamic thickness $x_H$ between the leading shock and the sonic surface. Note that $x_H$ was computed from the generalized ZND model with lateral losses \cite{klein1995, xiao2019}. Clearly, the boundary layer is much thinner than the channel width.
Since $ W\left(x'\right) = w + 2\delta^*\left(x'\right)$, where $w$ is the physical channel width of 19 mm, we can thus obtain
\begin{align}
\pde{}{x'}\left(lnA\right) = \dfrac{2}{w + 2\delta^*\left(x'\right)}\times\ode{\delta^*\left(x'\right)}{x'}
\end{align}
where $\delta^*\left(x'\right) \ll w$, Eq.\ \ref{DlnA/Dt} then changes to
\begin{align}
\dfrac{D}{Dt}\left( \ln A\right) = u' \dfrac{2}{w} \dfrac{K_M}{2}\sqrt{\dfrac{\mu_s }{\rho_0 D_s}}\left(x'\right)^{-0.5} \label{DlnA/Dt-2}
\end{align}
Since Mirels' model assumes that the post-shock state is uniform and steady, to the leading order, we can thus write $x'$ as $x' = u'\left(t-t_s\right)$, where $t_s$ is the time at which the particle crosses the shock. With the mass conservation across the shock $\rho_s u' = \rho_0 D_s$, where $\rho_s$ is the post-shock density, Eq.\ \ref{DlnA/Dt-2} can be greatly simplified as the following simple expression
\begin{align}
\dfrac{D}{Dt}\left( \ln A\right) =\dfrac{K_M}{w}\sqrt{\dfrac{\nu_s}{t-t_s}} \label{final-DADT}
\end{align}
where $\nu_s$ is the post-shock kinematic viscosity, and $t_{\textrm{elapse}} = t-t_s$ is the elapsed time since a particle has passed through the shock front. The shock time $t_s$ is recorded when the shock passes over, and convected with the motion of that particle:
\begin{align}
\pde{t_s}{t} + \vec{u}\boldsymbol{\cdot} \mathbb{\nabla}t_s = 0
\end{align}
\subsection{Two-step chemistry model}
\label{Two-step chemistry model}
In the experiment of Fig.\ \ref{ExpPhoto}c, we have calculated the post-shock temperatures using the experimentally measured shock speeds along the walls and cell axis. We found that the lowest post-shock temperature is about 900 K, which is still above its cross-over temperature of 800 K to 850 K. Thus, in this study, the two-step chain-branching reaction model \cite{short2003, leung2010} will be employed for describing the chemical kinetics. It consists of two components, i.e., a thermally neutral induction zone followed by an exothermic main reaction zone. The transport equations of the induction and reaction variables can be written as:
\begin{subequations}
\begin{align}
&\pde{\left(\rho \lambda_i \right)}{t} + \pde{\left(\rho u\lambda_i\right)}{x} +\pde{\left(\rho v \lambda_i\right)}{y} = -\rho \lambda_i \dfrac{1}{A}\dfrac{DA}{Dt} - \mathcal{H}\left(\lambda_i\right)k_i \rho^{\alpha+1} \textrm{exp}\left(-\dfrac{E_a}{R
T}\right) \label{induction}\\
&\pde{\left(\rho \lambda_r \right)}{t} + \pde{\left(\rho u\lambda_r\right)}{x} +\pde{\left(\rho v \lambda_r \right)}{y} = -\rho \lambda_r \dfrac{1}{A}\dfrac{DA}{Dt} - \left[1-\mathcal{H}\left(\lambda_i\right)\right]k_r \rho^{\beta+1}\lambda_r^{\nu} \label{reaction}
\end{align}\label{reaction-rate-equations}where $\lambda_i$ is the progress variable for the induction zone with a value of 1 in the reactants and 0 at the end of the induction zone, $\lambda_r$ \textcolor{black}{the reaction progress variable} with a value of 1 in the unburned zone and 0 in the burned products. $\mathcal{H}\left(\lambda_i\right)$ is the Heaviside function given as
\end{subequations}
\begin{align}
\textcolor{black}{\mathcal{H}\left(\lambda_i\right) =
\begin{cases}
0 & \text{if $\lambda_i=0$} \\
1 & \text{if $\lambda_i > 0$}
\end{cases}}
\end{align}which disables the progress of $\lambda_i$ at the end of the induction zone. $k_i$ and $k_r$ are rate constants, $E_a$ the activation energy controlling the temperature sensitivity of the induction zone duration, $\nu$ is the reaction order, while $\alpha$ and $\beta$ are further empirical reaction order parameters.
Table\ \ref{2-step-parameters} shows the non-dimensional parameters for the two-step model at three different initial pressures from experiments in Fig.\ \ref{ExpPhoto}. They were
\begin{table}[]
\footnotesize
\centering
\caption{The calibrated non-dimensional parameters for the two-step model from the detailed chemistry.}
\begin{tabular}{cccccc}
\toprule
$p_0$ (kPa) & $\gamma$ & $E_a/RT_0$ & $Q/RT_0$ & $k_i$ & $k_r$ \\
\midrule
4.1 & 1.5 & 31.2 & 11.5 & 45.6 & 0.078 \\
6.9 & 1.5 & 22.8 & 11.8 & 10.6 & 0.11 \\
10.3 & 1.5 & 24.2 & 12.0 & 12.7 & 0.14 \\
\bottomrule
\end{tabular}%
\label{2-step-parameters}%
\end{table}%
\begin{figure}[]
\centering
{\includegraphics[width=1.0\textwidth]{Pictures/ZND1.pdf}}
\caption{\textcolor{black}{ZND profiles of (a) density and (b) temperature obtained by using the 2-step chemistry model and the detailed chemistry at the initial pressure of 4.1 kPa.}} \label{ZND-structure}
\end{figure}calibrated from the detailed chemistry using the San Diego chemical reaction mechanism (Williams) \cite{Williams2014}, by using Shepherd's Shock and Detonation Toolbox (SDToolbox) \cite{SDtoolbox}. $\gamma$ was the post-shock isentropic exponent of the CJ detonation, while the heat release $Q$ was determined from the perfect gas relation recovering the correct Mach number \cite{fickett2000}. The effective activation energy $E_a$ was calculated from the \textcolor{black}{logarithmic} derivative of the ignition delay with respect to the inverse of post-shock temperature. Note that the present work adopts the initial state variables ($p_0, \rho_0, T_0$) and the ZND induction zone length ($\Delta_i$) as the normalization scales. As such, the rate constant $k_i$ can be directly obtained from Eq.\ (\ref{induction}) by scaling the dimensionless induction zone length to unity, while $k_r$ can be determined by recovering the correct induction/reaction time ratio from the detailed chemistry. Finally, $\alpha = 1.2, \beta = 1.0, \nu = 1.6$ were adopted for matching the detailed chemistry ZND structure, as can be easily seen from the \textcolor{black}{density and temperature profiles} in Fig.\ \ref{ZND-structure}. \textcolor{black}{For the minor discrepancies observed near the end of the reaction zone, it is due to the limitation of assuming the constant specific heats in the present work.}
\subsection {Computational details}
\label{Computational details}
The non-dimensional governing equations were solved employing the $MG$ code, developed by S. Falle of the University of Leeds, which uses a \textcolor{black}{second-order-accurate exact} Godunov solver \cite{falle1991} with adaptive mesh refinement. The computational domain height was held constant at the same height of experiments (203 mm in height), i.e., $72\Delta_i$ for $p_0 = 4.1$ kPa, $116\Delta_i$ for $p_0 = 6.9$ kPa, and $182\Delta_i$ for $p_0 = 10.3$ kPa. The domain length varied from $3000\Delta_i$ to $5000\Delta_i$. The detonation propagated from left to the right, with reflective boundary conditions imposed to the top and bottom sides, and zero-gradient boundary conditions applied to the left and right ends. The computations were started using a ZND profile placed $300\Delta_i$ in length from the left boundary. An initial density disturbance zone of $4\Delta_i$ was added ahead of the initial ZND solution for accelerating the evolution to cellular detonations. \textcolor{black}{This density perturbation method is the same as that of Maxwell et al. \cite{maxwell2017}, which is give by}
\begin{align}
\textcolor{black}{\rho\left(x, y, t = 0\right) =
\begin{cases}
\textnormal{ZND solution} & \text{if $x < 0$} \\
1.25 - 0.5n & \text{if $0 \le x \le 4$} \\
1 & \text{otherwise} \\
\end{cases}}
\end{align}\textcolor{black}{where $n$ is a random real number from 0 to 1.} As for the numerical resolution, 5 levels of mesh refinement were adopted with the coarsest and finest grid sizes of $1/2\Delta_i$ and $1/16\Delta_i$, respectively. Since the reaction zone length of the simulated cases in this work is in the order of $100\Delta_i$ to $1000\Delta_i$, as demonstrated in Fig.\ \ref{ZND-structure}, such resolution is adequate for obtaining reliable results. This has been verified by a resolution test using a higher level of mesh refinement for calculating the CJ detonation without loss, at the initial pressure of 4.1 kPa. We found that the final stable cell size does not change. In simulations, we ran all the cases for long enough time until we have obtained at least 10 repeated stable cycles of the detonation structures. Due to the large domain size, each case running with 100 to 200 cores in parallel in \textit{Cedar} of Compute Canada requires about one week to complete. More than 20 cases were involved in this study.
\subsection {Results and discussion}
\label{results}
\subsubsection{Comparison with experiments}
\label{Comparison with experiments}
Figure\ \ref{shutr-2} shows the numerically tracked maximum energy release rates for detonations with different losses, at the initial pressure of 4.1 kPa. This corresponds to the open shutter photograph in experiments. From the simulated results, it can be observed that the detonation cell size becomes larger as a result of increasing the Mirels' constant $K_M$, i.e., increasing the magnitude of boundary layer losses. As the ideal CJ detonation ($K_M = 0$) has four stable cells across the channel height, it can only accommodate a single-head detonation for $K_M = 2.0$. When $K_M $ is further increased to 2.5, the single-head detonation finally failed, as can be seen from Fig.\ \ref{shutr-2}e. According to the calculations performed by Xiao and Radulescu \cite{xiao2019}, the theoretical Mirels' constant for 2H$_2$/O$_2$/7Ar detonations is supposed to be $K_M \approx 4.0$. However, the present simulations show that $K_M = 1.75$ can very well recover the experiment (in Fig.\ \ref{ExpPhoto}c) in terms of the cell size and the velocity deficit. This also occurs for cases at other initial pressures of 6.9 kPa and 10.3 kPa, respectively, as shown in Fig. \ref{shutr-4}. While $K_M = 4.0$ results in a larger cell size and velocity deficit, $K_M \approx 2.5$ appears to be able to correctly recover them, when compared to the experiments from Fig.\ \ref{ExpPhoto}a and Fig.\ \ref{ExpPhoto}b. Such discrepancy from the theoretically computed $K_M$ presumably originates from Mirels' assumption of the uniform and steady state behind the shock. For detonations, significant gradients of pressure, temperature, \textcolor{black}{and velocity} exist. \textcolor{black}{Particularly, the flow acceleration (in the shock-attached reference) from being subsonic behind the detonation front to sonic in the reaction zone can contribute to thinned boundary layers, as already noted by Chinnayya et al. \cite{chinnayya2013}. In their 2D viscous simulations, they also found that the thickness of the computed boundary layers behind detonations is approximately half of that predicted by the uniform steady boundary layer theory.} Moreover, the theoretical calculations of $K_M$ assume the leading shock of the ideal CJ detonation speed, while the 2D simulations have detonations of significant velocity deficits. Thus, it results in a smaller $K_M$ than expected by theory. Future work should be devoted to refine the model to account for these non-idealities. Nevertheless, it is quite satisfying that the model works within a factor of 2, which can also be absorbed by uncertainties in chemical kinetics \cite{taylor2013}.
\begin{figure}[]
\centering
{\includegraphics[width=1.0\textwidth]{Pictures/shutr-2.pdf}}
\caption{The recorded maximum energy release rates of detonations at the initial pressure of 4.1 kPa with varied $K_M$. Their cell size and mean propagation speeds are: (a) $D/D_{CJ} = 1.0, \lambda = 51$ mm, (b) $D/D_{CJ} = 0.90, \lambda = 136$ mm, (c) $D/D_{CJ} = 0.85, \lambda = 203$ mm, (d) $D/D_{CJ} = 0.81, \lambda \approx 406$ mm , and (e) detonation failure. In the experiment, $D/D_{CJ} = 0.83, \lambda = 203$ mm . Note that the red symbol represents $50\Delta_i$ in length, and the length of the shown domain is $1500\Delta_i$.} \label{shutr-2}
\end{figure}
\begin{figure}[]
\centering
{\includegraphics[width=1.0\textwidth]{Pictures/shutr-4.pdf}}
\caption{The recorded maximum energy release rates of detonations at the initial pressure of 6.9 kPa and 10.3 kPa, respectively. Their cell size and mean propagation speeds are: (a) $D/D_{CJ} = 0.87, \lambda = 81$ mm, (b) $D/D_{CJ} = 0.78, \lambda = 203$ mm, (c) $D/D_{CJ} = 0.91, \lambda = 41$ mm, (d) $D/D_{CJ} = 0.87, \lambda \approx 58$ mm. Note that the red symbol represents $50\Delta_i$ in length, and the length of the shown domain is $1500\Delta_i$. } \label{shutr-4}
\end{figure}
Besides the velocity deficit and cell size, the experimentally visualized qualitative features of the detonation structure, as well as its cellular dynamics can be very well reproduced by the simulations, as shown in Fig.\ \ref{Exp-compare-0.6psi}. Figure\ \ref{Dspeed-compare-0.6psi} also shows the quantitative agreement in temporal velocity evolution at a single cell, when compared to experiments. This suggests the robustness of the proposed quasi-2D formulation as well as the two-step chemistry model in simulating the real detonations in experiments.
\begin{figure}[]
\centering
{\includegraphics[width=1.0\textwidth]{Pictures/Exp-compare-2.pdf}}
\caption{Comparisons of the gradient of the density (bottom column) from the quasi-2D simulation ( $p_0 = 4.1$ kPa, $K_M = 1.75$) with the schlieren photos (top column) from experiments at the initial pressure of 4.1 kPa. $H$ is the channel height of 203 mm. } \label{Exp-compare-0.6psi}
\end{figure}
\begin{figure}[]
\centering
\includegraphics[width=0.62\textwidth]{Pictures/Dspeed.pdf}
\caption{The evolution of detonation speeds in a cell at the initial pressure of 4.1 kPa. Note that the experimental speeds in a cell were reconstructed by Cheevers et al. \cite{cheevers2019}. The numerical data are the local time-average speed obtained with a running time average of 5 neighbouring points from more than 60 sampling points in a stable cell, for the case of $p_0 = 4.1$ kPa with $K_M = 1.75$. \textcolor{black}{$L$ is the cell length.}} \label{Dspeed-compare-0.6psi}
\end{figure}
\subsubsection{Dynamics of detonations with different losses}
The effect of boundary layer losses on detonation dynamics is shown in Fig.\ \ref{Dspeed-Km}, in terms of the normalized speed with respect to the position in a cell. More than 60 points were sampled in a cell for each case, and the local time-average speed was obtained with a running time average of 5 neighbouring points. Evidently, the presence of wall losses modifies the cellular dynamics. Compared to the CJ case, the cases with losses have a larger deviation from the larger maximum velocity at the beginning of the cell to the smaller minimum one before the collision of triple points. Whether it is the losses that directly modify the flow velocity inside the cellular structure, or the increased activation energy due to velocity deficits that results in such larger fluctuation, requires further confirmation.
\begin{figure}[]
\centering
{\includegraphics[width=0.62\textwidth]{Pictures/Dspeed-Km.pdf}}
\caption{Local time-average speed (along the cell axis) of detonations with different losses, at the initial pressure of 4.1 kPa. Note that the detonation speed is normalized by the mean propagation speed $D_{\textrm{avg}}$ in a cell, and $L$ is the cell length. } \label{Dspeed-Km}
\end{figure}
\subsubsection{The ${D}/{D_{CJ}}-K_M$ relationships}
The variation of the velocity deficit with respect to the Mirels' constant $K_M$ can be better appreciated from Fig.\ \ref{2step-ZND}. The theoretical predictions were obtained by solving the generalized ZND model with lateral flow divergence \cite{klein1995}, using the present two-step reaction model. Clearly, the ZND model underpredicts the velocity deficit obtained from the quasi-2D simulations. As the constant $K_M$ increases to the propagation limit, such discrepancy becomes more significant. These results are consistent with those reported previously by Sow et al. \cite{sow2014} and Reynaud et al. \cite{reynaud2017}, who also found the underprediction of the velocity deficit from the unsteady simulations by the analytical model for relatively regular detonations.
\begin{figure}[]
\centering
\includegraphics[width=0.6\textwidth]{Pictures/2step-ZND.pdf}
\caption{Comparisons of the quasi-2D simulation results with the steady 1D ZND model in terms of the ${D}/{D_{CJ}}-K_M$ relationships.} \label{2step-ZND}
\end{figure}
\subsubsection{The ${\lambda}/{\lambda_{CJ}} - {D}/{D_{CJ}}$ relationships}
We have further obtained the dimensionless relationships in terms of the detonation cell size ($\lambda$) and the characteristic induction zone length ($\Delta$) with respect to the velocity deficit, as shown in Fig.\ \ref{Cellsize-DDCJ}. $\lambda_{CJ}$ and $\Delta_{CJ}$ represent the corresponding length scales of the ideal CJ detonation. The induction length $\Delta$ was obtained through zero-dimensional constant-volume combustion calculations with the post-shock velocity (in the shock-attached frame of reference) multiplying by the time to the peak thermicity, as proposed by Shepherd \cite{shepherd1986}. The detailed chemistry mechanism was utilized in these calculations. The excellent agreement between the ${\lambda}/{\lambda_{CJ}}$ and ${\Delta}/{\Delta_{CJ}}$ correlations suggest that the increase in cell size due to wall losses is still controlled by the increase in the induction zone length as a result of the velocity deficits.
\begin{figure}[]
\centering
{\includegraphics[width=0.6\textwidth]{Pictures/Cellsize-DDCJ-2.pdf}}
\caption{The dimensionless cell size (${\lambda}/{\lambda_{CJ}}$) and characteristic induction zone length (${\Delta}/{\Delta_{CJ}}$) as a function of the velocity deficit. The symbols are from the quasi-2D simulations. The predictions (using Eq.\ (\ref{lamda-CJ-1})) with $E_a/RT_{s, CJ} = 5.0$ and $E_a/RT_{s, CJ} = 3.8$ correspond to initial pressures of 4.1 kPa and 10.3 kPa, respectively. \textcolor{black}{Note that the real-chemistry constant-volume combustion calculations ($\Delta / \Delta_{CJ}$) of 6.9 kPa (denoted by the purple dashed line) and 10.3 kPa (denoted by the green dashed line) follow almost the same curve.}} \label{Cellsize-DDCJ}
\end{figure}
Assuming that the cell size varies with the induction zone thickness \cite{shepherd1986} and that the ignition delay $t_{ig} \sim \textrm{exp}(E_a/RT_s)$, we can thus get
\begin{align}
\dfrac{\lambda}{\lambda_{CJ}} &\approxeq \dfrac{\Delta}{\Delta_{CJ}} = \dfrac{u'_s }{u'_{s, CJ}}\times \dfrac{t_{ig}}{t_{ig, CJ}} \notag \\
& \approxeq\left(\dfrac{u'_s}{u'_0}\right)\left(\dfrac{u'_0}{u'_{s, CJ}}\right)\textrm{exp}\left\lbrace \frac{E_a}{RT_{s, CJ}}\left[\left(\dfrac{T_{s, CJ}}{T_0}\right)\left(\dfrac{T_0}{T_{s}}\right) - 1\right] \right\rbrace \label{lamda-CJ-1}
\end{align}where $u'_s$ and $u'_0$ are the flow velocity behind and ahead of the shock in the shock-attached frame of reference, respectively, and $T_s$ the post-shock temperature. The terms in brackets can be readily evaluated from the shock-jump equations. In the limit of strong shock and high activation energy, we can further simplify Eq.\ (\ref{lamda-CJ-1}) to
${\lambda}/{\lambda_{CJ}} \approxeq \textrm{exp}\left\lbrace \left({2E_a}/{RT_{s, CJ}}\right)\left(1 - {D}/{D_{CJ}}\right) \right\rbrace \label{lamda-CJ-2}$. It thus highlights the exponential sensitivity of the cell size and induction zone length on velocity deficit, which is controlled by the global activation energy. This generalizes Desbordes' observations for overdriven detonations \cite{desbordes1988}. The results in Fig.\ \ref{Cellsize-DDCJ} show that the ${\lambda}/{\lambda_{CJ}} - {D}/{D_{CJ}}$ correlations can be well predicted by the simple expression \eqref{lamda-CJ-1}. The sensitivity of cell size on velocity deficits also highlights the importance of providing the detonation speed when reporting experimentally measured cell size, since the variation can be up to an order of magnitude, even for the weakly sensitive mixtures studied here.
\section{Conclusions}
\label{conclusion}
The present study has shown that the dynamics of 2D cellular detonations in narrow channels can be well captured using a quasi-2D approach modelling the lateral boundary layer losses using Mirels' theory. With an appropriate Mirels' constant, $K_M$, deviating by approximately a factor of 2 from the model proposed by Mirels for steady constant pressure boundary layers, the simulations are found in excellent agreement with experiment. \textcolor{black}{Compared to directly resolving the boundary layers of detonations in the present narrow channel experiments by calculating the 3D NS equations, this novel formulation can save the computational cost by up to five orders of magnitude.} We have also shown that the cellular cycle dynamics is also affected by the losses, which yield larger velocity fluctuations and more rapid decay rates of the lead shock. Finally, the increase in cell size with increasing velocity deficit follows the Arrhenius dependence of ignition delay on the temperature of an equivalent steady shock, in spite of the cellular dynamics, generalizing previous observations of Desbordes for overdriven detonations in generally regular mixtures.
\section*{Acknowledgments}
\label{Acknowledgments}
The authors wish to acknowledge the financial support from the Natural Sciences and Engineering
Research Council of Canada (NSERC) through the Discovery Grant ``Predictability of detonation wave dynamics in gases: experiment and model development", as well as the support of Compute Canada and \textcolor{black}{Core Facility for Advanced Research Computing at CWRU}.
\bibliographystyle{elsarticle-num-PROCI}
| -49,273.380217 |
[
-1.8134765625,
1.814453125
] | 28.899083 |
[
-3.326171875,
0.59033203125,
-1.7841796875,
-5.07421875,
-0.6630859375,
7.28515625
] |
[
1.2294921875,
6.5703125,
1.3544921875,
4.046875
] | 635 | 8,576 |
[
-3.30078125,
3.8828125
] | 26.522535 |
[
-5.84375,
-3.041015625,
-3.177734375,
-2.0390625,
1.595703125,
10.34375
] | 1.024469 | 20.977035 | 14.703825 | 2.863338 |
[
2.307766914367676
] | -32,277.081932 | 5.91861 | -47,700.481392 | 0.638323 | 5.992511 |
[
-3.134765625,
-3.546875,
-3.537109375,
-4.35546875,
2.48828125,
11.2890625
] |
[
-5.44140625,
-1.5654296875,
-1.9755859375,
-1.361328125,
3.15625,
3.794921875
] | |
BkiUckPxK7Dgs_cY5WH1
|
\section{Appendix}
In this section, we present supplementary information supporting the material reported above.
\subsection{Pulse shape discrimination}
The signal in this analysis is an argon nuclear recoil (NR) in liquid argon. These NR events are separated from electron recoils (ER) via pulse-shape discrimination (PSD). The PSD parameter, $\protect{F\ensuremath{{}_{90}}}\xspace=I_{90}/I$, is formed from the total number of photoelectrons, $I$, extracted from the corrected waveform in a 6~$\mu$s window after the initial pulse and the fast signal, $I_{90}$, dominated by singlet light, from the first $90$~ns of the pulse.
AmBe neutron source data are used to characterize the PSD response for NR events. The $\protect{F\ensuremath{{}_{90}}}\xspace$ distribution versus $E$ for these data as a function of energy is shown in Fig.~\ref{fig:f90distributions}a). This AmBe source provides both neutrons and photons which explains the NR band at $\protect{F\ensuremath{{}_{90}}}\xspace \approx 0.7$ and the ER band at $\protect{F\ensuremath{{}_{90}}}\xspace \approx 0.3$. The collection of events at $\protect{F\ensuremath{{}_{90}}}\xspace \approx 1.0$ are from beam-unrelated Cherenkov events. The behavior is broadly consistent with measurements from other LAr detectors~\cite{Hitachi:1983zz,Regenfus_2012}. The mean value of $\protect{F\ensuremath{{}_{90}}}\xspace$ for both NR and ER events is parameterized versus energy for use in the simulation and construction of the expected distributions that are shown in Figs.~\ref{fig:f90distributions}b)-\ref{fig:f90distributions}d).
\subsection{Liquid Argon Quenching Factor}
\begin{figure}
\includegraphics[width=0.9\columnwidth,page=9,trim={0 0 15 15},clip]{larqf_Edepfit_0_125keV.png}
\caption{\label{fig:larqf} Quenching factor fit with an error band constructed with a overall error scaling to yield $\chi^2/\mathrm{DOF}\approx 1$.}
\end{figure}
The predicted detector response to \protect{CEvNS}\xspace NR events relative to that for ER events is quantified with the so-called ``quenching factor'' \ensuremath{\mathrm{QF}}, defined as the ratio of the light output from NR events to that of ER events at the same kinetic energy. The most recent measurements of the \ensuremath{\mathrm{QF}}\ for liquid argon in the energy range $0$~--~$125$~keVnr~\cite{Agnes:2018mvl,Cao:2014gns,Creus:2015fqa,Gastler:2010sc} are shown in Figure~\ref{fig:larqf}. While there appears to be some tension between data sets, there is no \textit{a priori} reason to discard any particular measurement. Therefore, we decided to do a simultaneous fit of these data with a complete treatment of the errors. Since \cenns has little efficiency for $E<20$~keVnr, and any evidence for a more complex energy dependence is not clear, we assumed a linear model for the energy dependence of the \ensuremath{\mathrm{QF}}.
The simultaneous fit to this data utilized a standard least-squares method~\cite{PDG2018}, with an error matrix to handle any correlations in a particular data set. While it is reasonable to assume that all individual data sets contained correlated error, only Ref.~\cite{Gastler:2010sc} reported them, so the error matrix contained off-diagonal terms only for these data. We suggest here in passing that future \ensuremath{\mathrm{QF}}\ measurements report correlated errors (as would occur, for example, with an overall energy calibration uncertainty) with the \ensuremath{\mathrm{QF}}\ data, allowing for a more correct treatment of errors in fits to world data. The linear fit to this data with the reported errors yielded a $\chi^2/DOF$ of 138.1/36. Following the recommended method of Ref.~\cite{PDG2018}, the errors on all data points were simultaneously scaled by a factor of 2.0 such that $\chi^2/DOF=1$. This yields $\mathrm{QF}=(0.246 \pm 0.006) + ((7.8 \pm 0.9)\times10^{-4}\mathrm{~keVnr}^{-1})T$ with a correlation coefficient of -0.79 between the slope and intercept. This fit and resulting error band are shown in Figure~\ref{fig:larqf}. A factor of 2 increase in the errors brings the data into reasonable agreement when correlated errors are considered.
A 2\% error on the efficiency for acceptance of \protect{CEvNS}\xspace events resulted from this \ensuremath{\mathrm{QF}}\ uncertainty and was calculated by varying the \ensuremath{\mathrm{QF}}\ used in the simulation within the error band of Figure~\ref{fig:larqf}. Other scenarios for the energy dependence below 20~keVnr were also considered to quantify extreme possibilities. If the data only from Refs.~\cite{Agnes:2018mvl,Cao:2014gns} (\cite{Creus:2015fqa,Gastler:2010sc}) are used for the fit of \ensuremath{\mathrm{QF}}\ below 20~keVnr, a change in the \protect{CEvNS}\xspace acceptance of -1\% (+12\%) results.
\subsection{Maximum Likelihood Analysis}
A maximum likelihood fit was used to find the best estimate of $\ensuremath{N_{\mathrm{CEvNS}}}$ for the results reported here. The statistics-only null significance is determined by forming the quantity,
\begin{equation}
-2\Delta \ln L = -2(\ln L(\ensuremath{N_{\mathrm{CEvNS}}})-\ln L_\mathrm{best}),
\end{equation}
that depends on the difference between the likelihoods at a given value of $\ensuremath{N_{\mathrm{CEvNS}}}$ and at the best-fit value of \ensuremath{N_{\mathrm{CEvNS}}}, $L_\mathrm{best}$. The value of this quantity at $\ensuremath{N_{\mathrm{CEvNS}}}=0$ determines the statistics-only null-rejection significance with the assumption that it is distributed as a $\chi^2$ function with 1 degree of freedom. This assumption was tested with pseudo-data and supports our simple treatment of systematic errors in this analysis. Figure~\ref{fig:profilell} shows $-2\Delta \ln L$ profiled over the number of SS and BRN background events for the data sets in analyses A and B.
Figure~\ref{fig:AnalysisRprojs} shows the projections of the likelihood fit for analysis B. Figure~\ref{fig:unsubllprojections} shows the projections of the likelihood fit for analysis A, but without subtraction of the SS background distribution.
\begin{figure}
\includegraphics[width=0.9\columnwidth, trim={0 0 40 30},clip]{CombinedNLLPlots.pdf}
\caption{\label{fig:profilell} The likelihood function (curves) vs predicted number of \protect{CEvNS}\xspace events profiled over the number of SS and BRN background events for both Analysis A and B. The vertical lines show the predicted number of events from the SM \protect{CEvNS}\xspace cross section accounting for detector response.
}
\end{figure}
\begin{figure*}
\begin{minipage}{0.325\textwidth}
\includegraphics[width=\textwidth,trim={0 1 40 0},clip]{AnlBProjTime.pdf}
\end{minipage}
\hfill
\begin{minipage}{0.325\textwidth}
\includegraphics[width=\textwidth,trim={0 1 40 0},clip]{AnlBProjEnergy.pdf}
\end{minipage}
\hfill
\begin{minipage}{0.325\textwidth}
\includegraphics[width=\textwidth,trim={0 1 40 0},clip]{AnlBProjPSD.pdf}
\end{minipage}
\caption{\label{fig:AnalysisRprojs}
Projection of the maximum likelihood PDF from Analysis B on $t_\mathrm{trig}$ (left), reconstructed energy (center), and $\protect{F\ensuremath{{}_{90}}}\xspace$ (right). The fit SS background has been subtracted to better show the \protect{CEvNS}\xspace component. Bin-bin systematic errors were not calculated in this analysis.}
\end{figure*}
\begin{figure*}
\begin{minipage}{0.325\textwidth}
\includegraphics[width=\textwidth,trim={0 1 40 0},clip]{figures/FinalLikelihoodSSUnsubTime.pdf}
\end{minipage}
\hfill
\begin{minipage}{0.325\textwidth}
\includegraphics[width=\textwidth,trim={0 1 40 0},clip]{figures/FinalLikelihoodSSUnsubEnergy.pdf}
\end{minipage}
\hfill
\begin{minipage}{0.325\textwidth}
\includegraphics[width=\textwidth,trim={0 1 40 0},clip]{figures/FinalLikelihoodSSUnsubPSD.pdf}
\end{minipage}
\caption{\label{fig:unsubllprojections} Projection of the best-fit maximum likelihood probability density function (PDF) from Analysis A on $t_\mathrm{trig}$ (left), reconstructed energy (center), and $\protect{F\ensuremath{{}_{90}}}\xspace$ (right) along with selected data and statistical errors. The fit SS background is included in these projections. The green band shows the envelope of fit results resulting from the $\pm 1\sigma$ systematic errors on the PDF.}
\end{figure*}
\subsection{\protect{CEvNS}\xspace Cross Section $N$ Dependence}
With the result reported here, the COHERENT collaboration has measured the flux-weighted \protect{CEvNS}\xspace cross section with different nuclei. These results, along with the SM prediction, are shown in Figure~\ref{fig:xsectionvN}.
\begin{figure}
\includegraphics[width=0.99\columnwidth,trim={0 0 40 30},clip]{xsectionvN.pdf}
\caption{\label{fig:xsectionvN} The measured \protect{CEvNS}\xspace flux-weighted cross section from this analysis together with the previous results for CsI[Na]~\cite{Akimov:2017ade} and as expected in the SM as a function of neutron number. Expectations for planned COHERENT target nuclei are also computed. The form factor (FF) unity assumption is compared to the Klein-Nystrand~\cite{klein1999} value that is used for this analysis with the green band representing a $\pm3$\% variation on the neutron radius.
}
\end{figure}
\end{document}
| -9,251.373441 |
[
-2.21484375,
2.13671875
] | 36.25 |
[
-6.47265625,
-3.1640625,
-1.5048828125,
-7.2109375,
0.6923828125,
10.40625
] |
[
-0.495361328125,
4.296875,
2.009765625,
3.6640625
] | 81 | 1,074 |
[
-3.41796875,
4.12109375
] | 25.084413 |
[
-5.80859375,
-4.59375,
-2.75,
-1.08203125,
2.171875,
9.1875
] | 0.773341 | 24.529783 | 37.802607 | 3.390668 |
[
2.352501392364502
] | -7,177.227734 | 6.621974 | -8,994.982364 | 5.146232 | 5.272488 |
[
-3.439453125,
-4.03515625,
-4.0625,
-4.08203125,
2.79296875,
10.6953125
] |
[
-6.6875,
-4.1796875,
-3.228515625,
-2.486328125,
4.53125,
7.6796875
] | |
BkiUbAE5i7PA9OgJC22W
|
\section{Introduction}
The goal of this paper is to define and study the notion of the Frobenius-Perron dimension
of an integral $\Bbb Z_+$-ring $A$ which is not necessarily a fusion ring
(i.e., does not necessarily have a *-structure). Namely, if $N_i$ are the matrices of
multiplication by the basis vectors of $A$ and $d_i$ are their largest positive eigenvalues,
then we let $\bold p$ be the left eigenvector of $N_i$ with eigenvalues $d_i$, normalized to have positive
integer entries with greatest common divisor $1$. Then we set ${\rm FPdim}(A):=\sum_i p_id_i$.
For fusion rings $p_i=d_i$ so ${\rm FPdim}(A)=\sum_i d_i^2$, which coincides with the standard definition in \cite{EGNO}. Also, if $A$ is the Grothendieck ring of a finite tensor category $\mathcal{C}$ then ${\rm FPdim}(A)={\rm FPdim}(\mathcal{C})/D$, where $D$ is the greatest common divisor of the dimensions of the indecomposable projectives of $\mathcal{C}$.
We prove a number of properties of ${\rm FPdim}(A)$. In particular, we show that if $X\in A$ is a $\Bbb Z_+$-generator of $A$ and $r$ the rank of $A$ then $(2d)^{r-1}\ge {\rm FPdim}(A)$, where $d$ is the Frobenius-Perron dimension of a $\Bbb Z_+$-generator of $A$.
We then give two applications of this notion. The first application is to classification of quasi-Hopf and Hopf algebras with two simple modules. We give a number of examples of such quasi-Hopf and Hopf algebras and prove a number of restrictions on their structure. The second application is to quasi-Hopf and Hopf algebras $H$ of prime dimension $p$. It is conjectured that any such quasi-Hopf algebra is commutative and cocommutative, but this is not known even in characteristic zero, and we prove some restrictions on the structure of $H$ if it is not commutative and cocommutative; e.g., we show that $H$ has to have at least four simple modules. In particular, it follows that
a quasi-Hopf algebra of prime dimension $\le 31$ over any field is necessarily commutative and cocommutative. In the case of Hopf algebras, it is known that if $H$ is not commutative and cocommutative then $q>2$ and $\frac{p}{q}>4$ (\cite{NW}). We improve this bound to $\frac{p}{q+2}>\frac{14}{3}$ (Corollary \ref{9/2}). Also, we show that $\frac{p}{q+2}>{\rm min}(9,\phi(p))$, where $\phi$ is a certain function such that $\phi(p)\sim 2\sqrt{2} \left(\frac{\log p}{\log\log p}\right)^{1/2}$ as $p\to \infty$ (Theorem \ref{inner2}). This means that if $p$ is large enough, we must have $\frac{p}{q+2}>9$. Moreover, we show that if $H$ has a nontrivial simple module $X$ with $X\cong X^{**}$ then
$\frac{p}{q+2}>\phi(p)$.
The organization of the paper is as follows. In Section 2 we define the Frobenius-Perron dimension
of an integral $\Bbb Z_+$-ring $A$ and prove basic properties of this notion. In Section 3 we prove the lower bound on the dimension of a $\Bbb Z_+$-generator of $A$. In Section 4 we prove some auxiliary lemmas. In Section 5 we give applications to quasi-Hopf algebras with two simple modules. In Section 6 we prove some bounds for dimensions of
tensor categories with no nontrivial invertible objects.
In Section 7 we give applications to classification of quasi-Hopf algebras of prime dimension.
Finally, in Section 8 we give applications to classification of Hopf algebras of prime dimension, improving the bound on the characteristic from \cite{NW}.
\vskip .05in
{\bf Acknowledgements.} The author is grateful to C. Negron for useful discussions and to V. Ostrik, S.-H. Ng and X. Wang
for corrections and comments on the draft of this paper. The work of the author was partially supported by the NSF
grant DMS-1502244.
\section{The Frobenius-Perron dimension of an integral $\Bbb Z_+$-ring}
\subsection{Dimensions in integral $\Bbb Z_+$-rings.}
Let $A$ be a transitive unital $\Bbb Z_+$-ring of finite rank (\cite{EGNO}, Definitions 3.1.1, 3.3.1). This means that $A$ is a free finitely generated abelian group with basis $b_i, i\in I$
and a unital associative ring structure such that $b_ib_j=\sum_k N_{ij}^kb_k$ with $N_{ij}^k\in \Bbb Z_+$ and $b_0=1$ for some element $0\in I$, and for each $j,k$ there exists $i$ with $N_{ij}^k>0$. In this case we have a homomorphism ${\rm FPdim}: A\to \Bbb R$, determined uniquely by the condition that $d_i:={\rm FPdim}(b_i)>0$, called the {\it Frobenius-Perron dimension} (\cite{EGNO}, Subsection 3.3).
Let $N_i$ be the matrix with entries $N_{ij}^k$. Let $\bold d=(d_i)$ be the column vector with entries $d_i$. Then we have $N_i\bold d=d_i\bold d$, i.e., $\bold d$ is a common (right) eigenvector of the matrices $N_i$.
The Frobenius-Perron theorem implies that we also have a left eigenvector $\bold p=(p_i)$ with $p_i>0$ (defined uniquely up to a positive scalar)
such that $\bold p N_i=d_i\bold p$.
\begin{lemma}\label{easybound} For each $i,j,k$ one has $p_k\ge \frac{N_{ji}^k}{d_j}p_i$.
In particular, one has $p_k\ge p_0/d_k$.
\end{lemma}
\begin{proof} One has $d_jp_k=\sum_i N_{ji}^kp_i$, which implies the statement.
\end{proof}
If $A$ is a fusion ring then $N_{i^*}=N_i^T$, so $\bold p$ is proportional to $\bold d$ and it is natural to normalize
$\bold p$ by setting $\bold p=\bold d$; in this case the number ${\rm FPdim}(A):=\bold p\cdot \bold d=|\bold d|^2=\sum_i d_i^2$ is called the Frobenius-Perron dimension of $A$ (\cite{EGNO}, Section 3.3).
However, in general $\bold p$ does not come with a natural normalization, which is why in \cite{EGNO} the Frobenius-Perron dimension
of a general $\Bbb Z_+$-ring of finite rank is not defined.
Note that when $A$ is the Grothendieck ring of a finite tensor category $\mathcal{C}$ then one may take $p_i$ to be the Frobenius-Perron dimensions of the indecomposable projectives, and $\bold p\cdot\bold d=\sum_i p_id_i$ is then called the Frobenius-Perron dimension of $\mathcal{C}$. However,
this normalization depends on the choice of $\mathcal{C}$: e.g., the categories $\mathcal{C}_1$ of representations of $\Bbb Z/2$
and $\mathcal{C}_2$ of representations of Sweedler's 4-dimensional Hopf algebra categorify the same $\Bbb Z_+$-ring but give different normalizations of $\bold p$.
Now recall (\cite{EGNO}) that $A$ is said to be {\it integral} if ${\rm FPdim}$ lands in $\Bbb Z$.
In this case, the vector $\bold p$ can be uniquely normalized in such a way that $p_i$ are positive integers
whose greatest common divisor is $1$. Let us call this normalization the {\it canonical normalization}.
\begin{definition} The Frobenius-Perron dimension of $A$ is
${\rm FPdim}(A):=\bold p\cdot \bold d=\sum_i p_id_i$,
where $p_i$ are normalized canonically.
\end{definition}
In particular, we see that ${\rm FPdim}(A)\ge \sum_i d_i$.
Note that for fusion rings the canonical normalization just gives $\bold p=\bold d$ as above, so our definition of ${\rm FPdim}(A)$ agrees with the one in \cite{EGNO}. However, in general $p_0\ne 1$, as shown by the following example.
\begin{example}\label{X2=4} Consider the ring $A$ with basis $1,X$ with $X^2=4$.
Thus, $A$ is the Grothendieck ring of the representation category $\mathcal{C}=\mathop{\mathrm{Rep}}\nolimits(U)$
of the restricted enveloping algebra $U$ over a field $\mathbf{k}$
of characteristic $2$ generated by primitive elements $x,y,z$ with $z$ central,
$[x,y]=z$ and $x^2=y^2=0, z^2=z$. Then $d_0=1, d_1=2$, $p_0=2, p_1=1$.
This shows that the normalization with $p_0=1$ is not very natural: this would give $p_1=1/2$,
which is not an algebraic integer.
\end{example}
\subsection{Grothendieck rings of integral finite tensor categories}
Now assume that $A$ is the Grothendieck ring of a finite tensor category $\mathcal{C}$. Then $\mathcal{C}$ is integral, i.e., of the form $\mathop{\mathrm{Rep}}\nolimits H$, where $H$ is a finite dimensional quasi-Hopf algebra (\cite{EGNO}). Thus the Frobenius-Perron dimensions of objects of $\mathcal{C}$ are their usual vector space dimensions, and ${\rm FPdim}(\mathcal{C})=\dim H$. In this case $p_i$ divide the dimensions $\dim(P_i)$ of the indecomposable projectives $P_i$ but aren't necessarily equal to them. E.g., if $H$ is Sweedler's 4-dimensional Hopf algebra, then $p_0=p_1=1$ but the indecomposable projectives $P_0,P_1$ are 2-dimensional. It is easy to see that in general we have the following proposition.
\begin{proposition}\label{gcd} Let $\mathcal{C}$ be an integral finite tensor category with Grothendieck ring $A$, and $D$ be the greatest common divisor of dimensions of the projective objects of $\mathcal{C}$. Then ${\rm FPdim}(\mathcal{C})=D\cdot {\rm FPdim}(A)$.
\end{proposition}
We also have
\begin{proposition}\label{p1} Let $X_i$ be the simple objects of $\mathcal{C}$. Then one has $p_iD\ge d_i$, and the equality holds for a given $i$ if and only if the object $X_i$ is projective. Otherwise, one has $p_iD\ge 2d_i$.
\end{proposition}
\begin{proof} The first statement follows from the fact that the projective cover $P_i$ of $X_i$ contains
$X_i$ in its composition series. For the second statement, it is enough to note that by the Frobenius property of $\mathcal{C}$ we have $\dim P_i\ge 2\dim X_i$ if $X_i$ is not projective.
\end{proof}
\begin{example}\label{unip} A tensor category $\mathcal{C}$ is called {\it unipotent} if its only simple object is the unit object, i.e.
$A=\Bbb Z$ or, equivalently, ${\rm FPdim}(A)=1$ (\cite{EG1}). In other words, $\mathcal{C}={\rm Rep}H$, where $H$ is a local finite dimensional quasi-Hopf algebra. By \cite{EGNO}, Corollary 4.4.2, for $\dim H>1$ this can happen only in characteristic $p>0$. Moreover, in this case the associated graded ${\rm gr}(H)$ of $H$ with respect to the radical filtration is cocommutative \footnote{This is a straightforward generalization of \cite{W}, Proposition 2.2(7) to the quasi-Hopf case, and I thank
S. Gelaki for pointing this out.}, i.e. the group algebra of a finite unipotent group scheme over $k$, so
$\dim H={\rm FPdim}(\mathcal{C})$ is a power of $p$ (and clearly any power can arise this way).
\end{example}
\section{Lower bound on the dimension of a $\Bbb Z_+$-generator}
Let $A$ be an integral $\Bbb Z_+$-ring of rank $r>1$
and $b=\sum_i m_ib_i\in A$ with $m_i\ge 0$. Let us say that $b$ is a $\Bbb Z_+$-{\it generator} of $A$
if for some $n$ all coefficients of $1+b+...+b^n$ are positive. Let $d={\rm FPdim}(b)$ and $N={\rm FPdim}(A)$.
\begin{theorem}\label{bound} (i) Let $\chi$ be the characteristic polynomial of $b$ on $A$, and $Q(z)=(z-d)^{-1}\chi(z)$. Then $Q(d)$ is a positive integer divisible by $N$. In particular, $Q(d)\ge N$.
(ii) Let $s$ be the number of roots of $Q$ with positive imaginary part (counted with multiplicities). Then
$$
d\ge N^{\frac{1}{r-1}}\left(1-\frac{s+1}{r}\right)^{1-\frac{s}{r-1}},
$$
(iii)
We have
$$
d\ge N^{\frac{1}{r-1}}\left( \frac{\lceil \frac{r-1}{2}\rceil}{r}\right)^{\frac{\lceil \frac{r-1}{2}\rceil}{r-1}}.
$$
In particular, $d\ge \frac{1}{2}N^{\frac{1}{r-1}}$.
\end{theorem}
\begin{proof}
(i) First note that by the Frobenius-Perron theorem, $d$ is a simple eigenvalue of $b$, so $Q(d)$ is a positive integer. It remains to show that $Q(d)$ is divisible by $N$. Let $\bold v$ be an integral row vector such that $\bold p\cdot \bold v=1$ (it exists since $p_i$ are relatively prime). We have $Q(b)\bold v=m\bold d$, where $m=(Q(b)\bold v)_0$ is an integer. Therefore
$$
Q(d)=Q(d)\bold p\cdot \bold v=\bold p Q(b)\bold v=m\bold p\cdot \bold d=mN.
$$
This implies that $N$ divides $Q(d)$.
(ii) Let $\lambda_i$, $i=1,...,r-2s-1$ be the real roots of $Q$, and $\mu_1,...\mu_s$ the roots with positive imaginary part.
We have
$$
Q(d)=\prod_{i=1}^{r-2s-1} (d-\lambda_i)\prod_{j=1}^s |d-\mu_j|^2\ge N.
$$
Let $\alpha_i=\lambda_i/d$ and $\beta_j=\mu_j/d$. Then
$$
\prod_{i=1}^{r-2s-1} (1-\alpha_i)\prod_{j=1}^s |1-\beta_j|^2\ge \frac{N}{d^{r-1}}.
$$
This can be written as
$$
\prod_{i=1}^{r-2s-1} (1-\alpha_i)\prod_{j=1}^s (1-2{\rm Re}\beta_j+|\beta_j|^2)\ge \frac{N}{d^{r-1}}.
$$
By the Frobenius-Perron theorem, $|\beta_j|\le 1$, so we get
$$
\prod_{i=1}^{r-2s-1} (1-\alpha_i)\prod_{j=1}^s (2-2{\rm Re}\beta_j)\ge \frac{N}{d^{r-1}}.
$$
Thus by the arithmetic and geometric mean inequality,
$$
\sum_{i=1}^{r-2s-1} (1-\alpha_i)+\sum_{j=1}^s (2-2{\rm Re}\beta_j)\ge (r-s-1)\left(\frac{N}{d^{r-1}}\right)^{\frac{1}{r-s-1}}.
$$
But the left hand side is the trace of $1-b/d$, so it is $\le r$, as ${\rm Tr}(b)\ge 0$. Thus we get
$$
r\ge (r-s-1)\left(\frac{N}{d^{r-1}}\right)^{\frac{1}{r-s-1}}.
$$
This implies that
$$
d\ge N^{\frac{1}{r-1}}\left(1-\frac{s+1}{r}\right)^{1-\frac{s}{r-1}},
$$
as desired.
(iii) Analyzing the function of $s$ in (ii) and using that $2s\le r-1$, we find easily that the worst case scenario is the maximal possible value of $s$, i.e., $s=\lfloor \frac{r-1}{2}\rfloor$. This gives the result.
The last statement follows from the fact that
$$
\left( \frac{\lceil \frac{r-1}{2}\rceil}{r}\right)^{\frac{\lceil \frac{r-1}{2}\rceil}{r-1}}\ge \frac{1}{2}.
$$
\end{proof}
\section{Auxiliary lemmas}
\begin{lemma}\label{loca} Let $H$ be a local finite dimensional quasi-Hopf algebra, $M$ a finite dimensional $H$-module, and $a: M\to M^{**}$ a homomorphism. Then generalized eigenspaces of $a$ (regarded as a linear map $M\to M$) are $H$-submodules of $M$. In particular, if $M$ is indecomposable, then all eigenvalues of $a$ are the same.
\end{lemma}
\begin{proof} The associated graded ${\rm gr}(H)$ of $H$ with respect to the radical filtration
is a Hopf quotient of an enveloping algebra, so it has $S^2=1$. Thus the operator $S^2-1$ strictly preserves
the radical filtration. We have $ah=S^2(h)a$, so
$$
(a-\lambda)hv=h(a-\lambda)v+(S^2-1)(h)av
$$
We claim that
$$
(a-\lambda)^Nhv=\sum_{i=0}^N
\binom{N}{i}(S^2-1)^i(h)(a-\lambda)^{N-i}a^iv.
$$
The proof is by induction in $N$.
Namely, the case $N=0$ is clear, and for the induction step note that
$$
(a-\lambda)\sum_{i=0}^N
\binom{N}{i}(S^2-1)^i(h)(a-\lambda)^{N-i}a^iv=
$$
$$
=\sum_{i=0}^N
\binom{N}{i}(S^2-1)^i(h)(a-\lambda)^{N+1-i}a^iv+\sum_{i=0}^N
\binom{N}{i}(S^2-1)^{i+1}(h)(a-\lambda)^{N-i}a^{i+1}v,
$$
so the induction step follows from the identity $\binom{N}{i}+\binom{N}{i-1}=\binom{N+1}{i}$.
Therefore, since $(S^2-1)^i(h)$ is zero for sufficiently large $i$, the generalized eigenspaces
of $a$ are $H$-submodules of $M$. This implies the statement.
\end{proof}
\begin{remark} Note that in Lemma \ref{loca}, the ordinary eigenspaces of $a$ (as opposed to generalized eigenspaces)
don't have to be submodules of $M$, e.g. $M=H$ and $a=S^2$ when $S^2\ne 1$. An example of a local Hopf algebra
with $S^2\ne 1$ is given in \cite{NWW}, Theorem 1.3, case B3.
\end{remark}
\begin{lemma}\label{commu} Let $V$ be a simple object of a tensor category $\mathcal{C}$, and
$a: V\to V^{**}$ be an isomorphism. Then the morphism $a\otimes (a^*)^{-1}$ commutes
with the composition of $a\otimes 1$, evaluation and coevaluation
$$
V\otimes V^*\to V^{**}\otimes V^*\to \bold 1\to V\otimes V^*
$$
\end{lemma}
\begin{proof} It is easy to see that the morphism $a\otimes (a^*)^{-1}$
preserves the evaluation and the coevaluation moprhism, so it commutes with their composition.
It also obviously commutes with $a\otimes 1$.
\end{proof}
Now let $H$ be a finite dimensional Hopf algebra over an algebraically closed field $\mathbf{k}$ with antipode $S$
such that $S^2\ne 1$.
\begin{lemma}\label{nontri} Let $M$ be a finite dimensional $H$-module which tensor generates $\mathop{\mathrm{Rep}}\nolimits(H)$ and $a: M\to M^{**}$ be an isomorphism. Then $a$ is not a scalar (when regarded as a linear map $M\to M$).
\end{lemma}
\begin{proof} Assume the contrary. Then for any $h\in H$ we have $h|_{M}=S^2(h)|_{M}$. Hence the same holds for any tensor product of copies of $M$ and $M^*$. But since $M$ tensor generates $\mathop{\mathrm{Rep}}\nolimits(H)$, some such product contains the regular representation of $H$, which implies that $S^2=1$ on $H$, a contradiction.
\end{proof}
Recall that a Hopf algebra $H$ is called simple if it has no nontrivial Hopf quotients.
\begin{lemma}\label{no2} Let $H$ be a simple finite dimensional Hopf algebra
over an algebraically closed field $\mathbf{k}$ of characteristic $\ne 2$ with $S^2\ne 1$.
Let $V$ be an irreducible 2-dimensional $H$-module such that $V\cong V^{**}$.
Then $S^4\ne 1$. In particular, the distinguished grouplike element of $H$
is nontrivial.
\end{lemma}
\begin{proof} Since $H$ is simple and has a 2-dimensional simple module,
$H$ cannot have nontrivial 1-dimensional modules: otherwise $H$ would have a nontrivial commutative
Hopf quotient. For the same reason,
${\rm Ext}^1(\mathbf{k},\mathbf{k})=0$, since otherwise $H$ would have a
nontrivial local Hopf quotient. Also it is clear that $V$ tensor generates $\mathop{\mathrm{Rep}}\nolimits(H)$.
Assume the contrary, i.e. that $S^4=1$. Pick an isomorphism $a: V\to V^{**}$ such that $a^2=1$ (this can be done since $S^4=1$). Then $a$ is semisimple with eigenvalues $\pm 1$ (as ${\rm char}(\mathbf{k})\ne 2$)
and not a scalar by Lemma \ref{nontri}, so
${\rm Tr}(a)=0$. Since $H$ has no nontrivial 1-dimensional modules and ${\rm Ext}_H^1(\mathbf{k},\mathbf{k})=0$,
this implies that the Loewy series of $V\otimes V^*$ must be $\bold 1,W,\bold 1$, where
$W$ is an irreducible 2-dimensional module. By Lemma \ref{commu}, the
morphism $a\otimes (a^*)^{-1}$ commutes with the composition of $a\otimes 1$, evaluation and coevaluation
$V\otimes V^*\to V\otimes V^*$. Hence the trace of the block of the morphism $a\otimes (a^*)^{-1}$ on the constituent $W$
is $-2$, which implies that $a\otimes (a^*)^{-1}$ must act by $=-1$ on $W$. But since $H$ is simple,
$W$ must tensor generate $H$, which gives a contradiction
with Lemma \ref{nontri}.
\end{proof}
\begin{example} Let $n\ge 3$, $q$ be a primitive $n$-th root of unity, and
$H=\mathfrak{u}_q(\mathfrak{sl}_2)$ be the corresponding small quantum group
over $\Bbb C$, generated by a grouplike element $K$ and skew-primitive
elements $E$, $F$ with $K^n=1$, $E^n=0$, $F^n=0$ and the usual commutation relations
$KE=q^2EK$, $KF=q^{-2}EK$, $[E,F]=\frac{K-K^{-1}}{q-q^{-1}}$.
This Hopf algebra has a 2-dimensional tautological representation
$V$, which is simple, self-dual and tensor generates $\mathop{\mathrm{Rep}}\nolimits(H)$,
Also $S^2(a)=KaK^{-1}$, so the order of $S^2$ is
the order of $q^2$, which is $n$ for odd $n$ and $n/2$ for even $n$.
Thus $S^4=1$ for $n=4$. However, $H$ is simple if and only
if $n$ is odd: otherwise it has a 1-dimensional representation $\psi$
such that $\psi(E),\psi(F)=0$, $\psi(K)=-1$. So the only reason the proof
of Lemma \ref{no2} fails for $n=4$ is that the
representation $W$ is not irreducible but rather isomorphic to $\psi\oplus \psi$
and therefore does not tensor generate $\mathop{\mathrm{Rep}}\nolimits(H)$.
\end{example}
\begin{lemma}\label{tensprodproj}
Let $X$ be a simple object of a finite tensor category $\mathcal{C}$, such that $X\otimes X^*$ is projective.
Then $X$ is projective.
\end{lemma}
\begin{proof} Assume the contrary. Then
we have
$$
P_X\otimes X^*=X\otimes X^*\oplus Y.
$$
Hence $\mathrm{Hom}(X,P_X)=\mathrm{Hom}(\bold 1,P_X\otimes X^*)\ne 0$.
Thus $P_X$ is the injective hull of $X$, i.e. the socle of $P_X$ is $X$.
Hence $P_X\otimes X^*=2X\otimes X^*\oplus Z$, so
$\dim\mathrm{Hom}(X,P_X)=\dim\mathrm{Hom}(\bold 1,P_X,\otimes X^*)\ge 2$, a contradiction.
\end{proof}
\section{Finite dimensional quasi-Hopf algebras with two simple modules}
In this section we consider integral finite tensor categories of rank $r=2$ unless otherwise specified.
\subsection{The lower bound for the Frobenius-Perron dimension of a $\Bbb Z_+$-generator}
For $r=2$ the $\Bbb Z_+$-ring $A$ is $\Bbb Z_+$-generated by the nontrivial basis element $X$, so we may take $d={\rm FPdim}(X)$. Theorem \ref{bound} then gives the inequality $d\ge N/2$. More explicitly, let $b_0=1,b_1=X$. We have $X^2=aX+b$ for some $a\ge 0$, $b\ge 1$. Hence $d^2=ad+b$. It is easy to see that $p_0=d-a$ and $p_1=1$, so ${\rm FPdim}(A)=p_0+p_1d=2d-a$. Thus we get
\begin{lemma}\label{l1} Let $A$ be a transitive $\Bbb Z_+$-ring of rank $2$, and let the dimension of the nontrivial basis element $X$ be $d$. Then
$$
N-1\ge d\ge N/2,
$$
where $N={\rm FPdim}(A)$.
\end{lemma}
\subsection{The case of large characteristic}
Now let $\mathcal{C}=\mathop{\mathrm{Rep}}\nolimits H$ be the representation category of a quasi-Hopf algebra $H$ over $\mathbf{k}$ with two simple modules.
If $\mathcal{C}$ is semisimple, it is easy to see that $\mathcal{C}={\rm Vec}(\Bbb Z/2,\omega)$ is the category of $\Bbb Z/2$-graded vector spaces twisted by a $3$-cocycle $\omega$. So from now till the end of the section we assume that $\mathcal{C}$ is non-semisimple.
Let $C$ be the Cartan matrix of $\mathcal{C}$.
\begin{lemma}\label{detbound} $|\det C|\le \frac{{\rm FPdim}(\mathcal{C})^2}{4d^2}$.
\end{lemma}
\begin{proof} Using the arithmetic and geometric mean inequality, we have
$$
|\det C|=|c_{00}c_{11}-c_{10}c_{01}|\le {\rm max}(c_{00}c_{11},c_{10}c_{01})\le
$$
$$
\le \frac{1}{4d^2}{\rm max}((c_{00}+c_{11}d^2)^2,(c_{10}+c_{01})^2d^2)\le
$$
$$
\le \frac{1}{4d^2}(c_{00}+(c_{01}+c_{10})d+c_{11}d^2)^2=\frac{{\rm FPdim}(\mathcal{C})^2}{4d^2}.
$$
\end{proof}
Let $A$ be the Grothendieck ring of $\mathcal{C}$.
\begin{proposition}\label{largechar} Suppose that the characteristic of $\mathbf{k}$ is $0$
or \linebreak $p>\frac{{\rm FPdim}(\mathcal{C})^2}{4d^2}$, $\mathcal{C}$ is not pointed, and
${\rm Id}\cong **$ as an additive functor.
Then ${\rm FPdim}(\mathcal{C})=m{\rm FPdim}(A)^2$, where $m>1$ is an integer. In particular,
${\rm FPdim}(\mathcal{C})$ cannot be square free.
\end{proposition}
\begin{proof} By \cite{EGNO}, Theorem 6.6.1 and Lemma \ref{detbound}, $P_0$ and $P_1$ must be proportional in the Grothendieck ring
of $\mathcal{C}$, so $P_0=(d-a)P_1$. This implies that $P_1=m((d-a)\bold 1+X)$, where $m$ is a positive integer.
Hence ${\rm FPdim}(\mathcal{C})=\dim P_0+d\dim P_1=m(2d-a)^2$.
Assume that $m=1$. Since $\mathcal{C}$ is not pointed, it is unimodular. Also we have
$[P_1:X]=1$. By the Frobenius property of $\mathcal{C}$ this implies that $P_1=X$, so
$d-a=0$, a contradiction. Thus, $m>1$, as desired.
\end{proof}
\begin{remark} 1. Note that the statement that $m>1$ in Proposition \ref{largechar} is false in the pointed case, e.g.
for the representation category of Sweedler's 4-dimensional Hopf algebra.
2. Pointed finite tensor categories with two simple objects in characteristic zero
are classified in \cite{EG2}. They are either $\mathop{\mathrm{Rep}}\nolimits H$ where $H$ is a Nichols Hopf algebra of dimension $2^n$ (\cite{N}) or exceptional quasi-Hopf algebras of dimension $8$ or $32$.
\end{remark}
\subsection{Arbitrary characteristic}
\begin{corollary}\label{proje} In any characteristic one has ${\rm FPdim}(\mathcal{C})\ge d\cdot {\rm FPdim}(A)$, i.e., $D\ge d$, with the equality if and only if $X$ is projective. In particular, ${\rm FPdim}(\mathcal{C}) \ge \frac{1}{2}{\rm FPdim}(A)^2$. Moreover, if $X$ is not projective then
$D\ge 2d$ and ${\rm FPdim}(\mathcal{C}) \ge {\rm FPdim}(A)^2$.
\end{corollary}
\begin{proof} This follows from Proposition \ref{p1}, since $p_1=1$.
\end{proof}
\subsection{Minimal categories}
Let us say that $\mathcal{C}$ is {\it minimal} if $X$ is projective.
\begin{proposition}\label{mini} Suppose $\mathcal{C}$ is minimal and $d>1$. Then $d=p^r$ where $p={\rm char}(\mathbf{k})$, and $a=p^r-p^s$ for some $s\le r$, so ${\rm FPdim}(A)=p^{s}(p^{r-s}+1)$, ${\rm FPdim}(\mathcal{C})=p^{r+s}(p^{r-s}+1)$, $D=p^r$ and $\dim P_0=b=p^{r+s}$.
\end{proposition}
\begin{proof} We have $X^2=aX\oplus P_0$, and $P_0$ is an iterated extension of $\bold 1$, i.e. it coincides with the projective cover of $\bold 1$ in the unipotent subcategory of $\mathcal{C}$. This implies the statement, since the Frobenius-Perron dimension of any unipotent category is the power of the characteristic (Example \ref{unip}).
\end{proof}
\begin{corollary}\label{c1} Let $\mathcal{C}$ be a minimal category of Frobenius-Perron dimension $pn>2$ where
$p$ is a prime and $p>n$. Then $p=n+1$ and $n$ is a power of $2$, i.e.,
$p=2^{2^m}+1$ is a Fermat prime. Moreover, ${\rm char}(\mathbf{k})=2$, $d=p-1$ and $X^2=(p-2)X+p-1$.
\end{corollary}
\begin{proof} By Proposition \ref{mini} we have $pn=q^{r+s}(q^{r-s}+1)$, where
$q$ is a prime. Since $p>n$, we must have $s=0$, $n=q^r$ and $p=q^r+1$. Thus $q$ is a power of $2$ and $p$ is
a Fermat prime, as desired.
\end{proof}
\subsection{Examples}
\begin{example}
Minimal categories as in Corollary \ref{c1} exist for every Fermat prime $p$. Namely, let $G={\rm Aff}(p)$ be the group of affine linear transformations of the field $\Bbb F_{p}$, then we can take $\mathcal{C}={\rm Rep}_\mathbf{k}(G)$.
This shows that Proposition \ref{largechar} fails in small characteristic.
\end{example}
\begin{example}
Consider the case $a=0$, i.e., $X^2=d^2$ for some positive integer $d$. We claim that for $d>1$ such a ring $A$
admits a categorification by a finite tensor category in characteristic $p$ if and only if $d$ is a power of $p$, and in particular it never admits such categorification in characteristic $0$.
Indeed, let $\mathcal{C}$ be a finite tensor category with Grothendieck ring $A$. We may assume that $\mathcal{C}$ is tensor generated by $X$. Then $\mathcal{C}=\mathcal{C}_0\oplus \mathcal{C}_1$ is $\Bbb Z/2$-graded and ${\rm Ext}^1(\bold 1,X)={\rm Ext}^1(X,\bold 1)=0$. Thus $P_{\bold 1}$ is an iterated extension of $\bold 1$ and $\dim P_{\bold 1}={\rm FPdim}(\mathcal{C}_0)$ is a power of $p$, since $\mathcal{C}_0$ is unipotent (Example \ref{unip}). Also, $P_X$ is an iterated extension of $X$, and $X\otimes P_X=P_{\bold 1}$.
This implies that $d^2[P_X:X]=p^m$ for some $m$, which implies that $d=p^s$ for some $s$.
On the other hand, such categories exist for every $s$. Indeed, it is enough to construct such a category $\mathcal{C}$
for $s=1$, then for any $s$ we can take the subcategory $\mathcal{C}_s$ in $\mathcal{C}^{\boxtimes s}$ generated by $X^{\boxtimes s}$.
For $p=2$, an example of $\mathcal{C}$ is given in Example \ref{X2=4}. If $p>2$, let $H$ be the Hopf algebra $\Bbb Z/2{\triangleright\!\!\!<} \mathbf{k}[x,y]/(x^p,y^p)$, where $x,y$ are primitive and the generator $g\in \Bbb Z/2$ acts by $gxg^{-1}=-x, gyg^{-1}=-y$.
Let $J=\sum_{j=0}^{p-1}\frac{x^j\otimes y^j}{j!}$ be a twist for $H$, and $H^J$ be the corresponding twisted triangular Hopf algebra. Then we can take $\mathcal{C}$ to be the category of comodules over $H^J$.
\end{example}
\subsection{Categorifications of $X^2=X+d(d-1)$}\label{a=1} Suppose $a=1$, i.e., $X^2=X+d(d-1)$. Let $\mathcal{C}$ be a finite tensor category with Grothendieck ring $A$. Since $X$ is self-dual, this implies that any morphism $a: X\to X^{**}$ has zero trace. Indeed, otherwise $X\otimes X=\bold 1\oplus Y$ where $Y$ has only one copy of $X$ in the composition series. Thus $\mathrm{Hom}(Y,\bold 1)\ne 0$
(as $Y$ is self-dual), so $\dim\mathrm{Hom}(X\otimes X,\bold 1)\ge 2$, a contradiction.
\begin{proposition}\label{chardiv} The characteristic of $\mathbf{k}$ is a prime $p$ dividing \linebreak $d(d-1)$. Moreover, if $H$
is a Hopf algebra then $p$ divides $d$.
\end{proposition}
\begin{proof} Consider the Loewy series of $X\otimes X$. Exactly one of its terms contains $X$.
So we have a 3-step filtration of $X\otimes X$ with composition factors $M,X,N$ invariant under any automorphism of $X\otimes X$, where $M,N$ are indecomposable iterated self-extensions of $\bold 1$ (in fact, $M$ and $N^*$ are cyclic).
The $H$-modules $M$ and $N$ factor through a local quasi-Hopf algebra $H'$. Therefore, by Lemma \ref{loca}, for any isomorphism \linebreak $a: X\to X^{**}$ the morphism $a\otimes a$ on $X\otimes X$ has only one eigenvalue on $M$ and only one on $N$. Moreover, these two coincide by Lemma \ref{commu}; we will normalize $a$ in such a way that this eigenvalue is $1$.
Then, computing the trace of $a\otimes a$, we get $d(d-1)=0$ in $\mathbf{k}$, as claimed.
Now assume that $H$ is a Hopf algebra and let $a_1,...,a_d$ be the eigenvalues of $a$. We may assume that $d>2$. Then the eigenvalues of $a\otimes a$ are $a_ia_j$. At least $d(d-1)$ of these numbers are $1$, so at most $d$ of them are $\ne 1$. We claim that $a_i=1$ for all $i$ or $a_i=-1$ for all $i$. Indeed, if none of $a_i$ are $1$ or $-1$ then $a_i^2\ne 1$ for all $i$, so $a_ia_j=1$ for $i\ne j$, hence $a_i=a_j^{-1}$, which is impossible since $d>2$. So at least one $a_i$ is $1$ or $-1$, and we can assume it is $1$ by multiplying $a$ by $-1$ if needed. Let $n>0$ be the number of $i$ such that $a_i=1$. If $a_i=1,a_j\ne 1$ then $a_ia_j, a_ja_i\ne 1$, so $2n(d-n)\le d$. Thus for $d>2$ we get $n=d$, i.e. $a_i=1$ for all $i$. Thus, $d={\rm Tr}(a)=0$ in $\mathbf{k}$, as desired.
\end{proof}
\subsection{Categorifications of $X^2=X+2$}\label{X2=X+2}
Now consider the case $d=2$, i.e. the fusion rule $X^2=X+2$.
Let $\mathcal{C}_X$ be the tensor subcategory of $\mathcal{C}$ generated by $X$.
\begin{proposition}\label{tenpro}
One has $X\otimes X=P\oplus X$, where $P$ is a nontrivial extension of $\bold 1$ by $\bold 1$.
Moreover, $P$ and $X$ are the indecomposable projectives of $\mathcal{C}_X$, so ${\rm FPdim}(\mathcal{C}_X)=6$.
\end{proposition}
\begin{proof} Let us first show that $X\otimes X=P\oplus X$. Assume the contrary.
Since $X\otimes X$ is self-dual and $\dim\mathrm{Hom}(\bold 1,X\otimes X)=1$, we get that
$X\otimes X$ must be indecomposable with Loewy series $\bold 1,X,\bold 1$.
Then $\dim\mathrm{Hom}(X,X\otimes X\otimes X)=\dim{\rm End}(X\otimes X)=2$.
But $X\otimes X\otimes X$ is self-dual, has composition series $X,X\otimes X,X$, and $\mathrm{Hom}(X,X\otimes X)=0$.
This means that $X\otimes X\otimes X=2X\oplus X\otimes X$. Thus for each $n$, $X^{\otimes n}$
is a direct sum of copies of $X$ and $X\otimes X$. But for some $n$, the object $X^{\otimes n}$
has a direct summand which is projective in $\mathcal{C}_X$. Hence $X\otimes X$ is projective. Therefore, so is $X$
(as it is a direct summand in $X\otimes X\otimes X$). But then $X$ must be a direct summand in $X\otimes X$, a contradiction.
Note that $\dim\mathrm{Hom}(X,P\otimes X)=\dim\mathrm{Hom}(X\otimes X,P)=\dim\mathrm{Hom}(P,P)=2$. Thus, $P\otimes X=2X$.
Thus, for any $n$, $X^{\otimes n}$ is a direct sum of copies of $P$ and $X$. This implies that $P$ and $X$ are the indecomposable projectives of $\mathcal{C}_X$.
\end{proof}
We note that such a category of dimension $6$ does exist, e.g. ${\rm Rep}_\mathbf{k}(S_3)$, with ${\rm char}(\mathbf{k})=2$. It can also be realized as the $\Bbb Z/2$-equivariantization ${\rm Vec}_\mathbf{k}(\Bbb Z/3)^{\Bbb Z/2}$ of ${\rm Vec}_\mathbf{k}(\Bbb Z/3)$. A generalization
is ${\rm Vec}_\mathbf{k}(\Bbb Z/3,\omega)^{\Bbb Z/2}$, where $\omega\in H^3(\Bbb Z/3,\mathbf{k}^\times)$.
\subsection{Categories of dimension $sn$, where $s$ is square free and coprime to $n$}
\begin{proposition}\label{32} Let ${\rm char}(\mathbf{k})$ be zero or $p>\frac{{\rm FPdim}(\mathcal{C})^2}{4d^2}$, and
${\rm FPdim}(\mathcal{C})=sn$, where $s$ is a square free number coprime to $n$. Assume that $\mathcal{C}$ is not pointed, and ${\rm Id}\cong **$ as additive functors. Then $n=m\ell^2$, where $\ell\ge 4$ and $m\ge 2$, so $n\ge 32$.
\end{proposition}
\begin{proof} It follows from Proposition \ref{largechar} that $n=m\ell^2$ where $\ell={\rm FPdim}(A)$ and $m\ge 2$.
So it remains to show that $\ell\ge 4$. To do so, note that $\ell\ge 2$, and if $\ell=2$ then $\mathcal{C}$ is pointed.
Also, if $\ell=3$ then $X^2=X+2$, so ${\rm char}(\mathbf{k})=2$ (Proposition \ref{chardiv}).
\end{proof}
\begin{example}\label{32dim} (see \cite{Ne}, Subsection 9.4) The bound $n\ge 32$ in Proposition \ref{32} is sharp. Namely, let ${\rm char}(\mathbf{k})\ne 2$ and
let $B$ be the small quantum group $\mathfrak{u}_q(\mathfrak{sl}_2)$ at a primitive 8-th root of unity $q$.
It is a Hopf algebra of dimension $2^7=128$ generated by $E,F,K$ with relations $KE=q^2EK$, $KF=q^{-2}EK$, $[E,F]=\frac{K-K^{-1}}{q-q^{-1}}$, $E^4=F^4=K^8-1=0$. With its usual Hopf structure, $B$ does not admit an $R$-matrix, but
it admits a quasi-Hopf structure $\Phi$ with a factorizable R-matrix (\cite{CGR}). Let $\mathcal{D}=\mathop{\mathrm{Rep}}\nolimits(B^\Phi)$
be the corresponding nondegenerate braided category. This category contains a unique 1-dimensional nontrivial representation $\psi$ such that $\psi(E)=\psi(F)=0$, $\psi(K)=-1$ (it has highest weight $4$). It is easy to show using formula (4.10) of \cite{CGR} that the subcategory $\mathcal{E}$ generated by $\bold 1$ and $\psi$ has trivial braiding, i.e., it is equivalent to $\mathop{\mathrm{Rep}}\nolimits(\Bbb Z/2)$. Let $\mathcal{C}:=\mathcal{E}^\perp/\mathcal{E}$ be the de-equivariantization of the centralizer $\mathcal{E}^\perp$ of $\mathcal{E}$ by $\Bbb Z/2$. The centralizer consists of representations with even highest weights ($0,2,4,6$), so modding out by $\psi$ gives a category $\mathcal{C}$ with just two simple objects, $\bold 1$ and $X$ (the quantum adjoint representation).
The category $\mathcal{C}$ has Frobenius-Perron dimension $32$ and is nondegenerate braided. Its Grothendieck ring is generated by a basis element $X$ with $X^2=2X+3$. The indecomposable projectives $P_{\bold 1},P_X$ have the Loewy series $\bold 1,X\oplus X,\bold 1$ and
$X,\bold 1\oplus \bold 1,X$, and we have $X\otimes X=\bold 1\oplus P_X$.
Also $\mathcal{C}$ does not admit a pivotal structure: indeed, if $g$ is a pivotal structure and $u: {\rm Id}\to **$ the Drinfeld isomorphism then $g^{-1}u$ is an automorphism of the identity functor, hence acts by $1$ on $X$ (as $X$ is linked to $1$).
But then the squared braiding $c^2: X\otimes X\to X\otimes X$ is unipotent, which is a contradiction with formula (4.10) of \cite{CGR}, as this formula implies that $-1$ is an eigenvalue of $c^2$ on the tensor product of highest weight vectors.
Moreover, any {\it pivotal} finite nondegenerate braided tensor category $\mathcal{C}$ over $\Bbb C$ with two simple objects is equivalent to ${\rm Vec}(\Bbb Z/2)$ with braiding defined by the quadratic form $Q$ on $\Bbb Z/2$ such that $Q(1)=\pm i$. Indeed, by the result of \cite{GR}, a {\it pivotal} finite nondegenerate braided tensor category over $\Bbb C$ must contain a simple projective object, so $X$ is projective, which implies that $\mathcal{C}$ is semisimple. Thus, the above example shows that the pivotality assumption in \cite{GR} is essential.
\end{example}
\begin{remark} I don't know any other finite dimensional quasi-Hopf algebras over $\Bbb C$ with a unique nontrivial simple module $X$, such that $\dim X>1$. In particular, I don't know if there exist Hopf algebras with this property.
\end{remark}
\subsection{Non-minimal categories}
\begin{corollary}\label{c3} (i) Suppose $\mathcal{C}$ is non-minimal and ${\rm FPdim}(\mathcal{C})=pn$, where $p$ is a prime and $p>n$. Then ${\rm FPdim}(A)$ divides $n$ and the dimension of every projective object in $\mathcal{C}$ is divisible by $p$.
\end{corollary}
\begin{proof} By Corollary \ref{proje} we have ${\rm FPdim}(A)\le \sqrt{{\rm FPdim}(\mathcal{C})}<p$, so since ${\rm FPdim}(A)$ divides $pn$, it must divide $n$. Hence $D$ is divisible by $p$, as desired.
\end{proof}
Corollary \ref{c3} has strong implications for the structure of $\mathcal{C}$ if $n$ is small.
First of all note that if $\mathcal{C}$ is pointed then its Frobenius-Perron dimension has to be even.
\begin{proposition}\label{p4} Let $\mathcal{C}$ be an integral category with two simple objects of Frobenius-Perron dimension $pn$ where $p$ is a prime. If $n\le 3$ then $n=2$ and $\mathcal{C}$ is pointed or ${\rm FPdim}(\mathcal{C})=6$ and $\mathcal{C}$ is a minimal category which categorifies the fusion rule $X^2=X+2$ in characteristic $2$.
\end{proposition}
\begin{proof} By Corollary \ref{c1}, we may assume that $\mathcal{C}$ is non-minimal.
Suppose $n=1$. If $p>2$, Corollary \ref{c3} implies that ${\rm FPdim}(A)=1$, which is impossible, and if $p=2$, the category is semisimple, hence pointed.
Suppose $n=2$. If $p>2$, Corollary \ref{c3} implies that ${\rm FPdim}(A)=2$, i.e., $\mathcal{C}$ is pointed.
If $p=2$ then $d<2$, so $d=1$ and again $\mathcal{C}$ is pointed.
Suppose $n=3$. If $p>3$, Corollary \ref{c3} implies that ${\rm FPdim}(A)=2d-a=3>d$, which implies that $d=2$,
$a=1$, so $b=2$ and $X^2=X+2$. In the latter case,
${\rm char}(\mathbf{k})=2$ (Subsection \ref{X2=X+2}). Consider an isomorphism $g: P_{\bold 1}\to P_{\bold 1}^{**}$, and
let $w\in H$ be the element realizing the canonical isomorphism $V\to V^{****}$ such that $\varepsilon(w)=1$.
Then $g^{-2}w: P_{\bold 1}\to P_{\bold 1}$ is an automorphism, and we may normalize $g$ so that
this automorphism is unipotent. Then the eigenvalues of $g$ on the invariant part
of the graded pieces of the Loewy filtration of $P_{\bold 1}$ are all equal to $1$.
Since the trace of any morphism $X\to X^{**}$ is zero, we see that ${\rm Tr}(g)$
cannot be zero since $\dim P_{\bold 1}=p$ and hence $[P_{\bold 1}:\bold 1]$ is odd.
This is a contradiction, since this implies that $\bold 1$ is projective so $\mathcal{C}$ is semisimple.
\end{proof}
\subsection{Representations of finite groups}
Recall that ${\rm Aff}(q)$ denotes the group of affine transformation $x\to \alpha x+\beta$ over the finite field $\Bbb F_q$.
\begin{theorem}\label{figr} Let $G$ be a finite group. Then:
(i) If $G$ has exactly two irreducible representations over $\mathbf{k}$
then either $G=\Bbb Z/2$ or ${\rm char}(\mathbf{k})=p>0$.
(ii) Let ${\rm char}(\mathbf{k})=p>0$ and let $N$ be the largest normal $p$-subgroup of $G$. Then $G$ has exactly two irreducible representations over $\mathbf{k}$ if and only if $G/N$ is one of the following:
(1) $G={\rm Aff}(q)$ where $q$ is a Fermat prime, and $p=2$;
(2) $G={\rm Aff}(p+1)$ where $p$ is a Mersenne prime.
(3) $G={\rm Aff}(9)$ or $G=\Bbb Z/2{\triangleright\!\!\!<} {\rm Aff}(9)$ (where the generator of $\Bbb Z/2$ acts by the Galois automorphism of $\Bbb F_9$) and $p=2$.
(4) $G=\Bbb Z/2$, $p\ne 2$.
\end{theorem}
\begin{proof} (i) Let ${\rm char}(\mathbf{k})=0$ and $X$ be the nontrivial irreducible representation of $G$. Then $X\otimes X$ has to contain $\dim X$ copies of the trivial representation. Thus $\dim X=1$ and $G=\Bbb Z/2$.
(ii) This follows by specializing Theorem A in \cite{DN} to the case when the only linear representation of $G$ is trivial
and using the easy fact that consecutive prime powers are $q-1,q$ for a Fermat prime $q$,
or $p,p+1$ for a Mersenne prime $p$, or $8,9$. See e.g. \cite{CL} for a short proof of this statement.
\end{proof}
\section{Lower bounds for integral finite tensor categories with no nontrivial invertible objects}
\begin{proposition} \label{lowbo} Let $\mathcal{C}=\mathop{\mathrm{Rep}}\nolimits(H)$
be a non-semisimple integral finite tensor category of rank $r$ such that
${\rm Ext}^1(\bold 1,\bold 1)=0$ (e.g., ${\rm char}(\mathbf{k})=0$) and $\mathcal{C}$ has no nontrivial invertible objects. Then:
(i) $\mathop{\mathrm{FPdim}}\nolimits \mathcal{C}\ge 8r+3$;
(ii) If $r=2$ then $\mathop{\mathrm{FPdim}}\nolimits(\mathcal{C})\ge 32$.
In particular, these bounds hold if $\mathcal{C}$ is not pointed and simple (has no nontrivial tensor subcategories).
\end{proposition}
\begin{proof} (i) It is clear that $\dim P_{\bold 1}\ge 4$, since it has head and socle $\bold 1$.
Assume first that $\dim P_{\bold 1}=4$. Then $P_{\bold 1}$ is uniserial with
Loewy series $\bold 1,V,\bold 1$ for some 2-dimensional simple module $V$, so $V^*\cong V$ and
$\dim\mathop{\mathrm{Ext}}\nolimits^1(\bold 1,V)=1$, while ${\rm Ext}^1(\bold 1,Y)=0$ for any simple $Y\ne V$.
So if $V\otimes V$ has Loewy series $\bold 1,Y,\bold 1$, then $Y$ must be simple, and
we must have $Y\cong V$ (as otherwise ${\rm Ext}^1(\bold 1,Y)=0$). Then $V\otimes V=P_{\bold 1}$, so
$V$ is projective by Lemma \ref{tensprodproj}, a contradiction.
\footnote{Another way to get a contradiction is to note that $V^2=V+2$
in the Grothendieck ring and apply Proposition \ref{tenpro}.}
Thus, $V\otimes V=\bold 1\oplus X$, where $X$ is a self-dual simple 3-dimensional module.
Consider the tensor product $P_{\bold 1}\otimes V=P_V\oplus T$, which has composition factors $V,\bold 1, X,V$.
Note that $P_V$ necessarily includes composition factors $V,\bold 1,V$. If $P_V$ contains no other composition factors, i.e. it is 5-dimensional, then $T=X$, so $X=P_X$ is projective. But then $X\otimes V=P_V\oplus Q$ (as $\mathrm{Hom}(X\otimes V,V)\ne 0$), so $Q$ is projective and $\dim Q=1$, which is impossible. Thus, $P_V$ is indecomposable 8-dimensional with Loewy series $V,\bold 1\oplus X,V$.
Let $W$ be a 2-dimensional projective module. Then $W^*\otimes W=P_{\bold 1}$, so
$\mathrm{End}(W\otimes V)=\mathrm{Hom}(W^*\otimes W,V\otimes V)=\mathrm{Hom}(P_{\bold 1},\bold 1\oplus X)=\mathbf{k}$.
But $\mathcal{C}$ is unimodular, so the head and socle of any indecomposable projective coincide, and hence if $P$ is
a non-simple indecomposable projective then $\dim\mathrm{End}(P)\ge 2$.
Thus, $W\otimes V=Q$ is a 4-dimensional simple projective module. Moreover,
$Q\otimes V=W\otimes V\otimes V=W\oplus W\otimes X$, and
$$
{\rm End}(W\otimes X)=\mathrm{Hom}(W^*\otimes W,X\otimes X)=\mathrm{Hom}(P_{\bold 1},X\otimes X)=
\mathrm{Hom}(X\otimes P_{\bold 1},X).
$$
But ${\rm Ext}^1(V,X)\ne 0$, so $P_X$ includes composition factors
$X,V,X$, hence $\dim P_X\ge 8$. Thus $X\otimes P_{\bold 1}$, which has dimension $12$,
cannot contain more than one copy of $P_X$.
Hence ${\rm End}(W\otimes X)=\mathrm{Hom}(X\otimes P_{\bold 1},X)=\mathbf{k}.$
Thus $W\otimes X$ is a 6-dimensional simple projective module. So $W$ can be recovered from $Q$ as the unique 2-dimensional summand in $Q\otimes V$, i.e.,
the assignment $W\mapsto Q$ is injective.
Now suppose we have $s$ projective 2-dimensional objects.
Then they contribute $4s$ to ${\rm FPdim}(\mathcal{C})$, and the corresponding modules $Q$ contribute $16s$.
The rest of the simple modules except $\bold 1,V,X$ contribute at least $8$ each, so in total $8(r-2s-3)$.
So altogether we have (using that $\dim P_{\bold 1}=4$, $\dim P_V=8$, $\dim P_X\ge 8$):
$$
{\rm FPdim}(\mathcal{C})\ge 4+2\cdot 8+3\cdot 8+4s+16s+8(r-2s-3)=8r+20+4s\ge 8r+20,
$$
as claimed.
Now assume that $\dim P_{\bold 1}\ge 5$. Then by Lemma \ref{easybound}, for each
2-dimensional simple module $W$ we have $\dim P_W\ge 5/2$, so $W$ is not projective, and
$\dim P_W\ge 4$.
If $\dim P_{\bold 1}\ge 5$ then let $X$ be a simple submodule of $P_{\bold 1}/\bold 1$.
If $\dim X\ge 3$ then $\dim P_X\ge 7$ (as it has composition factors $X,\bold 1,X$). Hence ${\rm FPdim}(\mathcal{C})\ge 5+3\cdot 7+8(r-2)=8r+10$, as claimed.
It remains to consider the case $\dim X=2$, in which case we must have $\dim P_{\bold 1}\ge 6$.
Then $\dim P_X\ge 5$. If $\dim P_X=5$ then the Loewy series of $P_X$ is $X,\bold 1,X$,
so $X^*\otimes P_X$ has composition series $X^*\otimes X,X^*,X^*\otimes X$. But
$X^*\otimes P_X=P_{\bold 1}\oplus Q$, and $P_{\bold 1}$ has composition series $\bold 1,X,X',\bold 1$ where $\dim X'=2$, and $Q$ is a
4-dimensional projective, so cannot involve $\bold 1$. Thus $X^*\otimes X=\bold 1\oplus Y$, where $Y$ is a 3-dimensional simple,
i.e. we have only one 2-dimensional simple constituent in $X^*\otimes P_X$, a contradiction. Thus, $\dim P_X\ge 6$.
If $\dim P_X>6$ or $\dim P_{\bold 1}>6$ then they contribute at least $19$ into ${\rm FPdim}(\mathcal{C})$, so ${\rm FPdim}(\mathcal{C})\ge 19+8(r-2)=8r+3$, as claimed.
Thus it remains to consider the case $\dim P_{\bold 1}=\dim P_X=6$. Then the Loewy series of $P_X$ must have the form
$X,\bold 1\oplus \bold 1,X$ (so ${\rm Ext}^1(\bold 1,X)=\mathbf{k}^2$) and the Loewy series of $P_{\bold 1}$ therefore looks like $\bold 1,X\oplus X,\bold 1$, where $X$ is self-dual.
Now consider the module $X\otimes X$. If it has Loewy series $\bold 1,X',\bold 1$ where $X'$ is simple 2-dimensional then $\mathrm{Hom}(P_{\bold 1},X\otimes X)=\mathbf{k}$ (as then $X\otimes X$ is not a quotient of $P_{\bold 1}$), which is a contradiction since we have $[X\otimes X,\bold 1]=2$.
Thus, $X\otimes X=\bold 1\oplus Y$, where $Y$ is a 3-dimensional simple. So $Y$ contributes at least $9$ into ${\rm FPdim}(\mathcal{C})$.
Thus we get ${\rm FPdim}(\mathcal{C})\ge 6+2\cdot 6+9+8(r-3)=8r+3$, as claimed. This proves (i).
To prove (ii), note that $\dim P_{\bold 1}$ is divisible by ${\rm dim}P_X$. Let $X$ be the nontrivial simple object. If $\dim X\ge 4$ then $\dim P_X\ge 9$, so $\dim P_{\bold 1}\ge 9$ and $\mathop{\mathrm{FPdim}}\nolimits(\mathcal{C})\ge 9+4\cdot 9=45$, as claimed. If $\dim X=3$ then $\dim P_X\ge 7$, so $\dim P_{\bold 1}\ge 7$, which means that $[P_{\bold 1}:X]\ge 2$,
so $\dim P_{\bold 1}\ge 8$, $\dim P_X\ge 8$ and ${\rm FPdim}(\mathcal{C})\ge 32$, as claimed.
Finally, the case $\dim X=2$ is impossible by Proposition \ref{tenpro}, as ${\rm Ext}^1(\bold 1,\bold 1)\ne 0$.
\end{proof}
\begin{example} 1. The bound ${\rm FPdim}(\mathcal{C})\ge 27$ for $r=3$ in Proposition \ref{lowbo} is sharp in any characteristic. Namely, it is attained for the category
$\mathcal{C}$ of representations of the small quantum group $H=\mathfrak{u}_q(\mathfrak{sl}_2)$ where $q$ is a primitive cubic root of $1$ if ${\rm char}(\mathbf{k})\ne 3$ and of the restricted enveloping algebra $H=\mathfrak{u}(\mathfrak{sl}_2)$ in characteristic $3$.
In this case there are three simple modules: $\bold 1$, the 2-dimensional tautological module $V$ and the 3-dimensional Steinberg
module $X$, which is projective. Moreover, the projective covers of $\bold 1$ and $V$ have dimension $6$
and Loewy series $\bold 1,V\oplus V,\bold 1$ and $V,\bold 1\oplus \bold 1,V$.
2. The bound ${\rm FPdim}(\mathcal{C})\ge 32$ for $r=2$ in Proposition \ref{lowbo} is also sharp, as shown by Example \ref{32dim}.
3. There exists a category as in Proposition \ref{lowbo} with $\dim P_{\bold 1}=4$. Namely, let $\mathbf{k}=\Bbb C$,
$G$ be the binary icosahedral group (of order $120$), and $V$ its 2-dimensional tautological representation. Let $z$ be the nontrivial central element of $G$. Consider the supergroup $G{\triangleright\!\!\!<} V$ and the category $\mathcal{C}={\rm Rep}(G{\triangleright\!\!\!<} V,z)$ of representations of $G$ on superspaces such that $z$ acts by parity. Then $P_{\bold 1}=\wedge V={\rm Ind}_G^{G{\triangleright\!\!\!<} V}\bold 1$. The simple objects of this category are just simple $G$-modules, which are labeled by vertices of the affine Dynkin diagram $\widetilde{E}_8$ and have dimensions $1,2,3,4,5,6,3,4,2$, and ${\rm FPdim}(\mathcal{C})=480$.
4. Examples of categories of small dimension ${\rm FPdim}(\mathcal{C})=60$ satisfying the assumptions of Proposition \ref{lowbo} are given by the representation categories $\mathop{\mathrm{Rep}}\nolimits_p(A_5)$ of the alternating group $A_5$ in characteristics $p=2,3,5$. Namely:
$\mathop{\mathrm{Rep}}\nolimits_2(A_5)$ has simple objects of dimension $1,2,2,4$ with projective covers of dimension $12,8,8,4$;
$\mathop{\mathrm{Rep}}\nolimits_3(A_5)$ has simple objects of dimension $1,3,3,4$ with projective covers of dimension $6,3,3,9$;
The Loewy series of $P_\bold 1$ is $\bold 1,X,\bold 1$, where
$X$ is the 4-dimensional simple module.
$\mathop{\mathrm{Rep}}\nolimits_5(A_5)$ has simple objects of dimension $1,3,5$ with projective covers of dimension $5,10,5$.
The Loewy series of $P_\bold 1$ is $\bold 1,X,\bold 1$, where
$X$ is the 3-dimensional simple module.
In particular, these examples show that $P_{\bold 1}$ can have Loewy series $\bold 1,V,\bold 1$ with $V$ of dimensions
$2,3,4$.
\end{example}
\section{Tensor categories of dimension $p$}
Now let $\mathcal{C}$ be a finite tensor category over an algebraically closed field $\mathbf{k}$ of any characteristic $q\ge 0$ of Frobenius-Perron dimension $p>2$ (a prime). It is known that such a category is automatically integral (\cite{EGNO}). If $\mathcal{C}$ is pointed then either it is semisimple (hence $\mathcal{C}=\mathrm{Vec}(\Bbb Z/p,\omega)$) or $\mathcal{C}$ is unipotent and ${\rm char}(\mathbf{k})=p$.
It is conjectured that $\mathcal{C}$ is always pointed. The goal of this section is to prove it under some assumptions.
Suppose $\mathcal{C}$ is not pointed, and let $A$ be the Grothendieck ring of $\mathcal{C}$. Let $r$ be its rank, and $d$ the smallest dimension of a nontrivial simple object (call it $X$). Clearly $d>1$. Let $d_i$ be the dimensions of simple objects and $p_i$ the dimensions of the corresponding indecomposable projectives. So $\sum_i p_id_i=p$. Also, since ${\rm FPdim}(A)>1$ and divides $p$, we must have ${\rm FPdim}(A)=p$ and $D=1$. Also we have ${\rm Ext}^1(\bold 1,\bold 1)=0$,
since otherwise the tensor subcategory of $\mathcal{C}$ formed by iterated extensions of $\bold 1$ is nontrivial, and its dimension
must divide $p$ by \cite{EGNO}, Theorem 7.17.6.
\begin{proposition} \label{fourob}
(i) $\mathcal{C}$ has at least $4$ simple objects.
(ii) $p\ge 37$.
\end{proposition}
\begin{proof}
(i) Suppose $\mathcal{C}$ has two simple objects. Then by Theorem \ref{bound}, $d\ge p/2$, so $p\ge 1+d^2\ge 1+p^2/4$, hence
$p\le 2$, a contradiction.
Now assume that $\mathcal{C}$ has three simple objects, $\bold 1$ and $X,Y$ of Frobenius-Perron dimensions $d,m$.
By Theorem \ref{bound}, we have $d\ge (p/3)^{1/2}$, so $3d^2\ge p$.
If $P_X\ne X$ then $\dim P_X\ge 2d$ so
$$
p\ge 1+2d^2+m^2>3d^2\ge p,
$$
a contradiction. Similarly, if $P_Y\ne Y$ then $\dim P_Y\ge 2m$ so
$$
p\ge 1+d^2+2m^2>3d^2\ge p,
$$
again a contradiction. Thus $X$ and $Y$ are projective.
This means that the block of $\bold 1$ is a tensor subcategory of $\mathcal{C}$.
The Frobenius-Perron dimension of this subcategory must divide $p$, so this subcategory must be trivial.
Hence $\mathcal{C}$ is semisimple. Then $X^2=1+aX+bY$ with $b\ne 0$, so $d^2=1+ad+bm$, so $(d,m)=1$. Now
$X\otimes Y=sX+tY$, so $dm=sd+tm$, hence $sd=(d-t)m$. But $s,t>0$ since $X$ occurs in $Y\otimes Y$ and $Y$ occurs in $X\otimes X$. This gives a contradiction.
Thus $\mathcal{C}$ has at least $4$ simple objects.
(ii) It follows from (i) and Proposition \ref{lowbo} that $p\ge 8\cdot 4+3=35$. Hence $p\ge 37$.
\end{proof}
\begin{corollary}\label{bound2} One has
$$
d\ge 2^{-2/3}p^{\frac{1}{r-1}}.
$$
\end{corollary}
\begin{proof}
This follows from Proposition \ref{fourob}, Theorem \ref{bound}(iii) and the fact that the function
$$
f(r):=\left( \frac{\lceil \frac{r-1}{2}\rceil}{r}\right)^{\frac{\lceil \frac{r-1}{2}\rceil}{r-1}}
$$
satisfies $f(r)\ge f(4)=2^{-2/3}$ if $r\ge 4$.
\end{proof}
\begin{proposition}\label{5sim} Let $H$ be a quasi-Hopf algebra of prime dimension $p$ with $r$ simple modules, where $r>1$ is not of the form $4k+1$ with $k\in \Bbb Z_+, k\ge 2$ (e.g., $r\le 8$). Then there exists a simple $H$-module
$X\ne \bold 1$ such that $X\cong X^{**}$.
\end{proposition}
\begin{proof} Assume the contrary. Then the distinguished invertible object $\chi$ of $H$ must be $\bold 1$ (as $\chi^{**}\cong \chi$), so $X^{****}\cong X$ for all $X$ by categorical Radford's formula (\cite{EGNO}, Subsection 7.19). Thus the group $\Bbb Z/4$ acts freely on nontrivial simple $H$-modules by taking the dual. This implies that $r=4k+1$.
It remains to show that $r\ne 5$. Indeed, for $r=5$ the nontrivial simple objects are
$X_1=X$, $X_2=X^*$, $X_3=X^{**}$, $X_4=X^{***}$, so $d_j=d$ and $p_j=p_1$ for $1\le j\le 4$.
Consider the product $P_0\otimes X^*$, of dimension $dp_0$. It decomposes in a direct sum of $P_j$, $1\le j\le 4$.
Thus we have $dp_0=np_1$, where $n$ is the number of summands. But $p_0$ and $p_1$ are coprime (since $p_0+4dp_1=p$), so $d$ is divisible by $p_1$. Hence $d=p_1$ and $X_j$ are projective for $1\le j\le 4$. Thus ${\rm Ext}^1(\bold 1,\bold 1)$
cannot be zero, a contradiction.
\end{proof}
\section{Hopf algebras of dimension $p$}
Now let $H$ be a Hopf algebra of dimension $p$ (a prime) over a field $\mathbf{k}$ of characteristic $q$.
It was recently proved by R. Ng and X. Wang \cite{NW} that if $4q\ge p$ or $q=2$ then $H={\rm Fun}(\Bbb Z/p)$
or $q=p$ and $H=\mathbf{k}[x]/x^p$ with $\Delta(x)=x\otimes 1+1\otimes x$ or $\Delta(x)=x\otimes 1+1\otimes x+x\otimes x$.
This is conjectured to hold for any $q$, and our goal is to improve the bound $4q\ge p$.
Assume that $q>2$ and $H$ is not of this form. If $H$ is semisimple and cosemisimple then by \cite{EG3}, Theorem 3.4,
$H$ is commutative and cocommutative, so this is impossible. Thus, by replacing $H$ with $H^*$ if needed (as done in \cite{NW}), we may (and will) assume that $H$ is not semisimple. Also, by a standard argument, $H$ and $H^*$ have no nontrivial $1$-dimensional modules (as their order must divide $p$). In particular, the distinguished group-like elements of $H$ and $H^*$ must be trivial , so $S^4=1$. Thus $S^2\ne 1$ (as $\dim H=0$ in $\mathbf{k}$) and defines an involution on the simple $H$-modules $X_i$, sending $X_j$ to $X_j^{**}$. Also, $H$ is simple, so ${\rm Rep}(H)$ is tensor generated by any nontrivial simple module and ${\rm Ext}^1(\bold 1,\bold 1)=0$.
Let $r$ be the number of simple $H$-modules, $d_j$ be the dimension of $X_j$ and $p_j$ be the dimension of the projective cover $P_j$ of $X_j$.
\begin{proposition}\label{rankbound} (i) (cf. \cite{EG3}, Lemma 2.10) There exists $i$ such that $X_i^{**}=X_i$ and $p_i$
and $d_i$ are both odd. Moreover, if there exists $j\ne 0$ with $X_j\cong X_j^{**}$ then
there exists $i\ne 0$ with $p_i$ odd.
(ii) (\cite{NW}, Lemma 2.3) Suppose $p_i$ is odd and $X_i^{**}=X_i$. Then $p_i\ge q+2$.
(iii) Let $d_*$ be the maximum of all $d_j$, and let $i$ be such that $X_i\cong X_i^{**}$ and $p_i$ is odd, with largest possible $d_i$. Then we have
$$
p\ge (q+2)\left(d_i+\frac{1}{d_i}\sum_{k\ne i}{\rm max}(d_id_k/d_*,1)\right).
$$
(iv) If $d_i=d_*$ then
$$
p\ge (q+2)\left(d_i+\frac{\sum_{k\ne i}d_k}{d_i}\right).
$$
In particular,
$$
p\ge (q+2)\left(d_i+\frac{2r-3}{d_i}\right).
$$
If $d_i<d_*$ then
$$
p\ge (q+2)\left(d_i+1+\frac{r-2}{d_i}\right).
$$
(v) One has $p\ge 2(q+2)\sqrt{r-1}$.
(vi) If $d_i>1$ then $d_i\ge 3$.
\end{proposition}
\begin{proof} (i) Let $J$ be the set of $j$ such that $X_j^{**}\cong X_j$. Since double dual is an involution, we have
that $\sum_{j\notin J}p_jd_j$ is even. Hence $\sum_{j\in J}p_jd_j$ is odd (as they add up to $p$). So there exists
$i$ such that $p_id_i$ is odd, which proves the first statement.
To prove the second statement, we may assume that $d_j$ or $p_j$ is even, otherwise
we can take $i=j$. We may also assume that $p_0$ is odd; otherwise $i\ne 0$ automatically.
Now consider the even-dimensional object $X_j^*\otimes P_j=P_0\oplus \oplus_{k\ne 0}m_kP_k$. We see that
$\sum_{k\ne 0}m_kp_k$ is odd. So there exists $k$ such that $p_k$ is odd, and moreover, there is an odd number of
such $k$. Hence the involution $**$ has a fixed point on the set of such $k$, which gives the statement.
(ii) Let $a: P_i\to P_i^{**}$ be an isomorphism. We may choose $a$ so that $a^2=1$. Then ${\rm Tr}(a)=0$, otherwise $\bold 1$ is a direct summand in $P_i$, hence is projective and $H$ is semisimple, a contradiction.
Let $m_+,m_-$ be the multiplicities of the eigenvalues $1$ and $-1$ for $a$. Then $m_++m_-=p_i$,
$m_+-m_-=sq$ for some integer $s$. It is clear that $s$ has to be odd, so $|m_+-m_-|\ge q$, and $m_-,m_+>0$ by Lemma \ref{nontri}. Thus $p_i\ge q+2$.
(iii) By Lemma \ref{easybound}, $p_k\ge N_{ji}^kp_i/d_j\ge N_{ji}^k(q+2)/d_j$ for any $j,k$.
It is clear that for each $k$ there exists $j$ such that $X_k$ is a composition factor of $X_j\otimes X_i$ (so that $N_{ji}^k\ge 1$) and $d_j\le d_id_k$. Thus
$$
p_kd_k\ge (q+2){\rm max}(d_id_k/d_*,1)/d_i.
$$
Hence
$$
p=p_id_i+\sum_{k\ne i} p_kd_k\ge (q+2)\left(d_i+\frac{1}{d_i}\sum_{k\ne i}{\rm max}(d_id_k/d_*,1)\right),
$$
as desired.
(iv) The first inequality follows immediately from (iii). The second inequality follows from the first one, as $d_k\ge 2$ for
$k\ne 0$. To prove the third inequality, take $j$ such that $d_j=d_*$, and separate the corresponding term in the sum in (iii), while replacing all the other terms by $1$. This gives the statement.
(v) It follows from (iv) that
$$
p\ge (q+2)\left(d_i+\frac{r-1}{d_i}\right),
$$
so the statement follows by the arithmetic-geometric mean inequality.
(vi) This follows from Lemma \ref{no2}.
\end{proof}
The following corollary improves the bound $p>4q$ from \cite{NW}.
\begin{corollary}\label{9/2} One has $p>\frac{14}{3}(q+2)$.
\end{corollary}
\begin{proof} By Proposition \ref{fourob}, $r\ge 4$. Assume first that $r=4$. It is clear that $X^{**}=X$ for any
simple object $X$ (otherwise we would have objects $X,X^*,X^{**},X^{***}$
which are pairwise nonisomorphic, which would give $r\ge 5$). Thus by
Proposition \ref{rankbound}(i), we have $d_i\ge 2$, hence by Proposition \ref{rankbound}(vi)
$d_i\ge 3$. Thus, by Proposition \ref{rankbound}(iv) we get $p>14(q+2)/3$.
It remains to consider the case $r\ge 5$. In this case, if $d_i\ge 2$ then by Proposition \ref{rankbound}(vi) $d_i\ge 3$, so
if $d_i=d_*$ then by Proposition \ref{rankbound}(iv) we have $p>16(q+2)/3$, and
if $d_i\ne d_*$ then Proposition \ref{rankbound}(iv) yields $p>5(q+2)$.
On the other hand, if $d_i=1$ then Proposition \ref{rankbound}(iv) yields
$p>r(q+2)\ge 5(q+2)$.
So in all cases we have $p>14(q+2)/3$, which proves the statement.
\end{proof}
\begin{proposition}\label{inner} Suppose that for some $i$ we have $X_i\cong X_i^{**}$, $p_i$ is odd and $d_i>1$.
Then
$$
\frac{p}{q+2}> 2^{-2/3}p^{\frac{1}{r-1}}.
$$
\end{proposition}
\begin{proof} By Corollary \ref{bound2}, we have
$$
d_i\ge 2^{-2/3}p^{\frac{1}{r-1}}.
$$
Thus by Proposition \ref{rankbound}(ii) we have
$$
p> p_id_i\ge (q+2)d_i\ge 2^{-2/3}(q+2)p^{\frac{1}{r-1}},
$$
which implies the statement.
\end{proof}
\begin{corollary}\label{inner1} (i) Let $H$ be a non-semisimple Hopf algebra of dimension $p$ in characteristic $q$
for which there exists $i\ne 0$ such that $X_i^{**}\cong X_i$ (for instance, $**$ is isomorphic to ${\rm Id}$ as an additive functor on $\mathop{\mathrm{Rep}}\nolimits(H)$, i.e., $S^2$ is an inner automorphism).
Then if
$$
\frac{p}{q+2}\le 2^{-2/3}p^{\frac{1}{r-1}}.
$$
then $H$ is commutative and cocommutative.
(ii) Let $W$ be the Lambert $W$-function (the inverse to the function $f(z)=ze^z$), and
$$
\phi(x):=2^{-2/3}\exp\left(\frac{1}{2}W(2^{13/3}\log x)\right).
$$
If
$$
\frac{p}{q+2}\le \phi(p)
$$
then $H$ is commutative and cocommutative.
\end{corollary}
\begin{proof}
(i) By Proposition \ref{inner}, it suffices to show that there exists $i$ such that $X_i^{**}\cong X_i$,
$d_i>1$ and $p_i$ is odd. But this follows from Proposition \ref{rankbound}(i).
(ii) Fix $p$ and note that the function $2\sqrt{x}$ is increasing and $2^{-2/3}p^{1/x}$ is decreasing
in $x$. Thus if $s$ is the solution of the equation
$$
2\sqrt{s}=2^{-2/3}p^{1/s}
$$
then by Proposition \ref{rankbound}(v) and part (i),
$$
\frac{p}{q+2}>{\rm max}(2\sqrt{x},2^{-2/3}p^{1/x})|_{x=r-1}\ge 2\sqrt{s}.
$$
Let $y=2^{5/3}\sqrt{s}$. We have
$$
y=p^{2^{10/3}/y^2}.
$$
Thus
$$
2^{13/3}\log p=y^2\log(y^2).
$$
So $\log(y^2)=W(2^{13/3}\log p)$.
Thus we get
$$
y=\exp\left(\frac{1}{2}W(2^{13/3}\log p)\right),
$$
so
$$
2\sqrt{s}=2^{-2/3}\exp\left(\frac{1}{2}W(2^{13/3}\log p)\right).
$$
This implies (ii).
\end{proof}
\begin{remark} It is easy to show that
$$
\phi(x)\sim 2\sqrt{2} \left(\frac{\log x}{\log \log x}\right)^{1/2}
$$
as $x\to \infty$.
\end{remark}
\begin{theorem}\label{inner2}
Let $H$ be a Hopf algebra of prime dimension $p$ over a field $\mathbf{k}$ characteristic $q>2$, which is not commutative and cocommutative. Then
$$
\frac{p}{q+2}>{\rm max}\left(\frac{14}{3},{\rm min}(9,\phi(p))\right).
$$
\end{theorem}
\begin{proof} If there exists simple $X\ne \bold 1$ with $X\cong X^{**}$ then Corollary \ref{inner1} applies.
Otherwise, Proposition \ref{5sim} and Proposition \ref{rankbound}(v) imply that
$p> r(q+2)\ge 9(q+2)$. This together with Corollary \ref{9/2} implies the theorem.
\end{proof}
| -67,789.051603 |
[
-3.068359375,
2.673828125
] | 50.452489 |
[
-2.71484375,
0.17431640625,
-2.583984375,
-5.80859375,
-0.46875,
9.109375
] |
[
3.3984375,
9.46875,
1.947265625,
6.34765625
] | 629 | 8,238 |
[
-3.53125,
4.08203125
] | 35.374375 |
[
-5.171875,
-3.779296875,
-5.03515625,
-2.568359375,
1.7724609375,
13.03125
] | 0.393945 | 22.764165 | 18.184025 | 3.911275 |
[
1.1482605934143066
] | -40,460.729583 | 4.899369 | -67,206.262766 | 0.708605 | 5.936514 |
[
-1.41796875,
-3.068359375,
-3.921875,
-5.54296875,
1.7939453125,
12.5859375
] |
[
-5.640625,
-1.91015625,
-1.984375,
-1.189453125,
3.404296875,
4.16796875
] |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.